id
string | text
string | source
string | created
timestamp[s] | added
string | metadata
dict |
|---|---|---|---|---|---|
0910.3656
|
# The Asymptotic Falloff of Local Waveform Measurements in Numerical
Relativity
Denis Pollney Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-
Institut, Potsdam-Golm, Germany Christian Reisswig Max-Planck-Institut für
Gravitationsphysik, Albert-Einstein-Institut, Potsdam-Golm, Germany Nils
Dorband Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut,
Potsdam-Golm, Germany Erik Schnetter Center for Computation & Technology,
Louisiana State University, Baton Rouge, LA, USA Department of Physics &
Astronomy, Louisiana State University, Baton Rouge, LA, USA Peter Diener
Department of Physics & Astronomy, Louisiana State University, Baton Rouge,
LA, USA Center for Computation & Technology, Louisiana State University,
Baton Rouge, LA, USA
###### Abstract
We examine current numerical relativity computations of gravitational waves,
which typically determine the asymptotic waves at infinity by extrapolation
from finite (small) radii. Using simulations of a black hole binary with
accurate wave extraction at $r=1000M$, we show that extrapolations from the
near-zone are self-consistent in approximating measurements at this radius,
although with a somewhat reduced accuracy. We verify that $\psi_{4}$ is the
dominant asymptotic contribution to the gravitational energy (as required by
the peeling theorem) but point out that gauge effects may complicate the
interpretation of the other Weyl components.
###### pacs:
04.25.dg, 04.30.Db, 04.30.Tv, 04.30.Nk
I. Introduction. – Numerical relativity has made great strides in recent years
in the solution of the binary black hole (BH) problem. Since the original
breakthroughs by Pretorius Pretorius (2005) and the moving puncture approach
Baker et al. (2006); Campanelli et al. (2006), the calculation of long,
accurate gravitational waveforms (GWs) has become an almost routine procedure.
It is particularly satisfying that a variety of methods (numerical methods,
formulations of the Einstein equations, wave extraction techniques) are in use
and have been shown to produce consistent results (eg. Hannam et al. (2009)).
Certain systematic errors, however, are difficult to estimate. In particular,
current methods measure GWs at finite radii and extrapolate the results to
infinity. This extrapolation has been identified as one of the largest
remaining sources of systematic error within current extraction techniques,
particularly during the merger and ring-down Scheel et al. (2009); Boyle and
Mroue (2009). Potential ambiguities arise particularly at small radii where
gauge as well as nonlinear near-zone effects may dominate the expected
polynomial falloff of the amplitude.
In this paper, we verify the extrapolation procedure for GWs by performing
accurate wave extractions at large radii, out to $r=1000M$ from the source
(where $M$ is the mass of the spacetime). The waveforms have been calculated
using a new hybrid multi-patch/mesh-refinement algorithm, which allows for an
efficient discretisation of the wave zone so that high accuracy can be
obtained to large radii. We find that the measured waves between $r=100M$ and
$r=1000M$ are convergent and of good enough quality to extrapolate the phase
and amplitude accurately by low-order polynomial expansions. The measurements
at $r=1000M$ can be estimated to within $0.04\%$ in amplitude and
$0.001\text{rad}$ in phase, if the measurements out to $r=600M$ are used in
the extrapolation. This is true over the course of the evolution, including 8
orbits of inspiral, the merger, and ring-down. If only measurements within
$r=220M$ are used, as is common, then the errors increase by an order of
magnitude.
Finally, we note that the gravitational radiation is normally associated with
the leading order term in the falloff of the spacetime curvature. By the
peeling theorem, we expect this to be a polynomial in $1/r$ whose leading
coefficient is the Weyl component $\psi_{4}$. By measuring all of the Weyl
curvature components, we have been able to establish their respective falloff
rates, and verify that $\psi_{4}$ is indeed the leading order coefficient with
the expected $1/r$ falloff rate. The exponents for the $\psi_{1}$ and
$\psi_{0}$ components are less clear, however, and suggest that local gauge
effects influence their computation.
II. Computational setup. – A key feature of the calculations performed here is
the accuracy which we are able to achieve at large radii from the source
through the use of a newly implemented numerical scheme. The code makes use of
finite differences and standard mesh-refinement techniques, but incorporates
the use of multiple grid patches to cover the spacetime with flexible adapted
local coordinates. For the binary BH inspiral considered here, we consider two
regions, depicted in Fig 1. In the near-zone region where the BHs orbit, we
discretise the spacetime using standard Cartesian grids, applying 2:1 Berger-
Oliger mesh refinement in order to increase the resolution around each body
Schnetter et al. (2004). In the wave-zone, however, the dynamical fields are
essentially radially propagating waves. We cover this zone with six
overlapping patches, each of which incorporate a local radial coordinate $r$
and transverse angular coordinates $(\rho,\sigma)$. The use of six patches
avoids the problem of a coordinate singularity on the axis of a single
spherical-polar coordinate system, as well as providing a more uniform angular
resolution over the sphere. The particular coordinates which we have
implemented are the “inflated cube” coordinates, given explicitly in Thornburg
(2004).
Figure 1: Schematic depiction of the grid structure in the $z=0$ plane. Four
radial grids surround the equator. The right shows an expanded inset of the
Cartesian grid (shaded) covering the near-zone around the individual BHs.
Derivatives on each grid are locally computed using standard finite
differences at 8th-order. Data is passed between patches by interpolation,
typically via centred 5th-order Lagrange polynomials. Each patch is surrounded
by a boundary zone which is populated with data mapped from the neighbouring
patch, so that derivatives can be calculated up to the patch edge without the
need for off-centred stencils. A 4th-order Runge-Kutta integrator is used to
evolve the solution.
We write the Einstein equations in the commonly used BSSNOK form Nakamura et
al. (1987); Shibata and Nakamura (1995); Baumgarte and Shapiro (1998);
Alcubierre et al. (2000), adopting the particular variation proposed by
Marronetti et al. (2008), whereby the usual variable $\phi=\log\gamma/12$ is
replaced by $w=\gamma^{-2}$ (with $\gamma$ the 3-metric determinant). Gauges
are the commonly used $1+\log$ and $\tilde{\Gamma}$-driver conditions with
advection terms Alcubierre et al. (2003); Baker et al. (2006); Campanelli et
al. (2006).
The GWs are measured by evaluating the Weyl curvature tensor components,
$C_{\alpha\beta\gamma\delta}$, in a null frame
$(\mathbf{l},\mathbf{n},\mathbf{m},\mathbf{\bar{m}})$, oriented so that the
outgoing vector, $\mathbf{l}$, points along the coordinate $\hat{\mathbf{r}}$
direction while the other vectors determine an orthonormal null frame in the
local metric. The independent curvature components $\psi_{0}\ldots\psi_{4}$
are determined by the standard projections Newman and Penrose (1962). The Weyl
components are evaluated on spheres of fixed coordinate radius and projected
onto a basis of spherical harmonics, ${}_{-2}Y_{lm}$. For binary black hole
situation considered here, the dominant mode is $l=2$, $m=2$, which is used
for the results presented here.
To establish the numerical accuracy, we have performed evolutions at three
different resolutions, and find that the results converge at 4th-order during
the merger, and close to 8th-order during the inspiral, in both amplitude and
phase. (A more complete description of the code implementation and tests will
be given elsewhere Pollney et al. (2009).) Here we present results based on
the highest resolution evolved, for which the spatial resolution for all of
the GW measurements is uniformly $h_{r}=0.64M$ in the radial direction, and
$h_{\perp}\simeq 3^{\circ}$ in the angular directions. GW measurements are
taken every $0.144M$.
III. Extrapolation of waveforms. – We have evolved an equal-mass, non-spinning
binary from separation $d/M=11.0$ through approximately 8 orbits (a physical
time of around $1360M$), merger and ring-down. The masses of the punctures are
set to $m=(0.4872)$ and are initially placed on the $x$-axis with momenta
$p=(\pm 0.0903,\mp 0.000728,0)$, giving the initial slice an ADM mass
$M=0.99052$. These initial data parameters were determined using a post-
Newtonian evolution from large initial separation, following the procedure
outlined in Husa et al. (2008), with the conservative part of the Hamiltonian
accurate to 3PN and radiation-reaction to 3.5PN, and determine orbits with
eccentricity less than $0.2\%$.
We measure the Weyl components $\psi_{0}\ldots\psi_{4}$ every $20M$ from
$r=100M$ to $300M$, then at $400M,500M,600M$ and $1000M$. The radial grid
structure in the wave zone allows us to extend the outer boundary of the grid
at relatively little cost compared to Cartesian codes. For the runs presented
here, it is placed at $r=3600M$ with a resolution of $dr=2.56M$ at the outer
boundary so that the $l=2$, $m=2$ mode is reasonably well resolved throughout
the grid. This allows for $2600M$ of evolution time before a physical or
constraint violating mode traveling at the speed of light can reach the
outermost detector at $r=1000M$. That is, the outer boundary is effectively
causally disconnected from the wave measurements presented in this paper.
The Weyl components $\psi_{j}=A_{j}e^{i\phi_{j}}$ are assumed to fall-off as a
function of radius according to
$A_{j}(r,t^{*})=\sum_{i=0}^{n_{A}}\frac{A_{j}^{(i)}(t^{*})}{r^{i}},\qquad\phi_{j}(r,t^{*})=\sum_{i=0}^{n_{\phi}}\frac{\phi_{j}^{(i)}(t^{*})}{r^{i}}.$
(1)
The $r$ coordinate is that of the simulation coordinates, which we find to
differ by at most $0.1\%$ from the areal radius. The GWs are expressed in
terms of the retarded time $t^{*}=t-r^{*}$ where $t$ is the coordinate time
and $r^{*}=r+2M\ln[r/(2M)-1]$ is the tortoise coordinate Scheel et al. (2009).
We do not offset the retarded time to align the peaks of the waveforms.
It is generally difficult to estimate the error incurred when extrapolating.
Given the data at $r=1000M$, we can attempt to gauge an optimal choice of
extrapolation parameters by attempting to estimate this data from the
measurements at smaller radii. As test cases, we construct extrapolations
using four different sets of radii, $e_{1}\in\\{100M,200M\\}$,
$e_{2}\in\\{160M,280M\\}$, $e_{3}\in\\{200M,300M\\}$ and
$e_{4}\in\\{260M,600M\\}$. Each of these incorporates 6 data points, which
over-determines low-order polynomials. We evaluate the extrapolation
coefficients by a least-squares fit to these points, which can be important in
removing spurious oscillations that may arise fitting high-order polynomials
to noisy data.
Fitting the amplitude using various polynomial orders, $n_{A}$, suggests that
in all cases $n_{A}=3$ is optimal in predicting the amplitude of the measured
wave at $r=1000M$, with an error of approximately $0.02\%$ for $e_{4}$ and
$0.2\%$ for $e_{1}$. For the phase, we find that $n_{\phi}=3$ minimises the
error, at $6\times 10^{-4}\text{rad}$ and $5\times 10^{-3}\text{rad}$ for
$e_{4}$ and $e_{1}$, respectively. In both amplitude and phase, we note that
the error is reduced significantly if the outermost data $e_{4}$ is used. In
Fig. 2 we display the error in estimating the $r=1000M$ data using each of the
extrapolations at the optimal order. The maximum errors in both amplitude and
phase tend to occur during the late inspiral ($t^{*}=-200M$ to $t^{*}=0M$) and
ring-down, although there is no sign of a rapid growth of error during this
phase.
The corresponding extrapolations to $r\rightarrow\infty$ shows very similar
behaviour. In Fig. 3, we have compared each of the extrapolations with an
extrapolation obtained by including the $r=1000M$ data
($e_{5}\in\\{280M,1000M\\}$), evaluated at $r\rightarrow\infty$. The outermost
extrapolations differ by at most $\Delta A=0.03\%$ and
$\delta\phi=0.003\text{rad}$ over the course of the evolution.
Figure 2: Error in the extrapolated amplitude (top panel) and phase (bottom
panel) of the $\ell=2,m=2$ component of $\psi_{4}$ at $r=1000M$ as computed by
extrapolations $e_{1}\ldots e_{4}$ (defined in the text). Figure 3:
Differences between the $r\leq 600M$ extrapolations with an extrapolation
including $r=1000M$ data ($e_{5}$) evaluated in the limit
$r\rightarrow\infty$.
IV. Peeling properties. – The interpretation of $\psi_{4}$ as the radiated
gravitational energy is a consequence of the “peeling” property, which states
that for asymptotically flat spacetimes at large radii, the Weyl curvature
tensor can be represented schematically as
$C_{\alpha\beta\gamma\delta}\simeq\frac{\psi_{4}}{r}+\frac{\psi_{3}}{r^{2}}+\frac{\psi_{2}}{r^{3}}+\frac{\psi_{1}}{r^{4}}+\frac{\psi_{0}}{r^{5}}+O(1/r^{6}).$
(2)
That is, each component of the Weyl tensor falls off at a known fixed rate,
and at large radii, $\psi_{4}$ is the dominant component. At future null
infinity, $\mathcal{J}^{+}$, it can be related to $\dot{M}$, the change in
energy of the spacetime. We note, however, that the peeling theorem involves a
number of restrictions on the asymptotic form of the spacetime, and the
coordinates which are used there. A rigorous connection between finite radius
measurements and the asymptotic properties of the spacetime at
$\mathcal{J}^{+}$ is difficult to make.
Given the importance of the falloff of the curvature in the identification of
$\psi_{4}$ with the GW, it is useful to examine the behaviour of the other
Weyl components measured by the simulation. In Fig. 4, we have plotted their
falloff as a function of coordinate radius. The time series data for each
component is mapped to a scalar by integrating the amplitude over the interval
$t\in[-800M,50M]$. (Alternatively, one could obtain a scalar by taking the
measurements at a point such as the waveform peak. A similar plot results, but
the averaging effect of the integral reduces local noise slightly.)
For the cases of $\psi_{4}$ and $\psi_{3}$, we find that a straight line can
be fitted to each of the components, indicating a consistent exponent, with
measured values of $-0.99$ and $-1.99$ respectively, and a rather good
agreement with Eq. (2). Due to its small amplitude, the $\psi_{2}$ measurement
is dominated by numerical noise beyond a certain radius (clear from
examination of the time-series data), and as a result, the curve veers from a
straight line. However, if we fit a straight line to the five data points from
$r\leq 200M$, we find an exponent, $-2.99$, again agreeing well with the
expectation.
The $\psi_{1}$ and $\psi_{0}$ components present an interesting situation.
Particularly in the case of $\psi_{0}$, the amplitude is large enough that a
clear signal is present (of almost of the same amplitude as $\psi_{3}$). The
falloff, however, is of order $-2.00$, rather than the $-5$ which the peeling
theorem requires. Further, we note that the mode propagates outwards with a
peak coincident with that of $\psi_{4}$, in contrast to the interpretation of
$\psi_{0}$ as an “ingoing” component of radiation.
A possible explanation is that metric perturbations cause oscillations in the
frame in which the components are measured. As described above, we define the
null frame only with reference to the local space and time coordinates.
Attempts to modify the falloff of $\psi_{0}$ via frame rotations (spin-boosts
and null rotations) did not preserve the falloff of the other components.
However, other gauge effects are likely present. We note that measurements are
taken on spheres defined by the grid coordinates. The areal radius of these
spheres exhibits small (on the order of $0.1\%$) oscillations in the $\ell=2$,
$m=2$ mode. The finite-radius $\psi_{4}$ measurement is known to be
susceptible to pure gauge effects such as the presence of a non-zero shift
vector, which can produce spurious GW signals in static spacetimes Reisswig et
al. (2009a). Though these effects are small, so are the values of $\psi_{1}$
and $\psi_{0}$ and thus correspondingly sensitive compared to that of the
dominant component.
Figure 4: The radial falloff of the Weyl components. Lines are linear least-
squares fits to all of the points of $\psi_{0}$, $\psi_{3}$, and $\psi_{4}$,
and the $r\leq 200M$ points for $\psi_{1}$ and $\psi_{2}$. Measured slopes are
listed in the legend.
V. Discussion. – We have demonstrated a number of features related to the
measurement of GWs at finite radius. Our results suggest that polynomial
extrapolation of the $\psi_{4}$ component from small radii can provide an
accurate model for estimating the measurements at larger radii. Data measured
within $r=200M$ of the source have an error in amplitude and phase of $\Delta
A\simeq 0.2\%$ and $\Delta\phi\simeq 5\times 10^{-3}\text{rad}$ throughout the
evolution (including merger and ring-down) compared to the measurement at
$r=1000M$. This provides an important check on numerical relativity
measurements, which typically extrapolate from $r<200M$. Larger radius
measurements do, however, improve the extrapolation, and errors can be reduced
by a further order of magnitude if data to $r=600M$ is included. We also note
that while $\psi_{4}$ is dominated by the $1/r$ term beyond $r=300M$, at
smaller radii the higher order terms have a much larger contribution.
While the falloff of $\psi_{4}$ is the leading order contribution to the
curvature as expected, the results of Fig. 4 suggest that the picture may be
more complicated for the other Weyl components, and that care should be taken
in interpreting local variables according to their expected asymptotic
properties. With respect to the peeling property, we note that the asymptotic
evolution of generic initial data sets are likely to include polyhomogenous
terms, and even the situation for Bowen-York data with linear momentum is not
clear Chrusciel et al. (1992); Valiente Kroon (2007). While it seems most
likely that the effect we have observed in $\psi_{0}$ and $\psi_{1}$ is
related to gauge, this provides a strong caution regarding physical
predictions based on these quantities. Alternative gauge conditions may
alleviate (or exaggerate) the issues we have noted.
The mapping of finite radius results to asymptotic values at $\mathcal{J}^{+}$
needs to be considered with some care. While our results suggest that the
procedure of extrapolation is self-consistent and can be used to estimate the
results that would be obtained by direct measurement at large radii, they do
not establish the identification of the extrapolated quantities with
quantities that would be measured at $\mathcal{J}^{+}$. In a related paper
Reisswig et al. (2009b), we have demonstrated that the extrapolation procedure
does in fact reproduce results obtained at $\mathcal{J}^{+}$ to high accuracy,
though a small systematic error does remain. A number of local corrections
have been proposed to improve the rigor of $\psi_{4}$ measurements at finite
radius (cf. Nerozzi (2007); Lehner and Moreschi (2007); Deadman and Stewart
(2009)).
Finally, we note that our accurate measurements at $r=1000M$ are a result of a
new computational infrastructure making use of adapted coordinate grids in
conjunction with a finite difference, moving-puncture scheme. This is the
first demonstration that such methods can produce stable evolutions for
dynamical spacetimes. The efficiency gains allow the wave zone to be covered
with sufficient resolution to very large radii ($3600M$ in this case), which
has been a crucial in reducing boundary errors.
###### Acknowledgements.
Acknowledgments.$-$ The authors are pleased to thank: Ian Hinder, Sascha Husa,
Badri Krishnan, Philipp Moesta, Luciano Rezzolla, Juan Valiente-Kroon for
their helpful input; the developers of Cactus Goodale et al. (2003) and Carpet
Schnetter et al. (2004) for providing an open and optimised computational
infrastructure on which we have based our code; support from the DFG
SFB/Transregio 7, the VESF, and by the NSF awards no. 0701566 _XiRel_ and no.
0721915 _Alpaca_. Computations were performed at the AEI, at LSU, on LONI
(numrel03), on the TeraGrid (TG-MCA02N014), and the Leibniz Rechenzentrum
München (h0152).
## References
* Pretorius (2005) F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005).
* Baker et al. (2006) J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006).
* Campanelli et al. (2006) M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006).
* Hannam et al. (2009) M. Hannam et al., Phys. Rev. D79, 084025 (2009).
* Scheel et al. (2009) M. A. Scheel et al., Phys. Rev. D79, 024003 (2009).
* Boyle and Mroue (2009) M. Boyle and A. H. Mroue (2009).
* Schnetter et al. (2004) E. Schnetter, S. H. Hawley, and I. Hawke, Class. Quantum Grav. 21, 1465 (2004).
* Thornburg (2004) J. Thornburg, Class. Quantum Grav. 21, 3665 (2004).
* Nakamura et al. (1987) T. Nakamura, K. Oohara, and Y. Kojima, Prog. Theor. Phys. Suppl. 90, 1 (1987).
* Shibata and Nakamura (1995) M. Shibata and T. Nakamura, Phys. Rev. D 52, 5428 (1995).
* Baumgarte and Shapiro (1998) T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D 59, 024007 (1998).
* Alcubierre et al. (2000) M. Alcubierre, B. Brügmann, T. Dramlitsch, J. A. Font, P. Papadopoulos, E. Seidel, N. Stergioulas, and R. Takahashi, Phys. Rev. D 62, 044034 (2000).
* Marronetti et al. (2008) P. Marronetti, W. Tichy, B. Bruegmann, J. Gonzalez, and U. Sperhake, Phys. Rev. D77, 064010 (2008).
* Alcubierre et al. (2003) M. Alcubierre, B. Brügmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D 67, 084023 (2003).
* Newman and Penrose (1962) E. T. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962), erratum in J. Math. Phys. 4, 998 (1963).
* Pollney et al. (2009) D. Pollney, C. Reisswig, E. Schnetter, N. Dorband, and P. Diener (2009), submitted to PRL.
* Husa et al. (2008) S. Husa, M. Hannam, J. A. Gonzalez, U. Sperhake, and B. Bruegmann, Phys. Rev. D77, 044037 (2008).
* Reisswig et al. (2009a) C. Reisswig, N. T. Bishop, D. Pollney, and B. Szilágyi (2009a), in preparation.
* Chrusciel et al. (1992) P. T. Chrusciel, M. A. H. MacCallum, and D. B. Singleton, Proc. Roy. Soc. Lond. A436, 299 (1992).
* Valiente Kroon (2007) J. A. Valiente Kroon, Class. Quant. Grav. 24, 3037 (2007).
* Reisswig et al. (2009b) C. Reisswig, N. T. Bishop, D. Pollney, and B. Szilagyi (2009b).
* Nerozzi (2007) A. Nerozzi, Phys. Rev. D75, 104002 (2007).
* Lehner and Moreschi (2007) L. Lehner and O. M. Moreschi (2007).
* Deadman and Stewart (2009) E. Deadman and J. M. Stewart, Class. Quant. Grav. 26, 065008 (2009).
* Goodale et al. (2003) T. Goodale, G. Allen, G. Lanfermann, J. Massó, T. Radke, E. Seidel, and J. Shalf, in _Vector and Parallel Processing – VECPAR’2002, 5th International Conference, Lecture Notes in Computer Science_ (Springer, Berlin, 2003).
|
arxiv-papers
| 2009-10-19T19:32:17 |
2024-09-04T02:49:05.911603
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Denis Pollney, Christian Reisswig, Nils Dorband, Erik Schnetter, Peter\n Diener",
"submitter": "Denis Pollney",
"url": "https://arxiv.org/abs/0910.3656"
}
|
0910.3712
|
# Geometry of elastic hydrofracturing by injection of an over pressured non-
Newtonian Fluid
Mariano Cerca, Jazmin Chavez Alvarez, Bernardino Barrientos,
Enrique Soto and Carlos Mares
Universidad Nacional Autónoma de México,
Blvd Juriquilla 3001, Juriquilla, Querétaro, 76230, México
Centro de investigaciones en Óptica,
Loma del Bosque #115, León, 37150, México
###### Abstract
The nucleation and propagation of hydrofractures by injection of over
pressured fluids in an elastic and isotropic medium are studied
experimentally. Non-Newtonian fluids are injected inside a gelatine whose
mechanical properties are assumed isotropic at the experimental strain rates.
Linear elastic theory predicts that plastic deformation associated to breakage
of gelatin bonds is limited to a small zone ahead of the tip of the
propagating fracture and that propagation will be maintained while the fluid
pressure exceeds the normal stress to the fracture walls (Chávez-Álvarez,2008)
(i.e., the minimum compressive stress), resulting in a single mode I fracture
geometry. However, we observed the propagation of fractures type II and III as
well as nucleation of secondary fractures, with oblique to perpendicular
trajectories with respect to the initial fracture. In the Video experimental
evidence shows that the fracture shape depends on the viscoelastic properties
of gelatine coupled with the strain rate achieved by fracture propagation.
1. 1.
M.J. Chávez Álvarez, L.M. Cerca Martínez, _Analogue simulation of magama
rheology during dike emplacement: A preliminary study based on field
observations and rheological determinations of materials, Bolletino di
Geofisica, 49, 29-34 (2008)._
|
arxiv-papers
| 2009-10-19T21:56:13 |
2024-09-04T02:49:05.916894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mariano Cerca, Jazmin Chavez Alvarez, Bernardino Barrientos, Enrique\n Soto and Carlos Mares",
"submitter": "Enrique Soto",
"url": "https://arxiv.org/abs/0910.3712"
}
|
0910.3803
|
# High accuracy binary black hole simulations with an extended wave zone
Denis Pollney Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-
Institut, Potsdam-Golm, Germany Christian Reisswig Max-Planck-Institut für
Gravitationsphysik, Albert-Einstein-Institut, Potsdam-Golm, Germany Erik
Schnetter Center for Computation & Technology, Louisiana State University,
Baton Rouge, LA, USA Department of Physics & Astronomy, Louisiana State
University, Baton Rouge, LA, USA Nils Dorband Max-Planck-Institut für
Gravitationsphysik, Albert-Einstein-Institut, Potsdam-Golm, Germany Peter
Diener Department of Physics & Astronomy, Louisiana State University, Baton
Rouge, LA, USA Center for Computation & Technology, Louisiana State
University, Baton Rouge, LA, USA
(2009-10-13)
###### Abstract
We present results from a new code for binary black hole evolutions using the
moving-puncture approach, implementing finite differences in generalised
coordinates, and allowing the spacetime to be covered with multiple
communicating non-singular coordinate patches. Here we consider a regular
Cartesian near zone, with adapted spherical grids covering the wave zone. The
efficiencies resulting from the use of adapted coordinates allow us to
maintain sufficient grid resolution to an artificial outer boundary location
which is causally disconnected from the measurement. For the well-studied
test-case of the inspiral of an equal-mass non-spinning binary (evolved for
more than 8 orbits before merger), we determine the phase and amplitude to
numerical accuracies better than $0.010\%$ and $0.090\%$ during inspiral,
respectively, and $0.003\%$ and $0.153\%$ during merger. The waveforms,
including the resolved higher harmonics, are convergent and can be
consistently extrapolated to $r\rightarrow\infty$ throughout the simulation,
including the merger and ringdown. Ringdown frequencies for these modes (to
$(\ell,m)=(6,6)$) match perturbative calculations to within $0.01\%$,
providing a strong confirmation that the remnant settles to a Kerr black hole
with irreducible mass $M_{\rm irr}=0.884355\pm 20\times 10^{-6}$ and spin
$S_{f}/M_{f}^{2}=0.686923\pm 10\times 10^{-6}$.
###### pacs:
04.25.dg, 04.30.Db, 04.30.Tv, 04.30.Nk
## I Introduction
The numerical solution of Einstein’s equations has made great progress in
recent years. Orbits and mergers of binary systems of black holes and neutron
stars are now routinely published by a number of research groups, using
independent codes and methodologies Pretorius (2005); Baker et al. (2006a);
Campanelli et al. (2006); Scheel et al. (2009). A number of important
astrophysical phenomena associated with binary black hole mergers have been
studied in some detail. In particular, the recoil of the merger remnant has
been studied for a number of different initial data models Gonzalez et al.
(2007a, b); Campanelli et al. (2007a, b); Herrmann et al. (2007); Koppitz et
al. (2007); Pollney et al. (2007); Lousto and Zlochower (2008), and its final
mass and spin has been mapped out for fairly generic merger models involving
spinning and unequal mass black holes Rezzolla et al. (2008a); Rezzolla et al.
(2008b); Rezzolla et al. (2008c); Tichy and Marronetti (2007); Lousto and
Zlochower (2007); Barausse and Rezzolla (2009). Since these quantities are
determined by the last few quasi-circular orbits before merger, they can be
calculated to good approximation with fairly short evolutions, and with
minimal influence of an artificial outer boundary.
Of particular topical relevance, however, is the construction of long
waveforms which can be used for gravitational-wave analysis of the binary
Reisswig et al. (2009a), and also to construct a family of templates Ajith et
al. (2007, 2008); Ajith (2008); Ajith et al. (2009), so to inform and improve
gravitational wave detection algorithms. Here the requirements are
particularly challenging for numerical simulations, requiring waveforms which
are accurate in phase and amplitude over multiple cycles to allow for an
unambiguous matching to post-Newtonian waveforms at large separation. Some
recent studies have shown very promising results in this direction for
particular binary black hole models Baker et al. (2006b); Buonanno et al.
(2006); Hannam et al. (2008a, b); Damour et al. (2008a); Buonanno et al.
(2007); Damour et al. (2008b); Buonanno et al. (2009); Boyle et al. (2007).
However, they have also highlighted the problems associated with producing
long waveforms of sufficient accuracy.
First of all, for binaries with a larger separation, systematic errors
associated with gravitational waveform extraction at a finite radius become
more pronounced. Typically a number of extraction radii are used, and the
results extrapolated to infinite radius (assuming such a consistent
extrapolation exists given potential issues of gauge). In order to have some
confidence in the results, the outermost “extraction sphere” needs to be at a
large radius, say on the order of $150-200M$ (where $M$ is the mass of the
system and sets the fiducial length scale). Even at this radius, the amplitude
of the extrapolated waveform differs significantly from the measured waveform.
Unfortunately, extracting at larger radii comes at a computational expense.
One of the standard methods in use today is finite differencing in conjunction
with “mesh refinement”, in which the numerical resolution is chosen based on
the length scale of the problem. A minimum number of discrete data points are
required to resolve a feature of a given size accurately, which sets a limit
on the minimum resolution which should be applied in a region. Thus, even with
mesh refinement there is a limit on the coarseness of the grid which can be
allowed in the wave-zone. For a Cartesian grid, the number of computational
points scales as $r^{3}$, so that requiring a sufficient resolution to $200M$
already comes at significant expense, and increasing this distance further
becomes impractical.
An additional difficulty arises from the requirement that the outer boundary
have minimal influence on the interior evolution, since it is (in all current
binary black hole codes) an artificial boundary. This places an additional
requirement on the size of the computational grids, so that even outside the
wave-zone region where the physics is accurately resolved, it is conventional
to place several even coarser grids. This is done in the knowledge that the
physical variables can not be resolved in these regions, but the grids are
helpful as a numerical buffer between the outer boundary and interior domain.
Again, adding these outer zones comes at a computational expense. The
boundaries with under-resolved regions also lead to unphysical reflections
which can contaminate the solution. The problem of increasing the grid size
can be significantly reduced if, rather than a Cartesian coordinate system,
one uses a discretisation which has a radial coordinate. Then, for a fixed
angular resolution, the number of points on the discrete grid increases simply
as a linear function of $r$, rather than the $r^{3}$ of the Cartesian case.
This has two advantages. The gravitational wave-zone has spherical topology
and therefore, a numerical approximation would be most efficiently represented
by employing a spherical grid. A further computational motivation comes from
the fact that non-synchronous mesh-refinement (such as the Berger-Oliger
algorithm) can greatly complicate the parallelisation of an evolution scheme,
and thus having many levels of refinement clearly has an impact on the
efficiency of large scale simulations. This will become particularly relevant
for the coming generations of peta-scale machines which strongly emphasise
parallel execution (possibly over several thousand cores) over single
processor performance.
The use of spherical-polar coordinates has largely been avoided in
3-dimensional general relativity due to potential problems associated with the
coordinate singularity at the poles. Additionally, even if regularisation were
simple, the inhomogeneous areal distribution of latitude-longitude grid points
over the sphere make spherical-polar coordinates sub-optimal. A number of
alternative coordinate systems have been proposed and implemented for studies
of black holes in 3D. The Pittsburgh null code avoids the problem of
regularisation at the poles by implementing a 2D stereographic patch system
Bishop et al. (1997). Cornell/Caltech have developed a multipatch system which
has been used for long binary black hole evolutions Scheel et al. (2006, 2009)
111Multi-domain spectral methods have previously been applied to the problem
of generating initial data for binaries in Gourgoulhon et al. (2001, 2002);
Grandclement et al. (2002).. This code, using spectral spatial
differentiation, uses an intricate patch layout in order to reduce the overall
discretisation error. The boundary treatment between patches is based on the
transfer of characteristic variables. A similar approach was implemented by
the LSU group, for the case of finite differences with penalty boundary
conditions Schnetter et al. (2006a), and used to successfully evolve single
perturbed black holes with a fixed background Dorband et al. (2006) and have
recently been attempted for binary black hole systems Pazos et al. (2009).
In this paper we describe a binary black hole evolution code based on adapted
radial coordinates in the wave zone, for generic evolution systems. In
particular, we demonstrate stable and accurate binary black hole evolutions
using BSSNOK in conjunction with this coordinate system. The grids in the wave
zone follow a prescription which was first used by Thornburg Thornburg (2004),
in which six regular patches cover the sphere, and data at the boundaries of
the patches are filled by interpolation. The six patch wave zone is coupled to
an interior Cartesian code, which covers the domain in which the bodies move,
and optionally allows for mesh refinement around each of the individual
bodies. The resulting code has the advantages of making use of established
techniques for moving puncture evolutions on Cartesian grids, while having
excellent efficiency (and consequently accuracy) in the wave zone due to the
use of adapted radially-oriented grids.
In the following sections we detail the coordinate structures which we use. We
then describe our Einstein evolution code, which is largely based on
conventional techniques common to Cartesian puncture evolutions. Finally we
perform evolutions of a binary black hole system in order to validate the code
against known results, as well as demonstrate the ability to extract accurate
waves at a large radius with comparatively low computational cost.
## II Spacetime Discretisation
This section describes the implementation of a generic code infrastructure for
evolving spacetimes which are covered by multiple overlapping grid patches. A
key feature of our approach is its flexibility. It is not restricted to any
particular formulation of the Einstein equations; the mechanism for passing
data between patches (interpolation) is also formulation independent (though
characteristic Pfeiffer et al. (2003) or penalty-patch boundaries Carpenter et
al. (1994); Diener et al. (2007); Pazos et al. (2009) are also an option); the
size, placement and local coordinates of individual patches are completely
adaptable, provided that there is sufficient overlap between neighbours to
transfer boundary data. Further, each patch is a locally Cartesian grid with
the ability to perform mesh-refinement to better resolve localised steep
gradients, if necessary. The particular application demonstrated in this paper
is to provide a more efficient covering of the wave-zone of an isolated binary
black hole inspiral.
At the same time, we would like to take advantage of the fact that black hole
evolutions via the “moving puncture” approach are well established as a simple
and effective method for stably evolving black hole spacetimes Baker et al.
(2006a); Campanelli et al. (2006). By this method, gauge conditions are
applied to prevent the spacetime from reaching the curvature singularity, so
that an artificial boundary is not required within the horizons Hannam et al.
(2007). The usual approach is to discretise using Cartesian grids which cover
the black holes with an appropriate resolution, without special treatment or
boundary conditions for the black hole interiors, relying rather on the causal
structure of the evolution system to prevent error modes from emerging Brown
(2009). The Cartesian grids are extended to cover the wave zone (at reduced
resolution for the sake of efficiency), extending to a cubical grid outer
boundary where an artificial condition is applied.
A principal difficulty faced by this method is that the discretisation is not
well suited to model radial waves at large radii. In order to resolve the wave
profile, a certain minimum radial resolution is required and must be
maintained as the wave propagates to large radii. The angular resolution,
however, can remain fixed – if a wave is resolved at a certain angular
resolution as small radii, then it is unlikely to develop significant angular
features as it propagates to large distances from the isolated source.
Cartesian grids with fixed spacing, however, resolve spheres with an angular
resolution which scales according to $r^{2}$. Thus, to maintain a given
required radial resolution, the angular directions become extremely over-
resolved at large radii, and this comes at a large computational cost. Namely,
for a Cartesian grid to extend in size or increase it’s resolution by a factor
$n$, the cost in memory and number of computations per timestep increase by
$n^{3}$, while for a radial grid with fixed angular resolution, the increase
is linear, $n$ 222Note that the Courant limit introduces an additional factor
of $n$ in each case due to the requirement of a reduced timestep with
increasing resolution..
For the near-zone, in the neighbourhood of the orbits of the individual
bodies, the geometrical situation is not as straightforward, since there is no
clearly defined radial propagation direction between the bodies. If the local
geometry is reasonably well known (for instance, the location of horizon
surfaces), adapted coordinates can also be considered in this regime. The
technical requirements of such coordinate systems can, however, be high. Since
the bodies are moving, the grids must move with them, or dynamical gauges
chosen such that the bodies remain in place relative to the numerical
coordinates. Potential problems arise from the coordinate singularity if the
grids are extended to $r=0$, as is the case with the standard puncture
approach. Thus, in the near-zone, Cartesian coordinates can provide
significant simplification to the overall infrastructure requirements, while
the previously mentioned drawbacks of Cartesian coordinates are less
prevalent, as it is useful to have homogeneous resolution in each direction in
situations where there is no obvious symmetry.
The evolution code which we have constructed for the purpose of modelling
waveforms from an isolated system is based on a hybrid approach, involving a
Cartesian mesh-refined region covering the near zone in which the bodies
orbit, and a set of adapted radial grids which efficiently cover the wave
zone. The overall structure is illustrated in Fig. 1 (top), which shows an
equatorial slice of the numerical grid. Computations on each local patch are
carried out in a globally Cartesian coordinate system. In the particular
implementation considered here, the grids overlap by some distance so that
data at the boundaries between each local coordinate patch can be communicated
by interpolation from neighbouring patches. The resulting code is both
efficient, but also simple in structure and able to take advantage of well
established methods for evolving moving puncture black holes. If suitable
interpolation methods are used, then such a system can also be used for
solutions with discontinuities and shocks as are present in hydrodynamics.
The code has been implemented within the Cactus framework Goodale et al.
(2003); cac via extensions to the Carpet driver Schnetter et al. (2004);
Schnetter et al. (2006b); car , which handles parallelisation via domain
decomposition of grids over processors, as well as providing the required
interpolation operators for boundary communication and analysis tools.
### II.1 Coordinate systems
The configuration displayed in Fig. 1 consists of seven local coordinate
patches: an interior Cartesian grid, and six outer patches corresponding to
the faces of the interior cube. Each patch consists of a uniformly spaced (in
local coordinates) grid which can be refined (though in practise we will only
use this feature for the interior grid). The outer patches have a local
coordinate system $(\rho,\sigma,R)$ corresponding to the “inflated cube”
coordinates which have previously been used in relativity for single black
hole evolutions Thornburg (2004), and are displayed in the lower plot of Fig.
1. The local angular coordinates $(\rho,\sigma)$ range over
$(-\pi/4,+\pi/4)\times(-\pi/4,+\pi/4)$ and can be related to global angular
coordinates $(\mu,\nu,\phi)$ which are given by
$\displaystyle\mu\equiv\text{rotation angle about the x-axis}$
$\displaystyle=\arctan(y/z),$ (1a) $\displaystyle\nu\equiv\text{rotation angle
about the y-axis}$ $\displaystyle=\arctan(x/z),$ (1b)
$\displaystyle\phi\equiv\text{rotation angle about the z-axis}$
$\displaystyle=\arctan(y/x).$ (1c)
For example, on the $+z$ patch, the mapping between the local
$(\rho,\sigma,R)$ and Cartesian $(x,y,z)$ coordinates is given by:
$\displaystyle\rho\equiv\nu$ $\displaystyle=\arctan(x/z),$ (2a)
$\displaystyle\sigma\equiv\mu$ $\displaystyle=\arctan(y/z),$ (2b)
$\displaystyle R$ $\displaystyle=f(r),$ (2c)
with appropriate rotations for each of the other cube faces, and where
$r=\sqrt{x^{2}+y^{2}+z^{2}}$. As emphasised by Thornburg Thornburg (2004), in
addition to avoiding pathologies associated with the axis of standard
spherical polar coordinates, this choice of local coordinates has a number of
advantages. In particular, the angular coordinates on neighbouring patches
align so that interpolation is only 1-dimensional, in a line parallel to the
face of the patch. This potentially improves the efficiency of the
interpolation operation as well as the accuracy. The coordinates also cover
the sphere more uniformly than, say, a stereographic 2-patch system.
Figure 1: A schematic view of the $z=0$ slice of the grid setup that is used.
The upper plot shows the central Cartesian grid surrounded by six “inflated-
cube” patches (the four equatorial patches are shown, shaded). The boundaries
of the nominal grids owned by each patch are indicated by thick lines. The
lower plot shows an $r=\textrm{constant}$ surface of the exterior patches,
indicating their local coordinate lines.
The local radial coordinate, $R$, is determined as a function of the global
coordinate radius, $r$. We can use this degree of coordinate freedom to
increase the physical (global) extent of the wave-zone grids, at the cost of
some spatial resolution. In practise, we use a function of the form
$f(r)=A(r-r_{0})+B\sqrt{1+(r-r_{0})^{2}/\epsilon},$ (3a) with $R=f(r)-f(0).$
(3b)
in order to transition between two almost constant resolutions (determined by
the parameters $A$ and $B$) over a region whose width is determined by
$\epsilon$, centred at $r_{0}$.
The effect of the radial transformation is illustrated in Fig. 2. The
coordinate $R$ is a nearly constant rescaling of $r$ at small and large radii.
The change in the scale factor is largely confined to a transition region.
Note that since we apply the same global derivative operators (described
below) to analysis tools as are used for the the evolution, it is possible to
do analysis (e.g., measure waveforms, horizon finding) within regions where
the radial coordinate is non-uniform. The regions of near-constant resolution
are, however, useful in order to make comparisons of measurements at different
radii without the additional complication of varying numerical error due to
the underlying grid spacing.
Figure 2: The local radial coordinate, $R$ (solid line), can be stretched as a
function of the global coordinate, $r$, in order to increase the effective
size of the grid. The function specified by Eqs. (3b) transitions between two
almost constant radial resolutions over a distance $\epsilon$ centred at
$r_{0}$.
Data on each patch are evaluated at grid-points which are placed at uniformly
spaced points of a Cartesian grid. Thus, local derivatives can be calculated
via standard finite difference techniques. These are then transformed to a
common underlying Cartesian coordinate system by applying an appropriate
Jacobian which relates the local to global coordinates. That is, the global
(Cartesian) coordinates, $x_{i}$, are related to the local coordinates,
$a_{i}$, by
$x_{i}=x_{i}(a_{j}),\qquad i,j=0,1,2.$ (4)
and derivatives, $\partial/\partial a_{i}$, of fields are determined using
finite differences in the regularly spaced $a_{i}$ coordinates, which are then
transformed using
$\displaystyle\frac{\partial}{\partial x_{i}}$
$\displaystyle=\left(\frac{\partial a_{j}}{\partial
x_{j}}\right)\frac{\partial}{\partial a_{j}},$ (5a)
$\displaystyle\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}$
$\displaystyle=\left(\frac{\partial^{2}a_{k}}{\partial x_{i}\partial
x_{j}}\right)\frac{\partial^{2}}{\partial a_{k}^{2}}+\left(\frac{\partial
a_{k}}{\partial x_{i}}\frac{\partial a_{l}}{\partial
x_{j}}\right)\frac{\partial^{2}}{\partial a_{k}\partial a_{l}},$ (5b)
in order to determine their values in the global frame. We store and evaluate
tensor components and their evolution equations in the common global frame, so
that there is no need to apply transformations when relating data across patch
boundaries. In addition to the obvious simplification of the inter-patch
boundary treatment, this has a number of other advantages, particularly when
it comes to analysis tools (surface finding, gravitational wave measurements,
visualisation) which may reference data on multiple patches. Since the data is
represented in the common global basis, these tools do not need to know
anything about the local grid structures or coordinates.
### II.2 Inter-patch interpolation
Data is communicated between patches by interpolating onto overlapping points.
Each patch, $p$, is responsible for determining the numerical solution for a
region of the spacetime. The boundaries of these patches can overlap
neighbouring patches, $q$, (and in fact must do so for the case of the
interpolating boundaries considered here), creating regions of the spacetime
which are covered by multiple patches. We define three sets of points on a
patch. The _nominal_ regions, $\mathcal{N}_{p}$, contain the points where the
numerical solution is to be determined. The nominal regions of the patches do
not overlap, $\bigcap_{p}\mathcal{N}_{p}=\emptyset$, so that if data is
required at any point in the spacetime, it can be obtained without ambiguity
by referencing the single patch in whose nominal region it resides. A patch,
$p$, is bounded by a layer of _ghost_ points, $\mathcal{G}_{p}$, which overlap
the nominal zones of neighbouring patches, $q$,
$\mathcal{G}_{p}\cap\bigcup_{q}\mathcal{N}_{q}=\mathcal{G}_{p}$, and are
filled by interpolation. (These points are conceptually similar to the inter-
processor ghost-zones used by domain decomposition parallelisation algorithms
in order to divide grids over processors.) The size of these regions is
determined by the width of the finite difference stencil to be used in finite
differencing the evolution equations on the nominal grid. Finally, an
additional layer of _overlap_ points, $\mathcal{O}_{q}$, is required: i) to
ensure that the set of stencil points,
$\mathcal{S}_{q}\subset\mathcal{O}_{q}\cup\mathcal{N}_{q}$, used to
interpolated to the ghost zone does not itself originate from the ghost zone
of the interpolating patch, $\mathcal{S}_{q}\cap\mathcal{G}_{q}=\emptyset$,
$\mathcal{O}_{q}\cap\bigcup_{p}\mathcal{N}_{p}=\mathcal{O}_{q}$; and ii) to
compensate for any difference in the grid spacing between the local
coordinates on the two patches. An illustration of the scheme in 1-dimension
the scheme is provided in Fig. 3.
Figure 3: Schematic of interpolating patch boundaries in 1-dimension, assuming
4-point finite difference and interpolation stencils. Points in the nominal
zones, $\mathcal{N}_{p,q}$, are indicated by filled circles, points in ghost
zones, $\mathcal{G}_{p,q}$, by open squares, and points in overlap zones,
$\mathcal{O}_{p,q}$, by closed squares. The vertical dotted line demarcates
the boundary between nominal zones on each patch. Ghost points on patch $p$
are evaluated by centred interpolation operations from points in
$\mathcal{S}_{q}$ on the overlapping patch (arrows) and _vice versa_.
Note that points in
$\bigcup_{q}\mathcal{O}_{q}\subset\bigcup_{p}\mathcal{N}_{p}$ are not
interpolated, but rather are evolved in the same way as nominal grid points
within $\bigcup_{p}\mathcal{N}_{p}$. That is, in these regions points on each
grid are evolved independently, and is in principle multi-valued. However,
since the union of set of nominal points on each patch
$\bigcup_{p}\mathcal{N}_{p}$ uniquely and unambiguously covers the entire
simulation domain, i.e. $\bigcap_{p}\mathcal{N}_{p}=\emptyset$, and since the
overlap regions are a subset of the nominal grid, if data is required at a
point within these overlap zones, there is exactly one patch owing this point
on its nominal grid, and it will be returned uniquely from this patch. The
differences between evolved field values evaluated in these overlap points
converge away with the finite difference order of the evolution scheme.
The use of additional overlap points makes this inter-patch interpolation
algorithm somewhat simpler than the one implemented by Thornburg in Thornburg
(2004). That algorithm required inter-patch boundary conditions to be applied
in a specific order to ensure that all interpolation stencils are evaluated
without using undefined grid points, and requires off-centring interpolation
stencils under certain circumstances, which is not necessary when overlap
points are added. It also relies on the particular property of the inflated-
cube coordinates which ensured that the ghost-zones could be filled using
1-dimensional interpolation in a direction orthogonal to the boundary. This
property would be non-trivial (and often impossible) to generalise to match
arbitrary patch boundaries, such as that between the Cartesian and radially
oriented grids of Fig. 1.
Another significant difference between Thornburg’s approach and the approach
presented here is that former stores tensor components in the patch-local
frame, while we store them in the global coordinate frame. Evaluating
components in the patch-local frame requires a basis transformation while
interpolating. This is further complicated in the case of non-tensorial
quantities (such as the $\tilde{\Gamma}^{i}$ of the BSSNOK formulation) which
have quite complicated basis transformation rules involving spatial
derivatives. Instead, we store tensor components in the global coordinate
frame, which requires no basis transformation during inter-patch
interpolations.
The number of ghost points in $\mathcal{G}_{p}$ can be reduced using finite
difference stencils which become lop-sided towards the boundaries of the
patch, and may provide an important optimisation since interpolation between
grids can be expensive, particularly if processor communication is involved.
However, this tends to be at the cost of increased numerical error in the
finite difference operations towards the grid boundaries. We have generally
found it preferable to use centred stencils throughout the nominal,
$\mathcal{N}_{p}$, and overlap, $\mathcal{O}_{p}$, zones and have applied
certain optimisations to the interpolation operators as described below.
Another optimisation can be achieved by using lower order interpolation so
that it is possible to reduce the number of overlapping points in
$\mathcal{O}_{p}$.
The interpolation scheme for evaluating data in ghost zones is based on
Lagrange polynomials using data from the overlapping patch. In 1-dimension,
the Lagrange interpolation polynomial can be written as
$\mathcal{L}_{x}[\phi](x)=\sum_{i}^{N}b_{i}(x)\,\phi_{i}\,,$ (6a) where the
coefficients are $b_{i}(x)=\prod_{k\neq i}\frac{(x-x_{k})}{(x_{i}-x_{k})}\,.$
(6b)
In these expressions, $x\in\mathcal{G}_{p}$ is the coordinate of the
interpolation point and
$\phi_{i}\in\mathcal{S}_{q}\subset\mathcal{N}_{q}\cup\mathcal{O}_{q}$ are the
values at grid-points in the interpolation molecule surrounding $x$. The
number of grid-points in the interpolation molecule, $N$, determines the
interpolation order, and interpolation of $n$-th order accuracy is given by
$N=n+1$ stencil points in the molecule.
For interpolation in $d$-dimensions, the interpolation polynomial can be
constructed as a tensor product of 1-dimensional Lagrange interpolation
polynomials along coordinate directions, $\mathbb{x}=(x^{1},...,x^{d})$:
$\displaystyle\mathcal{L}[\phi](\mathbb{x})$
$\displaystyle=\mathcal{L}_{x^{1}}[\phi](x^{1})\otimes\ldots\otimes\mathcal{L}_{x^{d}}[\phi](x^{d})$
$\displaystyle=\left(\sum_{i}^{N}b_{i}(x^{1})\,\phi_{i}\,,\right)\cdots\left(\sum_{j}^{N}c_{j}(x^{d})\,\phi_{j}\right)\,.$
(7)
Therefore, for $d$-dimensional interpolation of order $n$, one has to
determine $N^{d}$ neighbouring stencil points and associated interpolation
coefficients, Eq. (6b), _for each_ point in the ghost-zone of a given patch.
Most generally, full 3-dimensional interpolation is required, though in
particular cases coordinates between two patches can be constructed such that
they align locally so that only 1-dimensional interpolation is needed. This
is, for instance, the case for the overlap region between the inflated-cube
spherical patches used here (see Fig. 1). We have optimised the current code
to automatically take advantage of this.
In order to interpolate to a point for which the coordinates $a^{p}_{i}$ given
in the basis of patch $p$ are given, we need to know the patch owning the
nominal region containing this point. For this we first convert $a^{p}_{i}$ to
the global coordinate basis $x_{i}$, then determine which patch $q$ owns the
corresponding nominal region $\mathcal{N}_{q}$, and then convert $x_{i}$ to
the local coordinate bases this patch $a^{q}_{i}$. By construction, patch $q$
has sufficient overlap points to evaluate the interpolation stencil there:
$\displaystyle x_{i}$ $\displaystyle:=\textrm{local-to-
global}_{p}(a^{p}_{i})\,,$ (8a) $\displaystyle q$
$\displaystyle:=\textrm{owning-patch}(x^{i})\,,$ (8b) $\displaystyle
a^{q}_{i}$ $\displaystyle:=\textrm{global-to-local}_{q}(x^{i})\,.$ (8c)
The three operations “local-to-global”, “owning-patch”, and “global-to-local”
depend on the patch system and their local coordinate systems.
We can then find the points of patch $q$ that are closest to the interpolation
point $a^{q}_{i}$ in the local coordinates this patch. In order to find these
points, we exploit the uniformity of the grid in local coordinates. The grid
indices of the stencil points in a given direction are determined via
$i\in\left\\{\text{floor}(j+k)\;\middle|\;j=\frac{x-x_{0}}{\Delta
x},\;k=-\frac{n}{2},\cdots,\frac{n}{2}\right\\}\,,$ (9)
where $x_{0}$ is the origin of the local grid, $n$ is the interpolation order,
and “floor” denotes rounding downwards to the nearest integer.
There remains to be determined the refinement level which contains the region
surrounding the interpolation point, as well as the processor that owns this
part of the grid. For this purpose, an efficient tree-search algorithm has
been implemented. In this algorithm, the individual patches and refinement
levels are defined as “super-regions”, i.e., bounding boxes that delineate the
global grid extent before processor decomposition. Each of these super-regions
is recursively split into smaller regions. The leaves of the resulting tree
structure represent the individual local components of the processor
decomposition. Locating a grid point in this tree structure requires $O(\log
n)$ operations on $n$ processors, whereas a linear search (that would be
necessary without a tree structure) would require $O(n)$ operations.
While the corresponding tree structure is generic, the actual algorithm used
in Carpet splits the domain into a $kd$ tree of depth $d$ in $d=3$ dimensions.
That is, the domain is first split into $k$ sub-domains in the $x$ direction,
each of these sub-domains is then independently split into several in the $y$
direction, and each of these is then split in the $z$ direction. This leads to
cuboid sub-domains for each processor, where the sub-domains do not overlap,
and where each sub-domain can have a different shape. Carpet balances the load
so that each processor receives approximately the same number of grid points,
while keeping the sub-domains’ shapes as close to a cube as possible.
Our implementation pre-calculates and stores most of the above information
when the grid structure is set up, saving a significant amount of time during
interpolation. In particular, the following are stored:
* •
For each ghost-point, the source patch (where the interpolation is performed),
and the local coordinates on this patch;
* •
For each ghost-point, the interpolation stencil coefficients (6b);
* •
For each processor, the communication schedule specifying which interpolation
points need to be sent to what other processor.
When the grid structure changes, for example, when a mesh-refinement grid is
moved or resized, these quantities have to be recalculated.
Altogether, the inter-patch interpolation therefore consists of applying
processor-local interpolation stencils, sending the results to other
processors, receiving results from other processors, and storing these results
in the local ghost-points. These are all operations requiring no look-up in
complex data structures, and which consequently execute very efficiently on
modern hardware.
### II.3 Finite differencing
Spatial derivatives are computed using standard finite difference stencils,
which have currently been implemented up to 8th-order Diener et al. (2007).
The stencils are centred, except for the terms corresponding to an advection
by the shift vector, of the form $\beta^{i}\partial_{i}u$ (see Sec. III,
below). These derivatives are calculated using an “upwind” stencil which is
shifted by one point in the direction of the shift, and of the same order. We
find that these upwind stencils provide a significant increase in the
numerical accuracy of the puncture motion at a given resolution (see Appendix
A). The particular stencils which we use can be generated via a recursion
algorithm, as outlined in Fornberg (1988).
On each patch we allow the local grids to be refined in order to increase the
accuracy of computations in localised regions. For the application of the
evolution of an isolated binary considered here, we only refine the central
Cartesian grid in the neighbourhood the bodies. This is done using standard
$2:1$ Berger-Oliger mesh refinement techniques via the Carpet infrastructure
Schnetter et al. (2004); Schnetter et al. (2006b); car . The time step for the
outer patches is taken to be the same as the coarse grid step of the interior
patch, so that no time-interpolation is required at inter-patch boundaries.
Time integration is carried out using standard method-of-lines integrators. We
find that for the time resolution we are using, a 4th-order Runge-Kutta (RK4)
method provides a good compromise between sufficient accuracy and a low memory
footprint. We set the time resolution of the outer grids according to the
timestep of the coarsest Cartesian grid, which is limited by the Courant
condition at the specified spatial resolution. By placing the Cartesian-
spherical boundary at a small radius (and thus extending to finer Cartesian
grids) we attain a high time resolution in the wave zone, potentially
important for determining higher harmonics.
### II.4 Surface integration and harmonic decomposition
A number of quantities of physical interest are measured by projecting them
onto closed surfaces surrounding the source. In particular, gravitational wave
measurements rely on computations on constant coordinate spheres $S^{2}$,
parameterized by local spherical-polar coordinates $(\theta,\phi)$ with
constant coordinate radius $r$. In principle, it would be possible to
construct coordinates on these 2-dimensional spheres which correspond to the
underlying grid coordinates of the evolution, for instance as portrayed in the
lower figure of Fig. 1. In this case, data can be mapped directly onto the
spheres. More generally, however, interpolation can be used to obtain data at
points on the measurement spheres, according to the procedure outlined in Sec.
II.2, above.
For the purpose of analysis, it is often convenient to decompose the data on
$S^{2}$ in terms of (spin-weighted) spherical harmonic modes,
$A_{\ell m}=\int d\Omega\sqrt{-g}A(\Omega){}_{s}\bar{Y}_{\ell m}(\Omega)\,,$
(10)
where $g$ is the determinant of the surface metric and $\Omega$ angular
coordinates. In standard spherical-polar coordinates $(\theta,\phi)$,
$\sqrt{-g}=\sin^{2}\theta\,.$ (11)
The integral, Eq. (10), is solved numerically as follows. In the spherical
polar case, we can take advantage of an highly accurate Gauss quadrature
scheme which is exact for polynomials of order up to $2N-1$, where $N$ is the
number of gridpoints. More specifically, we use Gauss-Chebyshev quadrature.
The scheme can be written out as
$\int d\Omega
f(\Omega)=\sum_{i}^{N_{\theta}}\sum_{j}^{N_{\phi}}f_{ij}w_{j}+\mathcal{O}(N_{\theta})\,,$
(12)
where $N_{\theta}$ and $N_{\phi}$ are the number of angular gridpoints and
$w_{j}$ are weight functions Driscoll and Healy (1994); Bateman (1955),
$\displaystyle w_{j}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{N_{\phi}}\frac{1}{N_{\theta}\sqrt{2\pi}}\sum_{l=0}^{N_{\theta}/2-1}\frac{1}{2l+1}\sin\left([2l+1]\frac{\pi
j}{N}\right)\,,$ (13) $\displaystyle\qquad j=0,...,N_{\theta}-1\,.$
In our implementation, the weight functions are pre-calculated for fast
surface integration.
## III Evolution system
We evolve the spacetime using a variant of the “BSSNOK” evolution system
Nakamura et al. (1987); Shibata and Nakamura (1995); Baumgarte and Shapiro
(1998); Alcubierre et al. (2000a) and a specific set of gauges Alcubierre et
al. (2003); van Meter et al. (2006), which have been shown to be effective at
treating the coordinate singularities of Bowen-York initial data.
The 4-geometry of a spacelike slice $\Sigma$ (with timelike normal,
$n^{\alpha}$) is determined by its intrinsic 3-metric, $\gamma_{ab}$ and
extrinsic curvature, $K_{ab}$, as well as a scalar lapse function, $\alpha$,
and shift vector, $\beta^{a}$ which determine the coordinate propagation. The
standard BSSNOK system defines modified variables by performing a conformal
transformation on the 3-metric,
$\phi:=\frac{1}{12}\ln\det\gamma_{ab},\qquad\tilde{\gamma}_{ab}:=e^{-4\phi}\gamma_{ab},$
(14)
subject to the constraint
$\det\tilde{\gamma}_{ab}=1,$ (15)
and by removing the trace of $K_{ab}$,
$\displaystyle K$
$\displaystyle:=\operatorname{\mathrm{tr}}K_{ij}=g^{ij}K_{ij},$ (16)
$\displaystyle\tilde{A}_{ij}$
$\displaystyle:=e^{-4\phi}(K_{ij}-\frac{1}{3}\gamma_{ij}K),$ (17)
with the constraint
$\tilde{A}:=\tilde{\gamma}^{ij}\tilde{A}_{ij}=0.$ (18)
Additionally, three new variables are introduced, defined in terms of the
Christoffel symbols of $\tilde{\gamma}_{ab}$ by
$\tilde{\Gamma}^{a}:=\tilde{\gamma}^{ij}\tilde{\Gamma}^{a}_{ij}.$ (19)
In principle the $\tilde{\Gamma}^{a}$ can be determined from the
$\tilde{\gamma}_{ab}$, on a slice however their introduction is key to
establishing a strongly hyperbolic (and thus stable) evolution system. In
practise, only the constraint Eq. (18) is enforced during evolution Alcubierre
et al. (2000b), while Eq. (15) and Eq. (19) are simply monitored as indicators
of numerical error. Thus, the traditional BSSNOK prescription evolves the
variables
$\phi,\quad\tilde{\gamma}_{ab},\quad
K,\quad\tilde{A}_{ab},\quad\tilde{\Gamma}^{a},$ (20)
according to evolution equations which have been written down a number of
times (see Baumgarte and Shapiro (2003); Alcubierre (2008) reviews).
In the context of puncture evolutions, it has been noted that alternative
scalings of the conformal factor may exhibit better numerical behaviour in the
neighbourhood of the puncture as compared with $\phi$. In particular, a
variable of the form
$\hat{\phi}_{\kappa}:=(\det\gamma_{ab})^{-1/\kappa},$ (21)
has been suggested Campanelli et al. (2006); Marronetti et al. (2008). In
Campanelli et al. (2006), it is noted that certain singular terms in the
evolution equations for Bowen-York initial data can be corrected using
$\chi:=\hat{\phi}_{3}$. Alternatively, Marronetti et al. (2008) notes that
$W:=\hat{\phi}_{6}$ has the additional benefit of ensuring $\gamma$ remains
positive, a property which needs to be explicitly enforced with $\chi$.
In terms of $\hat{\phi}_{\kappa}$, the BSSNOK evolution equations become:
$\displaystyle\partial_{t}\hat{\phi}_{\kappa}=$
$\displaystyle\frac{2}{\kappa}\hat{\phi}_{\kappa}\alpha
K+\beta^{i}\partial_{i}\hat{\phi}_{\kappa}-\frac{2}{\kappa}\hat{\phi}_{\kappa}\partial_{i}\beta^{i},$
(22a) $\displaystyle\partial_{t}\tilde{\gamma}_{ab}=$
$\displaystyle-2\alpha\tilde{A}_{ab}+\beta^{i}\partial_{i}\tilde{\gamma}_{ab}+2\tilde{\gamma}_{i(a}\partial_{b)}\beta^{i}$
(22b) $\displaystyle-\frac{2}{3}\tilde{\gamma}_{ab}\partial_{i}\beta^{i},$
$\displaystyle\partial_{t}K=$ $\displaystyle-
D_{i}D^{i}\alpha+\alpha(A_{ij}A^{ij}+\frac{1}{3}K^{2})+\beta^{i}\partial_{i}K,$
(22c) $\displaystyle\partial_{t}\tilde{A}_{ab}=$
$\displaystyle(\hat{\phi}_{\kappa})^{\kappa/3}(-D_{a}D_{b}\alpha+\alpha
R_{ab})^{\text{TF}}+\beta^{i}\partial_{i}\tilde{A}_{ab}$ (22d)
$\displaystyle+2\tilde{A}_{i(a}\partial_{b)}\beta^{i}-\frac{2}{3}A_{ab}\partial_{i}\beta^{i},$
$\displaystyle\partial_{t}\tilde{\Gamma}^{a}=$
$\displaystyle\tilde{\gamma}^{ij}\partial_{i}\beta_{j}\beta^{a}+\frac{1}{3}\tilde{\gamma}^{ai}\partial_{i}\partial_{j}\beta^{j}-\tilde{\Gamma}^{i}\partial_{i}\beta^{a}$
(22e)
$\displaystyle+\frac{2}{3}\tilde{\Gamma}^{a}\partial_{i}\beta^{i}-2\tilde{A}^{ai}\partial_{i}\alpha$
$\displaystyle+2\alpha(\tilde{\Gamma}^{a}_{ij}\tilde{A}^{ij}-\frac{\kappa}{2}\tilde{A}^{ai}\frac{\partial_{i}\hat{\phi}_{\kappa}}{\hat{\phi}_{\kappa}}-\frac{2}{3}\tilde{\gamma}^{ai}\partial_{i}K),$
where $D_{a}$ is the covariant derivative determined by $\tilde{\gamma}_{ab}$,
and “TF” indicates that the trace-free part of the bracketed term is used.
We have implemented the traditional $\phi$ form of the BSSNOK evolution
equations, as well as the $\chi$ and $W$ variants implicit in the evolution
system, Eqs. (22). All three evolution systems produce stable evolutions of
binary black holes, though the $\chi$ variant requires some special treatment
if, due to numerical error particularly in the neighbourhood of the punctures,
$\hat{\phi}_{3}\leq 0$ Bruegmann et al. (2008). Generally we find that the
advection of the puncture (and thus the phase accuracy of the simulation)
exhibits lower numerical error when using the $\chi$ and $W$ variants (see
Appendix C). Convergence properties of physical variables (such as measured
gravitational waves, or constraints measured outside of the horizons),
however, are not affected by the choice of conformal variable.
The Einstein equations are completed by a set of four constraints which must
be satisfied on each spacelike slice:
$\displaystyle\mathcal{H}$ $\displaystyle\equiv R^{(3)}+K^{2}-K_{ij}K^{ij}=0,$
(23a) $\displaystyle\mathcal{M}^{a}$ $\displaystyle\equiv
D_{i}(K^{ai}-\gamma^{ai}K)=0.$ (23b)
Again, we do not actively enforce these equations, but rather monitor their
magnitude in order to determine whether our numerical solution is deviating
from a solution to the Einstein equations.
The gauge quantities, $\alpha$ and $\beta^{a}$, are evolved using the
prescriptions that have been commonly applied to BSSNOK black hole, and
particularly puncture, evolutions. For the lapse, we evolve according to the
“$1+\log$” condition Bona et al. (1995),
$\partial_{t}\alpha-\beta^{i}\partial_{i}\alpha=-2\alpha K,$ (24)
while the shift is evolved using the hyperbolic “$\tilde{\Gamma}$-driver”
equation Alcubierre et al. (2003),
$\displaystyle\partial_{t}\beta^{a}-\beta^{i}\partial_{i}\beta^{a}$
$\displaystyle=\frac{3}{4}\alpha B^{a}\,,$ (25a)
$\displaystyle\partial_{t}B^{a}-\beta^{j}\partial_{j}B^{i}$
$\displaystyle=\partial_{t}\tilde{\Gamma}^{a}-\beta^{i}\partial_{i}\tilde{\Gamma}^{a}-\eta
B^{a}\,,$ (25b)
where $\eta$ is a parameter which acts as a (mass dependent) damping
coefficient, and is typically set to values on the order of unity for the
simulations carried out here. The advective terms in these equations were not
present in the original definitions of Alcubierre et al. (2003), where co-
moving coordinates were used, but have been added following the experience of
more recent studies using moving punctures Baker et al. (2006a); van Meter et
al. (2006).
### III.1 Wave extraction
The Newman-Penrose formalism Newman and Penrose (1962) provides a convenient
representation for a number of radiation related quantities as spin-weighted
scalars. In particular, the curvature component
$\psi_{4}\equiv-
C_{\alpha\beta\gamma\delta}n^{\alpha}\bar{m}^{\beta}n^{\gamma}\bar{m}^{\delta},$
(26)
is defined as a particular component of the Weyl curvature,
$C_{\alpha\beta\gamma\delta}$, projected onto a given null frame,
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\bar{\boldsymbol{m}}\\}$.
The identification of the Weyl scalar $\psi_{4}$ with the gravitational
radiation content of the spacetime is a result of the peeling theorem Sachs
(1961); Newman and Penrose (1962); Penrose (1963), which states that in an
appropriate frame and for sufficiently smooth and asymptotically flat initial
data near spatial infinity, the $\psi_{4}$ component of the curvature has the
slowest fall-off with radius, $\mathcal{O}(1/r)$.
The most straight-forward way of evaluating $\psi_{4}$ in numerical (Cauchy)
simulations is to define an orthonormal basis in the three space
$(\hat{\boldsymbol{r}},\hat{\boldsymbol{\theta}},\hat{\boldsymbol{\phi}})$,
centered on the Cartesian grid center and oriented with poles along
$\hat{\boldsymbol{z}}$. The normal to the slice defines a time-like vector
$\hat{\boldsymbol{t}}$, from which we construct the null frame
$\boldsymbol{l}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{t}}-\hat{\boldsymbol{r}}),\quad\boldsymbol{n}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{t}}+\hat{\boldsymbol{r}}),\quad\boldsymbol{m}=\frac{1}{\sqrt{2}}(\hat{\boldsymbol{\theta}}-{\mathrm{i}}\hat{\boldsymbol{\phi}})\
.$ (27)
Note that in order to make the vectors
$\\{\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\bar{\boldsymbol{m}}\\}$
null,
$(\hat{\boldsymbol{r}},\hat{\boldsymbol{\theta}},\hat{\boldsymbol{\phi}})$
have to be orthonormal relative to the spacetime metric. In practice, we fix
$\hat{\boldsymbol{r}}$ and then apply a Gram-Schmidt orthonormalization
procedure to determine $\hat{\boldsymbol{\theta}}$ and
$\hat{\boldsymbol{\phi}})$ 333Alternative frame constructions have also been
used, such as orthonormalising relative to one of the angular basis vectors
Baker et al. (2002), or omitting the orthonormalisation altogether Scheel et
al. (2009). We have generally found these modifications do not lead to
significantly different measurements. It is then possible to calculate
$\psi_{4}$ via a reformulation of (26) in terms of the geometrical variables
on the slice Gunnarsen et al. (1995) via the electric and magnetic parts of
the Weyl tensor,
$\psi_{4}=C_{ij}\bar{m}^{i}\bar{m}^{j}\,,$ (28)
where
$C_{ij}\equiv E_{ij}-iB_{ij}=R_{ij}-KK_{ij}+K_{i}{}^{k}K_{kj}-{\rm
i}\epsilon_{i}{}^{kl}\nabla_{l}K_{jk}\,.$ (29)
The remaining Weyl scalars can be similarly obtained and read
$\displaystyle\psi_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}C_{ij}\bar{m}^{i}e_{r}^{j}\,,$ (30a)
$\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}C_{ij}e_{r}^{i}e_{r}^{j}\,,$ (30b)
$\displaystyle\psi_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}C_{ij}m^{i}e_{r}^{j}\,,$ (30c)
$\displaystyle\psi_{0}$ $\displaystyle=$ $\displaystyle C_{ij}m^{i}m^{j}\,,$
(30d)
where $(e_{r}^{j})\equiv\hat{\boldsymbol{r}}$ is the unit radial vector.
In relating $\psi_{4}$ to the gravitational radiation, one is limited by the
fact that the measurements have been taken at a finite radius from the source.
Local coordinate and frame effects can complicate the interpretation of
$\psi_{4}$. These problems can largely be alleviated by taking measurements at
several radii and performing polynomial extrapolations to
$r\rightarrow\infty$. Procedures for doing so have been studied in Boyle and
Mroue (2009); Pollney et al. (2009). In Pollney et al. (2009) we have shown
that if a sufficiently large outermost extrapolation radius is used, the
variation in this procedure is of the order $\Delta A=0.03\%$ and
$\Delta\phi=0.003\,\text{rad}$ in amplitude and phase respectively, and is
consistent throught the evolution, including inspiral, merger and ringdown.
The extrapolation error is larger than the numerical error determined in Sec.
IV.3.2, below, even if it is performed using data at $r=1000M$ distant from
the source, highlighting the need for measurements at large radii. For the
“extrapolated” data plotted in this paper, we have performed polynomial
extrapolations as detailed in Pollney et al. (2009), using the six outermost
measurements at $r=\\{280M,300M,400M,500M,600M,1000M\\}$.
In a companion paper Reisswig et al. (2009b), we use the same dataset to
calculate $\psi_{4}$ directly at $\mathcal{J}^{+}$ using characteristic
extraction Bishop et al. (1999); Babiuc et al. (2009). Here the traditional
approach (which is gauge dependent and has a finite-radius cut-off error)
presented here is replaced by a characteristic formulation of the Einstein
equations in order to determine the fields out to future null infinity. In
this paper, we restrict ourselves to a discussion of the numerical error
inherent in the evolution procedure via the multi-patch code, and will report
in more detail on systematic measurement errors due to finite radius effects
and the characteristic extraction procedure elsewhere Reisswig et al. (2009b,
c).
## IV Code verification
### IV.1 Initial data
To demonstrate the efficacy of the infrastructure described in the previous
sections, we have carried out an evolution of the by now well-studied case of
the late-inspiral and merger of a pair of non-spinning equal-mass black holes
(see, for example, Hannam et al. (2009) for an extensive discussion of
numerical results involving this model). The particular numerical evolution
which we have carried out starts from an initial separation $d/M=11.0$ and
goes through approximately 8 orbits (a physical time of around $1360M$),
merger and ring-down. The masses of the punctures are set to $m=0.4872$ and
are initial placed on the $x$-axis with momenta $p=(\pm 0.0903,\mp
0.000728,0)$, giving the initial slice an ADM mass $M_{\rm ADM}=0.99051968\pm
2\times 10^{-8}$. These initial data parameters were determined using a post-
Newtonian evolution from large initial separation, following the procedure
outlined in Husa et al. (2008), with the conservative part of the Hamiltonian
accurate to 3PN, and radiation-reaction to 3.5PN, and determines orbits with a
measured eccentricity of $e=0.004\pm 0.0005$.
### IV.2 Grid setup
The binary black hole evolution was carried out on a 7-patch grid structure,
as described in Sec. II, incorporating a Cartesian mesh-refined region which
covers the near-zone, and six radially oriented patches covering the wave-
zone.
The inner boundary of the radial grids was placed at $r_{t}=35.2M$ relative to
the centre of the Cartesian grid. As a general rule, this boundary should be
made as small as possible to improve efficiency in terms of memory usage.
However other factors may make it preferable to move it further out. In
particular, since we do not perform time interpolation at grid boundaries, the
time step $dt$ of the coarsest Cartesian grid determines the timestep of the
radial grids, and thus the wave zone. Updates of the radial grids tend to be
expensive, as they are large, so that if $dt$ is too small, computation time
may be spent over-resolving (in time) the wave zone. Particularly if the
principle interest is in the lower order wave modes, it may be optimal to add
an additional Cartesian mesh-refinement grid with a coarser time-step, and
thus move $r_{t}$ outwards.
The outer boundary for the spherical grids was chosen based on the expected
time duration of the measurement and radius of the furthest detector, in order
to remove any influence of the artificial outer boundary condition. In
particular, given that the evolution takes a time $T_{m}$ for the entire
inspiral, merger and ringdown, and gravitational wave measurements taken at a
finite radius $r_{d}$, we would like to ensure that a disturbance travelling
at the speed of light from the outer boundary does not reach the measurement
radius (see Fig. 4). That is, noting that the physical modes travel at the
speed of light, $c=1$ Brown et al. (2007, 2009)444The $1+\log$ slicing
condition which we use propagates at $\sqrt{2}c$ Bona et al. (1995), however
this is a gauge mode and empirically we find it to have negligible effect on
measurements., we place the boundary at
$r_{b}>T_{m}+2r_{d}.$ (31)
For the particular evolution considered here, $T_{m}\simeq 1350M$, and our
outermost measurements are taken at $r_{d}=1000M$. We have placed the outer
boundary of the evolution domain at $r_{b}=3600M$.
The near-zone grids incorporate 5 levels of 2:1 mesh refinement, covering
regions centred around each of the black holes. For the highest resolution we
have considered here, the finest grid (covering the black hole horizon) has a
grid spacing of $dx=0.02M$. The wave-zone grids have an inner radial
resolution which is commensurate with the coarse Cartesian grid resolution,
$dr=0.64M$ in this case. This resolution is maintained essentially constant to
the outermost measurement radius ($r=1000M$), at which point we apply a
gradual decrease in resolution (as described in Sec. II.1) over a distance of
$r=500M$. From $r=1500M$ to the outer boundary, we maintain a resolution of
$dx=2.56M$, sufficient to resolve the inspiral frequencies of the dominant
$(\ell,m)=(2,2)$ mode of the gravitational wave signal. The transition between
the resolutions is performed over a distance of $500M$ between $r=1000M$ and
$r=1500M$. The angular coordinates have 31 points (30 cells) in $\nu$ and
$\phi$ on each of the 6 patches. The time-step of the wave-zone grids is
$dt=0.144$, and we take wave measurements at each iteration.
We have carried out evolutions at three resolutions in order to estimate the
convergence of our numerical methods. The grid described above is labelled
$h_{0.64}$. The lower resolutions, labelled $h_{0.80}$ and $h_{0.96}$ have
each of the specified grid spacings scaled by $0.80/0.64$ and $0.96/0.64$,
respectively.
Figure 4: Schematic of the causal propagation of information during the
evolution. The gravitational wave source is located in the vicinity of $r=0$,
with waves propagating outward at the speed of light $c=1$, and are measured
at radius $r_{d}$ for a time of interest which would include the inspiral,
merger and ringdown of a binary system. The unphysical outer boundary of the
grid is located at $r_{b}$, which is chosen to be sufficiently far removed
that the future Cauchy horizon of the domain of dependence of the initial
slice does not reach $r_{b}$ until the measurement is complete.
### IV.3 Results
Figure 5: The dominant spherical harmonic modes of $\psi_{4}$ for
$\ell=2,4,6,8$, measured at $r=200M$ from the coordinate centre. The plots on
the right show amplitude and frequency evolution during the late inspiral,
merger and ringdown..
The binary black hole initial data described in Sec. IV.1 evolves for about 8
orbits ($\simeq 1350M$) before merger. Various $(\ell,m)$ modes of $\psi_{4}$
are plotted in Fig. 5. We find that for the grids we have used, the modes to
$(\ell,m)=(4,4)$ mode are quite well resolved throughout the evolution. The
$(6,6)$ mode is also measurable, and shows a clear signal, particularly during
ringdown. The $(8,8)$ mode is dominated by noise for most of the inspiral,
though during the merger and ringdown phase, a clear signal is present and the
amplitude and frequency can be estimated. Tests with an analytical solution
confirm that the angular resolutions which we have used are at best marginal
for resolving this mode.
In the following sections, we report results regarding the convergence and
accuracy of these measurements, as well as determine the parameters of the
merger remnant. By analysing the ring-down behaviour of the waves we conclude
that the remnant is indeed a Kerr black hole (see Sec. IV.3.4, below).
#### IV.3.1 Numerical convergence
We can establish the consistency of our discretisation by showing that it does
indeed converge to a unique solution in the continuum limit. Ideally, an exact
solution can be used to test this. However, since there are no exact solutions
which adequately model the physical scenario which we wish to consider
(inspiralling black hole binaries), an alternative is to evaluate numerical
solutions at several (at least three) different resolutions and establish that
the differences decrease as resolution is increased. For an implementation in
which all of the discrete operations are carried out with the same order of
accuracy, the convergence test should yield a clear exponent corresponding to
that order.
The evolution code incorporates a number of discrete operations, which for
various practical reasons, are carried out to different orders of accuracy.
These are listed in Table 1. The primary operation which is carried out over
the bulk of the grid is the computation of finite difference derivative
operations in order to evaluate the right-hand side of the evolution equations
(22a)–(22e). For the tests carried out in this paper, 8th-order stencils are
used for this operation, including the upwinded advection terms. It is common
to apply a small amount of artificial dissipation in order to smooth high-
frequency effects. This is done at one higher order (9th) than the interior
finite differencing in order to maintain the correct continuum limit. (In our
experiments, however, we have noted that dissipation at this high order has a
negligible impact on the solution, and can effectively be omitted.) Various
boundary operations (inter-patch boundary communication, mesh-refinement
boundaries) are carried out at lower order. This is done largely for
efficiency reasons, as the communication involved in boundary interpolation
can be time-consuming if the stencil widths are large. Intuitively, the
numerical error associated with these operations may have reduced influence in
any case, as they are applied only at 2D interfaces. In practise this does
seem to be the case, for instance, as experiments with 4th and 5th order
interpolation operators between patches show similar accuracy in the final
solution. Similarly, operations involving different time-levels are at lower
order, again for efficiency reasons. The time resolution of our evolutions
tends to be high enough that one might expect a small error coefficient of the
RK4 integrator. The lowest order operation which we use is the 2nd-order time
interpolation at mesh-refinement boundaries. Applying higher order here would
require keeping more time levels in memory (currently we store three). Our
results are consistent with previous studies using mesh-refinement for black
hole evolution which suggest that the influence of the low order time-
interpolation boundary conditions is negligible for the time resolutions which
we apply (see, for example, Bruegmann et al. (2008)).
Numerical method | Order
---|---
Grid interior finite differencing | 8
Inter-patch interpolation | 5
Kreiss-Oliger dissipation | 9
Time integration (RK4) | 4
Mesh-refinement: |
Spatial prolongation | 5
Spatial restriction | n/a
Time interpolation | 2
Analysis tools: |
Interpolation | 4
Finite differencing | 8
Surface integration | $2N-1$
Table 1: Table of convergence order of various numerical aspects of the
evolution code. Spatial restriction is carried out by a direct copy. The
surface integration is exact for polynomials up to degree $2N-1$, where $N$ is
the number of grid-points along one direction on the sphere.
For test cases involving a single non-spinning black hole, in fact we find
8th-order convergence in the Hamiltonian constraints. This is likely due to
the relatively constant values (except for some gauge evolution) maintained by
the evolution variables during the evolution, which minimises error due to
time-integration or propagation across boundaries.
A more relevant situation is that of a binary black hole inspiral, which we
have tested using the parameters described above in Sec. IV.1. For this model,
we have measured the gravitational waveform, $\psi_{4}$, integrated over
spheres at radii from $r=100M$ to $r=1000M$, at the three resolutions
$h_{0.96}$, $h_{0.80}$ and $h_{0.64}$. Results for the $(\ell,m)=(2,2)$ mode
are shown in Fig. 6. The evolution lasts for about $1350M$ before merger, and
the plots encompass the inspiral, merger (at $t=0M$ on this time axis), and
ringdown. The figure plots the error in phase $\Delta\phi$ and relative
amplitude $\Delta A$ for the $(2,2)$ mode extracted at $r=100M$ and $r=1000M$,
respectively, between medium $h_{0.80}$ and low $h_{0.96}$ resolutions and
high $h_{0.64}$ and medium $h_{0.80}$ resolutions in the wave-zone. The latter
error is scaled such that the curves will overlap in the case of a 4th-order
convergent solution. At both radii, we find that during the inspiral phase,
the rescaled error of the higher resolutions lies below that of the lower
resolution, suggesting better than 4th-order convergence (in fact, closer to
8th-order over significant portions of the plot). At later times, around the
peak of the waveform, the curves are more closely aligned, indicating 4th-
order convergence. The plot suggests that during the very dynamical late
stages of the inspiral, the lower order boundary conditions and/or the time
integration, play a more important role relative to the early inspiral phase
of the evolution, where the convergence order is closer to that of the
interior finite differencing. The results are, however, convergent over the
entire evolution (including merger and ringdown). As we will see in the next
section, the accuracy is excellent for these resolutions so that the rate of
convergence is not a particular issue.
We have verified convergence for a number of different modes of the $\psi_{4}$
waveform at different radii. For instance, Fig. 7 shows similar results for
the $(\ell,m)=(6,6)$ mode, which is some two orders of magnitude smaller in
peak amplitude than the $(\ell,m)=(2,2)$ mode (see Fig. 5). During the early
inspiral, it is difficult to evaluate a convergence order due to high
frequency noise which is large relative to the waveform amplitude. However, a
measurable signal is clear in the last orbit, merger and ringdown phase, and
converges at a clear 3rd order.
Figure 6: Convergence in amplitude (top) and phase (bottom) of the
$(\ell,m)=(2,2)$ mode of $\psi_{4}$ for detectors at $r=100M$ and $r=1000M$.
The higher resolution difference, $h_{0.80}-h_{0.64}$, is scaled for 4th-order
convergence. Figure 7: Convergence in amplitude (top) and phase (bottom) of
the $(\ell,m)=(6,6)$ mode of $\psi_{4}$ for detector at $r=100M$ during the
late through merger. The higher resolution difference, $h_{0.80}-h_{0.64}$, is
scaled for 3rd-order convergence.
#### IV.3.2 Accuracy
We estimate the numerical phase and amplitude error by means of a Richardson
expansion at a given resolution $\Delta$,
$u_{\Delta}(t,x)=u(t,x)+\Delta
e_{1}(t,x)+\Delta^{2}e_{2}(t,x)+\cdot\cdot\cdot\,,$ (32)
where $u(t,x)$ is the solution of the original differential equation, and the
$e_{i}(t,x)$ are error terms at different orders in $\Delta$. Assuming
convergence at a fixed order, $n$, we can expect some of these error functions
to vanish. Using solutions, $u$, obtained at two resolutions, $\Delta_{1}$ and
$\Delta_{2}$, the Richardson expansion implies
$\displaystyle u_{\Delta_{1}}-u_{\Delta_{2}}$
$\displaystyle=e_{n}(\Delta_{1}^{n}-\Delta_{2}^{n})+\mathcal{O}(\Delta^{n+1})$
$\displaystyle=e_{n}\Delta_{2}^{n}(C^{n}-1)+\mathcal{O}(\Delta^{n+1})$
$\displaystyle\sim\epsilon_{\Delta_{2}}(C^{n}-1)\,,$ (33)
where $\epsilon_{\Delta_{2}}$ is the estimated solution error on the higher
resolution grid, and where
$C^{n}:=\left(\frac{\Delta_{1}}{\Delta_{2}}\right)^{n}\,.$ (34)
We thus obtain an estimate for the solution error that is at least accurate to
order $n+1$,
$\epsilon_{\Delta_{2}}\sim\frac{1}{C^{n}-1}(u_{\Delta_{1}}-u_{\Delta_{2}})\,,$
(35)
which we use as an estimate of the numerical error in our solutions.
During the inspiral phase (which for this purpose we regard as being the
period $t\leq-100M$), we have found roughly 8th-order convergence in the
amplitude and phase, as described above. The remaining relative error for the
$(\ell,m)=(2,2)$ mode can be estimated as
$\displaystyle\max_{T\in[-1350,-100]}{\text{err}(A)}_{\rm inspiral}$
$\displaystyle=0.090\%\,,$ (36a)
$\displaystyle\max_{T\in[-1350,-100]}{\text{err}(\phi)}_{\rm inspiral}$
$\displaystyle=0.010\%\,.$ (36b)
where $\text{err}(A):=\Delta A/A$ and $\text{err}(\phi):=\Delta\phi/\phi$,
i.e., the rate of loss of phase with $\phi$. During merger and ring-down
($t>-100M$), we observe 4th-order convergence in the amplitude, while
maintaining 8th-order convergence in the phase. This results in the estimate
$\displaystyle\max_{T\in(-100,150]}{\text{err}(A)}_{\rm merger}$
$\displaystyle=0.153\%\,,$ (37a)
$\displaystyle\max_{T\in(-100,150]}{\text{err}(\phi)}_{\rm merger}$
$\displaystyle=0.003\%\,.$ (37b)
The time evolution of the numerical error in phase and amplitude is shown in
Fig. 8.
We note that these errors are of comparable order to the errors inherent in
the extrapolation Pollney et al. (2009). Moreover, as is pointed out in
Reisswig et al. (2009b), the error between extrapolated waveforms and those
determined at future null infinity, $\mathcal{J}^{+}$, by characteristic
extraction, is an order of magnitude larger than the numerical error
determined here. This highlights the importance of reducing systematic errors
inherent in finite radius measurements of $\psi_{4}$.
Figure 8: Absolute numerical error in the amplitude (top) and phase (bottom)
accumulated over the course of the evolution for the highest resolution run,
determined according to Eq. (35) for the point-wise differences in amplitude
and phase between medium and high resolution runs. For the phase we assume the
measured 8th-order convergence over the entire evolution, while for the
amplitude we use 8th-order before $t\leq-100$, and 4th-order thereafter (see
text).
#### IV.3.3 Properties of the merger remnant
The merger remnant can be measured with high accuracy, using either the
isolated horizon formalism Dreyer et al. (2003); Ashtekar and Krishnan (2004),
or geometrical measures of the apparent horizon Brandt and Seidel (1995);
Alcubierre et al. (2005). Some results are reported in Table 2, along with
estimated numerical errors. The results agree well with previous high-accuracy
measurements, such as those obtained by spectral evolution Scheel et al.
(2009); Hannam et al. (2009), with the spin and irreducible mass agreeing
within three decimal and four decimal places, respectively. places. While this
is larger than the reported errors, we note that we have evolved a different
initial data set than Scheel et al. (2009). As reported in Sec. IV.1 our
evolution has somewhat more eccentricity, and the level of agreement can be
used to judge the influence of small amounts of eccentricity on the result.
By comparing the properties of the merger remnant with the integrated radiated
energy, $E_{\rm rad}$, and angular momentum, $J_{\rm rad}$, determined from
the gravitational waveforms, we find the residuals
$\displaystyle|M_{f}+M_{\rm rad}-M_{\rm ADM}|$ $\displaystyle=2.6\times
10^{-4},$ (38a) $\displaystyle|S_{f}+J_{\rm rad}-J_{\rm ADM}|$
$\displaystyle=3.1\times 10^{-4}.$ (38b)
Here we have used the extrapolations of the gravitational waveforms to
$r\rightarrow\infty$ based on the 6 outermost measurement radii. A more
detailed discussion of this procedure is given in Pollney et al. (2009). The
results can be further improved through taking measurements at
$\mathcal{J}^{+}$, as outlined in Reisswig et al. (2009b, c).
Total ADM mass, $M_{\rm ADM}$ | $0.99051968\pm 20\times 10^{-9}$
---|---
Total ADM angular momentum, $J_{\rm ADM}$ | $0.99330000\pm 10\times 10^{-17}$
Irreducible mass, $M_{\rm irr}$ | $0.884355\pm 20\times 10^{-6}$
Spin, $S_{f}/M_{f}^{2}$ | $0.686923\pm 10\times 10^{-6}$
Christodoulou mass, $M_{\rm f}$ | $0.951764\pm 20\times 10^{-6}$
Angular momentum, $S_{f}$ | $0.622252\pm 10\times 10^{-6}$
Radiated energy, $E_{\rm rad}$ | $0.038546\pm 51\times 10^{-6}$
Radiated angular momentum, $J_{\rm rad}$ | $0.370391\pm 17\times 10^{-6}$
Table 2: Properties of the merger remnant as measured on the apparent horizon
($M_{\rm irr}$, $S_{f}/M_{f}^{2}$) and from the gravitational radiation
($E_{\rm rad}$, $J_{\rm rad}$). Ranges indicate the estimated numerical error.
For the error in $J_{\rm ADM}$, we have simply quoted machine precision (it is
an analytical expression of the input momenta on the conformally flat initial
slice).
#### IV.3.4 Quasi-normal modes of the merger remnant
In Fig. 5, we have shown the late-time behaviour of the amplitude and
frequency for the dominant spherical harmonic modes of $\psi_{4}$, to
$(\ell,m)=(8,8)$. We note that during ring-down, the frequencies settle to a
constant value. If the final black hole is a Kerr black hole, these
frequencies are given by the quasi-normal modes of a Kerr black hole with
given spin $a$.
As reported in the previous section, our evolution leads to a merger remnant
with $a=0.686923\pm 1\times 10^{-5}$ (see Table 2), as measured on the
horizon. The real part of the prograde quasi-normal mode (QNM) frequencies for
modes up to $(\ell,m)=(7,7)$, can be found tabulated in Berti et al. (2006).
For example, $M\omega_{22}=0.526891$ for the $(\ell,m)=(2,2)$ mode, given a
final black hole of the measured mass $M_{f}$ and spin $S_{f}$.
At this point it is worth noting that the QNM determined from perturbations of
a Kerr black hole are most naturally expressed in terms of a basis of spin-
weighted spheroidal harmonics. By contrast, our waveforms have been decomposed
relative to a basis of spin-weighted spherical harmonics, which are easily
calculated via Legendre functions. In order to make an appropriate comparison
between these modes with the perturbative results we need to apply a
transformation to the wave-modes. We have
$\hat{\psi}_{4}^{\ell^{\prime}m^{\prime}}=\sum_{\ell,m}\psi_{4}^{\ell,m}\langle\ell,m|\ell^{\prime},m^{\prime}\rangle\,,$
(39)
where a dash denotes labelling of the spheroidal harmonic modes, and
$\langle\ell,m|\ell^{\prime},m^{\prime}\rangle$ is the overlap defined by
$\langle\ell,m|\ell^{\prime},m^{\prime}\rangle=\int_{\Omega}d\Omega{}_{-2}\bar{S}_{\ell^{\prime}m^{\prime}}(c_{\ell^{\prime}m^{\prime}}){}_{-2}Y_{\ell
m}\,.$ (40)
The spheroidal harmonics parameter
$c_{\ell^{\prime}m^{\prime}}=a\omega_{\ell^{\prime}m^{\prime}}$ depends on the
spin $a$ of the black hole and the corresponding prograde or retrograde QNM
frequency $\omega_{\ell^{\prime}m^{\prime}}$ of the
$(\ell^{\prime}m^{\prime})$ spheroidal harmonic mode555We restrict attention
to the $N=0$ harmonic only.. If $c=0$ (as is the case for non-spinning black
holes), the spheroidal harmonics reduce to the spherical harmonics. The spin-
weighted spheroidal harmonics used here have been implemented following Leaver
Leaver (1985) and are reviewed in Berti et al. (2006).
The frequencies measured during the ringdown are plotted in Fig. 9 for the
modes $(\ell,m)=(2,2)$,$(4,4)$ and $(6,6)$. We have plotted data for the
$r=1000M$ measurement, as well as the value obtained by extrapolating the
waveforms extracted at the outermost 6 measurement spheres to
$r\rightarrow\infty$, and find that in fact the extrapolation has little
effect on the frequency of the lower order modes at these distances from the
source. We note that there is a modulation of the ringdown frequency,
particularly apparent in the $(2,2)$ mode. This is a result of mode mixing,
which stems from the use of the spherical harmonic basis for the $\psi_{4}$
measurements. By transforming the $r=1000M$ result to spheroidal harmonics,
this modulation visible in the $t<40M$ signal is largely removed (dashed
line).
As the amplitude of the wave declines exponentially to the level of numerical
error, the frequencies become difficult to measure accurately. We estimate the
ringdown frequency for each mode by performing a least-squares fit of a
horizontal line through the measured spheroidal harmonic frequency over the
range $t\in[40,80]M$ (dotted line) with the standard deviation of the fit as a
gauge of the error (grey region). These constant lines represent the estimated
frequency of the associated QNM modes, and are tabulated as $\omega^{\rm NR}$
in Table 3. They agree to high precision with the prograde QNM frequencies,
$\omega^{\rm lit.}$, determined Kerr black holes by perturbative methods Berti
et al. (2006). We conclude that the merger remnant is compatible with a Kerr
black hole within the given error estimates.
$(\ell,m)$ | $M_{f}\omega^{\rm lit.}$ | $M_{f}\omega^{\rm NR}$ | $|M_{f}\omega^{\rm NR}-M_{f}\omega^{\rm lit.}|$
---|---|---|---
$(2,2)$ | $0.526891$ | $0.5267\pm 0.0011$ | $1.9\times 10^{-4}$
$(4,4)$ | $1.131263$ | $1.1312\pm 0.0028$ | $6.3\times 10^{-5}$
$(6,6)$ | $1.707630$ | $1.7074\pm 0.0662$ | $2.3\times 10^{-4}$
Table 3: Prograde $N=0$ QNM frequencies for different modes and spin
$a=0.6869$ as determined by perturbative methods Berti et al. (2006),
$\omega^{\rm lit.}$, and as measured during ringdown in the numerical
relativity simulation, $\omega^{\rm NR}$. Figure 9: The ringdown frequencies
for the dominant $\psi_{4}$ modes to $\ell=6$ of the merger remnant. From top
to bottom, the plots show the frequencies of the $(\ell,m)=(2,2)$, $(4,4)$ and
$(6,6)$ modes respectively, over a timescale from the $(2,2)$ waveform peak to
$100M$ later, at which point the waveform amplitude is too small to measure an
accurate frequency. The $\psi_{4}$ data measured at $r=1000M$ is plotted, in
addition to the value extrapolated to $r\rightarrow\infty$, and the
transformation to spheroidal harmonics. The expected quasi-normal mode
frequency is plotted as a dotted line, as well as a fit to the spheroidal
harmonic data over the range $t\in[40M,80M]$, with error-bars determined by
the standard deviation of the fit.
## V Discussion
The results of this paper provide a demonstration of the usefulness of adapted
coordinates in numerical relativity simulations. The precision of the
calculations have allowed us to obtain convergent modes to $\ell=6$, through
merger and ringdown, with accurate predictions of the quasi-normal ringdown
frequencies of the remnant.
Our implementation of non-singular radially adapted coordinates for the wave
zone is based on the use of multiple grid patches with interpolating
boundaries, coupled to a BSSNOK evolution code. Thornburg Thornburg (2004)
first demonstrated that such a setup could lead to stable evolutions in the
case of a spinning black hole in Kerr-Schild coordinates. We have demonstrated
that the approach is also effective and robust for dynamical puncture
evolutions, and in particular the problem of binary black holes.
The implementation described here has a number of advantages, principle among
them being its flexibility. While we have presented results for a particular
grid structure adapted to radially propagating waves, there are no principle
problems with restructuring the grids to cover any required domain, for
instance adapted to excision boundaries or toroidal fields. Since data is
stored in the underlying Cartesian basis, and passed by interpolation across
boundaries, the coordinates used on each patch are largely independent of the
others, and there is no need for numerical grid generating schemes. While we
have used the BSSNOK formalism to evolve the Einstein equations, in principle
any stable strongly hyperbolic system can be substituted. The BSSNOK system
has, however, proven particularly useful for evolving black holes via the
puncture approach, which itself has proven to be a very flexible methodology.
We have demonstrated results for the most well-studied test case, non-spinning
equal-mass black holes, the same techniques can be applied to different mass
ratios and spinning black holes, simply by changing the physical input
parameters. (The appendices include some examples of spinning black hole
evolutions.)
Finally, we emphasise again the accuracies which can be attained by this
approach. Our finite difference results show numerical error estimates which
are on par with those achieved using spectral spatial discretisation Scheel et
al. (2009). The adapted radial coordinate allows us to take measurements at
radii much larger than have been used before, as well as obtain accurate
measurements of higher $\ell$ modes during merger, which have an amplitude
more than two orders of magnitude smaller than the dominant $(\ell,m)=(2,2)$
mode. One of the aspects which makes this possible is the fact that we are
able to extend our grids to a distance such that the measurements are included
in the future domain of dependence of the initial data (causally disconnected
outer boundaries), and the waves are reasonably well resolved over this entire
domain so that internal reflections are minimised. Furher, we note that our
results are consistent with other puncture-method calculation in that the
results are convergent and can be consistently extrapolated to
$r\rightarrow\infty$ throughout the entire evolution, including late inspiral
and ringdown Pollney et al. (2009), where other approaches have had
difficulties.
The absence of artificial boundaries, as well as dissipative regions in the
wave zone, removes an important source of potential error in solving the
Einstein equations as an initial-boundary value problem. The remaining errors
can be categorised in three forms. First, numerical error due to the
discretisation. This can be reduced through the use of higher order methods
for the operations performed in various parts of the code, and fortunately is
also easy to quantify by performing tests at multiple resolutions. We note
that for finite differences, the largest improvement in accuracy occurs in
going from 2nd to 4th-order for the interior computations, and beyond that
there are diminishing returns Gustafsson et al. (1995). While it does not yet
seem to be a limiting factor, except possibly during the merger, the RK4 time-
stepping will at some level of resolution be a determining factor in the
accuracy regardless of the spatial order (and this is also the case for
current implementations of spectral methods). The second source of error is a
physical error, inherent in the choice of initial data parameters for the
binary evolution. At the separations which are practical for numerical
relativity (say $d<20M$), the physical model is expected to have shed all of
its eccentricity. We have used post-Newtonian orbital parameters to attempt to
place our black holes in low eccentricity trajectories, and this is quite
effective. Alternative approaches, involving iteratively correcting the
initial data parameters until a tolerable eccentricity has been reached, are
able to reduce the eccentricity still further Pfeiffer et al. (2007). This
technique can in principle also be adapted to the moving puncture approach.
The final source of error arises in the measurement of $\psi_{4}$, which is
done at a finite radius, and then extrapolated to $r\rightarrow\infty$ by some
procedure. We have attempted to minimise this error by placing detectors at
large radii, well into the region where the perturbations are linear, and have
shown that the extrapolations are consistent with measurements at larger
radii, as well as with each other in the $r\rightarrow\infty$ limit Pollney et
al. (2009). However, there remain ambiguities particularly in gauge-dependent
quantities such as the choice of surface on which measurements are taken, and
the definition of time and radial distance to be used in the extrapolation. In
a companion paper Reisswig et al. (2009b), we have demonstrated that these
ambiguities can be removed entirely by the procedure of _characteristic
extraction_ , whereby evolution data on a world-tube is used as an inner
boundary condition for a fully relativistic characteristic evolution,
extending to null infinity, $\mathcal{J}^{+}$. The results suggest that
systematic errors inherent in finite radius measurements of $\psi_{4}$ are
more than an order of magnitude larger than the numerical errors reported
here.
###### Acknowledgements.
We dedicate this paper to the memory of Thomas Radke, who has made invaluable
contributions to the development and optimisation of Cactus, Carpet and the
code described here. The authors are pleased to thank: Ian Hinder, Sascha
Husa, Badri Krishnan, Philipp Moesta, Christian D. Ott, Luciano Rezzolla,
Jennifer Seiler, Jonathan Thornburg, and Burkhard Zink for their helpful
input; the developers of Cactus Goodale et al. (2003); cac and Carpet
Schnetter et al. (2004); Schnetter et al. (2006b); car for providing an open
and optimised computational infrastructure on which we have based our code;
Nico Budewitz for optimisation work with our local compute cluster, damiana;
support from the DFG SFB/Transregio 7, the VESF, and by the NSF awards no.
0701566 _XiRel_ and no. 0721915 _Alpaca_. Computations were performed at the
AEI, at LSU, on LONI (numrel03), on the TeraGrid (TG-MCA02N014), and the
Leibniz Rechenzentrum München (h0152).
## Appendix A The influence of upwinded advection stencils
It has long been recognised that for BSSNOK evolutions employing a shift
vector, $\beta^{a}$, the overall accuracy can be improved by “upwinding” the
finite difference stencils for advective terms of the form
$\beta^{i}\partial_{i}u$ Alcubierre et al. (2003). The upwind derivatives
employ stencils which are off-centred by some number of grid points in the
direction of $\beta^{a}$. The drawback of the method is that in order to
maintain the same order of accuracy in the derivatives, the stencil must have
the same width as a centred stencil, but since it is offset in either a
positive or negative direction, it effectively requires an additional number
of points to be available to the derivative operator equal to the size of the
offset. For parallel codes which physically decompose the grid over processors
and communicate ghost-zone boundaries, this means that a larger number of
points must be communicated and can impact the overall efficiency. Further, a
larger number of points must be translated at inter-patch and refinement level
boundaries.
Figure 10: Trajectories of the two inspiralling punctures for a spinning
configuration $a_{1}=-a_{2}=0.8$, with upwinded advection terms (solid lines)
and without (dashed lines). In the case where no upwinding has been used, the
black holes do not inspiral, due to the accumulation of numerical error.
The original observation that upwinding is helpful was made with a code that
used 2nd-order spatial finite differences. In that case, the centred stencils
are small (three points) and the upwind derivatives correspond to sideways
derivatives in the direction of the shift, i.e., no “downwind” information is
used. For higher order schemes, the importance of upwinding may be less
significant, since the stencils are large relative to the size of the shift
vector. In practise, some implementations have empirically determined that
upwinding by 1 point at 6th-order is helpful Husa et al. (2008). However, this
is not done universally, particularly in conjunction with 8th-order centred
differencing Lousto and Zlochower (2008); Campanelli et al. (2009).
We have found upwinding to be important in reducing numerical error in the
black hole motion for every order of accuracy we have tried. The effect is
demonstrated in Fig. 10, which plots the motion of the black hole punctures
for a data set involving a pair of equal-mass binaries with spins
$a_{1}=-a_{2}=0.8$ evolved at a relatively low resolution with 8th-order
spatial finite differencing. The results of two evolutions are plotted, one
using fully centred stencils, and the other upwinding the advection terms with
a one-point offset. Whereas the latter evolution displays the expected
inspiral behaviour, at this resolution the binary evolved with centred
advection actually flies apart. The is purely a result of accumulated
numerical error, and at higher resolutions both tracks can be made to inspiral
and merge. Our observation, however, is that for a given fixed resolution, the
one-point offset advection has a significantly reduced numerical error in the
phase as compared to the fully centred derivatives.
Based on some limited experimentation with larger offsets, we have the general
impression that the one point offset provides the optimal accuracy for each of
the finite difference orders we have tried (4th, 6th, 8th). We do not exclude
the possibility that there may be situations in which the fully centred
stencils perform as well as upwinded advection, however we have not come
across a situation where the latter method performs worse.
As an alternative, we have also tested lower order upwinded derivatives as a
potential scheme which would allow us to maintain a smaller stencil width. We
generally find that the resultant numerical errors are of the same magnitude
or larger than if we had not done the upwind at all.
We note parenthetically the fact that the off-centering is most important in
the immediate neighbourhood of the black holes, where the shift has a non-
trivial amplitude. It is possible that a scheme where the stencils are off-
centred only on grids where the shift is larger than some threshold would also
be effective, and not suffer the drawbacks mentioned above over the bulk of
the grid. We have not experimented with such a scheme, however.
## Appendix B High order finite differencing
Figure 11: Phase evolution of the $(\ell,m)=(2,2)$ mode $\psi_{4}$ for the
aligned-spin model with $a_{1}=-a_{2}=0.8$ $h=0.64M$. The 6th-order case at
$h_{0.64}$ has a trajectory between the low resolution ($h_{0.80}$) and high
resolution ($h_{0.64}$) 8th-order evolution.
A recent trend in the implementation of finite difference codes for relativity
has been the push towards higher order spatial derivatives, It is now common
to use 6th or 8th-order stencils. The benefit of higher order stencils is that
the convergence rate can be dramatically increased, so that a small increase
in resolution leads to a large gain in accuracy. And while not guaranteed, it
is often the case that for a given fixed resolution, a higher order derivative
will be more accurate, requiring fewer points to accurately represent a
wavelength Gustafsson et al. (1995).
In moving to high order stencils, there is a trade-off between the possible
accuracy improvements, and the extra computational cost. High order stencils
generally involve two extra floating point operations per order. Since they
require a larger stencil width, they also incur a cost in communication of
larger ghost zones, as well as requiring wider overlap zones at grid
boundaries. In practice, we find that higher order stencils can also have a
more strict Courant limit, requiring a smaller timestep (and thus more
computation to reach a given physical time). While it is possible to
demonstrate a large gain in accuracy in switching from 2nd to 4th-order
operators, there are diminishing returns in the transition to 6th and higher
order Gustafsson et al. (1995).
We have experimented with 4th, 6th and 8th-order finite differencing for the
evolution equations. Generally we find that the 8th-order operators can indeed
provide a notable benefit, particularly in the phase accuracy, at low
resolution. In Fig. 11, we plot the phase evolution for an equal mass model
with spins $a_{1}=-a_{2}=0.8$. The evolution covers the last three orbits and
ringdown. We find that for this high-spin case, even over this short duration,
a significant dephasing takes place. Assuming 8th-order convergence, the 6th-
order evolution at the $h_{0.64}$ resolution would be comparable to the 8th-
order at approximately $h_{0.77}$ resolution. We can get some idea of the
relative amount of work required for each calculation by noting there would be
$N=(0.64/0.77)^{3}$ fewer grid points in the $h_{0.77}$ evolution, but the
8th-order derivatives require $9/7$ times as many floating point computations
for a derivative in one coordinate direction, and requires a Courant factor
which is $0.9$ times that of the 6th-order run. Taken together, this suggests
an 8th-order run at $h_{0.77}$ would require a factor $0.68$ of the amount of
work of the 6th-order case to achieve comparable accuracy. Note that this
computation does not take into account potential additional communication
overhead associated with the wider 8th-order stencils. But assuming this is
not dominant, the conclusion seems to be that for this level of accuracy, the
6th-order evolution is somewhat less efficient than the 8th-order version
would be.
For a given situation, it may be that these factors change significantly.
Implementation, and even hardware, details can shift the balance of costs
between various operations. Further, the test case considered here involves a
fairly high spin. Lower spin models (such as that considered in the main body
of the paper), are accurate at modest resolutions, and in such cases the 6th-
order evolutions may in fact prove to be relatively more efficient if the
accuracy is already sufficient for a given purpose. On the other hand, if grid
sizes and memory consumption are limiting factors, the 8th-order operators do
give a consistent accuracy benefit for a fixed grid size. Our expectation,
however, is that implementing yet higher order stencils (for example, 10th-
order) may not be justified on the basis of efficiency.
Figure 12: Amplitude and phase evolution of the $(\ell,m)=(2,2)$ mode of
$\psi_{4}$ for the equal-mass aligned-spin model, comparing 8th-order spatial
finite differencing with a scheme in which 8th-order is used only on the fine
meshes surrounding the bodies, and 4th-order on the wave-zone grids.
As a final point, we note that the required high-order accuracy appears to be
largely a consequence of the field gradients in the near-zone, immediately
surrounding the black holes. An alternative scheme, then, could be to apply
high-order finite differencing in this region, while using a lower order (and
thus more efficient) scheme in the wave zone. Results from such a test are
displayed in Fig. 12, where we have used 8th-order only on the finest
refinement level, i.e. , the mesh surrounding the black holes, but 4th-order
on all coarser Cartesian and radial wave-zone grids. This, in turn, allows for
a slightly less restrictive Courant limit, so that it becomes possible to run
with a slightly larger time-stepping. The phase evolution of $\psi_{4}$ is
almost identical to that of the fully 8th-order case, but the we found that
the speed of the run was increased by more than $25\%$ (similar to that of the
fully 6th-order evolution). Further optimisations, such as decreasing ghost-
zone sizes of the 4th-order grids and consequently the communication overhead,
might improve this further. While the errors and convergence order of this
scheme have not been tested in detail, we suggest it as a potentially quite
effective scheme for the impatient.
## Appendix C Choice of conformal variable
Figure 13: Differences in phase of a spinning configuration with resolution
$h=0.80M$ and conformal variables $\phi$ and $W$ against a simulation with
$h=0.64M$ and conformal variable $W$. The dephasing is significant as we are
on the coarse limit of resolution for this particular configuration.
In Sec. III, we have described our implementation of the BSSNOK evolution
system, and note that currently three variations are in use, based on the use
of different variables to represent the conformal scalar. The original
formulation is based on the use of $\phi:=\log\gamma/12$. An issue with this
variable in the context of puncture evolutions is that it has an $O(\ln r)$
singularity which can lead to large numerical error in finite differences
calculated in the neighbourhood of the puncture. More recently, the use of
alternative variables $\chi=\gamma^{-1/3}$ Campanelli et al. (2006) and
$W=\gamma^{-1/6}$ Marronetti et al. (2008) have been proposed as a means of
improving this situation by replacing $\phi$ with variables that are regular
everywhere on the initial data slice. In terms of the evolution system
outlined in Eqs. (22), the $\chi$ and $W$ options correspond to the choices
$\kappa=3$ and $\kappa=6$, respectively.
The influence of this change of variable can be seen in improved phase
accuracy of binary evolutions carried out with either $\chi$ or $W$. In Fig.
13, we show results from an evolution of the equal-mass aligned-spin (
$a_{1}=-a_{2}=0.8$) test case presented in the previous appendices, using
$\phi$ and $W$ as evolution variables. Plotted are the phase errors,
$\Delta\phi$, between runs at low resolution, $h_{0.80}$, using both $\phi$
and $W$ with a higher resolution, $h_{0.64}$, evolution using $W$. The
numerical error associated with the low resolution $\phi$ evolution is
significantly larger than that of the corresponding $W$ evolution.
The reason for this may be related to that of the benefit seen from upwind
advective differences in Appendix A. The phase accuracy of the waveforms is
crucially dependent on correctly modelling the motion of the bodies, and this
requires accurate advective derivatives in the neighbourhood of the punctures.
The reduced numerical error associated with the regular $\chi$ and $W$
variables is important.
Note that even in the $\phi$ case, numerical error generated at the puncture
seems to be confined to within the horizon. Quantities such as constraints
measured outside the horizon, or the horizon properties itself, are not
significantly affected. However, it seems that a clear reduction in phase
error can be attained through the use of either the $\chi$ or $W$ variants of
BSSNOK, and we have used the latter for the tests carried out in this paper.
## References
* Pretorius (2005) F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005).
* Baker et al. (2006a) J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006a).
* Campanelli et al. (2006) M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006).
* Scheel et al. (2009) M. A. Scheel et al., Phys. Rev. D79, 024003 (2009).
* Gonzalez et al. (2007a) J. A. Gonzalez, U. Sperhake, B. Bruegmann, M. Hannam, and S. Husa, Phys. Rev. Lett. 98, 091101 (2007a).
* Gonzalez et al. (2007b) J. A. Gonzalez, M. D. Hannam, U. Sperhake, B. Bruegmann, and S. Husa, Phys. Rev. Lett. 98, 231101 (2007b).
* Campanelli et al. (2007a) M. Campanelli, C. O. Lousto, Y. Zlochower, and D. Merritt, Astrophys. J. 659, L5 (2007a).
* Campanelli et al. (2007b) M. Campanelli, C. O. Lousto, Y. Zlochower, and D. Merritt, Phys. Rev. Lett. 98, 231102 (2007b).
* Herrmann et al. (2007) F. Herrmann, I. Hinder, D. Shoemaker, P. Laguna, and R. A. Matzner, Astrophys. J. 661, 430 (2007).
* Koppitz et al. (2007) M. Koppitz et al., Phys. Rev. Lett. 99, 041102 (2007).
* Pollney et al. (2007) D. Pollney, C. Reisswig, L. Rezzolla, B. Szilágyi, M. Ansorg, B. Deris, P. Diener, E. N. Dorband, M. Koppitz, A. Nagar, et al., Phys. Rev. D76, 124002 (2007).
* Lousto and Zlochower (2008) C. O. Lousto and Y. Zlochower, arXiv:0805.0159 (2008).
* Rezzolla et al. (2008a) L. Rezzolla et al., Astrophys. J679, 1422 (2008a).
* Rezzolla et al. (2008b) L. Rezzolla, E. Barausse, E. N. Dorband, D. Pollney, C. Reisswig, J. Seiler, and S. Husa, Astrophys. J. 674, L29 (2008b).
* Rezzolla et al. (2008c) L. Rezzolla et al., Phys. Rev. D78, 044002 (2008c).
* Tichy and Marronetti (2007) W. Tichy and P. Marronetti, Phys. Rev. D 78, 081501 (2007).
* Lousto and Zlochower (2007) C. O. Lousto and Y. Zlochower, Phys. Rev. D76, 041502 (2007).
* Barausse and Rezzolla (2009) E. Barausse and L. Rezzolla (2009).
* Reisswig et al. (2009a) C. Reisswig et al. (2009a).
* Ajith et al. (2007) P. Ajith et al., Class. Quant. Grav. 24, S689 (2007).
* Ajith et al. (2008) P. Ajith et al., Phys. Re. D 77, 104017 (2008).
* Ajith (2008) P. Ajith, Class. Quant. Grav. 25, 114033 (2008).
* Ajith et al. (2009) P. Ajith et al. (2009).
* Baker et al. (2006b) J. G. Baker, J. R. van Meter, S. T. McWilliams, J. Centrella, and B. J. Kelly (2006b).
* Buonanno et al. (2006) A. Buonanno, G. B. Cook, and F. Pretorius (2006).
* Hannam et al. (2008a) M. Hannam, S. Husa, U. Sperhake, B. Bruegmann, and J. A. Gonzalez, Phys. Rev. D77, 044020 (2008a).
* Hannam et al. (2008b) M. Hannam, S. Husa, B. Bruegmann, and A. Gopakumar, Phys. Rev. D78, 104007 (2008b).
* Damour et al. (2008a) T. Damour, A. Nagar, E. N. Dorband, D. Pollney, and L. Rezzolla, Phys. Rev. D77, 084017 (2008a).
* Buonanno et al. (2007) A. Buonanno et al., Phys. Rev. D76, 104049 (2007).
* Damour et al. (2008b) T. Damour, A. Nagar, M. Hannam, S. Husa, and B. Bruegmann, Phys. Rev. D78, 044039 (2008b).
* Buonanno et al. (2009) A. Buonanno et al. (2009).
* Boyle et al. (2007) M. Boyle et al., Phys. Rev. D76, 124038 (2007).
* Bishop et al. (1997) N. T. Bishop, R. Gómez, L. Lehner, M. Maharaj, and J. Winicour, Phys. Rev. D 56, 6298 (1997).
* Scheel et al. (2006) M. A. Scheel et al., Phys. Rev. D74, 104006 (2006).
* Gourgoulhon et al. (2001) E. Gourgoulhon, P. Grandclement, K. Taniguchi, J.-A. Marck, and S. Bonazzola, Phys. Rev. D63, 064029 (2001).
* Gourgoulhon et al. (2002) E. Gourgoulhon, P. Grandclément, and S. Bonazzola, Phys. Rev. D 65, 044020 (2002).
* Grandclement et al. (2002) P. Grandclement, E. Gourgoulhon, and S. Bonazzola, Phys. Rev. D65, 044021 (2002).
* Schnetter et al. (2006a) E. Schnetter, P. Diener, E. N. Dorband, and M. Tiglio, Class. Quant. Grav. 23, S553 (2006a).
* Dorband et al. (2006) E. N. Dorband, E. Berti, P. Diener, E. Schnetter, and M. Tiglio, Phys. Rev. D74, 084028 (2006).
* Pazos et al. (2009) E. Pazos, M. Tiglio, M. D. Duez, L. E. Kidder, and S. A. Teukolsky, Phys. Rev. D80, 024027 (2009).
* Thornburg (2004) J. Thornburg, Class. Quant. Grav. 21, 3665 (2004).
* Pfeiffer et al. (2003) H. P. Pfeiffer, L. E. Kidder, M. A. Scheel, and S. A. Teukolsky, Comput. Phys. Commun. 152, 253 (2003).
* Carpenter et al. (1994) M. Carpenter, D. Gottlieb, and S. Abarbanel, J. Comput. Phys. 111, 220 (1994).
* Diener et al. (2007) P. Diener, E. N. Dorband, E. Schnetter, and M. Tiglio, J. Sci. Comput. 32, 109 (2007).
* Hannam et al. (2007) M. Hannam, S. Husa, D. Pollney, B. Brugmann, and N. O’Murchadha, Phys. Rev. Lett. 99, 241102 (2007).
* Brown (2009) J. D. Brown (2009).
* Goodale et al. (2003) T. Goodale, G. Allen, G. Lanfermann, J. Massó, T. Radke, E. Seidel, and J. Shalf, in _Vector and Parallel Processing – VECPAR’2002, 5th International Conference, Lecture Notes in Computer Science_ (Springer, Berlin, 2003).
* (48) Cactus Computational Toolkit home page, URL http://www.cactuscode.org/.
* Schnetter et al. (2004) E. Schnetter, S. H. Hawley, and I. Hawke, Class. Quantum Grav. 21, 1465 (2004).
* Schnetter et al. (2006b) E. Schnetter, P. Diener, N. Dorband, and M. Tiglio, Class. Quantum Grav. 23, S553 (2006b).
* (51) Mesh Refinement with Carpet, URL http://www.carpetcode.org/.
* Fornberg (1988) B. Fornberg, Mathematics of Computation 51, 699 (1988).
* Driscoll and Healy (1994) J. R. Driscoll and D. M. Healy, Jr., Adv. Appl. Math. 15, 202 (1994), ISSN 0196-8858.
* Bateman (1955) H. Bateman, _Higher transcendental functions_ (1955).
* Nakamura et al. (1987) T. Nakamura, K. Oohara, and Y. Kojima, Prog. Theor. Phys. Suppl. 90, 1 (1987).
* Shibata and Nakamura (1995) M. Shibata and T. Nakamura, Phys. Rev. D 52, 5428 (1995).
* Baumgarte and Shapiro (1998) T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D 59, 024007 (1998).
* Alcubierre et al. (2000a) M. Alcubierre, B. Brügmann, T. Dramlitsch, J. A. Font, P. Papadopoulos, E. Seidel, N. Stergioulas, and R. Takahashi, Phys. Rev. D 62, 044034 (2000a).
* Alcubierre et al. (2003) M. Alcubierre, B. Brügmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D 67, 084023 (2003).
* van Meter et al. (2006) J. R. van Meter, J. G. Baker, M. Koppitz, and D.-I. Choi, Phys. Rev. D 73, 124011 (2006).
* Alcubierre et al. (2000b) M. Alcubierre, G. Allen, B. Brügmann, E. Seidel, and W.-M. Suen, Phys. Rev. D 62, 124011 (2000b).
* Baumgarte and Shapiro (2003) T. W. Baumgarte and S. L. Shapiro, Physics Reports 376, 41 (2003).
* Alcubierre (2008) M. Alcubierre, _Introduction to $3+1$ Numerical Relativity_ (Oxford University Press, Oxford, UK, 2008).
* Marronetti et al. (2008) P. Marronetti, W. Tichy, B. Bruegmann, J. Gonzalez, and U. Sperhake, Phys. Rev. D77, 064010 (2008).
* Bruegmann et al. (2008) B. Bruegmann et al., Phys. Rev. D77, 024027 (2008).
* Bona et al. (1995) C. Bona, J. Massó, E. Seidel, and J. Stela, Phys. Rev. Lett. 75, 600 (1995).
* Newman and Penrose (1962) E. T. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962), erratum in J. Math. Phys. 4, 998 (1963).
* Sachs (1961) R. Sachs, Proc. Roy. Soc. London A264, 309 (1961).
* Penrose (1963) R. Penrose, Phys. Rev. Lett. 10, 66 (1963).
* Baker et al. (2002) J. Baker, M. Campanelli, and C. O. Lousto, Phys. Rev. D 65, 044001 (2002).
* Gunnarsen et al. (1995) L. Gunnarsen, H. Shinkai, and K. Maeda, Class. Quantum Grav. 12, 133 (1995).
* Boyle and Mroue (2009) M. Boyle and A. H. Mroue (2009).
* Pollney et al. (2009) D. Pollney, C. Reisswig, N. Dorband, E. Schnetter, and P. Diener (2009), eprint arXiv:0910.3656 [gr-qc].
* Reisswig et al. (2009b) C. Reisswig, N. T. Bishop, D. Pollney, and B. Szilagyi (2009b).
* Bishop et al. (1999) N. Bishop, R. Isaacson, R. Gómez, L. Lehner, B. Szilágyi, and J. Winicour, in _Black Holes, Gravitational Radiation and the Universe_ , edited by B. Iyer and B. Bhawal (Kluwer, Dordrecht, The Neterlands, 1999), p. 393.
* Babiuc et al. (2009) M. C. Babiuc, N. T. Bishop, B. Szilágyi, and J. Winicour, Phys. Rev. D79, 084001 (2009).
* Reisswig et al. (2009c) C. Reisswig, N. T. Bishop, D. Pollney, and B. Szilágyi (2009c), in preparation.
* Hannam et al. (2009) M. Hannam et al., Phys. Rev. D79, 084025 (2009).
* Husa et al. (2008) S. Husa, M. Hannam, J. A. Gonzalez, U. Sperhake, and B. Bruegmann, Phys. Rev. D77, 044037 (2008).
* Brown et al. (2007) D. Brown, O. Sarbach, E. Schnetter, M. Tiglio, P. Diener, I. Hawke, and D. Pollney, Phys. Rev. D 76, 081503(R) (2007).
* Brown et al. (2009) D. Brown, P. Diener, O. Sarbach, E. Schnetter, and M. Tiglio, Phys. Rev. D 79, 044023 (2009).
* Dreyer et al. (2003) O. Dreyer, B. Krishnan, D. Shoemaker, and E. Schnetter, Phys. Rev. D 67, 024018 (2003).
* Ashtekar and Krishnan (2004) A. Ashtekar and B. Krishnan, Living Rev. Relativ. 7, 10 (2004).
* Brandt and Seidel (1995) S. Brandt and E. Seidel, Phys. Rev. D 52, 870 (1995).
* Alcubierre et al. (2005) M. Alcubierre, B. Brügmann, P. Diener, F. S. Guzmán, I. Hawke, S. Hawley, F. Herrmann, M. Koppitz, D. Pollney, E. Seidel, et al., Phys. Rev. D 72, 044004 (2005).
* Berti et al. (2006) E. Berti, V. Cardoso, and M. Casals, Phys. Rev. D 73, 024013 (2006).
* Leaver (1985) E. Leaver, Proc. R. Soc. London, Ser. A 402, 285 (1985).
* Gustafsson et al. (1995) B. Gustafsson, H.-O. Kreiss, and J. Oliger, _Time dependent problems and difference methods_ (Wiley, New York, 1995).
* Pfeiffer et al. (2007) H. P. Pfeiffer et al., Class. Quant. Grav. 24, S59 (2007).
* Campanelli et al. (2009) M. Campanelli, C. O. Lousto, H. Nakano, and Y. Zlochower, Phys. Rev. D79, 084010 (2009).
|
arxiv-papers
| 2009-10-20T10:41:33 |
2024-09-04T02:49:05.922711
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Denis Pollney, Christian Reisswig, Erik Schnetter, Nils Dorband, Peter\n Diener",
"submitter": "Denis Pollney",
"url": "https://arxiv.org/abs/0910.3803"
}
|
0910.3938
|
# A Blind Search for Magnetospheric Emissions from Planetary Companions to
Nearby Solar-type Stars
T. Joseph W. Lazio11affiliation: NASA Lunar Science Institute, NASA Ames
Research Center, Moffett Field, CA Naval Research Laboratory, 4555 Overlook
Avenue SW, Washington, DC 20375-5351 Joseph.Lazio@nrl.navy.mil S.
Carmichael,22affiliation: Current address: George Mason University J.
Clark,33affiliation: Current address: University of Notre Dame E.
Elkins,44affiliation: Current address: University of Virginia P.
Gudmundsen,55affiliation: Current address: Princeton University Z.
Mott,66affiliation: Current address: College of William & Mary M.
Szwajkowski,77affiliation: Current address: Northwestern University L. A.
Hennig Thomas Jefferson High School for Science and Technology, 6650 Braddock
Road, Alexandria, VA 22312
(Received 2009 April 28; Revised 2009 August 18; Accepted 2009 October 19)
###### Abstract
This paper reports a blind search for magnetospheric emissions from planets
around nearby stars. Young stars are likely to have much stronger stellar
winds than the Sun, and because planetary magnetospheric emissions are powered
by stellar winds, stronger stellar winds may enhance the radio luminosity of
any orbiting planets. Using various stellar catalogs, we selected nearby stars
($\lesssim 30$ pc) with relatively young age estimates ($<3$ Gyr). We
constructed different samples from the stellar catalogs, finding between 100
and several hundred stars. We stacked images from the 74-MHz (4-m wavelength)
VLA Low-frequency Sky Survey (VLSS), obtaining 3$\sigma$ limits on planetary
emission in the stacked images of between 10 and 33 mJy. These flux density
limits correspond to average planetary luminosities less than 5–10 $\times
10^{23}$ erg s-1. Using recent models for the scaling of stellar wind
velocity, density, and magnetic field with stellar age, we estimate scaling
factors for the strength of stellar winds, relative to the Sun, in our
samples. The typical kinetic energy carried by the stellar winds in our
samples is 15–50 times larger than that of the Sun, and the typical magnetic
energy is 5–10 times larger. If we assume that every star is orbited by a
Jupiter-like planet with a luminosity larger than that of the Jovian
decametric radiation by the above factors, our limits on planetary
luminosities from the stacking analysis are likely to be a factor of 10–100
above what would be required to detect the planets in a statistical sense.
Similar statistical analyses with observations by future instruments, such as
the Low Frequency Array (LOFAR) and the Long Wavelength Array (LWA), offer the
promise of improvements by factors of 10–100.
planetary systems
## 1 Introduction
In searches for extrasolar planets, the precision to which a planetary signal
can be extracted from the data can depend in part on the properties of the
host star. For instance, in radial velocity surveys, one of the limiting
factors in the velocity precision is intrinsic stellar “jitter,” caused by
starspots or other surface inhomogeneities. Such stellar jitter is well known
to be correlated with stellar activity, the level of which declines in age
(Butler et al., 1996; Saar & Donahue, 1997). Radial velocity surveys tend to
select stars that are chromospherically quiet (Saar et al., 1998; Cumming et
al., 2008), which is likely to introduce a bias toward older stars. Further,
the link between chromospheric activity and age means that distinguishing
planetary transits from stellar surface features will probably be easier for
older, less active stars (but see Jenkins, 2002). Consequently, there is a
selection bias _against_ planets around younger (“adolescent”) stars.
All of the solar system’s “magnetic” planets (Earth (catalog ), Jupiter
(catalog ), Saturn (catalog ), Uranus (catalog ), and Neptune (catalog ))
generate planetary-scale magnetic fields as the result of internal dynamo
currents within the planet. The solar wind incident on these planetary
magnetospheres is an energy source to the planetary magnetospheres, and the
magnetosphere-solar wind interaction produces energetic (keV) electrons that
then propagate along magnetic field lines into auroral regions, where electron
cyclotron masers are produced.
Specific details of the cyclotron maser emission vary from planet to planet,
depending upon such secondary effects as the planet’s magnetic field topology.
Nonetheless, applicable to all of the magnetic planets is a macroscopic
relation relating the incident solar wind power $P_{\mathrm{sw}}$, the
planet’s magnetic field strength, and the median radio luminosity
$L_{\mathrm{rad}}$. Various investigators (Desch & Barrow, 1984; Desch &
Kaiser, 1984; Desch & Rucker, 1985; Barrow et al., 1986; Rucker, 1987; Desch,
1988; Millon & Goertz, 1988) find
$L_{\mathrm{rad}}=\epsilon P_{\mathrm{sw}}^{x},$ (1)
with $\epsilon$ the efficiency at which the solar wind power is converted to
radio luminosity, and $x\approx 1$. The value for $\epsilon$ depends on
whether one considers the magnetic energy or kinetic energy, respectively,
carried by the stellar wind. The strong solar wind dependence is reflected in
the fact that the luminosity of the Earth is larger than that of either Uranus
or Neptune, even though their magnetic field are 10–50 times stronger than
that of the Earth.
The magnetospheric emissions from solar system planets and the discovery of
extrasolar planets has motivated both theoretical (Zarka et al., 1997; Farrell
et al., 1999; Zarka et al., 2001; Farrell et al., 2004; Lazio et al., 2004;
Stevens, 2005; Griessmeier et al., 2005; Zarka, 2006, 2007; Griessmeier et
al., 2007a, b) and observational work (Yantis et al., 1977; Winglee et al.,
1986; Bastian et al., 2000; Lazio et al., 2004; Ryabov et al., 2004; George &
Stevens, 2007; Lazio & Farrell, 2007; Smith et al., 2009) on magnetospheric
emissions from extrasolar planets, including some before the confirmed
discovery of any extrasolar planets.
Implicit in many of the early predictions for planetary magnetospheric
emissions was that the stellar winds of other stars are comparable to the
solar wind. Yet, from measurements of the sizes of astropauses (i.e., the
boundary between the stellar wind and the local interstellar medium), Wood et
al. (2002, 2005) find the mass loss rate as a function of age, $\dot{M}\propto
t^{x}$, with $x\approx-2$, a dependence probably linked to the decrease in
surface magnetic activity with stellar age. Thus, the stellar wind around a 1
Gyr old star may be 25 times as intense as the current solar wind (from a 4.5
Gyr old star).
As a specific illustration of the possible effect of considering younger stars
with more intense stellar winds, Farrell et al. (1999) and Lazio et al. (2004)
predicted that the magnetospheric emission from the planet orbiting $\tau$ Boo
would be of order 1–3 mJy at frequencies around 30 MHz (10 m wavelength),
assuming that the stellar wind of the star was comparable to that of the Sun.
Stevens (2005) and Griessmeier et al. (2007a), taking into account the likely
stellar wind strength of $\tau$ Boo, predicted that its emission would be at a
level of order 100–300 mJy. The former prediction (1–3 mJy) is below the
sensitivity of current instrumentation; the latter is not.
This paper reports a blind search for magnetospheric emissions from nearby
“adolescent” stars. In §2 we describe how we selected stars from existing
catalogs of nearby stars, in §3 we present the results from our stacking
analyses and as well as considering the stars individually, and in §4 we
present our conclusions.
## 2 Stellar Catalog Assembly
Three catalogs form the basis for our identification of nearby “adolescent”
stars.
1. 1.
The NStars program is part of the Space Interferometry Mission Preparatory
Science Program. Gray et al. (2003) and Gray et al. (2006) obtained
spectroscopic observations of the 3600 main-sequence and giant stars within 40
pc of the Sun with spectral types earlier than M0.
2. 2.
The Spectroscopic Properties of Cool Stars (SPOCS) is a compilation of stars
forming the basis of radial velocity planetary searches. From the series of
observations in the SPOCS program, Takeda et al. (2007) estimated various
physical parameters for the stars.
3. 3.
The Geneva-Copenhagen survey of the solar neighborhood (GCS) is an effort to
assemble a complete and consistent set of observational and physical
parameters for nearby F and G dwarfs (Holmberg et al., 2009).
We applied four selection criteria—spectral type, distance, age, and
declination—to the published catalogs to form samples for further study. For
spectral type, we selected main-sequence F, G, or K dwarfs. The electron
cyclotron maser—the same process by which planets generate radio emission—has
been detected from some stars. Typical detections are at frequencies of order
5000 MHz, from which (lower) coronal magnetic field strengths of order 300 G
or higher are inferred (Hallinan et al., 2008). For the electron maser
emission process, the quantity $B/\nu$ is approximately constant, where $B$ is
the magnetic field in the emitting region. At frequencies $\nu\approx 100$
MHz, we would expect magnetic field strengths of order 30 G; lower frequencies
would result in even lower field strengths. A field strength of 30 G is
typical of that inferred for very late M dwarfs and early L brown dwarfs
(Berger, 2002) and within a factor of a few of the field strength of Jupiter
at the cloud tops (4 G). Moreover, while the Sun generates intense emission at
these frequencies, notably Type III and IV radio bursts, even the strongest
such radio bursts are far too faint to be detected over interstellar
distances. Gergely (1986) has considered the detection of solar-type stars at
low radio frequencies and finds that current detection thresholds are at least
a factor of $10^{2}$ above what is required. If the targeted star is an F, G,
or K dwarf, the presence of low radio frequency emission would be a strong
indication that it is orbited by a sub-stellar companion.
Two of the catalogs, NStars and SPOCS, contain only relatively nearby stars,
with distances less than 40 pc and 130 pc, respectively; for the SPOCS
catalog, fully two-thirds of the stars are within 40 pc. The GCS catalog
extends to much larger distances, approximately 1000 pc. We impose a distance
constaint for the following reason.
In the general stacking analysis, as one stacks images from increasingly
distant sources, the signal-to-noise ratio in a given image is decreasing, but
is compensated by the increasing number of sources to stack. Considering stars
to lie in shells at distances $D$ with thickness $\Delta D$, if all stars host
planets, then the flux densities of planets at larger distances will be
smaller by a factor $D^{2}$ which will be exactly balanced by the increase in
volume provided by going to larger distances, assuming a uniform distribution
of stars. However, it is not yet known whether all stars host planets, and
current limits are that the fraction of stars with planets is $f_{p}\approx
0.19$, for planets with semi-major axes less than about 20 AU888 As
magnetospheric emissions are powered by the stellar wind, for the purposes of
this analysis, the presence of planets more distant from their host star are
not likely to be relevant. (Marcy et al., 2008). Consequently, as one stacks
images, an increasing fraction of the total number of images contain only
noise so that the effective “signal-to-noise” ratio is decreasing. For the GSC
catalog, we impose a distance constraint of 40 pc, for consistency with the
other catalogs.
Two of the catalogs (SPOCS and GCS) report the ages of the stars, which we
adopt. For both catalogs, not only are ages reported but also an estimated
uncertainty. Given that stellar ages often have large uncertainties associated
with them, in order not to exclude stars that might be younger than their
nominal ages, we construct two samples from both catalogs. One sample requires
that the reported age of the star be less than 3 Gyr (samples that we denote
SPOCS-age and GCS-age), while the other sample requires that the age taking
its uncertainty account is less than 3 Gyr (samples that we denote SPOCS-eage
and GCS-eage). The NStars catalog does not report the age of the star, but
does report chromospheric flux in the Ca II H and K lines ($\log
R_{HK}^{\prime}$). We convert these to estimated ages following the work of
Henry et al. (1996).
The final selection criterion, declination, is that we will focus on a
northern hemisphere survey for which the effective declination limit is
$-25\arcdeg$ (§3.1). More generally, we are not aware of an all-sky catalog,
at a common frequency and approximately constant rms image noise level, on
which to perform a similar analysis. Table 2 summarizes various properties of
the sub-samples from the three catalogs.
Table 1: Stellar Catalog Data
Catalog | Total Number | Magnetospheric | Median | Weighted Average | Stacked Image |
---|---|---|---|---|---|---
Name | in Catalog | Subset | Age | Distance | Noise Level | Reference
| | | (Gyr) | (pc) | (mJy beam-1) |
NStars | 664$+$1676 | 252$+$249 | 1.3 | 24.4 | 5.7 | 1,2
SPOCS-age | 1074 | 110 | 1.9 | 19.1 | 11 | 3
SPOCS-eage | 1074 | 176 | 1.4 | 14.3 | 9.3 | 3
GCS-age | 16682 | 355 | 1.6 | 21.4 | 6.0 | 4
GCS-Eage | 16682 | 656 | 0.7 | 23.0 | 4.8 | 4
Note. — The NStars catalog is published in two increments, a Northern Sample
and a Southern Sample. Both the SPOCS and GCS catalogs provide age estimates
and confidence intervals, from which we construct two measures of a star’s
age. The “age” samples adopt the nominal stellar age while the “eage” samples
use the lower limit on the age estimate. Column 3, “magnetospheric subset,”
lists the number of stars from each catalog passing the four selection
criteria of §2.
References. — (1) Gray et al. (2003); (2) Gray et al. (2006); (3) Takeda et
al. (2007); (4) Holmberg et al. (2009)
This search is “blind” in the sense that we have made no effort to select
stars already known to have planets, and even attempted to select stars whose
properties are such that they might not previously have been searched for
planets. Nonetheless, within our samples are small fractions of stars with
known planets. We have cross-correlated our samples with the Extrasolar Planet
Encyclopedia (Schneider, 2009, version July 1). The SPOCS-age sample has the
highest fraction of stars with known planets, 6%, with the remaining samples
all having fractions below 5%. The relatively large fraction of stars with
known planets in the SPOCS-age sample is consistent with the larger SPOCS
catalog being drawn from a set of stars which are or have been the focus of
radial velocity searches. The fraction of stars in our samples that are known
to have planets is smaller than the estimated fraction of stars with planets
(19%, Marcy et al., 2008), consistent with the notion that current planetary
searches continue to be affected by selection effects.
## 3 Magnetospheric Emission Searches
### 3.1 Statistical Analysis
Our search is based on the VLA Low-frequency Survey (VLSS, Cohen et al.,
2007). This survey imaged 95% the sky north of a declination limit
$\delta>-30\arcdeg$ at a frequency of 74 MHz (4 m wavelength) with a typical
rms noise level of 100 mJy beam-1. Our focus on this survey is two-fold.
First, the frequency of this survey is within a factor of 2 of the cutoff
frequency of Jupiter ($\simeq 40$ MHz), so that it is plausible that other
Jovian-like planets might emit at 74 MHz. Second, it is an electronically-
available survey covering a large fraction of the sky, so that many nearby
stars are potentially accessible. There are other surveys at lower frequencies
and which cover a significant fraction of the sky, to which this approach
might also be applied (e.g., Rees, 1990; Dwarakanath & Udaya Shankar, 1990)
However, the VLSS has the advantage of having images that are readily
available in an electronic format combined with a high angular resolution and
sensitivity.
Each individual VLSS image was obtained by combining a series of “snapshots”
acquired over a range of hour angles, with the time sampling within a snapshot
being 10 seconds. A snapshot was typically 15–25 minutes in length, with
snapshots separated typically by about 1 hour. For comparison, Jovian
decametric emission observed by the Nançay Decameter Array has been observed
to have an average duration of about 1 hr, with a range from about 0.5 hr to a
few hours, though these observations were probably dominated by the Io-
controlled component of Jovian decametric emission (Aubier et al., 2000).
Although it is too low in frequency to penetrate the Earth’s (catalog )
ionosphere, Saturian kilometric radiation observed by the Cassini spacecraft
shows similar temporal characteristics (Lamy et al., 2008), but is not
significantly affected by the presence of a major satellite. Assuming that
extrasolar planetary magnetospheric emissions are similar to those of Jupiter
(catalog ) and Saturn (catalog ), if a planet was emitting during the course
of a VLSS observation, it was likely to have been emitting for at least the
duration of one snapshot, and potentially all of them. Consequently, there is
a factor, potentially of order 1/3, in the luminosities that we derive below
that would account for the fact that a planet might only be emitting for a
fraction of the time that was used to acquire a VLSS observation. This factor
is sufficiently small, relative to other uncertainties, that we shall not
incorporate it explicitly into the analysis below.
For each of our samples (Table 2), we downloaded small images (“postage
stamps”, 1° in diameter) from the VLSS image server. Although the formal
declination limit of the VLSS is $-30\arcdeg$, the lowest declination fields
are the most incomplete and often have higher rms noise levels. Thus, we used
an effective declination limit of $-25\arcdeg$.
We aligned each postage stamp image so that the target star was in the central
pixel. The beam (point spread function) of the VLSS was 80″, with an image
pixel size of 25″. The coordinates of these stars are typically determined
from the Hipparcos astrometric mission, epoch 1991.25 (Perryman et al., 1997);
the VLSS observations were conducted between 2001 and 2007, with the majority
of the observations conducted between 2003 September and 2005 April. Lazio &
Farrell (2007) considered the possible astrometric uncertainties between the
Hipparcos and VLSS frames, for the relatively high proper motion star $\tau$
Boo (proper motion $\mu\simeq 0\farcs 5$ yr-1). They showed that the
combination of astrometric uncertainties, uncorrected ionospheric refraction
within the VLSS, and the proper motion of the star should have produced an
astrometric uncertainty in alignment of no more than 8″, a fraction of a
pixel. Thus, we are confident that alignment to the central pixel in the
postage stamp images is sufficient.
We examined each of the images for any sources that might be confused with a
target star or that would be close enough to the location of a target star to
affect the stacking process. As an example, the star HD 69582 (catalog ),
which appears in all of our samples, is approximately 75″ ($\approx 1$ beam)
from the radio source PKS 0814$-$029 (catalog ). This radio source is
generally identified as a QSO, although no redshift has been measured. The
offset between the star and radio source is significant enough that it is
unlikely that the two are the same. We return to the question of individual
stars below. In addition, we found a small number of stars that are located
close to the boundaries of the VLSS, particularly near the southern
declination limit. The noise level in the images for these stars was much
higher (and it is possible that the astrometry is not as precise), so we
excluded these. The total number of stars excluded on these considerations led
to the sample sizes being about 10 stars smaller than a straightforward
application of our initial selection criteria would suggest.
The number of stars $N$ in each of our samples ranges from approximately 100
to several hundred. Assuming that the noise in the VLSS images is gaussian
distributed, as generally expected for radio interferometric images,999 In
general, radio interferometric images of the sky are constructed using a Fast
Fourier Transform, with the flux density of the visibility function at the
spatial frequency origin taken to be identically zero. From Fourier Transform
properties, this so-called “zero-spacing” visibility value is equivalent to
the total flux density within the field of view. In the absence of a source,
the pixels in a thermal noise-limited image constructed in this manner will
have a zero-mean normal distribution. The VLSS images that we analyze were
constructed using this standard procedure. we anticipate that the noise in a
stacked image should be roughly 100$N^{-1/2}$ mJy beam-1, or about 5–10 mJy
beam-1. In our stacking analysis, we combined the images in a weighted sense,
weighting each image’s contribution to the final stacked image by its
individual rms noise level. Using the NStars sample as an illustration (Table
2), and taking the rms noise levels in the images into account, we expect that
the rms noise level in the stacked image should be 5.6 mJy beam-1. The actual
rms noise level is 5.7 mJy beam-1, indicating that the assumption of gaussian
noise-dominated images for the VLSS is justified, at least to the stacked
image noise levels we have obtained here. Table 2 also presents the rms noise
levels in the stacked images.
In none of the stacked images do we detect statistically significant emission.
In an area of approximately 1 beam in size, the strongest pixels range from
approximately 1.5$\sigma$ to 2.2$\sigma$. Table 2 presents 3$\sigma$ limits on
the average flux density of magnetospheric emissions from planets orbiting the
stars in our samples from the various catalogs. Table 2 presents the weighted
average distance for the stars in the various samples, with the weighted
average distance for the $N$ stars in a sample defined as
$\frac{1}{{\bar{D}}^{2}}\equiv\sum_{i=1}^{N}\frac{1}{D_{i}^{2}},$ (2)
where $D_{i}$ is the distance to the $i^{\mathrm{th}}$ star in the sample.
Using the weighted average distance, and assuming that the bandwidth of
planetary magnetospheric emissions is comparable to the observation frequency,
as it is in the case of Jupiter, we convert the flux density upper limits to
limits on average planetary luminosities. Table 2 also presents these
planetary luminosity limits.
### 3.2 Individual Stars
Prior to stacking the images in each sample, we determined the peak intensity
around each star relative to the _individual_ image noise level ($\approx
100\,\mbox{mJy~{}beam${}^{-1}$}$). Stars for which the peak intensity exceeded
2.5$\sigma$ were then re-examined.
One motivation for performing this check is that the VLSS catalog of sources
was constructed using a threshold test, relative to the image noise level,
with a relatively high signal-to-noise threshold in order to maintain a low
probability for a false detection. For the VLSS, the signal-to-noise
thresholds was 7$\sigma$. Thus, it is possible that there is stellar (or
planetary) emission that would not have been cataloged. We found no stellar
position for which radio emission above 3$\sigma$ ($\approx
300\,\mbox{mJy~{}beam${}^{-1}$}$) could be identified unambiguously. There
were stellar positions with radio emission above this level, but they could be
explained by other features, such as a sidelobe from another source.
As noted above, the positions of some stars were close to, if not coincident
with, radio sources, specifically the stars HD 38392 (catalog ), HD 49933
(catalog ), HD 79555 (catalog ), HD 143333 (catalog ), and HD 202575 (catalog
). These stars have been detected by ROSAT (Huensch et al., 1998, 1999), with
X-ray luminosities ranging from $6\times 10^{20}$ W to $3\times 10^{22}$ W.
The Benz-Güdel relation predicts that the centimeter-wavelength flux densities
of these stars should be of order 0.1 mJy. Scaling these flux densities to the
VLSS (74 MHz, $\lambda=4$ m), we expect no emission to be detectable, for any
reasonable radio spectral index. We also examined the NVSS (1400 MHz, 20 cm)
near the location of each star. The rms noise levels near these stars are
comparable, and a 3$\sigma$ upper limit to the radio emission on any of these
stars is 1.9 mJy, consistent with these stars not being radio sources,
Finally, we are aware of targeted radio observations of only one of these
stars, HD 143333 (catalog ), which placed the rather unconstraining limit of 2
Jy at 5 GHz (Blair et al., 1992) on the star or any associated planet.
## 4 Discussion and Conclusions
What do our limits imply about potential magnetospheric emissions from planets
orbiting any of the stars in our various samples? As noted in the discussion
following equation (1), the planetary luminosity can depend upon whether one
is considering the kinetic energy $P_{\mathrm{sw,kin}}$ or magnetic energy
$P_{\mathrm{sw,mag}}$ carried by the solar wind. The kinetic energy power
depends upon the stellar wind density $n$ and velocity $v$ as
$P_{\mathrm{sw,kin}}\propto nv^{3}$ while the magnetic energy power depends
upon the stellar wind velocity and magnetic field strength $B$ as
$P_{\mathrm{sw,mag}}\propto vB^{2}$ (Zarka et al., 2001; Griessmeier et al.,
2007b). The strengths of all of these quantities both depend upon distance
from the host star and are expected to evolve with stellar age.
For the specific case of $\tau$ Boo, Griessmeier et al. (2007b) illustrated
how one can use a stellar wind model for a star with a known age and a planet
at a known orbital distance to estimate the strength of the planetary radio
emission. A straightforward extension of their approach could be applied to
stars not yet known to be orbited by a planet(s) and even samples such as
those we have constructed here. The additional relevant quantities that are
needed would be the distribution of planetary semi-major axes and, in the case
of an individual star, the distribution of its age estimate or, in the case of
a sample of stars, the distribution of their age estimates. Combining these
distributions, one could estimate the appropriate scaling factor by which
planetary magnetospheric emission would be enhanced. However, in §1 we argued
that the current census of extragalactic planets is likely to be biased,
particularly with respect to those planets that might be most likely to be
radio emitting. For that reason, we do not consider the distribution of
planetary semi-major axes to be well enough constrained to incorporate it into
our analysis, and we shall adopt a somewhat more simplified approach below.
With respect to the stellar wind powers, Griessmeier et al. (2007a) have
synthesized various observations and models to determine functional
dependences for stellar wind velocity, density, and magnetic field strength as
a function of stellar age (their equations [15], [16], and [23]–[24]), for
planets that are not too close to their host star. Applying these relations,
and using the median age of the stars in the various samples (Table 2), we
determine a scaling factor, relative to the current solar value at 1 AU, for
each of these quantities. From the scaling factors for the individual
quantities ($v$, $n$, and $B$), the scaling factors for the kinetic energy and
magnetic energy powers are then determined. Table 2 (Columns 4–8) presents
these scaling factors.
These scaling factors are clearly only approximate and somewhat model
dependent. The number of stars for which stellar wind parameters have been
determined is small. Nonetheless, they serve as an indication of the potential
effect of stellar age. We see that powers delivered (at 1 AU) to potential
planetary magnetospheres around these stars, for most of these samples, may be
enhanced by factors of 10–50 for kinetic power and by factors of 5–10 for
magnetic power.
The one potential exception is the GCS3-eage sample, for which much larger
stellar wind amplification powers appear possible. These large factors result
from the relatively small median age for this sample (0.7 Gyr, Table 2). In
turn, this small median age likely reflects the relative lack of precision
with which stellar ages can be determined. Many stars in the GCS3-eage sample
have lower limits to their stellar ages around 0.1 Gyr, because their age
estimates have large uncertainties (approaching 100%).
These scaling factors for the stellar wind powers are for a fiducial distance
of 1 AU. As noted above, there is likely to be a strong star-planet distance
dependence on the luminosity, but we do not attempt to include a distribution
of planetary semi-major axes. Estimates of the star-planet distance dependence
are that it is $d^{x}$, with $x\lesssim 2$ (e.g., Farrell et al., 1999; Zarka,
2007), implying that Jupiter would be about 25 times more luminuous were it at
a distance of 1 AU instead of its current 5.2 AU distance. We therefore apply
an additional scaling factor of 25.
The final columns of Table 2 present Jupiter’s luminosity scaled by these
stellar wind kinetic energy and magnetic energy factors, respectively, and the
distance scaling factor of 25. Even if a planet with luminosity comparable to
Jupiter orbited every one of our target stars, our sensitivity limits remain
approximately a factor of 10–100 above what would be needed to detect such
planets. If the typical planet-star distance is larger than our fiducial 1 AU
value, the actual difference could be much larger.
We have not attempted to combine the stacked images from the various samples.
While the samples themselves are homogeneous, they obtain age estimates from
different methods. Further, there are many stars that are common to each
sample, so that the stacked images are not independent.
There are a number of next-generation, low radio frequency instruments under
development. Notable among these are the Low Frequency Array (LOFAR) and the
Long Wavelength Array (LWA). If they reach their design goals, both promise to
provide rms noise levels $\sigma\sim 3\,\mbox{mJy~{}beam${}^{-1}$}$, at
frequencies below 100 MHz, nearly two orders of magnitude better than the
74-MHz VLA system which was used to conduct the VLSS. A similar statistical
analysis applied to future LOFAR or LWA observations may improve significantly
upon the limits presented here, or, ideally, detect extrasolar planetary
emission.
Should either LOFAR or the LWA detect emission using a similar statistical
analysis, identifying the stacked emission as planetary rather than stellar
will be important. In targeted observations of an individual star or stars
(e.g., Bastian et al., 2000; Ryabov et al., 2004; George & Stevens, 2007;
Lazio & Farrell, 2007; Smith et al., 2009), an obvious distinguishing factor
would be whether the emission is modulated with the planetary orbit. For this
statistical analysis, alternate aspects of the stacked emission would have to
be examined. Both LOFAR and the LWA are being designed to be broadband (over
at least the 30–80 MHz frequency range), thus the radio spectrum of the
stacked emission could be determined. Further, any correlation between the
strength of stacked emission and the spectral type of the stars could be
useful.
Building upon LOFAR and the LWA will be the Square Kilometre Array (SKA) and
the Lunar Radio Array (LRA). While their designs will be influenced by the
work on LOFAR, LWA, and similar low radio frequency interferometers, both the
SKA and LRA currently anticipate operating at frequencies that would be
relevant for the detection of planetary magnetospheres; in the case of the
SKA, the design goal for its lower operational frequency limit is 70 MHz,
while, for the LRA, frequencies $\nu\sim 50$ MHz are envisioned. Both would
likely provide an order of magnitude sensitivity improvement upon LOFAR and
the LWA.
We thank N. Kassim, S. Kulkarni, and B. Farrell for discussions which spurred
this analysis, A. Cohen for discussions on the VLSS, B. Erickson and B. Slee
for discussions about solar decameter-wavelength emissions, and the referee
for several comments that improved the presentation of these results. This
research has made use of the SIMBAD database, operated at CDS, Strasbourg,
France, and NASA’s Astrophysics Data System. Basic research in radio astronomy
at the NRL is supported by 6.1 Base funding.
Table 2: Magnetospheric Emission Limits
| Flux Density | Planetary | Stellar Wind Amplification Factors | K. E. Scaled Jovian | M. E. Scaled Jovian
---|---|---|---|---|---
Sample | Upper Limit | Luminosity Limitaa$1\,\mathrm{W}=10^{7}\,\mathrm{erg}\,\mathrm{s}^{-1}$ | Velocity | Density | Magnetic Field | Kinetic Energy | Magnetic Energy | Luminosityaa$1\,\mathrm{W}=10^{7}\,\mathrm{erg}\,\mathrm{s}^{-1}$ | Luminosityaa$1\,\mathrm{W}=10^{7}\,\mathrm{erg}\,\mathrm{s}^{-1}$
| (mJy) | (erg s-1) | | | | | | (erg s-1) | (erg s-1)
NStars | 17 | $9.0\times 10^{23}$ | 1.7 | 9.8 | 2.4 | 48 | 9.5 | $1.2\times 10^{22}$ | $2.4\times 10^{21}$
SPOCS-age | 33 | $1.1\times 10^{24}$ | 1.4 | 4.9 | 1.8 | 15 | 4.8 | $3.7\times 10^{21}$ | $1.2\times 10^{21}$
SPOCS-eage | 28 | $5.1\times 10^{23}$ | 1.6 | 8.6 | 2.2 | 38 | 8.3 | $9.5\times 10^{21}$ | $2.1\times 10^{21}$
GCS-age | 18 | $7.3\times 10^{23}$ | 1.6 | 6.7 | 2.0 | 25 | 6.5 | $6.3\times 10^{21}$ | $1.6\times 10^{21}$
GCS-eage | 14 | $6.8\times 10^{23}$ | 2.2 | 30 | 3.6 | 319 | 28 | $8.0\times 10^{22}$ | $7.1\times 10^{21}$
Note. — The flux density upper limit and planetary luminosity limits are both
3$\sigma$. The flux density limits are converted to planetary luminosity
limits using the weighted average distances for the samples (Table 2) and
assuming a 74 MHz bandwidth for the planetary magnetospheric emissions. We
assume a fiducial planetary distance of 1 AU and a planetary luminosity
scaling with distance as $d^{2}$, so that the kinetic and magnetic energy
scaled Jovian luminosities include an additional factor of 25; if the typical
magnetospheric emitting planet is more distant from (closer) its host star
than 1 AU, these Jovian scaled luminosities would have to be adjusted downward
(upward).
## References
* Aubier et al. (2000) Aubier, A., Boudjada, M. Y., Moreau, P., Galopeau, P. H. M., Lecacheux, A., & Rucker, H. O. 2000, A&A, 354, 1101
* Barrow et al. (1986) Barrow, C. H., Genova, F., & Desch, M. D. 1986, A&A, 165, 244
* Bastian et al. (2000) Bastian, T. S., Dulk, G. A., & Leblanc, Y. 2000, ApJ, 545, 1058
* Berger (2002) Berger, E. 2002, ApJ, 572, 503
* Blair et al. (1992) Blair, D. G., Norris, R. P., Troup, E. R., et al. 1992, MNRAS, 257, 105
* Butler et al. (1996) Butler, R. P., Marcy, G. W., Williams, E., McCarthy, C., Dosanjh, P., & Vogt, S. S. 1996, PASP, 108, 500
* Cohen et al. (2007) Cohen, A. S., Lane, W. M., Cotton, W. D., Kassim, N. E., Lazio, T. J. W., Perley, R. A., Condon, J. J., & Erickson, W. C. 2007, AJ, 134, 1245
* Cumming et al. (2008) Cumming, A., Butler, R. P., Marcy, G. W., Vogt, S. S., Wright, J. T., & Fischer, D. A. 2008, PASP, 120, 531
* Desch (1988) Desch, M. D. 1988, Geophys. Res. Lett., 15, 114
* Desch & Rucker (1985) Desch, M. D. & Rucker, H. O. 1985, Adv. Space Res., 5, 333
* Desch & Barrow (1984) Desch, M. D. & Barrow, C. H. 1984, J. Geophys. Res., 89, 6819
* Desch & Kaiser (1984) Desch, M. D. & Kaiser, M. L. 1984, Nature, 310, 755
* Dwarakanath & Udaya Shankar (1990) Dwarakanath, K. S. & Udaya Shankar, N. 1990, J. Astrophys. Astron., 11, 323
* Farrell et al. (2004) Farrell, W. M., Lazio, T. J. W., Zarka, P., Bastian, T. J., Desch, M. D., & Ryabov, B. P. 2004, Plan. Space Sci., 52, 1469
* Farrell et al. (1999) Farrell, W. M., Desch, M. D., & Zarka, P. 1999, J. Geophys. Res., 104, 14025
* George & Stevens (2007) George, S. J., & Stevens, I. R. 2007, MNRAS, 382, 455
* Gergely (1986) Gergely, T. E. 1986, in Low Frequency Radio Astronomy, eds. W. C. Erickson & H. V. Cane (NRAO: Green Bank, WV) p. 97
* Gray et al. (2006) Gray, R. O., Corbally, C. J., Garrison, R. F., McFadden, M. T., Bubar, E. J., McGahee, C. E., O’Donoghue, A. A., & Knox, E. R. 2006, ApJS, 132, 161
* Gray et al. (2003) Gray, R. O., Corbally, C. J., Garrison, R. F., McFadden, M. T., & Robinson, P. E. 2003, ApJS, 126, 2048
* Griessmeier et al. (2007a) Griessmeier, J.-M., Zarka, P., & Spreeuw, H. 2007a, A&A, 475, 359
* Griessmeier et al. (2007b) Griessmeier, J.-M., Preusse, S., Khodachenko, M., Motschmann, U., Mann, G., & Rucker, H. O. 2007b, Plan. Space Sci., 55, 618
* Griessmeier et al. (2005) Griessmeier, J.-M., Motschmann, U., Mann, G., & Rucker, H. O. 2005, A&A, 437, 717
* Hallinan et al. (2008) Hallinan, G., Antonova, A., Doyle, J. G., Bourke, S., Lane, C., & Golden, A. 2008, ApJ, 684, 644
* Henry et al. (1996) Henry, T. J., Soderblom, D. R., Donahue, R. A., & Baliunas, S. L. 1996, AJ, 111, 439
* Holmberg et al. (2009) Holmberg, J., Nordström, B., & Andersen, J. 2009, A&A, in press
* Huensch et al. (1999) Huensch, M., Schmitt, H. H. M. M., Sterzik, M. F., & Voges, W. 1999, A&AS, 135, 319
* Huensch et al. (1998) Huensch, M., Schmitt, H. H. M. M., & Voges W. 1998, A&AS, 132, 155
* Jenkins (2002) Jenkins, J. M. 2002 ApJ, 575, 493
* Lamy et al. (2008) Lamy, L., Zarka, P., Cecconi, B., Prangé, R., Kurth, W. S., & Gurnett, D. A. 2008, J. Geophys. Res., 113, A07201
* Lazio & Farrell (2007) Lazio, T. J. W., & Farrell, W. M. 2007, ApJ, 668, 1182
* Lazio et al. (2004) Lazio, T. J. W., Farrell, W. M., Dietrick, J., Greenlees, E., Hogan, E., Jones, C., & Hennig, L. A. 2004, ApJ, 612, 511
* Marcy et al. (2008) Marcy, G. W., Butler, R. P., Vogt, S. S., et al. 2008, Phys. Scr., T130, 014001
* Millon & Goertz (1988) Millon, M. A. & Goertz, C. K. 1988, Geophys. Res. Lett., 15, 111
* Perryman et al. (1997) Perryman, M. A. C., Lindegren, L., Kovalevsky, J., et al. 1997, A&A, 323, L49
* Rees (1990) Rees, N. 1990, MNRAS, 244, 233
* Rucker (1987) Rucker, H. O. 1987, Annales Geophysicae, Series A, 5, 1
* Ryabov et al. (2004) Ryabov, V. B., Zarka, P., & Ryabov, B. P. 2004 Plan. Space Sci., 52, 1479
* Saar et al. (1998) Saar, S. H., Butler, R. P., & Marcy, G. W. 1998, ApJ, 498, L153
* Saar & Donahue (1997) Saar, S. H., & Donahue, R. A. 1997, ApJ, 485, 319
* Schneider (2009) Schneider, J. 2009, The Extra-Solar Planets Encyclopedia, http://www.exoplanet.eu/
* Smith et al. (2009) Smith, A. M. S., Collier Cameron, A., Greaves, J., Jardine, M., Langston, G., & Backer, D. MNRAS, 395, 335
* Stevens (2005) Stevens, I. R. 2005, MNRAS, 356, 1053
* Takeda et al. (2007) Takeda, G., Ford, E. B., Sills, A., Rasio, F. A., Fischer, D. A., & Valenti, J. A. 2007, ApJS, 168, 297
* Winglee et al. (1986) Winglee, R. M., G. A. Dulk, and T. S. Bastian, 1986, ApJ, 309, L59
* Wood et al. (2005) Wood, B. E., Müller, H.-R., Zank, G. P., Linsky, J. L., & Redfield, S. 2005, ApJ, 628, L143
* Wood et al. (2002) Wood, B. E., Müller, H.-R., Zank, G. P., & Linsky, J. L. 2002, ApJ, 574, 412
* Yantis et al. (1977) Yantis, W. F., Sullivan, W. T., III, & Erickson, W. C. 1977, Bull. Amer. Astron. Soc., 9, 453
* Zarka (2007) Zarka, P. 2007, Planet. Space Sci., 55, 598
* Zarka (2006) Zarka, P. 2006, in Planetary Radio Emissions VI, eds. H. O. Rucker et al. (Austrian Acad.: Vienna) p. 543
* Zarka et al. (2001) Zarka, P., Treumann, R. A., Ryabov, B. P., & Ryabov, V. B. 2001, Ap&SS, 277, 293
* Zarka et al. (1997) Zarka, P., Queinnec, J., Ryabov, B. P., et al. 1997, in Planetary Radio Astronomy IV, eds. H. O. Rucker, S. J. Bauer, & A. Lecacheux (Austrian Acad. Sci. Press: Vienna) p. 101
|
arxiv-papers
| 2009-10-20T19:37:51 |
2024-09-04T02:49:05.936164
|
{
"license": "Public Domain",
"authors": "T. Joseph W. Lazio (1 and 2), S. Carmichael (3), J. Clark (3), E.\n Elkins (3), P. Gudmundsen (3), Z. Mott (3), M. Szwajkowski (3), L. A. Hennig\n (3) ((1) NRL, (2) NLSI, (3) TJHSST)",
"submitter": "Joseph Lazio",
"url": "https://arxiv.org/abs/0910.3938"
}
|
0910.4075
|
# Universal bounds on eigenvalues of the buckling problem on spherical domains
111This research is supported by NSFC of China (No. 10671181), Project of
Henan Provincial department of Sciences and Technology (No. 092300410143), and
NSF of Henan Provincial Education department (No. 2009A110010).
Guangyue Huang, Xingxiao Li 222The corresponding author. Email:
xxl$@$henannu.edu.cn , Linfen Cao
Department of Mathematics, Henan Normal University
Xinxiang 453007, Henan, P.R. China
> Abstract. In this paper we study the eigenvalues of buckling problem on
> domains in a unit sphere. By introducing a new parameter and using Cauchy
> inequality, we optimize the inequality obtained by Wang and Xia in [12].
> Keywords: eigenvalue, universal bounds, buckling problem.
> Mathematics Subject Classification: Primary 35P15, Secondary 58G25.
## 1 Introduction
Let $\Omega$ be a connected bounded domain in an $n(\geq 2)$-dimensional
Euclidean space $\mathbb{R}^{n}$ and $\nu$ be the unit outward normal vector
field of $\partial\Omega$. The well-known eigenvalue problem
$\left\\{\begin{array}[]{ll}\Delta^{2}u=\Lambda(-\Delta)u&{\rm in}\ \Omega,\\\
u=\frac{\partial u}{\partial\nu}=0&{\rm on}\ \partial\Omega\end{array}\right.$
(1.1)
is called a buckling problem, which is used to describe the critical buckling
load of a clamped plate subjected to a uniform compressive force around its
boundary.
Let
$0<\Lambda_{1}\leq\Lambda_{2}\leq\Lambda_{3}\leq\cdots$
denote the successive eigenvalues for (1.1), where each eigenvalue is repeated
according to its multiplicity. In 1956, Payne-Pólya-Weinberger [11] proved
$\Lambda_{2}\leq 3\Lambda_{1}\ \ \ {\rm for}\ \Omega\in\mathbb{R}^{2}.$ (1.2)
Following the method of Payne-Pólya-Weinberger in [11], it reads that for
$\Omega\in\mathbb{R}^{n}$ (for the generalization of (1.2) to $n$ dimensions,
see [2]):
$\Lambda_{2}\leq\left(1+\frac{8}{n+2}\right)\Lambda_{1}.$
Subsequently, in 1984, Hile and Yeh [7] improved the above inequality as
follows:
$\Lambda_{2}\leq\frac{n^{2}+8n+20}{(n+2)^{2}}\Lambda_{1}.$
In 1998, Ashbaugh [1] obtained
$\sum_{i=1}^{n}\Lambda_{i+1}\leq(n+4)\Lambda_{1}.$
In a recent survey paper, answering a question of Ashbaugh in [1], Cheng-Yang
[4] proved the following universal inequalities on eigenvalues for the
eigenvalue problem (1.1):
$\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4(n+2)}{n^{2}}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\Lambda_{i}.$
It has become clear that many similar inequalities on eigenvalues of Laplacian
of Payne-Pólya-Weinberger rely on facts involving operators, their
commutators, and traces. For the related research and improvement in this
direction, see [6, 8, 10, 9, 5, 13, 3] and the references therein.
Let $x_{1},x_{2},\ldots,x_{n+1}$ be the standard Euclidean coordinate
functions of $\mathbb{R}^{n+1}$. Then the unit sphere is defined by
$\mathbb{S}^{n}=\left\\{(x_{1},x_{2},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\ ;\
\sum_{\alpha=1}^{n+1}x_{\alpha}^{2}=1\right\\}.$
In 2007, Wang and Xia [12] considered the buckling problem on domains in a
unit sphere $\mathbb{S}^{n}$ and obtained the following result:
###### Theorem 1.1.
[12] Let $\Lambda_{i}$ be the $i^{th}$ eigenvalue of the following eigenvalue
problem:
$\Delta^{2}u=\Lambda(-\Delta)u\ \ {\rm in}\ \Omega,\ \ \ \ u=\frac{\partial
u}{\partial\nu}=0\ \ {\rm on}\ \partial\Omega,$
where $\Omega$ is a connected domain in a unit sphere $\mathbb{S}^{n}(n\geq
2)$ with smooth boundary $\partial\Omega$ and $\nu$ is the unit outward normal
vector field of $\partial\Omega$. Then for any $\delta>0$,
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta\Lambda_{i}+\frac{\delta^{2}(\Lambda_{i}-(n-2))}{4(\delta\Lambda_{i}+n-2)}\right)$
(1.3)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
We remark that the right hand side of inequality (1.3) depends on $\delta$. In
the current paper, by introducing a new parameter and using Cauchy inequality,
we obtain a stronger inequality than (1.3) which is independent of $\delta$,
and derive an inequality of the type of Yang (see inequality (1.7)). Our main
results are stated as follows:
###### Theorem 1.2.
Let $\Lambda_{i}$ be the $i^{th}$ eigenvalue of the following eigenvalue
problem:
$\Delta^{2}u=\Lambda(-\Delta)u\ \ {\rm in}\ \Omega,\ \ \ \ u=\frac{\partial
u}{\partial\nu}=0\ \ {\rm on}\ \partial\Omega,$
where $\Omega$ is a connected domain in a unit sphere $\mathbb{S}^{n}(n\geq
2)$ with smooth boundary $\partial\Omega$ and $\nu$ is the unit outward normal
vector field of $\partial\Omega$. Then
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(2+\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
(1.4) $\displaystyle\leq$ $\displaystyle
2\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\right\\}^{1/2}$
(1.5)
$\displaystyle\qquad\qquad\qquad\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{1/2}.$
(1.6)
###### Corollary 1.3.
Under the assumptions of Theorem 1.2,
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right),$
(1.7) $\Lambda_{k+1}\leq S_{k+1}+\sqrt{S_{k+1}^{2}-T_{k+1}},$ (1.8)
$\Lambda_{k+1}-\Lambda_{k}\leq 2\sqrt{S_{k+1}^{2}-T_{k+1}},$ (1.9)
where
$S_{k+1}=\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}+\frac{1}{2k}\sum_{i=1}^{k}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right),$
(1.10)
$T_{k+1}=\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}^{2}+\frac{1}{k}\sum_{i=1}^{k}\Lambda_{i}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
(1.11)
###### Remark 1.4.
The inequality (1.4) is equivalent to
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle-\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\frac{n-2}{\Lambda_{i}-(n-2)}$
$\displaystyle+2\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\right\\}^{1/2}$
$\displaystyle\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{1/2}.$
From the inequality (2.34) in Section 2 and making use of Cauchy inequality,
the following inequality can be deduced:
$\displaystyle-\sum_{i=1}^{k}$
$\displaystyle(\Lambda_{k+1}-\Lambda_{i})^{2}\frac{n-2}{\Lambda_{i}-(n-2)}$
$\displaystyle+2\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\right\\}^{1/2}$
$\displaystyle\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{1/2}$
$\displaystyle\leq$
$\displaystyle-\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\frac{n-2}{\Lambda_{i}-(n-2)}$
$\displaystyle+\delta\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta\Lambda_{i}-\frac{(\delta+1)(n-2)}{\Lambda_{i}-(n-2)}\right)$
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta\Lambda_{i}+\frac{\delta^{2}(\Lambda_{i}-(n-2))}{4(\delta\Lambda_{i}+n-2)}\right)$
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right),$
which shows that the inequality (1.4) is sharper than inequality (1.3).
Therefore, Theorem 1.2 improves Theorem 1.1.
## 2 Proof of Theorem
By the method of constructing trial functions introduced by Cheng and Yang in
[4], for any $\alpha=1,\ldots,n+1$ and each $i=1,\ldots,k$, the vector-valued
functions $x_{\alpha}\nabla u_{i}$ can be decomposed as
$x_{\alpha}\nabla u_{i}=\nabla h_{\alpha i}+W_{\alpha i},$ (2.1)
where $h_{\alpha i}\in H^{2}_{2,D}(\Omega)$, $\nabla h_{\alpha i}$ is the
projection of $x_{\alpha}\nabla u_{i}$ in $\mathbf{H}^{2}_{1,D}(\Omega)$ and
$W_{\alpha i}\perp\mathbf{H}^{2}_{1,D}(\Omega)$ (for the definitions of
$H^{2}_{2,D}(\Omega)$ and $\mathbf{H}^{2}_{1,D}(\Omega)$, we refer to [12]).
Hence,
$W_{\alpha i}|_{\partial\Omega}=0$
and
$\int_{\Omega}\langle W_{\alpha i},\nabla u\rangle=0,\ \ \ \ {\rm for\ any}\
u\in H^{2}_{2,D}(\Omega).$
Define $\|f\|^{2}=\int_{\Omega}|f|^{2}$. Then (see inequalities (2.19) and
(2.40) in [12])
$\|x_{\alpha}\nabla u_{i}\|^{2}=\|\nabla h_{\alpha i}\|^{2}+\|W_{\alpha
i}\|^{2}$ (2.2)
and for any $\delta>0$,
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}r_{\alpha i}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+(\delta\Lambda_{i}+n-2)\|W_{\alpha i}\|^{2}$ (2.3)
$\displaystyle+\delta\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}\Big{)}+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2},$ (2.4)
where
$\displaystyle r_{\alpha i}=$ $\displaystyle 2\|\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}+\int_{\Omega}\langle\nabla
x_{\alpha}^{2},\Delta u_{i}\nabla u_{i}\rangle+(n-2)\|x_{\alpha}\nabla
u_{i}\|^{2},$ $\displaystyle p_{\alpha i}=$
$\displaystyle\int_{\Omega}\langle\nabla x_{\alpha}^{2},u_{i}\nabla(\Delta
u_{i})+\Lambda_{i}u_{i}\nabla u_{i}\rangle,$ $\displaystyle Z_{\alpha i}=$
$\displaystyle\nabla\langle\nabla x_{\alpha},\nabla
u_{i}\rangle-\frac{n-2}{2}x_{\alpha}\nabla u_{i}.$
Next, we are to optimize the inequality (2.41) in [12]. Let $C$ be a positive
constant. Then it follows from Cauchy inequality that
$\displaystyle C\|\langle\nabla x_{\alpha},\nabla u_{i}\rangle\|^{2}=$
$\displaystyle C\int_{\Omega}\langle\nabla x_{\alpha},\nabla u_{i}\rangle^{2}$
(2.5) $\displaystyle=$ $\displaystyle-C\int_{\Omega}x_{\alpha}{\rm
div}(\langle\nabla x_{\alpha},\nabla u_{i}\rangle\nabla u_{i})$ (2.6)
$\displaystyle=$ $\displaystyle-C\int_{\Omega}\langle x_{\alpha}\nabla
u_{i},\nabla\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\rangle-C\int_{\Omega}\langle\nabla x_{\alpha},\nabla u_{i}\rangle
x_{\alpha}\Delta u_{i}$ (2.7) $\displaystyle=$
$\displaystyle-C\int_{\Omega}\langle\nabla h_{\alpha i},\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\rangle-\frac{C}{2}\int_{\Omega}\langle\nabla
x_{\alpha}^{2},\nabla u_{i}\rangle\Delta u_{i}$ (2.8) $\displaystyle\leq$
$\displaystyle(\delta\Lambda_{i}+n-2)\|\nabla h_{\alpha
i}\|^{2}+\frac{C^{2}}{4(\delta\Lambda_{i}+n-2)}\|\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}$ (2.9)
$\displaystyle-\frac{C}{2}\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta u_{i},$ (2.10)
where ${\rm div}(Z)$ denotes the divergence of $Z$. Applying (2.5) to (2.3)
yields
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}r_{\alpha i}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+(\delta\Lambda_{i}+n-2)\|W_{\alpha i}\|^{2}$ (2.11)
$\displaystyle+(\delta-C)\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}+C\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}\Big{)}$ (2.12)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2}$ (2.13) $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+(\delta\Lambda_{i}+n-2)(\|W_{\alpha i}\|^{2}+\|\nabla h_{\alpha
i}\|^{2})$ (2.14) $\displaystyle+(\delta-C)\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}+\frac{C^{2}}{4(\delta\Lambda_{i}+n-2)}\|\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}$ (2.15)
$\displaystyle-\frac{C}{2}\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta
u_{i}\Big{)}+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2}$ (2.16) $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta
p_{\alpha i}+(\delta\Lambda_{i}+n-2)\|x_{\alpha}\nabla u_{i}\|^{2}$ (2.17)
$\displaystyle+(\delta-C)\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}+\frac{C^{2}}{4(\delta\Lambda_{i}+n-2)}\|\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}$ (2.18)
$\displaystyle-\frac{C}{2}\int_{\Omega}\langle\nabla x_{\alpha}^{2},\nabla
u_{i}\rangle\Delta
u_{i}\Big{)}+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\|Z_{\alpha
i}\|^{2},$ (2.19)
where in the last step in (2.11) we have used equality (2.2). A direct
calculation yields (see (2.44), (2.45), (2.46) and (2.47) in [12])
$\sum_{\alpha=1}^{n+1}r_{\alpha i}=n,$ (2.20) $\sum_{\alpha=1}^{n+1}p_{\alpha
i}=0,$ (2.21) $\sum_{\alpha=1}^{n+1}\|x_{\alpha}\nabla
u_{i}\|^{2}=\sum_{\alpha=1}^{n+1}\|\langle\nabla x_{\alpha},\nabla
u_{i}\rangle\|^{2}=1,$ (2.22) $\sum_{\alpha=1}^{n+1}\|\nabla\langle\nabla
x_{\alpha},\nabla u_{i}\rangle\|^{2}=\Lambda_{i}-(n-2)$ (2.23)
and
$\sum_{\alpha=1}^{n+1}\|Z_{\alpha i}\|^{2}=\Lambda_{i}+\frac{(n-2)^{2}}{4}.$
(2.24)
Therefore, summing up (2.11) over $\alpha$ from 1 to $n+1$, one gets
$\displaystyle n\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\Big{(}\delta\Lambda_{i}+n-2+(\delta-C)$
$\displaystyle+\frac{C^{2}}{4(\delta\Lambda_{i}+n-2)}(\Lambda_{i}-(n-2))\Big{)}$
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
That is,
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta\Lambda_{i}+(\delta-C)\right.$
(2.25)
$\displaystyle\left.+\frac{C^{2}}{4(\delta\Lambda_{i}+n-2)}(\Lambda_{i}-(n-2))\right)$
(2.26)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
(2.27)
Taking
$C=\frac{2(\delta\Lambda_{i}+n-2)}{\Lambda_{i}-(n-2)}$
in (2.25) yields
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta\Lambda_{i}-\frac{(\delta+1)(n-2)}{\Lambda_{i}-(n-2)}\right)$
(2.28)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right),$
(2.29)
and hence
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(2+\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
(2.30) $\displaystyle\leq$
$\displaystyle\delta\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
(2.31)
$\displaystyle+\frac{1}{\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
(2.32)
To complete the proof of Theorem 1.2, we need the following lemma:
###### Lemma 2.1.
Let $\Omega$ be a connected bounded domain in $\mathbb{S}^{n}$. Then
$\Lambda_{1}\geq n.$ (2.33)
Proof. Let $\Omega_{1}$, $\Omega_{2}$ be two connected bounded domains in
$\mathbb{S}^{n}$ and $\Omega_{1}\subset\Omega_{2}$. Let $u_{1}(\Omega_{1})$ be
the eigenfunction corresponding to $\Lambda_{1}(\Omega_{1})$. Then the
function defined by
$\widetilde{u_{1}}=\left\\{\begin{array}[]{ll}u_{1}&\ \ \ {\rm in}\ \
\Omega_{1},\\\ 0&\ \ \ {\rm in}\ \ \Omega_{2}-\Omega_{1}\end{array}\right.$
is a eigenfunction corresponding to $\Lambda_{1}(\Omega_{2})$. Hence,
$\Lambda_{1}(\Omega_{1})\geq\Lambda_{1}(\Omega_{2})$. Denote by $\lambda_{1}$
the first eigenvalue of Laplacian. It is easy to see
$\Lambda_{1}(\mathbb{S}^{n})=\lambda_{1}(\mathbb{S}^{n})=n$ because there are
no boundary conditions in this case. It follows that
$\Lambda_{1}(\Omega)\geq\Lambda_{1}(\mathbb{S}^{n})=n$ by setting
$\Omega=\Omega_{1}$ and $\Omega_{2}=\mathbb{S}^{n}$. This completes the proof
of Lemma 2.1. $\sqcup$$\sqcap$
Inequality (2.33) shows that for any $i$,
$\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}>0.$ (2.34)
Minimizing the right hand side of (2.30) as a function of $\delta$ by choosing
$\delta=\left(\frac{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)}{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)}\right)^{1/2},$
we obtain the inequality (1.4), completing the proof of Theorem 1.2.
Proof of Corollary 1.3. It is easy to see from (1.4) that
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\right\\}^{1/2}$
(2.35)
$\displaystyle\qquad\qquad\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{1/2}.$
(2.36)
One can check by induction that
$\displaystyle\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\right\\}\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}$
$\displaystyle\leq$
$\displaystyle\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\right\\}\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\},$
which together with inequality (2.35) yields inequality (1.7).
Solving the quadratic polynomial of $\Lambda_{k+1}$ in (1.7), we obtain
inequality (1.8).
Replacing $k+1$ with $k$ in (1.7), we obtain
$\sum_{i=1}^{k-1}(\Lambda_{k}-\Lambda_{i})^{2}\leq\sum_{i=1}^{k-1}(\Lambda_{k}-\Lambda_{i})\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
Therefore,
$\sum_{i=1}^{k}(\Lambda_{k}-\Lambda_{i})^{2}\leq\sum_{i=1}^{k}(\Lambda_{k}-\Lambda_{i})\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right).$
Solving this inequality of quadratic polynomial for $\Lambda_{k}$, we infer
$\Lambda_{k}\geq S_{k+1}-\sqrt{S_{k+1}^{2}-T_{k+1}},$ (2.37)
where $S_{k+1},T_{k+1}$ are given by (1.10) and (1.11) respectively.
Therefore, the inequality (1.9) follows from (2.37) and (1.8). Then Corollary
1.3 is completed.
## References
* [1] M.S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Edinburgh, 1998) E. B. Davies, and Yu Safarov eds., London Math. Soc. Lecture Notes vol. 273, Cambridge Univ. Press, Cambridge, 1999. pp. 95-139.
* [2] M.S. Ashbaugh, On universal inequalities for the low eigenvalues of the buckling problem. (English summary) Partial differential equations and inverse problems, 13-31, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, 2004.
* [3] Q.M. Cheng, H.C. Yang, Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc. 358 (2006) 2625-2635.
* [4] Q.M. Cheng, H.C. Yang, Universal bounds for eigenvalues of a buckling problem, Commun. Math. Phys. 262 (2006) 663-675.
* [5] A. EI Soufi, E.M. Harrell and S. Ilias, Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Amer. Math. Soc. 361 (2009) 2337-2350.
* [6] E.M. Harrell, J. Stubble, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997) 1797-1809.
* [7] G.N. Hile, R.Z. Yeh, Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math. 112 (1984) 115-133.
* [8] G.Y. Huang, X.X. Li and R.W. Xu, Extrinsic estimates for the eigenvalues of Schrödinger operator, to appear in Geom. Dedicata.
* [9] G.Y. Huang, W.Y. Chen, Inequalities of eigenvalues for the bi-kohn Laplacian on the Heisenberg group, to appear in Acta Math. Sci. Ser. B Engl. Ed. 2009, vol 29(6).
* [10] M. Levitin, L. Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues, J. Funct. Anal. 192 (2002) 425-445.
* [11] L.E. Payne, G. Pólya and H.F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. Phys. 35 (1956), 289-298.
* [12] Q.L. Wang, C.Y. Xia, Universal inequalities for eigenvalues of the buckling problem on spherical domains, Commun. Math. Phys. 270 (2007) 759-775.
* [13] Q.L. Wang, C.Y. Xia, Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Anal. 245 (2007) 334-352.
|
arxiv-papers
| 2009-10-21T13:37:48 |
2024-09-04T02:49:05.945994
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangyue Huang, Xingxiao Li, Linfen Cao",
"submitter": "Huang Guangyue",
"url": "https://arxiv.org/abs/0910.4075"
}
|
0910.4083
|
# Estimates on the first two buckling eigenvalues on spherical domains 111This
research is supported by National Natural Science Foundation of China (Project
No. 10671181) and National Natural Science Foundation of Henan (Project No.
092300410143; 2009A110010).
Guangyue Huang, Xingxiao Li 222The corresponding author. Email:
xxl$@$henannu.edu.cn, Xuerong Qi
Department of Mathematics, Henan Normal University
Xinxiang 453007, Henan, P.R. China
> Abstract. In this paper, we study the first two eigenvalues of the buckling
> problem on spherical domains. We obtain an estimate on the second eigenvalue
> in terms of the first eigenvalue, which improves one recent result obtained
> by Wang-Xia in [7].
> Keywords: eigenvalue, universal bound, spherical domain, buckling problem.
> Mathematics Subject Classification: Primary 35P15, Secondary 53C20.
## 1 Introduction
Let $\Omega$ be a connected bounded domain in an $n$-dimensional Euclidean
space $\mathbb{R}^{n}$ and $\nu$ be the unit outward normal vector field of
$\partial\Omega$. The well-known eigenvalue problem
$\left\\{\begin{array}[]{ll}\Delta^{2}u=-\Lambda\Delta u&{\rm in}\ \Omega,\\\
u={\partial u\over\partial\nu}=0&{\rm on}\ \partial\Omega\end{array}\right.$
(1.1)
is called a buckling problem, which is used to describe the critical buckling
load of a clamped plate subjected to a uniform compressive force around its
boundary.
Let
$0<\Lambda_{1}\leq\Lambda_{2}\leq\Lambda_{3}\leq\cdots$
denote the successive eigenvalues for (1.1), where each eigenvalue is repeated
according to its multiplicity. In 1956, Payne-Pólya-Weinberger [6] proved that
$\Lambda_{2}\leq\left(1+{4\over n}\right)\Lambda_{1}.$
Subsequently, Hile-Yeh [4] improved the above inequality as follows:
$\Lambda_{2}\leq{n^{2}+8n+20\over(n+2)^{2}}\Lambda_{1}.$
In 1994, Chen-Qian [2] considered the following more general eigenvalue
problem:
$\left\\{\begin{array}[]{ll}(-\Delta)^{p}u=\Lambda(-\Delta)^{q}u&{\rm in}\
\Omega,\\\ u={\partial
u\over\partial\nu}=\cdots={\partial^{p-1}u\over\partial\nu^{p-1}}=0&{\rm on}\
\partial\Omega\end{array}\right.$
with $p$ and $q$ are positive integers and $p>q$. They proved that
$\Lambda_{2}\leq{(n+2q)^{2}+4p(2p+n-2)-4q(2q+n-2)\over(n+2q)^{2}}\Lambda_{1}.$
In 1998, Ashbaugh [1] found that
$\sum_{i=1}^{n}\Lambda_{i+1}\leq(n+4)\Lambda_{1}.$
For answering a question of Ashbaugh in [1], Cheng-Yang [3] obtained in a
recent survey paper a universal inequality for higher eigenvalues of the
buckling problem. In fact, they proved that
$\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq{4(n+2)\over
n^{2}}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\Lambda_{i}.$
In 2007, Wang-Xia [7] considered the buckling problem on domains in a unit
sphere and obtained the following result:
Theorem A. Let $\Lambda_{i}$ be the $i^{th}$ eigenvalue of the following
eigenvalue problem:
$\Delta^{2}u=-\Lambda\Delta u\ \ {\rm in}\ \Omega,\ \ \ \ u={\partial
u\over\partial\nu}=0\ \ {\rm on}\ \partial\Omega,$
where $\Omega$ is a connected domain in an $n$-dimensional unit sphere with
smooth boundary $\partial\Omega$ and $\nu$ is the unit outward normal vector
field of $\partial\Omega$. Then for any $\delta>0$, it holds that
$\displaystyle 2\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq$
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\delta\Lambda_{i}+{\delta^{2}(\Lambda_{i}-(n-2))\over
4(\delta\Lambda_{i}+n-2)}\right)$ (1.2)
$\displaystyle+{1\over\delta}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+{(n-2)^{2}\over
4}\right).$
Recently, Huang-Li-Cao [5] improved the above result as follows:
Theorem B. Under the assumption of Theorem A. Then for any $\delta>0$,
$\displaystyle\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(2+\frac{n-2}{\Lambda_{i}-(n-2)}\right)$
(1.3) $\displaystyle\leq$ $\displaystyle
2\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\left(\Lambda_{i}-\frac{n-2}{\Lambda_{i}-(n-2)}\right)\right\\}^{\frac{1}{2}}$
$\displaystyle\qquad\qquad\qquad\times\left\\{\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\left(\Lambda_{i}+\frac{(n-2)^{2}}{4}\right)\right\\}^{\frac{1}{2}}.$
In the present paper, we consider the first two eigenvalues of the buckling
problem on spherical domains and obtain the following result:
Theorem 1.1. Let $\Lambda_{i}$ be the $i^{th}$ eigenvalue of the following
eigenvalue problem:
$\Delta^{2}u=-\Lambda\Delta u\ \ {\rm in}\ \Omega,\ \ \ \ u={\partial
u\over\partial\nu}=0\ \ {\rm on}\ \partial\Omega,$ (1.4)
where $\Omega$ is a connected domain in an $n$-dimensional unit sphere with
smooth boundary $\partial\Omega$ and $\nu$ is the unit outward normal vector
field of $\partial\Omega$. Then we have
$\displaystyle\Lambda_{2}\leq\Lambda_{1}+\left({n(n-\Lambda_{1})\over\Lambda_{1}}+2(n+2)\right){4\Lambda_{1}+(n-2)^{2}\over(n+2)^{2}}.$
(1.5)
Corollary 1.2. Under the same assumption of Theorem 1.1, we have
$\Lambda_{2}\leq\left(1+{8\over n+2}\right)\Lambda_{1}+{2(n-2)^{2}\over n+2}.$
(1.6)
Remark. From the inequality (2.18) in section 2, we have
$\Lambda_{1}\geq n.$
Hence, we derive from (1.3) that
$(\Lambda_{2}-\Lambda_{1})\leq\Lambda_{1}^{\frac{1}{2}}\left\\{(\Lambda_{2}-\Lambda_{1})\left(\Lambda_{1}+\frac{(n-2)^{2}}{4}\right)\right\\}^{\frac{1}{2}}.$
(1.7)
That is,
$\Lambda_{2}\leq\Lambda_{1}+\Lambda_{1}\left(\Lambda_{1}+{(n-2)^{2}\over
4}\right).$ (1.8)
Note that
$\left(1+{8\over n+2}\right)\Lambda_{1}+{2(n-2)^{2}\over
n+2}<\Lambda_{1}+\Lambda_{1}\left(\Lambda_{1}+{(n-2)^{2}\over 4}\right).$
Thus it is not hard to see that, for the first two eigenvalues of problem
(1.4), the inequality (1.6) is sharper than (1.3).
## 2 Proof of Theorem
Let $x_{1},x_{2},\ldots,x_{n+1}$ be the standard coordinate functions of the
Euclidean space $\mathbb{R}^{n+1}$. Define
$\mathbb{S}^{n}=\left\\{(x_{1},x_{2},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\ ;\
\sum_{\alpha=1}^{n+1}x_{\alpha}^{2}=1\right\\}.$
Denote by $u_{1}$ the eigenfunction corresponding to $\Lambda_{1}$ of the
eigenvalue problem (1.4), that is
$\left\\{\begin{array}[]{l}\Delta^{2}u_{1}=-\Lambda_{1}\Delta u_{1}\ \ {\rm
in}\ \Omega,\\\ u_{1}={\partial u_{1}\over\partial\nu}=0\ \ {\rm on}\
\partial\Omega,\\\ \int_{\Omega}\langle\nabla u_{1},\nabla
u_{1}\rangle=1.\end{array}\right.$
Let
$\varphi_{\alpha}=x_{\alpha}u_{1}-C_{\alpha}u_{1},$ (2.1)
where
$C_{\alpha}=\int_{\Omega}x_{\alpha}u_{1}(-\Delta)u_{1}.$
Then one gets $\varphi_{\alpha}=\partial\varphi_{\alpha}/\partial\nu=0$ on
$\partial\Omega$ and
$\int_{\Omega}\langle\nabla\varphi_{\alpha},\nabla u_{1}\rangle=0.$ (2.2)
It follows from the Rayleigh-Ritz inequality that
$\Lambda_{2}\leq{\int_{\Omega}\varphi_{\alpha}\Delta^{2}\varphi_{\alpha}\over\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}}.$
(2.3)
Using integration by parts and the definition of $\varphi_{\alpha}$, one finds
that
$\displaystyle\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}=$
$\displaystyle\int_{\Omega}\varphi_{\alpha}(-\Delta)\varphi_{\alpha}$ (2.4)
$\displaystyle=$
$\displaystyle\int_{\Omega}\varphi_{\alpha}(-\Delta)(x_{\alpha}u_{1})$
$\displaystyle=$
$\displaystyle\int_{\Omega}\varphi_{\alpha}x_{\alpha}(-\Delta)u_{1}-\int_{\Omega}\varphi_{\alpha}[\Delta(x_{\alpha}u_{1})+x_{\alpha}(-\Delta)u_{1}].$
Note that
$\displaystyle\int_{\Omega}\varphi_{\alpha}\Delta^{2}\varphi_{\alpha}=$
$\displaystyle\int_{\Omega}\varphi_{\alpha}\Delta^{2}(x_{\alpha}u_{1})$ (2.5)
$\displaystyle=$
$\displaystyle\Lambda_{1}\int_{\Omega}\varphi_{\alpha}x_{\alpha}(-\Delta)u_{1}+\int_{\Omega}\varphi_{\alpha}[\Delta^{2}(x_{\alpha}u_{1})-x_{\alpha}\Delta^{2}u_{1}].$
It follows from (2.3), (2.4) and (2.5) that
$\displaystyle(\Lambda_{2}-\Lambda_{1})\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}\leq$
$\displaystyle\Lambda_{1}\int_{\Omega}\varphi_{\alpha}[\Delta(x_{\alpha}u_{1})+x_{\alpha}(-\Delta)u_{1}]$
(2.6)
$\displaystyle+\int_{\Omega}\varphi_{\alpha}[\Delta^{2}(x_{\alpha}u_{1})-x_{\alpha}\Delta^{2}u_{1}].$
Again, by using integration by parts, one gets that
$\displaystyle\Lambda_{1}$
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}\varphi_{\alpha}[\Delta(x_{\alpha}u_{1})+x_{\alpha}(-\Delta)u_{1}]$
(2.7) $\displaystyle=$
$\displaystyle\Lambda_{1}\sum_{\alpha=1}^{n+1}\int_{\Omega}x_{\alpha}u_{1}[\Delta(x_{\alpha}u_{1})+x_{\alpha}(-\Delta)u_{1}]$
$\displaystyle=$
$\displaystyle\Lambda_{1}\sum_{\alpha=1}^{n+1}\int_{\Omega}x_{\alpha}u_{1}[-nx_{\alpha}u_{1}+2\langle\nabla
x_{\alpha},\nabla u_{1}\rangle]$ $\displaystyle=$
$\displaystyle-n\Lambda_{1}\int_{\Omega}u_{1}^{2}.$
Inserting (2.7) into (2.6) yields
$(\Lambda_{2}-\Lambda_{1})\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}\leq-n\Lambda_{1}\int_{\Omega}u_{1}^{2}\\\
+\int_{\Omega}\varphi_{\alpha}[\Delta^{2}(x_{\alpha}u_{1})-x_{\alpha}\Delta^{2}u_{1}].$
(2.8)
Let $\nabla$ and $\nabla^{2}$ be the gradient operator and the Hessian
operator of $\mathbb{S}^{n}$, respectively. Then we have
$\Delta(x_{\alpha}u_{1})=-nx_{\alpha}u_{1}+2\langle\nabla x_{\alpha},\nabla
u_{1}\rangle+x_{\alpha}\Delta u_{1},$ (2.9)
where we have used $\Delta x_{\alpha}=-nx_{\alpha}$. From the Ricci identity,
it follows that
$\Delta\langle\nabla x_{\alpha},\nabla
u_{1}\rangle=2\langle\nabla^{2}x_{\alpha},\nabla^{2}u_{1}\rangle+\langle\nabla
x_{\alpha},\nabla\Delta u_{1}\rangle+(n-2)\langle\nabla x_{\alpha},\nabla
u_{1}\rangle.$ (2.10)
It is well-known that
$\nabla^{2}x_{\alpha}=-x_{\alpha}\langle\ ,\ \rangle.$
Therefore, (2.10) becomes
$\Delta\langle\nabla x_{\alpha},\nabla u_{1}\rangle=-2x_{\alpha}\Delta
u_{1}+\langle\nabla x_{\alpha},\nabla\Delta u_{1}\rangle+(n-2)\langle\nabla
x_{\alpha},\nabla u_{1}\rangle.$ (2.11)
By virtue of (2.9) and (2.11), a direct calculation yields
$\Delta^{2}(x_{\alpha}u_{1})=x_{\alpha}\Delta^{2}u_{1}+n^{2}x_{\alpha}u_{1}-2(n+2)x_{\alpha}\Delta
u_{1}-4\langle\nabla x_{\alpha},\nabla u_{1}\rangle+4\langle\nabla
x_{\alpha},\nabla\Delta u_{1}\rangle.$ (2.12)
Then
$\displaystyle\sum_{\alpha=1}^{n+1}$
$\displaystyle\int_{\Omega}\varphi_{\alpha}[\Delta^{2}(x_{\alpha}u_{1})-x_{\alpha}\Delta^{2}u_{1}]$
(2.13) $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}x_{\alpha}u_{1}[\Delta^{2}(x_{\alpha}u_{1})-x_{\alpha}\Delta^{2}u_{1}]$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}x_{\alpha}u_{1}[n^{2}x_{\alpha}u_{1}-2(n+2)x_{\alpha}\Delta
u_{1}-4\langle\nabla x_{\alpha},\nabla u_{1}\rangle+4\langle\nabla
x_{\alpha},\nabla\Delta u_{1}\rangle]$ $\displaystyle=$ $\displaystyle
n^{2}\int_{\Omega}u_{1}^{2}-2(n+2)\int_{\Omega}u_{1}\Delta u_{1}$
$\displaystyle=$ $\displaystyle n^{2}\int_{\Omega}u_{1}^{2}+2(n+2).$
From (2.8) and (2.13), we arrive at
$(\Lambda_{2}-\Lambda_{1})\sum_{\alpha=1}^{n+1}\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}\leq
n(n-\Lambda_{1})\int_{\Omega}u_{1}^{2}+2(n+2).$ (2.14)
Let
$D_{\alpha}=\int_{\Omega}\langle\nabla\varphi_{\alpha},\nabla\langle\nabla
x_{\alpha},\nabla u_{1}\rangle-{n-2\over 2}x_{\alpha}\nabla u_{1}\rangle.$
Then
$\displaystyle\sum_{\alpha=1}^{n+1}D_{\alpha}=$
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}\langle\nabla(x_{\alpha}u_{1}),\nabla\langle\nabla
x_{\alpha},\nabla u_{1}\rangle-{n-2\over 2}x_{\alpha}\nabla u_{1}\rangle$
$\displaystyle-\sum_{\alpha=1}^{n+1}C_{\alpha}\int_{\Omega}\langle\nabla
u_{1},\nabla\langle\nabla x_{\alpha},\nabla u_{1}\rangle-{n-2\over
2}x_{\alpha}\nabla u_{1}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}\langle\nabla(x_{\alpha}u_{1}),\nabla\langle\nabla
x_{\alpha},\nabla u_{1}\rangle-{n-2\over 2}x_{\alpha}\nabla u_{1}\rangle$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}\langle\nabla(x_{\alpha}u_{1}),\nabla\langle\nabla
x_{\alpha},\nabla u_{1}\rangle\rangle-{n-2\over
2}\sum_{\alpha=1}^{n+1}\int_{\Omega}\langle\nabla(x_{\alpha}u_{1}),x_{\alpha}\nabla
u_{1}\rangle$ $\displaystyle=$ $\displaystyle-2-{n-2\over 2}$ $\displaystyle=$
$\displaystyle-{n+2\over 2}.$
Since
$\displaystyle\sum_{\alpha=1}^{n+1}\int_{\Omega}|\nabla\langle\nabla
x_{\alpha},\nabla u_{1}\rangle-{n-2\over 2}x_{\alpha}\nabla u_{1}|^{2}$
$\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{n+1}\left(\int_{\Omega}|\nabla\langle\nabla
x_{\alpha},\nabla
u_{1}\rangle|^{2}-(n-2)\int_{\Omega}\langle\nabla\langle\nabla
x_{\alpha}\nabla u_{1}\rangle,x_{\alpha}\nabla u_{1}\rangle\right.$
$\displaystyle\left.+\frac{(n-2)^{2}}{4}\int_{\Omega}|x_{\alpha}\nabla
u_{1}|^{2}\right)$ $\displaystyle=$ $\displaystyle\Lambda_{1}+{(n-2)^{2}\over
4}.$
It follows that
$\displaystyle{(n+2)^{2}\over 4}=$
$\displaystyle\left(\sum_{\alpha=1}^{n+1}D_{\alpha}\right)^{2}$ (2.15)
$\displaystyle\leq$
$\displaystyle\left(\sum_{\alpha=1}^{n+1}\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}\right)\left(\sum_{\alpha=1}^{n+1}\int_{\Omega}|\nabla\langle\nabla
x_{\alpha},\nabla u_{1}\rangle-{n-2\over 2}x_{\alpha}\nabla u_{1}|^{2}\right)$
$\displaystyle=$ $\displaystyle\left(\Lambda_{1}+{(n-2)^{2}\over
4}\right)\sum_{\alpha=1}^{n+1}\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}.$
By (2.15), we obtain
$\sum_{\alpha=1}^{n+1}\int_{\Omega}|\nabla\varphi_{\alpha}|^{2}\geq{(n+2)^{2}\over
4\Lambda_{1}+(n-2)^{2}}.$ (2.16)
Applying (2.16) to (2.14) yields
$\Lambda_{2}\leq\Lambda_{1}+\left(n(n-\Lambda_{1})\int_{\Omega}u_{1}^{2}+2(n+2)\right){4\Lambda_{1}+(n-2)^{2}\over(n+2)^{2}}.$
(2.17)
Lemma 2.1. Let $\Omega_{1}$, $\Omega_{2}$ be two connected bounded domains in
$\mathbb{S}^{n}$ and $\Omega_{1}\subset\Omega_{2}$. Then it holds that
$\Lambda_{1}(\Omega_{1})\geq\Lambda_{1}(\Omega_{2})$.
Proof. Let $u_{1}(\Omega_{1})$ be the eigenfunction corresponding to
$\Lambda_{1}(\Omega_{1})$. Then the function defined by
$\widetilde{u_{1}}=\left\\{\begin{array}[]{ll}u_{1}&\ \ \ {\rm in}\ \
\Omega_{1},\\\ 0&\ \ \ {\rm in}\ \ \Omega_{2}-\Omega_{1}\end{array}\right.$
is a eigenfunction corresponding to $\Lambda_{1}(\Omega_{2})$. This easily
proves Lemma 2.1.
Let $\lambda_{1}$ be the first eigenvalue of Laplacian. Then
$\Lambda_{1}(\mathbb{S}^{n})=\lambda_{1}(\mathbb{S}^{n})=n$ because there are
no boundary conditions in this case. It follows that
$\Lambda_{1}(\Omega_{1})\geq n$ (2.18)
by setting $\Omega=\Omega_{1}$ and $\Omega_{2}=\mathbb{S}^{n}$ in Lemma 2.1.
Using Schwarz inequality, we have
$1=\int_{\Omega}|\nabla
u_{1}|^{2}=\int_{\Omega}u_{1}(-\Delta)u_{1}\leq\left(\int_{\Omega}u_{1}^{2}\int_{\Omega}(\Delta
u_{1})^{2}\right)^{1/2}=\left(\Lambda_{1}\int_{\Omega}u_{1}^{2}\right)^{1/2}.$
Hence,
$\int_{\Omega}u_{1}^{2}\geq{1\over\Lambda_{1}}.$ (2.19)
Applying (2.18) and (2.19) into (2.17) yields
$\Lambda_{2}\leq\Lambda_{1}+\left({n(n-\Lambda_{1})\over\Lambda_{1}}+2(n+2)\right){4\Lambda_{1}+(n-2)^{2}\over(n+2)^{2}}.$
This completes the proof of Theorem 1.1.
Proof of Corollary 1.2. From the inequality (2.18), we have
${n(n-\Lambda_{1})\over\Lambda_{1}}+2(n+2)\leq 2(n+2).$ (2.20)
Applying (2.20) to (1.5) completes the proof of Corollary 1.2.
## References
* [1] M.S. Ashbaugh. Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Edinburgh, 1998) E. B. Davies, and Yu Safalov eds., London Math. Soc. Lecture Notes vol. 273, Cambridge Univ. Press, Cambridge, 1999. pp. 95-139.
* [2] Z.C. Chen and C.L. Qian. On the upper bound of eigenvalues for elliptic equations with higher orders, J. Math. Anal. Appl. 186 (1994), 821-834.
* [3] Q.M. Cheng and H.C. Yang. Universal bounds for eigenvalues of a buckling problem, Commun. Math. Phys. 262 (2006), 663-675.
* [4] G.N. Hile and R.Z. Yeh. Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math. 112 (1984), 115-133.
* [5] G.Y. Huang, X.X. Li and L.F. Cao. Universal bounds on eigenvalues of the buckling problem on spherical domains, to appear
* [6] L.E. Payne, G. Pólya and H. F. Weinberger. On the ratio of consecutive eigenvalues, J. Math. Phys. 35 (1956), 289-298.
* [7] Q.L. Wang and C.Y. Xia. Universal inequalities for eigenvalues of the buckling problem on spherical domains, Commun. Math. Phys. 270 (2007), 759-775.
|
arxiv-papers
| 2009-10-21T13:26:54 |
2024-09-04T02:49:05.950072
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangyue Huang, Xingxiao Li, Xuerong Qi",
"submitter": "Huang Guangyue",
"url": "https://arxiv.org/abs/0910.4083"
}
|
0910.4096
|
# Lower order eigenvalues of the poly-Laplacian with any order on spherical
domains 111This research is supported by Project of Henan Provincial
department of Sciences and Technology (No. 092300410143), and NSF of Henan
Provincial Education department (No. 2009A110010).
Guangyue Huang, Bingqing Ma 222The corresponding author. Email:
bqma$@$henannu.edu.cn
Department of Mathematics, Henan Normal University, Xinxiang 453007, Henan
People’s Republic of China
> Abstract. We consider the lower order eigenvalues of poly-Laplacian with any
> order on spherical domains. We obtain universal inequalities for them and
> show that our results are optimal.
> Keywords: eigenvalue; poly-Laplacian.
> Mathematics Subject Classification: Primary 35P15; Secondary 53C20.
## 1 Introduction
Let $\Omega$ be a connected bounded domain in an $n$-dimensional complete
Riemannian manifold $M$. In this paper, we consider the Dirichlet eigenvalue
problem of the poly-Laplacian with order $p$:
$\left\\{\begin{array}[]{ll}(-\Delta)^{p}u=\lambda u&\ \ \ {\rm in}\
\Omega,\\\ u={\partial
u\over\partial\nu}=\cdots={\partial^{p-1}u\over\partial\nu^{p-1}}=0&\ \ \ {\rm
on}\ \partial\Omega,\end{array}\right.$ (1.1)
where $\Delta$ is the Laplacian in $M$ and $\nu$ denotes the outward unit
normal vector field of $\partial\Omega$. Let
$0<\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\cdots\rightarrow+\infty$
denote the successive eigenvalues for (1.1), where each eigenvalue is repeated
according to its multiplicity.
When $p=1$, the eigenvalue problem (1.1) is called a fixed membrane problem.
For $M=\mathbb{R}^{2}$ and $p=1$, Payne-Pólya-Weinberger [10] proved
$\lambda_{2}+\lambda_{3}\leq 6\lambda_{1}.$
In 1993, for general dimensions $n\geq 2$, Ashbaugh and Benguria [2] proved
$\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})\leq 4\lambda_{1}.$
Recently, the inequalities of eigenvalues of the fixed membrane problem have
been generalized to some Riemannian manifolds. For the related research and
improvement in this direction, see [3, 4, 9, 11] and the references therein.
In particular, Sun-Cheng-Yang [11] proved that when $M$ is an $n$-dimensional
unit sphere $\mathbb{S}^{n}(1)$,
$\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})\leq 4\lambda_{1}+n^{2}.$ (1.2)
When $p=2$, the eigenvalue problem (1.1) is called a clamped plate problem.
For $M=\mathbb{R}^{n}$ and $p\geq 2$, Cheng-Ichikawa-Mametsuka proved in [8]
that
$\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})\leq 4p(2p-1)\lambda_{1},$ (1.3)
$\sum_{i=1}^{n}(\lambda_{i+1}^{\frac{1}{p}}-\lambda_{1}^{\frac{1}{p}})^{p-1}\leq(2p)^{p-1}\lambda_{1}^{\frac{p-1}{p}}.$
(1.4)
Inequalities (1.3) and (1.4) include two universal inequalities of the clamped
plate problem announced by Ashbauth in [1]. When $M$ is a general complete
Riemannian manifold, for $p=2$, Cheng-Huang-Wei [6] obtained
$\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{\frac{1}{2}}\leq\big{(}4\lambda_{1}^{\frac{1}{2}}+n^{2}H_{0}^{2}\big{)}^{\frac{1}{2}}\big{\\{}(2n+4)\lambda_{1}^{\frac{1}{2}}+n^{2}H_{0}^{2}\big{\\}}^{\frac{1}{2}},$
(1.5)
where $H_{0}^{2}$ is a nonnegative constant which depends only on $M$ and
$\Omega$. For $M=\mathbb{S}^{n}(1)$, we have $H_{0}^{2}=1$ such that (1.5)
becomes the following inequality:
$\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{\frac{1}{2}}\leq\big{(}4\lambda_{1}^{\frac{1}{2}}+n^{2}\big{)}^{\frac{1}{2}}\big{\\{}(2n+4)\lambda_{1}^{\frac{1}{2}}+n^{2}\big{\\}}^{\frac{1}{2}}.$
(1.6)
We remark that when $\Omega=\mathbb{S}^{n}(1)$, it holds that $\lambda_{1}=0$
and $\lambda_{2}=\cdots=\lambda_{n+1}=n^{2}$. Therefore, the inequality (1.6)
becomes equality. Hence, for $M=\mathbb{S}^{n}(1)$, the inequality (1.6) is
optimal.
In the present article, we consider the eigenvalue problem (1.1) with any $p$
when $M$ is a unit sphere $\mathbb{S}^{n}(1)$. we obtain the following result:
Theorem. Let $\Omega$ be a bounded domain in an $n$-dimensional unit sphere
$\mathbb{S}^{n}(1)$. Let $\lambda_{i}$ be the $i$-th eigenvlaue of the
eigenvalue problem (1.1). Then we have
$\displaystyle\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}\leq$
$\displaystyle\left\\{\left(\lambda_{1}^{1\over
p}+n\right)^{p}-\lambda_{1}+4[2^{p}-(p+1)]\lambda_{1}^{1\over
p}\left(\lambda_{1}^{1\over p}+n\right)^{p-2}\right\\}^{1\over 2}$ (1.7)
$\displaystyle\times\left\\{4\lambda_{1}^{1\over p}+n^{2}\right\\}^{1\over
2}.$
Remark 1. For $p=2$, the inequality (1.7) becomes the optimal inequality
(1.6).
Remark 2. For the unit sphere $S^{n}(1)$, by taking $\Omega=S^{n}(1)$, we know
$\lambda_{1}=0$ and $\lambda_{2}=\cdots=\lambda_{n+1}=n^{p}$. Hence, the
inequality (1.7) becomes equality. Therefore, our result is optimal.
Acknowledgement. The authors thank Professor Qing-Ming Cheng for his helpful
discussion and support.
## 2 Proof of Theorem
Let $u_{i}$ be the orthonormal eigenfunction corresponding to eigenvalue
$\lambda_{i}$, that is,
$\left\\{\begin{array}[]{ll}(-\Delta)^{p}u_{i}=\lambda_{i}u_{i}&\ \ \ {\rm
in}\ \Omega,\\\ u_{i}={\partial
u_{i}\over\partial\nu}=\cdots={\partial^{p-1}u_{i}\over\partial\nu^{p-1}}=0&\
\ \ {\rm on}\ \partial\Omega,\\\
\int\limits\limits_{\Omega}u_{i}u_{j}=\delta_{ij}.\end{array}\right.$
Let $x^{1},x^{2},\ldots,x^{n+1}$ be the standard Euclidean coordinate
functions of $\mathbb{R}^{n+1}$, then
$\mathbb{S}^{n}(1)=\left\\{(x^{1},x^{2},\ldots,x^{n+1})\in\mathbb{R}^{n+1}\ ;\
\sum_{i=1}^{n+1}(x^{i})^{2}=1\right\\}.$
It is well known that
$\Delta x^{i}=-nx^{i},\ \ \ i=1,2,\ldots,n+1.$
Assume that $B$ is an $(n+1)\times(n+1)$-matrix defined by $B=(b_{ij})$, where
$b_{ij}=\int\limits_{\Omega}x^{i}u_{1}u_{j+1}.$
Using the orthogonalization of Gram and Schmidt, we know that there exist an
upper triangle matrix $R=(r_{ij})$ and an orthogonal matrix $Q=(q_{ij})$ such
that $R=QB$, that is,
$r_{ij}=\sum\limits_{k=1}^{n+1}q_{ik}b_{kj}=\sum\limits_{k=1}^{n+1}\int\limits_{\Omega}q_{ik}x^{k}u_{1}u_{j}=0,\
\ 2\leq j\leq i\leq n+1.$
Defining $h_{i}=\sum\limits_{k=1}^{n+1}q_{ik}x^{k}$, one gets
$\int\limits_{\Omega}h_{i}u_{1}u_{j}=\sum\limits_{k=1}^{n+1}\int\limits_{\Omega}q_{ik}x^{k}u_{1}u_{j}=0,\
\ 2\leq j\leq i\leq n+1.$
Setting
$\varphi_{i}=h_{i}u_{1}-u_{1}\int\limits_{\Omega}h_{i}u_{1}^{2}.$
Then
$\int\limits_{\Omega}\varphi_{i}u_{j}=0,\ \ \ \ {\rm for\ any}\ j\leq i.$
It follows from Rayleigh-Ritz inequality that
$\lambda_{i+1}\leq{\int\limits_{\Omega}\varphi_{i}(-\Delta)^{p}\varphi_{i}\over\|\varphi_{i}\|^{2}},$
(2.1)
where $\|f\|^{2}=\int\limits_{\Omega}|f|^{2}$. By a direct calculation, we
derive at
$\displaystyle\int\limits_{\Omega}\varphi_{i}(-\Delta)^{p}\varphi_{i}=$
$\displaystyle\int\limits_{\Omega}\varphi_{i}(-\Delta)^{p}(h_{i}u_{1})$ (2.2)
$\displaystyle=$
$\displaystyle\int\limits_{\Omega}\varphi_{i}\\{[(-\Delta)^{p}(h_{i}u_{1})-h_{i}(-\Delta)^{p}u_{1}]+\lambda_{1}h_{i}u_{1}\\}$
$\displaystyle=$
$\displaystyle\lambda_{1}\|\varphi_{i}\|^{2}+\int\limits_{\Omega}\varphi_{i}[(-\Delta)^{p}(h_{i}u_{1})-h_{i}(-\Delta)^{p}u_{1}]$
$\displaystyle=$
$\displaystyle\lambda_{1}\|\varphi_{i}\|^{2}+\int\limits_{\Omega}h_{i}u_{1}[(-\Delta)^{p}(h_{i}u_{1})-h_{i}(-\Delta)^{p}u_{1}]$
$\displaystyle-\int\limits_{\Omega}h_{i}u_{1}^{2}\int\limits_{\Omega}u_{1}[(-\Delta)^{p}(h_{i}u_{1})-h_{i}(-\Delta)^{p}u_{1}]$
$\displaystyle=$
$\displaystyle\lambda_{1}\|\varphi_{i}\|^{2}+\int\limits_{\Omega}h_{i}u_{1}[(-\Delta)^{p}(h_{i}u_{1})-h_{i}(-\Delta)^{p}u_{1}].$
Defining
$\nabla^{r}=\left\\{\,\vbox{\openup
3.0pt\halign{$\displaystyle{#}$\hfil&\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr\Delta^{r/2}&\
\ {\rm when}\ r\ {\rm is\ even},\\\\\nabla(\Delta^{(r-1)/2})&\ \ {\rm when}\
r\ {\rm is\ odd}.\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$
Then (2.2) can be written as
$\int\limits_{\Omega}\varphi_{i}(-\Delta)^{p}\varphi_{i}=\lambda_{1}\|\varphi_{i}\|^{2}+\|\nabla^{p}(h_{i}u_{1})\|^{2}-\lambda_{1}\|h_{i}u_{1}\|^{2}.$
(2.3)
Putting (2.3) into (2.1) yields
$(\lambda_{i+1}-\lambda_{1})\|\varphi_{i}\|^{2}\leq\|\nabla^{p}(h_{i}u_{1})\|^{2}-\lambda_{1}\|h_{i}u_{1}\|^{2}.$
(2.4)
One gets from integration by parts that
$\displaystyle\int\limits_{\Omega}u_{1}h_{i}\langle\nabla h_{i},\nabla
u_{1}\rangle=$ $\displaystyle{1\over
4}\int\limits_{\Omega}\langle\nabla(h_{i}^{2}),\nabla(u_{1}^{2})\rangle=-{1\over
4}\int\limits_{\Omega}u_{1}^{2}\Delta(h_{i}^{2})$ $\displaystyle=$
$\displaystyle-{1\over 2}\int\limits_{\Omega}u_{1}^{2}h_{i}\Delta
h_{i}-{1\over 2}\int\limits_{\Omega}u_{1}^{2}|\nabla h_{i}|^{2}.$
Hence,
$\displaystyle\int\limits_{\Omega}\varphi_{i}\left(\langle\nabla h_{i},\nabla
u_{1}\rangle+{1\over 2}u_{1}\Delta h_{i}\right)$ (2.5) $\displaystyle=$
$\displaystyle\int\limits_{\Omega}u_{1}h_{i}\langle\nabla h_{i},\nabla
u_{1}\rangle+{1\over 2}\int\limits_{\Omega}u_{1}^{2}h_{i}\Delta
h_{i}-\int\limits_{\Omega}h_{i}u_{1}^{2}\left(\int\limits_{\Omega}u_{1}\langle\nabla
h_{i},\nabla u_{1}\rangle+{1\over 2}\int\limits_{\Omega}u_{1}^{2}\Delta
h_{i}\right)$ $\displaystyle=$
$\displaystyle\int\limits_{\Omega}u_{1}h_{i}\langle\nabla h_{i},\nabla
u_{1}\rangle+{1\over 2}\int\limits_{\Omega}u_{1}^{2}h_{i}\Delta h_{i}$
$\displaystyle=$ $\displaystyle-{1\over 2}\int\limits_{\Omega}u_{1}^{2}|\nabla
h_{i}|^{2}$ $\displaystyle=$ $\displaystyle-{1\over 2}\|u_{1}\nabla
h_{i}\|^{2}.$
By virtue of (2.4) and (2.5), it is easy to see
$\displaystyle(\lambda_{i+1}-\lambda_{1})^{1\over 2}\|u_{1}\nabla
h_{i}\|^{2}=$ $\displaystyle-2(\lambda_{i+1}-\lambda_{1})^{1\over
2}\int\limits_{\Omega}\varphi_{i}\left(\langle\nabla h_{i},\nabla
u_{1}\rangle+{1\over 2}u_{1}\Delta h_{i}\right)$ (2.6) $\displaystyle\leq$
$\displaystyle\delta(\lambda_{i+1}-\lambda_{1})\|\varphi_{i}\|^{2}+{1\over\delta}\left\|\langle\nabla
h_{i},\nabla u_{1}\rangle+{1\over 2}u_{1}\Delta h_{i}\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\delta\\{\|\nabla^{p}(h_{i}u_{1})\|^{2}-\lambda_{1}\|h_{i}u_{1}\|^{2}\\}+{1\over\delta}\left\|\langle\nabla
h_{i},\nabla u_{1}\rangle+{1\over 2}u_{1}\Delta h_{i}\right\|^{2},$
where $\delta$ is a positive constant. Summing over $i$ from 1 to $n+1$ for
(2.6), one finds that
$\displaystyle\sum_{i=1}^{n+1}(\lambda_{i+1}-\lambda_{1})^{1\over
2}\|u_{1}\nabla h_{i}\|^{2}\leq$
$\displaystyle\delta\sum_{i=1}^{n+1}\\{\|\nabla^{p}(h_{i}u_{1})\|^{2}-\lambda_{1}\|h_{i}u_{1}\|^{2}\\}$
(2.7) $\displaystyle+{1\over\delta}\sum_{i=1}^{n+1}\left\|\langle\nabla
h_{i},\nabla u_{1}\rangle+{1\over 2}u_{1}\Delta h_{i}\right\|^{2}$
$\displaystyle=$
$\displaystyle\delta\sum_{i=1}^{n+1}\\{\|\nabla^{p}(x_{i}u_{1})\|^{2}-\lambda_{1}\|x_{i}u_{1}\|^{2}\\}$
$\displaystyle+{1\over\delta}\sum_{i=1}^{n+1}\left\|\langle\nabla x_{i},\nabla
u_{1}\rangle+{1\over 2}u_{1}\Delta x_{i}\right\|^{2}$ $\displaystyle=$
$\displaystyle\delta\sum_{i=1}^{n+1}\int\limits\limits_{\Omega}u_{1}x_{i}\\{(-\Delta)^{p}(u_{1}x_{i})-x_{i}(-\Delta)^{p}u_{1}\\}$
$\displaystyle+{1\over\delta}\sum_{i=1}^{n+1}\left\|\langle\nabla x_{i},\nabla
u_{1}\rangle+{1\over 2}u_{1}\Delta x_{i}\right\|^{2}.$
Making use of the same method as proof of Lemma 1 in [5], it is easy to prove
$\int\limits_{\Omega}|\nabla u_{1}|^{2}\leq\lambda_{1}^{1\over p}.$
Thus,
$\displaystyle\sum_{i=1}^{n+1}\left\|\langle\nabla x_{i},\nabla
u_{1}\rangle+{1\over 2}u_{1}\Delta x_{i}\right\|^{2}=$
$\displaystyle\sum_{i=1}^{n+1}\int\limits_{\Omega}\left(\langle\nabla
x_{i},\nabla u_{i}\rangle+{1\over 2}u_{1}\Delta x_{i}\right)^{2}$ (2.8)
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n+1}\int\limits_{\Omega}\left({1\over
4}u_{1}^{2}(\Delta x_{i})^{2}+\langle\nabla x_{i},\nabla
u_{1}\rangle^{2}+{1\over 2}\Delta x_{i}\langle\nabla
x_{i},\nabla(u_{1}^{2})\rangle\right)$ $\displaystyle=$
$\displaystyle{n^{2}\over 4}+\int\limits_{\Omega}|\nabla u_{1}|^{2}$
$\displaystyle\leq$ $\displaystyle{n^{2}\over 4}+\lambda_{1}^{1\over p}.$
It has been shown in [7] (see Proposition 2.2 of [7]) that
$\displaystyle\sum_{i=1}^{n+1}$
$\displaystyle\int\limits\limits_{\Omega}u_{1}x_{i}\\{(-\Delta)^{p}(u_{1}x_{i})-x_{i}(-\Delta)^{p}u_{1}\\}$
(2.9) $\displaystyle\leq\left(\lambda_{1}^{1\over
p}+n\right)^{p}-\lambda_{1}+4[2^{p}-(p+1)]\lambda_{1}^{1\over
p}\left(\lambda_{1}^{1\over p}+n\right)^{p-2}.$
Inserting (2.8) and (2.9) into (2.7), we infer
$\displaystyle\sum_{i=1}^{n+1}(\lambda_{i+1}-\lambda_{1})^{1\over
2}\|u_{1}\nabla h_{i}\|^{2}\leq$
$\displaystyle\delta\left\\{\left(\lambda_{1}^{1\over
p}+n\right)^{p}-\lambda_{1}+4[2^{p}-(p+1)]\lambda_{1}^{1\over
p}\left(\lambda_{1}^{1\over p}+n\right)^{p-2}\right\\}$ (2.10)
$\displaystyle+{1\over\delta}\left\\{\lambda_{1}^{1\over p}+{n^{2}\over
4}\right\\}.$
Minimizing the right hand side of (2.10) as a function of $\delta$ by choosing
$\delta=\left(\lambda_{1}^{1\over p}+{n^{2}\over
4}\over\left(\lambda_{1}^{1\over
p}+n\right)^{p}-\lambda_{1}+4[2^{p}-(p+1)]\lambda_{1}^{1\over
p}\left(\lambda_{1}^{1\over p}+n\right)^{p-2}\right)^{1\over 2},$
we obtain
$\displaystyle\sum_{i=1}^{n+1}(\lambda_{i+1}-\lambda_{1})^{1\over
2}\|u_{1}\nabla h_{i}\|^{2}\leq$
$\displaystyle\left\\{\left(\lambda_{1}^{1\over
p}+n\right)^{p}-\lambda_{1}+4[2^{p}-(p+1)]\lambda_{1}^{1\over
p}\left(\lambda_{1}^{1\over p}+n\right)^{p-2}\right\\}^{1\over 2}$ (2.11)
$\displaystyle\times\left\\{4\lambda_{1}^{1\over p}+n^{2}\right\\}^{1\over
2}.$
By a transformation of coordinates if necessary, for any point $q$, one gets
$|\nabla h_{i}|^{2}\leq 1\ \ \ \ {\rm for\ any}\ i.$
It follows that
$\displaystyle\sum_{i=1}^{n+1}(\lambda_{i+1}-\lambda_{1})^{1\over 2}|\nabla
h_{i}|^{2}$ (2.12) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}|\nabla
h_{i}|^{2}+(\lambda_{n+1}-\lambda_{1})^{1\over 2}|\nabla h_{n+1}|^{2}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}|\nabla
h_{i}|^{2}+(\lambda_{n+1}-\lambda_{1})^{1\over 2}\left(n-\sum_{i=1}^{n}|\nabla
h_{i}|^{2}\right)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}|\nabla
h_{i}|^{2}+(\lambda_{n+1}-\lambda_{1})^{1\over 2}\sum_{i=1}^{n}\left(1-|\nabla
h_{i}|^{2}\right)$ $\displaystyle\geq$
$\displaystyle\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}|\nabla
h_{i}|^{2}+\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}\left(1-|\nabla
h_{i}|^{2}\right)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}.$
From (2.11) and (2.12), we obtain
$\displaystyle\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})^{1\over 2}\leq$
$\displaystyle\left\\{\left(\lambda_{1}^{1\over
p}+n\right)^{p}-\lambda_{1}+4[2^{p}-(p+1)]\lambda_{1}^{1\over
p}\left(\lambda_{1}^{1\over p}+n\right)^{p-2}\right\\}^{1\over 2}$
$\displaystyle\times\left\\{4\lambda_{1}^{1\over p}+n^{2}\right\\}^{1\over
2},$
which concludes the proof of Theorem.
## References
* [1] Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In spectral theory and geometry (Edinburgh, 1998, E. B. Davies and Yu Safarov, eds.), London Math. Soc. Lecture Notes, 273 (1999), 95-139
* [2] Ashbaugh, M.S., Benguria, R.D.: More bounds on eigenvalue ratios for Dirichlet Laplacians in $n$ dimensions. SIAM J. Math. Anal. 24, 1622-1651 (1993)
* [3] Brands, J.J.A.M.: Bounds for the ratios of the first three membrane eigenvalues. Arch. Rational Mech. Anal. 16, 265-268 (1964)
* [4] Chen, D.G., Cheng, Q.-M.: Extrinsic estimates for eigenvalues of the Laplace operator. J. Math. Soc. Japan 60, 325-339 (2008)
* [5] Chen, Z.C., Qian, C.L.: Estimates for discrete spectrum of Laplacian operator with any order. J. China Univ. Sci. Tech. 20, 259-266 (1990)
* [6] Cheng, Q.-M., Huang, G.Y., Wei, G.X.: Estimates for lower order eigenvalues of a clamped plate problem. arXiv: 0906.5192
* [7] Cheng, Q.-M., Ichikawa, T., Mametsuka, S.: Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere. To appear in Calc. Var.
* [8] Cheng, Q.-M., Ichikawa, T., Mametsuka, S.: Inequalities for eigenvalues of Laplacian with any order. To appear in Commun. Contemp. Math., 2009
* [9] Huang, G.Y., Li, X.X., Xu, R.W.: Extrinsic estimates for the eigenvalues of Schrödinger operator. To appear in Geom. Dedicata
* [10] Payne, L.E., Pólya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289-298 (1956)
* [11] Sun, H.J., Cheng, Q.-M., Yang, H.C.: Lower order eigenvalues of Dirichlet Laplacian. Manuscripta Math. 125, 139-156 (2008)
|
arxiv-papers
| 2009-10-21T14:10:56 |
2024-09-04T02:49:05.954171
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangyue Huang, Bingqing Ma",
"submitter": "Huang Guangyue",
"url": "https://arxiv.org/abs/0910.4096"
}
|
0910.4100
|
arxiv-papers
| 2009-10-21T14:24:56 |
2024-09-04T02:49:05.957526
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangyue Huang, Xingxiao Li",
"submitter": "Huang Guangyue",
"url": "https://arxiv.org/abs/0910.4100"
}
|
|
0910.4101
|
# Eigenvalue estimates for the higher order buckling problem 111This research
is supported by NSFC of China (No. 10671181), Project of Henan Provincial
department of Sciences and Technology (No. 092300410143), and NSF of Henan
Provincial Education department (No. 2009A110010).
Guangyue Huang, Xingxiao Li 222The corresponding author. Email:
xxl$@$henannu.edu.cn
Department of Mathematics, Henan Normal University
Xinxiang 453007, Henan, P.R. China
> Abstract. In this paper, we consider lower order eigenvalues of Laplacian
> operator with any order in Euclidean domains. By choosing special
> rectangular coordinates, we obtain two estimates for lower order
> eigenvalues.
> Mathematics Subject Classification 35P15.
> Keywords and phrases lower order eigenvalue, Rayleigh-Ritz inequality,
> buckling problem.
## 1 Introduction
Let $\Omega$ be a bounded domain in an $n(\geq 2)$-dimensional Euclidean space
$\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. In the present
article, we consider the eigenvalue estimate for the following problem:
$\left\\{\,\vbox{\openup
3.0pt\halign{$\displaystyle{#}$\hfil&\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr(-\Delta)^{p}u=\Lambda(-\Delta)u,&\
\ {\rm in}\ \Omega,\\\u=\frac{\partial
u}{\partial\nu}=\cdots=\frac{\partial^{p-1}u}{\partial\nu^{p-1}}=0&\ \ {\rm
on}\ \partial\Omega,\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$ (1.1)
where $\nu$ denotes the outward unit normal vector field of $\partial\Omega$
and $p$ is a positive integer. Let
$0<\Lambda_{1}<\Lambda_{2}\leq\Lambda_{3}\leq\cdots\rightarrow+\infty$ denote
the successive eigenvalues for (1.1), where each eigenvalue is repeated
according to its multiplicity.
When $p=2$, the eigenvalue problem (1.1) is called the buckling problem. For
the buckling problem, Payne-Pólya-Weinberger [11] proved, in the case of
$n=2$, that
$\Lambda_{2}\leq 3\Lambda_{1}.$ (1.2)
Following the method of Payne-Pólya-Weinberger in [11], the inequality (1.2)
can be generalized to $\Omega\subset\mathbb{R}^{n}$ as (see [2]):
$\Lambda_{2}\leq\left(1+\frac{8}{n+2}\right)\Lambda_{1}.$
In 1984, Hile and Yeh [10] improved the above results as follows:
$\Lambda_{2}\leq\frac{n^{2}+8n+20}{(n+2)^{2}}\Lambda_{1}.$
On the other hand, Ashbaugh [1] proved another inequality as the following
form:
$\sum_{i}^{n}(\Lambda_{i+1}-\Lambda_{1})\leq 4\Lambda_{1}.$ (1.3)
To answer a question of Ashbaugh given in [1], Cheng-Yang [4] obtained in 2006
a universal inequality for higher eigenvalues of (1.1) with $p=2$. In fact,
they proved that
$\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4(n+2)}{n^{2}}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\Lambda_{i}.$
(1.4)
As a generalization of (1.4), Huang and Li [7] proved the following inequality
of eigenvalue estimate for the problem (1.1) with $p\geq 2$:
$\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})^{2}\leq\frac{4(p-1)(n+2p-2)}{n^{2}}\sum_{i=1}^{k}(\Lambda_{k+1}-\Lambda_{i})\Lambda_{i}.$
(1.5)
Estimates for higher order eigenvalues of (1.1) has been recently studied by
many mathematicians. For the other related development in this direction, we
refer to [12, 6, 7, 8, 3, 5, 13] and the references therein.
In particular, Cheng-Ichikawa-Mametsuka considered in [9] the eigenvalue
estimate for the problem
$\left\\{\,\vbox{\openup
3.0pt\halign{$\displaystyle{#}$\hfil&\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr(-\Delta)^{p}u=\lambda
u,&\ \ {\rm in}\ \Omega,\\\u=\frac{\partial
u}{\partial\nu}=\cdots=\frac{\partial^{p-1}u}{\partial\nu^{p-1}}=0&\ \ {\rm
on}\ \partial\Omega,\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$ (1.6)
and proved the following inequalities:
$\sum_{i=1}^{n}(\lambda_{i+1}-\lambda_{1})\leq 4p(2p-1)\lambda_{1}\ \ \ {\rm
for}\ p\geq 2,$ (1.7)
$\sum_{i=1}^{n}(\lambda_{i+1}^{\frac{1}{p}}-\lambda_{1}^{\frac{1}{p}})^{p-1}\leq(2p)^{p-1}\lambda_{1}^{\frac{p-1}{p}}\
\ \ {\rm for}\ p\geq 2.$ (1.8)
Inspired by [9], we consider the eigenvalue problem (1.1) with $p\geq 2$ and
wish to obtain the similar results as (1.7) and (1.8). Our main results of
this paper are stated as follows:
###### Theorem 1.1.
Let $\Omega$ be a bounded domain in an $n(\geq 2)$-dimensional Euclidean space
$\mathbb{R}^{n}$. Assume that $\Lambda_{i}$ is the $i$-th eigenvalue of the
problem (1.1) with $p\geq 2$. Then,
$\sum_{i=1}^{n}(\Lambda_{i+1}-\Lambda_{1})\leq 4[p(2p+n-2)-n]\Lambda_{1}.$
(1.9)
###### Theorem 1.2.
Let $\Omega$ be a bounded domain in an $n(\geq 2)$-dimensional Euclidean space
$\mathbb{R}^{n}$. Assume that $\Lambda_{i}$ is the $i$-th eigenvalue of the
problem (1.1) with $p\geq 3$. Then,
$\sum_{i=1}^{n}(\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}})^{p-2}\leq(2p)^{p-2}\Lambda_{1}^{\frac{p-2}{p-1}}.$
(1.10)
## 2 Proof of Theorem 1.1
Let $u_{i}$ be the orthonormal eigenvalue function of the problem (1.1) with
respect to $L^{2}$ inner product corresponding to $\Lambda_{i}$, that is,
$\left\\{\,\vbox{\openup
3.0pt\halign{$\displaystyle{#}$\hfil&\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr(-\Delta)^{p}u_{i}=\Lambda_{i}(-\Delta)u_{i}&\
\ {\rm in}\ \Omega,\\\u_{i}=\frac{\partial
u_{i}}{\partial\nu}=\cdots=\frac{\partial^{p-1}u_{i}}{\partial\nu^{p-1}}=0&\ \
{\rm on}\ \partial\Omega,\\\\\int_{\Omega}\langle\nabla u_{i},\nabla
u_{j}\rangle=\delta_{ij}.&\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$
We first choose rectangular coordinates for $\mathbb{R}^{n}$ by taking as
origin the center of gravity of $\Omega$ with mass-distribution $|\nabla
u_{1}|^{2}$ such that
$\int_{\Omega}\langle\nabla(x_{i}u_{1}),\nabla u_{1}\rangle=0\ \ \ \ \ {\rm
for}\ i=1,2,\cdots,n.$
Then, by a rotation of the coordinate system if necessary, we may also assume
$\int_{\Omega}\langle\nabla(x_{i}u_{1}),\nabla u_{j}\rangle=0\ \ \ \ \ {\rm
for}\ 2\leq j\leq i\leq n,$
and hence we arrive at
$\int_{\Omega}\langle\nabla(x_{i}u_{1}),\nabla u_{j}\rangle=0\ \ \ \ \ {\rm
for}\ 1\leq j\leq i\leq n.$
Let $\varphi_{i}=x_{i}u_{1}$. Then
$\varphi_{i}=\frac{\partial\varphi_{i}}{\partial\nu}=\cdots=\frac{\partial^{p-1}\varphi_{i}}{\partial\nu^{p-1}}=0$
on $\partial\Omega$ and
$\int_{\Omega}\langle\nabla\varphi_{i},\nabla u_{j}\rangle=0\ \ \ \ \ {\rm
for}\ 1\leq j\leq i\leq n.$
From the Rayleigh-Ritz inequality, one gets
$\Lambda_{i+1}\leq\frac{\int_{\Omega}\varphi_{i}(-\Delta)^{p}\varphi_{i}}{\int_{\Omega}|\nabla\varphi_{i}|^{2}}.$
(2.1)
Note that
$(-\Delta)^{p}\varphi_{i}=(-\Delta)^{p}(x_{i}u_{1})=\Lambda_{1}x_{i}(-\Delta)u_{1}-2p(-\Delta)^{p-1}u_{1,x_{i}},$
where $u_{1,x_{i}}={\partial u_{1}}/{\partial x_{i}}$. It follows that
$\displaystyle\int_{\Omega}\varphi_{i}(-\Delta)^{p}\varphi_{i}=$
$\displaystyle\int_{\Omega}\varphi_{i}[\Lambda_{1}x_{i}(-\Delta)u_{1}-2p(-\Delta)^{p-1}u_{1,x_{i}}]$
(2.2) $\displaystyle=$
$\displaystyle\int_{\Omega}\varphi_{i}[\Lambda_{1}(-\Delta)(x_{i}u_{1})+2\Lambda_{1}u_{1,x_{i}}-2p(-\Delta)^{p-1}u_{1,x_{i}}]$
(2.3) $\displaystyle=$
$\displaystyle\Lambda_{1}\int_{\Omega}|\nabla\varphi_{i}|^{2}+2\Lambda_{1}\int_{\Omega}x_{i}u_{1}u_{1,x_{i}}-2p\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}$
(2.4) $\displaystyle=$
$\displaystyle\Lambda_{1}\int_{\Omega}|\nabla\varphi_{i}|^{2}-\Lambda_{1}\int_{\Omega}u_{1}^{2}-2p\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}.$
(2.5)
Combining with (2.1) and (2.2) yields
$(\Lambda_{i+1}-\Lambda_{1})\int_{\Omega}|\nabla\varphi_{i}|^{2}\leq-\Lambda_{1}\int_{\Omega}u_{1}^{2}-2p\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}.$
(2.6)
Using integration by parts, we have
$\displaystyle\int_{\Omega}\langle\nabla\varphi_{i},\nabla
u_{1,x_{i}}\rangle=$
$\displaystyle-\int_{\Omega}\langle\nabla(x_{i}u_{1})_{x_{i}},\nabla
u_{1}\rangle=-\int_{\Omega}|\nabla
u_{1}|^{2}-\int_{\Omega}\langle\nabla(x_{i}u_{1,x_{i}}),\nabla u_{1}\rangle$
$\displaystyle=$
$\displaystyle-1-\int_{\Omega}u_{1,x_{i}}^{2}-\int_{\Omega}\langle[\nabla(x_{i}u_{1})-u_{1}\nabla
x_{i}],\nabla u_{1,x_{i}}\rangle$ $\displaystyle=$
$\displaystyle-1-2\int_{\Omega}u_{1,x_{i}}^{2}-\int_{\Omega}\langle\nabla\varphi_{i},\nabla
u_{1,x_{i}}\rangle.$
Hence,
$1\leq
1+2\int_{\Omega}u_{1,x_{i}}^{2}=-2\int_{\Omega}\langle\nabla\varphi_{i},\nabla
u_{1,x_{i}}\rangle.$ (2.7)
By Cauchy inequality one knows from (2.7) that
$1\leq 4\left(\int_{\Omega}\langle\nabla\varphi_{i},\nabla
u_{1,x_{i}}\rangle\right)^{2}\leq
4\int_{\Omega}|\nabla\varphi_{i}|^{2}\int_{\Omega}|\nabla u_{1,x_{i}}|^{2}.$
(2.8)
Then from (2.6) and (2.8), it is easily seen that
$\displaystyle\sum_{i=1}^{n}(\Lambda_{i+1}-\Lambda_{1})\leq
4\sum_{i=1}^{n}\left\\{(\Lambda_{i+1}-\Lambda_{1})\int_{\Omega}|\nabla\varphi_{i}|^{2}\int_{\Omega}|\nabla
u_{1,x_{i}}|^{2}\right\\}$ (2.9) $\displaystyle\leq$ $\displaystyle
4\sum_{i=1}^{n}\left\\{\left(-\Lambda_{1}\int_{\Omega}u_{1}^{2}-2p\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}\right)\int_{\Omega}|\nabla
u_{1,x_{i}}|^{2}\right\\}$ (2.10) $\displaystyle\leq$ $\displaystyle
4\left\\{\sum_{i=1}^{n}\left(-\Lambda_{1}\int_{\Omega}u_{1}^{2}-2p\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}\right)\right\\}\left\\{\sum_{i=1}^{n}\int_{\Omega}|\nabla
u_{1,x_{i}}|^{2}\right\\}.$ (2.11)
Denote
$\nabla^{r}=\left\\{\,\vbox{\openup
3.0pt\halign{$\displaystyle{#}$\hfil&\quad$\displaystyle{{}#}$\hfil\cr\mathstrut\crcr\kern-12.0pt\cr\Delta^{r/2}&\
\ {\rm when}\ r\ {\rm is\ even},\\\\\nabla(\Delta^{(r-1)/2})&\ \ {\rm when}\
r\ {\rm is\ odd}.\crcr\mathstrut\crcr\kern-12.0pt\cr}}\,\right.$
Then we have the following lemma:
###### Lemma 2.1.
[7] Let $u_{1}$ be the eigenfunction of the problem (1.6) corresponding to the
eigenvalue $\Lambda_{1}$. Then we have
$\int_{\Omega}|\nabla^{r}u_{1}|^{2}\leq\Lambda_{1}^{(r-1)/(p-1)}\ \ \ \ \ {\rm
for}\ \ r=2,3,\ldots,p.$ (2.12)
Proof. First we prove the inequality
$\left(\int_{\Omega}u_{1}(-\Delta)^{r}u_{1}\right)^{1/(r-1)}\leq\left(\int_{\Omega}u_{1}(-\Delta)^{r+1}u_{1}\right)^{1/r}.$
(2.13)
For $r=2$, we have
$\displaystyle\int_{\Omega}u_{1}(-\Delta)^{2}u_{1}=$
$\displaystyle\int_{\Omega}\langle\nabla u_{1},\nabla(-\Delta)u_{1}\rangle$
$\displaystyle\leq$ $\displaystyle\left(\int_{\Omega}|\nabla
u_{1}|^{2}\right)^{1/2}\left(\int_{\Omega}|\nabla(-\Delta)u_{1}|^{2}\right)^{1/2}$
$\displaystyle=$
$\displaystyle\left(\int_{\Omega}u_{1}(-\Delta)^{3}u_{1}\right)^{1/2}.$
Suppose that inequality (2.12) holds for $r-1$, that is,
$\left(\int_{\Omega}u_{1}(-\Delta)^{r-1}u_{1}\right)^{1/(r-2)}\leq\left(\int_{\Omega}u_{1}(-\Delta)^{r}u_{1}\right)^{1/(r-1)}.$
Then, for integer $r$,
$\displaystyle\int_{\Omega}u_{1}(-\Delta)^{r}u_{1}=$
$\displaystyle-\int_{\Omega}\langle\nabla^{r-1}u_{1},\nabla^{r+1}u_{1}\rangle$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}|\nabla^{r-1}u_{1}|^{2}\right)^{1/2}\left(\int_{\Omega}|\nabla^{r+1}u_{1}|^{2}\right)^{1/2}$
$\displaystyle=$
$\displaystyle\left(\int_{\Omega}u_{1}(-\Delta)^{r-1}u_{1}\right)^{1/2}\left(\int_{\Omega}u_{1}(-\Delta)^{r+1}u_{1}\right)^{1/2}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}u_{1}(-\Delta)^{r}u_{1}\right)^{(r-2)/2(r-1)}\left(\int_{\Omega}u_{1}(-\Delta)^{r+1}u_{1}\right)^{1/2},$
which gives
$\left(\int_{\Omega}u_{1}(-\Delta)^{r}u_{1}\right)^{1/(r-1)}\leq\left(\int_{\Omega}u_{1}(-\Delta)^{r+1}u_{1}\right)^{1/r}.$
This means that inequality (2.13) holds. Repeatedly using inequality (2.13),
we deduce
$\displaystyle\left(\int_{\Omega}u_{1}(-\Delta)^{r}u_{1}\right)^{1/(r-1)}\leq$
$\displaystyle\left(\int_{\Omega}u_{1}(-\Delta)^{r+1}u_{1}\right)^{1/r}$
$\displaystyle\leq$
$\displaystyle\cdots\leq\left(\int_{\Omega}u_{1}(-\Delta)^{p}u_{1}\right)^{1/(p-1)}=\Lambda_{1}^{1/(p-1)}.$
This concludes the proof of Lemma 2.1. $\sqcup$$\sqcap$
From (2.12) and Schwarz inequality it follows that
$\displaystyle 1=$ $\displaystyle\left(\int_{\Omega}|\nabla
u_{1}|^{2}\right)^{2}=\left(\int_{\Omega}u_{1}(-\Delta)u_{1}\right)^{2}$
(2.14) $\displaystyle\leq$
$\displaystyle\int_{\Omega}u_{1}^{2}\int_{\Omega}u_{1}(-\Delta)^{2}u_{1}\leq\Lambda_{1}^{\frac{1}{p-1}}\int_{\Omega}u_{1}^{2}.$
A direct computation yields
$\displaystyle\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}=\int_{\Omega}u_{1,x_{i}}(-\Delta)^{p-1}(x_{i}u_{1})$
(2.15) $\displaystyle=$
$\displaystyle\int_{\Omega}u_{1,x_{i}}\Big{(}x_{i}(-\Delta)^{p-1}u_{1}-2(p-1)(-\Delta)^{p-2}u_{1,x_{i}}\Big{)}$
(2.16) $\displaystyle=$
$\displaystyle-\int_{\Omega}u_{1}\Big{(}(-\Delta)^{p-1}u_{1}+x_{i}(-\Delta)^{p-1}u_{1,x_{i}}\Big{)}$
(2.17) $\displaystyle+2(p-1)\int_{\Omega}u_{1}(-\Delta)^{p-2}u_{1,x_{i}x_{i}}$
(2.18) $\displaystyle=$
$\displaystyle-\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}-\int_{\Omega}u_{1}(-\Delta)^{p-1}u_{1}$
(2.19)
$\displaystyle+2(p-1)\int_{\Omega}u_{1}(-\Delta)^{p-2}u_{1,x_{i}x_{i}},$
(2.20)
which shows that
$\displaystyle\sum_{i=1}^{n}\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}$
(2.21) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(-\frac{1}{2}\int_{\Omega}u_{1}(-\Delta)^{p-1}u_{1}+(p-1)\int_{\Omega}u_{1}(-\Delta)^{p-2}u_{1,x_{i}x_{i}}\right)$
(2.22) $\displaystyle=$
$\displaystyle-\frac{2p+n-2}{2}\int_{\Omega}u_{1}(-\Delta)^{p-1}u_{1}$ (2.23)
$\displaystyle\geq$
$\displaystyle-\frac{2p+n-2}{2}\Lambda_{1}^{\frac{p-2}{p-1}}.$ (2.24)
On the other hand, it easy to see that
$\sum_{i=1}^{n}\int_{\Omega}|\nabla
u_{1,x_{i}}|^{2}=-\sum_{i=1}^{n}\int_{\Omega}u_{1}(-\Delta)u_{1,x_{i}x_{i}}=\int_{\Omega}u_{1}(-\Delta)^{2}u_{1}\leq\Lambda_{1}^{\frac{1}{p-1}}.$
(2.25)
Finally, applying (2.14), (2.21) and (2.25) to (2.8), one finds
$\displaystyle\sum_{i=1}^{n}(\Lambda_{i+1}-\Lambda_{1})\leq
4[p(2p+n-2)-n]\Lambda_{1},$
completing the proof of Theorem 1.1. $\sqcup$$\sqcap$
## 3 Proof of Theorem 1.2
By virtue of (2.14), it holds that
$\Lambda_{1}\int_{\Omega}u_{1}^{2}\geq\Lambda_{1}^{\frac{p-2}{p-1}}.$ (3.1)
And from (2.15) one finds that
$\displaystyle\int_{\Omega}\varphi_{i}(-\Delta)^{p-1}u_{1,x_{i}}=$
$\displaystyle-\frac{1}{2}\int_{\Omega}u_{1}(-\Delta)^{p-1}u_{1}+(p-1)\int_{\Omega}u_{1}(-\Delta)^{p-2}u_{1,x_{i}x_{i}}$
(3.2) $\displaystyle\geq$
$\displaystyle-\frac{1}{2}\Lambda_{1}^{\frac{p-2}{p-1}}-(p-1)\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2},$
(3.3)
where (2.12) has been used in the last inequality. Putting (3.1) and (3.2)
into (2.6) yields
$(\Lambda_{i+1}-\Lambda_{1})\int_{\Omega}|\nabla\varphi_{i}|^{2}\leq(p-1)\Lambda_{1}^{\frac{p-2}{p-1}}+2p(p-1)\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}.$
(3.4)
In order to complete the proof of Theorem 1.2, we need the following lemmas.
###### Lemma 3.1.
Let $\Lambda_{i}$ be the $i$-th eigenvalue of problem (1.1) with $p\geq 2$,
and $u_{i}$ be the orthonormal eigenfunction corresponding to $\Lambda_{i}$.
Then for $1\leq i\leq n$, either
$\sum_{k=1}^{p-2}\Lambda_{1}^{\frac{k-1}{p-1}}\left(\Lambda_{i+1}^{\frac{p-1-k}{p-1}}-\Lambda_{1}^{\frac{p-1-k}{p-1}}\right)\leq
2p(p-1)\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2},$ (3.5)
or
$\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\leq
4\int_{\Omega}|\nabla u_{1,x_{i}}|^{2}.$ (3.6)
Proof. Suppose that there exists an $i$ such that neither (3.5) nor (3.6)
holds. Then by (3.4)
$\displaystyle(\Lambda_{i+1}-\Lambda_{1})\int_{\Omega}|\nabla\varphi_{i}|^{2}\leq$
$\displaystyle(p-1)\Lambda_{1}^{\frac{p-2}{p-1}}+2p(p-1)\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}$
$\displaystyle<$
$\displaystyle(p-1)\Lambda_{1}^{\frac{p-2}{p-1}}+\sum_{k=1}^{p-2}\Lambda_{1}^{\frac{k-1}{p-1}}\left(\Lambda_{i+1}^{\frac{p-1-k}{p-1}}-\Lambda_{1}^{\frac{p-1-k}{p-1}}\right)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{p-1}\Lambda_{i+1}^{\frac{p-1-k}{p-1}}\Lambda_{1}^{\frac{k-1}{p-1}}=\frac{\Lambda_{i+1}-\Lambda_{1}}{\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}},$
which shows that
$(\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}})\int_{\Omega}|\nabla\varphi_{i}|^{2}<1.$
(3.7)
On the other hand, it follows from (2.8) that
$1\leq 4\int_{\Omega}|\nabla\varphi_{i}|^{2}\int_{\Omega}|\nabla
u_{1,x_{i}}|^{2}<(\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}})\int_{\Omega}|\nabla\varphi_{i}|^{2},$
which contradicts with (3.7). This concludes the proof of Lemma 3.1.
$\sqcup$$\sqcap$
###### Lemma 3.2.
Let $\Lambda_{i}$ be the $i$-th eigenvalue of problem (1.1) with $p\geq 2$,
and $u_{i}$ be the orthonormal eigenfunction corresponding to $\Lambda_{i}$.
Then
$\left(\int_{\Omega}|\nabla^{r}u_{1,x_{i}}|^{2}\right)^{\frac{1}{r}}\leq\left(\int_{\Omega}|\nabla^{r+1}u_{1,x_{i}}|^{2}\right)^{\frac{1}{r+1}}\
\ \ {\rm for}\ r=1,2,\ldots,p-2.$ (3.8)
Proof. For $r=1$, we have
$\displaystyle\int_{\Omega}|\nabla u_{1,x_{i}}|^{2}=$
$\displaystyle-\int_{\Omega}\langle\nabla^{2}u_{1,x_{i}},u_{1,x_{i}}\rangle\leq\left(\int_{\Omega}|\nabla^{2}u_{1,x_{i}}|^{2}\int_{\Omega}|u_{1,x_{i}}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}|\nabla^{2}u_{1,x_{i}}|^{2}\int_{\Omega}|\nabla
u_{1}|^{2}\right)^{\frac{1}{2}}=\left(\int_{\Omega}|\nabla^{2}u_{1,x_{i}}|^{2}\right)^{\frac{1}{2}}.$
Assume that (3.8) is true for $r-1$, that is,
$\left(\int_{\Omega}|\nabla^{r-1}u_{1,x_{i}}|^{2}\right)^{\frac{1}{r-1}}\leq\left(\int_{\Omega}|\nabla^{r}u_{1,x_{i}}|^{2}\right)^{\frac{1}{r}}.$
Then for $r$
$\displaystyle\int_{\Omega}|\nabla^{r}u_{1,x_{i}}|^{2}=$
$\displaystyle-\int_{\Omega}\langle\nabla^{r-1}u_{1,x_{i}},\nabla^{r+1}u_{1,x_{i}}\rangle$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}|\nabla^{r-1}u_{1,x_{i}}|^{2}\right)^{\frac{1}{2}}\left(\int_{\Omega}|\nabla^{r+1}u_{1,x_{i}}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}|\nabla^{r}u_{1,x_{i}}|^{2}\right)^{\frac{r-1}{2r}}\left(\int_{\Omega}|\nabla^{r+1}u_{1,x_{i}}|^{2}\right)^{\frac{1}{2}},$
which gives that
$\left(\int_{\Omega}|\nabla^{r}u_{1,x_{i}}|^{2}\right)^{\frac{1}{r}}\leq\left(\int_{\Omega}|\nabla^{r+1}u_{1,x_{i}}|^{2}\right)^{\frac{1}{r+1}},$
and Lemma 3.2 is obtained. $\sqcup$$\sqcap$
Making use of Lemma 3.1 and Lemma 3.2, we can prove the following lemma:
###### Lemma 3.3.
If $p\geq 3$, then
$\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\leq
2p\left(\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\right)^{\frac{1}{p-2}}$
(3.9)
holds for $1\leq i\leq n$.
Proof. By Lemma 3.1, either (3.5) or (3.6) holds.
(1) If (3.5) holds, then
$\displaystyle 2p(p-1)\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\geq$
$\displaystyle\sum_{k=1}^{p-2}\Lambda_{1}^{\frac{k-1}{p-1}}\left(\Lambda_{i+1}^{\frac{p-1-k}{p-1}}-\Lambda_{1}^{\frac{p-1-k}{p-1}}\right)$
(3.10) $\displaystyle\geq$
$\displaystyle\sum_{k=1}^{p-2}\Lambda_{1}^{\frac{k-1}{p-1}}\left(\Lambda_{1}^{\frac{p-2-k}{p-1}}\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{p-1-k}{p-1}}\right)$
(3.11) $\displaystyle=$
$\displaystyle\sum_{k=1}^{p-2}\Lambda_{1}^{\frac{p-3}{p-1}}\left(\Lambda_{i+1}^{{1\over
p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\right)$ (3.12) $\displaystyle=$
$\displaystyle(p-2)\Lambda_{1}^{\frac{p-3}{p-1}}\left(\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\right).$
(3.13)
Since
$\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\leq\sum_{i=1}^{n}\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}=\int_{\Omega}|\nabla^{p-1}u_{1}|^{2}\leq\Lambda_{1}^{\frac{p-2}{p-1}},$
we obtain from (3.10) that
$\displaystyle 2p(p-1)\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\geq$
$\displaystyle(p-2)\Lambda_{1}^{\frac{p-3}{p-1}}\left(\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\right)$
$\displaystyle\geq$
$\displaystyle(p-2)\left(\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\right)^{\frac{p-3}{p-2}}\left(\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\right).$
Thus, for $p\geq 3$, one gets
$\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\leq\frac{2p(p-1)}{p-2}\left(\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\right)^{\frac{1}{p-2}}\leq
2p\left(\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\right)^{\frac{1}{p-2}}.$
(3.14)
(2) If (3.6) holds, then using (3.8), it is easy to see
$\displaystyle\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}}\leq$
$\displaystyle 4\int_{\Omega}|\nabla u_{1,x_{i}}|^{2}\leq
2p\int_{\Omega}|\nabla u_{1,x_{i}}|^{2}$ (3.15) $\displaystyle\leq$
$\displaystyle\ldots\leq
2p\left(\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}\right)^{\frac{1}{p-2}}.$
Thus (3.9) holds anyway. $\sqcup$$\sqcap$
Now summing up (3.9) over $i$ from 1 to $n$ yields
$\displaystyle\sum_{i=1}^{n}(\Lambda_{i+1}^{\frac{1}{p-1}}-\Lambda_{1}^{\frac{1}{p-1}})^{p-2}\leq$
$\displaystyle(2p)^{p-2}\sum_{i=1}^{n}\int_{\Omega}|\nabla^{p-2}u_{1,x_{i}}|^{2}=(2p)^{p-2}\sum_{i=1}^{n}\int_{\Omega}|\nabla^{p-1}u_{1}|^{2}$
$\displaystyle\leq$ $\displaystyle(2p)^{p-2}\Lambda_{1}^{\frac{p-2}{p-1}},$
concluding the proof of Theorem 1.2. $\sqcup$$\sqcap$
## References
* [1] M.S. Ashbaugh. Isoperimetric and universal inequalities for eigenvalues. In: Davies, E.B., Safarov, Y.(eds.) Spectral theory and geometry, 1998. London Math. Soc. Lecture Notes, vol 273, pp 95-139. Cambridge University Press, Cambridge(1999)
* [2] M.S. Ashbaugh. On universal inequalities for the low eigenvalues of the buckling problem. (English summary) Partial differential equations and inverse problems, 13-31, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, 2004
* [3] Z.C. Chen and C.L. Qian. On the upper bound of eigenvalues for elliptic equations with higher orders. J. Math. Anal. Appl. 186 (1994), 821-834
* [4] Q.M. Cheng and H.C.Yang. Universal bounds for eigenvalues of a buckling problem. Commun. Math. Phys. 262 (2006), 663-675
* [5] G.Y. Huang and W.Y. Chen. Universal bounds for eigenvalues of Laplacian operator with any order. To appear in Acta Math. Sci. Ser. B Engl. Ed. 2010, vol 30(1)
* [6] G.Y. Huang, X.X. Li and L.F. Cao. Universal bounds on eigenvalues of the buckling problem on spherical domains (preprint)
* [7] G.Y. Huang and X.X. Li. Universal inequalities for eigenvalues of Laplacian with any order (preprint)
* [8] G.Y. Huang, X.X. Li and X.R. Qi. Estimates on the first two buckling eigenvalues on spherical domains (preprint)
* [9] Q.M. Cheng, T. Ichikawa and S. Mametsuka. Inequalities for eigenvalues of Laplacian with any order (preprint)
* [10] G.N. Hile and R.Z. Yeh. Inequalities for eigenvalues of the biharmonic operator. Pacific J. Math. 112(1984), 115-133
* [11] L.E. Payne, G. Pólya and H.F. Weinberger. On the ratio of consecutive eigenvalues. J. Math. Phys. 35(1956), 289-298
* [12] Q.L. Wang and C.Y. Xia. Universal inequalities for eigenvalues of the buckling problem on spherical domains. Commun. Math. Phys. 270(2007), 759-775
* [13] F.E. Wu and L.F. Cao. Estimates for eigenvalues of Laplacian operator with any order. Sci. China Ser. A, 50(2007), 1078-1086
|
arxiv-papers
| 2009-10-21T14:29:59 |
2024-09-04T02:49:05.960358
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangyue Huang, Xingxiao Li",
"submitter": "Huang Guangyue",
"url": "https://arxiv.org/abs/0910.4101"
}
|
0910.4265
|
# Shape oscillation of bubbles in the acoustic field
Keishi Matsumoto1, Ichiro Ueno2
1Graduate School, Tokyo University of Science,
2641 Yamazaki, Noda, Chiba 278-8510, JAPAN
2Tokyo University of Science, Noda, JAPAN
###### Abstract
The authors introduce dynamics of multiple air bubbles exposed to ultrasonic
wave while ascending in water in the present fluid dynamics video. The authors
pay attention to the shape oscillation and the transition from the volume to
the shape oscillations of the bubble. Correlation between the bubble size and
the mechanism of the excitation of the shape oscillation is introduced.
## 1 Introduction
The video is Video1.
To understand a behavior of the bubbles, their shape and volume oscillations,
under periodic external force is of one of great importance in order to
control heat/mass transfer in multiphase flows. A large number of researches
have been carried out on the volume oscillation by theoretical and
experimental approaches (reviewed by Plesset & Prosperetti (1) and Feng & Leal
(2)). On the shape oscillation, however, there exist few experimental works
although there have been accumulated theoretical predictions. In the present
study, The authors introduce the effect of the preceding bubble in the
acoustic field on the shape oscillation arisen on the following bubble.
REFERENCES
(1) Pleset, M. S. Prosperetti,A., Bubble Dynamics and Cavitaion,Ann. Rev.
Fluid Mech.,29(1977),145-185
(2) Feng, Z. C. & Leal, L. G., Nonlinear Bubble Dynamics, Ann. Rev. Fluid
Mech., 29(1997), 201-243
|
arxiv-papers
| 2009-10-22T09:37:05 |
2024-09-04T02:49:05.967915
|
{
"license": "Public Domain",
"authors": "Keishi Matsumoto, Ichiro Ueno",
"submitter": "Keishi Matsumoto",
"url": "https://arxiv.org/abs/0910.4265"
}
|
0910.4276
|
Stochastic local operations and classical communication
equations and classification of even $n$ qubits111The paper was supported by
NSFC(Grants No. 10875061,60433050, and 60673034 ) and Tsinghua National
Laboratory for Information Science and Technology.
Xiangrong Lia, Dafa Lib
a Department of Mathematics, University of California, Irvine, CA 92697-3875,
USA
b Department of mathematical sciences, Tsinghua University, Beijing 100084
CHINA
###### Abstract
For any even $n$ qubits we establish four SLOCC equations and construct four
SLOCC polynomials (not complete) of degree $2^{n/2}$, which can be exploited
for SLOCC classification (not complete) of any even $n$ qubits. In light of
the SLOCC equations, we propose several different genuine entangled states of
even $n$ qubits and show that they are inequivalent to the $|GHZ\rangle$,
$|W\rangle$, or $|l,n\rangle$ (the symmetric Dicke states with $l$
excitations) under SLOCC via the vanishing or not of the polynomials. The
absolute values of the polynomials can be considered as entanglement measures.
Keywords: entanglement measure, SLOCC entanglement classification.
PACS numbers: 03.67.Mn, 03.65.Ud
## 1 Introduction
A fundamental concept in quantum information theory is the understanding of
entanglement. Quantum entanglement can be viewed as a crucial resource in
quantum information. The key question is how to quantify and classify
entanglement of quantum states. Polynomial functions in the coefficients of
pure states which are invariant under stochastic local operations and
classical communication (SLOCC) transformations have been studied extensively
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and exploited to
construct entanglement measures [1, 2, 5, 12, 13, 14]. The concurrence [1] and
three-tangle [2], which measure entanglement of two-qubit and three-qubit
states, are polynomial invariants of degrees 2 and 4 respectively. It is known
that the concurrence and three-tangle are the absolute values of
hyperdeterminants for two and three qubits respectively [3]. An expression has
recently been derived for four-tangle, which is a polynomial invariant and a
measure of genuine entanglement of four-qubit states [4]. Polynomial
invariants of degrees 2, 4 and 6 for four and five qubits have been
constructed from classical invariant theory [5, 6]. The absolute values of the
polynomial invariants obtained in [5] may be used to construct entanglement
measures of four-qubit states. Further, polynomial invariants of degrees 2, 4,
6, 8, 10 and 12 for four and five qubits have been obtained using local
invariant operators [7]. Despite these efforts, few attempts have so far been
made towards the generalization to higher number of qubits. Three-tangle has
been generalized to $n$-tangle for even $n$ qubits [8] and has been shown to
be equal to the square of the polynomial invariant of degree 2 [9]. A
generalization of three-tangle to odd $n$ qubits has been recently proposed in
[10]. In [11], polynomial invariants of degree 2 for even $n$ qubits and
degree 4 for odd $n$ qubits have been derived by induction based on the
definition of SLOCC.
SLOCC classification of pure states has been under intensive research [3, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. For three qubits, two genuine
entanglement states, namely the $|GHZ\rangle$ and $|W\rangle$ states, have
been distinguished and characterized by the vanishing or not of the three-
tangle [17]. For four or more qubits, the number of SLOCC classes is infinite.
It is highly desirable to divide these infinite SLOCC classes into a finite
number of families. Central to the issue is the criteria to determine which
family an arbitrary state belongs to. Various methods have been undertaken to
tackle the classification of four-qubit states, including those based on Lie
group theory [18], on hyperdeterminant [3], on inductive approach [19], on
string theory [20], and on polynomials (algebraic) invariants [21, 22, 23, 24,
25, 26]. Recently, the Majorana representation has been used for SLOCC
entanglement classification of $n$-qubit symmetric states [27]. For $n$
qubits, it is known that the $|l,n\rangle$ states (symmetric Dicke states with
$l$ excitations) are inequivalent to the $|GHZ\rangle$ state or the
$|W\rangle$ state under SLOCC [28]. Therefore it is necessary to develop
schemes to find other genuine entangled states which are inequivalent to the
$|GHZ\rangle$, $|W\rangle$, or $|l,n\rangle$ states.
In this paper, we establish four SLOCC equations and construct four SLOCC
polynomials (not complete) of degree $2^{n/2}$ for any even $n$ qubits. The
equations are obtained from the polynomials (determinants) of the coefficients
of the two SLOCC equivalent states by induction via direct manipulation of
SLOCC definition. For $n=4$, the SLOCC polynomials of degree $2^{n/2}$ reduce
to the polynomials of degree 4 in [5]. In light of the SLOCC equations, we
propose several different genuine SLOCC entanglement classes of even $n$
qubits and show that they are inequivalent to the $|GHZ\rangle$, $|W\rangle$,
or $|l,n\rangle$ (the symmetric Dicke states with $l$ excitations) SLOCC
classes via the vanishing or not of the polynomials.
The manuscript is organized as follows. In Sections 2, 3, 4 and 5, we
construct SLOCC polynomials and present SLOCC equations of type I, II, III,
and IV, respectively. We also discuss SLOCC classifications by means of these
polynomials. In Section 6, we draw our conclusions.
## 2 SLOCC equation and polynomial of type I
Let $|\psi\rangle$ and $|\psi^{\prime}\rangle$ be any states of $n$ qubits.
Then we can write
$|\psi^{\prime}\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle,|\psi\rangle=\sum_{i=0}^{2^{n}-1}b_{i}|i\rangle,$
where $\sum_{i=0}^{2^{n}-1}|a_{i}|^{2}=1$ and
$\sum_{i=0}^{2^{n}-1}|b_{i}|^{2}=1$. Two states $|\psi\rangle$ and
$|\psi^{\prime}\rangle$ are equivalent under SLOCC if and only if there exist
invertible local operators $\mathcal{A}_{1}$,
$\mathcal{A}_{2},\cdots,\mathcal{A}_{n}$ such that
$|\psi^{\prime}\rangle=\underbrace{\mathcal{A}_{1}\otimes\mathcal{A}_{2}\otimes\cdots\otimes\mathcal{A}_{n}}_{n}|\psi\rangle.$
(2.1)
For the state $|\psi^{\prime}\rangle$ of even $n$ qubits, let $\Theta(a,n)$ be
the determinant of the coefficient matrix ($2^{n/2}$ by $2^{n/2})$ which is
partitioned into blocks, _i.e._
$\Theta(a,n)=\left|\left(\begin{tabular}[]{llll}$\Theta_{1}$&$\Theta_{2}$&$\cdots$&$\Theta_{2^{n/2}}$\end{tabular}\right)\right|,$
(2.2)
where the blocks $\Theta_{i}$, $i=1,\cdots,2^{n/2}$, are the columns of the
matrix and $\bigl{\\{}\Theta_{1}^{T}$
$\Theta_{2}^{T}\cdots\Theta_{n}^{T}\bigr{\\}}$ is just the coefficient vector
$\bigl{\\{}a_{0}$ $a_{1}\cdots a_{2^{n}-1}\bigr{\\}}$.
To understand the structure of $\Theta(a,n)$, we list $\Theta(a,4)$ below:
$\Theta(a,4)=\left|\begin{tabular}[]{llll}$a_{0}$&$a_{4}$&$a_{8}$&$a_{12}$\\\
$a_{1}$&$a_{5}$&$a_{9}$&$a_{13}$\\\ $a_{2}$&$a_{6}$&$a_{10}$&$a_{14}$\\\
$a_{3}$&$a_{7}$&$a_{11}$&$a_{15}$\end{tabular}\right|,$ (2.3)
which turns out to be the determinant $L$ in [5].
Now, suppose that $|\psi^{\prime}\rangle$ and $|\psi\rangle$ are equivalent
under SLOCC. Then we get the following result:
$\Theta(a,n)=\Theta(b,n)\bigl{[}\det(\mathcal{A}_{1})\cdots\det(\mathcal{A}_{n})\bigr{]}^{2^{(n-2)/2}},$
(2.4)
where $\Theta(b,n)$ is obtained from $\Theta(a,n)$ by replacing $a$ by $b$.
Eq. (2.4) and $\Theta(a,n)$ are referred to as SLOCC equation and polynomial
of type I for even $n$ qubits, respectively. The proof of Eq. (2.4) for $n=2$
can be seen as follows. Solving Eq. (2.1) yields $a_{0}a_{3}-a_{1}a_{2}=$
$(b_{0}b_{3}-b_{1}b_{2})\det(\mathcal{A}_{1})\det(\mathcal{A}_{2})$ [11]. The
desired result then follows by noting that $(a_{0}a_{3}-a_{1}a_{2})$ is the
determinant $\Theta(a,2)$ of the coefficients of states for two qubits. For
$n\geq 4$, we refer the reader to Appendix A for the proof.
It follows from Eq. (2.4) that if one of $\Theta(a,n)$ and $\Theta(b,n)$
vanishes while the other does not, then the state $|\psi^{\prime}\rangle$ is
not equivalent to the state $|\psi\rangle$ under SLOCC.
We next demonstrate that $\Theta(a,n)$ vanishes for the $|GHZ\rangle$,
$|W\rangle$ and Dicke states for $n>2$. It is trivial to see that
$\Theta(a,n)$ vanishes for the $|GHZ\rangle$ and $|W\rangle$ states. Recall
that the $n$-qubit symmetric Dicke states with $l$ excitations, where $1\leq
l\leq(n-1)$, were defined as [29]
$|l,n\rangle=\sum_{i}P_{i}|1_{1}1_{2}\cdots 1_{l}0_{l+1}\cdots 0_{n}\rangle,$
(2.5)
where $\\{P_{i}\\}$ is the set of all the distinct permutations of the qubits.
Note that $|1,n\rangle$ is just $|W\rangle$. For Dicke states $|l,n\rangle$,
it is known that $|l,n\rangle$ and $|(n-l),n\rangle$ are equivalent to each
other under SLOCC. Hence we only need to consider $2\leq l\leq n/2$.
Inspection of the binary form of the subscripts of the entries in the second
and third columns of $\Theta(a,n)$ reveals that those two columns are equal.
Indeed, for $l<n/2$, we see that all the entries in the last column of
$\Theta(a,n)$ vanish. It follows that $\Theta(a,n)$ vanishes for Dicke states
as well.
Consider the following two states
$\displaystyle|\chi_{1}\rangle$ $\displaystyle=$
$\displaystyle(1/\sqrt{2^{n/2}})\biggl{[}\sum_{m=0}^{2^{n/2}-2}|(2^{n/2}+1)m\rangle-|2^{n}-1\rangle\biggr{]},$
(2.6) $\displaystyle|\chi_{2}\rangle$ $\displaystyle=$
$\displaystyle(1/\sqrt{2^{n/2}})\biggl{[}\sum_{m=1}^{2^{n/2}-1}|(2^{n/2}-1)m\rangle-|2^{n}-2^{n/2}\rangle\biggr{]}.$
(2.7)
We observe that all the non-zero coefficients of $|\chi_{1}\rangle$ lie on the
diagonal of $\Theta(a,n)$. This leads to non-vanishing $\Theta(a,n)$ for
$|\chi_{1}\rangle$. Similary, all the non-zero coefficients of
$|\chi_{2}\rangle$ lie on the antidiagonal of $\Theta(a,n)$ and therefore $\
\Theta(a,n)$ does not vanish for $|\chi_{2}\rangle$. In light of Eq. (2.4),
for $n>2$, $|\chi_{1}\rangle$ and $|\chi_{2}\rangle$ are both different from
the $|GHZ\rangle$, $|W\rangle$ and Dicke states under SLOCC. It can be further
demonstrated that $|\chi_{1}\rangle$ and $|\chi_{2}\rangle$ are entangled, and
that $|\chi_{2}\rangle$ is equivalent to $|\chi_{1}\rangle$ under SLOCC. We
exemplify the result for the case of four qubits. We find that
$|\chi_{1}\rangle=(1/2)\bigl{(}|0\rangle+|5\rangle+|10\rangle-|15\rangle\bigr{)}$
and it was shown in [23] that $|\chi_{1}\rangle$ is different from the
$|GHZ\rangle$, $|W\rangle$, and Dicke states under SLOCC.
Remark 2.1. In $|\chi_{1}\rangle$ SLOCC entanglement class, the states
$|\chi_{1}\rangle$ and $|\chi_{2}\rangle$ have the minimal number of product
terms (_i.e._ $2^{n/2}$ product terms).
## 3 SLOCC equation and polynomial of type II
For the state $|\psi^{\prime}\rangle$ of even $n$ qubits, let $\Pi(a,n)$ be
the determinant of the coefficient matrix ($2^{n/2}$ by $2^{n/2})$ which is
partitioned into blocks, _i.e._
$\Pi(a,n)=\left|\left(\begin{tabular}[]{c}$\Pi_{1}$\\\ $\Pi_{2}$\\\
$\vdots$\\\ $\Pi_{2^{n/2}}$\end{tabular}\right)\right|,$ (3.1)
where the blocks $\Pi_{i}$, $i=1,\cdots,2^{n/2}$, are the rows of the matrix,
$\bigl{\\{}\Pi_{1}$ $\Pi_{3}\cdots\Pi_{2^{n/2}-1}\bigr{\\}}$ is just the
coefficient vector $\bigl{\\{}a_{0}$ $a_{2}\cdots a_{2k}\cdots
a_{2^{n}-2}\bigr{\\}}$, and $\bigl{\\{}\Pi_{2}$
$\Pi_{4}\cdots\Pi_{2^{n/2}}\bigr{\\}}$ is just the coefficient vector
$\bigl{\\{}a_{1}$ $a_{3}\cdots a_{2k+1}\cdots a_{2^{n}-1}\bigr{\\}}$.
To understand the structure of $\Pi(a,n)$, we list $\Pi(a,4)$ below:
$\Pi(a,4)=\left|\begin{tabular}[]{llll}$a_{0}$&$a_{2}$&$a_{4}$&$a_{6}$\\\
$a_{1}$&$a_{3}$&$a_{5}$&$a_{7}$\\\ $a_{8}$&$a_{10}$&$a_{12}$&$a_{14}$\\\
$a_{9}$&$a_{11}$&$a_{13}$&$a_{15}$\end{tabular}\right|,$ (3.2)
which is equal to the determinant $N$ in [5].
Now, suppose that $|\psi^{\prime}\rangle$ and $|\psi\rangle$ are equivalent
under SLOCC. Then we get the following result:
$\Pi(a,n)=\Pi(b,n)\bigl{[}\det(\mathcal{A}_{1})\cdots\det(\mathcal{A}_{n})\bigr{]}^{2^{(n-2)/2}},$
(3.3)
where $\Pi(b,n)$ is obtained from $\Pi(a,n)$ by replacing $a$ by $b$. Eq.
(3.3) and $\Pi(a,n)$ are referred to as SLOCC equation and polynomial of type
II for even $n$ qubits, respectively. For $n=2$, Eq. (3.3) can be verified by
directly solving Eq. (2.1). For $n\geq 4$, we refer the reader to Appendix B
for the proof.
It follows from Eq. (3.3) that if one of $\Pi(a,n)$ and $\Pi(b,n)$ vanishes
while the other does not, then the state $|\psi^{\prime}\rangle$ is not
equivalent to the state $|\psi\rangle$ under SLOCC.
Furthermore, it is trivial to see that $\Pi(a,n)$ vanishes for the
$|GHZ\rangle$ and $|W\rangle$ states for $n>2$. For Dicke states $|l,n\rangle$
($l\geq 2$) for $n>2$, $\Pi(a,n)$ vanishes as well owing to the fact that the
second and third rows of $\Pi(a,n)$ are equal.
Consider the following two states
$\displaystyle|\chi_{3}\rangle$ $\displaystyle=$
$\displaystyle(1/\sqrt{2^{n/2}})\biggl{[}\sum_{m=0}^{2^{n/2-1}-2}(|2^{n/2+1}m+4m\rangle+|2^{n/2+1}m+4m+3\rangle)+|2^{n}-4\rangle-|2^{n}-1\rangle\biggr{]},$
(3.4) $\displaystyle|\chi_{4}\rangle$ $\displaystyle=$
$\displaystyle(1/\sqrt{2^{n/2}})\biggl{[}\sum_{m=1}^{2^{n/2-1}-1}(|2^{n/2+1}m-4m+2\rangle+|2^{n/2+1}m-4m+1\rangle)+|2^{n}-2^{n/2+1}+2\rangle$
(3.5) $\displaystyle-$ $\displaystyle|2^{n}-2^{n/2+1}+1\rangle\biggr{]}.$
An argument analogous to the one in section 2 shows that $\Pi(a,n)$ does not
vanish for $|\chi_{3}\rangle$ or for $|\chi_{4}\rangle$. In light of Eq.
(3.3), for $n>2$, the states $|\chi_{3}\rangle$ and $|\chi_{4}\rangle$ are
both different from the $|GHZ\rangle$, $|W\rangle$, and Dicke states under
SLOCC. It can be further demonstrated that $|\chi_{3}\rangle$ and
$|\chi_{4}\rangle$ are entangled, and that $|\chi_{4}\rangle$ is equivalent to
$|\chi_{3}\rangle$ under SLOCC. We exemplify the result for the case of four
qubits. We find that
$|\chi_{3}\rangle=(1/2)\bigl{(}|0\rangle+|3\rangle+|12\rangle-|15\rangle\bigr{)}$
and it was shown in [23] that $|\chi_{3}\rangle$ is different from the
$|GHZ\rangle$, $|W\rangle$, and Dicke states under SLOCC.
Remark 3.1. In light of Eq. (2.4), for $n>2$, $|\chi_{3}\rangle$ is
inequivalent to $|\chi_{1}\rangle$ under SLOCC, since we can show that
$\Theta(a,n)=0$ for $|\chi_{3}\rangle$ and $\Theta(a,n)\neq 0$ for
$|\chi_{1}\rangle$.
Remark 3.2. For $|\chi_{3}\rangle$ SLOCC entanglement class, the states
$|\chi_{3}\rangle$ and $|\chi_{4}\rangle$ have the minimal number of product
terms (_i.e._ $2^{n/2}$ product terms).
## 4 SLOCC equation and polynomial of type III
For the state $|\psi^{\prime}\rangle$ of even $n$ qubits, let $\Gamma(a,n)$ be
the determinant of the coefficient matrix ( $2^{n/2}$ by $2^{n/2})$ which is
partitioned into $2^{n/2+1}$ $1$ by $2^{n/2-1}$ blocks, _i.e._
$\Gamma(a,n)=\left|\left(\begin{tabular}[]{cc}$\Gamma_{1}$&$\Gamma_{1}^{\prime}$\\\
$\Gamma_{2}$&$\Gamma_{2}^{\prime}$\\\ $\vdots$&$\vdots$\\\
$\Gamma_{2^{n/2}}$&$\Gamma_{2^{n/2}}^{\prime}$\end{tabular}\right)\right|,$
(4.1)
where the blocks $\Gamma_{i}$ and $\Gamma_{i}^{\prime}$,
$i=1,2,\cdots,2^{n/2}$, satisfy that $\bigl{\\{}\Gamma_{1}$
$\Gamma_{2}\cdots\Gamma_{2^{n/2}}\bigr{\\}}$ is just the coefficient vector
$\bigl{\\{}a_{0}$ $a_{1}\cdots a_{2^{n-1}-1}\\}$, and $\\{\Gamma_{1}^{\prime}$
$\Gamma_{2}^{\prime}\cdots\Gamma_{2^{n/2}}^{\prime}\bigr{\\}}$ is just the
coefficient vector $\bigl{\\{}a_{2^{n-1}}$ $a_{2^{n-1}+1}\cdots
a_{2^{n}-1}\bigr{\\}}$.
To understand the structure of $\Gamma(a,n)$, we list $\Gamma(a,6)$ below:
$\left|\begin{tabular}[]{llllllll}$a_{0}$&$a_{1}$&$a_{2}$&$a_{3}$&$a_{32}$&$a_{33}$&$a_{34}$&$a_{35}$\\\
$a_{4}$&$a_{5}$&$a_{6}$&$a_{7}$&$a_{36}$&$a_{37}$&$a_{38}$&$a_{39}$\\\
$a_{8}$&$a_{9}$&$a_{10}$&$a_{11}$&$a_{40}$&$a_{41}$&$a_{42}$&$a_{43}$\\\
$a_{12}$&$a_{13}$&$a_{14}$&$a_{15}$&$a_{44}$&$a_{45}$&$a_{46}$&$a_{47}$\\\
$a_{16}$&$a_{17}$&$a_{18}$&$a_{19}$&$a_{48}$&$a_{49}$&$a_{50}$&$a_{51}$\\\
$a_{20}$&$a_{21}$&$a_{22}$&$a_{23}$&$a_{52}$&$a_{53}$&$a_{54}$&$a_{55}$\\\
$a_{24}$&$a_{25}$&$a_{26}$&$a_{27}$&$a_{56}$&$a_{57}$&$a_{58}$&$a_{59}$\\\
$a_{28}$&$a_{29}$&$a_{30}$&$a_{31}$&$a_{60}$&$a_{61}$&$a_{62}$&$a_{63}$\end{tabular}\right|.$
(4.2)
Now, suppose that $|\psi^{\prime}\rangle$ and $|\psi\rangle$ are equivalent
under SLOCC. Then we get the following result:
$\Gamma(a,n)=\Gamma(b,n)\bigl{[}\det(\mathcal{A}_{1})\cdots\det(\mathcal{A}_{n})\bigr{]}^{2^{(n-2)/2}},$
(4.3)
where $\Gamma(b,n)$ is obtained from $\Gamma(a,n)$ by replacing $a$ by $b$.
Eq. (4.3) and $\Gamma(a,n)$ are referred to as SLOCC equation and polynomial
of type III for even $n$ qubits, respectively. For $n=2$, Eq. (4.3) can be
verified by directly solving Eq. (2.1). For $n\geq 4$, we refer the reader to
Appendix C for the proof.
It follows from Eq. (4.3) that if one of $\Gamma(a,n)$ and $\Gamma(b,n)$
vanishes while the other does not, then the state $|\psi^{\prime}\rangle$ is
not equivalent to the state $|\psi\rangle$ under SLOCC.
Furthermore, it is trivial to see that $\Gamma(a,n)$ vanishes for the
$|GHZ\rangle$ and $|W\rangle$ states for $n>2$. For Dicke states $|l,n\rangle$
($l\geq 2$) for $n>2$, $\Gamma(a,n)$ vanishes as well owing to the fact that
the second and third columns of $\Gamma(a,n)$ are equal.
Consider the following two states
$\displaystyle|\chi_{5}\rangle$ $\displaystyle=$
$\displaystyle(1/\sqrt{2^{n/2}})\biggl{[}\sum_{m=0}^{2^{n/2-1}-1}|(2^{n/2-1}+1)m\rangle+\sum_{m=0}^{2^{n/2-1}-2}|(2^{n/2-1}+1)m+3\cdot
2^{n-2}\rangle-|2^{n}-1\rangle\biggr{]},$ (4.4)
$\displaystyle|\chi_{6}\rangle$ $\displaystyle=$
$\displaystyle(1/\sqrt{2^{n/2}})\biggl{[}\sum_{m=1}^{2^{n/2-1}}|2^{n-1}+(2^{n/2-1}-1)m\rangle+\sum_{m=1}^{2^{n/2-1}-1}|2^{n-2}+(2^{n/2-1}-1)m\rangle$
(4.5) $\displaystyle-$ $\displaystyle|2^{n-1}-2^{n/2-1}\rangle\biggr{]}.$
An argument analogous to the one in section 2 shows that $\Gamma(a,n)$ does
not vanish for $|\chi_{5}\rangle$ or for $|\chi_{6}\rangle$. In light of Eq.
(4.3), for $n>2$, $|\chi_{5}\rangle$ and $|\chi_{6}\rangle$ are both different
from the $|GHZ\rangle$, $|W\rangle$, and Dicke states under SLOCC. It can be
further demonstrated that $|\chi_{5}\rangle$ and $|\chi_{6}\rangle$ are
entangled, and that $|\chi_{6}\rangle$ is equivalent to $|\chi_{5}\rangle$
under SLOCC. We exemplify the result for the case of four qubits. We find that
$\Gamma(a,4)=\Pi(a,4)$ and $|\chi_{5}\rangle=|\chi_{3}\rangle$.
Remark 4.1. In light of Eqs. (2.4) and (3.3), $|\chi_{5}\rangle$ is
inequivalent to $|\chi_{1}\rangle$ for $n>2$ or $|\chi_{3}\rangle$ for $n>4$
under SLOCC, since we can show that $\Theta(a,n)=\Pi(a,n)=0$ for
$|\chi_{5}\rangle$, $\Theta(a,n)\neq 0$ for $|\chi_{1}\rangle$ and
$\Pi(a,n)\neq 0$ for $|\chi_{3}\rangle$.
Remark 4.2. For $|\chi_{5}\rangle$ SLOCC entanglement class, the states
$|\chi_{5}\rangle$ and $|\chi_{6}\rangle$ have the minimal number of product
terms (_i.e._ $2^{n/2}$ product terms).
## 5 SLOCC equation and polynomial of type IV
For the state $|\psi^{\prime}\rangle$ of even $n$ qubits, let $\Omega(a,n)$ be
the determinant of the coefficient matrix ($2^{n/2}$ by $2^{n/2})$ which is
partitioned into $2^{n/2+1}$ $1$ by $2^{n/2-1}$ blocks, _i.e._
$\Omega(a,n)=\left|\left(\begin{tabular}[]{cc}$\Omega_{1}$&$\Omega_{1}^{\prime}$\\\
$\Omega_{2}$&$\Omega_{2}^{\prime}$\\\ $\vdots$&$\vdots$\\\
$\Omega_{2^{n/2}}$&$\Omega_{2^{n/2}}^{\prime}$\end{tabular}\right)\right|,$
(5.1)
where the blocks $\Omega_{i}$ and $\Omega_{i}^{\prime}$ satisfy
$\displaystyle\bigl{\\{}\Omega_{1}\Omega_{2}\Omega_{5}\Omega_{6}\cdots\Omega_{4k+1}\Omega_{4k+2}\cdots\Omega_{2^{n/2}-3}\Omega_{2^{n/2}-2}\bigr{\\}}$
$\displaystyle=$ $\displaystyle\bigl{\\{}a_{0},a_{2},\cdots
a_{2^{n-1}-2}\bigr{\\}},$ (5.2)
$\displaystyle\bigl{\\{}\Omega_{3}\Omega_{4}\Omega_{7}\Omega_{8}\cdots\Omega_{4k+3}\Omega_{4k+4}\cdots\Omega_{2^{n/2}-1}\Omega_{2^{n/2}}\bigr{\\}}$
$\displaystyle=$
$\displaystyle\bigl{\\{}a_{1},a_{3},\cdots,a_{2^{n-1}-1}\bigr{\\}},$ (5.3)
$\displaystyle\bigl{\\{}\Omega_{1}^{\prime}\Omega_{2}^{\prime}\Omega_{5}^{\prime}\Omega_{6}^{\prime}\cdots\Omega_{4k+1}^{\prime}\Omega_{4k+2}^{\prime}\cdots\Omega_{2^{n/2}-3}^{\prime}\Omega_{2^{n/2}-2}^{\prime}\bigr{\\}}$
$\displaystyle=$
$\displaystyle\bigl{\\{}a_{2^{n-1}},a_{2^{n-1}+2},\cdots,a_{2^{n}-2}\bigr{\\}},$
(5.4)
$\displaystyle\bigl{\\{}\Omega_{3}^{\prime}\Omega_{4}^{\prime}\Omega_{7}^{\prime}\Omega_{8}^{\prime}\cdots\Omega_{4k+3}^{\prime}\Omega_{4k+4}^{\prime}\cdots\Omega_{2^{n/2}-1}^{\prime}\Omega_{2^{n/2}}^{\prime}\bigr{\\}}$
$\displaystyle=$
$\displaystyle\bigl{\\{}a_{2^{n-1}+1},a_{2^{n-1}+3},\cdots,a_{2^{n}-1}\bigr{\\}},$
(5.5)
for $0\leq k\leq 2^{n/2-2}-1$.
To understand the structure of $\Omega(a,n)$, we list $\Omega(a,6)$ below:
$\left|\begin{tabular}[]{llllllll}$a_{0}$&$a_{2}$&$a_{4}$&$a_{6}$&$a_{32}$&$a_{34}$&$a_{36}$&$a_{38}$\\\
$a_{8}$&$a_{10}$&$a_{12}$&$a_{14}$&$a_{40}$&$a_{42}$&$a_{44}$&$a_{46}$\\\
$a_{1}$&$a_{3}$&$a_{5}$&$a_{7}$&$a_{33}$&$a_{35}$&$a_{37}$&$a_{39}$\\\
$a_{9}$&$a_{11}$&$a_{13}$&$a_{15}$&$a_{41}$&$a_{43}$&$a_{45}$&$a_{47}$\\\
$a_{16}$&$a_{18}$&$a_{20}$&$a_{22}$&$a_{48}$&$a_{50}$&$a_{52}$&$a_{54}$\\\
$a_{24}$&$a_{26}$&$a_{28}$&$a_{30}$&$a_{56}$&$a_{58}$&$a_{60}$&$a_{62}$\\\
$a_{17}$&$a_{19}$&$a_{21}$&$a_{23}$&$a_{49}$&$a_{51}$&$a_{53}$&$a_{55}$\\\
$a_{25}$&$a_{27}$&$a_{29}$&$a_{31}$&$a_{57}$&$a_{59}$&$a_{61}$&$a_{63}$\end{tabular}\right|.$
(5.6)
Now, suppose that $|\psi^{\prime}\rangle$ and $|\psi\rangle$ are equivalent
under SLOCC. Then we get the following result:
$\Omega(a,n)=\Omega(b,n)\bigl{[}\det(\mathcal{A}_{1})\cdots\det(\mathcal{A}_{n})\bigr{]}^{2^{(n-2)/2}},$
(5.7)
where $\Omega(b,n)$ is obtained from $\Omega(a,n)$ by replacing $a$ by $b$.
Eq. (5.7) and $\Omega(a,n)$ are referred to as SLOCC equation and polynomial
of type IV for even $n$ qubits, respectively. For $n=2$, Eq. (5.7) can be
verified by directly solving Eq. (2.1). For $n\geq 4$, we refer the reader to
Appendix D for the proof.
It follows from Eq. (5.7) that if one of $\Omega(a,n)$ and $\Omega(b,n)$
vanishes while the other does not, then the state $|\psi^{\prime}\rangle$ is
not equivalent to the state $|\psi\rangle$ under SLOCC.
Furthermore, it is trivial to see that $\Omega(a,n)$ vanishes for the
$|GHZ\rangle$ and $|W\rangle$ states for $n>2$. For Dicke states $|l,n\rangle$
($l\geq 2$) for $n>2$, $\Omega(a,n)$ vanishes as well owing to the fact that
the second and third columns of $\Omega(a,n)$ are equal.
Consider the following state
$\displaystyle|\chi_{7}\rangle$ $\displaystyle=$
$\displaystyle(1/\sqrt{2^{n/2}})\biggl{[}\sum_{m=0}^{2^{n/2-3}-1}(|2^{n/2+1}m+8m\rangle+|2^{n/2+1}m+8m+3\cdot
2^{n-2}\rangle)$ (5.8) $\displaystyle+$
$\displaystyle\sum_{m=0}^{2^{n/2-3}-1}(|(2m+1)2^{n/2}+8m+2\rangle+|(2m+1)2^{n/2}+8m+2+3\cdot
2^{n-2}\rangle)$ $\displaystyle+$
$\displaystyle\sum_{m=0}^{2^{n/2-3}-1}(|2^{n/2+1}m+8m+5\rangle+|2^{n/2+1}m+8m+5+3\cdot
2^{n-2}\rangle)$ $\displaystyle+$
$\displaystyle\sum_{m=0}^{2^{n/2-3}-1}(|(2m+1)2^{n/2}+8m+7\rangle+|(2m+1)2^{n/2}+8m+7+3\cdot
2^{n-2}\rangle)\biggr{]}$ $\displaystyle-$
$\displaystyle(2/\sqrt{2^{n/2}})|2^{n}-1\rangle,$
for $n\geq 6$ and
$|\chi_{7}\rangle=(1/2)\bigl{(}|0\rangle+|6\rangle+|9\rangle-|15\rangle\bigr{)}$
for $n=4$. An argument analogous to the one in section 2 shows that
$\Omega(a,n)$ does not vanish for $|\chi_{7}\rangle$. In light of Eq. (5.7),
for $n>2$, $|\chi_{7}\rangle$ is different from the $|GHZ\rangle$,
$|W\rangle$, and Dicke states under SLOCC. It can be further demonstrated that
the state $|\chi_{7}\rangle$ is entangled. In particular, for four qubits, it
was shown in [23] that $|\chi_{7}\rangle$ is different from the $|GHZ\rangle$,
$|W\rangle$ and Dicke states under SLOCC. We further note that
$|\chi_{7}\rangle=$ $|\chi_{5}\rangle$ for the case of six qubits.
Remark 5.1. In light of Eqs. (2.4), (3.3), and (4.3), for $n>2$,
$|\chi_{7}\rangle$ is inequivalent to $|\chi_{1}\rangle$, $|\chi_{3}\rangle$,
or $|\chi_{5}\rangle$ ($n\neq 6$ for $|\chi_{5}\rangle$) under SLOCC, since we
can show that $\Theta(a,n)=\Pi(a,n)=\Gamma(a,n)=0$ for $|\chi_{7}\rangle$.
Remark 5.2. For $|\chi_{7}\rangle$ SLOCC entanglement class, the state
$|\chi_{7}\rangle$ has the minimal number of product terms (_i.e._ $2^{n/2}$
product terms).
## 6 Conclusion
In this paper, for even $n$ qubits we have established four SLOCC equations
and constructed four SLOCC polynomials of degree $2^{n/2}$. For $n=4$, the
SLOCC polynomials of degree $2^{n/2}$ reduce to the polynomials of degree 4 in
[5]. For $n\geq 6$, the four SLOCC polynomials are linearly independent. The
SLOCC equations can be exploited for SLOCC classification of any even $n$
qubits. In light of the SLOCC equations, we have proposed several different
genuine SLOCC entanglement classes of even $n$ qubits and showed that they are
inequivalent to the $|GHZ\rangle$, $|W\rangle$, or $|l,n\rangle$ (the
symmetric Dicke states with $l$ excitations) via the vanishing or not of the
polynomials.
The concurrence and three-tangle, which measure entanglement of two-qubit and
three-qubit states, have been known to be the absolute values of
hyperdeterminants for two and three qubits respectively [3]. Recently,
polynomial invariants have been proposed to construct entanglement monotones.
The absolute values of the polynomial invariants obtained in [5] may be used
to construct entanglement measures of four-qubit states. We expect that the
absolute values of the polynomials in this paper can be considered as
entanglement measures.
## Appendix A. The proof for SLOCC equation of type I
Proof. We will prove Eq. (2.4) by induction principle. For the base case,
letting $\mathcal{A}_{1}=\mathcal{A}_{2}=\cdots=\mathcal{A}_{n}=I$ in Eq.
(2.1) yields $\Theta(a,n)=\Theta(b,n)$.
Let $|\phi\rangle=\sum_{i=0}^{2^{n}-1}c_{i}|i\rangle$ and
$|\phi\rangle=\underbrace{I\otimes\cdots\otimes
I\otimes\mathcal{A}_{r+1}\otimes\cdots\otimes\mathcal{A}_{n}}_{n}|\psi\rangle.$
(A1)
Assume that
$\Theta(c,n)=\Theta(b,n)\bigl{[}\det(\mathcal{A}_{r+1})\cdots\det(\mathcal{A}_{n})\bigr{]}^{2^{(n-2)/2}}$,
where $\Theta(c,n)$ is obtained from $\Theta(a,n)$ by replacing $a$ by $c$.
Next we will show that when
$|\psi^{\prime}\rangle=\underbrace{I\otimes\cdots\otimes
I\otimes\mathcal{A}_{r}\otimes\cdots\otimes\mathcal{A}_{n}}_{n}|\psi\rangle,$
(A2)
then
$\Theta(a,n)=\Theta(b,n)\bigl{[}\det(\mathcal{A}_{r})\cdots\det(\mathcal{A}_{n})\bigr{]}^{2^{(n-2)/2}}.$
(A3)
It is easy to see that $|\psi^{\prime}\rangle$
$=\underbrace{I\otimes\cdots\otimes I\otimes\mathcal{A}_{r}\otimes
I\cdots\otimes I}_{n}|\phi\rangle$. If we can prove that
$\Theta(a,n)=\Theta(c,n)\bigl{[}\det(\mathcal{A}_{r})\bigr{]}^{2^{(n-2)/2}}$,
then we can finish the induction.
For readability, let $\mathcal{A}_{l+1}=$
$\tau=\left(\begin{tabular}[]{ll}$\tau_{1}$&$\tau_{2}$\\\
$\tau_{3}$&$\tau_{4}$\end{tabular}\right)$. Thus, we only need to prove that
$\Theta(a,n)=\Theta(c,n)\bigl{[}\det(\tau)\bigr{]}^{2^{(n-2)/2}},$ (A4)
whenever $|\psi^{\prime}\rangle$ and $|\phi\rangle$ satisfy the following
equation
$|\psi^{\prime}\rangle=\underbrace{I\otimes\cdots\otimes
I}_{l}\otimes\tau\otimes\underbrace{I\otimes\cdots\otimes
I}_{n-l-1}|\phi\rangle.$ (A5)
From Eq. (A5), we obtain
$\displaystyle a_{2^{n-l}k+s}$ $\displaystyle=$
$\displaystyle\tau_{1}c_{2^{n-l}k+s}+\tau_{2}c_{2^{n-l}k+2^{n-l-1}+s},$ (A6)
$\displaystyle a_{2^{n-l}k+2^{n-l-1}+s}$ $\displaystyle=$
$\displaystyle\tau_{3}c_{2^{n-l}k+s}+\tau_{4}c_{2^{n-l}k+2^{n-l-1}+s},$ (A7)
where $0\leq k\leq 2^{l}-1$ and $0\leq s\leq 2^{n-l-1}-1$.
We distinguish two cases.
Case 1. $0\leq l\leq n/2-1$.
Let $A_{k,j}$ be a column of $\Theta(a,n)$ with entries
$a_{2^{n-l}k+2^{n/2}j+q}$ where $0\leq q\leq(2^{n/2}-1)$, and let
$A_{k,j}^{\ast}$ be a column obtained from $A_{k,j}$ by replacing each entry
$a_{\eta}$ by $a_{\eta+2^{n-l-1}}$. Then, the columns of $\Theta(a,n)$ are
(from left to right)
$\cdots,A_{k,j},A_{k,j+1},\cdots,A_{k,j}^{\ast},A_{k,j+1}^{\ast},\cdots,A_{k+1,j},A_{k+1,j+1},\cdots,A_{k+1,j}^{\ast},A_{k+1,j+1}^{\ast},\cdots,$
(A8)
where $0\leq k\leq 2^{l}-1$ and $0\leq j\leq 2^{n/2-l-1}-1$.
Note that $2^{n/2}j+q\leq 2^{n-l-1}-1$. Substituting Eqs. (A6) and (A7) into
$A_{k,j}$ and $A_{k,j}^{\ast}$ yields
$A_{k,j}=\tau_{1}C_{k,j}+\tau_{2}C_{k,j}^{\ast}$ and
$A_{k,j}^{\ast}=\tau_{3}C_{k,j}+\tau_{4}C_{k,j}^{\ast}$, where $C_{k,j}$ and
$C_{k,j}^{\ast}$ are obtained from $A_{k,j}$ and $A_{k,j}^{\ast}$ by replacing
$a$ by $c$ respectively. We see that $C_{k,j}$ and $C_{k,j}^{\ast}$ are
columns of $\Theta(c,n)$.
To compute $\Theta(a,n)$, we first let $\mathcal{T}_{k,j}$ be either
$\tau_{1}$ or $\tau_{2}$, and let $\mathcal{T}_{k,j}^{\ast}$ be either
$\tau_{3}$ or $\tau_{4}$. Let $U_{k,j}=C_{k,j}$ if
$\mathcal{T}_{k,j}=\tau_{1}$, and $U_{k,j}=C_{k,j}^{\ast}$ otherwise. Further,
let $U_{k,j}^{\ast}=C_{k,j}$ if $\mathcal{T}_{k,j}^{\ast}=\tau_{3}$, and
$U_{k,j}^{\ast}=C_{k,j}^{\ast}$ otherwise. Due to the multilinear property of
determinant, $\Theta(a,n)$ is the sum of $2^{2^{n/2}}$ determinants, each of
which consists of columns (from left to right):
$\displaystyle\cdots,\mathcal{T}_{k,j}U_{k,j},\mathcal{T}_{k,j+1}U_{k,j+1},\cdots,\mathcal{T}_{k,j}^{\ast}U_{k,j}^{\ast},\mathcal{T}_{k,j+1}^{\ast}U_{k,j+1}^{\ast},\cdots,\mathcal{T}_{k+1,j}U_{k+1,j},\mathcal{T}_{k+1,j+1}U_{k+1,j+1},$
$\displaystyle\hskip
216.81pt\cdots,\mathcal{T}_{k+1,j}^{\ast}U_{k+1,j}^{\ast},\mathcal{T}_{k+1,j+1}^{\ast}U_{k+1,j+1}^{\ast},\cdots,$
where $0\leq k\leq 2^{l}-1$ and $0\leq j\leq 2^{n/2-l-1}-1$.
Denote $t$ the product
$\cdots\mathcal{T}_{k,j}\mathcal{T}_{k,j+1}\cdots\mathcal{T}_{k,j}^{\ast}\mathcal{T}_{k,j+1}^{\ast}\cdots\mathcal{T}_{k+1,j}\mathcal{T}_{k+1,j+1}\cdots\mathcal{T}_{k+1,j}^{\ast}\mathcal{T}_{k+1,j+1}^{\ast}\cdots.$
Clearly, each of the $2^{2^{n/2}}$ determinants can be written in the form
$t\cdot\Delta$. Associated with each $t$ is a determinant $\Delta$ which
consists of columns (from left to right):
$\cdots,U_{k,j},U_{k,j+1},\cdots,U_{k,j}^{\ast},U_{k,j+1}^{\ast},\cdots,U_{k+1,j},U_{k+1,j+1},\cdots,U_{k+1,j}^{\ast},U_{k+1,j+1}^{\ast},\cdots,$
where $0\leq k\leq 2^{l}-1$ and $0\leq j\leq 2^{n/2-l-1}-1$.
We illustrate with an example. Let
$t=\cdots\tau_{1}\tau_{1}\cdots\tau_{4}\tau_{4}\cdots\tau_{1}\tau_{1}\cdots\tau_{4}\tau_{4}\cdots$,
whose power form is $(\tau_{1}\tau_{4})^{2^{(n-2)/2}}$, then
$\Delta=\Theta(c,n)$.
For Eq. (A4) to hold, we need the following 3 results.
Result 1. Given $t$ such that for some $k,j$, $\mathcal{T}_{k,j}=\tau_{1}$ and
$\mathcal{T}_{k,j}^{\ast}=\tau_{3}$, or $\mathcal{T}_{k,j}=\tau_{2}$ and
$\mathcal{T}_{k,j}^{\ast}=\tau_{4}$, then $\Delta$ vanishes.
Proof. If $\mathcal{T}_{k,j}=\tau_{1}$ and
$\mathcal{T}_{k,j}^{\ast}=\tau_{3}$, then by definition
$U_{k,j}=U_{k,j}^{\ast}=C_{k,j}$. We immediately see that $\Delta$ vanishes
since $\Delta$ has two equal columns. Likewise, if
$\mathcal{T}_{k,j}=\tau_{2}$ and $\mathcal{T}_{k,j}^{\ast}=\tau_{4}$, then by
definition $U_{k,j}=U_{k,j}^{\ast}=C_{k,j}^{\ast}$ and therefore $\Delta$
vanishes.
Result 2. Given $t$ such that for $0\leq k\leq 2^{l}-1$ and $0\leq j\leq
2^{n/2-l-1}-1$, $\mathcal{T}_{k,j}=\tau_{1}$ and
$\mathcal{T}_{k,j}^{\ast}=\tau_{4}$, or $\mathcal{T}_{k,j}=\tau_{2}$ and
$\mathcal{T}_{k,j}^{\ast}=\tau_{3}$ with $m$ occurrences for each of
$\tau_{2}$ and $\tau_{3}$, then $\Delta=(-1)^{m}\Theta(c,n)$.
Proof. If $\mathcal{T}_{k,j}=\tau_{1}$ and $\mathcal{T}_{k,j}^{\ast}=\tau_{4}$
for some $k,j$, then by definition $U_{k,j}=C_{k,j}$ and
$U_{k,j}^{\ast}=C_{k,j}^{\ast}$. These two columns of $\Delta$ are already in
order and nothing needs to be done here. If $\mathcal{T}_{k,j}=\tau_{2}$ and
$\mathcal{T}_{k,j}^{\ast}=\tau_{3}$, then by definition
$U_{k,j}=C_{k,j}^{\ast}$ and $U_{k,j}^{\ast}=C_{k,j}$. To obtain $\Theta(c,n)$
from $\Delta$, we need to interchange these two columns. It turns out that
$\Theta(c,n)$ can be obtained from $\Delta$ by interchanging two coulmns for
$m$ times, _i.e._ $\Delta=(-1)^{m}\Theta(c,n)$.
Result 3. The number of $t$ such that its power form is
$(\tau_{1}\tau_{4})^{i}(\tau_{2}\tau_{3})^{2^{(n-2)/2}-i}$ is given by
$\left(\begin{tabular}[]{c}$2^{(n-2)/2}$\\\ $i$\end{tabular}\right)$.
Proof. With the help of Result 1, we only need to consider those $t$ in which
$\mathcal{T}_{k,j}=\tau_{1}$ and $\mathcal{T}_{k,j}^{\ast}=\tau_{4}$, or
$\mathcal{T}_{k,j}=\tau_{2}$ and $\mathcal{T}_{k,j}^{\ast}=\tau_{3}$. In fact,
we only need to count the number of occurrences of $\tau_{1}$ and $\tau_{2}$
in
$\mathcal{T}_{k,0}\cdots\mathcal{T}_{k,j}\cdots\mathcal{T}_{k,2^{n/2-l-1}-1}$,
where $0\leq k\leq 2^{l}-1$ and $0\leq j\leq 2^{n/2-l-1}-1$. It is readily
seen that there are $\left(\begin{tabular}[]{c}$2^{(n-2)/2}$\\\
$i$\end{tabular}\right)$ such cases, each of which contains $i$ occurrences of
$\tau_{1}$ and $(2^{(n-2)/2}-i)$ occurrences of $\tau_{2}$.
It follows immediately from Result 2 and Result 3 that the sum of the
$2^{2^{n/2}}$ determinants is given by
$\Theta(c,n)\bigl{[}\det(\mathcal{\tau})\bigr{]}^{2^{(n-2)/2}}$. Therefore Eq.
(A4) holds.
Case 2. $n/2\leq l\leq(n-1)$.
Results analogous to Result 1, Result 2 and Result 3 can be derived by
replacing “column” by “row”.
Combining the above two cases, Eq. (A4) holds, and the proof is complete.
## Appendix B. The proof for SLOCC equation of type II
Proof. By induction principle and the argument in Appendix A, we only need to
prove that
$\Pi(a,n)=\Pi(c,n)\bigl{[}\det(\mathcal{\tau})\bigr{]}^{2^{(n-2)/2}}$ when
$|\psi^{\prime}\rangle$ and $|\phi\rangle$ satisfy Eq. (A5).
We distinguish three cases.
Case 1. $0\leq l\leq n/2-2$.
The proof is analogous to that in case 2 in Appendix A by investigating the
rows of $\Pi(a,n)$.
Case 2. $n/2-1\leq l\leq n-2$.
The proof is analogous to that in case 1 in Appendix A by investigating the
columns of $\Pi(a,n)$.
Case 3. $l=n-1$.
In this case, Eqs. (A6) and (A7) become
$\displaystyle a_{2k}$ $\displaystyle=$
$\displaystyle\tau_{1}c_{2k}+\tau_{2}c_{2k+1},$ (B1) $\displaystyle a_{2k+1}$
$\displaystyle=$ $\displaystyle\tau_{3}c_{2k}+\tau_{4}c_{2k+1},$ (B2)
where $0\leq k\leq 2^{n-1}-1$. The $(2r)th$ row of $\Pi(a,n)\ $is given by
$\bigl{(}a_{2^{n/2}2r}$, $a_{2^{n/2}2r+2},\cdots,a_{2^{n/2}(2r+2)-2}\bigr{)}$,
and the $(2r+1)th$ row of $\Pi(a,n)$ can be obtained from the $(2r)th$ row by
replacing each entry $a_{\eta}$ by $a_{\eta+1}$. The rest of the proof is
analogous to that in case 2 in Appendix B.
## Appendix C. The proof for SLOCC equation of type III
Proof. By induction principle and the argument in Appendix A, we only need to
prove that
$\Gamma(a,n)=\Gamma(c,n)\bigl{[}\det(\mathcal{\tau})\bigr{]}^{2^{(n-2)/2}}$
when $|\psi^{\prime}\rangle$ and $|\phi\rangle$ satisfy Eq. (A5).
We distinguish three cases.
Case 1. $l=0$.
The proof is analogous to that in case 1 in Appendix A by investigating the
columns of $\Gamma(a,n)$.
Case 2. $1\leq l\leq n/2$.
The proof is analogous to that in case 2 in Appendix A by investigating the
rows of $\Gamma(a,n)$.
Case 3. $n/2+1\leq l\leq n-1$.
The proof is analogous to that in case 1 in Appendix A by investigating the
columns of $\Gamma(a,n)$.
## Appendix D. The proof for SLOCC equation of type IV
Proof. By induction principle and the argument in Appendix A, we only need to
prove that
$\Omega(a,n)=\Omega(c,n)\bigl{[}\det(\mathcal{\tau})\bigr{]}^{2^{(n-2)/2}}$
when $|\psi^{\prime}\rangle$ and $|\phi\rangle$ satisfy Eq. (A5).
We distinguish four cases.
Case 1. $l=0$.
The proof is analogous to that in case 1 in Appendix A by investigating the
columns of $\Omega(a,n)$.
Case 2. $1\leq l\leq n/2-1$.
The proof is analogous to that in case 2 in Appendix A by investigating the
rows of $\Omega(a,n)$.
Case 3. $n/2\leq l\leq n-2$.
The proof is analogous to that in case 1 in Appendix A by investigating the
columns of $\Omega(a,n)$.
Case 4. $l=n-1$.
In this case, Eqs. (A6) and (A7) become Eqs. (B1) and (B2). Consider the
$(4k+1)th$, the $(4k+2)th$, the $(4k+3)th$, and the $(4k+4)th$ ($0\leq k\leq
2^{n/2-2}-1$) rows of $\Omega(a,n)$, respectively. The rest of the proof is
analogous to that in case 2 in Appendix A.
## References
* [1] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
* [2] V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A 61, 052306 (2000).
* [3] A. Miyake, Phys. Rev. A 67, 012108 (2003).
* [4] S.S. Sharma and N.K. Sharma, Phys. Rev. A 82, 012340 (2010).
* [5] J.-G. Luque and J.-Y. Thibon, Phys. Rev. A 67, 042303 (2003).
* [6] J.-G. Luque and J.-Y. Thibon, J. Phys. A: Math. Gen. 39, 371 (2006).
* [7] D.Z. Djoković and A. Osterloh, J. Math. Phys. 50, 033509 (2009).
* [8] A. Wong and N. Christensen, Phys. Rev. A 63, 044301 (2001).
* [9] X. Li and D. Li, Quantum Inf. Comput. 10, 1018 (2010).
* [10] D. Li, arXiv:quant-ph/0912.0812.
* [11] D. Li, X. Li, H. Huang, and X. Li, Phys. Rev. A 76, 032304 (2007) [arXiv:quant-ph/0704.2087].
* [12] D. Li, X. Li, H. Huang, and X. Li, J. Math. Phys. 50, 012104 (2009).
* [13] A. Osterloh and J. Siewert, Phys. Rev. A 72, 012337 (2005).
* [14] A. Osterloh and J. Siewert, Int. J. Quant. Inf. 4, 531 (2006).
* [15] M.S. Leifer, N. Linden, and A. Winter, Phys. Rev. A 69, 052304 (2004).
* [16] P. Lévay, J. Phys. A: Math. Gen. 39, 9533 (2006).
* [17] W. Dür, G. Vidal, and J.I. Cirac, Phys. Rev. A 62, 062314 (2000).
* [18] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A 65, 052112 (2002).
* [19] L. Lamata, J. León, D. Salgado, and E. Solano, Phys. Rev. A 75, 022318 (2007).
* [20] L. Borsten, D. Dahanayake, M.J. Duff, A. Marrani, and W. Rubens, Phys. Rev. Lett. 105, 100507 (2010).
* [21] O. Chterental and D.Z. Djoković, in Linear Algebra Research Advances, edited by G.D. Ling (Nova Science Publishers, Inc., Hauppauge, NY, 2007), Chap. 4, 133.
* [22] Y. Cao and A.M. Wang, Eur. Phys. J. D 44, 159 (2007).
* [23] D. Li, X. Li, H. Huang, and X. Li, Phys. Rev. A 76, 052311 (2007).
* [24] D. Li, X. Li, H. Huang, and X. Li, Quantum Inf. Comput. 9, 0778 (2009).
* [25] R.V. Buniy and T.W. Kephart, arXiv:quant-ph/1012.2630.
* [26] O. Viehmann, C. Eltschka, and J. Siewert, arXiv:quant-ph/1101.5558.
* [27] T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano, Phys. Rev. Lett. 103, 070503 (2009).
* [28] D. Li, X. Li, H. Huang, and X. Li, EPL 87, 20006 (2009).
* [29] J.K. Stockton, J.M. Geremia, A.C. Doherty, and H. Mabuchi, Phys. Rev. A 67, 022112 (2003).
|
arxiv-papers
| 2009-10-22T10:12:19 |
2024-09-04T02:49:05.971414
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "X. Li, D. Li",
"submitter": "Dafa Li",
"url": "https://arxiv.org/abs/0910.4276"
}
|
0910.4327
|
# Search for neutrinos from Gamma-Ray Bursts with the Baikal neutrino
telescope NT200
A. Avrorin1, V. Aynutdinov1, V. Balkanov1, I. Belolaptikov4, D. Bogorodsky2,
N. Budnev2,
I. Danilchenko1, G. Domogatsky1, A. Doroshenko1, A. Dyachok2, Zh.-A.
Dzhilkibaev1,
S. Fialkovsky6, O. Gaponenko1, K. Golubkov4, O. Gress2, T. Gress2, O.
Grishin2,
A. Klabukov1, A. Klimov8, A. Kochanov2, K. Konischev4, A. Koshechkin1, V.
Kulepov6,
D. Kuleshov1, L. Kuzmichev3, V. Lyashuk1, E. Middell5, S. Mikheyev1, M.
Milenin6,
R. Mirgazov2, E. Osipova3, G. Pan’kov2, L. Pan’kov2, A. Panfilov1, D.
Petukhov1,
E. Pliskovsky4, P. Pokhil1, V. Poleschuk1, E. Popova3, V. Prosin3, M.
Rozanov7,
V. Rubtzov2, A. Sheifler1, A. Shirokov3, B. Shoibonov4, Ch. Spiering5, O.
Suvorova1,
B. Tarashansky2, R. Wischnewski5, I. Yashin3, V. Zhukov1
1Institute for Nuclear Research of Russian Academy of Sciences,
117312, Moscow, 60-th October Anniversary pr. 7a, Russia 2Irkutsk State
University, Irkutsk, Russia 3Skobeltsyn Instutute of Nuclear Physics MSU,
Moscow, Russia 4Joint Institute for Nuclear Research, Dubna, Russia 5DESY,
Zeuthen, Germany 6Nizhni Novgorod State Technical University, Nizhnij
Novgorod, Russia 7St.Petersburg State Marine University, St.Petersburg, Russia
8Kurchatov Institute, Moscow, Russia
###### Abstract
We present an analysis of neutrinos detected with the Baikal neutrino
telescope NT200 for correlations with gamma-ray bursts (GRB). No neutrino
events correlated with GRB were observed. Assuming a Waxman-Bahcall spectrum,
a neutrino flux upper limit of $E^{2}\Phi<1.1\times
10^{-6}cm^{-2}s^{-1}sr^{-1}GeV$ was obtained. We also present the Green’s
Function fluence limit for this search, which extends two orders of magnitude
beyond the energy range of the Super-Kamiokande limit.
Neutrino telescope, BAIKAL, Gamma- ray burst
## 1 Introduction
The Baikal neutrino telescope NT200 [1, 2] is operating in Lake Baikal,
Siberia, at a depth 1.1 km since April, 1998. NT200 consists of 8 strings of
70 m length: 7 peripheral strings and a central one. Interstring distances are
about 20 m. Each string includes 24 pairwise arranged optical modules (OM).
Each OM contains a 37-cm diameter hybrid photodetector QUASAR-370.
A number of relevant physics results has been obtained so far with the NT200
telescope, e.g. limits on the diffuse flux of extraterrestrial high energy
neutrinos, limits on neutrino fluxes from Dark Matter annihilation (Sun,
Earth), and on the flux of relativistic and slow magnetic monopoles [2, 3, 4].
This work is devoted to the search of neutrino events correlated with
observations of more than 300 gamma-ray bursts (GRBs) reported from 1998 to
2000 by the Burst and Transient Source Experiment (BATSE) [5].
The detection strategy for neutrino events with the NT200 telescope is based
on a search for Cherenkov light from relativistic up-going muons produced by
neutrino interactions. Information about the GRB time and location on the sky
allows to reduce the atmospheric muon background and, as a result,
significantly increases the sensitivity of the neutrino telescope to neutrino
events correlated with GRB.
## 2 Experimental data
For the present analysis, the experimental data obtained with NT200 from April
1998 to May 2000 were used. The selected data sample contains those events
which were formally reconstructed as up-going muons. Taking into account the
high level of background for directions close to the horizon, only events with
zenith angles larger than $100^{\circ}$ were selected. The average rate of
such events was 0.037 Hz. Most fake events are due to misreconstructed muons
close to horizon and to muon bundles.
For the present analysis of time and directional correlations with NT200
events we used information about GRB location, time, duration $T_{90}$, and
location error from the basic BATSE 4B catalog [5] (triggered bursts) and from
the catalog of non-triggered GRB [6]. The error distribution of BATSE GRB
locations was taken from [7]. A total of 303 GRBs (155 triggered and 148 non-
triggered) at zenith angles larger than $100^{\circ}$ and occurring during
periods of stable operation of NT200 have been selected.
## 3 Data selection criteria and detector effective area
The optimization of the data selection criteria was performed on the basis of
simulated neutrino events [8] and events of atmospheric muon background [9] in
NT200. Taking into account the varying NT200 configurations during the
considered time period, calculations have been performed for nine basic
detector configurations, most of them closely corresponding to the real status
of the detector.
The results of the reconstruction of simulated events were used to estimate
the reconstruction efficiency and to calculate the background. Event
reconstruction and data selection for NT200 are described in detail in [8].
There, selection criteria were designed and optimized for atmospheric neutrino
separation. They provide a rejection factor of atmospheric muons larger than
$10^{7}$. For GRB, the aditional information about detection time and location
on sky, however, allows softening the requirements to the background
rejection. This increases the registration probability for useful events and
therefore greatly increases the effective neutrino detection area. Following
the approach of [8], $P_{hit}\times P_{nohit}$ and $Z_{dist}$ were chosen as
basic parameters for event selection. $Z_{dist}$ is the maximal distance
between all projections of the triggered OM coordinates onto the reconstructed
muon trajectory. $P_{hit}$ is the normalized probability of fired channels to
be hit, and $P_{nohit}$ is the probability of non-fired channel not to be hit.
For the present correlation analysis, two sets of criteria for event selection
were chosen:
Cut-A:
$(Z_{dist}>30m)\&(P_{hit}\times P_{nohit}>0.1)\&(\Psi<10^{\circ})$,
Cut-B:
$(Z_{dist}>30m)\&(\Psi<5^{\circ})$,
were $\Psi$ is the angle between up-going muon and GRB-direction.
Cut-A dominantly selects neutrinos with energies below $\sim 10^{6}$ GeV.
Cut-B allows a significant extension of the energy range, but the expected
background is approximately four times larger than for Cut-A.
Calculating the effective area of NT200 for the two sets of criteria, we took
into account the absorption of neutrinos passing through the Earth, as well as
the production, propagation, detection and reconstruction within a given
angular cut $\Psi$ of muons. The calculated effective areas for the Cut-A and
Cut-B samples are presented in Fig. 1 as a function of neutrino energy.
The effective areas for the two sets are close to each other up to $\sim
10^{5}$ GeV. For larger energies, the effective area for Cut-A stays
essentially constant. The behavior for $E>10^{6}$ GeV is largely defined by
neutrino absorption in the Earth.
The energy range of the NT200 sensitivity was estimated for an $E^{-2}$
neutrino spectrum. The 90% sensitivity range of NT200 extends up to $\sim
10^{6}$ GeV and $\sim 10^{7}$ GeV for selection criterion A and B,
respectively.
Figure 1: NT200 effective area averaged over zenith angles between
$100^{\circ}$ and $180^{\circ}$ as a function of neutrino energy, for
selection criteria Cut-A and Cut-B.
A global estimation of the background, expected for the GRB search time
window, was obtained from the NT200 raw event rate (0.037 Hz), the calculated
atmospheric muon rejection factor for Cut-A and Cut-B, and the total GRB
duration $T_{GRB}$ ($\sim 1.8\times 10^{4}$ s). $T_{GRB}$ was calculated as
the sum of the $T_{90}$ intervals for all 303 triggered and non-triggered GRB.
For compensation of possible event time uncertainties, five seconds were added
at both sides of the time intervals $T_{90}$. For cases of missing information
on $T_{90}$ (about 25% of the triggered GRB), a fixed time interval was used.
The expected background values, obtained from the full detector livetime
interval, are 0.8 and 3.1 events for Cut-A and Cut-B, respectively.
## 4 Data analysis and results
The basic objective of this study was to verify a sufficiently precise
detector responce simulation, to search for events correlated with GRB, and to
provide a solid background estimation.
To check the simulation procedures, the calculated atmospheric muon rejection
factors were compared to experimental values. The results are presented in
Table I for different criteria $P_{hit}\times P_{nohit}$ ($Z_{dist}$ is set to
30 m). The simulated values are in agreement with the experimental results
within the systematic error of our calculation, about 20%.
Table 1: Atmospheric Muon Rejection Factor: Experiment and Simulation $P_{hit}\times P_{nohit}$ | $\geq 0.1$ | $\geq 0.2$ | $\geq 0.3$
---|---|---|---
Experiment | 0.053 | 0.012 | 0.0035
Model | 0.062 | 0.014 | 0.0040
This search for correlation with NT200 neutrino events uses 303 GRBs
(triggered and non-triggered) selected from the total sample of 736 BATSE GRBs
recorded in 1998-2000. The search for events selected according to criteria A
and B was done for the time intervals $T_{GRB}$ (see section III). For a
detailed background estimation, we used a time interval $\pm 1000$ s with
respect to the start time of the GRB (excluding the $T_{GRB}$ window) and an
angle between GRB and up-going muon $\Psi\leq 10^{\circ}$.
No events were found according to criterion A and one event by criterion B.
Table II shows the number of signal and background events, the event upper
limit $\mu_{90}$ that was obtained in accordance with [10], the number of GRB
corrected to coefficient $\beta$ (the probability to detect events associated
with gamma-ray burst in the given angle interval $\Psi$, given the location
error of the GRB), and the 90% C.L. upper limit on the number of events per
GRB $N_{90}=\mu_{90}$ / $(N_{GRB}\times\beta)$. Data are presented for all GRB
(triggered and non-triggered), as well as separatly for triggered GRB selected
according to Cut-B.
Table 2: Results of GRB Analysis Selection | Signal | Backgr | $\mu_{90}$ | $N_{GRB}\times\beta$ | $N_{90}$
---|---|---|---|---|---
Cut-A, all | 0 | 0.56 | 1.9 | 236 | 0.0085
Cut-B, all | 1 | 2.7 | 2.1 | 199 | 0.010
Cut-B, trig | 1 | 1.6 | 2.8 | 120 | 0.023
No excess of events associated with a GRB was observed. The limit on the
neutrino flux associated with gamma-ray bursts was obtained using the approach
from [11]. According to this approach, the limit $F(E_{\nu})$ is presented as
a function on neutrino energy, the ”Green’s function”:
$F(E_{\nu})=N_{90}/S_{eff}(E_{\nu}),$ (1)
$S_{eff}(E_{\nu})$ is the detector effective area, $N_{90}$ the 90% C.L. upper
limit on number of events per GRB. The main advantage of this approach is that
its result does not depend on assumptions about the neutrino energy spectrum.
Figure 2: $90\%$ C.L. upper limits on the GRB neutrino fluence Green’s
function $F(E_{\nu})$ for NT200, Super-Kamiokande and AMANDA.
Figure 2 shows the 90% C.L. upper limits of the GRB neutrino fluence Green’s
function $F(E_{\nu})$ for NT200 (for Cut-B, all), Super-Kamiokande [11] and
AMANDA [12]. The Super-Kamiokande and NT200 limits are mainly for GRB from the
southern sky, while the AMANDA limit is for the northern sky.
Since predictions for the energy spectrum of neutrinos from GRB differ from
model to model, we prefer to present our basic experimental result as a
Green’s function $F(E_{\nu})$, which allows to calculate limits for any
neutrino energy spectrum. In addition, we translate our result to a benchmark
spectrum. We have chosen that of E.Waxman and J.Bahcall [14], and E.Waxman
[15, 16]. Following these works, the muon neutrino differential flux
$\Phi_{W-B}(E_{\nu})$ in the energy range up to 10 PeV is:
$E_{\nu}^{2}\,\Phi_{\nu}^{W-B}(E_{\nu})=A^{W-B}\times\min(1,E_{\nu}/E_{\nu
b}),$ (2)
$E_{\nu b}$=100 TeV, $A_{W-B}\approx 8\times 10^{-9}GeVcm^{-2}s^{-1}sr^{-1}$.
The Model Rejection Factor MRF for the Waxman-Bahcall spectrum was calculated
as:
$MRF=N_{90}/N_{ex},$ (3)
where $N_{90}$ is the upper limit on the number of events per GRB and $N_{ex}$
the expected number of events, calculated for the given spectrum as:
$N_{ex}=\int\Phi_{\nu}^{Earth}(E_{\nu})\times
S_{eff}(E_{\nu})\times(4\pi/n)dE_{\nu}.$ (4)
Here, $n\approx 2.2\times 10^{-5}s^{-1}$ is the average GRB rate in $4{\pi}sr$
($\sim 700$ events within the detection range of the BATSE per year) and
$\Phi_{\nu}^{Earth}(E_{\nu})$=$0.5\times\Phi_{\nu}^{W-B}(E_{\nu})$ the
neutrino flux at the Earth.
Taking into account that the estimation of the expected event number in the
given approach is made for for the BATSE burst detection rate, the MRF was
calculated only for triggered GRB ($N_{GRB}\times\beta=120$, see Table II).
From that Green’s function (approximately twice that of the (Cut-B, all)
sample, see Fig.2), the resulting MRF value is $2.8\times 10^{2}$, and the
corresponding GRB neutrino flux limit is
$E_{\nu}^{2}\,\Phi_{\nu}\leq 1.1\times 10^{-6}GeVcm^{-2}s^{-1}sr^{-1}$ (5)
This diffuse limit is considerably weaker that of the AMANDA muon analysis
[12], and two times higher that their cascade analysis [13]. In view of a
search for bright individual GRBs, our result may be considered as
complementary to AMANDA since variations in absolute energy output, Lorentz
factor and distance might lead to a GRB neutrino detection with a less
sensitive detector, while that source was outside the other detector field of
view. NT200+ is presently complementing the ANTARES detector, in particular
for those cases where the GRB is above horizon with respect to this
instrument. We also not that, normalized to a single GRB, NT200 exceeds the
sensitivity of Super-Kamiokande by a factor of 2 for neutrino energy above 1
TeV.
## 5 Conclusion
We have presented results of a search for neutrino induced muons detected with
Baikal Telescope NT200 in coincidence with 303 gamma-ray bursts recorded from
1998 to 2000 by BATSE. NT200’s field of view covers most part of the Southern
hemisphere. No evidence for neutrino-induced muons from gamma-ray bursts is
found. The resulting Green’s Function fluence limit for this search extends
that of Super-Kamiokande by two orders of magnitude in energy. Assuming a
Waxman-Bahcall spectrum, a neutrino flux upper limit of
$E_{\nu}^{2}\,\Phi_{\nu}\leq 1.1\times 10^{-6}GeVcm^{-2}s^{-1}sr^{-1}$ is
obtained.
## 6 Acknowledgments
This work was supported in part by the Russian Ministry of Education and
Science, by the German Ministry of Education and Research, by the Russian
Found for Basic Research (grants 08-02-00432-a, 07-02-00791, 08-02-00198,
09-02-10001-k, 09-02-00623-a), by the grant of the President of Russia
NSh-321.2008-2 and by the program ”Development of Scientific Potential in
Higher Schools” (projects 2.2.1.1/1483, 2.1.1/1539, 2.2.1.1/5901).
## References
* [1] V. Aynutdinov et al., Izvestia Akademii Nauk (Izvestia Russ. Academy Science), Ser. Phys., vol. 71, N. 4, 2007;
V. Aynutdinov et al., Nucl. Instrum. Methods, vol A602, p. 14, 2009.;
* [2] V. Avrorin et al., these proceedings, paper-109.
* [3] V. Aynutdinov et al., Astropart. Phys., vol. 25, p. 140, 2006.
* [4] V. Aynutdinov et al., Astropart. Phys. vol. 29, p. 366, 2008.
* [5] W.S. Paciesas at al., Astrophys.J.Suppl. vol. 122, p. 465, 1999: arXiv:astro-ph/9903205.
* [6] B. Stern and Ya. Tikhomirova,
$www.astro.su.se/groups/head/grb\\_archive.html$, 2002.
* [7] M.S. Briggs at al., arXiv:astro-ph/9901111, 1999.
* [8] I. Belolaptikov, preprint INR RAS - 1178, March, 2007.
* [9] E. Bugaev, S. Klimushin, I. Sokalsky, Phys. Rev. D, vol. 64, 2001.
* [10] G. Feldman and R. Cousins, Phys. Rev. D, vol. 57, p. 3873, 1998.
* [11] S. Fukuda et al., Astrophys. J., vol. 578, p.317, 2002: arXiv:astro-ph/0205304.
* [12] A. Achterberg et al., Astrophys. J., vol. 674, p.357, 2008: arXiv:astro-ph/0705.1186.
* [13] A. Achterberg et al., Astrophys. J., vol. 664, p. 397, 2007: arXiv:astro-ph/0702265.
* [14] E. Waxman and J. Bahcall, Phys. Rev. Lett., vol. 78, p. 2292, 1997: arXiv:astro-ph/9701231.
* [15] E. Waxman, Astroparticle physics and cosmology: arXiv:astro-ph/0103186, 2001.
* [16] E.Waxman. Phil. Trans. Roy. Soc. Lond., vol. A365, p. 1323, 2007: arXiv:astro-ph/0701170.
|
arxiv-papers
| 2009-10-22T13:44:53 |
2024-09-04T02:49:05.978133
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Avrorin (for the Baikal Collaboration)",
"submitter": "Dzhilkibaev Zhan-Arys",
"url": "https://arxiv.org/abs/0910.4327"
}
|
0910.4334
|
# Estimates for solutions of KDV on the phase space of periodic distributions
in terms of action variables
Evgeny L. Korotyaev
###### Abstract.
We consider the KdV equation on the Sobolev space of periodic distributions.
We obtain estimates of the solution of the KdV in terms of action variables.
###### Key words and phrases:
periodic KDV, action variables, estimates
###### 1991 Mathematics Subject Classification:
37K05, (35Q53, 37K10)
## 1\. Introduction and main results
Consider the KdV equation
$\partial_{t}\psi=-\psi_{xxx}^{\prime\prime\prime}+6\psi\psi_{x}^{\prime}$
(1.1)
on the Sobolev space of zero-meanvalue 1-periodic distributions
$H_{-1}=\\{\psi=q^{\prime}:q\in H\\}$, where the real Hilbert space $H=H_{0}$
consists of zero-meanvalue functions $q\in L^{2}({\mathbb{T}})$,
${\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$. The space $H_{-1}$ is equipped with
the norm $\|\psi\|_{{-1}}^{2}=\|q\|^{2}=\int_{0}^{1}q^{2}(x)dx$ for
$\psi=q^{\prime}\in H_{-1}$. The initial value problem for KdV in the phase
space of periodic distributions was solved by Kappeler and Topalov [KT], see
also [B], [CT]. That problem in various Sobolev spaces was studied by many
authors, see references in [B], [CT], [KT]. The action-angle variables for the
periodic KdV are studied by Veselov-Novikov [VN], Kuksin [Ku], Kappeler-
Pöschel [KP]. The action-angle variables for the case $\psi\in H_{-1}$ were
constructed by Kappeler-Möhr-Topalov [KMT] and were essentially used in [KT].
In Sobolev spaces estimates for the potential $\psi$ and for the
KdV–Hamiltonian in terms of the action variables were obtained by Korotyaev
[K2].
We describe the motivation of the present paper. Introduce the real Hilbert
spaces $\ell^{2}_{m},m\in{\mathbb{R}}$, of sequences $(f_{n})_{1}^{\infty}$,
equipped with the norm $\|f\|_{m}^{2}=\sum_{n\geqslant 1}(2\pi
n)^{2m}f_{n}^{2}$. Recall that the KDV equation on $H_{-1}$ admits action-
angle variables $A_{n}\geqslant 0,\phi_{n}\in[0,2\pi),n\geqslant 1$ such that
(see [KT]):
(1) for each $q\in H_{-1}$ there exist actions $A_{n}\geqslant 0$ such that
$\sum_{n\geqslant 1}{A_{n}\over n}<\infty$ and angles
$\phi_{n}\in[0,2\pi),n\geqslant 1$.
(2) The mapping $\Psi:H_{-1}\to\ell_{-{1\over 2}}^{2}\oplus\ell_{-{1\over
2}}^{2}$ given by $q\to\Psi(q)=(|A_{n}|^{1\over 2}\cos\phi_{n},|A_{n}|^{1\over
2}\sin\phi_{n})_{1}^{\infty}$ is a real analytic isomorphism between $H_{-1}$
and $\ell_{-{1\over 2}}^{2}\oplus\ell_{-{1\over 2}}^{2}$.
(3) This mapping is symplectic.
For the case of periodic distributions no estimates for the potential $q$ in
terms of action variables $A_{n}$ were known. Our main goal in this paper is
to obtain them.
We recall some results on the action variables for the KdV. If $\psi\in
H_{0}$, then the following identity hold true (see [MT], [K4]):
$\|\psi\|^{2}=4\sum_{n\geqslant 1}(\pi n)A_{n}.$ (1.2)
Moreover, if $\psi^{\prime}\in H$, then the Hamiltonian
${\mathscr{H}}(\psi)={1\over 2}\int_{0}^{1}({\psi^{\prime}}^{2}+2\psi^{3})dx$
obey the following estimates:
$8P_{3}-8P_{1}P_{-1}\leqslant{\mathscr{H}}(\psi)\leqslant 8P_{3},\qquad$ (1.3)
where $P_{j}=\sum_{n\geqslant 1}(\pi n)^{j}A_{n},j\in{\mathbb{R}}$, see [K6]
and see (2.8) for definition of the actions $A_{n}$.
We formulate our main result.
###### Theorem 1.1.
Let $\psi\in H_{-1}$ and let $P_{-1}=\sum_{n\geqslant 1}{A_{n}\over\pi n}$.
Then the following estimates hold true:
$\|\psi\|_{-1}^{2}\leqslant 3P_{-1}(1+P_{-1}),$ (1.4)
$P_{-1}\leqslant\|\psi\|_{-1}^{2}(1+\|\psi\|_{-1})^{3\over 2}.$ (1.5)
Conjecture. The estimate (1.4) is sharp. It means that an estimate
$\|\psi\|_{-1}^{2}\leqslant CP_{-1}(1+P_{-1})^{\beta}$ with some $C>0$ and
$\beta<1$ is not correct.
We formulate a simple corollary, which follows directly from estimate (1.4),
(1.5).
###### Corollary 1.2.
Let $\psi(x,t)$ be a solution of (1.1) such that $\psi(\cdot,0)\in H_{-1}$.
Then for all time $t$ the following estimates hold true:
$\|\psi(\cdot,t)\|_{-1}\leqslant
3\|\psi(\cdot,0)\|_{-1}(1+\|\psi(\cdot,0)\|_{-1})^{5\over 2},$ (1.6)
$\|\psi(\cdot,0)\|_{-1}\leqslant
14\max\\{\|\psi(\cdot,t)\|_{-1},\|\psi(\cdot,t)\|_{-1}^{5\over 2}\\}.$ (1.7)
Example. We now discuss relation of estimates (1.6) with the inverse cascade
of energy in the KdV equation. Let an initial condition $\psi(\cdot,0)\in
H_{0}$ satisfies
$\|\psi(\cdot,0)\|_{-1}=\varepsilon\in[0,1/4],\qquad\qquad\|\psi(\cdot,0)\|=C=\mathop{\mathrm{const}}\nolimits.$
(1.8)
Then for any $N\geqslant 1$ and every $t$ estimate (1.6) yields
$\|\psi(\cdot,t)\|_{-1}\leqslant
6\varepsilon,\qquad\qquad\|{\mathcal{P}}_{N}\psi(\cdot,t)\|\leqslant 6(2\pi
N)\varepsilon,$ (1.9)
where ${\mathcal{P}}_{N}f,f\in H_{0}$ is given by
${\mathcal{P}}_{N}f=\sum_{|n|\leqslant N}e^{i2\pi nx}\int_{0}^{1}f(s)e^{-i2\pi
ns}ds.$ Let in addition, $\delta=6(2\pi N)\varepsilon$ be small enough. Then
(1.9) gives
$\|(I-{\mathcal{P}}_{N})\psi(\cdot,t)\|^{2}\geqslant
C^{2}-\delta^{2},\qquad\|{\mathcal{P}}_{N}\psi(\cdot,t)\|\leqslant\delta,\quad
any\quad t\geqslant 0.$ (1.10)
Thus we deduce that in our case the inverse cascade of energy is impossible.
It means that if the initial condition is such that $\|\psi(\cdot,0)\|=1$ and
$\|{\mathcal{P}}_{N_{0}}\psi(\cdot,0)\|=0$ for some $N_{0}\gg N$, then
$\|{\mathcal{P}}_{N}\psi(\cdot,t)\|\leqslant{6N\over N_{0}}$ will be small for
all time $t$, since $\|\psi(\cdot,0)\|_{-1}\leqslant{1\over 2\pi N}$. That is,
if the energy of a solution was initially concentrated in high modes, then a
substantial part of the energy cannot flow to low modes.
Note that the function $\psi(x,0)$ with the property (1.8) may be a finite
trigonometric polynomial.
## 2\. Proof of the main theorem
Our main ingredients to study the KdV equation (similar to [KT]) are the
spectral properties of the Schrödinger operator $T=-{d^{2}\over
dx^{2}}+\psi+q_{0}$, where $\psi\in H_{-1}$ is a 1-periodic distribution with
zero mean-value and $q_{0}\in{\mathbb{R}}$ is a constant. It is well known
[K3] that the spectrum of $T$ is absolutely continuous and consists of
intervals $\mathfrak{S}_{n}=[\lambda^{+}_{n-1},\lambda^{-}_{n}]$, where
$\lambda^{+}_{n-1}<\lambda^{-}_{n}\leqslant\lambda^{+}_{n},\ n\geqslant 1$. We
take a constant $q_{0}$ such that $\lambda_{0}^{+}=0$. The intervals
$\mathfrak{S}_{n}$ and $\mathfrak{S}_{n+1}$ are separated by the gap
$\gamma_{n}=(\lambda^{-}_{n},\lambda^{+}_{n})$. If a gap degenerates, that is
$\gamma_{n}=\emptyset$, then the corresponding segments $\mathfrak{S}_{n}$ and
$\mathfrak{S}_{n+1}$ merge. The sequence
$\lambda_{0}^{+}<\lambda_{1}^{-}\leqslant\lambda_{1}^{+}\ <\dots$ is the
spectrum of the equation $-y^{\prime\prime}+(\psi+q_{0})y=\lambda y$ with the
2-periodic boundary conditions, i.e., $y(x+2)=y(x),x\in{\mathbb{R}}$. If
$\lambda_{n}^{-}=\lambda_{n}^{+}$ for some $n$, then this number
$\lambda_{n}^{\pm}$ is the double eigenvalue of this equation with the
2-periodic boundary conditions. The lowest eigenvalue $\lambda_{0}^{+}$ is
always simple and the corresponding eigenfunction is 1-periodic. The
eigenfunctions, corresponding to the eigenvalue $\lambda_{n}^{\pm}$, are
1-periodic, when $n$ is even and are antiperiodic, i.e., $y(x+1)=-y(x),\ \
x\in{\mathbb{R}}$, when $n$ is odd.
We can not introduce the standard fundamental solutions for the operator $T$
since the perturbation $\psi\in H_{-1}$ is very strong. But we can do this
using another representation of $T$ given by
$T={\mathscr{U}}T_{w}{\mathscr{U}}^{-1}$. Here $T_{w}$ is the self-adjoint
periodic operator acting in $L^{2}({\mathbb{R}},w^{2}(x)dx)$ and given by
$\qquad\qquad T_{w}f=-{1\over
w^{2}}(w^{2}f^{\prime})^{\prime}=-f^{\prime\prime}-2pf^{\prime},\
w(x)=e^{\int_{0}^{x}p(s)ds},\ \ p\in H.$ (2.1)
${\mathscr{U}}$ is the unitary transformation
${\mathscr{U}}:L^{2}({\mathbb{R}},w^{2}dx)\to L^{2}({\mathbb{R}},dx)$, given
by the multiplication by $w$. Note that
$T=-{d^{2}\over dx^{2}}+q^{\prime}+q_{0}\geqslant 0,\qquad
q^{\prime}=p^{\prime}(x)+p^{2}(x)-\|p\|^{2},\qquad
q_{0}=\|p\|^{2}=\int_{0}^{1}p^{2}(x)dx,$ (2.2)
where $\psi=q^{\prime}$ is a 1-periodic potential (distribution). Thus, if
$p^{\prime}\in H$, then $T_{w}$ corresponds to the Hill operator $T$ with
$L^{2}$-potential. The operator $T_{w}$ is well studied, see [K5] and
references therein. In fact the direct spectral problem for $T_{w}$ is
equivalent to that for $T$ [K3].
The operator $T_{w}$ has the standard fundamental solutions
$\varphi(x,\lambda),\vartheta(x,\lambda)$, which satisfy the equation
$-y^{\prime\prime}-2py^{\prime}=\lambda y,\ \lambda\in{\mathbb{C}}$ and the
conditions $\varphi^{\prime}(0,\lambda)=\vartheta(0,\lambda)=1,\
\varphi(0,\lambda)=\vartheta^{\prime}(0,\lambda)=0$. Here and below we use the
notation $f^{\prime}={\partial\over\partial x}f$. Introduce the Lyapunov
function $\Delta(\lambda)={1\over
2}(\varphi^{\prime}(1,\lambda)+\vartheta(1,\lambda))$. Note that
$\Delta(\lambda_{n}^{\pm})=(-1)^{n},\ n\geqslant 1,$ and that for each
$n\geqslant 1$ there exists a unique point
$\lambda_{n}\in[\lambda^{-}_{n},\lambda^{+}_{n}]$ such that
$\Delta^{\prime}(\lambda_{n})=0$.
Now we recall results, crucial for the present paper. For each $\psi\in
H_{-1}$ there exists a unique conformal mapping (the quasimomentum)
$k:{\mathcal{Z}}\to{\mathcal{K}}(h)$ with asymptotics $k(z)=z+o(1)$ as
$|z|\to\infty$ (see Fig. 1 and 2) and such that (see [K3])
$\cos k(z)=\Delta(z),\ \
z\in{\mathcal{Z}}={\mathbb{C}}\setminus\cup\overline{g}_{n},\quad
g_{n}=(z^{-}_{n},z^{+}_{n})=-g_{-n},\quad
z_{n}^{\pm}=\sqrt{\lambda_{n}^{\pm}}\geqslant 0,\quad n\geqslant 1,\\\
{\mathcal{K}}(h)={\mathbb{C}}\setminus\cup\Gamma_{n},\ \ \ \Gamma_{n}=(\pi
n-ih_{n},\pi n+ih_{n}),\qquad h_{0}=0,\qquad h_{n}=h_{-n}\geqslant 0,\qquad\\\
h_{n}\geqslant 0\qquad{\rm is\ defined\ by\ the\ equation}\qquad\cosh
h_{n}=(-1)^{n}\Delta(\lambda_{n})\geqslant 1.$ (2.3)
Here $g_{0}=\emptyset$ and $\Gamma_{n}$ is the vertical cut,
$z_{n}=\sqrt{\lambda_{n}}\in[z_{n}^{-},z_{n}^{+}],n\geqslant 1$,
$\Delta^{\prime}(z_{n}^{2})=0$. Moreover, we have
$(h_{n})_{1}^{\infty}\in\ell^{2}$ iff $\psi\in H_{-1}$ (and
$(nh_{n})_{1}^{\infty}\in\ell^{2}$ iff $\psi\in H$), see [K3], [K1].
Due to [MO1], the quantities $v=\mathop{\mathrm{Im}}\nolimits k(z)$ and
$u=\mathop{\mathrm{Re}}\nolimits k(z),z\in{\mathcal{Z}}$, possess the
following properties:
1) $v(z)\geqslant\mathop{\mathrm{Im}}\nolimits z>0$ and
$v(z)=-v(\overline{z})$ for all
$z\in{\mathbb{C}}_{+}=\\{\mathop{\mathrm{Im}}\nolimits z>0\\}$.
2) $v(z)=0$ for all
$z\in\sigma_{n}=[z^{+}_{n-1},z^{-}_{n}]=-\sigma_{-n},n\geqslant 1$.
3) If some $g_{n}\neq\emptyset,n\in{\mathbb{Z}}$, then the function
$v(z+i0)>0$ for all $z\in g_{n}$, and $v(z+i0)$ has a maximum at $z_{n}\in
g_{n}$ such that $\Delta^{\prime}(z_{n}^{2})=0$ and
$v(z_{n}+i0)=h_{n}>0,v^{\prime}(z_{n})=0$, and
$v(z+i0)=-v(z-i0)>v_{n}(z)=|(z-z_{n}^{-})(z-z_{n}^{+})|^{1\over 2}>0,\qquad
v^{\prime\prime}(z+i0)<0,$ (2.4)
for all $z\in g_{n}\neq\emptyset$, see Fig. 3.
4) $u^{\prime}(z)>0$ on ${\mathbb{R}}\setminus\cup\overline{g}_{n}$ and
$u(z)=\pi n$ for all $z\in g_{n}\neq\emptyset,n\in{\mathbb{Z}}$.
5) The function $k(z)$ maps a horizontal cut (a ”gap” ) $\overline{g}_{n}$
onto a vertical cut $\Gamma_{n}$ and a spectral band $\sigma_{n}$ onto the
segment $[\pi(n-1),\pi n]$ for all $\pm n\in{\mathbb{N}}$.
The heights $h_{n},\ n\geqslant 1$ are so-called Marchenko-Ostrovski
parameters [MO1]. In spirit, such result goes back to the classical Hilbert
Theorem (for a finite number of cuts, see e.g. [J]) in the conformal mapping
theory. A similar theorem for the Hill operator is technically more
complicated (there is a infinite number of cuts) and was proved by Marchenko-
Ostrovski [MO1] for the case $\psi\in H$. For additional properties of the
conformal mapping we also refer to our previous papers [K1]-[K6]. Note that
the inverse problems for the operator $H$ with $\psi\in H_{-1}$ in terms of
the Marchenko-Ostrovski parameters $h_{n},n\geqslant 1$ and gap-lengths were
solved by Korootyev in [K3].
For the sake of the reader, we briefly recall the results existing in the
literature about estimates. In the case $h=(h_{n})_{1}^{\infty}$ and $\psi\in
H$ Marchenko and Ostrovki [MO1-2] obtained the estimates: $\|\psi\|\leqslant
C(1+\sup_{n\geqslant 1}h_{n})\|h\|_{1}$ and $\|h\|_{1}\leqslant
C\|\psi\|\exp(\ C_{1}\|\psi\|\ )$ for some absolute constants $C,C_{1}$. These
estimates are not sharp since they used the Bernstein inequality. Using the
harmonic measure argument Garnett and Trubowitz [GT] obtained
$\|\gamma\|\leqslant(4+\|h\|_{1})\|h\|_{1}$ for the case $\psi\in H$ and
$\gamma=(|\gamma_{n}|)_{1}^{\infty}$, where $|\gamma_{n}|$ is a gap length.
Using the conformal mapping theory, Korotyaev [K1]-[K6] obtained estimates of
potentials (and the Hamiltonian of the KDV) in terms of gap lengths, actions
variables, effective masses, the heights $h=(h_{n})_{1}^{\infty}$ for large
class of potentials. In fact in order to get new estimates new results from
the comformal mapping theory were obtained. Note that estimates simplify the
proof for the inverse spectral theory, see [KK], [K3]. We recall only few
results from these estimates:
I). Let $\psi\in H_{-1}$. Then the following estimates hold true (see [K3]):
$\|\gamma\|_{-1}\leqslant\sqrt{2}\|\psi\|_{-1}(1+\|\psi\|_{-1}),\ \ \ \ \
\|\psi\|_{-1}\leqslant 8\pi\|\gamma\|_{-1}(1+\|\gamma\|_{-1}),$ (2.5)
${\sqrt{\pi}\over\sqrt{8}}\|\psi\|_{-1}\leqslant\|h\|_{0}\leqslant{\pi\over
2}\|\psi\|_{-1}(1+\|\psi\|_{-1})^{1\over 2}.$ (2.6)
II) If $\psi\in H$, then the following estimates hold true (see [K1]):
$\|\psi\|\leqslant 2\|\gamma\|_{0}(1+\|\gamma\|_{0}^{1\over
3}),\qquad\|\gamma\|_{0}\leqslant 2\|\psi\|(1+\|\psi\|^{1\over 3}).$
If $\psi\in H_{-1}$, then the quasimomentum $k(\cdot)$ has asymptotics
$k(z)=z-{Q_{0}+o(1)\over z}\qquad as\quad z\to+i\infty,$
where $Q_{0}={1\over\pi}\int_{\mathbb{R}}v(z+i0)dz\geqslant 0$ and $p$
(defined in (2.2)) satisfy the identities from [K5]:
$Q_{0}={1\over\pi}\int_{\mathbb{R}}v(z+i0)dz={1\over
2\pi}\int\\!\\!\int_{\mathbb{C}}|z^{\prime}(k)-1|^{2}dudv={\|p\|^{2}\over
2},\quad k=u+iv.$ (2.7)
Due to [FM] we define the action $A_{n},n\geqslant 1$ by
$A_{n}={(-1)^{n+1}2\over\pi}\int_{\gamma_{n}}{\lambda\Delta^{\prime}(\lambda)d\lambda\over|\Delta^{2}(\lambda)-1|^{1\over
2}}\geqslant 0.$ (2.8)
We rewrite $A_{n}$ in the more convenient form. The differentiation of
$\Delta(z^{2})=\cos k(z)$ gives
$k^{\prime}(z)=-{\Delta^{\prime}(z^{2})2z\over\sin k(z)}$, which together with
$\sin k(z)=\sqrt{1-\Delta^{2}(z^{2})}$ yield
$A_{n}=-{1\over i\pi}\int_{c_{n}}z^{2}{\Delta^{\prime}(z^{2})2z\over\sin
k(z)}dz={1\over i\pi}\int_{c_{n}}z^{2}k^{\prime}(z)dz=-{2\over
i\pi}\int_{c_{n}}zk(z)dz={4\over\pi}\int_{g_{n}}zv(z+i0)dz\geqslant 0,$
which gives
$A_{n}={4\over\pi}\int_{g_{n}}zv(z+i0)dz\geqslant 0.$ (2.9)
$0$$\mathop{\mathrm{Re}}\nolimits z$$\mathop{\mathrm{Im}}\nolimits
z$$z_{1}^{-}$$z_{1}^{+}$$z_{2}^{-}$$z_{2}^{+}$$z_{3}^{-}$$z_{3}^{+}$$-z_{1}^{-}$$-z_{1}^{+}$$-z_{2}^{-}$$-z_{2}^{+}$$-z_{3}^{-}$$-z_{3}^{+}$
Figure 1. Domain ${\mathcal{Z}}={\mathbb{C}}\setminus\cup g_{n}$, where
$z=\sqrt{\lambda}$ and momentum gaps $g_{n}=(z_{n}^{-},z_{n}^{+})$
$0$$\mathop{\mathrm{Re}}\nolimits k$$\mathop{\mathrm{Im}}\nolimits
k$$\pi$$-\pi$$2\pi$$-2\pi$$3\pi$$-3\pi$$\pi+ih_{1}$$-\pi+ih_{1}$$2\pi+ih_{2}$$-2\pi+ih_{2}$$3\pi+ih_{3}$$-3\pi+ih_{3}$
Figure 2. $k$-plane and cuts $\Gamma_{n}=(\pi n-ih_{n},\pi
n+ih_{n}),n\in{\mathbb{Z}}$
$z_{n}^{-}$$z_{n}^{+}$$v_{n}$$v$$z_{n-1}^{+}$$z_{n+1}^{-}$$z_{n}$ Figure 3.
The graph of $v(z+i0),\ z\in g_{n}\cup\sigma_{n}\cup\sigma_{n+1}$ and
$|h_{n}|=v(z_{n}+i0)>0$
Below we need results about the Riccati mapping (see Theorem 1.2 in [K3]).
###### Theorem 2.1.
The Riccati map $R:H\to H$ given by $p\to
q=R(p),q^{\prime}=p^{\prime}(x)+p^{2}(x)-\|p\|^{2}$ is a real analytic
isomorphism of $H$ onto itself. Moreover, the following estimates hold true:
$\|q\|\leqslant\|p\|(1+2\|p\|),\ \ \ \ \ $ (2.10)
$\|p\|\leqslant\sqrt{2}\|q\|(1+2\|q\|).\ \ \ \ \ $ (2.11)
In order to show (1.4), we need
###### Lemma 2.2.
The following estimate holds true:
$\|p\|^{2}\leqslant\sum_{n\geqslant 1}{A_{n}\over\pi n}=P_{-1}.$ (2.12)
Proof. Using the following identity for
$Q_{0}={1\over\pi}\int_{\mathbb{R}}v(z+i0)dz$ (see Theorem 2.3 from [K4])
$Q_{0}={2\over\pi}\int_{0}^{\infty}{u(z)v(z)\over z}dz,$ (2.13)
we obtain
$Q_{0}^{2}\leqslant{2\over\pi}\int_{g_{+}}{uvdz\over
z}{2\over\pi}\int_{g_{+}}{zvdz\over u}=Q_{0}{2\over\pi}\int_{g_{+}}{zvdz\over
u},\qquad g_{+}=\cup_{n\geqslant 1}g_{n},$
which together with the identity for $A_{n}$ (2.9) yields
$Q_{0}\leqslant{2\over\pi}\int_{g_{+}}{zvdz\over u}={1\over 2}\sum_{n\geqslant
1}{A_{n}\over\pi n},$
since $u|_{g_{n}}=\pi n$ and the identity (2.7) gives (2.12).
We show the estimate (1.4). Using (2.10), (2.12) we obtain
$\|\psi\|_{-1}^{2}=\|q\|^{2}\leqslant\|p\|^{2}(1+2\|p\|)^{2}\leqslant\|p\|^{2}(1+\|p\|^{2})\leqslant
5P_{-1}(1+P_{-1}),$
which gives (1.4).
We show the estimate (1.5). The estimate $v|_{g_{n}}\leqslant h_{n}$ and the
identity for $A_{n}$ (2.9) gives
$A_{n}={4\over\pi}\int_{g_{n}}zv(z)dz\leqslant{4h_{n}\over\pi}\int_{g_{n}}zdz={4h_{n}\over\pi}|\gamma_{n}|,$
and then
$P_{-1}=\sum_{n\geqslant 1}{A_{n}\over\pi n}\leqslant\sum_{n\geqslant
1}{4h_{n}\over\pi}{|\gamma_{n}|\over\pi
n}\leqslant{4\over\pi}\|h\|_{0}\|\gamma\|_{-1}.$ (2.14)
Substituting estimates (2.5), (2.6) into (2.14) we obtain (1.5).
Acknowledgments. The author is grateful to Sergei Kuksin (Ecole Polytechnique,
Paris) for stimulating discussions and useful comments.
## References
* [B] Bourgain, J. Periodic Korteweg - de Vries equation with measures as initial data, Selecta Math., 3(1997), 115-159.
* [CT] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 16(2003), 705-749.
* [FM] Flaschka H., McLaughlin D. Canonically conjugate variables for the Korteveg- de Vries equation and the Toda lattice with periodic boundary conditions. Prog. of Theor. Phys. 55(1976), 438-456.
* [GT] Garnett J., Trubowitz E.: Gaps and bands of one dimensional periodic Schrödinger operators. Comment. Math. Helv. 59(1984), 258-312
* [J] Jenkins A.: Univalent functions and conformal mapping. Berlin, Göttingen, Heidelberg: Springer, 1958.
* [KT] Kappeler, T.; Topalov, P. Global wellposedness of KdV in $H^{-1}({\mathbb{T}},{\mathbb{R}})$. Duke Math. J. 135 (2006), no. 2, 327–360.
* [KP] T.Kappeler; J. Pöschel, Kdv $\&$ Kam. Springer, 2003.
* [KMT] T. Kappeler, C. Möhr, P. Topalov, Birkhoff coordinates for KdV on phase spaces of distributions, Selecta Math. (N.S.), 11(2005), 37–98.
* [KK] Kargaev P., Korotyaev E. The inverse problem for the Hill operator, a direct method. Invent. Math. 129(1997), 567-593.
* [K1] Korotyaev E. Estimates for the Hill operator. I, Journal Diff. Eq. 162(2000), 1–26.
* [K2] Korotyaev E. Estimate for the Hill operator.II, Journal Diff. Eq. 223 (2006), no. 2, 229–260.
* [K3] Korotyaev E. Characterization of the spectrum of Schrödinger operators with periodic distributions. Int. Math. Res. Not. (2003) no. 37, 2019–2031.
* [K4] Korotyaev E. The estimates of periodic potentials in terms of effective masses. Commun. Math. Phys. 183(1997), 383–400.
* [K5] Korotyaev, E. Periodic ”weighted” operators. J. Differential Equations 189 (2003), no. 2, 461–486.
* [K6] Korotyaev, E. A priori estimates of KdV Hamiltomian in terms of actions, preprint, 2009.
* [Ku] Kuksin, S. Analysis of Hamiltonian PDEs. Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford, 2000.
* [MT] McKean H., Trubowitz E. Hill’s surfaces and their theta functions, Bull. Am. Math. Soc. 84, 1978, 1042-1085.
* [MO1] Marchenko V.; Ostrovski I. A characterization of the spectrum of the Hill operator. Math. USSR Sb. 26(1975), 493-554 .
* [MO2] Marchenko V.; Ostrovski I. Approximation of periodic by finite-zone potentials. Selecta Math. Sovietica. 6(1987), no 2, 101-136.
* [VN] Veselov, A.; Novikov, S. Poisson brackets and complex tori. Proc. Steklov Inst. Math. 165(1985), 53 -65.
|
arxiv-papers
| 2009-10-22T14:19:33 |
2024-09-04T02:49:05.982242
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Evgeny L. Korotyaev",
"submitter": "Evgeny Korotyaev",
"url": "https://arxiv.org/abs/0910.4334"
}
|
0910.4384
|
# Superconducting RF cavity R&D for future accelerators
C. M. Ginsburg Fermilab, Batavia, IL 60510 USA
###### Abstract
High-beta superconducting radiofrequency (SRF) elliptical cavities are being
developed for several accelerator projects including Project X, the European
XFEL, and the International Linear Collider (ILC). Fermilab has recently
established an extensive infrastructure for SRF cavity R&D for future
accelerators, including cavity surface processing and testing and cavity
assembly into cryomodules. Some highlights of the global effort in SRF R&D
toward improving cavity performance, and Fermilab SRF cavity R&D in the
context of global projects are reviewed.
## I Introduction
Substantial effort is being expended in the quest for high gradients in
superconducting radiofrequency (SRF) cavities for ongoing and proposed
projects: (1) the test/user facilities STF (KEK), NML (Fermilab), and FLASH
(DESY), (2) the European XFEL currently under construction, (3) Project X at
Fermilab, and (4) the International Linear Collider (ILC). The common
requirements or choices for the high-gradient cavities in these projects are
gradients of at least 23 MV/m, $\beta$=1, elliptical shape and an accelerating
mode frequency of 1.3 GHz. Recent SRF R&D highlights in the pursuit of high
cavity gradients and the status of Fermilab SRF infrastructure development are
reviewed.
## II Achieving High Gradient
Achieving high gradient in SRF cavities for current projects requires pure
niobium, an excellent surface quality, and a good geometry.
For the ILC ilc_rdr , the niobium sheets from which cavities are formed must
have a residual resistivity ratio (RRR) of at least 300, consistent with very
high purity niobium, to ensure good superconducting properties.
Because RF fields occupy the first $\sim$40 nm of the inner cavity surface for
these cavities, as shown in Fig. 1, the quality of the innermost surface is
critical and must be carefully controlled both during cavity fabrication and
test preparation. To avoid the inclusion of contaminants or defects apparent
after the sheet manufacturing, eddy current scanning (ECS) of all sheets is
performed before cavity fabrication. ECS has proven to be a very useful
feedback mechanism for material vendors, constributing substantially to an
overall material quality improvement in recent years. After fabrication, the
cavity inner surface must be very smooth, with no inclusion of foreign
particles, or topological defects such as bumps or pits or sharp niobium grain
boundaries, which could increase surface resistance locally. In addition to
the strict quality assurance for the material, the introduction of dust or
other microscopic contaminants must be avoided after the final surface
preparation, to prevent field emission which can become dark current during
machine operation and cause undesirable heating in superconducting elements
and damage to accelerator components.
Figure 1: Simplified drawing of an SRF cavity surface. [Courtesy of Hui Tian.]
In selecting a cavity geometry for optimum performance, cavity cell shapes are
typically optimized for low peak surface magnetic field (Hpeak) and low peak
surface electric field (Epeak) relative to the gradient (Eacc), as well as
ease of surface processing and fabrication.
Many cavities have reached 35 MV/m or more in the last decade SRF2007_Saito ;
MAC2007_Lilje , particularly single-cell cavities of varying elliptical shapes
and 9-cell Tesla-shape cavities, as shown in Fig. 2.
Figure 2: Maximum cavity gradient as a function of time for 1-cell cavities of
non-standard shapes (top, courtesy of Kenji Saito) and of 9-cell cavities of
Tesla shape (bottom, courtesy of Lutz Lilje). All tests were performed at 2K.
(Data are from KEK, Cornell, JLab, and DESY.)
## III Surface Processing
The surface treatment intended to maximize cavity performance, developed for
the ILC TTC_surfaceprep , includes initial preparation steps to remove
$\sim$$150\ \mu$m from the inner surface using electropolishing (EP). This
initial removal step may be performed with centrifugal barrel polishing (CBP)
or buffered chemical polishing (BCP) at some labs; however, the maximum
gradients reached with BCP as the initial preparation step are lower than
those achieved using the other methods. Cavities then undergo an 800 C
annealing step, to drive hydrogen from the surface. The final preparation
steps include degreasing with detergent, another light electropolishing
($\sim$$20\ \mu$m), a high pressure rinse (HPR) with ultrapure water, drying
in a class-10 cleanroom, and then evacuation and low-temperature baking (120
C) for about 48 hours after the final assembly with couplers. Additional
surface treatments to address field emission, which occur after the final EP,
will be described later. The primary methods for material removal during
surface preparation are CBP, EP and BCP.
CBP is a standard technique developed for cavities at KEK in which abrasive
small stones are placed into a cavity with water to form a slurry and the
cavity is rotated. The KEK CBP machine and a diagram showing the CBP process
are shown in Fig. 3. As a centrifugal process, material is preferentially
removed from the equator region. Since standard cavities have an equator weld,
CBP is very effective in smoothing the weld.
Figure 3: The KEK CBP machine (top) and a diagram showing the CBP process
(bottom). [Courtesy of Kenji Saito.]
EP is an electrolytic current-supported material removal, and has been
developed for use on cavities by KEK in collaboration with industrial
partners, and adopted at most labs. The EP process is complementary to CBP
because the material removal is preferentially on the iris. The ANL EP machine
and a diagram showing the EP process are shown in Fig. 4. In this case, the
niobium cavity functions as an anode, and an aluminum cathode is inserted on
the cavity axis. The electrolyte is HF(40%):H2SO4 in a ratio of 1:9 by volume.
Some sulfur remains on the surface after EP and will cause field emission
unless it is removed.
Figure 4: The ANL EP machine (top, courtesy of Michael Kelly), and a diagram
showing the EP process (bottom, courtesy of Lutz Lilje).
BCP involves filling a cavity with a combination of
HF(40%):HNO3(65%):H3PO4(85%) in a 1:1:2 ratio by volume. BCP has been shown to
cause hydrogen contamination at the surface; this problem can be mitigated by
using the appropriate proportion of acid and buffer and by keeping the
temperature below about 15 C. BCP is rather less expensive than EP and is
often sufficient to produce cavities attaining gradients as high as about 25
MV/m. However, it tends to enhance grain boundaries which may degrade the
performance of standard fine-grain cavities.
An improvement to the average maximum gradient was seen after the previous BCP
standard was replaced with EP, as shown in Fig. 2. The comparison of best test
results of DESY cavities treated with BCP and EP is shown in Fig. 5, revealing
that the maximum gradient achieved with BCP is $\sim$25-30 MV/m, whereas the
maximum gradient achieved with EP is rather higher at $\sim$35-40 MV/m. A
comparison of the resulting surface smoothness is shown in Fig. 6. Although it
is possible now to achieve gradients in 9-cell cavities higher than 35 MV/m,
there are several factors which limit cavity performance.
Figure 5: Comparison of cavities processed with BCP (top) and EP (bottom). The
EP’d cavities reach noticeably higher gradients. [Courtesy of DESY.] Figure 6:
Comparison of a sample surface after BCP (left) or EP (right). The EP’d
surface is noticeably smoother than the BCP’d surface. [Courtesy of DESY.]
## IV Reducing Field Emission
One of the factors which limits cavity performance is the presence of field
emission. Four recent field emission studies have been performed: flash EP
(also known as fresh EP), dry ice cleaning, degreasing, and ethanol rinsing.
In tests at KEK using six single-cell Ichiro-shape cavities, a 3 $\mu$m EP
using fresh acid after the final EP was studied flashEP . A gradient
improvement to both average and rms was observed. Furthermore, the treatment
was found to increase the gradient at which field emission turns on. Dry ice
cleaning is a less developed but promising alternative being developed at DESY
dry_ice . It reduces or eliminates contaminating particles in several ways:
they become brittle upon rapid cooling, they encounter pressure and shearing
forces as CO2 crystals hit the surface, and they are rinsed due to the 500
times increased volume after sublimation. Also, LCO2 is a good solvent and
detergent for hydrocarbons and silicones, etc. Dry ice cleaning is a dry
process which leaves no residues, because the loosened contaminants are blown
out the ends of the cavities by the positive pressure, and can be performed in
a horizontal orientation. The DESY dry ice nozzle is shown in Fig. 7. Dry ice
cavity cleaning might be possible after coupler installation, a procedure
which risks introducing field emission. Improved field emission
characteristics have been seen in single-cell cavity tests, and an extension
of the system to 9-cell cavities is planned.
Figure 7: The DESY dry ice nozzle. [Courtesy of DESY.]
One of several degreasing R&D studies is that of a KEK 9-cell Ichiro-shape
cavity which was processed and tested at KEK and JLab geng_detergent . In an
initial test at JLab, the cavity produced substantial field emission which was
observed at gradients above 15 MV/m. After the cavity was ultrasonically
cleaned with a 2% Micro-90 solution and standard HPR, the field emission was
substantially reduced or even eliminated.
An extensive study of ethanol rinsing after the final EP was carried out at
DESY DESY_ethanol . Out of 33 cavities used in the study, 20 were treated
using the standard procedure and 13 with an additional ethanol rinse after the
final EP. The number of tests in which field emission was observed was
substantially reduced. In addition, the maximum gradient was somewhat
improved, from $27\pm 4$ MV/m without ethanol to $31\pm 5$ MV/m with ethanol
rinse. Since field emission was found to be reduced so substantially, the
ethanol rinse is now part of the standard DESY cavity treatment.
## V Manufacturing and Shape Studies
The vast majority of cavities which have been processed and tested in the last
decade are Tesla-shape fine-grain cavities TESLA_cavities . Some fundamental
changes to the standard Tesla-shape fine-grain cavities which have been under
investigation include changes to the fabrication techniques, material
composition and cavity shape.
Because electron-beam welding is a substantial portion of the cavity
fabrication cost, and may be a source of surface defects which cause premature
quenches (described later), reducing the number of equator welds required in
cavity fabrication would be advantageous. Recently, a 9-cell cavity was
fabricated at DESY using a hydroforming technique Singer_hydroforming . The
cavity was built from three 3-cell hydroformed units, so it had only two iris
welds and two beampipe welds. The surface was treated with the standard
procedure including ethanol rinse. The cavity performance was comparable to
cavities built using standard techniques.
Material is wasted, and a lot of time is required, to roll sheets and stamp
out the disks from which fine-grain cavities are made. In principle, it should
be possible to save manufacturing costs by slicing an ingot directly into
large-grain sheets. It may also be possible to achieve high gradients using
BCP only, since the performance of large-grain cavities should be less
sensitive to the grain boundary enhancement seen in fine-grain cavities, as
long as the grain boundaries are strategically located in low surface-field
regions. Recent experience at DESY with large-grain cavities shows their
performance is comparable to fine-grain cavities. It is still unclear whether
BCP will be sufficient or whether EP will be necessary. Many 1-cell tests have
shown high gradients, in a range comparable to fine-grain cavities. Recent
tests on three 9-cell large-grain cavities Singer_LG showed gradients
comparable to fine-grain cavities as well. Effective large-grain ingot cutting
methods are being pursued in industry. Single-crystal niobium cavities have
been difficult to produce, because it is difficult to produce large diameter
single-grain ingots. Six single-cell single-grain cavities of varying shape
and fabrication technique were fabricated, processed and tested recently by a
JLab/DESY collaboration Kneisel_singlecrystal , with performance found to be
comparable to that of fine-grain cavities. Unless substantial performance
improvement is seen, the difficulty of producing single-grain cavities may not
justify the effort required. In both the large-grain and single-grain cases,
further study of crystal orientation effects is needed. For a summary of
recent large-grain and single-grain cavity work, see Ref. Kneisel_LG .
It may be possible to increase the RF breakdown magnetic field of
superconducting cavities by creating a multilayer coating of alternating
insulating layers and thin superconducting layers Gurevich . By using
multilayers thinner than the RF penetration depth, the critical magnetic field
can be increased from niobium Hc1 to one similar to high-Tc superconductors,
thereby significantly improving the maximum gradient achievable. Recently, a
cavity was prepared with such a composite surface and tested Proslier . A 10
nm layer of Al2O3 was chemically bonded to the niobium surface of an existing
single-cell cavity using atomic layer deposition. The surface was then covered
by a 3 nm layer of Nb2O5. The cavity showed promising early results, and
further study is underway.
Cell accelerating length and equator diameter are fixed by $\beta$ and
frequency respectively. However, the details of the shape may be optimized for
low field emission (low Epeak/Eacc) and reduced sensitivity to the fundamental
maximum surface magnetic field (low Hpeak/Eacc); see, e.g., Ref. Jacek_shapes
. Excellent results on single-cell elliptical cavities have recently been
obtained. A comparison of three shapes of elliptical single-cell cavities is
shown in Fig. 8. A re-entrant shape cavity built and tested by Cornell
University recently reached 59 MV/m Cornell_record , as shown in Fig. 9,
setting a world record for the type of cavities described in this paper.
Excellent results have also been achieved with an Ichiro shape cavity at KEK,
with a record of 53.5 MV/m SRF2007_Saito . Furthermore, $46.7\pm 1.9$ MV/m was
reached on six single-cell cavities with optimized surface treatment
parameters flashEP . Another low-loss shape single-cell cavity which was
processed and tested by a DESY/KEK collaboration reached 47.3 MV/m
Furuta_EPAC06 . It is rather more difficult to manufacture an excellent multi-
cell cavity than an excellent single-cell cavity, and the very high gradients
seen in single-cell alternative-shape cavities have not been achieved in
9-cell versions yet. One 9-cell Ichiro-shape cavity, without endgroups, which
was processed and tested by a KEK/JLab collaboration reached up to 36 MV/m
Geng_LINAC08 .
Figure 8: Single-cell 1.3 GHz elliptical cavities: Cornell re-entrant (left),
Tesla (center) and Ichiro (right). Figure 9: The record SRF 1-cell cavity
gradient. [Courtesy of Cornell University.]
## VI Understanding Cavity Behavior
Quenches and field emission appear as hot spots on the outer cavity surface.
Temperature mapping (T-map) systems have been used at nearly all of the labs
active in this field for many years to study these phenomena Padamsee . The
existing T-map systems primarily use Allen-Bradley carbon resistors as thermal
sensors, as developed at Cornell Cornell_thermometry . T-map systems commonly
in use for many years at DESY include a fixed type for single-cell cavities
with 768 sensors DESY_fixed_thermometry , and a rotating system for 9-cell
cavities with 128 sensors DESY_rotating_thermometry . A fixed thermometry
system for 9-cell cavities with four sensors around each equator and a few
sensors on the endgroups has been developed at KEK and was used for tests of
STF Tesla-like cavities KEK_diagnostics . Second sound detection has been used
at ANL for quench location on split-ring resonators Shepard_secondsound . A
selection of new hot spot detection systems are described; these systems vary
in coverage, flexibility, and in the number of cavity tests required to
extract useful T-map information.
A system of Cernox temperature sensors has been developed at Fermilab
fast_thermometry and used for quench location. Up to 32 sensors may be
attached as needed to suspect locations; therefore the system is very flexible
but also time consuming to install, typically requiring several cavity test
cycles to conclusively locate quenches. It is highly portable and suitable for
any cavity shape. An example installation of this system on a 9-cell cavity is
shown in Fig. 10, in which a hard quench at 16 MV/m was observed, accompanied
by a temperature rise of about 100 mK, above the 2K operation, over about 2
sec in sensors #3 and #4 before the quench. The quench was seen on all
sensors.
Figure 10: Fermilab Cernox thermometry system installed on a 9-cell cavity in
the region of a pre-determined hot spot.
New multi-cell T-map systems have also been developed. A new 2-cell system at
JLab was recently commissioned JLab_2-cell_thermometry , with 160 Allen-
Bradley sensors installed around each of the two equators. Understanding
cavity behavior requires two cooldowns: one to measure the TM010 passband
modes and determine within two the limiting cells, then a second after the
T-map installation to measure the temperature distribution of the limiting
cell. At LANL, a new 9-cell T-map system has been developed
LANL_9-cell_thermometry which employs 4608 Allen-Bradley sensors and a
multiplexing scheme to map an entire 9-cell cavity in a single test yet limit
the number of cables leaving the cryostat. The preliminary results are very
promising. A 9-cell T-map system at Fermilab using 8640 diodes as a
multiplexed thermometer array is under development diode_thermometry . All of
these multi-cell T-maps are specifically designed for the Tesla cavity shape.
Cornell has developed a new quench location system using second sound sensors
Cornell_secondsound . Second sound is a thermal wave which can propagate only
in superfluid helium. It is generated when a heat pulse is transmitted from a
heat source, such as a quench, through superfluid helium. In this system,
eight sensors detect the arrival of the temperature oscillation, and the
location is determined from the relative timing of the arrival of the
oscillation from different sensors. This simple system is suitable for any
cavity shape or number of cells and can locate quenches in a single cavity
test. The detector is shown in Fig. 11, and an example of the data used to
triangulate the location of the second sound source is shown in Fig. 12.
Figure 11: The Cornell second sound sensor (left, courtesy of Cornell
University) and a diagram showing the function of the sensor (right, courtesy
of Genfa Wu). Figure 12: Cornell data showing the arrival of second sound
waves at three different sensors. [Courtesy of Zachary Conway.]
A resurgence of interest in optical inspection occurred recently when several
cavities with hard quench limitation in the 15-20 MV/m range, were observed to
have surface defects correlated with hot spots, using a new optical inspection
system developed by Kyoto University/KEK with a clever lighting technique and
excellent resolution. This optical system Kyoto_camera consists of an
integrated camera, mirror, and lighting system on a fixed rod; the cavity is
moved longitudinally and axially with respect to the camera system. The
lighting is provided by a series of electroluminescent strips, which provide
lighting with low glare on the highly polished surface. These strips can be
turned off and on, and the resulting shadows studied for an effective 3D image
of the depth or height of the defect. The camera resolution is about 7 $\mu$m.
A photograph of the Kyoto/KEK camera system is shown in Fig. 13. An optical
inspection system employed regularly at JLab Geng_LINAC08 includes a Questar
long-distance microscope and mirror to inspect any cavity inner surface. This
system uses electroluminescent lighting. Additional optical inspection systems
are in use or under development at other labs.
Figure 13: The KEK/Kyoto University cavity inspection camera system. [Courtesy
of Yoshihisa Iwashita.]
Because many of the cavity-limiting surface defects found by thermometry and
optical inspection are located in the heat-affected zone of the electron-beam
welds, weld properties are under intense scrutiny. Understanding which weld
parameters might contribute to creating such defects, and ultimately
eliminating the defect occurrence, is critical to improving the yield of high-
gradient cavities. Note that cavities with such defects are largely, but not
exclusively, from new cavity manufacturers. One group has attempted to
reproduce these features on samples Cooley , to permit their systematic study.
In time, study of cavity surface quality may improve cavity quality in a
manner similar to the improvement seen in niobium sheet quality resulting from
regular eddy current scanning.
## VII Fermilab Infrastructure for SRF Development
At Fermilab, the SRF infrastructure has been substantially built up in recent
years. The key program goals are the achievement of high gradients with high
yields for future accelerators, the testing of cavities, and the ramp up of
the production rates needed to construct cryomodules for Project X and ILC
R&D. The infrastructure which has been built to achieve these goals includes
facilities for single-cell cavity R&D, an ANL/FNAL joint cavity processing
facility, a vertical cavity test facility, a horizontal test stand, a
cryomodule assembly facility, and a cryomodule test facility.
The ANL/FNAL cavity processing facility provides surface processing and
assembly of cavities for vertical test, for both 1-cell and 9-cell elliptical
cavities. Recent photographs of the primary facility components: EP,
ultrasonic degreasing, high-pressure ultrapure water rinsing, and assembly
support and vacuum leak testing, are shown in Fig. 14. All of these components
are now in routine operation.
Figure 14: ANL/FNAL join facility infrastructure: (clockwise from top left)
EP, ultrasonic degreasing, HPR and assembly stands. [Courtesy of Michael Kelly
and Damon Bice.]
At the FNAL vertical cavity test facility (VCTF), more than 40 cavity tests
have been performed in FY08/FY09, where a test is defined as a cryogenic
cycle. The test cavities have been primarily 9-cell and single-cell 1.3 GHz
elliptical cavities, with two 325 MHz single-spoke resonators included. The
tests have primarily been for instrumentation development, e.g., variable
input coupler, thermometry, cavity vacuum pumping system, and for cavity
diagnostic tests for vendor development. Upgrades are planned to increase the
cavity test throughput to more than 200 cavity tests per year by October 2011,
to support Project X and ILC R&D. Some of the cavities recently tested at the
VCTF are shown in Fig. 15. A summary of the vertical test results of 9-cell
cavities is shown in Fig. 16.
Figure 15: Some of the cavities recently tested at the Fermilab VCTF:
(clockwise from top left) SSR1, 9-cell Tesla-shape, and 1-cell Tesla-shape
(with diode thermometry) cavities. [Courtesy of Bill Mumper.] Figure 16: A
summary of maximum gradients from 9-cell vertical tests in the ILC Americas
region. The cavities tested at the VCTF are shown by the black histogram.
The horizontal test stand (HTS), which is used to test 9-cell cavities which
have been welded into a helium tank, i.e., dressed, using pulsed high power,
is now operational. It has been commissioned for both 1.3 and 3.9 GHz
cavities. Four 3.9 GHz cavities were tested in 2008; they have been installed
in a cryomodule which is currently at DESY. Upgrades are planned to expand the
capacity with the addition of a second cryostat by 2012. A photograph of the
HTS is shown in Fig. 17.
Figure 17: A photograph of the HTS, showing a 9-cell 1.3 GHz cavity being
installed. [Courtesy of Fermilab Visual Media Services.]
Two cryomodule assembly facilities (CAF’s) have been built at Fermilab, CAF-
MP9 and CAF-ICB. The CAF-MP9 facility receives dressed cavities and their
peripheral parts, and assembles these into a cavity string in a class-10
cleanroom. Then, the string assembly is installed onto the cold mass and
transported to CAF-ICB. At CAF-ICB, the cavity string is aligned and assembled
into the vacuum vessel. After the cryomodule is complete, it is shipped to the
cryomodule test facility (NML) for testing. One cryomodule, CM1, built with
cavities from DESY, has been assembled in the CAF facilities. Recent
photographs of the CAF-MP9 and CAF-ICB, showing steps in the assembly of CM1,
are shown in Figs. 18 and 19, respectively.
Figure 18: Recent photographs of the CAF-MP9 facility, showing stages in the
assembly of the first cryomodule, CM1. [Courtesy of Tug Arkan.]
Figure 19: Recent photographs of the CAF-ICB facility, showing stages in the
assembly of the first cryomodule, CM1. [Courtesy of Tug Arkan.]
The cryomodule test facility, NML, will test one RF unit, i.e., 3 cryomodules,
using a 10 MW RF system and an electron beam with ILC parameters. Various
Project-X parameters will also be tested with beam in this facility. NML will
provide the capability to conduct advanced accelerator R&D for future
accelerator components. Through the end of FY09, the facility is being
prepared to test the first Fermilab-assembled cryomodule, CM1, without beam,
requiring completion of the RF system and the cryogenic system. A recent
photograph of the NML facility is shown in Fig. 20.
Figure 20: A recent photograph of the NML facility, indicating where the first
cryomodule, CM1, has been installed. [Courtesy of Jerry Leibfritz.]
## VIII Conclusions and Outlook
Highlights of the rich R&D activity in the quest for high gradient and reduced
cost have been described. Very high gradients have been measured in niobium
SRF cavities: more than 50 MV/m in single-cell cavities of various shapes, and
more than 35 MV/m in several 9-cell Tesla-shape cavities. Several promising
new results from large-grain 9-cell cavities and hydroformed cavities address
the cost and reliability issues of these cavities. Several cavities which have
been limited to 15-20 MV/m by hard quench have been studied using various
techniques, and the outlook is good that useful information can be fed back to
cavity manufacturers to improve the yield of high-gradient cavities over time.
Surface treatment is crucial for optimum performance, and several promising
studies on final preparation methods have been found to sharply reduce field
emission. Fundamental SRF studies are also underway, many showing promising
results.
An overview of Fermilab infrastructure in pursuit of SRF R&D has been given.
Among the recent accomplishments, one 1.3 GHz cryomodule using DESY dressed
cavities has been built and transported to the cryomodule test facility for
testing. Most key infrastructure components are now in place, with final
commissioning underway.
###### Acknowledgements.
I am grateful to my colleagues for their contributions to the content for this
talk, as referenced. I especially thank those who graciously permitted me to
show their unpublished data. This manuscript has been authored by Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S.
Department of Energy. The United States Government retains and the publisher,
by accepting the article for publication, acknowledges that the United States
Government retains a nonexclusive, paid-up, irrevocable, worldwide license to
publish or reproduce the published form of this manuscript, or allow others to
do so, for United States Government purposes.
## References
* (1) International Linear Collider RDR, 2007, http://ilcdoc.linearcollider.org/record/6321/files/ ILC_RDR_Volume_3-Accelerator.pdf.
* (2) K. Saito, ”Gradient Yield Improvement Efforts for Single- and Multicells and Progress for Very High Gradient Cavities,” SRF2007, Beijing, Oct. 2007, TU202; http://cern.ch/AccelConf/srf2007/PAPERS/ TU202.pdf.
* (3) L. Lilje, ILC MAC Meeting (FNAL), April 26, 2007; http://ilcagenda.linearcollider.org/ materialDisplay.py?contribId=4&materialId=slides&confId =1388 .
* (4) G. Ciovati et al., “Final Surface Preparation for Superconducting Cavities,” TTC-Report 2008-05, http://flash.desy.de/sites/site_vuvfel/content/ e403/e1644/e2271/e2272/infoboxContent2354/ TTC-Report2008-05.pdf.
* (5) F. Furuta et al., “High Reliable Surface Treatment Recipe of High Gradient Single Cell Superconducting Cavities at KEK,” SRF 2007, Beijing, Oct. 2007, TUP10; http://cern.ch/AccelConf/srf2007/PAPERS/ TUP10.pdf.
* (6) D. Reschke et al., “Dry-Ice Cleaning: The Most Effective Cleaning Process for SRF Cavities?” SRF 2007, Beijing, Oct. 2007, TUP48; http://cern.ch/AccelConf/srf2007/PAPERS/ TUP48.pdf.
* (7) R.L. Geng, ILC SCRF Meeting at Fermilab, Apr.21-25, 2008; http://ilcagenda.linearcollider.org/getFile.py/ access?contribId=1&sessionId=5&resId=1& materialId=slides&confId=2650 .
* (8) D. Reschke and L. Lilje, ”Recent Experience with Nine-Cell Cavity Performance at DESY,” SRF 2007, Beijing, Oct. 2007, TUP77; http://cern.ch/AccelConf/srf2007/PAPERS/ TUP77.pdf.
* (9) B. Aune et al., “The superconducting TESLA cavities,” Phys. Rev. STAB 3,092001 (2000).
* (10) W. Singer et al., ”Preliminary Results from Multi-Cell Seamless Niobium Cavities,” LINAC08, Victoria, Sep. 2008, THP043; http://www.JACoW.org.
* (11) W. Singer (DESY), private communication.
* (12) P. Kneisel et al., “Performance of Single Crystal Niobium Cavities,” EPAC’08, Genoa, Jun. 2008, MOPP136; http://www.JACoW.org.
* (13) P. Kneisel, “Progress on Large Grain and Single Grain Niobium - Ingots and Sheet and Review of Progress on Large Grain and Single Grain Niobium Cavities,” SRF 2007, Beijing, Oct. 2007, TH102; http://cern.ch/AccelConf/srf2007/PAPERS/ TH102.pdf
* (14) A. Gurevich, “Enhancement of rf breakdown field of superconductors by multilayer coating,” Appl. Phys. Lett. 88, 012511 (2006).
* (15) T. Proslier et al., “Tunneling Study of SRF Cavity-Grade Niobium,” ASC08, Chicago, Aug. 2008; 2LX05, submitted to IEEE Transactions on Applied Superconductivity.
* (16) J. Sekutowicz, “New Geometries: Elliptical Cavities,” ICFA Beam Dyn. Newslett. 39, 112 (2006).
* (17) R.L. Geng et al., “High Gradient Studies for ILC with Single-Cell Re-entrant Shape and Elliptical Shape Cavities Made of Fine-grain and Large-grain Niobium,” PAC’07, Albuquerque, Jun. 2007, WEPMS006, p.2337-(2007), http://www.JACoW.org.
* (18) F. Furuta et al., ”Experimental Comparison at KEK of High Gradient Performance of Different Single Cell Superconducting Cavity Designs, EPAC’06, Edinburgh, Jun. 2006; http://www.JACoW.org.
* (19) R.L. Geng, “High Gradient SRF R&D for ILC at Jefferson Lab,” LINAC08, Victoria, Sep. 2008, THP043; http://www.JACoW.org.
* (20) H. Padamsee, J. Knoboch and T. Hays, RF Superconductivity for Accelerators, (Wiley, New York 2008), pp.164-169.
* (21) J. Knobloch, H. Muller, and H. Padamsee, “Design of a High-Speed, High-Resolution Thermometry System for 1.5-GHz Superconducting Radio-Frequency Cavities, Rev. Sci. Instrum. 65, 3521 (1994).
* (22) M. Pekeler et al., “Thermometric study of electron emission in a 1.3-GHz superconducting cavity,” SRF1995, Gif-sur-Yvette, Oct. 1995; Part. Accel. 53, 35 (1996).
* (23) Q.S Shu et al., “A novel rotating temperature and radiation mapping system in superfluid He and its successful diagnostics,” CEC/ICMC 1995, Columbus, Ohio, Jul. 1995; Adv. Cryog. Eng. 41, 895 (1996).
* (24) Y. Yamamoto et al., “Cavity Diagnostic System for the Vertical Test of the Baseline SC Cavity in KEK-STF,” SRF 2007, Beijing, Oct. 2007, WEP13; http://cern.ch/AccelConf/srf2007/PAPERS/ WEP13.pdf .
* (25) K.W. Shepard et al., IEEE Trans.Nucl.Sci.24, 1147 (1977); K.W. Shepard et al., IEEE Trans. Nucl. Sci. 30, 3339 (1983).
* (26) D. Orris et al., “Fast Thermometry for Superconducting RF Cavity Testing,” WEPMN105, PAC’07, Albuquerque, NM, USA, Jun. 2007.
* (27) G. Ciovati et al., Jefferson Lab TN-08-012.
* (28) A. Canabal et al., “Full Real-Time Temperature Mapping System for 1.3 GHz 9-cell Cavities,” EPAC’08, Genoa, June 2008, MOPP121; http://www.JACoW.org.
* (29) N. Dhanaraj et al., “Multiplexed Diode Array for Temperature Mapping of ILC 9-cell Cavities,” ASC08, Chicago, Aug.2008; 3LPB07, submitted to IEEE Transactions on Applied Superconductivity.
* (30) Z.A. Conway et al., “Oscillating Superleak Transducers for Quench Detection in Superconducting ILC Cavities Cooled with He-II,” TTC-Report 2008-06; http://flash.desy.de/reports_publications/ .
* (31) Y. Iwashita et al., “Development of high resolution camera for observations of superconducting cavities,” Phys. Rev. STAB 11, 093501 (2008).
* (32) L. Cooley, private communication.
|
arxiv-papers
| 2009-10-22T19:16:01 |
2024-09-04T02:49:05.987715
|
{
"license": "Public Domain",
"authors": "C.M. Ginsburg",
"submitter": "Camille Ginsburg",
"url": "https://arxiv.org/abs/0910.4384"
}
|
0910.4469
|
# Investigating the robustness of the classical enzyme kinetic equations in
small intracellular compartments
Ramon Grima
School of Biological Sciences
Centre for Systems Biology at Edinburgh, University of Edinburgh, UK
* •
Corresponding author: Ramon Grima, email: ramon.grima@ed.ac.uk
## Abstract
Background: Classical descriptions of enzyme kinetics ignore the physical
nature of the intracellular environment. Main implicit assumptions behind such
approaches are that reactions occur in compartment volumes which are large
enough so that molecular discreteness can be ignored and that molecular
transport occurs via diffusion. Though these conditions are frequently met in
laboratory conditions, they are not characteristic of the intracellular
environment, which is compartmentalized at the micron and submicron scales and
in which active means of transport play a significant role.
Results: Starting from a master equation description of enzyme reaction
kinetics and assuming metabolic steady-state conditions, we derive novel
mesoscopic rate equations which take into account (i) the intrinsic molecular
noise due to the low copy number of molecules in intracellular compartments
(ii) the physical nature of the substrate transport process, i.e. diffusion or
vesicle-mediated transport. These equations replace the conventional
macroscopic and deterministic equations in the context of intracellular
kinetics. The latter are recovered in the limit of infinite compartment
volumes. We find that deviations from the predictions of classical kinetics
are pronounced (hundreds of percent in the estimate for the reaction velocity)
for enzyme reactions occurring in compartments which are smaller than
approximately 200nm, for the case of substrate transport to the compartment
being mediated principally by vesicle or granule transport and in the presence
of competitive enzyme inhibitors.
Conclusions: The derived mesoscopic rate equations describe subcellular enzyme
reaction kinetics, taking into account, for the first time, the simultaneous
influence of both intrinsic noise and the mode of transport. They clearly show
the range of applicability of the conventional deterministic equation models,
namely intracellular conditions compatible with diffusive transport and simple
enzyme mechanisms in several hundred nanometre-sized compartments. An active
transport mechanism coupled with large intrinsic noise in enzyme
concentrations is shown to lead to huge deviations from the predictions of
deterministic models. This has implications for the common approach of
modeling large intracellular reaction networks using ordinary differential
equations and also for the calculation of the effective dosage of competitive
inhibitor drugs.
## Background
The inside of a cell is a highly complex environment. In the past two decades,
detailed measurements of the chemical and biophysical properties of the
cytoplasm have established that the conditions in which intracellular
reactions occur are, by and large, very different than those typically
maintained in laboratory conditions. One of the outstanding differences
between _in vivo_ and _in vitro_ conditions, is that in the former,
biochemical reactions typically occur in minuscule reaction volumes [1]. For
example, in eukaryotic cells, many biochemical pathways are sequestered within
membrane-bound compartments, ranging from $\sim 50$nm diameter vesicles to the
nucleus, which can be several microns in size [2]. It is also found that the
total concentration of macromolecules inside both prokaryotic and eukaryotic
cells is very large [3, 4], of the order of $50-400$ mg/ml which implies that
between $5\%$ and $40\%$ of the total intracellular volume is physically
occupied by these molecules [5]. The concentration of these crowding molecules
is highly heterogeneous (see for example [6]), meaning that typically one will
find small pockets of intracellular space, characterized by low macromolecular
crowding, surrounded by a “sea” of high crowding; such pockets of space may
serve as effective compartments where reactions may occur more easily than in
the rest of the cytosol. Analysis of experimental data for the dependence of
diffusion coefficients with molecular size suggests the length scale of such
effective compartments is in the range 35-50nm [7], a size comparable to that
of the smallest vesicles. The significant crowding also suggests that
frequently an active means of transport such as vesicle-mediated transport,
may be more desirable than simple diffusion as a means of intracellular
transport.
The volume of a spherical cavity of space of diameter 50nm is merely $\sim
6.5\times 10^{-20}$ liters, an extremely small number compared to the typical
macroscopic reaction volumes of _in vitro_ experiments (experimental attolitre
biochemistry is still in its infancy - see for example [8]). These very small
reaction volumes imply that at physiologically relevant concentrations (nano
to millimolar), the copy number of a significant number of intracellular
molecules is very small [1] and consequently that intrinsic noise cannot be
ignored; for example $255\mu M$ corresponds to an average of just 10 molecules
in a 50nm vesicle and fluctuations about this mean of the order of 3 molecules
[9].
The traditional mathematical framework of physical chemistry ignores the basic
physical properties of the intracellular environment. Kinetics are described
by a set of coupled ordinary differential equations which implicitly assume
(i) that the reaction compartment is so large that molecular discreteness can
be ignored and that hence integer numbers of molecules per unit volume can be
replaced by a continuous variable, the molar concentration. Since the number
of molecules is assumed to be very large, stochastic fluctuations are deemed
negligible and the equations are hence deterministic; (ii) the reaction
compartment is well-stirred so that homogeneous conditions prevail throughout
[9]. Both assumptions can be justified for reactions occurring in a constantly
stirred reactor of macroscopic dimensions. However if diffusion is the
dominant transport process inside the compartment then the homogeneity
assumption holds only if the volume is small enough so that in the time
between successive reactions, a molecule will diffuse a distance much larger
than the size of the compartment. This comes at the expense of the first
assumption. It hence appears natural that for intracellular applications, the
first assumption, namely that of deterministic kinetics cannot be justified _a
priori_. The second assumption can be justified if reactions are localized in
sufficiently small parts of the cell and in particular for reaction-limited
processes i.e. those for which the typical time for two molecules to meet each
other via diffusion is much less than the typical time for them to react if
they are in close proximity. For such conditions, a molecule will come within
reaction range several times before participating in a successful reaction, in
the process sampling the compartment many times which naturally leads to well-
mixed conditions [9, 10, 11].
In this article we seek to understand the magnitude of deviations from the
classical kinetic equations in small intracellular compartment volumes. We
specifically focus on the case of reaction-limited enzyme reactions which
allows us to relax the first assumption of physical chemistry while keeping
the second one; this makes the mathematics tractable. We quantify deviations
from classical kinetics in the context of the Michaelis-Menten (MM) equation;
this is the cornerstone of present day enzyme kinetics and is a derivative of
the traditional deterministic mathematical framework based on ordinary
differential equations. In steady-state metabolic conditions, it is predicted
to be exact. Thus this equation is ideal as a means to accurately test the
effects of small-scale compartmentation on chemical kinetics. We consider
three successive biological models of intracellular enzyme kinetics, each one
building on the biological detail and realism from the previous one (Figure
1). The models incorporate the intrinsic noisiness of kinetics in small
compartments, the details of the substrate transport process to the
compartment (diffusion or active transport) and the presence of intra-
compartmental molecules other than substrate molecules which may modulate the
enzyme-catalyzed process e.g. inhibitors. On the macroscopic level, i.e. for
large volumes, the steady-state kinetics of all models conform to the MM
equation. We test whether this equation holds on the on the length scale of
small intracellular compartments by deriving the dependence of the ensemble
averaged rate of product formation on the ensemble-averaged substrate
concentration from the corresponding stochastic models in the steady-state. It
is shown via both calculation and stochastic simulation that at these small
length scales the MM equation breaks down, being replaced by a new more
general equation. Practical consequences of this breakdown are illustrated in
the context of the calculation of the effective dosage of enzyme inhibitor
drug needed to suppress intra-compartmental enzyme activity by a given amount.
To make our approach accessible to readers not familiar with stochastic
equations and their analysis, in the Results/Discussion sections we mainly
focus on the biological/biophysical context and implications of the models
together with the main mathematical results which are verified by simulation.
Detailed mathematical derivations and the methods of simulation are relegated
to the Methods section.
## Results
### Model I: Michaelis-Menten reaction occurring in a compartment volume of
sub-micron dimensions. Substrate input into compartment occurs via a Poisson
process
This is the simplest, biologically-relevant case (Figure 1(A)). The reaction
scheme is $\xrightarrow{k_{in}}S+E\xrightleftharpoons[k_{1}]{k_{0}}\
C\xrightarrow{k_{2}}E+P$. Substrate molecules (_S_) are continuously supplied
inside the compartment at some rate $k_{in}$, they reversibly bind to enzyme
molecules (_E_) with rate constants $k_{0}$ (forward reaction) and $k_{1}$
(backward reaction) to form transitory enzyme-substrate complex molecules
(_C_) which then decay with rate $k_{2}$ into enzyme and product molecules
(_P_). The substrate input is assumed to be governed by a Poisson process with
mean $k_{in}$; this is consistent with substrate transport to the compartment
being dominated by normal diffusion. The enzyme acts as a catalyst,
effectively speeding up the reaction by orders of magnitude. It is assumed
that diffusion inside the compartment is normal and not rate-limiting on the
catalytic process i.e. well-mixed conditions or rate-limited kinetics inside
the compartment. Given these conditions we ask ourselves what is the
relationship between the reaction velocity and the substrate concentration
inside the compartment. The simplest approach consists of writing down the
rate equations of traditional physical chemistry:
$\displaystyle[E_{T}]$ $\displaystyle=[E]+[C]=constant,$ (1)
$\displaystyle{d[S]}/{dt}$ $\displaystyle=k_{in}-k_{0}[E][S]+k_{1}[C],$ (2)
$\displaystyle{d[C]}/{dt}$ $\displaystyle=k_{0}[E][S]-(k_{1}+k_{2})[C],$ (3)
$\displaystyle\frac{d[P]}{dt}$ $\displaystyle=k_{2}[C].$ (4)
By imposing steady-state conditions we get the sought-after relationship which
is simply the well-known MM equation:
$\frac{d[P]}{dt}=v=\frac{v_{max}[S]}{K_{M}+[S]},$ (5)
where $K_{M}=(k_{1}+k_{2})/k_{0}$ is the MM constant, $v_{max}=k_{2}[E_{T}]$
is the maximum reaction velocity and square brackets indicate the macroscopic
concentrations. We note that steady-state conditions for substrate necessarily
require that $k_{in}\leq v_{max}$ otherwise the substrate will continuously
accumulate with time. Though this approach is simple and straightforward, as
mentioned in the introduction, the assumptions behind the formulation of the
rate equations are not consistent with the known physical properties of the
cytoplasm. In particular it is clear that if the volume of our compartment is
very small (as is the case), the numbers of particles is also quite small,
meaning that the concept of a continuous variable such as the average
macroscopic concentration has little meaning. Rather we require a mathematical
description in terms of discrete, integer numbers of particles and which is
stochastic. The natural description of such cases is a master equation which
is a differential equation in the joint probability function $\pi$ describing
the system [12, 13, 14, 15]:
$\displaystyle\frac{d\pi}{dt}$
$\displaystyle=k_{in}\Omega(\Theta_{S}^{-1}-1)\pi+\frac{k_{0}}{\Omega}(\Theta_{S}\Theta_{C}^{-1}-1)n_{S}n_{E}\pi\
$
$\displaystyle+k_{1}(\Theta_{C}\Theta_{S}^{-1}-1)n_{C}\pi+k_{2}(\Theta_{C}\Theta_{P}^{-1}-1)n_{C}\pi,$
(6)
where $\pi=\pi(n_{S},n_{C},n_{P})$, $n_{Y}$ is the integer number of molecules
of type $Y$, $\Omega$ is the compartment volume, and $\Theta_{X}^{\pm 1}$ are
step operators defined in the Methods section. This equation cannot be solved
exactly. However it can be solved perturbatively using the system-size
expansion due to van Kampen [12]. This expansion is one in powers of the
inverse square root of the compartment volume. In the Methods section, we
calculate the first three terms of the expansion, namely those proportional to
$\Omega^{1/2}$, $\Omega^{0}$ and $\Omega^{-1/2}$. The first term, being the
dominant one for large volumes, gives back as expected, the rate equations
Eqs. (1)-(4). The second term gives the magnitude of stochastic fluctuations
about the macroscopic concentrations. Corrections to the rate equations and to
the MM equation (due to small compartment volumes or equivalently due to
intrinsic noise) are found by considering the third term. In the rest of the
article, instead of using the reaction velocity $v$, we use the normalized
reaction velocity, $\alpha$, which is simply the velocity of the reaction,
$v$, divided by the maximum reaction velocity, $v_{max}$. Given some measured
intracompartmental substrate concentration, $[S^{*}]=\langle
n_{S}/\Omega\rangle$ (angled brackets imply average), the relationship between
the normalized reaction velocity predicted by the MM equation
($\alpha_{M}=[S^{*}]/(K_{M}+[S^{*}])$) and the actual normalized reaction
velocity ($\alpha$), as predicted by our theory, is given by:
$\alpha+(1-\alpha_{M})f(\alpha)\Omega^{-1}=\alpha_{M},$ (7)
where,
$f(\alpha)=\frac{\alpha^{2}}{K_{M}+[E_{T}](1-\alpha)^{2}}.$ (8)
Hence the prediction of the MM equation is only correct, i.e.
$\alpha=\alpha_{M}$, in the limit of infinitely large compartment volumes, in
which case the second term on the left hand side of Eq. (7) will become
vanishingly small and can be neglected. For finite compartment volumes, the MM
equation is not exact (except in the two limiting cases of
$\alpha_{M}\rightarrow 0$ and $\alpha_{M}\rightarrow 1$) but is at best an
approximation, even though steady-state conditions are imposed; this is at
odds with the prediction of the conventional deterministic theory. An
inspection of Eqs. (7) and (8) shows that the magnitude of the deviations from
the MM equation depends on the two non-dimensional quantities: (i)
$K_{M}\Omega$, a measure of the rate at which enzyme-substrate combination
events occur relative to the rate of decay of complex molecules; (ii)
$[E_{T}]\Omega$, the total integer number of enzyme molecules in the
compartment.
As shown in the Methods section, the MM equation is found to implicity assume
that the noise about the macroscopic substrate and enzyme concentrations is
uncorrelated (this assumption has generally been found to be at the heart of
many macroscopic models - for example see [16]); properly taking into account
these non-zero correlations leads to the corrections encapsulated by Eqs. (7)
and (8). These correlations are expected to be small in two particular cases:
(i) if $K_{M}$ is large; in this case when substrate molecules combine with an
enzyme to form a complex, the latter dissociates very quickly back into free
enzyme and thus successive enzyme-substrate events to the same enzyme molecule
are bound to be almost independent of each other. The opposite situation of
small $K_{M}$ would imply that the bottleneck in the catalytic process is the
decay of complex rather than enzyme-substrate combination; if a successful
combination occurs, the next substrate to arrive to the same enzyme molecule
would have to wait until the complex decays, naturally leading to correlations
between successive enzyme-substrate combination events. (ii) if the total
number of enzyme molecules is large; in such a case, at any one time, the
noise about the macroscopic concentrations will be the sum total from a large
number of enzymes, each at a different stage in the catalytic process and each
independent from all others, which naturally dilutes any temporal
correlations.
To estimate the magnitude of the deviations from the MM equation inside cells,
we use the above two equations, Eqs. (7) and (8), to compute the absolute
percentage error $R_{p}=100|1-\alpha_{M}/\alpha|$. These estimates are also
computed by stochastic simulation of the master Eq. (6), using the exact
stochastic simulation algorithm of Gillespie [10] (see Methods for details
regarding the method of simulation); this provides a direct test of the
theory. Figure 2 shows the typical dependence of $R_{p}$ on $\alpha_{M}$, as
predicted by both theory (solid lines) and simulation (data points). Generally
the agreement between the two is found to be very good; discrepancies increase
as $K_{M}$ and compartment volume decrease but are small for parameter values
realistic for intracellular conditions. The maxima of such plots gives the
maximum absolute percentage error which is a measure of the maximum expected
deviations from the MM equation. Table 1 summarizes these estimates (theory
and simulation) over wide ranges of the parameters typical of _in vivo_
conditions: $K_{M}=10\mu M-1000\mu M$ [17], enzyme copy numbers of ten and one
hundred per compartment which correspond to enzyme concentrations ranging from
$4\mu M$ to 2.5 _m_ M and compartment diameters ranging from 50nm to 200nm.
Note that the maximum deviations from the MM equation are estimated to be less
than approximately $20\%$ and typically just a few percent over large ranges
of parameter values – this robustness of the MM equation with respect to
intrinsic molecular noise is indeed surprising, since strictly speaking it is
only valid for infinite compartment volumes.
The theory is always found to underestimate the actual deviations predicted by
simulations; hence the theoretical expressions provide a quick, convenient way
by which one can generally estimate a lower bound on the deviations to be
expected from the MM equation without the need to perform extensive stochastic
simulation.
### Model II: Michaelis-Menten reaction occurring in a compartment volume of
sub-micron dimensions. Substrate is input into compartment in groups or bursts
of M molecules at a time
Model I captures the basics of a general enzyme-catalyzed process occurring in
a small intracellular compartment. In this section we build upon this model to
incorporate further biological realism. In particular, in the previous model
we assumed that substrate input can be well described by a Poisson process,
where one molecule at a time is fed into the compartment with some average
rate $k_{in}$. This is the simplest possible assumption and approximates well
the situation in which molecules are brought to the compartment via normal
diffusion. However there are many situations where this may not be the case;
we now describe two such cases.
The intracellular condition of macromolecular crowding limits the Brownian
motion of molecules in the cytoplasm, this being reflected in the relatively
small diffusion coefficients measured _in vivo_ compared to those known _in
vitro_ for moderately to relatively large molecules. Experiments with inert
tracer particles in the cytoplasm of Swiss 3T3 cells show that the _in vivo_
diffusion coefficient is an order of magnitude less than that _in vitro_ for
molecules with hydrodynamic radius $14$nm and diffusion becomes negligibly
small for molecules larger than approximately $25$nm [7]; similar results have
been obtained in Xenopus neurons [18] and skeletal muscle myotubes [19]. If
diffusion is considerably hindered, one expects active transport to become a
more desirable mode of transport. Indeed there exists ample evidence for the
active transport of macromolecules: they are typically packaged in a vesicle
or a granule which is then transported along microtubules or by some other
means. It is also found that each vesicle or granule typically contains
several of these molecules (examples are: mRNA molecules - several estimated
per granule [20, 21]; cholesterol molecules which are transported in low-
density lipoproteins [2] \- approximately 1500 per lipoprotein).
Generally an active means of transport is not exclusively linked with the
transport of large substrate molecules. The cell being a highly
compartmentalized and dynamic entity requires for its survival the precise
transport of certain molecules from one compartment to another and a
regulation of this transport depending on its current physiological state.
Brownian motion leads to an isotropic movement of molecules down the
concentration gradient and to a consequent damping of the substrate
concentration with distance. In contrast active transport provides a directed
(anisotropic) means of transport with little or no loss of substrate with
distance, is independent of the concentration gradient and it is also easily
amenable to modulation.
Hence it follows that a more general process modeling molecular entry into an
intracellular compartment is one in which $M$ molecules are fed into the
compartment at a rate $k_{in}^{0}$; the latter rate constant is the rate at
which vesicles or granules arrive to the site of the compartment (Figure
1(B)). The total mean substrate input rate is then $k_{in}=Mk_{in}^{0}$. The
special case $M=1$ corresponds to Model I. We construct the relevant master
equation and employ the system-size expansion as for the previous model (see
Methods for details); it is found that the deterministic rate equations are
exactly Eqs. (1)-(4) i.e. at the macroscopic level, given two reactions
occurring in two different compartments, both with the same total mean
substrate input rate $k_{in}$ but one occurring via diffusion (e.g.
$M=1,k_{in}^{0}=1$) and the other via active transport (e.g.
$M=10,k_{in}^{0}=0.1$) , cannot be distinguished. However if the compartment
volumes become small, then once again we find corrections to the MM equation
and interestingly these corrections are sensitive to the mode of transport.
The relationship between the normalized reaction velocity predicted by the MM
equation ($\alpha_{M}$) and the actual normalized reaction velocity
($\alpha$), as predicted by our theory, is given by Eq. (7) together with:
$f(\alpha)=\frac{\alpha[\alpha+\frac{1}{2}(M-1)]}{K_{m}+[E_{T}](1-\alpha)^{2}}.$
(9)
This suggests that generally deviations from the predictions of the MM
equation increase with the carrying capacity, $M$, of the vesicle or granule.
To compare the effects of active transport and diffusion on the kinetics, we
set $M=50$ and adjusted $k_{in}^{0}$ so that in all cases, the total mean
substrate input rate for model II is equal to $k_{in}$, the input rate of
Model I (i.e. the two models would be indistinguishable from a macroscopic
point of view). Using the same procedure as for Model I, we computed the
maximum percentage error using Eqs. (7) and (9) and also from simulations. The
results are summarized in Table 2. Notice that now the deviations from the MM
equation are much larger than before, running into hundreds of percent rather
than the tens as for Model I. Because of the increase in substrate
fluctuations, the quantitative accuracy of the theory is now less than before;
it generally fares very well for compartments with diameters larger than $\sim
100$nm and $K_{M}$ larger than $\sim 100\mu M$. Nevertheless in all cases
theory does correctly predict a large increase in discrepancy between the
reaction velocities given by the deterministic MM equation and those from
stochastic simulation compared to the case of Model I. The intuitive reason
behind these increases in discrepancy is that substrate which is input in
bursts enhances correlations between successive enzyme-substrate events.
The explicit dependence of the reaction velocity on substrate concentration is
complex and generally requires the solution of the cubic polynomial
encapsulated by Eqs. (7) and (9). However for small substrate concentrations,
the equations simplify to a simple linear equation:
$\alpha=[S^{*}]\left(K_{M}\left[1+\frac{M-1}{2\Omega(K_{M}+[E_{T}])}\right]\right)^{-1}$
(10)
Note that if the MM equation was correct, one would expect
$\alpha=[S^{*}]/K_{M}$. Indeed Eq. (10) reduces to the latter prediction in
the limit of large volumes. Note also that this renormalization of the
proportionality constant occurs only if the substrate input occurs in bursts,
i.e. $M>1$. These predictions of our theory are verified by simulations
(Figure 3).
### Model III: Michaelis-Menten reaction with competitive inhibitor occurring
in a compartment volume of sub-micron dimensions. Substrate input as in
previous models
In this last section, we further build on the previous two models by adding
competitive inhibitors to the intracellular compartment in which enzymes are
localized. A competitive inhibitor, $I$, is one which binds reversibly to the
active site of the enzyme (forming a complex $EI$), thus preventing substrate
molecules from binding to the enzyme and slowing down catalysis (Figure 1(C)).
In standard textbooks and in the literature, this is typically modeled by the
set of reactions (see for example [22]):
$\xrightarrow{k_{in}}S+E\xrightleftharpoons[k_{1}]{k_{0}}\
C\xrightarrow{k_{2}}E+P,\ E\xrightleftharpoons[k_{3}]{k_{4}}EI$, where
$k_{4}=k_{4}^{0}[I]$ and $[I]$ is the inhibitor concentration. Note that it is
implicitly assumed that inhibitor is in such abundance that the second-order
bimolecular reaction between inhibitor and enzyme can be replaced by a pseudo
first-order reaction with constant inhibitor concentration. We shall assume
the same in our model. Substrate input into the compartment is considered to
occur as in Model II since this encapsulates that of Model I as well. The
deterministic model of this set of reactions leads to a MM equation of the
form:
$\frac{d[P]}{dt}=\frac{v_{max}[S]}{K_{m}(1+\beta)+[S]},$ (11)
where $\beta=[I]/K_{i}$ and $K_{i}=k_{3}/k_{4}^{0}$ is the dissociation
constant of the inhibitor. The perturbative solution of the master equation
describing the system is now significantly more involved than in previous
models; the underlying reason for this is that the computation of the noise
correlators to order $\Omega^{0}$ requires the inversion of a $6\times 6$
matrix as opposed to a $3\times 3$ one in previous models (see Methods for
details). The analysis predicts corrections to the MM equation by postulating
a new mesoscopic rate equation having the form of Eq. (7) together with:
$f(\alpha)=\frac{1+\beta}{K_{m}[E_{T}]}\frac{\sum_{i=0}^{4}c_{i}(1-\alpha)^{i}}{\sum_{i=0}^{4}d_{i}(1-\alpha)^{i}},$
(12)
where $c_{i}$ and $d_{i}$ are coefficients with a complex dependence on the
various enzyme parameters (these are given in full in the Methods Section).
Table 3 shows the maximum percentage error computed using Eqs. (7) and (12)
and also from simulations for the cases in which substrate input occur a
molecule at a time and in bursts of 50 at a time. The parameter values chosen
in the simulations and calculations (see caption of Table 3) are typical for
many enzymatic processes: the bimolecular rate coefficients span the range
$10^{6}-10^{9}s^{-1}M^{-1}$ [22], the backward decay processes are in the
middle of the range $10-10^{5}s^{-1}$ [22], the inhibitor concentration is ten
times larger than the total enzyme concentration (satisfying the implicit
assumption that the inhibitor is in significantly larger concentration than
free enzyme), and the intracompartmental enzyme concentrations are in the
range $4-255\mu M$. The deviations from the MM equation in this case are more
severe than in the previous two models, this being due to non-zero
correlations between substrate and the complex $EI$ in addition to the already
present correlations between substrate and complex $C$. Note that the
agreement between theory and simulations is overall better than in previous
models, even when the burst size is large, $M=50$. As mentioned in the section
for Model I, discrepancies between theory and simulation are generally found
to decrease with increasing $K_{M}$; for the case of competitive inhibition,
the effective $K_{M}$ of the reaction is significantly larger than that of the
enzyme (see Eq. (11)), which explains the increased agreement between theory
and simulations for Model III compared to the previous two models.
A significant number of drugs suppress a chain of biochemical reactions by
reducing the activity of key enzymes in the pathway via competitive inhibition
[17]. The conventional method to estimate the required concentrations of these
inhibitors involves plotting the variation of the enzyme activity with
inhibitor concentration, $[I]$, using the MM equation; the substrate
concentration is kept fixed and is chosen so that at $[I]=0$, the reaction
velocity is close to the maximum, $v_{max}$. Since there are significant
corrections to the MM equation when reactions occur in intracellular
compartments, it is not clear how accurate are estimates of $[I]$ based upon
it. Figure 4 compares the enzyme activity curve based on the MM equation with
the theoretical predictions for the corrected enzyme activity curves based on
the mesoscopic rate equation embodied by Eqs. (7) and (12), for compartments
of diameter 50nm and 100nm (inset) and for substrate input burst sizes of
$M=20$ and $50$. The substrate concentration is chosen so that at $[I]=0$,
$v/v_{max}=0.909$ in all cases. We find that generally as the burst size
increases, the actual inhibitor concentration needed to suppress enzyme
activity by a given amount is larger than that estimated by the MM equation;
this discrepancy decreases with increasing compartment volume. For the example
in Figure 4, for the case in which substrate is input into the compartment in
bursts of $M=50$, the actual inhibitor concentration needed to decrease the
enzyme activity from $0.909$ to $0.1$ is approximately 7 times larger than the
MM estimate; if the compartment diameter is doubled (inset of Figure 4), the
actual inhibitor concentration needed is less than twice that of the MM
estimate. Generally we find that for the typical parameter values of enzymatic
reactions, the corrections to the enzyme-activity curves can be neglected for
compartments larger than about $200$nm in diameter.
## Discussion and Conclusion
In this last section we discuss some fine points regarding: (i)the assumptions
behind the use of master equations which throws light on the range of use of
the derived mesoscopic equations, (ii) the use of the system-size expansion to
perturbatively solve the master equation and (iii) the assumption of steady-
state metabolic conditions. We conclude by placing our work in the context of
previous recent studies of stochastic enzyme kinetics and discuss possible
experiments to verify some of the conclusions we have reached.
We have implicitly assumed throughout the article that a single (global)
master equation model suffices to capture the deviations from classical
kinetics due to fluctuations in chemical concentrations inside a single
subcellular compartment. As noted by Baras and Mansour [23], “the global
master equation selects the very limited class of exceptionally large
fluctuations that appear at the level of the entire system, disregarding
important nonequilibrium features originated by local fluctuations.” Hence the
results presented here necessarily underestimate the possible deviations from
classical kinetics, in particular the local fluctuations due to diffusion of
molecules inside the compartment. These local fluctuations are typically small
for reaction-limited processes (as in this article) but significant for
diffusion-limited ones. To capture them effectively, one would be required to
spatially discretize the compartment into many small elements and describe the
reaction-diffusion processes between these elements by means of a multivariate
master equation [12, 23]. The latter is known as a reaction-diffusion master
equation; typically it does not allow detailed analytical investigation as for
a global master equation and one is limited to stochastic simulation. Use of
the global master equation is also restricted for compartments which are not
too small: in particular the linear dimensions of the compartment should be
larger than the average distance traveled by a molecule before undergoing a
successful reaction with another molecule i.e. the length scale is much larger
than that inherent in molecular dynamics simulation [23].
We have applied the systematic expansion due to van Kampen to perturbatively
solve the master equation. It is sometimes _a priori_ assumed that because
this expansion is about the macroscopic concentrations, it cannot give
information regarding the stochastic kinetics of few particle / small volume
systems. This is true if one restricts oneself to the expansion to order
$\Omega^{0}$ i.e. the linear-noise approximation; this is commonly the case
found in the literature since the algebra becomes tedious if one considers
more terms. However we note that as argued and shown by van Kampen himself
[12], terms beyond the linear-noise approximation in the system-size expansion
add terms to the fluctuations that are of order of a single particle relative
to the macroscopic quantities and are essential to understanding how
fluctuations are affected by the presence of non-linear terms in the
macroscopic equation (substrate-enzyme binding in our case). In our theory we
went beyond the linear-noise approximation. We find that the predicted
theoretical results are in reasonable agreement, in many cases (comparison of
bold and italic values in Tables 1, 2 and 3), with stochastic simulations of
just a few tens of enzyme molecules in sub-micron compartments, which
justifies our methodology.
We have also imposed metabolic steady-state conditions inside the subcellular
compartment. Technically this is convenient since in such a case one does not
deal with complex transients. Also since under such conditions the MM equation
is exact from a deterministic point of view, it provides a very useful
reference point versus which to accurately compute deviations due to intrinsic
noise. In reality one may not always have steady-state conditions inside
cells, this depending strongly on the rate of substrate input relative to the
maximum rate at which the enzyme can process substrate. Another possibility is
that one is dealing with a batch reaction i.e. one in which a number of
substrate molecules are transported at one go and just once to the subcellular
compartment (e.g. via vesicle-mediated transport) and the reaction proceeds
thereafter without any further substrate replenishment. This latter scenario
is compatible with the presentation of the MM equation typical in standard
physical chemistry textbooks. The MM equation is then an approximation (not
exact as in steady-state case) to the deterministic kinetics, when substrate
is present in much larger concentration than enzyme. This case is currently
under investigation using the same perturbative framework used in this
article.
We note that this is not the first attempt to study stochastic enzyme
kinetics. The bulk of recent studies [24, 25, 26, 27] have focused on
understanding the kinetics of a Michaelis-Menten type reaction catalyzed by a
single enzyme molecule. Deviations from classical kinetics were found to be
most pronounced when one takes into account substrate fluctuations [26]. These
pioneering studies were restricted to a single-enzyme assisted reaction which
reduces complexity thereby making it ideal from a theoretical perspective;
since the reaction is dependent on just a single enzyme molecule one also
finds maximum deviations from deterministic kinetics. In reality, it is
unlikely to find just one enzyme molecule inside a subcellular compartment -
as mentioned in the introduction a physiological concentration of just a few
hundred micromolar would correspond to few tens inside the typically smallest
subcellular compartment. It is also the case that diffusion may not always be
the main means of substrate transport to the compartment and that the reaction
maybe more complex than the simple Michaelis-Menten type reaction of these
previous studies. The present study fills in these gaps by using a systematic
method to derive approximate and relatively simple analytic expressions for
mesoscopic rate equations describing the kinetics of the general case of $N$
enzyme molecules in a subcellular compartment with or without active transport
of substrate and in the presence of enzyme inhibitors. Most importantly our
approach shows the effects of intrinsic noise on the kinetics can be captured
via effective ordinary differential equations. This enables quick estimation
of the magnitude of stochastic effects on reaction kinetics and thus gives
insight into whether a model or parts of a model should be designed to be
stochastic or deterministic without the need for extensive stochastic
simulation. In the present study, this approach enabled us to readily compute,
for the first time, the deviations from deterministic kinetics for a broad
range of realistic _in vivo_ parameter constants (Tables 1, 2 and 3), a task
which would be considerably lengthy if one had to rely solely on data obtained
from ensemble-averaged stochastic simulations.
We conclude by briefly discussing possible experiments which can verify the
predictions made in this article. It is arguably not an easy task to perform
the required experiments in real-time in a living cell. A viable alternative
would consist of monitoring reaction kinetics inside single artificially-made
vesicles. Pick et al [8] have shown that the addition of cytochalasin to
mammalian cells induces them to extrude from their plasma membrane minuscule
vesicles of attolitre volume with fully functional cell surface receptors and
also retaining cytosolic proteins in their interior. The change in the intra-
vesicular calcium ion concentration in response to surface ligand binding was
measured using fluorescence confocal microscopy (FCM). Since the vesicle sizes
are of typical small sub-cellular compartment dimensions (1 attolitre
corresponds to a spherical vesicle of approximate diameter 120nm) and FCM
allows the measurement of the concentration of a fluorescent probe (via a
calibration procedure), this experimental technique appears ideal to verify
the predictions of Model I and of Model III for the case of diffusive
substrate transport. Model II and Model III with vesicle-transport of
substrate are probably much more challenging to verify since one then needs to
construct the _in vitro_ equivalent of microtubules. This is within the scope
of synthetic biology and may be a possibility in the next few years.
## Methods
We here provide full details of the calculations reported in the Results
section. The system size-expansion which is at the heart of the analysis has
to-date not been applied extensively to biological problems and thus we go
into some detail in its elucidation in Subsection I, which is dedicated
exclusively to Model I. For other recent applications of the general method in
the context of reaction kinetics, see for example [28] and [29]. Subsections
II and III (treating Model II and Model III, respectively) naturally build on
the results of the first subsection and thus we only give the main steps of
the calculations in these last two cases. Subsection IV has a brief discussion
of the simulation methods used to verify the theoretical results.
### Model I: Michaelis-Menten reaction occurring in a compartment volume of
sub-micron dimensions. Substrate input into compartment is modeled as a
Poisson process
The reaction scheme is
$\xrightarrow{k_{in}}S+E\xrightleftharpoons[k_{1}]{k_{0}}\
C\xrightarrow{k_{2}}E+P$. The stochastic description of this system is
encapsulated by the master equation which is a differential equation in the
joint probability function $\pi$ describing the system:
$\displaystyle\frac{d\pi}{dt}$
$\displaystyle=k_{in}\Omega(\Theta_{S}^{-1}-1)\pi+\frac{k_{0}}{\Omega}(\Theta_{S}\Theta_{C}^{-1}-1)n_{S}n_{E}\pi$
(13)
$\displaystyle+k_{1}(\Theta_{C}\Theta_{S}^{-1}-1)n_{C}\pi+k_{2}(\Theta_{C}\Theta_{P}^{-1}-1)n_{C}\pi,$
where $\pi=\pi(n_{C},n_{P},n_{S})$, $n_{X}$ is the integer number of molecules
of type $X$ (where $X={C,P,S}$), $\Omega$ is the compartment volume, and
$\Theta_{X}^{\pm 1}$ are the step operators defined by their action on a
general function $g(n_{X})$ as: $\Theta_{X}^{\pm 1}g(n_{X})=g(n_{X}\pm 1)$.
Note that the relevant variables are three, not four: the integer number of
molecules of free enzyme ($n_{E}$) is not an independent variable due to the
fact that the total amount of enzyme is conserved. The master equation cannot
be solved exactly but it is possible to systematically approximate it by using
an expansion in powers of the inverse square root of the volume of the
compartments. This is generally called the system-size expansion [12].
The method proceeds as follows. The stochastic quantity, $n_{X}/\Omega$,
fluctuates about the macroscopic concentrations [X]; these fluctuations are of
the order of the square root of the number of particles:
$n_{X}=\Omega[X]+\Omega^{1/2}\epsilon_{X}.$ (14)
Note that since $n_{E}+n_{C}=constant$, it follows that
$n_{E}=\Omega[E]-\Omega^{1/2}\epsilon_{C}$. The joint distribution function
and the operators can now be written as functions of the new variables,
$\epsilon_{X}$, giving: $\pi=\Pi(\epsilon_{C},\epsilon_{P},\epsilon_{S},t)$
and $\Theta_{X}^{\pm
1}=1\pm\Omega^{-1/2}{\partial}/{\partial\epsilon_{X}}+\frac{1}{2}\Omega^{-1}{\partial^{2}}/{\partial\epsilon_{X}^{2}}+O(\Omega^{-3/2})$;
using these new variables the master equation Eq. (13) takes the form:
$\displaystyle\frac{\partial\Pi}{\partial
t}-\Omega^{1/2}\biggl{(}\frac{d[C]}{dt}\frac{\partial\Pi}{\partial\epsilon_{C}}+\frac{d[P]}{dt}\frac{\partial\Pi}{\partial\epsilon_{P}}+\frac{d[S]}{dt}\frac{\partial\Pi}{\partial\epsilon_{S}}\biggr{)}=\Omega^{1/2}a_{1}\Pi+\Omega^{0}a_{2}\Pi+\Omega^{-1/2}a_{3}\Pi+O(\Omega^{-1})$
(15)
where
$\displaystyle
a_{1}=-(k_{in}+k_{1}[C]-k_{0}[E][S])\frac{\partial}{\partial\epsilon_{S}}+((k_{1}+k_{2})[C]-k_{0}[E][S])\frac{\partial}{\partial\epsilon_{C}}-k_{2}[C]\frac{\partial}{\partial\epsilon_{P}},$
(16)
$\displaystyle a_{2}=$
$\displaystyle\frac{1}{2}k_{in}\frac{\partial^{2}}{\partial\epsilon_{S}^{2}}+\frac{1}{2}\biggl{(}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}(k_{0}[S][E]+k_{1}[C])+k_{2}\biggl{[}\frac{\partial}{\partial\epsilon_{C}}-\frac{\partial}{\partial\epsilon_{P}}\biggr{]}\epsilon_{C}$
$\displaystyle+\biggl{[}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{]}[k_{0}(\epsilon_{S}[E]-\epsilon_{C}[S])-k_{1}\epsilon_{C}]+\frac{1}{2}k_{2}\biggl{(}\frac{\partial}{\partial\epsilon_{P}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}[C],$
(17)
$\displaystyle a_{3}=$
$\displaystyle\frac{1}{2}\biggl{(}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}(k_{0}\epsilon_{S}[E]-k_{0}\epsilon_{C}[S]+k_{1}\epsilon_{C})$
$\displaystyle-
k_{0}\biggl{[}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{]}\epsilon_{S}\epsilon_{C}+\frac{1}{2}k_{2}\biggl{(}\frac{\partial}{\partial\epsilon_{P}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}\epsilon_{C}.$
(18)
Note that in Eq. (18) terms which involve products of first and second-order
derivatives, third-order derivatives or higher have been omitted - these do
not affect the low-order moment equations which we will be calculating.
#### Analysis of $\Omega^{1/2}$ terms
The terms of order $\Omega^{1/2}$ are the dominant ones in the limit of large
volumes. By equating both terms of this order on the right and left hand sides
of Eq. (15) and using Eq. (16), one gets the deterministic rate equations:
$\displaystyle{d[S]}/{dt}$ $\displaystyle=k_{in}-k_{0}[E][S]+k_{1}[C],$ (19)
$\displaystyle{d[C]}/{dt}$ $\displaystyle=k_{0}[E][S]-(k_{1}+k_{2})[C],$ (20)
$\displaystyle{d[P]}/{dt}$ $\displaystyle=k_{2}[C].$ (21)
These are exactly those which one would write down based on the classical
approach whereby one ignores molecular discreteness and fluctuations. This is
an important benchmark of the method since it shows that it gives the correct
result in the limit of large volumes. On a more technical note, the
cancelation of these two terms of order $\Omega^{1/2}$ makes Eq. (15) a proper
expansion in powers of $\Omega^{-1/2}$. By imposing steady-state conditions we
have the Michaelis-Menten (MM) equation:
$\frac{d[P]}{dt}=\frac{v_{max}[S]}{K_{M}+[S]},$ (22)
where $v_{max}=k_{2}[E_{T}]$ is the maximum reaction velocity,
$[E_{T}]=[E]+[C]$ is the total enzyme concentration which is a constant at all
times and $K_{M}=(k_{1}+k_{2})/k_{0}$ is the Michaelis-Menten constant.
#### Analysis of $\Omega^{0}$ terms
To this order, the master equation is a multivariate Fokker-Planck equation
whose solution is Gaussian and thus fully determined by its first and second
moments. The equations of motion for these moments can be straightforwardly
obtained from the master equation to this order, leading to a set of coupled
but solvable ordinary differential equations:
$\partial_{t}\left[\begin{array}[]{c}\langle\epsilon_{S}\rangle\\\
\langle\epsilon_{C}\rangle\end{array}\right]=\left(\begin{array}[]{cc}-k_{0}[E]&k_{1}+k_{0}[S]\\\
k_{0}[E]&-k_{0}(K_{M}+[S])\end{array}\right)\left[\begin{array}[]{c}\langle\epsilon_{S}\rangle\\\
\langle\epsilon_{C}\rangle\end{array}\right]$ (23)
$\partial_{t}\left[\begin{array}[]{c}\langle\epsilon_{S}^{2}\rangle\\\
\langle\epsilon_{C}^{2}\rangle\\\
\langle\epsilon_{S}\epsilon_{C}\rangle\end{array}\right]=A\cdot\left[\begin{array}[]{c}\langle\epsilon_{S}^{2}\rangle\\\
\langle\epsilon_{C}^{2}\rangle\\\
\langle\epsilon_{S}\epsilon_{C}\rangle\end{array}\right]+B,$ (24)
where,
$A=\left(\begin{array}[]{ccc}-2k_{0}[E]&0&2(k_{1}+k_{0}[S])\\\
0&-2k_{0}(K_{M}+[S])&2k_{0}[E]\\\
0&-2k_{2}&-2k_{2}\end{array}\right),B=\left(\begin{array}[]{c}k_{in}+k_{1}[C]+k_{0}[S][E]\\\
k_{0}([S][E]+K_{M}[C])\\\ k_{in}+k_{2}[C]\end{array}\right).$ (25)
Note that the matrices and vectors in the above equations have been reduced to
a simpler form by the application of a few row operations. Note also that
these equations are independent of $\epsilon_{p}$ since the product-forming
step is irreversible and hence the fluctuations in substrate and complex are
necessarily decoupled from its fluctuations. At the steady-state, it is found
that $\langle\epsilon_{S,C}\rangle\rightarrow 0$. From Eq. (14), it is clear
that this implies that to this order the average number of substrate molecules
per unit volume, $\langle n_{S}/\Omega\rangle$, is simply equal to the
macroscopic concentration, $[S]$. The same applies for complex molecules.
Hence to this order in the system-size expansion there cannot be any
corrections to the macroscopic equations or to the MM equation. By writing the
macroscopic concentrations in Eqs. (24) and (25) in terms of $k_{in}$ and
solving, one obtains the variance and covariance of the fluctuations about the
steady-state macroscopic concentrations. We here only give the result for the
covariance since this will be central to our discussion later on:
$\langle\epsilon_{C}\epsilon_{S}\rangle=\frac{K_{M}[E_{T}]\alpha^{2}}{K_{M}+[E_{T}](1-\alpha)^{2}},$
(26)
where $\alpha=k_{in}/v_{max}$ is the normalized reaction velocity of the
enzyme.
#### Analysis of $\Omega^{-1/2}$ terms
The system-size expansion is almost never carried out to this order because of
the algebraic complexity typically involved, however it is crucial to find
finite volume corrections to the deterministic rate equations and in
particular to the MM equation. Using the master equation to this order, the
first moment of the complex concentration is governed by the equation of
motion:
$\displaystyle{d\langle\epsilon_{C}\rangle}/{dt}=-k_{0}([S]+K_{M})\langle\epsilon_{C}\rangle+k_{0}[E]\langle\epsilon_{S}\rangle-
k_{0}\Omega^{-1/2}\langle\epsilon_{S}\epsilon_{C}\rangle.$ (27)
Now the production of product _P_ from complex occurs through a decay process
which necessarily has to be described by a linear term of the form:
$k_{in}=k_{2}\langle n_{C}/\Omega\rangle$ (the steady-state condition). Since
the steady-state macroscopic complex concentration is equal to
$[C]=k_{in}/k_{2}$, then it follows that to any order in the expansion we have
$\langle\epsilon_{C}\rangle=0$. This is always found to be the case in
simulations as well. Hence it immediately follows from Eq. (27) that the
average of fluctuations about the macroscopic substrate concentration are non-
zero and given by:
$\langle\epsilon_{S}\rangle=\frac{\langle\epsilon_{S}\epsilon_{C}\rangle}{\Omega^{1/2}[E]}.$
(28)
From a physical point of view, this indicates that the steady-state
concentration of substrate in the compartment is not equal to the value
predicted by the MM equation (i.e. [S]) and hence the non-zero value of the
average of the fluctuations about [S]. The real substrate concentration inside
the compartment is obtained by substituting Eqs. (28) and (26) in Equation
(14), leading to:
$\left\langle\frac{n_{S}}{\Omega}\right\rangle=[S]+\frac{K_{M}\alpha^{2}}{(1-\alpha)[K_{M}+[E_{T}](1-\alpha)^{2}]\Omega}$
(29)
#### An alternative mesoscopic rate equation replacing the MM equation
The renormalization of the steady-state substrate concentration indicates the
breakdown of the MM equation; this phenomenon occurs because of non-zero
correlations between noise in the substrate and enzyme concentrations,
$\langle\epsilon_{S}\epsilon_{C}\rangle$, which the MM equation implicity
neglects. To obtain the alternative to the latter, one needs to obtain a
relationship between the normalized reaction velocity, $\alpha$ and the real
substrate concentration $\langle n_{S}/\Omega\rangle$; writing $[S]$ in terms
of $\alpha$ and substituting in Eq. (29), one obtains this new relation:
$\displaystyle\alpha+\left(1-\frac{\langle n_{S}/\Omega\rangle}{K_{M}+\langle
n_{S}/\Omega\rangle}\right)f(\alpha)\Omega^{-1}=\frac{\langle
n_{S}/\Omega\rangle}{K_{M}+\langle n_{S}/\Omega\rangle},$ (30) $\displaystyle
f(\alpha)=\frac{\alpha^{2}}{K_{M}+[E_{T}](1-\alpha)^{2}}$ (31)
Note that in the limit of large volumes, the second term on the left hand side
of Eq. (30) becomes vanishingly small and we are left with the MM equation. In
the results section the quantity on the right hand side of Eq. (30) is
referred to as $\alpha_{M}$ since this is the normalized reaction velocity
which would be predicted by the MM equation given the measured substrate
concentration $\langle n_{S}/\Omega\rangle$ inside the compartment. A quick
estimate of the magnitude of the error that one is bound to incur by using the
conventional MM equation can be obtained by substituting $\alpha=1/2$ (i.e.
enzyme is half saturated with substrate) in Eqs. (30) and (31), solving for
$\alpha_{M}$ and then using this value to compute the fractional error
$e=1-\alpha_{M}/\alpha$. This leads to the simple expression:
$e=[1+\Omega([E_{T}]+4K_{M})]^{-1}$ (32)
We finish this section by noting that Eq. (30) will be found to be valid
generally and not only for the simple Michaelis-Menten scheme treated in this
section; the details of the reaction network come in through the form of Eq.
(31) which is reaction-specific.
### Model II: Michaelis-Menten reaction occurring in a compartment volume of
sub-micron dimensions. Substrate is input into compartment in groups or bursts
of $M$ molecules at a time.
A natural generalization of Model I which has direct biological application is
when substrate molecules are fed into the compartment $M$ at a time with mean
rate $k_{in}^{0}$. The total mean substrate input rate is then equal to
$k_{in}=Mk_{in}^{0}$. The master equation for this process is Eq. (13) with
the first term on the right hand side replaced by
$\Omega(\Theta_{S}^{-M}-1)k_{in}^{0}$. This leads to the following change in
the expression for $a_{2}$ (Eq. 17):
$\displaystyle\frac{1}{2}k_{in}\frac{\partial^{2}}{\partial\epsilon_{S}^{2}}$
$\displaystyle\rightarrow\frac{1}{2}k_{in}M\frac{\partial^{2}}{\partial\epsilon_{S}^{2}}.$
(33)
Note that since the expression for $a_{1}$ (Eq. 16) is unchanged, the
deterministic equations are precisely the same as those of Model I. However
now the fluctuations about the macroscopic substrate concentration are
enhanced by a factor $M$; consequently the entries in the vector B in Eq. (25)
need the change $k_{in}\rightarrow k_{in}M$. The analysis proceeds in the same
manner as before. The mesoscopic rate equation replacing the MM equation is
now given by Eq. (30) together with:
$f(\alpha)=\frac{\alpha[\alpha+\frac{1}{2}(M-1)]}{K_{M}+[E_{T}](1-\alpha)^{2}}.$
(34)
The fractional error rate evaluated at $\alpha=1/2$ gives:
$e=\frac{M}{M+\Omega([E_{T}]+4K_{M})}$ (35)
This clearly shows that generally larger deviations from the predictions of
the MM equation are expected in this case compared to those computed for Model
I.
### Model III: Michaelis-Menten reaction with competitive inhibitor occurring
in a compartment volume of sub-micron dimensions. Substrate input as in two
previous models.
Competitive inhibition is modeled by the set of reactions:
$\xrightarrow{k_{in}}S+E\xrightleftharpoons[k_{1}]{k_{0}}\
C\xrightarrow{k_{2}}E+P,\ E\xrightleftharpoons[k_{3}]{k_{4}}EI$, where
$k_{4}=k_{4}^{0}[I]$ and $[I]$ is the inhibitor concentration (similar models
have been studied by Roussel and collaborators [30, 31] in the context of
biochemical oscillators though these assume $M=1$). In the rest of this
section, we change the notation of enzyme-inhibitor complex from $EI$ to $V$,
just to make the math notation easier to read. The substrate input into the
compartment is considered to occur as in Model II since this encapsulates that
of Model I as well. The master equation for this system is:
$\displaystyle\frac{d\pi}{dt}$
$\displaystyle=k_{in}^{0}\Omega(\Theta_{S}^{-M}-1)\pi+\frac{k_{0}}{\Omega}(\Theta_{S}\Theta_{C}^{-1}-1)n_{S}n_{E}\pi+k_{1}(\Theta_{C}\Theta_{S}^{-1}-1)n_{C}\pi$
$\displaystyle+k_{2}(\Theta_{C}\Theta_{P}^{-1}-1)n_{C}\pi+k_{3}(\Theta_{V}-1)n_{V}+k_{4}(\Theta_{V}^{-1}-1)n_{E}.$
(36)
The change of variables from $n_{X}$ to $\epsilon_{X}$ is done as before,
however note that now the conservation law for enzyme is different than in the
two previous models. The total enzyme concentration is now equal to
$[E_{T}]=[E]+[C]+[V]$ from which it follows that
$n_{E}=\Omega[E]-\Omega^{1/2}(\epsilon_{C}+\epsilon_{V})$. The description is
chosen to be in terms of numbers of molecules of types $C$, $S$ and $V$ and
thus $E$ being a dependent variable does not show up explicitly in the step
operators of the master equation above.
Due to the significant number of changes in the terms of the expansion from
those of previous models, we will show the equivalent of Eqs. (15)-(18) in
full. The master equation in the new variables $\epsilon_{X}$ is given by:
$\displaystyle\frac{\partial\Pi}{\partial
t}-\Omega^{1/2}\biggl{(}\frac{d[C]}{dt}\frac{\partial\Pi}{\partial\epsilon_{C}}+\frac{d[P]}{dt}\frac{\partial\Pi}{\partial\epsilon_{P}}+\frac{d[S]}{dt}\frac{\partial\Pi}{\partial\epsilon_{S}}$
$\displaystyle+\frac{d[V]}{dt}\frac{\partial\Pi}{\partial\epsilon_{V}}\biggr{)}=$
$\displaystyle\Omega^{1/2}a_{1}\Pi+\Omega^{0}a_{2}\Pi+\Omega^{-1/2}a_{3}\Pi+O(\Omega^{-1})$
(37)
where
$\displaystyle a_{1}=$
$\displaystyle-(k_{in}+k_{1}[C]-k_{0}[E][S])\frac{\partial}{\partial\epsilon_{S}}+((k_{1}+k_{2})[C]-k_{0}[E][S])\frac{\partial}{\partial\epsilon_{C}}$
(38)
$\displaystyle+(k_{3}[V]-k_{4}[E])\frac{\partial}{\partial\epsilon_{V}}-k_{2}[C]\frac{\partial}{\partial\epsilon_{P}},$
$\displaystyle a_{2}=$
$\displaystyle\frac{1}{2}k_{in}M\frac{\partial^{2}}{\partial\epsilon_{S}^{2}}+\frac{1}{2}\biggl{(}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}(k_{0}[S][E]+k_{1}[C])+k_{2}\biggl{[}\frac{\partial}{\partial\epsilon_{C}}-\frac{\partial}{\partial\epsilon_{P}}\biggr{]}\epsilon_{C}$
$\displaystyle+\biggl{[}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{]}[k_{0}(\epsilon_{S}[E]-(\epsilon_{C}+\epsilon_{V})[S])-k_{1}\epsilon_{C}]+k_{3}\biggl{(}\frac{\partial}{\partial\epsilon_{V}}\epsilon_{V}+\frac{1}{2}\frac{\partial^{2}}{\partial\epsilon_{V}^{2}}[V]\biggr{)}$
$\displaystyle+k_{4}\left(\frac{1}{2}[E]\frac{\partial^{2}}{\partial\epsilon_{V}^{2}}+\frac{\partial}{\partial\epsilon_{V}}(\epsilon_{C}+\epsilon_{V})\right)+\frac{1}{2}k_{2}\biggl{(}\frac{\partial}{\partial\epsilon_{P}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}[C],$
(39)
$\displaystyle a_{3}=$
$\displaystyle\frac{1}{2}\biggl{(}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}(k_{0}\epsilon_{S}[E]-k_{0}(\epsilon_{C}+\epsilon_{V})[S]+k_{1}\epsilon_{C})-k_{0}\biggl{[}\frac{\partial}{\partial\epsilon_{S}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{]}\epsilon_{S}(\epsilon_{C}+\epsilon_{V})$
$\displaystyle+\frac{1}{2}k_{2}\biggl{(}\frac{\partial}{\partial\epsilon_{P}}-\frac{\partial}{\partial\epsilon_{C}}\biggr{)}^{2}\epsilon_{C}+\frac{1}{2}k_{3}\frac{\partial^{2}}{\partial\epsilon_{V}^{2}}\epsilon_{V}-\frac{1}{2}k_{4}\frac{\partial^{2}}{\partial\epsilon_{V}^{2}}(\epsilon_{C}+\epsilon_{V}).$
(40)
#### Analysis of $\Omega^{1/2}$ terms
As for previous models, these terms give the macroscopic equations. Equating
both terms of this order on the right and left hand sides of Eq. (37) and
using Eq. (38), one obtains:
$\displaystyle{d[S]}/{dt}$ $\displaystyle=k_{in}-k_{0}[E][S]+k_{1}[C],$ (41)
$\displaystyle{d[C]}/{dt}$ $\displaystyle=k_{0}[E][S]-(k_{1}+k_{2})[C],$ (42)
$\displaystyle{d[P]}/{dt}$ $\displaystyle=k_{2}[C],$ (43)
$\displaystyle{d[V]}/{dt}$ $\displaystyle=k_{4}[E]-k_{3}[V].$ (44)
In the steady-state we have the Michaelis-Menten (MM) equation:
$\frac{d[P]}{dt}=\frac{v_{max}[S]}{K_{M}(1+\beta)+[S]},$ (45)
where $\beta=[I]/K_{i}$ and $K_{i}=k_{3}/k_{4}^{0}$ is the dissociation
constant of the inhibitor.
#### Analysis of $\Omega^{0}$ and $\Omega^{-1/2}$ terms
The equations for the first moments are easily obtained and we shall not
reproduce them here; suffice to say that at steady-state, it is found that
$\langle\epsilon_{S,C,V}\rangle\rightarrow 0$ which implies that to this order
in the system-size expansion there cannot be any corrections to the
macroscopic equations or to the MM equation. The addition of a new species,
$V$, does however substantially increase the algebraic complexity in the
equations of motion for the second moments computed using terms up to order
$\Omega^{0}$. In particular the matrix A is now a 6 $\times$ 6 matrix, rather
than the 3 $\times$ 3 matrix of the previous two models.
$\partial_{t}\left[\begin{array}[]{c}\langle\epsilon_{S}^{2}\rangle\\\
\langle\epsilon_{C}^{2}\rangle\\\ \langle\epsilon_{V}^{2}\rangle\\\
\langle\epsilon_{S}\epsilon_{V}\rangle\\\
\langle\epsilon_{C}\epsilon_{V}\rangle\\\
\langle\epsilon_{S}\epsilon_{C}\rangle\end{array}\right]=A\cdot\left[\begin{array}[]{c}\langle\epsilon_{S}^{2}\rangle\\\
\langle\epsilon_{C}^{2}\rangle\\\ \langle\epsilon_{V}^{2}\rangle\\\
\langle\epsilon_{S}\epsilon_{V}\rangle\\\
\langle\epsilon_{C}\epsilon_{V}\rangle\\\
\langle\epsilon_{S}\epsilon_{C}\rangle\end{array}\right]+B,$ (46)
where,
$A=\left(\begin{array}[]{cccccc}-2k_{0}[E]&0&0&2k_{0}[S]&0&2(k_{1}+k_{0}[S])\\\
0&-2k_{0}(K_{M}+[S])&0&0&-2k_{0}[S]&2k_{0}[E]\\\
0&0&-2k^{\prime}&0&-2k_{4}&0\\\
0&-k_{4}&0&-k^{\prime}&-(k_{2}+k^{\prime})&-k_{4}\\\
0&-k_{4}&-k_{0}[S]&k_{0}[E]&-k_{0}(K_{M}+[S])-k^{\prime}&0\\\
0&-k_{2}&0&0&0&-k_{2}\end{array}\right),$ (47)
and
$B=\left(\begin{array}[]{c}k_{in}M+k_{1}[C]+k_{0}[S][E]\\\
k_{0}([S][E]+K_{M}[C])\\\ k_{4}[E]+k_{3}[V]\\\ 0\\\ 0\\\
\frac{1}{2}(k_{in}M+k_{2}[C])\end{array}\right).$ (48)
In the above equations we have defined $k^{\prime}=k_{3}+k_{4}$. Note also
that the system of equations has been simplified through the application of a
few row operations.
Now to next order, i.e. $\Omega^{-1/2}$, the first moments of the
concentrations of molecules of type $C$ and $V$ are governed by the equation
of motions:
$\displaystyle{d\langle\epsilon_{C}\rangle}/{dt}$
$\displaystyle=-k_{0}([S]+K_{M})\langle\epsilon_{C}\rangle-
k_{0}[S]\langle\epsilon_{V}\rangle+k_{0}[E]\langle\epsilon_{S}\rangle-
k_{0}\Omega^{-1/2}(\langle\epsilon_{S}\epsilon_{C}\rangle+\langle\epsilon_{S}\epsilon_{V}\rangle)$
(49) $\displaystyle{d\langle\epsilon_{V}\rangle}/{dt}$
$\displaystyle=-k_{3}\langle\epsilon_{V}\rangle-
k_{4}(\langle\epsilon_{C}\rangle+\langle\epsilon_{V}\rangle)$ (50)
As in previous models, since the production of product _P_ from complex occurs
through a decay process, it follows that at steady-state,
$\langle\epsilon_{C}\rangle=0$ which also implies
$\langle\epsilon_{V}\rangle=0$ from Eq. (50). Hence it follows from Eq. (49)
that
$\langle\epsilon_{S}\rangle=[\langle\epsilon_{S}\epsilon_{C}\rangle+\langle\epsilon_{S}\epsilon_{V}\rangle]/\Omega^{1/2}[E]$.
The two cross correlators can be estimated to order $\Omega^{0}$ by solving
Eqs. (46)-(48). The non-zero value of $\langle\epsilon_{S}\rangle$ implies a
renormalization of the substrate concentration inside the compartment and
hence to a new rate equation replacing the MM equation. This is obtained
exactly in the same manner as previously shown for Model I. The mesoscopic
rate equation is found to be given by Eq. (30) together with:
$f(\alpha)=\frac{(1+\beta)}{K_{M}[E_{T}]}\frac{\sum_{i=0}^{4}c_{i}(1-\alpha)^{i}}{\sum_{i=0}^{4}d_{i}(1-\alpha)^{i}},$
(51)
where the numerator coefficients are given by:
$\displaystyle c_{0}=$
$\displaystyle+k_{3}(\beta+1)^{3}K_{M}^{2}k_{0}[E_{T}],$ (52) $\displaystyle
c_{1}=$
$\displaystyle+K_{M}(\beta+1)^{2}[(\beta+1)[E_{T}]k_{3}^{2}-(3\beta+2)[E_{T}]k_{0}K_{M}k_{3}+k_{0}\beta
v_{max}K_{M}],$ (53) $\displaystyle c_{2}=$ $\displaystyle-
K_{M}(\beta+1)[2(2\beta+\beta^{2}+1)[E_{T}]k_{3}^{2}-((3\beta^{2}+4\beta+1)[E_{T}]k_{0}K_{M}+$
(54)
$\displaystyle-\beta(\beta+1)v_{max}-k_{0}[E_{T}]^{2})k_{3}+\beta(1+2\beta)k_{0}v_{max}K_{M}-(\beta+1)[E_{T}]k_{0}v_{max}],$
$\displaystyle c_{3}=$
$\displaystyle+[(1+3\beta+\beta^{3}+3\beta^{2})[E_{T}]K_{M}k_{3}^{2}-(\beta(\beta+1)^{2}[E_{T}]k_{0}K_{M}^{2}+$
(55)
$\displaystyle(-\beta(\beta+1)^{2}v_{max}-2(1+\beta)[E_{T}]^{2}k_{0})K_{M}+\beta(\beta+1)[E_{T}]v_{max})k_{3}+$
$\displaystyle\beta^{2}(1+\beta)k_{0}v_{max}K_{M}^{2}-(2+3\beta+2\beta^{2})[E_{T}]k_{0}v_{max}K_{M}],$
$\displaystyle c_{4}=$
$\displaystyle-[(-(\beta+1)[E_{T}]^{2}k_{0}K_{M}+\beta(\beta+1)[E_{T}]v_{max})k_{3}+$
(56) $\displaystyle-[E_{T}](\beta+\beta^{2}+1)k_{0}v_{max}K_{M}],$
and the denominator coefficients by:
$\displaystyle d_{0}=$ $\displaystyle+K_{M}^{2}k_{0}k_{3}(1+\beta)^{4},$ (57)
$\displaystyle d_{1}=$
$\displaystyle+K_{M}k_{3}(\beta+1)^{3}[\beta(k_{3}-k_{0}K_{M})+k_{3}],$ (58)
$\displaystyle d_{2}=$
$\displaystyle+K_{M}k_{0}(\beta+1)^{2}[k_{3}[E_{T}](\beta+2)+v_{max}],$ (59)
$\displaystyle d_{3}=$
$\displaystyle+(\beta+1)[k_{3}^{2}\beta^{2}[E_{T}]-k_{0}\beta^{2}k_{3}K_{M}[E_{T}]+2k_{3}^{2}\beta[E_{T}]$
(60) $\displaystyle-k_{0}k_{3}\beta K_{M}[E_{T}]-k_{0}\beta
v_{max}K_{M}+k_{3}^{2}[E_{T}]],$ (61) $\displaystyle d_{4}=$
$\displaystyle+[E_{T}]k_{0}[k_{3}[E_{T}]+k_{3}\beta[E_{T}]+v_{max}].$ (62)
Note that $\sum_{i=0}^{4}c_{i}=0$ such that at $\alpha=0$, there is no
correction to the MM equation i.e. $\alpha_{M}=0$ also. The case $\beta=0$
reduces to Model II, i.e. $f(\alpha)$ is given by Eq. (34).
### Stochastic simulation
In this section we briefly describe the simulation methods used to verify the
theoretical results which are described in detail in the Results section. All
simulations were carried out using Gillespie’s exact stochastic simulation
algorithm, conveniently implemented in the standard simulation platform, Dizzy
[32].
The data points in Figure 2 were generated by iterating the following four-
step procedure: (i) pick a value for $\alpha$ between 0 and 1. This gives the
substrate input rate $k_{in}=\alpha v_{max}$; (ii) run the simulation and
measure the ensemble-averaged substrate concentration, $\langle
n_{s}/\Omega\rangle=[S^{*}]$ at long times; (iii) compute $\alpha_{M}$ using
the MM equation, $\alpha_{M}=[S^{*}]/([S^{*}]+K_{M})$; (iv) compute the
absolute percentage error $R_{p}=100|(1-\alpha_{M}/\alpha)|$. The solid curves
in Figure 2 were obtained by numerically solving the cubic polynomial in
$\alpha$ given by Eqs. (7) and (8) in the Results section for given values of
$\alpha_{M}$ and then using the above expression for $R_{p}$. Figure 3 is
generated in the same manner as Figure 2, except that: in step (i) we fix $M$
and pick a value for $\alpha$ between 0 and 1. Since $k_{in}=Mk_{in}^{0}$, the
required simulation parameter is $k_{in}^{0}=\alpha v_{max}/M$; step (iv) is
not needed. The solid curves were obtained by numerically solving the cubic
polynomial in $\alpha$ given by Eqs. (7) and (9) in the Results section for
given values of $[S^{*}]$. The y-axis for this figure is
$v/v_{max}=\alpha_{M}$ for the MM equation and $v/v_{max}=\alpha$ for the
stochastic model. Figure 4 is obtained by numerically solving the quintic
polynomial in $\alpha$ given by Eqs. (7) and (12) in the Results section
together with the coefficients given by Eqs. (52)-(62) in the present section;
the inhibitor concentration, $[I]$, is varied while the substrate
concentration, $[S^{*}]$, is kept fixed. The substrate concentration is chosen
so that at $[I]=0$, $v/v_{max}=0.909$ in all cases. Note that for models I and
II, $\alpha_{M}=[S^{*}]/([S^{*}]+K_{M})$ while for Model III,
$\alpha_{M}=[S^{*}]/([S^{*}]+(1+\beta)K_{M})$. Note that the error bars are
very small on the scale of the figures and thus are not shown.
## Acknowledgments
It is a pleasure to thank Arthur Straube and Philipp Thomas for interesting
discussions. The author gratefully acknowledges support from SULSA (Scottish
Universities Life Sciences Alliance).
## References
* [1] Luby-Phelps K: Cytoarchitecture and Physical properties of Cytoplasm: Volume, Viscosity, Diffusion, Intracellular Surface Area. Int Rev Cytology 1999, 192:189–221
* [2] Alberts B et al: Molecular Biology of the Cell. Garland Publishing; 1994
* [3] Minton AP: How can biochemical reactions within cells differ from those in test tubes? J Cell Sci 2006, 119:2863 -2869
* [4] Trepat X et al: Universal physical responses to stretch in the living cell. Nature 2007, 447:592 -596
* [5] Schnell S, Turner TE: Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws. Prog Biophys Mol Biol 2004, 85:235–260
* [6] Medalia O et al: Macromolecular Architecture in Eukaryotic Cells Visualized by Cryoelectron tomography. Science 2004, 298:1209–1213
* [7] Luby-Phelps K, Castle PE, Taylor DL, Lanni F: Hindered diffusion of inert tracer particles in the cytoplam of mouse 3T3 cells. Proc Natl Acad Sci USA 1987, 84:4910–4913
* [8] Pick H et al: Investigating Cellular Signaling Reactions in Single Attoliter vesicles. J Am Chem Soc 2005, 127:2908–2912
* [9] Grima R, Schnell S: Modelling reaction kinetics inside cells. Essays in Biochemistry 2008, 45:41–56
* [10] Gillespie DT: Exact stochastic simulation of coupled chemical reactions, J Phys Chem 1977, 81:2340–2361
* [11] Ben-Avraham D, Havlin S: Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press; 2000
* [12] van Kampen NG: Stochastic processes in physics and chemistry. Amsterdam: Elsevier; 2007
* [13] Bartholomay AF: A Stochastic Approach to Statistical Kinetics with Application to Enzyme Kinetics, Biochemistry 1962, 1:223–230
* [14] Bartholomay AF: Enzymatic Reaction-Rate Theory - A Stochastic Approach, Annals of the New York Academy of Sciences 1962, 96:897
* [15] Jachimowski CJ, McQuarrie DA, Russell ME: A Stochastic Approach to Enzyme-Substrate Reactions, Biochemistry 1964, 3:1732–1736
* [16] Grima R: Multiscale modeling of biological pattern formation Curr Top Dev Biol 2008, 81:435–460
* [17] Berg JM, Tymoczko JL, Stryer L Biochemistry $5^{th}$ edition. New York: Freeman; 2002
* [18] Popov S, Poo MM: Diffusional transport of macromolecules in developing nerve processes. J Neurosci 1992, 12:77–85
* [19] Arrio-Dupont M, Cribier S, Foucault G, Devaux PF, d’Albis A: Diffusion of fluorescently labelled macromolecules in cultured muscle cells. Biophys J 1996, 70:2327–2332
* [20] Ainger K et al: Transport and localization of exogenous myelin basic protein mRNA microinjected into Oligodendrocytes. J Cell Biol 1993, 123:431 -441
* [21] Bassell G, Singer RH: mRNA and cytoskeletal filaments. Curr Opin Cell Biol 1997, 9:109–115
* [22] Fersht A Structure and mechanism in protein science. New York: Freeman; 1998
* [23] Baras F, Mansour MM: Reaction-diffusion master equation: A comparison with microscopic simulations. Phys Rev E 1996, 54:6139–6148
* [24] English BP, Min W, van Oijen AM, Lee KT, Luo G et al:Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited. Nat Chem Biol 2005, 2:87–94
* [25] Kou SC, Cherayil BJ, Min W, English BP, Xie SX: Single-molecule Michaelis-Menten equations. J. Phys. Chem B 2005, 109:19068–19081
* [26] Stefanini MO, McKane AJ, Newman TJ: Single enzyme pathways and substrate fluctuations. Nonlinearity 2005, 18:1575–1595
* [27] Qian H, Elson EL: Single-molecule enzymology: stochastic Michaelis-Menten kinetics. Biophys Chem 2002, 101-102:565–576
* [28] Grima R: Noise-induced breakdown of the Michaelis-Menten equation in steady-state conditions. Phys Rev Letts 2009, 102:218103
* [29] McKane AJ, Nagy J, Newman TJ, Stefanini M: Amplified biochemical oscillations in cellular systems J Stat Phys 2007, 128:165–191
* [30] Ngo LG, Roussel MR: A new class of biochemical oscillators based on competitive binding Eur J Biochem 1997, 245:182–190
* [31] Davis KL, Roussel MR: Optimal observability of sustained stochastic competitive inhibition oscillations at organellar volumes FEBS journal 2006, 273:84–95
* [32] Ramsey S, Orrell D, Bolouri H: Dizzy: Stochastic simulation of large-scale genetic regulatory networks J Bioinform Comput Biol 2005, 3: 415–436
Figure 1: Schematic illustrating the three models considered in this article.
(A) Model I: Michaelis-Menten reaction occurring in a compartment volume of
sub-micron dimensions (shown by dashed rectangle). Substrate input into
compartment occurs via a Poisson process i.e. diffusion-mediated substrate
transport. (B) Model II: As for Model I but now substrate is input into
compartment in groups or bursts of _M_ molecules at a time i.e. vesicle-
mediated substrate transport along microtubules (MT). (C) Model III:
Michaelis-Menten reaction with competitive inhibitor (_I_) occurring in a
small subcellular compartment. Substrate transport as in previous two models.
Figure 2: Deviations from the predictions of the MM equation for diffusion-
mediated substrate transport. (Model I) Plot of the Percentage Error in
reaction velocity, $R_{p}=100|1-\alpha_{M}/\alpha|$, versus the normalized
reaction velocity of the MM equation, $\alpha_{M}$ for 10 enzymes (green) and
100 enzymes (red) with $K_{M}=10\mu M$ in compartments with diameter $100$nm
(A) and $50$nm (B) . The solid lines show the theoretical predictions, as
encapsulated by Eqs. (7) and (8); the data points are obtained by stochastic
simulation (see Methods for details). Figure 3: Deviations from the
predictions of the MM equation for vesicle-mediated substrate transport.
(Model II) Testing the validity of the MM relationship at small substrate
concentrations for the case in which substrate input into compartments occurs
in bursts. The data is for 10 enzymes with $K_{M}=100\mu M$ in compartments of
diameter (A) 200nm (circles), (B) 100nm (diamonds) and (C) 50nm (crosses);
substrate is input $M=50$ molecules at a time. The deterministic prediction
for all three cases is the same MM equation shown by the green curve. In
contrast, the stochastic models, [Eqs. (7) and(9)], predict different rate
equations for each case (red solid lines). Data points are obtained by
stochastic simulation (see Methods for details). Note that
$v/v_{max}=\alpha_{M}$ and $\alpha$ for solid green and red lines
respectively. Figure 4: Effects of intrinsic noise on the inhibition of enzyme
activity in small compartments. (Model III) Plots of normalized enzyme
activity versus normalized inhibitor concentration (measured in units of the
total enzyme concentration $[E_{T}]$) for 10 enzymes with $K_{M}=100\mu M$ in
compartments of diameter 50nm and 100nm (inset). The colors correspond to:
(red) MM equation; (green) stochastic model, $M=20$; (blue) stochastic model,
$M=50$. The latter two curves are those predicted by theory [Eqs. (7)
and(12)]. Parameters same as mentioned in caption of Table 3 (except for [I],
which is a variable in the present case). Substrate concentrations chosen so
that at $[I]=0$, $v/v_{max}=0.909$ in all cases. Black dashed lines contrast
the inhibitor concentration required to decrease enzyme activity from 0.909 to
0.1 as predicted by the MM equation and the stochastic models. Note that
$v/v_{max}=\alpha_{M}$ and $\alpha$ for solid red and blue/green lines
respectively.
Table 1: Maximum Percentage error in reaction velocity from prediction of the MM equation for Model I. The copy number indicates the total number of enzyme molecules per compartment. Values in bold and in square brackets are those estimated by simulation; the italic values are obtained from the derived theoretical expressions, Eqs. (7) and (8). D/nm | $K_{M}=10\mu M$ | $100\mu M$ | $1000\mu M$ | Copy No.
---|---|---|---|---
50 | _11.83_ [17.00] | _4.09_ [4.33] | _0.59_ | 10
100 | _4.74_ [5.00] | _0.73_ [0.74] | _0.08_ | 10
200 | _0.90_ | _0.10_ | _0.01_ | 10
50 | _3.98_ [5.33] | _1.88_ [2.02] | _0.43_ | 100
100 | _2.10_ [2.23] | _0.52_[0.52] | _0.07_ | 100
200 | _0.61_ | _0.09_ | _0.01_ | 100
Table 2: Maximum Percentage error in reaction velocity from prediction of the MM equation for Model II. The copy number indicates the total number of enzyme molecules per compartment. Values in bold and in square brackets are those estimated by simulation; the italic values are obtained from the theoretical expressions, Eqs. (7) and (9). D/nm | $K_{M}=10\mu M$ | $100\mu M$ | $1000\mu M$ | Copy No.
---|---|---|---|---
50 | _225.40_ | _152.83_ [291.56] | _45.43_ | 10
100 | _161.59_ [331.66] | _52.74_ [58.39] | _6.82_ [6.99] | 10
200 | _65.09_ | _8.45_ [8.50] | _0.88_ | 10
50 | _32.97_ | _30.17_ [61.66] | _18.14_ | 100
100 | _30.78_ [66.03] | _19.76_[24.52] | _5.57_ [6.06] | 100
200 | _21.27_ | _6.61_ [6.91] | _0.85_ | 100
Table 3: Maximum Percentage error in reaction velocity from prediction of the MM equation for Model III. The total number of enzyme molecules per compartment is ten in all cases. Values in bold and in square brackets are those estimated by simulation; the italic values are obtained from the theoretical expressions, Eqs. (7) and (12). The parameters are: $k_{0}=10^{9}s^{-1}M^{-1},k_{1}=k_{3}=1000s^{-1},k_{4}^{0}=10^{7}s^{-1}M^{-1}$, and $[I]=10[E_{T}]$. D/nm | $K_{M}=10\mu M$ | $100\mu M$ | $1000\mu M$ | M (burst size)
---|---|---|---|---
50 | _67.8_ [76.8] | _67.8_ [76.5] | _67.8_ | 1
100 | _20.8_ [26.4] | _20.6_ [26.1] | _20.6_ | 1
200 | _2.8_ | _2.7_ | _2.7_ | 1
50 | _1001.8_ | _234.9_ [169.4] | _86_ | 50
100 | _343.7_ [345.5] | _73.4_[75.2] | _26.2_ [31.5] | 50
200 | _71.4_ | _11.3_ [11.5] | _3.6_ | 50
|
arxiv-papers
| 2009-10-23T13:42:20 |
2024-09-04T02:49:05.995942
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ramon Grima",
"submitter": "Ramon Grima",
"url": "https://arxiv.org/abs/0910.4469"
}
|
0910.4490
|
# Maximum entropy principle and the form of source in non-equilibrium
statistical operator method
V Ryazanov Institute for Nuclear Research, pr.Nauki, 47, Kiev, Ukraine
vryazan@kinr.kiev.ua
###### Abstract
It is supposed that the exponential multiplier in the method of the non-
equilibrium statistical operator (Zubarev‘s approach) can be considered as a
distribution density of the past lifetime of the system, and can be replaced
by an arbitrary distribution function. To specify this distribution the method
of maximum entropy principle as in [Schönfeldt J-H, Jiminez N, Plastino A R,
Plastino A, Casas M 2007 Physica A 374 573] is used. The obtained distribution
is close to exponential one. Another approach to the maximum entropy
principle, as in [Van der Straeten E and Beck C 2008 Phys. Rev. E 78 051101],
except exponential distributions yields power-like, log-normal distributions,
as well as distributions of other kind and transitions between them.
###### pacs:
05.70.Ln, 05.40.-a
## 1 Introduction
Among possible approaches to the description of non-equilibrium systems the
Non-equilibrium Statistical Operator Method (NESOM) especially demonstrated
its efficiency [1, 2, 3]. NESOM provides a very promising technique that
implies in a far-reaching generalization the statistical methods developed by
Boltzmann and Gibbs. NESOM was initially built on intuitive and heuristic
arguments, apparently it can be incorporated within an interesting approach to
the rationalization of statistical mechanics, as contained in the maximization
of (informational statistical) entropy (MaxEnt for short) and Bayesian
methods. NESOM appears as a very powerful, concise, based on sound principles,
and elegant formalism of a broad scope to deal with systems arbitrarily far
from equilibrium. The non-equilibrium statistical operator (NSO) introduced in
[1, 2, 3] has a form
$\ln\varrho(t)=\int_{0}^{\infty}p_{qzub}(u)\ln\varrho_{q}(t-u,-u)du,\quad\ln\varrho_{q}(t,0)=-\Phi(t)-\sum_{n}F_{n}(t)P_{n};\\\
$ (1)
$\ln\varrho_{q}(t,t_{1})=\rme^{\left\\{-t_{1}H/i\hbar\right\\}}\ln\varrho_{q}(t,0)\rme^{\left\\{t_{1}H/i\hbar\right\\}};\quad\Phi(t)=\ln
Sp\exp\\{-\sum_{n}F_{n}(t)P_{n}\\}\,,$
where
$p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon u}\,,\quad u=t-t_{0}\,,$
(2)
$H$ is Hamiltonian, $\ln\rho(t)$ is the logarithm of the NSO,
$\ln\rho_{q}(t_{1},t_{2})$ is the logarithm of the quasi-equilibrium (or
relevant) distribution; the first time argument indicates the time dependence
of the values of the thermodynamic parameters $F_{n}$; the second time
argument $t_{2}$ in $\ln\rho_{q}(t_{1},t_{2})$ denotes the time dependence
through the Heizenberg representation for dynamical variables $P_{n}$ on which
$\ln\rho_{q}(t,0)$ can depend [1, 2, 3].
In [1, 2, 3] $p_{q}(u)=p_{qzub}(u)=\varepsilon\exp\\{-\varepsilon u\\}$; after
the thermodynamic limiting transition $N\rightarrow\infty$,
$V\rightarrow\infty$, $N/V=const$, $\varepsilon\rightarrow 0$. From the
complete group of solutions of Liouville equation (symmetric in time) the
subset of retarded ”unilateral” in time solutions is selected by means of
introducing a source $K$ in the Liouville equation for $\ln\rho(t)$ ($L$ is
Liouville operator;
$iL=-\\{H,\varrho\\}=\sum_{k}{\displaystyle\left[\frac{\partial H}{\partial
p_{k}}\frac{\partial\varrho}{\partial q_{k}}-\frac{\partial H}{\partial
q_{k}}\frac{\partial\varrho}{\partial p_{k}}\right]}$; $p_{k}$, and $q_{k}$
are pulses and coordinates of particles; $\\{\dots\\}$ are Poisson brackets)
$\frac{\partial\ln\rho(t)}{\partial
t}+iL\ln\rho(t)=-\varepsilon(\ln\rho(t)-\ln\rho_{q}(t,0))=K_{zub}.$ (3)
In [4, 5] a convenient redefinition of the source term is proposed. Although
infinitesimally small, the source term introduced by Zubarev into the
Liouville equation is shown to influence the macroscopic behaviour of the
system in the sense that the corresponding evolution equations do not coincide
exactly with those obtained from an initial-value problem which corresponds to
a definite experimental situation and a physical set of macroobservables.
In [6, 7] it was noted that in place of the function
$p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon u}$ in (1) arbitrary (but
having certain properties [8]) weigth functions $w(t,t_{0})$ can be used.
Zubarev’s nonequilibrium statistical operator does satisfy Liouville equation,
but it must be borne in mind that the group of its solutions is composed of
two subsets, one corresponding to the retarded and second one to the advanced
solutions. The presence of the weight function $w(t,t_{0})$ (Abel’s kernel (2)
in Zubarev’s approach) in the time-smoothing or quasi-average procedure that
has been introduced selects the subset of retarded solutions from the total
group of solutions of the Liouville equation. This consideration is related to
the question: how to obtain an irreversible behavior in the evolution of the
macroscopic state of the system? In the MaxEnt-NESOM approach the
irreversibility is incorporated from the outset using an ad hoc non-mechanical
hypothesis. MaxEnt-NESOM yields information on the macrostate of the system at
time $t$, when a measurement is performed, including the evolutionary history
(in the interval from the initial time of preparation $t_{0}$ up to time $t$)
by which the system came into that state (which introduced a generalization of
Kirkwood‘s time-smoothing formalism [9]). Functions $w(t,t_{0})$ are typically
kernels [1, 2, 3, 8] that appear in the mathematical theory of convergence of
integrals. In [10, 11, 12] other interpretation of the functions $w(t,t_{0})$,
denoted as $p_{q}(u)$, is given. With the change of function
$w(t,t_{0})=p_{q}(u)$ the form of source (3) in the Liouville equation also
changes. For an arbitrary function $p_{q}(u)$ it looks like (4).
In [10, 11, 12] it was noted that the function
$p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon u}$ in NESOM [1, 2, 3] for
the non-equilibrium distribution function can be interpreted as the
exponential probability distribution of the lifetime $\Gamma$ of a system.
$\Gamma$ is a random variable of lifetime (time span) from the moment $t_{0}$
of its birth till the current moment $t$; $\varepsilon^{-1}=\langle
t-t_{0}\rangle=\langle\Gamma\rangle$, where $\langle\Gamma\rangle=\int
up_{q}(u)du$ is the average lifetime of the system. This time period can be
called the time period of getting information about system from its past.
Instead of the exponential distribution $p_{qzub}(u)$ (2) in (1) used in [1,
2, 3] any other sample distribution $p_{q}(u)$ could be taken; integration by
parts in time is performed at $\int p_{q}(u)du_{|u=0}=-1$; $\int
p_{q}(u)du_{|u\rightarrow\infty}=0$. If $p_{q}(u)=p_{qzub}(u)=\varepsilon
e^{-\varepsilon u}$ by (2), $\varepsilon=1/\langle\Gamma\rangle$ the
expression for NSO passes in (1) from [1, 2, 3].
The same interpretation of the distribution $p_{q}(u)$ is given in [2], where
this value is understood as the distribution of the initial moment of time
$t_{0}$. Since the random (past) lifetime is equal to $\Gamma=u=t-t_{0}$, the
distribution of the past lifetime ${u}$ coincides with the distribution of the
initial time values $t_{0}$. The moment $t_{0}$ will be the moment of the
first passage in the inverse time, if the moment $t$ is taken as initial. In
[2] the uniform distribution for an initial moment $t_{0}$ is chosen, which
after the transition from Abel integration to Cesàro integration passes to the
exponential distribution $p_{q}(u)=p_{qzub}(u)=\varepsilon\rme^{-\varepsilon
u}$. Such distribution serves as the limiting distribution of the lifetime
[13], the first-passage time of a certain level. In the general case it is
possible to choose a lot of functions for the obvious type of distribution
$p_{q}(u)$, which was noted in [10, 11, 12].
In [13] the lifetimes of the system are introduced as random moments of the
first-passage time till the moment when a random process describing system
reaches a certain limit, for example, a zero value. In [13] approximate
exponential expressions for the probability density function (with a single
parameter) and probability distribution of lifetime are obtained, and the
accuracy of these expressions is estimated.
In [14] it was noted, that the role of the form of the source term in the
Liouville equation in the NSO method has never been investigated. In [15] it
is stated that the exponential distribution is the only one which possesses
the Markovian property of the absence of afteraction, that is whatever is the
actual age of a system, the remaining time does not depend on the past and has
the same distribution as the lifetime itself. It is known [1, 2, 3] that the
Liouville equation for NSO contains the source
$K=K_{zub}=-\varepsilon[\ln\rho(t)-\ln\rho_{q}(t,0)]$ (3) which becomes
vanishingly small after taking the thermodynamic limit and setting
$\varepsilon\rightarrow 0$, which in the spirit of [10] corresponds to the
infinitely large lifetime value of an infinitely large system. For a system
with finite size this source is not equal to zero. In [8] this term enters the
modified Liouville operator and coincides with the form of Liouville equation
suggested by Prigogine [16] (the Boltzmann-Prigogine symmetry), when the
irreversibility is introduced in the theory at the microscopic level.
In [10] a new interpretation of the method of the NSO is given, in which the
operation of taking the invariant part [1, 2] or the use of an auxiliary
”weight function” (in the terminology of [6, 7, 8]) in NSO are treated as
averaging the quasi-equilibrium statistical operator over the distribution of
past lifetime of a system. This approach agrees with the approach of the
general theory of random processes, the renewal theory, and also with the
conception of Zubarev work [2] where the NSO is conceived as some averaging
over the initial moment of time.
The statistical operator depends on the information-gathering interval
$(t_{0},t)$, but it must be borne in mind that this is the formal point
consisting in that (as Kirkwood pointed out) that the description to be built
must contain all the previous history in the development of the macrostate of
the system. In [6, 7, 8] several basic steps for the construction of the NESOM
formalism are indicated: a third basic step has just been introduced, namely,
the inclusion of the past history (other terms used are retro-effects or
historicity) of the macrostate of the dissipative system. A fourth basic step
needs now to be considered, which is a generalization of Kirkwood’s time-
smoothing procedure: the one that accounts for the past history and future
dissipative evolution. The time-smoothing procedure introduces a kind of
Prigogine’s dynamical condition for dissipativity. The procedure introduces a
kind of evanescent history as the system macrostate evolves toward future from
the initial condition at time $t_{0}$. The function $w(t;t_{0})$ [6, 7, 8]
introduces the time-smoothing procedure. In principle, any kernel provided by
the mathematical theory of convergence of trigonometrical series and transform
integrals provides is acceptable for these purposes. Kirkwood, Green, Mori [9,
17, 18] and others have chosen what in mathematical terminology is Fejèr (or
Cesàro-1) kernel. Meanwhile Zubarev introduced the one consisting in Abel’s
kernel for $w$ in Eq. (1) - which apparently appears to be the best choice,
either mathematically but mostly physically: that is, taking
$w(t;t_{0})=\varepsilon\rme^{\varepsilon(t_{0}-t)}$, where $\varepsilon$ is a
positive infinitesimal value which tends to zero after the calculation of
averages has been performed, and with $t_{0}$ going to $-\infty$. Therefore a
process with fading memory is introduced. In Zubarev’s approach this fading
process occurs in an adiabatic-like form towards the remote past: as time
evolves memory decays exponentially with lifetime $\varepsilon^{-1}$ [8]. The
approach suggested in [10, 11, 12] and in the present work enables to use a
family of functions $w(t,t_{0})=p_{q}(u)$ and makes clear both their physical
sense and those physical situations in which one or another function
$w(t,t_{0})=p_{q}(u)$ can be used.
Besides the Zubarev’s form of NSO [1, 2, 3], the NSO formulation in the Green-
Mori form [17, 18] is known, where one assumes the auxiliary weight function
[6, 7, 8] to be equal to $W(t,t_{0})=1-(t-t_{0})/\tau$;
$w(t,t_{0})=dW(t,t_{0})/dt_{0}=1/\tau$; $\tau=t-t_{0}$. After averaging one
sets $\tau\rightarrow\infty$. This choice at $p_{q}(u=t-t_{0})=w(t,t_{0})$
coincides with the uniform lifetime distribution. The source in the Liouville
equation takes the form $K=\ln\rho_{q}/\tau$. In [1] this form of NSO is
compared to the Zubarev’s form. One could name many examples of explicitely
setting the function $p_{q}(u)$. Each and every definition implies some
specific form of the source term $K$ in the Liouville equation, some specific
form of the modified Liouville operator and NSO [10, 11, 12]. Thus the whole
family of NSO is defined.
It is possible to make different assumptions about the form of the function of
$p_{q}(u)$, getting different expressions for the source in the Liouville
equation and for non-equilibrium characteristics of the system. It is possible
to show [12] that certain choices of the function of $p_{q}(u)$ result in the
changes in non-equilibrium characteristics in the limit of infinitely large
average lifetimes as well. In [10, 11, 12] an analogy is traced to the passage
to the thermodynamic limit of systems of infinite size. So explicit form of
the function $p_{q}(u)$ is important for describing non-equilibrium systems by
the NSO method.
Setting the form of the function $p_{q}(u)$ reflects not only the internal
properties of a system, but also the influence of the environment on an open
system, the particular character of its interaction with the environment [8].
In [2] a physical interpretation of the exponential distribution for the
function $p_{q}(u)$ is given: a system evolves freely like an isolated system
governed by the Liouville operator. Besides that the system undergoes random
transitions, and the phase point representing the system switches from one
trajectory to another one with an exponential probability under the influence
of the ”thermostat”; the average intervals between successive push events
increase infinitely. This takes place if the parameter of the exponential
distribution tends to $0$ after the transition to the thermodynamic limit.
Real physical systems have finite sizes. The exponential distribution
describes completely random systems. The influence of the environment on a
system can have organized character as well, for example, this is the case of
systems in a stationary non-equilibrium state with input and output fluxes.
The character of the interaction with the environment can also vary; therefore
different forms of the function $p_{q}(u)$ can be used.
The adequate choice of the function $p_{q}(u)$ is important for correct
description of the non-equilibrium properties of statistical systems. To find
the type of function $p_{q}(u)$, it is necessary to resort to some general
principles, such as MaxEnt principle. In this work two variants of MaxEnt are
used for this purpose: the one introduced in [19] for the Liouville equation
with a source (Section 2) and that suggested in [26] for the superstatistics
(Section 3).
## 2 Maximum entropy principle for Liouville equations with source
In this paper we apply the maximum entropy principle for the determination of
the function $p_{q}(u)$. The same approach was applied in [19] for the
evolution equations with source terms. In [7, 10, 11, 12] a general form for
the source in the Liouville equation for $\ln\rho(t)$ (3) is obtained. For our
case the source term has the following form
$K=p_{q}(0)\ln\rho_{q}(t,0)+\int_{0}^{\infty}\frac{\partial p_{q}(u)}{\partial
u}\ln\rho_{q}(t-u,-u)du.$ (4)
In [19] in the Liouville equation the distribution function ${\rho}$(z,t) is
written in a form
$\rho(\overrightarrow{z},t)=Nf_{ME}(\overrightarrow{z},t)=\frac{N}{Z}\exp\\{-\sum_{i=1}^{M}\lambda_{i}A_{i}\\},$
(5)
where $A_{i}(\vec{z})$ are $M$ appropriate quantities that are functions of
the phase space point ${\vec{z}}$; the quantities ${A_{i}(\vec{z})}$
correspond to the values $P_{i}$ from (1). The partition function $Z$ is given
by
$Z=\int\exp\\{-\sum_{i=1}^{M}\lambda_{i}A_{i}\\}d^{N}z.$ (6)
The function ${f_{ME}(\vec{z},t)}$ is normalized to unity:
$\int f_{ME}(\overrightarrow{z},t)d^{N}z=1;$ (7)
$\int\rho(\overrightarrow{z},t)d^{N}z=N(t);\quad\frac{dN}{dt}=\int Kd^{N}z;$
(8) $\frac{\partial\rho}{\partial
t}+\overrightarrow{w}\nabla\rho=\frac{d\rho}{dt}=K;\quad\nabla\overrightarrow{w}=0.$
The probability distribution ${f_{ME}(\vec{z},t)}$ is the one that maximizes
the entropy $S[f]$ under the constraints imposed by normalization and relevant
mean values $\langle A_{i}\rangle=\int A_{i}\rho d^{N}z$ (or $a_{i}=\langle
A\rangle_{i}/N)$. The re-scaled mean values $a_{i}$ and the associated
Lagrange multipliers $\lambda_{i}$ are related by the Jayne’s relations [20,
21]
$\lambda_{i}=\frac{\partial S}{\partial a_{i}},\quad\ a_{i}=\frac{\langle
A_{i}\rangle}{N}=-\frac{\partial}{\partial\lambda_{i}}(\ln Z)\,,$ (9) $S=-\int
f\ln fd^{N}z=\ln Z+\sum_{i}\lambda_{i}a_{i}.$
If we choose for $\ln\rho(z,t)$ the function
$\ln\rho=\ln\varrho(t)=\int_{0}^{\infty}p_{q}(u)\ln\varrho_{q}(t-u,-u)du=\int_{-\infty}^{t}p_{q}(t-t_{0})\ln{\rho}_{q}(t_{0},t_{0}-t){dt}_{0}$
(10)
from (1), which is included in the Liouville equation (3) (without integrating
over time) with $p_{qzub}\rightarrow p_{q},K_{zub}\rightarrow K$, and,
following [1], choose
$\mathit{\lambda}_{i}=p_{q}(t-t_{0})F_{i}(t_{0}),$ (11)
then
$\displaystyle\ln{\rho}(\vec{z},t)=\int_{-\infty}^{t}p_{q}(t-t_{0})\ln{\rho}_{q}(t_{0},t_{0}-t){dt}_{0}=$
(12)
$\displaystyle-\int_{-\infty}^{t}(\sum_{i}\lambda_{i}A_{i}+p_{q}(t-t_{0})\ln
Z_{1}){dt}_{0}=\ln f^{\prime}_{ME}+\ln N\,,$
where
$\ln f^{\prime}_{ME}=-\int_{-\infty}^{t}\sum_{i}\lambda_{i}A_{i}{dt}_{0}-\ln
Z_{\lambda};$ $\ln N={\Delta}Z=\ln
Z_{\lambda}-\int_{-\infty}^{t}p_{q}(t-t_{0})\ln Z_{1}{dt}_{0};\ $ (13)
$Z_{\lambda}=\int\exp[-\int_{-\infty}^{t}\sum_{i}\lambda_{i}A_{i}{dt}_{0}]d^{N}z,\quad
Z_{1}=\int\exp[-\sum_{i}F_{i}A_{i}]d^{N}z$
($F_{i}$ are taken from (1)). The values $Z_{\lambda}$ and $Z_{1}$ in the
terminology of [1] are related to the partition functions for a non-
equilibrium and relevant statistical operator accordingly.
In [19] for the Liouville equation of the kind (3) with constant sources
equation one gets for $d\lambda_{i}/dt$
$\frac{d\lambda_{i}}{dt}=(\sum_{i=1}^{M}C_{ji}{\lambda}_{j})-\frac{1}{N}\frac{\partial}{\partial
a_{i}}\int{K\ln f_{ME}}d^{N}z,$
where the Zubarev-Peletminskiy selection rule [22, 23, 1, 7, 8]
$\vec{{w}}\vec{\nabla}A_{i}=\sum_{j=1}^{M}{C_{ij}}A_{j},\ (i=1,\dots,M),\
\frac{d\vec{z}}{dt}=\vec{w}(\vec{z});\ \frac{\partial\mathit{\rho}}{\partial
t}+\vec{\nabla}(\rho\vec{w})=K$ (14)
is used; $i,j=1,2...$; the $C_{ij}$ are c-numbers. In other representations
the quantities $A_{i}$ can depend on the space variable, that is, when
considering local densities of dynamical variables, and then the $C_{ij}$ can
depend on the space variable as well or be differential operators.
If more complex shape of the source (4) is considered, the equation for
$d\lambda_{i}/dt$ takes on the form
$\displaystyle\frac{d\lambda_{i}}{dt}=\left(\sum_{j=1}^{M}C_{ji}{\lambda}_{j}\right)-\frac{\partial}{\partial
a_{i}}\left(\frac{1}{N}\int{K\ln f_{ME}d^{N}z}\right)-$ (15)
$\displaystyle\sum_{j}{\lambda}_{j}\frac{\partial}{\partial
a_{i}}\left({\displaystyle\frac{1}{N}}\int A_{j}Kd^{N}z\right)-\ln
Z_{1}{\displaystyle\frac{\partial}{\partial
a_{i}}}\left({\displaystyle\frac{\dot{N}}{N}}\right).$
We replace the operators ${\displaystyle\frac{\partial}{\partial a_{i}}}$ and
${\displaystyle\frac{\partial}{\partial\mathit{\lambda}_{i}}}$ taking into
account (9)-(13) by the functional differentiation of the kind
$\frac{\partial}{\partial a_{i}}\rightarrow\frac{\delta}{\delta
a_{i}}=N\frac{\delta}{\delta\langle
A_{i}\rangle},\quad\frac{\partial}{\partial\lambda_{i}}\rightarrow\frac{\delta}{\delta\lambda_{i}},$
which takes off the integration over time. For example
$\frac{\delta}{\delta\langle A_{i}\rangle}\ln
Z_{\lambda}=-p_{q}(t-t_{0})\sum_{k}\frac{\partial F_{k}}{\partial\langle
A_{i}\rangle}\langle A_{k}\rangle;\quad\frac{\delta\ln
Z_{\lambda}}{\delta\lambda_{i}}=-\langle A_{i}\rangle.$
The relations (5-8) and (9) are thus hold. If to take into account that
$\int\dots\rho_{q}d^{N}z=\int\dots\rho d^{N}z=\langle\dots\rangle$ in the NSO
method, then
$\frac{\partial}{\partial a_{i}}N=\frac{\partial}{\partial
a_{i}}\left(\frac{\dot{N}}{N}\right)=0\,.$ (16)
Let us consider the integrals in the rhs of (15) of the form $\int KB(z)dz$,
$B$ being an arbitrary function of the dynamic variables $z$, and the source
term $K$ taken from (4); for Eq. (8), (14)
$K=[p_{q}(0)\ln\rho_{q}(t,0)+{\displaystyle\int_{-\infty}^{t}\frac{\partial
p_{q}(t-t_{0})}{\partial t}}\ln\rho_{q}(t_{0},t_{0}-t)dt_{0}]\rho$. Assume
that ${p_{q}(u)}$ does not depend on $z$. Integrating by parts and assuming
${p_{q}(u)_{u\rightarrow\infty}\rightarrow 0}$, we get:
$\int KBd^{N}z=-\int_{0}^{\infty}p_{q}(u)\frac{d}{du}\langle
B\ln\rho_{q}(t-u,-u)\rangle du;\quad u=t-t_{0}.$ (17)
Taking into account (16) and the fact that the operation
${\displaystyle\frac{\partial}{\partial
a_{i}}\rightarrow\frac{\mathit{\delta}}{\mathit{{\delta}a}_{i}}}$ eliminates
the integration by time, the equation (15) takes on the form
${F_{i}(t_{0})\frac{dp_{q}(t-t_{0})}{dt}=-p_{q}(t-t_{0})C_{i}-p_{q}(t-t_{0})r_{1}-p_{q}^{2}}(t-t_{0})r_{2},$
(18)
where $C_{i}=\sum_{j}C_{ji}F_{j}(t_{0})$,
$r_{1}=-\frac{\partial}{\partial\langle
A_{i}\rangle}\frac{d}{dt}\langle(\ln{\rho}(t)-\ln
N)\ln{\rho}_{q}(t_{0},t_{0}-t)\rangle;$ (19)
$r_{2}=-\sum_{j}{F_{j}(t_{0})r_{2j};\;\quad
r_{2j}=\frac{\partial}{\partial\langle A_{i}\rangle}\frac{d}{dt}\langle
A_{j}(t)\ln{\rho}_{q}(t_{0},t_{0}-t)}\rangle.$ (20)
An unknown function $p_{q}(u)$ enters the expression (19) through the terms
$\ln\rho(t)$ and $\ln N$. To get rid of this dependence, we use the averaging
theorem. For the expressions for $\ln{\rho}(t)$ and $\ln N$ in (19) we take
all terms besides $p_{q}(u)$ out of the time integration. For each of these
function however a different effective average time value should be used. The
remaining integrals over ${p_{q}(u)}$ are equal to unity. We get:
$\ln N\simeq\ln Z_{1}(c_{3})-\ln Z_{1}(c_{4}),$ (21)
$\ln{\rho}(t)\simeq\ln{\rho}_{q}(c_{1},c_{1}-t)=-(\sum_{m}F_{m}(c_{1})A_{m}(c_{1}-t)+\ln
Z(c_{1})).$ (22)
Let us make another approximation and change the order of the operations
${\partial/\partial\langle A_{i}\rangle}$ and $d/dt$ in the expressions
(19)-(20). The value $\int\limits_{t_{0}}^{t}r_{1}dt=D(t)-D(t_{0})$ enters the
expression (18), where
$\displaystyle D(t)=-\frac{\partial}{\partial\langle
A_{i}\rangle}\left\langle(\ln{\rho}(t)-\ln
N)\ln{\rho}_{q}(t_{0},t_{0}-t)\right\rangle$ $\displaystyle=F_{i}(t_{0})[\ln
Z(c_{3})-\ln Z(c_{4})-\ln Z(c_{1})]-$ (23) $\displaystyle F_{i}(c_{1})\ln
Z(t_{0})+\sum_{m,n}\big{(}\langle A_{m}A_{n}\rangle-\langle
A_{m}\rangle\langle A_{n}\rangle\big{)}\left[\frac{F_{n}(t_{0})}{\langle
A_{m}A_{i}\rangle-\langle A_{m}\rangle\langle A_{i}\rangle}-\right.$
$\displaystyle\left.-\frac{F_{m}(c_{1})}{\langle A_{i}A_{n}\rangle-\langle
A_{i}\rangle\langle
A_{n}\rangle}\right]-\sum_{m,n}F_{m}(c_{1})F_{n}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{m}A_{n}\rangle-\langle A_{m}A_{n}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle A_{k}\rangle};$
$\displaystyle r_{2j}=\frac{\partial}{\partial
t}\left[\sum_{m}F_{m}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{j}A_{m}\rangle-\langle A_{j}A_{m}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle A_{k}\rangle}+\delta_{ij}\ln
Z(t_{0})\right.$ $\displaystyle\left.-\sum_{m}\frac{\langle
A_{j}A_{m}\rangle-\langle A_{j}\rangle\langle A_{m}\rangle}{\langle
A_{i}A_{m}\rangle-\langle A_{i}A_{m}\rangle}\right]\,.$ (24)
The values of the correlators $\langle A_{i}\rangle$, $\langle
A_{i}A_{k}\rangle$, $\langle A_{j}A_{k}A_{m}\rangle$ are averaged with
$\rho(t)$ and are $t$-dependent. In deriving (23), (24) we used the relations
like
$\frac{\partial\ln Z(c)}{\partial\langle A\rangle}=\sum_{n}\frac{\partial\ln
Z(c)}{\partial F_{n}(c)}\frac{\partial F_{n}(c)}{\partial\langle
A\rangle}=\sum_{n}\frac{\langle A_{n}\rangle}{\langle AA_{n}\rangle-\langle
A\rangle\langle A_{n}\rangle};\quad A_{n}(-c)=e^{\textstyle-icL}A_{n};$
$\frac{\partial F_{m}}{\partial\langle A_{i}\rangle}=\frac{1}{\langle
A_{i}A_{m}\rangle-\langle A_{i}\rangle\langle A_{m}\rangle}.$
One can proceed with the expression (24) using the relations
$\frac{\partial F_{i}}{\partial t}=\sum_{k}\frac{\partial
F_{i}}{\partial\langle A_{k}\rangle}\frac{\partial\langle
A_{k}\rangle}{\partial t}=\sum_{k}{\frac{\partial F_{k}}{\partial\langle
A_{i}\rangle}\frac{\partial\langle A_{k}\rangle}{\partial
t};\;\;\frac{\partial F_{i}}{\partial\langle A_{k}\rangle}=\frac{\partial
F_{k}}{\partial\langle A_{i}\rangle}=\frac{\partial^{2}S}{\partial\langle
A_{k}\rangle\partial\langle A_{i}\rangle}\,;}$
$\frac{\partial\ln Z}{\partial t}=-\sum_{m}\frac{\partial F_{m}}{\partial
t}\langle A_{m}\rangle.$
The solution to (18) has the form
$p_{q}(t-t_{0})=\frac{p_{q}(0)M}{1+p_{q}(0){\displaystyle\int\limits_{t_{0}}^{t}}{\displaystyle\frac{1}{F_{i}}}r_{2}Mdt}\,.$
(25)
Since $C_{i}$ and $F_{i}(t_{0})$ does not depend on $t$,
$\displaystyle
M(t)=\exp\left\\{-\frac{C_{i}}{F_{i}}(t-t_{0})-\frac{1}{F_{i}}\int_{t_{0}}^{t}r_{1}dt\right\\}=$
$\displaystyle\exp\left\\{-\frac{C_{i}}{F_{i}}(t-t_{0})-\frac{1}{F_{i}}(D(t)-D(t_{0}))\right\\}\
,$ (26)
$M(t_{0})=1$, where $\int_{t_{0}}^{t}r_{1}dt$ is written in (23).
Integrating by parts the second term in the denominator of (25) write it in
the following form:
$p_{q}(t-t_{0})=\frac{p_{q}(0)M(t)}{1-L};$ (27)
$L=p_{q}(0)\frac{1}{F_{i}}\left[\left(\int r_{2}dt\right)_{|t_{0}}-\left(\int
r_{2}dt\right)_{|t}M(t)-\int_{t_{0}}^{t}\left(\int
r_{2}dt\right)\left(\frac{C_{i}}{F_{i}}+\frac{1}{F_{i}}\frac{dD(t)}{dt}\right)M(t)dt\right],$
where
$\displaystyle\int
r_{2}{dt}=\sum_{j}\sum_{m}F_{m}(t_{0})F_{j}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{j}A_{m}\rangle-\langle A_{j}A_{m}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle A_{k}\rangle}+$ (28)
$\displaystyle F_{i}\ln Z(t_{0})-\sum_{m}\sum_{j}F_{j}(t_{0})\frac{\langle
A_{j}A_{m}\rangle-\langle A_{j}\rangle\langle A_{m}\rangle}{\langle
A_{i}A_{m}\rangle-\langle A_{i}\rangle\langle A_{m}\rangle};$
$\displaystyle\frac{dD(t)}{dt}=\sum_{m,n}\left\\{F_{n}(t_{0})\frac{\langle
A_{m}A_{n}\rangle-\langle A_{m}\rangle\langle A_{n}\rangle}{\langle
A_{m}A_{i}\rangle-\langle A_{m}\rangle\langle
A_{i}\rangle}\left[\frac{d}{dt}\ln(\langle A_{m}A_{n}\rangle-\langle
A_{m}\rangle\langle A_{n}\rangle)-\right.\right.$ (29)
$\displaystyle\left.-\frac{d}{dt}\ln(\langle A_{m}A_{i}\rangle-\langle
A_{m}\rangle\langle A_{i}\rangle)\right]+F_{m}(c_{1})\frac{\langle
A_{m}A_{n}\rangle-\langle A_{m}\rangle\langle A_{n}\rangle}{\langle
A_{i}A_{n}\rangle-\langle A_{i}\rangle\langle A_{n}\rangle}\times$
$\displaystyle\left[\frac{d}{dt}\ln(\langle A_{m}A_{n}\rangle-\langle
A_{m}\rangle\langle A_{n}\rangle)-\frac{d}{dt}\ln(\langle
A_{i}A_{n}\rangle-\langle A_{i}\rangle\langle A_{n}\rangle)\right]-$
$\displaystyle-F_{m}(c_{1})F_{n}(t_{0})\sum_{k}\frac{\langle
A_{k}A_{m}A_{n}\rangle-\langle A_{m}A_{n}\rangle\langle A_{k}\rangle}{\langle
A_{i}A_{k}\rangle-\langle A_{i}\rangle\langle
A_{k}\rangle}\left[\frac{d}{dt}\ln(\langle A_{k}A_{m}A_{n}\rangle-\langle
A_{m}A_{n}\rangle\langle A_{k}\rangle)-\right.$
$\displaystyle\left.\left.-\frac{d}{dt}\ln(\langle A_{i}A_{k}\rangle-\langle
A_{i}\rangle\langle A_{k}\rangle)\right]\right\\}\,.$
The expression for $M(t)$ is given in (26), and $p_{q}(0)$ is determined from
the conditions for the norm $\int_{0}^{\infty}p_{q}(u)du=1$ and for the
average lifetime $\langle\Gamma\rangle=\int_{0}^{\infty}up_{q}(u)du$.
If one either considers the stationary case or assumes a weak time dependence
in the correlators in (23)-(24), $D(t)\simeq D(t_{0}),r_{2}\simeq 0$, and
(25)-(27) take on the form
$p_{q}(t-t_{0})=p_{q}(0)\exp\\{-{\frac{C_{i}}{F_{i}}}(t-t_{0})\\}$ (30)
with $C_{i}/F_{i}=p_{q}(0)=1/\langle\Gamma\rangle$. In (30) an exponential
distribution for $p_{q}(u)$, used in [1, 2, 3] is obtained. However generally
the correlators in (23)-(24) are time dependent. Applying the full form of
(27) for concrete systems it is possible to state the conditions where the
denominator in (27) is considerably different from unity, hence the lifetime
distribution essentially deviates from the exponential one (30).
The term $L$ in the denominator of (27) is small, since $p_{q}(0)\approx
1/\langle\Gamma\rangle\ll 1$. In the denominator of (27) it stands in the
combination with 1, $1-L\approx 1$ . Therefore one can write
$p_{q}=p_{q}(0)M[1+L+L^{2}+\dots]$ , and the distribution (27) is close to the
exponential distribution (30) used in [1, 2, 3, 7, 8]. This results agrees
with the results of [13], where the exponential distribution for the lifetime
is shown to be a limiting distribution. But the expression for
$\langle\Gamma\rangle$ is explicitely given in (30), and in (23)-(29). It was
already pointed above that the situations where L is comparable to 1 can arise
as well.
For the maximum entropy principle with the Shannon measure of the information
entropy the exponential distribution used in [1, 2, 3], is basic. Choosing
another measures for the information entropy (e.g. Tsallis and Renyi measures
[24, 25]) changes the function $f_{ME}$ (5), which yields another forms of the
lifetime distributions. For the NSO method the functions $f_{ME}$ are in the
form (5), and the information entropy is given by the Shannon measure, hence
basic distribution is the exponential one.
The expressions (25), (27) for $p_{q}(u)$ depend on $F_{i}$, the functions
$C_{i}=\sum_{j}C_{ji}F_{j}$, $r_{k},\;k=1,2$, in (18)-(19) depend on the index
$i$, the function $p_{q}(u,i)$ depends on $i$ as well. One can get an
$i$-independent function $p_{q}(u)$ by symmetrizing the distribution, for
example, using the operation
$p_{q}(u)=\left[\prod_{i=1}^{M}p_{q}(u,i)\right]^{\textstyle\frac{1}{M}}.$
Such formulation of maxent principle, as in [19], gives the distribution for
the lifetime, related to the exponential distribution which serves in this
case as the base one. Distributions of other type can be obtained using some
other form of maxent principle.
## 3 Another approach to the method of maximum entropy
Yet another approach to the determination of the type of function of
distribution of lifetime is related to the method of maximal entropy inference
(”maxent”), developed in [26] for the determination of the distribution of
superstatistics. We note a formal similarity between the superstatistics
method where the averaging is performed over the parameter $\beta$, (for
example, the inverse temperature)
$p(E)\propto\int_{0}^{\infty}f(\beta)e^{-\beta E}d\beta\,,$
and the NSO method where the averaging is performed over the past life spans
$u=t-t_{0}$,
$\ln\rho(t)=\int_{0}^{\infty}p_{q}(u)\ln\rho_{q}(t-u,-u)du;\ u=t-t_{0},$
which was already used in [27]. Therefore the maxent method of [26] can be
applied for determining the function ${p_{q}(u)}$. The analogy here is not
merely formal. The principal assumption of [26], that is the separation of the
time scales is essential for the NSO method as well [1, 2, 3]. In the approach
of superstatistics [28, 29, 30] the system is split into cells and local
fluctuations of the value $\beta$ are considered; the fluctuations of the
value $u=t-t_{0}$ affect the complete system.
Closely related is also the research of Crooks [31]. He studies general non-
equilibrium systems, without assuming that the system can be divided into
different cells that are at local equilibrium. Crooks claims that instead of
trying to obtain the probability distribution of the entire non-equilibrium
system, one has to try to estimate the “metaprobability,” the probability of
the microstate distribution. Crooks also uses the maximum entropy principle
but sets in (31) $\lambda_{3}=0$. The main difference is that Crooks does not
assume a local equilibrium in the cells, hence his approach, though being an
interesting theoretical construction, does not give a straightforward physical
interpretation to the fluctuating parameter $\beta$. However such an approach
can be applied to the fluctuations of the value $u=t-t_{0}$. In the approach
of [26] one obtains a local fluctuating temperature that coincides with the
thermodynamic temperature and which can in principle be measured. The work of
Crooks is used by Naudts [32] to describe equilibrium systems. The author
shows that some well-known results of the equilibrium statistical mechanics
can be reformulated in a very general context with the use of the concepts
introduced in [26, 29, 31].
In [26] following expression is obtained for the distribution function of the
superstatistics $f(\beta;\lambda_{i})$
$f(\beta;\lambda_{i})=\frac{Z(\beta)^{-\lambda_{1}/V}}{Z(\lambda_{i})}\exp(-\beta\lambda_{2}\frac{E(\beta)}{V}-\lambda_{3}g(\beta)),$
where $\lambda_{i}$ are Lagrange multipliers, $V$ being an arbitrary constant
(taking out a common factor out of the definition of $\lambda_{1}$ and
$\lambda_{2}$ will turn out to be useful in the following). Using the well-
known formula $S_{\beta}=\ln Z(\beta)+\beta E(\beta)$ with $Z(\lambda_{i})$
being a normalization constant that is fixed by the condition $\langle
1\rangle_{\beta}=1$.
The same approach with said limitations used for the function $p_{q}(u)$,
yields
$p_{q}(u;\lambda_{i})=\frac{Z(t-u)^{-\lambda_{1}/V}}{Z(\lambda_{i})}\exp(-\lambda_{2}\frac{\sum_{m}F_{m}(t-u)\langle
A_{m}\rangle}{V}-\lambda_{3}{g(u))},$ (31)
where $g(u)$ is an arbitrary function of $u$, which is determined by the
physical peculiarities of the behaviour of the system in one or another period
of its history. Expression (31) is obtained by the optimization of the entropy
$S(\lambda_{i})=\int p_{q}(u)\ln p_{q}(u)du$ with the constraints for entropy
$\int p_{q}(u)S(u)du=\int
p_{q}(u)\int\rho_{q}(t-u,-u)\ln\rho_{q}(t-u,-u)dzdu=$ $\int
p_{q}(u)[-\sum_{m}F_{m}(t-u)\langle A_{m}\rangle-\ln Z(t-u)]du$
and parameters $\int p_{q}(u)\sum_{m}F_{m}(t-u)\langle A_{m}\rangle du$.
Similarly to [26], one can set the functions $g(u)$ in a different fashion.
For example, for
$g(u)=u,\lambda_{1}=\lambda_{2}=0,\lambda_{3}=1/\langle\Gamma\rangle$, where
$\langle\Gamma\rangle$ is the average span of past life of a system (till the
present time moment), we get the exponential distribution used in [1, 2, 3].
Setting $g(u)=\ln u$ with appropriate corresponding values of $\lambda$ one
gets the power-like distribution for ${p_{q}(u)}$; setting $g(u)=(\ln u)^{2}$
with corresponding $\lambda$ results in the log-normal distribution. Thus
setting the function $g(u)$ and $\lambda$ accordingly it is possible to obtain
various distributions for the lifetime considered, for example, in [33].
It is possible to examine more difficult cases when the behaviour of the
system changes at different stages of its evolution, when, for example the
function $g(u)=\ln u$ yields the power-like function $p_{q}(u)$ at $u<c$, and
$g(u)$ gives an exponential shape of $p_{q}(u)$ at $u>c$.
## 4 Conclusion
For the determination of the lifetime distribution in the NSO method the
method of maximum entropy principle as in [19] is used. The obtained
distribution is close to exponential $p_{qzub}(u)$ (2), but does not coincide
with it. It is possible to find conditions at which this difference is
essential. Using other variants of the maximum entropy principle, as in [26],
it is possible to obtain other distributions except exponential one, in
particular, power-like and log-normal distributions, transitory ones between
them, as well as distributions of other classes.
In the interpretation of [2] it is the random value $t_{0}$ in $u=t-t_{0}$
that fluctuates. In [2] the limiting transition is performed for the parameter
$\varepsilon,\varepsilon\rightarrow 0$ in the exponential distribution
$p_{q}(u)=\varepsilon\exp\\{-\varepsilon u\\}$ after the thermodynamic
limiting transition. In the interpretation of [10] it corresponds to the
average lifetime of a system tending to infinity:
$\langle\Gamma\rangle=\langle t-t_{0}\rangle=1/\varepsilon\rightarrow\infty$.
But the average intervals between successive random jumps grow infinitely,
getting larger than the lifetime of a system. Therefore the source term in the
Liouville equation turns to 0. If however the distribution ${p_{q}(u)}$
changes over the interval of the lifetime, the influence of the environment
which caused this change, remains within the life span even if the lifetime
tends to infinity [12].
There are numerous experimental confirmations for such change of the lifetime
distribution $p_{q}(u)$ over the interval of the system lifetime. The examples
thereto are the transition to chaos and the transition from laminar to
turbulent flow which are accompanied by the change of the distribution of
$p_{q}(u)$. In [34, 35] the passage from Gaussian to non-Gaussian behaviour of
the distribution of the first-passage time for some time moment is
demonstrated. Besides the real systems possess finite sizes and finite
lifetime. Therefore influence of surroundings on them is always present.
Slow change of the function $g(u)=u=t-t_{0}$ corresponds in the interpretation
of [2] to the slow change of the random value $t_{0}$ on a temporal scale.
Accordingly slow change of the function $g(u)=\ln u$ corresponds to the slow
change of $\ln(t-t_{0})$. Setting other functions $g(u)$, for example,
$g(u)=(\ln u)^{2}$ and so on is explained on the same footing.
In the present work by means of two variants of the method of maximum entropy
we obtained the expressions for the distribution of the lifetime value. It is
noted that the choice of the form of the distribution function for the
lifetime value can affect the non-equilibrium behavior of a system even after
performing the thermodynamic limiting transition.
## References
## References
* [1] Zubarev D N 1974 _Nonequilibrium statistical thermodynamics_ (New York: Plenum-Consultants Bureau)
* [2] Zubarev D N 1980 in: Reviews of Science and Technology: _Modern Problems of Mathematics_ , 15, 131-226, (in Russian) ed. by R.B. Gamkrelidze (Moscow: Nauka) [English Transl.: J. Soviet Math. 16 (1981) 1509]
* [3] Zubarev D N, Morozov V and Röpke G 1996 Statistical mechanics of nonequilibrium processes vol 1 Basic Concepts, Kinetic Theory (Berlin: Akad. Verl.)
* [4] Der R, Röpke G 1983 Phys. Lett.A 95A 347
* [5] Der R 1985 PhysicaA 132A 74
* [6] Algarte A C, Vasconcellos A R, Luzzi R and Sampaio A J 1985 Revista Brasileira de Flsica 15 106
* [7] Ramos J G, Vasconcellos A R and Luzzi R 1999 Fortschr. Phys./Progr. Phys. 47 937
* [8] Luzzi R, Vasconcellos A R and Ramos J G 2000 Statistical Foundations of Irreversible Thermodynamics (Stuttgart: Teubner-BertelsmannSpringer)
* [9] Kirkwood J G, 1946 J.Chem.Phys. 14 180; J.Chem.Phys. 15 72\.
* [10] Ryazanov V V 2001 Fortschritte der Phusik/Progress of Physics 49 885
* [11] Ryazanov V V 2007 Low Temperature Physics 33 1049
* [12] Ryazanov V V 2009 cond-mat:0902.1454
* [13] Stratonovich R L 1967 The elected questions of the fluctuations theory in a radio engineering (New York: Gordon and Breach)
* [14] Morozov V G and Röpke G, 1998 Condensed Matter Physics 1 673
* [15] Feller W 1971 An Introduction to Probability Theory and its Applications vol.2 (New York: J.Wiley)
* [16] Prigogine I 1980 From Being to Becoming (San Francisco: Freeman)
* [17] Green M 1952 J. Chem. Phys. 20 1281; 1954 ibid. 22 398
* [18] Mori H, Oppenheim I and Ross J 1962 in Studies in Statistical Mechanics I, edited by J. de Boer and G.E. Uhlenbeck, (Amsterdam: North-Holland) pp. 217-298
* [19] Schönfeldt J-H, Jiminez N, Plastino A R, Plastino A and Casas M 2007 Physica A 374 573
* [20] Jaynes E T 1957 Phys. Rev. 106 620
* [21] Jaynes E T 1957 Phys. Rev. 108 171
* [22] Peletminskii S V and Yatsenko A A 1968 Soviet Phys JETP26 773; 1967 Zh. Eksp. Teor. Fiz. 53, 1327
* [23] Akhiezer A I and Peletminskii S V 1981 Methods of statistical physics (Oxford: Pergamon)
* [24] Tsallis C 1988 J. Stat.Phys. 52 479; http://tsallis.cat.cbpf.br/biblio.htm .
* [25] Rényi A 1961 Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 547
* [26] Van der Straeten E and Beck C 2008 Phys. Rev. E 78 051101
* [27] Ryazanov V V 2007 cond-mat:07101764
* [28] Beck C and Cohen E G D 2003 Physica A 322 267
* [29] Beck C and Cohen E G D and Swinney H L 2005 Phys. Rev. E 72 056133
* [30] Beck C and Cohen E G D 2004 Physica A 344 393
* [31] Crooks G E 2007 Phys. Rev. E 75 041119
* [32] Naudts J 2007 AIP Conf. Proc. No. 965 (NY: AIP, Melville) p. 84
* [33] Cox D R and Oakes D 1984 Analysis of Survival Data (London, New York: Chapman and Hall)
* [34] Inoue Jun-ichi and Sazuka N 2007 Physical Review E 76 021111(9)
* [35] Mantegna R N and Stanley H E 1994 Phys. Rev. Lett, 73 2946
|
arxiv-papers
| 2009-10-23T12:10:53 |
2024-09-04T02:49:06.005322
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. V. Ryazanov",
"submitter": "Vasiliy Ryazanov",
"url": "https://arxiv.org/abs/0910.4490"
}
|
0910.4523
|
# Flavourful hadronic physics
B. El-Bennich, M. A. Ivanov and C. D. Roberts[ANL] Physics Division, Argonne
National Laboratory, Argonne, Illinois 60439, USA Bogoliubov Laboratory of
Theoretical Physics, JINR, 141980 Dubna, Russia Department of Physics, Peking
University, Beijing 100871, China
###### Abstract
We review theoretical approaches to form factors that arise in heavy-meson
decays and are hadronic expressions of non-perturbative QCD. After motivating
their origin in QCD factorisation, we retrace their evolution from quark-model
calculations to non-perturbative QCD techniques with an emphasis on
formulations of truncated heavy-light amplitudes based upon Dyson-Schwinger
equations. We compare model predictions exemplarily for the
$F^{B\to\pi}(q^{2})$ transition form factor and discuss new results for the
$g_{D^{*}\\!D\pi}$ coupling in the hadronic $D^{*}$ decay.
## 1 Flavour physics and strong phases
The last two decades have witnessed important advances in flavour physics and
in particular heavy-meson decays. From the first observation of a $B$ meson by
the CLEO Collaboration in 1981 at the Cornell Electron Storage Ring [1] (and
their ongoing $D$-meson research program) to the dedicated $B$-physics
facilities at SLAC in California and KEK in Japan, much progress has been
made. Of course, while $B$ physics is the main focus of the Collaborations
Belle at KEK and BaBar at SLAC, and of the CDF experiment at Fermilab,
considerable efforts have also been devoted to studies of $D$-meson decays,
charmonium and $\tau$ physics.
Naturally, the driving force is to confirm the electroweak sector of the
Standard Model which has established itself as the foremost paradigm to
describe $CP$ violation; in other words, the main task experimentalists strive
for is the precise area of the Cabibbo-Kobayashi-Maskawa (CKM) triangle and
the weak $CP$ violating phase codified within its angles. In order to measure
the size and exact form of this triangle, the angles are derived from
branching fractions in a nowadays large variety of decay channels. We focus on
non-leptonic decays, $B\to M_{1}M_{2}$, but our discussion also applies to the
cleaner semi-leptonic $B\to M\ell\bar{\nu}_{\ell}$ case.
From a theoretical point of view, heavy mesons can be used to test
simultaneously all the manifestations of the Standard Model, namely the
interplay between electroweak and strong interactions. They also provide an
excellent playground to examine non-perturbative QCD effects already much
studied in hadronic physics. It is noteworthy to remind that no $CP$-violating
amplitude can be generated without strong phases. Suppose a heavy particle $H$
decays into a mesonic final state $M=M_{1}M_{2}...$, $H\to M$, and that the
Standard Model lagrangian contributes two terms (two Feynman diagrams) to this
process. Then, the decay amplitude and its corresponding $CP$ conjugate,
written most generally, are
$\displaystyle\mathcal{A}(H\to M)$ $\displaystyle=$
$\displaystyle\lambda_{1}A_{1}e^{i\varphi_{1}}+\lambda_{2}A_{2}e^{i\varphi_{2}}$
(1) $\displaystyle\bar{\mathcal{A}}(\bar{H}\to\bar{M})$ $\displaystyle=$
$\displaystyle\lambda_{1}^{*}A_{1}e^{i\varphi_{1}}+\lambda_{2}^{*}A_{2}e^{i\varphi_{2}}\
.$ (2)
The weak coupling $\lambda_{i}$ is a combination of possibly complex CKM
matrix elements and $Ae^{i\varphi}$ denotes the strong (hadronic) parts of the
transition amplitude, where we emphasise that they too can have both a real
part, or magnitude, and a phase, or absorptive part, due to multiple
rescattering of the final-state quarks and mesons. These $CP$-related
intermediate states must contribute the same absorptive part to the two
decays, therefore the strong phases $\varphi_{i}$ are the same in Eqs. (1) and
(2). Taking the difference of the absolute squares, known as direct $CP$
violation,
$|\mathcal{A}|^{2}-|\bar{\mathcal{A}}|^{2}=2A_{1}A_{2}\,\mathrm{Im}(\lambda_{1}\lambda_{2}^{*})\sin(\varphi_{1}-\varphi_{2})\
,$ (3)
one sees that no such violation, $|\bar{\mathcal{A}}/\mathcal{A}|\neq 1$, can
occur if the weak couplings contain only real phases or the strong phases are
the same. Hence, in order to extract the weak CKM phases with precision from
the decay amplitudes, it is crucial to evaluate the QCD contributions
reliably.
## 2 QCD factorisation
As simple as these mesons appear to be—a bound colourless heavy-light
$\bar{q}q$ pair—the difference in quark masses and the energetic light mesons
produced in their decays lead to a disparate array of energy scales. A central
aspect of heavy-meson phenomenology are factorisation theorems which allow for
a disentanglement of short-distance or hard physics, which encompasses
electroweak interactions and perturbative QCD, from long-distance or soft
physics, governed by non-perturbative hadronic effects. In the following, we
illustrate the factorisation with non-leptonic decays of a heavy meson $H$.
In the hamiltonian formulation of heavy-quark effective theory (HQET) [2], in
which amplitudes are expanded in powers of $\Lambda_{\mathrm{QCD}}/m_{h}$ and
the heavy quark is a static particle in the limit $m_{h}\to\infty$, the $H\to
M_{1}M_{2}$ decay amplitude is given by
$\mathcal{A}=\frac{G_{F}}{\sqrt{2}}\sum_{p}\lambda_{p}\sum_{i}\,C_{i}(\zeta)\langle
M_{1}M_{2}|O_{i}|H\rangle(\zeta)\ ,$ (4)
where $\lambda_{p}=V_{pb}V^{*}_{pk}$ $(p=u,c;k=d,s)$ are products of CKM
matrix elements and $G_{F}$ is the Fermi coupling constant. The dimension-six
four-quark operators $O_{i}$ result from integrating out the weak gauge bosons
$W^{\pm}$ in the operator product expansion and the Wilson coefficients
$C_{i}(\zeta)$ encode perturbative QCD effects above the renormalisation point
$\zeta$.
In what is called “naive” factorisation, the hadronic matrix element $\langle
M_{1}M_{2}|O_{i}|H\rangle$ is approximated by the product of two bilinear
currents, $\langle M_{1}|\bar{q}\gamma^{\mu}(1-\gamma_{5})b|B\rangle$
$\otimes$ $\langle M_{2}|\bar{q}^{\prime}\gamma_{\mu}(1-\gamma_{5})q|0\rangle$
\+ $(M_{1}\leftrightarrow M_{2})$, where colour indices have been omitted.
This factorisation simply expresses the matrix element of a local four-quark
operator as a product of a decay constant and a transition form factor.
However, as has long been known, the saturation by vacuum insertion fails in
the case of most $D$ decay modes and is largely insufficient to reproduce the
experimentally more precise data on $B\to M_{1}M_{2}$ branching fractions. In
fact, any hard final-state gluon interaction has been neglected and soft-gluon
exchange is at best incorporated into an effective colour parameter or form
factors. Moreover, the renormalisation scale and scheme dependence of
$C_{i}(\zeta)$ are not cancelled by those of the matrix element $\langle
M_{1}M_{2}|O_{i}|H\rangle(\zeta)$.
A major improvement over this simple factorisation Ansatz comes from the
systematic reorganisation of weak and QCD interactions in the HQET. Three
distinctive approaches have emerged in recent years: QCD factorisation (QCDF)
[3], perturbative QCD (pQCD) [4] and soft-collinear effective theory (SCET)
[5]. We here focus on the form factors that emerge in these factorisation
approaches and solely remark that QCD corrections beyond naive factorisation
entail, in the limit $m_{h}\gg\Lambda_{\mathrm{QCD}}$, the $B\to M_{1}M_{2}$
decay amplitude can be schematically written as
$\displaystyle\langle M_{1}M_{2}|O_{i}|B\rangle\ =\ \langle
M_{1}|j_{1}|B\rangle\langle M_{2}|j_{2}|0\rangle$ (5) $\displaystyle\hskip
14.22636pt\times\
\left[1+\sum_{n}r_{n}\alpha_{s}^{n}+\mathcal{O}(\Lambda_{\mathrm{QCD}}/m_{b})\right],$
where $j_{1}$ and $j_{2}$ are the bilinear currents. This has been shown
explicitly to leading order in $\alpha_{s}$ [3, 6] and including the one-loop
correction $(\alpha_{s}^{2})$ to the tree-diagram scattering between the
emitted meson and the one containing the spectator quark [7].
The factorisation theorem derived using SCET agrees with QCDF if perturbation
theory is applied at the hard $m^{2}_{b}$ and hard-collinear $m_{b}\Lambda$
scales, with $\Lambda$ typically of the order of 100 MeV. It is evident from
Eq. (5) that higher orders in $\alpha_{s}$ break the factorisation but these
corrections can be systematically supplemented; the analogy with perturbative
factorisation for exclusive processes in QCD at large-momentum transfer is not
accidental [8]. Further contributions that break the factorisation, formally
suppressed in $\Lambda_{\mathrm{QCD}}/m_{b}$ yet not irrelevant, are weak
annihilation decay amplitudes and final-state interactions between daughter
hadrons [9]. Neglecting power corrections in $\alpha_{s}$ and taking the limit
$m_{b}\to\infty$, the naive factorisation is recovered.
## 3 Separating scales: the softer the harder
While factorisation theorems elaborated with SCET provide the means to
systematically integrate out energy scales in the perturbative domain,
yielding approximations valid in the heavy-quark limit for a given decay in
terms of products of hard and soft matrix elements, a reliable evaluation of
the latter is notoriously difficult. In fact, it is the soft physics of the
bound states that renders the task hard, as it implies non-perturbative QCD
contributions. Full ab initio calculations are currently out of reach and for
the time being one is left with modelling the heavy-to-light amplitudes with
as much input from non-perturbative QCD as possible. Just how much soft
physics is included depends on the theoretical Ansatz and techniques employed.
A variety of theoretical approaches have been applied to this problem, recent
amongst which are analyses using light-front and relativistic quark models,
light-cone sum rules (LCSR) and lattice-QCD simulations. In Section 4 we
briefly summarise these approaches while Section 5 deals in more detail with
studies of heavy-to-light form factors within the framework of the Dyson-
Schwinger equation (DSE). We refer to a recent review [10] for a summary of
transition form factor data from lattice-regularised QCD and just note that
current results are obtained at large squared momentum transfer, i.e.,
$q^{2}\simeq 16$ GeV2 for $B\to\pi$ transitions. Hence, values at low $q^{2}$
must necessarily be extrapolated by means of a (pole-dominance)
parametrisation [11, 23].
$q^{2}$ [GeV2] | Ref. [12] | Ref. [16] | Ref. [18] | Ref. [20] | Ref. [22] | Ref. [23] | Ref. [21]
---|---|---|---|---|---|---|---
$0$ | 0.217 | 0.29 | 0.247 | 0.24 | $0.25\pm 0.05$ | $0.26\pm 0.03$ | $0.27\pm 0.02\pm 0.07$
$10$ | 0.41 | 0.54 | 0.49 | 0.53 | 0.54 | 0.51 | –
$15$ | 0.67 | 0.84 | 0.97 | 0.91 | 0.90 | 0.81 | –
$20$ | 1.40 | 1.56 | $>10$ | 1.75 | 1.83 | 1.58 | –
Table 1: Numerical comparison for the transition form factor,
$F_{+}^{B\to\pi}(q^{2})$; the $q^{2}$ values of Ref. [20] are calculated,
whereas for Refs. [12, 16, 18, 22, 23] the value $F_{+}^{B\to\pi}(0)$ and the
corresponding extrapolation in these references are employed.
## 4 Hadronic transition form factors
Quark models: relativistic quark models [12, 13, 14, 15, 16, 17, 18] have in
common that their only degrees of freedom are constituent quarks whose masses
are parameters of the hamiltonian. The hadronisation of the two valence quarks
is described by vertex wave functions or phenomenological Bethe-Salpeter
amplitudes (BSA). The approaches in [13, 14, 15] represent heavy-to-light
transition amplitudes by triangle diagrams, a 3-point function between two
meson BSA and the weak coupling, which yields the transition amplitude
$\langle M(p_{2})|\bar{q}\,\Gamma_{I}h|H(p_{1})\rangle$ and reads generally,
$\displaystyle\mathcal{A}(p_{1},p_{2})=\mathrm{tr}_{\mathrm{CD}}\\!\
\\!\int\\!\frac{d^{4}k}{(2\pi)^{4}}\,\bar{\Gamma}_{M}^{(\mu)}(k;-p_{2})S_{q}(k+p_{2})$
(6) $\displaystyle\times$
$\displaystyle\Gamma_{I}(p_{1},p_{2})S_{Q}(k+p_{1})\Gamma_{H}(k;p_{1})S_{q^{\prime}}(k)\
,$
where $S(k)$ are quark propagators, $Q=c,b$; $q=q^{\prime}=u,d,s$; $M=S,P,V,A$
and the index $\mu$ indicates a possible vector structure in the final-state
BSA. $\Gamma_{I}$ is the interaction vertex whose Lorentz structure depends on
the operator $O_{i}$ in the HQET and $\Gamma_{H}$ is the heavy meson BSA. The
trace is over Dirac and colour indices.
An analogous approach represents the amplitude in Eq. (6) by relativistic
double-dispersion integrals over the initial- and final-mass variables
$p_{1}^{2}$ and $p_{2}^{2}$, where the integration kernel arises from the
double discontinuity of the triangle diagram (putting internal quark
propagators on-shell via the Landau-Cutkosky rule). The meson-vertex functions
are given by one-covariant BSA [16, 17]. Other quark models [12] represent
$B\to M$ form factors by overlap integrals of meson wave functions, obtained
from confining potential models, and appropriate weak interaction vertices.
Similar quark model calculations were performed on the light cone [18].
All these approaches model soft contributions with vertex functions, while the
propagation of the constituent quark, $S(k)=(/\\!\\!\\!k-m_{q})^{-1}$, is
scale independent and does not describe confinement and dynamical chiral
symmetry breaking (DCSB). As noted in Refs. [17, 19, 20], this can lead to
considerable model dependance at larger momentum transfer.
Light-cone sum rules: In a LCSR the operator-product expansion of a given
correlation function is combined with hadronic dispersion relations. The
quark-hadron duality is invoked: the correlator function is calculated twice,
as a hadronic object and with subhadronic degrees of freedom. After separation
in HQET of the heavy meson’s static part, $P_{H}=p+q=m_{h}v_{h}+k$, where
$v_{h}$ is the four-velocity and $k$ is the residual momentum, and likewise
redefinition of the momentum transfer $q=m_{h}v_{h}+\tilde{q}\Rightarrow
p+\tilde{q}=k$, one obtains the heavy-limit correlation function,
$\displaystyle\Pi^{H}(p,q)=\tilde{\Pi}^{H_{v}}(p,\tilde{q})+\mathcal{O}(1/m_{h});$
(7)
$\displaystyle\tilde{\Pi}^{H_{v}}(p,\tilde{q})=i\\!\\!\int\\!d^{4}\\!x\,e^{ipx}\langle
0|T[J_{M}(x)J_{h_{v}}(0)]|H_{v}\rangle,$
whereas a hadronic correlator can be written,
$\displaystyle\tilde{\Pi}^{\mathrm{had.}}(p,q)=\frac{\langle
0|J_{M}|M(p)\rangle\langle M(p)|J_{h}|H(P_{H})\rangle}{m_{M}^{2}-p^{2}},$ (8)
where $J_{h}(0)$ and $J_{h_{v}}(0)$ are heavy-light currents and $J_{M}(x)$
the interpolating current for a pseudoscalar or vector meson.
In Eq. (8) only the light-meson contribution is represented but higher and
continuum states can also be taken into account. In Eqs. (7) and (8), the
usual role of the correlation functions has been reversed [21, 22]: the
correlation function is taken between the vacuum and the on-shell $B$-state
vector using its light-cone distribution amplitude (DA) expansion and the pion
is interpolated with the light-quark (axialvector) current $J_{M}(x)$. The
$B$-meson DAs are universal non-perturbative objects introduced within HQET.
In Ref. [23], however, the correlation function is taken between the vacuum
and the light-meson state, whereas the $B$ meson is interpolated by the heavy-
light quark current $J_{h}(0)$. As a result, the long-distance dynamics in the
correlation function is described by a set of light-meson ($\pi,K,\rho,K^{*}$)
DA. In the last step, a Borel transformations is applied to both, Eqs. (7) and
(8), from which one derives a transition form factor expressed as a sum rule.
The transformation introduces a scale via the Borel parameter which, in turn,
is fixed with sum rules for light-meson decay constants.
Besides a systematic uncertainty owing to the duality assumption, the main
incertitude lies within the DAs which encode the relevant non-perturbative
effects. Only their asymptotic form is known exactly from perturbative QCD. As
of yet, the first two Gegenbauer moments of the DA for various light
pseudoscalar and vector mesons have been obtained from QCD sum rules with very
large errors, though the first moment is consistent with lattice calculations
[24]. Moreover, the transition form factors must be extrapolated to space-like
momenta.
For purpose of comparison, we list $B\to\pi$ transitions form factors,
$F_{+}(q^{2})$, for various models in Table 1. As observed therein, there is a
30% variation within the quark models [12, 16, 18] at $q^{2}=0$ which
increases at larger $q^{2}$ values. The LCSR predictions [22, 23] agree at
$q^{2}=0$ but their respective slopes for $q^{2}>0$ vary by $12\%$. The form
factors obtained with the DSE model Ref. [20] are calculated on the entire
physical momentum domain and the chiral limit is directly accessible.
## 5 Flavourful Dyson-Schwinger equations
The elements entering the amplitude in Eq. (6) can be motivated by the
solutions of DSEs applied to QCD. A general review of the DSEs can be found in
Refs. [25, 26] and their applications to heavy-light transition form factors
have been investigated in [20, 27, 28].
Dressed quark propagator: The mesons are bound states of a quark and antiquark
pair, where for a given quark flavour their dressing is described by the DSE
(in Euclidean metric),
$S^{-1}(p)=Z_{2}(i\gamma\cdot p+m^{\mathrm{bm}})+\Sigma(p^{2})\ ,$ (9)
with the dressed quark self energy,
$\Sigma(p^{2})=Z_{1}g^{2}\\!\\!\int^{\Lambda}_{k}\\!\\!\\!D^{\mu\nu}(p-k)\frac{\lambda^{a}}{2}\gamma_{\mu}S(k)\Gamma^{a}_{\nu}(k,p),$
(10)
where $\int_{k}^{\Lambda}$ represents a Poincaré invariant regularisation of
the integral with the regularisation mass scale $\Lambda$. The current quark
bare mass $m^{\mathrm{bm}}$ receives corrections from the self energy
$\Sigma(p^{2})$ in which the integral is over the dressed gluon propagator,
$D_{\mu\nu}(k)$, the dressed quark-gluon vertex, $\Gamma^{a}_{\nu}(k,p)$, and
$\lambda^{a}$ are the usual SU$(3)$ colour matrices. The solution to the gap
equation (9) reads
$\displaystyle S(p)$ $\displaystyle=$ $\displaystyle-i\gamma\cdot p\
\sigma_{V}(p^{2})+\sigma_{S}(p^{2})$ (11) $\displaystyle=$
$\displaystyle\left[i\gamma\cdot p\ A(p^{2})+B(p^{2})\right]^{-1}.$
The renormalisation constants for the quark-gluon vertex,
$Z_{1}(\zeta,\Lambda^{2})$, and quark-wave function,
$Z_{2}(\zeta,\Lambda^{2})$, depend on the renormalisation point, $\zeta$, the
regularisation scale, $\Lambda$, and the gauge parameter, whereas the mass
function $M(p^{2})=B(p^{2})/A(p^{2})$ is independent of $\zeta$. Since QCD is
asymptotically free, it is useful to impose at large spacelike $\zeta^{2}$ the
renormalisation condition,
$S^{-1}(p)|_{p^{2}=\zeta^{2}}=i\gamma\cdot p+m(\zeta^{2})\ ,$ (12)
where $m(\zeta^{2})$ is the renormalised running quark mass, so that for
$\zeta^{2}\gg\Lambda_{\mathrm{QCD}}^{2}$ quantitative matching with pQCD
results can be made.
Infrared dressing of light quarks has profound consequences for hadron
phenomenology [29]: the quark-wave function renormalisation,
$Z(p^{2})=1/A(p^{2})$, is suppressed whereas the dressed quark-mass function,
$M(p^{2})=B(p^{2})/A(p^{2})$, is enhanced in the infrared which expresses
dynamical chiral symmetry breaking (DCSB) and is crucial to the emergence of a
constituent quark mass scale. Both, numerical solutions of the quark DSE and
simulations of lattice-regularised QCD [30], predict this behaviour of
$M(p^{2})$ and pointwise agreement between DSE and lattice results has been
explored in Refs. [31, 32]. Studies that do not implement light-quark dressing
run into artefacts caused by rather large light-quark masses [17, 19] to
emulate confinement. This is because unphysical thresholds in transition
amplitudes can only be overcome with the prescription that
$m_{H}<m_{q_{1}}+m_{q_{2}}$, which poses problems for a description of light
vector mesons ($\rho,K^{*}$), heavy flavoured vector mesons ($D^{*},B^{*}$)
and for $P$-wave and excited charmonium states.
Whereas the impact of gluon dressing is striking for light quarks, its effect
on the heavy quarks is barely notable. This can be appreciated, for instance,
in Fig. 1 of Ref. [28]: for light quarks, mass can be generated from nothing,
i.e., the Higgs mechanism is irrelevant to their acquiring of a constituent-
like mass.
Bethe-Salpeter amplitudes: The BSA can be determined reliably by solving the
Bethe-Salpeter equation (BSE) in a truncation scheme consistent with that
employed in the gap equation (9). Consider the inhomogeneous BSE for the
axialvector vertex $\Gamma^{fg}_{5\mu}$ in which pseudoscalar and axialvector
mesons appear as poles:
$\displaystyle\Gamma^{fg}_{5\mu}(k;P)=Z_{2}\gamma_{5}\gamma_{\mu}-g^{2}\\!\int_{q}^{\Lambda}\\!\\!D^{\alpha\beta}(k-q)\frac{\lambda^{a}}{2}\gamma_{\alpha}$
$\displaystyle\times\
S_{f}(q_{+})\,\Gamma^{fg}_{5\mu}(q;P)S_{g}(q_{-})\frac{\lambda^{a}}{2}\Gamma^{g}_{\beta}(q_{-},k_{-})$
(13) $\displaystyle+\
g^{2}\\!\int_{q}^{\Lambda}\\!\\!\\!D^{\alpha\beta}(k-q)\frac{\lambda^{a}}{2}\gamma_{\alpha}S_{f}(q_{+})\frac{\lambda^{a}}{2}\Lambda^{fg}_{5\mu\beta}(k,q;P)\,.$
In Eq. (5), $P$ is the total meson momentum, $q_{\pm}=q\pm P/2,k_{\pm}=k\pm
P/2$, $\Lambda^{fg}_{5\mu\beta}$ is a 4-point Schwinger function entirely
defined via the quark self energy [33] and $f,g$ denote the flavour indices of
a light-light or heavy-light bound state. The solutions of the vertex
$\Gamma^{fg}_{5\mu}$ must satisfy the axial-vector Ward-Takahashi identity,
$\displaystyle
P^{\mu}\,\Gamma^{fg}_{5\mu}(k;P)=S_{f}^{-1}(k_{+})i\gamma_{5}+i\gamma_{5}S_{g}^{-1}(k_{-})$
$\displaystyle-\ i[m_{f}(\zeta)+m_{g}(\zeta)]\ \Gamma^{fg}_{5}(k;P)\ ,$ (14)
where $\Gamma^{fg}_{5}$ solves the pseudoscalar analogue to Eq. (5). A
systematic, symmetry-preserving truncation of the DSE and BSE is given by the
Rainbow ladder [34] which is their leading-order term with the dressed quark-
gluon vertex, $\Gamma^{f}_{\mu}$, replaced by $\gamma_{\mu}$. It can be shown
that $\Lambda^{fg}_{5\mu\beta}\equiv 0$ in this approximation.
The above Ward-Takahashi identity possesses another remarkable property; the
set of quark-level Goldberger-Treiman relations that follow from it reveal the
full structure of the light-meson BSA [35]. In particular, it enables one to
relate the leading covariant, $\varepsilon_{5}(k;P)$, of the light
pseudoscalar BSA with the scalar part, $B(p^{2})$, of the dressed-quark
propagator (11) in the chiral limit. This motivates an effective
parametrisation of the light mesons ($M=\pi,K$),
$\Gamma_{M}(k;P)=i\gamma_{5}\,\varepsilon_{M}(k^{2})=i\gamma_{5}\,B_{M}(k^{2})/\hat{f}_{M}\
,$ (15)
$\hat{f}_{M}=f_{M}/\sqrt{2}$, which has been capitalised on in transition form
factor calculations [20, 27, 28].
Simultaneous solutions of the quarks’s DSE and the heavy meson’s BSA with
renormalisation-group improved ladder truncations, obtained for the kaon [35],
prove to be difficult. The truncations do not yield the Dirac equation when
one of the quark masses is large. A recent attempt to calculate BSA for $D$
and $B$ mesons [36] reproduces well the respective masses but underestimates
experimental leptonic decay constants by $30-50$%. With a consistent
derivation of the heavy meson BSA pending, simple one-covariant forms for
$\Gamma_{H}(k;P)$ are currently employed in Eq. (6), which reproduce leptonic
decay constants in a simultaneous calculation.
## 6 Hadronic decays
The decay $D^{*}\to D\pi$ can be used to extract the strong coupling $\hat{g}$
between heavy vector and pseudoscalar mesons to a low-momentum pion in the
heavy meson chiral lagrangian [37]. One considers the matrix element,
$\langle D(p_{2})\pi(q)|D^{*}(p_{1},\lambda)\rangle=g_{D^{*}\\!D\pi}\
\bm{\epsilon}_{\lambda}\\!\cdot q\ ,$ (16)
where the coupling, $g_{D^{*}\\!D\pi}=17.9\pm 0.3\pm 1.9$, is experimentally
known [38] and related to $\hat{g}$. Similarly, one may also extract $\hat{g}$
from the unphysical decay $B^{*}\to B\pi$ in the chiral limit and where
$m_{b}/\Lambda_{\mathrm{QCD}}$ corrections are better controlled.
The coupling $g_{D^{*}\\!D\pi}$ is related to a heavy-to-heavy transition form
factor via the LSZ reduction of the pion and the use of PCAC,
$\pi(x)=\partial^{\mu}A_{\mu}(x)/(f_{\pi}m_{\pi}^{2})$, which leads to:
$\displaystyle\langle D(p_{2})\pi(q)|D^{*}(p_{1})\rangle\ =\
q^{\mu}\frac{(m_{\pi}^{2}-q^{2})}{f_{\pi}m_{\pi}^{2}}$ (17)
$\displaystyle\times$ $\displaystyle\\!\\!\\!\int d^{4}x\,e^{iq\cdot x}\langle
D(p_{2})|A_{\mu}(x)|D^{*}(p_{1})\rangle.$
Hence, the matrix element in Eq. (16) has been reduced to the Fourier
transform of a transition matrix element between the $D^{*}$ and $D$ mesons in
the chiral limit with the axial QCD current
$A_{\mu}(x)=\bar{q}^{a}\gamma_{\mu}\gamma_{5}q^{b}$. Results from this
reduction procedure have been obtained on the lattice [39, 40] and most
recently in a simulation with $n_{f}=2$ [41] which yields
$g_{D^{*}\\!D\pi}=20\pm 2$.
This form factor can also be calculated straightforwardly without reduction of
the pion employing Eq. (6) with the dressed quark propagators in Eq. (11) and
substituting the pion’s BA (15) for $\Gamma_{I}$. In a reassessment and
improvement of a calculation of $g_{D^{*}\\!D\pi}$ within a Dyson-Schwinger
model [28], we obtain $g_{D^{*}\\!D\pi}=21$ [42] in agreement with the lattice
result [41] and about $16\%$ larger than the experimental value.
## 7 Conclusive remarks
We have stressed the importance of hadronic effects in decays of heavy-
flavoured mesons and portrayed the various theoretical Ansätze for the heavy-
to-light transition form factors. In short, the main obstacle to their precise
calculation, which veraciously reproduces the infrared features of QCD, are
the uncertainties of the light-cone DA in the case of LCSR and the lack of
model-independent wave functions in relativistic quark model calculations. We
have argued that the running quark mass of the DSE quark propagators is
crucial to include confinement and DCSB effects in the transition amplitudes;
an unfinished task are consistent solutions of the BSE for the $D$ and $B$
mesons within the DSE formalism, which will reduce model dependence.
## Acknowledgments
Based on the talk given at Light Cone 2009: Relativistic Hadronic and Particle
Physics, 8–13 July 2009, São José dos Campos, São Paulo, Brazil. B. E. thanks
the organisers at the Instituto Tecnológico de Aeronáutica for the welcoming
atmosphere and in particular Tobias Frederico and João Pacheco de Melo for
their hospitality. Several stimulating discussions with Arlene Aguilar,
Adriano Natale, Fernando Navarra, Marina Nielsen, Joannis Papavassiliou and
Lauro Tomio were greatly appreciated. This work was supported by the
Department of Energy, Office of Nuclear Physics, No. DE-AC02-06CH11357.
## References
* [1] C. Bebek et al. [CLEO Collaboration], Phys. Rev. Lett. 46, 84 (1981); K. Chadwick et al. [CLEO Collaboration], ibid. 46, 88 (1981).
* [2] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
* [3] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); id., Nucl. Phys. B 591, 313 (2000).
* [4] Y. Y. Keum, H. N. Li and A. I. Sanda, Phys. Rev. D 63, 054008 (2001).
* [5] C. W. Bauer et al., Phys. Rev. D 63, 114020 (2001); C. W. Bauer, D. Pirjol and I. W. Stewart, ibid. 65, 054022 (2002); C. W. Bauer et al., ibid. 66, 014017 (2002).
* [6] C. W. Bauer et al., Phys. Rev. D 70, 054015 (2004).
* [7] M. Beneke and S. Jäger, Nucl. Phys. B 751, 160 (2006); id., Nucl. Phys. B 768, 51 (2007).
* [8] G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
* [9] B. El-Bennich et al., Phys. Rev. D 79, 094005 (2009); B. El-Bennich et al., Phys. Rev. D 74, 114009 (2006).
* [10] M. Artuso et al., Eur. Phys. J. C 57, 309 (2008).
* [11] D. Bećirević and A. B. Kaidalov, Phys. Lett. B 478, 417 (2000).
* [12] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D 75, 074008 (2007).
* [13] M. A. Ivanov, J. G. Körner and P. Santorelli, Phys. Rev. D 63, 074010 (2001); id. 71, 094006 (2005); 75, 019901(E) (2007).
* [14] A. Faessler et al., Eur. Phys. J. direct C 4, 18 (2002).
* [15] M. A. Ivanov, J. G. Körner and O. N. Pakhomova, Phys. Lett. B 555, 189 (2003).
* [16] D. Melikhov, N. Nikitin and S. Simula, Phys. Rev. D 57, 6814 (1998); D. Melikhov, Eur. Phys. J. direct C 2, 1 (2002).
* [17] B. El-Bennich, O. Leitner, J. P. Dedonder and B. Loiseau, Phys. Rev. D 79, 076004 (2009).
* [18] C. D. Lu, W. Wang and Z. T. Wei, Phys. Rev. D 76 014013 (2007).
* [19] B. El-Bennich et al., Braz. J. Phys. 38, 465 (2008).
* [20] M. A. Ivanov et al., Phys. Rev. D 76, 034018 (2007).
* [21] F. De Fazio, T. Feldmann and T. Hurth, Nucl. Phys. B 733, 1 (2006) [Erratum-ibid. B 800, 405 (2008)].
* [22] A. Khodjamirian, T. Mannel and N. Offen, Phys. Lett. B 620, 52 (2005); id., Phys. Rev. D 75, 054013 (2007).
* [23] P. Ball and R. Zwicky, JHEP 0110, 019 (2001); id., Phys. Rev. D 71, 014015 (2005).
* [24] P. A. Boyle et al., [UKQCD Collab.], Phys. Lett. B 641, 67 (2006).
* [25] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994).
* [26] C. D. Roberts, M. S. Bhagwat, A. Holl and S. V. Wright, Eur. Phys. J. ST 140, 53 (2007).
* [27] M. A. Ivanov et al., Phys. Rev. C 57, 1991 (1998).
* [28] M. A. Ivanov, Yu. L. Kalinovsky and C. D. Roberts, Phys. Rev. D 60, 034018 (1999).
* [29] C. D. Roberts, Nucl. Phys. A 605, 475 (1996); C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45, S1 (2000); R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001).
* [30] J. B. Zhang et al., Phys. Rev. D 71, 014501 (2005).
* [31] M. S. Bhagwat, M. A. Pichowsky, C. D. Roberts and P. C. Tandy, Phys. Rev. C 68, 015203 (2003); M. S. Bhagwat and P. C. Tandy, Phys. Rev. D 70, 094039 (2004).
* [32] R. Alkofer, W. Detmold, C. S. Fischer and P. Maris, Phys. Rev. D 70, 014014 (2004).
* [33] L. Chang and C. D. Roberts, Phys. Rev. Lett. 103, 081601 (2009).
* [34] H. J. Munczek, Phys. Rev. D 52, 4736 (1995); A. Bender, C. D. Roberts and L. Von Smekal, Phys. Lett. B 380, 7 (1996).
* [35] P. Maris, C. D. Roberts and P. C. Tandy, Phys. Lett. B 420, 267 (1998); P. Maris and C. D. Roberts, Phys. Rev. C 56, 3369 (1997).
* [36] T. Nguyen, N. A. Souchlas and P. C. Tandy, AIP Conf. Proc. 1116, 327 (2009).
* [37] M. B. Wise, Phys. Rev. D 45, 2188 (1992).
* [38] A. Anastassov et al. [CLEO Collaboration], Phys. Rev. D 65, 032003 (2002).
* [39] G. M. de Divitiis et al. [UKQCD Collaboration], JHEP 9810, 010 (1998).
* [40] A. Abada et al., Phys. Rev. D 66, 074504 (2002).
* [41] D. Bećirević and B. Haas, arXiv:0903.2407 [hep-lat]; D. Bećirević, B. Blossier, E. Chang and B. Haas, Phys. Lett. B 679, 231 (2009).
* [42] B. El-Bennich, M. A. Ivanov and C. D. Roberts, in preparation.
|
arxiv-papers
| 2009-10-23T14:48:12 |
2024-09-04T02:49:06.013778
|
{
"license": "Public Domain",
"authors": "B. El-Bennich, M.A. Ivanov and C.D. Roberts",
"submitter": "Bruno El-Bennich",
"url": "https://arxiv.org/abs/0910.4523"
}
|
0910.4613
|
# Fading Cognitive Multiple-Access Channels With Confidential Messages
Ruoheng Liu, Yingbin Liang and H. Vincent Poor The work of R. Liu and H. V.
Poor was supported by the National Science Foundation under Grant
CNS-09-05398, and by the Air Force Office of Scientific Research under Grant
FA9550-08-1-0480, and the work of Y. Liang was supported by a National Science
Foundation CAREER Award under Grant CCF-08-46028 and under Grant
CCF-09-15772.Ruoheng Liu and H. Vincent Poor are with the Department of
Electrical Engineering, Princeton University, Princeton, NJ 08544, USA (email:
{rliu,poor}@princeton.edu).Yingbin Liang is with the Department of Electrical
Engineering and Computer Science, Syracuse University, Syracuse, NY 13244, USA
(email: yliang06@syr.edu).
###### Abstract
The fading cognitive multiple-access channel with confidential messages (CMAC-
CM) is investigated, in which two users attempt to transmit common information
to a destination and user 1 also has confidential information intended for the
destination. User 1 views user 2 as an eavesdropper and wishes to keep its
confidential information as secret as possible from user 2. The multiple-
access channel (both the user-to-user channel and the user-to-destination
channel) is corrupted by multiplicative fading gain coefficients in addition
to additive white Gaussian noise. The channel state information (CSI) is
assumed to be known at both the users and the destination. A parallel CMAC-CM
with independent subchannels is first studied. The secrecy capacity region of
the parallel CMAC-CM is established, which yields the secrecy capacity region
of the parallel CMAC-CM with degraded subchannels. Next, the secrecy capacity
region is established for the parallel Gaussian CMAC-CM, which is used to
study the fading CMAC-CM. When both users know the CSI, they can dynamically
change their transmission powers with the channel realization to achieve the
optimal performance. The closed-form power allocation function that achieves
every boundary point of the secrecy capacity region is derived.
###### Index Terms:
Secure communication, fading channel, multiple-access channel, equivocation,
secrecy capacity.
## I Introduction
Wireless transmissions lack physical boundaries and so any adversary within
range can receive them. Thus, security is one of the most important issues in
wireless communications. One approach to security involves applying encryption
algorithms to make messages unintelligible to adversaries. Unfortunately,
these security methods are often designed without consideration of the
specific properties of wireless networks. More specifically, encryption
methods tend to be layer-specific and ignore the most fundamental
communication layer, i.e., the physical-layer, whereby devices communicate
through the encoding and modulation of information into waveforms.
The first study of secure communication via physical layer approaches was
captured by a basic wiretap channel introduced by Wyner in [1]. In this model,
a single source-destination communication link is eavesdropped upon by an
eavesdropper via a degraded channel. The source node wishes to send
confidential information to the destination node in a reliable manner as well
as to keep the eavesdropper as ignorant of this information as possible. The
performance measure of interest is the secrecy capacity which characterizes
the largest possible communication rate from the source node to the
destination node with the eavesdropper obtaining no source information.
Wyner’s formulation was generalized by Csiszár and Körner who determined the
secrecy capacity region of a more general model referred to as the broadcast
channel with confidential messages (BCC) [2].
More recently, multi-terminal communication with confidential messages has
been studied intensively. (See [3] for a recent survey of progress in this
area.) Among these studies, a generalization of both the wiretap channel and
the classical multiple-access channel (MAC) was studied in [4], in which each
user also receives channel outputs, and hence may obtain the confidential
information sent by the other user from the channel output it receives. In
this communication scenario, each user views the other user as an
eavesdropper, and wishes to keep its confidential information as secret as
possible from the other user. The authors of [4] investigated the rate-
equivocation region and secrecy capacity region for this channel. Some other
related studies on secure communication over multiple access channels can be
found in [5, 6, 7].
Fading has traditionally been considered to be an obstacle to providing
reliable wireless communication. However, over the past decade, it has been
demonstrated that fading can help improve capacity, reliability, and
confidentiality of wireless networks. The impact of fading on secure
communication was studied in, e.g., [8, 9, 10]. More specifically, [8] studied
the secrecy capacity of ergodic fading BCCs when the channel state information
(CSI) is known at all communicating nodes; [9] considered the ergodic scenario
of fading wiretap channel in which the transmitter has no CSI about the
eavesdropper channel; and [10] studied the outage preference of secure
communication over wireless channels, in which the transmitter has no CSI
about either the legitimate receiver’s channel or the eavesdropper’s channel.
In this paper, we investigate the fading cognitive multiple-access channel
with both common and confidential messages, a problem which is inspired by the
studies of secure communication over MACs in [4]. In our communication
scenario, we assume that two users (users 1 and 2) have common information,
while user 1 has confidential information intended for a destination and
treats user 2 as an eavesdropper. Hence, user 1 wishes to keep its
confidential messages as secret as possible from user 2. We refer to this
model as the cognitive MAC with one confidential message (CMAC-CM); (see Fig.
1.(a)), because this channel also models cognitive communication in which the
secondary user (user 1) helps the primary user (user 2) to send a common
message $W_{0}$, and also has a confidential message $W_{1}$ intended for the
destination, which needs to be kept secret from the primary user. Furthermore,
we consider the situation in which both the user-to-user and the user-to-
destination channels are corrupted by multiplicative fading gain coefficients
in addition to additive white Gaussian noise. The fading CMAC-CM model
captures the basic time-varying and superposition properties of wireless
channels, and thus, understanding this channel plays an important role in
solving security issue in wireless application. For the fading CMAC-CM, we
assume that the fading gain coefficients are stationary and ergodic over time
and that the CSI is known at both users and the destination. Note that
knowledge of the user-to-destination CSI is necessary in order to
cooperatively transmit the common message, and thus should be provided through
state feedback from the destination terminal to the user terminals. Knowledge
of CSI between the user terminals can be obtained via the reciprocity property
of those channels. Users are motivated to do so in order to enable better
cooperation for sending the common message.
(a) CMAC-CM
(b) Parallel CMAC-CM
Figure 1: Cognitive multiple-access channel with confidential messages.
To solve the fading CMAC-CM problem, we first consider a general information-
theoretic model, i.e., the parallel MAC with $L$ independent subchannels. As
shown in Fig. 1.(b), the two users communicate with the destination over $L$
parallel links and each of the $L$ links is eavesdropped upon by user 2. We
establish the secrecy capacity region for the parallel CMAC-CM. In particular,
we provide a converse proof to show that having independent inputs for each
subchannel is optimal to achieve the secrecy capacity region. The secrecy
capacity region of the parallel CMAC-CM further gives the secrecy capacity
region of the parallel CMAC-CM with degraded subchannels. Next, we consider
the parallel Gaussian CMAC-CM, which is an example parallel CMAC-CM with
degraded subchannels. Based on the maximum-entropy theorem [11] and the
extremal inequality [12], we show that the secrecy capacity region of the
parallel Gaussian CMAC-CM is achievable by using jointly Gaussian inputs and
optimizing power allocations at two users among the parallel subchannels. We
then apply this result to investigate the fading CMAC-CM. We study the ergodic
performance, where no delay constraint on message transmission is assumed and
the secrecy capacity region is averaged over all channel states. In fact, the
fading CMAC-CM can be viewed as the parallel Gaussian CMAC-CM with each fading
state corresponding to one subchannel. Hence, the secrecy capacity region of
the parallel Gaussian CMAC-CM applies to the fading CMAC-CM. Since both users
know the CSI, users can dynamically change their transmission powers with the
channel realization to achieve the optimal performance. The optimal power
allocation that achieves every boundary point of the secrecy capacity region
can be characterized as a solution to a non-convex problem. The Karush-Kuhn-
Tucker (KKT) conditions (as necessary conditions) greatly facilitate
exploitation of the specific structure of the problem, and enable us to obtain
a closed-form solution for the optimal power allocation strategy for the two
users.
The remainder of this paper is organized as follows. We first study the
parallel CMAC-CM with independent subchannels and its special case of the
parallel CMAC-CM with degraded subschannels in Section II. Next, we
investigate the secrecy capacity region of the parallel Gaussian CMAC-CM in
Section III and the ergodic performance of the fading CMAC-CM in Section IV.
We then provide some numerical examples in Section V. Finally, we summarize
our results in Section VI.
## II Parallel CMAC-CM
### II-A Channel Model
We consider the discrete memoryless parallel CMAC-CM with $L$ independent
subchannels (see Fig. 1.(b)). Each subchannel is assumed to connect users 1
and 2 to the destination, and user 2 can also receive the channel output from
each subchannel, and hence may obtain information sent by user 1. The channel
transition probability distribution is given by
$\displaystyle p(y_{[1,L]},y_{2,[1,L]}$
$\displaystyle|x_{1,[1,L]},x_{2,[1,L]})=\prod_{j=1}^{L}p(y_{j},y_{2,j}|x_{1,j},x_{2,j}),$
(1)
where $y_{[1,L]}:=(y_{1},...,y_{L})$.
In this model, a common message $W_{0}$ is known to both the primary user
(user 2) and the secondary user (user 1), and hence both users cooperate to
transmit $W_{0}$ to the destination. Moreover, the secondary user (user 1)
also has confidential message $W_{1}$ intended for the destination. User 1
views user 2 as an eavesdropper and wishes to keep its confidential
information as secret as possible from user 2. In this paper, we focus on the
case in which perfect secrecy is achieved, i.e., user 2 should not obtain any
information about the message $W_{1}$. More formally, this condition is
characterized by (e.g., see [1, 2, 4]):
$\displaystyle\frac{1}{n}I(W_{1};Y_{2}^{n},X_{2}^{n},W_{0})\rightarrow 0$ (2)
where $X_{2}^{n}:=(X_{2,1},\dots,X_{2,n})$ and
$Y_{2}^{n}:=(Y_{2,1},\dots,Y_{2,n})$ are the input and output sequences of
user 2, respectively, and the limit is taken as the block length
$n\rightarrow\infty$. The goal is to characterize the secrecy capacity region
${\mathcal{C}}_{s}$ that contains rate pairs achievable by some coding scheme
(more detailed definitions for the rates of the messages and encoding and
decoding schemes can be found in [4]).
### II-B Secrecy Capacity Region of the Parallel CMAC-CM
For the parallel CMAC-CM, we obtain the following secrecy capacity region.
###### Theorem 1
For the parallel CMAC-CM, the secrecy capacity region is given by
$\displaystyle{\mathcal{C}}_{s}^{\rm[P]}=$
$\displaystyle\bigcup_{\begin{subarray}{c}\prod_{j}p(q_{j},x_{2,j})p(u_{j}|q_{j})p(x_{1,j}|u_{j})\\\
p(y_{j},y_{2,j}|x_{1,j},x_{2,j})\end{subarray}}\left\\{\begin{array}[]{l}(R_{0},\,R_{1}):\\\
~{}~{}R_{0}\geq 0,\;R_{1}\geq 0;\\\
~{}~{}R_{1}\leq\sum_{j=1}^{L}[I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})]\\\
~{}~{}R_{0}\leq\sum_{j=1}^{L}I(Q_{j},X_{2,j};Y_{j})\end{array}\right\\}$ (7)
where $Q_{j}$ and $U_{j}$’s are auxiliary random variables, and $Q_{j}$ can be
chosen to be a deterministic function of $U_{j}$ for $j=1,\dots,L$.
###### Proof:
See Appendix -A. ∎
Theorem 1 implies that having independent inputs for each subchannel is
optimal. This fact does not follow directly from the single-letter result on
the secrecy capacity region of the CMAC-CM given in [4]. Hence, a converse
proof is needed, which is provided in Appendix -A.
### II-C Parallel CMAC-CM with Degraded Subchannels
We consider the parallel CMAC-CM with degraded subchannels, in which each
subchannel is either degraded such that given the input of user 2, the output
at user 2 is a conditionally degraded version of the output at the
destination, or reversely degraded such that given the input of user 2, the
output at the destination is a conditionally degraded version of the output at
user 2.
Following [4], we define the conditionally degraded subchannels as follows.
Let ${\mathcal{A}}$ denote the index set that includes all indices of
subchannels such that given $x_{2,j}$, the output at user 2 is a conditionally
degraded version of the output at the destination, i.e., for
$j\in{\mathcal{A}}$,
$\displaystyle
p(y_{j},y_{2,j}|x_{1,j},x_{2,j})=p(y_{j}|x_{1,j},x_{2,j})p(y_{2,j}|y_{j},x_{2,j}).$
(8)
We further define $\bar{{\mathcal{A}}}$ to be the complement of the set
${\mathcal{A}}$, and $\bar{{\mathcal{A}}}$ includes all indices of subchannels
such that given $x_{2,j}$, the output at the destination is a conditionally
degraded version of the output at user 2, i.e., for $j\in\bar{{\mathcal{A}}}$,
$\displaystyle
p(y_{j},y_{2,j}|x_{1,j},x_{2,j})=p(y_{2,j}|x_{1,j},x_{2,j})p(y_{j}|y_{2,j},x_{2,j}).$
(9)
Hence, the channel transition probability distribution is given by
$\displaystyle p(y_{[1,L]},y_{2,[1,L]}|$ $\displaystyle
x_{1,[1,L]},x_{2,[1,L]})$ $\displaystyle=$
$\displaystyle\prod_{j\in{\mathcal{A}}}p(y_{j}|x_{1,j},x_{2,j})p(y_{2,j}|y_{j},x_{2,j})\prod_{j\in\bar{{\mathcal{A}}}}p(y_{2,j}|x_{1,j},x_{2,j})p(y_{j}|y_{2,j},x_{2,j}).$
(10)
For the parallel CMAC-CM with degraded subchannels, we apply Theorem 1 and
obtain the following secrecy capacity region.
###### Theorem 2
For the parallel CMAC-CM with degraded subchannels, the secrecy capacity
region is given by
$\displaystyle{\mathcal{C}}_{s}^{\rm[D]}=$
$\displaystyle\bigcup_{\begin{subarray}{c}\prod_{j}p(q_{j},x_{2,j})p(x_{1,j}|q_{j})\\\
p(y_{j},y_{2,j}|x_{1,j},x_{2,j})\end{subarray}}\left\\{\begin{array}[]{l}(R_{0},\,R_{1}):\\\
R_{0}\geq 0,\;R_{1}\geq 0;\\\
~{}~{}R_{1}\leq\sum_{j\in{\mathcal{A}}}[I(X_{1,j};Y_{j}|X_{2,j},Q_{j})-I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j})]\\\
~{}~{}R_{0}\leq\sum_{j\in{\mathcal{A}}}I(Q_{j},X_{2,j};Y_{j})+\sum_{j\in\bar{{\mathcal{A}}}}I(X_{1,j},X_{2,j};Y_{j})\end{array}\right\\}$
(15)
where $Q_{j}$, for $j=1,\dots,L$, are auxiliary random variables that satisfy
the Markov chain relationship
$\displaystyle Q_{j}\rightarrow(X_{1,j},X_{2,j})\rightarrow(Y_{j},Y_{2,j}).$
(16)
###### Proof:
See Appendix -B. ∎
It can be seen that the common message $W_{0}$ is sent over all subchannels,
and the confidential message $W_{1}$ of user 1 is sent only over the
subchannels for which the output at user 2 is a conditionally degraded version
of the output at the destination. Furthermore, user 1 sends the common message
$W_{0}$ and the confidential message $W_{1}$ by using superposition encoding.
## III Parallel Gaussian CMAC-CM
### III-A Channel Model
In this section, we consider the parallel Gaussian CMAC-CM in which the
channel outputs at the destination and user 2 are corrupted by additive
Gaussian noise terms. The channel input-output relationship is given by
$\displaystyle Y_{j,i}$ $\displaystyle=X_{1,j,i}+X_{2,j,i}+Z_{j,i}$
$\displaystyle\text{and}\qquad Y_{2,j,i}$
$\displaystyle=X_{1,j,i}+X_{2,j,i}+Z_{2,j,i}$ (17)
where $i$ is the time index, and for $j=1,\dots,L$, the noise processes
$\\{Z_{j,i}\\}$ and $\\{Z_{2,j,i}\\}$ are independent and identically
distributed (i.i.d.) with the components being zero-mean Gaussian random
variables with variances $\nu_{j}$ and $\mu_{j}$, respectively. We assume
$\nu_{j}<\mu_{j}$ for $j\in{\mathcal{A}}$ and $\nu_{j}\geq\mu_{j}$ for
$j\in\bar{{\mathcal{A}}}$. The channel input sequences $X^{n}_{1,[1,L]}$ and
$X^{n}_{2,[1,L]}$ are subject to average power constraints $P_{1}$ and
$P_{2}$, respectively, i.e.,
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{L}{\sf E}[X_{1,j,i}^{2}]$
$\displaystyle\leq P_{1}$
$\displaystyle\text{and}\qquad\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{L}{\sf
E}[X_{2,j,i}^{2}]$ $\displaystyle\leq P_{2}.$ (18)
### III-B Secrecy Capacity Region
We now apply Theorem 2 to obtain the secrecy capacity region of the parallel
Gaussian MAC. It can be seen from (17) that the subchannels of the parallel
Gaussian MAC are not physically degraded. We consider the following
subchannels, for $j\in{\mathcal{A}}$:
$\displaystyle Y_{j,i}$
$\displaystyle=X_{1,j,i}+X_{2,j,i}+Z_{j,i},~{}~{}Y_{2,j,i}=Y_{j,i}+Z^{\prime}_{2,j,i};$
(19)
and, for $j\in\bar{{\mathcal{A}}}$:
$\displaystyle Y_{j,i}$
$\displaystyle=Y_{2,j,i}+Z^{\prime}_{j,i},~{}~{}Y_{2,j,i}=X_{1,j,i}+X_{2,j,i}+Z_{2,j,i}$
(20)
where $\\{Z^{\prime}_{j,i}\\}$ and $\\{Z^{\prime}_{2,j,i}\\}$ are i.i.d.
random processes with components being zero-mean Gaussian random variables
with variances $\nu_{j}-\mu_{j}$ for $j\in\bar{{\mathcal{A}}}$ and
$\mu_{j}-\nu_{j}$ for $j\in{\mathcal{A}}$, respectively. Moreover,
$\\{Z^{\prime}_{j,i}\\}$ is independent of $\\{Z_{2,j,i}\\}$, and
$\\{Z^{\prime}_{2,j,i}\\}$ is independent of $\\{Z_{j,i}\\}$. We notice that
the channel defined in (19)-(20) is a parallel Gaussian MAC with physically
degraded subchannels. Since the channel (19)-(20) has the same marginal
distributions $p(y|x_{1},x_{2})$ and $p(y_{2}|x_{1},x_{2})$ as the parallel
Gaussian MAC defined in (17), these two channels have the same secrecy
capacity region.111This argument is in fact identical to the so-called
degraded, same-marginals technique; e.g., see [4] for further details.
For the channel defined in (19)-(20), we can apply Theorem 2 to obtain he
following secrecy capacity region. In particular, the degradedness of the
subchannels allows the use of the entropy power inequality in the proof of the
converse. We can thus obtain the secrecy capacity region for the parallel
Gaussian CMAC-CM.
###### Theorem 3
For the parallel Gaussian CMAC-CM, the secrecy capacity region is given by
$\displaystyle{\mathcal{C}}_{s}^{\rm[G]}=$
$\displaystyle\bigcup_{\underline{p}\in{\mathcal{P}}}\left\\{\begin{array}[]{l}(R_{0},\,R_{1}):\\\
~{}~{}R_{0}\geq 0,\;R_{1}\geq 0;\\\
~{}~{}R_{1}\leq\sum_{j\in{\mathcal{A}}}\left[\frac{1}{2}\log\left(1+\frac{b_{j}}{\nu_{j}}\right)-\frac{1}{2}\log\left(1+\frac{b_{j}}{\mu_{j}}\right)\right]\\\
~{}~{}R_{0}\leq\sum_{j\in{\mathcal{A}}}\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}{b_{j}+\nu_{j}}\right)\\\
\qquad+\sum_{j\in\bar{{\mathcal{A}}}}\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}{\nu_{j}}\right)\end{array}\right\\}$
(26)
where $\underline{p}$ is the power allocation vector, which consists of
$(a_{j},b_{j},p_{2,j})$ for $j\in{\mathcal{A}}$ and $(a_{j},0,p_{2,j})$ for
$j\in\bar{{\mathcal{A}}}$ as components, and the set ${\mathcal{P}}$ includes
all power allocation vectors $\underline{p}$ that satisfy the power constraint
$\displaystyle{\mathcal{P}}:=\left\\{\underline{p}:\sum_{j=1}^{L}(a_{j}+b_{j})\leq
P_{1}~{}\text{and}~{}\sum_{j=1}^{L}p_{2,j}\leq P_{2}\right\\}.$ (27)
###### Proof:
See Appendix -C. ∎
We notice that $\underline{p}$ denotes the power allocation among all
subchannels. In particular, for $j\in{\mathcal{A}}$, since user 1 needs to
transmit both common and confidential information, the pair $(a_{j},b_{j})$
controls the power allocation between the common message $W_{0}$ and the
confidential message $W_{1}$. For $j\in\bar{{\mathcal{A}}}$, user 1 transmits
only the common information, and $b_{j}=0$ indicates that the power is
allocated to transmit the common message $W_{0}$ only.
### III-C Optimal Power Allocation
To characterize the secrecy capacity region of the parallel Gaussian CMAC-CM
given in (26), we need to characterize every boundary point and the power
allocation vector that achieve each boundary point. Since the secrecy capacity
region ${\mathcal{C}}_{s}^{\rm[G]}$ is convex, for every boundary point
$(R_{0}^{\star},R_{1}^{\star})$, there exists $\gamma_{1}\geq 0$ such that
$(R_{0}^{\star},R_{1}^{\star})$ is the solution to the optimization problem
$\displaystyle\max_{(R_{0},R_{1})\in{\mathcal{C}}_{s}^{\rm[G]}}\left[R_{0}+\gamma_{1}R_{1}\right].$
(28)
Note that the optimization problem (28) serves as a complete characterization
of the corresponding boundary of the secrecy capacity region, and the solution
to (28) provides the power allocations that achieve the boundary of the
secrecy capacity region. Let $(x)^{+}=\max(0,\,x)$. We obtain the optimal
power allocation $\underline{p}$ that solves (28).
###### Theorem 4
Let $\underline{p}^{\star}$ be an optimal solution to the optimization problem
of (28) that achieves the boundary of the secrecy capacity region of the
parallel Gaussian CMAC-CM. Then, $\underline{p}^{\star}$ can be written as
follows.
For $j\in{\mathcal{A}}$, if
$\displaystyle\frac{2\lambda_{1}^{2}\ln
2}{\lambda_{1}+\lambda_{2}}<\frac{\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j}}{\mu_{j}\nu_{j}},$
(29)
then
$\displaystyle a_{j}^{\star}$
$\displaystyle=\frac{\lambda_{2}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}-\phi_{j}\right)^{+},$
$\displaystyle b_{j}^{\star}$
$\displaystyle=\left(\min\left[s_{2,j},\,\phi_{j}\right]\right)^{+}$
$\displaystyle\text{and}\qquad p_{2,j}^{\star}$
$\displaystyle=\frac{\lambda_{1}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}-\phi_{j}\right)^{+};$
(30)
alternatively, if
$\displaystyle\frac{2\lambda_{1}^{2}\ln
2}{\lambda_{1}+\lambda_{2}}\geq\frac{\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j}}{\mu_{j}\nu_{j}},$
(31)
then
$\displaystyle a_{j}^{\star}$
$\displaystyle=\frac{\lambda_{2}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}\right)^{+},$
$\displaystyle b_{j}^{\star}$ $\displaystyle=0$ $\displaystyle\text{and}\qquad
p_{2,j}^{\star}$
$\displaystyle=\frac{\lambda_{1}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}\right)^{+};$
(32)
for $j\in\bar{{\mathcal{A}}}$,
$\displaystyle
a_{j}^{\star}=\frac{\lambda_{2}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}\right)^{+}\quad\text{and}\quad
p_{2,j}^{\star}$
$\displaystyle=\frac{\lambda_{1}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}\right)^{+};$
(33)
where $\gamma_{1}\geq 0$,
$\displaystyle s_{1,j}$
$\displaystyle=\frac{\lambda_{1}+\lambda_{2}}{2\lambda_{1}\lambda_{2}\ln
2}-\nu_{j},$ $\displaystyle s_{2,j}$
$\displaystyle=\frac{1}{2}\left[\sqrt{(\mu_{j}-\nu_{j})\left(\mu_{j}-\nu_{j}+\frac{2\gamma_{1}}{\lambda_{1}\ln
2}\right)}-(\mu_{j}+\nu_{j})\right],$ $\displaystyle\phi_{j}$
$\displaystyle=-\frac{1}{2}\left(\mu_{j}+\nu_{j}+\frac{1}{\omega}\right)+\frac{1}{2}\sqrt{\left(\mu_{j}+\nu_{j}+\frac{1}{\omega}\right)^{2}-4\left[\mu_{j}\nu_{j}-\frac{\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j}}{\omega}\right]},$
$\displaystyle\omega$ $\displaystyle=(2\ln
2)\frac{\lambda_{1}^{2}}{\lambda_{1}+\lambda_{2}}$ (34)
and the pair $(\lambda_{1},\lambda_{2})$ is chosen to satisfy the power
constraint
$\displaystyle\sum_{j=1}^{L}(a_{j}+b_{j})\leq
P_{1}~{}\text{and}~{}\sum_{j=1}^{L}p_{2,j}\leq P_{2}.$ (35)
###### Proof:
The optimization problem is non-convex. Our proof technique involves applying
KKT conditions (as necessary conditions), which help express the Lagrangian in
the form of an integral. This specific structure of the problem is then
exploited to obtain a closed-form solution for the optimal power allocation
strategy. The details can be found in Appendix -D. ∎
## IV Fading CMAC-CM
In this section, we study the fading CMAC-CM, where both the user-to-
destination and the user-to-user channels are corrupted by multiplicative
fading gain processes in addition to additive white Gaussian processes. The
channel input-output relationship is given by
$\displaystyle Y_{i}$ $\displaystyle=h_{1,i}X_{1,i}+h_{2,i}X_{2,i}+Z_{i}$
$\displaystyle\text{and}\quad Y_{2,i}$
$\displaystyle=g_{1,i}X_{1,i}+g_{2,i}X_{2,i}+Z_{2,i}$ (36)
where $i$ is the time index, $X_{1,i}$ and $X_{2,i}$ are channel inputs at the
time instant $i$ from user 1 and user 2, respectively, $Y_{i}$ and $Y_{2,i}$
are channel outputs at the time instant $i$ at the destination and the
receiver of user 2, respectively; $\underline{h}_{i}:=(h_{1,i},h_{2,i})$ and
$\underline{g}_{i}:=(g_{1,i},g_{2,i})$ are proper complex random channel
attenuation pairs imposed on the destination and the receiver of user 2; and
the noise processes $\\{Z_{i}\\}$ and $\\{Z_{2,i}\\}$ are i.i.d. with the
components being zero-mean proper complex Gaussian random variables with
variances $\nu$ and $\mu$, respectively. The input sequences $\\{X_{1,i}\\}$
and $\\{X_{2,i}\\}$ are subject to the average power constraint $P_{1}$ and
$P_{2}$, i.e.,
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}{\sf E}[X_{1,i}^{2}]\leq
P_{1}\quad\text{and}\quad\frac{1}{n}\sum_{i=1}^{n}{\sf E}[X_{2,i}^{2}]\leq
P_{2}.$ (37)
Figure 2: Fading CMAC-CM.
We assume that the CSI (i.e., the realization of
$(\underline{h},\underline{g})$) is known at both the transmitters and the
receivers instantaneously. Depending on the CSI, each user can dynamically
change its transmission power and rate to achieve better performance. In this
section, we assume that there is no delay constraint on the transmitted
messages, and that the secrecy capacity region is an average over all channel
states, which is referred to as the ergodic secrecy capacity region.
We notice that for a given fading state, i.e., a realization of
$(\underline{h},\underline{g})$, the fading CMAC-CM is a Gaussian CMAC-CM.
Hence, the fading CMAC-CM can be viewed as a parallel Gaussian CMAC-CM with
each fading state corresponding to one subchannel. Thus, the following secrecy
capacity region of the fading CMAC-CM follows from Theorem 3.
In the following, for each channel state $(\underline{h},\underline{g})$, we
use $p_{1}(\underline{h},\underline{g})$ and
$p_{2}(\underline{h},\underline{g})$ to denote the powers allocated at users 1
and 2, respectively. We further define
$\displaystyle
p(\underline{h},\underline{g}):=\left(a(\underline{h},\underline{g}),b(\underline{h},\underline{g}),p_{2}(\underline{h},\underline{g})\right).$
(38)
Let ${\mathcal{P}}$ denote the set that includes all power allocations that
satisfy the power constraint
$\displaystyle{\mathcal{P}}:=\bigl{\\{}p(\underline{h},\underline{g}):~{}$
$\displaystyle{\sf
E}[a(\underline{h},\underline{g})+b(\underline{h},\underline{g})]\leq
P_{1}\quad\text{and}\quad{\sf E}[p_{2}(\underline{h},\underline{g})]\leq
P_{2}\bigr{\\}},$ (39)
and ${\mathcal{A}}$ denote the set of channel states as follows:
$\displaystyle{\mathcal{A}}:=\left\\{(\underline{h},\underline{g}):~{}\frac{|h_{1}|^{2}}{\nu}>\frac{|g_{1}|^{2}}{\mu}\right\\}.$
(40)
###### Corollary 1
The secrecy capacity region of the fading CMAC-CM is given by (45)
$\displaystyle{\mathcal{C}}_{s}^{\rm[F]}=$
$\displaystyle\bigcup_{p(\underline{h},\underline{g})\in{\mathcal{P}}}\left\\{\begin{array}[]{l}(R_{0},\,R_{1}):\\\
~{}~{}R_{0}\geq 0,\;R_{1}\geq 0;\\\ ~{}~{}R_{1}\leq{\sf
E}_{(\underline{h},\underline{g})\in{\mathcal{A}}}\left[\log\left(1+\frac{b(\underline{h},\underline{g})|h_{1}|^{2}}{\nu}\right)-\log\left(1+\frac{b(\underline{h},\underline{g})|g_{1}|^{2}}{\mu}\right)\right]\\\
~{}~{}R_{0}\leq{\sf
E}_{(\underline{h},\underline{g})\in{\mathcal{A}}}\log\left(1+\frac{\chi(\underline{h},\underline{g})}{b(\underline{h},\underline{g})|h_{1}|^{2}+\nu}\right)+{\sf
E}_{(\underline{h},\underline{g})\in\bar{{\mathcal{A}}}}\log\left(1+\frac{\chi(\underline{h},\underline{g})}{\nu}\right)\end{array}\right\\}$
(45)
where
$\displaystyle\chi(\underline{h},\underline{g})$
$\displaystyle=\left[\sqrt{a(\underline{h},\underline{g})}|h_{1}|+\sqrt{p_{2}(\underline{h},\underline{g})}|h_{2}|\right]^{2}$
(46)
and the random vector pair $(\underline{h},\underline{g})$ has the same
distribution as the marginal distribution of the process
$\\{(\underline{h}_{i},\underline{g}_{i})\\}$ at a single time instant.
The secrecy capacity region given in Corollary 1 is established for fading
processes $(\underline{h},\underline{g})$ where only ergodic and stationary
conditions are assumed. The fading process $(\underline{h},\underline{g})$ can
be correlated across time, and is not necessarily Gaussian.
Since users are assumed to know the CSI, they can allocate their powers
according to the instantaneous channel realization to achieve the optimal
performance, i.e., the boundary of the secrecy capacity region. The optimal
power allocation that achieves the boundary of the secrecy capacity region for
the fading CMAC-CM can be derived from Theorem 4 and is given in the
following.
###### Corollary 2
Let $p(\underline{h},\underline{g})^{\star}$ be an optimal power allocation
that achieves the boundary of the secrecy capacity region of the fading CMAC-
CM. Then, $p(\underline{h},\underline{g})^{\star}$ is given as follows:
* •
for $(\underline{h},\underline{g})\in{\mathcal{A}}$, if
$\displaystyle\frac{\lambda_{1}^{2}|h_{2}|^{2}\ln
2}{\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}}<\frac{\gamma_{1}\left(\mu|h_{1}|^{2}-\nu|g_{1}|^{2}\right)-\mu|h_{1}|^{2}}{\mu\nu},$
(47)
then
$\displaystyle a^{\star}(\underline{h},\underline{g})$
$\displaystyle=\frac{\lambda_{2}^{2}|h_{1}|^{2}}{\left(\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}\right)^{2}}\left[s_{1}(\underline{h},\underline{g})-\phi(\underline{h},\underline{g})\right]^{+},$
$\displaystyle b^{\star}(\underline{h},\underline{g})$
$\displaystyle=\left(\min\left[s_{2}(\underline{h},\underline{g}),\,\phi(\underline{h},\underline{g})\right]\right)^{+}$
$\displaystyle\text{and}\qquad p_{2}^{\star}(\underline{h},\underline{g})$
$\displaystyle=\frac{\lambda_{1}^{2}|h_{2}|^{2}}{\left(\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}\right)^{2}}\left[s_{1}(\underline{h},\underline{g})-\phi(\underline{h},\underline{g})\right]^{+};$
(48)
alternatively, if
$\displaystyle\frac{\lambda_{1}^{2}|h_{2}|^{2}\ln
2}{\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}}\geq\frac{\gamma_{1}\left(\mu|h_{1}|^{2}-\nu|g_{1}|^{2}\right)-\mu|h_{1}|^{2}}{\mu\nu},$
(49)
then
$\displaystyle a^{\star}(\underline{h},\underline{g})$
$\displaystyle=\frac{\lambda_{2}^{2}|h_{1}|^{2}}{\left(\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}\right)^{2}}\left[s_{1}(\underline{h},\underline{g})\right]^{+},$
$\displaystyle b^{\star}(\underline{h},\underline{g})$ $\displaystyle=0$
$\displaystyle\text{and}\qquad p_{2}^{\star}(\underline{h},\underline{g})$
$\displaystyle=\frac{\lambda_{1}^{2}|h_{2}|^{2}}{\left(\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}\right)^{2}}\left[s_{1}(\underline{h},\underline{g})\right]^{+};$
(50)
* •
for $(\underline{h},\underline{g})\in\bar{{\mathcal{A}}}$,
$\displaystyle a^{\star}(\underline{h},\underline{g})$
$\displaystyle=\frac{\lambda_{2}^{2}|h_{1}|^{2}}{\left(\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}\right)^{2}}\left[s_{1}(\underline{h},\underline{g})\right]^{+}$
$\displaystyle\text{and}\qquad p_{2}^{\star}(\underline{h},\underline{g})$
$\displaystyle=\frac{\lambda_{1}^{2}|h_{2}|^{2}}{\left(\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}\right)^{2}}\left[s_{1}(\underline{h},\underline{g})\right]^{+};$
(51)
where $\gamma_{1}\geq 0$,
$\displaystyle s_{1}(\underline{h},\underline{g})$
$\displaystyle=\frac{\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}}{\lambda_{1}\lambda_{2}\ln
2}-\nu,$ $\displaystyle s_{2}(\underline{h},\underline{g})$
$\displaystyle=\frac{1}{2}\left[\sqrt{\left(\frac{\mu}{|g_{1}|^{2}}-\frac{\nu}{|h_{1}|^{2}}\right)\left(\frac{\mu}{|g_{1}|^{2}}-\frac{\nu}{|h_{1}|^{2}}+\frac{2\gamma_{1}}{\lambda_{1}\ln
2}\right)}-\left(\frac{\mu}{|g_{1}|^{2}}+\frac{\nu}{|h_{1}|^{2}}\right)\right],$
$\displaystyle\phi(\underline{h},\underline{g})$
$\displaystyle=-\frac{1}{2}\left(\frac{\mu}{|g_{1}|^{2}}+\frac{\nu}{|h_{1}|^{2}}+\frac{1}{\omega(\underline{h},\underline{g})}\right)$
$\displaystyle\quad+\frac{1}{2}\sqrt{\left(\frac{\mu}{|g_{1}|^{2}}+\frac{\nu}{|h_{1}|^{2}}+\frac{1}{\omega(\underline{h},\underline{g})}\right)^{2}-4\left[\frac{\mu}{|g_{1}|^{2}}\frac{\nu}{|h_{1}|^{2}}-\frac{\gamma_{1}\left(\frac{\mu}{|g_{1}|^{2}}-\frac{\nu}{|h_{1}|^{2}}\right)-\frac{\mu}{|g_{1}|^{2}}}{\omega(\underline{h},\underline{g})}\right]}$
$\displaystyle\omega(\underline{h},\underline{g})$ $\displaystyle=(\ln
2)\frac{\lambda_{1}^{2}|h_{2}|^{2}}{\lambda_{1}|h_{2}|^{2}+\lambda_{2}|h_{1}|^{2}}$
(52)
and the pair $(\lambda_{1},\lambda_{2})$ is chosen to satisfy the power
constraint
$\displaystyle{\sf
E}[a(\underline{h},\underline{g})+b(\underline{h},\underline{g})]\leq
P_{1}\quad\text{and}\quad{\sf E}[p_{2}(\underline{h},\underline{g})]\leq
P_{2}.$ (53)
## V Numerical Examples
In this section, we study two numerical examples to illustrate the secrecy
capacity regions of the parallel Gaussian CMAC-CM and the fading CMAC-CM,
respectively.
Figure 3: Secrecy capacity region vs. asynchronous secrecy rate region for the
example $L=10$ parallel Gaussian CMAC-CM.
We first consider an $L=10$ parallel Gaussian CMAC-CM. We assume that the
source power constraints of users 1 and 2 are
$P_{1}=12~{}\text{dB}\quad\text{and}\quad P_{2}=10~{}\text{dB},$
and the noise variances at the receivers of the destination and of user 2 are
given by
$\displaystyle\underline{\nu}$ $\displaystyle=[1,2,3,4,5,6,7,8,9,10]$
$\displaystyle\text{and}\qquad\underline{\mu}$
$\displaystyle=[5,3,4,9,1,10,8,7,2,6].$
Fig. 3 illustrates the boundary of the secrecy capacity region for this
channel. For comparison, we also consider the asynchronous case, in which
users 1 and 2 send the common message $W_{0}$ in a asynchronous transmission
mode. In this case, the secrecy rate region is given by
$\displaystyle{\mathcal{R}}_{s}^{\rm[G]}=$
$\displaystyle\bigcup_{\underline{p}\in{\mathcal{P}}}\left\\{\begin{array}[]{l}(R_{0},\,R_{1}):\\\
~{}~{}R_{0}\geq 0,\;R_{1}\geq 0;\\\
~{}~{}R_{1}\leq\sum_{j\in{\mathcal{A}}}\left[\frac{1}{2}\log\left(1+\frac{b_{j}}{\nu_{j}}\right)\frac{1}{2}\log\left(1+\frac{b_{j}}{\mu_{j}}\right)\right]\\\
~{}~{}R_{0}\leq\sum_{j\in{\mathcal{A}}}\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}}{b_{j}+\nu_{j}}\right)+\sum_{j\in\bar{{\mathcal{A}}}}\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}}{\nu_{j}}\right)\end{array}\right\\}$
(58)
where $\underline{p}$ is the power allocation vector, which consists of
$(a_{j},b_{j},p_{2,j})$ for $j\in{\mathcal{A}}$ and $(a_{j},0,p_{2,j})$ for
$j\in\bar{{\mathcal{A}}}$ as components, and the set ${\mathcal{P}}$ includes
all power allocation vector $\underline{p}$ that satisfy the power constraint
(35). We observe that the synchronous transmission mode significantly
increases the rate $R_{0}$ of the common message since coherent combining
detection can be employed at the destination.
Next, we consider the Rayleigh-fading CMAC-CM, where $h_{1}$, $h_{2}$ and
$g_{1}$ are zero-mean proper complex Gaussian random variables. Hence,
$|h_{1}|^{2}$, $|h_{2}|^{2}$ and $|g_{1}|^{2}$ are exponentially distributed
with means $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$. We assume that the
power constraints of users 1 and 2 are $P_{1}=P_{2}=10~{}\text{dB}$, and the
noise variances at the receivers of the destination and of user 2 are
$\nu=\mu=2$. In Fig. 4, we plot the boundaries of the secrecy capacity regions
corresponding to $\sigma_{1}=0.5,\;1,\;2$ and fixed $\sigma_{2}=\sigma_{3}=1$.
It can been seen that as $\sigma_{1}$ increases, both the secrecy rate $R_{1}$
of the confidential message $W_{1}$ and the rate $R_{0}$ of the common message
$W_{0}$ improve. This is because larger $\sigma_{1}$ implies a better channel
from user 1 to the destination. In Fig. 5, we plot the boundaries of the
secrecy capacity regions corresponding to $\sigma_{2}=0.5,\;1,\;2$ and fixed
$\sigma_{1}=\sigma_{3}=1$. It can been seen that as $\sigma_{2}$ increases,
only the rate $R_{0}$ of the common message $W_{0}$ improves. In Fig. 6, we
plot the boundaries of the secrecy capacity regions corresponding to
$\sigma_{3}=0.5,\;1,\;2$ and fixed $\sigma_{1}=\sigma_{2}=1$. It can been seen
that as $\sigma_{3}$ decreases, only the rate $R_{1}$ of the confidential
message $W_{1}$ improves.
Figure 4: Secrecy capacity regions for the example fading CMAC-CMs
($P_{1}=P_{2}=10~{}\text{dB}$, $\nu=\mu=2$, and $\sigma_{2}=\sigma_{3}=1$).
Figure 5: Secrecy capacity regions for the example fading CMAC-CMs
($P_{1}=P_{2}=10~{}\text{dB}$, $\nu=\mu=2$, and $\sigma_{1}=\sigma_{3}=1$).
Figure 6: Secrecy capacity regions for the example fading CMAC-CMs
($P_{1}=P_{2}=10~{}\text{dB}$, $\nu=\mu=2$, and $\sigma_{1}=\sigma_{2}=1$).
## VI Conclusion
We have established the secrecy capacity region of the parallel CMAC-CM, in
which it is seen that having independent inputs to each subchannel is optimal.
From this result, we have derived the secrecy capacity region for the parallel
Gaussian CMAC-CM and the ergodic secrecy capacity region for the fading CMAC-
CM. We have illustrated that, when both users know the CSI, they can
dynamically adapt their transmission powers with the channel realization to
achieve the optimal performance.
### -A Proof of Theorem 1
#### Achievability
The achievability follows from [4, Corollary 3] by setting
$\displaystyle Q$ $\displaystyle:=(Q_{1},\dots,Q_{L}),\quad
U:=(U_{1},\dots,U_{L})$ $\displaystyle X_{1}$
$\displaystyle:=(X_{1,1},\dots,X_{1,L}),\quad X_{2}:=(X_{2,1},\dots,X_{2,L})$
$\displaystyle Y$ $\displaystyle:=(Y_{1},\dots,Y_{L}),\quad\text{and}\quad
Y_{2}:=(Y_{2,1},\dots,Y_{2,L})$ (59)
with $Q$, $U$, $X_{1}$, and $X_{2}$ having independent components.
Furthermore, we choose the components of these random vectors to satisfy the
condition
$\displaystyle
p(q_{j},u_{j},x_{1,j},x_{2,j},y_{j},y_{2,j})=p(q_{j},x_{2,j})p(u_{j}|q_{j})p(x_{1,j}|u_{j})p(y_{j},y_{2,j}|x_{1,j},x_{2,j})~{}~{}\text{for}~{}j=1,\dots,L.$
(60)
Using the above definition, we have the following achievable region
$\displaystyle{\mathcal{R}}_{s}^{P}:=\bigcup_{\begin{subarray}{c}\prod_{j}p(q_{j},x_{2,j})p(u_{j}|q_{j})p(x_{1,j}|u_{j})\\\
p(y_{j},y_{2,j}|x_{1,j},x_{2,j})\end{subarray}}\left\\{(R_{0},\,R_{1})\;\left|\begin{array}[]{l}R_{0}\geq
0,\;R_{1}\geq 0;\\\
R_{1}\leq\sum_{j=1}^{L}\left[I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})\right]\\\
R_{0}+R_{1}\leq\sum_{j=1}^{L}\left[I(U_{j},X_{2,j},Q_{j};Y_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})\right]\end{array}\right.\right\\}.$
(64)
Note that
$\displaystyle\left[I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})\right]+I(X_{2,j},Q_{j};Y_{j})=I(U_{j},X_{2,j},Q_{j};Y_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})$
(65)
and hence, any rate pair $(R_{0},R_{1})\in{\mathcal{C}}_{s}^{P}$ must also
satisfies $(R_{0},R_{1})\in{\mathcal{R}}_{s}^{P}$. This implies that the
secrecy rate region ${\mathcal{C}}_{s}^{\rm[P]}$ is achievable.
#### Converse
By Fano’s inequality [11, Chapter 2.11], we have
$\displaystyle H\left(W_{0},W_{1}|Y^{n}_{[1,L]}\right)$ $\displaystyle\leq
n(R_{0},+R_{1})\epsilon+1:=n\delta$ (66)
where $\delta\rightarrow 0$ if $\epsilon\rightarrow 0$. On the other hand, the
information theoretic secrecy implies that
$\displaystyle H(W_{1})\leq
H\left(W_{1}|Y^{n}_{2,[1,L]},X^{n}_{2,[1,L]},W_{0}\right)+n\epsilon.$ (67)
Now, we consider the upper bound on the secrecy rate $R_{1}$ as
$\displaystyle nR_{1}$ $\displaystyle=H(W_{1})$ $\displaystyle\leq
H\left(W_{1}|Y^{n}_{2,[1,L]},X^{n}_{2,[1,L]},W_{0}\right)+n\epsilon$ (68)
$\displaystyle\leq
H\left(W_{1}|Y^{n}_{2,[1,L]},X^{n}_{2,[1,L]},W_{0}\right)-H\left(W_{1}|Y^{n}_{[1,L]},X^{n}_{2,[1,L]},W_{0}\right)+n(\epsilon+\delta)$
(69)
$\displaystyle=I\left(W_{1};Y^{n}_{[1,L]}|X^{n}_{2,[1,L]},W_{0}\right)-I\left(W_{1};Y^{n}_{2,[1,L]}|X^{n}_{2,[1,L]},W_{0}\right)+n(\epsilon+\delta)$
$\displaystyle=\sum_{j=1}^{L}\left[I\left(W_{1};Y^{n}_{j}|Y^{n}_{[1,j-1]},X^{n}_{2,[1,L]},W_{0}\right)-I\left(W_{1};Y^{n}_{2,j}|Y^{n}_{2,[j+1,L]},X^{n}_{2,[1,L]},W_{0}\right)\right]+n(\epsilon+\delta)$
(70)
$\displaystyle=\sum_{j=1}^{L}\sum_{i=1}^{n}\left[I\left(W_{1};Y_{j,i}|Y_{j}^{i-1},Y^{n}_{[1,j-1]},X^{n}_{2,[1,L]},W_{0}\right)-I\left(W_{1};Y_{2,j,i}|Y^{n}_{2,j,i+1},Y^{n}_{2,[j+1,L]},X^{n}_{2,[1,L]},W_{0}\right)\right]+n(\epsilon+\delta)$
(71)
where (68) follows from the secrecy constraint (67), (69) follows from Fano’s
inequality (66), and (70) and (71) follow from the chain rule of mutual
information [11, Chapter 2.5]. Let
$\displaystyle
Q_{j,i}:=\left(Y_{j}^{i-1},Y^{n}_{[1,j-1]},Y^{n}_{2,j,i+1},Y^{n}_{2,[j+1,L]},X^{n}_{2,[1,L]},W_{0}\right).$
(72)
We notice that this definition implies the Markov chain relationship
$\displaystyle X_{2,j,i}\rightarrow Q_{j,i}\rightarrow W_{1}\rightarrow
X_{1,j,i}.$ (73)
Then, following from [2, Lemma 7], we have
$\displaystyle nR_{1}$
$\displaystyle\leq\sum_{j=1}^{L}\sum_{i=1}^{n}\left[I\left(W_{1};Y_{j,i}|X_{2,j,i},Q_{j,i}\right)-I\left(W_{1};Y_{2,j,i}|X_{2,j,i},Q_{j,i}\right)\right]+n(\epsilon+\delta).$
(74)
We also can write
$\displaystyle nR_{0}$ $\displaystyle=H(W_{0})$ $\displaystyle\leq
I(W_{0};Y^{n}_{[1,L]})+n\delta$ (75)
$\displaystyle=\sum_{j=1}^{L}\sum_{i=1}^{n}I(W_{0};Y_{j,i}|Y_{j}^{i-1},Y^{n}_{[1,j-1]})+n\delta$
(76)
$\displaystyle\leq\sum_{j=1}^{L}\sum_{i=1}^{n}I(W_{0},Y_{j}^{i-1},Y^{n}_{[1,j-1]},Y^{n}_{2,j,i+1},Y^{n}_{2,[j+1,L]},X^{n}_{2,[1,L]};Y_{j,i})+n\delta$
$\displaystyle=\sum_{j=1}^{L}\sum_{i=1}^{n}I(Q_{j,i},X_{2,j,i};Y_{j,i})+n\delta$
(77)
where (75) follows from Fano’s inequality (66), (76) follows from the chain
rule, and (77) follows from the definition of $Q_{j,i}$ in (72).
We introduce a time-sharing random variable $T$ [11, Chapter 14.3] that is
independent of all other random variables in the model, and uniformly
distributed over $\\{1,\dots,n\\}$. Define $Q_{j}=(T,Q_{i},j)$,
$U_{j}=(Q_{j},W_{1})$, $X_{1,j}=X_{1,T,j}$, $X_{2,j}=X_{2,T,j}$,
$Y_{2,j}=X_{2,T,j}$, and $Y_{j}=Y_{T,j}$ for $j=1,\dots,L$. Note that
$(Q_{j},X_{1,j},X_{2,j},Y_{j},Y_{2,j})$ satisfies the following Markov chain
relationship
$\displaystyle Q_{j}\rightarrow
U_{j}\rightarrow(X_{1,j},X_{2,j})\rightarrow(Y_{j},Y_{2,j}),\quad\text{for}~{}j=1,\dots,L.$
(78)
Using the above definition, (74) and (77) become
$\displaystyle R_{1}$
$\displaystyle\leq\sum_{j=1}^{L}\left[I\left(U_{j};Y_{j}|X_{2,j},Q_{j}\right)-I\left(U_{j};Y_{2,j}|X_{2,j},Q_{j}\right)\right]+(\epsilon+\delta).$
$\displaystyle\text{and}\qquad R_{0}$
$\displaystyle\leq\sum_{j=1}^{L}I\left(X_{2,j},Q_{j};Y_{j}\right)+\delta.$
(79)
### -B Proof of Theorem 2
The achievability follows from Theorem 1 by setting
$\displaystyle U_{j}$ $\displaystyle=X_{1,j}\qquad\text{for}\quad
j\in{\mathcal{A}}$ $\displaystyle\text{and}\qquad Q_{j}=U_{j}$
$\displaystyle=X_{1,j}\qquad\text{for}\quad j\in\bar{{\mathcal{A}}}.$ (80)
To show the converse, we first consider the upper bound on $R_{0}$. By using
(7) in Theorem 1, we have
$\displaystyle R_{0}$ $\displaystyle\leq\sum_{j=1}^{L}I(Q_{j},X_{2,j};Y_{j})$
$\displaystyle=\sum_{j\in{\mathcal{A}}}I(Q_{j},X_{2,j};Y_{j})+\sum_{j\in\bar{{\mathcal{A}}}}I(Q_{j},X_{2,j};Y_{j})$
$\displaystyle\leq\sum_{j\in{\mathcal{A}}}I(Q_{j},X_{2,j};Y_{j})+\sum_{j\in\bar{{\mathcal{A}}}}I(X_{1,j},X_{2,j};Y_{j})$
(81)
where (81) follows from the Markov chain relationships
$\displaystyle Q_{j}\rightarrow(X_{1,j},X_{2,j})\rightarrow Y_{j}.$ (82)
Now, we consider the upper bound on $R_{1}$. By applying Theorem 1, we obtain
$\displaystyle R_{1}$
$\displaystyle\leq\sum_{j=1}^{L}[I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})]$
$\displaystyle=\sum_{j\in{\mathcal{A}}}[I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})]+\sum_{j\in\bar{{\mathcal{A}}}}[I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})].$
(83)
For $j\in\bar{\mathbf{A}}$, the subchannel satisfies
$\displaystyle
p(y_{j},y_{2,j}|x_{1,j},x_{2,j})=p(y_{2,j}|x_{1,j},x_{2,j})p(y_{j}|y_{2,j},x_{2,j}),\quad\text{for}~{}j\in\bar{{\mathcal{A}}}.$
(84)
This implies that
$\displaystyle I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})$
$\displaystyle\leq I(U_{j};Y_{j}|X_{2,j},Q_{j},Y_{2,j})$ $\displaystyle\leq
I(Q_{j},U_{j};Y_{j}|X_{2,j},Y_{2,j})$
$\displaystyle=0\quad\text{for}~{}j\in\bar{{\mathcal{A}}}$ (85)
where the last equality of (85) follows from the Markov chain relationship
$\displaystyle(Q_{j},U_{j})\rightarrow(X_{1,j},X_{2,j})\rightarrow(Y_{2,j},X_{2,j})\rightarrow
Y_{j}\quad\text{for}~{}j\in\bar{{\mathcal{A}}}.$ (86)
On the other hand, for $j\in\mathbf{A}$, the subchannel satisfies
$\displaystyle
p(y_{j},y_{2,j}|x_{1,j},x_{2,j})=p(y_{j}|x_{1,j},x_{2,j})p(y_{2,j}|y_{j},x_{2,j}),\quad\text{for}~{}j\in{\mathcal{A}}.$
(87)
Hence, we obtain
$\displaystyle I(U_{j};Y_{j}|X_{2,j},Q_{j})-I(U_{j};Y_{2,j}|X_{2,j},Q_{j})$
$\displaystyle\leq I(U_{j};Y_{j}|X_{2,j},Q_{j},Y_{2,j})$ $\displaystyle\leq
I(U_{j},X_{1,j};Y_{j}|X_{2,j},Q_{j},Y_{2,j})$
$\displaystyle=I(X_{1,j};Y_{j}|X_{2,j},Q_{j},Y_{2,j})$ (88)
$\displaystyle=I(X_{1,j};Y_{j},Y_{2,j}|X_{2,j},Q_{j})-I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j})$
(89)
$\displaystyle=I(X_{1,j};Y_{j}|X_{2,j},Q_{j})-I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j})\quad\text{for}~{}j\in{\mathcal{A}}$
(90)
where (88) follows from the Markov chain relationship
$\displaystyle(Q_{j},U_{j},Y_{2,j})\rightarrow(X_{1,j},X_{2,j})\rightarrow
Y_{j},$ (91)
(89) follows from the chain rule of mutual information, and (90) follows from
the conditional degradedness (87). Now, substituting (85) and (90) into (83),
we obtain the bound on $R_{1}$ given in (15). This concludes the proof of the
converse.
### -C Proof of Theorem 3
By the degraded, same-marginals argument (see [4]), we need to prove Theorem 3
only for the channel defined by (19)-(20).
#### Achievability
The achievability follows by applying Theorem 2 and choosing the input
distribution as follows
$\displaystyle Q_{j}$
$\displaystyle=\text{constant},~{}X_{2,j}\sim{\mathcal{N}}(0,p_{2,j}),$
$\displaystyle X^{\prime}_{1,j}$
$\displaystyle\sim{\mathcal{N}}(0,(1-\alpha_{j})p_{1,j}),~{}X^{\prime}_{1,j}~{}\text{is
independent of}~{}X_{2,j}$ $\displaystyle\text{and}\quad X_{1,j}$
$\displaystyle=\sqrt{\frac{\alpha_{j}p_{1,j}}{p_{2,j}}}X_{2,j}+X^{\prime}_{1,j}.$
(92)
Moreover, by the fact $\alpha_{j}=1$ for $j\in\bar{{\mathcal{A}}}$, we obtain
the secrecy rate region ${\mathcal{C}}_{s}^{\rm[G]}$ is achievable.
#### Converse
Here, we derive a tight upper bound on the achievable weighted sum rate
$\displaystyle R_{0}+\gamma_{1}R_{1}$
using Theorem 2 as the starting point. Since a capacity region is always
convex (via a time-sharing argument), an exact characterization of all the
achievable weighted sum rates for all nonnegative $\gamma_{1}$ provides an
exact characterization of the entire secrecy capacity region. By Theorem 2,
any achievable rate pair $(R_{0},R_{1})$ must satisfy:
$\displaystyle R_{0}+\gamma_{1}R_{1}$
$\displaystyle\leq\sum_{j\in{\mathcal{A}}}\left[I(Q_{j},X_{2,j};Y_{j})+\gamma_{1}I(X_{1,j};Y_{j}|X_{2,j},Q_{j})-\gamma_{1}I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j})\right]$
$\displaystyle\quad+\sum_{j\in\bar{{\mathcal{A}}}}I(X_{1,j},X_{2,j};Y_{j}).$
(93)
For the subchannel $j\in\bar{{\mathcal{A}}}$, we are concerned only with the
term
$\displaystyle I(X_{1,j},X_{2,j};Y_{j}).$ (94)
The maximum-entropy theorem [11] implies that (94) is maximized when $X_{1,j}$
and $X_{2,j}$ are jointly Gaussian with variance $p_{1,j}$ and $p_{2,j}$
repetitively, and are aligned, i.e., $X_{1,j}=\sqrt{p_{1,j}/p_{2,j}}X_{2,j}$.
Hence, we have
$\displaystyle
I(X_{1,j},X_{2,j};Y_{j})\leq\frac{1}{2}\log\left(1+\frac{p_{1,j}+p_{2,j}+2\sqrt{p_{1,j}p_{2,j}}}{\nu_{j}}\right)\quad\text{for}~{}j\in\bar{{\mathcal{A}}}.$
(95)
For the subchannel $j\in{\mathcal{A}}$, we focus on the term
$\displaystyle
I(Q_{j},X_{2,j};Y_{j})+\gamma_{1}I(X_{1,j};Y_{j}|X_{2,j},Q_{j})-\gamma_{1}I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j}).$
Based on the channel model defined in (19)-(20), we have
$\displaystyle
I(Q_{j},X_{2,j};Y_{j})+\gamma_{1}I(X_{1,j};Y_{j}|X_{2,j},Q_{j})-\gamma_{1}I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j})$
$\displaystyle=h(Y_{j})+(\gamma_{1}-1)h(Y_{j}|X_{2,j},Q_{j})-\gamma_{1}h(Y_{2,j}|X_{2,j},Q_{j})+\frac{\gamma_{1}}{2}\log\frac{\mu_{j}}{\nu_{j}}.$
(96)
Now, we consider the following two cases.
Case 1: $\gamma_{1}\leq 1$. In this case, note that
$\displaystyle h(Y_{j}|X_{2,j},Q_{j})$ $\displaystyle\geq
h(Y_{j}|X_{1,j},X_{2,j},Q_{j})=\frac{1}{2}\log 2\pi e\nu_{j}$ $\displaystyle
h(Y_{j}|X_{2,j},Q_{j})$ $\displaystyle\geq
h(Y_{2,j}|X_{1,j},X_{2,j},Q_{j})=\frac{1}{2}\log 2\pi e\mu_{j}$
$\displaystyle\text{and}\qquad\qquad\qquad h(Y_{j})$
$\displaystyle\leq\frac{1}{2}\log\left(p_{1,j}+p_{2,j}+2\sqrt{p_{1,j}p_{2,j}}+\nu_{j}\right).$
(97)
Hence, we have
$\displaystyle
I(Q_{j},X_{2,j};Y_{j})+\gamma_{1}I(X_{1,j};Y_{j}|X_{2,j},Q_{j})-\gamma_{1}I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j})$
$\displaystyle\leq\frac{1}{2}\log\left(1+\frac{p_{1,j}+p_{2,j}+2\sqrt{p_{1,j}p_{2,j}}}{\nu_{j}}\right)\quad\text{for}~{}j\in{\mathcal{A}}~{}\text{and}~{}\gamma_{1}\leq
1.$ (98)
This result implies that when the weight of the confidential-message rate is
less than the weight of the common-message rate, the optimum solution is to
allocate all possible power to transmit the common message.
Case 2: $\gamma_{1}>1$. Without loss of generality, we assume that the
conditional covariance of $X_{1,j}$ given $(X_{2,j},Q_{j})$ is given by
$\displaystyle{\sf cov}(X_{1,j}|X_{2,j},Q_{j})=\rho_{j}p_{1,j}$ (99)
where $0\leq\rho_{j}\leq 1$. By applying the extremal inequality [12, Theorem
8], we have
$\displaystyle(\gamma_{1}-1)h(Y_{j}|X_{2,j},Q_{j})-\gamma_{1}h(Y_{2,j}|X_{2,j},Q_{j})\leq\frac{\gamma_{1}-1}{2}\log
2\pi e\left(\rho_{j}p_{1,j}+\nu_{j}\right)-\frac{\gamma_{1}}{2}\log 2\pi
e\left(\rho_{j}p_{1,j}+\mu_{j}\right).$ (100)
Moreover, for a given $\rho_{j}$,
$\displaystyle h(Y_{j})$
$\displaystyle\leq\frac{1}{2}\log\left(p_{1,j}+p_{2,j}+2\sqrt{(1-\rho_{j})p_{1,j}p_{2,j}}+\nu_{j}\right).$
(101)
Substituting (100) and (101) into (96), we obtain
$\displaystyle
I(Q_{j},X_{2,j};Y_{j})+\gamma_{1}I(X_{1,j};Y_{j}|X_{2,j},Q_{j})-\gamma_{1}I(X_{1,j};Y_{2,j}|X_{2,j},Q_{j})$
$\displaystyle\leq\max_{0\leq\rho_{j}\leq
1}\left[\frac{1}{2}\log\left(1+\frac{p_{1,j}+p_{2,j}+2\sqrt{(1-\rho_{j})p_{1,j}p_{2,j}}}{\nu_{j}}\right)\right.$
$\displaystyle\qquad~{}~{}\left.+\frac{\gamma_{1}-1}{2}\log 2\pi
e\left(1+\frac{\rho_{j}p_{1,j}}{\nu_{j}}\right)-\frac{\gamma_{1}}{2}\log 2\pi
e\left(1+\frac{\rho_{j}p_{1,j}}{\mu_{j}}\right)\right]$
$\displaystyle=\max_{0\leq\alpha_{j}\leq
1}\left[\frac{\gamma_{1}}{2}\log\left(1+\frac{(1-\alpha_{j})p_{1,j}}{\nu_{j}}\right)-\frac{\gamma_{1}}{2}\log\left(1+\frac{(1-\alpha_{j})p_{1,j}}{\mu_{j}}\right)\right.$
$\displaystyle\qquad~{}~{}\left.+\frac{1}{2}\log\left(1+\frac{\alpha_{j}p_{1,j}+p_{2,j}+2\sqrt{\alpha_{j}p_{1,j}p_{2,j}}}{(1-\alpha_{j})p_{1,j}+\nu_{j}}\right)\right]\quad\text{for}~{}j\in{\mathcal{A}}~{}\text{and}~{}\gamma_{1}>1.$
(102)
Finally, combining (95), (98) and (102), we complete the converse proof.
### -D Proof of Theorem 4
We need fine the optimal $\underline{p}^{\star}\in{\mathcal{P}}$ that
maximizes
$\displaystyle R_{0}+\gamma_{1}R_{1}$ (103)
where $\gamma_{1}\geq 0$. The Lagrangian is given by
$\displaystyle{\mathcal{L}}$
$\displaystyle=\sum_{j\in{\mathcal{A}}}\left[\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}{b_{j}+\nu_{j}}\right)+\frac{\gamma_{1}}{2}\log\left(1+\frac{b_{j}}{\nu_{j}}\right)-\frac{\gamma_{1}}{2}\log\left(1+\frac{b_{j}}{\mu_{j}}\right)\right]$
$\displaystyle\qquad+\sum_{j\in\bar{{\mathcal{A}}}}\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}{\nu_{j}}\right)-\lambda_{1}\left[\sum_{j\in{\mathcal{A}}}(a_{j}+b_{j})+\sum_{j\in\bar{{\mathcal{A}}}}a_{j}\right]-\lambda_{2}\sum_{j=1}^{L}p_{2,j}$
(104)
where $\lambda_{1}$ and $\lambda_{2}$ are Largrange multiplier.
For $j\in\bar{{\mathcal{A}}}$, $(a_{j}^{\star},p_{2,j}^{\star})$ needs to
maximize the following ${\mathcal{L}}_{j}$,
$\displaystyle{\mathcal{L}}_{j}=\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}{\nu_{j}}\right)-\lambda_{1}a_{j}-\lambda_{2}p_{2,j}.$
(105)
Taking derivative of the Lagrangian in (105) over $a_{j}$ and $p_{2,j}$, the
KKT conditions can be written as follows:
$\displaystyle\frac{1}{2\ln
2}\frac{\theta_{1,j}(a_{j},p_{2,j})}{\sqrt{a_{j}}}$
$\displaystyle=\lambda_{1}$ $\displaystyle\text{and}\qquad\frac{1}{2\ln
2}\frac{\theta_{1,j}(a_{j},p_{2,j})}{\sqrt{p_{2,j}}}$
$\displaystyle=\lambda_{2}$ (106)
where
$\displaystyle\theta_{1,j}(a_{j},p_{2,j})$
$\displaystyle=\frac{\sqrt{a_{j}}+\sqrt{p_{2,j}}}{\nu_{j}+a_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}.$
(107)
This implies that the pair $(a_{j}^{\star},p_{2,j}^{\star})$ that optimizes
${\mathcal{L}}_{j}$ must satisfy
$\displaystyle
p_{2,j}^{\star}=\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{2}a_{j}^{\star}.$
(108)
Let us define
$\displaystyle\beta=\lambda_{1}/\lambda_{2}.$ (109)
On substituting (108) into (105), we obtain that
$\displaystyle{\mathcal{L}}_{j}$
$\displaystyle=\frac{1}{2}\log\left[1+\frac{a_{j}(1+\beta)^{2}}{\nu_{j}}\right]-\lambda_{1}a_{j}(1+\beta)$
$\displaystyle=\int_{0}^{a_{j}(1+\beta)^{2}}t_{1,j}(s)\,ds$ (110)
where
$\displaystyle t_{1,j}(s)=\frac{1}{(2\ln
2)}\frac{1}{(\nu_{j}+s)}-\frac{\lambda_{1}}{1+\beta}.$ (111)
We define $s_{1,j}$ to be the root of the equation $t_{1,j}(s)=0$, i.e.,
$\displaystyle s_{1,j}$ $\displaystyle=\frac{1+\beta}{2\lambda_{1}\ln
2}-\nu_{j}$
$\displaystyle=\frac{\lambda_{1}+\lambda_{2}}{2\lambda_{1}\lambda_{2}\ln
2}-\nu_{j}.$ (112)
Hence, we obtain, for $j\in\bar{{\mathcal{A}}}$,
$\displaystyle a_{j}^{\star}$
$\displaystyle=\frac{1}{(1+\beta)^{2}}(s_{1,j})^{+}$
$\displaystyle=\frac{\lambda_{2}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(\frac{\lambda_{1}+\lambda_{2}}{2\lambda_{1}\lambda_{2}\ln
2}-\nu_{j}\right)^{+}$ (113)
and
$\displaystyle p_{2,j}^{\star}$ $\displaystyle=\beta^{2}a_{j}^{\star}$
$\displaystyle=\frac{\lambda_{1}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(\frac{\lambda_{1}+\lambda_{2}}{2\lambda_{1}\lambda_{2}\ln
2}-\nu_{j}\right)^{+}.$ (114)
For $j\in{\mathcal{A}}$, $(a_{j}^{\star},b_{j}^{\star},p_{2,j}^{\star})$ needs
to maximize the following ${\mathcal{L}}_{j}$:
$\displaystyle{\mathcal{L}}_{j}$
$\displaystyle=\frac{1}{2}\log\left(1+\frac{a_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}{b_{j}+\nu_{j}}\right)+\frac{\gamma_{1}}{2}\log\left(1+\frac{b_{j}}{\nu_{j}}\right)-\frac{\gamma_{1}}{2}\log\left(1+\frac{b_{j}}{\mu_{j}}\right)-\lambda_{1}(a_{j}+b_{j})-\lambda_{2}p_{2,j}.$
(115)
Taking derivative of the Lagrangian in (115) over $a_{j}$ and $p_{2,j}$, the
KKT conditions can be written as follows:
$\displaystyle\frac{1}{2\ln
2}\frac{\theta_{2,j}(a_{j},b_{j},p_{2,j})}{\sqrt{a_{j}}}$
$\displaystyle=\lambda_{1}$ $\displaystyle\text{and}\qquad\frac{1}{2\ln
2}\frac{\theta_{2,j}(a_{j},b_{j},p_{2,j})}{\sqrt{p_{2,j}}}$
$\displaystyle=\lambda_{2}$ (116)
where
$\displaystyle\theta_{2,j}(a_{j},b_{j},p_{2,j})$
$\displaystyle=\frac{\sqrt{a_{j}}+\sqrt{p_{2,j}}}{\nu_{j}+a_{j}+b_{j}+p_{2,j}+2\sqrt{a_{j}p_{2,j}}}.$
(117)
This implies that the pair $(a_{j}^{\star},p_{2,j}^{\star})$ that optimizes
${\mathcal{L}}_{j}$ must satisfy
$\displaystyle
p_{2,j}^{\star}=\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{2}a_{j}^{\star}=\beta^{2}a_{j}^{\star}.$
(118)
On substituting (118) into (115), we obtain that
$\displaystyle{\mathcal{L}}_{j}$
$\displaystyle=\frac{1}{2}\log\left[1+\frac{a_{j}(1+\beta)^{2}}{b_{j}+\nu_{j}}\right]+\frac{\gamma_{1}}{2}\log\left(1+\frac{b_{j}}{\nu_{j}}\right)-\frac{\gamma_{1}}{2}\log\left(1+\frac{b_{j}}{\mu_{j}}\right)-\lambda_{1}[a_{j}(1+\beta)+b_{j}]$
$\displaystyle=\int_{b_{j}}^{b_{j}+a_{j}(1+\beta)^{2}}t_{1,j}(s)\,ds+\int_{0}^{b_{j}}t_{2,j}(s)\,ds$
$\displaystyle\leq\int_{0}^{\infty}\left(\max\\{t_{1,j}(s),t_{2,j}(s)\\}\right)^{+}\,ds$
(119)
where $t_{1,j}(s)$ is defined in (111) and
$\displaystyle t_{2,j}(s)$ $\displaystyle=\frac{\gamma_{1}}{2\ln
2}\left(\frac{1}{\nu_{j}+s}-\frac{1}{\mu_{j}+s}\right)-\lambda_{1}.$ (120)
Next, we will derive $(a_{j}^{\star},b_{j}^{\star},p_{2,j}^{\star})$ that
achieves the upper bound on ${\mathcal{L}}_{j}$ in (119). We consider the
point of intersection between $t_{1,j}(s)$ and $t_{2,j}(s)$. By using the
definitions of $t_{1,j}(s)$ in (111) and $t_{2,j}(s)$ in (120), the point of
intersection must satisfy
$\displaystyle\frac{1}{2\ln
2}\frac{1}{\nu_{j}+s}-\frac{\lambda_{1}}{1+\beta}=\frac{\gamma_{1}}{2\ln
2}\left(\frac{1}{\nu_{j}+s}-\frac{1}{\mu_{j}+s}\right)-\lambda_{1},$ (121)
i.e.,
$\displaystyle
s^{2}+\left(\mu_{j}+\nu_{j}+\frac{1}{\omega}\right)s+\left[\mu_{j}\nu_{j}-\frac{\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j}}{\omega}\right]=0$
(122)
where
$\displaystyle\omega$ $\displaystyle=(2\lambda_{1}\ln 2)\frac{\beta}{1+\beta}$
$\displaystyle=(2\ln 2)\frac{\lambda_{1}^{2}}{\lambda_{1}+\lambda_{2}}.$ (123)
In the following, we consider two cases based on the relationship between
$\omega$ and $(\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j})/(\mu_{j}\nu_{j})$.
#### -D1
$\omega\geq\frac{\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j}}{\mu_{j}\nu_{j}}$
In this case, (122) implies that the point of intersection between
$t_{1,j}(s)$ and $t_{2,j}(s)$ is either zero or negative. Moreover, it is easy
to see, for $s\geq 0$,
$\displaystyle
t_{1,j}(s)-t_{2,j}(s)=\frac{(\nu_{j}+s)(\mu_{j}+s)\omega-[\gamma_{1}(\mu_{j}-\nu_{j})-(\mu_{j}+s)]}{(2\ln
2)(\nu_{j}+s)(\mu_{j}+s)}\geq 0.$ (124)
Hence, the upper bound on ${\mathcal{L}}_{j}$ in (119) is achieved by
$b_{j}^{\star}=0$,
$\displaystyle a_{j}^{\star}$
$\displaystyle=\frac{1}{(1+\beta)^{2}}(s_{1,j})^{+}$
$\displaystyle=\frac{\lambda_{2}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(\frac{\lambda_{1}+\lambda_{2}}{2\lambda_{1}\lambda_{2}\ln
2}-\nu_{j}\right)^{+}$ (125)
and
$\displaystyle p_{2,j}^{\star}$ $\displaystyle=\beta^{2}a_{j}^{\star}$
$\displaystyle=\frac{\lambda_{1}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(\frac{\lambda_{1}+\lambda_{2}}{2\lambda_{1}\lambda_{2}\ln
2}-\nu_{j}\right)^{+}$ (126)
where $s_{1,j}$ is defined in (112).
#### -D2 $\omega<\frac{\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j}}{\mu_{j}\nu_{j}}$
In this case, (122) implies that, for $s>0$, $t_{1,j}(s)$ and $t_{2,j}(s)$
intersect only once at
$\displaystyle\phi_{j}=-\frac{1}{2}\left(\mu_{j}+\nu_{j}+\frac{1}{\omega}\right)+\frac{1}{2}\sqrt{\left(\mu_{j}+\nu_{j}+\frac{1}{\omega}\right)^{2}-4\left[\mu_{j}\nu_{j}-\frac{\gamma_{1}(\mu_{j}-\nu_{j})-\mu_{j}}{\omega}\right]}.$
(127)
Moreover, it is easy to see that $t_{1,j}(0)<t_{2,j}(0)$. Hence, we have
$\displaystyle t_{1,j}(s)$
$\displaystyle<t_{2,j}(s)\qquad\text{for}~{}~{}0\leq s<\phi_{j}$
$\displaystyle\text{and}\qquad t_{1,j}(s)$ $\displaystyle\geq
t_{2,j}(s)\qquad\text{for}~{}~{}s\geq\phi_{j}.$ (128)
Let $s_{2,j}$ denote the largest root of $t_{2,j}(s)=0$, i.e.,
$\displaystyle s_{2,j}$
$\displaystyle=\frac{1}{2}\left[\sqrt{(\mu_{j}-\nu_{j})\left(\mu_{j}-\nu_{j}+\frac{2\gamma_{1}}{\lambda_{1}\ln
2}\right)}-(\mu_{j}+\nu_{j})\right].$ (129)
The optimal $(a_{j}^{\star},b_{j}^{\star},p_{2,j}^{\star})$ depends on the
values $t_{2,j}(0)$, $s_{1,j}$ and $\phi_{j}$, and falls into the following
three possibilities.
(2.a) If $t_{2,j}(0)<0$, then both $t_{1,j}(s)$ and $t_{2,j}(s)$ are negative
for $s\geq 0$ (since both $t_{1,j}(s)$ and $t_{2,j}(s)$ are decreasing
functions for $s\geq 0$). Then, the upper bound on ${\mathcal{L}}_{j}$ in
(119) is achieved by $b_{j}^{\star}=0$, $a_{j}^{\star}=0$ and
$p_{2,j}^{\star}=0$.
(2.b) If $t_{2,j}(0)\geq 0$ and $s_{1,j}<\phi_{j}$, then the upper bound on
${\mathcal{L}}_{j}$ in (119) is achieved by $b_{j}^{\star}=s_{2,j}$,
$a_{j}^{\star}=0$ and $p_{2,j}^{\star}=0$.
(2.c) If $t_{2,j}(0)\geq 0$ and $s_{1,j}\geq\phi_{j}$, then the upper bound on
${\mathcal{L}}_{j}$ in (119) is achieved by $b_{j}^{\star}=\phi_{j}$,
$\displaystyle a_{j}^{\star}$
$\displaystyle=\frac{1}{(1+\beta)^{2}}\left(s_{1,j}-\phi_{j}\right)\quad\text{and}\quad
p_{2,j}^{\star}=\frac{\beta^{2}}{(1+\beta)^{2}}\left(s_{1,j}-\phi_{j}\right).$
(130)
Combing the cases (2.a), (2.b) and (2.c), we obtain
$\displaystyle a_{j}^{\star}$
$\displaystyle=\frac{\lambda_{2}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}-\phi_{j}\right)^{+}$
$\displaystyle b_{j}^{\star}$
$\displaystyle=\left(\min\left[\phi_{j},\,s_{2,j}\right]\right)^{+}$
$\displaystyle\text{and}\qquad p_{2,j}^{\star}$
$\displaystyle=\frac{\lambda_{1}^{2}}{(\lambda_{1}+\lambda_{2})^{2}}\left(s_{1,j}-\phi_{j}\right)^{+}.$
(131)
Finally, the Lagrange parameters $\lambda_{1}\geq 0$ and $\lambda_{2}\geq 0$
are chosen to satisfy the power constraint (35).
## References
* [1] A. D. Wyner, “The wire-tap channel,” _Bell Syst. Tech. J._ , vol. 54, no. 8, pp. 1355–1387, Oct. 1975.
* [2] I. Csiszár and J. Körner, “Broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 24, no. 3, pp. 339–348, May 1978\.
* [3] Y. Liang, H. V. Poor, and S. Shamai (Shitz), “Information theoretic security,” _Foundations and Trends in Communications and Information Theory_ , vol. 5, pp. 355–580, 2008.
* [4] Y. Liang and H. V. Poor, “Multiple access channels with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 3, pp. 976–1002, Mar. 2008.
* [5] R. Liu, I. Maric, R. D. Yates, and P. Spasojevic, “The discrete memoryless multiple access channel with confidential messages,” in _Proc. IEEE Int. Symp. Information Theory_ , Seattle, WA, Jul. 2006, pp. 957 – 961.
* [6] E. Tekin and A. Yener, “The Gaussian multiple access wire-tap channel with collective secrecy constraints,” in _Proc. IEEE Int. Symp. Information Theory_ , Seattle, WA, Jul. 2006.
* [7] O. Simeone and A. Yener, “The cognitive multiple access wire-tap channel,” in _Proc. Conference on Information Sciences and Systems_ , Baltimore, MA, Mar. 2009.
* [8] Y. Liang, H. V. Poor, and S. Shamai (Shitz), “Secure communication over fading channels,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 6, pp. 2470–2492, Jun. 2008.
* [9] P. Gopala, L. Lai, and H. El Gamal, “On the secrecy capacity of fading channels,” in _Proc. IEEE Int. Symp. Information Theory (ISIT)_ , Nice, France, June 24-29, 2007.
* [10] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “On the throughput of secure Hybrid-ARQ protocols for Gaussian block-fading channels,” _IEEE Trans. Inf. Theory_ , vol. 55, pp. 1575–1591, Apr. 2009.
* [11] T. Cover and J. Thomas, _Elements of Information Theory_. New York: John Wiley Sons, Inc., 1991.
* [12] T. Liu and P. Viswanath, “An extremal inequality motivated by multiterminal information-theoretic problems,” _IEEE Trans. Inf. Theory_ , vol. 53, pp. 1839–1851, May 2007.
|
arxiv-papers
| 2009-10-24T02:43:30 |
2024-09-04T02:49:06.021326
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruoheng Liu, Yingbin Liang and H. Vincent Poor",
"submitter": "Ruoheng Liu",
"url": "https://arxiv.org/abs/0910.4613"
}
|
0910.4670
|
# Angular performance measure for tighter uncertainty relations
Z. Hradil Department of Optics, Palacky University, 17. listopadu 50, 772 00
Olomouc, Czech Republic J. Řeháček Department of Optics, Palacky University,
17. listopadu 50, 772 00 Olomouc, Czech Republic A. B. Klimov Departamento
de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico I.
Rigas Departamento de Óptica, Facultad de Física, Universidad Complutense,
28040 Madrid, Spain L. L. Sánchez-Soto Departamento de Óptica, Facultad de
Física, Universidad Complutense, 28040 Madrid, Spain
###### Abstract
The uncertainty principle places a fundamental limit on the accuracy with
which we can measure conjugate quantities. However, the fluctuations of these
variables can be assessed in terms of different estimators. We propose a new
angular performance that allows for tighter uncertainty relations for angle
and angular momentum. The differences with previous bounds can be significant
for particular states and indeed may be amenable to experimental measurement
with the present technology.
###### pacs:
03.65.Ta, 42.50.Dv, 42.50.Lc, 42.50.Tx
Apart from interpretational issues, the main goal of quantum mechanics is to
make predictions on the outcomes of experiments. In fact, in many modern
setups one is led to measurements that simultaneously estimate two
noncommuting variables. The precision with which they are jointly estimated
obey a fundamental constraint dictated by the uncertainty principle Peres
(1993).
The archetypal example is the case of continuous variables, such as position
and linear momentum of a single particle. The standard formalization of the
uncertainty principle is presented in terms of the associated variances
[defined as $(\Delta A)^{2}=\langle\hat{A}^{2}\rangle-\langle\hat{A}\rangle$],
and it reads Heisenberg (1967) (with $\hbar=1$ throughout)
$(\Delta x)^{2}\,(\Delta p)^{2}\geq\frac{1}{4}\,.$ (1)
These variances are a measure of the width of the corresponding probability
distributions in the quantum state. However, it has long been argued that some
experiments do not measure variances and encouraging reformulations of Eq. (1)
have been proposed in terms of other resolution measures Heinonen and Lahti
(2007); Schürmann and Hoffmann (2009). In other words, one can assign
different measures of inaccuracy (each one with its own pros and cons) to a
particular measurement and this proves crucial to properly set its ultimate
resolution limits. The price one has to pay is that establishing an
uncertainty principle in terms of these measures can turn out to be very
intricate Wootters and Zurek (1979); Englert (1996); Dür (2001); Hall (2004);
Ozawa (2004). The situation is even more ambiguous for magnitudes that cannot
be measured, but must be only inferred, as it happens with, e. g.,
entanglement Horodecki et al. (2009).
Angular variables are also riddled with the same kind of problems, but
aggravated by the peculiarities of their periodic character Lynch (1995);
Peřinova et al. (1998); Luis and Sánchez-Soto (1998, 2000). Though this is an
old question, it experiences periodic revivals in connection with some hot
topics. Nowadays, a renewed interest in these features has been triggered by
the treatment of rotating Bose-Einstein condensates Leggett (2001); Cornell
and Wieman (2002) and the quantum optics of vortex beams Allen et al. (2003).
It is worth remarking that we have at hand very simple experimental schemes to
test in practice ideal angle concepts.
There is agreement in using the variance $(\Delta L)^{2}$ to characterize
fluctuations in angular momentum (although, since this variable is unbounded,
the variance may fail in some instances to provide a satisfactory expression
for the uncertainty principle Uffink (1990)). In contrast, there is no wide
consensus concerning the proper assessment of the conjugate angle
fluctuations. Periodicity may lead to serious troubles when using variance,
since the powers of the angle are not periodic functions, so that their mean
values depend on the origin chosen. There are several proposals that avoid
these problems, such as the Süssmann measure Bialynicki-Birula et al. (1993);
Hall (1999); Luis (2006), circular variance Lévy-Leblond (1976); Breitenberger
(1985); Bandilla et al. (1991); Hradil (1992); Opatrný (1994), entropies
Bialynicki-Birula and Mycielski (1975); Deutsch (1983); Maassen and Uffink
(1988); Abe (1992); Brukner and Zeilinger (2001), reciprocal peak height
Shapiro and Shepard (1991); Schleich et al. (1991); Hradil and Shapiro (1992),
origin-optimized angle variance Trifonov (2003); Pegg and Barnett (1997), and
other nonstandard quantities Landau and Pollak (1961); Kowalski and
Rembieliński (2002). In short, for periodic variables there are a lot of
candidates for assessing fluctuations, each one surely with its virtues, but
no undisputed champion.
As commented before, if we decide to choose, e. g., the circular variance
(which is computed as the standard one, but using the moments of the complex
exponential of the angle rather than the angle itself and is the simplest
natural choice from a pure statistical viewpoint Mardia and Jupp (2000)), the
resulting uncertainty relation is rather involved and cannot be saturated,
except in very trivial cases Hradil et al. (2006); Řeháček et al. (2008).
All these difficulties motivate this paper. We shall seek for a new angular
performance measure that, apart from properly quantifying angle fluctuations,
provides simple and feasible bounds for the conjugate variable.
To be as self-contained as possible, we first introduce some basic notions for
the problem at hand. We are concerned (assuming cylindrical symmetry) with the
planar rotations by an angle $\phi$ generated by the angular momentum along
the $z$ axis, which for simplicity will be denoted henceforth as $\hat{L}$.
Classically, a point particle is necessarily located at a single value of the
periodic angular coordinate $\phi$, defined within a chosen window. The
corresponding quantum wave function, however, is an object extended around the
unit circle and so can be directly affected by the nontrivial topology.
One may be tempted to think that angle should stand in the same relationship
to angular momentum as ordinary position stands to linear momentum. This would
prompt to interpret the angle operator as multiplication by $\phi$ while
$\hat{L}$ is the differential operator $\hat{L}=-i\partial_{\phi}$. However,
the use of this operator may entail many pitfalls for the unwary: in
particular, single-valuedness restricts the Hilbert space to the subspace of
$2\pi$-periodic functions, which, among other things, rules out the angle
coordinate as a bona fide observable Carruthers and Nieto (1968); Emch (1972).
Many of these difficulties can be avoided by simply selecting angular
coordinates that are both periodic and continuous instead. A single such
quantity cannot uniquely specify a point on the circle because periodicity
implies extrema, which excludes a one-to-one correspondence and hence is
incompatible with uniqueness. Perhaps the simplest choice Louisell (1963);
Mackey (1963) is to adopt two angular coordinates, such as, e. g., cosine and
sine. In classical mechanics this is indeed of a good definition, while in
quantum mechanics one would have to show that these variables, we shall denote
by $\hat{C}$ and $\hat{S}$ to make no further assumptions about the angle
itself, form a complete set of commuting operators. One can concisely condense
all this information using the complex exponential of the angle
$\hat{E}=\hat{C}-i\hat{S}$, which satisfies the commutation relation
$[\hat{E},\hat{L}]=\hat{E}\,.$ (2)
In mathematical terms, this defines the Lie algebra of the two-dimensional
Euclidean group E(2). Interestingly enough, E(2) is the canonical symmetry of
the cylinder, which is the phase space for our system.
The action of $\hat{E}$ on the angular momentum basis $|\ell\rangle$ is
$\hat{E}|\ell\rangle=|\ell-1\rangle$, and it possesses then a simple
implementation by means of phase mask removing a unit charge from a vortex
state Hradil et al. (2006). Since the integer $\ell$ runs from $-\infty$ to
$+\infty$, $\hat{E}$ is a unitary operator whose eigenvectors
$|\phi\rangle=\frac{1}{\sqrt{2\pi}}\sum_{\ell\in\mathbb{Z}}e^{i\ell\phi}\,|\ell\rangle$
(3)
describe states with well-defined angle. Although the proposal that this
operator represents the angle conflicts with the orthodox view of describing
observables by Hermitian operators, the option for $\hat{E}$ is actually very
natural. Note that one could expect a Fourier relationship between angle and
angular momentum. In this context, this can be expressed as
$e^{-i\phi^{\prime}\hat{L}}|\phi\rangle=|\phi-\phi^{\prime}\rangle\,,$ (4)
which can be easily verified by using the explicit form in Eq. (3).
Let us turn to the corresponding uncertainty relations. The Robertson
inequality Robertson (1929); Sudarshan et al. (1995) (which remains valid for
unitary operators) can be applied to obtain
$\ (\Delta L)^{2}\geq\frac{1}{4}\frac{[1-(\Delta E)^{2}]}{(\Delta E)^{2}}\,,$
(5)
where we have rearranged terms to facilitate comparison with the next steps in
our analysis. Here we have used the natural extension of variance for unitary
operators Lévy-Leblond (1976)
$(\Delta
E)^{2}=\langle\hat{E}^{\dagger}\hat{E}\rangle-\langle\hat{E}^{\dagger}\rangle\langle\hat{E}\rangle=1-|\langle\hat{E}\rangle|^{2}\,,$
(6)
which it exactly agrees with the circular variance Mardia and Jupp (2000). The
form (5) has been advocated by many authors. However, although correct, it
does not provide the tightest lower bound and equality cannot be attained
except for some trivial states Kowalski and Rembieliński (2002).
To face this disadvantage, let us first recast Eq. (2) in terms of the
corresponding Hermitian components
$[\hat{C},\hat{L}]=i\hat{S},\qquad[\hat{S},\hat{L}]=-i\hat{C}\,,$ (7)
while $[\hat{C},\hat{S}]=0$. Moreover, for reasons that will be apparent soon,
we look at their rotated versions
$\hat{C}_{\alpha}=\hat{C}\cos\alpha-\hat{S}\sin\alpha\,,\qquad\hat{S}_{\alpha}=\hat{S}\cos\alpha+\hat{C}\sin\alpha\,.$
(8)
This means that we allow the reference frame in which we compute the
trigonometric functions to be rotated by an angle $\alpha$. One can check that
they satisfy a commutation relation identical to Eq. (7). Therefore, the
associated uncertainty relations are
$(\Delta S_{\alpha})^{2}(\Delta
L)^{2}\geq\frac{1}{4}|\langle\hat{C}_{\alpha}\rangle|^{2}\,,\quad(\Delta
C_{\alpha})^{2}(\Delta
L)^{2}\geq\frac{1}{4}|\langle\hat{S}_{\alpha}\rangle|^{2}\,.$ (9)
Since Eqs. (9) are fully equivalent to Eq. (5), they cannot be saturated
simultaneously. In fact, there are further unfavorable aspects of them that
have been reviewed in Ref. Uffink (1990).
A common way of going on is to look for intelligent states minimizing, e. g.,
the first one of these equations. Although this can be seen as dealing only
with “half” the uncertainty principle, the resulting states are often referred
to as circular squeezed states Kowalski and Rembieliński (2002) and exhibit
amazing properties. They are defined by
$(\hat{L}-i\kappa\hat{C}_{\alpha})|\Psi\rangle=\lambda|\Psi\rangle,$ (10)
where $\kappa$ and $\lambda$ are real parameters. Using the angle
representation, this extremal equation reads as
$-i\frac{d}{d\phi}\Psi(\phi)=[\lambda+i\kappa\cos(\phi+\alpha)]\Psi(\phi)\,,$
(11)
whose integration yields the normalized solution
$\Psi(\phi)=\frac{1}{\sqrt{2\pi
I_{0}(2\kappa)}}\exp[i\lambda\phi+\kappa\cos(\phi+\alpha)]\,,$ (12)
$I_{0}$ being the modified Bessel function of order 0. These are called von
Mises states, since the associated probability distribution is precisely the
von Mises, a very close analog of the Gaussian distribution on the circle
Řeháček et al. (2008). The meaning of the parameters is clear: $\lambda$ is
the mean value of the angular momentum, whereas $\kappa$ determines the
angular spread.
Next, we observe that the associated uncertainty relation in Eq. (9) can be
cast in the form
$(\Delta L)^{2}\geq
U^{2}\equiv\frac{1}{4}\max_{\alpha}\frac{|\langle\hat{C}_{\alpha}\rangle|^{2}}{(\Delta
S_{\alpha})^{2}}\,.$ (13)
Let us introduce the following vectors
$\mathbf{x}=\left(\begin{array}[]{c}\cos\alpha\\\ \sin\alpha\\\
\end{array}\right)\,,\qquad\mathbf{c}=\left(\begin{array}[]{c}\langle
C\rangle\\\ \langle S\rangle\\\ \end{array}\right)\,,$ (14)
and the covariance matrix
$\mathbf{\Gamma}=\left(\begin{array}[]{cc}(\Delta S)^{2}&\Delta(SC)\\\
\Delta(CS)&(\Delta C)^{2}\\\ \end{array}\right),$ (15)
where
$\Delta(CS)=\langle\hat{C}\hat{S}\rangle-\langle\hat{C}\rangle\langle\hat{S}\rangle$.
Then, $U^{2}$ can be written as
$U^{2}=\frac{1}{4}\max_{|\mathbf{x}|=1}\frac{(\mathbf{c}^{t}\,\mathbf{x})^{2}}{\mathbf{x}^{t}\,\mathbf{\Gamma}\,\mathbf{x}}\,,$
(16)
and the superscript $t$ denotes the transpose. The optimization over
$\mathbf{x}$ can be easily performed, getting
$\frac{\mathbf{c}^{t}\,\mathbf{x}}{\mathbf{x}^{t}\,\mathbf{\Gamma}\,\mathbf{x}}\,\mathbf{c}-\left(\frac{\mathbf{c}^{t}\,\mathbf{x}}{\mathbf{x}^{t}\,\mathbf{\Gamma}\,\mathbf{x}}\right)^{2}\mathbf{\Gamma}\,\mathbf{x}=0\,,$
(17)
whose solution gives the optimal value
$U^{2}=\frac{1}{4}\mathbf{c}^{t}\mathbf{\Gamma}^{-1}\mathbf{c}\,.$ (18)
We stress that while the variances $(\Delta C)^{2}$ and $(\Delta S)^{2}$ are
not invariant under rotations of the state around the $z$ axis, this is not
the case with $U^{2}$, which constitutes a major advantage. In addition,
$U^{2}$ combines the moments of $\hat{C}$ and $\hat{S}$ in a rather nontrivial
way, since
$\displaystyle|\langle\hat{E}\rangle|^{2}$ $\displaystyle=$ $\displaystyle
1-\mathop{\mathrm{tr}}\nolimits\mathbf{\Gamma}\,,$
$\displaystyle|\langle(\Delta E)^{2}\rangle|^{2}$ $\displaystyle=$
$\displaystyle(\mathop{\mathrm{tr}}\nolimits\mathbf{\Gamma})^{2}-4\det\mathbf{\Gamma}\,.$
The performance measure $U^{2}$ can be interpreted as a projection of the
noise into the direction of the preferred angle, analogously to what was done
for the ellipse representing a squeezed state in phase space Hradil (1991).
Denoting by $\gamma_{-}$ and $\gamma_{+}$ the smaller and larger eigenvalues
of $\mathbf{\Gamma}$, a simple calculation allows us to estimate
$U^{2}\geq\frac{1}{4\gamma_{+}}|\mathbf{c}|^{2}\equiv
V^{2}\geq\frac{1}{4}\frac{[1-(\Delta E)^{2}]}{(\Delta E)^{2}}\,,$ (20)
where we have introduced a new resolution performance
$V^{2}=\frac{1}{4}\frac{2(1-\mathop{\mathrm{tr}}\nolimits\mathbf{\Gamma})}{\mathop{\mathrm{tr}}\nolimits\mathbf{\Gamma}+\sqrt{(\mathop{\mathrm{tr}}\nolimits\mathbf{\Gamma})^{2}-4\det\mathbf{\Gamma}}}\,,$
(21)
that combines the two basic invariants of $\mathbf{\Gamma}$. Notice that
$V^{2}$ is related to the covariance matrix (15) pretty much in the same way
as the degree of polarization is linked to the polarization matrix. As we can
see, it gives intermediate values between the bound in Eq. (5) (which cannot
be attained for nontrivial states) and the one in Eq. (13) (which is saturated
by all the von Mises states).
The second inequality in Eq. (20) is saturated only in trivial instances, such
as, e. g., the eigenstates of $\hat{L}$ Řeháček et al. (2008). A condition for
the first inequality to be saturated is
$\Delta(CS)=0\,.$ (22)
This holds if the associated probability distribution is symmetrical about
some reference angle $\phi_{0}$, that is, $P(\phi_{0}+\phi)=P(\phi_{0}-\phi)$.
In addition, $U^{2}=V^{2}$ also implies the additional constraint
$(\Delta S)^{2}\geq(\Delta C)^{2}\,.$ (23)
The Von Mises states are among those satisfying conditions (22) and (23). For
the other cases, one has $U^{2}>V^{2}$.
Figure 1: (Color online) Plot of the different bounds for $(\Delta L)^{2}$ in
terms of the dispersion $\Delta E$ for the state (25). From bottom to top, we
show the equality in Eq. (5) (blue dashed-dotted line), in Eq. (24) (thick red
line), and in Eq. (13) (black dashed line).
In consequence, the inequality
$(\Delta L)^{2}\geq{V}^{2}$ (24)
is always true, significantly improves the standard bound in Eq. (5), and the
right-hand side is saturable. Given that a useful performance should be a
simple expression of measurable quantities, we opt for using $V^{2}$, which
depends on the two basic invariants of the covariance matrix as one might
expect, instead of $U^{2}$. This latter quantity, in general, provides a
slightly tighter bound. However, as we have shown above, for the majority of
states of interest the two bounds Eq. (13) and (24) coincide, and both are
saturated by the von Mises states. The difference between $U^{2}$ and $V^{2}$
is in most cases unimportant and more than compensated by the utility and
feasibility of the proposed uncertainty relation Eq. (24).
In Fig. 1 we have condensed all this information for the state
$\Psi(\phi)=\frac{1}{\sqrt{4\pi
I_{0}(2\kappa)}}\left[\exp(\kappa\cos\phi)-i\exp(i\phi+\kappa\cos\phi)\right]\,,$
(25)
which corresponds to the superposition of two von Mises states with
$\langle\hat{L}\rangle=0$ and $\langle\hat{L}\rangle=1$. This can be seen as
an angular counterpart of a cat state, with a probability distribution
$P(\phi)=(1+\sin\phi)\exp(-2\kappa\cos\phi)\,,$ (26)
displaying a lack of symmetry. The proposed bound (24) constitutes a good
improvement over the standard one (5), as we can see in the figure: all the
area shaded corresponds to the values of $(\Delta L)^{2}$ that, for a given
angular fluctuation $(\Delta E)^{2}$, are permitted by the standard
uncertainty relation but not allowed according to our proposal. Obviously, the
strongest bound in Fig. 1 is provided by relation (13). The family of states
(26) was deliberately chosen so as to make the difference between $V^{2}$ and
$U^{2}$ large; but even in that case the improvement of both (24) and (13)
over the standard bound is seen to be much larger than the difference between
them.
Our arguments support the role of von Mises distribution on the circle as an
analog of the Gaussian distribution on the line, at least as far as the
uncertainty product is concerned. However, things may be not that simple with
other aspects of quantum behavior. Indeed, our latest research indicates that
the Wigner function of von Mises states is not positive (as it happens for
Gaussian states in the line), since this property is reserved exclusively to
the angular momentum eigenstates Rigas et al. (2008, 2009).
Finally, we observe that one could introduce a ladder operator Kowalski et al.
(1996)
$\hat{X}=e^{-\hat{L}-1/2}\,\hat{E}\,.$ (27)
Since this can be expressed also in terms of the non-unitary transformation
$\hat{X}=e^{\hat{L}^{2}/2}\hat{E}e^{-\hat{L}^{2}/2}$ Kastrup (2006), the
commutator $[\hat{X},\hat{L}]=\hat{X}$ still remains valid. The construction
of the accessible lower bound in this paper can be thus repeated, provided the
role of $\hat{C}$ and $\hat{S}$ is now taken by the quadrature-like operators
$\hat{Q}=\frac{1}{2}(\hat{X}+\hat{X}^{\dagger})\,,\qquad\hat{P}=\frac{1}{2i}(\hat{X}-\hat{X}^{\dagger})\,.$
(28)
Obviously, the (unnormalized) extremal states for these operators are given by
the von Mises states, but transformed by $e^{-\hat{L}^{2}/2}$.
In summary, what we expect to have accomplished here is to present convincing
arguments for the use of a new angular resolution measure that involves only
invariant and measurable quantities, has no problem with periodicity, and
gives an improved feasible criterion to assess minimal angle fluctuations.
We acknowledge discussions with Hubert de Guise. This work was supported by
the Research Project of the Czech Ministry of Education “Measurement and
Information in Optics” MSM 6198959213 and the Spanish Research Directorate,
Grant No FIS2008-04356.
## References
* Peres (1993) A. Peres, _Quantum Theory: Concepts and Methods_ (Kluwer, Dordrecht, 1993).
* Heisenberg (1967) W. Heisenberg, _The Physical Principles of the Quantum Theory._ (Dover, New York, 1967).
* Heinonen and Lahti (2007) T. Heinonen and P. J. Lahti, Phys. Rep. 452, 155 (2007).
* Schürmann and Hoffmann (2009) T. Schürmann and I. Hoffmann, Found. Phys. 39, 958 (2009).
* Wootters and Zurek (1979) W. K. Wootters and W. H. Zurek, Phys. Rev. D 19 (1979).
* Englert (1996) B.-G. Englert, Phys. Rev. Lett. 77, 2154 (1996).
* Dür (2001) S. Dür, Phys. Rev. A 64, 042113 (2001).
* Hall (2004) M. J. W. Hall, Phys. Rev. A 69, 052113 (2004).
* Ozawa (2004) M. Ozawa, Ann. Phys. 311, 350 (2004).
* Horodecki et al. (2009) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
* Lynch (1995) R. Lynch, Phys. Rep. 256, 367 (1995).
* Peřinova et al. (1998) V. Peřinova, A. Lukš, and J. Peřina, _Phase in Optics_ (World Scientific, Singapore, 1998).
* Luis and Sánchez-Soto (1998) A. Luis and L. L. Sánchez-Soto, Eur. Phys. J. D 3, 195 (1998).
* Luis and Sánchez-Soto (2000) A. Luis and L. L. Sánchez-Soto, Prog. Opt. 44, 421 (2000).
* Leggett (2001) A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).
* Cornell and Wieman (2002) E. A. Cornell and C. E. Wieman, Rev. Mod. Phys 74, 875 (2002).
* Allen et al. (2003) L. Allen, S. M. Barnett, and M. J. Padgett, _Optical Angular Momentum_ (Institute of Physics Publishing, Bristol, 2003).
* Uffink (1990) J. B. M. Uffink, Ph.D. thesis, University of Utrecht (1990).
* Bialynicki-Birula et al. (1993) I. Bialynicki-Birula, M. Freyberger, and W. Schleich, Phys. Scr. T48, 113 (1993).
* Hall (1999) M. J. W. Hall, Phys. Rev. A 59, 2602 (1999).
* Luis (2006) A. Luis, Phys. Lett. A 354, 71 (2006).
* Lévy-Leblond (1976) J. M. Lévy-Leblond, Ann. Phys. 101, 319 (1976).
* Breitenberger (1985) E. Breitenberger, Found. Phys. 15, 353 (1985).
* Bandilla et al. (1991) A. Bandilla, H. Paul, and H.-H. Ritze, Quantum Opt. 3, 267 (1991).
* Hradil (1992) Z. Hradil, Phys. Rev. A 46, R2217 (1992).
* Opatrný (1994) T. Opatrný, J. Phys. A 27, 7201 (1994).
* Bialynicki-Birula and Mycielski (1975) I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975).
* Deutsch (1983) D. Deutsch, Phys. Rev. Lett. 50, 831 (1983).
* Maassen and Uffink (1988) H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).
* Abe (1992) S. Abe, Phys. Lett. A 166, 163 (1992).
* Brukner and Zeilinger (2001) Č. Brukner and A. Zeilinger, Phys. Rev. A 63, 022113 (2001).
* Shapiro and Shepard (1991) J. H. Shapiro and S. R. Shepard, Phys. Rev. A 43, 3795 (1991).
* Schleich et al. (1991) W. P. Schleich, J. P. Dowling, and R. J. Horowicz, Phys. Rev. A 44, 3365 (1991).
* Hradil and Shapiro (1992) Z. Hradil and J. H. Shapiro, Quantum Opt. 4, 31 (1992).
* Trifonov (2003) D. A. Trifonov, J. Phys. A 36, 11873 (2003).
* Pegg and Barnett (1997) D. T. Pegg and S. M. Barnett, J. Mod. Opt. 44, 225 (1997).
* Landau and Pollak (1961) H. J. Landau and H. O. Pollak, Bell Syst. Tech. J. 40, 65 (1961).
* Kowalski and Rembieliński (2002) K. Kowalski and J. Rembieliński, J. Phys. A 35, 1405 (2002).
* Mardia and Jupp (2000) K. V. Mardia and P. E. Jupp, _Directional Statistics_ (Wiley, Chichester, 2000).
* Hradil et al. (2006) Z. Hradil, J. Rehacek, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Phys. Rev. Lett. 97, 243601 (2006).
* Řeháček et al. (2008) J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
* Carruthers and Nieto (1968) P. Carruthers and M. M. Nieto, Rev. Mod. Phys 40, 411 (1968).
* Emch (1972) G. G. Emch, _Algebraic Methods in Statistical Mechanics and Quantum Field Theory_ (Wiley, New York, 1972).
* Louisell (1963) W. H. Louisell, Phys. Lett. 7, 60 (1963).
* Mackey (1963) G. W. Mackey, _Mathematical Foundations of Quantum Mechanics_ (Benjamin, New York, 1963).
* Robertson (1929) H. P. Robertson, Phys. Rev. 34, 163 (1929).
* Sudarshan et al. (1995) E. C. G. Sudarshan, C. B. Chiu, and G. Bhamathi, Phys. Rev. A 52, 43 (1995).
* Hradil (1991) Z. Hradil, Phys. Rev. A 44, 792 (1991).
* Rigas et al. (2008) I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A 78, 060101 (R) (2008).
* Rigas et al. (2009) I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Phys. Rev. A (2009).
* Kowalski et al. (1996) K. Kowalski, J. Rembieliński, and L. C. Papaloucas, J. Phys. A 29, 4149 (1996).
* Kastrup (2006) H. A. Kastrup, Phys. Rev. A 73, 052104 (2006).
|
arxiv-papers
| 2009-10-24T17:45:41 |
2024-09-04T02:49:06.031664
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Z. Hradil, J. Rehacek, A. B. Klimov, I. Rigas, L. L. Sanchez-Soto",
"submitter": "Luis L. Sanchez. Soto",
"url": "https://arxiv.org/abs/0910.4670"
}
|
0910.4695
|
††thanks: This project was initiated at the workshop WIN Women in Numbers in
November 2008. The authors would like to thank the Banff International
Research Station for hosting the workshop and the National Security Agency,
the Fields Institute, the Pacific Institute for the Mathematical Sciences,
Microsoft Research, and University of Calgary for their financial support.
Author Pries was partially supported by NSF grant 07-01303. Author Im was
partially supported by the Korea Science and Engineering Foundation (KOSEF)
grant (No. R01-2007-000-10660-0) funded by the Korea government (MOST). The
authors would also like to thank the referee for helpful comments
# Semi-direct Galois covers of the affine line
Linda Gruendken Department of Mathematics, University of Pennsylvania, David
Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104-6395
lindagr@math.upenn.edu Laura Hall-Seelig Department of Mathematics and
Statistics, Lederle Graduate Research Tower, University of Massachusetts,
Amherst, MA 01003-9305 hall@math.umass.edu Bo-Hae Im Department of
Mathematics, Chung-Ang University, 221, Heukseok-dong, Dongjak-gu, Seoul
156-756
South Korea imbh@cau.ac.kr Ekin Ozman Department of Mathematics, University
of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706
ozman@math.wisc.edu Rachel Pries Department of Mathematics, Colorado State
University, Fort Collins, CO 80523-1874, USA pries@math.colostate.edu
Katherine Stevenson Department of Mathematics, California State University,
18111 Nordhoff St, Northridge, CA 91330-8313 katherine.stevenson@csun.edu
###### Abstract.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a
semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ where
$b$ is a positive integer and $\ell$ is a prime distinct from $p$. In this
paper, we study Galois covers $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over
$\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ which
admits a covering map of this form and we give an explicit formula for this
genus in terms of $\ell$ and $p$. The minimal genus occurs when $b$ equals the
order $a$ of $\ell$ modulo $b$ and we also prove that the number of curves $Z$
of this minimal genus which admit such a covering map is at most $(p-1)/a$
when $p$ is odd.
###### 1991 Mathematics Subject Classification:
14H30, 14E20, 14F40
## 1\. Introduction
Let $k$ be an algebraically closed field of characteristic $p>0$. In sharp
contrast with the situation in characteristic $0$, there exist Galois covers
$\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over infinity. By Abhyankar’s
Conjecture [2], proved by Raynaud and Harbater [9], [4], a finite group $G$
occurs as the Galois group of such a cover $\psi$ if and only if $G$ is
quasi-$p$, i.e., $G$ is generated by $p$-groups. This result classifies all
the finite quotients of the fundamental group $\pi_{1}(\mathbb{A}^{1}_{k})$.
It does not, however, determine the profinite group structure of
$\pi_{1}(\mathbb{A}^{1}_{k})$ because this fundamental group is an infinitely
generated profinite group.
There are many open questions about Galois covers
$\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over infinity. For example, given
a finite quasi-$p$ group $G$, what is the smallest integer $g$ for which there
exists a cover $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over infinity with
$Z$ of genus $g$? As another example, suppose $G$ and $H$ are two finite
quasi-$p$ groups such that $H$ is a quotient of $G$. Given an unramified
Galois cover $\phi$ of $\mathbb{A}^{1}_{k}$ with group $H$, under what
situations can one dominate $\phi$ with an unramified Galois cover $\psi$ of
$\mathbb{A}^{1}_{k}$ with Galois group $G$? Answering these questions will
give progress towards understanding how the finite quotients of
$\pi_{1}(\mathbb{A}^{1}_{k})$ fit together in an inverse system. These
questions are more tractible for quasi-$p$ groups that are $p$-groups since
the maximal pro-$p$ quotient $\pi_{1}^{p}(\mathbb{A}^{1}_{k})$ is free (of
infinite rank) [10].
In this paper, we study Galois covers $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ whose Galois group is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$, where
$\ell$ is a prime distinct from $p$. Such a cover $\psi$ must be a composition
$\psi=\phi\circ\omega$ where $\omega:Z\to Y$ is unramified and
$\phi:Y\to\mathbb{P}^{1}_{k}$ is an Artin-Schreier cover ramified only over
$\infty$. The cover $\phi$ has an affine equation $y^{p}-y=f(x)$ for some
$f(x)\in k[x]$ with degree $s$ prime-to-$p$. The $\ell$-torsion
$\mathrm{Jac}\,(Y)[\ell]$ of the Jacobian of $Y$ is isomorphic to
$(\mathbb{Z}/\ell\mathbb{Z})^{2g_{Y}}$. When $f(x)=x^{s}$, we determine how an
automorphism $\tau$ of $Y$ of order $p$ acts on $\mathrm{Jac}\,(Y)[\ell]$.
This allows us to construct a Galois cover
$\psi_{a}:Z_{a}\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ which
dominates $\phi$, such that the Galois group of $\psi_{a}$ is
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ where
$a$ is the order of $\ell$ modulo $p$ (Section 3). We prove that the genus of
$Z_{a}$ is minimal among all natural numbers that occur as the genus of a
curve $Z$ which admits a covering map $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. We
also prove that the number of curves $Z$ of this minimal genus which admit
such a covering map is at most $(p-1)/a$ when $p$ is odd (Section 4).
## 2\. Quasi-$p$ semi-direct products
We recall which groups occur as Galois groups of covers of
$\mathbb{P}^{1}_{k}$ ramified only over $\infty$.
###### Definition 2.1.
A finite group is a quasi $p$-group if it is generated by all of its Sylow
$p$-subgroups.
It is well-known that there are other equivalent formulations of the quasi-$p$
property, such as the next result.
###### Lemma 2.2.
A finite group is a quasi $p$-group if and only if it has no nontrivial
quotient group whose order is relatively prime to $p$.
The importance of the quasi-$p$ property is that it characterizes which finite
groups occur as Galois groups of unramified covers of the affine line.
###### Theorem 2.3.
A finite group occurs as the Galois group of a Galois cover of the projective
line $\mathbb{P}^{1}_{k}$ ramified only over infinity if and only if it is a
quasi-$p$ group.
###### Proof.
This is a special case of Abhyankar’s Conjecture [2] which was jointly proved
by Harbater [4] and Raynaud [9]. ∎
We now restrict our attention to groups $G$ that are semi-direct products of
the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. The
semi-direct product action is determined by a homomorphism
$\iota:\mathbb{Z}/p\mathbb{Z}\to\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{b})$.
###### Lemma 2.4.
Suppose a quasi-$p$ group $G$ is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ for a
positive integer $b$.
1. (1)
Then $G$ is not a direct product.
2. (2)
Moreover, $b\geq\mathrm{ord}_{p}(\ell)$ where $\mathrm{ord}_{p}(\ell)$ is the
order of $\ell$ modulo $p$.
###### Proof.
Part (1) is true since $(\mathbb{Z}/\ell\mathbb{Z})^{b}$ cannot be a quotient
of the quasi-$p$ group $G$. For part (2), the structure of a semi-direct
product $({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$
depends on a homomorphism
$\iota:\mathbb{Z}/p\mathbb{Z}\to\mathrm{Aut}\,(\mathbb{Z}/\ell\mathbb{Z})^{b}$.
By part (1), $\iota$ is an inclusion. Thus
$\mathrm{Aut}\,(\mathbb{Z}/\ell\mathbb{Z})^{b}\simeq\mathrm{GL}_{b}({\mathbb{Z}}/\ell{\mathbb{Z}})$
has an element of order $p$. Now
$|\mathrm{GL}_{b}({\mathbb{Z}}/\ell{\mathbb{Z}})|=(\ell^{b}-1)(\ell^{b}-\ell)\cdots(\ell^{b}-\ell^{b-1}).$
Thus $\ell^{\beta}\equiv 1\bmod p$ for some positive integer $\beta\leq b$
which implies $b\geq\mathrm{ord}_{p}(\ell)$. ∎
###### Lemma 2.5.
If $a=\mathrm{ord}_{p}(\ell)$, then there exists a semi-direct product of the
form $({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$
which is quasi-$p$. It is unique up to isomorphism.
###### Proof.
If $a=\mathrm{ord}_{p}(\ell)$, then there is an element of order $p$ in
$\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{a})$ and so there is an injective
homomorphism
$\iota:\mathbb{Z}/p\mathbb{Z}\to\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{a})$.
Thus there exists a non-abelian semi-direct product $G$ of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. To
show that $G$ is quasi-$p$, suppose $N$ is a normal subgroup of $G$ whose
index is relatively prime to $p$. Then $N$ contains an element $\tau$ of order
$p$. By [3, 5.4, Thm. 9], since $G$ is not a direct product and
$(\mathbb{Z}/\ell\mathbb{Z})^{a}$ is normal in $G$, the subgroup
$\langle\tau\rangle$ is not normal in $G$. Thus $\langle\tau\rangle$ is a
proper subgroup of $N$. It follows that $\ell$ divides $|N|$ and so $N$
contains an element $h$ of order $\ell$ by Cauchy’s theorem. Recall that
$\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{\beta})$ contains no element of
order $p$ for any positive integer $\beta<a$. Thus the group generated by the
conjugates of $h$ under $\tau$ has order divisible by $\ell^{a}$. Thus $N=G$
and $G$ has no non-trivial quotient group whose order is relatively prime to
$p$. By Lemma 2.2, $G$ is quasi-$p$.
The uniqueness follows from [8, Lemma 6.6]. ∎
## 3\. Explicit construction of
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$-Galois
covers of $\mathbb{A}^{1}_{k}$
In this section, we give concrete examples of Galois covers
$\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ with Galois group of
the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. To
compute the genus of the covering curve $Z$, we will need to determine the
higher ramification groups of $\psi$.
###### Definition 3.1.
Let $L/K$ be a Galois extension of function fields of curves with Galois group
$G$ and let $P,P^{\prime}$ be primes of $K$ and $L$ such that $P^{\prime}|P$.
Let $\nu_{P^{\prime}}$ and ${\mathcal{O}}_{P^{\prime}}$ be the corresponding
valuation function and valuation ring for $P^{\prime}$. For any integer
$i\geq-1$, the $i$th ramification group of $P^{\prime}|P$ is
$I_{i}(P^{\prime}|P)=\\{\sigma\in G\ |\ \nu_{P^{\prime}}(\sigma(z)-z)\geq
i+1,\forall z\in{\mathcal{O}}_{P^{\prime}}\\}.$
###### Lemma 3.2.
Suppose $f(x)\in k[x]$ is a polynomial of degree $s$ for a positive integer
$s$ prime to $p$. Let $\phi:Y\to\mathbb{P}^{1}_{k}$ be the cover of curves
corresponding to the field extension
$k(x)\hookrightarrow k(x)[y]/(y^{p}-y-f(x)).$
1. (1)
Then $\phi:Y\to\mathbb{P}^{1}_{k}$ is a Galois cover with Galois group
$\mathbb{Z}/p\mathbb{Z}$ ramified only at the point $P_{\infty}$ over
$\infty$.
2. (2)
The $i$th ramification group at $P_{\infty}$ satisfies
$I_{i}=\left\\{\begin{array}[]{ll}\mathbb{Z}/p\mathbb{Z}&\textrm{if }i\leq
s\\\ 0&\textrm{if }i>s.\end{array}\right.$
3. (3)
The genus $g_{Y}$ of $Y$ is equal to
$g_{Y}=(p-1)(s-1)/2.$
###### Proof.
For part (1), note that the extension $k(x)\hookrightarrow
k(x)[y]/(y^{p}-y-f(x))$ is cyclic of degree $p$, with Galois group generated
by the automorphism $\tau:y\mapsto y+1$ of order $p$. Let $P$ be a finite
prime of $k(x)$ and let $\nu_{P}$ be the corresponding valuation. Then
$\nu_{P}(f(x))\geq 0$, hence $P$ is unramified by [12, Prop. III.7.8(b)]. For
the infinite prime $\infty$ with corresponding valuation $\upsilon_{\infty}$,
we have
$\nu_{{\infty}}(f(x)-(z^{p}-z))\leq 0$
for all $z\in k[x]$ thus $P_{\infty}$ is totally ramified by [12, Prop.
III.7.8(c)].
To prove part (2), we note that furthermore
$\upsilon_{P_{\infty}}(y^{p}-y)=\upsilon_{P_{\infty}}(f(x))=\upsilon_{P_{\infty}}(x^{s})=-sp,$
which implies that
$\upsilon_{P_{\infty}}(y)=-s.$
Now let $\widehat{\theta}$ be the completion of the valuation ring of
$k(x)[y]/(y^{p}-y-f(x))$ at $P_{\infty}$, and let $\pi_{\infty}$ be a
generator of the unique prime in $\widehat{\theta}$. Then write
$y=\pi_{\infty}^{-s}u$, where $u$ is a unit in $\widehat{\theta}\simeq
k[[\pi_{\infty}]]$. Since $k$ is algebraically closed,
$\sqrt[s]{u}\in\widehat{\theta}$, and so $\sqrt[s]{y}\in\widehat{\theta}$.
After possibly changing $\pi_{\infty}$, we can assume without loss of
generality that $\sqrt[s]{y}=\pi_{\infty}^{-1}$. Recalling that $\tau$ acts on
$y$ by $\tau(y)=y+1$, we have
$\displaystyle\tau(\pi_{\infty})$ $\displaystyle=$
$\displaystyle\tau\left(1/y\right)^{1/s}=\left(\pi_{\infty}^{s}/(1+\pi_{\infty}^{s})\right)^{1/s}$
$\displaystyle=$
$\displaystyle\pi_{\infty}(1-\pi_{\infty}^{s}+\pi_{\infty}^{2s}-+\ldots)^{\frac{1}{s}}$
$\displaystyle=$
$\displaystyle\pi_{\infty}-(1/s)\pi_{\infty}^{s+1}+a_{2s+1}\pi_{\infty}^{2s+1}-+\ldots.$
Thus $\upsilon_{P_{\infty}}(\tau(\pi_{\infty})-\pi_{\infty})=s+1$, which
completes the proof of part (2).
To find the genus $g_{Y}$ of $Y$ for part (3), we make use of the Riemann-
Hurwitz formula
$2g_{Y}-2=p(-2)+\sum_{i=0}^{\infty}\left(|I_{i}|-1\right),$
where $I_{i}$ denotes the $i$th ramification group at $P_{\infty}$, [5, Thms.
7.27 & 11.72]). From part (2), we then obtain that $g_{Y}=(p-1)(s-1)/2$. ∎
Recall the following facts about the $p$th cyclotomic polynomial
$\Phi_{p}(t):=t^{p-1}+\cdots+1$, which is the minimal polynomial over
${\mathbb{Q}}$ of a primitive $p$th root of unity $\zeta_{p}$. Now
${\mathbb{Q}}(\zeta_{p})$ is a Galois extension of $\mathbb{Q}$, unramified
over $\ell$ since $\ell\not=p$, and all primes over $\ell$ have the same
residue field degree. The irreducible factors of $\Phi_{p}(t)$ modulo $\ell$
are in one-to-one correspondence with the primes of ${\mathbb{Z}}[\zeta_{p}]$
over $\ell$, and each of their degrees is equal to the residue field degree of
the corresponding prime over $\ell$. The latter equals the order
$a=\mathrm{ord}_{p}(\ell)$ of $\ell$ modulo $p$ [3, Ch. 12.2, Exercise #20].
We shall soon explicitly construct a cover of $\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ with Galois group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. But
before we do so, we start with a specific example.
###### Example 3.3.
Let $p$ be an odd prime. Consider the Artin-Schreier cover
$\phi:Y_{2}\to\mathbb{P}^{1}_{k}$ corresponding to the field extension
$k(x)\hookrightarrow k(x)[y]/(y^{p}-y-x^{2})$. By Lemma 3.2(3), the genus of
$Y_{2}$ is $g_{Y}=(p-1)/2$.
Let $\mathrm{Jac}\,(Y)$ be the Jacobian of $Y$. The automorphism $\tau$ of $Y$
given by $\tau(y)=y+1$ defines an automorphism of $\mathrm{Jac}\,(Y)$ of order
$p$.
Now we describe the action of $\tau$ on the subgroup $\mathrm{Jac}\,(Y)[2]$ of
2-torsion points of $\mathrm{Jac}\,(Y)$ explicitly. Note that since
$2g_{Y}=(p-1)$, then $\mathrm{Jac}\,(Y)[2]$ is isomorphic to
$(\mathbb{Z}/2\mathbb{Z})^{p-1}$ by [7, pg. 64]. Thus we can represent $\tau$
as an element of $\mathrm{GL}_{p-1}(\mathbb{Z}/2\mathbb{Z})$.
For $0\leq i\leq p-1$, let $P_{i}$ denote the closed point of $Y$ at which the
function $y-i$ vanishes. For each $i$, the divisors $P_{i}$ and
$D_{i}=P_{i}-P_{\infty}$ on $Y$ can be identified with elements of
$\mathrm{Jac}\,(Y)$. Let $O$ be the identity element of $\mathrm{Jac}\,(Y)$,
i.e., the linear equivalence class of principal divisors. Then the divisor
$2D_{i}$ is equivalent to $O$ since ${\rm div}(y-i)=2D_{i}$. Moreover since
${\rm div}(x)=D_{0}+D_{1}+\cdots+D_{p-1}$ is equivalent to $0$, we have
$D_{i}\in\mathrm{Jac}\,(Y)[2]$ with the only relation
$D_{p-1}=-(D_{0}+D_{1}+\cdots+D_{p-2})$. In particular, $D_{0},\ldots,D_{p-2}$
form a basis of $\mathrm{Jac}\,(Y)[2]$. With respect to this basis, the action
of $\tau$ can be represented by the $(p-1)\times(p-1)$-matrix
$\left(\begin{array}[]{ccccc}0&0&\ldots&0&-1\\\ 1&0&\ldots&0&-1\\\
0&1&\ldots&0&-1\\\ \vdots&\vdots&\ddots&0&-1\\\ 0&0&\ldots&1&-1\\\
\end{array}\right).$
The characteristic polynomial of $\tau$ is $\Phi_{p}(t)=1+t+\ldots
t^{p-1}\in(\mathbb{Z}/2\mathbb{Z})[t]$, which factors into irreducible
polynomials each of degree equaling the order of $2$ modulo $p$. In
particular, $\tau$ acts irreducibly on $\mathrm{Jac}\,(Y)[2]$ if and only if
$2$ is a primitive root modulo $p$, i.e., if and only if $p$ is an Artin
prime.
For example, if $p=3$, then $\tau$ acts irreducibly on $\mathrm{Jac}\,(Y)[2]$
with minimal polynomial $\Phi_{3}(t)=t^{2}+t+1$. If $p=7$, then $2$ has order
$3$ modulo $7$ and the factorization of $\Phi_{7}(t)$ into irreducible
polynomials is $\Phi_{7}(t)\equiv(x^{3}+x^{2}+1)(x^{3}+x+1)$ modulo $2$. Thus
the action of $\tau$ on $\mathrm{Jac}\,(Y)[2]$ can be represented by the
$6\times 6$-matrix
$\left(\begin{array}[]{cc}A_{1}&0\\\ 0&A_{2}\\\ \end{array}\right)$
where $A_{1}$ and $A_{2}$ are the irreducible $3$-dimensional companion
matrices of $x^{3}+x^{2}+1$ and $x^{3}+x+1$ respectively.
For the rest of the paper, let $\phi_{s}:Y_{s}\to\mathbb{P}^{1}_{k}$ be the
Artin-Schreier cover corresponding to the field extension
$k(x)\hookrightarrow k(x)[y]/(y^{p}-y-x^{s}).$
We show that $\phi_{s}$ can be dominated by a Galois cover of
$\mathbb{P}^{1}_{k}$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ for $a$
equal to the order of $\ell$ modulo $p$.
###### Proposition 3.4.
Let $s$ and $\ell$ be primes distinct from $p$. Let
$\phi_{s}:Y_{s}\to\mathbb{P}^{1}_{k}$ be the Artin-Schreier cover with affine
equation $y^{p}-y=x^{s}$. Let $a=\mathrm{ord}_{p}(\ell)$ be the order of
$\ell$ modulo $p$. Then there exists an unramified Galois cover
$\omega:Z_{a}\to Y_{s}$ with Galois group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$ such that
$\psi_{a}=\phi_{s}\circ\omega:Z_{a}\rightarrow\mathbb{P}^{1}_{k}$ is a Galois
cover of $\mathbb{P}^{1}_{k}$ ramified only over $\infty$ whose Galois group
is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes\mathbb{Z}/p\mathbb{Z}$.
###### Proof.
By Lemma 3.2(1), $\phi_{s}:Y_{s}\rightarrow\mathbb{P}^{1}_{k}$ is a Galois
cover with Galois group $\mathbb{Z}/p\mathbb{Z}$ ramified only at the point
$P_{\infty}$ over $\infty$. The genus $g_{s}$ of $Y_{s}$ is $(p-1)(s-1)/2$.
Consider two commuting automorphisms of $Y_{s}$ defined by
$\tau:\begin{cases}x\mapsto x,\\\ y\mapsto
y+1,\end{cases}~{}~{}\sigma:\begin{cases}x\mapsto\zeta_{s}x,\text{ where
}\zeta_{s}\text{ is a primitive $s$th root of unity,}\\\ y\mapsto
y.\end{cases}$
Let $\mathrm{Jac}\,(Y_{s})$ be the Jacobian of $Y_{s}$. Then $\tau$ and
$\sigma$ define commuting automorphisms of $\mathrm{Jac}\,(Y_{s})$ of orders
$p$ and $s$ respectively. Therefore, $\mathrm{End}\,(\mathrm{Jac}\,(Y_{s}))$
contains a ring isomorphic to
$\mathbb{Z}[\zeta_{p},\zeta_{s}]\cong\mathbb{Z}[\zeta_{ps}]$, which is a
$\mathbb{Z}$-module of rank $\phi(ps)=(p-1)(s-1)=2g_{s}$. Then
${\mathbb{Q}}(\zeta_{ps})$ is contained in
$\mathrm{End}\,(\mathrm{Jac}\,(Y_{s}))\otimes{\mathbb{Q}}$. In other words,
$\mathrm{Jac}\,(Y_{s})$ has complex multiplication by
${\mathbb{Q}}(\zeta_{ps})$.
For a prime $\ell$ distinct from $p$, the automorphism $\tau$ induces an
action on the subgroup $\mathrm{Jac}\,(Y_{s})[\ell]$ of $\ell$-torsion points
of $\mathrm{Jac}\,(Y_{s})$. Recall that there is a bijection between
$\ell$-torsion points $D$ of $\mathrm{Jac}\,(Y_{s})$ and unramified
$(\mathbb{Z}/\ell\mathbb{Z})$-Galois covers $\omega_{D}:Z_{D}\to Y_{s}$ [6,
Prop. 4.11]. Also $D$ has order $\ell$ if and only if $Z_{D}$ is connected.
This induces a bijection between orbits of $\tau$ on the set of unramified
$(\mathbb{Z}/\ell\mathbb{Z})$-Galois covers $\omega_{D}:Z_{D}\to Y_{s}$ and on
the set of $\ell$-torsion points of $\mathrm{Jac}\,(Y_{s})$. For a point $D$
of order $\ell$ of $\mathrm{Jac}\,(Y_{s})$, consider the compositum
$\omega:Z\to Y_{s}$ of all of the conjugates
$\omega_{\tau^{j}(D)}:Z_{\tau^{i}(D)}\to Y_{s}$ for $0\leq j\leq p-1$:
$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{Z_{\tau(D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{Z_{\tau^{2}(D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{\ldots}$$\textstyle{Z_{\tau^{p-1}(D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{Y_{s}}$
Then $Z$ is invariant under $\tau$ and so
$\phi_{s}\circ\omega:Z\to\mathbb{P}^{1}_{k}$ is Galois. Moreover,
$\phi_{s}\circ\omega$ is the Galois closure of
$\phi_{s}\circ\omega_{D}:Z_{D}\to\mathbb{P}^{1}_{k}$.
Suppose there is a non-trivial one-dimensional $\tau$-invariant subspace of
$\mathrm{Jac}\,(Y_{s})[\ell]$ with eigenvalue $1$; i.e. $\tau$ acts trivially
on this subgroup of order $\ell$. This yields a cover
$\psi_{s}\circ\omega_{1}:Z_{1}\to Y_{s}\to\mathbb{P}^{1}_{k}$. Since the
action of $\tau$ is trivial, $\psi_{s}\circ\omega_{1}$ is Galois, ramified
only over $\infty$, with abelian Galois group
$\mathbb{Z}/\ell\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$. This contradicts
Lemma 2.4.
Since $\tau$ has order $p$, the minimal polynomial $m_{\tau}(t)$ of $\tau$
divides $t^{p}-1=(t-1)(t^{p-1}+\cdots+1)$ in $(\mathbb{Z}/\ell\mathbb{Z})[t]$.
From the preceding paragraph, there is no non-trivial one-dimensional
$\tau$-invariant subspace of $\mathrm{Jac}\,(Y_{s})[\ell]$ with eigenvalue
$1$. This implies that $m_{\tau}(t)$ divides the $p$th cyclotomic polynomial
$\Phi_{p}(t)=t^{p-1}+\cdots+1$ in $(\mathbb{Z}/\ell\mathbb{Z})[t]$. The
irreducible factors of $\Phi_{p}(t)$ in $(\mathbb{Z}/\ell\mathbb{Z})[t]$ all
have degree $a$. Thus the degree of $m_{\tau}(t)$ equals $a$.
Since $2g_{s}=(p-1)(s-1)$, we have
$\mathrm{Jac}\,(Y_{s})[\ell]\cong(\mathbb{Z}/\ell\mathbb{Z})^{(p-1)(s-1)}$, so
we can represent $\tau$ as an element of
$\mathrm{GL}_{(p-1)(s-1)}(\mathbb{Z}/\ell\mathbb{Z})$. We can choose a basis
of $\mathrm{Jac}\,(Y_{s})[\ell]$ such that $\tau$ is represented as an element
of $\mathrm{GL}_{(p-1)(s-1)}(\mathbb{Z}/\ell\mathbb{Z})$ in block form. The
first irreducible subrepresentation of $\tau$ has dimension $a$. Moreover,
since ${\mathbb{Q}}(\zeta_{ps})$ is a Galois extension of ${\mathbb{Q}}$, the
block form of $\tau$ consists entirely of irreducible blocks of the same size.
In particular, the number of irreducible blocks is $(p-1)(s-1)/a$. In other
words, $\tau$ can be represented by an element of
$\mathrm{GL}_{(s-1)(p-1)}(\mathbb{Z}/\ell\mathbb{Z})$ of the form
$\left(\begin{array}[]{ccc}A_{1}&&0\\\ &A_{2}&\\\ &\ddots&\\\
0&&A_{(p-1)(s-1)/a}\end{array}\right),$
where $A_{i}$ is an $a\times a$ matrix representing an $a$-dimensional
irreducible subrepresentation of $\tau$ on $\mathrm{Jac}\,(Y_{s})[\ell]$.
Using the bijection between orbits of $\mathrm{Jac}\,(Y_{s})[\ell]$ and orbits
of $(\mathbb{Z}/\ell\mathbb{Z})$-covers of $Y_{s}$ under $\tau$ and the above
observation for the action of $\tau$ on $\mathrm{Jac}\,(Y_{s})[\ell]$, there
exists an unramified $(\mathbb{Z}/\ell\mathbb{Z})^{a}$-Galois cover
$\omega:Z_{a}\to Y_{s}$ such that
$\psi_{a}=\phi_{s}\circ\omega:Z_{a}\rightarrow\mathbb{P}^{1}_{k}$ is a Galois
cover of $\mathbb{P}^{1}_{k}$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes\mathbb{Z}/p\mathbb{Z}$. Also
$\psi_{a}$ is ramified only over infinity since $\phi_{s}$ is ramified only
over $\infty$ and since $\omega$ is unramified. ∎
## 4\. Minimal genus of
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$-Galois
covers of $\mathbb{A}^{1}_{k}$
In this section, we find the minimal genus of a curve $Z$ that admits a
covering map $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$, with
Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. The
minimal genus depends only on $\ell$ and $p$. We consider the cases $p$ odd
and $p=2$ separately. We also prove that the number of curves $Z$ of this
minimal genus which admit such a covering map is at most $(p-1)/a$ when $p$ is
odd and at most $\ell+1$ when $p=2$. The following lemma will be useful.
###### Lemma 4.1.
Let $G$ be a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ where
$\ell$ is a prime distinct from $p$. If $\psi:Z\to\mathbb{P}^{1}_{k}$ is a
Galois cover ramified only over $\infty$ with Galois group $G$, then the
subcover $\omega:Z\to Y$ with Galois group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}$ is unramified.
###### Proof.
The quotient of $G$ by the normal subgroup
$N=({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}$ is ${\mathbb{Z}}/p{\mathbb{Z}}$. Thus
the cover $\psi$ is a composition $\psi=\phi\circ\omega$ where
$\phi:Y\to\mathbb{P}^{1}_{k}$ has Galois group ${\mathbb{Z}}/p{\mathbb{Z}}$
and is totally ramified at the unique point $P_{\infty}$ over $\infty$ and
where $\omega:Z\to Y$ has Galois group $N$ and is branched only over
$P_{\infty}$. Then $\omega$ is a prime-to-$p$ abelian cover of $Y$. Let $g$ be
the genus of $Y$. Then by [1, XIII, Cor. 2.12], the prime-to-$p$ fundamental
group of $Y-\\{P_{\infty}\\}$ is isomorphic to the prime-to-$p$ quotient
$\Gamma$ of the free group on generators
$\\{a_{1},b_{1},\ldots,a_{g},b_{g},c\\}$ subject to the relation
$\prod_{i=1}^{g}[a_{i},b_{i}]=c^{-1}$. The cover $\omega$ corresponds to a
surjection of $\Gamma$ onto $N$ where $c$ maps to the canonical generator of
inertia $\gamma$ of a point of $Z$ over $P_{\infty}$. Thus $N$ is generated by
elements $\\{\alpha_{1},\beta_{1},\ldots,\alpha_{g},\beta_{g},\gamma\\}$
subject to the relation $\prod_{i=1}^{g}[\alpha_{i},\beta_{i}]=\gamma^{-1}$.
Then $\gamma=1$ since $N$ is abelian and so $\omega$ is unramified. ∎
###### Theorem 4.2.
Let $p$ be an odd prime. Let $\ell$ be a prime distinct from $p$ and let $a$
be the order of $\ell$ modulo $p$. Then:
1. (1)
There exists a Galois cover $\psi_{a}:Z_{a}\to\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ whose Galois group is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ such
that $g_{Z_{a}}=1+\ell^{a}(p-3)/2$.
2. (2)
The integer $g_{Z_{a}}$ is the minimal genus of a curve $Z$ which admits a
covering map $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ with
Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ for any
positive integer $b$.
3. (3)
There are at most $(p-1)/a$ isomorphism classes of curves $Z$ which admit a
Galois covering map as in part (1) with minimal genus $g_{Z_{a}}$.
###### Proof.
By the construction in Proposition 3.4, there exists a Galois cover
$\psi_{a}:Z_{a}\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ whose Galois
group is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. We
compute the genus of the curve $Z_{a}$. Recall that $\psi_{a}$ is a
composition $\psi=\phi_{2}\circ\omega$ where $\omega:Z\to Y_{2}$ is an
unramified $({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois cover and
$\phi_{2}:Y_{2}\to\mathbb{P}^{1}_{k}$ has Artin-Schreier equation
$y^{p}-y=x^{2}$. Then $Y_{2}$ has genus $g_{Y_{2}}=(p-1)/2$ by Lemma 3.2(3).
By the Riemann-Hurwitz formula,
$2g_{Z_{a}}-2=\ell^{a}(2g_{Y_{2}}-2)=\ell^{a}(p-3)$, i.e.,
$g_{Z_{a}}=1+\ell^{a}(p-3)/2$. This completes part (1).
For part (2), suppose $\psi:Z\to\mathbb{P}^{1}_{k}$ is a Galois cover ramified
only over $\infty$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. If $g$
is the genus of $Z$, we will show that $g\geq g_{Z_{a}}$. As described in the
proof of Lemma 4.1, the cover $\psi$ is a composition $\psi=\phi\circ\omega$
where $\phi:Y\to\mathbb{P}^{1}_{k}$ has Galois group $\mathbb{Z}/p\mathbb{Z}$
and is ramified only over $\infty$ and where $\omega$ is unramified with group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}$. By the Riemann-Hurwitz formula,
$2g-2=\ell^{b}(2g_{Y}-2)$.
By Artin-Schreier theory, $\phi$ is given by an equation $y^{p}-y=f(x)$ where
$f\in k[x]$ has degree $s$ for some integer $s$ relatively prime to $p$. Since
the genus $g_{Y}$ of $Y$ is $(p-1)(s-1)/2$ by Lemma 3.2 (3), we should make
$s$ as small as possible. The value $s=1$ is impossible since then $Y$ is a
projective line and there do not exist Galois covers of the projective line
ramified only over one point with Galois group $\mathbb{Z}/\ell\mathbb{Z}$.
Thus $s=2$ yields the smallest possible value for $g_{Y}$, namely $(p-1)/2$.
Recall that $b\geq a$ by Lemma 2.4. Thus $g\geq 1+\ell^{a}(p-3)/2=g_{Z_{a}}$.
For part (3), suppose $\psi:Z\to\mathbb{P}^{1}_{k}$ is a Galois cover ramified
only over $\infty$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ and the
genus of $Z$ satisfies $g_{Z}=1+\ell^{a}(p-3)/2$. As in part (2), $\psi$
factors as $\phi\circ\omega$ where $\omega:Z\to Y$ is an unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois cover, where
$\phi:Y\to\mathbb{P}^{1}_{k}$ is an Artin-Schreier cover ramified only over
$\infty$, and where $Y$ has genus $(p-1)/2$. By Lemma 3.2(3), $Y$ has an
affine equation $y^{p}-y=a_{2}x^{2}+a_{1}x+a_{0}$ for some $a_{0},a_{1}\in k$
and $a_{2}\in k^{*}$. Since $p$ is odd and $k$ is algebraically closed, it is
possible to complete the square and write
$a_{2}x^{2}+a_{1}x+a_{0}=x_{1}^{2}+\epsilon$ where
$x_{1}=\sqrt{a_{2}}x+a_{1}/2\sqrt{a_{2}}$. After modifying by an automorphism
of the projective line, specifically by the affine linear transformation
$x\mapsto x_{1}$, the equation for $Y$ can be rewritten as
$y^{p}-y=x_{1}^{2}+\epsilon$. Since $k$ is algebraically closed, there exists
$\delta\in k$ such that $\delta^{p}-\delta=\epsilon$. Let $y_{1}=y-\delta$.
After the change of variables $y\mapsto y_{1}$, the curve $Y$ is isomorphic to
the curve $Y_{2}$ with affine equation $y_{1}^{p}-y_{1}=x_{1}^{2}$. Thus there
is a unique possibility for the isomorphism class of the curve $Y$.
From the proof of Proposition 3.4, there is a bijection between
$\tau$-invariant connected unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois covers of $Y_{2}$ and orbits of
$\tau$ on points $D$ of order $\ell$ on $\mathrm{Jac}\,(Y_{2})$. The action of
$\tau$ on $\mathrm{Jac}\,(Y_{2})[\ell]$ decomposes into $(p-1)/a$ irreducible
subrepresentations. Each of these is distinct, because the irreducible factors
of $\Phi_{p}(t)\in(\mathbb{Z}/\ell\mathbb{Z})[t]$ are distinct. Thus there are
$(p-1)/a$ choices for a $\tau$-invariant unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois cover of $Y_{2}$. Thus there are
at most $(p-1)/a$ isomorphism classes of curves $Z$ which admit a Galois
covering map as in part (1) with minimal genus $g_{Z_{a}}$. ∎
We note that the set of curves which are unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois covers of $Y_{2}$ may contain
fewer than $(p-1)/a$ isomorphism classes of curves.
###### Theorem 4.3.
Let $p=2$ and let $\ell$ be an odd prime. Then:
1. (1)
There exists a Galois cover $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over
$\infty$ with Galois group of the form
$\mathbb{Z}/\ell\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$.
2. (2)
The minimal genus of a curve $Z$ which admits a covering map as in part (1) is
$g_{Z}=1$.
3. (3)
There are at most $\ell+1$ isomorphism classes of curves $Z$ which admit a
Galois covering map as in part (1) with minimal genus $g_{Z}=1$.
###### Proof.
Note that the order of $\ell$ modulo $2$ is $a=1$. For part (1), Lemma 2.5
shows that there exists a semi-direct product of the form
$\mathbb{Z}/\ell\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$ which is quasi-$2$.
The result is then immediate from Theorem 2.3.
Suppose $\psi:Z\to\mathbb{P}^{1}_{k}$ is a Galois cover ramified only over
$\infty$ with Galois group as in part (1). As before, $\psi$ factors as a
composition $\phi\circ\omega$. where $\omega:Z\to Y$ has Galois group
$\mathbb{Z}/\ell\mathbb{Z}$ and $\phi:Y\to\mathbb{P}^{1}_{k}$ is an Artin-
Schreier extension with affine equation $y^{2}-y=f(x)$ for some $f(x)\in k[x]$
of odd degree $s$. By Lemma 4.1, $\omega$ is unramified. The minimal genus for
$Z$ will thus occur when $s$ is as small as possible. As before, $s=1$ is
impossible, and so $s=3$ is the smallest choice. In this case, by Lemma
3.2(3), $g_{Y}=1$, i.e., $Y$ is an elliptic curve. By the Riemann-Hurwitz
formula, the minimal genus for $Z$ is $g_{Z}=1+\ell(g_{Y}-1)=1$, which
completes part (2).
For part (3), since $k$ is algebraically closed, we can complete the cube of
$f(x)$ and make the corresponding change of variables, which is a scaling and
translation of $x$. So we can assume that $Y$ has affine equation
$y^{2}-y=x^{3}+a_{1}x+a_{0}$ for some $a_{0},a_{1}\in k$. Then it follows from
[11, Appendix A, Prop. 1.1c] that the $j$-invariant of $Y$ is $j(Y)=0$ and
that the discriminant is $\Delta(Y)=(-1)^{4}=1$. Since $k$ is algebraically
closed, by [11, Appendix A, Prop. 1.2b], all elliptic curves $Y$ with $j(Y)=0$
are isomorphic over $k$. Thus there is a unique choice for $Y$ up to
isomorphism. Without loss of generality, we may assume that $Y=Y_{3}$ has
affine equation $y^{2}-y=x^{3}$.
From the proof of Proposition 3.4, the action of $\tau$ on
$\mathrm{Jac}\,(Y_{3})[\ell]$ decomposes into the direct sum of two
$1$-dimensional subrepresentations. In other words, the action of $\tau$ is
diagonal with both eigenvalues equal to $-1$. The number of non-trivial
$\tau$-invariant subgroups of $\mathrm{Jac}\,(Y_{3})[\ell]$ is the number of
subgroups of order $\ell$ in $(\mathbb{Z}/\ell\mathbb{Z})^{2}$, which is
$\ell+1$. As in Theorem 4.2, this implies that there are at most $\ell+1$
isomorphism classes of curves $Z$ which admit a Galois covering map as in part
(1) with minimal genus $g_{Z}=1$. ∎
We note that the set of curves which are unramified
$\mathbb{Z}/\ell\mathbb{Z}$-Galois covers of $Y_{3}$ may contain fewer than
$\ell+1$ isomorphism classes of curves.
## References
* [1] Revêtements étales et groupe fondamental. Springer-Verlag, Berlin, 1971. Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud, Lecture Notes in Mathematics, Vol. 224.
* [2] Shreeram Abhyankar. Coverings of algebraic curves. Amer. J. Math., 79:825–856, 1957.
* [3] David S. Dummit and Richard M. Foote. Abstract algebra. John Wiley & Sons Inc., Hoboken, NJ, third edition, 2004.
* [4] David Harbater. Abhyankar’s conjecture on Galois groups over curves. Invent. Math., 117(1):1–25, 1994.
* [5] J. W. P. Hirschfeld, G. Korchmáros, and F. Torres. Algebraic curves over a finite field. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2008.
* [6] James S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.
* [7] David Mumford. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay, 1970.
* [8] Rachel Pries and Katherine Stevenson. A survey of Galois theory of curves in characteristic $p$. WIN - Women In Numbers, Fields Communication Volume.
* [9] M. Raynaud. Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar. Invent. Math., 116(1-3):425–462, 1994.
* [10] I. Shafarevitch. On $p$-extensions. Rec. Math. [Mat. Sbornik] N.S., 20(62):351–363, 1947.
* [11] Joseph H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original.
* [12] Henning Stichtenoth. Algebraic function fields and codes, volume 254 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, second edition, 2009.
|
arxiv-papers
| 2009-10-25T18:51:27 |
2024-09-04T02:49:06.037450
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Linda Gruendken, Laura Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel\n Pries, Katherine Stevenson",
"submitter": "Ekin Ozman",
"url": "https://arxiv.org/abs/0910.4695"
}
|
0910.4716
|
# Bounds for the relative n–th nilpotency degree in compact groups
Rashid Rezaei Department of Mathematics, University of Malayer, P. O. Box
657719, 95863, Malayer, Iran ras$\\_$rezaei@yahoo.com and Francesco G. Russo
Structural Geotechnical Dynamics Laboratory StreGa, University of Molise, Via
Duca degli Abruzzi, 86039, Termoli (CB), Italy
and
Department of Mathematics, University of Palermo, Via Archirafi 34, 90123,
Palermo, Italy. francescog.russo@yahoo.com
(Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
∗ Corresponding author)
###### Abstract.
The line of investigation of the present paper goes back to a classical work
of W. H. Gustafson of the 1973, in which it is described the probability that
two randomly chosen group elements commute. In the same work, he gave some
bounds for this kind of probability, providing information on the group
structure. We have recently obtained some generalizations of his results for
finite groups. Here we improve them in the context of the compact groups.
###### Key words and phrases:
Commutativity degree, Haar measure, n–th nilpotency degree
###### 2010 Mathematics Subject Classification:
Primary 22A05, 28C10; Secondary 22A20, 43A05.
## 1\. Introduction
A compact group $G$ admits a unique left Haar measure $\mu_{G}$ which is
normalized and left-invariant (see [11, Sections 18.1, 18.2, Proposition
18.2.1]). This allows us to assume that $G$ has a unique probability measure
space with respect to $\mu_{G}$ (see [11, Sections 18.1, 18.2] or [10, Section
2]). On the product measure space $G\times G$, it is possible to consider the
product measure $\mu_{G}\times\mu_{G}$ which is a probability measure. If
$C_{2}=\\{(x,y)\in G\times G\ |\ [x,y]=1\\},$
then $C_{2}=f^{-1}(1)$, where
$f:(x,y)\in G\times G\mapsto f(x,y)=[x,y]\in G.$
Clearly, $f$ is continuous and $C_{2}$ is a compact measurable subset of
$G\times G$. Therefore it is possible to define
$d(G)=(\mu_{G}\times\mu_{G})(C_{2})$
as the $commutativity$ $degree$ of $G$. In the finite case $d(G)$ is described
in [1, 2, 3, 5, 12, 13, 14]. We may extend the notion of $d(G)$ as follows.
Suppose that $n\geq 1$, $G^{n}$ is the product of $n$-copies of $G$ and
$\mu^{n}_{G}$ that of $n$-copies of $\mu_{G}$. We define
$d^{(n)}(G)=\mu^{n+1}_{G}(C_{n+1})$
as the $n$-$th$ $nilpotency$ $degree$ of $G$, where
$C_{n+1}=\\{(x_{1},\ldots,x_{n+1})\in G^{n+1}\ |\
[x_{1},x_{2},...,x_{n+1}]=1\\}.$
Obviously, if $G$ is finite, then $G$ is a compact group with the discrete
topology and so the Haar measure of $G$ is the counting measure. Then, for a
finite group $G$, we have
$d^{(n)}(G)=\mu^{n+1}_{G}(C_{n+1})=\frac{|C_{n+1}|}{|G|^{n+1}}.$
See for details [5, 13].
More generally, let $H$ be a closed subgroup of a compact group $G$. We define
$D_{2}=\\{(h,g)\in H\times G\ |\ [h,g]=1\\}$
and note that $D_{2}=\phi^{-1}(1)$, where
$\phi:(h,g)\in H\times G\mapsto\phi(h,g)=[h,g]\in G.$
Clearly, $\phi$ is continuous and $D_{2}$ is a compact measurable subset of
$H\times G$. Note that $\phi$ is the restriction of $f$ to $H\times G$ and
this shows that $H$ has to be closed subgroup of $G$, if we want to preserve
the topological structure. Then we define
$d(H,G)=(\mu_{H}\times\mu_{G})(D_{2})$
as the $relative$ $commutativity$ $degree$ of $H$ with respect to $G$.
Considering
$D_{n+1}=\\{(h_{1},,...,h_{n},g)\in H^{n}\times G\ |\
[h_{1},h_{2},...,h_{n},g]=1\\},$
we define
$d^{(n)}(H,G)=(\mu^{n}_{H}\times\mu_{G})(D_{n+1})$
as the $relative$ $n$-$th$ $nilpotency$ $degree$ of $H$ with respect to $G$.
As already noted, [1, 5, 12, 13, 14] give contributions to the knowledge of
the $n$-th nilpotency degree in case of finite groups. Recently, the case of
infinite groups can be found in [4, 6, 7, 8, 9, 15, 16]. We will try to extend
the results in [5, Sections 3,4,5] looking at the methods in [4, 6, 7, 8, 15,
16].
## 2\. Relative commutativity degree
The next statement is useful for proving most of our results.
###### Lemma 2.1.
Assume that $G$ is a compact group, $H$ is a closed subgroup of $G$ and
$C_{G}([h_{1},...,h_{n}])$ is the centralizer of the commutator
$[h_{1},...,h_{n}]$ in $G$ for some elements $h_{1},...,h_{n}$ in $H$. Then
$d^{(n)}(H,G)=\int_{H}\ldots\left(\int_{H}\mu_{G}(C_{G}([h_{1},...,h_{n}]))d\mu_{H}(h_{1})\right)\ldots
d\mu_{H}(h_{n}),$
where
$\mu_{G}(C_{G}([h_{1},...,h_{n}]))=\int_{G}\chi_{{}_{D_{n+1}}}(h_{1},...,h_{n},g)d\mu_{G}(g)$
and $\chi_{{}_{D_{n+1}}}$ denotes the characteristic map of the set $D_{n+1}$.
###### Proof.
Since
$\mu_{G}(C_{G}([h_{1},...,h_{n}]))=\int_{G}\chi_{{}_{D_{n+1}}}(h_{1},...,h_{n},g)d\mu_{G}(g),$
we have by Fubini-Tonelli’s Theorem:
$d^{(n)}(H,G)=(\mu^{n}_{H}\times\mu_{G})(D_{n+1})=\int_{H^{n}\times
G}\chi_{{}_{D_{n+1}}}(d\mu^{n}_{H}\times d\mu_{G})$
$=\int_{H}...\left(\int_{H}\left(\int_{G}\chi_{{}_{D_{n+1}}}(h_{1},...,h_{n},g)d\mu_{G}(g)\right)d\mu_{H}(h_{1})\right)...\
d\mu_{H}(h_{n})$
$=\int_{H}...\left(\int_{H}\mu_{G}(C_{G}([h_{1},...,h_{n}])d\mu_{H}(h_{1})\right)...\
d\mu_{H}(h_{n}).$
∎
We recall the following elementary fact, which can be found in [10]. See also
[6, Lemma 3.1].
###### Lemma 2.2.
Assume $H$ is a closed subgroup of a compact group $G$. If $|G:H|=n<\infty$,
then $\mu_{G}(H)=\frac{1}{n}$. If $|G:H|=\infty$, then $\mu_{G}(H)=0$.
###### Proof.
Assume that $|G:H|=n$ is finite. Then
$G={\overset{n}{\underset{i=1}{\bigcup}}}g_{i}H$. So we have
$1=\mu_{G}(G)=\mu_{G}(\bigcup^{n}_{i=1}g_{i}H)=\sum^{n}_{i=1}\mu_{G}(g_{i}H)=\sum^{n}_{i=1}\mu_{G}(H)=n\mu_{G}(H)$
and therefore $\mu_{G}(H)=\frac{1}{n}$ . Now assume that
$\alpha=|G:H|=\infty$. Of course, $\alpha>0$, then $t\alpha>1$ for some
positive integer $t$. By assumption, $G={\underset{i\in I}{\bigcup}}g_{i}H$,
where $I$ is an infinite set. Choose a subset $J$ of $I$ of cardinality $t$.
It follows that
$1=\mu_{G}(G)\geq\mu_{G}(\bigcup_{j\in J}g_{j}H)\geq\sum_{j\in
J}\mu_{G}(g_{j}H)=t\alpha>0.$
This contradicts $\mu_{G}(H)=0$ and the proof of the lemma follows. ∎
Lemma 2.2 will be used in most of our proofs, even if the following form is
more suitable.
###### Lemma 2.3.
Assume $H$ is a closed subgroup of a compact group $G$. If $|G:H|\geq n$, then
$\mu_{G}(H)\leq\frac{1}{n}$. If $|G:H|\leq n$, then
$\mu_{G}(H)\geq\frac{1}{n}$. In particular, $|G:H|=n$ if and only if
$\mu_{G}(H)=\frac{1}{n}$.
###### Proof.
This follows from an argument as in Lemma 2.2. ∎
Lemma 2.3 allows us to reformulate [5, Theorem 3.10] for infinite groups in
terms of the following result. The reader may find exactly the same proof in
[15]: here we repeat it, just for sake of completeness and because we want to
point out the methods and the ideas which are often used in similar
circumstances.
###### Theorem 2.4.
Let $H$ be a closed subgroup of a compact group $G$.
* (i)
If $d(H,G)=\frac{3}{4}$, then $H/(Z(G)\cap H)$ is cyclic of order 2.
* ii)
If $d(H,G)=\frac{5}{8}$ and $H$ is nonabelian, then $H/(Z(G)\cap H)$ is
2-elementary abelian of rank 2.
###### Proof.
(i). Assume that $d(H,G)=\frac{3}{4}$ and let $K=H\cap Z(G)$. If $h$ is a
element of $H$ not belonging to $K$, then $|G:C_{G}(h)|\geq 2$ and so
$\mu_{G}(C_{G}(h))\leq\frac{1}{2}$ by Remark 2.3. On the other hand, if $h$ is
an element of $K$, then $\mu_{G}(C_{G}(h))=1$. From these facts and Lemma 2.1,
we have
$\frac{3}{4}=d(H,G)=\int_{H}\mu_{G}(C_{G}(h))d\mu_{H}(h)$
$=\int_{K}\mu_{G}(C_{G}(h))d\mu_{H}(h)+\int_{H-K}\mu_{G}(C_{G}(h))d\mu_{H}(h)$
$\leq\int_{K}d\mu_{H}(h)+\frac{1}{2}\int_{H-K}d\mu_{H}(h)=\mu_{H}(K)+\frac{1}{2}(1-\mu_{H}(K)).$
Therefore, $\mu_{H}(K)\geq\frac{1}{2}$. On the other hand, $K$ is a closed
subgroup of the abelian group $H$ such that $\mu_{H}(K)\leq\frac{1}{2}$. Then
$\mu_{H}(K)=\frac{1}{2}$ and so $|H:K|=2$. This means that $H/K$ is cyclic of
order 2, as claimed.
(ii). Assume that $d(H,G)=\frac{5}{8}$ and let $K=H\cap Z(G)$. We may argue as
in the previous statement (i). On a hand, we have
$\frac{5}{8}=d(H,G)\leq\frac{1}{2}+\frac{1}{2}\mu_{H}(K).$ Therefore,
$\mu_{H}(K)\geq\frac{1}{4}$. On the other hand, $K$ is a closed subgroup of
the nonabelian group $H$ so that $\mu_{H}(K)\leq\frac{1}{4}$, still by Lemma
2.3 . This gives $\mu_{H}(K)=\frac{1}{4}$ so that $|H:K|=4$. This means that
$H/K$ has order 4. Since $H$ is nonabelian, $H/K$ cannot be cyclic. From this,
$H/K$ is 2-elementary abelian of rank 2, as claimed. ∎
Note that [5, Theorem 3.10] follows from Theorem 2.4 when we consider a finite
group with the counting measure on it. Now we extend [5, Lemma 3.2] to the
case of infinite groups. The next result overlaps [6, Lemma 3.2].
###### Lemma 2.5.
Let $H$ be a closed subgroup of a compact group $G$. Then
$\mu_{G}(C_{G}(x))\leq\mu_{H}(C_{H}(x))$
for all $x\in G$.
###### Proof.
Consider the map
$f:hC_{H}(x)\in\\{hC_{H}(x)\ |\ h\in H\\}\mapsto
f(hC_{H}(x))=hC_{G}(x)\in\\{gC_{G}(x)\ |\ g\in G\\}.$
$f$ is one–to–one and so $|H:C_{H}(x)|\leq|G:C_{G}(x)|$. This implies
$\mu_{G}(C_{G}(x))\leq\mu_{H}(C_{H}(x))$. ∎
An important dominance condition is the following.
###### Theorem 2.6.
Let $H$ be a closed subgroup of a compact group $G$. Then
$d(G)\leq d(H,G)\leq d(H).$
###### Proof.
From Lemma 2.5, $\mu_{H}(C_{H}(x))\geq\mu_{G}(C_{G}(x))$. Integrating over $H$
and keeping in mind Lemma 2.1, we have
$d(H)=\int_{H}\mu_{H}(C_{H}(x))d\mu_{H}(x)\geq\int_{H}\mu_{G}(C_{G}(x))d\mu_{H}(x)=d(H,G).$
On the other hand, Lemmas 2.1 and 2.5 give
$d(H,G)=\int_{G}\mu_{H}(C_{H}(x))d\mu_{G}(x)\geq\int_{G}\mu_{G}(C_{G}(x))d\mu_{G}(x)=d(G).$
∎
###### Theorem 2.7.
Let $H$ be a closed subgroup of a compact group $G$. Then
* (i)
$d(G)\leq\frac{1}{2}+\frac{1}{2}\mu_{G}(Z(G))$;
* (ii)
$d(H,G)\leq\frac{1}{2}+\frac{1}{2}\mu_{H}(K)$, where $K=H\cap Z(G)$.
###### Proof.
(i). By Lemma 2.1 and noting that $\mu_{G}(C_{G}(x))\leq\frac{1}{2}$ for each
noncentral element $x$ of $G$, we have
$d(G)=\int_{G}\mu_{G}(C_{G}(x))d\mu_{G}(x)$
$=\int_{Z(G)}\mu_{G}(C_{G}(x))d\mu_{G}(x)+\int_{G-Z(G)}\mu_{G}(C_{G}(x))d\mu_{G}(x)$
$=\mu_{G}(Z(G))+\int_{G-Z(G)}\mu_{G}(C_{G}(x))d\mu_{G}(x)$
$\leq\mu_{G}(Z(G))+\frac{1}{2}(1-\mu_{G}(Z(G)))=\frac{1}{2}+\frac{1}{2}\mu_{G}(Z(G)).$
(ii). By Lemma 2.1 and noting that $\mu_{G}(C_{G}(h))\leq\frac{1}{2}$ for each
element $h$ of $H-K$,
$d(H,G)=\int_{H}\mu_{G}(C_{G}(h))d\mu_{H}(h)$
$=\int_{K}\mu_{G}(C_{G}(h))d\mu_{H}(x)+\int_{H-K}\mu_{G}(C_{G}(h))d\mu_{H}(h)$
$=\mu_{H}(K)+\int_{H-K}\mu_{G}(C_{G}(h))d\mu_{H}(h)$
$\leq\mu_{H}(K)+\frac{1}{2}(1-\mu_{H}(K))=\frac{1}{2}+\frac{1}{2}\mu_{H}(K).$
∎
Note that the upper bounds in [5, Theorem 3.5] follow from Theorem 2.7 when we
consider a finite group with the counting measure on it. The lower bounds in
[5, Theorem 3.5] cannot be true in the infinite case, as the infinite dihedral
group shows.
###### Corollary 2.8.
Assume that $H$ is a closed subgroup of a nonabelian compact group $G$.
* i)
If $H\leq Z(G)$, then $d(H,G)=1.$
* (ii)
If $H\not\leq Z(G)$ and $H$ is abelian, then $d(H,G)\leq\frac{3}{4}.$
* (iii)
If $H\not\leq Z(G)$ and $H$ is nonabelian, then $d(H,G)\leq\frac{5}{8}.$
###### Proof.
(i). Obvious.
(ii). Since $H\not\leq Z(G)$, $K=H\cap Z(G)\not\leq H$. As in the proof of
Theorem 2.4 (ii), we have $\mu_{H}(K)\leq\frac{1}{4}$. Theorem 2.5 (ii)
implies $d(H,G)\leq\frac{1}{2}+\frac{1}{2}(\frac{1}{4})=\frac{3}{4}$.
(iii). We know from Theorem 2.6 and [10] that $d(H,G)\leq
d(H)\leq\frac{5}{8}$. ∎
Note that [5, Theorem 3.6] follows from Corollary 2.8 when we consider a
finite group with the counting measure on it.
###### Corollary 2.9.
Let $A$ and $B$ be two closed subgroups of a compact group $G$ such that
$A\leq B$. Then $d(A,B)\geq d(A,G)\geq d(B,G).$
###### Proof.
As in the proof of Theorem 2.6, the condition
$|A:C_{A}(x)|\leq|B:C_{B}(x)|\leq|G:C_{G}(x)|$
implies the condition
$\mu_{A}(C_{A}(x))\geq\mu_{B}(C_{B}(x))\geq\mu_{G}(C_{G}(x))$ for every
element $x$ of $G$. Integrating and keeping in mind Lemma 2.1, we have
$d(A,B)=\int_{A}\mu_{B}(C_{B}(x))d\mu_{A}(x)\geq$
$d(A,G)=\int_{A}\mu_{G}(C_{G}(x))d\mu_{A}(x)\geq\int_{B}\mu_{G}(C_{G}(x))d\mu_{B}(x)=d(B,G).$
∎
Note that [5, Theorem 3.7] follows from Corollary 2.8 when we consider a
finite group with the counting measure on it. We recall to convenience of the
reader [5, Lemma 3.8].
###### Lemma 2.10.
Let $H$ and $N$ be two closed subgroups of $G$ such that $N\leq H$ and $N$ is
normal in $G$. Then $C_{H}(x)N/N\leq C_{H/N}(xN)$ for every element $x$ of
$G$. Moreover, the equality holds if $N\cap[H,G]$ is trivial.
Then we may formulate another interesting dominance condition as follows.
###### Theorem 2.11.
Let $H$ and $N$ be two closed subgroups of a compact group $G$ such that
$N\leq H$ and $N$ is normal in $G$. Then $d(H,G)\leq d(H/N,G/N)d(N).$ In
particular, the equality holds if $N\cap[H,G]$ is trivial.
###### Proof.
Consider $S=\\{g\in G\ |\ |H:C_{H}(g)|\textrm{ is finite}\\}$. We have
$d(H,G)=\int_{G}\mu_{H}(C_{H}(g))d\mu_{G}(g)=\int_{S}\mu_{H}(C_{H}(g))d\mu_{G}(g)$
$=\int_{S}\frac{\mu_{H}(C_{H}(g)N)}{|C_{H}(g)N:C_{H}(g)|}d\mu_{G}(g)=\int_{S}\mu_{H}(C_{H}(g)N)\mu_{N}(C_{N}(g))d\mu_{G}(g).$
In the last equality we have used the argument just before Theorem 2.4 and the
fact that $|C_{H}(g)N:C_{H}(g)|$ is finite, getting
$|C_{H}(g)N:C_{H}(g)|=|N:C_{H}(g)\cap N|=\frac{1}{\mu_{N}(C_{N}(g))}.$
Now we get:
$d(H,G)\leq\int_{G}\mu_{H}(C_{H}(g)N)\mu_{N}(C_{N}(g))d\mu_{G}(g)$
$=\int_{\frac{G}{N}}\left(\int_{N}\mu_{H}(C_{H}(gx)N)\mu_{N}(C_{N}(gx))d\mu_{N}(x)\right)d\mu_{G/N}(gN).$
By Lemma 2.10,
$\mu_{H}(C_{H}(gx)N)=\mu_{\frac{H}{N}}\Big{(}\frac{C_{H}(gx)N}{N}\Big{)}\leq\mu_{\frac{H}{N}}(C_{\frac{H}{N}}(gN))$,
then
$d(H,G)\leq\int_{\frac{G}{N}}\mu_{G/N}(C_{G/N}(gN))\left(\int_{N}\mu_{N}(C_{N}(gx))d\mu_{N}(x)\right)d\mu_{G/N}(gN).$
On another hand,
$C_{2}=\\{(x,y)\in N\times N\ |\ [gx,y]=1\\}=\\{(x,y)\in N\times N\ |\ gx\in
C_{G}(y)\cap gN\\}.$
If $x_{0}\in C_{G}(y)\cap gN\neq\varnothing$, then either $gN=g_{0}N$ or
$g=g_{0}t$ for some $t\in N$, whence $C_{G}(y)\cap gN=g_{0}(C_{G}(y)\cap
N)=g_{0}C_{N}(y)$ and so
$C_{2}=\\{(x,y)\in N\times N\ |\ gx\in g_{0}C_{N}(y)\\}=\\{(x,y)\in N\times N\
|\ x\in tC_{N}(y)\\}.$
Therefore
$\int_{N}\mu_{N}(C_{N}(gx))d\mu_{N}(x)\leq\int_{N}\mu_{N}(tC_{N}(y))d\mu_{N}(y)$
$=\int_{N}\mu_{N}(C_{N}(y))d\mu_{N}(y)=d(N).$
Hence
$d(H,G)\leq
d(N)\int_{\frac{G}{N}}\mu_{\frac{H}{N}}(C_{\frac{H}{N}}(gN))d\mu_{G/N}(gN)=d(N)d(H/N,G/N).$
In particular, if $N\cap[H,G]=1$, then $C_{H}(g)=C_{H}(g)N$ and so
$\mu_{H}(C_{G}(g))=\mu_{H}(C_{H}(g)N)$ for all $g\in G.$ Furthermore, we have
$\mu_{H}(C_{H}(gn)N)=\mu_{G/N}\Big{(}\frac{C_{H}(gn)N}{N}\Big{)}=\mu_{G/N}(C_{H/N}(gN)).$
Therefore each inequality becomes equality and so $d(H,G)=d(H/N,G/N)d(N)$. ∎
## 3\. Relative n–th commutativity degree
The present section is devoted to extend some results of Section 2. For
instance, the next statement extends the upper bound in Theorem 2.6.
###### Theorem 3.1.
If $H$ is a closed subgroup of a compact group $G$, then
$d^{(n)}(H,G)\leq d^{(n)}(H).$
###### Proof.
We may argue as in the proof of Theorem 2.6 in order to get
$\mu_{H}(C_{H}([h_{1},...,h_{n}]))\geq\mu_{G}(C_{G}([h_{1},...,h_{n}])).$
Integrating over $H$ and keeping in mind Lemma 2.1, we have
$d^{(n)}(H)=\int_{H}\ldots\left(\int_{H}\mu_{H}(C_{H}([h_{1},...,h_{n}]))d\mu_{H}(h_{1})\right)\ldots
d\mu_{H}(h_{n})$
$\geq\int_{H}\ldots\left(\int_{H}\mu_{G}(C_{G}([h_{1},...,h_{n}]))d\mu_{H}(h_{1})\right)\ldots
d\mu_{H}(h_{n})=d^{(n)}(H,G).$
∎
Note that Theorem 3.1 informs us that the sequence $\\{d^{(n)}(H,G)\\}_{n\geq
1}$ is increasing for any compact group $G$ and any closed subgroup $H$ of
$G$.
The evidences of the finite case and the considerations of many situations in
the infinite case can be summarized in the following result.
###### Theorem 3.2.
If $H$ is a closed subgroup of a compact group $G$ and $K=H\cap Z(G)$, then
$d^{(n+1)}(H,G)\leq\frac{1}{2}\Big{(}1+d^{(n)}(H/K)\Big{)}.$
###### Proof.
Let $A=\\{(h_{1},...,h_{n+1})\in H^{n+1}\ |\ [h_{1},...,h_{n+1}]\in Z(G)\cap
H\\}$ and $B=H^{n+1}-A$. Then
$d^{(n+1)}(H,G)=\int_{H^{n+1}}\mu_{G}([h_{1},...,h_{n+1}])d(\mu_{H})^{n+1}$
$=\int_{A}\mu_{G}([h_{1},...,h_{n+1}])d(\mu_{H})^{n+1}+\int_{B}\mu_{G}([h_{1},...,h_{n+1}])d(\mu_{H})^{n+1}$
$\leq\mu_{H}^{n+1}(A)+\frac{1}{2}\mu_{H}^{n+1}(B)\leq\mu_{H}^{n+1}(A)+\frac{1}{2}(1-\mu_{H}^{n+1}(A))=\frac{1}{2}(1+\mu_{H}^{n+1}(A)).$
On the other hand,
$\mu_{H}^{n+1}(A))=\int_{H}...\int_{H}\mu_{\frac{H}{K}}(C_{\frac{H}{K}}([\bar{h_{1}},...,\bar{h_{n}}]))d\mu_{H}(h_{1})...d\mu_{H}(h_{n})$
$=\int_{H}...\left(\int_{\frac{H}{K}}\int_{K}\mu_{\frac{H}{K}}(C_{\frac{H}{K}}([\bar{h_{1}},...,\bar{h_{n}}]))d\mu_{K}(k)d\mu_{H}(\bar{h_{1}})\right)...d\mu_{H}(h_{n})$
$=\int_{H}...\left(\int_{\frac{H}{K}}\mu_{\frac{H}{K}}(C_{\frac{H}{K}}([\bar{h_{1}},...,\bar{h_{n}}]))d\mu_{H}(\bar{h_{1}})\right)...d\mu_{H}(h_{n})$
$=\int_{\frac{H}{K}}...\int_{\frac{H}{K}}\mu_{\frac{H}{K}}(C_{\frac{H}{K}}([\bar{h_{1}},...,\bar{h_{n}}]))d\mu_{\frac{H}{K}}(\bar{h_{1}})...d\mu_{\frac{H}{K}}(\bar{h_{n}})=d^{(n)}(H/K).$
and the result follows.∎
Note that [5, Theorem 4.3] follows from Theorem 3.2 when we consider a finite
group with the counting measure on it. Note that Theorem 3.2 is true also for
groups of the form $A_{i}\times B_{j}$, where $A_{i}$ is a compact abelian
(infinite) group, $B_{j}$ is a finite group, $i\in I$ and $j\in J$.
###### Corollary 3.3.
If $G$ is a compact group, then
$d^{(n+1)}(G)\leq\frac{1}{2}(1+d^{(n)}(G/Z(G))).$
###### Proof.
This follows from Theorem 3.2 with $H=G$. ∎
It is possible to bound $d^{n+1}(G)$ as follows.
###### Theorem 3.4.
If $G$ is a compact group, then
$d^{(n+1)}(G)\leq\frac{1}{2^{n}}(2^{n}-1+d(G/Z_{n}(G))).$
###### Proof.
We may repeat the proof of [5, Theorem 4.5], since we do not need that $G$ is
finite. We should only note that $Z_{n}(G)/Z(G)=Z_{n-1}(G/Z(G))$ is a closed
subgroup of $G/Z(G)$. ∎
Note that [5, Theorem 4.5] follows from Theorem 3.4 when we consider a finite
group with the counting measure on it. Furthermore Theorem 3.4 is true for
compact groups of the form $A\times B$, where $A$ is a compact abelian
(infinite) group and $B$ is a finite group. In such a case [5, Theorem 4.5]
cannot be applied.
###### Theorem 3.5.
Let $H$ and $N$ be two closed subgroups of $G$ such that $N\leq H$ and $N$ is
normal in $G$. Then $d^{(n)}(H,G)\leq d^{(n)}(H/N,G/N).$ In particular, the
equality holds if $N\cap[_{n}H,G]$ is trivial.
###### Proof.
Let $\lambda$, $\mu$ and $\nu$ be corresponding Haar measures on $N$, $G$ and
$G/N$ respectively. Consider $S=\\{(h_{1},...,h_{n})\ |\
|G:C_{G}([h_{1},...,h_{n}])|\textrm{ is finite}\\}$. Then
$d^{(n)}(H,G)=\int_{H^{n}}\mu_{G}(C_{G}([h_{1},...,h_{n}]))d\mu^{n}_{H}=\int_{S}\mu_{G}(C_{G}([h_{1},...,h_{n}]))d\mu^{n}_{H}$
$=\int_{S}\frac{\mu_{G}(C_{G}([h_{1},...,h_{n}])N)}{|C_{G}([h_{1},...,h_{n}])N:C_{G}([h_{1},...,h_{n}])|}d\mu^{n}_{H}$
$=\int_{S}\mu_{G}(C_{G}([h_{1},...,h_{n}])N)\mu_{N}(C_{G}([h_{1},...,h_{n}]))d\mu^{n}_{H}$
$\leq\int_{H^{n}}\mu_{G}(C_{G}([h_{1},...,h_{n}])N)\mu_{N}(C_{G}([h_{1},...,h_{n}]))d\mu^{n}_{H}$
$=\int_{\frac{H}{N}}\int_{N}...\int_{\frac{H}{N}}\int_{N}\mu_{G}(C_{G}([h_{1}a_{1},...,h_{n}a_{n}])N)\mu_{N}(C_{G}([h_{1}a_{1},...,h_{n}a_{n}]))$
$d\mu_{N}(a_{1})d\mu_{\frac{H}{N}}(h_{1}N)...d\mu_{N}(a_{n})d\mu_{\frac{H}{N}}(h_{n}N).$
On the other hand,
$\mu_{G}(C_{G}([h_{1}a_{1},...,h_{n}a_{n}])N)=\mu_{\frac{G}{N}}(\frac{C_{G}([h_{1}a_{1},...,h_{n}a_{n}])N}{N})$
$\leq\mu_{\frac{G}{N}}(C_{\frac{G}{N}}([h_{1}N,...,h_{n}N])).$
Therefore
$d^{(n)}(H,G)\leq\int_{\frac{H}{N}}...\int_{\frac{H}{N}}\mu_{G/N}(C_{\frac{G}{N}}([h_{1}N,...,h_{n}N]))$
$\int_{N}...\int_{N}\mu_{N}(C_{G}([h_{1}a_{1},...,h_{n}a_{n}]))d\mu_{N}(a_{1})d...d\mu_{N}(a_{n})\mu_{\frac{H}{N}}(h_{1}N)...d\mu_{\frac{H}{N}}(h_{n}N)$
$\leq\int_{\frac{H}{N}}...\int_{\frac{H}{N}}\mu_{\frac{H}{N}}(C_{\frac{G}{N}}([h_{1}N,...,h_{n}N]))\mu_{\frac{H}{N}}(h_{1}N)...d\mu_{\frac{H}{N}}(h_{n}N)=d^{(n)}(\frac{H}{N},\frac{G}{N})$
∎
An easy consequence is the following.
###### Corollary 3.6.
If $N$ is a closed normal subgroup of a compact group $G$, then
$d^{(n)}(G)\leq d^{(n)}(G/N).$
###### Proof.
This follows from Theorem 3.5 with $H=G$. ∎
## 4\. Weakening some bounds
In the present section we will give some upper and lower bounds for
$d^{(n)}(G)$ and $d^{(n)}(H,G)$ by means of the results which have been
previously found.
###### Corollary 4.1.
If G is a compact group which is not nilpotent of class at most $n$, then
$d^{(n)}(G)\leq\frac{2^{n+2}-3}{2^{n+2}}.$
###### Proof.
$G/Z_{n-1}(G)$ is a nonabelian group by the assumptions. From [10],
$d(G/Z_{n-1}(G))\leq\frac{5}{8}.$
Now Theorem 3.4 gives
$d^{(n)}(G)\leq\frac{1}{2^{n-1}}(2^{n-1}-1+\frac{5}{8})=\frac{2^{n+2}-3}{2^{n+2}}.$∎
Note that [5, Theorem 5.1] follows from Corollary 4.1 when we consider a
finite group with the counting measure on it.
###### Corollary 4.2.
If G is a nontrivial compact group with trivial center, then
$d^{(n)}(G)\leq\frac{2^{n}-1}{2^{n}}.$
###### Proof.
Of course, $Z_{n}(G)$ is trivial for each $n\geq 1$. Thus $G$ is a
nonnilpotent group. In particular $\mu_{G}(Z(G))=0$ so Theorem 2.7 (i) implies
$d(G)\leq\frac{1}{2}.$ Now the result follows by Theorem 3.4 by induction on
$n$. ∎
Our last result improves Corollary 2.8 and extends [5, Theorem 5.5].
###### Theorem 4.3.
Assume that $H$ is a proper closed subgroup of a nonabelian compact group $G$
such that $n\geq 1$ and $K=Z(G)\cap H$.
* (i)
If $H\leq Z_{n}(G)$, then $d^{(n)}(H,G)=1.$
* (ii)
If $H\not\leq Z_{n}(G)$ and $H/K$ is a nilpotent group of class at most
$n$-$1$, then $d^{(n)}(H,G)=1.$
* (iii)
If $H\not\leq Z(G)$ and $H/K$ is a nonnilpotent group of class at most
$n$-$1$, then
$d^{(n)}(H,G)\leq\frac{2^{n+2}-3}{2^{n+2}}.$
###### Proof.
(i). This is obvious.
(ii). Of course, if $H/K$ is nilpotent of class at most $n$-$1$, then
$[\bar{x}_{1},\ldots,\bar{x}_{n}]=K$ for some elements
$\bar{x}_{1},\ldots,\bar{x}_{n}$ of $H/K$. Therefore,
$G=C_{G}([\bar{x}_{1},\ldots,\bar{x}_{n}])$ and we may argue as in Theorem
3.2, getting $d^{(n)}(H,G)=1.$
(iii). Using Corollary 4.1 and the fact that $H/K$ is nonnilpotent of class at
most $n$-$1$, we get $d^{(n-1)}(H/K)\leq\frac{2^{n+1}-3}{2^{n+1}}$. By Theorem
3.2, we get
$d^{(n)}(H,G)\leq\frac{1}{2}(1+d^{(n-1)}(H/K))\leq\frac{1}{2}(1+\frac{2^{n+1}-3}{2^{n+1}})=\frac{2^{n+2}-3}{2^{n+2}}.$
∎
## References
* [1] K. Chiti, M.R.R. Moghaddam and A.R. Salemkar, n–isoclinism classes and n–nilpotency degree of finite groups, Algebra Colloq. 12 (2005), 225–261.
* [2] A.K. Das and R.K. Nath, On the generalized relative commutative degree of a finite group, Int. Electr. J. Algebra 7 (2010), 140–151.
* [3] A.K. Das and R.K. Nath, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo 59 (2010), 137–142.
* [4] A. Erfanian and R. Kamyabi–Gol, On the mutually $n$-tuples in compact groups, Int. J. Algebra 6 (2007), 251–262.
* [5] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra 35 (2007), 4183–4197.
* [6] A. Erfanian and R. Rezaei, On the commutativity degree of compact groups, Arch. Math. (Basel) 93 (2009), 201–212.
* [7] A. Erfanian and F.G. Russo, Probability of mutually commuting $n$-tuples in some classes of compact groups, Bull. Iran. Math. Soc. 24 (2008), 27–37.
* [8] A. Erfanian and F.G. Russo, Isoclinism in probability of commuting n–tuples, Ital. J. Pure Appl. Math. 25 (2009), 27–36.
* [9] A. Erfanian and F.G Russo, Detecting the commuting probability of the derived subgroup, Int. J. Math. Game Theory Algebra 19 (2010), 7–14.
* [10] W.H. Gustafson, What is the probability that two groups elements commute?, Amer. Math. Monthly 80 (1973), 1031–1304.
* [11] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Springer Verlag, New York, 1963.
* [12] P. Lescot, Sur certains groupes finis, Rev. Math. Spéciales 8 (1987), 276–277.
* [13] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177 (1995), 847–869.
* [14] P. Lescot, Central extensions and commutativity degree, Comm. Algebra 29 (2001), 4451–4460.
* [15] R. Rezaei and F.G. Russo, A note on relative isoclinism classes of compact groups, Proceedings of the conference Ischia Group Theory 2008, 233–235, Eds. P. Longobardi et al., World Scientific Press, Singapore, 2009.
* [16] F.G. Russo, A probabilistic meaning of certain quasinormal subgroups, Int. J. Algebra 1 (2007), 385–392.
|
arxiv-papers
| 2009-10-25T13:05:45 |
2024-09-04T02:49:06.043387
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rashid Rezaei (University of Malayer, Malayer, Iran) and Francesco G.\n Russo (Universita' degli Studi di Palermo, Palermo, Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/0910.4716"
}
|
0910.4793
|
# Unbroken supersymmetry, without new particles
Alejandro Rivero Institute for Biocomputation and Physics of Complex Systems,
University of Zaragoza, Spain al.rivero@gmail.com
###### Abstract
We consider Koide’s equation for charged leptons jointly with the hypotheses
implied in the ancient Dual Quark-Gluon model. The focus is the possibility of
motivating a supersymmetric framework akin to the one of D=11 maximal
supergravity.
The dual Quark-Gluon Model of Hadrons, where mesons and quarks are in equal
footing, was proposed early schw1 ; schw2 as a consequence of Ramond model of
fermionsramond . This model was obvious if one is willing to encompass all
composite QCD strings (mesons and diquarks) and fermions (baryions) in a
superstring model, as the emission of a diquark barionic string from a baryion
will produce a single quark state.
It was forgotten after the discovery of supergravity and the reinterpretation
of dual models as a theory of gravity. Also, the theoretical dificulties of
working with a model both elementary and composite of itself was evident, and
it was advocated to see it as ”a quark model without quarks”, neglecting the
fermion side. As it has been told elsewherearivero , the model only closes on
itself for three generations and one highly massive top quark, or for more
than three generations with an absurd number of massive tops and bottoms. Back
in 1971, there was no experimental support for a third generation, so this
prediction (never written down nor proposed, as far as I know) was a fail of
the model.
## I Koide’s model of subquarks.
In 1981, Y. Koide koide suggested some models where quarks and leptons were
composed of more elementary bosons and fermions having global $SU(3)$ color
and $SU(3)$ ”generations” symmetry. With an adequate choosing of the
representations for these subparticles, plus a relatively adhoc symmetry
breaking scheme, it was possible to derive some parameters of the CKM matrix
and, more interestingly, to predict the mass of the tau lepton which was
measured years later. The prediction has today an astonishing precision within
the current experimental error.
Koide’s symmetry breaking scheme constrains the square root of the mass matrix
$M_{l}$ of charged leptons. When $M^{1/2}_{l}$ is decomposed in a central part
$U$ (a multiple of the identity) plus a traceless part $V$,
$(U+V)^{2}=M_{l}$ (1)
the symmetry breaking scheme imposes a relationship between traces
$Tr[U^{2}]-Tr[V^{2}]=0$ (2)
And it works: with PDG 2009 data, the LHS is between -0.05 and 0.09 MeV. Or,
if you prefer an adimensional quantity, the quotient $Tr[U^{2}]/Tr[V^{2}]$ is
$1.00002\pm 0.00008$.
Recently, Koide has produced some ways to the same formula without asking for
compositeness, but we keep an eye on it because of the next section. It must
be stressed that in 1981 the value of the mass of the tau lepton was far away
from its current measurement.
I want to do two observations here:
1) If we think of mass as a component of the momentum operator $P_{\mu}$, the
fact of having a condition in its square root smells to a condition on
supersymmetry generators.
2) If supersymmetry were unbroken, the same condition should appear in the
scalar partners of the charged fermions. Furthermore, we could think that the
sfermions are composite. Even we could think that the sfermions are the only
composites, where Koide’s breaking scheme applies, and that Koide’s formula in
the leptons is only a reflection of the actual formula for sfermions.
## II sBootstrap.
A motivation to pursue the above idea is that the masses of muon and tau,
which are free parameters in the standard model, are very near of similarly
charged (pseudo)scalar particles, the pion and the D meson. In fact the former
case is so near than historically the muon was first interpreted, mistakenly,
as a pion.
We advance one step over Koide’s models of subquarks and do a bold suggestion:
that the particles labeling the sfermions are really quarks themselves.
Astonishingly we exceed expectations when we notice that there are actually
five ”light” quarks, on the QCD-Chiral-Electromagnetic mass scales, and a
massive quark, in the electroweak mass scale. And this quark is as massive as
to be unable to bind into mesons. So the flavour global symmetry of our
Standard Model quarks is SU(5). It can be labeled under SU(3)xSU(2) in order
to separate the (d,s,b) and (u,c) kinds of charge.
The decomposition of SU(5) is well known. Take the 24 of
$5\otimes\bar{5}=24\oplus 1$ and the $15$ of $\bar{5}\otimes\bar{5}=15\oplus
10$. Then under $SU(5)\supset SU(2)\times SU(3)\times U(1)$ we have
$\displaystyle 15$ $\displaystyle=$ $\displaystyle(3,1)(6)+(2,3)(1)+(1,6)(-4)$
(3) $\displaystyle 24$ $\displaystyle=$
$\displaystyle(1,1)(0)+(3,1)(0)+(2,3)(-5)+(2,\bar{3})(5)+(1,8)(0)$ (4)
The $(2,3)(1)$ and $(1,6)(-4)$ are the partners of quarks down and up
respectively. The antiparticles are provided in the $\bar{15}$ representation.
The $(2,3)(-5)$ and $(2,\bar{3})(5)$ are the partners of positron and
electron. The other 12 particles of this multiplet are neutral.
So our arivero basic observation here is that
3) For three generations and a single ”massive” quark, the system closes on
itself: the degrees of freedom generated in the product are the ones needed
for the sfermions of a supersymmetric standard model, which in turn are
transformed by susy into the original fermions we need to generate them.
It is not possible to do the same trick with any other number of generations,
so in some sense the sBootstrap fixes the number of generations.
A major objection against the sBootstrap mechanism is that, having paired the
mesons from QCD with the elementary quarks and leptons, we should expect to
detect an string structure in the electron spin and charge, which we do no
detect. On the other hand, it is more satisfactory to think that there is some
principle forbidding fermions to have extended non-pointset structure, that to
think that string theory has provided us with a local extended model of the
electron. It could be worth to remember (Susskind et al.) that the initial
research in dual models run into problems when calculating the sizes of
hadronic strings.
## III D=11 supergravity multiplet.
There are some general motivations to look for susy in D=11 instead of down-
to-earth in D=4. First, it is well known that the minumum number of extra
dimensions to allow for SU(3)xSU(2)xU(1) symmetry is 7. It is known that
superstring theory lives in D=10 and develops an extra dimension in some
limits. It is less known that Connes’ geometric version of the standard model
lives in dimension 2 mod 8.
Superstring theory is a motivation for us from the point of view of our
composites: we really would like to interpret each of our sfermions as quarks
at the ends of a string. Really we know from Sagnotti and Marcus that the
SO(32) symmetry of the quantised superstring is produced with only a set of 5
“fermion flavour” in the worldsheet.
Also there is an intriguing “reggeization” of the process of $Z^{0}$ decay,
which happens to show a total rate coinciding with the scaling of
electromagnetic decay of QCD strings (refer to the dimensionless quantity
$\Gamma^{e.m.}_{X}/m^{3}_{X}$, for X a QCD meson).
Supergravity is a motivation because its minimal multiplet has barely the
number of degrees of freedom to store the information of the supersymmetric
standard model, except for the Higgs. See boya : after accounting for all the
standard model, their gauge forces and their superparners, plus the 2 degrees
of freedom of the 4-D gravitino, only 6 scalars, from the 128 of the full
multiplet, are left. And we need 8 for the Higgs of the MSSM. Yet, the Higgs
mechanism is still undiscovered. Or, the graviton could be exiled in favour of
a 4D, $N=4\times N=4$, superYangMils theory.
D=11 supergravity has a natural way to produce D=4 because of the structure of
supersymmetry multiplets. Its 128-component fermion can not be matched with a
44 component graviton only, so an extra field is needed to hold the extant 84
bosons. This field happens to be a tensor with three indices and thus induces
a 4D uncompactified sector in some models.
Amusingly, we could be interested on two numerological happenings of the
number ”84” in the standard model:
\- the ”charged fermions” of the standard model amount to 84 degrees of
freedom (plus 12 neutrinos = 96). They could be organised in three families of
multiplets of 28 components each.
\- the ”light fermions” of the standard model amount to 84 degrees of freedom
(plus 3 colors of the top quark = 96). They could be organised in two chiral
multiplets of 42 components each, or four multiplets of 21 components.
The first realisation seems easier to look for. Barring the neutrinos, we
expect a non chiral theory. Strings of type IIA, which are known to recover
11D supergravity in the strong coupling regime, have a 28 in the NS-NS sector
and a 56 in the R-R sector, coming both from the decomposition of the 11D
3-form.
If we refine down to 9D via compactification in $S^{1}$, and even having the
fact that 9D is non-chiral, we have some scent of the organisation on
21-plets: the 28 decomposes in a 21 + 7, and the 56 in 21+35. Of course we
want the 9D theory to be non chiral: it has the minimum number of extra
dimensions required to fit in Kaluza Klein a SU(3)xU(1) theory, which is the
non-chiral part of the standard model.
In the standard model, the mass of the top quark seems linked to the EW
breaking scale, having an experimental yukawa coupling equal to one, within a
few percent. So if we take the EW scale to infinity, completely breaking the
group, then also the top is removed from the scenary. This provides some
motivation to look for the second realisation.
A more than bold conjecture should be that this second realisation hides in
IIB. Being chiral, IIB theory does not descent from 11D SUGRA. The 84 degrees
of freedom are hidden in the sum of a 28 in the NS-NS sector and a 28 plus a
35 in the R-R sector, ie two 2-forms and one self-dual 4-form. To produce them
explicitly, it is needed to consider the 9D theory and to discard a mix of the
9th component of both 2-forms, accounting for 7 components. In this way we are
left with a 35, two 21s, and the surviving 7 components of the mix. Of course,
noticing that IIA amounts to include the 7 components from the R-R sector, in
this case we could be tempted to unmix and directly exclude them, taking the
ones from the NS-NS sector.
What is left to find is a way to produce the SU(2) group in string theory. It
could be time to revisit the symmetry enhancement condition which happens when
the compactifyied radius is equal to the string scale. Fixing
$R=\alpha^{\prime 1/2}$ and moving R to zero needs a careful consideration of
the limit process, because at the same time we enter into the strong coupling
regime, where IIA grows an extra dimension.
## IV Koide’s relationship for mesons
We can think that the supersymmetry in the bosonic part has got an extra
contribution from the breaking of flavour symmetry. So lets see what happens
if we restore some of this breaking.
If we set the masses of the bottom and strange quark to the same value that
the mass of the down quark, we are left with only three mass levels: 1870 MeV,
139.5 MeV and 0. The zero level should be inhabited by the “$\eta_{1}$
singlet” combinations of $\pi^{-},K^{-},B^{-}$ and
$D^{-},D^{-}_{s},B^{-}_{c}$, while the other levels are inhabited by the
“$\eta_{8},\pi_{8}$” combinations.
SUSY, of course, is still slightly broken, if we compare with the lepton
triple (0.511, 105.66, 1777 MeV). But Koide’s relationship fares better, and
the quotient $Tr[U^{2}]/Tr[V^{2}]$ is about 1.005. It seems that the breaking
of SUSY aims to preserve Koide’s, contrary to our initial expectation (or,
again, exceeding it)
If instead restoring SU(3)-down flavour we restore SU(2)-up flavour (or none
at all) we still have that the masses of Kaon and Upsilon, jointly with a zero
mass, also form a Koide “triple”. In fact there are no more triples with a
zero mass.
In the neutral case the fit is not so good because the Kaon is a highly mixed
particle. The $\eta_{8}$ of the classical SU(3) “u,d,s” flavour group is a
better candidate.
Other triples in the meson and baryon sectors have been explored by Carl
Brannen brannen .
## V Puzzling chirality
Of course, there are no chiral fermions in D=11, thus no way to implement
SU(2) in the way it works in the standard model. It seems that the chiral part
of the standard model, $SU(2)\times U(1)_{ew}$, interpolates, as we move the
mass of W from zero to infinity, between two non-chiral theories in D=9 and
D=11.
The neutrinos in the 24 of SU(5) appear in a very irregular way: a triplet, a
singlet and an octuplet. We need to mix them if we want to recover a grouping
in three generations. From the point of view of composites $D\bar{U}$, the
charged leptons can contemplate two ways to decay to neutrinos: either a decay
$D$ to $U$, landing in the triplet, or a decay $\bar{U}$ to $\bar{D}$, landing
in the combination of octet and singlet. So perhaps the (3,1) triplet of the
$24$ has some special role to build chiral interactions.
The (3,1) triplet from the $15$, and its antitriplet from the $\bar{1}5$, are
the only predicted particles not in the standard model. They should have
electric charge 4/3 so it seems that we can not accomodate them as the
partners of (three generations of) a two-component fermion. But we can not
arrange them in Dirac fermions, as they amount to 6 degrees of freedom, or 18
if we consider that they should be coloured as the rest of the $15$ multiplet.
More, if they are coloured they overcrown the D=11 multiplet. So it seems that
the (3,1) triplet in $15$ asks for a chiral fermion in each family, but the
sBootstrap, charge, and other considerations, ask for this chiral fermion to
dissapear.
## References
* (1) L. J. Boya, “Supersymmetry and Polytopes,” AIP Conf. Proc. 1093 (2009) 28 [arXiv:0808.3667 [hep-th]].
* (2) http://www.brannenworks.com/koidehadrons.pdf
* (3) http://dorigo.wordpress.com/2007/10/16/guest-post-alejandro-rivero-sbootstrap/
* (4) Y. Koide, “A New Formula For The Cabibbo Angle And Composite Quarks And Leptons,” Phys. Rev. Lett. 47 (1981) 1241.
* (5) Y. Koide, “Quark And Lepton Masses Speculated From A Subquark Model,” (1981)
* (6) A. Neveu and J. H. Schwarz, Phys. Rev. D 4 (1971) 1109.
* (7) P. Ramond, Phys. Rev. D 3 (1971) 2415.
* (8) A. Rivero, “Supersymmetry with composite bosons,” [arXiv:hep-ph/0512065].
* (9) J. H. Schwarz, Phys. Lett. B 37 (1971) 315.
|
arxiv-papers
| 2009-10-26T02:17:13 |
2024-09-04T02:49:06.050240
|
{
"license": "Public Domain",
"authors": "Alejandro Rivero (Universidad de Zaragoza)",
"submitter": "Alejandro Rivero",
"url": "https://arxiv.org/abs/0910.4793"
}
|
0910.4936
|
# Annihilation cross sections and interaction couplings of the dark matter
candidates in the warped and flat extra dimensions
E. O. Iltan
Physics Department, Middle East Technical University
Ankara, Turkey
E-mail address: eiltan@newton.physics.metu.edu.tr
###### Abstract
We consider a scenario with an additional scalar standard model singlet
$\phi_{S}$, living in a single extra dimension of the RS1 background. The zero
mode of this scalar which is localized in the extra dimension is a dark matter
candidate and the annihilation cross section is strongly sensitive to its
localization parameter. As a second scenario, we assume that the standard
model Higgs field is accessible to the fifth flat extra dimension. At first we
take the additional standard model singlet scalar field as accessible to the
sixth extra dimension and its zero mode is a possible dark matter candidate.
Second, we consider that the new standard model singlet, the dark matter
candidate, lives in four dimensions. In both choices the KK modes of the
standard model Higgs field play an observable role for the large values of the
compactification radius $R$ and the effective coupling $\lambda_{S}$ is of the
order of $10^{-2}-10^{-1}$ ($10^{-6}$) far from (near to) the resonant
annihilation.
The missing matter which is required holds almost $23\%$ [1, 2, 3] of present
Universe and it is called dark matter (DM) since it is not detectable by the
radiation emitted. The evidence of the existence of DM comes from numerous
observations: the galactic rotation curves [4], galaxies orbital velocities
[5], the cosmic microwave background anisotrophy [6], observations of type Ia
supernova [3]. However the nature of DM is still a mystery. The DM problem can
not be solved in the framework of the standard model (SM) and it is inevitable
to search new physics beyond in order to provide a dark matter candidate. In
the literature, there are many studies which are based on the models beyond
the SM in order to understand the nature of DM; DM in the framework of the
supersymmetric models [7], the universal and non universal extra dimension
(UED and NUED) models [8]-[21], the split UED models [22, 23, 24], the Private
Higgs model [25], the Inert doublet model [26]-[33], the Little Higgs model
[34], the Heavy Higgs model [35]. The common idea is that a large amount of DM
is in the class of nonrelativistic cold DM and the Weakly Interacting Massive
Particles (WIMPs) belong to this class. WIMPs, having masses in the range $10$
GeV- a few TeV, involve in the weak and gravitational interactions and they
are stable in the sense that they do not decay in to SM particles and they
play a crucial role in the structure formation of Universe. On the other hand
they disappear by pair annihilation (see for example [36, 37] for further
discussion). Notice that the stability of WIMPs are ensured by an appropriate
discrete symmetry, in various models (for details see for example [38] and
references therein).
From the experimental point of view there are two possibilities to detect the
DM candidate WIMP: The direct detection of DM, the search for the scattering
of DM particles off atomic nuclei within a detector, and the indirect one, the
search for products of WIMP annihilations. An upper limit of the order of
$10^{-7}-10^{-6}\,pb$ [39] for the WIMP-nucleon cross section has been
obtained in the direct detection experiments. On the other hand since the
current relic density could be explained by thermal freeze-out of their pair
annihilation the present DM abundance by the WMAP collaboration [40] leads to
the bounds for the annihilation cross section.
In the present work we study the annihilation cross section and the related
coupling of the DM candidates in the warped and flat extra dimensions. At
first, we consider that all SM particles live on the 4 dimensional brane and
there exists an additional scalar SM singlet $\phi_{S}$ which is accessible to
a single extra dimension in the RS1 background and its zero mode is a possible
DM candidate. As a second scenario we choose the flat extra dimension(s) where
the SM Higgs doublet, necessarily the gauge fields, are accessible to a single
extra dimension (the fifth one) with two possibilities: the additional SM
model singlet scalar field lives in the sixth extra dimension and its zero
mode is a possible DM candidate; the new SM singlet, the DM candidate, lives
in four dimensions.
DM as the zero mode of SM singlet $\phi_{S}$ which is accessible to a single
extra dimension in the RS1 background
The RS1 background111Here, the extra dimension, having two boundaries, the
hidden (Planck) brane and the visible (TeV) brane with opposite and equal
tensions, is compactified onto $S^{1}$ orbifold with the compactification
radius $R$. In this case the low energy effective theory has flat 4D
spacetime, even if the 5D cosmological constant is non vanishing. The gravity,
having an extension into the bulk with varying strength, is taken to be
localized on the hidden brane. [41, 42] is based on the curved extra dimension
with the metric
$\displaystyle
ds^{2}=e^{-2\,\sigma}\,\eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}-dy^{2}\,,$ (1)
where $\sigma=k\,|y|$, $k$ is the bulk curvature constant, the exponential
$e^{-\sigma}$, with $y=R\,|\theta|$, is the warp factor. Now we consider that
an additional SM singlet $\phi_{S}$ is accessible to the extra dimension. The
compactification of the extra dimension onto $S^{1}$ orbifold with radius $R$
results in the appearance of KK modes as
$\displaystyle\phi_{S}(x,y)=\sum_{n=0}^{\infty}\,\phi_{S}^{(n)}(x)\,f_{n}(y)\,,$
(2)
where the non-vanishing zero mode exists if the fine tuning of the parameters
(see [43, 44] and [45] for details)222There is another possibility of fine
tuning of the parameters $b$ and $a$ for the non-vanishing zero mode, namely
$b=2+\sqrt{4+a}$. However we ignore this choice since it is not appropriate
for our case since it leads to an effective coupling which is not valid for
the perturbative calculation. $b$ and $a$
$\displaystyle b=2-\sqrt{4+a}\,,$ (3)
is reached, and it reads
$\displaystyle
f_{0}(y)=\frac{e^{b\,k\,y}}{\sqrt{\frac{e^{2\,(b-1)\,k\,\pi\,R}-1}{(b-1)\,k}}}\,\,.$
(4)
If one respects the existence of the ad-hoc discrete $Z_{2}$ symmetry
$\phi_{S}\rightarrow-\phi_{S}$ and considers that $\phi_{S}$ has no vacuum
expectation value, the stability of the zero mode scalar $\phi^{(0)}_{S}$ is
guaranteed333The scalar field $\phi_{S}$ has no SM decay products. and it can
be taken as a DM candidate which disappears by pair annihilation with the help
of the exchange particle. The possible interaction which drives the pair
annihilation is
$\displaystyle{\cal{S}}_{Int}=\int
d^{5}x\sqrt{g}\,\Bigg{(}\lambda_{5\,S}\,(\phi_{S}^{(0)})^{2}\,(\Phi_{1}^{\dagger}\,\Phi_{1})\,\Bigg{)}\,\delta(y-\pi
R)\,\,\,,$ (5)
where $\Phi_{1}$ is the SM Higgs field
$\displaystyle\Phi_{1}=\frac{1}{\sqrt{2}}\left[\left(\begin{array}[]{c c}0\\\
v+H^{0}\end{array}\right)\;+\left(\begin{array}[]{c c}\sqrt{2}\chi^{+}\\\
i\chi^{0}\end{array}\right)\right]\,,$ (10)
with the vacuum expectation value
$\displaystyle<\Phi_{1}>=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c c}0\\\
v\end{array}\right)\,.$ (13)
Here the SM Higgs boson $H^{0}$ is the exchange particle and the pair
annihilation occurs after the electroweak symmetry breaking. In this part of
the work we will study the effects of the zero mode scalar localization
parameter $a$ and the curvature $k$ on the annihilation cross section.
DM as the SM singlet (or the zero mode of the SM singlet living in the sixth
flat extra dimension) and the Higgs field living in the fifth flat extra
dimension
At first, we assume that the SM Higgs doublet and the additional SM model
singlet scalar field are accessible to fifth and sixth extra dimension
respectively, however the SM fields, except gauge fields, live in four
dimensions. The compactification of the extra dimensions on $S_{1}\times
S_{1}$ with radii $R$ results in the expansion of the SM Higgs doublet
$\Phi_{1}$ (see eq.(10)) and the new SM singlet $\phi_{S}$ into their KK modes
as
$\displaystyle\Phi_{1}(x,y)$ $\displaystyle=$ $\displaystyle{1\over{\sqrt{2\pi
R}}}\left\\{\Phi_{1}^{(0)}(x)+\sqrt{2}\sum_{n=1}^{\infty}\Phi_{1}^{(n)}(x)\cos(ny/R)\right\\}\,,$
$\displaystyle\phi_{S}(x,y)$ $\displaystyle=$ $\displaystyle{1\over{\sqrt{2\pi
R}}}\left\\{\phi_{S}^{(0)}(x)+\sqrt{2}\sum_{n=1}^{\infty}\phi_{S}^{(n)}(x)\cos(nz/R)\right\\}\,,$
(14)
where $y$ and $z$ are the coordinates of the fifth and sixth extra dimensions.
Now, we consider the interaction of the additional scalar singlet with the SM
Higgs doublet as
$\displaystyle{\cal{L}}_{Int}=\Bigg{(}\lambda_{6\,S}\,\phi_{S}^{2}\,(\Phi_{1}^{\dagger}\,\Phi_{1})\,\Bigg{)}_{y=0,z=0}\,.$
(15)
After the electroweak symmetry one gets the interaction term
$\displaystyle{\cal{L^{\prime}}}_{Int}=\frac{\lambda_{6\,S}\,v}{(2\,\pi\,R)^{2}}\,(\phi_{S}^{(0)})^{2}\,\Bigg{(}H^{0\,(0)}+\sqrt{2}\,\sum_{n=1}\,H^{0\,(n)}\Bigg{)}\,,$
(16)
which is responsible for the the annihilation process of $\phi_{S}^{(0)}$
which we consider as a DM candidate. Here the zero mode and KK mode Higgs
fields are intermediate particles which carry the annihilation process. Notice
that the stability of the DM candidate under consideration is ensured by
respecting that the SM singlet scalar $\phi_{S}$, having no vacuum expectation
value, obeys the discrete $Z_{2}$ symmetry $\phi_{S}\rightarrow-\phi_{S}$.
Second, we consider that the SM Higgs doublet is accessible to fifth extra
dimension, however, the additional SM model singlet scalar field, the DM
candidate, lives in four dimensions. This is the case that the interaction of
the additional scalar singlet with the SM Higgs doublet reads
$\displaystyle{\cal{L}}_{Int}=\Bigg{(}\lambda_{5\,S}\,\phi_{S}^{2}\,(\Phi_{1}^{\dagger}\,\Phi_{1})\,\Bigg{)}_{y=0}\,,$
(17)
and the interaction term, which is responsible for the annihilation process,
becomes
$\displaystyle{\cal{L^{\prime}}}_{Int}=\frac{\lambda_{5\,S}\,v}{2\,\pi\,R}\,\phi_{S}^{2}\,\Bigg{(}H^{0\,(0)}+\sqrt{2}\,\sum_{n=1}\,H^{0\,(n)}\Bigg{)}\,,$
(18)
after the electroweak symmetry breaking. Similar to the previous case the zero
mode and KK mode Higgs fields play the role of intermediate particles which
drive the annihilation process. The stability of the DM candidate is ensured
with the above ad-hoc $Z_{2}$ symmetry and with vanishing vacuum expectation
value.
Now, we present the total averaging annihilation rate of DM which is obtained
by the annihilation process DM DM$\rightarrow H^{0}\,\rightarrow X_{SM}$
$\displaystyle<\sigma\,v_{r}>$ $\displaystyle=$
$\displaystyle\frac{4\,\lambda_{S}^{2}\,v^{2}}{m_{S}}\,\frac{1}{(4\,m_{S}^{2}-m_{H^{0}}^{2})^{2}+m^{2}_{H^{0}}\,\Gamma^{2}_{H^{0}}}\,\Gamma(\tilde{h}\rightarrow
X_{SM})\,,$ (19)
where $\Gamma(\tilde{h}\rightarrow
X_{SM})=\sum_{i}\,\Gamma(\tilde{h}\rightarrow X_{i\,SM})$ with virtual Higgs
$\tilde{h}$ having mass $2\,m_{S}$ (see [46, 47]) and
$v_{r}=\frac{2\,p_{CM}}{m_{S}}$ is the average relative speed of two zero mode
scalars (see for example [48]). Here the effective coupling $\lambda_{S}$
model dependent and, for the case that the DM is the zero mode SM singlet
$\phi_{S}$ which is accessible to a single extra dimension in the RS1
background, it reads
$\displaystyle\lambda_{S}=\lambda_{5\,S}\,e^{-2\,k\,\pi\,R}\,f^{2}_{0}(\pi\,R)\,,$
(20)
where $f_{0}(y)$ is given in eq.(4). In the case that the SM Higgs field is
accessible to the fifth flat extra dimension and the DM candidate is the zero
mode of the SM singlet, which is accessible to the sixth one, the total
averaging annihilation rate of DM reads
$\displaystyle<\sigma\,v_{r}>$ $\displaystyle=$
$\displaystyle\frac{4\,\lambda_{S}^{2}\,v^{2}}{m_{S}}\,\Bigg{|}\,\frac{1}{(4\,m_{S}^{2}-m_{H^{0\,(0)}}^{2})+i\,m_{H^{0\,(0)}}\,\Gamma_{H^{0\,(0)}}}$
(21) $\displaystyle+$
$\displaystyle\sqrt{2}\,\sum_{n=1}\,\frac{1}{(4\,m_{S}^{2}-m_{H^{0\,(n)}}^{2})+i\,m_{H^{0\,(n)}}\,\Gamma_{H^{0\,(n)}}}\,\Bigg{|}^{2}\,\Gamma(\tilde{h}\rightarrow
X_{SM})\,,$
where $m_{H^{0\,(n)}}^{2}=m_{H^{0\,(0)}}^{2}+\frac{n^{2}}{R^{2}}$ and
$\lambda_{S}$ is
$\displaystyle\lambda_{S}=\frac{\lambda_{6\,S}}{(2\,\pi\,R)^{2}}\,.$ (22)
If the DM candidate is the SM singlet, living in four dimensions,
$\lambda_{S}$ becomes
$\displaystyle\lambda_{S}=\frac{\lambda_{5\,S}}{2\,\pi\,R}\,,$ (23)
and the annihilation cross section is given in eq.(21).
For the the annihilation cross section $<\sigma\,v_{r}>$ we respect the
restriction
$\displaystyle<\sigma\,v_{r}>=0.8\pm 0.1\,pb\,,$ (24)
which is constructed in the case that s-wave annihilation is dominant (see
[49] for details.). These bounds are coming from the relic density
$\displaystyle\Omega\,h^{2}=\frac{x_{f}\,10^{-11}\,GeV^{-2}}{<\sigma\,v_{r}>}\,,$
(25)
where $x_{f}\sim 25$ [2, 22, 48, 50, 51] and, by the WMAP collaboration [40],
the present DM abundance reads
$\displaystyle\Omega\,h^{2}=0.111\pm 0.018\,.$ (26)
Discussion
The present work is devoted to the analysis of the annihilation cross sections
of some DM candidates and the couplings that drive these cross sections. Here
we consider two scenarios. As a first one, we assume that all SM particles
live on the 4 dimensional brane and there exists an additional scalar SM
singlet $\phi_{S}$ which is accessible to a single extra dimension in the RS1
background. In this case the zero mode of the scalar singlet is a candidate of
DM and it is localized in the extra dimension (see eq.(4)). The interaction
term, which is represented by the action in eq.(5), is responsible for the
existence of the vertex DM DM $H^{0}$ which appears after the elecroweak
symmetry breaking. This term drives the annihilation cross section which
should be compatible with the present observed DM relic density (eq.(26)) and
the strength of the interaction is regulated by the the effective coupling
$\lambda_{S}$ (eq.(20)). The free parameters in this scenario are the Higgs
mass $m_{H^{0}}$, the zero mode scalar mass $m_{S}$, the curvature $k$ and the
parameter $a$ which plays an essential role in the localization of DM. In our
numerical calculations we take Higgs mass around $110-120\,GeV$, the DM
candidate mass in the range of $10-80\,GeV$ and we choose two different values
for the curvature $k$, $k=10^{7}\,GeV$ and $k=10^{8}\,GeV$. Now we study the
localization parameter $a$ dependence of the annihilation cross section
$<\sigma\,v_{r}>$ and we estimate the range of $a$ by respecting the upper and
lower bounds of the current experimental value of the relic abundance, namely
$0.7\,pb\leq\,\,<\sigma\,v_{r}>\,\,\,\leq 0.9\,pb$.
In Figs.1 and 2 we plot the localization parameter $a$ dependence of the
annihilation cross section $<\sigma\,v_{r}>$ for $m_{H^{0}}=110\,GeV$ and
$m_{H^{0}}=120\,GeV$. Here the left-right solid (long dashed; dashed; dotted)
line represents $<\sigma\,v_{r}>$ for $k=10^{17}-10^{18}\,GeV$
$m_{S}=80\,(m_{R};50;10)\,GeV$ where $m_{R}=55(60)\,GeV$ for
$m_{H^{0}}=110\,GeV$ ($m_{H^{0}}=120\,GeV$). We observe that the annihilation
cross section strongly depends on the parameter $a$. The $0.5\,\%$ variation
in $a$ results in that $<\sigma\,v_{r}>$ changes between the estimated upper
and lower bounds. If the mass of the scalar is $m_{S}=55\,GeV$ the resonant
annihilation occurs and $a$ reaches the greatest value so that the increase in
$<\sigma\,v_{r}>$ is appropriately suppressed in order to set it in the
estimated range. For $m_{S}=50\,GeV$ $a$ is still larger compared to ones for
$m_{S}=80\,GeV$ and $m_{S}=10\,GeV$ since this is the case that the scalar
mass is near to the resonant annihilation mass. For far from resonant
annihilation, heavy scalar mass causes that the ratio in the definition of the
annihilation cross section decreases and the parameter $a$ must increase to
set the cross section in the region which is restricted by the experimental
result (see the curves for $m_{S}=80\,GeV$ and $m_{S}=10\,GeV$). On the other
hand the increase in the compactification radius $R$ results in suppression in
the parameter $a$. For the SM Higgs mass $m_{H^{0}}=120\,GeV$ the behavior of
the annihilation cross section $<\sigma\,v_{r}>$ is similar to the previous
case. Here the curve for $m_{S}=80\,GeV$ lags the one for $m_{S}=50\,GeV$
since, in this case, the DM scalar with mass $m_{S}=50\,GeV$ is relatively far
from the resonant annihilation.
As a second scenario we take the extra dimension(s) flat and, at first, we
assume that the SM Higgs doublet and the additional SM model singlet scalar
field are accessible to fifth and sixth extra dimension respectively. Here the
zero mode of SM singlet is considered as the DM candidate. This is the case
that the annihilation of the DM occurs with the help of the SM Higgs boson and
its KK modes after the electroweak symmetry breaking (see eq.(16)). In this
scenario we study the behavior of the coupling $\lambda_{6S}$ in six
dimensions with respect to the compactification radius $R$, by respecting the
current average value of the annihilation cross section,
$<\sigma\,v_{r}>=0.8\,pb$. In the numerical calculations we take the
compactification radius $R$ in the range
$0.00001\,GeV^{-1}\leq\,R\,\leq\,0.005\,GeV^{-1}$.
Fig.3 represents $R$ dependence of the coupling $\lambda_{6S}$ in six
dimension for444Notice that, in the following, we denote the zero mode
$H^{0\,(0)}$ as $H^{0}$. $m_{H^{0}}=110\,GeV$ and $m_{H^{0}}=120\,GeV$. Here
the upper-lower solid line represents $\lambda_{6S}$ for
$m_{H^{0}}=110-120\,GeV$ $m_{S}=80\,GeV$ and the upper-lower long dashed
(dashed; dotted) line represents $\lambda_{6S}$ for $m_{H^{0}}=120-110\,GeV$
$m_{S}=60-55\,(50;10)\,GeV$. $\lambda_{6S}$ lies in the range of
$10^{-10}-10^{-6}\,GeV^{-2}$ for the interval of the compactification radius
$10^{-5}-10^{-3}\,GeV^{-1}$ for DM that is far from the resonant annihilation
case. In the case of resonant annihilation, $\lambda_{6S}$ is suppressed and
it is in the range of $10^{-14}-10^{-10}\,GeV^{-2}$. $\lambda_{6S}$ is weakly
sensitive to the SM Higgs mass for the interval under consideration, i.e.
$110\,GeV\leq m_{H^{0}}\leq 120\,GeV$. It is observed that the coupling
$\lambda_{6S}$ changes its behavior for the large values of $R$,
$R>0.004\,GeV^{-1}$, especially in the case that the mass of the DM scalar is
far from the resonant annihilation. This variation comes from the readable
effects of the intermediate SM Higgs KK modes for the compactification radius
in this range. Notice that the Higgs KK mode contribution is suppressed with
the decreasing values of the radius $R$.
As a second choice, we assume that the SM Higgs doublet is accessible to fifth
extra dimension and the additional SM model singlet lives in four dimensions.
Here we study the behavior of the coupling $\lambda_{5S}$ with respect to the
compactification radius $R$, by respecting the current average value of the
annihilation cross section, $<\sigma\,v_{r}>=0.8\,pb$, similar to the previous
case. In Fig.4 we show the $R$ dependence of the coupling $\lambda_{5S}$ for
$m_{H^{0}}=110\,GeV$ and $m_{H^{0}}=120\,GeV$. Here the upper-lower solid line
represents $\lambda_{5S}$ for $m_{H^{0}}=110-120\,GeV$ $m_{S}=80\,GeV$ and the
upper-lower long dashed (dashed; dotted) line represents $\lambda_{5S}$ for
$m_{H^{0}}=120-110\,GeV$ $m_{S}=60-55\,(50;10)\,GeV$. $\lambda_{5S}$ is in the
range of $10^{-6}-10^{-3}\,GeV^{-1}$ for the interval of the radius $R$
$10^{-5}-10^{-3}\,GeV^{-1}$ in the case that the DM mass is far from the
resonant annihilation. If the resonant annihilation occurs $\lambda_{5S}$
decreases up to the range of $10^{-10}-10^{-8}\,GeV^{-1}$. The effects of
Higgs KK modes are observed if the mass of the DM scalar is far from the
resonant annihilation and the radius $R$ is large, $R>0.003\,GeV^{-1}$.
Finally, we plot the the effective coupling $\lambda_{S}$ for both choices in
the second scenario in Fig 5. Here the upper-lower solid line represents
$\lambda_{S}$ for $m_{H^{0}}=110-120\,GeV$ $m_{S}=80\,GeV$ and the upper-lower
long dashed (dashed; dotted) line represents $\lambda_{S}$ for
$m_{H^{0}}=120-110\,GeV$ $m_{S}=60-55\,(50;10)\,GeV$. $\lambda_{S}$ is at the
order of magnitude of $10^{-2}-10^{-1}$ ($10^{-6}$) far from (near to) the
resonant annihilation. The effects of intermediate KK modes appear for
$R>0.002\,GeV^{-1}$ and these effects are negligible for the resonant
annihilation case.
At this stage we would like to present our results
* •
In the first scenario, the annihilation cross section is strongly sensitive to
the localization parameter $a$ and $a$ reaches its greatest value in the
resonant annihilation case. The increase in curvature $k$ (or the decrease in
the compactification radius $R$) forces $a$ to be suppressed.
* •
In the second scenario, we choose the extra dimension(s) flat and we assume
that the SM Higgs doublet is accessible to fifth dimension. Here we consider
two different possibilities. In the first we take the additional SM model
singlet scalar field which is accessible to the sixth extra dimension and its
zero mode is a possible DM candidate. In the second we consider that the new
SM singlet, the DM candidate, lives in four dimensions. In both possibilities
the KK modes of SM Higgs field play an observable role for large values of the
compactification radius $R$, $R>0.003\,GeV^{-1}$. On the other hand the
dimensionfull couplings ($\lambda_{6S}$ for the first choice and
$\lambda_{5S}$ for the second choice) are weak and the effective coupling
$\lambda_{S}$, which is the same for both choices, is of the order of
$10^{-2}-10^{-1}$ ($10^{-6}$) far from (near to) the resonant annihilation.
The forthcoming more accurate experimental measurements and the possible
observation of the SM Higgs boson at LHC will shed light on the nature of the
DM and its annihilation mechanism.
## References
* [1] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267, 195 (1996).
* [2] G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405, 279 (2005).
* [3] E. Komatsu et al., Astrophys. J. Suppl. Ser. 180, 330 (2009).
* [4] A. Borriello and P. Salucci, Mon. Not. Roy. Astron. Soc. 323, 285 (2001).
* [5] F. Zwicky, Helv. Phys. Acta. 6, 110 (1993).
* [6] D. N. Spergel et.al (WAMP Collaboration), Astrophys. J. Suppl. Ser. 170 377 (2007).
* [7] M. Asano, S. Matsumoto, M. Senami, H. Sugiyama, Phys. Lett. B663, 330 (2008).
* [8] T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D64, 035002 (2001).
* [9] H. C. Cheng, K. T. Matchev and M. Schmaltz, Phys. Rev. D66, 056006 (2002).
* [10] I. Antoniadis, Phys. Lett. B246, 377 (1990).
* [11] D. Hooper and S. Profumo, Phys. Rept. 453, 29 (2007).
* [12] F. Burnell and G. D. Kribs, Phys. Rev. D73, 015001 (2006).
* [13] K. Kong and K. T. Matchev, JHEP 0601, 038 (2006).
* [14] D. Hooper and G. D. Kribs, Phys. Rev. D70, 115004 (2004).
* [15] E. A. Baltz and D. Hooper, JCAP 0507, 001 (2005).
* [16] L. Bergstrom, T. Bringmann, M. Eriksson and M. Gustafsson, Phys. Rev. Lett. 94, 131301 (2005).
* [17] L. Bergstrom, T. Bringmann, M. Eriksson and M. Gustafsson, JCAP 0504, 004 (2005).
* [18] A. Barrau, P. Salati, G. Servant, F. Donato, J. Grain, D. Maurin and R. Taillet, Phys. Rev. D72, 063507 (2005).
* [19] A. Birkedal, K. T. Matchev, M. Perelstein and A. Spray, hep-ph/0507194 (2005).
* [20] S. Matsumoto, J. Sato, M. Senami, M. Yamanaka, hep-ph/0903.3255 (2009).
* [21] E.O. Iltan, hep-ph/0907.1391 (2009).
* [22] G. Servant and T. M. P. Tait, Nucl. Phys. B650, 391 (2003).
* [23] S. C. Park and J. Shu, Phys. Rev. D79, 091702(R) (2009).
* [24] C. R. Chen, M. M. Nojiri, S. C. Park, J. Shu, M. Takeuchi, hep-ph/0903.1971 (2009).
* [25] C.B. Jackson, hep-ph/0804.3792 (2008).
* [26] E. Ma, Phys. Rev. D73, 077301 (2006).
* [27] R. Barbieri, L. J. Hall, and V. S. Rychkov, Phys. Rev. D74, 015007 (2006).
* [28] M. Cirelli, N. Fornengo, and A. Strumia, Nucl. Phys. B753, 178 (2006).
* [29] N. G. Deshpande and E. Ma, Phys. Rev D18, 2574 (1978).
* [30] J. A. Casas, J. R. Espinosa, and I. Hidalgo, Nucl. Phys. B777, 226 (2007).
* [31] L. L. Honorez, E. Nezri, J. F. Oliver, M. H. G. Tytgat, JCAP 0702, 028 (2007).
* [32] S. Andreas, T. Hambye, M. H.G. Tytgat, JCAP 0810, 034 (2008).
* [33] X. Calmet and J. F. Oliver, Europhys. Lett. B77, 51002 (2007).
* [34] Y. Bai Phys.Lett. B666, 332 (2008).
* [35] D. Majumdar and A. Ghosal, Mod. Phys. Lett. A23, 2011 (2008).
* [36] F. D. Eramo Phys. Rev. D76, 083522 (2007).
* [37] W. L. Guo, X. Zhang, Phys. Rev. D79, 115023 (2009).
* [38] C.-R. Chen, M. M. Nojiri, S. C. Park, J. Shu, IPMU09-0101, Aug 2009\. 19pp., hep-ph/0908.4317 (2009).
* [39] D. S. Akerib et.al. [CDMS Collaboration], Phys. Rev. Lett. 96, 011302 (2006).
* [40] D. N. Spergel et.al (WAMP Collaboration), Astrophys. J. Suppl. Ser. 148 175 (2003).
* [41] L. Randall, R.Sundrum, Phys. Rev. Lett. 83 3370, (1999).
* [42] L. Randall, R.Sundrum, Phys. Rev. Lett. 83 4690 (1999).
* [43] W. D. Goldberger, M. B. Wise, Phys. Rev. Lett. D 83 4922, (1999).
* [44] I. I. Kogan, S. Mouslopolous, A. Papazoglou, G. G. Ross, Nucl. Phys. B615 191, (2001).
* [45] E.O. Iltan, hep-ph/0806.2478 (2008).
* [46] C. Bird, P. Jackson, R. Kowalewski and M. Pospelov, Phys. Rev. Lett. 93, 201803 (2004).
* [47] C. Bird, R. Kowalewski and M. Pospelov, Mod. Phys. Lett. A21, 457 (2006).
* [48] X. G. He, T. Li, X. Q. Li, and H.-C. Tsai, Mod. Phys. Lett. A22, 117 (2005).
* [49] E. W. Kolb and M. S. Turner, The Early Universe (Addison- Wesley, Reading, MA, 1990).
* [50] S. Gopalakrishna, S. Gopalakrishna, A. de Gouvea, W. Porod, JCAP 0605, 005 (2006).
* [51] S. Gopalakrishna, S. J. Lee, J. D. Wells, hep-ph/0904.2007.
Figure 1: $<\sigma\,v_{r}>$ as a function of $a$ for $m_{H^{0}}=110\,GeV$.
Here the left-right solid (long dashed; dashed; dotted) line represents
$<\sigma\,v_{r}>$ for $k=10^{17}-10^{18}\,GeV$ $m_{S}=80\,(55;50;10)\,GeV$.
Figure 2: The same as Fig1 but for $m_{H^{0}}=120\,GeV$ and
$m_{S}=80\,(60;50;10)\,GeV$. Figure 3: $\lambda_{6S}$ as a function of $R$.
Here the upper-lower solid line represents $\lambda_{6S}$ for
$m_{H^{0}}=110-120\,GeV$ $m_{S}=80\,GeV$ and the upper-lower long dashed
(dashed; dotted) line represents $\lambda_{6S}$ for $m_{H^{0}}=120-110\,GeV$
$m_{S}=60-55\,(50;10)\,GeV$. Figure 4: The same as Fig3 but for
$\lambda_{5S}$. Figure 5: The same as Fig3 but for $\lambda_{S}$.
|
arxiv-papers
| 2009-10-26T17:44:59 |
2024-09-04T02:49:06.056751
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. O. Iltan",
"submitter": "Erhan Iltan",
"url": "https://arxiv.org/abs/0910.4936"
}
|
0910.5038
|
# LOCAL RIGIDITY OF PARTIALLY HYPERBOLIC ACTIONS
Zhenqi WANG Department of Mathematics
The Pennsylvania State University
University Park, PA,16802 wang_z@math.psu.edu
###### Abstract.
We consider partially hyperbolic abelian algebraic high-rank actions on
compact homogeneous spaces obtained from simple indefinite orthogonal and
unitary groups. In the first part of the paper, we show local differentiable
rigidity for such actions. The conclusions are based on progress towards
computations of the Schur multipliers of these non-split groups, which is the
main aim of the second part.
###### Contents
1. 1 Introduction
2. 2 Setting and results
1. 2.1 Generic restrictions of split Cartan actions
2. 2.2 Cocycles and rigidity
3. 2.3 Formulation of results
4. 2.4 Comments on related problems
3. 3 Cocycle rigidity for the actions $\alpha_{0,G}$
1. 3.1 Preliminaries
2. 3.2 Paths and cycles for a collection of foliations
3. 3.3 Split Cartan actions on $SO^{+}(m,n)/\Gamma$ and $SU(m,n)/\Gamma$
4. 3.4 Generating relations and Steinberg symbols
5. 3.5 Proof of Theorem 3
4. 4 Proof of Theorems 1 and 2
1. 4.1 Proof of Corollary 2.1
5. 5 Schur multipliers of non-split groups
1. 5.1 Preliminaries and notations from K-theory
2. 5.2 Construction of universal central extension
6. 6 Generating relations of $SO^{+}(m,n)$
1. 6.1 Basic settings for $SO^{+}(m,n)$
2. 6.2 “Chains” in $SO^{+}(m,n)$
3. 6.3 Basic relations
4. 6.4 Structure of $\tilde{H}$
5. 6.5 Construction of a $S^{1}$-symbol
6. 6.6 Structure of $\ker(\pi_{1})\cap\tilde{H}_{s_{0}}$
7. 6.7 Proof of Proposition 6.1
8. 6.8 Proof of Theorem 7
9. 6.9 Proof of Theorem 4
7. 7 Generating relations of $SU(m,n)$
1. 7.1 Basic settings for $SU(m,n)$
2. 7.2 “Chain” in $SU(m,n)$ and basic relations
3. 7.3 Structure of $\ker(\pi_{1})$
4. 7.4 Construction of $S_{\mathbb{R}}^{1}$-symbols
5. 7.5 Structure of $H^{i}$
6. 7.6 Proof of Lemma 9
7. 7.7 Structure of $\ker(\pi_{1})\cap\tilde{H}_{s_{0}}$
8. 7.8 Proof of Theorem 10
9. 7.9 Proof of Theorem 5
## 1\. Introduction
In this paper we extend results of D.Damjanovic and A.Katok about rigidity of
certain diagonal actions on compact homogeneous spaces [2, 3, 5] from split to
some non-split Lie groups.
A. Katok and R.Spatzier considered the differentiable rigidity of Weyl chamber
flows on symmetric spaces: let $G$ be a connected semisimple Lie group of real
rank $\geq 2$, $\Gamma$ a cocompact torsion-free lattice in $G$; let $A$ be a
maximal split Cartan subgroup of $G$, and let $K$ be the compact part of the
centralizer of $A$ which intersects with $A$ trivially. If $G$ is of
$\mathbb{R}$-rank greater than one, the action of $A$ on the space
$K\backslash G/\Gamma$ is an Anosov (normally hyperbolic) action, and it is
locally differentiably rigid [15]. The method of proof can be called a priori
regularity since it is based on showing smoothness of the Hirsch-Pugh-Shub
orbit equivalence [10]. Another ingredient in the Katok-Spatzier method is
cocycle rigidity used to “straighten out” a time change; it is proved by a
harmonic analysis method.
In [2, 3], a special case $G=SL(n,\mathbb{R})(n\geq 3)$ is considered. In this
case, rather than the full $A$ action by left translations on
$SL(n,\mathbb{R})/\Gamma$, they considered the restrictions of the full
diagonal action to subgroups in the acting group $\mathbb{R}^{n-1}$ that
contain lattices in 2-planes in general position. Those actions are partially
hyperbolic rather than Anosov and a priori regularity methods is not
applicable. The proof of cocycle rigidity and differentiable rigidity in [2,
3] is “geometric”, in contrast with earlier proofs in [15]. The totally new
manner is based on geometry and combinatorics of invariant foliations and
using insights from algebraic $K$-theory as an essential tool.
The approach of [2, 3] was further employed in [5], for extending cocycle
rigidity and differentiable rigidity from $SL(n,\mathbb{R})/\Gamma$ and
$SL(n,\mathbb{C})/\Gamma$ to compact homogeneous spaces obtained from some
simple split Lie groups of nonsymplectic type.
The purpose of this paper is to further extend the results of the above-
mentioned authors to higher rank partially hyperbolic actions on compact
homogeneous spaces obtained from indefinite orthognal and unitary groups. In
the present work we derive information about generators and relations in these
non-split groups which are not readily available from the literature. As soon
as this algebraic information is obtained, the geometric scheme developed by
Katok and Damjanovic essentially applies to non-split cases. However, there
are still at some places significant technical differences which require some
new arguments to handle them.
In Section 3, based on the conclusions about Schur multipliers proved in
Section 5-7, we apply the approach of [2] to prove trivialization of small,
non-abelian, group valued cocycles over partially hyperbolic abelian algebraic
acrion described in Section 2.2. The key points are: (i) local transitivity of
Lyapunov foliations, (ii) the fact that some Lyapunov cycle can be
approximated by a composition of conjugates to stable cycles, and (iii)
vanishing of periodic cycle functionals on broken paths along leaves of stable
and unstable foliations generated by multiple-dimension root spaces. Once the
cocycle rigidity is obtained in Section 3 via geometric method, proving that
cocycle rigidity is robust under $C^{2}$-small perturbations is similar to the
case of $G=SL(n,\mathbb{R})$, which is treated in [3] (See Section 4).
In Section 5-7, we make sufficient progress towards the computations of the
Schur multipliers of $SO^{+}(m,n)$ and $SU(m,n)$ where $m\geq n\geq 3$ to
obtain information needed for the proofs in Section 3. This work is extension
of the ideas of R. Steinberg [23] and V. Deodhar [6]. Steinberg considered the
so-called Schur multipliers (cf. Steinberg [[24], Section 7]) of rational
points of simply connected Chevalley groups and obtained results of
importance, especially in the case of split semisimple algebraic groups.
Deodhar extended Steinberg’s theory to a more general class of quasi-split
groups. In the present work, our main aim is to obtain similar results for
some non-split groups that are not quasi-split. Deodhar’s construction carries
carry over to the non-split groups with minor changes. However, it does not
provide sufficient information and needs to be supplemented by some new
method.
Let $k$ denote an arbitrary local field, and $G$ denote a connected, simply
connected algebraic group which is defined and absolutely almost simple over
$k$. In Section 5 we use Deodhar’s results freely in the general case: We
construct explicitly, in terms of generators and relations, a “universal
central extension” for $G^{+}_{k}$ generated by $k$-rational unipotent
elements which belong to the radical of a parabolic subgroup defined over $k$
in $G$.
In Section 6 and 7, our main aim is to make some progress towards the
computation of the fundamental group $\pi_{1}(G_{k})$ (= Schur multiplier) of
$G_{k}$ when $G_{\mathbb{R}}=SO^{+}(m,n)(m\geq n\geq 3)$ and
$G_{\mathbb{R}}=SU(m,n)(m\geq n\geq 3)$. For this purpose, we introduce a way
to tackle the “rotation” and “reflection” in root spaces with dimensions
greater than 1.
For definitions and general background on partially hyperbolic dynamical
systems see [22]; all necessary background on algebraic actions can be found
in [14]. We will also strongly rely on definitions, constructions and results
from the earlier papers on the subject [2, 3, 5]. In the present paper we
consider algebraic actions of
$\mathbb{Z}^{k}\times\mathbb{R}^{\ell},k+\ell\geq 2$. We treat generic
restrictions of full split Cartan actions on $SO^{+}(m,n)/\Gamma$ and on
$SU(m,n)/\Gamma$(where $\Gamma$ is a cocompact lattice).
I’d like to thank Nigel Higson for pointing out to some relevant sources and
results in algebraic $K$-theory and many stimulating discussions. I am
grateful to Grigory Margulis and Yuri Zarhin who kindly helped me in algebraic
area. My sincere thanks are also due to my advisor Anatole Katok, who
suggested the problem to me. He also looked through several preliminary
versions of this paper, suggested some changes, made important comments and
encouraged me a lot.
## 2\. Setting and results
### 2.1. Generic restrictions of split Cartan actions
Let $Q$ be a non-degenerate standard bilinear form on $\mathbb{R}^{m+n}$ of
signature $(n,m)$. The group $SO^{+}(m,n)$ is then the connected Lie group of
$(m+n)\times(m+n)$ matrices that preserve $Q$ with determinant 1. Then we can
choose a base of $\mathbb{R}^{m+n}$ in terms of which the quadratic form $Q$
is given by $Q(e_{i},e_{i+n})=1$, if $1\leq i\leq n$; $Q(e_{j},e_{j})=1$, if
$2n+1\leq j\leq m+n$ and $Q(e_{i},e_{j})=0$ otherwise.
Using this base, the Lie algebra $so(m,n)$ of $SO^{+}(m,n)$ can be represented
as $(m+n)\times(m+n)$ matrices
$\begin{pmatrix}A_{1}&A_{2}&\vline&B_{1}\\\ A_{3}&A_{4}&\vline&B_{2}\\\
\hline\cr C_{1}&C_{2}&\vline&D\end{pmatrix},$
where $A_{1},A_{2},A_{3},A_{4}$ are $n\times n$ matrices, $B_{1},B_{2}$ are
$n\times(m-n)$ matrices, $C_{1},C_{2}$ are $(m-n)\times n$ matrices and $D$ is
a $(m-n)\times(m-n)$ matrix satisfying
$\displaystyle A_{1}$ $\displaystyle=-A^{\tau}_{4},$ $\displaystyle\qquad
A^{\tau}_{2}$ $\displaystyle=-A_{2},$ $\displaystyle\qquad A^{\tau}_{3}$
$\displaystyle=-A_{3},$ $\displaystyle D^{\tau}$ $\displaystyle=-D,$
$\displaystyle\qquad B_{1}$ $\displaystyle=-C^{\tau}_{2},$
$\displaystyle\qquad B_{2}$ $\displaystyle=-C^{\tau}_{1}.$
Here $M^{\tau}$ denotes the transpose of the matrix $M$.
Let $H$ be a non-degenerate standard Hermitian form of signature $(n,m)$. Then
we can choose a base of $\mathbb{C}^{m+n}$(under a linear transformation with
real coefficients) in terms of which the quadratic form $H$ is given by
$H(e_{i},e_{i+n})=1$, $1\leq i\leq n$, $H(e_{j},e_{j})=1$, $2n+1\leq j\leq
m+n$ and $H(e_{i},e_{j})=0$, otherwise 0. The group $SU(m,n)$ is then the
connected Lie group of $(m+n)\times(m+n)$ matrices that preserve $H$ with
determinant 1.
Using this base the Lie algebra $su(m,n)$ of $SU(m,n)$ can be expressed as
$(m+n)\times(m+n)$ matrices
$\begin{pmatrix}A_{1}&A_{2}&\vline&B_{1}\\\ A_{3}&A_{4}&\vline&B_{2}\\\
\hline\cr C_{1}&C_{2}&\vline&D\end{pmatrix},$
where $A_{1},A_{2},A_{3},A_{4}$ are $n\times n$ matrices, $B_{1},B_{2}$ are
$n\times(m-n)$ matrices, $C_{1},C_{2}$ are $(m-n)\times n$ matrices and $D$ is
a $(m-n)\times(m-n)$ matrix satisfying
$\displaystyle A_{1}$ $\displaystyle=-\overline{A}^{\tau}_{4},$
$\displaystyle\qquad\overline{A}^{\tau}_{2}$ $\displaystyle=-A_{2},$
$\displaystyle\qquad\overline{A}^{\tau}_{3}$ $\displaystyle=-A_{3},$
$\displaystyle\overline{D}^{\tau}$ $\displaystyle=-D,$ $\displaystyle\qquad
B_{1}$ $\displaystyle=-\overline{C}^{\tau}_{2},$ $\displaystyle\qquad B_{2}$
$\displaystyle=-\overline{C}^{\tau}_{1}.$
Here $\overline{M}^{\tau}$ denotes the complex conjugate transpose of the
matrix $M$.
Let $X:=SO^{+}(m,n)/\Gamma,$ (corr. $SU(m,n)/\Gamma$) with $m\geq n\geq 3$ and
$\Gamma$ a cocompact lattice in $X$. Let
$\displaystyle D_{+}=\exp\mathbb{D_{+}}=$
$\displaystyle\\{\text{diag}\bigl{(}\exp t_{1},\exp t_{2},\dots,\exp
t_{n},\exp(-t_{1}),\exp(-t_{2}),\dots,$
$\displaystyle\text{exp}(-t_{n}),1,\dots,1\bigl{)}:(t_{1},\dots,t_{n})\in\mathbb{R}^{n}\\}$
be the group of diagonal matrices with lower $(m-n)\times(m-n)$ matrix
identity. In fact, $D_{+}$ is the maximal split Cartan subgroup both in
$SO^{+}(m,n)$ and $SU(m,n)$.
We denote the action of $D^{+}$ on $X$ by left translations by $\alpha_{0}$
and call it the split Cartan action.
For $1\leq i\neq j\leq n$ the hyperplanes in $\mathbb{D_{+}}$ defined by
$\mathbb{H}_{i-j}=\\{(t_{1},\dots,t_{n})\in\mathbb{D_{+}}:t_{i}=t_{j}\\},$
$\mathbb{H}_{i+j}=\\{(t_{1},\dots,t_{n})\in\mathbb{D_{+}}:t_{i}+t_{j}=0\\}\quad\text{and}$
$\mathbb{H}_{i}=\\{(t_{1},\dots,t_{n})\in\mathbb{D_{+}}:t_{i}=0\\}$
(exist if $m-n\geq 1$) are _Lyapunov hyperplanes_ for the action $\alpha_{0}$,
i.e. kernels of Lyapunov exponents of $\alpha_{0}$. Elements of
$\mathbb{D_{+}}\backslash\bigcup\mathbb{H}_{r}$(where $r=i\pm j,i$) are
_regular_ elements of the action. Connected components of the set of regular
elements are $Weyl$ $chambers$.
The smallest non-trivial intersections of stable foliations of various
elements of the action $\alpha_{0}$ are $Lyapunov$ $foliations$. Each regular
element either exponentially expands or exponentially contracts each of those
leaves. For more details, see Section 3.3.
###### Definition 2.1.
A two-dimensional plane $\mathbb{P}\subset\mathbb{D_{+}}$ is in $general$
$position$ if it intersects any two distinct Lyapunov hyperplanes along
distinct lines.
Let $\mathbb{G}\subset\mathbb{D_{+}}$ be a closed subgroup which contains a
lattice $\mathbb{L}$ in a plane in general position and let
$G=\exp\mathbb{G}$. One can naturally think of $G$ as the image of an
injective homomorphism
$i_{0}:\mathbb{Z}^{k}\times\mathbb{R}^{\ell}\rightarrow D_{+}$ (where
$k+\ell\geq 2)$.
###### Definition 2.2.
The action $\alpha_{0,G}$ of $G$ by left translations on $X$ is given by
(2.1) $\displaystyle\alpha_{0,G}(a,x)=i_{0}(a)\cdot x$
and will be referred to as a higher-rank generic restriction of split Cartan
actions or just a generic restriction for short.
### 2.2. Cocycles and rigidity
Let $\alpha:A\times M\rightarrow M$ be an action of a topological group $A$ on
a compact Riemannian manifold M by diffeomorphisms. For a topological group
$Y$ a $Y$-valued cocycle (or an one-cocycle) over $\alpha$ is a continuous
function $\beta:A\times M\rightarrow Y$ satisfying:
(2.2) $\displaystyle\beta(ab,x)=\beta(a,\alpha(b,x))\beta(b,x)$
for any $a,b\in A$. A cocycle is cohomologous to a constant cocycle (cocycle
not depending on $x$) if there exists a homomorphism $s:A\rightarrow Y$ and a
continuous transfer map $H:M\rightarrow Y$ such that for all $a\in A$
(2.3) $\displaystyle\beta(a,x)=H(\alpha(a,x))s(a)H(x)^{-1}$
In particular, a cocycle is a coboundary if it is cohomologous to the trivial
cocycle $\pi(a)=id_{Y}$, $a\in A$, i.e. if for all $a\in A$ the following
equation holds:
(2.4) $\displaystyle\beta(a,x)=H(\alpha(a,x))H(x)^{-1}.$
For more detailed information on cocycles adapted to the present setting see
[3]. Let
$Y=\begin{pmatrix}A_{1}&0\\\ 0&A_{2}\\\ \end{pmatrix},$
be the subgroup of $SU(m,n)$ with
$\displaystyle A_{1}=$ $\displaystyle\\{\operatorname{diag}\bigl{(}\exp
z_{1},\dots,\exp
z_{n},\exp(-\overline{z_{1}}),\dots,\exp(-\overline{z_{n}})\bigl{)}:$
$\displaystyle(z_{1},\dots,z_{n})\in\mathbb{C}^{n}\\}$
and $A_{2}$ $(m-n)\times(m-n)$ unitary matrices. Let $Y_{X}=Y\cap SO^{+}(m,n)$
if $X=SO^{+}(m,n)/\Gamma$(corr. $Y_{X}=Y$ if $X=SU(m,n)/\Gamma$). $Y_{X}$ is
isomorphic to $\mathbb{R}^{n}\times SO(m-n)$ when $X=SO^{+}(m,n)/\Gamma$ and
isomorphic to $\mathbb{R}^{n}\times\mathbb{T}^{n-1}\times U(m-n)$ if
$X=SU(m,n)/\Gamma$.
Let $\mathbb{P}\subset\mathbb{D_{+}}$ be a 2-dimensional plane in general
position. We will show (Theorem 3 in Section 3) that every small Hölder
cocycle with values in $Y_{X}$ over the action $\alpha_{0,G}$, where $G$ is
any subgroup of $D_{+}$ which contains $\exp\mathbb{P}$, is cohomologous to a
constant cocycle. Similarly to the proofs in [2, 3] we use the geometric
structure of Lyapunov foliations of the action. By applying the method of [2,
13] we show that a cocycle over a partially hyperbolic action with locally
transitive Lyapunov foliations is cohomologous to a constant cocycle if and
only if the periodic cycle functional (PCF) vanishes on all closed broken
paths whose pieces lie in leaves of Lyapunov foliations of the action.
Furthermore, the presentations of Schur multipliers of $SO^{+}(m,n)$ and
$SU(m,n)$ that we will construct in Sections 5-7, give explicit description of
closed broken paths along Lyapunov foliations which leads to vanishing of the
PCF on all such paths and to cocycle rigidity. Smoothness of the transfer map
for smooth cocycles is a consequence of the fact that for a generic
restriction, the Lyapunov distributions along with their Lie brackets generate
the tangent space at every point.
### 2.3. Formulation of results
Our main results are contained in the following two theorems.
###### Theorem 1 (Differentiable rigidity of generic restrictions).
Let $\alpha_{0,G}$ be a high rank generic restriction of the action of a
maximal split Cartan subgroup on $SO^{+}(m,n)/\Gamma$ or $SU(m,n)/\Gamma$
where $m\geq n\geq 3$.
If $\tilde{\alpha}$ is $C^{\infty}$ action of
$\mathbb{Z}^{k}\times\mathbb{R}^{\ell}$ sufficiently $C^{2}$-close to
$\alpha_{0,G}$, then there exists a homomorphism
$i:\mathbb{Z}^{k}\times\mathbb{R}^{\ell}\rightarrow Y_{X}$ close to $i_{0}$
and a $C^{\infty}$ diffeomorphism $h:X\rightarrow X$ such that
$\tilde{\alpha}(a,h(x))=h(i(a)\cdot x)$ for all
$\mathbb{Z}^{k}\times\mathbb{R}^{\ell}$.
The principal ingredient in the proof of Theorem 1 is the next theorem which
is the main technical result of the present paper. It extends the cocycle
rigidity result from [3, 5].
###### Theorem 2 (Cocycle rigidity for perturbations).
Let $\alpha_{0,G}$ be a generic restriction of the action of a maximal split
Cartan subgroup on $SO^{+}(m,n)/\Gamma$ or $SU(m,n)/\Gamma$ where $m\geq n\geq
3$. Let $\tilde{\alpha}$ be a sufficiently $C^{2}$-small $C^{1}$ perturbation
of $\alpha_{0,G}$.
If $\beta$ is a $H\tilde{o}lder$ cocycle over $\tilde{\alpha}$ with values in
$Y_{X}$ then $\beta$ is cohomologous to a constant cocycle given by a
homomorphism $s:\mathbb{Z}^{k}\times\mathbb{R}^{\ell}\rightarrow Y_{X}$ via a
continuous transfer function. Furthermore, if the cocycle $\beta$ is
sufficiently small in a $H\tilde{o}lder$ norm the transfer map is $C^{0}$
arbitrary small.
Let $X_{1}:=M\backslash SO^{+}(m,n)/\Gamma,$ with $m\geq n\geq 3$, where
$M=SO(m-n)$ and $\Gamma$ a cocompact lattice in $SO^{+}(m,n)$.
###### Corollary 2.1.
Let $\alpha_{0,G}$ be a high rank generic restriction of the action of a
maximal split Cartan subgroup on $M\backslash SO^{+}(m,n)/\Gamma$ where $m\geq
n\geq 3$.
If $\tilde{\alpha}$ is $C^{\infty}$ action of
$\mathbb{Z}^{k}\times\mathbb{R}^{\ell}$ sufficiently $C^{2}$-close to
$\alpha_{0,G}$, then there exists a homomorphism
$i:\mathbb{Z}^{k}\times\mathbb{R}^{\ell}\rightarrow Y_{X}$ close to $i_{0}$
and a $C^{\infty}$ diffeomorphism $h:X\rightarrow X$ such that
$\tilde{\alpha}(a,h(x))=h(i(a)\cdot x)$ for all
$\mathbb{Z}^{k}\times\mathbb{R}^{\ell}$.
The action by left translations of $D_{+}$ on $X_{1}$ is the Weyl chamber flow
(WCF) and we denote this action by $\alpha_{0}$. The following result is a
special case of Corollary 2.1.
###### Corollary 2.2.
Let $\alpha_{0}$ be the WCF on $M\backslash SO^{+}(m,n)/\Gamma$ where $m\geq
n\geq 3$. If $\tilde{\alpha}$ is $C^{\infty}$ action sufficiently
$C^{2}$-close to $\alpha_{0}$, then there exists a homomorphism
$i:\mathbb{Z}^{k}\times\mathbb{R}^{\ell}\rightarrow D_{+}$ close to identity
and a $C^{\infty}$ diffeomorphism $h:X\rightarrow X$ such that
$\tilde{\alpha}(a,h(x))=h(i(a)\cdot x)$.
### 2.4. Comments on related problems
Results of this paper belong to the general program of establishing various
flavors of local differentiable rigidity for partially hyperbolic algebraic
actions of higher rank abelian groups. For general comments on that program
see [4, Section 1]. We do not attempt a comprehensive overview of the current
state of the program but restrict ourselves to few comments on those aspects
that are closely related to our results.
First let us discuss generic restrictions of Cartan actions on other higher
rank simple Lie groups.
The condition $n\geq 3$ for $SO^{+}(m,n)$ and $SU(m,n)$ is necessary for the
method used in this paper. Algebraically, the complications in these cases are
similar to those encountered in the case $n=2$ for the linear Steinberg groups
([7] Section 1.4E). Geometrically, homotopy classes can’t be reduced to each
other using allowable substitution. Thus the cases of groups $SO^{+}(m,2)$ and
$SU(m,2)$ remain open.
Using the technique in dealing with an infinite fundamental group similar to
that we use in case of $SU(m,n)$, we can solve the cocycle rigidity and
differentiable rigidity problem of generic restrictions for Cartan action on
the split groups $Sp(n,\mathbb{R})$ that is left out in [5]. This result will
appear in a separate paper.
Extension to generic restrictions of split Cartan actions for other classical
non-split simple groups requires information about generators and relations
not available from the literature. This is not surprising that this is already
the case with the groups $SO(m,n)$ and $SU(m,n)$. While our general approach
may (and probably should) work more techniques are needed for calculations of
generators and relations of matrix groups of $SL(n,\mathbb{H})$, $SP(m,n)$ and
$SO^{*}(2n)$. Those are defined over quaternions and one has to double the
sizes of matrices to represent them by complex matrices which makes the
“rotations” much more complex.
We have not looked into exceptional groups and are not aware of any effective
way to solve the generators and relations problem for those groups. Similar to
the quaternionic cases, one hopes that with sufficiently hard work those
groups (with the possible exception of the rank two case) may be amenable to
our method.
A necessary condition for applicability of the Damjanovic–Katok geometric
method (although not for local rigidity) is that contracting distributions of
various action elements and their brackets of all orders generate the tangent
space to the phase space. Generic restrictions for Cartan actions satisfy that
condition. Naturally one may look at non-generic restrictions of Cartan
actions. Some of those still satisfy it but nothing is known even for the
$SL(n,\mathbb{R})$ case since one cannot use the full force of the algebraic
$K$-theory machinery. More detailed analysis of generators and relations may
help resolve some of those cases.
The next natural step is to consider a similar problem on products of simple
groups factored by irreducible lattices. There are certain cases that look
amenable to the method. Finally, one may consider actions of higher-rank
abelian subgroups on homogeneous spaces of compact extensions of simple or
semisimple Lie groups. Since the compact fibers are included into the neutral
directions one should consider cocycles with values in more general groups
that are extensions of abelian groups by compact groups. Our methods can be
extended to at least some of those cases.
## 3\. Cocycle rigidity for the actions $\alpha_{0,G}$
The purpose of this section is to describe a geometric method for proving
cocycle rigidity for this action following [2, 3].
###### Theorem 3.
Any $Y_{X}$-valued Hölder cocycle over the generic restriction of the split
Cartan action on $X$ is cohomologous to a constant cocycle via a Hölder
$C^{\infty}$ transfer function.
Any $Y_{X}$-valued $C^{\infty}$ cocycle over the generic restriction of the
split Cartan action on $X$ is cohomologous to a constant cocycle via a
$C^{\infty}$ transfer function.
### 3.1. Preliminaries
Let $\alpha:A\rightarrow\text{Diff}(M)$ be an action of $A:=\mathbb{R}^{k}$,
$k\in\mathbb{N}$ on a compact manifold $M$ by diffeomorphisms of $M$
preserving an ergodic probability measure . Then there are finitely many
linear functionals $\lambda$ on $A$, called $Lyapunov$ $exponents$, a set of
full measure $\Lambda$ and a measurable splitting of the tangent bundle
$T_{\Lambda}M=\bigoplus_{\lambda}E^{\lambda}$, such that for $v\in
E^{\lambda}$ and $a\in A$ the Lyapunov exponent of $v$ with respect to
$\alpha(a)$ is $\lambda(a)$.
If $\chi$ is a non-zero Lyapunov exponent then we define its $coarse$
$Lyapunov$ $subspace$ by
$E_{\chi}:=\bigoplus_{\lambda=c\chi:c>0}E^{\lambda}.$
For every $a\in A$ one can define stable, unstable and neutral subspaces for a
by: $E^{s}_{a}=\bigoplus_{\lambda(a)<0}E^{\lambda}$,
$E^{u}_{a}=\bigoplus_{\lambda(a)>0}E^{\lambda}$ and
$E^{0}_{a}=\bigoplus_{\lambda(a)=0}E^{\lambda}$. In particular, for any $a\in
A:=\bigcap_{\chi\neq 0}(Ker\chi)^{c}$ the subspace $E^{0}_{a}$ is the same and
thus can be denoted simply by $E^{0}$. Hence we have for any such $a:$
$TM=E^{s}_{a}\oplus E^{0}\oplus E^{u}_{a}.$
See [[12], Section 5.2] for more details. If in addition $E^{0}$ is a
continuous distribution uniquely integrable to a foliation $\mathcal{N}$ with
smooth leaves, and if there exists $a\in A$ such that $\alpha(a)$ is uniformly
normally hyperbolic with respect to $\mathcal{N}$ (in the sense of the Hirsch-
Pugh-Shub [10]) then $\alpha$ is a $partially$ $hyperbolic$ $action$. Elements
in $A$ which are uniformly normally hyperbolic with respect to $\mathcal{N}$
are called regular. Let $\widetilde{A}$ be the set of regular elements. We
call an action a partially hyperbolic $A$ action if the set $\widetilde{A}$ is
dense in $A$. In particular, if $E^{0}$ is the tangent distribution to the
orbit foliation of a normally hyperbolic action, then the action is called
Anosov.
If the set $\widetilde{A}$ is dense in $A$, then for each non-zero Lyapunov
exponent $\chi$ and every $p\in M$ the coarse Lyapunov distribution is:
$E_{\chi}(p)=\bigcap_{a\in\widetilde{A},\chi(a)<0}E^{s}_{a}(p).$
The right-hand side is Hölder continuous and $E^{\chi}$ can be extended to a
Hölder distribution tangent to the foliation
$\mathcal{T}_{\chi}:=\bigcap_{a\in\widetilde{A},\chi(a)<0}\mathcal{W}^{s}_{a}(p)$
with $C^{\infty}$ leaves. This is the $coarse$ $Lyapunov$ $foliation$
corresponding to $\chi$ (See [[2], Section 2] and [11]).
We denote by $\chi_{1},\dots,\chi_{r}$ a maximal collection of non-zero
Lyapunov exponents that are not positive multiples of one another and by
$\mathcal{T}_{1},\dots,\mathcal{T}_{r}$ the corresponding coarse Lyapunov
foliations.
Given a foliation $\mathcal{T}_{i}$ and $x\in M$ we denote by
$\mathcal{T}_{i}(x)$ the leaf of $\mathcal{T}_{i}$ through $x$.
### 3.2. Paths and cycles for a collection of foliations
. In this section we recall some notation and results from [3]. Let
$\mathcal{T}_{1},\dots,\mathcal{T}_{r}$ be a collection of mutually
transversal continuous foliations on $M$, with smooth simply connected leaves.
For $N\in\mathbb{N}$ and $j_{k}\in\\{1,\dots,r\\},k\in\\{1,\dots,N-1\\}$ an
ordered set of points $p(j_{1},\dots,j_{N-1}):x_{1},\dots,x_{N}\in M$ is
called an $\mathcal{T}$-path of length $N$ if for every
$k\in\\{1,\dots,N-1\\},x_{i+1}\in\mathcal{T}_{j_{k}}(x_{k})$. A closed
$\mathcal{T}$-path(i.e., when $x_{N}=x_{1}$) is a $\mathcal{T}$-cycle.
A $\mathcal{T}$-cycle $p(j_{1},\dots,j_{N-1}):x_{1},\dots,x_{N}=x_{1}\in M$ is
called $stable$ for the $A$ action $\alpha$ if there exists a regular element
$a\in A$ such that the whole cycle $p$ is contained in a leaf of the stable
foliations for the map $\alpha(a,\cdot)$, i.e., if
$\displaystyle\bigcap_{k=1}^{N}\\{a:\chi_{j_{k}}(a)<0\\}\neq\phi.$
###### Definition 3.1.
Let $p(j_{1},\dots,j_{N-1}):x_{1},\dots,x_{N}$ and
$p_{n}(j_{1},\dots,j_{N-1}):x^{n}_{1},\dots,x^{n}_{N}$ be two
$\mathcal{T}$-paths. Then $p=\lim_{n\rightarrow\infty}p_{n}$ if for all
$k\in\\{1,\dots,N\\}$
$\displaystyle x_{k}=\lim_{n\rightarrow\infty}x^{n}_{k}.$
Limits of $\mathcal{T}$-cycles are defined similarly.
Two $\mathcal{T}$-cycles,
$p(j_{1},\dots,j_{N+1}):x_{1},\dots,x_{k},y,x_{k},\dots,x_{N}=x_{1}$ and
$p(j_{1},\dots,j_{N-1}):x_{1},\dots,x_{k},x_{k+1},\dots x_{N}$ are said to be
conjugate if $y\in\mathcal{T}_{i}(x_{k})$ for some $i\in\\{1,\dots,r\\}$. For
$\mathcal{T}$-cycles, $p(j_{1},\dots,j_{N-1}):x_{1},\dots,x_{N}=x_{1}$ and
$p^{\prime}(j_{1}^{\prime},\dots,j_{K-1}^{\prime}):x_{1}=x_{1}^{\prime},\dots,x_{K}^{\prime}=x_{1}$
define their composition or concatenation $p\ast p^{\prime}$ by
$p\ast
p^{\prime}(j_{1},\dots,j_{N-1},j_{1}^{\prime},\dots,j_{K-1}^{\prime}):x_{1},\dots
x_{N},x_{1}^{\prime},\dots,x_{K}^{\prime}=x_{1}.$
Let $\mathcal{A}\mathcal{S}^{s}_{\mathcal{T}}(\alpha)$ denote the collection
of stable $\mathcal{T}$-cycles. Let
$\mathcal{A}\mathcal{S}_{\mathcal{T}}(\alpha)$ denote the collection of
$\mathcal{T}$-cycles which contains
$\mathcal{A}\mathcal{S}^{s}_{\mathcal{T}}(\alpha)$ and is closed under
conjugation, concatenation of cycles, and under the limitation procedure
defined above. $\mathcal{A}\mathcal{S}_{\mathcal{T}}^{x}(\alpha)$ denotes the
subset of $\mathcal{A}\mathcal{S}_{\mathcal{T}}(\alpha)$ which contain point
$x$.
A path $p:x_{1},\dots,x_{k},\dots,x_{N}$ reduces to a path
$p^{{}^{\prime}}:x_{1},x^{{}^{\prime}}_{2},\dots,x^{{}^{\prime}}_{k},\dots,x_{N}^{{}^{\prime}}$
via an $\alpha$-$allowable$ $\mathcal{T}$-substitution if the
$\mathcal{T}$-cycle
$\displaystyle p\ast
p^{{}^{\prime}}:x_{1},\dots,x_{k},\dots,x_{N-1},x_{N},x_{N-1}^{{}^{\prime}},\dots,x^{{}^{\prime}}_{2},x_{1}$
obtained by concatenation of $p$ and $p^{{}^{\prime}}$ is in the collection
$\mathcal{A}\mathcal{S}_{\mathcal{T}}(\alpha)$.
Two $\mathcal{T}$-cycle $c_{1}$ and $c_{2}$ are $\alpha$-equivalent if $c_{1}$
reduces to $c_{2}$ via a finite sequence of $\alpha$-allowable $\mathcal{T}$
-substitutions. A $\mathcal{T}$-cycle we call $\alpha$-reducible if it is in
$\mathcal{A}\mathcal{S}_{\mathcal{T}}(\alpha)$.
###### Definition 3.2.
For $N\in\mathbb{N}$ and $j_{k}\in\\{1,\dots,r\\},k\in\\{1,\dots,N\\}$ an
ordered set of points $p(j_{1},\dots,j_{N}):x_{1},\dots,x_{N},x_{N+1}=x_{1}\in
M$ is called an $\mathcal{T}$-cycle of length $N$ if for every
$k\in\\{1,\dots,N\\},x_{i+1}\in\mathcal{T}_{j_{k}}(x_{k})$. A $\mathcal{T}$
cycle which consists of a single point is a trivial $\mathcal{T}$-cycle.
###### Definition 3.3.
Foliations $\mathcal{T}_{1},\dots,\mathcal{T}_{r}$ are locally transitive if
there exists $N\in\mathbb{N}$ such that for any $\varepsilon>0$ there exists
$\delta>0$ such that for every $x\in M$ and for every $y\in B_{X}(x,\delta)$
(where $B_{M}(x,\delta)$ is a $\delta$ ball in $M$) there is a
$\mathcal{T}$-path
$p(j_{1},\dots,j_{N-1}):x=x_{1},x_{2},\dots,x_{N-1},x_{N}=y$ in the ball
$B_{M}(x,\varepsilon)$ such that $x_{k+1}\in\mathcal{T}_{j_{k}}(x_{k})$ and
$d_{\mathcal{T}_{j_{k}}(x_{k})}(x_{k+1},x_{k})<2\varepsilon$.
In other words, any two sufficiently close points can be connected by a
$\mathcal{T}$-path of not more than $N$ pieces of a given bounded length.
Here, for a submanifold $Y$ in $M$, $d_{Y}(x,y)$ denotes the infimum of
lengths of smooth curves in $Y$ connecting $x$ and $y$.
###### Definition 3.4.
For a partially hyperbolic $A$-action $\alpha$ on a compact manifold $M$ with
coarse Lyapunov foliations $\mathcal{T}_{1},\dots,\mathcal{T}_{r}$ and for a
cocycle $\beta:AM\rightarrow Y$ over $\alpha$, where $Y$ is a Lie group, we
define $Y$-valued potential of $\beta$ as
$\displaystyle\left\\{\begin{aligned}
&P^{j}_{a}(y,x)=\lim_{n\rightarrow+\infty}\beta(na,y)^{-1}\beta(na,x),\qquad\chi_{j}(a)<0\\\
&P^{j}_{a}(y,x)=\lim_{n\rightarrow-\infty}\beta(na,y)^{-1}\beta(na,x),\qquad\chi_{j}(a)>0\end{aligned}\right.$
Now for any $\mathcal{T}$-cycle $\mathfrak{c}:x_{1},\dots,x_{N+1}=x_{1}$ on
$M$, we can define the corresponding periodic cycle functional:
(3.1)
$\displaystyle\text{PCF}(\mathfrak{c})(\beta)=\prod_{i=1}^{N}P^{j(i)}_{a}(x_{i},x_{i+1})(\beta).$
It is proved in [2] that the expression for (PCF) does not depend on the
choice of $a$. For a general Lie group H there may be a “competition” between
the exponential speed of decay for the distance between $nax$ and $nay$ on the
one hand, and the exponential growth of the cocycle norm on the other. More
information about guaranteeing convergence of non-abelian potentials can be
found in [2].
In this paper, we only consider $Y=Y_{X}$ which possesses a bi-invariant
metric, so the limits in the right hand part of 3.4 always exist.
Two essential properties of the PCF which are crucial for our purpose are that
PCF is continuous and that it is invariant under the operation of moving
cycles around by elements of the action $\alpha$. We end this section with an
important proposition which is the base of our further proof.
###### Proposition 3.1.
_(Proposition 4.[2])_ Let $\alpha$ be an $\mathbb{R}^{k}$ action by
diffeomorphisms on a compact Riemannian manifold $M$ such that a dense set of
elements of $\mathbb{R}^{k}$ acts normally hyperbolically with respect to an
invariant foliation. If the foliations $\mathcal{F}_{1},\dots,\mathcal{F}_{r}$
are locally transitive and if $\beta$ is a Hölder cocycle over the action
$\alpha$ such that $F(\mathcal{C})(\beta)=0$ for any cycle $\mathcal{C}$ then:
$\beta$ is cohomologous to a constant cocycle via a continuous map
$h:M\rightarrow Y$.
### 3.3. Split Cartan actions on $SO^{+}(m,n)/\Gamma$ and $SU(m,n)/\Gamma$
We use notations from Section 2.1. Let $d(\cdot,\cdot)$ denote a right
invariant metric on $SO^{+}(m,n)$ and the induced metric on
$SO^{+}(m,n)/\Gamma$. We use $e_{k,\ell}$ to denote the matrix with the
$(k,\ell)$ element equal to 1, and all other elements equal to 0. Let $1\leq
i,j\leq n,i\neq j$ be two distinct indices, $\ell\leq m-n$, and let $\exp$ be
the exponentiation map for matrices.
Let $\Phi$ be the root system of $SO^{+}(m,n)$ with respect to $D_{+}$. The
roots are $\pm L_{i}\pm L_{j}(i<j\leq n)$, whose dimensions are one and $\pm
L_{i}(1\leq i\leq n)$ are also roots if $m\geq n+1$ with dimensions $m-n$. The
corresponding root spaces are
$\displaystyle\mathfrak{g}_{L_{i}+L_{j}}=\mathbb{R}(e_{i,j+n}-e_{j,i+n})_{i<j},\qquad\mathfrak{g}_{L_{i}-L_{j}}=\mathbb{R}(e_{i,j}-e_{j+n,i+n})_{i\neq
j},$
$\displaystyle\mathfrak{g}_{-L_{i}-L_{j}}=\mathbb{R}(e_{j+n,i}-e_{i+n,j})_{i<j},$
$\displaystyle\mathfrak{g}_{L_{i}}=\bigoplus_{\ell\leq
m-n}\mathbb{R}f_{L_{i}}^{\ell},\text{ where
}f_{L_{i}}^{\ell}=e_{i,2n+\ell}-e_{2n+\ell,i+n},$
$\displaystyle\mathfrak{g}_{-L_{i}}=\bigoplus_{\ell\leq
m-n}\mathbb{R}f_{-L_{i}}^{\ell},\text{ where
}f_{-L_{i}}^{\ell}=e_{i+n,2n+\ell}-e_{2n+\ell,i}.$
Let
$\displaystyle f_{L_{i}+L_{j}}=(e_{i,j+n}-e_{j,i+n})_{i<j},\qquad
f_{L_{i}-L_{j}}=(e_{i,j}-e_{j+n,i+n})_{i\neq j},$ $\displaystyle
f_{-L_{i}-L_{j}}=(e_{j+n,i}-e_{i+n,j})_{i<j}.$
With these notations for $t\in\mathbb{R}$,
$a=(a_{1},...,a_{m-n})\in\mathbb{R}^{m-n}$ we define foliations $F_{r}$ for
$r=\pm L_{i}\pm L_{j},i\neq j$ and $F_{\rho}$ for $\rho=\pm L_{i}$ for which
the leaf through $x$
(3.2) $\displaystyle F_{r}(x)=\exp(tf_{r})x,\qquad
F_{\rho}(x)=\bigl{(}\prod_{j}\exp(a_{j}f^{j}_{\rho})\bigl{)}x$
consists of all left multiples of $x$ by matrices of the form $F_{r}(t)$ or
$F_{\rho}(a)$.
The foliations $F_{r}$ and $F_{\rho}$ are invariant under $\alpha_{0}$. In
fact, let $\mathfrak{t}=(t_{1},t_{2},\dots,t_{n})\in\mathbb{D_{+}},$ for
$\forall t\in\mathbb{R}$ we have Lie bracket relations
$[\mathfrak{t},tf_{r}]=r(\mathfrak{t})tf_{r},\qquad[\mathfrak{t},tf_{\rho}^{\ell}]=\alpha(\mathfrak{t})tf_{\rho}^{\ell}$
where $r(\mathfrak{t})=\pm t_{i}\pm t_{j}$ if $r=\pm L_{i}\pm L_{j}$;
$\rho(\mathfrak{t})=\pm t_{i}$ if $\rho=\pm L_{i}$.
Using the basic identity for any square matrices $X,Y$:
$\exp X\exp Y=\exp(e^{s}Y)\exp X,\text{ if }[X,Y]=sY,$
it follows
(3.3)
$\displaystyle\alpha_{0}(\mathfrak{t})\bigl{(}\exp(tf_{r})\bigl{)}x=\exp(te^{r(\mathfrak{t})}f_{r})\alpha_{0}(\mathfrak{t})x,$
(3.4)
$\displaystyle\alpha_{0}(\mathfrak{t})\bigl{(}\prod_{j}\exp(a_{j}f^{j}_{\rho})\bigl{)}x=\bigl{(}\prod_{j}\exp(a_{j}e^{\rho(\mathfrak{t})}f^{j}_{\rho})\bigl{)}\alpha_{0}(\mathfrak{t})x$
where $r(\mathfrak{t})=\pm t_{i}\pm t_{j}$ if $r=\pm L_{i}\pm L_{j}$;
$\rho(\mathfrak{t})=\pm t_{i}$ if $\rho=\pm L_{i}$. Hence the leaf $F_{r}(x)$
is mapped into $F_{r}(\alpha_{0}(\mathfrak{t})x)$ and $F_{\rho}(x)$ is mapped
into $F_{\rho}(\alpha_{0}(\mathfrak{t})x)$. Consequently the foliation $F_{r}$
and $F_{\rho}$ are contracted (corr. expanded or neutral) under $\mathfrak{t}$
if $r(\mathfrak{t})<0$ (corr. $r(\mathfrak{t})>0$ or $r(\mathfrak{t})=0$). If
the foliation $F_{r}$ and $F_{\rho}$ are neutral under
$\alpha_{0}(\mathfrak{t})$, it is in fact isometric under
$\alpha_{0}(\mathfrak{t})$. The leaves of the orbit foliation is
$\mathcal{O}(x)=\\{\alpha_{0}(\mathfrak{t})x:\mathfrak{t}\in\mathbb{D_{+}}\\}$.
The tangent vectors to the leaves in (3.2) for various $r$ and $\rho$ together
with their length one Lie brackets form a basis of the tangent space at every
$x\in X$.
Let $\Phi^{\prime}$ be the root system of $SU(m,n)$ with respect to $D_{+}$.
The roots are $\pm L_{i}\pm L_{j}(i<j\leq n)$, whose dimensions are 2 and $\pm
2L_{i}(i\leq n)$ whose dimension is 1. Also the $\pm L_{i}(i\leq n)$ are roots
if $m\neq n$ with dimensions $2(m-n)$. The corresponding root spaces are
$\displaystyle\mathfrak{g}_{L_{i}+L_{j}}=\mathbb{R}(e_{i,j+n}-e_{j,i+n})_{i<j}\oplus\mathbb{R}\textrm{i}(e_{i,j+n}+e_{j,i+n})_{i<j},$
$\displaystyle\mathfrak{g}_{L_{i}-L_{j}}=\mathbb{R}(e_{i,j}-e_{j+n,i+n})_{i\neq
j}\oplus\mathbb{R}\textrm{i}(e_{i,j}+e_{j+n,i+n})_{i\neq j},$
$\displaystyle\mathfrak{g}_{-L_{i}-L_{j}}=\mathbb{R}(e_{j+n,i}-e_{i+n,j})_{i<j}\oplus\mathbb{R}\textrm{i}(e_{j+n,i}+e_{i+n,j})_{i<j},$
$\displaystyle\mathfrak{g}_{L_{i}}=\bigoplus_{\ell\leq
m-n}\bigl{(}\mathbb{R}(e_{i,2n+\ell}-e_{2n+\ell,i+n})\oplus\mathbb{R}\textrm{i}(e_{i,2n+\ell}+e_{2n+\ell,i+n})\bigl{)},$
$\displaystyle\mathfrak{g}_{-L_{i}}=\bigoplus_{\ell\leq
m-n}\bigl{(}\mathbb{R}(e_{i+n,2n+\ell}-e_{2n+\ell,i})\oplus\mathbb{R}\textrm{i}(e_{i+n,2n+\ell}+e_{2n+\ell,i})\bigl{)},$
$\displaystyle\mathfrak{g}_{-2L_{i}}=\mathbb{R}\textrm{i}e_{i+n,i},\qquad\mathfrak{g}_{2L_{i}}=\mathbb{R}\textrm{i}e_{i,i+n}.$
For $z\in\mathbb{C}$ and $t\in\mathbb{R}$, let
$\displaystyle f_{L_{i}+L_{j}}(z)$
$\displaystyle=(ze_{i,j+n}-\overline{z}e_{j,i+n})_{i<j},\qquad$ $\displaystyle
f_{L_{i}-L_{j}}(z)$ $\displaystyle=(ze_{i,j}-\overline{z}e_{j+n,i+n})_{i\neq
j},$ $\displaystyle f_{-L_{i}-L_{j}}(z)$
$\displaystyle=(ze_{j+n,i}-\overline{z}e_{i+n,j})_{i<j},\qquad$ $\displaystyle
f_{L_{i}}^{\ell}(z)$
$\displaystyle=ze_{i,2n+\ell}-\overline{z}e_{2n+\ell,i+n},$ $\displaystyle
f_{-L_{i}}^{\ell}(z)$
$\displaystyle=ze_{i+n,2n+\ell}-\overline{z}e_{2n+\ell,i},\qquad$
$\displaystyle f_{2L_{i}}(t)$ $\displaystyle=t\textrm{i}e_{i,i+n},$
$\displaystyle f_{-2L_{i}}(t)$ $\displaystyle=t\textrm{i}e_{i+n,i}.$
With these notations, for $z\in\mathbb{C}$, $t\in\mathbb{R}$,
$a=(a_{1},\dots,a_{m-n})\in\mathbb{C}^{m-n}$ individual expanding and
contracting foliations are similarly given by $F_{r}$ for $r=\pm L_{i}\pm
L_{j},i\neq j$ and $F_{\rho}$ for $\rho=\pm L_{i}$ for which the leaf through
$x$
(3.5) $\displaystyle F_{r}(z)=\exp\bigl{(}f_{r}(z)\bigl{)}x,\qquad
F_{\rho}(t,a)=\exp\bigl{(}f_{2\rho}(t)\bigl{)}\exp\bigl{(}\sum_{j}f^{j}_{\rho}(a_{j})\bigl{)}x$
consists of all left multiples of $x$ by matrices of the form $F_{r}(t)$ or
$F_{\rho}(a)$.
The foliations $F_{r}$ and $F_{\rho}$ are invariant under $\alpha_{0}$. In
fact, let $\mathfrak{t}=(t_{1},t_{2},\dots,t_{n})\in\mathbb{D_{+}},$ for
$\forall z\in\mathbb{C}$ and $t\in\mathbb{R}$ we have Lie bracket relations
$\displaystyle[\mathfrak{t},f_{r}(z)]$
$\displaystyle=r(\mathfrak{t})f_{r}(z),\qquad[\mathfrak{t},f_{\rho}^{\ell}(z)]=\alpha(\mathfrak{t})f_{\rho}^{\ell}(z),$
$\displaystyle[\mathfrak{t},f_{2\rho}(t)]$
$\displaystyle=2\rho(\mathfrak{t})f_{2\rho}(t)$
where $r(\mathfrak{t})=\pm t_{i}\pm t_{j}$ if $\rho=\pm L_{i}\pm L_{j}$;
$\rho(\mathfrak{t})=\pm t_{i}$ if $\rho=\pm L_{i}$.
Using the basic identity for any square matrices $X,Y$:
$\exp X\exp Y=\exp(e^{s}Y)\exp X,\text{ if }[X,Y]=sY,$
it follows
(3.6)
$\displaystyle\alpha_{0}(\mathfrak{t})\exp\bigl{(}f_{r}(z)\bigl{)}x=\exp\bigl{(}f_{r}(e^{r(\mathfrak{t})}z)\bigl{)}\alpha_{0}(\mathfrak{t})x,$
$\displaystyle\alpha_{0}(\mathfrak{t})\exp\bigl{(}f_{2\rho}(t)\bigl{)}\exp\bigl{(}\sum_{j}f^{j}_{\rho}(a_{j})\bigl{)}x$
(3.7)
$\displaystyle=\exp\bigl{(}f_{2\rho}(e^{2\rho(\mathfrak{t})}t)\bigl{)}\exp\bigl{(}\sum_{j}f^{j}_{\rho}(e^{\rho(\mathfrak{t})}a_{j})\bigl{)}\alpha_{0}(\mathfrak{t})x$
where $r(\mathfrak{t})=\pm t_{i}\pm t_{j}$ if $r=\pm L_{i}\pm L_{j}$;
$\rho(\mathfrak{t})=\pm t_{i}$ if $\rho=\pm L_{i}$. Hence the leaf $F_{r}(x)$
is mapped into $F_{r}(\alpha_{0}(\mathfrak{t})x)$ and $F_{\rho}(x)$ is mapped
into $F_{\rho}(\alpha_{0}(\mathfrak{t})x)$. Consequently the foliation $F_{r}$
and $F_{\rho}$ are contracted (corr. expanded or neutral) under $\mathfrak{t}$
if $r(\mathfrak{t})<0$ (corr. $r(\mathfrak{t})>0$ or $r(\mathfrak{t})=0$). If
the foliation $F_{r}$ and $F_{\rho}$ are neutral under
$\alpha_{0}(\mathfrak{t})$, they are in fact isometric under
$\alpha_{0}(\mathfrak{t})$. The leaves of the orbit foliation are
$\mathcal{O}(x)=\\{\alpha_{0}(\mathfrak{t})x:\mathfrak{t}\in\mathbb{D_{+}}\\}$.
The tangent vectors to the leaves in (3.5) for various $r$ and $\rho$ together
with their length one Lie brackets form a basis of the tangent space at every
$x\in X$.
If $\mathbb{P}$ is a $2$-plane in general position then the foliations $F_{r}$
and $F_{\rho}$ are also Lyapunov foliations for $\alpha_{0,\mathbb{P}}$. The
leaves of $F_{r}$ and $F_{\rho}$ are intersections of the leaves of stable
manifolds of the action by different elements of $\mathbb{P}$. The same holds
for the action by any regular lattice in $\mathbb{P}$ and thus for any generic
restriction $\alpha_{0,G}$. The neutral foliation for a generic restriction
$\alpha_{0,G}$ will be denoted by $\mathcal{N}_{0}$.
###### Remark 3.1.
If $m\geq n+1$, neutral foliation of the full split Cartan action, as well as
any generic restriction contains not only the orbit foliation, but also
compact part of the centralizer of the maximal split Cartan subgroup. In fact,
the neutral foliation of the Cartan action is given by
$\mathcal{N}_{0}(x)=\\{Y_{X}\cdot x:x\in X\\}.$
The above discussion can be summarized as follows.
###### Proposition 3.2.
(1) Non-zero Lyapunov exponents for the full Cartan action on
$SO^{+}(m,n)/\Gamma$ are $\pm t_{i}\pm t_{j}$(each has multiplicity 1)and $\pm
t_{i}$(each has multiplicity $m-n$) where $i\neq j$ and $1\leq i,j\leq n$.
Zero Lyapunov exponent comes from the neutral foliation and has multiplicity
$n+\frac{(m-n-1)(m-n)}{2}$. Consequently any matrix $d\in D_{+}$ whose first
$n$ diagonal entries are pairwise different acts normally hyperbolically on
$SO^{+}(m,n)/\Gamma$ with respect to the neutral foliation and hence is only
partially hyperbolic.
(2) If $m\geq n+1$, non-zero Lyapunov exponents for the full Cartan action on
$SU(m,n)/\Gamma$ are $\pm t_{i}\pm t_{j}$(each has multiplicity 2) and $\pm
t_{i}$(each has multiplicity $2m$) where $i\neq j$ and $1\leq i,j\leq n$; if
$m=n$, non-zero Lyapunov exponents are $\pm t_{i}\pm t_{j}$, $i\neq j$(each
has multiplicity 2) and $\pm 2t_{i}$(each has multiplicity 1) where $i\neq j$
and $1\leq i,j\leq n$. Zero Lyapunov exponent comes from the neutral foliation
and has multiplicity $2n+(m-n)^{2}-1$. Any matrix $d\in D_{+}$ whose first $n$
diagonal entries are pairwise different acts normally hyperbolically on
$SU(m,n)/\Gamma$ with respect to the neutral foliation.
### 3.4. Generating relations and Steinberg symbols
In this section we state two theorems which play a crucial role in proofs of
Theorem 3. The proof of those theorems are given in Section 5–7 which comprise
the algebraic part of the paper.
We use notation set in Section 3.3. Since $\mathbb{R}$ is embedded in
$\mathbb{R}^{m-n}$ in a obvious way, there is no confusion if we write
$F_{r}(t,0,\dots,0)=F_{r}(t)$ for $r=\pm L_{i}\pm L_{j}$. On the other hand,
if we write $F_{r}(a)$ where $a\in\mathbb{R}^{m-n}$, then
$a=(a_{1},0,\dots,0)$ for some $a_{1}\in\mathbb{R}$.
###### Theorem 4.
$SO^{+}(m,n)$, $3\leq n\leq m$ is generated by $F_{r}(a)$, where $r=\pm
L_{i}\pm L_{j},\pm L_{i}$, $0\leq i\neq j\leq n$ and $a\in\mathbb{R}^{m-n}$
subject to the relations:
(3.8) $\displaystyle F_{r}(a)F_{r}(b)=F_{r}(a+b),$ (3.9)
$\displaystyle[F_{r}(a),F_{p}(b)]=\prod_{ir+jp\in\Phi,i,j>0}F_{ir+jp}(g_{ijpr}(a,b)),r+p\neq
0$ (3.10) $\displaystyle[F_{r}(a),F_{p}(b)]=\emph{id},\qquad 0\neq
r+p\notin\Phi,$
here $a,b\in\mathbb{R}^{m-n}$ and $g_{ijpr}$ are functions of $a,b$ depending
only on the structure of $SO^{+}(m,n)$;
(3.11) $\displaystyle
h_{L_{1}-L_{2}}(t)h_{L_{1}-L_{2}}(s)=h_{L_{1}-L_{2}}(ts),$
where
$h_{L_{1}-L_{2}}(t)=F_{L_{1}-L_{2}}(t)F_{L_{2}-L_{1}}(-t^{-1})F_{L_{1}-L_{2}}(t)F_{L_{1}-L_{2}}(-1)F_{L_{2}-L_{1}}(1)F_{L_{1}-L_{2}}(-1)$
for each $t\in\mathbb{R}^{*}$;
(3.12) $\displaystyle h_{L_{1}-L_{2}}(-1)h_{L_{1}+L_{2}}(-1)=\emph{id},$
where
$h_{L_{1}+L_{2}}(t)=F_{L_{1}+L_{2}}(t)F_{-L_{1}-L_{2}}(-t^{-1})F_{L_{1}+L_{2}}(t)F_{L_{1}+L_{2}}(-1)F_{-L_{1}-L_{2}}(1)F_{L_{1}+L_{2}}(-1)$
for each $t\in\mathbb{R}^{*}$;
(3.13) $\displaystyle
h^{1}_{L_{n}}(\sqrt{2}a,\sqrt{2}b)h^{1}_{L_{n}}(\sqrt{2}c,\sqrt{2}d)=h^{1}_{L_{n}}\bigl{(}\sqrt{2}(ac-
bd),\sqrt{2}(ad+bc)\bigl{)},$
where
$\displaystyle
h^{1}_{L_{n}}(\sqrt{2}a,\sqrt{2}b)=F^{1}_{L_{n}}(\sqrt{2}a)F^{2}_{L_{n}}(\sqrt{2}b)F^{1}_{-L_{n}}(\sqrt{2}a)F^{2}_{-L_{n}}(\sqrt{2}b)F^{1}_{L_{n}}(\sqrt{2}a)F^{2}_{L_{n}}(\sqrt{2}b)$
$\displaystyle\cdot
F^{1}_{L_{n}}(-\sqrt{2})F^{1}_{-L_{n}}(-\sqrt{2})F^{1}_{L_{n}}(-\sqrt{2})$
for each $(a,b)\in S^{1}$.
If $n\leq m\leq n+1$, there is no relation 3.13.
Now we consider the group $SU(m,n)$. We write $F_{r}(0,z,0,\dots,0)=F_{r}(z)$
for $r=\pm L_{i}\pm L_{j}$ where $z\in\mathbb{C}$. On the other hand, if we
write $F_{r}(a)$ where $a\in\mathbb{R}\times\mathbb{C}^{m-n}$, then
$a=(0,a_{1},0,\dots,0)$ for some $a_{1}\in\mathbb{C}$.
###### Theorem 5.
Let $\Phi$ be the root system of $SU(m,n)$ $3\leq n\leq m$. Then $SU(m,n)$ is
generated by $F_{r}(a)$ where $r=\pm L_{i}\pm L_{j},\pm L_{i}$, $i\neq j$ and
$a\in\mathbb{R}\times\mathbb{C}^{m-n}$ subject to the relations:
(3.14) $\displaystyle F_{r}(a)F_{r}(b)=F_{r}(a+b),$ (3.15)
$\displaystyle[F_{r}(a),F_{p}(b)]=\prod_{\begin{subarray}{c}i,j>0\\\
ir+jp\in\Phi\end{subarray}}F_{ir+jp}(N_{r,p,i,j}(a,b)),r+p\neq 0$ (3.16)
$\displaystyle[F_{r}(a),F_{p}(b)]=e,\qquad 0\neq r+p\notin\Phi,$
here $a,b\in\mathbb{C}^{m-n}$, $N_{r,p,i,j}$ are funtions depend only on the
structure of $SU(m,n)$;
and the following relations:
(3.17) $\displaystyle
h_{L_{1}-L_{2}}(z_{1})h_{L_{1}-L_{2}}(z_{2})=h_{L_{1}-L_{2}}(z_{1}z_{2})$
where
$\displaystyle h_{L_{1}-L_{2}}(z)$
$\displaystyle=F_{L_{1}-L_{2}}(z)F_{L_{2}-L_{1}}(-z^{-1})F_{L_{1}-L_{2}}(z)$
$\displaystyle\cdot F_{L_{1}-L_{2}}(-1)F_{L_{2}-L_{1}}(1)F_{L_{1}-L_{2}}(-1)$
for each $z\in\mathbb{C}^{*}$;
(3.18) $\displaystyle h_{2L_{n}}(-1)h_{2L_{n}}(-1)=\emph{id},$
where
$\displaystyle h_{2L_{n}}(-1)$
$\displaystyle=\bigl{(}F_{L_{n}}(-1,0,\dots,0)F_{-L_{n}}(-1,0,\dots,0)F_{L_{n}}(-1,0,\dots,0)\bigl{)}^{2};$
and
(3.19) $\displaystyle
h^{1}_{L_{n}}(\sqrt{2}a,\sqrt{2}b)h^{1}_{L_{n}}(\sqrt{2}c,\sqrt{2}d)$
$\displaystyle=h^{1}_{L_{n}}\bigl{(}\sqrt{2}(ac-bd),\sqrt{2}(ad+bc)\bigl{)}$
(3.20) $\displaystyle
h^{1}_{L_{n}}(\sqrt{2}a,\sqrt{2}b\emph{i})h^{1}_{L_{n}}\bigl{(}\sqrt{2}c,\sqrt{2}d\emph{i})$
$\displaystyle=h^{1}_{L_{n}}(\sqrt{2}(ac-bd),\sqrt{2}(ad+bc)\emph{i}\bigl{)}$
for each $(a,b),(c,d)\in S^{1}$ where
$\displaystyle h^{1}_{L_{n}}(\sqrt{2}z,\sqrt{2}w)$
$\displaystyle=F_{L_{n}}(0,\sqrt{2}z,\sqrt{2}w,0,\dots,0)F_{-L_{n}}(0,\sqrt{2}z,\sqrt{2}w,0,\dots,0)$
$\displaystyle\cdot
F_{L_{n}}(0,\sqrt{2}z,\sqrt{2}w,0,\dots,0)F_{L_{n}}(0,-\sqrt{2},0,\dots,0)$
$\displaystyle\cdot F_{-L_{n}}(0,-\sqrt{2},0,\dots,0)\cdot
F_{L_{n}}(0,-\sqrt{2},0,\dots,0)$
for $(z,w)\in\mathbb{C}^{2}$ with $\lvert z\rvert^{2}+\lvert w\rvert^{2}=1$.
If $n\leq m\leq n+1$ there are no relations 3.19 and 3.20.
Relations 3.8–3.12 in Theorem 4 and 3.14–3.18 in Theorem 5 are similar to
those in split groups [23]; while relation 3.19 and 3.20 are dealing with
“rotations” inside the compact part of the centralizer of the maximal split
Cartan subgroups.
Any bi-multiplicative map $c:\mathbb{K}^{*}\times\mathbb{K}^{*}\rightarrow B$
into an abelian group $B$ satisfying $c(t,1-t)=1_{B}$ is called a Steinberg
symbol on the field $\mathbb{K}$. We will use the following result about
continuous Steinberg symbols for the field $\mathbb{R}$ and $\mathbb{C}$ [21]:
###### Theorem 6 (Milnor).
a)Every continuous Steinberg symbol on the field $\mathbb{C}$ of complex
numbers is trivial.
b) If $c(t,s)$ is a continuous Steinberg symbol on the field $\mathbb{R}$,
then $c(t,s)=1$ if $s$ or $t$ are positive, and $c(t,s)=c(-1,-1)$ has order at
most 2 if $s$ and $t$ are both negative.
The following Lemma treats a case which occurs in the proof of Theorem 3 when
one can study reducible classes within homotopy classes of Lyapunov-cycles.
###### Lemma 3.1.
Let $L$ be an irreducible lattice in $\widetilde{SO}^{+}(m,n)$(corr.
$\widetilde{SU}(m,n)$) where $\widetilde{SO}^{+}(m,n)$ is the universal cover
of $SO^{+}(m,n)$(corr. $SU(m,n)$. Then for any homomorphism from $L$ to
$Y_{X}$, the order of image $h(L)$ is bounded by a number only dependent on
$m-n$.
###### Proof.
We first consider the case of $\widetilde{SO}^{+}(m,n)$. Let $h$ be a
homomorphism from $L$ to $Y_{X}$. We decompose $h=(h_{1},h_{2})$ where $h_{1}$
maps $L$ to $\mathbb{R}^{n}$ and $h_{2}$ maps $L$ to $SO(m-n)$. By the
Margulis Normal Subgroup Theorem [17, 4’ Theorem], $h_{1}$ is trivial. Thus
$h$ can be considered as a homomorphism from $L$ to $SO(m-n)$. By [17,
(3)Theorem], Zariski closure of $\overline{h(L)}$ is a semisimple
$\mathbb{Q}$-algebraic group. We first show that $h(L)$ is finite. Suppose it
is not finite. Since $\overline{h(L)}/\overline{h(L)}^{0}$ is finite, we can
assume $\overline{h(L)}$ is connected. Then $\overline{h(L)}$ decomposes
uniquely into (up to permutation of the factors) a direct product of
$\mathbb{Q}$-almost simple algebraic linear groups. We can assume
$\overline{h(L)}$ is almost simple. Compose $h$ with a Galois automorphism
$\sigma$ of $\mathbb{C}$ over $\mathbb{Q}$ to matrix coefficients of elements
from $h(L)$, then $\sigma h(L)$ is a non-compact subgroup of
$\overline{h(L)}=\sigma\overline{h(L)}$. By finiteness of
$Z(\overline{h(L)})$, we can assume $(\sigma h)^{\prime}$ is from $L$ to
$\overline{h(L)}/Z(\overline{h(L)})$. By Margulis lattice superrigidity
Theorem [17, 2’Theorem], $(\sigma h)^{\prime}$ can be extended to a continuous
homomorphism $\widetilde{h}$ from $\widetilde{SO}^{+}(m,n)$ to
$\overline{h(L)}/Z(\overline{h(L)})$. We can assume $\widetilde{h}$ is from
$\widetilde{SO}^{+}(m,n)$ to $\overline{h(L)}$. By simplicity of
$\widetilde{SO}^{+}(m,n)$(in the meaning of Lie groups),
$\ker(\tilde{h})\subseteq Z\bigl{(}\widetilde{SO}^{+}(m,n)\bigl{)}$ or
$\ker(\tilde{h})=\widetilde{SO}^{+}(m,n)$. While
$\ker(\tilde{h})=\widetilde{SO}^{+}(m,n)$ contradicts the infiniteness of
$h(L)$. Notice $\overline{h(L)}\subseteq SO(m-n,\mathbb{C})$. Thus we get a
continues isomorphism from $\widetilde{SO}^{+}(m,n)/\ker(\tilde{h})$ to its
image, which contradicts the fact the the maximal connected compact subgroup
in $\widetilde{SO}^{+}(m,n)$ is the universal cover of $SO(n)\times SO(m)$
while in $SO(m-n,\mathbb{C})$ is $SO(m-n)$. Hence we proved that $h(L)$ is
finite.
Next, we show this number is bounded independence of the homomorphism. By
Jordan’s theorem which claims that any finite group $G\subseteq
GL(\ell,\mathbb{C})$ contains a normal abelian subgroup whose index is at most
$j(\ell)$, we let the biggest normal abelian subgroup in $h(L)$ be $A$.
Consider the restriction of $h$ from $h^{-1}(A)$ to $A$. The index of
$[L:h^{-1}(A)]$ is bounded by $j(m-n)$. There are only finitely many
sublattices $L^{\prime}$ in $L$ with the index smaller than $j(m-n)$, the
arguments go as follows: first, every subgroup of finite index in $L$ contains
a normal subgroup of $L$ with index dividing $j(m-n)!$. So, it suffices to
check that the number of normal subgroups of index smaller than $j(m-n)$ is
finite. Such normal subgroups are exactly the kernels of (surjective)
homomorphisms of $L$ into a finite group of order smaller than $j(m-n)$.
Recall that the set of finite groups of order small than $j(m-n)$ is finite
(up to an isomorphism). On the other hand, since $L$ is finitely generated (as
a group)[17], the set of homomorphisms of $L$ into a given finite group is
finite.
Since $[h^{-1}(A):[h^{-1}(A),h^{-1}(A)]]$ is finite[17, 4’ Theorem], it is
bounded by a number $i(m-n)$ which depends only on $m-n$ by above analysis.
The order of $A$ is bounded by $i(m-n)$ by its abelian property. Hence the
order of $h(L)$ is bounded by $i(m-n)j(m-n)$.
For $\widetilde{SU}(m,n)$, Notice here $h=(h_{1},h_{2},h_{3})$ where $h_{1}$
maps $L$ to $\mathbb{R}^{n}$, $h_{2}$ maps $L$ to $\mathbb{T}^{n-1}$ and
$h_{3}$ maps $L$ to $U(m-n)$. $h_{1}$ is trivial by Margulis Normal Subgroup
Theorem; order of $h_{2}(L)$ is bounded by $[L,[L,L]]$. To prove finiteness of
$h_{3}(L)$, we can assume it is from $L$ to $SU(m-n)$. Similar arguments hold
to get a continuous isomorphism from $\widetilde{SU}(m-n)/D$ to a subgroup
inside $SL(m-n,\mathbb{C})$, since the real locus of $SL(m-n,\mathbb{C})$ is
$SU(m-n)$, where $D$ is inside the center of $\widetilde{SU}(m-n)$. Thus
$[Z(\widetilde{SU}(m-n)):D]$ is finite by the fact that every simple matrix
group has finite center, which contradicts the fact that the maximal connected
compact subgroups in $\widetilde{SU}(m,n)/D$ is a finite lift of
$S\bigl{(}U(m)\times U(n)\bigl{)}$, while in $SL(m-n,\mathbb{C})$ is
$SU(m-n)$. The next step to prove the uniform bound of order of $h_{3}(L)$ is
exactly the same as previous case. ∎
### 3.5. Proof of Theorem 3
Notice $\alpha_{0,G}$ can be lifted to a $G$-action on
$\widetilde{SO}^{+}(m,n)$(corr. $\widetilde{SU}(m,n)$) where
$\widetilde{SO}^{+}(m,n)$ is the universal cover of $SO^{+}(m,n)$(corr.
$\widetilde{SU}(m,n)$ is the universal cover of $SU(m,n)$). We denote the new
action by $\widetilde{\alpha}_{0,G}$ and the projection from
$\widetilde{SO}^{+}(m,n)$(corr. $\widetilde{SU}(m,n)$) to $SO^{+}(m,n)$(corr.
$SU(m,n)$) by $p$. We proceed in exactly the same manner as in [2].
At first we show the cocycle rigidity for Hölder cocycles. The invariant
foliations that we considered in section 3.3 are $F_{r}$ and $F_{\rho}$ where
$r=\pm L_{i}\pm L_{j},i\neq j$ and $\rho=\pm L_{i}$. Notice that those
foliations are smooth and their Lie brackets at length one generate the whole
tangent space. This implies that this system of foliations is locally
$1/2-H\ddot{o}lder$ transitive ([[13], Section 4, Proposition 1]). Hence the
lifted foliations which we still denote by $F_{r}$ and $F_{\rho}$ without
confusion are locally transitive on the universal covering spaces. Every such
cycle represents a relation in the group. The word represented by this cycle
can be written as a product of conjugates of basic relations in Theorem 4 and
5 that can be lifted to closed cycles in the universal covering spaces.
At first we consider $\widetilde{SO}^{+}(m,n)$ which is a 4-fold covering
space of $SO^{+}(m,n)$ if $m\geq n\geq 3$. Since $F_{r}$ and $F_{\rho}$ are
Lyapunov foliations for the full Cartan action and therefore for any generic
restriction $\alpha_{0,\mathbb{P}}$(see Propostion 7 in [2]), which implies
these relations of the type (3.8), (6.9) and (3.10) are contained in a leaf of
the stable manifold for some element of the action.
For relation (3.12), in proof of Theorem (4), we showed that if doubled, it is
lifted to a closed cycle in the universal cover and afteran allowable
substitution, it is reducible.
For relation (3.11) follow exactly the same way as in Milnor s proof in [[21],
Theorem A1] or in [3], we can show that if doubled, they are contractble and
in $\mathcal{A}\mathcal{S}_{F}(\alpha)$(defined in Definition 3.1), thus these
doubled relation is lifted to closed cycles in $\widetilde{SO}^{+}(m,n)$ and
after an allowable substitution, it is reducible.
For relations (3.13), since it is a symbol defined on $S^{1}$, follow exactly
the same way as in Milnor s proof in [[21], Theorem A1], we can show they are
contractble and in $\mathcal{A}\mathcal{S}_{F}(\alpha)$, thus these relations
are lifted to closed cycles in $\widetilde{SO}^{+}(m,n)$ and after allowable
substitution, they are reducible.
Now we consider $\widetilde{SU}(m,n)$. Follow exactly the same way as in the
proof of the previous case, we are though all the relations except (3.18).
Notice
$h_{2L_{n}}(-1)=$diag($1,\dots,\underset{n}{-}1,1,\dots,\underset{2n}{-}1,\dots,1$),
thus homotopy classes of
$\bigl{(}h_{2L_{n}}(-1)h_{2L_{n}}(-1)\bigl{)}^{k}(k\in\mathbb{Z})$ generate
the fundermental group of $SU(m,n)$ which is isomorphic to $\mathbb{Z}$. Hence
we don’t need to consider this relation in $\widetilde{SU}(m,n)$.
Finally, to cancel conjugations one notices that canceling
$F_{r}(t)F_{r}(t)^{-1}=\textrm{id}$ or
$F_{\rho}(a)F_{\rho}(a)^{-1}=\textrm{id}$ are also an allowed substitution and
each conjugation can be canceled inductively using that.
Thus, the value of the periodic cycle functional for any Hölder cocycle
$\beta$ depends only on the element of $p^{-1}(\Gamma)$ this cycle represents.
Notice $p^{-1}(\Gamma)$ is an irreducible lattice also. Furthermore, these
values provide a homomorphism from $p^{-1}(\Gamma)$ to $Y_{X}$. By Lemma 3.1,
orders of images of any homomorphism are bounded by a number depending only on
$m-n$, which means no non-trivial homomorphism or it contradicts the smallness
of the cocyle.
Hence all periodic cycle functionals vanish on $\beta$. Now Proposition 3.1
implies that $\beta$ is cohomologous to a constant cocycle via a
H$\ddot{o}$lder transfer function.
Now consider the case of $C^{\infty}$ cocycles. Notice that the transfer
function $H$ constructed using periodic cycle functionals is $C^{\infty}$
along the stable foliations of various elements of the action. Now a general
result stating that in case the smooth distributions along with their Lie
brackets generate the tangent space at any point of a manifold a function
smooth along corresponding foliations is necessarily smooth (see [16] for a
detailed discussion and references to proofs), implies that the transfer map
$H$ is $C^{\infty}$.
## 4\. Proof of Theorems 1 and 2
The neutral foliation for a generic restriction $\alpha_{0,G}$ is a smooth
foliation, we may use the Hirsch-Pugh-Shub structural stability theorem [[10],
Chapter 6]. Namely if $\widetilde{\alpha}_{G}$ is a sufficiently $C^{1}$-small
perturbation of $\alpha_{0,G}$ then for all elements $a\in A$ which are
regular for $\alpha_{0,G}$ and sufficiently away from non-regular ones (denote
this set by $\overline{A}$) are also regular for $\widetilde{\alpha}_{G}$. The
central distribution is the same for any $a\in\overline{A}$ and is uniquely
integrable to an $\widetilde{\alpha}_{G}(a,\cdot)$-invariant foliation which
we denote by $\mathcal{N}$. Moreover, there is a Hölder homeomorphism
$\widetilde{h}$ of $X$, $C^{0}$ close to the $id_{X}$, which maps leaves of
$\mathcal{N}_{0}$ to leaves of $\mathcal{N}$:
$\widetilde{h}\mathcal{N}_{0}=\mathcal{N}$. This homeomorphism is uniquely
defined in the transverse direction, i.e. up to a homeomorphism preserving
$\mathcal{N}$. Furthermore, $\widetilde{h}$ can be chosen smooth and $C^{1}$
close to the identity along the leaves of $\mathcal{N}_{0}$ although we will
not use the latter fact. Clearly the leaves of the foliation $\mathcal{N}_{0}$
are preserved by every $a\in\overline{A}$. The action $\alpha_{G}$ is Hölder
but it is smooth and $C^{1}$-close to $\alpha_{0,G}$ along the leaves of the
neutral foliation $\mathcal{N}_{0}$.
Let us define an action $\alpha_{G}$ of $G$ on $X$ as the conjugate of
$\widetilde{\alpha}_{G}$ by the map $\widetilde{h}$ obtained from the Hirsch-
Pugh- Shub stability theorem:
$\alpha_{G}:=\widetilde{h}^{-1}\circ\widetilde{\alpha}_{G}\circ\widetilde{h}$
Since the action $\alpha_{G}$ is a $C^{0}$ small perturbation of
$\alpha_{0,G}$ along the leaves of the neutral foliation of $\alpha_{0,G}$
whose leaves are $\\{Y_{X}\cdot x:x\in X\\}$, we have that $\alpha_{G}$ is
given by a map $\beta:(\mathbb{Z}^{k}\times\mathbb{R}^{\ell})\times
X\rightarrow Y$ by
(4.1) $\displaystyle\alpha_{G}(a,x)=\beta(a,x)\cdot\alpha_{0,G}(a,x)$
for $a\in\mathbb{Z}^{k}\times\mathbb{R}^{\ell}$ and $x\in X$. Notice that
since $\alpha_{G}$ is a small perturbation of the action by left translations
$\alpha_{0,G}$, it can be lifted to a $G$-action $\overline{\alpha}_{G}$ on
$\widetilde{X}=\widetilde{SO}^{+}(m,n)$(corr. $\widetilde{SU}(m,n)$) commuting
with the right $p^{-1}(\Gamma)$ action on $\widetilde{SO}^{+}(m,n)$ (corr.
$\widetilde{SU}(m,n)$), and $\beta$ is lifted to a cocycle $\overline{\beta}$
over $\overline{\alpha}_{G}$ (for more details see [[18], example 2.3]). In
particular we have:
$\overline{\beta}(ab,x)=\overline{\beta}(a,\overline{\alpha}_{G}(b,x))\overline{\beta}(b,x).$
Let $U:U_{1},\dots,U_{r}$ denote the invariant unipotent foliations for the
lifted action $\overline{\alpha}_{0,G}$ of $\alpha_{0,G}$ on $\widetilde{X}$
which projects to invariant Lyapunov foliations for $\alpha_{0,G}$; and let
$T:T_{1},\dots,T_{r}$ denote invariant Lyapunov foliations for lifted
$\overline{\alpha}_{G}$ which projects to invariant Lyapunov foliations for
$\alpha_{G}$. Notice that the latter foliations have only Hölder leaves but we
are justified in calling them Lyapunov foliations since they are images of
Lyapunov foliations for a smooth perturbed action under a Hölder conjugacy.
Denote the neutral foliation $\mathcal{N}_{0}$ on the covering space by
$N_{0}$. An immediate corollary of the result of Brin and Pesin [1] on
persistence of local transitivity of stable and unstable foliations of a
partially hyperbolic diffeomorphisms and the fact that the collection of
homogeneous Lyapunov foliations $U:U_{1},\dots,U_{r}$ is locally transitive
and $T:T_{1},\dots,T_{r}$ is transitive and they are leafwise $C^{0}$ close.
Following the proof line closely with only trivial modifications from those of
[Section 6.2, 6.2 and 6.4 [3]], and [Section 5.3,5.4, [5]], we can show
$U$-cycles and $T$-cycles project to each other along the neutral foliations
(precise definitions are in [Section 6.2,[3]]), which implies:
###### Proposition 4.1.
The lifted cocycle for the perturbed action $\overline{\alpha}_{G}$ is
cohomologous to a constant cocycle.
By Proposition 4.1, the value of the periodic cycle functional for
$H\tilde{o}lder$ cocycle $\beta$ over $\widetilde{\alpha}_{G}$ or its
$H\tilde{o}lder$ conjugate $\alpha_{G}$ depends only on the element of
$p^{-1}(\Gamma)$ this cycle represents. Using the same trick as in proof of
Theorem 3, we can show every homomorphism from $p^{-1}(\Gamma)$ to $Y_{X}$ is
trivial. Thus we proved Theorem 2.
Thus by Theorem 2, $\beta$ is cohomologous to a small constant cocycle
$s:\mathbb{Z}^{k}\times\mathbb{R}^{\ell}\rightarrow Y_{X}$ via a continuous
transfer map $H:X\rightarrow Y_{X}$ which can be chosen close to identity in
$C^{0}$ topology if the perturbation $\widetilde{\alpha}_{G}$ small in $C^{2}$
topology.
Let us consider the map $h^{\prime}(x):=H^{-1}(x)\cdot x$. We have from the
cocycle equation 4.1 and the cohomology equation 2.3
$h^{\prime}({\alpha}_{G}(a,x))=\alpha_{0,\widetilde{G}}(a,h^{\prime}(x))$
where $\alpha_{0,\widetilde{G}}(a,x):=i(a)\cdot x$, where
$i(a):=s(a)i_{0}(a),a\in A$ and $i_{0}$ is as in 2.1. Since the map
$h^{\prime}$ is $C^{0}$ close to the identity it is surjective and thus the
action $\alpha_{G}$ is semi-conjugate to the standard perturbation
$\alpha_{0,\widetilde{G}}$ of $\alpha_{0,G}$, i.e. $\alpha_{0,\widetilde{G}}$
is a factor of $\alpha_{G}$. It is enough to prove that $h^{\prime}$ is
injective. By simple transitivity of $U$-holonomy group and the fact that
there is no non-trivial element in $Y_{X}$ such that all its powers are small
[Section 7.1 [3]] we have:
###### Proposition 4.2.
_(Section 6.1[3])_ The map $h^{\prime}$ is a homeomorphism and hence provides
a topological conjugacy between $\alpha_{G}$ and $\alpha_{0,\widetilde{G}}$.
Now by letting $h:=h^{\prime}\widetilde{h}^{-1}$ we have
$h\circ\widetilde{\alpha}_{G}h^{-1}=\alpha_{0,\widetilde{G}}$
thus there is a topological conjugacy between $\widetilde{\alpha}_{G}$ and
$\alpha_{0,\widetilde{G}}$. The smoothness of this homeomorphism follows as in
[15], [3] or [18], by the general Katok-Spatzier theory of non-stationary
normal forms for partially hyperbolic abelian actions.
### 4.1. Proof of Corollary 2.1
Proofs of Theorems 1 and 2 apply to this case with minor changes when proving
$T$-cycles are projected to $U$-cycles when $SO^{+}(m,n)$ is not split.
Similar to proof of Lemma 6.5[3], we need to show that a $T$-cycle at $x$
projected to a $U$-path starting at $x$ gives a $U$-cycle which is either
contractible or its fourth power, after adding a $U$-path of bounded length
which connects the 2 endpoints and closed up the $U$-path. It is due to
Theorem 3 and the fact that $h^{j}_{L_{n}}(\sqrt{2}a,\sqrt{2}b)(a,b)\in
S^{1}$, $j\leq m-n-1$ generate $M$.
## 5\. Schur multipliers of non-split groups
### 5.1. Preliminaries and notations from K-theory
In this part, we follow nations and quote conclusions without proof fairly
close to [6]. Let $k$ be any arbitrary field. Let $\Omega$ denote its
algebraic closure in a ”universal domain.” Let
$G\hookrightarrow\operatorname{GL}(n,\Omega)$ be a connected simply connected
algebraic group which is of $k$-rank $\geq 2$. We also assume that $G$ is
absolutely simple over $\Omega$. Let $G_{k}=G\bigcap GL(n,k)$ be the group of
$k$ rational points of $G$. For a subgroup $H$ of $G$, let $H_{k}$ denote set
$H\bigcap G_{k}$. Let $\mathfrak{g}$ be the Lie algebra of $G$, $S\subset G$
be the $k$-split torus. Let $\Phi$ be the $k$-root system of $G$ with respect
to $S$, and $\mathfrak{g}_{\alpha}$ the corresponding root space. Let
$\Phi^{+}$ be the set of positive roots and $\Delta$ the system of simple
roots with respect to $\Phi^{+}$. Define
$\displaystyle\Phi_{1}=\\{\alpha\in\Phi|\alpha/2\notin\Phi\\}\qquad\text{ and
}\qquad\Phi_{2}=\\{\alpha\in\Phi|2\alpha\notin\Phi\\}.$
For $\alpha\in\Phi$, let $u^{\alpha}=\sum_{k>0}\mathfrak{g}_{k\alpha}$, and
$U^{\alpha}$ the corresponding algebraic subgroup of $G$. Let $U^{+}$ be the
algebraic subgroup of $G$ whose Lie algebra is
$\sum_{\alpha\in\Phi^{+}}\mathfrak{g}_{\alpha}$. $U^{-}$ is defined as the
subgroup corresponding to $\sum_{-\alpha\in\Phi^{+}}\mathfrak{g}_{\alpha}$.
For $\alpha\in\Phi_{1}$, let $G^{\alpha}$ be the connected algebraic subgroup
generated by $U^{\alpha}$ and $U^{-\alpha}$. Let $Z(S)$ be the centralizer of
$S$ in $G$ and $N(S)$ the normalizer of $S$ in $G$ and $W_{0}=N(S)/Z(S)$ be
the Weyl group. Let $W\subset N(S)_{k}$ be a complete representatives. We also
assume that $w_{\alpha}$ is so chosen that for any $\alpha\in\Phi$,
$w_{\alpha}\in N(S)\bigcap G^{\alpha}_{k}$ and has order 2. Next, let
$G^{+}_{k}$ be the group generated by $k$-rational unipotent elements which
belong to the radical of a parabolic subgroup defined over $k$ in $G$. It is
known that for a wide class of $G$, $G^{+}_{k}=G_{k}$. Moreover, the only
proper normal subgroups of $G^{+}_{k}$ are central (and finite). Also,
$G^{+}_{k}=[G^{+}_{k},G^{+}_{k}]$. We start with a technical lemma whose role
will be clear from the subsequent development.
###### Lemma 5.1.
(_The Chain Lemma_) For $\alpha\in\Phi_{1}$, let $(e\neq)x\in U^{\alpha}_{k}$
be any element. Then there exists elements $x_{i}\in U^{\alpha}_{k}$,
$y_{i}\in U^{-\alpha}_{k}(i\in\mathbb{Z})$ such that:
* 1
. $\qquad x_{0}=x$.
* 2
. $x_{i}y_{i}x_{i+1}=y_{j}x_{j+1}y_{j+1}$ $\forall i,j\in\mathbb{Z}$; we
denote this element by $w$.
* 3
. The element $w$ belongs to $N(S)_{k}$ and “acts” on $S$ as the reflection
with respect to $\alpha$, i.e., $ww^{-1}_{\alpha}\in Z(S)_{k}(w_{\alpha}\in
W)$.
* 4
. Given any $x_{i}$ or $y_{i}$ the remaining elements of the chain
$\\{x_{n},y_{m}\\}$ are uniquely determined.
###### Definition 5.1.
We define $w_{\alpha}(x)$, for $(e\neq)x\in U^{\alpha}_{k}$ to be the element
$w$ in the chain lemma. Thus, $w_{\alpha}(x)\in N(S)_{k}$ and “acts as the
reflection with respect to $\alpha$”. Further, we have
$w_{\alpha}(x)=w_{\alpha}(x_{i})=w_{-\alpha}(y_{j})$ and
$w_{\alpha}(x^{-1})=w_{\alpha}(x)^{-1}$ $\forall i,j\in\mathbb{Z}$. For
$\alpha\in\Phi_{1}$, $x,x_{1}$(both$\neq e$)$\in U^{\alpha}_{k}$, consider the
element $h_{\alpha}(x,x_{1})=w_{\alpha}(x)w_{\alpha}(x_{1})^{-1}$. It is clear
that $h_{\alpha}(x,x_{1})\in Z(S)_{k}\bigcap G_{k}^{+}$. Let $H_{k}$ be the
subgroup generated by these elements.
###### Remark 5.1.
If $\text{char}k=0,$ then we have the exponential map
$\exp:\mathfrak{g}\rightarrow G$. Let $X\in\mathfrak{g}_{\alpha}$ be a
$k$-rational(nilpotent) element. Then $x=\exp X\in U^{\alpha}_{k}$. The chain
associated with $x$ can also be obtained in the following way: We have the
Jacobson-Morosov theorem which asserts the existence of an element
$Y\in(\mathfrak{g}_{-\alpha})_{k}$ such that $\\{X,Y,[X,Y]\\}$ span a three-
dimension split Lie algebra over $k$. It is then easy to prove that
$x_{i}=x=\exp X,\forall i\in\mathbb{Z}$ and $y_{j}=\exp(-Y),\forall
j\in\mathbb{Z}$. If we let $y=$exp$(-Y)$, then we denote $xyx$ by
$w_{\alpha}(x)$.
### 5.2. Construction of universal central extension
Let $N$ be an abstract group. A central extension of $N$ is a pair
$(\pi,N^{\prime})$ where $N^{\prime}$ is a group, $\pi$ is a homomorphism of
$N^{\prime}$ onto $N$ and $\ker\pi\subseteq$(center of $N^{\prime}$). A
central extension $(\pi,N^{\prime})$ of $N$ is said to be universal if for any
central extension $(\eta,E^{\prime})$ of $N$, there exists a unique
homomorphism $\phi:N^{\prime}\rightarrow E^{\prime}$ such that
$\eta\circ\phi=\pi$. A necessary and sufficient condition for $N$ to have a
universal central extension is that $N=[N,N]$. (For the proof of this and
other elementary properties of a universal central extension, one may refer to
[[24], Section 7].)
We now construct the universal central extension $(\textrm{u.c.e.})$ of the
group $G^{+}_{k}$. (Such an extension exists since
$G^{+}_{k}=[G^{+}_{k},G^{+}_{k}]$.) For $\alpha,\beta\in\Phi_{1}$ such that
$\alpha\neq-\beta$, it is known that
$[U^{\alpha},U^{\beta}]\subset\prod_{\gamma=i\alpha+j\beta,i,j\geq
1}U^{\gamma}.$
This clearly gives rise to relations $R_{\alpha,\beta}$ between commutators of
above form and elements which belongs to
$\prod_{\gamma=i\alpha+j\beta,i,j\geq 1}U^{\gamma}.$
Let $\widetilde{G}^{\prime}=U^{+}_{k}*U^{-}_{k}$, the free product of groups.
Now for $\alpha,\beta\in\Phi,\alpha\neq-\beta$, $R_{\alpha,\beta}$ has a
natural meaning in $\widetilde{G}^{\prime}$. We now quotient
$\widetilde{G}^{\prime}$ by the relations $\\{R_{\alpha,\beta}\\}$ to get a
group we denote by $\widetilde{G}$. It is clear that there exists a well-
defined homomorphism $\pi_{1}:\widetilde{G}\rightarrow G^{+}_{k}$. Further, it
is clear that $\pi_{1}$ is surjective. We write for $u\in U^{\alpha}_{k}$, the
corresponding element in $\widetilde{G}$ by $\tilde{x}_{\alpha}(u)$. Then
$\tilde{w}_{\alpha}(u)$, $\tilde{h}_{\alpha}(u,u_{1})$ for $u,u_{1}\in
U^{\alpha}_{k}$ are obviously defined elements of $\widetilde{G}$.
###### Lemma 5.2.
$(\widetilde{G},\pi_{1})$ is a $(\emph{u.c.e.})$ of $G^{+}_{k}$
Now we list the following lemma which is very important for the sequel.
###### Lemma 5.3.
If $\alpha,\beta\in\Phi_{1},(e\neq)u\in
U^{\alpha}_{\mathbb{R}},(e\neq)v,v_{1}\in U^{\beta}_{\mathbb{R}}$, then
* 1
$\tilde{w}_{\alpha}(u)\tilde{x}_{\beta}(v)\tilde{w}_{\alpha}(u)^{-1}=\tilde{x}_{w_{\alpha}(\beta)}\bigl{(}w_{\alpha}(u)vw_{\alpha}(u)^{-1}\bigl{)}$
* 2
$\tilde{w}_{\alpha}(u)\tilde{w}_{\beta}(v)\tilde{w}_{\alpha}(u)^{-1}=\tilde{w}_{w_{\alpha}(\beta)}\bigl{(}w_{\alpha}(u)x_{\beta}(v)w_{\alpha}(u)^{-1}\bigl{)}$
* 3
$\tilde{w}_{\alpha}(u)\tilde{h}_{\beta}(v,v_{1})\tilde{w}_{\alpha}(u)^{-1}$
$=\tilde{h}_{w_{\alpha}(\beta)}\bigl{(}w_{\alpha}(u)x_{\beta}(v)w_{\alpha}(u)^{-1},w_{\alpha}(u)x_{\beta}(v_{1})w_{\alpha}(u)^{-1}\bigl{)}$
* 4
$\tilde{h}_{\beta}(v,v_{1})\tilde{x}_{\alpha}(u)\tilde{h}_{\beta}(v,v_{1})^{-1}$
$=\tilde{x}_{\alpha}\bigl{(}h_{\beta}(v,v_{1})x_{\alpha}(u)h_{\beta}(v,v_{1})^{-1}\bigl{)}$.
###### Lemma 5.4.
Let $\widetilde{N}$ be the subgroup of $G$ generated by
$\\{\tilde{w}_{\alpha}(u),\alpha\in\Phi_{1},(e\neq)u\in U_{k}^{\alpha}\\}$.
For $\alpha\in\Phi^{+}$, denote by $\tilde{H}_{\alpha}$ the subgroup generated
by $\tilde{h}_{\alpha}(v,v_{1})$, $(e\neq)v,v_{1}\in U^{\alpha}_{k}$. let
$\tilde{H}$ be he subgroup generated by
$\\{\tilde{H}_{\alpha},\alpha\in\Phi_{1}\\}$. Then
* 1
$\tilde{H}_{\alpha},\alpha\in\Phi_{1}$, is normal in $\widetilde{H}$, and
$\widetilde{H}$ is normal in $\widetilde{N}$.
* 2
$\widetilde{H}$ normalizes each $\tilde{U}_{k}^{\alpha}$, and hence
$\tilde{U}_{k}^{+}$.
* 3
$\tilde{H}=\prod_{\alpha\in\Delta}\tilde{H}_{\alpha}$
* 4
$\ker(\pi_{1})\subset\tilde{H}$.
###### Remark 5.2.
We now consider the condition under which
$\tilde{h}=\prod_{\alpha\in\Delta}\tilde{h}_{\alpha}$ is in the kernel of
$\pi_{1}$. Using the simple connectedness of $G$ over $\Omega^{-}$, it is easy
to see that $\tilde{h}\in\ker\pi_{1}$ iff
$\tilde{h}_{\alpha}\in\ker\pi_{1}\forall\alpha\in\Delta$. In other words, the
Schur multiplier $\pi_{1}$ of $G_{k}$ is generated by
$\\{\pi_{1}\bigcap\tilde{H}_{\alpha},\alpha\text{ simple}\\}$. If $G$ is not
simply connected, we only have $\ker(\pi_{1})\subset\tilde{H}$.
## 6\. Generating relations of $SO^{+}(m,n)$
### 6.1. Basic settings for $SO^{+}(m,n)$
In this part we study the generators of $SO^{+}(m,n)(m\geq n\geq 3)$. We use
notations as in Section 3.3 and Section 5. Explicitly, this is the case where
$G=SO(m+n,\mathbb{C})$ defined by a non-degenerate standard bilinear form of
signature $(m,n)$.
We denote by $S$ the set of $(m+n)\times(m+n)$ diagonal matrices in
$G_{\mathbb{R}}$ with lower-right $(m-n)\times(m-n)$ block identity. Let
$\Phi$ be the root system with respect to $S$. The roots are $\pm L_{i}\pm
L_{j}(i<j\leq n)$, whose dimensions are one and $\pm L_{i}(1\leq i\leq n)$ are
also roots if $m\geq n+1$ with dimensions $m-n$. We easily see
$\Phi=\Phi_{1}$. If $m\geq n+1$, the set of positive roots $\Phi^{+}$ and the
corresponding set of simple roots $\Delta$ are
$\displaystyle\Phi^{+}=\\{L_{i}-L_{j}\\}_{i<j}\cup\\{L_{i}+L_{j}\\}_{i<j}\cup\\{L_{i}\\}_{i},$
$\displaystyle\Delta=\\{L_{i}-L_{i+1}\\}_{i}\cup\\{L_{n}\\};$
if $m=n$, the set of positive roots $\Phi^{+}$ and the corresponding set of
simple roots $\Delta$ are
$\displaystyle\Phi^{+}=\\{L_{i}-L_{j}\\}_{i<j}\cup\\{L_{i}+L_{j}\\}_{i<j},$
$\displaystyle\Delta=\\{L_{i}-L_{i+1}\\}_{i}\cup\\{L_{n-1}+L_{n}\\}.$
With notations in Section 3.3 we have
$\displaystyle U^{r}_{\mathbb{R}}=\\{\exp(tf_{r})\mid t\in\mathbb{R}\\}\text{
for }r=\pm L_{i}\pm L_{j}(i<j),$ $\displaystyle
U^{\alpha}_{\mathbb{R}}=\\{\prod_{j}\exp\left(a_{j}f^{j}_{\alpha}\right)\mid
a=(a_{1},\dots,a_{m-n})\in\mathbb{R}^{m-n}\\}\text{ for }\alpha=\pm L_{i}.$
Correspondingly, for $t\in\mathbb{R}$ and
$a=(a_{1},\dots,a_{m-n})\in\mathbb{R}^{m-n}$ we write
$\displaystyle x_{r}(t)=\exp(tf_{r})\in U^{r}_{\mathbb{R}}\qquad\text{ for
}r=\pm L_{i}\pm L_{j}(i<j),$ $\displaystyle
x_{\alpha}(a)=\prod_{j}\exp(a_{j}f^{j}_{\alpha})\in
U^{\alpha}_{\mathbb{R}}\qquad\text{ for }\alpha=\pm L_{i}.$
### 6.2. “Chains” in $SO^{+}(m,n)$
Our next step is to determine explicitly the “chain” (cf. Lemma 5.1)
corresponding to the element $x_{\alpha}(a)(\neq e)\in
U^{\alpha}_{\mathbb{R}}(\alpha=\pm L_{i})$. For this, define
$f:\mathbb{R}^{m-n}\backslash 0\rightarrow\mathbb{R}^{m-n}\backslash 0$ by
$f(a)=\bigl{(}\frac{2a_{1}}{\sum a_{i}^{2}},\dots,\frac{2a_{m-n}}{\sum
a_{i}^{2}}\bigl{)}$ for
$a=\left(a_{1},\dots,a_{m-n}\right)\in\mathbb{R}^{m-n}\backslash 0$. With this
notation, we have:
###### Lemma 6.1.
For $x_{\alpha}(a)(\neq e)\in U^{\alpha}_{\mathbb{R}}(\alpha=\pm L_{i})$, the
“chain” corresponding to it is given by
$\displaystyle x_{i}=x_{\alpha}(a),i\in\mathbb{Z};\qquad
y_{i}=x_{-\alpha}(f(a)),i\in\mathbb{Z}.$
Denoting the element $w_{\alpha}(x_{\alpha}(a))$ by $w_{\alpha}(a)$, we have
$\displaystyle
w_{\alpha}(a)=x_{\alpha}(a)x_{-\alpha}\left(f(a)\right)x_{\alpha}(a).$
###### Proof.
It is easy to check
$\\{\sum_{j}a_{j}f^{j}_{\alpha},\sum_{j}\big{(}-\frac{2a_{j}}{\sum
a_{j}^{2}}f^{j}_{-\alpha}\big{)},[\sum_{j}a_{j}f^{j}_{\alpha},\sum_{j}\big{(}-\frac{2a_{j}}{\sum
a_{j}^{2}}f^{j}_{-\alpha}\big{)}]\\}$
span a three-dimensional Lie algebra isomorphic to $SL_{2}(\mathbb{R})$. By
Remark 5.1 we get the conclusion. ∎
###### Remark 6.1.
Similar computations can be made for the other roots, such as $\pm L_{i}\pm
L_{j}(i<j)$. We record the results here:
$\displaystyle w_{r}(t)=x_{r}(t)x_{-r}(-t^{-1})x_{r}(t),\qquad
t\in\mathbb{R}^{*}$
where
$\displaystyle x_{i}=x_{r}(t)\forall i,\qquad y_{i}=x_{-r}(-t^{-1})\forall i.$
Correspondingly, we define
$\displaystyle h_{r}(t)=w_{r}(t)w_{r}(1)^{-1},\qquad t\in\mathbb{R}^{*},r=\pm
L_{i}\pm L_{j}(i<j),$ $\displaystyle
h_{\alpha}(a,b)=w_{\alpha}(a)w_{\alpha}(b)^{-1},\qquad
a,b\in\mathbb{R}^{m-n}\backslash 0,\alpha=\pm L_{i}.$
Let us write $p(\pi)$ the permutation matrix corresponding to the permutation
$\pi$, that is, the $i,j$ entry of $p(\pi)$ is $1$ if $i=\pi(j)$ and zeros
otherwise. Let $A,B,C,\cdots$ be square matrices(not necessary the same size),
we use diag$\left(A_{j_{1}},B_{j_{2}},C_{j_{3}},\cdots\right)$ to denote the
$(m+n)\times(m+n)$ matrix that’s constructed in the following way. First, the
matrix $A$ is placed as a block in
diag$\left(A_{j_{1}},B_{j_{2}},C_{j_{3}},\cdots\right)$ with its upper left
corner positioned at the $(j_{1},j_{1})$ entry. Matrices $B,C,\cdots$ are
placed similarly. Then we fill the rest of diagonal blocks of
diag$\left(A_{j_{1}},B_{j_{2}},C_{j_{3}},\cdots\right)$ with identity matrices
of suitable sizes and the off-diagonal blocks with zero matrices. For example,
let $A=\begin{pmatrix}2&3\\\ -1&4\\\ \end{pmatrix}$ and $B=(3)$. Then
$\operatorname{diag}\\{A_{2},B_{5}\\}$ is the following matrix
$\begin{pmatrix}1&0&0&0&0\\\ 0&2&3&0&0\\\ 0&-1&4&0&0\\\ 0&0&0&1&0\\\
0&0&0&0&3\end{pmatrix}.$
With these notations we have:
$\displaystyle
w_{L_{i}-L_{j}}(t)=p(\pi)\text{diag}\left((-t^{-1})_{i},t_{j},(-t)_{i+n},(t^{-1})_{j+n}\right),\qquad\text{
for }t\in\mathbb{R}^{*}$
where $\pi$ only permutes $(i,j)$ and $(i+n,j+n)$ while fixes other numbers.
$\displaystyle
w_{L_{i}+L_{j}}(t)=p(\pi)\text{diag}\left((-t^{-1})_{i},(t^{-1})_{j},(-t)_{i+n},t_{j+n}\right),\qquad\text{
for }t\in\mathbb{R}^{*}$
where $\pi$ only permutes $(i,j+n)$ and $(j,i+n)$ while fixes other numbers.
$\displaystyle w_{L_{i}}(a)=p(\pi)\text{diag}\bigl{(}(-2\lvert
a\rvert^{-2})_{i},(-\frac{1}{2}\lvert
a\rvert^{2})_{i+n},B_{2n+1}\bigl{)},\qquad\text{ for
}a\in\mathbb{R}^{m-n}\backslash 0,$
where $B\in O(m-n)$ and $\pi$ only permutes $(i,i+n)$ while fixes other
numbers.
### 6.3. Basic relations
We can now define elements $\tilde{x}_{r}(t)$, $\tilde{x}_{\alpha}(a)$,
$\tilde{w}_{r}(t)$, $\tilde{w}_{\alpha}(a)$, $\tilde{h}_{r}(t)$,
$\tilde{h}_{\alpha}(a,b)$ etc. as was done in Section 5.2. We denote by
$\tilde{W}_{r}$ the subgroup of $\widetilde{G}$ generated by
$\tilde{w}_{r}(t)$, $\tilde{W}_{\alpha}$ the subgroup generated by
$\tilde{w}_{\alpha}(a)$, $\tilde{H}_{r}$ the subgroup generated by
$\tilde{h}_{r}(t)$, $\tilde{H}_{\alpha}$ the subgroup generated by
$\tilde{h}_{\alpha}(a,b)$. Also, from Lemma 5.3, it is clear that certain
relations hold both in $\widetilde{G}$ and $G_{\mathbb{R}}$. We record these
results in two separate lemmas(Lemma 6.2 and Lemma 6.3), since they will serve
as ready references later.
###### Lemma 6.2.
If $a\in\mathbb{R}^{m-n}$, $t\in\mathbb{R}\backslash 0$, the following hold
$\widetilde{G}$(and hence in $G_{\mathbb{R}}$ too).
* 1
$\tilde{w}_{L_{n}}(a)\tilde{w}_{L_{n-1}-L_{n}}(t)\tilde{w}_{L_{n}}(a)^{-1}=\tilde{w}_{L_{n-1}+L_{n}}\left(-\frac{1}{2}\lvert
a\rvert^{2}t\right)$,
* 2
$\tilde{w}_{L_{n}}(a)\tilde{w}_{L_{n-1}+L_{n}}(t)\tilde{w}_{L_{n}}(a)^{-1}=\tilde{w}_{L_{n-1}-L_{n}}\left(-2\lvert
a\rvert^{-2}t\right)$,
* 3
$\tilde{w}_{L_{n-1}-L_{n}}(t)\tilde{w}_{L_{n}}(a)\tilde{w}_{L_{n-1}-L_{n}}(t)^{-1}=\tilde{w}_{L_{n-1}}(at)$,
* 4
$\tilde{w}_{L_{n-1}-L_{n}}(t)\tilde{w}_{L_{n-1}}(a)\tilde{w}_{L_{n-1}-L_{n}}(t)^{-1}=\tilde{w}_{L_{n}}\left(-at^{-1}\right)$.
Hence,
* 5
$\tilde{h}_{L_{n-1}-L_{n}}(t)\tilde{w}_{L_{n}}(a)\tilde{h}_{L_{n-1}-L_{n}}(t)^{-1}=\tilde{w}_{L_{n}}(at^{-1})$,
* 6
$\tilde{w}_{L_{n}}(a)\tilde{h}_{L_{n-1}-L_{n}}(t)\tilde{w}_{L_{n}}(a)^{-1}\\\
=\tilde{h}_{L_{n-1}+L_{n}}\left(-\frac{1}{2}\lvert
a\rvert^{2}t\right)\tilde{h}_{L_{n-1}+L_{n}}\left(-\frac{1}{2}\lvert
a\rvert^{2}\right)^{-1}$.
We denote by $S^{i}$ the sphere in $\mathbb{R}^{i+1}$, denote by
$\tilde{W}_{s}$ the subgroup generated by
$\tilde{w}_{L_{n}}\left(\sqrt{2}a\right)$, $a\in S^{m-n-1}$. If
$a=\left(a_{1},...,a_{n}\right)$, then
$\pi_{1}(\tilde{w}_{L_{n}}(\sqrt{2}a))=p(\pi)\text{diag}\bigl{(}(-1)_{n},(-1)_{2n},B_{2n+1}\bigl{)},$
where $\pi$ permutes n and 2n while fixes other numbers and $B\in O(m-n)$ with
entries $B_{i,j}=-2a_{i}a_{j},i\neq j$ and $B_{i,i}=1-2a_{i}^{2}$. Then $B$ is
a reflection in the hyperplane orthogonal to $a$. Thus for any
$w\in\tilde{W}_{s}$,
$\pi_{1}(w)=p(\pi)\text{diag}\bigl{(}(-1)^{\delta}_{n},(-1)^{\delta}_{2n},B_{2n+1}\bigl{)},$
where $\delta=2$ if $p(\pi)=I_{m+n}$ and $B\in SO(m-n)$; $\delta=1$ if
$p(\pi)$ permutes $n$ and $2n$ and and $B\in O(m-n)$. Without confusion, we
identify $\pi_{1}(w_{L_{n}}(\sqrt{2}a))$ with $B$. The following holds:
###### Lemma 6.3.
If $w\in\tilde{W}_{s}$,
$\pi_{1}(w)=p(\pi)\operatorname{diag}\bigl{(}(-1)^{\delta}_{n},(-1)^{\delta}_{2n},B_{2n+1}\bigl{)}$,
$\delta=1$ or $2$, $B\in O(m-n)$, $a\in S^{m-n-1}$, we have
$\displaystyle
w\tilde{w}_{L_{n}}\bigl{(}\sqrt{2}a\bigl{)}w^{-1}=\left\\{\begin{aligned}
&\tilde{w}_{L_{n}}\bigl{(}\sqrt{2}B\cdot a\bigl{)},&\qquad&\text{ if
}p(\pi)=I_{m+n},\\\ &\tilde{w}_{L_{-n}}\bigl{(}-\sqrt{2}B\cdot
a\bigl{)},&\qquad&\text{ if }p(\pi)\neq I_{m+n},\\\ \end{aligned}\right.$
where $\cdot$ means linear operation on vectors. Using above formula and
Definition 5.1 one further has
$\displaystyle
w\tilde{w}_{L_{n}}\bigl{(}\sqrt{2}a\bigl{)}w^{-1}=\left\\{\begin{aligned}
&\tilde{w}_{L_{n}}\bigl{(}\sqrt{2}B\cdot a\bigl{)},&\qquad&\text{ if
}p(\pi)=I_{m+n},\\\ &\tilde{w}_{L_{n}}\bigl{(}-\sqrt{2}B\cdot
a\bigl{)}.&\qquad&\text{ if }p(\pi)\neq I_{m+n}\\\ \end{aligned}\right.$
### 6.4. Structure of $\tilde{H}$
We recall the notation set in Section 6.3. We prove the following:
###### Lemma 6.4.
If $m=n$, $\tilde{H}$ is generated by $\prod_{r\in\Delta}\tilde{H}_{r}$, where
$\Delta=\\{L_{i}-L_{i+1},L_{n-1}+L_{n}\\}$; if $m\geq n+1$, $\tilde{H}$ is
generated by
$\bigl{(}\prod_{r\in\Delta}\tilde{H}_{r}\bigl{)}\cdot\tilde{H}_{s}$, where
$\Delta=\\{L_{i}-L_{i+1},L_{n-1}+L_{n}\\}$ and $\tilde{H}_{s}$ is generated by
$\tilde{h}_{L_{n}}(\sqrt{2}a,\sqrt{2}b),a,b\in S^{m-n-1}$.
###### Proof.
By Lemma 5.4, the case for $m=n$ is obvious. If $m\geq n+1$, $\tilde{H}$ is
generated by $\prod_{r\in\Delta}\tilde{H}_{r}$, where
$\Delta=\\{L_{i}-L_{i+1},L_{n}\\}$.
For any $a,b\in\mathbb{R}^{m-n}\backslash 0$, denote $\sum a_{i}^{2}=a_{0}$,
$\sum b_{i}^{2}=b_{0}$. Using Lemma 6.2, we have:
$\displaystyle\tilde{h}_{L_{n}}(a,b)$
$\displaystyle=\tilde{w}_{L_{n}}(a)\tilde{w}_{L_{n}}(-b)$
$\displaystyle=\tilde{h}_{L_{n-1}-L_{n}}\big{(}\frac{\sqrt{2}}{\sqrt{a_{0}}}\big{)}\tilde{w}_{L_{n}}\big{(}\frac{\sqrt{2}a}{\sqrt{a_{0}}}\big{)}\tilde{h}_{L_{n-1}-L_{n}}\big{(}\frac{\sqrt{2}}{\sqrt{a_{0}}}\big{)}^{-1}$
$\displaystyle\cdot\tilde{h}_{L_{n-1}-L_{n}}\big{(}\frac{\sqrt{2}}{\sqrt{b_{0}}}\big{)}\tilde{w}_{L_{n}}\big{(}-\frac{\sqrt{2}b}{\sqrt{b_{0}}}\big{)}\tilde{h}_{L_{n-1}-L_{n}}\big{(}\frac{\sqrt{2}}{\sqrt{b_{0}}}\big{)}^{-1}$
$\displaystyle=\tilde{h}_{L_{n-1}-L_{n}}\big{(}\frac{\sqrt{2}}{\sqrt{a_{0}}}\big{)}\tilde{h}_{L_{n-1}+L_{n}}(-1)\tilde{h}_{L_{n-1}+L_{n}}\big{(}-\frac{\sqrt{2}}{\sqrt{a_{0}}}\big{)}^{-1}$
$\displaystyle\cdot\tilde{h}_{L_{n-1}+L_{n}}\big{(}-\frac{\sqrt{2}}{\sqrt{b_{0}}}\big{)}\tilde{h}_{L_{n-1}+L_{n}}(-1)^{-1}\tilde{h}_{L_{n}}\big{(}\frac{\sqrt{2}a}{\sqrt{a_{0}}},\frac{\sqrt{2}b}{\sqrt{b_{0}}}\big{)}$
$\displaystyle\cdot\tilde{h}_{L_{n-1}-L_{n}}\big{(}\frac{\sqrt{2}}{\sqrt{b_{0}}}\big{)}^{-1}$
Thus $\tilde{H}_{L_{n}}$ is generated by $\tilde{H}_{s}$,
$\tilde{H}_{L_{n-1}-L_{n}}$ and $\tilde{H}_{L_{n-1}+L_{n}}$. Hence we proved
the lemma. ∎
If $m\geq n+2$, for any $(a,b)\in S^{1},$ $1\leq j\leq m-n-1$, let
$\displaystyle\tilde{h}^{j}_{L_{n}}\bigl{(}\sqrt{2}a,\sqrt{2}b\bigl{)}$
$\displaystyle=\tilde{w}_{L_{n}}\bigl{(}0,\dots,0,\underset{j}{\sqrt{2}a},\underset{j+1}{\sqrt{2}b},0,\dots,0\bigl{)}$
$\displaystyle\cdot\tilde{w}_{L_{n}}\bigl{(}0,\dots,0,\underset{j}{-\sqrt{2}},0,\dots,0\bigl{)}.$
###### Corollary 6.1.
If $m\geq n+2$, $\tilde{H}$ is generated by
$\bigl{(}\prod_{r\in\Delta}\tilde{H}_{r}\bigl{)}\tilde{H}_{s_{0}}$, where
$\Delta=\\{L_{i}-L_{i+1},L_{n-1}+L_{n}\\}$ and $\tilde{H}_{s_{0}}$ is
generated by $\tilde{h}^{j}_{L_{n}}(\sqrt{2}a,\sqrt{2}b),(a,b\in S^{1})$.
###### Proof.
By Lemma 6.4, we just need to show $\tilde{H}_{s}=\tilde{H}_{s_{0}}$. Observe
that
$\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(\sqrt{2}a,\sqrt{2}b)\bigl{)}$=diag$(R_{2n+j})$,
where
$R=\left(\begin{array}[]{ccc}a^{2}-b^{2}&-2ab\\\ 2ab&a^{2}-b^{2}\\\
\end{array}\right).$
Then it is clear that $\pi_{1}(\tilde{h}^{j}_{L_{n}}(a,b))(j\leq m-n-1)$
generate a subgroup isomorphic to $SO(m-n)$.
Hence for any $a\in S^{m-n-1}$ we can find $(e_{i},f_{i})\in S^{1}$ such that
$\bigl{(}\prod_{i}\pi_{1}(\tilde{h}^{i}_{L_{n}}(\sqrt{2}e_{i},\sqrt{2}f_{i}))\bigl{)}\cdot(\sqrt{2},0,...,0)=\sqrt{2}a.$
Denote $\prod_{i}\tilde{h}^{i}_{L_{n}}(\sqrt{2}e_{i},\sqrt{2}f_{i})$ by $A$.
By Lemma 6.3 we have
$\displaystyle
A\tilde{w}_{L_{n}}(\sqrt{2},0,...,0)A^{-1}=\tilde{w}_{L_{n}}(\sqrt{2}a).$
Similarly, for any $b\in S^{m-n-1}$, we can find $B\in\tilde{H}_{s_{0}}$ such
that
$\displaystyle
B\tilde{w}_{L_{n}}(-\sqrt{2},0,...,0)B^{-1}=\tilde{w}_{L_{n}}(\sqrt{2}b).$
It follows
$\displaystyle\tilde{w}_{L_{n}}(\sqrt{2}a)\tilde{w}_{L_{n}}(\sqrt{2}b)$
$\displaystyle=A\bigl{(}\tilde{w}_{L_{n}}(\sqrt{2},0,...,0)A^{-1}B\tilde{w}_{L_{n}}(-\sqrt{2},0,...,0)\bigl{)}B^{-1}.$
By Lemma 6.3, it is easy to check
$\tilde{w}_{L_{n}}(\sqrt{2},0,...,0)\tilde{H}_{s_{0}}\tilde{w}_{L_{n}}(-\sqrt{2},0,...,0)^{-1}\subseteq\tilde{H}_{s_{0}}.$
Thus we get the conclusion. ∎
###### Lemma 6.5.
$\displaystyle(1)\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}=\\{\prod_{i}\tilde{h}_{L_{1}-L_{2}}(t_{i})\mid\text{
\emph{with} }\prod_{i}t_{i}=1\\}.$
$\displaystyle(2)\ker(\pi_{1})\cap\tilde{H}_{r}=\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}},\qquad\text{
\emph{for} }r=\pm L_{i}\pm L_{j}(i\neq j).$
###### Proof.
(1). Notice
$\pi_{1}(\tilde{h}_{L_{1}-L_{2}}(t))=$diag$\left(t_{1},(t^{-1})_{2},(t^{-1})_{1+n},t_{2+n}\right)$.
Thus (1) is clear.
It follows from (1) that $\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$ is
generated by elements
$\tilde{h}_{L_{1}-L_{2}}(t_{1})\tilde{h}_{L_{1}-L_{2}}(t_{2})\tilde{h}_{L_{1}-L_{2}}(t_{1}t_{2})^{-1},\text{
where }t_{1},t_{2}\in\mathbb{R}^{*}.$
(2) We can prove similarly that $\ker(\pi_{1})\cap\tilde{H}_{r}$($r=\pm
L_{i}\pm L_{j}$) is generated by elements
$\tilde{h}_{r}(t_{1})\tilde{h}_{r}(t_{2})\tilde{h}_{r}(t_{1}t_{2})^{-1}$.
Since these simple roots belong to the same orbit under the Weyl group, an
argument similar to one in [[20], Lemma 8.2] shows that
$\ker(\pi_{1})\cap\tilde{H}_{r}\subseteq\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$
for all roots $r=\pm L_{i}\pm L_{j}$. This proves (2). ∎
For $t_{1},t_{2}\in\mathbb{R}^{*}$, we define:
$\displaystyle\\{t_{1},t_{2}\\}=\tilde{h}_{L_{1}-L_{2}}(t_{1})\tilde{h}_{L_{1}-L_{2}}(t_{2})\tilde{h}_{L_{1}-L_{2}}(t_{1}t_{2})^{-1}.$
Now in exactly the same manner as the proof in the appendix of [20], we prove
that these $\\{t_{1},t_{2}\\}$’s satisfy the conditions
###### Lemma 6.6.
$\displaystyle\\{t_{1},t_{2}\\}$
$\displaystyle=\\{t_{2},t_{1}\\}^{-1}\qquad\forall
t_{1},t_{2}\in\mathbb{R}^{*},$ $\displaystyle\\{t_{1},t_{2}\cdot t_{3}\\}$
$\displaystyle=\\{t_{1},t_{2}\\}\cdot\\{t_{1},t_{3}\\}\qquad\forall
t_{1},t_{2},t_{3}\in\mathbb{R}^{*},$ $\displaystyle\\{t_{1}\cdot
t_{2},t_{3}\\}$
$\displaystyle=\\{t_{1},t_{3}\\}\cdot\\{t_{2},t_{3}\\}\qquad\forall
t_{1},t_{2},t_{3}\in\mathbb{R}^{*},$ $\displaystyle\\{t,1-t\\}$
$\displaystyle=1\qquad\forall t\in\mathbb{R}^{*},t\neq 1,$
$\displaystyle\\{t,-t\\}$ $\displaystyle=1\qquad\forall t\in\mathbb{R}^{*}.$
Thus we define a symbol on $\mathbb{R}$.
### 6.5. Construction of a $S^{1}$-symbol
We construct a new symbol on $S^{1}$ to get prepared for further study of
$\ker{\pi_{1}}\cap\tilde{H}_{s_{0}}$. Up to Section 6.8, we will doing
calculation inside $\tilde{W}_{s}$(defined after Lemma 6.2). For the sake of
simplicity, henceforth we denote $\tilde{w}_{L_{n}}(\sqrt{2}a)(a\in
S^{m-n-1})$ by $\tilde{w}_{L_{n}}(a)$,
$\tilde{h}_{L_{n}}(\sqrt{2}a,\sqrt{2}b)(a,b\in S^{m-n-1})$ by
$\tilde{h}_{L_{n}}(a,b)$ and
$\tilde{h}^{i}_{L_{n}}(\sqrt{2}b,\sqrt{2}c)((b,c)\in S^{1})$ by
$\tilde{h}^{i}_{L_{n}}(b,c)$ until the end of Section 6.8.
###### Lemma 6.7.
For $\forall(a,b),(c,d)\in S^{1}$ we have:
$\displaystyle[\tilde{h}^{i}_{L_{n}}(a,b),\tilde{h}^{i}_{L_{n}}(c,d)]$
$\displaystyle=\tilde{h}^{i}_{L_{n}}\bigl{(}(a^{2}-b^{2},2ab)\cdot(c,d)\bigl{)}\tilde{h}^{i}_{L_{n}}(a^{2}-b^{2},2ab)^{-1}\tilde{h}^{i}_{L_{n}}(c,d)^{-1}$
$\displaystyle=\tilde{h}^{i}_{L_{n}}(a,b)\tilde{h}^{i}_{L_{n}}(c^{2}-d^{2},2cd)\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b)\cdot(c^{2}-d^{2},2cd)\bigl{)}^{-1}$
where the $\cdot$ is multiplication among complex numbers.
###### Proof.
Using Lemma 6.3, it follows
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b)\tilde{w}_{L_{n}}(0,\dots,c_{i},d_{i+1},0\dots,0)\tilde{h}^{i}_{L_{n}}(a,b)^{-1}$
$\displaystyle=\tilde{w}_{L_{n}}\bigl{(}0,\dots,(ca^{2}-cb^{2}-2dba)_{i},(da^{2}-db^{2}+2cba)_{i+1},\dots,0\bigl{)}.$
Thus we have:
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b)\tilde{h}^{i}_{L_{n}}(c,d)\tilde{h}^{i}_{L_{n}}(a,b)^{-1}$
$\displaystyle=\tilde{h}^{i}_{L_{n}}\bigl{(}(a^{2}-b^{2},2ab)\cdot(c,d)\bigl{)}\tilde{h}^{i}_{L_{n}}(a^{2}-b^{2},2ab)^{-1},$
and
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b)^{-1}\tilde{h}^{i}_{L_{n}}(c,d)\tilde{h}^{i}_{L_{n}}(a,b)$
$\displaystyle=\tilde{h}^{i}_{L_{n}}(c^{2}-d^{2},2cd)\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b)\cdot(c^{2}-d^{2},2cd)\bigl{)}^{-1}.$
We have thus proved the lemma. ∎
###### Lemma 6.8.
For $\forall a\in S^{m-n-1}$,
$\tilde{w}_{L_{n}}(a)=\tilde{w}_{L_{n}}(-a)\tilde{h}^{1}_{L_{n}}(-1,0)$
and
$\tilde{h}^{1}_{L_{n}}(-1,0)\tilde{h}^{1}_{L_{n}}(-1,0)=e.$
###### Proof.
Notice $\pi_{1}(\tilde{h}^{1}_{L_{n}}(-1,0))=I_{m+n}$. By Lemma 6.3, for any
$a=(a_{1},a_{2},\dots,a_{m-n})\in S^{m-n-1}$ we have
$\displaystyle\tilde{h}^{1}_{L_{n}}(-1,0)=\tilde{w}_{L_{n}}(a)\tilde{h}^{1}_{L_{n}}(-1,0)\tilde{w}_{L_{n}}(a)^{-1}$
$\displaystyle=\tilde{w}_{L_{n}}\left(1-2a_{1}^{2},-2a_{1}a_{2},-2a_{1}a_{3},\dots,-2a_{1}a_{m-n}\right)$
$\displaystyle\cdot\tilde{w}_{L_{n}}\left(1-2a_{1}^{2},-2a_{1}a_{2},-2a_{1}a_{3},\dots,-2a_{1}a_{m-n}\right).$
Let $f:S^{m-n-1}\rightarrow S^{m-n-1}$ be
$f(a)=\left(1-2a_{1}^{2},-2a_{1}a_{2},-2a_{1}a_{3},\dots,-2a_{1}a_{m-n}\right)$
for $\forall a=\left(a_{1},a_{2},\dots,a_{m-n}\right)\in S^{m-n-1}$.
It is easy to check $f$ is surjective. Hence we proved the first par of the
lemma. For the second part, notice
$\pi_{1}(\tilde{h}^{1}_{L_{n}}(-1,0))=I_{m+n}$ and use Lemma 6.2 we have
$\displaystyle\tilde{h}^{1}_{L_{n}}(-1,0)=$
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{h}^{1}_{L_{n}}(-1,0)\tilde{h}_{L_{n-1}-L_{n}}(-1)^{-1}$
$\displaystyle=$
$\displaystyle\tilde{w}_{L_{n}}(1,\dots,0)\tilde{w}_{L_{n}}(1,\dots,0)$
$\displaystyle=$ $\displaystyle\tilde{h}^{1}_{L_{n}}(-1,0)^{-1}.$
Hence we proved the lemma. ∎
Let $H^{i}$ be the subgroup generated by $\tilde{h}^{i}_{L_{n}}(a,b)((a,b)\in
S^{1})$. We prove the following:
###### Lemma 6.9.
$\displaystyle(1)\ker(\pi_{1})\cap
H^{1}=\\{\prod_{j}\tilde{h}^{1}_{L_{n}}(a_{j},b_{j})\mid\text{ \emph{with}
}\prod_{j}(a_{j}^{2}-b_{j}^{2}+2a_{j}b_{j}\emph{i})=1\\},$
$\displaystyle(2)\ker(\pi_{1})\cap H^{j}=\ker(\pi_{1})\cap H^{1},\qquad\text{
\emph{for} }j\leq m-n-1.$
###### Proof.
(1) Notice $\pi_{1}(\tilde{h}^{1}_{L_{n}}(a,b))$=diag$(R_{2n+1})$, where
$R=\left(\begin{array}[]{ccc}a^{2}-b^{2}&-2ab\\\ 2ab&a^{2}-b^{2}\\\
\end{array}\right).$
We identify the above matrix with a complex number
$a_{j}^{2}-b_{j}^{2}+2a_{j}b_{j}\textrm{i}$, then (1) is clear.
It follows from (1) that $\ker(\pi_{1})\cap H^{1}$ is generated by elements
$\tilde{h}^{1}_{L_{n}}(a,b)\tilde{h}^{1}_{L_{n}}(c,d)\tilde{h}^{1}_{L_{n}}\bigl{(}(a,b)\cdot(c,d)\bigl{)}^{-1}\text{
and }\tilde{h}^{1}_{L_{n}}(-1,0).$
where $(a,b),(c,d)\in S^{1}$ and $\cdot$ means multiplication among complex
numbers.
(2) We can prove similarly that $\ker(\pi_{1})\cap H^{i}$ is generated by
elements
$\tilde{h}^{i}_{L_{n}}(a,b)\tilde{h}^{i}_{L_{n}}(c,d)\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b)\cdot(c,d)\bigl{)}^{-1}\text{
and }\tilde{h}^{i}_{L_{n}}(-1,0).$
Let $i,j$ be distinct, let
$w=\tilde{w}_{L_{n}}\bigl{(}0,-\frac{\sqrt{2}}{2},\dots,(\frac{\sqrt{2}}{2})_{i+1},\dots,0\bigl{)}\cdot\tilde{w}_{L_{n}}\bigl{(}-\frac{\sqrt{2}}{2},\dots,(\frac{\sqrt{2}}{2})_{i},\dots,0\bigl{)}.$
By Lemma 6.3 we have
$\displaystyle w\tilde{h}^{i}_{L_{n}}(a,b)w^{-1}=\tilde{h}^{1}_{L_{n}}(a,b).$
Since $\tilde{h}^{i}_{L_{n}}(-1,0)\in Z(\widetilde{G}),$ we have
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b)\tilde{h}^{i}_{L_{n}}(c,d)\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b)\cdot(c,d)\bigl{)}^{-1}$
$\displaystyle=w\tilde{h}^{i}_{L_{n}}(a,b)\tilde{h}^{i}_{L_{n}}(c,d)\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b)\cdot(c,d)\bigl{)}^{-1}w^{-1}$
$\displaystyle=\tilde{h}^{1}_{L_{n}}(a,b)\tilde{h}^{1}_{L_{n}}(c,d)\tilde{h}^{1}_{L_{n}}\bigl{(}(a,b)\cdot(c,d)\bigl{)}^{-1},$
and
$\displaystyle\tilde{h}^{i}_{L_{n}}(-1,0)=w\tilde{h}^{i}_{L_{n}}(-1,0)w^{-1}=\tilde{h}^{1}_{L_{n}}(-1,0).$
Thus we have proved (2). ∎
For $(a,b),(c,d)\in S^{1}$, we define:
$\displaystyle\\{(a,b),(c,d)\\}=\tilde{h}^{1}_{L_{n}}\bigl{(}(a,b)\cdot(c,d)\bigl{)}\tilde{h}^{1}_{L_{n}}(a,b)^{-1}\tilde{h}^{1}_{L_{n}}(c,d)^{-1},$
By Lemma 6.7, in exactly the same manner as the proof in the appendix of [20],
these $\\{(a,b),(c,d)\\}$’s satisfy the conditions
###### Lemma 6.10.
For all $(a,b),(c,d),(a_{1},b_{1}),(c_{1},d_{1})\in S^{1}$, we have:
$\displaystyle\\{(a,b),(c,d)\\}$ $\displaystyle=\\{(c,d),(a,b)\\}^{-1},$
$\displaystyle\\{(a,b),(c,d)\cdot(c_{1},d_{1})\\}$
$\displaystyle=\\{(a,b),(c,d)\\}\cdot\\{(a,b),(c_{1},d_{1})\\},$
$\displaystyle\\{(a,b)\cdot(a_{1},b_{1}),(c,d)\\}$
$\displaystyle=\\{(a,b),(c,d)\\}\cdot\\{(a_{1},b_{1}),(c,d)\\},$
$\displaystyle\\{(c,d),(-c,-d)\\}$ $\displaystyle=1.$
Thus we define a symbol on $S^{1}$.
### 6.6. Structure of $\ker(\pi_{1})\cap\tilde{H}_{s_{0}}$
We recall the notation set in Corollary 6.1. $\tilde{H}_{s_{0}}$ is the
subgroup generated by all $\tilde{h}^{j}_{L_{n}}(a,b)$($(a,b)\in S^{1}$). We
focus on studying $\ker(\pi_{1})\cap\tilde{H}_{s_{0}}$. The crucial step in
proving the main Theorem 4 is:
###### Theorem 7.
$\ker(\pi_{1})\cap\tilde{H}_{s_{0}}=\ker(\pi_{1})\cap H^{1}$.
The ensuing discussion (up to Lemma 6.2) proves this theorem. Henceforth we
consider the quotient group $\tilde{W}_{s}/(\ker(\pi_{1})\cap H^{1})$ until
the end of Section 6.8.(Note that $\ker(\pi_{1})\cap H^{1}$ being central,
this is well defined.) We continue to write $\tilde{h}^{j}_{L_{n}}(a,b)$ for
$(a,b)\in S^{1}$ and $\tilde{w}_{L_{n}}(a)$ for $a\in S^{m-n-1}$ for their
images in $\tilde{W}_{s}/(\ker(\pi_{1})\cap H^{1})$. However, there is no
confusion in doing so.
As a first step towards the proof of Theorem 7, we prove:
###### Proposition 6.1.
$\displaystyle H^{i}\cdot H^{i+1}\cdot H^{i}=H^{i+1}\cdot H^{i}\cdot H^{i+1}.$
Let $(a,b,c)\in S^{2}$, define
$\tilde{w}_{L_{n}}^{i}(a,b,c)=\tilde{w}_{L_{n}}(0,\ldots,a_{i},b_{i+1},c_{i+2},0,\ldots,0).$
Lemma 6.11–6.12 are preparations for proof of Proposition 6.1.
###### Lemma 6.11.
For $\forall$ $(a,b,c)\in S^{2}$, $\forall x\in\mathbb{R}$, $\forall j$, there
exist $(d,g,f)\in S^{2}$, $(d_{1},g_{1},f_{1})\in S^{2}$, $y\in\mathbb{R}$ and
$y_{1}\in\mathbb{R}$ such that
$\displaystyle(1)\tilde{w}^{j}_{L_{n}}(a,b,c)\tilde{w}^{j}_{L_{n}}(\cos x,\sin
x,0)=\tilde{w}^{j}_{L_{n}}(d,g,f)\tilde{w}^{j}_{L_{n}}(0,\cos y,\sin y),$
$\displaystyle(2)\tilde{w}^{j}_{L_{n}}(a,b,c)\tilde{w}^{j}_{L_{n}}(0,\cos
x,\sin x)=\tilde{w}^{j}_{L_{n}}(d_{1},g_{1},f_{1})\tilde{w}^{j}_{L_{n}}(\cos
y_{1},\sin y_{1},0).$
###### Proof.
(1) If $(a,b,c)$ and $(\cos x,\sin x,0)$ are collinear, then $(a,b,c)=\pm(\cos
x,\sin x,0)$. We just let $-(d,g,f)=(0,\cos y,\sin y)=(0,-1,0)$. By Lemma 6.8,
we get the conclusion.
Suppose $(a,b,c)$ and $(\cos x,\sin x,0)$ are not collinear. Then we choose
$y\in\mathbb{R}$ and $(d,g,f)\in S^{2}$ such that $(0,\cos y,\sin y)$ and
$(d,g,f)$ are both on the plane generated by $(a,b,c)$ and $(\cos x,\sin x,0)$
and satisfying
$\angle\bigl{(}(a,b,c),(\cos x,\sin x,0)\bigl{)}=\angle\bigl{(}(d,g,f),(0,\cos
y,\sin y)\bigl{)},$
where $\angle$ means the angle between 2 vectors in $\mathbb{R}^{3}$. Choose
an $h$ in the subgroup generated by $H^{j}$ and $H^{j+1}$ such that
$\pi_{1}(h)$ maps the 4 vectors to $xy$-plane(with last coordinate 0 in
$\mathbb{R}^{3}$). Denote
$\displaystyle\pi_{1}(h)\cdot(a,b,c)$ $\displaystyle=(a_{1},b_{1},0),$
$\displaystyle\qquad\pi_{1}(h)\cdot(\cos x,\sin x,0)$ $\displaystyle=(\cos
x_{1},\sin x_{1},0),$ $\displaystyle\pi_{1}(h)\cdot(d,g,f)$
$\displaystyle=(d_{1},g_{1},0),$ $\displaystyle\qquad\pi_{1}(h)\cdot(0,\cos
y,\sin y)$ $\displaystyle=(\cos y_{1},\sin y_{1},0).$
We have
$\angle\left((a_{1},b_{1},0),(\cos x_{1},\sin
x_{1},0)\right)=\angle\left((d_{1},g_{1},0),(\cos y_{1},\sin y_{1},0)\right),$
since $\pi_{1}(h)\in SO(3)$.
From the above equation and the fact that
$\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(a_{1},b_{1},0)\tilde{w}^{j}_{L_{n}}(\cos
x_{1},\sin x_{1},0)\bigl{)}$ is a rotation in $xy$-plane with angle 2 times
the one between $(\cos x_{1},\sin x_{1},0)$ and $(a_{1},b_{1},0)$, it follows
$\displaystyle\pi_{1}\left(\tilde{w}^{j}_{L_{n}}(a_{1},b_{1},0)\tilde{w}^{j}_{L_{n}}(\cos
x_{1},\sin x_{1},0)\right)$
$\displaystyle=\pi_{1}\left(\tilde{w}^{j}_{L_{n}}(d_{1},g_{1},0)\tilde{w}^{j}_{L_{n}}(\cos
y_{1},\sin y_{1},0)\right).$
Hence we get
$\displaystyle h\tilde{w}^{j}_{L_{n}}(a,b,c)\tilde{w}^{j}_{L_{n}}(\cos x,\sin
x,0)h^{-1}$
$\displaystyle=\tilde{w}^{j}_{L_{n}}(a_{1},b_{1},0)\tilde{w}^{j}_{L_{n}}(\cos
x_{1},\sin x_{1},0)$
$\displaystyle=\tilde{w}^{j}_{L_{n}}(d_{1},g_{1},0)\tilde{w}^{j}_{L_{n}}(\cos
y_{1},\sin y_{1},0)\qquad(\text{ by Lemma }\ref{le:12})$
$\displaystyle=h\tilde{w}^{j}_{L_{n}}(d,g,f)\tilde{w}^{j}_{L_{n}}(0,\cos
y,\sin y)h^{-1}.$
Thus we proved
$\tilde{w}^{j}_{L_{n}}(a,b,c)\tilde{w}^{j}_{L_{n}}(\cos x,\sin
x,0)=\tilde{w}^{j}_{L_{n}}(d,g,f)\tilde{w}^{j}_{L_{n}}(0,\cos y,\sin y).$
(2) A similar argument holds for (2).
We have thus proved the lemma completely. ∎
###### Lemma 6.12.
For $\forall\theta_{1},\theta_{2},\theta_{3}\in\mathbb{R}$, there exist
$\beta_{1},\beta_{2},\beta_{3},$ $\beta_{4},\beta_{5},\beta_{6}\in\mathbb{R}$,
such that
$\displaystyle(1)$
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{h}^{j+1}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})$
$\displaystyle=\tilde{h}^{j+1}_{L_{n}}(\cos\beta_{1},\sin\beta_{1})\tilde{h}^{j}_{L_{n}}(\cos\beta_{2},\sin\beta_{2})\tilde{h}^{j+1}_{L_{n}}(\cos\beta_{3},\sin\beta_{3});$
$\displaystyle(2)$
$\displaystyle\tilde{h}^{j+1}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{h}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j+1}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos\beta_{4},\sin\beta_{4})\tilde{h}^{j+1}_{L_{n}}(\cos\beta_{5},\sin\beta_{5})\tilde{h}^{j}_{L_{n}}(\cos\beta_{6},\sin\beta_{6}).$
###### Proof.
(1) Using Lemma 6.3, it follows
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{h}^{j+1}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{w}^{j}_{L_{n}}(0,\cos\theta_{2},\sin\theta_{2})\tilde{w}^{j}_{L_{n}}(0,-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{w}^{j}_{L_{n}}(0,\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})^{-1}$
$\displaystyle\cdot\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{w}^{j}_{L_{n}}(0,-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})$
$\displaystyle=\tilde{w}^{j}_{L_{n}}\left(-\sin 2\theta_{1}\cos\theta_{2},\cos
2\theta_{1}\cos\theta_{2},\sin\theta_{2}\right)\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})$
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(0,-1,0)\tilde{h}^{j}_{L_{n}}\left(\cos\theta_{3},\sin\theta_{3}\right)\tilde{w}^{j}_{L_{n}}(-1,0,0)\tilde{w}^{j}_{L_{n}}(1,0,0).$
Since
$\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{w}^{j}_{L_{n}}(0,-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})\tilde{w}^{j}_{L_{n}}(-1,0,0)\in
H^{j},$
and
$\displaystyle\pi_{1}\left(\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{w}^{j}_{L_{n}}(0,-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})\tilde{w}^{j}_{L_{n}}(-1,0,0)\right)$
$\displaystyle=\pi_{1}\left(\tilde{h}^{j}_{L_{n}}\left(-\sin(\theta_{1}-\theta_{3}),\cos(\theta_{1}-\theta_{3})\right)\right),$
by Lemma 6.9, we have
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{w}^{j}_{L_{n}}(0,-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})\tilde{w}^{j}_{L_{n}}(-1,0,0)$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(-\sin(\theta_{1}-\theta_{3}),\cos(\theta_{1}-\theta_{3})).$
It follows
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{h}^{j+1}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})$
$\displaystyle=\tilde{w}^{j}_{L_{n}}\left(-\sin 2\theta_{1}\cos\theta_{2},\cos
2\theta_{1}\cos\theta_{2},\sin\theta_{2}\right)$
$\displaystyle\cdot\tilde{h}^{j}_{L_{n}}(-\sin(\theta_{1}-\theta_{3}),\cos(\theta_{1}-\theta_{3}))\tilde{w}^{j}_{L_{n}}(1,0,0)$
$\displaystyle=\tilde{w}^{j}_{L_{n}}(-\sin 2\theta_{1}\cos\theta_{2},\cos
2\theta_{1}\cos\theta_{2},\sin\theta_{2})$
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(-\sin(\theta_{1}-\theta_{3}),\cos(\theta_{1}-\theta_{3}),0).$
By Lemma 6.11, there exist $(a,b,c)\in S^{2}$ and $\alpha\in\mathbb{R}$
satisfying
$\displaystyle\tilde{w}^{j}_{L_{n}}(-\sin 2\theta_{1}\cos\theta_{2},\cos
2\theta_{1}\cos\theta_{2},\sin\theta_{2})$
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(-\sin(\theta_{1}-\theta_{3}),\cos(\theta_{1}-\theta_{3}),0)$
(6.1)
$\displaystyle=\tilde{w}^{j}_{L_{n}}(a,b,c)\tilde{w}^{j}_{L_{n}}(0,\cos\alpha,\sin\alpha).$
Using a similar argument , for
$\forall\gamma_{1},\gamma_{2},\gamma_{3}\in\mathbb{R}$ we have
$\displaystyle\tilde{h}^{j+1}_{L_{n}}(\cos\gamma_{1},\sin\gamma_{1})\tilde{h}^{j}_{L_{n}}(\cos\gamma_{2},\sin\gamma_{2})\tilde{h}^{j+1}_{L_{n}}(\cos\gamma_{3},\sin\gamma_{3})$
$\displaystyle=\tilde{w}^{j}_{L_{n}}\left(-\sin\gamma_{2},\cos
2\gamma_{1}\cos\gamma_{2},\sin 2\gamma_{1}\cos\gamma_{2}\right)$ (6.2)
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}\left(0,\cos(\gamma_{1}-\gamma_{3}),\sin(\gamma_{1}-\gamma_{3})\right).$
It is easy to see that there exist
$\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$ satisfying
$\displaystyle(-\sin\beta_{2},\cos 2\beta_{1}\cos\beta_{2},\sin
2\beta_{1}\cos\beta_{2})$ $\displaystyle=(a,b,c)$
and
$\beta_{1}-\beta_{3}=\alpha.$
Let $\gamma_{1}=\beta_{1},$ $\gamma_{2}=\beta_{2}$ and $\gamma_{3}=\beta_{3},$
combine (6.6) and (6.6), we thus proved (1).
A similar argument holds for (2). We have thus proved the lemma completely. ∎
### 6.7. Proof of Proposition 6.1
By Lemma 6.9, every element of $H^{i}$ can be expressed as
$\tilde{h}^{i}_{L_{n}}(a,b)$ where $(a,b)\in S^{1}$. Thus by Lemma 6.12, it
follows
$\displaystyle H^{i}\cdot H^{i+1}\cdot H^{i}\subseteq H^{i+1}\cdot H^{i}\cdot
H^{i+1}$
and
$\displaystyle H^{i+1}\cdot H^{i}\cdot H^{i+1}\subseteq H^{i}\cdot
H^{i+1}\cdot H^{i},$
which completes the proof.
We denote $H^{1}\cdot H^{2},\dotsc,\cdot H^{j}$ by $\prod_{j=1}^{i}H^{j}$.
With this notation we have:
###### Proposition 6.2.
$\displaystyle\tilde{H}_{s_{0}}=\bigl{(}\prod_{i=1}^{m-n-1}H^{i}\bigl{)}\cdot\bigl{(}\prod_{i=1}^{m-n-2}H^{i}\bigl{)},\dotsc,\cdot\bigl{(}H^{1}\bigl{)}.$
###### Proof.
We write $\tilde{h}^{i}_{L_{n}}$ for the set of all
$\tilde{h}^{i}_{L_{n}}(a,b)$, $(a,b)\in S^{1}$. It is sufficient to prove:
$\displaystyle\tilde{H}_{s_{0}}=\bigl{(}\prod_{i=1}^{m-n-1}\tilde{h}^{i}_{L_{n}}\bigl{)}\cdot\bigl{(}\prod_{i=1}^{m-n-2}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\cdot\bigl{(}\tilde{h}^{1}_{L_{n}}\bigl{)}.$
We use induction on $m-n=2+k$. For $k=0$ we are though by Lemma 6.9.
If $k=1$, any element in $\tilde{H}_{s_{0}}$ can be written as
$\tilde{h}^{1}_{L_{n}}\tilde{h}^{2}_{L_{n}},\dotsc,\tilde{h}^{1}_{L_{n}}\tilde{h}^{2}_{L_{n}}$.
Keep using Proposition 6.1 and Lemma 6.9, then we are through.
Now assume the lemma is correct for $m-n=2+k$. We will show it is correct for
$m-n=2+k+1$. Denote $Y_{k}$ the subgroup generated by $H^{j}$ where $j\leq
k+1$. Then $\tilde{H}_{s_{0}}$ is generated by
$\\{Y_{k},\tilde{h}^{k+2}_{L_{n}}\\}$. By induction, any $y_{k}\in Y_{k}$ can
be written as
$y_{k}=\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\cdot\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)}\cdot,\dotsc,\cdot\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.$
Next we will see what changes are brought to this express after adding
$\tilde{h}^{i}_{L_{n}}(i\leq 2+k)$ from both sides of $y_{k}$. Obviously,
adding $\tilde{h}^{i}_{L_{n}}(i\leq 1+k)$ from either side makes no changes.
One considers adding $\tilde{h}^{k+2}_{L_{n}}$ from both sides. Notice if
$2\leq|i-j|$,
(6.3)
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b)\tilde{h}^{j}_{L_{n}}(c,d)\tilde{h}^{i}_{L_{n}}(a,b)^{-1}=\tilde{h}^{j}_{L_{n}}(c,d),$
thus for any $y_{k}$, we have
$\displaystyle\tilde{h}^{k+2}_{L_{n}}y_{k}\tilde{h}^{k+2}_{L_{n}}$
$\displaystyle=\tilde{h}^{k+2}_{L_{n}}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+2}_{L_{n}}$
$\displaystyle=\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+2}_{L_{n}}\tilde{h}^{k+1}_{L_{n}}\tilde{h}^{k+2}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\qquad(\text{
by }(\ref{for:9}))$
$\displaystyle=\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\tilde{h}^{k+2}_{L_{n}}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\qquad(\text{
by Proposition }\ref{po:1})$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.$
Since
$\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\in
Y_{k}$, by induction, we have
$\displaystyle\tilde{h}^{k+2}_{L_{n}}y_{k}\tilde{h}^{k+2}_{L_{n}}$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.$
Next we consider the changes after adding $\tilde{h}^{i}_{L_{n}}(i\leq 2+k)$
from left side of the above expression. It is clear adding
$\tilde{h}^{1}_{L_{n}}$ makes no changes. Neither for
$\tilde{h}^{k+2}_{L_{n}}$, since a same argument holds as in previous proof.
For $\tilde{h}^{j}_{L_{n}}(1<j<k+2)$ we have:
$\displaystyle\tilde{h}^{j}_{L_{n}}\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}$
$\displaystyle=\bigl{(}\prod_{i=1}^{j-2}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{j}_{L_{n}}\tilde{h}^{j-1}_{L_{n}}\tilde{h}^{j}_{L_{n}}\bigl{(}\prod_{i=j+1}^{k+2}\tilde{h}^{j}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}(\text{
by }(\ref{for:9}))$
$\displaystyle=\bigl{(}\prod_{i=1}^{j-2}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{j-1}_{L_{n}}\tilde{h}^{j}_{L_{n}}\tilde{h}^{j-1}_{L_{n}}\bigl{(}\prod_{i=j+1}^{k+2}\tilde{h}^{j}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)},\dots,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}(\text{
by Proposition }\ref{po:1})$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{j-1}_{L_{n}}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dots,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.\qquad(\text{
by }(\ref{for:9}))$
Since
$\tilde{h}^{j-1}_{L_{n}}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\in
Y_{k}$, by induction we have
$\displaystyle\tilde{h}^{j}_{L_{n}}\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.$
Hence we are through left side case. Now consider adding elements from right
side. In this case, we are left with adding $\tilde{h}^{k+2}_{L_{n}}$. Then
$\displaystyle\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+2}_{L_{n}}$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+2}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\qquad(\text{
by }(\ref{for:9}))$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+2}_{L_{n}}\tilde{h}^{k+1}_{L_{n}}\tilde{h}^{k+2}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\tilde{h}^{k+2}_{L_{n}}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.\qquad(\text{
by Proposition }\ref{po:1}).$
Since
$\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\in
Y_{k}$, by induction we have
$\displaystyle\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\tilde{h}^{k+2}_{L_{n}}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+2}_{L_{n}}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}$
$\displaystyle=\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.\qquad(\text{
by }(\ref{for:9}))$
Notice
$\displaystyle\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\in
Y_{k},$
by induction we have
$\displaystyle\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}\tilde{h}^{k+1}_{L_{n}}\bigl{(}\prod_{i=1}^{k}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)},$
$\displaystyle=$
$\displaystyle\bigl{(}\prod_{i=1}^{k+2}\tilde{h}^{i}_{L_{n}}\bigl{)}\bigl{(}\prod_{i=1}^{k+1}\tilde{h}^{i}_{L_{n}}\bigl{)},\dotsc,\bigl{(}\prod_{i=1}\tilde{h}^{i}_{L_{n}}\bigl{)}.$
Thus we have proved adding elements from right leaves the expression
unchanged. Then we finally proved this proposition. ∎
### 6.8. Proof of Theorem 7
By Proposition 6.2, $\forall h\in\tilde{H}_{s_{0}}$ can be written as
$\displaystyle
h=\bigl{(}\prod_{i=1}^{m-n-1}\tilde{h}^{i}_{L_{n}}(a^{m-n-1}_{i},b^{m-n-1}_{i})\bigl{)}\bigl{(}\prod_{i=1}^{m-n-2}\tilde{h}^{i}_{L_{n}}(a^{m-n-2}_{i},b^{m-n-2}_{i})\bigl{)}$
$\displaystyle,\dots,\tilde{h}^{1}_{L_{n}}\left(a^{1}_{1},b^{1}_{1}\right)h_{0}$
where $(a^{j}_{i},b^{j}_{i})\in S^{1}$ for $i,j\leq m-n-1$ and
$h_{0}\in\ker(\pi_{1})\cap H^{1}$. Notice the element in the lower right
corner of matrix $\pi_{1}(h)$ is
$\left(a^{m-n-1}_{m-n-1}\right)^{2}-\left(b^{m-n-1}_{m-n-1}\right)^{2}$.
If $\pi_{1}(h)=I_{m+n}$, we have $a^{m-n-1}_{m-n-1}=\pm 1$,
$b^{m-n-1}_{m-n-1}=0$. Thus it follows
$\tilde{h}^{m-n-1}_{L_{n}}(a^{m-n-1}_{m-n-1},b^{m-n-1}_{m-n-1})\in\ker(\pi_{1})\cap
H^{1}$. By induction, we have $h\in\ker(\pi_{1})\cap H^{1}$. We thus proved
$\ker(\pi_{1})\cap\tilde{H}_{s_{0}}\subseteq\ker(\pi_{1})\cap H^{1}$. The
other side inclusion is obvious. Hence we finished the proof completely.
### 6.9. Proof of Theorem 4
We first consider $m=n$. By the fact coming from Lemma 6.4 that
$\ker(\pi_{1})\subseteq\prod_{r\in\Delta}\tilde{H}_{r}$, where
$\Delta=\\{L_{i}-L_{i+1},L_{n-1}+L_{n}\\}$, we only need to consider the
elements in $\prod_{r\in\Delta}\tilde{H}_{r}$. By Lemma 6.5, $\forall
h\in\prod_{r\in\Delta}\tilde{H}_{r}$ can be written as
$h=\tilde{h}_{L_{1}-L_{2}}(a_{1})\tilde{h}_{L_{2}-L_{3}}(a_{2}),\dots,\tilde{h}_{L_{n-1}-L_{n}}(a_{n-1})\tilde{h}_{L_{n-1}+L_{n}}(a_{n})h_{0}$
where $h_{0}\in\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$.
If $\pi_{1}(h)=I_{m+n}$, we have $a_{1}=a_{2}=\dots=a_{n-2}=1$ and
$a_{n-1}=a_{n}=\pm 1$. Thus we have
$\displaystyle h=h_{0}\qquad\text{ or }\qquad
h=\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{h}_{L_{n-1}+L_{n}}(-1).$
Notice
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{h}_{L_{n-1}+L_{n}}(-1)$
$\displaystyle=h_{L_{1}-L_{n}}^{\sim}(-1)\bigl{(}\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{h}_{L_{n-1}+L_{n}}(-1)\bigl{)}h_{L_{1}-L_{n}}^{\sim}(-1)^{-1}$
$\displaystyle=\tilde{w}_{L_{n-1}-L_{n}}(1)\tilde{w}_{L_{n-1}-L_{n}}(1)\tilde{w}_{L_{n-1}+L_{n}}(1)\tilde{w}_{L_{n-1}-L_{n}}(1)\tilde{w}_{L_{n-1}+L_{n}}(1)$
$\displaystyle=\bigl{(}\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{h}_{L_{n-1}+L_{n}}(-1)\bigl{)}^{-1}$
Thus we have
$\bigl{(}\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{h}_{L_{n-1}+L_{n}}(-1)\bigl{)}^{2}=e$.
Thus we proved the case for $m=n$.
If $m=n+1$ by Lemma 6.4,
$\ker(\pi_{1})\subseteq\bigl{(}\prod_{r\in\Delta}\tilde{H}_{r}\bigl{)}\cdot\tilde{H}_{s}$,
where $\Delta=\\{L_{i}-L_{i+1},L_{n-1}+L_{n}\\}$. Further, by Lemma 6.5,
$\forall
h\in\bigl{(}\prod_{r\in\Delta}\tilde{H}_{r}\bigl{)}\cdot\tilde{H}_{s}$ can be
written as
$h=\tilde{h}_{L_{1}-L_{2}}(a_{1})\tilde{h}_{L_{2}-L_{3}}(a_{2}),\dots,\tilde{h}_{L_{n-1}-L_{n}}(a_{n-1})\tilde{h}_{L_{n-1}+L_{n}}(a_{n})h_{1}h_{0}$
where $h_{0}\in\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$ and
$h_{1}\in\tilde{H}_{s}$.
If $\pi_{1}(h)=I_{m+n}$, we have $a_{1}=a_{2}=\dots=a_{n-2}=1$,
$a_{n-1}=a_{n}=\pm 1$ and $\pi_{1}(h_{1})=I_{m+n}$.
If $a_{n-1}=a_{n}=1$, we get
$h\in\left(\ker(\pi_{1})\cap\tilde{H}_{s}\right)\cdot\left(\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\right)$.
If $a_{n-1}=a_{n}=-1$, we have
$\displaystyle\tilde{h}^{1}_{L_{n}}(-1,0)$
$\displaystyle=\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{w}_{L_{n}}(\sqrt{2},0,\ldots,0)$
$\displaystyle\cdot\tilde{h}_{L_{n-1}-L_{n}}(-1)^{-1}\tilde{w}_{L_{n}}(-\sqrt{2},0,\ldots,0)$
(6.4)
$\displaystyle=\tilde{h}_{L_{n-1}-L_{n}}(-1)\tilde{h}_{L_{n-1}+L_{n}}(-1).$
Thus we still get
$h\in\left(\ker(\pi_{1})\cap\tilde{H}_{s}\right)\cdot\left(\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\right)$.
Notice $\ker(\pi_{1})\cap\tilde{H}_{s}$ is the 2-cyclic group generated by
$h^{1}_{L_{n}}(-1)$ by the fact $\bigl{(}h^{1}_{L_{n}}(-1)\bigl{)}^{2}=e$ from
lemma 6.8. Thus we proved the case for $m=n+1$.
If $m\geq n+2$, by Corollary 6.1,
$\ker(\pi_{1})\subseteq\left(\prod_{r\in\Delta}\tilde{H}_{r}\right)\tilde{H}_{s_{0}}$,
where $\Delta=\\{L_{i}-L_{i+1},L_{n-1}+L_{n}\\}$. Further, by Lemma 6.5,
$\forall h\in\left(\prod_{r\in\Delta}\tilde{H}_{r}\right)\tilde{H}_{s_{0}}$
can be written as
$h=\tilde{h}_{L_{1}-L_{2}}(a_{1})\tilde{h}_{L_{2}-L_{3}}(a_{2}),\dots,\tilde{h}_{L_{n-1}-L_{n}}(a_{n-1})\tilde{h}_{L_{n-1}+L_{n}}(a_{n})h_{1}h_{0}$
where $h_{0}\in\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$ and
$h_{1}\in\tilde{H}_{s_{0}}$. If $\pi_{1}(h)=I_{m+n}$, we have
$a_{1}=a_{2}=\dots=a_{n-2}=1$, $a_{n-1}=a_{n}=\pm 1$ and
$\pi_{1}(h_{1})=I_{m+n}$.
If $a_{n-1}=a_{n}=1$, we get
$h\in\left(\ker(\pi_{1})\cap\tilde{H}_{s_{0}}\right)\cdot\left(\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\right)$.
If $a_{n-1}=a_{n}=-1$, by (6.9) we still get
$h\in\left(\ker(\pi_{1})\cap\tilde{H}_{s_{0}}\right)\cdot\left(\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\right).$
By Theorem 7, $\ker(\pi_{1})\cap\tilde{H}_{s_{0}}=\ker(\pi_{1})\cap H^{1}$.
Thus we have proved
$\ker(\pi_{1})=\left(\ker(\pi_{1})\cap
H^{1}\right)\cdot\left(\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\right).$
Hence we proved Theorem 4 completely.
## 7\. Generating relations of $SU(m,n)$
### 7.1. Basic settings for $SU(m,n)$
In this part, we study the generators of $SU(m,n)$ where $m\geq n\geq 3$. we
use $\overline{a}$ to denote complex conjugate of complex numbers, vectors or
matrices. We use notations as in Section 3.3 and Section 5. Explicitly, this
is the case where $G=SU(m,n)(\mathbb{C},H)$ with $H$ a non-degenerate standard
hermitian form of signature $(m,n)$.
In the sequel we freely use the notation of previous part without confusion.
We denote by set $S$ the the $(m+n)\times(m+n)$ real diagonal matrices in
$G_{\mathbb{R}}$ with lower-right $(m-n)\times(m-n)$ block identity. Let
$\Phi$ be the root system of $G$ with respect to $S$. The roots are $\pm
L_{i}\pm L_{j}(i<j\leq n)$, whose dimensions are 2 and $\pm 2L_{i}(i\leq n)$
whose dimension is 1. Also the $\pm L_{i}(i\leq n)$ are roots if $m\neq n$
with dimensions $2(m-n)$. If $m-n\geq 1$, the set of positive roots $\Phi^{+}$
and the corresponding set of simple roots $\Delta$ are
$\displaystyle\Phi^{+}=\\{L_{i}-L_{j}\\}_{i<j}\cup\\{L_{i}+L_{j}\\}_{i<j}\cup\\{L_{i}\\}_{i}\cup\\{2L_{i}\\}_{i},$
$\displaystyle\Delta=\\{L_{i}-L_{i+1}\\}_{i<j}\cup\\{L_{n}\\};$
if $m=n$, the set of positive roots $\Phi^{+}$ and the corresponding set of
simple roots $\Delta$ are
$\displaystyle\Phi^{+}=\\{L_{i}-L_{j}\\}_{i<j}\cup\\{L_{i}+L_{j}\\}_{i<j}\cup\\{2L_{i}\\}_{i},$
$\displaystyle\Delta=\\{L_{i}-L_{i+1}\\}_{i}\cup\\{L_{n-1}-L_{n}\\}_{i}.$
Correspondingly, if $m-n\geq 1$, the set of $\Phi_{1}$ are $\pm L_{i}\pm
L_{j}(i\neq j),\pm L_{i}$; if $m=n$, the set of $\Phi_{1}$ are $\pm L_{i}\pm
L_{j}(i\neq j)$.
We use $e_{k,\ell}$ to denote matrix with $(k,\ell)$ element 1, otherwise 0.
We denote
$\displaystyle f^{1}_{L_{i}+L_{j}}$
$\displaystyle=(e_{i,j+n}-e_{j,i+n})_{i<j},$ $\displaystyle\quad
f^{2}_{L_{i}+L_{j}}$ $\displaystyle=\textrm{i}(e_{i,j+n}+e_{j,i+n})_{i<j},$
$\displaystyle f^{1}_{L_{i}-L_{j}}$
$\displaystyle=(e_{i,j}-e_{j+n,i+n})_{i\neq j},$ $\displaystyle\quad
f^{2}_{L_{i}-L_{j}}$ $\displaystyle=\textrm{i}(e_{i,j}+e_{j+n,i+n})_{i\neq
j},$ $\displaystyle f^{1}_{-L_{i}-L_{j}}$
$\displaystyle=(e_{j+n,i}-e_{i+n,j})_{i<j},$ $\displaystyle\quad
f^{2}_{-L_{i}-L_{j}}$ $\displaystyle=\textrm{i}(e_{j+n,i}+e_{i+n,j})_{i<j},$
$\displaystyle(f_{L_{i}}^{\ell})_{1}$
$\displaystyle=(e_{i,2n+\ell}-e_{2n+\ell,i+n})_{\ell\leq m-n},$
$\displaystyle\quad(f_{L_{i}}^{\ell})_{2}$
$\displaystyle=\textrm{i}(e_{i,2n+\ell}+e_{2n+\ell,i+n})_{i\leq m-n},$
$\displaystyle(f_{-L_{i}}^{\ell})_{1}$
$\displaystyle=(e_{i+n,2n+\ell}-e_{2n+\ell,i})_{\ell\leq m-n},$
$\displaystyle\quad(f_{-L_{i}}^{\ell})_{2}$
$\displaystyle=\textrm{i}(e_{i+n,2n+\ell}+e_{2n+\ell,i})_{i\leq m-n},$
$\displaystyle f_{2L_{i}}$ $\displaystyle=\textrm{i}e_{i,i+n},$
$\displaystyle\quad f_{-2L_{i}}$ $\displaystyle=\textrm{i}e_{i+n,i}.$
For any complex number $z$, we use $r(z)$ to denote the real part and $i(z)$
to denote the imaginary part. Thus we have
$\displaystyle U^{r}_{\mathbb{R}}=$
$\displaystyle\\{\exp\left(r(z)f^{1}_{r}\right)\exp\left(i(z)f^{2}_{r}\right)\mid
z\in\mathbb{C}\\}\text{ for }r=\pm L_{i}\pm L_{j}(i<j),$ $\displaystyle
U^{\alpha}_{\mathbb{R}}=$
$\displaystyle\\{\exp\bigl{(}\bigl{(}t-\sum_{j}r(a_{j})i(a_{j})\bigl{)}f_{2\alpha}\bigr{)}\exp\bigl{(}r(a_{1})(f^{1}_{\alpha})_{1}\bigl{)}\exp\bigl{(}i(a_{1})(f^{1}_{\alpha})_{2}\bigl{)}$
$\displaystyle,\dots,\exp\bigl{(}r(a_{m-n})(f^{m-n}_{\alpha})_{1}\bigl{)}\exp\bigl{(}i(a_{m-n})(f^{m-n}_{\alpha})_{2}\bigl{)}$
$\displaystyle\mid
a=(a_{1},\dots,a_{m-n})\in\mathbb{C}^{m-n},t\in\mathbb{R}\\}\qquad\text{ for
}\alpha=\pm L_{i}.$
Correspondingly, for $t\in\mathbb{R}$, $z\in\mathbb{C}$ and
$a=(a_{1},\dots,a_{m-n})\in\mathbb{C}^{m-n}$ we write
$\displaystyle x_{r}(z)=$
$\displaystyle\exp\left(r(z)f^{1}_{r}\right)\exp\left(i(z)f^{2}_{r}\right)\in
U^{r}_{\mathbb{R}}\qquad\text{ for }r=\pm L_{i}\pm L_{j}(i<j),$ $\displaystyle
x_{\alpha}(t,a)=$
$\displaystyle\exp\bigl{(}\bigl{(}t-\sum_{j}r(a_{j})i(a_{j})\bigr{)}f_{2\alpha}\bigr{)}\exp\bigl{(}r(a_{1})(f^{1}_{\alpha})_{1}\bigl{)}\exp\left(i(a_{1})(f^{1}_{\alpha})_{2}\right)$
$\displaystyle,\dots,\exp\bigl{(}r(a_{m-n})(f^{m-n}_{\alpha})_{1}\bigl{)}\exp\left(i(a_{m-n})(f^{m-n}_{\alpha})_{2}\right)\in
U^{\alpha}_{\mathbb{R}}$ $\displaystyle\qquad\text{ for }\alpha=\pm L_{i}.$
Notice if $a=0$, $x_{\alpha}(t,0)=\exp(tf_{2\alpha})$.
### 7.2. “Chain” in $SU(m,n)$ and basic relations
Our next step is to determine explicitly the “chain” (cf. Lemma 5.1)
corresponding to the element $x_{\alpha}(t,a)(\neq e)\in
U^{\alpha}_{\mathbb{R}}(\alpha=\pm L_{i})$ where
$a=(a_{1},\dots,a_{m-n})\in\mathbb{C}^{m-n}$.
We determine the “chain” for $x_{\alpha}(0,a)$ at first. For this, define
$f:\mathbb{C}^{m-n}\backslash 0\rightarrow\mathbb{C}^{m-n}\backslash 0$ by
$f(a)=\bigl{(}\frac{2a_{1}}{\sum\lvert
a_{i}\rvert^{2}},\dots,\frac{2a_{m-n}}{\sum\lvert a_{i}\rvert^{2}}\bigr{)}$
for $a=(a_{1},\dots,a_{m-n})\in\mathbb{C}^{m-n}\backslash 0$. With this
notation, we have:
###### Lemma 7.1.
For $x_{\alpha}(0,a)(\neq e)\in U^{\alpha}_{\mathbb{R}}(\alpha=\pm L_{i})$,
the “chain” corresponding to it is given by
$\displaystyle x_{i}=x_{\alpha}(0,a),i\in\mathbb{Z};\qquad
y_{i}=x_{-\alpha}(0,f(a)),i\in\mathbb{Z}.$
Denoting the element $w_{\alpha}(x_{\alpha}(0,a))$ by $w_{\alpha}(0,a)$, we
have
$\displaystyle w_{\alpha}(0,a)$
$\displaystyle=x_{\alpha}(0,a)x_{-\alpha}(0,f(a))x_{\alpha}(0,a).$
###### Proof.
It is easy to check $\\{X,Y,[X,Y]\\}$ span a three-dimensional Lie algebra
isomorphic to $SL_{2}(\mathbb{R})$, where
$\displaystyle
X=\sum_{j}(r(a_{j})(f^{j}_{\alpha})_{1}+i(a_{j})(f^{j}_{\alpha})_{2}),$
$\displaystyle
Y=\sum_{j}-\frac{2r(a_{j})(f^{j}_{-\alpha})_{1}+2i(a_{j})(f^{j}_{-\alpha})_{2}}{\sum\lvert
a_{i}\rvert^{2}}.$
And we have
$\displaystyle\exp(X)=x_{\alpha}(0,a),\qquad\exp(-Y)=x_{\alpha}(0,f(a)).$
Thus by Remark 5.1 we get the conclusion. ∎
###### Remark 7.1.
Similar computations can be made for the other roots, such as $\pm L_{i}\pm
L_{j}(i<j)$, $\pm 2L_{i}$. We record the results here:
$\displaystyle w_{r}(z)=x_{r}(z)x_{-r}(-z^{-1})x_{r}(z),\qquad
z\in\mathbb{C}^{*},r=\pm L_{i}\pm L_{j}(i<j),$
where
$\displaystyle x_{i}=x_{r}(z)\forall i,\qquad y_{i}=x_{-r}(-z^{-1})\forall i.$
$\displaystyle
w_{\beta}(t)=w_{\alpha}(t,0)=x_{\alpha}(t,0)x_{-\alpha}(t^{-1},0)x_{\alpha}(t,0),\qquad
t\in\mathbb{R}^{*},\beta=2\alpha,\alpha=\pm L_{i},$
where
$\displaystyle x_{i}=x_{\alpha}(t,0)\forall i,\qquad
y_{i}=x_{-\alpha}(t^{-1},0)\forall i.$
Correspondingly, we define
$\displaystyle h_{r}(z)=w_{r}(z)w_{r}(1)^{-1},\qquad z\in\mathbb{C}^{*},r=\pm
L_{i}\pm L_{j}(i<j),$ $\displaystyle
h_{\beta}(t)=w_{\beta}(t)w_{\beta}(1)^{-1},\qquad t\in\mathbb{R}^{*},\beta=\pm
2L_{i},$ $\displaystyle
h_{\alpha}((0,a),(0,b))=w_{\alpha}(0,a)w_{\alpha}(0,b)^{-1},\qquad
a,b\in\mathbb{C}^{m-n}\backslash 0,\alpha=\pm L_{i}.$
Using the same notations as in Remark 6.1 we have:
$\displaystyle
w_{L_{i}-L_{j}}(z)=p(\pi)\text{diag}\bigl{(}(-z^{-1})_{i},z_{j},(-\overline{z})_{i+n},(\overline{z}^{-1})_{j+n}\bigr{)},\qquad\text{
for }z\in\mathbb{C}^{*}$
where $\pi$ only permutes $(i,j)$ and $(i+n,j+n)$ while fixes other numbers.
$\displaystyle
w_{L_{i}+L_{j}}(z)=p(\pi)\text{diag}\bigl{(}(-z^{-1})_{i},(\overline{z}^{-1})_{j},(-\overline{z})_{i+n},z_{j+n}\bigr{)},\qquad\text{
for }z\in\mathbb{C}^{*}$
where $\pi$ only permutes $(i,j+n)$ and $(j,i+n)$ while fixes other numbers.
$\displaystyle w_{L_{i}}(0,a)=p(\pi)\text{diag}\bigl{(}(-2\lvert
a\rvert^{-2})_{i},(-\frac{1}{2}\lvert
a\rvert^{2})_{i+n},B_{2n+1}\bigr{)},\qquad\text{ for
}a\in\mathbb{C}^{m-n}\backslash 0,$
where $B\in U(m-n)$ and $\pi$ only permutes $(i,i+n)$ while fixes other
numbers.
$\displaystyle
w_{2L_{i}}(t)=p(\pi)\text{diag}\bigl{(}(t^{-1}\textrm{i})_{i},(t\textrm{i})_{i+n}\bigl{)},\qquad\text{
for }t\in\mathbb{C}^{*},$
where $\pi$ only permutes $(i,i+n)$ while fixes other numbers.
###### Definition 7.1.
We can now define elements $\tilde{x}_{r}(z)$, $\tilde{x}_{\beta}(t)$,
$\tilde{x}_{\alpha}(t,a)$, $\tilde{w}_{r}(z)$, $\tilde{w}_{\beta}(t)$,
$\tilde{w}_{\alpha}(0,a)$, $\tilde{h}_{r}(z)$, $\tilde{h}_{\beta}(t)$,
$\tilde{h}_{\alpha}\bigl{(}(0,a),(0,b)\bigr{)}$ etc. as was done in Section
5.2. We denote by $\tilde{W}_{r}(r=\pm L_{i}\pm L_{j},i<j)$ the subgroup of
$\widetilde{G}$ generated by $\tilde{w}_{r}(z)$, $\tilde{H}_{r}(r=\pm L_{i}\pm
L_{j},i<j)$ the subgroup generated by $\tilde{h}_{r}(z)$.
Also, by Lemma 5.3, it is clear that certain relations hold both in
$\widetilde{G}$ and $G_{\mathbb{R}}$. We record these results in 2 separate
lemmas(Lemma 7.2 and Lemma 7.3), since they will serve as ready references
later.
###### Lemma 7.2.
If $a\in\mathbb{C}^{m-n}\backslash 0$, $z\in\mathbb{C}^{*}$,
$t\in\mathbb{R}^{*}$ the following hold in $\widetilde{G}$:
* 1
$\tilde{w}_{L_{n}}(0,a)\tilde{w}_{L_{n-1}-L_{n}}(z)\tilde{w}_{L_{n}}(0,a)^{-1}=\tilde{w}_{L_{n-1}+L_{n}}(-\frac{1}{2}\lvert
a\rvert^{2}z)$,
* 2
$\tilde{w}_{L_{n}}(0,a)\tilde{w}_{L_{n-1}+L_{n}}(z)\tilde{w}_{L_{n}}(0,a)^{-1}=\tilde{w}_{L_{n-1}-L_{n}}(-2\lvert
a\rvert^{-2}z)$,
* 3
$\tilde{w}_{L_{n-1}-L_{n}}(z)\tilde{w}_{L_{n}}(0,a)\tilde{w}_{L_{n-1}-L_{n}}(z)^{-1}=\tilde{w}_{L_{n-1}}(0,az)$,
* 4
$\tilde{w}_{L_{n-1}-L_{n}}(z)\tilde{w}_{L_{n-1}}(0,a)\tilde{w}_{L_{n-1}-L_{n}}(z)^{-1}=\tilde{w}_{L_{n}}(0,-az^{-1})$,
* 5
$\tilde{w}_{L_{n-1}-L_{n}}(z)\tilde{w}_{2L_{n}}(t)\tilde{w}_{L_{n-1}-L_{n}}(z)^{-1}=\tilde{w}_{2L_{n-1}}(t\lvert
z\rvert^{2})$,
* 6
$\tilde{w}_{2L_{n}}(t)\tilde{w}_{L_{n-1}-L_{n}}(z)\tilde{w}_{2L_{n}}(t)^{-1}=\tilde{w}_{L_{n-1}+L_{n}}(-tz\emph{i})$.
Hence,
* 5
$\tilde{h}_{L_{n-1}-L_{n}}(z)\tilde{w}_{L_{n}}(0,a)\tilde{h}_{L_{n-1}-L_{n}}(z)^{-1}=\tilde{w}_{L_{n}}(0,az^{-1})$,
* 6
$\tilde{h}_{L_{n-1}-L_{n}}(z)\tilde{w}_{2L_{n}}(t)\tilde{h}_{L_{n-1}-L_{n}}(z)^{-1}=\tilde{w}_{2L_{n}}(t\lvert
z\rvert^{-2})$,
* 7
$\tilde{w}_{L_{n}}(0,a)\tilde{h}_{L_{n-1}-L_{n}}(z)\tilde{w}_{L_{n}}(0,a)^{-1}\\\
=\tilde{h}_{L_{n-1}+L_{n}}\left(-\frac{1}{2}\lvert
a\rvert^{2}z\right)\tilde{h}_{L_{n-1}+L_{n}}\left(-\frac{1}{2}\lvert
a\rvert^{2}\right)^{-1}$,
* 8
$\tilde{w}_{2L_{n}}(t)\tilde{h}_{L_{n-1}-L_{n}}(z)\tilde{w}_{2L_{n}}(t)^{-1}=\tilde{h}_{L_{n-1}+L_{n}}(-tz\emph{i})\tilde{h}_{L_{n-1}+L_{n}}(-t\emph{i})^{-1}$.
We denote by $S_{\mathbb{R}}^{i}$ the sphere in $\mathbb{R}^{i+1}$ and by
$S_{\mathbb{C}}^{i}$ the sphere in $\mathbb{C}^{i+1}$. Let $\tilde{W}_{s}$ be
the subgroup generated by $\tilde{w}_{L_{n}}(0,\sqrt{2}a)$, $a\in
S_{\mathbb{C}}^{m-n-1}$. If $a=(a_{1},\dots,a_{n})$, then
$\pi_{1}\bigl{(}\tilde{w}_{L_{n}}(0,\sqrt{2}a)\bigr{)}=p(\pi)\text{diag}\bigl{(}(-1)_{n},(-1)_{2n},B_{2n+1}\bigr{)},$
where $\pi$ permutes n and 2n while fixes other numbers and $B\in U(m-n)$ with
entries $B_{i,j}=-2\overline{a_{i}}a_{j}$, for $i\neq j$ and
$B_{i,i}=1-2\lvert a_{i}\rvert^{2}$. Then $B$ is a reflection in the
hyperplane orthogonal to $\overline{a}$. Thus for any $w\in\tilde{W}_{s}$,
$\pi_{1}(w)=p(\pi)\text{diag}\bigl{(}(-1)^{\delta}_{n},(-1)^{\delta}_{2n},B_{2n+1}\bigr{)},$
where $\delta=2$ if $p(\pi)=I_{m+n}$ and $B\in SU(m-n)$; $\delta=1$ if
$p(\pi)$ permutes $n$ and $2n$ and and $B\in U(m-n)$ with determinant $-1$.
Without confusion, we identify $\pi_{1}(\tilde{w}_{L_{n}}(0,\sqrt{2}a))$ and
$B$.
Arguments similar to those in Lemma 6.3 show the following lemma:
###### Lemma 7.3.
If $w\in\tilde{W}_{s}$,
$\pi_{1}(w)=p(\pi)\operatorname{diag}((-1)^{\delta}_{n},1,(-1)^{\delta}_{2n},B_{2n+1})$,
$\delta=1$ or $2$, $B\in U(m-n)$, $a\in S_{\mathbb{C}}^{m-n-1}$,
$b\in\mathbb{C}^{m-n}$, $t\in\mathbb{R}^{*}$ then
(7.1) $\displaystyle w\tilde{w}_{L_{n}}(0,\sqrt{2}a)w^{-1}$
$\displaystyle=\left\\{\begin{aligned}
\tilde{w}_{L_{n}}(0,\sqrt{2}\overline{B}\cdot a),&\quad&\text{ if
}p(\pi)=I_{m+n},\\\ \tilde{w}_{L_{n}}(0,-\sqrt{2}\overline{B}\cdot
a),&\quad&\text{ if }p(\pi)\neq I_{m+n},\\\ \end{aligned}\right.$ (7.2)
$\displaystyle w\tilde{x}_{L_{n}}(t,b)w^{-1}$
$\displaystyle=\left\\{\begin{aligned} \tilde{x}_{L_{n}}(t,\overline{B}\cdot
b),&\quad&\text{ if }p(\pi)=I_{m+n},\\\
\tilde{x}_{-L_{n}}(-t,-\overline{B}\cdot b),&\quad&\text{ if }p(\pi)\neq
I_{m+n}.\\\ \end{aligned}\right.$
where $\cdot$ means linear operation on vectors.
We now determine the “chain” for $x_{\alpha}(t,a)$.
###### Lemma 7.4.
For $x_{\alpha}(t,a)(\neq e)\in U^{\alpha}_{\mathbb{R}}(\alpha=\pm L_{n})$,
the “chain” corresponding to it is given by
$\displaystyle
x_{i}=x_{\alpha}\bigl{(}t,\overline{a_{0}^{i}}a_{0}^{-i}a\bigr{)},i\in\mathbb{Z};$
$\displaystyle y_{i}=x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i}}a_{0}^{-i-1}a\bigr{)},i\in\mathbb{Z},$
where $a_{0}=-\frac{1}{2}\lvert a\rvert^{2}+t\emph{i}$. Denoting the element
$w_{\alpha}(x_{\alpha}(t,a))$ by $w_{\alpha}(t,a)$, we have
$\displaystyle w_{\alpha}(t,a)$
$\displaystyle=x_{\alpha}(t,a)x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-a_{0}^{-1}a\bigr{)}x_{\alpha}\bigl{(}t,\overline{a_{0}}a_{0}^{-1}a\bigr{)}.$
###### Proof.
For $a$, we can find $B\in SU(m-n)$ such that $B\cdot a=(\lvert
a\rvert,\dots,0)$. Denote $(\lvert a\rvert,\dots,0)$ by $a^{\prime}$. By
remarks after Lemma 7.2, we can find $b_{i}\in S_{\mathbb{C}}^{m-n-1}$ such
that
$\pi_{1}\bigl{(}\prod_{i}w_{\alpha}(\sqrt{2}b_{i})\bigr{)}=\overline{B}.$
Let $w=\prod_{i}w_{\alpha}(\sqrt{2}b_{i})$. Using Lemma 7.3 we have
$\displaystyle
wx_{\alpha}\bigl{(}t,\overline{a_{0}^{i}}a_{0}^{-i}a\bigr{)}x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i}}a_{0}^{-i-1}a\bigr{)}x_{\alpha}\bigl{(}t,\overline{a_{0}^{i+1}}a_{0}^{-i-1}a\bigr{)}w^{-1}$
$\displaystyle=x_{\alpha}\bigl{(}t,\overline{a_{0}^{i}}a_{0}^{-i}a^{\prime}\bigr{)}x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i}}a_{0}^{-i-1}a^{\prime}\bigr{)}x_{\alpha}\bigl{(}t,\overline{a_{0}^{i+1}}a_{0}^{-i-1}a^{\prime}\bigr{)}.$
In [[6], p. 30], it was proved
$w_{\alpha}(t,a^{\prime})=x_{\alpha}\bigl{(}t,\overline{a_{0}^{i}}a_{0}^{-i}a^{\prime}\bigr{)}x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i}}a_{0}^{-i-1}a^{\prime}\bigr{)}x_{\alpha}\bigl{(}t,\overline{a_{0}^{i+1}}a_{0}^{-i-1}a^{\prime}\bigr{)}\in
N(S)_{\mathbb{R}}$
and in fact acts as the reflection with respect to $\alpha$. Since $w\in
Z(S)_{\mathbb{R}}$, it follows that $x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i}}a_{0}^{-i-1}a\bigr{)}=y_{i}$ and
$x_{\alpha}\bigl{(}t,\overline{a_{0}^{i+1}}a_{0}^{-i-1}a\bigr{)}=x_{i}$ are
the “right” elements in the chain of $x_{\alpha}(t,a)$ and
$w_{\alpha}(t,a)=x_{\alpha}\bigl{(}t,\overline{a_{0}^{i}}a_{0}^{-i}a\bigr{)}x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i}}a_{0}^{-i-1}a\bigr{)}x_{\alpha}\bigl{(}t,\overline{a_{0}^{i}}a_{0}^{-i}a\bigr{)}.$
And we have
$\displaystyle w_{\alpha}(t,a)=w^{-1}w_{\alpha}(t,a^{\prime})w.$
Repeating the argument, we find that
$\displaystyle w_{\alpha}(t,a)=x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i}}a_{0}^{-i-1}a\bigr{)}x_{\alpha}\bigl{(}t,\overline{a_{0}^{i+1}}a_{0}^{-i-1}a\bigr{)}x_{-\alpha}\bigl{(}\lvert
a_{0}\rvert^{-2}t,-\overline{a_{0}^{i+1}}a_{0}^{-i-2}a\bigr{)}.$
This proves the lemma completely. ∎
###### Definition 7.2.
Denote $w_{L_{n}}(t,z)=w_{L_{n}}(t,z,0,\dots,0)$ for
$(t,z)\in(\mathbb{R}\times\mathbb{C})\backslash 0$. Let $W_{L_{n}}$ be the
subgroup generated by all $w_{L_{n}}(t,a)$ where
$(t,a)\in(\mathbb{R}\times\mathbb{C}^{m-n})\backslash 0$ and $H_{L_{n}}$ the
subgroup generated by all $w_{L_{n}}(t_{1},a_{1})w_{L_{n}}(t_{2},a_{2})^{-1}$
where $(t_{1},a_{1})$,
$(t_{2},a_{2})\in(\mathbb{R}\times\mathbb{C}^{m-n})\backslash 0$. Denote by
$W_{u}$ the subgroup generated by $w_{L_{n}}(t,z)$, where
$(t,z)\in(\mathbb{R}\times\mathbb{C})\backslash 0$ and denote by $H_{u}$ the
subgroup generated by all $w_{L_{n}}(t_{1},z_{1})w_{L_{n}}(t_{2},z_{2})^{-1}$
where $(t_{1},z_{1}),(t_{2},z_{2})\in(\mathbb{R}\times\mathbb{C})\backslash
0$. Denote by $W_{v}$ the subgroup generated by $w_{L_{n}}(0,a)$ where
$a\in\mathbb{C}^{m-n}\backslash 0$ and denote by $H_{v}$ the subgroup
generated by $w_{L_{n}}(0,a)w_{L_{n}}(0,b)^{-1}$ where
$a,b\in\mathbb{C}^{m-n}\backslash 0$. Let $\tilde{W}_{L_{n}}$,
$\tilde{H}_{L_{n}}$, $\tilde{W}_{u}$, $\tilde{W}_{v}$, $\tilde{H}_{v}$ and
$\tilde{H}_{u}$ be the corresponding subgroups in $\widetilde{G}$.
For $(t,a)\in(\mathbb{R}\times\mathbb{C}^{m-n})\backslash 0$, where
$a=(a_{1},...,a_{n})$. $a_{0}=-\frac{1}{2}\lvert a\rvert^{2}+t\textrm{i}$ then
$\pi_{1}\bigl{(}\tilde{w}_{L_{n}}(t,a)\bigr{)}=p(\pi)\text{diag}\bigl{(}(\overline{a_{0}^{-1}})_{n},(a_{0})_{2n},B_{2n+1}\bigr{)},$
where $\pi$ permutes n and 2n while fixes other numbers and $B\in
U(m-n)$(determinant of $B$ is $-\overline{a_{0}}a_{0}^{-1}$) with entries
$B_{i,j}=\overline{a_{i}}a_{j}a_{0}^{-1}$, for $i\neq j$ and $B_{i,i}=1+\lvert
a_{i}\rvert^{2}a_{0}^{-1}$. Without confusion, we identify
$\pi_{1}(w_{L_{n}}(t,a))$ and $B$. The following lemma are proved easily by
computations using the Steinberg’s relations [[24], p. 40] or by using Lemma
5.3:
###### Lemma 7.5.
For $(t,a),(t_{1},b)\in(\mathbb{R}\times\mathbb{C}^{m-n})\backslash 0$,
$z\in\mathbb{C}^{*}$,
$\pi_{1}\bigl{(}\tilde{w}_{L_{n}}(t,a)\bigr{)}=p(\pi)\operatorname{diag}\bigl{(}(\overline{a_{0}^{-1}})_{n},(a_{0})_{2n},B_{2n+1}\bigr{)}$
where $a_{0}=(-\frac{1}{2}\lvert a\rvert^{2}+t\emph{i})$. we have
* 1
$\tilde{w}_{L_{n}}(t,a)\tilde{w}_{L_{n}}(t_{1},b)\tilde{w}_{L_{n}}(t,a)^{-1}=\tilde{w}_{-L_{n}}(t_{1}\lvert
a_{0}\rvert^{-2},\overline{a_{0}^{-1}B}\cdot b)$,
* 2
$\tilde{w}_{L_{n}}(t,a)\tilde{w}_{L_{n-1}-L_{n}}(z)\tilde{w}_{L_{n}}(t,a)^{-1}=\tilde{w}_{L_{n-1}+L_{n}}(z\overline{a_{0}})$,
* 3
$\tilde{w}_{L_{n}}(t,a)\tilde{w}_{L_{n-1}+L_{n}}(z)\tilde{w}_{L_{n}}(t,a)^{-1}=\tilde{w}_{L_{n-1}-L_{n}}(z_{1}a_{0}^{-1})$,
* 4
$\tilde{w}_{L_{n-1}+L_{n}}(z)\tilde{w}_{L_{n}}(t,a)\tilde{w}_{L_{n-1}+L_{n}}(z)^{-1}=\tilde{w}_{-L_{n-1}}(t\lvert
z\rvert^{-2},\overline{z_{1}^{-1}}a)$,
* 5
$\tilde{w}_{L_{n-1}-L_{n}}(z)\tilde{w}_{L_{n}}(t,a)\tilde{w}_{L_{n-1}-L_{n}}(z)^{-1}=\tilde{w}_{L_{n-1}}(t\lvert
z\rvert^{2},za)$.
Notice when $m\geq n+1$, $\tilde{w}_{L_{n}}(t,0)=\tilde{w}_{2L_{n}}(t)$ where
$t\in\mathbb{R}^{*}$.
For $z_{1},z_{2}\in\mathbb{C}^{*}$, we define:
$\displaystyle\\{z_{1},z_{2}\\}=\tilde{h}_{L_{1}-L_{2}}(z_{1})\tilde{h}_{L_{1}-L_{2}}(z_{2})\tilde{h}_{L_{1}-L_{2}}(z_{1}z_{2})^{-1}.$
In exactly the same manner as proofs of Lemma 6.5 and 6.6, we have the
followings:
###### Lemma 7.6.
$\displaystyle(1)\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}=\\{\prod_{i}\tilde{h}_{L_{1}-L_{2}}(z_{i})\mid\text{
\emph{with} }\prod_{i}z_{i}=1\\}.$
$\displaystyle(2)\ker(\pi_{1})\cap\tilde{H}_{r}=\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}},\qquad\text{
\emph{for} }r=\pm L_{i}\pm L_{j}(i\neq j).$
###### Lemma 7.7.
$\displaystyle\\{z_{1},z_{2}\\}$
$\displaystyle=\\{z_{2},z_{1}\\}^{-1}\qquad\forall
z_{1},z_{2}\in\mathbb{C}^{*},$ $\displaystyle\\{z_{1},z_{2}\cdot z_{3}\\}$
$\displaystyle=\\{z_{1},z_{2}\\}\cdot\\{z_{1},z_{3}\\}\qquad\forall
z_{1},z_{2},z_{3}\in\mathbb{C}^{*},$ $\displaystyle\\{z_{1}\cdot
z_{2},z_{3}\\}$
$\displaystyle=\\{z_{1},z_{3}\\}\cdot\\{z_{2},z_{3}\\}\qquad\forall
z_{1},z_{2},z_{3}\in\mathbb{C}^{*},$ $\displaystyle\\{z,1-z\\}$
$\displaystyle=1\qquad\forall z\in\mathbb{C}^{*},z\neq 1,$
$\displaystyle\\{z,-z\\}$ $\displaystyle=1\qquad\forall z\in\mathbb{C}^{*}.$
Thus we define a symbol on $\mathbb{C}$.
### 7.3. Structure of $\ker(\pi_{1})$
If $m\geq n+2$, for any $(a,b)\in S^{1}_{\mathbb{C}}$, $j\leq m-n-1$, we
define
$\displaystyle\tilde{h}^{j}_{L_{n}}(0,\sqrt{2}a,\sqrt{2}b)$
$\displaystyle=\tilde{w}_{L_{n}}(0,\dots,0,\underset{j+1}{\sqrt{2}a},\underset{j+2}{\sqrt{2}b},0,\dots,0)$
$\displaystyle\cdot\tilde{w}_{L_{n}}(0,\dots,0,\underset{j+1}{-\sqrt{2}},0,\dots,0).$
Let $\tilde{H}_{s_{0}}$ denote the subgroup generated by
$\tilde{h}^{j}_{L_{n}}(0,\sqrt{2}a,\sqrt{2}b),((a,b)\in S_{\mathbb{C}}^{1})$
and $\tilde{H}_{0}$ denote the cyclic group generated by
$\tilde{h}_{2n}(-1)\tilde{h}_{2n}(-1)$ and $\tilde{H}_{c}$ denote the cyclic
group generated by $\tilde{h}_{2n}(-1)$.
An important step in proving Theorem 5 is:
###### Theorem 8.
If $n\leq m\leq n+1$,
$\ker(\pi_{1})=\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\bigl{)}\cdot\tilde{H}_{0}$;
if $m\geq n+2$,
$\ker(\pi_{1})=\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\bigr{)}\cdot\tilde{H}_{0}\cdot\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{s_{0}}\bigr{)}$.
The proof of this theorem relies on the following result. Recall the notation
set in Definition 7.2. We have
###### Lemma 7.8.
* (i)
$\tilde{H}_{2L_{n}}\subseteq\tilde{H}_{L_{n-1}-L_{n}}\cdot\tilde{H}_{L_{n-1}+L_{n}}\cdot\tilde{H}_{c}$.
* (ii)
$\ker(\pi_{1})\cap\tilde{H}_{2L_{n}}\subseteq(\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}})\cdot\tilde{H}_{0}$.
* (iii)
$\tilde{H}_{v}\subseteq\tilde{H}_{L_{n-1}-L_{n}}\cdot\tilde{H}_{L_{n-1}+L_{n}}\cdot\tilde{H}_{s_{0}}$.
* (iv)
$\ker(\pi_{1})\cap\tilde{H}_{v}\subseteq(\ker(\pi_{1})\cap\tilde{H}_{L_{n-1}-L_{n}})\cdot(\ker(\pi_{1})\cap\tilde{H}_{s_{0}})$.
* (v)
$\ker(\pi_{1})\cap\tilde{H}_{L_{n}}\subseteq\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\bigr{)}\cdot\tilde{H}_{0}\cdot\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{s_{0}}\bigr{)}$
###### Proof.
(i) Using Lemma 7.2, for $\forall t\in\mathbb{R}^{*}$, let
$z\in\mathbb{C}^{*}$ such that $\lvert z\rvert=\lvert t\rvert$. We have
$\displaystyle\tilde{h}_{2L_{n}}(t)$
$\displaystyle=\tilde{w}_{2L_{n}}(t)\tilde{w}_{2L_{n}}(-1)$
$\displaystyle=\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})\tilde{w}_{2L_{n}}(t\lvert
z\rvert^{-1})\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})^{-1}\tilde{w}_{2L_{n}}(-1)$
$\displaystyle=\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})\tilde{w}_{2L_{n}}(t\lvert
t\rvert^{-1})\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})^{-1}\tilde{w}_{2L_{n}}(-1).$
If $t>0$ we have
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})\tilde{w}_{2L_{n}}(t\lvert
t\rvert^{-1})\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})^{-1}\tilde{w}_{2L_{n}}(-1)$
$\displaystyle=$
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})(\tilde{w}_{2L_{n}}(1)\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})^{-1}\tilde{w}_{2L_{n}}(-1))$
$\displaystyle=$
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})\tilde{h}_{L_{n-1}+L_{n}}(-\textrm{i})\tilde{h}_{L_{n-1}+L_{n}}(-z^{-\frac{1}{2}}\textrm{i})^{-1}.$
If $t<0$ we have
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})\tilde{w}_{2L_{n}}(t\lvert
t\rvert^{-1})\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})^{-1}\tilde{w}_{2L_{n}}(-1)$
$\displaystyle=$
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})\tilde{w}_{2L_{n}}(-1)\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})^{-1}\tilde{w}_{2L_{n}}(-1)$
$\displaystyle=$
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(z^{-\frac{1}{2}})\tilde{h}_{L_{n-1}+L_{n}}(\textrm{i})\tilde{h}_{L_{n-1}+L_{n}}(z^{-\frac{1}{2}}\textrm{i})^{-1}\tilde{h}_{2L_{n}}(-1).$
Especially, if $t=-1$, we have
$e=\tilde{h}_{L_{n-1}-L_{n}}(z)\tilde{h}_{L_{n-1}+L_{n}}(\textrm{i})\tilde{h}_{L_{n-1}+L_{n}}(z\textrm{i})^{-1},$
for $\forall z\in S^{0}_{\mathbb{C}}$. By Lemma 7.6, we have
$\tilde{h}_{L_{n-1}+L_{n}}(\textrm{i})\tilde{h}_{L_{n-1}+L_{n}}(z\textrm{i})^{-1}\in\tilde{h}_{L_{n-1}+L_{n}}(z^{-1})\cdot\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\bigr{)},$
then it follows
(7.3)
$\displaystyle\tilde{h}_{L_{n-1}-L_{n}}(z)\tilde{h}_{L_{n-1}+L_{n}}(z^{-1})\in\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$
for $\forall z\in S^{0}_{\mathbb{C}}$. Hence we proved (i).
(ii) By Lemma 7.6 and (i), any $h\in\tilde{H}_{2L_{n}}$ can be written as
$h=\tilde{h}_{L_{n-1}-L_{n}}(z_{1})\tilde{h}_{L_{n-1}+L_{n}}(z_{2})h_{1}h_{2}$
where $z_{1},z_{2}\in\mathbb{C}^{*}$,
$h_{1}\in\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$ and
$h_{2}\in\tilde{H}_{c}$.
If $\pi_{1}(h)=I_{m+n}$, we have
$z_{1}=\overline{z_{2}}=\overline{z_{1}^{-1}}$, and $\pi_{1}(h_{2})=I_{m+n}$.
By (7.3) we have
$\tilde{h}_{L_{n-1}-L_{n}}(z_{1})\tilde{h}_{L_{n-1}+L_{n}}(z_{2})\in\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}.$
Notice $\pi_{1}(\tilde{h}_{2n}(-1))=$diag$((-1)_{n},(-1)_{2n})$, it follows
$h_{2}=\bigl{(}\tilde{h}_{2n}(-1)\bigl{)}^{2k}$, $k\in\mathbb{Z}$. Hence we
proved (ii).
(iii) Notice for $(a,b)\in S_{\mathbb{C}}^{1}$, we have
$\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(0,\sqrt{2}a,\sqrt{2}b)\bigr{)}=\text{diag}(R_{2n+j}),$
where
$R_{j}=\left(\begin{array}[]{ccc}\lvert a\rvert^{2}-\lvert
b\rvert^{2}&-2\overline{a}b\\\ 2\overline{b}a&\lvert a\rvert^{2}-\lvert
b\rvert^{2}\\\ \end{array}\right).$
Then $\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(0,\sqrt{2}a,\sqrt{2}b)\bigl{)}$,
$j\leq m-n-1$ generate a subgroup isomorphic to $SU(m-n)$. Using Lemma 7.5,
similar to proofs in Lemma 6.4 and Corollary 6.1, it follows
$\displaystyle\tilde{H}_{v}\subseteq\tilde{H}_{L_{n-1}-L_{n}}\cdot\tilde{H}_{L_{n-1}+L_{n}}\tilde{H}_{s_{0}}.$
(iv) By Lemma 7.6 and (iii), any $h\in\tilde{H}_{v}$ can be written as
$h=\tilde{h}_{L_{n-1}-L_{n}}(z_{1})\tilde{h}_{L_{n-1}+L_{n}}(z_{2})h_{1}h_{2}$
where $z_{1},z_{2}\in\mathbb{C}^{*}$,
$h_{1}\in\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$ and
$h_{2}\in\tilde{H}_{s_{0}}$.
If $\pi_{1}(h)=I_{m+n}$, we have
$z_{1}=\overline{z_{2}}=\overline{z_{1}^{-1}}$, and $\pi_{1}(h_{2})=I_{m+n}$.
Along the line in proof of (ii), we get (iv).
(v) By Lemma 7.4, for
$\forall(t,a)\in(\mathbb{R}\times\mathbb{C}^{m-n})\backslash 0$ we can find
$w\in\tilde{H}_{v}$ such that
$\displaystyle\tilde{w}_{L_{n}}(t,a)=w^{-1}\tilde{w}_{L_{n}}(t,\lvert
a\rvert)w.$
Using Lemma 7.5, for any $w_{1}\in\tilde{W}_{u}$ we have
$\displaystyle w_{1}\tilde{W}_{v}w_{1}^{-1}\subseteq\tilde{W}_{v}\qquad\text{
and }\qquad w_{1}\tilde{H}_{v}w_{1}^{-1}\subseteq\tilde{H}_{v}.$
Thus it follows
$\displaystyle\tilde{H}_{L_{n}}\subseteq\tilde{H}_{v}\cdot\tilde{H}_{u}.$
Then any $h\in\tilde{H}_{L_{n}}$ can be written as $h=h_{v}h_{u}$ where
$h_{v}\in\tilde{H}_{v}$ and $h_{u}\in\tilde{H}_{u}$.
If $\pi_{1}(h)=I_{m+n}$ we have $\pi_{1}(h_{u})=\pi_{1}(h_{v})=I_{m+n}$. Thus
it follows that
$\ker(\pi_{1})\cap\tilde{H}_{L_{n}}\subseteq(\ker(\pi_{1})\cap\tilde{H}_{v})\cdot(\ker(\pi_{1})\cap\tilde{H}_{u})$.
In [[6], p37-p59] Theorem 2.13 asserts that if $m=n+1$,
$\ker(\pi_{1})=\ker\pi_{1})\cap\tilde{H}_{2L_{n}}$. It follows
$\ker(\pi_{1})\cap\tilde{H}_{u}\subseteq\ker(\pi_{1})\cap\tilde{H}_{2L_{n}}$.
By (ii) and (iv) we get (v). ∎
Proof of Theorem 8
If $m=n$, by Remark 5.2,
$\ker(\pi_{1})\subseteq(\prod_{r\in\Delta}\ker(\pi_{1})\cap\tilde{H}_{r})$
where $\Delta=\\{L_{i}-L_{i+1},2L_{n}\\}$. By Lemma 7.6,
$\ker(\pi_{1})\cap\tilde{H}_{L_{i}-L_{i+1}}=\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}$.
By Lemma 7.8 (2), we are thus though this case.
We are though the case $m=n+1$ by referring to [[6],Theorem 2.13] which
asserts that if $m=n+1$, $\ker(\pi_{1})=\ker(\pi_{1})\cap\tilde{H}_{2L_{n}}$.
When $m\geq n+2$, by Remark 5.2,
$\ker(\pi_{1})\subseteq(\prod_{r\in\Delta}\ker(\pi_{1})\cap\tilde{H}_{r})$
where $\Delta=\\{L_{i}-L_{i+1},L_{n}\\}$. Along the line in proof of the case
$m=n$, and by Lemma 7.8 (5), we are though this case.
### 7.4. Construction of $S_{\mathbb{R}}^{1}$-symbols
We now proceed with the study of $\ker(\pi_{1})\cap\tilde{H}_{s_{0}}$ when
$m-n\geq 2$. Up to Lemma 7.10, we study properties of
$\tilde{w}_{L_{n}}(0,\sqrt{2}a)(a\in S_{\mathbb{C}}^{m-n-1})$ and
$\tilde{h}^{i}_{L_{n}}(0,\sqrt{2}b,\sqrt{2}c)\bigl{(}(b,c)\in S^{1}\bigr{)}$
which build up the whole $\tilde{H}_{s_{0}}$, and construct new symbols on
$S_{\mathbb{R}}^{1}$ to get prepared for further study of
$\ker(\pi_{1})\cap\tilde{H}_{s_{0}}$.
###### Definition 7.3.
Recall that $\tilde{W}_{s}$ the subgroup generated by
$\tilde{w}_{L_{n}}(0,\sqrt{2}a)$, where $a\in S_{\mathbb{C}}^{m-n-1}$. We will
do calculations inside $\tilde{W}_{s}$ until the end of Section 7.6. For the
sake of simplicity, we denote $\tilde{w}_{L_{n}}(0,\sqrt{2}a)(a\in S^{m-n-1})$
by $\tilde{w}_{L_{n}}(a)$,
$\tilde{h}^{j}_{L_{n}}(0,\sqrt{2}a,\sqrt{2}b)(a,b\in S^{1}_{\mathbb{C}})$ by
$\tilde{h}^{j}_{L_{n}}(a,b)$.
###### Lemma 7.9.
For $(a,b),(c,d)\in S_{\mathbb{R}}^{1}$, $v\in S_{\mathbb{C}}^{0}$, we have
$\displaystyle(1)$ $\displaystyle\tilde{h}^{i}_{L_{n}}(v,0)=e,$
$\displaystyle(2)$
$\displaystyle[\tilde{h}^{i}_{L_{n}}(a,b),\tilde{h}^{i}_{L_{n}}(c,d)]$
$\displaystyle=\tilde{h}^{i}_{L_{n}}\bigl{(}(a^{2}-b^{2},2ab)\cdot(c,d)\bigl{)}\tilde{h}^{i}_{L_{n}}(a^{2}-b^{2},2ab)^{-1}\tilde{h}^{i}_{L_{n}}(c,d)^{-1}$
$\displaystyle=\tilde{h}^{i}_{L_{n}}(a,b)\tilde{h}^{i}_{L_{n}}(c^{2}-d^{2},2cd)\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b)\cdot(c^{2}-d^{2},2cd)\bigl{)}^{-1},$
$\displaystyle(3)[\tilde{h}^{i}_{L_{n}}(a,b\emph{i}),\tilde{h}^{i}_{L_{n}}(c,d\emph{i})]$
$\displaystyle=$
$\displaystyle\tilde{h}^{i}_{L_{n}}\bigl{(}(a^{2}-b^{2},-2ab\emph{i})\cdot(c,d\emph{i})\bigl{)}\tilde{h}^{i}_{L_{n}}(a^{2}-b^{2},-2ab\emph{i})^{-1}\tilde{h}^{i}_{L_{n}}(c,d\emph{i})^{-1}$
$\displaystyle=$
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b\emph{i})\tilde{h}^{i}_{L_{n}}(c^{2}-d^{2},-2cd\emph{i})\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b\emph{i})\cdot(c^{2}-d^{2},-2cd\emph{i})\bigl{)}^{-1},$
where the $\cdot$ follows multiplication rule among quarternions if we
identify any quarternion $(x+y\emph{i}+z\emph{j}+w\emph{k})$ with
$(x+y\emph{i},z+w\emph{i})$.
###### Proof.
(1) Notice $\pi_{1}(\tilde{h}^{i}_{L_{n}}(v,0))=I_{m+n}$ for any $v\in
S_{\mathbb{C}}^{0}$. For $\forall u\in S^{0}_{\mathbb{C}}$, choose
$h\in\tilde{H}_{s_{0}}$ such that
$\pi_{1}(h)$=diag$(\overline{u}_{2n+i},u_{2n+i+1})$, we have
$\displaystyle\tilde{h}^{i}_{L_{n}}(v,0)=\tilde{h}\tilde{h}^{i}_{L_{n}}(v,0)\tilde{h}^{-1}$
$\displaystyle=\tilde{w}_{L_{n}}(0,\dots,(uv)_{i},\dots,0)\tilde{w}_{L_{n}}(0,\dots,-u_{i},\dots,0)$
$\displaystyle=\tilde{h}^{i}_{L_{n}}(uv,0)\tilde{h}^{i}_{L_{n}}(u,0)^{-1}.$
Thus for any $u,v\in S_{\mathbb{C}}^{0}$, we have
$\tilde{h}^{i}_{L_{n}}(uv,0)=\tilde{h}^{i}_{L_{n}}(u,0)\tilde{h}^{i}_{L_{n}}(v,0).$
Thus we have
$\displaystyle\tilde{h}^{i}_{L_{n}}(-v^{2},0)=\tilde{h}^{i}_{L_{n}}(v,0)\tilde{h}^{i}_{L_{n}}(-v,0)$
$\displaystyle=\tilde{w}_{L_{n}}(0,\dots,v_{i},\dots,0)\tilde{w}_{L_{n}}(0,\dots,(-1)_{i},\dots,0)$
$\displaystyle\cdot\tilde{w}_{L_{n}}(0,\dots,(-v)_{i},\dots,0)\tilde{w}_{L_{n}}(0,\dots,(-1)_{i},\dots,0)$
$\displaystyle=\tilde{w}_{L_{n}}(0,\dots,(-1)_{i},\dots,0)\tilde{w}_{L_{n}}(0,\dots,(-1)_{i},\dots,0)$
$\displaystyle=\tilde{h}^{i}_{L_{n}}(-1,0).$
Since $\tilde{h}^{i}_{L_{n}}(-\textrm{i}^{2},0)=\tilde{h}^{i}_{L_{n}}(1,0)=e$,
we thus proved (1).
(2) and (3) By Lemma 7.3 we have
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b)\tilde{w}_{L_{n}}(0,\dots,c_{i},d_{i+1},0\dots,0)\tilde{h}^{i}_{L_{n}}(a,b)^{-1}$
$\displaystyle=\tilde{w}_{L_{n}}(ca^{2}-cb^{2}-2dba,da^{2}-db^{2}+2cba)$
$\displaystyle\tilde{h}^{i}_{L_{n}}(a,b\textrm{i})\tilde{w}_{L_{n}}(0,\dots,c_{i},d_{i+1}\textrm{i},0,\dots,0)\tilde{h}^{i}_{L_{n}}(a,b\textrm{i})^{-1}$
$\displaystyle=\tilde{w}_{L_{n}}(ca^{2}-cb^{2}+2dba,(da^{2}-db^{2}-2cba)\textrm{i}).$
Similar to the proof of Lemma 6.7, we get (2) and (3). ∎
###### Lemma 7.10.
For $\forall v\in S_{\mathbb{C}}^{0}$, $\forall a\in S_{\mathbb{C}}^{m-n-1}$
we have $\tilde{w}_{L_{n}}(a)=\tilde{w}_{L_{n}}(va)$.
###### Proof.
For $v\in S_{\mathbb{C}}^{0}$ $a=(a_{1},\dots,a_{n})\in
S_{\mathbb{C}}^{m-n-1}$, define $f:S_{\mathbb{C}}^{m-n-1}\times
S_{\mathbb{C}}^{0}\rightarrow S_{\mathbb{C}}^{m-n-1}$ to be $f(a,g)=((2\lvert
a_{1}\rvert^{2}-1)g,2ga_{2}\overline{a_{1}},\dots,2ga_{n}\overline{a_{1}})$.
It is easy to check $f$ is surjective. Using Lemma 7.3 and Lemma 7.9, for any
$u,v\in S_{\mathbb{C}}^{0}$ we have
$\displaystyle e$
$\displaystyle=\tilde{h}^{1}_{L_{n}}(u,0)(\tilde{h}^{1}_{L_{n}}(-v,0))^{-1}$
$\displaystyle=\tilde{w}_{L_{n}}(u,\dots,0)\tilde{w}_{L_{n}}(v,\dots,0)$
$\displaystyle=\tilde{w}_{L_{n}}(a)\tilde{w}_{L_{n}}(u,\dots,0)\tilde{w}_{L_{n}}(v,\dots,0)\tilde{w}_{L_{n}}(a)^{-1}$
$\displaystyle=\tilde{w}_{L_{n}}\bigl{(}(2\lvert
a_{1}\rvert^{2}-1)u,2ua_{2}\overline{a_{1}},\dots,2ua_{n}\overline{a_{1}}\bigr{)}$
$\displaystyle\cdot\tilde{w}_{L_{n}}\bigl{(}(2\lvert
a_{1}\rvert^{2}-1)v,2va_{2}\overline{a_{1}},\dots,2va_{n}\overline{a_{1}}\bigr{)}.$
Thus we proved the lemma. ∎
For $(a,b),(c,d)\in S_{\mathbb{R}}^{1}$ we define:
$\displaystyle\\{(a,b),(c,d)\\}^{1}_{i}=\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b)\cdot(c,d)\bigr{)}\tilde{h}^{i}_{L_{n}}(a,b)^{-1}\tilde{h}^{i}_{L_{n}}(c,d)^{-1},$
$\displaystyle\\{(a,b),(c,d)\\}^{2}_{i}=\tilde{h}^{i}_{L_{n}}\bigl{(}(a,b\textrm{i})\cdot(c,d\textrm{i})\bigr{)}\tilde{h}^{i}_{L_{n}}(a,b\textrm{i})^{-1}\tilde{h}^{i}_{L_{n}}(c,d\textrm{i})^{-1}.$
Let $i,j$ be distinct, and let
$\displaystyle w$
$\displaystyle=\tilde{w}_{L_{n}}(0,\dots,(-\frac{\sqrt{2}}{2})_{i+1},\dots,(\frac{\sqrt{2}}{2})_{j+1},0\dots,0)$
$\displaystyle\cdot\tilde{w}_{L_{n}}(0,\dots,(-\frac{\sqrt{2}}{2})_{i},\dots,(\frac{\sqrt{2}}{2})_{j},0,\dots,0).$
By Lemma 7.3, for $\forall(u,v)\in S^{1}_{\mathbb{C}}$ we have
$w\tilde{h}^{i}_{L_{n}}(u,v)w^{-1}=\tilde{h}^{j}_{L_{n}}(u,v).$
Since $\\{(a,b),(c,d)\\}^{\delta}_{i}\in Z(\tilde{G})$($\delta=1,2$), it
follows that
###### Lemma 7.11.
$\displaystyle\\{(a,b),(c,d)\\}^{\delta}=\\{(a,b),(c,d)\\}^{\delta}_{i}\qquad\delta=1,2,$
are well defined.
Using Lemma 7.9, in exactly the same manner as the proof in the appendix of
[8], we prove that these $\\{(a,b),(c,d)\\}$’s satisfy the conditions
###### Lemma 7.12.
For all $(a,b),(c,d),(a_{1},b_{1}),(c_{1},d_{1})\in S_{\mathbb{R}}^{1}$ and
$\delta=1,2$ we have:
$\displaystyle\\{(a,b),(c,d)\\}^{\delta}$
$\displaystyle=(\\{(c,d),(a,b)\\}^{\delta})^{-1},$
$\displaystyle\\{(a,b),(c,d)\cdot(c_{1},d_{1})\\}^{\delta}$
$\displaystyle=\\{(a,b),(c,d)\\}^{\delta}\cdot\\{(a,b),(c_{1},d_{1})\\}^{\delta},$
$\displaystyle\\{(a,b)\cdot(a_{1},b_{1}),(c,d)\\}^{\delta}$
$\displaystyle=\\{(a,b),(c,d)\\}^{\delta}\cdot\\{(a_{1},b_{1}),(c,d)\\}^{\delta},$
$\displaystyle\\{(c,d),(-c,-d)\\}^{\delta}$ $\displaystyle=1.$
Thus we define 2 symbols on $S^{1}_{\mathbb{R}}$. Denote by $H_{sym}$ The
subgroup generated by these symbols.
### 7.5. Structure of $H^{i}$
Let $H^{i}_{0}$ be the subgroup generated by
$\tilde{h}^{i}_{L_{n}}(a,b)\bigl{(}(a,b)\in S^{1}_{\mathbb{R}}\bigr{)}$,
$H^{i}_{1}$ be the subgroup generated by
$\tilde{h}^{i}_{L_{n}}(a,b\textrm{i})\bigl{(}(a,b)\in
S^{1}_{\mathbb{R}}\bigr{)}$ and $H^{i}$ the subgroup generated by
$\tilde{h}^{i}(u,v)\bigl{(}(u,v)\in S^{1}_{\mathbb{C}}\bigr{)}$. Using Lemma
7.11, along the line of proof of Lemma 6.9, We get the following:
###### Lemma 7.13.
$\displaystyle(1)\ker(\pi_{1})\cap H^{j}_{\delta}=\ker(\pi_{1})\cap
H^{1}_{\delta}\qquad j\leq m-n-1,\delta=0,1,$
$\displaystyle(2)\ker(\pi_{1})\cap H^{1}_{\delta}=H_{sym}\cap
H^{1}_{\delta}\qquad\delta=0,1.$
We now make a slight digression to state a fact and prove a lemma whose roles
will be clear from the subsequent development. However, the fact and the lemma
in themselves seems to be interesting.
###### Fact 7.1.
If $\pi_{1}\bigl{(}\tilde{w}_{L_{n}}(a)\tilde{w}_{L_{n}}(b)\bigl{)}=C$, where
$a,b\in S_{\mathbb{C}}^{m-n-1}$. Then we have
$4\lvert\langle a,b\rangle\rvert^{2}+m-n-4=\textrm{trace}(C),$
where $\langle a,b\rangle=a\cdot\overline{b}$ is the inner product of $a$ and
$b$.
Denote
$\tilde{w}^{j}_{L_{n}}(a,b)=\tilde{w}_{L_{n}}(0,\dots,a_{j},b_{j+1},0,\dots,0)$
where $(a,b)\in S^{1}_{\mathbb{C}}$.
###### Lemma 7.14.
If
$\langle(a_{1},a_{2}),(b_{1},b_{2})\rangle=\langle(c_{1},c_{2}),(d_{1},d_{2})\rangle$,
where $(a_{1},a_{2})$, $(b_{1},b_{2})$, $(c_{1},c_{2})$, $(d_{1},d_{2})\in
S^{1}_{\mathbb{C}}$, there exists $g\in S^{0}_{\mathbb{C}}$ and $h\in SU(2)$
such that
$h\cdot(a_{1},a_{2})=(gc_{1},gc_{2}),\qquad
h\cdot(b_{1},b_{2})=(gd_{1},gd_{2}).$
###### Proof.
We first consider the case $(c_{1},c_{2})=(1,0)$. Direct calculation shows for
any $g\in S^{0}_{\mathbb{C}}$, if we let
$h=\begin{pmatrix}\overline{a_{1}}g&\overline{a_{2}}g\\\
-a_{2}\overline{g}&a_{1}\overline{g}\\\ \end{pmatrix},$
then
$\displaystyle h\cdot(a_{1},a_{2})$ $\displaystyle=(gc_{1},0),$ $\displaystyle
h\cdot(b_{1},b_{2})$
$\displaystyle=\bigl{(}g(\overline{a_{1}}b_{1}+\overline{a_{2}}b_{2}),g(-a_{2}\overline{g}^{2}b_{1}+a_{1}\overline{g}^{2}b_{2})\bigl{)}.$
Since $h$ preserves inner product, we have
$\displaystyle\langle
h\cdot(a_{1},a_{2}),h\cdot(b_{1},b_{2})\rangle=\langle(c_{1},c_{2}),(d_{1},d_{2})\rangle=\overline{d_{1}}.$
Thus it follows $\overline{a_{1}}b_{1}+\overline{a_{2}}b_{2}=d_{1}$, $\lvert-
a_{2}b_{2}+a_{1}b_{2}\rvert=\sqrt{1-\lvert d_{1}\rvert^{2}}=\lvert
d_{2}\rvert$. Hence we can choose right $g$ such that
$-a_{2}\overline{g}^{2}b_{2}+a_{1}\overline{g}^{2}b_{2}=d_{2}$. We thus proved
the case for $(c_{1},c_{2})=(1,0)$.
If $(c_{1},c_{2})\neq(1,0)$, there exists $h^{\prime}\in SU(2)$ such that
$h^{\prime}\cdot(c_{1},c_{2})=(1,0)$. Then we reduce it to previous case. We
hence proved the lemma. ∎
An important step in proving the main Theorem 5 is:
###### Theorem 9.
$\ker(\pi_{1})\cap H^{j}=H_{sym}$ for $\forall j\leq m-n-1$.
The ensuring discussion up to Lemma 7.16 proves the theorem. Recall the
definition for $\tilde{W}_{s}$ in Definition 7.3. Consider the quotient group
$\tilde{W}_{s}/H_{sym}$ until the end of Lemma 7.16. Note that $H_{sym}$ being
central, this is well defined. We continue to write
$\tilde{h}^{j}_{L_{n}}(a,b)\bigl{(}(a,b)\in S^{1}_{\mathbb{C}}\bigr{)}$ and
$\tilde{w}^{j}_{L_{n}}(a)(a\in S^{m-n-1}_{\mathbb{C}})$ for their images in
$\tilde{W}_{s}/H_{sym}$. However, there is no confusion in doing so.
###### Lemma 7.15.
Let $g_{1},g_{2}:[0,2\pi]^{2}\rightarrow S^{1}_{\mathbb{C}}$ be defined as
follows:
$\displaystyle g_{1}(a,b)=(\cos a\cos b-\sin a\sin b\emph{i},\cos a\sin b-\sin
a\cos b\emph{i}),$ $\displaystyle g_{2}(a,b)=(\cos a\cos b-\sin a\sin
b\emph{i},\sin a\cos b+\cos a\sin b\emph{i}).$
For any $a,b,x\in[0,2\pi]$, there exist $c,d,y,$
$c_{1},d_{1},y_{1}\in[0,2\pi]$ such that
$\displaystyle(1)\tilde{w}^{j}_{L_{n}}\bigl{(}g_{2}(a,b)\bigr{)}\tilde{w}^{j}_{L_{n}}(\cos
x,\sin
x)=\tilde{w}^{j}_{L_{n}}\bigl{(}g_{1}(c,d)\bigr{)}\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\emph{i}),$
$\displaystyle(2)\tilde{w}^{j}_{L_{n}}\bigl{(}g_{1}(a,b)\bigl{)}\tilde{w}^{j}_{L_{n}}(\cos
x,\sin
x\emph{i})=\tilde{w}^{j}_{L_{n}}\bigl{(}g_{2}(c_{1},d_{1})\bigr{)}\tilde{w}^{j}_{L_{n}}(\cos
y_{1},\sin y_{1}).$
###### Proof.
(1) Fix $j$. At first, we want to show that there exist $(z_{1},z_{2})\in
S^{1}_{\mathbb{C}}$ and $y\in[0,2\pi]$ such that
(7.4)
$\displaystyle\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin
x)\bigr{)}=\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(z_{1},z_{2})\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\textrm{i})\bigr{)}.$
Suppose
$\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)\bigr{)}$ is given by the following matrix
$\begin{pmatrix}\alpha&\beta\\\ -\overline{\beta}&\overline{\alpha}\\\
\end{pmatrix}.$
If $\alpha\in\mathbb{R}$, $\beta\textrm{i}\in\mathbb{R}$, there is nothing to
prove.
Now suppose $(\alpha,\beta)\notin\mathbb{R}\times\mathbb{R}\textrm{i}$. If
$\alpha\notin\mathbb{R}$, $\beta\notin\mathbb{R}\textrm{i}$, let
$u=t(1-\alpha^{2}+\beta^{2})$,
$v=\frac{u\overline{\beta}-\beta\overline{u}}{\alpha-\overline{\alpha}}$ where
$t\in\mathbb{R}$ satisfying $\lvert v\rvert^{2}+\lvert u\rvert^{2}=1$; if
$\alpha\notin\mathbb{R}$, $\beta\textrm{i}\in\mathbb{R}$ let
$u=t\alpha\textrm{i}$,
$v=\frac{u\overline{\beta}-\beta\overline{u}}{\alpha-\overline{\alpha}}$ where
$t\in\mathbb{R}$ satisfying $\lvert v\rvert^{2}+\lvert u\rvert^{2}=1$; if
$\alpha\in\mathbb{R}$, $\beta\notin\mathbb{R}\textrm{i}$, let $u=t\beta$,
$v=-\frac{u\overline{\alpha}+\alpha\overline{u}}{\overline{\beta}+\beta}$
where $t\in\mathbb{R}$ satisfying $\lvert v\rvert^{2}+\lvert u\rvert^{2}=1$.
Let $R$ be the following matrix
$\begin{pmatrix}v&u\\\ \overline{u}&-v\\\ \end{pmatrix},$
then we have
$R\cdot(\alpha,-\overline{\beta})\in\mathbb{R}\times\mathbb{R}\textrm{i}$.
Thus there exists $y\in\mathbb{R}$ such that
$R\cdot(\alpha,-\overline{\beta})=(-\cos 2y,\sin 2y\textrm{i})$.
Let
$(z_{1},z_{2})=(\sqrt{\frac{1}{2}(1-v)},-\frac{u}{2\sqrt{\frac{1}{2}(1-v)}})$
if $v\neq 1$; $(z_{1},z_{2})=(0,1)$ if $v=1$, then we have
$\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(z_{1},z_{2})\bigl{)}=R$ and
$\displaystyle\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin
x)\bigr{)}=\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(z_{1},z_{2})\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\textrm{i})\bigr{)}.$
Next, we want to show there exists $(c,d)\in\mathbb{R}^{2}$ such that
$\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{1}(c,d))\bigl{)}=\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(z_{1},z_{2})\bigl{)}.$
By Lemma 7.10, we just need to show there exist $(c,d)$ and $z\in
S^{0}_{\mathbb{C}}$ such that $g_{1}(c,d)=(zz_{1},zz_{2})$. Notice
$z_{1}\in\mathbb{R}$. Let $A$ denote the following matrix:
$\begin{pmatrix}\cos c&\sin c\textrm{i}\\\ \sin c\textrm{i}&\cos c\\\
\end{pmatrix}.$
If $z_{2}\in\mathbb{R}$, let $\sin c=0$, $z=1$, then $A\cdot(zz_{1},zz_{2})\in
S^{1}_{\mathbb{R}}$. If $z_{1}=0$, let $\sin c=0$,
$z=\frac{\overline{z_{2}}}{\lvert z_{2}\rvert}$, then
$A\cdot(zz_{1},zz_{2})\in S^{1}_{\mathbb{R}}$.
Now suppose $z_{1}z_{2}\neq 0$ and $z_{2}\notin\mathbb{R}$. If
$z_{2}\in\mathbb{R}\textrm{i}$, then $(z_{1},z_{2})=(\cos r,\sin r\textrm{i})$
for some $r\in\mathbb{R}$. Let $z=1$, $c=-r$, then $A\cdot(zz_{1},zz_{2})\in
S^{1}_{\mathbb{R}}$.
Now suppose $z_{1}z_{2}\neq 0$, $z_{2}\notin\mathbb{R}$ and
$z_{2}\notin\mathbb{R}\textrm{i}$. Let
$z=\frac{\sqrt{z_{1}^{2}-\overline{z_{2}}^{2}}}{\lvert\sqrt{z_{1}^{2}-\overline{z_{2}}^{2}}\rvert}$,
$\cot
c=\frac{(zz_{2}+\overline{zz_{2}})\textrm{i}}{z_{1}\overline{z}-zz_{1}}$(notice
in this case $z_{1}^{2}-\overline{z_{2}}^{2}\neq 0$ and
$\overline{z}z_{1}-zz_{1}\neq 0$), then we have $A\cdot(zz_{1},zz_{2})\in
S^{1}_{\mathbb{R}}$. Let $d\in\mathbb{R}$ satisfying $(\cos d,\sin
d)=A\cdot(zz_{1},zz_{2})$, then $g_{1}(c,d)=(zz_{1},zz_{2})$.
Hence we proved that for any given $a,b$, any $x$ we can find
$(c,d)\in[0,2\pi]$ and $y\in[0,2\pi]$ such that
$\displaystyle\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin
x)\bigr{)}=\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{1}(c,d))\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\textrm{i})\bigr{)}.$
By Fact 7.1 we have
$\displaystyle\lvert\bigl{\langle}g_{2}(c,d),(\cos x,\sin
x)\bigr{\rangle}\rvert=\lvert\bigl{\langle}g_{1}(a,b),(\cos y,\sin
y\textrm{i})\bigr{\rangle}\rvert.$
There exists $z_{1}\in S_{\mathbb{C}}^{1}$ such that
$\displaystyle\bigl{\langle}g_{2}(c,d),(\cos x,\sin
x)\bigr{\rangle}=\bigl{\langle}z_{1}g_{1}(a,b),(\cos y,\sin
y\textrm{i})\bigr{\rangle}.$
By Lemma 7.14, there exist $z_{0}\in S^{0}_{\mathbb{C}}$ and $h^{\prime}\in
H^{j}$ such that
$\displaystyle\overline{\pi_{1}(h^{\prime})}(g_{2}(c,d))=z_{0}z_{1}g_{1}(a,b)\text{
and }\overline{\pi_{1}(h^{\prime})}(\cos x,\sin x)=(z_{0}\cos y,z_{0}\sin
y\textrm{i}).$
Then it follows
$\displaystyle
h^{\prime}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)\bigr{)}h^{\prime-1}$
$\displaystyle=\tilde{w}^{j}_{L_{n}}(z_{0}z_{1}g_{1}(a,b))\tilde{w}^{j}_{L_{n}}(z_{0}\cos
y,z_{0}\sin y\textrm{i})$ (7.5)
$\displaystyle=\tilde{w}^{j}_{L_{n}}\bigl{(}g_{1}(a,b)\bigr{)}\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\textrm{i}).$ $\displaystyle(\text{ by }\text{ Lemma }\ref{pop:2})$
If $\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)\bigr{)}=I_{m+n}$, there is nothing to prove.
Now suppose
$\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)\bigr{)}\neq I_{m+n}$.
Let $x_{1},x_{2}\in\mathbb{R}$ such that
$\lvert\cos x_{1}\cos x_{2}+\sin x_{1}\sin
x_{2}\rvert=\lvert\bigl{\langle}g_{2}(a,b),(\cos x,\sin
x)\bigr{\rangle}\rvert.$
Then there exists $z\in S^{0}_{\mathbb{C}}$ such that
$\displaystyle\bigl{\langle}(\cos x_{1},\sin x_{1}),(\cos x_{2},\sin
x_{2})\bigr{\rangle}=\bigl{\langle}zg_{2}(a,b),(\cos x,\sin x)\bigr{\rangle},$
and by Lemma 7.14, there exist $h_{1}\in H^{j}$ and $z_{2}\in
S^{0}_{\mathbb{C}}$ satisfying
$\displaystyle\overline{\pi_{1}(h_{1})}(zg_{2}(a,b))=(z_{2}\cos
x_{1},z_{2}\sin x_{1}),$ $\displaystyle\overline{\pi_{1}(h_{1})}(\cos x,\sin
x)=(z_{2}\cos x_{2},z_{2}\sin x_{2}).$
Thus following (7.5) and Lemma 7.10 we have
$\displaystyle
h_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{1}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\textrm{i})\bigr{)}h_{1}^{-1}$
$\displaystyle=h_{1}h^{\prime}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)\bigr{)}h^{\prime-1}h_{1}^{-1}$
$\displaystyle=(h_{1}h^{\prime}h_{1}^{-1})\bigl{(}h_{1}\tilde{w}^{j}_{L_{n}}(zg_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)h_{1}^{-1}\bigr{)}\cdot(h_{1}h^{\prime}h_{1}^{-1})^{-1}$
$\displaystyle=(h_{1}h^{\prime}h_{1}^{-1})\bigl{(}\tilde{w}^{j}_{L_{n}}(z_{2}\cos
x_{1},z_{2}\sin x_{1})\tilde{w}^{j}_{L_{n}}(z_{2}\cos x_{2},z_{2}\sin
x_{2})\bigr{)}(h_{1}h^{\prime}h_{1}^{-1})^{-1}$
$\displaystyle=(h_{1}h^{\prime}h_{1}^{-1})\bigl{(}\tilde{w}^{j}_{L_{n}}(\cos
x_{1},\sin x_{1})\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin
x_{2})\bigr{)}(h_{1}h^{\prime}h_{1}^{-1})^{-1}.$
Notice
$\displaystyle\pi_{1}\bigl{(}h_{1}\tilde{w}^{j}_{L_{n}}(g_{1}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\textrm{i})h_{1}^{-1}\bigr{)}$
$\displaystyle=\pi_{1}\bigl{(}h_{1}\tilde{w}^{j}_{L_{n}}(zg_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)h_{1}^{-1}\bigr{)}$
$\displaystyle=\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(\cos x_{1},\sin
x_{1})\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2})\bigr{)}.$
Since $\pi_{1}(h_{1}h^{\prime}h_{1}^{-1})$ commute with
$\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(\cos x_{1},\sin
x_{1})\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2})\bigr{)}$ which is conjugate
with
$\pi_{1}\bigl{(}\tilde{w}^{j}_{L_{n}}(g_{1}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
y,\sin y\textrm{i})\bigr{)}\neq I_{m+n}$ and is in $\pi_{1}(H^{j}_{0})$, hence
there exists $h\in H^{j}_{0}$ and $h_{0}\in\ker(\pi_{1})$ such that
$h_{1}h^{\prime}h_{1}^{-1}=hh_{0}$.
Hence we have
$\displaystyle
h_{1}\tilde{w}^{j}_{L_{n}}(g_{1}(a,b))\tilde{w}^{j}_{L_{n}}(\cos y,\sin
y\textrm{i})h_{1}^{-1}$
$\displaystyle=(h_{1}h^{\prime}h_{1}^{-1})\bigl{(}\tilde{w}^{j}_{L_{n}}(\cos
x_{1},\sin x_{1})\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin
x_{2})\bigr{)}(h_{1}h^{\prime}h_{1}^{-1})^{-1}$
$\displaystyle=hh_{0}\bigl{(}\tilde{w}^{j}_{L_{n}}(\cos x_{1},\sin
x_{1})\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2})\bigl{)}(hh_{0})^{-1}$
$\displaystyle=\tilde{w}^{j}_{L_{n}}(\cos x_{1},\sin
x_{1})\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2})(\text{ by Lemma
}\ref{le:32})$
$\displaystyle=h_{1}\tilde{w}^{j}_{L_{n}}(g_{2}(a,b))\tilde{w}^{j}_{L_{n}}(\cos
x,\sin x)h_{1}^{-1}.$
Hence we have proved (1).
(2) Similar arguments hold for (2). ∎
As a second step toward the proof of Theorem 9, we prove
###### Lemma 7.16.
For any $\theta_{1},\theta_{2},\theta_{3}$, we can find
$\beta_{1},\beta_{2},\beta_{3}$, $\alpha_{1},\alpha_{2},\alpha_{3}$, such that
$\displaystyle(1)$
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\emph{i})\tilde{h}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\emph{i})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos\beta_{1},\sin\beta_{1})\tilde{h}^{j}_{L_{n}}(\cos\beta_{2},\sin\beta_{2}\emph{i})\tilde{h}^{j}_{L_{n}}(\cos\beta_{3},\sin\beta_{3}),$
$\displaystyle(2)$
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1})\tilde{h}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2}\emph{i})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos\alpha_{1},\sin\alpha_{1}\emph{i})\tilde{h}^{j}_{L_{n}}(\cos\alpha_{2},\sin\alpha_{2})\tilde{h}^{j}_{L_{n}}(\cos\alpha_{3},\sin\alpha_{3}\emph{i}).$
###### Proof.
(1) Let
$\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\bigl{)}=A$,
we have
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\tilde{h}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\tilde{w}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{w}^{j}_{L_{n}}(-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})$
$\displaystyle=\tilde{w}^{j}_{L_{n}}\bigl{(}\overline{A}\cdot(\cos\theta_{2},\sin\theta_{2})\bigl{)}\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})$
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})\tilde{w}^{j}_{L_{n}}(-1,0)\tilde{w}^{j}_{L_{n}}(1,0)$
$\displaystyle=\tilde{w}^{j}_{L_{n}}\bigl{(}g_{1}(-2\theta_{1},\theta_{2})\bigl{)}\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})$
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})\tilde{w}^{j}_{L_{n}}(-1,0)\tilde{w}^{j}_{L_{n}}(1,0).$
Since
$\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\tilde{w}^{j}_{L_{n}}(-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})\tilde{w}^{j}_{L_{n}}(-1,0)\in
H^{j}_{1},$
there exists $x_{1}\in\mathbb{R}$ such that
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\tilde{w}^{j}_{L_{n}}(-1,0)\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})\tilde{w}^{j}_{L_{n}}(1,0)$
$\displaystyle=\tilde{w}^{j}_{L_{n}}(\cos x_{1},\sin
x_{1}\textrm{i})\tilde{w}^{j}_{L_{n}}(-1,0).$
Thus we have
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\tilde{h}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})$
$\displaystyle=\tilde{w}^{j}_{L_{n}}\bigl{(}g_{1}(-2\theta_{1},\theta_{2})\bigl{)}\tilde{w}^{j}_{L_{n}}(\cos
x_{1},\sin x_{1}\textrm{i}).$
By Lemma 7.15, there exist $a,b,x_{2}\in[0,2\pi]$ such that
$\tilde{w}^{j}_{L_{n}}\bigl{(}g_{1}(-2\theta_{1},\theta_{2})\bigl{)}\tilde{w}^{j}_{L_{n}}(\cos
x_{1},\sin
x_{1}\textrm{i})=\tilde{w}^{j}_{L_{n}}\bigl{(}g_{2}(2a,b)\bigl{)}\tilde{w}^{j}_{L_{n}}(\cos
x_{2},\sin x_{2}).$
It follows
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\tilde{h}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})$
$\displaystyle=\tilde{w}^{j}_{L_{n}}\bigl{(}g_{2}(2a,b)\bigl{)}\tilde{w}^{j}_{L_{n}}(\cos
x_{2},\sin x_{2})$ $\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos a,\sin
b)\tilde{w}^{j}_{L_{n}}(\cos b,\sin b\textrm{i})\tilde{h}^{j}_{L_{n}}(\cos
a,\sin b)^{-1}$ $\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin
x_{2})$ $\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos a,\sin
b)\tilde{w}^{j}_{L_{n}}(\cos b,\sin b\textrm{i})\tilde{w}^{j}_{L_{n}}(-1,0)$
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(1,0)\tilde{h}^{j}_{L_{n}}(\cos a,\sin
b)^{-1}\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos a,\sin b)\tilde{h}^{j}_{L_{n}}(\cos
b,\sin b\textrm{i})$
$\displaystyle\cdot\tilde{w}^{j}_{L_{n}}(1,0)\tilde{h}^{j}_{L_{n}}(\cos a,\sin
b)^{-1}\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2}).$
Notice
$\tilde{w}^{j}_{L_{n}}(1,0)\tilde{h}^{j}_{L_{n}}(\cos a,\sin
b)^{-1}\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2})\in H^{j}_{0}$
by Lemma 7.13 there exists $x\in[0,2\pi]$ such that
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos x,\sin x)$
$\displaystyle=\tilde{w}^{j}_{L_{n}}(1,0)\tilde{h}^{j}_{L_{n}}(\cos a,\sin
b)^{-1}\tilde{w}^{j}_{L_{n}}(\cos x_{2},\sin x_{2}).$
Thus we have proved
$\displaystyle\tilde{h}^{j}_{L_{n}}(\cos\theta_{1},\sin\theta_{1}\textrm{i})\tilde{h}^{j}_{L_{n}}(\cos\theta_{2},\sin\theta_{2})\tilde{h}^{j}_{L_{n}}(\cos\theta_{3},\sin\theta_{3}\textrm{i})$
$\displaystyle=\tilde{h}^{j}_{L_{n}}(\cos a,\sin a)\tilde{h}^{j}_{L_{n}}(\cos
b,\sin b\textrm{i})\tilde{h}^{j}_{L_{n}}(\cos x,\sin x).$
Let $\beta_{1}=a$, $\beta_{2}=b$ and $\beta_{3}=x$, we proved (1).
(2) It follows almost the same manner as the proof of (1).
We thus completely proved the lemma. ∎
### 7.6. Proof of Lemma 9
Observe that for $\forall j$, $\pi_{1}(H^{j}_{0})$ and $\pi_{1}(H^{j}_{1})$
generate a subgroup isomorphic to $SU(2)$. Using the same trick as we used in
proof of Corollary 6.1 and Lemma 7.4 etc., it follows $H^{j}$ is generated by
$H^{j}_{0}$ and $H^{j}_{1}$. By Lemma 7.13 and Lemma 7.16, every element
$h_{0}\in H_{j}$ can be express as
$h=\tilde{h}^{j}_{L_{n}}(\cos x_{1},\sin x_{1})\tilde{h}^{j}_{L_{n}}(\cos
x_{2},\sin x_{2}\textrm{i})\tilde{h}^{j}_{L_{n}}(\cos x_{3},\sin x_{3})h_{0}$
for some $x_{1},x_{2},x_{3}\in\mathbb{R}$ and $h_{0}\in H_{sym}$.
If $\pi_{1}(h)=I_{m+n}$, we have
$\displaystyle\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(\cos x_{2},\sin
x_{2}\textrm{i})\bigl{)}=I_{m+n},$
$\displaystyle\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(\cos x_{3},\sin
x_{3})\tilde{h}^{j}_{L_{n}}(\cos x_{1},\sin x_{1})\bigl{)}=I_{m+n}.$
Or
$\displaystyle\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(\cos x_{2},\sin
x_{2}\textrm{i})\bigl{)}=\text{diag}(-1_{2n+j},-1_{2n+j+1}),$
$\displaystyle\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(\cos x_{3},\sin
x_{3})\tilde{h}^{j}_{L_{n}}(\cos x_{1},\sin
x_{1})\bigl{)}=\text{diag}(-1_{2n+j},-1_{2n+j+1}).$
For the former case, it is clear $h\in H_{sym}$. For the latter one, we have
$\cos x_{2}=\cos(x_{1}+x_{3})=0$. Using Lemma 7.10 we have
$\tilde{h}^{j}_{L_{n}}(\cos x_{2},\sin
x_{2}\textrm{i})=\tilde{h}^{j}_{L_{n}}(0,\pm\textrm{i})=\tilde{h}^{j}_{L_{n}}(0,1).$
Hence we also get $h\in H_{sym}$ for this case. We thus have prove the theorem
completely.
###### Corollary 7.1.
If $m-n=2$,
$\ker(\pi_{1})=\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\bigr{)}\cdot\tilde{H}_{0}\cdot
H_{sym}$.
###### Proof.
The conclusion is clear by Theorem 8 and Theorem 9. ∎
### 7.7. Structure of $\ker(\pi_{1})\cap\tilde{H}_{s_{0}}$
Let $L_{i}=\\{(t,a)|(t,a)\in\mathbb{R}\times\mathbb{C}^{i},\frac{1}{4}\lvert
a\rvert^{4}+t^{2}=1\\}$. Let
$H=\bigl{(}\ker(\pi_{1})\cap\tilde{H}_{L_{1}-L_{2}}\bigr{)}\cdot\tilde{H}_{0}\cdot
H_{sym}$. Denote
$\tilde{w}^{j}(t,a,b,c)=\tilde{w}_{L_{n}}(t,\dots,a_{j+1},b_{j+2},c_{j+3},\dots,0)$
where $(t,a,b,c)\in L_{3}$. Let
$\tilde{h}^{j}(0,a,b)=\tilde{w}^{j}(0,a,b,0)\tilde{w}^{j}(t,-\sqrt{2},0,0)$
where $(0,a,b)\in L_{2}$. Notice if $t=0$, then $\lvert a\rvert=\sqrt{2}$.
Thus the notation set here coincide with what was defined in Definition 7.3.
The crucial step in proving Theorem 5 is:
###### Theorem 10.
$\ker(\pi_{1})\cap\tilde{H}_{s_{0}}\subseteq H$.
The ensuring discussion up to Lemma 7.20 proves the theorem. We prove a
technical lemma at first.
###### Lemma 7.17.
If
$(t_{1},a_{1},b_{1},0),(t_{2},a_{2},b_{2},0),(t_{3},a_{3},b_{3},0),(t_{4},a_{4},b_{4},0)\in
L_{3}$, satisfying
$\pi_{1}\bigl{(}\tilde{w}^{j}(t_{1},a_{1},b_{1},0)\tilde{w}^{j}(t_{2},a_{2},b_{2},0)\bigl{)}=\pi_{1}\bigl{(}\tilde{w}^{j}(t_{3},a_{3},b_{3},0)\tilde{w}^{j}(t_{4},a_{4},b_{4},0)\bigl{)},$
there exists $h_{0}\in H$ such that
$\displaystyle\tilde{w}^{j}(t_{1},a_{1},b_{1},0)\tilde{w}^{j}(t_{2},a_{2},b_{2},0)=\tilde{w}^{j}(t_{3},a_{3},b_{3},0)\tilde{w}^{j}(t_{4},a_{4},b_{4},0)h_{0}.$
###### Proof.
Let
$w=\tilde{w}_{L_{n}}(0,0,-1,\dots,1_{j+2},0\dots,0)\cdot\tilde{w}_{L_{n}}(0,-1,\dots,1_{j+1},0,\dots,0).$
By Lemma 7.5 we have
$\displaystyle\tilde{w}^{j}(t_{1},a_{1},b_{1},0)\tilde{w}^{j}(t_{2},a_{2},b_{2},0)\bigl{(}\tilde{w}^{j}(t_{3},a_{3},b_{3},0)\tilde{w}^{j}(t_{4},a_{4},b_{4},0)\bigl{)}^{-1}$
$\displaystyle=w\tilde{w}^{j}(t_{1},a_{1},b_{1},0)\tilde{w}^{j}(t_{2},a_{2},b_{2},0)w^{-1}$
$\displaystyle\cdot
w\bigl{(}\tilde{w}^{j}(t_{3},a_{3},b_{3},0)\tilde{w}^{j}(t_{4},a_{4},b_{4},0)\bigl{)}^{-1}w^{-1}$
$\displaystyle=\tilde{w}^{1}(t_{1},a_{1},b_{1},0)\tilde{w}^{1}(t_{2},a_{2},b_{2},0)$
$\displaystyle\cdot\bigl{(}\tilde{w}^{1}(t_{3},a_{3},b_{3},0)\tilde{w}^{1}(t_{4},a_{4},b_{4},0)\bigl{)}^{-1}.$
Using Corollary 7.1 we get the conclusion. ∎
Now we consider the quotient group $\tilde{W}_{L_{n}}(t,a)/H$ where $(t,a)\in
L$ until Lemma 7.20. We continue to write
$\tilde{h}^{j}_{L_{n}}(0,a,b)\bigl{(}(0,a,b)\in L\bigr{)}$ and
$\tilde{w}^{j}_{L_{n}}(t,a)\bigl{(}(t,a)\in L\bigl{)}$ for their images in
$\tilde{W}_{L_{n}}(t,a)/H$ without confusion.
###### Lemma 7.18.
For $\forall(0,a,b,c)\in L_{3}$, $\forall(0,u,v)\in L_{2}$, we can find
$(t_{1},a_{1},b_{1},c_{1})$, $(t_{2},a_{2},b_{2},c_{2})\in L_{3}$,
$(t_{1},u_{1},v_{1})$, $(t_{2},u_{2},v_{2})\in L_{2}$ such that
$\displaystyle(1)\tilde{w}^{j}(0,a,b,c)\tilde{w}^{j}(0,0,u,v)=\tilde{w}^{j}(t_{1},a_{1},b_{1},c_{1})\tilde{w}^{j}(-t_{1},u_{1},v_{1},0),$
$\displaystyle(2)\tilde{w}^{j}(0,a,b,c)\tilde{w}^{j}(0,u,v,0)=\tilde{w}^{j}(t_{2},a_{2},b_{2},c_{2})\tilde{w}^{j}(-t_{2},0,u_{2},v_{2}).$
###### Proof.
(1) If $(a,b,c)$ and $(0,u,v)$ are collinear, then $(a,b,c)=\pm(0,u,v)$. We
can assume $(a,b,c)=(0,-u,-v)$ by Lemma 7.10. Let $t_{1}=0$,
$(a_{1},b_{1},c_{1})=(-u_{1},-v_{1},0)$ we get the conclusion.
If they are not collinear, there exists $(u_{1}^{\prime},v_{1}^{\prime})\in
S^{1}_{\mathbb{C}}$ such that $(u_{1}^{\prime},v_{1}^{\prime},0)$ is on the
plane generated by $(a,b,c)$ and $(0,u,v)$. Let $h$ be an element of the
subgroup generated by $H^{j}$ and $H^{j+1}$ such that $\pi_{1}(h)$ maps
$(a,b,c)$ and $(0,u,v)$ to $xy$-plane in $\mathbb{C}^{3}$ with last coordinate
$0$ and $\pi_{1}(h)\cdot(u_{1}^{\prime},v_{1}^{\prime},0)=(0,1,0)$. Let
$(u_{1},v_{1})=(\sqrt[4]{4-4t^{2}}u_{1}^{\prime},\sqrt[4]{4-4t^{2}}v_{1}^{\prime})$
where $0\leq t\leq 1$.
Let $f:(L_{2}\cap\mathbb{R}^{3})\times S^{0}_{\mathbb{C}}\rightarrow
S^{1}_{\mathbb{C}}$ defined as follows:
$f(t,t_{1},t_{2},g)=\bigl{(}(a+t_{1}^{2})\overline{a},-t_{1}t_{2}ga)$
where $a=-\sqrt{1-t^{2}}+t\textrm{i}$. It is clear that $f$ is surjective.
Since
$\pi_{1}\bigl{(}\tilde{w}^{j}(t,t_{1},t_{2}g,0)\bigl{)}\cdot\pi_{1}\bigl{(}\tilde{w}^{j}(-t,\overline{\pi_{1}(h)}\cdot(u_{1},v_{1},0)\bigl{)}$
is given by the following matrix
$\begin{pmatrix}(a+t_{1}^{2})\overline{a}&-t_{1}t_{2}ga\\\
t_{1}t_{2}\overline{ga}&(\overline{a}+t_{1}^{2})a\\\ \end{pmatrix},$
if we denote $\pi_{1}(h)=B$, by assumption about $h$, there exists
$(t,t_{1},t_{2},g)$ such that
$\displaystyle\pi_{1}\bigl{(}\tilde{w}^{j}(0,\overline{B}\cdot(a,b,c))\tilde{w}^{j}(0,\overline{B}\cdot(0,u,v))\bigl{)}$
$\displaystyle=\pi_{1}\bigl{(}\tilde{w}^{j}(t,t_{1},t_{2}g,0)\bigl{)}\pi_{1}\bigl{(}\tilde{w}^{j}(-t,\overline{\pi_{1}(h)}\cdot(u_{1},v_{1},0)\bigl{)}.$
Hence by Lemma 7.17, it follows
$\tilde{w}^{j}(t,t_{1},t_{2}g,0)\tilde{w}^{j}\bigl{(}-t,\overline{B}\cdot(u_{1},v_{1},0)\bigl{)}=\tilde{w}^{j}\bigl{(}0,\overline{B}\cdot(a,b,c)\bigl{)}\tilde{w}^{j}\bigl{(}0,\overline{B}\cdot(0,u,v)\bigl{)}.$
Let $(a_{1},b_{1},c_{1})=\overline{\pi_{1}(h)^{-1}}(t_{1},t_{2}g,0)$, then we
have
$\displaystyle
h\tilde{w}^{j}(t,a_{1},b_{1},c_{1})\tilde{w}^{j}(-t,u_{1},v_{1},0)h^{-1}$
$\displaystyle=\tilde{w}^{j}(t,t_{1},t_{2}g,0)\tilde{w}^{j}\bigl{(}-t,\overline{B}\cdot(u_{1},v_{1},0)\bigl{)}$
$\displaystyle=\tilde{w}^{j}\bigl{(}0,\overline{B}\cdot(a,b,c)\bigl{)}\tilde{w}^{j}\bigl{(}0,\overline{B}\cdot(0,u,v)\bigl{)}$
$\displaystyle=h\tilde{w}^{j}(0,a,b,c)\tilde{w}^{j}(0,0,u,v)h^{-1}.$
Thus we get
$\tilde{w}^{j}(t,a_{1},b_{1},c_{1})\tilde{w}^{j}(-t,u_{1},v_{1},0)=\tilde{w}^{j}(0,a,b,c)\tilde{w}^{j}(0,0,u,v).$
Hence we proved (1).
(2) Similar arguments hold for (2). ∎
Let $H_{j}^{0}$ denote the subgroup generated by
$\tilde{w}^{j}(0,a,b,0)\tilde{w}^{j}(0,1,1,0)$ where $\lvert a\rvert=\lvert
b\rvert=1$. Observe that $\pi_{1}(H_{j}^{0})$ is isomorphic to set of all
diagonal matrices in $SU(2)$.
###### Lemma 7.19.
$\displaystyle H^{j}H^{j+1}H^{j}=H^{j+1}H^{j}H^{j+1}.$
###### Proof.
Fix $j$. Observe that $\pi_{1}\bigl{(}\tilde{h}^{j}_{L_{n}}(0,a,b)\bigl{)}$
where $(0,a,b)\in L_{2}$ and $\pi_{1}(H_{j}^{0})$ generate a subgroup
isomorphic to $SU(2)$, by Theorem 9, every element in $H_{j}$ can be expressed
as $\tilde{h}^{j}_{L_{n}}(0,a,b)h$ where $(0,a,b)\in L_{2}$ and $h\in
H_{j}^{0}$.
We now prove
$H^{j}H^{j+1}H^{j}\subseteq H^{j+1}H^{j}H^{j+1}.$
Let $h_{1}^{j},h_{3}^{j}\in H^{j}$, $h_{2}^{j+1}\in H^{j+1}$. By above
analysis, there exist $(0,a_{2},b_{2})\in L_{2}$ and $h_{2}\in H_{j+1}^{0}$
such that $h_{2}^{j+1}=\tilde{h}^{j+1}_{L_{n}}(0,a_{2},b_{2})h_{2}$. Let
$\pi_{1}(h_{1}^{j})=A$.
Keep using Lemma 7.3, we have
$\displaystyle h_{1}^{j}h_{2}^{j+1}h_{3}^{j}$
$\displaystyle=h_{1}^{j}\tilde{h}^{j+1}_{L_{n}}(0,a_{2},b_{2})h_{2}h_{3}^{j}$
$\displaystyle=h_{1}^{j}\tilde{w}^{j}(0,0,a_{2},b_{2})\tilde{w}^{j}(0,0,-1,0)h_{2}h_{3}^{j}$
$\displaystyle=\tilde{w}^{j}\bigl{(}0,\overline{A}\cdot(0,a_{2},b_{2})\bigl{)}h_{1}^{j}\tilde{w}^{j}(0,0,-1,0)h_{2}h_{3}^{j}$
$\displaystyle=\tilde{w}^{j}\bigl{(}0,\overline{A}\cdot(0,a_{2},b_{2})\bigl{)}h_{1}^{j}\tilde{w}^{j}(0,0,-1,0)(h_{3}^{j})^{\prime}h_{2}$
for some $(h_{3}^{j})^{\prime}\in H^{j}$ since $h_{2}H^{j}h_{2}^{-1}\subseteq
H^{j}.$ Notice
$h_{1}^{j}\tilde{w}^{j}(0,0,-1,0)(h_{3}^{j})^{\prime}\tilde{w}^{j}(0,0,-1,0)\in
H^{j},$
by Theorem 9 there exist $(0,a_{1},b_{1})\in L_{2}$ and $h^{\prime}\in
H_{j}^{0}$ such that
$\displaystyle
h_{1}^{j}\tilde{w}^{j}(0,0,-1,0)(h_{3}^{j})^{\prime}\tilde{w}^{j}(0,0,-1,0)=\tilde{w}^{j}(0,a_{1},b_{1},0)h^{\prime}\tilde{w}^{j}(0,0,-1,0)$
Continue, we have
$\displaystyle h_{1}^{j}h_{2}^{j+1}h_{3}^{j}$
$\displaystyle=\tilde{w}^{j}\bigl{(}\overline{A}\cdot(0,a_{2},b_{2})\bigl{)}h_{1}^{j}\tilde{w}^{j}(0,0,-1,0)(h_{3}^{j})^{\prime}\tilde{w}^{j}(0,0,-1,0)$
$\displaystyle\cdot\tilde{w}^{j}(0,0,1,0)h_{2}$
$\displaystyle=\tilde{w}^{j}\bigl{(}\overline{A}\cdot(0,a_{2},b_{2})\bigl{)}\tilde{w}^{j}(0,a_{1},b_{1},0)h^{\prime}h_{2}.$
By Lemma 7.18, there exist $(t,a,b,c)\in L_{3}$ and $(t,u,v)\in L_{2}$ such
that
$\displaystyle\tilde{w}^{j}\bigl{(}\overline{A}\cdot(0,a_{2},b_{2})\bigl{)}\tilde{w}^{j}(0,a_{1},b_{1},0)=\tilde{w}^{j}(t,a,b,c)\tilde{w}^{j}(-t,0,u,v).$
Hence we have
$\displaystyle
h_{1}^{j}h_{2}^{j+1}h_{3}^{j}=\tilde{w}^{j}(t,a,b,c)\tilde{w}^{j}(-t,0,u,v)h^{\prime}h_{2}.$
If $b=c=0$, let $t_{1}=-\frac{1}{2}\lvert a\rvert^{2}+t\textrm{i}$, by Lemma
7.3 it follows
$\displaystyle
h_{1}^{j}h_{2}^{j+1}h_{3}^{j}=\tilde{w}^{j}(t,a,b,c)\tilde{w}^{j}(-t,0,u,v)h^{\prime}h_{2}$
$\displaystyle=\bigl{(}\tilde{w}^{j}(t,a,b,c)\tilde{w}^{j}(-t,0,u,v)\tilde{w}^{j}(t,a,b,c)^{-1}\bigl{)}\tilde{w}^{j}(t,a,b,c)h^{\prime}h_{2}$
$\displaystyle=\tilde{w}^{j}(-t,0,t_{1}u,t_{1}v)\tilde{w}^{j}(t,a,0,0)h^{\prime}h_{2}$
$\displaystyle=\tilde{w}^{j}(-t,0,t_{1}u,t_{1}v)\tilde{w}^{j}(t,0,a,0)\tilde{w}^{j}(-t,0,-a,0)\tilde{w}^{j}(t,a,0,0)h^{\prime}h_{2}.$
Let $h^{j+1}_{4}=\tilde{w}^{j}(-t,0,t_{1}u,t_{1}v)\tilde{w}^{j}(t,0,a,0)$,
$h^{j}_{5}=\tilde{w}^{j}(-t,0,-a,0)\tilde{w}^{j}(t,a,0,0)h^{\prime}$,
$h^{j+1}_{6}=h_{2}$, then we have
$h_{1}^{j}h_{2}^{j+1}h_{3}^{j}=h^{j+1}_{4}h^{j}_{5}h^{j+1}_{6}.$
Notice $\pi_{1}(h^{j+1}_{4}),\pi_{1}(h^{j}_{5})\in SU(2)$, by Lemma 7.17,
$h^{j+1}_{4}\in H^{j+1}$, $h^{j}_{5}\in H^{j}$. Hence we are though the case
$b=c=0$.
Suppose $b\neq 0$. Let $t_{1}=\sqrt{\lvert c\rvert^{2}+\lvert b\rvert^{2}}$,
$\alpha=t_{1}^{-1}b$, $\beta=-t_{1}^{-1}\overline{c}$,
$(x,y)=(a,b\alpha^{-1})$ and $h_{1}\in H^{j+1}$ such that $\pi_{1}(h_{1})$
given by the following matrix
$\begin{pmatrix}\alpha&\beta\\\ -\overline{\beta}&\overline{\alpha}\\\
\end{pmatrix}.$
Thus we have $\overline{\pi_{1}(h_{1})}\cdot(x,y,0)=(a,b,c)$. We have
$\displaystyle h_{1}^{j}h_{2}^{j+1}h_{3}^{j}$
$\displaystyle=\tilde{w}^{j}(t,a,b,c)\tilde{w}^{j}(-t,0,u,v)h^{\prime}h_{2}$
$\displaystyle=h_{1}\tilde{w}^{j}(t,x,y,0)h_{1}^{-1}\tilde{w}^{j}(-t,0,u,v)h^{\prime}h_{2}$
$\displaystyle=h_{1}\tilde{w}^{j}(t,x,y,0)\tilde{w}^{j}(-t,0,\sqrt[4]{4-4t^{2}},0)$
$\displaystyle\cdot\tilde{w}^{j}(t,0,-\sqrt[4]{4-4t^{2}},0)h_{1}^{-1}\tilde{w}^{j}(-t,0,u,v)h^{\prime}h_{2}.$
Notice $h^{\prime}H^{j+1}(h^{\prime})^{-1}\subseteq H^{j+1}$, hence if we let
$\displaystyle
h^{j}=\tilde{w}^{j}(t,x,y,0)\tilde{w}^{j}(-t,0,\sqrt[4]{4-4t^{2}},0)h^{\prime},$
$\displaystyle
h^{j+1}=(h^{\prime})^{-1}\tilde{w}^{j}(t,0,-\sqrt[4]{4-4t^{2}},0)h_{1}^{-1}\tilde{w}^{j}(-t,0,u,v)h^{\prime}h_{2},$
we have
$h_{1}^{j}h_{2}^{j+1}h_{3}^{j}=h_{1}h^{j}h^{j+1}.$
Notice $\pi_{1}(h^{j}),\pi_{1}(h^{j+1})\in SU(2)$, by Lemma 7.17, $h^{j}\in
H^{j}$ and $h^{j+1}\in H^{j+1}$. Hence we are though the case $b\neq 0$. For
case $c\neq 0$, we follow exactly the same way as the proof of (1) except
changing $(x,y)=(a,b\alpha^{-1})$ to $(x,y)=(a,-c\overline{\beta^{-1}})$.
Thus we have proved
$H^{j}H^{j+1}H^{j}\subseteq H^{j+1}H^{j}H^{j+1}.$
The proof of inverse containment is similar.
∎
The last tool we need is the following
###### Lemma 7.20.
$\displaystyle\tilde{H}_{s_{0}}=\bigl{(}\prod_{i=1}^{m-n-1}H^{i}\bigl{)}\bigl{(}\prod_{i=1}^{m-n-2}H^{i}\bigl{)},\dots,\bigl{(}\prod_{i=1}H^{i}\bigl{)}$
###### Proof.
The proof is exactly the same as the proof of Lemma 6.1 after changing
$h^{j}_{L_{n}}$ with $H^{j}$. ∎
### 7.8. Proof of Theorem 10
By Lemma 7.20, any element of $\tilde{h}_{s_{0}}$ can be written as
$\displaystyle
h=\bigl{(}\prod_{i=1}^{m-n-1}\tilde{h}^{i}_{m-n-1}\bigl{)}\bigl{(}\prod_{i=1}^{m-n-2}\tilde{h}^{i}_{m-n-2}\bigl{)},\dots,\tilde{h}^{1}_{1}h_{0},$
where $\tilde{h}^{i}_{k}\in H^{i}$, $k\leq m-n-1$ and $h_{0}\in H$. Notice the
the element in the lower right corner of matrix $\pi_{1}(h)$ is the same as
that of matrix $\pi_{1}(\tilde{h}^{m-n-1}_{m-n-1})$. If $\pi_{1}(h)=I_{m+n}$,
we have $\pi_{1}(\tilde{h}^{m-n-1}_{m-n-1})=I_{m+n}$, which means
$\tilde{h}^{m-n-1}_{m-n-1}\in H_{sym}$ by Theorem 9. By induction it follows
$h\in H$. Hence we proved the theorem.
### 7.9. Proof of Theorem 5
###### Proof.
It is a direct conclusion of Theorem 8 and Theorem 10. ∎
## References
* [1] M. Brin, Y. Pesin, Partially hyperbolic dynamical systems. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.
* [2] D. Damjanovic and A. Katok, Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $\mathbb{R}^{k}$ actions, Discr. Cont. Dyn.Syst., 13, (2005), 985–1005.
* [3] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions.II. The geometric method and restrictions of Weyl Chamber flows on $SL(n,\mathbb{R})/\Gamma$.
* [4] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. I. KAM method and $\mathbb{Z}^{k}$ actions on the torus, www.math.psu.edu/katok a/papers.html.
* [5] D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Modern Dyn. 1 (2007), 665–688.
* [6] Vinay V. Deodhar, On Central Extensions of Rational Points of Algebraic Groups, Amer. J. Math. 100 (1978), 303–386.
* [7] A.J.Hahn and O.T.O’Meara, The classical groups and K-theory, Springer Verlag, Berlin, 1980, 55–58.
* [8] W. Fulton, J. Harris, Representation Theory, A First Course, 101, GTM 129, Springer Verlag, New York, 1991.
* [9] S. Helgason, Differential geometry, Lie groups, and symmetric spaces
* [10] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds. Lecture notes in mathematics, 583, Springer Verlag, Berlin, 1977.
* [11] B. Kalinin, R. Spatzier, On the Classification of Cartan Actions, GAFA.
* [12] B. Kalinin, A. Katok, Invariant measures for actions of higher rank abelian groups, in Smooth Ergodic Theory and its applications, Proc. Symp. Pure Math., 69 (2001), 593 C637.
* [13] A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math. Res. Letters, 3 (1996), 191–210.
* [14] A. Katok and V. Nitica, Differentiable rigidity of higher rank abelian group actions, Cambridge University Press, to appear.
* [15] A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math. 216 (1997), 287–314.
* [16] A. Katok, R. Spatzier, Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Letters, 1 (1994), 193–202.
* [17] G.A. Margulis, Discrete subgroups of semisimple Lie groups, Springer-Verlag, 1991\.
* [18] G.A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, Ergodic Theory Dynam. Systems, 21 (2001), no. 1, 121 C164.
* [19] H. Matsumoto, Sur les sous-groupes arithmetiques des groupes semisimples deployes, These, Univ. de Paris, 1968.
* [20] C. Moore, Group extensions of p-adic and adelic linear groups, Inst. Hautes Etudes Sci. Publ. Math., No. 35, 1969, pp. 157-222.
* [21] J. Milnor, Introduction to algebraic K-theory, Princeton University Press, 1971\.
* [22] Ya. Pesin, Lectures on partial hyperbolicity and stable ergodicity, European mathematical society, 2004.
* [23] R. Steinberg, Generateurs, relations et revetements de groupes algebriques, Colloque de Bruxelles, 1962, 113–127.
* [24] R. Steinberg, Lecture Notes on Chevalley Groups, Yale Univ., 1967.
|
arxiv-papers
| 2009-10-27T04:00:45 |
2024-09-04T02:49:06.066984
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhenqi Wang",
"submitter": "Zhenqi Wang",
"url": "https://arxiv.org/abs/0910.5038"
}
|
0910.5047
|
arxiv-papers
| 2009-10-27T09:07:50 |
2024-09-04T02:49:06.083569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sam Lewallen",
"submitter": "Sam Lewallen",
"url": "https://arxiv.org/abs/0910.5047"
}
|
|
0910.5055
|
# An Efficient Algorithm for approximating 1D Ground States
Dorit Aharonov dorit.aharonov@gmail.com School of Computer Science and
Engineering,
Hebrew University, Jerusalem, Israel Itai Arad arad.itai@gmail.com School of
Computer Science,
Tel-Aviv University, Tel-Aviv, Israel Sandy Irani irani@ics.uci.edu.
Computer Science Department,
University of California, Irvine , CA, USA
###### Abstract
The density-matrix renormalization-group method is very effective at finding
ground states of one-dimensional (1D) quantum systems in practice, but it is a
heuristic method, and there is no known proof for when it works. In this
article we describe an efficient classical algorithm which provably finds a
good approximation of the ground state of 1D systems under well defined
conditions. More precisely, our algorithm finds a matrix product state of bond
dimension $D$ whose energy approximates the minimal energy such states can
achieve. The running time is exponential in $D$, and so the algorithm can be
considered tractable even for $D$ which is logarithmic in the size of the
chain. The result also implies trivially that the ground state of any local
commuting Hamiltonian in 1D can be approximated efficiently; we improve this
to an exact algorithm.
## I Introduction
Finding ground states of local one-dimensional (1D) Hamiltonian systems is a
major problem in physics. The most commonly used method is the density-matrix
renormalization-group (DMRG) White92 ; White93 ; ostlund1 ; ostlund2 ; peschel
; DMRG-overview , discovered in 1992. DMRG can be cast in the form of matrix
product states (MPSs) which are succinct representations of 1D quantum states
using $D\times D$ matrices, where the coefficients in the state can be written
in terms of products of these matrices. The number of matrices is $dn$, where
$d$ is the dimension of each individual particle and $n$ is the number of
particles in the system. The parameter $D$ is called the _bond dimension_.
DMRG works essentially as follows: The algorithm starts with some initial MPS
and sweeps from one end of the chain to the other, optimizing the entries of
the matrices at one site with the other parameters fixed. Some versions allow
optimizing over two neighboring sites at once, which enables the algorithm to
increase the bond dimension in the course of the algorithm for improved
accuracy. In all cases, the approach is to apply local optimizations
iteratively. It is thus easy to construct examples in which the DMRG algorithm
gets trapped in a local minimum. To illustrate this, think of a 1D spin chain
whose Hamiltonian consists of two types of interactions: One type consists of
interactions which force the spins to be aligned; every two neighboring sites
gain an energy penalty of say $4$ if they are not aligned. The other type of
term gives every spin an energy penalty of $1$ if it points upward. Starting
from the all-up string, a local move only increases the energy; thus, local
update rules cannot take the system to its ground state, the all-down string.
This example can of course be handled by randomizing the initial string, for
example, or increasing the window size; however, it demonstrates that DMRG has
a fundamental difficulty in addressing non local characteristics of the
system. It is natural to ask if there is a general algorithm that does not get
stuck in local minima as DMRG does and provably always find a good
approximation of the ground state of a given 1D system in a reasonable amount
of time.
To answer this question, we first ask what is known regarding the analogous
question in the easier, classical, case. It was Kitaev kitaev:book who drew
the important connection between the problem of finding ground energy and
ground states of local Hamiltonians, and the well-known classical _constraint
satisfaction problem_ (CSP). The input to a CSP consists of constraints
$\\{H_{c}\\}_{c}$ on $n$ $q$-state classical particles. Each $H_{c}$ acts on
$k$ particles (for some constant $k$) and is given as a Boolean function on
the possible assignments to those $k$ particles; when $H_{c}=1$ the
configuration is forbidden and when $H_{c}=0$ it is allowed. The problem is to
determine the maximum number of constraints that can be satisfied, or
alternatively, to minimize $\sum_{c}H_{c}$. The decision version of this
problem is to determine whether it is possible to satisfy more than some given
number of constraints. This is one of the most well-known NP-complete
problems. CSP can clearly be seen as a special case of the problem of finding
ground states and ground energies of local Hamiltonians, in which the terms in
the Hamiltonian are projections on local forbidden configurations. This
analogy has led over the past few years to many interesting insights regarding
the local Hamiltonian problem (see, e.g.,
LABEL:kitaev:book,_bravyi,_adiabatic,_focsversion,_detectability,_randomsat).
Let us therefore see what the known classical results regarding CSP in 1D can
teach us about 1D local Hamiltonians and their ground states. We recall that
in the classical case, 1D CSPs (in which the particles are arranged in a line
and constraints are between $k$ adjacent neighbors) are dramatically easier
than their higher-dimensional counterparts. While even the 2D case is NP
complete, the 1D problem can be solved in polynomial time. The reason for the
tractability of the problem in 1D is essentially that the problem can be
divided into sub problems, namely, the left- and the right-hand sides of the
chain, which interact only via the $k$ particles on the border. The fact that
these particles can only be assigned a small number of possible values makes
it possible to handle the problem by solving each sub problem separately for
each fixed possible assignment to the border particles and then gluing the sub
solutions together by picking the best choice for the middle particles. We
explain the algorithm in detail later; the outcome is an algorithm which is
linear in the number of particles in the chain and quadratic in the number of
states per particle.
Unfortunately, there is no hope of getting such a general result for the 1D
quantum problem. Aharonov _et al_ focsversion have shown that approximating
the ground energy for general 1D quantum systems is as hard as quantum-NP.
Even when restricted to ground states that are well-approximated by MPSs of
polynomial bond dimension, the problem is NP-hard, as was shown by Schuch _et
al_ schuch08 . A related earlier result due to Eisert eisert06 showed that
optimizing a constant number of matrices in the MPS representation subject to
fixed values in the other matrices is NP-hard. These results indicate that the
dichotomy between the computational difficulty of 1D and 2D classical systems
does not carry over to the quantum setting, and it is highly unlikely that the
quantum 1D problem is tractable. Nevertheless, we show here that using the
classical 1D algorithm as a template for an algorithm for the quantum problem
leads to a solution for a wide and interesting class of local Hamiltonian
problems, namely, for those cases in which we can assume that the bond
dimension is small.
### I.1 Main Result
We derive an efficient algorithm for approximating the minimal energy of a 1D
system among all states of a bounded bond dimension $D$. The algorithm is
exponential in $D$ and thus can be considered reasonable, though maybe not
practical, even for $D$, which is logarithmic in the size of the chain. The
algorithm also provides a description of an MPS with the approximate minimal
energy.
###### Theorem 1
Let $H$ be a nearest-neighbor Hamiltonian on a 1D system of $n$
$d$-dimensional particles. Let $J$ be a bound on the operator norm of each
local term. There is an algorithm that takes as input $\epsilon$, $H$ and $D$
and produces an MPS $|\Omega\rangle$ of bond dimension $D$, such that for any
MPS $|\psi\rangle$ of bond dimension $D$ with $nD^{2}\geq 12$,
$\displaystyle\langle\Omega|H|\Omega\rangle\leq\langle\psi|H|\psi\rangle+2JD^{2}n^{2}\epsilon\
.$ (1)
The algorithm runs in time $n\cdot poly(d,D,N)$, where
$N=\mathcal{O}\left(\frac{144dD}{\epsilon}\right)^{D+2dD^{2}}.$
Several remarks are in place here. First, note that the restriction that the
interactions are nearest neighbor is done without loss of generality since any
1D system can be reduced to a 2-local 1D system with nearest neighbor
interactions by grouping neighboring particles together.
Note also that the running time in the above theorem is phrased as a linear
function in $n$, the size of the system, times some fixed amount of time spent
per particle. The error, however, scales with $n^{2}$. One may want to apply
the theorem to derive an approximation with a fixed additive error $\delta$,
in which case simply set $\epsilon=\delta n^{-2}$ in the above theorem to get
the running time as a function of $\delta$.
This result shows that the problem of finding bounded bond dimension MPSs can
be done in polynomial time. Unfortunately, the running time, though efficient
in theory, is quite impractical, as even for $D=2$ and the error
$\epsilon/n^{2}$ a constant, we get a running time which scales like $n^{16}$.
It is hard to imagine that these running times are practical. Nevertheless, it
is very likely that the running time can be improved; in particular, when
solving specific problems with certain symmetries, dramatic improvements may
be possible. Moreover, it is possible that this algorithm can be used to boost
DMRG in certain cases where it gets stuck or to create the initial state of
DMRG. All these improvements are left for further research.
We now provide an overview of the algorithm. To understand the general idea,
we first recall how the classical 1D algorithm works in detail. Consider the
case of the classical CSP on a line with $k=2$, namely the problem of
minimizing the energy function $H=\sum_{i=1}^{n}H_{i,i+1}$. An optimal
assignment can be found efficiently by a standard algorithmic technique called
_dynamic programming_. Define the partial problem up to the $(r+1)$th
particle, $H_{r}=\sum_{i=1}^{r}H_{i,i+1}$. The algorithm starts with the
partial problem defined for $r=1$ and creates a list $L_{2}$ of possible
assignments to the first two particles as follows: For each of the $q$
possible assignments $\sigma_{2}$ to particle $2$, the algorithm finds an
assignment $\sigma_{1}$ to particle $1$ which minimizes
$H_{1}(\sigma_{1},\sigma_{2})$. That optimal $\sigma_{1}$ is called the tail
of $\sigma_{2}$. For each $\sigma_{2}$ the algorithm keeps its tail
$\sigma_{1}$ and also the energy of this partial assignment,
$H_{1}(\sigma_{1},\sigma_{2})$. $L_{2}$ thus contains the best possible
partial assignment with each possible ending. After $r-1$ iterations, we
assume the algorithm has a list $L_{r}$ consisting of an optimal tail
$\sigma_{1},...,\sigma_{r-1}$ for each of the $q$ possible assignments
$\sigma_{r}$ to the $r$th particle, where optimality is measured with respect
to $H_{r-1}$. In other words, the algorithm has a solution to the subproblem
confined to the first $r$ particles, with any possible ending. To include the
next particle, and create the next list $L_{r+1}$, the algorithm finds the
optimal tail of each assignment $\sigma_{r+1}$. This is done by considering
all items in the list $L_{r}$ as possible tails for $\sigma_{r+1}$ and taking
the tail which minimizes $H_{r}(\sigma_{1},...,\sigma_{r+1}).$ In each of the
$n-1$ iterations, the algorithm checks for each of the $q$ possible
assignments $\sigma_{r}$, all $q$ items in the list $L_{r-1}$. Thus, in time
which is linear in $n$ and quadratic in $q$, we can derive the final list
$L_{n-1}$. The final solution is an assignment of minimal energy in that list.
The main idea in this article is to generalize the above algorithm to MPSs by
replacing assignments to particles by possible values of MPS matrices. Since
matrices are continuous objects, we use an $\epsilon$-net over all possible
matrices of bond dimension $D$. The number of possible assignments to one
variable, $q$, will now be replaced by the number of points in the
$\epsilon$-net, denoted as $N$. We will move from one site to the next,
keeping track of the minimum-energy MPS state, which ends in each MPS matrix
for the right most particle that the algorithm has reached.
In order to carry out this idea, it must not happen that the choice of the MPS
matrix of a later iteration can change the optimality of the partial MPS state
found in an earlier iteration. To avoid this, we work with a restricted form
of MPSs called _canonical MPSs_ , in which the energy of each term in the
Hamiltonian depends only on MPS matrices associated with nearby particles.
There are, however, various technical issues we need to handle. In particular,
we cannot use perfectly canonical MPSs but only an approximated version of
those, which imposes further technicalities, and in particular, the
neighboring MPS matrices do not match perfectly (we call this _imperfect
stitching_). These technicalities make the error analysis a bit subtle. Before
we formally define canonical MPSs and provide the details of the algorithm, we
mention an implication for a related problem.
### I.2 Commuting Hamiltonians in 1D
A problem related to finding minimum-energy MPS states is the complexity of
calculating the ground energy of commuting Hamiltonians in which all the local
terms commute. Bravyi and Vyalyi proved that for 2-local Hamiltonians the
problem lies inside NP bravyi . For $k$-local commuting Hamiltonians with
$k>2$, the complexity of the problem is still open. The complexity of the 1D
case was not studied before as far as we know; an immediate corollary of
Theorem 1 is that there is an efficient classical algorithm for approximating
the ground energy of commuting Hamiltonians in 1D to within $1/poly(n)$. This
is because the ground state of a commuting Hamiltonian in 1D is an MPS of
constant $D$ (this is a well-known fact that we explain later for
completeness), and therefore Theorem 1 can be applied. In fact, the result can
be improved to an exact algorithm (up to exponentially good approximations due
to truncations of real numbers) for a certain general class of problems. We
prove the following.
###### Theorem 2
Given is a 1D Hamiltonian whose terms commute. There is an efficient algorithm
that can compute the ground energy of this Hamiltonian to within any desired
accuracy $\epsilon$ in time polynomial in $n$ and in $\frac{1}{\epsilon}$. If
we may assume also that the ground space of the total Hamiltonian is well
separated from the higher excited states, by a spectral gap which is at least
$1/poly(n)$, then the algorithm can find both the ground energy and a
description of an MPS for the ground state exactly (i.e., up to exponentially
small errors due to handling of real numbers).
The basic idea for the exact algorithm can be illustrated when the terms in
the Hamiltonian are all projections and the ground state is unique. Since the
terms commute, the ground state is an eigenstate of each term separately, with
eigenvalue either $0$ or $1$. We start by applying the dynamic programming
algorithm, to create a good approximation of the ground state. From this
approximation we can deduce the correct eigenvalue ($0$ or $1$) for each of
the terms. The projections on the relevant eigenspaces can then be applied to
the MPS of the approximate state to make it exact. One gets a tensor network
of small depth, which can be converted into an MPS again. It can be shown that
applying the projections does not increase the bond dimension of the MPS too
much with respect to the approximating state. The details are fleshed out in
the proof (Sec. V).
Handling the degenerate case is very easy; essentially, we force the dynamic
algorithm to choose one state of the various possible states. The assumption
on the spectral gap ensures that the errors created by the epsilon net
approximations would not cause a confusion between the ground space and some
excited states.
We provide an alternative proof of Theorem 2, which also uses dynamic
programming. In fact, this proof holds for a somewhat stronger version of the
theorem, in which the conditions on the spectrum are far less restrictive. In
the algorithm given by this approach, the state is not provided as an MPS but
rather as a tensor product of two-particle states. The construction is based
on the work of LABEL:bravyi in which it is proved that the ground states of
$2$-local commuting Hamiltonians have this special structure. Bravyi and
Vyalyi use this structure to show that general $2$-local commuting
Hamiltonians problem is in NP. Since 1D chains with $k$-local interactions can
always be made $2$-local by treating nearby particles as one particle of a
larger dimension, LABEL:bravyi implies that the 1D commuting problem lies in
NP. However, by exploiting the special form of these ground states, dynamic
programming can be applied to find the solution efficiently in a very similar
manner to the 1D CSP, in which the NP witness is found using the 1D structure.
Unfortunately, in this approach too, it seems that one cannot avoid some
assumption on the spectrum of the total Hamiltonian, albeit a significantly
less restrictive one. Throughout its execution, the dynamic programming
algorithm compares various partial energies. If these are too close, and
cannot be distinguished even by computations performed with exponentially good
precision, then the algorithm might get confused between the ground energy and
a slightly excited state. A sketch of the alternative proof of Theorem 2,
providing the stronger version of it, and a discussion of the above precision
issue are given in Sec. V.
We mention that this latter proof (and in particular the observation that
dynamic programming can be useful for 1D quantum systems and not only for 1D
classical systems) was the inspiration for the current article, rather than
its corollary.
### I.3 Discussion and Open Questions
It is natural to ask how much the results in this article can be improved. By
LABEL:schuch08, we know that no polynomial algorithm exists for finding
optimal approximations of _polynomial_ bond dimension (unless P=NP). However,
the difficult instances of LABEL:schuch08 have a spectral gap of $1/poly(n)$.
Hastings has shown that ground state of 1D quantum system with a constant gap
can be approximated by a MPS with polynomial bond dimension hastings .
However, this is too large to immediately yield an efficient algorithm from
our result. It may still be true, however, that under the additional
restriction that the Hamiltonian has a constant gap, a polynomial time
algorithm exists, even when the bond dimension is as large as polynomial.
It is very likely that the efficiency of our algorithm can be significantly
improved even for the general case. In particular, a factor of $n$ would be
shaved from the error in Theorem 1 if we could use an $\epsilon$-net which is
both exactly canonical and enables perfect overlap between matrices at
neighboring particles, as we later explain. Unfortunately, even if this can be
done, the running time for this general algorithm is still quite large.
As mentioned earlier, we leave for further research the question of how this
algorithm can be used in combination with DMRG, and how certain symmetries in
the problem can be utilized to enhance its performance time for specific
interesting cases.
We note that very similar results to those presented in this article were
derived independently by Schuch and Cirac NI .
### I.4 Paper Organization
Section II starts by defining tensor networks, MPSs and canonical MPSs. In
Sec. III we describe the algorithm. This is where the $\epsilon$-nets are
defined and an algorithm to generate them is given. Also in Sec. III, we show
how they are used in the dynamic programming algorithm. Sec. IV provides an
exact analysis of the error accumulated in the algorithm. The complexity is
analyzed as a function of the desired error. In Sec. V we provide the proof
regarding the approximate and exact solutions for the commuting 1D case. We
defer several technical lemmas to the Appendix.
## II Tensor Networks and Matrix Product States
### II.1 Tensor Networks
We start with some background on tensor networks, since MPSs are a special
case of those. A detailed introduction to the use of tensor networks in the
context of quantum computation can be found in Refs markov2008simulating ;
aharonov2006quantum ; arad2008quantum .
A tensor network is a graph in which we allow some of the edges to be incident
to only one node. These edges are called the _legs_ of the network. Each node
is assigned a tensor whose rank (number of indices) is equal to the degree of
the node. Each index of the tensor corresponds to one edge that is incident to
that node. To each edge (or index) we also assign a positive integer which
indicates the range of the index. The indices associated with some of the
edges in the tensor network may be assigned fixed values. The other edges are
called _free_ edges.
We call an assignment of values to the indices of the free edges in the
network a _configuration_. With all the indices fixed, the tensor at each node
in the network yields a particular value. We say that the value of the
configuration is the product of the values for each of the nodes.
The value of the network is in general a tensor, whose rank is equal to the
number of legs in the network. If there are no such legs, the value is simply
a number (a scalar). Each assignment of values to the indices associated with
the legs of the network gives rise to a value for the network tensor. We
compute the tensor value for this assignment by summing over all
configurations which are consistent with that assignment the value of each
such configuration.
We note that often in the literature, one assigns values not to entire edges
but to the two sides of an edge separately (where each side inherits its range
of indices from the tensor associated with the node on that side). In the
evaluation of the network, we require that the values on the two sides of one
edge are equal, or else the entire configuration contributes zero to the sum.
Tensors will be denoted as bold-face fonts:
$\bm{\lambda},\bm{\Gamma},\bm{\mu}$. Their contraction will be denoted as an
expression like $\bm{\lambda}\bm{\Gamma}\bm{\mu}$, when it is clear from the
context along which indices the contraction is performed.
It is possible to restrict a tensor of rank $k$ to a tensor of rank $k-1$ by
assigning a fixed value to one of its legs. For example,
$\bm{\Gamma}_{\alpha}$ is the restriction of the tensor $\bm{\Gamma}$ to the
case in which the relevant edge associated with the index $\alpha$ is given
some value (which, by the usual abuse of notation of variables and their
values, will also be denoted as $\alpha$).
It is convenient to associate with every tensor (which can be given as a
contraction of a tensor network) a quantum state. For example, let
$\bm{\Gamma}=\Gamma^{i}_{\alpha,\beta}$ be a rank-$3$ tensor. Then we define
$|\bm{\Gamma}\rangle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i,\alpha,\beta}\Gamma^{i}_{\alpha,\beta}|\alpha\rangle\otimes|i\rangle\otimes|\beta\rangle$.
### II.2 Matrix Product States
We work in the notation of Vidal vidal1 for MPSs, with minor changes. A MPS
of a chain of $n$ $d$ dimensional particles, with bond dimension $D$, is a
tensor network with a 1D structure as in Fig. 1. Horizontal edges correspond
to indices ranging from $1$ to the bond dimension $D$ and are denoted with
$\alpha,\beta$,…, while vertical edges correspond to indices ranging from $1$
to the physical dimension $d$. (In our description, the end particles will
actually have a different physical dimension, denoted $d_{end}$. This is
required due to a technical reason described in Sec. II.3) The indices of
vertical edges are denoted with $i$,$j$,…The figures show two types of nodes:
black and white. The tensors of black nodes are typically of rank $3$ (except
for the boundary tensors, which are of rank $2$), and we denote them with
$\bm{\Gamma}$’s. For example, when the tensor that is second from left is
written with its indices, it is denoted as
$\Gamma^{{[2]}^{i}}_{\alpha_{2},\alpha_{3}}$, where the index $[2]$ in
brackets corresponds to its location in the graph. The tensors associated with
white nodes are always of rank $2$ and are denoted with $\bm{\lambda}$’s. They
are required to be diagonal and hence are given only one index (i.e.,
$\lambda^{[2]}_{\alpha_{2}}$). Without loss of generality, we will also demand
that the entries of $\bm{\lambda}$ are non negative since the phases can be
absorbed in the neighboring $\bm{\Gamma}$ tensors.
The MPS defined by this network is
$|\psi\rangle=\sum_{i_{1},\ldots,i_{n}}C_{i_{1}\cdots
i_{n}}|i_{1}\rangle\cdots|i_{n}\rangle$ with
$\displaystyle C_{i_{1}\cdots
i_{n}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{\alpha_{2},\ldots,\alpha_{n}}{\Gamma^{[1]}}_{\alpha_{2}}^{i_{1}}\lambda^{[2]}_{\alpha_{2}}{\Gamma^{[2]}}_{\alpha_{2}\alpha_{3}}^{i_{2}}\lambda^{[3]}_{\alpha_{3}}\cdots\lambda^{[n]}_{\alpha_{n}}{\Gamma^{[n]}}_{\alpha_{n}}^{i_{n}}\
.$
Figure 1: MPS as a tensor network. Figure 2: A description of a canonical
MPS. The tensors are chosen such that cutting a MPS between the $j-1$th and
$j$th particles corresponds to the Schmidt decomposition between the left and
right parts:
$|\psi\rangle=\sum_{\beta}\lambda^{[j]}_{\beta}|\mathbf{L}^{[j]}_{\beta}\rangle\otimes|\mathbf{R}^{[j]}_{\beta}\rangle$.
In the language of tensor states, $|\psi\rangle$ is exactly the tensor state
of the contraction
$\bm{\Gamma}^{[1]}\bm{\lambda}^{[2]}\bm{\Gamma}^{[2]}\cdots\bm{\lambda}^{[n]}\bm{\Gamma}^{[n]}$.
### II.3 Canonical MPSs
An MPS is in _canonical form_ if every cut in the chain induces a Schmidt
decomposition (as in Fig. 2). In other words, we can rewrite the MPS by
changing the order of summation to sum last over the index $\beta$ of the
$j$th $\bm{\lambda}$ tensor:
$|\psi\rangle=\sum_{\beta}\lambda^{[j]}_{\beta}|\mathbf{L}^{[j]}_{\beta}\rangle\otimes|\mathbf{R}^{[j]}_{\beta}\rangle$,
where $\mathbf{L}^{[j]}_{\beta}$ ($\mathbf{R}^{[j]}_{\beta}$) denote the
contraction of the all the tensors to the left (right) of the cut with fixed
$\beta$ and $|\mathbf{L}^{[j]}_{\beta}\rangle$
($|\mathbf{R}^{[j]}_{\beta}\rangle$) are their corresponding states. Then the
canonical conditions are that for all $j$ from $2$ to $n$,
$\sum_{\beta}\big{(}\lambda^{[j]}_{\beta}\big{)}^{2}=1$ and
$\langle\mathbf{L}^{[j]}_{\alpha}|\mathbf{L}^{[j]}_{\beta}\rangle=\langle\mathbf{R}^{[j]}_{\alpha}|\mathbf{R}^{[j]}_{\beta}\rangle=\delta_{\alpha\beta}$.
In addition, for normalization, we require that the entire MPS state is
normalized, which is guaranteed by the normalization requirement on the
$\bm{\lambda}^{[j]}$ tensors.
There is a small technical issue that needs attention: The canonical
conditions cannot be satisfied at the boundaries if $d<D$. Consider for
example the left boundary; there are not enough dimensions in the Hilbert
space of the left particle for an orthonormal set of vectors
$|\mathbf{L}^{[2]}_{\alpha}\rangle$ to exist. This issue remains a problem
even as we move away from the boundary by one particle, as the dimension of
the left-side Hilbert space increases to $d^{2}$ which may still be smaller
than $D$. There are many ways of handling this technicality; here we choose to
assume that the particles at the end of the chain have dimension of at least
$D$. This will ensure that at any cut along the chain, the Hilbert space of
the subsystems on each side have dimension of at least $D$. We can achieve
this by grouping $s$ particles at each end of the chain into a single
particle, where $s$ is chosen to be the smallest integer such that $d^{s}\geq
D$. Denote $d^{s}$ as $d_{end}$, the dimensionality of each of those end
particles. Note that $d_{end}=d^{s}\leq Dd$. The dimension of the rest of the
particles will remain $d$. We renumber the particles after the grouping, so
that the new $H_{1,2}$ is now the sum of the old $H_{i,i+1}$ for $i$ ranging
from $1$ to $s$. The term in the Hamiltonian for the last two particles is
adjusted in a similar manner. We will assume from now on that the Hamiltonian
is given in this form.
Let us now see how the canonical conditions can be stated in a local manner.
Graphically, the second condition is equivalent to
(2)
and similarly from the other side. Here the upper part of the network
corresponds to $|\mathbf{L}^{[j]}_{\alpha}\rangle$, and the lower part
corresponds to $\langle\mathbf{L}^{[j]}_{\beta}|$. Notice that the canonical
conditions imply that we can “collapse” the network both from the left side
and from the right side. Moreover, as this condition holds at every bond, it
is not difficult to see that a necessary and sufficient condition for an MPS
to be canonical consists of the following _local_ conditions on
$(\bm{\lambda}^{[j]},\bm{\Gamma}^{[j]},\bm{\lambda}^{[j+1]})$: For every
$j=2,\ldots,n-1$,
$\displaystyle\langle(\bm{\lambda}^{[j]}\bm{\Gamma}^{[j]})_{\alpha}|(\bm{\lambda}^{[j]}\bm{\Gamma}^{[j]})_{\beta}\rangle$
$\displaystyle=\delta_{\alpha\beta}\ \ \text{\small(left canonical)},$ (3)
$\displaystyle\langle(\bm{\Gamma}^{[j]}\bm{\lambda}^{[j+1]})_{\alpha}|(\bm{\Gamma}^{[j]}\bm{\lambda}^{[j+1]})_{\beta}\rangle$
$\displaystyle=\delta_{\alpha\beta}\ \ \text{\small(right canonical)}.$ (4)
For $j=1$ and $j=n$, for $1\leq\alpha,\beta\leq D$:
$\displaystyle\langle\bm{\Gamma}^{[1]}_{\alpha}|\bm{\Gamma}^{[1]}_{\beta}\rangle=\langle\bm{\Gamma}^{[n]}_{\alpha}|\bm{\Gamma}^{[n]}_{\beta}\rangle=\delta_{\alpha\beta}$
(5) $\displaystyle\qquad\text{(boundary canonical conditions)}\ .$
We also require that the $\bm{\lambda}^{\prime}s$ are normalized, namely, that
for every $j$ from $2$ to $n$,
$\displaystyle\langle\bm{\lambda}^{[j]}|\bm{\lambda}^{[j]}\rangle=1\ .$ (6)
Graphically, these conditions are summarized in Fig. 3.
Figure 3: (a) The normalization condition for $j=2,\ldots,n$. (b) The
left/right canonical conditions for $j=2,\ldots,n-1$ [see Eqs. (3) and (4)].
(c) The boundary canonical conditions for $j=1$ and $j=n$ [see Eq. (5)].
Any triplet
$(\bm{\lambda}^{[j]},\bm{\Gamma}^{[j]},\bm{\lambda}^{[j+1]})=(\bm{\lambda},\bm{\Gamma},\bm{\mu})$
that satisfies the normalization and the left and right canonical conditions
[Eqs. (3), (4), and (6)] is called a _canonical triplet_. Such a triplet can
be associated with a quantum state on three particles
$|\psi\rangle=|\bm{\lambda}\bm{\Gamma}\bm{\mu}\rangle=\sum_{\alpha,i,\beta}\lambda_{\alpha}\Gamma^{i}_{\alpha\beta}\mu_{\beta}|\alpha\rangle|i\rangle|\beta\rangle$,
with the following properties: ${\|\psi\|}=1$; the Schmidt basis of the first
particle is the standard basis, with Schmidt coefficients
$\\{\lambda_{\alpha}\\}$; and the Schmidt basis of the third particle is the
standard basis, with Schmidt coefficients $\\{\mu_{\beta}\\}$. A canonical MPS
can thus be described as a set of canonical triplets (or equivalently
$3$-particle states) such that the right $\bm{\mu}$ tensor of one state is
equal to the left $\bm{\lambda}$ tensor of the next canonical triplet.
Instead of describing a canonical MPS in terms of canonical triplets
$(\bm{\lambda},\bm{\Gamma},\bm{\mu})$, we will often describe it using
_canonical pairs_ $(\bm{\lambda},\bm{B})$, where
$\displaystyle\bm{B}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bm{\Gamma}\bm{\mu}\
.$
The advantage is that for canonical MPSs, the elements in $\bm{B}$ are always
bounded (since the $L_{2}$ norm of $\bm{B}$ satisfies ${\|\bm{B}\|}=\sqrt{D}$;
see Sec. II.4), unlike $\bm{\Gamma}$ whose entries can approach infinity when
the corresponding $\bm{\mu}$ entries approach zero.
An MPS that is described by the contraction
$\bm{\Gamma}^{[1]}\bm{\lambda}^{[2]}\bm{\Gamma}^{[2]}\bm{\lambda}^{[3]}\cdots\bm{\lambda}^{[n]}\bm{\Gamma}^{[n]}$
can also be denoted as
$\bm{\Gamma}^{[1]}\bm{\lambda}^{[2]}\bm{B}^{[2]}\bm{B}^{[3]}\cdots\bm{B}^{[n-1]}\bm{\Gamma}^{[n]}$.
No information is lost since $\bm{\mu}$ can always be recovered from
$(\bm{\lambda},\bm{B})$: $\mu_{\beta}$ is the norm (see Sec. II.4) of the
tensor state $(\bm{\lambda}\bm{B})_{\beta}$:111Recall that $\mu_{\beta}$
corresponds to a Schmidt coefficient in a Schmidt decomposition that coincides
with the standard basis.
$\displaystyle\mu_{\beta}=\left(\sum_{i,\alpha}|\lambda_{\alpha}B_{\alpha\beta}^{i}|^{2}\right)^{1/2}\
.$
We define
$\bm{\mu}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\bm{\mu}(\bm{\lambda},\bm{B})$
this way also for non-canonical pairs.
The advantage of working with the canonical form is that the energy of local
Hamiltonians involves only the local tensors, as the following figure
illustrates:
The above equality was obtained using the canonical conditions that are
described in Eq. (2). Consequently, the energy
$\langle\psi|H_{j-1,j}|\psi\rangle$ only involves five tensors:
$\bm{\lambda}^{[j-1]},\bm{\Gamma}^{[j-1]}$,
$\bm{\lambda}^{[j]},\bm{\Gamma}^{[j]}$, and $\bm{\lambda}^{[j+1]}$. Similarly,
$H_{1,2}$ only depends on $\bm{\Gamma}^{[1]}$, $\bm{\lambda}^{[2]}$,
$\bm{\Gamma}^{[2]},\bm{\lambda}^{[3]}$, and $H_{n-1,n}$ only depends on
$\bm{\lambda}^{[n-1]},\bm{\Gamma}^{[n-1]}$,
$\bm{\lambda}^{[n]},\bm{\Gamma}^{[n]}$. It is important that each energy term
does not involve tensors further to the right in the chain since the algorithm
attempts to compute (or approximate) the optimal MPS up to a certain point. We
would like to be able to grow the description of the state from left to right,
without affecting the energies we have already computed. If matrices in the
right side of the chain affected energies of terms in the left side, we would
need to go back and change the MPS matrices of the particles we have already
handled after we make new assignments to particles on the right. This would
ruin the entire idea of dynamic programming.
Fortunately, any MPS representing a normalized state can be written as a
canonical MPS with no increase in bond dimension. This follows from
LABEL:vidal1, in which it is shown that any state with Schmidt rank of at most
$D$ across any cut can be written as a canonical MPS with bond dimension $D$.
### II.4 Tensor Norms and Distances
We use the $L_{2}$ norm on tensors
${\|\bm{X}\|}^{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{i_{1}\ldots
i_{k}}|X_{i_{1}\ldots i_{k}}|^{2}$. This norm of course induces a metric,
namely, a way of defining the distance between tensors of the same rank. It is
easy to see that the norm of a tensor $\bm{C}$ is equal to the Eucledian norm
of its corresponding state $|\bm{C}\rangle$. Also, for a rank-2 tensor (which
can be viewed as a matrix), it is known that its _operator_ norm is not larger
than its tensor norm (which in this case is simply the Frobenious norm).
It is true that for any three tensors, $\bm{B}_{1},\bm{B}_{2},\bm{B}$, we have
$\|\bm{B}_{1}\bm{B}-\bm{B}_{2}\bm{B}\|\leq\|\bm{B}_{1}-\bm{B}_{2}\|\cdot\|\bm{B}\|$.
In fact, many times in the context of MPSs, a much stronger inequality holds.
Assume $\bm{B}$ connects with $\bm{B}_{1}$ or $\bm{B}_{2}$ along one edge,
indexed by $\alpha$. Assume further that $\|\bm{B}_{\alpha}\|=1$ for every
$\alpha$ (in the context of canonical MPSs, it will often be the case that we
consider the contraction of one side of the chain with a fixed index $\alpha$
of the cut edge, and this contraction is indeed of norm $1$ by the canonical
conditions). In this case, we have a much stronger inequality, which can be
easily verified:
$\displaystyle{\|\bm{B}_{1}\bm{B}-\bm{B}_{2}\bm{B}\|}={\|\bm{B}_{1}-\bm{B}_{2}\|}\
.$ (7)
We can apply this to cases of interest, when we compare contractions of
tensors which differ in only a single term. For example, consider vector
$\lambda_{\alpha}$ with norm $1$ and two tensors
$B_{\alpha,i_{1},\ldots,i_{k}}$ and $A_{\alpha,j_{1},\ldots,j_{l}}$ such that
when $\alpha$ is fixed, the resulting tensors $\bm{A}_{\alpha}$ and
$\bm{B}_{\alpha}$ have norm 1. Let $\hat{\bm{\lambda}}$, $\hat{\bm{A}}$ and
$\hat{\bm{B}}$ be tensors with the same rank and dimensions as $\bm{\lambda}$,
$\bm{A}$ and $\bm{B}$. We have, by Eq. (7),
$\displaystyle{\|\bm{A}\bm{\lambda}\bm{B}-\bm{A}\hat{\bm{\lambda}}\bm{B}\|}={\|\bm{\lambda}-\hat{\bm{\lambda}}\|}\
,$ (8)
and also
$\displaystyle{\|\bm{A}\bm{\lambda}\bm{B}-\hat{\bm{A}}\hat{\bm{\lambda}}\bm{B}\|}={\|\bm{A}\bm{\lambda}-\hat{\bm{A}}\hat{\bm{\lambda}}\|}\
.$ (9)
And similarly,
$\displaystyle{\|\bm{A}\bm{\lambda}\bm{B}-\bm{A}\bm{\lambda}\hat{\bm{B}}\|}$
$\displaystyle={\|\bm{\lambda}\bm{B}-\bm{\lambda}\hat{\bm{B}}\|}$
$\displaystyle=\left[\sum_{\alpha}|\lambda_{\alpha}|^{2}{\|(\bm{B}_{\alpha}-\hat{\bm{B}}_{\alpha})\|}^{2}\right]^{1/2}$
$\displaystyle\leq\max_{\alpha}\|\bm{B}_{\alpha}-\hat{\bm{B}}_{\alpha}\|\ .$
(10)
## III The Algorithm
As discussed earlier, in order to carry out the outline described in Sec. I.1,
we would like to work with canonical MPSs. Additionally, since the tensor
pairs $(\bm{\lambda},\bm{B})$ for neighboring nodes overlap, we would like an
$\epsilon$ net over canonical pairs such that $\bm{\mu}(\bm{\lambda},\bm{B})$
could be equal to the $\bm{\lambda}$ of the next pair (we call this _perfect
stitching_). We do not know how to efficiently construct an $\epsilon$ net
that satisfies those conditions exactly; we resort to approximately canonical
MPSs with approximate stitching.
### III.1 $\epsilon$ nets
We fix $\epsilon>0$ (to be determined later) and define two $\epsilon$ nets.
We start with discretizing $\bm{\Gamma}^{[1]}$ and $\bm{\Gamma}^{[n]}$.
###### Definition 1 (the $G_{end}$ $\epsilon$ net)
$G_{end}$ is a set of canonical boundary tensors [see Eq. (5)] such that, for
any canonical boundary tensor $\hat{\bm{\Gamma}}$ there is $\bm{\Gamma}\in
G_{end}$ such that for each $\alpha$,
$\|\hat{\bm{\Gamma}}_{\alpha}-\bm{\Gamma}_{\alpha}\|\leq\epsilon$.
We now define an $\epsilon$ net over the intermediate tensors, or more
precisely, for the pairs $(\bm{\lambda},\bm{B})$.
###### Definition 2 (the $G$ $\epsilon$ net)
$G$ is a set of pairs of tensors $(\bm{\lambda},\bm{B})$ such that:
1. 1.
$\bm{\lambda}$ is positive and normalized: For all $\alpha$
$\lambda_{\alpha}\geq 0$ and $\langle{\bm{\lambda}}|{\bm{\lambda}}\rangle=1.$
2. 2.
$G$ is an $\epsilon$ net: For every canonical triplet
$(\hat{\bm{\lambda}},\hat{\bm{\Gamma}},\hat{\bm{\mu}})$ there is
$(\bm{\lambda},\bm{B})\in G$ such
$\|\hat{\bm{\lambda}}\hat{\bm{\Gamma}}\hat{\bm{\mu}}-\bm{\lambda}\bm{B}\|\leq\epsilon$.
3. 3.
$\bm{B}$ is perfectly right canonical: For every $\alpha,\alpha^{\prime}$,
$\langle{\bm{B}_{\alpha}}|\bm{B}_{\alpha^{\prime}}\rangle=\delta_{\alpha\alpha^{\prime}}$
(here $\alpha,\alpha^{\prime}$ are the left Greek indices of $\bm{B}$).
4. 4.
$(\bm{\lambda},\bm{B})$ are approximately left canonical: For every
$\beta\neq\beta^{\prime}$,
$\displaystyle|\langle{(\bm{\lambda}\bm{B})_{\beta}}|(\bm{\lambda}\bm{B})_{\beta^{\prime}}\rangle|\leq
3\epsilon\ .$ (11)
### III.2 $\epsilon$ net Generators
We now explain how to construct such nets efficiently. Both generators for the
$\epsilon$ nets will make use of the following general lemma
###### Lemma 3
For any positive integers $a\leq b$ and any $\nu$ in the range
$\left(0,1/\sqrt{a}\right]$, we can generate a set of $a\times b$ matrices
$S_{ab}$ over the complex numbers such that for any $A\in S_{ab}$, the rows of
$A$ are an ortho-normal set of length $b$ vectors. Furthermore, for any
$a\times b$ matrix $B$ whose rows form a set of orthonormal vectors, there is
a matrix $A\in S_{ab}$ such that each row of $A-B$ has $L_{2}$ norm at most
$\nu$. The size of $S_{ab}$ is at most $\mathcal{O}((72b/\nu)^{2ab})$. The
time to generate $S_{ab}$ is $\mathcal{O}(a^{2}b(72b/\nu)^{2ab})$. If $a=1$,
we can generate a set of vectors with real non-negative entries, rather than
complex. The size of the net is $\mathcal{O}((72b/\nu)^{b})$ and the time to
generate it is $\mathcal{O}(b(72b/\nu)^{b})$.
The proof appears in the Appendix.
#### III.2.1 Generating $G_{end}$:
Invoke Lemma 3 with $\nu=\epsilon$, $a=D$, and $b=d_{end}$. For every $A\in
S_{D,d_{end}}$, add a $\bm{\Gamma}$ to the $\epsilon$ net, where
$A_{\alpha,i}=\Gamma^{i}_{\alpha}$. Note that the conditions of Lemma 3, are
satisfied if $\epsilon\leq 1/\sqrt{D}$. Since $d_{end}\leq Dd$, the size of
the net is at most $(72Dd/\epsilon)^{2dD^{2}}$ and the time to generate it is
$O(dD^{3})$ times the size of the set.
#### III.2.2 Generating $G$:
We generate $G$ by first generating an $\epsilon/2$-net over the
$\bm{\lambda}$’s and an $\epsilon/2$-net over the $\bm{B}$’s. To generate the
net of the $\bm{\lambda}$’s, invoke Lemma 3 with $a=1$, $b=D$ and the $\nu$ in
the lemma set to $\epsilon/2$. Note that we would like to have a
$\bm{\lambda}$ with non negative real entries. According to Lemma 3, this
actually requires fewer items in our net since we are omitting the phases in
each entry in the tensor. The resulting net for the $\bm{\lambda}$’s has size
$(144D/\epsilon)^{D}$ and can be generated in time $O(D(144D/\epsilon)^{D})$.
To generate the net over the $\bm{B}$’s, we invoke Lemma 3 with $a=D$, $b=dD$,
and $\nu=\epsilon/2$. Note that in order to invoke Lemma 3, we require that
$\epsilon\leq 2/\sqrt{D}$. For any matrix $A_{\alpha,(i,\beta)}$ in the set,
we generate a tensor $\bm{B}$ where
$B^{i}_{\alpha,\beta}=A_{\alpha,(i,\beta)}$. This way we generate a set of
pairs $(\bm{\lambda},\bm{B})$ which satisfies both the normalization condition
[condition (1) of Definition 2] and the condition of being perfect right
canonical [condition (3) of Definition 2].
To see that we in fact have an $\epsilon$ net [i.e. condition (2) is
satisfied], consider a perfectly canonical pair $(\bm{\lambda},\bm{B})$, and
let us find a pair $(\hat{\bm{\lambda}},\hat{\bm{B}})$ in the net that is
$\epsilon$-close to it. We first replace $\bm{\lambda}$ with a
$\hat{\bm{\lambda}}$ from the first net and then replace $\bm{B}$ with a
$\hat{\bm{B}}$ from the second net. Using Eq. (8), we have that
$\displaystyle{\|\bm{\lambda}\bm{B}-\hat{\bm{\lambda}}\bm{B}\|}={\|\bm{\lambda}-\hat{\bm{\lambda}}\|}\leq\frac{\epsilon}{2}\
,$
Using Eq. (10), we also have
$\displaystyle{\|\hat{\bm{\lambda}}\bm{B}-\hat{\bm{\lambda}}\hat{\bm{B}}\|}\leq\max_{\alpha}{\|\bm{B}_{\alpha}-\hat{\bm{B}}_{\alpha}\|}\leq\frac{\epsilon}{2}\
.$
Next, we discard all tensors $(\bm{\lambda},\bm{B})$ that are not
approximately left canonical, namely, those that violate condition $(4)$. It
remains to show that the remaining tensors still satisfy condition $(2)$, that
is, the $\epsilon$ net condition. We do that by showing that a pair
$(\bm{\lambda},\bm{B})$ that is $\epsilon$ close to a canonical triplet must
necessarily be approximately left canonical. Therefore, such a pair would not
have been eliminated.
To see this, let the tensor $\bm{A}=\bm{\lambda}\bm{\Gamma}\bm{\mu}$ be the
contraction of the canonical triplet and $\bm{C}$ be the contraction of
$\hat{\bm{\lambda}}\hat{\bm{B}}$ from the net such that
${\|\bm{A}-\bm{C}\|}\leq\epsilon$. The fact that $\bm{A}$ is perfectly left
canonical is expressed in the fact that for every $\beta\neq\beta^{\prime}$,
$\langle\bm{A}_{\beta}|\bm{A}_{\beta^{\prime}}\rangle=0$. To prove that
$\bm{C}$ is approximately left canonical, we need to show
$|\langle\bm{C}_{\beta}|\bm{C}_{\beta^{\prime}}\rangle|\leq 3\epsilon$.
Indeed, ${\|\bm{A}-\bm{C}\|}\leq\epsilon$ implies
${\|\bm{A}_{\beta}-\bm{C}_{\beta}\|}\leq\epsilon$ for every $\beta$. Assume
$\beta\neq\beta^{\prime}$. Then
$\displaystyle|\langle{\bm{C}_{\beta}}|{\bm{C}_{\beta^{\prime}}}\rangle|$
$\displaystyle=|\langle{\bm{A}_{\beta}+(\bm{C}_{\beta}-\bm{A}_{\beta})}|{\bm{A}_{\beta^{\prime}}+(\bm{C}_{\beta^{\prime}}-\bm{A}_{\beta^{\prime}})}\rangle|$
$\displaystyle\leq|\langle{\bm{A}_{\beta}}|{\bm{A}_{\beta^{\prime}}}\rangle|+|\langle{\bm{A}_{\beta}}|{\bm{C}_{\beta^{\prime}}-\bm{A}_{\beta^{\prime}}}\rangle|$
$\displaystyle\ \ +\
|\langle{\bm{C}_{\beta}-\bm{A}_{\beta}}|{\bm{A}_{\beta^{\prime}}}\rangle|+|\langle{\bm{C}_{\beta}-\bm{A}_{\beta}}|{\bm{C}_{\beta^{\prime}}-\bm{A}_{\beta^{\prime}}}\rangle|$
$\displaystyle\leq{\|\bm{A}_{\beta}\|}{\|\bm{C}_{\beta^{\prime}}-\bm{A}_{\beta^{\prime}}\|}+{\|\bm{A}_{\beta^{\prime}}\|}{\|\bm{C}_{\beta}-\bm{A}_{\beta}\|}$
$\displaystyle\ +\
{\|\bm{C}_{\beta}-\bm{A}_{\beta}\|}{\|\bm{C}_{\beta}-\bm{A}_{\beta}\|}$
$\displaystyle\leq 2\epsilon+\epsilon^{2}\leq 3\epsilon\ .$
This concludes the proof that $G$ is indeed an $\epsilon$ net according to
Definition 2.
#### III.2.3 Complexity of Generating $G$ and $G_{end}$:
By Lemma 3, $N\stackrel{{\scriptstyle\mathrm{def}}}{{=}}|G|$, the size of the
$\epsilon$ net $G$ is
$\displaystyle N=\mathcal{O}\left(\frac{144dD}{\epsilon}\right)^{D+2dD^{2}}.$
(12)
This is the size of the set formed by taking all pairs
$(\bm{\lambda},\bm{B})$, where each $\bm{\lambda}$ and $\bm{B}$ come from
their respective nets. The time required to generate the original net (before
tensors are discarded) is $O(dD^{3}N)$. The cost of checking whether a
$(\lambda,B)$ pair is approximately left canonical is $\mathcal{O}(dD^{3})$,
so the total cost of generating the net is $\mathcal{O}(dD^{3}N)$.
For $G_{end}$, both the number of points and the running time which were
determined in Sec. III.2.1, are bounded above by the corresponding bounds of
$G$.
### III.3 The algorithm
When processing particle $j$, the algorithm creates a list $L_{j}$ of partial
solutions, one for each $(\bm{\lambda},\bm{B})$ pair in $G$. For each such
partial solution, a tail (i.e., the tensors to the left of the $j$th particle)
and energy is kept.
First step:
Create the first list $L_{2}$: For each $(\bm{\lambda}^{[2]},\bm{B}^{[2]})\in
G$, find its tail, namely the $\bm{\Gamma}^{[1]}\in G_{end}$ which minimizes
the energy with respect to $H_{1,2}$ of the tensor
$\bm{\Gamma}^{[1]}\bm{\lambda}^{[2]}\bm{B}^{[2]}$. Denote this minimal energy
by $E_{2}(\bm{\lambda}^{[2]},\bm{B}^{[2]})$. We keep both the tail and the
computed energy, for each pair $(\bm{\lambda}^{[2]},\bm{B}^{[2]})\in G$.
Going from $j=3$ to $j=n-1$:
we assume we have created the list $L_{j-1}$. For each pair
$(\bm{\lambda}^{[j-1]},\bm{B}^{[j-1]})\in G$ there is a tail in $L_{j-1}$:
$\displaystyle\bm{\Gamma}^{[1]},(\bm{\lambda}^{[2]},\bm{B}^{[2]}),(\bm{\lambda}^{[3]},\bm{B}^{[3]}),\ldots,(\bm{\lambda}^{[j-2]},\bm{B}^{[j-2]})$
and an energy value that we denote by
$E_{j-1}(\bm{\lambda}^{[j-1]},\bm{B}^{[j-1]})$. To create $L_{j}$, we find a
tail for each $(\bm{\lambda}^{[j]},\bm{B}^{[j]})\in G$. We require that the
tail for a given $(\bm{\lambda}^{[j]},\bm{B}^{[j]})$ is an item in $L_{j-1}$
which satisfies the “stitching” condition:
$\displaystyle{\|\bm{\mu}(\bm{\lambda}^{[j-1]},\bm{B}^{[j-1]})-\bm{\lambda}^{[j]}\|}\leq
2\epsilon\ .$ (13)
We pick the tail for $(\bm{\lambda}^{[j]},\bm{B}^{[j]})$ to be an item in
$L_{j-1}$ which satisfies the stitching condition and minimizes
$H_{j-1,j}(\bm{\lambda}^{[j-1]}\bm{B}^{[j-1]}\bm{B}^{[j]})+E_{j-1}(\bm{\lambda}^{[j-1]},\bm{B}^{[j-1]})$.
The minimum such value is defined to be
$E_{j}(\bm{\lambda}^{[j]},\bm{B}^{[j]})$.
Final step:
The final step, $j=n$, is exactly as in the intermediate steps except the
algorithm goes over $\bm{\Gamma}^{[n]}\in G_{end}$, rather than over pairs
from $G$ and there is no stitching constraint. More precisely, we pick the
tail for $\bm{\Gamma}^{[n]}$ to be the item in $L_{n-1}$ which minimizes
$H_{n-1,n}(\bm{\lambda}^{[n-1]}\bm{B}^{[n-1]}\bm{\Gamma}^{[n]})+E_{n-1}(\bm{\lambda}^{[n-1]},\bm{B}^{[n-1]})$.
The minimal value is defined to be $E_{n}(\bm{\Gamma}^{[n]})$.
Finally, we choose $\bm{\Gamma}^{[n]}$ which minimizes
$E_{n}(\bm{\Gamma}^{[n]})$. We output the MPS that is defined by
$\bm{\Gamma}^{[n]}$ and its tail:
$\displaystyle|\Omega\rangle\stackrel{{\scriptstyle\mathrm{def}}}{{=}}|\bm{\Gamma}^{[1]}\bm{\lambda}^{[2]}\bm{B}^{[2]}\bm{B}^{[3]}\cdots\bm{B}^{[n-1]}\bm{\Gamma}^{[n]}\rangle\
,$ (14)
together with the energy which the algorithm calculated:
$\displaystyle
E_{alg}(\Omega)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}E_{n}(\bm{\Gamma}^{[n]})\
.$ (15)
Note that since each $(\bm{\lambda}^{[j]},\bm{B}^{[j]})$ is perfectly right
canonical, the state $|\Omega\rangle$ is normalized. This can be seen by
contracting the tensor network corresponding to the inner product
$\langle{\Omega}|{\Omega}\rangle$ from right to left.
Unlike in the classical case, our algorithm does not search all states due to
the discretization. Moreover, it does not optimize over the real energy of the
states that it does check, but rather over
$E_{alg}(\Omega)=\sum_{j}H_{j-1,j}(\bm{\lambda}^{[j-1]}\bm{B}^{[j-1]}\bm{B}^{[j]})$.
$E_{alg}$ is different from the true energy $E$ because the states are not
exactly canonical. Note that the output $E_{alg}(\Omega)$ is thus just an
approximation of the real energy $E(\Omega)$ of the output MPS
$|\Omega\rangle$. We output $E_{alg}(\Omega)$ anyway, since our guarantee on
its error is somewhat better than on the error for $E(\Omega)$, as we will see
in Sec. IV.
The following claim easily follows from the same reasoning as for the
classical dynamic programming algorithm:
###### Claim 4
The algorithm finds the state which minimizes $E_{alg}$ among all MPSs of the
form
$\bm{\Gamma}^{[1]}\bm{\lambda}^{[2]}\bm{B}^{[2]}\bm{B}^{[3]}\cdots\bm{B}^{[n-1]}\bm{\Gamma}^{[n]}$,
such that $\bm{\Gamma}^{[1]},\bm{\Gamma}^{[n]}\in G_{end}$,
$(\bm{\lambda}^{[j]},\bm{B}^{[j]})\in G$ for all $j\in\\{2,...,n-1\\}$, and
the stitching conditions (Eq. (13)) are all satisfied.
## IV Error and Complexity Analysis
In order to finish the proof of Theorem 1, we will prove the theorem below. As
noted above, this theorem actually gives a better error bound on
$E_{alg}(\Omega)$ than the bound on $E(\Omega)$ that is given in Theorem 1.
###### Theorem 5 (Error bound)
Let $E_{0}$ be the minimal energy that can be achieved by a state with bond
dimension $D$, and $J$ the maximal operator norm $\|H_{j,j+1}\|$ over all
terms. Then:
$\displaystyle E_{alg}(\Omega)-6Jn\epsilon\leq E_{0}\leq E(\Omega)\leq
E_{alg}(\Omega)+\tfrac{3}{2}JD^{2}n^{2}\epsilon\ .$ (16)
It is easy to verify that as long as $nD^{2}\geq 12$, Eq. (16) implies Eq. (1)
of Theorem 1.
Proof:
By definition, $E_{0}\leq E(\Omega)$. We first prove that
$E_{alg}(\Omega)-6Jn\epsilon\leq E_{0}$. Let:
$\displaystyle|\psi\rangle=|\hat{\bm{\Gamma}}^{[1]}\hat{\bm{\lambda}}^{[2]}\hat{\bm{\Gamma}}^{[2]}\cdots\hat{\bm{\lambda}}^{[n]}\hat{\bm{\Gamma}}^{[n]}\rangle\
.$
be a state with $E(\psi)=E_{0}$ of bond dimension $D$, written as a canonical
MPS. For every triplet
$(\hat{\bm{\lambda}}^{[j]},\hat{\bm{\Gamma}}^{[j]},\hat{\bm{\lambda}}^{[j+1]})$
for $j=2,\ldots n-1$, we associate a pair
$(\tilde{\bm{\lambda}}^{[j]},\tilde{\bm{B}}^{[j]})\in G$ which is
$\epsilon$-close to that triplet. In addition, we find
$\tilde{\bm{\Gamma}}_{1}\in G_{end}$ close to $\hat{\bm{\Gamma}}_{1}$ and
$\tilde{\bm{\Gamma}}_{n}\in G_{end}$ close to $\hat{\bm{\Gamma}}_{n}$. We
define the state:
$\displaystyle|\phi\rangle=|\tilde{\bm{\Gamma}}_{1}\tilde{\bm{\lambda}}_{2}\tilde{\bm{B}}_{2}\tilde{\bm{B}}_{3}\cdots\tilde{\bm{B}}_{n-1}\tilde{\bm{\Gamma}}_{n}\rangle\
.$
Just like $|\Omega\rangle$, this state is normalized due to the fact that the
tensors in $G_{end}$ and $G$ are perfectly right canonical.
We first claim that $E_{alg}(\Omega)\leq E_{alg}(\phi)$. This follows from the
fact that $|\phi\rangle$ belongs to the set of states over which the dynamic
algorithm searches (see Claim 4), since the
$\tilde{\bm{\lambda}}^{[j-1]}\tilde{\bm{B}}^{[j-1]}$ and
$\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}$ satisfy the stitching
condition (13), as promised by the following lemma:
###### Lemma 6
For every $j=3,\ldots,n-1$,
$\displaystyle{\|\mu(\tilde{\bm{\lambda}}^{[j-1]},\tilde{\bm{B}}^{[j-1]})-\tilde{\bm{\lambda}}^{[j]}\|}\leq
2\epsilon\ .$ (17)
Proof: We use the fact (established in Lemma 8 in the Appendix) that for any
two bipartite states $|A\rangle=\sum_{i}a_{i}|i\rangle|A_{i}\rangle$, with
normalized $|A_{i}\rangle$, $|B\rangle=\sum_{i}b_{i}|i\rangle|B_{i}\rangle$
with normalized $|B_{i}\rangle$, we have
$\sum_{i}|a_{i}-b_{i}|^{2}\leq\|A-B\|^{2}.$
The tensors
$\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\hat{\bm{\lambda}}^{[j+1]}$
and $\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}$ represent two quantum
states on 3 particles, where in both states, the Schmidt basis of the _first_
particle is the standard basis, and the perfect right canonical condition of
Definition 2 (or alternatively, the condition of Equation 4) holds. The
Schmidt coefficients are given by $\\{\hat{\lambda}^{[j]}_{\alpha}\\}$ and
$\\{\tilde{\lambda}^{[j]}_{\alpha}\\}$, respectively. According to the above
fact (Lemma 8)
$\displaystyle{\|\hat{\bm{\lambda}}^{[j]}-\tilde{\bm{\lambda}}^{[j]}\|}\leq{\|\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\hat{\bm{\lambda}}^{[j+1]}-\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}\|}\leq\epsilon\
.$ (18)
Similarly, we know that
${\|\hat{\bm{\lambda}}^{[j-1]}\hat{\bm{\Gamma}}^{[j-1]}\hat{\bm{\lambda}}^{[j]}-\tilde{\bm{\lambda}}^{[j-1]}\tilde{\bm{B}}^{[j-1]}\|}\leq\epsilon$.
Consider now these 3-particle states expanded in terms of the basis vectors
$|\beta\rangle$ of the _third_ particle. Denote these expansions by
$\sum_{\beta}a_{\beta}|v_{\beta}\rangle|\beta\rangle$, with normalized
$|v_{\beta}\rangle$, and $\sum_{\beta}b_{\beta}|w_{\beta}\rangle|\beta\rangle$
with normalized $|w_{\beta}\rangle$, respectively. Then by definition,
$a_{\beta}=\hat{\lambda}^{[j]}_{\beta}$, and
$b_{\beta}=\mu_{\beta}(\tilde{\bm{\lambda}}^{[j-1]},\tilde{\bm{B}}^{[j-1]})$.
We can therefore apply again Lemma 8 and get:
${\|\mu(\tilde{\bm{\lambda}}^{[j-1]},\tilde{\bm{B}}^{[j-1]})-\hat{\bm{\lambda}}^{[j]}\|}\leq\epsilon$.
Together with Eq. (18), we therefore obtain
${\|\mu(\tilde{\bm{\lambda}}^{[j-1]},\tilde{\bm{B}}^{[j-1]})-\tilde{\bm{\lambda}}^{[j]}\|}\leq
2\epsilon$.
Thus far, we have established that $E_{alg}(\Omega)\leq E_{alg}(\phi)$. We
will therefore prove the inequality $E_{alg}(\Omega)-6Jn\epsilon\leq E_{0}$ by
showing that $|E_{alg}(\phi)-E_{0}|\leq 6nJ\epsilon$. Observe that each energy
term in $E_{0}$ depends solely on two overlapping triplets
$\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\hat{\bm{\lambda}}^{[j+1]}\hat{\bm{\Gamma}}^{[j+1]}\hat{\bm{\lambda}}^{[j+2]}$.
The corresponding energy term in $E_{alg}(\phi)$ depends only on
$\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}\tilde{\bm{B}}^{[j+1]}$. We now
bound the distance between these two tensors. We have
$\displaystyle\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}\tilde{\bm{B}}^{[j+1]}-\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\hat{\bm{\lambda}}^{[j+1]}\hat{\bm{\Gamma}}^{[j+1]}\hat{\bm{\lambda}}^{[j+2]}$
$\displaystyle=\big{(}\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}-\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\hat{\bm{\lambda}}^{[j+1]})\tilde{\bm{B}}^{[j+1]}$
$\displaystyle+\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\big{(}\hat{\bm{\lambda}}^{[j+1]}-\tilde{\bm{\lambda}}^{[j+1]}\big{)}\tilde{\bm{B}}^{[j+1]}$
$\displaystyle+\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\big{(}\tilde{\bm{\lambda}}^{[j+1]}\tilde{\bm{B}}^{[j+1]}-\hat{\bm{\lambda}}^{[j+1]}\hat{\bm{\Gamma}}^{[j+1]}\hat{\bm{\lambda}}^{[j+2]}\big{)}\
$
Taking the LHS and RHS sides of the above equation, and using Eq. (8) and Eq.
(9), we have that
$\displaystyle{\|\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}\tilde{\bm{B}}^{[j+1]}-\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\hat{\bm{\lambda}}^{[j+1]}\hat{\bm{\Gamma}}^{[j+1]}\hat{\bm{\lambda}}^{[j+2]}\|}$
$\displaystyle\leq\|\tilde{\bm{\lambda}}^{[j]}\tilde{\bm{B}}^{[j]}-\hat{\bm{\lambda}}^{[j]}\hat{\bm{\Gamma}}^{[j]}\hat{\bm{\lambda}}^{[j+1]}\|$
$\displaystyle+{\|\hat{\bm{\lambda}}^{[j+1]}-\tilde{\bm{\lambda}}^{[j+1]}\|}$
$\displaystyle+{\|\tilde{\bm{\lambda}}^{[j+1]}\tilde{\bm{B}}^{[j+1]}-\hat{\bm{\lambda}}^{[j+1]}\hat{\bm{\Gamma}}^{[j+1]}\hat{\bm{\lambda}}^{[j+2]}\|}\
.$
The first and third term in the above sum can be bounded by $\epsilon$ because
of the condition of the $\epsilon$ net $G$. The norm of the middle term is
bounded in Eq. (18). Therefore the norm of the difference between the tensors
is at most $3\epsilon$. It follows that the difference between the two energy
contributions is at most $6\epsilon{\|H_{j,j+1}\|}\leq 6\epsilon J$.
We illustrate the boundary cases by working through the analysis for the left
end of the chain. We want to bound
$\|\hat{\bm{\Gamma}}^{[1]}\hat{\bm{\lambda}}^{[2]}\hat{\bm{\Gamma}}^{[2]}\hat{\bm{\lambda}}^{[3]}-\tilde{\bm{\Gamma}}^{[1]}\tilde{\bm{\lambda}}^{[2]}\tilde{\bm{B}}^{[2]}\|.$
Note that
$\|\hat{\bm{\Gamma}}^{[1]}(\hat{\bm{\lambda}}^{[2]}\hat{\bm{\Gamma}}^{[2]}\hat{\bm{\lambda}}^{[3]}-\tilde{\bm{\lambda}}^{[2]}\tilde{\bm{B}}^{[2]})\|$
is bounded by $\epsilon$ because of the conditions on the $\epsilon$ net and
Eq. (7). Using Eq. (10), we have that
$\displaystyle{\|(\hat{\bm{\Gamma}}^{[1]}-\tilde{\bm{\Gamma}}^{[1]})\tilde{\bm{\lambda}}^{[2]}\tilde{\bm{B}}^{[2]}\|}\leq\max_{\alpha}{\|\hat{\bm{\Gamma}}^{[1]}_{\alpha}-\tilde{\bm{\Gamma}}^{[1]}_{\alpha}\|}\leq\epsilon\
.$
Hence, the overall bound on the difference is $2\epsilon$. It follows that the
difference between the two energy contributions is it most
$4\epsilon{\|H_{1,2}\|}\leq 4\epsilon J$. A similar argument holds for
$H_{n-1,n}$.
Now we turn to the right inequality in Theorem 5 and show
$|E(\Omega)-E_{alg}(\Omega)|\leq\tfrac{3}{2}JD^{2}n^{2}\epsilon$. We bound the
difference in energy for each term $H_{j-1,j}$. The contribution of this term
to $E_{alg}(\Omega)$ is calculated from
$\bm{\lambda}^{[j-1]}\bm{B}^{[j-1]}\bm{B}^{[j]}$. The true energy, however,
depends on
$\bm{\Gamma}^{[1]}\bm{\lambda}^{[2]}\bm{B}^{[2]}\bm{B}^{[3]}\cdots\bm{B}^{[j]}$
since $|\Omega\rangle$ is only approximately left canonical. We will show that
the error accumulates linearly as we sweep from left to right, summing up to
$3jJD^{2}\epsilon$ for $H_{j-1,j}$. Therefore, the total error is
$|E_{alg}(\Omega)-E(\Omega)|\leq\tfrac{3}{2}JD^{2}n^{2}\epsilon$.
We now provide a more accurate argument. The energy estimate for the term
$H_{j-1,j}$ is calculated from the contraction
$\bm{\lambda}^{[j-1]}\bm{B}^{[j-1]}\bm{B}^{[j]}$. Graphically, this
contribution is given by
The true energy, however, is calculated from the contraction of
$\bm{\lambda}^{[2]}\bm{B}^{[2]}\bm{B}^{[3]}\cdots\bm{B}^{[j]}$. Graphically,
this is given by
(Notice that we have collapsed the $\bm{\Gamma}^{[1]}$ terms because of the
canonical condition 5 – see Fig. 3 (b)).
Had the state $|\Omega\rangle$ been perfectly left canonical, the two would
have been the same. But since it is only approximately canonical from the
left, there is some difference that can be bounded. The analysis is done
iteratively from left to right. We start by writing
In this picture, the tensor $R_{\beta\beta^{\prime}}$ is defined to be off
diagonal (i.e., equal to zero on the diagonal: $R_{\beta\beta}=0$) and for the
$\beta\neq\beta^{\prime}$ terms, it is defined by
$R_{\beta\beta^{\prime}}=\langle{(\bm{\lambda}^{[2]}\bm{B}^{[2]})_{\beta}}|{(\bm{\lambda}^{[2]}\bm{B}^{[2]})_{\beta^{\prime}}}\rangle=\sum_{\alpha,i}|\lambda^{[2]}_{\alpha}|^{2}B^{[2]i}_{\alpha\beta}(B^{[2]i}_{\alpha\beta^{\prime}})^{*}$.
$\bm{\Delta}$ is defined by:
$\displaystyle\Delta_{\beta\beta^{\prime}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}R_{\beta\beta^{\prime}}+\delta_{\beta\beta^{\prime}}(|\lambda^{[3]}_{\beta}|^{2}-|\mu^{[2]}_{\beta}|^{2})\
.$
Using the fact that $(\bm{\lambda}^{[2]},\bm{B}^{[2]})$ is approximately left
canonical (see Eq. (11)), and the stitching conditions of $\bm{\lambda}^{[3]}$
and $\bm{\mu}^{[2]}$ (see Eq. (13)), it is easy to see that for every
$\beta,\beta^{\prime}$,
$\displaystyle|\Delta_{\beta\beta^{\prime}}|\leq 3\epsilon\ .$ (19)
We may therefore write the true energy contribution as the sum of
and
The analysis of the first term is done in the next iteration step. The second
term can be seen as the error introduced by the fact that
$(\bm{\lambda}^{[2]}\bm{B}^{[2]})$ is approximately left canonical. To
estimate its size, notice that it can be viewed as the expectation value of
the operator $\bm{\Delta}\otimes H_{j-1,j}$ (here $\bm{\Delta}$ is viewed as a
matrix), with respect to the MPS that is described by
$|\bm{B}^{[3]}\bm{B}^{[4]}\cdots\bm{B}^{[j]}\rangle$. Using Eq. (19) and the
assumption ${\|H_{j-1,j}\|}\leq J$, it is easy to see that
${\|\bm{\Delta}\otimes H_{j-1,j}\|}\leq 3JD\epsilon$. Here, in both cases, we
used ${\|\cdot\|}$ to denote the _operator_ norm of $\bm{\Delta}\otimes
H_{j-1,j}$, instead of the usual tensor norm; we can do this since the
operator norm is at most as large as the tensor norm, and the tensor norm of
$\bm{\Delta}$ is at most $3D\epsilon$. Moreover, the norm of the MPS
$|\bm{B}^{[3]}\bm{B}^{[4]}\cdots\bm{B}^{[j]}\rangle$ is exactly $\sqrt{D}$ (it
would have been exactly 1 had there been a $\bm{\lambda}^{[3]}$ term before
$B^{[3]}$), and therefore the amplitude of second term is upper bounded by
$3JD^{2}\epsilon$.
Carrying the same analysis all way to $(\bm{\lambda}^{[j-2]},\bm{B}^{[j-2]})$,
we end up with a term that is identical to the energy estimation of the
algorithm, plus some error term whose amplitude is at most $3jJD^{2}\epsilon$.
Therefore, by simple algebra, we have that for the total system,
$\displaystyle|E_{alg}-E(\Omega)|\leq\tfrac{3}{2}JD^{2}n^{2}\epsilon\ .$ (20)
For a target error $\delta$, we select
$\epsilon\leq\frac{\delta}{2JD^{2}n^{2}}\ .$ Using the bound from Eq. (12), we
get that the size of the net for the interior particles is
$\displaystyle
N=\mathcal{O}\left(\left(\frac{144JdD^{3}n^{2}}{\delta}\right)^{D+2dD^{2}}\right).$
(21)
Note that in using Lemma 3, we required that $\epsilon\leq 1/\sqrt{D}$. It is
reasonable to expect that $\delta/Jn<1$ ( meaning that the desired error is at
most the maximum energy in the system) which implies that this condition is
met. The algorithm has $n$ iterations in which $\mathcal{O}(N^{2})$ possible
extensions for the MPS are considered. For each such possibility, we perform a
contraction of tensors $(\bm{\lambda},\bm{B},\bm{B}^{\prime})$ in order to
evaluate the energy of a particular term. This contraction takes time
$\mathcal{O}(D^{3}d^{2})$. Thus the total running time is
$\mathcal{O}(nN^{2}D^{3}d^{2})$.
## V Commuting Hamiltonian in 1D
We now prove Theorem 2. Let us first notice that Theorem 1 immediately implies
the first claim in Theorem 2, namely that approximating the ground state and
ground energy of a commuting Hamiltonian in 1D to within polynomially good
accuracy can be done efficiently. This follows from the well known fact that
the ground state of a commuting Hamiltonian in 1D can be described by an MPS
of constant bond dimension. We can therefore apply Theorem 1 to the problem,
and hence approximate both the ground state and ground energy efficiently.
For completeness, here is a sketch of a proof of this fact: assume we have a
$2$-local commuting Hamiltonian in 1D. If the Hamiltonian is $k$-local for
$k>2$, just combine adjacent particles together. To see that there is a ground
state which is described by an MPS of constant bond dimension, notice that for
any commuting Hamiltonian, there is a ground state $|\psi\rangle$ which is an
eigenvector of each of the terms in the Hamiltonian, with some well defined
eigenvalue for each term. For each term, consider the projection onto the
eigenspace corresponding to that eigenvalue. For any state with non-zero
projection on the ground state, applying these projections (no matter the
order) would result in a ground state. Since there is always a computational
basis state $|w\rangle$ that has a non-zero projection on the ground state, we
can express a ground state as the projection of all these local terms applied
to $|w\rangle$. We first apply the projections which interact the pairs of
particles $(1,2)$, $(3,4)$, etc; we then apply the projections that interact
the pairs of particles $(2,3)$, $(4,5)$, etc. This sequence of operations can
be viewed as a tensor network of depth $2$. We can thus represent the ground
state as the contraction of a tensor network of depth $2$. It can be easily
seen that such a state must have a constant Schmidt rank along any cut between
the left and right sides; to move to an MPS of a constant bond dimension, use
Vidal’s result vidal1 .
Let us now provide the proof of the improvement to an exact algorithm, for the
case that the Hamiltonian has a polynomial spectral gap. In other words, we
are promised that the ground energy is separated from the rest of the
eigenvalues of the Hamiltonian by a gap $\Delta\geq 1/n^{c}$ for some constant
$c$. Notice that we don’t assume a unique ground state.
The first step of the proof would be to use Theorem 1 to find an MPS
$|\Omega\rangle$ of constant bond dimension such that
$\langle\Omega|H|\Omega\rangle\leq E_{0}+\Delta/3$. From the discussion above,
it is clear that this can be done in polynomial time. Next, we would like to
project this MPS sequentially on some chosen eigenspaces of the Hamiltonians
along the chain. As we are in a commuting system, this would result in a
common eigenvector of all Hamiltonians, and therefore an eigenvector of $H$
itself. If we manage to do this without increasing the energy above
$E_{0}+\Delta$, then by the existence of the gap, we are promised to have
reached a ground state.
To do this, we rely on the following lemma:
###### Lemma 7
Let $H=\sum_{i}H_{i}$ be a commuting local Hamiltonian system with ground
energy $E_{0}$, and let $|\psi\rangle$ be a state such that
$\langle\psi|H|\psi\rangle=E_{0}+h$. Consider one term $H_{i}$ in $H$ with $k$
eigenvalues and projections $P_{1},\ldots,P_{k}$ into the corresponding
eigenspaces. For every $j=1,\ldots,k$, let $|\psi_{j}\rangle$ be the
_normalization_ of $P_{j}|\psi\rangle$, and let
$c_{j}=\langle\psi|P_{j}|\psi\rangle$. Then for any $n>2$ there is always a
$j$ such that $c_{j}\geq\frac{1}{kn^{2}}$ and
$\langle\psi_{j}|H|\psi_{j}\rangle\leq E_{0}+(1+\frac{1}{n})h$.
Proof: As the $\\{H_{i}\\}$ terms are commuting, it follows that
$\displaystyle\langle\psi|H|\psi\rangle$
$\displaystyle=\langle\psi|P_{1}HP_{1}|\psi\rangle+\langle\psi|P_{2}HP_{2}|\psi\rangle$
$\displaystyle\ \ +\ldots+\langle\psi|P_{k}HP_{k}|\psi\rangle$
$\displaystyle=c_{1}\langle\psi_{1}|H|\psi_{1}\rangle+c_{2}\langle\psi_{2}|H|\psi_{2}\rangle$
$\displaystyle\ \ +\ldots+c_{k}\langle\psi_{k}|H|\psi_{k}\rangle\ ,$
with $\sum_{j=1}^{k}c_{j}=1$. Assume, by contradiction, that for every $j$,
either $c_{j}<\frac{1}{kn^{2}}$ or
$\langle\psi_{j}|H|\psi_{j}\rangle>E_{0}+(1+\frac{1}{n})h$. Then partition the
$k$ eigenspaces into two subsets: subset $A$ in which the first condition
holds, and subset $B$ in which the second condition holds. Then
$\displaystyle E_{0}+h$ $\displaystyle=\langle\psi|H|\psi\rangle$
$\displaystyle=\sum_{A}c_{j}\langle\psi_{j}|H|\psi_{j}\rangle+\sum_{B}c_{j}\langle\psi_{j}|H|\psi_{j}\rangle$
$\displaystyle\geq
E_{0}\sum_{A}c_{j}+\left(E_{0}+(1+\frac{1}{n})h\right)\sum_{B}c_{j}$
$\displaystyle=E_{0}+(1+\frac{1}{n})h\sum_{B}c_{j}\ ,$
using $\sum_{j}c_{j}=1$. Since
$\sum_{A}c_{j}\leq\frac{k}{kn^{2}}=\frac{1}{n^{2}}$, we have that
$\sum_{B}c_{j}\geq 1-\frac{1}{n^{2}}$. Plugging this into the above equality
implies $h>h(1+\frac{1}{n})(1-\frac{1}{n^{2}})$ which is a contradiction for
$n>2$.
We now apply the lemma sequentially to project the approximate state
$|\Omega\rangle$ on the relevant local eigenspaces. We start with $H_{1,2}$,
where we use $h=\Delta/3$ in the lemma. The lemma promises the existence of a
subspace indexed $j$ (out of $k$ possible $j$s) which, if we project
$|\Omega\rangle$ onto that subspace, the projection will not have too large
energy. We denote $c_{j}$ and $P_{j}$ by $c_{12}$ and $P_{12}$ respectively
(We will shortly explain how all calculations required for finding the
promised $j$ can be done efficiently). We proceed to find $c_{23}$ and
$P_{23}$ for the next term $H_{2,3}$, using the newly projected state, and so
on up to $H_{n,n-1}$. After applying the $n-1$ projections, using the lemma
$n-1$ times, we arrive to a state $|\psi\rangle$ given by
$\displaystyle|\psi\rangle=\frac{1}{\sqrt{c_{12}c_{23}\cdots
c_{n-1,n}}}P_{12}P_{23}\cdots P_{n-1,n}|\Omega\rangle\ ,$
which satisfies
$\displaystyle\langle\psi|H|\psi\rangle\leq
E_{0}+\left(1+\frac{1}{n}\right)^{n-1}\frac{\Delta}{3}\leq
E_{0}+\frac{e\Delta}{3}\ .$
Using the assumption of the gap and the fact that $|\psi\rangle$ is an
eigenvector of $H$, it must be that $|\psi\rangle$ is a ground state and
$\langle\psi|H|\psi\rangle=E_{0}$.
We now argue why finding the $j$ whose existence is promised by the lemma can
be done efficiently. Consider for example the term $H_{m,m+1}$. To find the
relevant $j$ we have to compute, for the current state $|\psi\rangle$, both
the norms squared $c_{j}=\langle\psi|P_{j}|\psi\rangle$ as well as the
expectation values
$\langle\psi_{j}|H|\psi_{j}\rangle=\frac{1}{c_{j}}\langle\psi|P_{j}HP_{j}|\psi\rangle$,
for all eigenspaces $P_{j}$ of $H_{m,m+1}$. Note first that we are handling
here real numbers; the projections $P_{j}$ on the eigenspaces of $H_{m,m+1}$
may require infinite precision to describe exactly in binary (or any other)
representation. We truncate the entries in the projections to exponentially
good precision, using polynomially many bits, so that all the calculations can
be performed efficiently. This introduces an exponentially small error.
The expressions we are interested in calculating are all of the form
$\displaystyle\langle\Omega|P_{12}\cdots P_{m,m-1}\cdot P_{j}OP_{j}\cdot
P_{m,m-1}\cdots P_{12}|\Omega\rangle\ ,$ (22)
where $O$ can be either a local Hamiltonian $H_{i,i+1}$ or the identity, and
the $P_{i,i+1}$ are projections on eigenspaces of the local terms. Recalling
that $|\Omega\rangle$ is a constant-bond MPS, and using the fact that the
projections commute between themselves, we can write Eq. (22) as a constant
depth-tensor network. This is done by partitioning the projections into two
layers: in one layer the projections that work on the sites
$(1,2),(3,4),(5,6),\ldots$ and on the other, the projections that act on the
sites $(2,3),(4,5),(6,7),\ldots$. The resultant tensor-network is shown in
Fig. 4. One dimensional tensor-networks with constant depth can be efficiently
calculated on a classical computer because their bubble width is constant when
swallowed from left to right aharonov2006quantum .222This can also be seen by
analyzing the tree-width of that tensor-network and using the analysis of
LABEL:markov2008simulating
Thus, all calculations (under our assumptions of polynomially many bits of
precision of the $P_{j}$’s) can be performed efficiently. The resulting state
is given by a tensor network of constant depth (namely the original
$|\Omega\rangle$ on which the chosen projections are applied.) As before, this
can be modified to a MPS of constant bond dimension using Vidal’s result
vidal1 , concluding the proof.
Figure 4: An illustration of how the expression in Eq. (22) is given as a
tensor-network with a constant number of horizontal layers. Specifically, the
figure describes the tensor-network of
$\langle\Omega|P_{12}P_{23}\cdots\,H_{j,j+1}\,\cdots
P_{23}P_{12}|\Omega\rangle$
### V.1 A Proof for the Commuting Hamiltonians case, based on Ref. bravyi
First we describe the alternate algorithm assuming we have the ability to
perform arithmetic operations with infinite precision and then discuss the
consequences of limited precision. LABEL:bravyi prove certain properties about
the ground states of $2$-local commuting Hamiltonians in which the interaction
graph is a general graph. We express those properties for the special case of
interest here in which the graph is a line. Let ${\cal{H}}_{j}$ be the Hilbert
space of particle $j$. It is shown in LABEL:bravyi that when the terms of the
Hamiltonian commute, the Hilbert space of each particle can be expressed as a
direct sum, ${\cal{H}}_{j}=\oplus_{\alpha_{j}}{\cal{H}}_{j}^{(\alpha_{j})}$,
such that each ${\cal{H}}_{j}^{(\alpha_{j})}$ can then be expressed as a
tensor product of three spaces
$\displaystyle{\cal{H}}_{j}^{(\alpha_{j})}={\cal{H}}_{L,j}^{(\alpha_{j})}\otimes{\cal{H}}_{C,j}^{(\alpha_{j})}\otimes{\cal{H}}_{R,j}^{(\alpha_{j})}\
.$
This structure has the property that $H_{j,j+1}$ leaves the subspaces
${\cal{H}}_{j}^{(\alpha_{j})}\otimes{\cal{H}}_{j+1}^{(\alpha_{j+1})}$
invariant, and moreover, when restricted to such a subspace, $H_{j,j+1}$ acts
non-trivially only on
${\cal{H}}_{R,j}^{(\alpha_{j})}\otimes{\cal{H}}_{L,j+1}^{(\alpha_{j+1})}$ (the
right part of ${\cal{H}}_{j}^{(\alpha_{j})}$ and the left part of
${\cal{H}}_{j+1}^{(\alpha_{j+1})}$). Consequently, there exists a ground state
which resides in some subspace
${\cal{H}}^{(\alpha)}=\otimes_{j}{\cal{H}}_{j}^{(\alpha_{j})}$, for some
choice of $\alpha_{1},\ldots,\alpha_{n}$. Moreover, within the subspace
${\cal{H}}^{(\alpha)}$ the state can be written as a tensor product of $2$
particle states living in the spaces of the form
${\cal{H}}_{R,j}^{(\alpha_{j})}\otimes{\cal{H}}_{L,j+1}^{(\alpha_{j+1})}$,
tensored with some arbitrary single particle states living in the
${\cal{H}}_{C,j}^{(\alpha_{j})}$ spaces.
If the algorithm knows the correct choice of indices
$\alpha_{1},\ldots,\alpha_{n}$, it can find such a ground state efficiently,
as follows. Note that the descriptions of both the spaces
${\cal{H}}_{j}^{(\alpha_{j})}$ and their divisions
${\cal{H}}_{j}^{(\alpha_{j})}={\cal{H}}_{L,j}^{(\alpha_{j})}\otimes{\cal{H}}_{C,j}^{(\alpha_{j})}\otimes{\cal{H}}_{R,j}^{(\alpha_{j})}$
are derived from local properties of ${\cal{H}}_{j}$ imposed by the two local
Hamiltonians $H_{j-1,j}$ and $H_{j,j+1}$. The subdivision of ${\cal{H}}_{j}$
in this way can be expressed as a solution to a set of quadratic homogeneous
constraints. Since the dimension of ${\cal{H}}_{j}$ and hence the number of
variables is constant, it can be efficiently computed. If the algorithm knows
the $\alpha_{j}$’s, it therefore knows the description of the subspaces
${\cal{H}}_{L,j}^{(\alpha_{j})}\otimes{\cal{H}}_{C,j}^{(\alpha_{j})}\otimes{\cal{H}}_{R,j}^{(\alpha_{j})}$,
and the restriction of the $H_{j,j+1}$ to those spaces; it therefore just
needs to find a ground state of linearly many 2-particle Hamiltonians, which
is an easy task. It is therefore enough for the algorithm to find the correct
$\alpha_{1},\ldots,\alpha_{n}$ indices.
We will do this using dynamic programming. The critical point in using dynamic
programming here is that the energy contribution of $H_{\ell,\ell+1}$ depends
only on the choice of $\alpha_{\ell}$ and $\alpha_{\ell+1}$, so the choice of
$\alpha_{k}$ for $k\leq j-1$ does not affect the energy of the
$H_{\ell,\ell+1}$ terms for any $\ell\geq j$. Using this observation, the
algorithm proceeds from left to right as follows. For the first term
$H_{1,2}$, the algorithm finds the division into a direct sum of subspaces for
particles $1$ and $2$. The algorithm keeps an optimal state (choice of
$\alpha_{1}$) and energy for each possible $\alpha_{2}$.
Then, in a general step, we assume at particle $i$ we have the following
information for each index $\alpha_{i}$: a list of indices
$\alpha_{1},\ldots,\alpha_{i-1}$ such that the ground energy of the
Hamiltonian of particles $1,\ldots,i$ restricted to the subspaces
${\cal{H}}_{1}^{(\alpha_{1})}\otimes\ldots\otimes{\cal{H}}_{i}^{(\alpha_{i})}$
is minimal. To continue to the next particle, we first compute the division
into subspaces for particle $i+1$, indexed by $\alpha_{i+1}$, and optimize for
each subspace in turn. For each subspace, we consider all items in the
previous list; for each item, we have a list of subspaces, one for each
particle. We compute the minimal energy for each such restriction, including
now the $H_{i,i+1}$ term in the calculation of the energy, restricted
according to subspaces $\alpha_{i+1}$ and $\alpha_{i}$, the last choice coming
from the list. We pick the tail of the subspace of the $i+1$ particle to be
the one which minimizes the terms up to that point.
Notice that in each step the dynamic program compares partial energies
emerging from restricting the state to a different sector in the Hilbert
space. These energies can be computed efficiently with polynomially many bits,
namely up to exponentially good precision. Thus, this second algorithm
achieves exact results for a somewhat larger set of Hamiltonians than our
first algorithm, namely those for which the partial energies will not be
confused if the computations are done with exponentially good precision.
Note that even with this extremely good resolution, it might be the case that
the ground energy is confused with a slightly excited energy which is, say,
_doubly_ exponentially close. We do not know of a good condition which would
rule out the possibility of such very close energies, except for some very
trivial assumptions such as requiring that all eigenvalues are integer
numbers. For example, even if we require that the different entries in the
terms in the Hamiltonian are all rationals smaller than $1$ with denominator
upper bounded by a constant, it is still not known how to rule out the
possibility that two eigenvalues of the overall Hamiltonian are doubly
exponentially close. This issue touches upon an open question in number theory
related to sums of algebraic numbers – see the open problem described in
LABEL:p33, which can be traced back to LABEL:or (if not earlier), and also
LABEL:qian and references therein.
## VI Acknowledgment
Dorit Aharonov is partially supported by ISF Grants 039-7549 and 039-8066, ARO
Grant 030-7799, and SCALA Grant 030-7811. Sandy Irani Partially supported by
NSF Grant CCF-0916181. Itai Arad acknowledges support by the ERC Starting
grant of Julia Kempe (PI).
The main progress on the results reported on in this paper was made while the
three authors were visiting the Erwin Schrödinger International Institute for
mathematical Physics (ESI) in Vienna, Austria.
*
## Appendix A Proofs of lemmas
_Proof of Lemma 3:_
Let $\delta=\nu/72b$. We will occasionally use the assumption that $\delta\leq
1/72b$.
First we create a set $R(\delta)$ of real numbers in the interval $[0,1]$ such
that for any real number in the range $[0,1]$, it is within $\delta$ of some
element in $R(\delta)$. $R(\delta)$ will have $\lceil 1/2\delta\rceil$
elements. To create $R(\delta)$, we add $(2j+1)\delta$ for each integer j in
the range from $0$ through $\lceil 1/2\delta\rceil-2$. Note that the largest
point in $R(\delta)$ so far is in the range $[1-3\delta,1-\delta)$. Then we
add $1-\delta$ to $R(\delta)$.
Then using $R(\delta)$, we create a set $C(\delta)$ which is a set of complex
scalars which form a net over all complex scalars with norm at most $1$.
Include $xe^{i2\pi y}$, for every $x,y\in R(\delta)$. There are $\lceil
1/2\delta\rceil^{2}\leq(1/\delta)^{2}$ points in $C(\delta)$. For any complex
number $c$ if norm at most $1$, there is a number $c^{\prime}$ in $C(\delta)$
such that $|c-c^{\prime}|\leq 2\delta$.
To generate $S_{a,b}$, consider first the set $S_{1}$ of of all possible
$a\times b$ matrices with entries from $C(\delta)$. This set contains
$|C(\delta)|^{ab}$ matrices. In the case where $a=1$ and we only want entries
with real, non-negative coefficients, we use $R(\delta)$ for the entries
instead of $C(\delta)$ and the set contains $|R(\delta)|^{b}$ matrices (in
fact, vectors). Then:
1. 1.
Remove any matrix from $S_{1}$ which has a row whose norm is greater than
$1+\sqrt{b}2\delta$ or less than $1-\sqrt{b}2\delta$, to get $S_{2}$.
2. 2.
Renormalize each row in every matrix in $S_{2}$ to get $S_{3}$.
3. 3.
Remove any matrix from $S_{3}$ which has any two rows whose inner product is
more than $9\sqrt{b}\delta$.
4. 4.
For any matrix in $S_{3}$, Apply the Gram-Schmidt procedure to the rows to
form an orthonormal set.
We claim that the final set is the desired $S_{a,b}$. Note that the number of
matrices is $O((1/\delta)^{2ab})=O((72b/\nu)^{2ab})$, and the running time to
produce the set is $O(a^{2}b(1/\delta)^{2ab})=O(a^{2}b(72b/\nu)^{2ab})$ as
required. What remains to show is that if $A$ is any $a\times b$ matrix whose
rows form an ortho-normal set then we can find a matrix $B$ in $S_{a,b}$ which
is close to it.
Let $W$ be an $a\times b$ matrix. We will denote it’s $i^{th}$ row by $W_{i}$.
Define the distance between two matrices $d(W,W^{\prime})$ to be
$\max_{i}\|W_{i}-W^{\prime}_{i}\|$. Let $X$ be the matrix obtained by rounding
every entry in $A$ to the nearest complex number in $C(\delta)$. Let $Y$ be
the matrix obtained after the rows of $X$ are normalized and let $Z$ be the
matrix obtained after the rows of $Y$ are transformed into an ortho-normal set
via the Gram-Schmidt procedure. We need to prove that $d(A,Z)\leq\nu$, and to
show that $Z\in S_{a,b}$, which would imply together that we can choose $B$ in
the lemma to be equal to $Z$.
We will now prove both of the above claims. For the second part we need to
prove that $X$ survives step $1$ and $Y$ survives step $3$.
$X$ survives step $1$: Since each entry in $A-X$ has magnitude at most
$2\delta$, we know that $d(A,X)\leq\sqrt{b}2\delta$. In order to bound the
norm of $X_{i}$, observe that
$\sqrt{b}2\delta\geq{\|A_{i}-X_{i}\|}\geq|{\|A_{i}\|}-{\|X_{i}\|}|.$
Since ${\|A_{i}\|}=1$, it follows that ${\|X_{i}\|}$ lies in the range from
$1-\sqrt{b}2\delta$ to $1+\sqrt{b}2\delta$ and it will survive Step $1$. We
have:
$\displaystyle d(X,Y)$
$\displaystyle\leq\max_{i}\|A^{i}-\frac{1}{1-\sqrt{b}2\delta}A^{i}\|$
$\displaystyle=\frac{\sqrt{b}2\delta}{1-\sqrt{b}2\delta}\leq\sqrt{b}2\delta(36/35)\
.$
The latter inequality uses the assumption that $\delta\leq 1/72\sqrt{b}$.
Using the triangle inequality for our distance $d(\cdot)$, we have that for
any $i$ ${\|A_{i}-Y_{i}\|}\leq(4+\frac{2}{35})\sqrt{b}\delta$.
$Y$ survives step $3$: Now we need to bound the inner product of any two rows
of $Y$ in order to establish that it is not removed in Step $3$:
$\displaystyle|\langle{Y_{i}}|{Y_{j}}\rangle|$
$\displaystyle=|\langle{A_{i}+(Y_{i}-A_{i})}|{A_{j}+(Y_{j}-A_{j})}\rangle|$
$\displaystyle\leq|\langle{A_{i}}|{A_{j}}\rangle|+|\langle{Y_{i}-A_{i}}|{Y_{j}-A_{j}}\rangle|$
$\displaystyle\ \
+|\langle{Y_{i}-A_{i}}|{A_{j}}\rangle|+|\langle{A_{i}}|{Y_{j}-A_{j}}\rangle|$
$\displaystyle\leq{\|Y_{i}-A_{i}\|}{\|Y_{j}-A_{j}\|}+{\|Y_{i}-A_{i}\|}{\|A_{j}\|}$
$\displaystyle\ \ +{\|A_{i}\|}{\|Y_{j}-A_{j}\|}$
$\displaystyle\leq\sqrt{b}\delta\left[\left(4+\frac{2}{35}\right)^{2}\sqrt{b}\delta+2\left(4+\frac{2}{35}\right)\right]$
$\displaystyle\leq 9\sqrt{b}\delta\ .$
The second inequality uses the Chauchy-Schwartz inequality. The last
inequality uses the fact that $\sqrt{b}\delta\leq 1/72$.
Bounding the distance $d(A,Z)$: Finally, we need to consider how much the
matrix shifts as a result of the Gram-Schmidt procedure, to bound $d(Y,Z)$.
Let $\mu=9\sqrt{b}\delta=9\nu/72\sqrt{b}$. Since $a\leq b$, by assumption in
the lemma, we know that $\mu\leq 9\nu/72\sqrt{a}$. We use this latter bound in
the next part of the proof since we are bounding quantities by a function of
$a$ instead of $b$. Since we assume that $\nu\leq 1/\sqrt{a}$, we can assume
that $a\mu\leq 9/72$. Recall that the Gram-Schmidt procedure starts with
$Z_{1}=Y_{1}$. Then each $Z_{i}$ is determined by first creating an
unnormalized state $\tilde{Z}_{i}$:
$\tilde{Z}_{i}=Y_{i}-\sum_{j=1}^{i-1}\langle{Z_{j}}|{Y_{i}}\rangle Z_{j}.$
Then $\tilde{Z}_{i}$ is normalized to $1$. We will prove the following two
properties by induction in $i$,
1. 1.
$|\langle{Z_{i}}|{Y_{j}}\rangle|\leq 2\mu$ for all $j$ such that $j>i$
2. 2.
$1-2\sqrt{a}\mu\leq{\|\tilde{Z}_{i}\|}\leq 1+2\sqrt{a}\mu$.
$\tilde{Z}_{1}$ is not defined, but we can take it to be $Z_{1}$. The two
properties clearly hold for $Z_{1}$. Now by induction
$\displaystyle{\|\tilde{Z}_{i}\|}=$
$\displaystyle{\|Y_{i}-\sum_{j=1}^{i-1}\langle{Z_{j}}|{Y_{i}}\rangle Z_{j}\|}$
$\displaystyle\leq$
$\displaystyle{\|Y_{i}\|}+{\|\sum_{j=1}^{i-1}\langle{Z_{j}}|{Y_{i}}\rangle
Z_{j}\|}$ $\displaystyle=$ $\displaystyle
1+\left(\sum_{j=1}^{i-1}\langle{Z_{j}}|{Y_{i}}\rangle\langle{Y_{i}}|{Z_{j}}\rangle\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle 1+2\sqrt{a}\mu$
A similar argument can be used to show that
$1-2\sqrt{a}\mu\leq{\|\tilde{Z_{i}}\|}$. Next we establish Property $1$:
$\displaystyle|\langle{Y_{k}}|{Z_{i}}\rangle|=$
$\displaystyle\frac{1}{{\|\tilde{Z}_{i}\|}}\left|\langle{Y_{k}}|{Y_{i}}\rangle-\sum_{j=1}^{i}\langle{Z_{j}}|{Y_{i}}\rangle\langle{Y_{k}}|{Z_{j}}\rangle\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{1-2\sqrt{a}\mu}\left[\mu+\sum_{j=1}^{i-1}4\mu^{2}\right]$
$\displaystyle\leq$ $\displaystyle\frac{\mu(1+4a\mu)}{1-2\sqrt{a}\mu}\leq
2\mu$
The first inequality follows from the inductive hypothesis. The last
inequality make use of the fact that $a\mu\leq 9/72$. Finally to bound
${\|Y_{i}-Z_{i}\|}$, we have
$\displaystyle|{\|Y_{i}-Z_{i}\|}|\leq$
$\displaystyle\left(1-\frac{1}{{\|\tilde{Z}_{i}\|}}\right){\|Y_{i}\|}+\frac{1}{{\|\tilde{Z}_{i}\|}}{\|\sum_{j=1}^{i-1}\langle{Z_{j}}|{Y_{i}}\rangle
Z_{j}\|}$ $\displaystyle\leq$
$\displaystyle\frac{2\sqrt{a}\mu{\|Y_{i}\|}}{1-2\sqrt{a}\mu}+\frac{1}{1-2\sqrt{a}\mu}\left(\sum_{j=1}^{i-1}|\langle{Z_{j}}|{Y_{i}}\rangle|^{2}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle\frac{4\sqrt{a}\mu}{1-2\sqrt{a}\mu}\leq
6\sqrt{a}\mu$
The last inequality uses again the fact that $\sqrt{a}\mu\leq 9/72$. The total
distance between $A$ and $Z$ is at most $5\sqrt{b}\delta+6\sqrt{a}\mu$.
Plugging in $\mu=9\sqrt{b}\delta$ and using the fact that $a\leq b$, we get an
upper bound of $59b\delta\leq\nu$ on the distance of $A$ to $Z$, using the
definition of $\delta$.
###### Lemma 8
Let $|A\rangle$, $|B\rangle$ be two two-particles states that, and expand them
in the standard basis of the first particle:
$\displaystyle|A\rangle$ $\displaystyle=\sum_{i}a_{i}|i\rangle|A_{i}\rangle\
,$ $\displaystyle|B\rangle$
$\displaystyle=\sum_{i}b_{i}|i\rangle|B_{i}\rangle\ ,$
such that $|A_{i}\rangle$ are normalized but not-necessarily orthogonal to
themselves and similarly the $|B_{i}\rangle$. Then
$\displaystyle{\|a-b\|}=\left(\sum_{i}|a_{i}-b_{i}|^{2}\right)^{1/2}\leq{\|A-B\|}\
.$ (23)
Proof:
$\displaystyle{\|a-b\|}^{2}$ $\displaystyle=\sum_{i}|a_{i}-b_{i}|^{2}$
$\displaystyle\leq\sum_{i}{\|a_{i}|A_{i}\rangle-b_{i}|B_{i}\rangle\|}^{2}$
$\displaystyle={\|\sum_{i}|i\rangle(a_{i}|A_{i}\rangle-
b_{i}|B_{i}\rangle)\|}^{2}$ $\displaystyle={\||A\rangle-|B\rangle\|}^{2}\ .$
## References
* (1) S. R. White, Phys. Rev. Lett. 69, 2863 (1992)
* (2) S. R. White, Phys. Rev. B 48, 10345 (1993)
* (3) S. Östlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (Nov 1995)
* (4) S. Rommer and S. Östlund, Phys. Rev. B 55, 2164 (Jan 1997)
* (5) I. Peschel, X. Wang, M. Kaulke, and K. H. (Eds), _Density-Matrix Renormalization – A New Numerical Method in Physics_ , Lecture Notes in Physics, Vol. 528 (Springer-Verlag, 1998) ISBN 3-540-66129-8
* (6) U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005)
* (7) A. Kitaev, A. Shen, and M. Vyalyi, _Classical and Quantum Computation_ (AMS, Providence, RI, 2002)
* (8) S. Bravyi and M. Vyalyi, Arxiv preprint quant-ph/0308021(2003)
* (9) D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, SIAM JOURNAL OF COMPUTING 37, 166 (2007), http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0405098
* (10) D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, in _FOCS_ (2007) pp. 373–383, arXiv:0705.4077v2
* (11) D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, in _Proceedings of the 41st annual ACM symposium on Symposium on theory of computing_ (2009) pp. 417–426
* (12) C. R. Laumann, R. Moessner, A. Scardicchio, and S. L. Sondhi, Quant. Inf. and Comp. 10, 0001 (2010), 0903.1904
* (13) N. Schuch, I. Cirac, and F. Verstraete, Phys. Rev. Lett. 100, 250501 (2008), arXiv:0802.3351
* (14) J. Eisert, Physical Review Letters 97, 260501 (Dec. 2006), arXiv:quant-ph/0609051
* (15) M. B. Hastings, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007), arXiv:0705.2024
* (16) N. Schuch and I. Cirac, Arxiv preprint(2009), arXiv:0910.4264
* (17) I. Markov and Y. Shi, SIAM Journal on Computing 38, 963 (2008), arXiv: quant-ph/0511069
* (18) D. Aharonov, Z. Landau, and J. Makowsky, Arxiv preprint quant-ph/0611156(2006)
* (19) I. Arad and Z. Landau, SIAM Journal on Computing 39, 3089 (2010), arXiv:0805.0040
* (20) G. Vidal, Phys. Rev. Lett. 91, 147902 (2003)
* (21) Recall that $\mu_{\beta}$ corresponds to a Schmidt coefficient in a Schmidt decomposition that coincides with the standard basis.
* (22) This can also be seen by analyzing the tree-width of that tensor-network and using the analysis of Ref. markov2008simulating
* (23) J. D. Demaine, J. S. B. Mitchell, and J. O’Rourke, “The Open Problems Project, Problem 33,” http://maven.smith.edu/~orourke/TOPP/Welcome.html
* (24) J. O’Rourke, Amer. Math. Monthly 88, 769 (1981)
* (25) J. Qian and C. Wang, Information Processing Letters 100, 194 (2006)
|
arxiv-papers
| 2009-10-27T19:58:27 |
2024-09-04T02:49:06.087913
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dorit Aharonov, Itai Arad, Sandy Irani",
"submitter": "Itai Arad",
"url": "https://arxiv.org/abs/0910.5055"
}
|
0910.5208
|
# Optimal Decoherence Control in non-Markovian Open, Dissipative Quantum
Systems
Wei Cui Zairong Xi zrxi@iss.ac.cn Yu Pan Key Laboratory of Systems and
Control, Institute of Systems Science, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100080, P. R. China
###### Abstract
We investigate the optimal control problem for non-Markovian open, dissipative
quantum system. Optimal control using Pontryagin maximum principle is
specifically derived. The influences of Ohmic reservoir with Lorentz-Drude
regularization are numerically studied in a two-level system under the
following three conditions: $\omega_{0}\ll\omega_{c}$,
$\omega_{0}\approx\omega_{c}$ or $\omega_{0}\gg\omega_{c}$, where $\omega_{0}$
is the characteristic frequency of the quantum system of interest, and
$\omega_{c}$ the cut-off frequency of Ohmic reservoir. The optimal control
process shows its remarkable influences on the decoherence dynamics. The
temperature is a key factor in the decoherence dynamics. We analyze the
optimal decoherence control in high temperature, intermediate temperature, and
low temperature reservoirs respectively. It implies that designing some
engineered reservoirs with the controlled coupling and state of the
environment can slow down the decoherence rate and delay the decoherence time.
Moreover, we compare the non-Markovian optimal decoherence control with the
Markovian one and find that with non-Markovian the engineered artificial
reservoirs are better than with the Markovian approximation in controlling the
open, dissipative quantum system’s decoherence.
###### pacs:
03.65Yz, 03.67.Lx, 03.67.Pp
††preprint: APS/123-QED
## I Introduction
The theory of open quantum systems deals with the systems that interact with
their surrounding environments H.P.Breuer ; zhang ; slotine ; zanzrdi ; chuang
; Alicki ; p.zanardi . Such systems are of great interest, and these open
quantum systems have been extensively studied since the origin of quantum
theory J.Von. Neumann . Despite of the noticeable progresses in the theory,
many fundamental difficulties still remain. One of the problem is decoherence
(or loss of coherence) due to the interactions between system and environment.
Recently, it received intense considerations in quantum information and
quantum computation, where decoherence is regarded as a bottleneck to the
construction of quantum information processor Nielsen ; Mensky ; zhang . The
persistence of quantum coherence is relied on in quantum computer, quantum
cryptography and quantum teleportation. And it is also fundamental in
understanding the quantum world for the interpretation that the emergence of
the classical world from the quantum world can be seen as a decoherence
process due to the interaction between system and environment.
Various methods have been proposed to reduce this unexpected effect, such as
the quantum error-correction code shor ; slotine , error-avoiding code zanzrdi
; chuang , minimal decoherence model Alicki , Bang-Bang techniques p.zanardi ,
quantum Zeno effect (QZE) viola and decoherence-free subspaces (DFS) elattari
. Unfortunately, all of these schemes cannot suppress the unexpect effect
successfully for accessorial conditions are needed. Altafini altafini pointed
out that the irreversible decohering dynamics is uncontrollable under coherent
control. Optimal control technique, which has been successfully studied in
chemical systems rabitz ; levis ; rabitz2 and classical systems krotov , has
been exploited to control the quantum decoherence Jirari ; Sugny ; zhang ,
where an optimal control law was designed to effectively suppress decoherence
effects in Markovian open quantum systems, dynamic coupling in the spin-boson
model, and time optimal control respectively. In this paper, we consider the
optimal decoherence control problem in non-Markovian quantum open system.
Markovian approximation is used under the assumption that the correlation time
between the systems and environments is infinitely short H.P.Breuer ;
C.W.Gardiner ; zhang . For neglecting the memory effect, the Lindblad master
equation has been built. However, in some cases, such as quantum Brownian
motion(QBM) zurek and a two-level atom interacting with a thermal reservoir
with Lorentzian spectral density Garraway , an exactly analytic description of
the open quantum system dynamic is needed. Especially in high-speed
communication the characteristic time scales become comparable with the
reservoir correlation time, and in solid state devices memory effects are
typically non negligible. So it is necessary to extensively study the non-
Markovian master equation. We briefly compare the non-Markovian dynamics (non-
Markovian master equation) with Markovian process (Markovian master equation)
in Appendix A. For details one can refer to Gardiner’s book C.W.Gardiner
or/and Breuer’s H.P.Breuer .
In this paper the focus will be on the optimal decoherence control of non-
Markovian quantum system, particularly the simplest system possible, a two-
level system governed by the time-convolutionless (TCL) equation. We determine
control fields which minimize the cost functional suppressing the decoherence
process by applying the Pontryagin maximum principle (PMP) in Ohmic reservoir
with Lorentz-Drude regularization in the following three conditions:
$\omega_{0}\ll\omega_{c}$, $\omega_{0}\approx\omega_{c}$,
$\omega_{0}\gg\omega_{c}$, where $\omega_{0}$ is the characteristic frequency
of the quantum system of interest and $\omega_{c}$ the cut-off frequency of
Ohmic reservoir. Thus, $\omega_{c}\ll\omega_{0}$ implies that the spectrum of
the reservoir does not completely overlap with the frequency of the system
oscillator and $\omega_{0}\gg\omega_{c}$ implies the converse case. With this
it is possible to engineer different types of artificial reservoir, and couple
them to the system in a controlled way. We also compare our results with no-
control system evolution and the optimal control of the open system with
Markovian approximation. The main result of the paper is that decoherence
phenomenon can be successfully suppressed in the $\omega_{0}\ll\omega_{c}$
case. Then this explores the coupling of the system to engineered reservoirs
Myatt ; Turchette , in which the coupling and state of the environment are
controllable. This may pave a newly way to the realization of the first basic
elements of quantum computers.
The paper is organized as follows. We first introduce quantum decoherence and
the quantum master equation for driven open quantum systems. In Sec. III we
formulate the optimal control formalism and deduced PMP with a minimum cost
functional. Moreover, we consider the non-Markovian two-level optimal control
problem. In Sec. IV, we numerically analyze the optimal control of decoherence
to the two-level system, and analyze the difference between Markovian optimal
control and non-Markovian optimal control from both the system time evolution
and the power spectrum. Conclusions and prospective views are given in Sec.V.
## II Modeling the quantum decoherence control system
Consider a quantum system $S$ embedded in a dissipative environment $B$ and
interacting with a time-dependent classical external field, i.e., the control
field. The total Hamiltonian has the general form
$\begin{array}[]{rcl}H_{tot}&=&H_{0}+H_{B}+H_{int}\\\
&=&H_{S}+H_{C}(t)+H_{B}+H_{int},\end{array}$ (1)
where $H_{S}$ is the Hamiltonian of the system, $H_{C}(t)$ the Hamiltonian of
the control field, and $H_{B}$ the bath and $H_{int}$ their interaction that
is responsible for decoherence. The operators $H_{S}$ and $H_{B}$ act on
$\mathcal{H}_{S}$ and $\mathcal{H}_{B}$, respectively. The operator $H_{C}(t)$
contains a time-dependent external field to adjust the quantum evolution of
the system. One of the central goals of the theoretical treatment is then the
analysis of the dynamical behavior of the populations and coherences, which
are given by the elements of the reduced density matrix, defined as
$\begin{split}\rho_{S}(t)=tr_{B}[\rho_{tot}(t)],\end{split}$ (2)
where $\rho_{tot}$ is the total density matrix for both the system and the
environment, $tr_{B}$ the partial trace taken over the environment. The driven
model consists of a $N$-level system interacting with a thermal bath in the
presence of external control field zhang ; zhang2 , and the Hamiltonian is
$\begin{split}H_{C}(t)=\sum_{i}u_{i}(t)H_{i}\end{split}$ (3)
$H_{i}$ is the control Hamiltonian adjusted by the control parameters
$u_{i}(t)$, and $u_{i}(t)$ represents the control field. The Hamiltonian of
the environment is assumed to be composed of harmonic oscillators with natural
frequencies $\omega_{i}$ and masses $m_{i}$,
$\begin{split}H_{B}=\sum_{i=1}^{N}(\frac{p_{i}^{2}}{2m_{i}}+\frac{m_{i}}{2}x_{i}^{2}\omega_{i}^{2}),\end{split}$
(4)
where $(x_{1},x_{2},\cdots,x_{N},p_{1},p_{2},\cdots,p_{N})$ are the
coordinates and their conjugate momenta, and the Planck constant $\hbar$ is
assigned to be $1$. The interaction Hamiltonian between the system $S$ and the
environment $B$ is assumed to be bilinear H.P.Breuer ,
$\begin{split}H_{int}=\alpha\sum_{n}A_{n}\otimes B_{n}.\end{split}$ (5)
The interaction Hamiltonian in the interaction picture therefore takes the
form
$\begin{array}[]{rcl}H_{int}(t)&=&e^{i(H_{S}+H_{B})t}H_{int}e^{-i(H_{S}+H_{B})t}\\\
&=&\alpha\sum_{n}A_{n}(t)\otimes B_{n}(t),\end{array}$ (6)
where
$\begin{array}[]{rcl}A_{n}(t)&=&e^{iH_{S}t}A_{n}e^{-iH_{S}t},\\\
B_{n}(t)&=&e^{iH_{B}t}B_{n}e^{-iH_{B}t}.\end{array}$
The effect of the environment on the dynamics of the system can be seen as a
interplay between the dissipation and fluctuation phenomena. And it is the
general environment that makes the quantum system loss of coherence
(decoherence). In general the decoherence can be demonstrated as the
interaction between the system and environment. Then the reduced density
matrix of the system can evolve into the form
$\begin{split}\rho_{T}\simeq\sum_{n}|c_{n}|^{2}|a_{n}\rangle\langle
a_{n}|\end{split}$ (7)
which describes a statistical mixture of noninterfering states. Thus, a
commonly proposed way to analyze decoherence is by examining how the
nondiagonal elements of the reduced density matrix evolve under the master
equation.
In the present work, we shall concentrate on optimal control of the
decoherence effect in open quantum system. The kinetic equation of strong
coupling non-Markovian quantum system is the following exact time-
convolutionless (TCL) form of the master equation,
$\begin{split}\frac{d}{dt}\mathcal{P}\rho(t)=\sum_{i}u_{i}\tilde{\mathcal{K}}_{i}(t)\mathcal{P}\rho(t)+\mathcal{K}(t)\mathcal{P}\rho(t)+\mathcal{I}(t)\mathcal{Q}\rho(t_{0}),\end{split}$
(8)
with the time-local generator, called the TCL generator
$\begin{split}\tilde{\mathcal{K}}_{i}(t)=e^{iH_{S}t}H_{i}e^{-iH_{S}t},~{}~{}~{}~{}\mathcal{K}(t)=\alpha\mathcal{P}\mathcal{L}(t)[1-\Sigma(t)]^{-1}\mathcal{P},\end{split}$
(9)
and the inhomogeneity
$\begin{split}\mathcal{I}(t)=\alpha\mathcal{P}\mathcal{L}(t)[1-\Sigma(t)]^{-1}g(t,t_{0})\mathcal{Q},\end{split}$
(10)
where $\Sigma(t)$ is the superoperator
$\Sigma(t)=\alpha\int_{t_{0}}^{t}ds\mathcal{G}(t,s)\mathcal{Q}\mathcal{L}(s)\mathcal{P}G(t,s).$
For detail one see Appendix A and/or H.P.Breuer
In order to facilitate the calculations, we will convert the differential
equation (8) from the complex density matrix representation into the so-called
coherent vector representation Blum ; altafini ; zhang . Firstly, we choose an
orthonormal basis of $N\times N$ matrices
$\\{(I,\Omega_{j})\\}_{j=1,2,\cdots,N^{2}-1}$ with respect to the inner
product $\langle X,Y\rangle=tr(X^{{\dagger}}Y)$, where $I$ is the
N-dimensional identity matrix and $\Omega_{j}$ are $N\times N$ Hermitian
traceless matrices. In particular, the Hermitian density matrix $\rho$ can be
represented as $\rho=\frac{1}{N}I+\sum_{i}x_{i}\cdot\Omega_{i}$, where
$\vec{x}=(x_{1},x_{2},\cdots x_{N^{2}-1})^{T}$ is a real $(N^{2}-1)$
dimensional vector, called the coherent vector of $\rho$. This is the well-
known Bloch vector representation of quantum systems. Thus the master equation
(8) can be rewritten as a differential equation of the coherent vector:
$\begin{array}[]{rcl}\dot{x}(t)&=&O_{0}x(t)+\sum_{i=1}^{k}u_{i}(t)O_{i}x(t)+L_{1}(t)x(t)+L_{2}(t),\\\
\end{array}$ (11)
with the initial condition,
$x(t_{0})=x_{0}$
where $O_{0},O_{i}\in\text{so}(N^{2}-1)$ are the adjoint representation
matrices of $-iH_{0},-iH_{i}$ respectively, and $x_{0}$ is the coherence
vector of $\rho_{0}$, and the term $L_{1}(t)x(t)$ represents the decoherence
process, $k$ is the number of control fields, $\sum_{i}^{k}u_{i}H_{i}$ adjusts
the quantum evolution such that the coherence is conserved.
## III quantum optimal control problem
### III.1 General Formalism
As well known, the evolution of the state variable $x(t)$ governed by the
master equation (11) depends not only on the initial state $x_{0}$ but also on
the choice of the time-dependent control variable $u(t)$. Some earlier works
to these control problem are listed in the reference Thorwart ; Grifoni ; Shao
. Especially, the exact result was considered of the quantum two-state
dynamics driven by stationary non-Markovian discrete noise in Goychuk . In
this section, we are going to suppress the unexpected effect of decoherence by
optimal control technique that wants to force the system evolving along some
prescribed cohering trajectories. The target state chosen is the free
evolution of the closed system:
$\begin{split}\dot{\rho}_{T}(t)=-i[H_{0},\rho_{T}(t)],\end{split}$ (12)
which is equivalent to $x^{0}(t)=e^{O_{0}(t-t_{0})}x_{0}$. The cost functional
is
$\begin{split}J[u(t)]=\Psi[x(t_{f}),x^{0}(t_{f})]+\int_{t_{o}}^{t_{f}}\Theta(x(t),x^{0}(t_{f}),u(t))dt,\end{split}$
(13)
where the functional $\Psi[x(t_{f}),x^{0}(t_{f})]$ represents distance between
the system and objects at final time and the functional
$\int_{t_{o}}^{t_{f}}\Theta(x(t),x^{0}(t_{f}),u(t))$ accounts for the
transient response with $\Theta(x(t),x^{0}(t_{f}),u(t))\geq 0$.
The optimal control problem considered in this paper is to minimize the cost
functional $J[u(t)]$ with some dynamical constraints. That is, our problem is
$\begin{array}[]{rcl}\min_{u\in\mathcal{U}_{[t_{0},t_{f}]}}J[u(t)]&=&\Psi[x(t_{f}),x^{0}(t_{f})]+\int_{t_{o}}^{t_{f}}\Theta(t)dt,\\\
\dot{x}(t)&=&O_{0}x(t)+\sum_{i=1}^{k}u_{i}(t)O_{i}x(t)\\\
&+&L_{1}(t)x(t)+L_{2}(t),\\\
x(t_{0})&=&x_{0},~{}t\in[t_{0},t_{f}],\end{array}$ (14)
where
$\mathscr{U}_{[t_{0},t_{f}]}=\\{u(\cdot):[t_{0},t_{f}]\longrightarrow\mathbb{R}^{k}\\}$
and $u(\cdot)$ piecewise continuous.
Using the Pontryagin’s maximum principle krotov , the optimal solution to this
problem is characterized by the following Hamilton-Jacobi-Bellman(HJB)
Equation
$\begin{cases}\frac{\partial J}{\partial
t}+min_{u\in\mathscr{U}_{[t_{0},t_{f}]}}\\{O_{0}x(t)+\sum_{i=1}^{k}u_{i}(t)O_{i}x(t)+L_{1}(t)x(t)+L_{2}(t)+\Theta(x(t),x^{0}(t_{f}),u(t))\\}=0,\\\
J(x(t_{f}),t_{f})=\Psi[x(t_{f})].\end{cases}$ (15)
In general, it is usually difficult to obtain the analytic solution.
Nevertheless, one can always have numerical solution. To illustrate this
method and give more insight, we will consider this problem for the non-
Markovian two-level system in the following.
### III.2 Optimal Control of non-Markovian Two-Level System
In this subsection we consider the decoherence of two-level system whose
controlled Hamiltonian is
$H_{0}=\frac{1}{2}\\{\omega_{0}\sigma_{z}+u_{x}(t)\sigma_{x}+u_{y}(t)\sigma_{y}\\},$
(16)
where $\sigma_{k}$ with $k=x,y,z$ are the Pauli matrices; $\omega_{0}$ is the
transition frequency of the two-level system, and $u(t)$ is the modulation by
the time-dependent external control field. In fact, the free Hamiltonian is
$H_{S}=\frac{1}{2}\omega\sigma_{z}$. Then the control Hamiltonian can be
described by $\sigma_{x},\sigma_{y}$ according to Cartan decomposition of the
Lie algebra $\text{su}(2)$, which was discussed by Zhang et. al in details
zhang2 . This is the standard model for atom-field interaction scully ;
meystre ; Anastopoulos ; Shresta .
In our two-level system the assumed bilinear interaction between the system
$S$ and the environment $B$ can be written as
$H_{int}=\alpha\left(\sigma_{+}\otimes B+\sigma_{-}\otimes
B^{{\dagger}}\right)~{}~{}~{}\text{with}~{}~{}B=\sum_{i}k_{i}a_{i},$ (17)
where $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$, the raising and lowering
operator respectively, $k_{i}$ is the coupling constant between the spin
coordinate and the $i$th environment oscillator, and $a_{i}$ is the
annihilation operator of the $i$th harmonic oscillators of the environment.
The coupling constants enter the spectral density function $J(\omega)$ of the
environment defined by
$J(\omega)=\frac{\pi}{2}\sum_{i}\frac{k_{i}}{m_{i}\omega_{i}}\delta(\omega-\omega_{i})$
(18)
and the index $i$ labels the different field models of the reservoir with
frequencies $\omega_{i}$. In the continuum limit the spectral density has the
form
$\begin{split}J(\omega)=\eta\omega(\frac{\omega}{\omega_{c}})^{n-1}\exp(-\frac{\omega}{\omega_{c}}),\end{split}$
(19)
where $\omega_{c}$ is a cutoff frequency, and $\eta$ a dimensionless coupling
constant. The environment is classified as Ohmic, sub-Ohmic, and sup-Ohmic
according to $n=1$, $0<n<1$, and $n>1$, respectively An ; Leggett ; weiss .
In this case, the open quantum system can be written as follows H.P.Breuer ;
Maniscalco ; Maniscalco2
$\begin{array}[]{rcl}\dot{\rho}_{S}=-\frac{i}{2}\omega_{0}[\sigma_{z},\rho_{S}]-\frac{i}{2}u_{x}(t)[\sigma_{x},\rho_{S}]-\frac{i}{2}u_{y}(t)[\sigma_{y},\rho_{S}]\\\
+\frac{\Delta(t)+\gamma(t)}{2}\\{2\sigma_{-}\rho_{S}\sigma_{+}-\sigma_{+}\sigma_{-}\rho_{S}-\rho_{S}\sigma_{+}\sigma_{-}\\}\\\
+\frac{\Delta(t)-\gamma(t)}{2}\\{2\sigma_{+}\rho_{S}\sigma_{-}-\sigma_{-}\sigma_{+}\rho_{S}-\rho_{S}\sigma_{-}\sigma_{+}\\}.\end{array}$
(20)
For convenience we map the density matrix of the two-level system onto the
Bloch vector $x(t)=(x_{1}(t),x_{2}(t),x_{3}(t))^{T}\in\mathbb{R}^{3}$ defined
by $x(t)=Tr[\sigma\rho(t)]$, which implies that
$\begin{array}[]{rcl}x_{1}(t)&\equiv&\rho_{01}(t)+\rho_{10}(t),\\\
x_{2}(t)&\equiv&i(\rho_{01}(t)-\rho_{10}(t)),\\\
x_{3}(t)&\equiv&\rho_{00}(t)-\rho_{11}(t).\end{array}$ (21)
Then the explicit equations of motion for the components of the Bloch vector
read
$\left\\{\begin{array}[]{rcl}\dot{x_{1}}(t)&=&-\Delta(t)x_{1}(t)-\omega_{0}x_{2}(t)+x_{3}(t)u_{y}(t),\\\
\dot{x_{2}}(t)&=&\omega_{0}x_{1}(t)-\Delta(t)x_{2}(t)-x_{3}(t)u_{x}(t),\\\
\dot{x_{3}}(t)&=&-2\Delta(t)x_{3}(t)-2\gamma(t)+x_{2}(t)u_{x}(t)-x_{1}(t)u_{y}(t),\end{array}\right.$
(22)
where the expressions for the relevant time dependent coefficients, up to the
second order in the system-reservoir coupling constant, are given by
Maniscalco ; H.P.Breuer
$\begin{array}[]{rcl}\Delta(t)&=&\int_{0}^{t}d\tau
k(\tau)\cos(\omega_{0}\tau)\\\
\gamma(t)&=&\int_{0}^{t}d\tau\mu(\tau)\sin(\omega_{0}\tau)\end{array}$ (23)
with
$\begin{array}[]{rcl}k(\tau)&=&2\int_{0}^{\infty}d\omega
J(\omega)\coth[\hbar\omega/2k_{B}T]\cos(\omega\tau),\\\
\mu(\tau)&=&2\int_{0}^{\infty}d\omega J(\omega)\sin(\omega\tau),\end{array}$
(24)
being the noise and the dissipation kernels, respectively. The equation (22)
can be written compactly as
$\begin{split}\dot{x}(t)=A(t)x(t)+B(t),\end{split}$ (25)
where
$A(t)=\left(\begin{array}[]{ccc}-\Delta(t)&-\omega_{0}&u_{y}(t)\\\
\omega_{0}&-\Delta(t)&-u_{x}(t)\\\
-u_{y}(t)&u_{x}(t)&-2\Delta(t)\end{array}\right)$
and
$B(t)=\left(\begin{array}[]{ccc}0\\\ 0\\\ -2\gamma(t)\end{array}\right).$
Let the Ohmic spectral density with a Lorentz-Drude cutoff function,
$\begin{split}J(\omega)=\frac{2\gamma_{0}}{\pi}\omega\frac{\omega_{c}^{2}}{\omega_{c}^{2}+\omega^{2}},\end{split}$
(26)
where $\gamma_{0}$ is the frequency-independent damping constant and usually
assumed to be $1$. $\omega$ is the frequency of the bath, and $\omega_{c}$ is
the high-frequency cutoff. For this type of spectral density the bath
correlations can be determined analytically as
$\begin{split}k(\tau)=4k_{B}T{\omega_{c}}^{2}\sum_{n=-\infty}^{+\infty}\frac{\omega_{c}e^{-{\omega_{c}}|\tau|}-|\nu_{n}|e^{-|\nu_{n}||\tau|}}{\omega_{c}^{2}-\nu_{n}^{2}}\end{split}$
(27)
where $\nu_{n}=2\pi nk_{B}T$ and
$\begin{split}\mu(\tau)=2\hbar\omega_{c}^{2}e^{-\omega_{c}|\tau|}sign~{}\tau.\end{split}$
(28)
Then the analytic expression for the dissipation coefficient $\gamma(t)$
appearing in the equation (23) is
$\begin{split}\gamma(t)=\frac{\alpha^{2}\omega_{0}r^{2}}{1+r^{2}}[1-e^{-r\omega_{0}t}\cos(\omega_{0}t)-re^{-r\omega_{0}t}\sin(\omega_{0}t)],\end{split}$
(29)
and the closed analytic expression for $\Delta(t)$ is Maniscalco2
$\begin{array}[]{rcl}\Delta(t)&=&\alpha^{2}\omega_{0}\frac{r^{2}}{1+r^{2}}\\{\coth(\pi
r_{0})-\cot(\pi
r_{c})e^{-\omega_{c}t}[r\cos(\omega_{0}t)-\sin(\omega_{0}t)]+\frac{1}{\pi
r_{0}}\cos(\omega_{0}t)[\bar{F}(-r_{c},t)\\\
&&+\bar{F}(r_{c},t)-\bar{F}(ir_{0},t)-\bar{F}(-ir_{0},t)]-\frac{1}{\pi}\sin(\omega_{0}t)[\frac{e^{-\nu_{1}t}}{2r_{0}(1+r_{0}^{2})}[(r_{0}-i)\bar{G}(-r_{0},t)\\\
&&+(r_{0}+i)\bar{G}(r_{0},t)]+\frac{1}{2r_{c}}[\bar{F}(-r_{c},t)-\bar{F}(r_{c},t)]]\\},\end{array}$
(30)
where $r_{0}=\omega_{0}/2\pi k_{B}T$, $r_{c}=\omega_{c}/2\pi k_{B}T$,
$r=\omega_{c}/\omega_{0}$, and
$\begin{split}\bar{F}(x,t)\equiv_{2}F_{1}(x,1,1+x,e^{-\nu_{1}t}),\end{split}$
(31)
$\begin{split}\bar{G}(x,t)\equiv_{2}F_{1}(2,1+x,2+x,e^{-\nu_{1}t}).\end{split}$
(32)
${}_{2}F_{1}(a,b,c,z)$ is the hypergeometric function and takes the form
${}_{2}F_{1}(a,b,c,z)$ $\displaystyle=$ $\displaystyle
1+\frac{ab}{1!c}z+\frac{a(a+1)b(b+1)}{2!c(c+1)z^{2}}z^{2}+\cdots$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!},$
where $(a)_{n}$ is a Pochhammer symbol. Under the high temperature limit, we
have
$\begin{split}\Delta(t)=2\alpha^{2}\kappa
T\frac{r^{2}}{1+r^{2}}\\{1-e^{-r\omega_{0}t}[\cos(\omega_{0}t)-\frac{1}{r}\sin(\omega_{0}t)]\\}.\end{split}$
(33)
In the following we consider the optimal control formalism of our two-level
system. For simplicity we define the cost functional as:
$\begin{split}J[u(t)]=\int_{t_{0}}^{t_{f}}[(x(t)-x^{0}(t))^{2}+\theta
u^{T}(t)u(t)]dt\end{split}$ (34)
where $\theta>0$ is a weighting factor used to achieve a balance between the
tracking precision and the control constraints. The corresponding Hamiltonian
function is
$\displaystyle\mathcal{H}(x(t),u(t),\lambda(t),t)$ $\displaystyle=$
$\displaystyle[(x(t)-x^{0}(t))^{2}+\theta
u^{T}(t)u(t)]+\lambda(t)^{T}[A(t)x(t)+B(t)]$ $\displaystyle=$
$\displaystyle[(x_{1}(t)-x_{1}^{0}(t))^{2}+(x_{2}(t)-x_{2}^{0}(t))^{2}+(x_{3}(t)-x_{3}^{0}(t))^{2}+\theta(u_{1}^{2}(t)+u_{2}^{2}(t))]$
$\displaystyle+\lambda_{1}(t)[-\Delta(t)x_{1}(t)-\omega_{0}x_{2}(t)+x_{3}(t)u_{y}(t)]+\lambda_{2}(t)[\omega_{0}x_{1}(t)-\Delta(t)x_{2}(t)$
$\displaystyle-
x_{3}(t)u_{x}(t)]+\lambda_{3}(t)[-2\Delta(t)x_{3}(t)-2\gamma(t)+x_{2}(t)u_{x}(t)-x_{1}(t)u_{y}(t)],$
where $\lambda(t)=(\lambda_{1}(t),\lambda_{2}(t),\lambda_{3}(t))^{T}$ is the
so-called Lagrange multiplier and
$x^{0}(t)=(x^{0}_{1}(t),x^{0}_{2}(t),x^{0}_{2}(t))$ is the target trajectory
defined by $\dot{\rho}=-\frac{i}{2}[H_{0},\rho]$. It is easy to see that
$x^{0}(t)=(x^{0}_{1}\cos\omega t-x^{0}_{2}\sin\omega t,x^{0}_{1}\sin\omega
t+x^{0}_{2}\cos\omega,x^{0}_{3})$. The optimal solution can be solved by the
following differential equation with two-sided boundary values,
$\begin{cases}\dot{x}^{*}(t)=\frac{\partial\mathcal{H}}{\partial\lambda}=A(t)x(t)+B(t)\\\
\dot{\lambda}(t)=-\frac{\partial\mathcal{H}}{\partial
x}=-2[x(t)-x^{0}(t)]-A(t)^{T}\lambda(t)\\\ x^{*}(0)=x_{0}\\\
\lambda(t_{f})=0\end{cases}$ (35)
together with
$\begin{cases}\frac{\partial\mathcal{H}}{\partial
u}|_{*}=\frac{\partial\mathcal{H}(x^{*}(t),u^{*}(t),\lambda(t),t)}{\partial
u}=0,\\\ \frac{\partial^{2}\mathcal{H}}{\partial
u^{2}}|_{*}=\frac{\partial^{2}\mathcal{H}(x^{*}(t),u^{*}(t),\lambda(t),t)}{\partial
u^{2}}\leq 0,\end{cases}$ (36)
which implies that
$\begin{cases}u_{x}(t)=\frac{1}{2\theta}\\{\lambda_{2}x_{3}-\lambda_{3}x_{2}\\},\\\
u_{y}(t)=\frac{1}{2\theta}\\{\lambda_{3}x_{1}-\lambda_{1}x_{3}\\}.\end{cases}$
(37)
The minimum principle requires the solution of the complicated nonlinear
equations. When there is one and only one solution $\\{x(t),~{}\lambda(t)\\}$
it is the required optimal solution krotov . In general, it is difficult to
obtain the analytic solution, if possible existence, to the above optimal
control problem. So numerical demonstration to this problem will be considered
in the next section.
## IV numerical demonstration and discussions
In this section, we use the formalism of the preceding section to determine
the optimal control of the decoherence. Though the spin-bath models of real
systems are expected to be more complicated than the two-level Hamiltonians
considered here, we study the system in various aspects to understand the
effect of this simple system on the decoherence control.
In our simulations, the system parameters are chosen as following,
$x(0)=(\frac{\sqrt{3}}{2},\frac{-\sqrt{2}}{4},\frac{-\sqrt{2}}{4})$, strong
coupling constant $\alpha^{2}=0.01$, weighting factor $\theta=1$,
$\omega_{0}=1$ as the norm unit. Moreover, we regard the temperature as a key
factor in decoherence process. For high temperature $k_{B}T=300\omega_{0}$,
intermediate temperature $k_{B}T=3\omega_{0}$, and low temperature
$k_{B}T=0.3\omega_{0}$. Another reservoir parameter playing a key role in the
dynamics of the system is the ratio $r=\omega_{c}/\omega_{0}$ between the
reservoir cutoff frequency $\omega_{c}$ and the system oscillator frequency
$\omega_{0}$. As we will see in this section, by varying these two parameters
$k_{B}T$ and $r=\omega_{c}/\omega_{0}$, both the time evolution and the
optimal control of the open system vary prominently from Markovian to non-
Markovian.
### IV.1 High temperature reservoir
For high reservoir temperature, diffusion coefficient $\Delta(t)$ (30) has the
approximation form (33), which plays a dominant role since
$\Delta(t)\gg\gamma(t)$. Note that, for time $t$ large enough, the
coefficients $\Delta(t)$ and $\gamma(t)$ can be approximated by their
Markovian stationary values $\Delta_{M}=\Delta(t\rightarrow\infty)$ and
$\gamma_{M}=\gamma(t\rightarrow\infty)$. From eqs.(29) and (30) we have
$\gamma_{M}=\frac{\alpha^{2}\omega_{0}r^{2}}{1+r^{2}},$ (38)
and
$\Delta_{M}=\alpha^{2}\omega_{0}\frac{r^{2}}{1+r^{2}}\coth(\pi r_{0}).$ (39)
Then, under high temperature, noting
$\coth(\pi r_{0})\simeq 1+\frac{1}{\pi r_{0}}\simeq\frac{2kT}{\omega_{0}},$
$\Delta_{M}^{HT}=2\alpha^{2}kT\frac{r^{2}}{1+r^{2}}.$ (40)
Inserting Eqs.(38) and (40) into Eqs.(35) one can easily get the Markovian
optimal decoherence control.
Figure 1: (Color online)Surviving coherence in off-diagonal matrix elements vs
time $t$ [Eq.35] under high temperature environment, without control
action(black solid line), Markovian optimal control(blue dashed line), non-
Markovian optimal control(red dotted line), and target trajectory(crimson
dash-dotted line) at $r=0.1$, $r=1$, $r=10$ respectively.
Figure 1 shows optimal control of decoherence for $r\ll 1,~{}~{}r=1$, and
$r\gg 1$ in high temperature reservoir. All of these contain solid line for
free evolution, dashed line for Markovian optimal control, dotted line for
non-Markovian optimal control, and dash-dotted line for target trajectory. We
can see clearly that the decoherence can be controlled perfectly in $r\ll 1$
reservoir. From Figure 2 we can see that the decoherence time $\tau_{D}$ can
be delayed for a long time and its amplitude amplified heavily with the non-
Markovian control. On the other hand, Figure 3 shows that the non-Markovian
control field is changed more rapidly than the Markovian control field and the
frequency of non-Markovian is more plenty than the Markovian, which helps to
understand that the non-Markovian case is done better than the Markovian case
and implies that it is necessary to consider the non-Markovian case.
Figure 2: (Color online)Comparing Markovian optimal control, non-Markovian
optimal control with no control under high temperature environment, without
control action(black solid line), Markovian optimal control(blue dashed line),
non-Markovian optimal control(red dotted line), and target trajectory(crimson
dash-dotted line) at $r=0.1$.
Figure 3: (Color online)Comparing Markovian optimal control with non-Markovian
one in r=0.1 high temperture reservoir. Markovian optimal control $u_{x}$(blue
solid line) $u_{y}$(blue dash-dotted line) , non-Markovian optimal control
$u_{x}$(red solid line) $u_{y}$ (red dash-dotted line).
From Figure 1 we can also see that either Markovian or non-Markovian optimal
control cannot do well when $r=1$ or $r=10$. As we discussed before, diffusion
is always dominant under the high temperature. In the case $r\ll 1$,
$\Delta(t)>0$ is always true Maniscalco2 . However, Maniscalco, et. al.
Maniscalco2 showed that if $r>0.27$ the diffusion coefficient $\Delta(t)<0$,
and the system becomes non-Lindblad. It implies that the environment induced
fluctuations will be large enough. So our control field is negligible when
comparing with the high-frequency harmonic oscillators of the reservoir.
### IV.2 Lower temperature reservoir
As decreasing temperature, the amplitude of $\Delta(t)$ becomes smaller and
smaller and $\gamma(t)$ becomes larger and larger, which is not negligible
anymore. There exists a time which relate to both the temperature and the
ratio such that after the time the combination of dissipation and diffusion
coefficient $\Delta(t)-\gamma(t)<0$, which changes the properties of the
control system (35).
Figure 4 shows the non-Markovian optimal control and Figure 5 their power
spectrum for intermediate temperature, and Figure 6 and Figure 7 for low
temperature. At intermediate temperature the non-Markovian optimal control
plays little role especially in Figure 4(b)and Figure 4(c). Note that in
Figure 4(a) and 6(a) the free evolution is with little decoherence. We note
that the optimal control does well at low temperature in Figure 6. In Figure
6, both Markovian and non-Markovian play an important role in controlling the
decoherence in both $r=1$ and $r=10$. They can make the quantum coherence
persistence for a long time.
Figure 4: (Color online)Surviving coherence in off-diagonal matrix elements vs
time $t$ [Eq.35] under medium temperature environment, without control
action(black solid line), Markovian optimal control(blue dashed line), non-
Markovian optimal control(red dotted line), and target trajectory(crimson
dash-dotted line) at $r=0.1$, $r=1$, $r=10$ respectively.
Figure 5: (Color online)non-Markovian optimal controls and their power
spectrum in medium temperature reservoir for $r=0.1$, $r=1$, and $r=10$
respectively. Non-Markovian optimal control $u_{x}$(red solid line)
$u_{y}$(red dash-dotted line).
Figure 6: (Color online)Surviving coherence in off-diagonal matrix elements vs
time $t$ [Eq.35] under low temperature environment, without control
action(black solid line), Markovian optimal control(blue dashed line), non-
Markovian optimal control(red dotted line), and target trajectory(crimson
dash-dotted line) at $r=0.1$, $r=1$, $r=10$ respectively.
Figure 7: (Color online)non-Markovian optimal controls and their power
spectrum in low temperature reservoir for $r=0.1$, $r=1$, and $r=10$
respectively. Non-Markovian optimal control $u_{x}$(red solid line)
$u_{y}$(red dash-dotted line).
### IV.3 Engineering Reservoirs
During the last two decades, great advances in laser cooling and trapping
experimental techniques have made it possible to trap a single ion and cool it
down to very low temperature. These cold trapped ions are the favorite
candidates for a physical implementation of quantum computers and realization
of the quantum cryptography and quantum teleportation. All of these rely on
the persistence of quantum coherence. Myatt ; Turchette are the recent
experimental procedures for engineering artificial reservoirs. They showed how
to couple a properly engineered reservoirs with the trapped atomic ion’s
harmonic motion. They measured the decoherence of superpositions of coherent
states and two-Fock-state superpositions in the engineering artificial
reservoirs. Several type of engineering artificial reservoirs are simulated,
e.g., a high-temperature amplitude reservoir, a zero-temperature amplitude
reservoir, and a high-temperature phase reservoir.
From above discussions we find that our optimal decoherence control fields do
well in the engineering artificial reservoirs.
TABLE I. Controllability
T&r | Low T | Med T | High T
---|---|---|---
r=0.1 | slow decay | slow decay | controllable(non)
r=1 | controllable | uncontrollable | uncontrollable
r=10 | controllable | uncontrollable | uncontrollable
Table I shows the controllable property of non-Markovian open, dissipative
quantum system. When $r\ll 1$ the system free evolution is with little
decoherence at low and intermediate temperature and our optimal control plays
an important role in controlling the decoherence phenomenon at high
temperature. Moreover, when $r\gg 1$ and $r=1$ our optimal control also plays
an important role in controlling the decoherence phenomenon at low
temperature. They indicate that these engineered reservoirs could be designed
that the coupling and state of the environment can be controlled to slow down
the decoherence rate and delay decoherence time.
## V conclusions
In the present work, we have studied the optimal control of the decoherence
for the non-Markovian open quantum system. In the general formalism we
proposed the optimal control problem and derived the corresponding Hamilton-
Jacobi-Bellman equation. Usually this kind of problem is difficult to be
analytically solved. Then we considered this problem in the non-Markovian two-
level system. Through transforming its master equation into the Bloch vector
representation we obtained the corresponding differential equation with two-
sided boundary values.
Finally, we numerically studied the non-Markovian decoherence control for
three different conditions, i.e., $\omega_{0}\ll\omega_{c}$,
$\omega_{0}\approx\omega_{c}$, $\omega_{0}\gg\omega_{c}$ in the Ohmic
environment whose spectral density is with a Lorentz-Drude cuttoff function.
Our numerical results indicated that the decoherence dynamics behaves
differently for the different environmental condition which leads to
significant distinctness in the time dependent behavior of the dissipation
function $\gamma(t)$ and $\Delta(t)$. We regarded temperature as a key factor
in the decoherence effect and showed that the decoherence can’t be controlled
effectively in high temperature for both the Markovian and non-Markovian.
Comparing with the Markovian approximation we believed that it is necessary to
consider the non-Markovian quantum system. Most of all, we analyzed the short
time, moderate time, and long time decoherence control behaviors for $r=0.1$,
which implies $\omega_{c}\ll\omega_{0}$. In this case the decoherence can be
controlled effectively, which may indicates that the decoherence rate can be
slowed down and decoherence time can be delayed through designing some
engineered reservoirs proposed by Myatt et. al.
###### Acknowledgements.
This research is supported by the National Natural Science Foundation of China
(No. 60774099, No. 60221301) and by the Chinese Academy of Sciences
(KJCX3-SYW-S01). And the first author would like to thank Dr. J. Zhang for
many fruitful discussions.
### V.1 Comparing the non-Markovian dynamics with the Markovian dynamics
#### V.1.1 Quantum Markovian Process and Markovian Master Equation
Quantum Markovian process or Markovian approximation is widely used in open
quantum system, typically in interaction of radiation with matter (weak
coupling); quantum optics and cavity-QED (weak damping); quantum decoherence;
quantum Brownian motion (high temperatures); quantum information; quantum
error correction; stochastic unravelling (Monte Carlo simulations); laser
cooling (Lévy statistics of quantum jumps), and so on. The essence of quantum
Markovian process contains three assumptions:
* •
(i)The initial factorization ansatz (Feynman-Vernon approximation). At time
$t=0$ the bath $B$ is in thermal equilibrium and uncorrelated with the system
$S$
$\begin{split}\rho_{tot}(0)=\rho_{S}(0)\otimes\rho_{B};\end{split}$ (41)
* •
(ii)Weak system-bath interaction (Born approximation).
* •
(iii)Markovian approximation. The relaxation time $\tau_{B}$ of the heat bath
is much shorter than the time scale $\tau_{R}$ ($\tau_{B}\ll\tau_{R}$) over
which the state of the system varies appreciably.
Then it induced the dynamical map $\Phi_{t}$:
$\begin{split}\rho_{S}(0)\rightarrow\rho_{S}(t)=\Phi_{t}\rho_{s}(0)=tr_{B}{[U_{t}(\rho_{S}(0)\otimes\rho_{B})U^{{\dagger}}_{t}]}.\end{split}$
(42)
With some conditions, like completely positive and Hermiticity and trace
preservation we get a quantum dynamical semigroup:
$\Phi_{t}=exp[\mathcal{L}t],$ which implies the Markovian master equation:
$\begin{split}\frac{d}{dt}\rho_{s}(t)=\mathcal{L}\rho_{S}(t),\end{split}$ (43)
where generator of time evolution is in Lindblad form:
$\begin{split}\mathcal{L}\rho_{S}(t)=-\frac{i}{\hbar}[H_{S},\rho_{S}]+\sum_{i}\gamma_{i}[a_{i}\rho_{S}a_{i}^{{\dagger}}-\frac{1}{2}\\{a^{{\dagger}}_{i}a_{i},\rho_{S}\\}].\end{split}$
(44)
#### V.1.2 Non-Markovian Dynamics and Non-Markovian Master Equation
Non-Markovian dynamics system is not a new research problem, but recently it
received considerable consideration H.P.Breuer ; Maniscalco2 ; Breuer1 ;
Breuer2 ; Breuer3 . Comparing with the Markovian dynamics it has three
properties: (i)Semigroup property violated: slow decay of correlations, strong
memory effects; (ii)Initial correlations: classically correlated or entangled
initial states; (iii)Strong couplings and low temperatures, with which we can
studied the short-time behavior and exact evolution of quantum decoherence.
With the help of these three properties we can derive effective equations
(Master equations). As far as we known, there are two ways to derive the
master equation. One is called the path-integral method by Halliwell et.al
Halliwell96 , Hu et. al Hu92 , Ford et. alford01 , and Karrlein et.alkarrlein
, the other is the projection operator method by H.P. Breuer H.P.Breuer ;
Breuer1 ; Breuer2 ; Breuer3 .
The projection operator method is also called the Nakajima-Zwanzig projection.
The basic idea of the technique is to define a map $\mathcal{P}$ as
$\begin{split}\mathcal{P}{\rho}=tr_{B}\\{\rho\\}\otimes\rho_{B},\end{split}$
(45)
where $\rho_{B}$ is a fixed environment state and the map $\mathcal{P}$ is a
projection super-operator acting on operators, i.e.,
$\mathcal{P}^{2}=\mathcal{P}.$ Its complementary projection is
$\begin{split}\mathcal{Q}=\mathcal{I}-\mathcal{P},\end{split}$ (46)
where $\mathcal{I}$ is the identity map. Thus the Nakajima-Zwanzig equation
can be derived Breuer3 :
$\begin{split}\frac{d}{dt}\mathcal{P}\rho(t)=\int_{0}^{t}dsK(t,s)\mathcal{P}\rho(s)+\mathcal{I}(t)\mathcal{Q}\rho(0),\end{split}$
(47)
where $K(t,s)$ is the memory kernel. To second order in the coupling constant
the general form of the master equation can be approximated by
$\begin{split}\frac{d}{dt}\mathcal{P}\rho(t)=\mathcal{K}(t)\mathcal{P}\rho(t)+\mathcal{I}(t)\mathcal{Q}\rho(0).\end{split}$
(48)
In general, the TCL generator is
$\begin{split}\mathcal{K}(t)\rho_{S}=-\frac{i}{\hbar}[H_{S}(t),\rho_{S}]+\sum_{i}[C_{i}(t)\rho_{S}D_{i}^{{\dagger}}(t)+D_{i}(t)\rho_{S}C_{i}^{{\dagger}}(t)]-\frac{1}{2}\sum_{i}\\{D_{i}^{{\dagger}}(t)C_{i}(t)+C_{i}^{{\dagger}}D_{i}(t),\rho_{S}\\},\end{split}$
(49)
where $C_{i}(t)\neq D_{i}(t)$, which means that it is not in the Lindblad
form.
## References
* (1) H.P. Breuer, and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2002\.
* (2) J. Zhang, C.W. Li, R.B. Wu, T.J. Tarn, and X.S. Liu, J. Phys. A: Math. Gen. 38, 6587(2005).
* (3) S. Lloyd, and J-J. E. Slotine, Phys. Rev. Lett. 80, 4088(1998)
* (4) P. Zanardi, and M. Rasetti, Phys. Rev. Lett. 79, 3306(1997)
* (5) I. L. Chuang, and Y. Yamamoto, Phys. Rev. A. 52, 3489(1995)
* (6) R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A, 65, 062101(2002).
* (7) P. Zanardi, Phys. Rev. A 63, 012301(2001).
* (8) J.Von. Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
* (9) M.A. Nielsen, and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000\.
* (10) M.B. Mensky, Quantum Menseurements and Decoherence: Models and Phenomenology, Kluwer Academic Publishers, Dordrecht, 2000.
* (11) P.W. Shor, Phys. Rev. A 52, 2493(1995)
* (12) L. Viola, and S. Lloyd, Phys. Rev. A 58, 2733(1998).
* (13) B. Elattari, and S.A. Gurvitz, Phys. Rev. Lett. 84, 2047(2000).
* (14) C. Altafini, J. Math. Phys. 44, 2357 (2003).
* (15) H. Rabitz, R. de ViVie-Riedle, M. Motzkus, and K. Kompa, Science 288, 824(2000).
* (16) R.J. Levis, G.M. Menkir, and H. Rabitz, Science 292, 709(2001).
* (17) H. Rabitz, Science 299, 525(2003).
* (18) V.F. Krotov, Global Methods in Optimal Control Theory, Marcel Dekker INC, New York, (1996).
* (19) H. Jirari, and W. Pötz, Phys. Rev. A 74, 022306(2006).
* (20) D. Sugny, C. Kontz, and H. R. Jauslin, Phys. Rev. A 76, 023419(2007).
* (21) C.W. Gardiner, and P. Zoller, Quantum Noise (2nd Edition), Springer-Verlag, Berlin Heidelberg New York, 2000\.
* (22) W.H. Zurek, Rev. Mod. Phys. 75, 715(2003).
* (23) B.M. Garraway, Phys. Rev. A 55, 2290(1997).
* (24) C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, Nature(London) 403, 269(2000).
* (25) Q. A. Turchette, C. J. Myatt, B. E. King, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. A 62, 053807(2000).
* (26) A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(2005).
* (27) J.J. Halliwell and T. Yu, Phys. Rev. D 53, 2012 (1996).
* (28) J. Zhang, R.B. Wu, C.W. Li, T.J. Tarn, and J.W. Wu, Phy. Rev. A75, 022324(2007).
* (29) K. Blum, Density Matrix Theory and Applications, Plenum Press, New York, (1981).
* (30) M. Thorwart, L. Hartmann, I. Goychuk, and P. Hänggi, J. Mod. Opt. 47,(2000).
* (31) M. Grifoni, and P. Hänggi, Phys. Rep. 304, (1998).
* (32) J. S. Shao, C. Zerbe, and P. Hänggi, Chem. Phys. 235,(1998).
* (33) I. Goychuk, and P. Hänggi, Chem. Phys. 324, (2006).
* (34) M. Ban, S.Kitajima, and F. Shibata, Phys. Rev. A 76, 022307(2007)
* (35) M. O. Scully, and M. S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, (1997).
* (36) P. Meystre, Atom Optics, Springer-Verlag, New York, (2001).
* (37) C. Anastopoulos, and B. L. Hu, Phys. Rev. A 62, 033821 (2000).
* (38) S. Shresta, C. Anastopoulos, A. Dragulescu, and B. L. Hu, Phys. Rev. A 71, 022109 (2005).
* (39) B.L. Hu, J.P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992).
* (40) U. Weiss, Quantum Dissipative System, World Scientific Publishing, Singapore (1993).
* (41) A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1(1987).
* (42) J. H. An, W. M. Zhang, Phys. Rev. A 76, 042127 (2007).
* (43) S. Maniscalco, S. Olivares, and M. G. A. Paris, Phys. Rev. A 75, 062119 (2007).
* (44) H.P. Breuer, B.Kappler, and F. Petruccione, Ann. Phys.(N.Y.)291, 36(2001).
* (45) H.P. Breuer, D. Burgarth, and F. Petruccione, Phys. Rev. B70, 045323(2004).
* (46) H.P. Breuer, Phy. Rev. A75, 022103(2007).
* (47) G.W. Ford, and R.F. O’Connel, Phys. Rev. D 64, 105020(2001).
* (48) S. Maniscalco, J.Piilo, F. Petruccione, and A. Messina Phys. Rev. A 70, 032113(2004).
* (49) R. Karrlein and H. Grabert Phys. Rev. E 55, 153(1997).
|
arxiv-papers
| 2009-10-27T18:50:05 |
2024-09-04T02:49:06.103729
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Cui, Zairong Xi, and Yu Pan",
"submitter": "Wei Cui",
"url": "https://arxiv.org/abs/0910.5208"
}
|
0910.5210
|
# The entanglement dynamics of bipartite quantum system: Towards entanglement
sudden death
Wei Cui1,2, Zairong Xi1∗ and Yu Pan1,2 1Key Laboratory of Systems and Control,
Institute of Systems Science, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
2Graduate University of Chinese Academy of Sciences, Beijing 100039, People’s
Republic of China zrxi@iss.ac.cn
###### Abstract
We investigate the entanglement dynamics of bipartite quantum system between
two qubits with the dissipative environment. We begin with the standard
Markovian master equation in the Lindblad form and the initial state which is
prepared in the extended Werner-like state: $\rho^{\Phi}_{AB}(0)$. We examine
the conditions for entanglement sudden death (ESD) and calculate the
corresponding ESD time by the Wootters’ concurrence. We observe that ESD is
determined by the parameters like the mean occupation number of the
environment $N$, amount of initial entanglement $\alpha$, and the purity $r$.
For $N=0$, we get the analytical expression of both ESD condition and ESD
time. For $N>0$ we give a theoretical analysis that ESD always occurs, and
simulate the concurrence as a function of $\gamma_{0}t$ and one of the
parameters $N,\alpha$, and $r$.
###### pacs:
03.65.Ud, 03.65.Yz, 03.67.Mn, 05.40.Ca
## 1 Introduction
Entanglement is responsible for the most counterintuitive aspects of quantum
mechanics [1], and motivated many philosophical discussions in the early days
of quantum physics [2]. Recently it has been regarded as a resource for
quantum information processing [3]. In fact, entanglement is one of the key
ingredient for quantum teleportation [4, 5, 6], quantum cryptography [7] and
is believed to be the origin of the power of quantum computers, etc. However,
a quantum system used in quantum information processing inevitably interacts
with the surrounding environments (or the thermal reservoirs), which takes the
pure state of the quantum system into a mixed state [8]. Thus, it is an
important subject analyzing the entanglement decay induced by the unavoidable
interaction of the interested systems with the environment [9]. In one-party
quantum system, this process is called decoherence, and various methods have
been proposed to reduce this unexpected effect [10, 11, 12, 13]. In multiparty
systems with non-local quantum correlations much interest has been arisen in
the dynamics of entanglement. For example, entanglement sudden death (ESD),
which means that it disappears at finite time, was discovered by Yu and Eberly
[14, 15]. It differs remarkably from the single qubit coherent evolution. The
interesting phenomenon has been experimentally observed for entanglement
photon pairs [16] and atomic ensembles [17].
ESD puts a limitation on the time when entanglement must be exploited. The
evolution of the entanglement and ESD have been analyzed and various
interesting results obtained [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].
Ref. [23] investigated the time evolution of entanglement of various entangled
states of a two-qubit system. For four different initial states they analyzed
the entanglement sudden death conditions and time. From the simulation they
got the conclusion that the ESD always exists except for the vacuum reservoir.
However, it is still a question when the ESD occurs and what ESD time is in
the general cases and universal initial states. The aim of this paper is to
discuss the above problems for the standard Markovian master equation in the
Lindblad form and the initial states, the extended Werner-like states:
$\rho^{\Phi}_{AB}(0)$.
The paper is organized as follows. We first introduce the Wootters’
concurrence [29] and the extended Werner-like states: $\rho^{\Phi}_{AB}(0)$.
In Section III we give a standard Markovian master equation [30, 31]. The
master equation is equivalent to a first order coupled differential equations
when the initial state is the extended Werner like state
$\rho^{\Phi}_{AB}(0)$. In sections IV, we analyze the ESD conditions and ESD
time for the initial state $\rho^{\Phi}_{AB}(0)$. Conclusions and prospective
views are given in Section V.
## 2 Concurrence and initial states
A useful measure of entanglement is the Wootters’ concurrence [29]. For a
bipartite system described by the density matrix $\rho$, the concurrence
$\mathcal{C}(\rho)$ is
$\mathcal{C}(\rho)=\max(0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}),$
(1)
where $\lambda_{1},\lambda_{2},\lambda_{3}$, and $\lambda_{4}$ are the
eigenvalues (with $\lambda_{1}$ the largest one) of the “spin-flipped” density
operator $\zeta$, and
$\zeta=\rho(\sigma_{y}^{A}\otimes\sigma_{y}^{B})\rho^{*}(\sigma_{y}^{A}\otimes\sigma_{y}^{B}),$
(2)
where $\rho^{*}$ denotes the complex conjugate of $\rho$ and $\sigma_{y}$ is
the usual Pauli matrix. $\mathcal{C}$ ranges in magnitude from 0 for a
disentanglement state to 1 for a maximally entanglement state.
The general structure of an “X” density matrix [14, 15] is as follows
$\hat{\rho}=\left(\begin{array}[]{cccc}x&0&0&v\\\ 0&y&u&0\\\ 0&u^{*}&z&0\\\
v^{*}&0&0&w\end{array}\right),$ (3)
with $x,y,z,w$ real positive and $u,v$ complex quantities. Such states are
general enough to include states such as the Werner states, the Bell states,
_et al._. A remarkable aspect of the “X” states is that the initial “X”
structure is maintained during the Lindblad master equation evolution. This
particular form of the density matrix allows us to analytically express the
concurrence at time $t$ as [14]
$\mathcal{C}_{\rho}^{X}(t)=2\max\\{0,|u|-\sqrt{xw},|v|-\sqrt{yz}\\}.$ (4)
In the present work, we will analyze in detail the two-qubit entanglement
dynamics in Markovian environment starting from the initial “X” states defined
in Eq. (3). We will examine the exactly ESD time and entanglement evolution
for two kind of special states, the extended Werner-like states,
$\rho^{\Phi}_{AB}(0)=r|\Phi\rangle_{ABAB}\langle\Phi|+\frac{1-r}{4}\emph{{I}}_{AB},$
(5)
where $r$ the purity of the initial states, $\emph{{I}}_{AB}$ the $4\times 4$
identity matrix and
$\Phi_{AB}=(\cos(\alpha)|10\rangle+\sin(\alpha)|01\rangle)_{AB},$ (6)
with $\alpha$ measuring the amount of initial entanglement. The state
$\rho^{\Phi}_{AB}(0)$ has the following form
$\rho_{AB}^{\Phi}(0)=\left(\begin{array}[]{cccc}\frac{1-r}{4}&0&0&0\\\
0&r\cos^{2}(\alpha)+\frac{1-r}{4}&r\sin(\alpha)\cos(\alpha)&0\\\
0&r\sin(\alpha)\cos(\alpha)&r\sin^{2}(\alpha)+\frac{1-r}{4}&0\\\
0&0&0&\frac{1-r}{4}\end{array}\right).$ (7)
Figure 1: (Color online)Plot the initial entangled area (colored) of the
extended Werner-like states for $\alpha\in[0,2\pi]$. The blue line is
$r=\frac{1}{1+2|\sin(2\alpha)|}$.
Obviously, the state in Eq. (5) is reduced to the standard Werner state when
$\alpha=\pi/4$, and the Werner-like state become totally mixed state for
$r=0$, while the well-known Bell state for $r=1$. Note that, the extended
Werner-like state $\rho^{\Phi}_{AB}(0)$ contain both separate state and
entangled state. According to Peres’ criterion [32], when $\frac{1}{1+2|\sin
2\alpha|}\leq r\leq 1$ the extended Werner-like state would be entangled,
otherwise it would be separated. Fig.1 shows the entangled and separated
areas.
## 3 The master equation
The standard two-qubit Markovian master equation is the following Lindblad
form [8, 30, 31],
$\frac{d\rho}{dt}=\frac{\gamma_{0}(N+1)}{2}\sum_{j=1}^{2}\\{2\sigma_{j}^{-}\rho\sigma_{j}^{+}-\sigma_{j}^{+}\sigma_{j}^{-}\rho-\rho\sigma_{j}^{+}\sigma_{j}^{-}\\}\\\
+\frac{\gamma_{0}N}{2}\sum_{j=1}^{2}\\{2\sigma_{j}^{+}\rho\sigma_{j}^{-}-\sigma_{j}^{-}\sigma_{j}^{+}\rho-\rho\sigma_{j}^{-}\sigma_{j}^{+}\\},$
(8)
where $N=1/(e^{\frac{\hbar\omega_{0}}{K_{B}T}}-1)$, the mean occupation number
of the environment oscillators, $\gamma_{0}$ is the spontaneous emission rate.
As introduced before, we assume the initial state to be a “X” state. When
substituting (3) into the master equation (8) we obtain the following first-
order coupled differential equations,
$\left(\begin{array}[]{ccc}\dot{x}(t)\\\ \dot{y}(t)\\\ \dot{z}(t)\\\
\dot{w}(t)\end{array}\right)=\left(\begin{array}[]{cccc}-2\gamma_{0}(N+1)&\gamma_{0}N&\gamma_{0}N&0\\\
\gamma_{0}(N+1)&-\gamma_{0}(2N+1)&0&\gamma_{0}N\\\
\gamma_{0}(N+1)&0&-\gamma_{0}(2N+1)&\gamma_{0}N\\\
0&\gamma_{0}(N+1)&\gamma_{0}(N+1)&-2\gamma_{0}N\end{array}\right)\left(\begin{array}[]{ccc}x(t)\\\
y(t)\\\ z(t)\\\ w(t)\end{array}\right),$ (9)
and
$\begin{array}[]{rcl}\dot{u}(t)&=&-(1+2N)\gamma_{0}u(t),\\\
\dot{v}(t)&=&-(1+2N)\gamma_{0}v(t).\end{array}$ (10)
The solution of the previous master equation can be found by solving the
system of differential equations. The reduced density matrix elements
$x(t),y(t),z(t),w(t),u(t)$ and $v(t)$ are given
(i) $N>0$
$\begin{array}[]{rcl}x(t)&=&c_{1}\frac{N}{N+1}-c_{2}\Gamma^{2}(t)-c_{4}\gamma_{0}N\Gamma(t),\\\
y(t)&=&c_{1}+c_{2}\Gamma^{2}(t)+c_{3}\Gamma(t),\\\
z(t)&=&c_{1}+c_{2}\Gamma^{2}(t)-c_{3}\Gamma(t)-c_{4}\gamma_{0}\Gamma(t),\\\
w(t)&=&c_{1}\frac{N+1}{N}-c_{2}\Gamma^{2}(t)+c_{4}\gamma_{0}(N+1)\Gamma(t),\\\
u(t)&=&c_{5}\Gamma(t),\\\ v(t)&=&c_{6}\Gamma(t)\par\end{array}$ (11)
(ii) $N=0$
$\begin{array}[]{rcl}x(t)&=&d_{4}\Upsilon^{2}(t),\\\
y(t)&=&d_{2}\Upsilon(t)+d_{3}\Upsilon(t)-d_{4}\Upsilon^{2}(t),\\\
z(t)&=&-d_{3}\Upsilon(t)-d_{4}\Upsilon^{2}(t),\\\
w(t)&=&d_{1}-d_{2}\Upsilon(t)+d_{4}\Upsilon^{2}(t),\\\
u(t)&=&d_{5}\Upsilon(t),\\\ v(t)&=&d_{6}\Upsilon(t)\par\end{array}$ (12)
where $\Gamma(t)=e^{-(1+2N)\gamma_{0}t}$, $\Upsilon(t)=e^{-\gamma_{0}t}$, and
the coefficients $c_{1},c_{2},c_{3},c_{4},c_{5}$,$c_{6}$ in Eq. (11), and
$d_{1},d_{2},d_{3},d_{4},d_{5}$, $d_{6}$ in Eq. (12) are determined by the
corresponding initial conditions.
## 4 Entanglement dynamics with the initial conditions
### 4.1 $N>0$
We now analyze the entanglement dynamics. Starting from the initial states
$\rho_{AB}^{\Phi}(0)$ in Eq. (7), the coefficients of Eq.(11) are determined
as
$\begin{array}[]{rcl}c_{1}&=&\frac{N(N+1)}{(2N+1)^{2}},\\\
c_{2}&=&\frac{N(N+1)}{(2N+1)^{2}}-\frac{1-r}{4},\\\
c_{3}&=&\frac{1}{2(2N+1)^{2}}+\frac{r}{2}(\cos^{2}(\alpha)-\sin^{2}(\alpha)),\\\
c_{4}&=&-\frac{1}{\gamma_{0}(2N+1)^{2}},\\\
c_{5}&=&r\sin(\alpha)\cos(\alpha),\\\ c_{6}&=&0\end{array}$ (13)
Then the concurrence of $\rho_{AB}^{\Phi}(t)$ is
$C(\rho_{AB}^{\Phi}(t))=2\max\\{0,|u(t)|-\sqrt{x(t)w(t)}\\}.$ (14)
So the ESD appears when,
$|u(t)|-\sqrt{x(t)w(t)}\leq 0\Leftrightarrow u^{2}(t)-x(t)w(t)\leq 0.$ (15)
Obviously,
$t\rightarrow+\infty,~{}~{}~{}~{}u^{2}(t)-x(t)w(t)=-\frac{N^{2}(N+1)^{2}}{(2N+1)^{4}}<0,$
and
$t=0,~{}~{}~{}~{}~{}u^{2}(t)-x(t)w(t)=r^{2}\sin^{2}(\alpha)\cos^{2}(\alpha)-\frac{(1-r)^{2}}{16}.$
(16)
Thus, ESD always occurs if the initial state is entangled, which means
$\frac{1}{1+2|\sin 2\alpha|}\leq r\leq 1$, according to Peres’ criterion.
Note that Ref. [23] found that when the mean thermal photon number is not
zero, in the thermal reservoir the entanglement sudden death always happen
from the simulation. However here we get the same result from the theoretic
analysis. In Fig. 2, 3, and 4 we simulate the concurrence as a function of
$\gamma_{0}t$ and one of the parameters $N,\alpha$, and $r$, respectively.
Fig. 2 is the concurrence $\mathcal{C}_{\rho}^{\Phi}(t)$ as function of
$\gamma_{0}t$ and $N$, fixing the purity $r=1$ and initial degree of
entanglement $\alpha=\frac{\pi}{4}$, the Bell-like states. It shows that ESD
time is affected by $N$. The smaller $N$, the longer its ESD time. In Fig. 3,
we plot the concurrence $\mathcal{C}_{\rho}^{\Phi}(t)$ as function of
$\gamma_{0}t$ and initial entanglement $\alpha$ when $r=1$. When
$\alpha=\frac{\pi}{4}$ or $\alpha=\frac{3\pi}{4}$ the initial states are
reduced to Bell-like states. The ESD time is sensibly affected by $\alpha$.
Fig. 4 is the concurrence as function of $\gamma_{0}t$ and purity $r$. As
exemplified before, when $0\leq r\leq\frac{1}{1+2|\sin(2\alpha)|}$ the initial
state is separated, so we choose $r$ from $1/3$ to $1$. It shows that the
larger the purity $r$ the longer the ESD time.
Here we study the entanglement dynamics of bipartite quantum system in the
global environment effect. However, Ref. [26] had shown that under the
classical niose effect entanglement may experience a sudden death process even
if the local coherence of one participating particle is well preserved and the
other one decays to zero asymptotically. How do the local thermal reservoirs
influence the entanglement dynamics? Up to my knowledge, the master equation
needs to be reconstructed and the local environment effect embodied by the
spectral density of the thermal reservoir $J(\omega,T)$. The one-body
decoherence dynamics was studied in [12, 33] under the local environment
effect. Whether does the multipartite entanglement dynamics under the local
thermal reservoirs hold under the classical noise effect proved by Yu etc al
[26]? We will study it in our further work.
Figure 2: (Color online)($r=1,\alpha=\pi/4$)Plot of concurrence of
$\mathcal{C}_{\rho}^{\Phi}(t)$ vs “N” and $\gamma_{0}t$.
Figure 3: (Color online)($r=1,N=0.1$)Plot of concurrence of
$\mathcal{C}_{\rho}^{\Phi}(t)$ vs $\alpha$ and $\gamma_{0}t$.
Figure 4: (Color online)($\alpha=\pi/4,N=0.25$)Plot of concurrence
$\mathcal{C}_{\rho}^{\Phi}(t)$ vs “r” and $\gamma_{0}t$.
### 4.2 $N=0$
The coefficients in Eq. (12) have the form
$\begin{array}[]{rcl}d_{1}&=&1,\\\ d_{2}&=&1,\\\
d_{3}&=&-\frac{1-r}{2}-r\sin^{2}(\alpha),\\\ d_{4}&=&\frac{1-r}{4},\\\
d_{5}&=&r\sin(\alpha)\cos(\alpha),\\\ d_{6}&=&0.\end{array}$ (17)
Thus
$\begin{array}[]{rcl}&&u^{2}(t)-x(t)w(t)\leq 0\\\
&\Leftrightarrow&\Upsilon^{2}(t)-\frac{4}{1-r}\Upsilon(t)+\frac{4}{1-r}-\frac{16r^{2}}{(1-r)^{2}}\sin^{2}(\alpha)\cos(\alpha)\geq
0.\end{array}$ (18)
So the ESD occurs when,
$\frac{1}{1+2|\sin(2\alpha)|}<r<\frac{-1+\sqrt{1+4\sin^{2}(2\alpha)}}{2\sin^{2}(2\alpha)},$
(19)
and the ESD time is
$\gamma_{0}t^{*}=\ln(1-r)-\ln[2(1-\sqrt{r+r^{2}\sin^{2}(2\alpha)})].$ (20)
When $\gamma_{0}t\in[\gamma_{0}t^{*},+\infty)$, the concurrence
$C^{\Phi}_{AB}=0$. In Fig. 5 we plot the ESD area. If the initial state is
pure, i.e., $r=1$, we find that $x(t)\equiv 0$. Then the concurrence is
$C(\rho_{AB}^{\Phi}(t))=2|u(t)|=|\sin(2\alpha)|e^{-\gamma_{0}t}$, which
implies that the nonexistence of ESD. After we submitted our paper, a closely
related work appeared in [28]. Ikram et al extended their former result [23]
to more general state for the zero temperature limit and provided some
discussions on the thermal case. Comparing with us, there are three main
differences. Firstly, although both papers studied the same system, a two
2-level atom system, different initial states had been chosen. These two
initial states are independent, and the results are complementary. Secondly,
for pure 2-qubit entangled states in a thermal environment, Ref. [28] says
that “We see in plots of figure 3 that entanglement sudden death always
happens for non-zero average photon number in the two cavities and
entanglement sudden death time depends upon the initial preparation of the
entangled states.” Here, a theoretical analysis is provided by Eqs. (14, 15,
16). Finally, in the vacuum reservoir we give the sufficient and necessary
condition Eq.(19) for the entanglement sudden death, and answer the question
when the ESD occurs and what ESD time is. Furthermore, the ESD area relies on
the initial purity $r$ and initial entanglement $\alpha$, see Fig. 5. Here we
give a more detail examination and analysis.
The ESD condition and corresponding time for initial state
$\rho_{AB}^{\Phi}(0)$ at $N=0$ are summarized in the following table.
TABLE I. ESD condition and time for initial state $\rho_{AB}^{\Phi}(0)$ at
$N=0$
$\alpha$ | (0, $2\pi$)
---|---
Condition | $\frac{1}{1+2|\sin(2\alpha)|}<r<\frac{-1+\sqrt{1+4\sin^{2}(2\alpha)}}{2\sin^{2}(2\alpha)}$
Time | $\gamma_{0}t^{*}=\ln(1-r)-\ln[2(1-\sqrt{r+r^{2}\sin^{2}(2\alpha)})]$
Figure 5: (Color online)Plot the ESD area of the extended Werner-like states
$\rho_{AB}^{\Phi}(t)$ with $N=0$ for $\alpha\in[0,2\pi]$. The red line is
$r=\frac{-1+\sqrt{1+4\sin^{2}(2\alpha)}}{2\sin^{2}(2\alpha)}$, and the blue
line is $r=\frac{1}{1+2|\sin(2\alpha)|}.$
## 5 Conclusions
In summary, we have analyzed the interesting phenomenon of entanglement sudden
death determined by the dimensionless parameters
$\hbar\omega_{0}/k_{B}T,\alpha$, and $r$. Sufficient conditions for ESD have
been given for initial state $\rho^{\Phi}_{AB}(0)$. We examine the conditions
for ESD and calculate the corresponding ESD time by the Wootters’ concurrence.
We observe that ESD is determined by the parameters like $N$, amount of
initial entanglement $\alpha$, and the purity $r$. For $N=0$, we get the
analytical expressions of the ESD condition and ESD time. For $N>0$ we give a
theoretical analysis that ESD always occurs, and simulates the concurrence as
a function of $\gamma_{0}t$ and one of the parameters $N,\alpha$, and $r$. In
fact, these above results can also be obtain for other “X” initial states, for
example
$\rho^{\Psi}_{AB}(0)=r|\Psi\rangle_{ABAB}\langle\Psi|+\frac{1-r}{4}\emph{{I}}_{AB},$
where $\Psi_{AB}=(\cos(\alpha)|00\rangle+\sin(\alpha)|11\rangle)_{AB}.$
We analyze the dynamic behavior of entanglement in the open quantum system But
our ultimate aim is to control the entanglement such that it be a resource for
practical realization. Despite of the noticeable progresses of entanglement,
many fundamental difficulties still remain. One of the problem is ESD due to
the interactions between system and environment; the other is that as the
$N$-particle increased, the entanglement becomes arbitrarily small, and
therefore useless as a resource [34]. Here we indicate that preparing some
initial states can help prolong the ESD time. However, we think that it is a
long way how to design some effective control field to make the $N$-particle
large enough and protect the entanglement that make it practically useful.
## Acknowledgments
We thank the referees for improving the manuscript. This research is supported
by the National Natural Science Foundation of China (No. 60774099, No.
60221301) and by the Chinese Academy of Sciences (KJCX3-SYW-S01).
## References
## References
* [1] Bell J S 1964 Physics 1 195
* [2] Einstein A, Podolsky B and Rosen R 1935 Phys. Rev. 47 777
* [3] Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
* [4] Bennett C H and DiVincenzo D P 2000 Nature(London) 404 247
* [5] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A and Wootters W K 1993 Phys. Rev. Lett. 70 1895
* [6] Bouwmeester D, Pan J W, Weinfurter M, and Zeilinger A 1997 Nature(London) 390 575
* [7] Ekert A K 1991 Phys. Rev. Lett. 67 661
* [8] Breuer H P and Petruccione F 2002 The Theory of Open Quantum Systems (Oxford: Oxford University Press)
* [9] Weiss U 1999 Quantum Dissipative Systems (Second Edition) (Singapore: World Scientific Publishing)
* [10] Mintert F, _et al_. 2005 Physics Reports 415 207
* [11] Branderhorst M P A, _et al_. 2008 Science 320, 638
* [12] Cui W, Xi Z R and Pan Y 2008 Phys. Rev. A 77 032117
* [13] Zhang J, Li C W, Wu R B, Tarn T J, and Liu X S 2005 J. Phys. A: Math. Gen. 38 6587
* [14] Yu T and Eberly J H 2004 Phys. Rev. Lett. 93 140404
* [15] Yu T and Eberly J H 2006 Phys. Rev. Lett. 97 140403
* [16] Almeida M P, _et al_. 2007 Science 316 579
* [17] Laurat J, _et al_. 2007 Phys. Rev. Lett. 99 180504
* [18] Wang J, _et al_. 2006 J. Phys. B 39 4343
* [19] Dajka J, Mierzejewski M and Łuczka J 2008 Phys. Rev. A 77 042316
* [20] Dajka J, and Łuczka J 2008 Phys. Rev. A 77 062303
* [21] Roos C 2008 Nature Physics 4 97
* [22] Konrad T, _et al_. 2008 Nature Physics 4 99
* [23] Ikram M, Li F L , and Zubairy M S 2007 Phys. Rev. A 75 062336
* [24] Al-Qasimi A and James Daniel F V 2008 Phys. Rev. A 77, 012117
* [25] Huang J H and Zhu S Y 2008 Opt. Comm. 281, 2156
* [26] Yu T and Eberly J H 2006 Opt. Comm. 264 393
* [27] Huang J H and Zhu S Y 2007 Phys. Rev. A 76 062322
* [28] Tahir R, Ikram M, Aazim T, and Zubairy M 2008 J. Phys. B 41 062332
* [29] Wootters W K 1998 Phys. Rev. Lett. 80 2245
* [30] Turchette Q A, Myatt C J, King B E, Sackett C A, Kielpinski D, Itano W M, Monroe C and Wineland D J 2000 Phys. Rev. A 62 053807
* [31] Maniscalco S and Petruccione F 2006 Phys. Rev. A 73, 012111
* [32] Peres A 1996 Phys. Rev. Lett. 77 1413
* [33] Villares Ferrer A, and Morais Smith C 2007 Phys. Rev. B 76 214304
* [34] Aolita L, Chaves R, Cavalcanti, Acín A, and Davidovich L 2008 Phys. Rev. Lett. 100 080501
|
arxiv-papers
| 2009-10-27T19:04:36 |
2024-09-04T02:49:06.110098
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Cui, Zairong Xi, and Yu Pan",
"submitter": "Wei Cui",
"url": "https://arxiv.org/abs/0910.5210"
}
|
0910.5279
|
# The mystery of lost energy in ideal capacitors
A. P. James1
1Queensland Micro-nanotechnology Center, Griffith University
Nathan, QLD 4111, Australia
E-mail: a.james@griffith.edu.au
###### Abstract
The classical two-capacitor problem shows a mysterious lose of energy even
under lossless conditions and questions the basic understanding of energy
relation in a capacitor. Here, we present a solution to the classical two-
capacitor problem. We find that by reinterpreting the energy calculations we
achieve no lose of energy thereby obeying the conservation of energy law.
The introductory books in electronic circuits and physics [1, 2, 3, 4], often
put forward an energy paradox on idealised capacitor switching. This paradox
[5] is described in Fig. 1a, where energy before and after the switch become
closed does not seem to be same. The main issue here is the mysterious loss of
50% energy, despite all the components being ideal (i.e. ideal capacitor,
ideal wires, and ideal switches). Further, in Fig. 1a, the total charge, total
voltage, and total power in the circuit is conserved, so having energy reduced
by half questions the primary idea of conservation principle. For the past
several decades, as this paradox had no explanation or solution under
“idealistic” conditions, much focus has been on rationalising the lose of
energy by using non-ideal or realistic conditions such as by the addition of
resistors and inductors in the capacitor circuits [6, 5, 7, 8, 9, 10, 11, 12].
We start with redefining the basic energy-voltage relation for the capacitor
circuit as:
$E_{total}=\frac{1}{2}Q_{total}V_{total}$ (1)
where, $E_{total}$ is the total energy in the circuit, $Q_{total}$ is the
total charge in the circuit and $V_{total}$ is the total voltage (provided by
the source) in the circuit. We also reexpress the relation of charge and
voltage for any $N$ number of capacitors with equal capacitance $C$ as:
$Q_{total}=NC|V_{c}|$ (2)
where $|V_{c}|$ is the magnitude of the voltage across each capacitor with
capacitance $C$. This idea can be expressed using a water-charge analogy (Fig.
1b), the question in general is to distribute the volume of water from a main
tank (the source) equally among storage tanks with same volume and dimensions.
Here charge in a capacitor corresponds to volume of water, voltage corresponds
to the level (height) of water and capacitance is related to the capacity of
the tank.
Using this redefined view on the energy-voltage relation, we attempt to solve
the two-capacitor problem. The equivalent circuit for this problem can be
drawn as shown in Fig. 1c. In analogy to the water-charge model in Fig 1b,
irrespective of how the capacitors are tied up, the total charge (or volume of
water) before and after should remain same, in other words, the volume of
water pumped from the main storage tank should be equal to sum of volume of
water received at the storage tanks. In the two-capacitor problem, we can
think this analogy as the following: (1) the water from the main tank is
pumped to a first tank till it reaches a specified height (voltage =$V$), and
(2) the main supply is removed and the water from the first tank is now pumped
to a second tank until both tanks have equal volume of water. Here, it maybe
noted that we do not take any non-ideal assumptions on the storage tank
mechanisms and keep all components models in the circuit as ideal. The
leftmost circuit in Fig. 1c shows the equivalent circuit for charge storage
mechanism that occur as a result of pumping the charge from a main tank
(source) to a storage tank with capacitance $C$. The storage tank is analogous
to a capacitor with capacitance $C$ and main tank analogous to the source
voltage with voltage $V_{total}=V$, we can calculate the initial total charge
$Q_{initial}=CV=Q_{total}$, initial total voltage $V_{initial}=V=V_{total}$
and initial total energy $E_{inital}$ as:
$E_{inital}=\frac{1}{2}Q_{inital}V_{inital}=\frac{1}{2}CV^{2}$ (3)
When the switch is closed (Fig. 1a) at time $t=0^{+}$, both capacitors (with
equal capacitances $C$) become connected, and the charged capacitor now
charges the newly connected capacitor until the voltage and charge across each
capacitor at equilibrium reaches a value of $V/2$ and $Q_{initial}/2$.
According to the principle of conservation of charge, and from the water-
charge analogy, irrespective of how the capacitors are connected the total
charge before and after the switch is closed should remain the same. However,
we find that direct application of $Q=CV$ relation contradicts this very basic
assumption. This can be understood better through two examples of series and
parallel combination of capacitors. As a general case for a series combination
of capacitors, there are two possible ways by which charge reflects on the
individual capacitors having a capacitance $C$ with potential difference
$\frac{V}{2}$. Depending on polarity of the charge, the voltage across the
equivalent capacitance $C/2$ can be $\frac{V}{2}+\frac{V}{2}=V$ or
$\frac{V}{2}-\frac{V}{2}=0$, thereby reducing the total charge to $Q/2$ or 0
respectively. The equivalent circuit models in Fig 2a illustrates these two
situations. In analogy to the water-charge model, two individual storage tanks
are placed one above the other (Fig. 2a). Through the top view, the level
information of the water (charge) is not visible and is lost
($\frac{V}{2}-\frac{V}{2}=0$) , while from the side-view individual level
information of the water (charge) is visible and so added
($\frac{V}{2}+\frac{V}{2}=V$). The equivalent capacitance of the storage tank
is proportional to the ratio between total area and total height. Irrespective
of whether its a side-view or top-view, since in Fig. 2a the total area remain
constant while total height of the combined tank doubles, the overall
capacitance decreases by a factor of two. Now, although the equivalent
capacitance seems to reduce the total capacitance by a factor of two, the
capacity of the individual storage tanks do not change and so stores the same
amount of charge. It can be further observed that in the calculation of
charge, the use of top view capacitor model can be straight away avoided as it
loses the information on charge levels (voltage $=0$) and causes the total
charge to reflect as zero. On the other hand, if we consider the side view
model where the total voltage is $V$ and the equivalent capacitance $C/2$.
When calculating the charge, the reduction in capacitance reduces the total
charge by a factor of two, which is physically incorrect and again contradicts
the total amount of already available charge in the tank. So the only possible
solution in both cases (side view and top view) is to calculate the total
charge as a sum of charges contributed by individual tanks, which if using
$Q=CV$ relation for each capacitor with capacitance $C$ would mean to always
take the magnitude of individual voltage (as $Q=C|V|$).
Figure 2b shows the situation when the capacitors are connected in parallel.
The tanks are now placed one beside the other. Irrespective of whether its a
side-view or top-view, the total area of this combination increases by a
factor of two and therefore the equivalent capacitance increases by a factor
of two. In the side view, the voltage level does not change, and remain at
$Vc=V/2$. In the case of the circuit models illustrated for side view (Fig.
2b), putting a probe across the capacitor would yield a value of $V/2$, its
equivalent capacitance will be $2C$ and charge $Q=CV$. Here, the charge
relation work fine as the capacity of the tanks become additive and the levels
of voltages are not disturbed. However, the top view of the water-charge model
(Fig. 2b) result in loss of depth information, $\frac{V}{2}-\frac{V}{2}=0$ and
cannot be used for charge calculations. Here, again the only possible solution
is to account for charge in each storage tanks individually by considering the
magnitude of individual tank voltages for the charge calculations. Clearly,
the idea of equivalent capacitances to calculate the total charge do not
comply properly with the physical meaning of charge storage. Individual
treatment of capacitor is needed for preserving the physical meaning of charge
for energy calculations. Since the polarity of voltage is not important for
charge calculation, it should not be important for energy calculations as well
(which is in agreement with the $\frac{CV^{2}}{2}$ relation).
Using these ideas, we revisit the two capacitor problem when the two
capacitors are connected as shown in Fig. 1c. Each capacitors have a
capacitance of $C$ and voltage of $V/2$ across it. We can calculate the final
total charge $Q_{final}=C|\frac{V}{2}|+C|\frac{V}{2}|=CV=Q_{total}$ and show
conservation of charge as, $Q_{final}=Q_{initial}$. We can calculate the final
total voltage provided by the voltage source
$V_{final}=|\frac{V}{2}|+|\frac{V}{2}|=V=V_{total}$ as the magnitude sum of
potential difference that occur across the individual charge tanks, this
results in the conservation of voltage as, $V_{final}=V_{initial}$.
Substituting the values of $Q_{final}$ and $V_{final}$ in Eq. (1), we
calculate the final energy $E_{final}$ after the switch is closed as:
$E_{final}=\frac{1}{2}Q_{final}V_{final}=\frac{1}{2}CV^{2}$ (4)
From (3) and (4), it can be seen that energy is also conserved as,
$E_{final}=E_{initial}$.
In this letter, we present a simple and general solution to the energy
conservation paradox shown by classical two-capacitor problem. As shown by
Fig. 2, although the total charge should remain the same in all the possible
capacitor combinations, but application of conventional $CV$ relation using
equivalent circuit models makes conservation of charge fail, henceforth energy
is also not conserved. However, such a situation is physically impossible in
“idealistic” or “realistic” conditions. When the physical meaning of the
charge and voltage is preserved, by applying the conservation of voltage and
charge separately, we result in an energy equation that is general to use in
any type of capacitor based circuit analysis. By this approach we are able to
completely resolve the paradox that exists in the “idealistic” conditions.
## References
* [1] F. W. Sears and M. W. Zemansky. University Physics, page 601. (Addison-Wesley, Reading, MA, 1964).
* [2] H. Semat. Fundementals of Physics, page 370. (Holt, Rinehart and Winston, New York, 1966).
* [3] D. Halliday and R. Resnick. Physics, volume 2, page 656. (Wiley, New York, 1978).
* [4] M. A. Plonus. Applied Electromagnetics, page 178. (McGraw-Hill, New York, 1978).
* [5] R. A. Powell. Two-capacitor problem: A more realistic view. Am. J. Phys., 47, (1979).
* [6] T. B. Boykin, D. Hite, and N. Singh. The two-capacitor problem with radiation. Am. J. Phys., 70, (2002).
* [7] K. Mita and M. Boufaida. Ideal capacitor circuits and energy conservation. Am. J. Phys., 67, (1999).
* [8] Keeyung Lee. The two-capacitor problem revisited: a mechanical harmonic oscillator model approach. Eur. J. Phys., 30, (2009).
* [9] Sami M Al-Jaber and Subhi K. Salih. Energy consideration in the two-capacitor problem. Eur. J. Phys., (2000).
* [10] T. C. Choy. Capacitors can radiate: Further results for the two-capacitor problem. Am. J. Phys., (2004).
* [11] A. M. Sommariva. Solving the two capacitor paradox through a new asymptotic approach. IEE proceedings. Circuits, devices and systems, 150(3):227–231, (2003).
* [12] H. L. Neal. Kirchhoff’s Rule and the Two Capacitor Paradox. In 74th Annual Meeting of the Southeastern Section, (2007).
Figure 1: The circuit diagrams for illustrating the two-capacitor problem. (a)
shows the circuit diagram for the problem, switch is open initially at time
$t=0$ and closed soon after at time $t=0^{+}$, (b) analogy of water-storage
tank to a capacitor charge tank. The water from a main storage tank is equally
pumped to two storage tank. The main storage tank is analogous to a voltage
source, and (c) an equivalent circuit using the capacitor charge tank model.
Figure 2: Illustration shows the equivalent capacitor models using the
proposed water charge analogy. (a) shows the analogy of series capacitor
configurations. Side view preserves the depth information and the individual
voltage level contributions are added in the process. Top view results in
total loss of depth information and the individual voltage level contributions
are lost in the process, and (b) shows the analogy of parallel capacitor
configurations. Side view preserves the depth information but the individual
voltage level contributions are not added and partly lost in the process. Top
view results in total loss of depth information and the individual voltage
level contributions are lost in the process.
|
arxiv-papers
| 2009-10-28T02:16:57 |
2024-09-04T02:49:06.117031
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. P. James",
"submitter": "Alex James Dr",
"url": "https://arxiv.org/abs/0910.5279"
}
|
0910.5294
|
# DNA Breathing Dynamics in the Presence of a Terahertz Field
B. S. Alexandrov Theoretical Division and Center for Nonlinear Studies, Los
Alamos National Laboratory, Los Alamos, New Mexico 87545 V. Gelev Harvard
Medical School, Boston, Massachusetts 02215 A. R. Bishop Theoretical
Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545 A. Usheva Harvard Medical School, Boston,
Massachusetts 02215 K. Ø. Rasmussen Theoretical Division and Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico
87545
###### Abstract
We consider the influence of a terahertz field on the breathing dynamics of
double-stranded DNA. We model the spontaneous formation of spatially localized
openings of a damped and driven DNA chain, and find that linear instabilities
lead to dynamic dimerization, while true local strand separations require a
threshold amplitude mechanism. Based on our results we argue that a specific
terahertz radiation exposure may significantly affect the natural dynamics of
DNA, and thereby influence intricate molecular processes involved in gene
expression and DNA replication.
Spectroscopic techniques in the terahertz range are currently emerging as new
tools for the investigation of biological macromolecules Markelz . In spite of
the experimental difficulties (caused by the water’s giant Debye dipole
moment, which leads to a substantial dielectric relaxational loss in the THz
range) a notable amount of research effort is being devoted to the development
of sophisticated THz bio-imaging Nature Photonics . Nevertheless, very little
is known about THz-radiation’s influence on biological systems, and the
mechanisms that govern this influence. The possibility that low frequency
electromagnetic radiation may affect genetic material, enzymatic reactions,
etc. was introduced long ago Frohlich , and since then has been a subject of
constant debate. The energy of such radiation is too low to directly disrupt
any chemical bonds or cause electronic transitions. Only a resonance-type
interaction might lead to an appreciable, biological effect. In biomolecules
such interactions are possible through the ubiquitous hydrogen bonds that have
energies in the THz range. Numerous in vivo and in vitro experiments have been
conducted to clarify low frequency radiation’s ability to cause biological
effects, such as chromosomal aberration, genetic damage etc. The experimental
studies have been conducted under a variety of conditions, but mostly at
frequencies below 0.01 THz, power below 1 $mW/cm^{2}$, and short exposure
times. The data collected in these conditions led to mixed conclusions: some
studies reported significant genetic damages while others, although similar,
showed none Mutation Research . The major international research project,
”THz-bridge” THz-Bridge , which was specifically concerned with THz radiation
genotoxicity concluded that: under some specific conditions of exposure,
change in membrane permeability of liposomes was detected and an induction of
genotoxicity was observed to occur in lymphocytes. Hence, this project
confirmed the existence of THz genotoxicity, but it remains unclear under
which specific conditions such effects occur.
Recent measurements confirm that only extended (6 hours) exposure to a weak
THz field can cause genomic instability in human lymphocytes Korenstein .
Independently, it was reported that neurons exposed in vitro to powerful THz
radiation (over 30 $mW/cm^{2}$ ) cause infringement of the morphology of the
cellular membranes and intracellular structures Russ1 . The same work also
showed that at decreasing power and/or at different frequencies the
morphological changes do not occur. Recently, it was further pointed out that
exposure to a low level THz radiation can interfere with the protein-
recognition processes Korenstein II . It was also shown that exposure of mice
to 3.6 THz high power (15 $mW/cm^{2}$) radiation, for 30 min caused behavioral
changes Russ2 , while under short (5 min) exposure the changes could not be
detected.
Figure 1: Spontaneous formation of a localized permanent opening in dsDNA in
the presence of a THz field with amplitude $A=144$ pN, and $\Omega=2$ and a
spatial fluctuation with high amplitude (see text).
Thus the available experimental data strongly suggest that THz-radiation can
affect biological function, but only under specific conditions, viz. high
power, or/and extended exposure, or/and specific THz frequency.
The present work is devoted to developing a qualitative description of the
existence of such THz effects. We argue that the appropriate conditions
correspond to irradiation parameters that assure the existence of linear and
nonlinear resonances between DNA conformational dynamics and THz field Fig.1.
We will base our analysis on the Peyrard-Bishop-Dauxois (PBD) model of dsDNA 2
, which is arguably the most successful available model for describing the
local DNA pairing/unpairing (breathing) dynamics 6 . Of particular importance
for the THz effect is our previous demonstration of strong correlations
between regulatory activity, such as protein-DNA binding and transcription,
and the equilibrium propensity of dsDNA for local strand separation -8 ; 3 ; 4
; 5 . With this background it is natural to consider the influence of THz
radiation on dsDNA dynamics within the PBD modeling framework. One
complication is that the specific physical nature of the interactions between
DNA and the THz electromagnetic field is not known in detail. However, given
the sensitivity of THz radiation to the strand pairing state of DNA 15 , we
will here simply augment the PBD to include a drive in the THz frequency
range, without specifying the precise nature of the underlying physical
coupling. Since the wavelength of the THz field is larger than 100 $\mu m$ and
the characteristic size the investigated dsDNA sequences is less than 20 $nm$
it is reasonable to consider an uniform THz field. In this fashion the
breathing dynamics (represented by the normalized separation $y_{n}$ of
complementary bases) of the $n$’th base pair in the presence of a
monochromatic spatially homogeneous external field is described by
$\displaystyle m\ddot{y}_{n}=$ $\displaystyle-$ $\displaystyle
U^{\prime}(y_{n})-W^{\prime}(y_{n+1},y_{n})-W^{\prime}(y_{n},y_{n-1})$ (1)
$\displaystyle-$ $\displaystyle m\gamma\dot{y}_{n}+A\cos{(\Omega t)},$
where, the Morse potential $U(y_{n})=D_{n}(\exp{(-a_{n}y_{n})}-1)^{2}$
represents the hydrogen bonding of the complementary bases. The parameters
$D_{n}$ and $a_{n}$ depend on the type of the base pair (A-T or G-C).
Similarly,
$W(y_{n},y_{n-1})=\frac{1}{2}\chi[y_{n},y_{n-1}](y_{n}-y_{n-1})^{2}$ with
$\chi[y_{n},y_{n-1}]=k(1+\rho\exp{(-\beta(y_{n}-y_{n-1}))})$ represents the
stacking energy between consecutive base pairs. The term $m\gamma\dot{y}_{n}$
is the drag caused by the solvent while $A\cos{(\Omega t)}$ is the (THz)
drive. For simplicity, we consider here a homogeneous poly(A)
note_on_parameters DNA molecule with 64 base pairs. Our numerical simulations
of this set of nonlinear coupled equations (using periodic boundary
conditions) showed that the primary response of the system is spatially
homogeneous (Fig.1 the first 40 $ps$). However, we observed that a presence of
fluctuations may lead to the creation of persistent spatially localized
openings of the double-stranded molecule, Fig.1. Because, such openings
(bubbles) are known to functionally affect dsDNA we focus here on
characterizing the conditions under which these states appear.
Since, the primary response of of the driven system is spatially uniform it
can be understood in terms of a single classical damped and driven Morse
oscillator. Such systems have been studied quite extensively in many other
contexts 7 , and are known to display rich dynamical behavior including limit
cycles, period-doubling bifurcations etc. An illustration of this is shown in
Fig.2, A, for our
Figure 2: A) Bifurcation diagram for a poly(A) molecule in an external THz
field with amplitude $A=144$ pN. B) The instability curves for wave vector
$k=0$ (solid curve) and $k=\pi$ (dashed curve) are plotted. On the vertical
axis is the amplitude of the external field, and on the horizontal axis is the
frequency $\Omega$ of the THz field.
system parameters, in terms of the Poincaré section $\dot{y}_{n}(t=mT)$, where
$T=2\pi/\Omega$, versus the driving frequency $\Omega$ for several integer
values of $m$. Feigenbaum’s period-doubling route to chaos is clearly seen in
this figure, which demonstrates that this system’s main instability mechanism
is through period-doubling. As described earlier, our interest is to
investigate whether the presences of a THz field can lead to the creation of
spatially localized unbinding of the DNA double-strand. Such states are most
easily created in connection with the period-doubling events, as a result of a
spatially inhomogeneous perturbation. In order to investigate this phenomenon
we adopt a method developed in Ref. 9 . In this work the linear stability
analysis (based on the Floquet theorem) of a single Morse oscillator was
performed, and it was shown that in a rotating-wave approximation Bessel the
dynamics ensuing from a given perturbation is governed by the extended Hill
equation. The stability analysis of the Hill equation has been developed by
Ince 11 based on his extension of the Whittaker’s method of solving Mathieu’s
equation 20 . Since, our system is spatially extended, a perturbation may be
spatially inhomogeneous. In this case the response of the system is
characterized by $y_{n}(t)=y^{0}(t)+z_{n}(t)$, where $y^{0}(t)$ is the
spatially homogeneous and temporally periodic solution described above (see
Fig. 2), A, and $z_{n}(t)$ represents the perturbations in whose dynamics we
are interested. In the most general form the perturbation is given in terms of
Fourier modes: $z_{n}(t)=\sum_{k}\exp[ikn]\xi_{k}(t)$ with wavenumber, $k$,
and amplitude $\xi_{k}(t)$. As with the single oscillator, the dynamics of the
Fourier amplitudes $\xi_{k}(t)$ of the perturbation are governed by set of
uncoupled extended Hill equations. However, the coefficients of these
equations depend on the wave-number $k$. The Ince stability analysis allows us
to analytically determine note1 the Floquet exponents (i.e. the stability
criteria) as functions of the spatial profile as given by $k$. For more
technical details on the method see Ref. 12 . The result of our analysis are
the stability curves dividing the parameter space $(A/m,\Omega,k)$ into
stability and instability zones. Figure 2, panel B shows the stability
boundaries of a spatially homogeneous perturbation ($k=0$, solid curve) and of
a staggered ($k=\pi$, dashed curve) perturbation. Perturbations represented by
other wave-numbers have stability boundaries in-between those shown in Fig. 2,
B. Below these curves the spatially homogeneous states perform stable temporal
oscillations at the frequency of the external drive and its higher harmonics,
while above the curve the perturbation leads to period-doubled oscillations.
It is clearly seen that for driving frequencies below $\Omega\sim 2.5$ THz
spatially homogeneous ($k=0$) perturbations lead to the strongest instability
(occurring at the lowest value of the driving amplitude). However, above
$\Omega\sim 2.5$ THz the staggered ($k=\pi$) perturbation becomes the primary
instability mode. In the region above the stability boundary the period-
doubled state will therefore tend to adopt the spatial profile of the most
unstable Fourier-mode (homogeneous or staggered). The presence and location of
the
Figure 3: The $k=\pi$ spatial pattern, at $\Omega=2.52$ THz, is presented. The
upper panel shows evolution of the system. The lower panel shows the power
spectrum of the breather motion. Figure 4: Breather formation at $\Omega=2.00$
THz. The upper panel shows the evolution of the breather. The lower panel
shows the power spectrum of the breather motion.
obtained instability curves (Fig. 2, B) were verified by exploring the
parameter-space $(A/m,\Omega,k)$ by direct simulations of Eq. (1). The
dynamics of small spatially random (uniformly distributed in the interval
$[-\epsilon,\epsilon]$ and thus containing all wave-numbers) perturbation
$z_{n}$ introduced on top of the spatially homogeneous and temporally periodic
solution $y^{0}(t)$ was followed numerically. As this perturbation injects
energy into all wave-numbers, the ensuing dynamics will emphasize the mode
with the strongest instability. For point A (Fig. 2, B), the results are shown
in Figure 3. The upper panel shows that after the introduction of the
perturbation (at 40 ps) a spatially staggered structure forms. The power
spectrum of this dynamics (lower panel in Fig. 3), shows that the frequency of
the vibrations is mainly $\Omega/2$, i.e. a period-doubling transition has
occurred as expected from the above analysis. The power spectrum (lower panel)
shows that the signature of the drive persists at $\Omega$, in addition to a
component at zero frequency, which results from the asymmetry of the base-pair
potential $U(y_{n})$. A similar scenario is observed for driving frequencies
below $\Omega\sim 2.5$ THz, but in these cases the solutions remain spatially
homogeneous after the introduction of the perturbation. The fact that the
instability curves for $0<k<\pi$ lie between the two curves in Fig. 2, B shows
that it is, in general, impossible for any spatially localized feature with a
finite ($2~{}\mbox{base pairs}<\lambda<\infty$) length scale $\lambda$ to
emerge as a result of a linear instability. This is in contrast with many
other nonlinear condensed matter models, where spatially localized and
temporally periodic modes (intrinsic localized modes or breathers) commonly
arise through linear modulational instabilities. For our model of dsDNA a
perturbation of finite amplitude $\epsilon$ is required to create localized
unbindings of the DNA double strand through nonlinear mechanisms. An
illustration of this is given in Fig. 4 for $\Omega=2.0$ THz, where the
dynamics caused by a random perturbation with $\epsilon>0.14$ is shown to
result in a localized opening of the DNA double strand. Again the power
spectrum, given in the lower panel of Fig. 4 shows, as expected, that the
localized state vibrates at the frequency $\Omega/2$ in contrast to the
uniform background, which remains synchronized to the external drive at
$\Omega$. A more detailed study of the nonlinear mechanisms underlying the
creation of such localized unbinding states reveals that a localized injection
of a certain amount of energy is required. Due to the asymmetry of the base
pair potential, an effective way to achieve this is a localized compression of
the double strand. Using this insight we found the perturbation
$z_{n}=0.42\cos{(\frac{\pi}{4}(n-n_{0}))}$, for $-4\leq n-n_{o}\leq 4$
($z_{n}=0$ otherwise) to be an effective perturbation for the creation of a
localized unbinding state at $n_{0}$, Fig.1. The phase portrait (not shown) of
the base pair $n_{0}$, at the transient time period just before the breather
is established, reveals the breather formation mechanism. First, the locally
injected energy from the compression must be sufficient for a few consecutive
base pairs to undergo a temporary local melting transition, i.e. reach the
plateau of the Morse potential. Subsequently, the stacking interactions and
friction dissipate the energy to gradually reach a steady state. At given
driving amplitude and frequency, the dsDNA molecule has two steady
conformational states with different average energies and amplitudes. Only in
these states is the externally pumped energy exactly compensated by the energy
dissipated by the friction. The first state is vibrating predominantly at the
driving frequency $\Omega$, while the second mainly oscillates at $\Omega/2$.
Hence, the breather formation proceeds in three stages: 1) all the base pairs
are in the first state then; 2) a few of them undergo a temporary melting
transition as a result of the perturbational energy injection. Finally, 3) the
perturbed base pairs reach the second steady conformational state, after
dissipating their additional energy.
In summary, we have found that a THz field may cause dynamical separations of
the DNA double-strand. In the presence of weak perturbations e.g. thermal
fluctuations, small amplitude response occurs (at half the driving frequency)
either in a spatially uniform manner or, at higher-frequencies, in an unusual
spatially dimerized form. In the latter case, neighboring base pairs oscillate
in an out-of-phase fashion. However, large localized openings (bubbles) in the
DNA double strand can only occur via a nonlinear mechanism requiring a spatial
perturbation above a certain amplitude threshold that is determined by the
intensity and the frequency of the THz field.
We previously showed experimentally that the introduction of an artificial
permanent bubble in dsDNA, via a short mismatched segment, is in fact
sufficient for transcription even in the absence of any auxiliary factors 5 .
Our present finding of large localized openings resulting from an external THz
field therefore underscores the importance of including the interactions of
genomic DNA with the surrounding environment in DNA models. The amplitudes of
the resulting openings observed in the presence of a THz field are significant
compared to the dynamic signatures of protein binding sites and transcription
start sites 4 . This suggests that THz radiation may significantly interfere
with the naturally occurring local strand separation dynamics of double-
stranded DNA, and consequently, with DNA function.
Based on the model results present here, we believe that the main effect of
THz radiation is to resonantly influence the dynamical stability of the dsDNA
system. Hence, our instability curves define the parameters space (i.e. the
amplitude of the field (or the power), and frequency) in which THz radiation
can have an immediate effect. In contrast, nonlinear instability may occur at
any point of the instability diagram but it requires fluctuations with
significant amplitudes. In biological systems, such fluctuations are generated
thermally. Hence, the occurrence of a fluctuation with sufficiently large
amplitude is very rare, and therefore extended exposure is required for the
THz effect to take place, via nonlinear instabilities, especially if the power
is small. In this framework, it is than natural that the character of THz
genotoxic effects are probabilistic rather than deterministic.
We acknowledge Dr. Voulgarakis for initial discussions regarding the subject
of this work.This research was carried out under the auspices of the U.S.
Department of Energy at Los Alamos National Laboratory under Contract No. DE-
AC52-06NA25396. and it was supported by the National Institutes of Health (R01
GM073911 to A.U.).
## References
* (1) A. G Markelz, IEEE Journal of Selected Topics in Quantum Electronics, 14, 180 (2008).
* (2) L. Ho, M. Pepper, P. Taday, Nature Photonics, $\bf{2}$, 541 (2008). A. P. George et al. Opt. Exp. $\bf{16}$, 1578 (2008).
* (3) H. Frohlich, Proc. Nat. Acad. Set. USA, $\bf{72}$, 4211 (1975).
* (4) L. Verschaeve, Mutation Research, $\bf{681}$, 259 (2009).
* (5) http://www.frascati.enea.it/THz-BRIDGE/, (2004).
* (6) A. Korenstein-Ilan et al. Rad. Res. $\bf{170}$, 224 (2008).
* (7) J. S. Olshevskaya et al., Proc. IEEE Sibircon, 210 (2008);J. S. Olshevskaya et al., Zhurn. Vyss. Nervn. Deyat. Im. P. Pavlova, $\bf{59}$(3), 353 (2009).
* (8) A. Homenko et al.,Bioelectromagnetics $\bf{30}$,167 (2009).
* (9) N. P. Bondar et al., Bull. of Exp. Biol. and Med., $\bf{145}$, 4 (2008).
* (10) M. Peyrard, and A.R. Bishop, Phys. Rev. Lett. $\bf{62}$, 2755, (1989);T. Dauxois, M. Peyrard, A.R. Bishop, Phys. Rev. E $\bf{47}$, R44 (1993).
* (11) B. S. Alexandrov et al., J. Phys. Cond. Matter $\bf{21}$, 034107 (2009), and references therein.
* (12) B. S. Alexandrov et al., Nucleic Acids Res. 37, 2405 (2009).
* (13) C.H. Chu et al.,Nucleic Acids Res. $\bf{32}$, 1584 (2004).
* (14) C.H. Chu et al., B. Phys. J. $\bf{95}$, 597 (2008).
* (15) B. S. Alexandrov et al., PLoS Comp. Biol. $\bf{5(3)}$, e1000313 (2009); A. Usheva, T. Shenk, Proc Natl Acad Sci US $\bf{93}$, 13571, (1996).
* (16) M. Brucherseifer et al., Appl. Phys. Lett. $\bf{77}$, 11 (2000).
* (17) The specific parameter values are: $D_{n}$ =0.05 eV, $a_{n}$ = 4.2 $\AA^{-1}$, k=0.025 eV$\AA^{-2}$, $\beta$=0.35 $\AA^{-1}$, and $\rho$=2. The linear resonance frequency is $\omega=$1.22 THz. For the solvent drag we use $\gamma=1$ps-1, consistent with a typical relaxation time of water.
* (18) C. Lie George and Yuan Jian-Min J. Chem. Phys. $\bf{84}$ (10) (1986).
* (19) A.M. Samson et al., Radiofizika $\bf{33}$, 49 (1990).
* (20) In this specific rotating-wave approximation the Morse potential is expanded using the equality: $\exp{(r\cos(\phi))}=I_{0}(r)+2\sum^{\infty}_{n=1}{I_{n}(r)\cos(n\phi)}$ where $I_{n}(r)$ are modified Bessel functions of the first kind.
* (21) E. L. Ince, Roy. Astronom. Soc. $\bf{75}$, 436 (1915).
* (22) E. T. Whittaker, Proc. Edin. Math. Soc. $\bf{xxxii}$, 76 (1913).
* (23) For the analytical work we used a harmonic stacking potential, $\chi=const$.
* (24) A. Vanossi et al., Phys. Rev. E $\bf{62}$, 7353 (2000).
|
arxiv-papers
| 2009-10-28T05:20:33 |
2024-09-04T02:49:06.120689
|
{
"license": "Public Domain",
"authors": "B. S. Alexandrov, V. Gelev, A. R. Bishop, A. Usheva, and K. O.\n Rasmussen",
"submitter": "Boian Alexandrov S",
"url": "https://arxiv.org/abs/0910.5294"
}
|
0910.5339
|
# On Physically Secure and Stable Slotted ALOHA System
Yunus Sarikaya, Ozgur Ercetin
Faculty of Engineering and Natural Sciences Sabanci University, Istanbul,
Turkey
###### Abstract
In this paper, we consider the standard discrete-time slotted ALOHA with a
finite number of terminals with infinite size buffers. In our study, we
jointly consider the stability of this system together with the physical layer
security. We conduct our studies on both dominant and original systems, where
in a dominant system each terminal always has a packet in its buffer unlike in
the original system. For $N=2$, we obtain the secrecy-stability regions for
both dominant and original systems. Furthermore, we obtain the transmission
probabilities, which optimize system throughput. Lastly, this paper proposes a
new methodology in terms of obtaining the joint stability and secrecy regions.
## I INTRODUCTION
Wireless multiple-access broadcast networks have received significant interest
from researchers in the past. Slotted ALOHA is one of the basic class of such
networks, and a large number of random multiple access algorithms are devised
as modifications of this basic system. The two important issues of wireless
systems, i.e., the stability and security, have been separately studied in the
context of slotted ALOHA and the wireless broadcast networks. Basically,
stability requires that queue sizes remain finite when time goes to infinity.
Stability in slotted ALOHA has been investigated in [1], [2]. These results
have led to further studies, where various bounds and stability regions are
obtained for which the queues are stable [3], [4]. In [5], sufficient and
necessary conditions for the stability of the system are obtained and for two
user case $(N=2)$, the stability region is identified. More recent studies
have aimed at obtaining tighter bounds for the stability [6]. On the other
hand, secure communication over physical layer was first introduced by Shannon
[7] and Wyner [8]. Recently, physically secure communication is gaining more
attention and a plethora of work have emerged on this issue. The security of a
single broadcast channel is investigated in [9], [10]. The secure rate
allocation vectors are determined for gaussian channels in [11], [12]. [13]
considers fading channels for which the perfect secrecy regions and the
optimal power allocation vectors maximizing secrecy region, are obtained.
These works investigated problems only in the security context. In [14],
stability and security are combined in the context of wireless broadcast
networks, where a base station sends confidential messages to the users. In
this system, only downlink channels are considered, where no contention is
taking place.
In this paper, we jointly consider the stability and security issues for
slotted ALOHA systems, where each user wants to communicate with a single base
station. The broadcast channel is modeled as Rayleigh fading channel, where
$N$ user nodes send their confidential messages to a base station as shown in
Figure 1. Each message should be kept secret from other users. Hence, all
users except the transmitting one are eavesdroppers. In the meantime, the
stability of the system should be maintained, i.e., the sizes of the queue for
each user should be finite when time goes to infinity. All transmitters are
assumed to be synchronized, and the reception of a packet starts at the
beginning of a slot and ends at the end of a slot, i.e., each packet
transmission occupies exactly one time slot. Also, each user transmits with a
probability, $q_{i}$, at each time slot.
Figure 1: Fading Broadcast Network
In prior studies, the secrecy region has always been studied at the symbol
level, and thus the secrecy region with respect to transmission probabilities
has not been examined before and this perspective is introduced in our paper.
Let us define “secrecy-stability region” as the collection of transmission
probabilities, $q_{i}$, that satisfies both the stability and the security
conditions. Our goal is to find the secrecy-stability region and obtain the
optimal transmission probabilities, which maximize the system throughput. For
$N=2$, we specify the optimal transmission probabilities for the dominant
system, where it is assumed that users always have a packet to send in their
buffer. In the original system; however the queues may not always have a
packet to transmit, and in this case, we show that the maximized system
throughput does not depend on the transmission probabilities.
The rest of the paper is organized as follows: In Section II, we introduce the
channel model, and give the definitions of the secrecy and the stability. In
Section III and IV, we present our results for the dominant and original
systems. In Section V, we conclude the paper by summarizing our contributions.
## II Channel Model and Secrecy Capacity for Downlink Channels
We consider a wireless broadcast network operating on a single frequency
channel. We assume that the channels from each user to base station and other
users are Rayleigh fading broadcast channels, in which each output signals
obtained by the base station and other users are corrupted by multiplicative
fading gains in addition to an additive Gaussian noise as:
$Y_{jn}=h_{ijn}X_{i}+w_{ijn},\mbox{ for }1<i<N,$ (1)
where $X_{i}$ denotes the message transmitted by $i^{th}$ user, $Y_{jn}$ is
the channel output at user $j$, $h_{ijn}$ is the fading coefficient for the
channel between $i^{th}$ user and $j^{th}$ user, and $w_{ijn}$ is Gaussian
noise term with zero mean and unit variance at the $n^{th}$ symbol time.
The secrecy level of confidential message, $W_{i}$, transmitted from user $i$
to the base station is measured by the following equivocation rate [14]:
$R_{i}\leq
lim_{n\rightarrow\infty}\frac{1}{n}H(W_{i}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})$
(2)
The perfect secrecy is achieved when the transmission rate satisfies (2) and
the secrecy region is defined as the set of all achievable rate vectors such
that the perfect secrecy is achieved [8]. In [14], the secrecy region of
fading broadcast channels in the downlink is obtained as:
$\displaystyle R_{s}=\begin{cases}&\bigcup(R_{1},R_{2},...,R_{N}):\\\
&R_{i}\leq R_{s,i}=\underset{j\neq i,1\leq j\leq
N}{\min}E_{\overline{h}\epsilon A(i)}\begin{aligned}
&\left[\log(1+P|h_{i}|^{2})\right.\\\
&\left.-\log(1+P|h_{ij}|^{2})\right]\end{aligned},\\\ &\mbox{for }1\leq i\leq
N\end{cases}$ (3)
where $\overline{h}$ defines channel state, $h_{i}$ denotes the channel gain
between $i^{th}$ user and the base station, and $h_{ij}$ denotes channel gain
between $i^{th}$ and $j^{th}$ users. $A(i)$ is the set of all channel states
for which the channel gain between $i^{th}$ user and the base station is the
largest. In addition, each user transmits with the same power level denoted as
$P$.
In addition, we define the stability of a queue as in [6], i.e., a queue is
stable if it satisfies the following
$\displaystyle lim_{t\rightarrow\infty}Pr[s_{i}(t)<x]=F(x),\mbox{ and }$
$\displaystyle lim_{x\rightarrow\infty}F(x)=1,$ (4)
where $s_{i}(t)$ is the size of the queue at time $t$. Namely, the queue size
should be finite at any time to achieve stability.
In the following, we investigate the secrecy-stability regions for dominant
and original systems.
## III Dominant Aloha Uplink Channel
The analysis of dominant systems was previously investigated in [1]-[6]. For
the dominant system, it is shown that the stability condition is as follows:
###### Lemma 1
[3] If
$\lambda_{i}<Q_{i}$ (5)
for all $i$ $(i=1,...,N)$, then the system is stable, where $\lambda_{i}$ is
the arrival rate and $Q_{i}$ is the successful transmission probability for
fading channels calculated as:
$Q_{i}=(1-p_{f,i})q_{i}\prod_{j=1\\\ j\neq i}^{N}(1-q_{j}),$ (6)
where $p_{f,i}$ is the average failure probability of user $i$ due to fading.
Since $p_{f,i}$ is constant, we define $\lambda_{i}^{\prime}$ as
$\lambda_{i}/p_{f,i}$. Then, the stability condition can be rewritten as:
$\lambda_{i}^{\prime}<q_{i}\prod_{j=1\\\ j\neq i}^{N}(1-q_{j})$ (7)
Lemma 1 specifies the stability condition, but does not give any result about
which specified arrival rates lead to a positive stability region.
###### Lemma 2
[3] The following condition should be satisfied to have positive stability
region:
$x^{N-1}-\prod_{i=1}^{N}(x+\lambda_{i}^{\prime})>0\mbox{ for any }x>0$ (8)
For $N=2$, expression in (8) can be written as:
$\sqrt{\lambda_{1}^{\prime}}+\sqrt{\lambda_{2}^{\prime}}<1$ (9)
The proofs of Lemma 1 and 2 can be found in [3].
Now, we are ready to derive the perfect secrecy condition for dominant ALOHA
systems.
###### Theorem 1
If
$\displaystyle\rho_{i}\geq q_{i}\prod_{j=1,j\neq i}^{N}(1-q_{j})$ (10)
$\displaystyle\mbox{for }\rho_{i}=\frac{R_{s,i}}{R_{i}}$
for all $i$ $(i=1,...,N)$, then the system is secure. Note that $\rho_{i}$
defines the ratio between the perfect secrecy capacity, $R_{s,i}$, defined as
in (3) and the capacity of fading channels, $R_{i}$.
Proof
The security region given in (3) is computed for downlink channels, where
there is no channel contention. However, in our model, we consider uplink
channels, where packets when transmitted simultaneously collide with each
other. We assume that collision results in scrambled bits and thus the
received packets cannot be correctly decoded. Therefore, we assume that the
packets in collision have no information value. Let us define new events
$Z_{1}^{i}$, $Z_{2}^{i}$ and $W_{i}$ as:
$\displaystyle Z_{1}^{i}$ $\displaystyle=$
$\displaystyle\begin{cases}1,&\mbox{ transmission for }i^{th}\mbox{ user }\\\
0,&\mbox{ no transmission for }i^{th}\mbox{ user }\end{cases}$ $\displaystyle
Z_{2}^{i}$ $\displaystyle=$ $\displaystyle\begin{cases}1,&\mbox{ no collision
for }i^{th}\mbox{ user }\\\ 0,&\mbox{ collision for }i^{th}\mbox{ user
}\end{cases}$ (11) $\displaystyle W_{i}$ $\displaystyle=$
$\displaystyle\begin{cases}\hat{W_{i}},&\mbox{ if }Z_{1}^{i}=1\mbox{ and
}Z_{2}^{i}=1\\\ 0,&\mbox{ otherwise }\end{cases}$
where $\hat{W_{i}}$ defines the event where messages are transmitted with no
collision.
The relationship between the equivocation rates of $W_{i}$ and $\hat{W_{i}}$
is:
$\displaystyle
H(\hat{W_{i}}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})=$
$\displaystyle
P(Z_{1}^{i}=1,Z_{2}^{i}=1)H(W_{i}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})$
(12)
Note that, the equivocation rate of $\hat{W_{i}}$ is the same as in the
downlink channels given in (3).
We now determine a bound on transmission probabilities, $q_{i}$, as follows:
$\displaystyle\small R_{i}$ $\displaystyle\overset{(a)}{\leq}$
$\displaystyle\frac{H(W_{i}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})}{n}$
(13) $\displaystyle\overset{(b)}{=}$
$\displaystyle\frac{H(\hat{W_{i}}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})}{n}\frac{1}{P(Z_{1}^{i}=1,Z_{2}^{i}=1)}$
$\displaystyle=$
$\displaystyle\frac{H(\hat{W_{i}}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})}{P(Z_{1}^{i}=1)P(Z_{2}^{i}=1|Z_{1}^{i}=1)n}$
$\displaystyle\overset{(c)}{=}$ $\displaystyle\frac{1}{q_{i}\prod_{j\neq
i}{(1-q_{j})}}\frac{H(\hat{W_{i}}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})}{n}$
$\displaystyle\overset{(d)}{=}$ $\displaystyle\frac{1}{q_{i}\prod_{j\neq
i}{(1-q_{j})}}R_{s,i}$
where (a) follows from the perfect secrecy condition, (b) follows from the
branching property of entropy as in (12), (c) is obtained by inserting the
collision probability, (d) follows from the fact that a successful
transmission on the uplink has a secrecy capacity equal to the secrecy
capacity over a downlink channel.
Figure 2: Time Scaling for Secrecy Regions of Uplink and Downlink Channels
Furthermore, in (13), $R_{i}$ is the capacity of fading channels, which can be
obtained by Shannon capacity, $\log(1+P|h_{i}|^{2})$ and $R_{s,i}$ is secrecy
capacity of downlink channels given by (3).
By taking the ratio of secrecy capacity, $R_{s,i}$, and the shannon capacity,
$R_{i}$, we obtain the following perfect secrecy condition in terms of the
transmission probabilities:
$\rho_{i}\geq q_{i}\prod_{j=1\\\ j\neq i}^{N}(1-q_{j})$ (14)
Intuitively, as shown in Figure 2, the condition in (14) can be interpreted as
the scaled version of a downlink channel, when all slots with collisions are
removed. In this case, out of $n$ slots only $q_{i}\prod(1-q_{j})$ portion of
them carry packets that can be correctly decoded, and the security condition
for this system is the same as the condition for downlink channel with no
contention.
After we obtain the secrecy and stability regions, the intersection of these
will give us the joint secrecy-stability region, where both secrecy and
stability are achieved.
###### Corollary 1
If $\lambda_{i}^{\prime}<\rho_{i}$, for all $i$ $(i=1,...,N)$, then there
exists positive secrecy-stability region.
Proof of this corollary can easily be derived from the secrecy and the
stability condition (10) and (7) respectively.
Overall, Lemma 2 and Corollary 1 should be satisfied to have positive secrecy-
stability region.
For $N=2$, there are three different secrecy-stability regions as illustrated
in Figure 3, which lead to different solutions to the maximization of system
throughput. As seen in Figure 3(a), case 1 suggests more tighter secrecy bound
and as a result we obtain smaller secrecy-stability region. In this case, the
channel with eavesdropper is better compared to other cases. However, in case
3, the secrecy condition loses its influence on the secrecy-stability region,
which is only equal to the stability region. In real life, we may encounter
with this case when there is a wall between eavesdropper and transmitter,
which worsens the channel.
---
(a) Case 1
---
(b) Case 2
---
(a) Case 3
Figure 3: Secrecy-Stability Regions for Dominant System
###### Theorem 2
For $N=2$, the optimal transmission probabilities are as follows:
(1) when
$\sqrt{\rho_{1}}+\sqrt{\rho_{2}}\leq 1$ (15)
$\displaystyle
q_{1}=\frac{(1+\rho_{1})}{2}\bar{+}\frac{\sqrt{(1+\rho_{1})^{2}-4(\rho_{1}+\rho_{2})}}{2}$
$\displaystyle
q_{2}=\frac{(1+\rho_{2})}{2}\bar{+}\frac{\sqrt{(1+\rho_{2})^{2}-4(\rho_{1}+\rho_{2})}}{2}$
(16)
(2) when
$\displaystyle\sqrt{\rho_{1}}+\sqrt{\rho_{2}}\geq 1$
$\displaystyle\sqrt{\rho_{1}}+\sqrt{\lambda_{2}^{\prime}}<1$ (17)
$\displaystyle\sqrt{\rho_{2}}+\sqrt{\lambda_{1}^{\prime}}<1$
$\displaystyle q_{1}=\sqrt{\rho_{1}},\mbox{ }q_{2}=1-\sqrt{\rho_{1}}\mbox{ or
}$ $\displaystyle q_{1}=1-\sqrt{\rho_{2}},\mbox{ }q_{2}=\sqrt{\rho_{2}}$ (18)
(3) when
$\displaystyle\sqrt{\rho_{1}}+\sqrt{\rho_{2}}\geq 1$
$\displaystyle\sqrt{\rho_{1}}+\sqrt{\lambda_{2}^{\prime}}>1$ (19)
$\displaystyle\sqrt{\rho_{2}}+\sqrt{\lambda_{1}^{\prime}}>1$
$\displaystyle q_{1}=\sqrt{\lambda_{1}^{\prime}},\mbox{
}q_{2}=1-\sqrt{\lambda_{1}^{\prime}}\mbox{ or }$ $\displaystyle
q_{1}=1-\sqrt{\lambda_{2}^{\prime}},\mbox{ }q_{2}=\sqrt{\lambda_{2}^{\prime}}$
(20)
Proof
The throughput optimization problem can be formulated as follows:
$\displaystyle\mbox{max }S$ $\displaystyle=$ $\displaystyle
q_{1}(1-q_{2})(1-p_{f,1})$ $\displaystyle+q_{2}(1-q_{1})(1-p_{f,2})$ s.t.
$\displaystyle q_{1}(1-q_{2})\leq\rho_{1}$ (26) $\displaystyle
q_{2}(1-q_{1})\leq\rho_{2}$ $\displaystyle\lambda_{1}^{\prime}<q_{1}(1-q_{2})$
$\displaystyle\lambda_{2}^{\prime}<q_{2}(1-q_{1})$ $\displaystyle 0\leq
q_{1},q_{2}\leq 1,$
where $p_{f,1}$ and $p_{f,2}$ denote the probability of channel failure of the
first and second users respectively.
The objective function in (III) can be rewritten as a sum of two linear
variables, e.g., ($X(\mbox{for }q_{1}(1-q_{2}))+Y(\mbox{for
}q_{2}(1-q_{1}))$), while these linear variables are also constrained by
linear inequalities. From the basic knowledge of linear programming, the
optimal solution is known to be located at the corners of feasible region.
Thus, as long as $q_{1}$ and $q_{2}$ are in $[0,1]$, we expect that the
optimal solution is to appear on the boundary of the feasible region.
(1) First, we consider the case when the optimal solution is achieved at the
boundary of the secrecy region given in Figure 3(a). Then, the Lagrangian to
solve optimization problem in (III)-(26) is given by:
$\displaystyle L$ $\displaystyle=$ $\displaystyle
q_{1}(1-q_{2})(1-p_{f,1})+q_{2}(1-q_{1})(1-p_{f,2})$
$\displaystyle-\beta_{1}(q_{1}(1-q_{2})-\rho_{1})-\beta_{2}(q_{2}(1-q_{1})-\rho_{2})$
$\displaystyle+\alpha_{1}(q_{1}(1-q_{2})-\lambda_{1}^{\prime})+\alpha_{2}(q_{2}(1-q_{1})-\lambda_{2}^{\prime}),$
where $\beta_{1}$ and $\beta_{2}$ are lagrange multipliers for inequalities in
(26) and (26), and $\alpha_{1}$ and $\alpha_{2}$ for inequalities in (26) and
(26). Since the solution is assumed to be at the boundary of secrecy region,
$\alpha_{1}=0$ and $\alpha_{2}=0$. We take the derivative of the lagrangian
with respect to non-zero lagrange multipliers and transmission probabilities,
and equate to zero as:
$\displaystyle\frac{\partial L}{\partial q_{1}}$ $\displaystyle=$
$\displaystyle(1-q_{2})(1-p_{f,1})-q_{2}(1-p_{f,2})$
$\displaystyle-\beta_{1}(1-q_{2})+\beta_{2}q_{2}=0$
$\displaystyle\frac{\partial L}{\partial q_{2}}$ $\displaystyle=$
$\displaystyle(1-q_{1})(1-p_{f,2})-q_{1}(1-p_{f,1})$
$\displaystyle-\beta_{2}(1-q_{1})+\beta_{1}q_{1}=0$
$\displaystyle\frac{\partial L}{\partial\beta_{1}}$ $\displaystyle=$
$\displaystyle q_{1}(1-q_{2})-\rho_{1}=0$ $\displaystyle\frac{\partial
L}{\partial\beta_{2}}$ $\displaystyle=$ $\displaystyle
q_{2}(1-q_{1})-\rho_{2}=0$ (28)
By simple manipulations, we obtain $\beta_{1}$ and $\beta_{2}$ as 1, which
satisfies the condition that lagrange multipliers should be greater than zero.
For this case, we found $q_{1}$ as
$\frac{(1+\rho_{1})}{2}\bar{+}\frac{\sqrt{(1+\rho_{1})^{2}-4(\rho_{1}+\rho_{2})}}{2}$
and $q_{2}$ as
$\frac{(1+\rho_{2})}{2}\bar{+}\frac{\sqrt{(1+\rho_{2})^{2}-4(\rho_{1}+\rho_{2})}}{2}$.
Note that this solution attains a real root, when the following conditions are
satisfied : $(1+\rho_{1})^{2}-4(\rho_{1}+\rho_{2})\geq 0$ and
$(1+\rho_{2})^{2}-4(\rho_{1}+\rho_{2})\geq 0$. After some manipulations, we
see that a real solution is realized when
$\sqrt{\rho_{1}}+\sqrt{\rho_{2}}\leq 1$ (29)
(2) Figure 3(b) shows the secrecy-stability region, when the condition in (29)
does not hold, i.e., $\beta_{1}>0$ and $\beta_{2}>0$ jointly cannot be
satisfied.
First, Let $\beta_{1}\geq 0$ and $\beta_{2}=0$, then we have the following
derivatives:
$\displaystyle\frac{\partial L}{\partial q_{1}}$ $\displaystyle=$
$\displaystyle(1-q_{2})(1-p_{f,1})-q_{2}(1-p_{f,2})-\beta_{1}(1-q_{2})=0$
$\displaystyle\frac{\partial L}{\partial q_{2}}$ $\displaystyle=$
$\displaystyle(1-q_{1})(1-p_{f,2})-q_{1}(1-p_{f,1})+\beta_{1}q_{1}=0$
$\displaystyle\frac{\partial L}{\partial\beta_{1}}$ $\displaystyle=$
$\displaystyle q_{1}(1-q_{2})-\rho_{1}=0$ (30)
From the first two equations in (30), we find $q_{1}=1-q_{2}$ and by using the
third equation in (30), we obtain the solution as: $q_{1}=\sqrt{\rho_{1}}$ and
$q_{2}=1-\sqrt{\rho_{1}}$. However, this solution should satisfy the stability
condition in (26) as well:
$\displaystyle q_{2}(1-q_{1})$ $\displaystyle>$
$\displaystyle\lambda_{2}^{\prime}$
$\displaystyle(1-\sqrt{\rho_{1}})(1-\sqrt{\rho_{1}})$ $\displaystyle>$
$\displaystyle\lambda_{2}^{\prime}$
$\displaystyle\sqrt{\rho_{1}}+\sqrt{\lambda_{2}^{\prime}}$ $\displaystyle<$
$\displaystyle 1$ (31)
Similarly, when $\beta_{2}\geq 0$ and $\beta_{1}=0$, we can follow the same
discussion as before to obtain the solution as: $q_{2}=\sqrt{\rho_{2}}$ and
$q_{1}=1-\sqrt{\rho_{2}}$ when
$\sqrt{\rho_{2}}+\sqrt{\lambda_{1}^{\prime}}<1$.
Now, the optimal solution is one of these two solutions; however, since the
secrecy-stability region is not convex, we cannot determine the optimal closed
form solution.
(3) When the conditions in (29) and (31) do not hold, the secrecy-stability
region only consists of the stability region as shown in Figure 3(c). Then, we
have the following optimization problem:
max $\displaystyle S=q_{1}(1-q_{2})(1-p_{f,1})+q_{2}(1-q_{1})(1-p_{f,2})$ s.t.
$\displaystyle\lambda_{1}^{\prime}<q_{1}(1-q_{2})$ (32)
$\displaystyle\lambda_{1}^{\prime}<q_{2}(1-q_{1})$ $\displaystyle 0\leq
q_{1},q_{2}\leq 1,$
As before, we expect that the optimal solution is to appear at the boundary.
Both constraints cannot be active, so we select only one of them as active.
First, we consider the first constraint as the active constraint: The lagrange
multipliers are: $\alpha_{1}\leq 0$ and $\alpha_{2}=0$. Then, the lagrange
function is as follows:
$\displaystyle L$ $\displaystyle=$ $\displaystyle
q_{1}(1-q_{2})(1-p_{f,1})+q_{2}(1-q_{1})(1-p_{f,2})$
$\displaystyle+\alpha_{1}(q_{1}(1-q_{2})-\lambda_{1}^{\prime})$
Then, we have the following derivatives:
$\displaystyle\frac{\partial L}{\partial q_{1}}$ $\displaystyle=$
$\displaystyle(1-q_{2})(1-p_{f,1})-q_{2}(1-p_{f,2})+\alpha_{1}(1-q_{2})=0$
$\displaystyle\frac{\partial L}{\partial q_{2}}$ $\displaystyle=$
$\displaystyle(1-q_{1})(1-p_{f,2})-q_{1}(1-p_{f,1})-\alpha_{1}q_{1}=0$
$\displaystyle\frac{\partial L}{\partial\alpha_{1}}$ $\displaystyle=$
$\displaystyle q_{1}(1-q_{2})-\lambda_{1}^{\prime}=0$
If we solve these equations, we obtain the solution as:
$q_{1}=\sqrt{\lambda_{1}^{\prime}}$ and $q_{2}=1-\sqrt{\lambda_{1}^{\prime}}$.
Similarly, if we let $\alpha_{2}\leq 0$ and $\alpha_{1}=0$, then we get the
solution as follows: $q_{1}=1-\sqrt{\lambda_{2}^{\prime}}$ and
$q_{2}=\sqrt{\lambda_{2}^{\prime}}$.
## IV Original Aloha Uplink Channel
In this section, we consider systems where the buffers of the users do not
always have packets. Let $p_{e,i}$ be the probability of queue of user $i$
being empty. Then, the secrecy condition is defined as follows:
###### Theorem 3
If
$q_{i}\prod_{j=1,j\neq i}^{N}((1-p_{e,j})(1-q_{j})+p_{e,j})\leq\rho_{i}$ (35)
for all $i$ $(i=1,...,N)$, then the system is secure.
Proof
In Theorem 1, we obtained the secrecy condition for dominant systems, where
there are no empty queues. The method of the proof of Theorem 3 is the same,
where we want to determine the portion of time when there is a single
transmission and no collision. However, in the original system the probability
of this event is different from the one in a dominant system. Let us define
new event, $E_{i}$ as:
$\displaystyle E_{i}=\begin{cases}1,&\mbox{ queue of }i^{th}\mbox{ user is not
empty}\\\ 0,&\mbox{ queue of }i^{th}\mbox{ user is empty}\end{cases}$ (36)
Then, the equivocation rate for the original system is
$\displaystyle
H(\hat{W_{i}}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n})=$ (37)
$\displaystyle
P(Z_{1}^{i}=1,Z_{2}^{i}=1|E_{i}=1)H(W_{i}|Y_{1}^{n},...,Y_{i-1}^{n},Y_{i+1}^{n},...,Y_{N}^{n}),$
where
$\displaystyle P(Z_{1}^{i}=1,Z_{2}^{i}=1|E_{i}=1)=$ $\displaystyle
q_{i}\prod_{j=1,j\neq i}^{N}((1-p_{e,j})(1-q_{j})+p_{e,j})$ (38)
If we make the same mathematical operations as in (13), we obtain the
following secrecy condition:
$\displaystyle q_{i}\prod_{j=1,j\neq
i}^{N}((1-p_{e,j})(1-q_{j})+p_{e,j})\leq\rho_{i}$ (39)
Note that $\rho_{i}$ can be interpreted as proportion of time in all occupied
slots with successful transmissions with no collisions.
###### Theorem 4
For $N=2$, the secrecy condition is as follows:
$\displaystyle\frac{1+\lambda_{1}^{\prime}-\lambda_{2}^{\prime}-\sqrt{\rho_{1}^{2}((\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})^{2}-4\lambda_{1}^{\prime})}}{2\lambda_{1}^{\prime}}=q_{1}^{*}\leq
q_{1}$ $\displaystyle\mbox{ for
}q_{2}\geq\frac{\lambda_{2}^{\prime}\rho_{1}}{\rho_{1}-q_{1}^{*}\lambda_{1}^{\prime}}=q_{2}^{**}$
(40)
Due to the symmetric behavior of the system, the secrecy condition for the
second user is obtained by replacing $\rho_{1}$ by $\rho_{2}$,
$\lambda_{1}^{\prime}$ by $\lambda_{2}^{\prime}$ and $\lambda_{2}^{\prime}$ by
$\lambda_{1}^{\prime}$ in (40).
Proof
In Theorem 3, we have shown that the system is secure when
$q_{1}\leq\frac{\rho_{1}}{(1-p_{e,2})(1-q_{2})+p_{e,2}}$ (41)
Also by Little’s theorem [2], we know that
$p_{e,1}=1-\frac{\lambda_{1}}{\mu_{1}},$ (42)
where $\mu_{1}$ is the average service rate of the first user. We have the
following relationship between the service rate, $\mu_{1}$, and $\rho_{1}$:
$\mu_{1}=(1-p_{f,1})q_{1}((1-p_{e,2})(1-q_{2})+p_{e,2})\leq\rho_{1}(1-p_{f,1})$
(43)
Thus, by substituting (43) into (42), we obtain
$p_{e,1}\leq
1-\frac{\lambda_{1}}{\rho_{1}(1-p_{f,1})}=1-\frac{\lambda_{1}^{\prime}}{\rho_{1}}$
(44)
By the symmetric behavior of the system, we know that
$\displaystyle p_{e,2}$ $\displaystyle=$ $\displaystyle
1-\frac{\lambda_{2}^{\prime}}{q_{2}((1-p_{e,1})(1-q_{1})+p_{e,1})}$ (45)
$\displaystyle\geq$ $\displaystyle
1-\frac{\lambda_{2}^{\prime}}{q_{2}(\frac{\lambda_{1}^{\prime}}{\rho_{1}}(1-q_{1})+1-\frac{\lambda_{1}^{\prime}}{\rho_{1}})}$
By substituting (45) into (41), we obtain the following quadratic equation:
$\lambda_{1}^{\prime}q_{1}^{2}+\rho_{1}(\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})q_{1}+\rho_{1}^{2}\leq
0$ (46)
Interestingly, the above equation does not depend on the transmission
probability of eavesdropper, $q_{2}$. From (46), we obtain a bound on $q_{1}$
as:
$\displaystyle\mbox{max}(0,\frac{1+\lambda_{1}^{\prime}-\lambda_{2}^{\prime}-\sqrt{\rho_{1}^{2}((\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})^{2}-4\lambda_{1}^{\prime})}}{2\lambda_{1}^{\prime}})\leq
q_{1}$
$\displaystyle\leq\mbox{min}(1,\frac{1+\lambda_{1}^{\prime}-\lambda_{2}^{\prime}+\sqrt{\rho_{1}^{2}((\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})^{2}-4\lambda_{1}^{\prime})}}{2\lambda_{1}^{\prime}})$
(47)
Also note that, the term,
$\frac{1+\lambda_{1}^{\prime}-\lambda_{2}^{\prime}-\sqrt{\rho_{1}^{2}((\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})^{2}-4\lambda_{1}^{\prime})}}{2\lambda_{1}^{\prime}}$,
is positive, since $\rho_{1}\leq 1$ and $\lambda_{1}^{\prime}\geq 0$. In
addition, the term,
$\frac{1+\lambda_{1}^{\prime}-\lambda_{2}^{\prime}+\sqrt{\rho_{1}^{2}((\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})^{2}-4\lambda_{1}^{\prime})}}{2\lambda_{1}^{\prime}}$,
is always bigger than one, since from lemma 2 we know that
$\sqrt{\lambda_{1}^{\prime}}+\sqrt{\lambda_{2}^{\prime}}<1$ and so
$\lambda_{1}^{\prime}+\lambda_{2}^{\prime}<1$. Then, the solution in (47)
becomes:
$\frac{1+\lambda_{1}^{\prime}-\lambda_{2}^{\prime}-\sqrt{\rho_{1}^{2}((\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})^{2}-4\lambda_{1}^{\prime})}}{2\lambda_{1}^{\prime}})=q_{1}^{*}\leq
q_{1}$ (48)
Also, $p_{e,2}\geq 0$ which results in (49) by substituting $q_{1}^{*}$ in
(45)
$q_{2}\geq\frac{\lambda_{2}^{\prime}\rho_{1}}{\rho_{1}-q_{1}^{*}\lambda_{1}^{\prime}}=q_{2}^{**},$
(49)
Note that, when $q_{1}$ is equal to $q_{1}^{*}$ and $q_{2}$ is $q_{2}^{**}$,
then $p_{e,2}$ is zero, which means that the second user always has a packet
to transmit as in a dominant system.
Finally, we attain the following condition:
$\displaystyle\frac{1+\lambda_{1}^{\prime}-\lambda_{2}^{\prime}-\sqrt{\rho_{1}^{2}((\lambda_{2}^{\prime}-1-\lambda_{1}^{\prime})^{2}-4\lambda_{1}^{\prime})}}{2\lambda_{1}^{\prime}})=q_{1}^{*}\leq
q_{1}$ $\displaystyle\mbox{ for
}q_{2}\geq\frac{\lambda_{2}^{\prime}\rho_{1}}{\rho_{1}-q_{1}^{*}\lambda_{1}^{\prime}}$
(50)
###### Lemma 3
In order to have a stable system, the average service rate, $\mu_{i}$, should
be greater than the arrival rate. Then, we have the following stability
condition:
$\displaystyle\mu_{i}=(1-p_{f,i})q_{i}\prod_{j=1,j\neq
i}^{N}((1-p_{e,j})(1-q_{j})+p_{e,j})>\lambda_{i}$ $\displaystyle
q_{i}\prod_{j=1,j\neq
i}^{N}((1-p_{e,j})(1-q_{j})+p_{e,j})>\frac{\lambda_{i}}{1-p_{f,i}}=\lambda_{i}^{\prime}$
(51)
The secrecy and stability regions for the original system are shown in Figure
4(a) and Figure 4(b) respectively. The proof for the stability region can be
found in [3].
By combining both regions, we obtain the secrecy-stability region as
illustrated in Figure 4(c). At the points, $(q_{1}^{*},q_{2}^{**})$ and
$(q_{1}^{**},q_{2}^{*})$, the probability of queue 2 and queue 1 being empty
are zero, which is the same in the dominant system. Thus, the point
$(q_{1}^{**},q_{2}^{**})$ is located on the intersection of two stability
curves as seen in Figure 4(c).
---
(a) Secrecy Region
---
(b) Stability Region
---
(c) Secrecy-Stability Region
Figure 4: Secrecy and Stability Regions for Original System
###### Theorem 5
The optimum throughput, $S^{*}$, for any transmission probabilities in the
secrecy-stability region is equal to sum of arrival rates:
$S^{*}=\sum_{i=1}^{N}\lambda_{i}$ (52)
Proof
The system throughput is formulated as:
$S=\sum_{i=1}^{N}(1-p_{f,i})(1-p_{e,i})q_{i}\prod_{j\neq
i}\left[(1-p_{e,j})(1-q_{j})+p_{e,j}\right]$ (53)
We know that $1-p_{e,i}$ is equal to $\lambda_{i}/\mu_{i}$ and the term,
$(1-p_{f,i})q_{i}\prod_{j\neq i}(1-p_{e,j})(1-q_{j})+p_{e,j}$, is defined as
the average service rate, then for $\mu_{i}>0$ we obtain the following result:
$\displaystyle S$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\frac{\lambda_{i}}{\mu_{i}}\mu_{i}$ (54)
$\displaystyle=$ $\displaystyle\sum_{i=1}^{N}\lambda_{i}$
In theorem 5, we find out that the optimum throughput can be any point in the
secrecy-stability region. That is because, increasing the transmission
probabilities leads to a decrease in the probability of having empty queue,
and this results in a decrease in successful transmission probability. Thus,
even if we increase the transmission opportunities, success out of these
opportunities will not change.
## V Conclusion
In this paper, we have studied slotted ALOHA network, for which we have
obtained secrecy-stability conditions for the dominant and original system. We
have further obtained the optimal transmission probabilities for $N=2$. This
is the first work that jointly addresses both the secrecy and stability of a
wireless network with contention.
## References
* [1] N.Abramson, “The ALOHA system-another alternative for computer communication,” in AFIPS Conf. Proc., vol. 37, pp. 281-285, 1970.
* [2] D. Bertsekas and R. Gallager, Data Network, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1992.
* [3] B. S. Tsybakov, V. A. Mikhailov, “Ergodicity of a Slotted ALOHA System”, Problemy Preedachi Inf., vol. 15, no. 4, pp. 73-87, March 1979\.
* [4] R. Rao and A. Ephredemis, “On the stability of interacting queues in a multi-access system,” IEEE Trans. Inform. Theory, vol. 34, pp. 918-930, Sept. 1988
* [5] W. Malyshev, “A classification of two-dimensional markov chains and piecewise-linear martingales,” Dokl. Akad. Nauk USSR, vol. 3, pp. 526-528,1972.
* [6] W. Luo abd A. Ephredemis, “Stability of N interacting queues in random-access Systems”, IEEE Trans. Inform. Theory, vol. 45, pp. 1579-1587, July 1999.
* [7] C. E. Shannon, “Communication theory of secrecy systems,” Bell Syst. Tech. J., vol. 28, pp. 656-715, Oct. 1949
* [8] A. D. Wyner, ”The wire-tap channel, “Bell Syst. Tech. J., vol. 54, no. 8, pp. 1355-1387, Oct. 1975.
* [9] I. Csiszar, J. K rner, “Broadcast Channels with Confidential Messages”, IEEE Transaction on Information Theory, Vol 24,No. 3, pp. 339-348, May 1978.
* [10] R. Liu, I. Maric, P. Spasojevic, R. Yates, “Discrete memoryless interference and broadcast channels with confidential messages: Secrecy capacity regions,” IEEE Transaction on Information Theory, Vol. 54, No. 6, pp. 2493-2507, June 2008.
* [11] T. Liu, V. Prabhakaran, S. Vishwanath, “The Secrecy Capacity of a Class of Parallel Gaussian Compound Wiretap Channels”, International Symposium on Information Theory 2008, ISIT 2008, pp. 116-120, July 2008\.
* [12] S. Shafiee, N. Liu, S. Ulukus, “Secrecy capacity of the 2-2-1 gaussian MIMO wire-tap channel”, 3rd International Symposium on Communications, Control and Signal Processing, ISCCSP 2008, Vol. 12, No. 14, pp. 207-212, March 2008
* [13] Y. Liang, H. V. Poor., Shamai S., “Secure Communication Over Fading Channels”, IEEE Transaction on Information Theory, Vol. 54, No. 6, pp. 2470-2492, June 2008.
* [14] Y. Liang, H. V. Poor, L. Ying, “Wireless Broadcast Networks: Reliability, Security and Stability ”, Information Theory and Applications Workshop 2008, ITA 2008, pp. 249-255, Feb. 2008.
* [15] J. C. Arnbak and W. van Blitterswijk, “Capacity of Slotted Aloha in Rayleigh-fading channels,” IEEE J. Select. Areas Commun., vol. SAC-5, pp. 261-269, Feb. 1987.
|
arxiv-papers
| 2009-10-28T10:55:14 |
2024-09-04T02:49:06.126388
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yunus Sarikaya, Ozgur Ercetin",
"submitter": "Yunus Sarikaya",
"url": "https://arxiv.org/abs/0910.5339"
}
|
0910.5341
|
# Quantum field theoretical study of an effective spin model in coupled
optical cavity arrays
Sujit Sarkar Poornaprajna Institute of Scientific Research, 4 Sadashivanagar,
Bangalore 5600 80, India.
E-Mail: sujit@physics.iisc.ernet.in
Phone: 091 80 23612511/23619034, Fax: 091-80-2360-0228
###### Abstract
Atoms trapped in microcavities and interacting through the exchange of virtual
photons can model an anisotropic Heisenberg spin-1/2 lattice. We do the
quantum field theoretical study of such a system using the Abelian
bosonization method followed by the renormalization group analysis. We present
interesting physics due to the presence of exchange anisotropy. An infinite
order Kosterliz-Thouless-Berezinskii transition is replaced by second order XY
transition even an infinitesimal a small anisotropy in exchange coupling is
introduced. We predict a quantum phase transition between Mott insulating and
photonic superfluid phase due to detuning between the cavity and laser
frequency, a large detuning favours the photonic superfluid phase. We also do
the analysis of Jaynes and Cumming Hamiltonian to support results of quantum
field theoretical study.
Pacs: 42.50.Dv, 42.50.Pq, 03.67.Bg, 75.10.Jm
Keywords:Quantum Many Body Models, Polariton, Cavity QED and Spin Chain Model
## I I. Introduction
The physics of strongly correlated system is interesting in its own right and
manifests in different branches of physics. Some of the important correlated
physics appears in natural oxide materials rao and some of them are in
engineered materials, like the correlated physics in Josephson junction array
lik1 , Bose-Einstein condensation and optical lattice dens ; zoller .
Therefore one can raise the question what is the further source of correlated
physics in the state of enginnering ?. The recent experimental success in
engineering strong interaction between the photons and atoms in high quality
micro-cavities opens up the possibility to use light matter system as quantum
simulators for many body physics green ; hart1 ; hart2 ; ji ; byrn ; caru ;
bhas ; toma ; zhao ; pipp ; rossi ; horn ; blat .
Here we would like to discuss the basic physics of micro-optical cavity very
briefly. A micro-cavity can be created in a photonic band gap material by
producing a localised defect in the structure of the crystal, in such a way
that light of a particular frequency can not propage outside the defect area.
Large arrays of such micro-cavities have been produce. Photon hopping between
neighboring cavities has been observed in the microwave and optical domains.
Many body Hamiltonians can be created and probed in coupled cavity arrays.
There atoms are used for detection and also for the generation of interaction
between photons in the same cavity. As the distance between the adjacent
cavities is considerably larger than the optical wave length of the resonant
mode, individual cavities can be addressed. These artificial system could act
as a quantum simulator. In this optical cavities system we study the different
quantum phase of polariton ( a combined excitations of atom-photon
interactions. ) by using spin models that conserve the total number of
excitations. Best of our knowledge, at first we have done the explcit quantum
field theoretical calculations of this type of system.
At first we would like to discuss the generation of the spin model for such
type of systems. It has already been discussed in the literature but we
mention this process very briefly for the sake of completeness. In this
description we will follow the Ref. hart1 and Ref. hart2 .
Micro-cavities of the photonic crystal are coupled through the exchange of
photons. Each cavity consists of one atom with three levels in the energy
spectrum, two of them are long lived and represents two spin states of the
system and the other is excited. Externally applied laser and cavity modes
couple to each atom of the cavity. It may induce the Raman transition between
these two long lived levels. Under a suitable detuning between the laser and
the cavity modes virtual photons are created in the cavity which mediate an
interaction with another atom in a neighboring cavities. One can eliminated
the excited states of atomic level photon states by choosing the appropriate
detuning between the applid laser and cavity modes. Then one can achieve only
two states per atom in the long lived state and the system can be described by
a spin-1/2 Hamiltonian hart1 ; hart2 .
Fig. 1 shows the schematic phase diagram of our description to generate the xy
spin interaction of the system. The Hamiltonian of the system consists of
three parts:
$H~{}=~{}{H_{A}}~{}+~{}{H_{B}}~{}+~{}{H_{C}}$ (1)
Hamiltonians are following
${H_{A}}~{}=~{}\sum_{j=1}^{N}{{\omega}_{e}}|e_{j}><e_{j}|~{}+~{}{\omega}_{ab}|b_{j}><b_{j}|$
(2)
$j$ is the cavity index. ${\omega}_{ab}$ and ${\omega}_{e}$ is the energy of
the state $|b>$ and the excited state respectively. The energy level of state
$|a>$ is set zero. The following Hamiltonian describes photons in the cavity,
${H_{C}}~{}=~{}{{\omega}_{C}}\sum_{j=1}^{N}{{a_{j}}}^{\dagger}{a_{j}}~{}+~{}{J_{C}}\sum_{j=1}^{N}({{a_{j}}}^{\dagger}{a_{j+1}}+h.c),$
(3)
where ${a_{j}}^{\dagger}$ creates a photon in cavity $j$, ${\omega}_{C}$ is
the energy of photons and $J_{C}$ is the tunneling rate of photons between
neighboring cavities. Interaction between the atoms and photons and also by
the driving lasers are described
${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+{g_{a}}{a_{j}})|e_{j}><a_{j}|+h.c]+[a\leftrightarrow
b].$ (4)
Here ${g_{a}}$ and ${g_{b}}$ are the couplings of the cavity mode for the
transition from energy states $|a>$ and $|b>$ to the excited state.
${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of lasers with
frequencies ${\omega}_{a}$ and ${\omega}_{b}$ respectively.
They have derived an effective spin model by considering the following
physical processes: A virtual process regarding emission and absorption of
photons between the two stable states of neghiboring cavities. The resulting
effective Hamiltonian is
${H_{xy}}=\sum_{j=1}^{N}B{{\sigma}_{j}}^{z}~{}+~{}\sum_{j=1}^{N}({J_{1}}{{\sigma}_{j}}^{\dagger}{{\sigma}_{j+1}}^{-}~{}+~{}{J_{2}}{{\sigma}_{j}}^{-}{{\sigma}_{j+1}}^{-}+h.c)$
(5)
When $J_{2}$ is real then this Hamiltonian reduces to the XY model. Where
${{\sigma}_{j}}^{z}=|b_{j}><b_{j}|~{}-~{}|a_{j}><a_{j}|$,
${{\sigma}_{j}}^{+}=|b_{j}><a_{j}|$, ${{\sigma}_{j}}^{-}=|a_{j}><b_{j}|$
$H_{xy}~{}=~{}\sum_{i=1}^{N}B({{\sigma}_{i}}^{z}~{}+~{}{J_{x}}{{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}~{}+~{}{J_{y}}{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}).$
(6)
With ${J_{x}}=(J_{1}+J_{2})/2$ and ${J_{y}}=(J_{1}-J_{2})/2$.
Here we discuss very briefly about an effective
${{\sigma}_{i}}^{z}{{\sigma}_{i}}^{z}$ in such a system. Authors of Ref.hart1
; hart2 have proposed the same atomic level configuration but having only one
laser of frequency ${\omega}$ that mediates atom-atom coupling through virtual
photons. Another laser field with frequency $\nu$ is used to tune the
effective magnetic field. They described the one-dimensional case. In this
case the Hamiltonian ${H_{AC}}$ will change but the Hamiltonians $H_{A}$ and
$H_{C}$ will not.
${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+\frac{{\Lambda}_{a}}{2}e^{-i{{\nu}_{a}}t}{g_{a}}{a_{j}})|e_{j}><a_{j}|+h.c]+[a\leftrightarrow
b].$ (7)
Here, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the
driving laser with frequency ${\omega}$ on transition $|a>\rightarrow|e>$
$|b>\rightarrow|e>$, whereas ${\Lambda}_{a}$ and ${\Lambda}_{b}$ driving laser
with frequency ${\nu}$ on transition $|a>\rightarrow|e>$ $|b>\rightarrow|e>$.
They have eliminated adiabatically the excited atomic levels and photons by
considering the interaction picture with respect to $H_{0}=H_{A}~{}+~{}H_{C}$.
They have considered the detuning parameter in such a way that the Raman
transitions between two level supressed. They have also chosen the parameter
in such a way that the dominant two-photon processes are those that involve
one laser photon and one cavity photon each but the atom does no transition
between levels a and b. Whenever two atoms exchange a virtual photon then both
of them experience a Stark shift plays the role of an efective
${{\sigma}^{z}}{{\sigma}^{z}}$ interaction. Then the effective Hamiltonian
reduce to
${H_{zz}}~{}=~{}\sum_{j=1}^{N}(\tilde{B}{{\sigma}_{j}}^{z}~{}+~{}{J_{z}}{{\sigma}_{j}}^{z}{{\sigma}_{j+1}}^{z})$
(8)
Analytical expressions for $\tilde{B}$, $J_{1}$, $J_{2}$ and $J_{z}$ has given
in Ref. hart2 . These two parameters can be tuned independently by varying the
laser frequencies. They have obtained an effective model by combining
Hamiltonians $H_{xy}$ and $H_{zz}$ by using Suzuki-Trotter formalism. The
effective Hamiltonian simulated by this procedure is
$H_{spin}~{}=~{}\sum_{j=1}^{N}(B_{tot}{{\sigma}_{j}}^{z}~{}+~{}\sum_{{\alpha}=x,y,z}J_{\alpha}{{\sigma}_{j}}^{\alpha}{{\sigma}_{j+1}}^{\alpha})$
(9)
where $B_{tot}=B+\tilde{B}$. It has been shown in Ref. hart2 that $J_{y}$ is
less than than $J_{x}$. It is clear from analytical expressions for $J_{x}$
and $J_{y}$ that the magnitudes of ${J_{1}}$ and $J_{2}$ are different. This
result of numerical simulations trigger us to define a model, which has given
below to study the quantum phases of this system and also the transition among
them and at the same time this subject is in the state of art of engineering.
## II II. Renormalization Group study of model Hamiltonian
We consider the anisotropic Heisenberg spin-1/2 Hamiltonian on a one
dimensional lattice. The XYZ Heisenberg Hamiltonians is defined as:
$\displaystyle H_{XYZ}~{}=~{}\sum_{n}~{}[$
$\displaystyle(1+a)~{}S_{n}^{x}S_{n+1}^{x}~{}+~{}(1-a)~{}S_{n}^{y}S_{n+1}^{y}$
(10)
$\displaystyle+~{}\Delta~{}S_{n}^{z}S_{n+1}^{z}~{}+~{}h~{}S_{n}^{z}~{}]~{},$
where $S_{n}^{\alpha}$ are the spin-1/2 operators. We assume that the $XY$
anisotropy $a$ and the $zz$ coupling $\Delta$ satisfy $-1\leq\Delta\leq 1$,
and $0<a\leq 1$ and magnetic field strength $h\geq 0$. The Hamiltonian
$H_{XYZ}$ is invariant under the transformation $S_{n}^{x}\rightarrow-
S_{n}^{x}$, $S_{n}^{y}\rightarrow-S_{n}^{y}$, $S_{n}^{z}\rightarrow-
S_{n}^{z}$, actually it is a $Z_{2}$ symmetry. For finite $h$, $Z_{2}$
symmetry is absent when $S_{n}^{z}\rightarrow-S_{n}^{z}$. Here $h\sim
B_{tot}$, ${\Delta=J_{z}}$, ${J_{1}=1}$ and ${J_{2}}=a$.
Spin operators can be recasted in terms of spinless fermions through Jordan-
Wigner transformation and then finally one can express the spinless fermions
in terms of bosonic fields gia . We recast the spinless fermions operators in
terms of field operators by this relation gia .
${\psi}(x)~{}=~{}~{}[e^{ik_{F}x}~{}{\psi}_{R}(x)~{}+~{}e^{-ik_{F}x}~{}{\psi}_{L}(x)]$
(11)
where ${\psi}_{R}(x)$ and ${\psi}_{L}(x)$ describe the second-quantized fields
of right- and the left-moving fermions respectively. $k_{F}$ is Fermi wave
vector. Therefore, one can study the effect of gate voltage through arbitrary
$k_{F}$. We would like to express the fermionic fields in terms of bosonic
field by the relation
${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))},$
(12)
$r$ is denoting the chirality of the fermionic fields, right (1) or left
movers (-1). The operators $U_{r}$ commutes with the bosonic field. $U_{r}$ of
different species commute and $U_{r}$ of the same species anti-commute. $\phi$
field corresponds to the quantum fluctuations (bosonic) of spin and $\theta$
is the dual field of $\phi$. They are related by the relations
${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$.
Hamiltonian $H_{0}$ is non-interacting part of $H_{XYZ}$.
$H_{0}~{}=~{}\frac{v}{2}~{}\int~{}dx~{}[~{}(\partial_{x}\theta)^{2}~{}+~{}(\partial_{x}\phi)^{2}~{}]~{},$
(13)
where $v$ is the velocity of the low-energy excitations. It is one of the
Luttinger liquid parameters and the other is $K$, which is related to $\Delta$
by gia
$K~{}=~{}\frac{\pi}{\pi+2\sin^{-1}(\Delta)}~{}.$ (14)
$K$ takes the values 1 and 1/2 for $\Delta=0$ (free field), and $\Delta=1$
(isotropic anti-ferromagnet), respectively. The relation between $K$ and
$\Delta$ is not preserved under the renormalization, so this relation is only
correct for initial Hamiltonian. The analytical form of the spin operators in
terms of the bosonic fields are
$\displaystyle S_{n}^{x}~{}$ $\displaystyle=$
$\displaystyle~{}[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\cos({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{y}~{}$ $\displaystyle=$
$\displaystyle~{}-[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\sin({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{z}~{}$ $\displaystyle=$
$\displaystyle~{}{\sqrt{\frac{\pi}{K}}}~{}\partial_{x}\phi~{}+~{}(-1)^{n}c_{1}\cos(2{\sqrt{\pi
K}}\phi)~{},$ (15)
where $c_{i}$s are constants as given in Ref. zamo . The Hamiltonian $H_{XYZ}$
in terms of bosonic fields is the following,
$\displaystyle H_{XYZ}$ $\displaystyle=$ $\displaystyle
H_{0}+a\int\cos(2{\sqrt{\frac{\pi}{K}}}\theta(x))dx+\Delta\int\cos(4{\sqrt{\pi
K}}\phi(x))dx$ (16) $\displaystyle-h\int{{\partial}_{x}}{\phi(x)}dx$
One can get the $H_{XY}$ Hamiltonian by simply putting $\Delta=0$ in the above
Hamiltonian. In this derivation, different powers of coefficients $c_{i}$ have
been absorbed in the definition of $a,h$ and $\Delta$. The integration of the
oscillatory terms in the Hamiltonian yield negligible small ontributions, the
origin of the oscillatory terms are the spin operators. So it’s a resonably
good approximation to keep only the non-oscillatory terms in the Hamiltonian .
The Gaussian scaling dimension of these coupling terms, $a,\Delta$ are $1/K$,
$4K$ respectively. The third term ($\Delta$) of the Hamiltonian tends to order
the system into density wave phase, whereas the second term ($a$) of the
Hamiltonian favours the staggered order in the xy plane. Two sine-Gordon
couplings terms are from two dual fields. Therefore the model Hamiltonian
consists of two competing interactions between the ordered phase and the $XY$
order. This Hamiltonian contains two strongly relevant and mutually nonlocal
perturbation over the Gaussian (critical) theory. In such situation strong
coupling fixed point is usually determined by the most relevant perturbation
whose amplitude grows up according to its Gaussian scaling dimensions and it
is not much affected by the less relevant coupling terms. However this is not
the general rule, if the two operators exclude each other, i.e., if the field
configurations which minimize one perturbation term do not minimize the other.
In this case interplay between the two competing relevant operators can
produce a novel quantum phase transition through a critical point or a
critical line. Therefore we would like to study the RG euation to interpret
the quantum phases of the system.
We will now study how the parameters $a$, $\Delta$ and $K$ flow under RG. The
operators in Eq. 17 are related to each other through the operator product
expansion; the RG equations for their coefficients will therefore be coupled
to each other. Here we derive the RG equations by using perturbative
renormalization group approach scheme. We use operator product expansion to
derive these RG equations which is independent of boundary condition cardy .
RG equations themselves have been established in a perturbative expansion in
coupling constant ($g(l)$), they cease to be valid beyond the certain length
scale, where $g(l)\sim 1$ gia . The RG equations for the coefficients of
Hamiltonian $H_{XYZ}$ are
$\displaystyle\frac{da}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}(2-\frac{1}{K})a,$ $\displaystyle\frac{d{\Delta}}{dl}~{}$
$\displaystyle=$ $\displaystyle~{}(2-4K){\Delta}$
$\displaystyle\frac{dK}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}\frac{a^{2}}{4}~{}-~{}K^{2}{O}^{2}~{},$ (17)
These RG equations have trivial (${a^{*}}=0={{\Delta}^{*}}$) fixed points for
any arbitrary $K$. Apart from that these RG equations have also two non-
trivial fixed lines, $a=\Delta$ and $a=-{{\Delta}}$ for $K=1/2$. In our study,
there are critical surfaces on which the system flows onto the non-trivial
fixed lines (${a}=\pm{{\Delta}}$). A density wave states can be characterized
when $K\rightarrow 0$ or the staggered ordered when ($K\rightarrow\infty$).
Note that the transition occuring on them are second order. Infinitesimal
amount of anisotropy will change the situation drastically, a gapless phase in
absence of anisotropy will change to the gapped phase in presence of
anisotropy. Since this gapped excitation is not directly related to
magnetization, therefore it will not favour to create the plateau phase. When
the system is in the plateau phase the transition driven by the magnetic field
is always of $z=2$ ( z is the dynamical critical exponent) and thus the
plateau shows a square-root behaviour of magnetization. When $a$ is
increasing, a second order transition drives system to the in plane XY ordered
phase , whose exponents depend on initial couplings and hence are
nonuniversal. In absence of planar anisotropy the transition to plateau state
is Kosterliz- Thouless-Brezinskii (KTB). When the inplane anisotropy is
present, then $z=1$ phase . Please see Refs. subir ; sondhi for detail
understanding of this subject. A magnetic field larger than the relevant gap
of the system drives the system to a gapless phase. This transition is from
commensurate phase to incommensurate phase transition.
We have seen the analytical expression of $B_{tot}$ from Ref. hart2 that the
total magnetic field increase for the larger values of detuning, therefore
larger detuning drives the system from gapped (Mott-insulating) state to
gapless superfluid state.
We now discuss, how the effective repulsion will decrease as we increase the
detuning between the atomic and laser frequency. It can be explained starting
from the Jaynes-Cummings Hamiltonian jay ; horo . Janes-Cummings Hamiltonian
for a single atom is
$H_{JC}~{}=~{}{{\omega}_{C}}{{a}^{\dagger}a}+{{\omega}_{0}}|e><e|+{\lambda}({a}^{\dagger}|g><e|~{}+~{}a|e><g|)$
(18)
${{\omega}_{C}}$ and ${{\omega}_{0}}$ are the frequency of the resonant mode
of the cavity and of the atomic transition, respectively. ${\lambda}$ is the
Jaynes-Cumming coupling between the cavity mode and the two level system.
${a}^{\dagger}$ (${a}$) is the creation (annihilation) operator of a photon
inside a cavity. $|g>$ and $|e>$ is respectively the ground state and excited
states of the two level system respectively. When we consider large values of
photon and atomic frequencies compare to atom-photon coupling $\lambda$, the
number of excitations is conserved for this Hamiltonian. Suppose we consider
fixed numbers of excitation, $n$. The energy eigenvalues for $n$ excitations
is
${E_{n}}^{\pm}=n{{\omega}_{C}}+\frac{\Delta}{2}\pm\sqrt{n{\lambda}^{2}~{}+~{}\frac{{\Delta}^{2}}{4}}$
(19)
Here $\Delta~{}=~{}({\omega}_{0}-{\omega}_{c})$ and $n\geq 1$. Now we consider
an array of cavities, the basic Hamiltonian for each cavity is the same as
that of Eq.19 . Here we consider the system with one excitation of energy
$E_{1}$ in each cavity and the lowest energy of two excitation in each cavity
is $E_{2}$. Therefore to create one additional excitation in each cavity
requires energy
${E_{2}-2E_{1}}=2\sqrt{{\lambda}^{2}~{}+~{}\frac{{\Delta}^{2}}{4}}-\sqrt{2{\lambda}^{2}~{}+~{}\frac{{\Delta}^{2}}{4}}-\frac{\Delta}{2}.$
Which one may consider as an effective on-site repulsion because it measures
the difference between the energy of two and single excitation (polariton in
each cavity). This effective repulsion decreases as we increase the detune
factor. Therefore we can conclude that for $\Delta=0$, double occupation never
occurs, indicating a Mott insulating behaviour. When $\Delta$ is much larger
than the coupling $\lambda$ then the occupation number larger than one occurs
as to be expected for a photonic superfluid regime. In our quantum field
theoretical calculations, we have also predicted that the large detuning
drives the system from gapped Mott insulating phase to the gapless superfluid
phase.
## III Conclusions
At first we have done the quantum field theoretical analysis of an effective
spin model in coupled optical cavity arrays. We have predicted two quantum
phases, Mott insulator and photonic superfludity. Anisotropy in the exchange
interaction has also created a gapped phase. An infinite order KTB transition
has been replaced by the second order XY transition. The rigorous quantum
field theoretical derivation of this manuscript is absent in all previous
studies and also we provide physical explanation of the transition process
based on Jaynes-Cummings Hamiltonian.
Acknowledgments
The author would like to acknowledge The Center for Condensed Matter Theory of
the Physics Department of IISc for providing working space and also Dr. R. C.
Sarasij for reading the manuscript very critically.
Figure 1: Level structure, driving laserrs, and relevant couplings to the
cavity mode to generate effective ${{\sigma}^{x}}{{\sigma}^{x}}$ and
${{\sigma}^{y}}{{\sigma}^{y}}$ couplings for one atom. The cavity mode couples
with strength ${g_{a}}$ and ${g_{b}}$ to transition $|a>\leftrightarrow|e>$
and $|b>\leftrightarrow|e>$ respectively. One laser frequency ${\omega}_{a}$
couples to the transition $|a>\leftrightarrow|e>$ with Rabi frequency
${\Omega}_{a}$ and another with frequency ${\omega}_{b}$ to
$|b>\leftrightarrow|e>$ with ${\Omega}_{b}$. The dominant two-photon processes
are indicated in faint arrows. Reprinted with permission from American
Physical Society. Analytical expressions and physical meaning of different
symbols have given in Ref.hart2 . Figure 2: Level structure, driving laserrs,
and relevant couplings to the cavity mode to generate effective
${{\sigma}^{z}}{{\sigma}^{z}}$ couplings for one atom. The cavity mode couples
with strength ${g_{a}}$ and ${g_{b}}$ to transition $|a>\leftrightarrow|e>$
and $|b>\leftrightarrow|e>$ respectively. Two lasers with frequencies $\omega$
and $\nu$ couple with Rabi frequencies ${\Omega}_{a}$, respectively,
${\Lambda}_{a}$ to transition $|a>\leftrightarrow|e>$ and ${\Omega}_{b}$,
respectively, ${\Lambda}_{b}$ to transition $|b>\leftrightarrow|e>$ . The
dominant two-photon processes are indicated in faint arrows. Reprinted with
permission from American Physical Society. Analytical expressions and physical
meaning of different symbols have given in Ref.hart2 .
## References
* (1) C. N. R. Rao and T. V. Ramakrishnan in Superconductivity Today (Universities Press, Hyderabad, 1999).
* (2) K. K. Likharev in Dynamics of Josephson junction and circuits (Gordon and Breach 1988).
* (3) J. Hecker. Denschlog $et~{}al.$, J. Phys. B: At. Mol. Opt Phys, 35, 3095 (2002).
* (4) D. Jaksch and P. Zoller, Annals of Physics 315, 52 (2005).
* (5) A. D. Greentree $et~{}al.$, Nature Phys. 466, 856 (2006).
* (6) Michael J. Hartmann, Fernando G. S. L Brando and Martin B. Plenio, Nature Phys. 462, 849 (2006); Michael J. Hartmann, Fernando G. S. L Brando and Martin B. Plenio, Laser and Photonics Rev. 2, 527 (2008).
* (7) Michael J. Hartmann, Fernando G. S. L Brando and Martin B. Plenio, Phys. Rev. Lett 99, 160501 (2007).
* (8) A-C.Ji, X. C. Xie, and W. M. Liu, Phys. Rev.Lett. 99, 183602 (2007).
* (9) T. Byrnes, N. Y. Kim, K. Kusudo, and Y. Yamamoto, Phys. Rev. B 78, 075320 (2008).
* (10) I. Carusotto $et~{}al.$, arXiv:0812.4195 (2008).
* (11) M. J. Bhaseen, M. Hohenadler, A. O. Silver, and B. D. Simons, Phys. Rev. Lett. 102
* (12) A. Tomadin $et~{}al.$ arXiv:0904.4437 (2009).
* (13) J. Zhao, A. W. Sandvik, and K. Ueda, arXiv:0806.3603 (2008).
* (14) P. Pippan, H. G. Evertz, and M. Hohenadler, arXiv: 0904.1350 (2009).
* (15) D. Rossini, and R. Fazio, Phys. Rev. Lett. 99, 186401 (2007).
* (16) M. Aichhorn $et~{}al.$, Phys. Rev. Lett. 100, 216401 (2008).
* (17) S. Schmidt, and G. Blatter, arXiv:0905.3344 .
* (18) D. G. Angelakis, M. F. Santos and S. Bose, Phys. Rev. A 76, R031805 (2007).
* (19) T. Giamarchi in Quantum Physics in One Dimension (Clarendon Press, Oxford 2004).
* (20) S. Lukyanov and A. Zamolodchikov, Nucl. Phys. B 493, 571 (1997).
* (21) In quantum phase transitions, in addition to the standard critical exponent it is useful to define an additional exponent z, called the dynamical exponent, which tells us how a characteristic length in the time direction is related to a length in the spatial direction ${\xi}_{\tau}\sim{{\xi}_{x}}^{z}$. For quantum problem time plays a special role and this special direction has no reasons to have the same exponent as the spatial one. Deep inside the Mott insulating phase, particle and hole excitations are gapped. The system is almost alike to the atomic limit in the deep Mott insulating phase. When one approaches the phase boundary from the deep Mott insulating phase then the dispersion relation is quadratic ${\omega}\sim{k^{2}}$ ($z=2$). But the situation is different at the end of CDW phase and the starting point of XY staggered order phase. At around the multical critical point $z$ is $1$. Please see the Refs. subir ; sondhi and for a detailed understanding of this subject.
* (22) Subir Sachdev in ”Quantum Phase Transition” (Cambridge University Press Cambridge, 1998).
* (23) S. L. Sondhi, $et~{}al.$ Rev. Mod. Phys. 69, 315 (1997).
* (24) J. Cardy in Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge 1996); I. Affleck in Fields, Strings and and Critical Phenomena, ed E. Brezin and J. Zinn-Justin (North-Holland, Amstardam 1989).
* (25) Jaynes, E. T. and F. W. Cummings, Proc IEEE, 51, 89 (1963).
* (26) S. Horoche and J. M. Raimond in Exploring the Quantum Atoms, Cavities, and Photons, (Oxford University Press, 2006).
|
arxiv-papers
| 2009-10-28T11:02:55 |
2024-09-04T02:49:06.131110
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sujit Sarkar",
"submitter": "Sujit Sarkar",
"url": "https://arxiv.org/abs/0910.5341"
}
|
0910.5346
|
# Josephson decoupling phase in inhomoheneous arrays of superconducting
quantum dots
Sujit Sarkar PoornaPrajna Institute of Scientific Research, 4 Sadashivanagar,
Bangalore 5600 80, India.
###### Abstract
We present a study of quantum phase analysis of inhomogeneous and homogeneous
arrays of superconducting quantum dots (SQD). We observe the existance of
Josephson decouple (JD) phase only at the half filling for inhomogeneous array
of SQD due to the fluctuation of Josephson couplings over the sites at half
filling. In JD phase superconductivity disappears even in the absence of
Coulomb blockade phase. We also observe that fluctuation of on-site Coulomb
charging energy produces the relevant coupling term that yields Coulomb
blockade gapped phase. The presence of nearest-neighbor and next-nearest-
neighbor Coulomb interaction yields the same physics for inhomogeneous and
homogeneous SQD.
Introduction: It is well known that the quantal phase ($\phi$) of
superconductor is coherent over the superconducting system; therefore, we
expect the quantum properties of the electron to be visible at a macroscopic
level jose1 ; jose2 ; bard1 ; bard2 ; coo ; pde . The Josephson effect is
nothing but the manifestation of coherence of the superconducting quantal
phase in the system. In this effect, the system is gaining energy to stabilize
the superconducting phase. A superconducting phase is stable when the
Josephson coupling ($E_{J}$) between two superconductors separated by a
junction, is larger than the Cooper pair charging energy ($E_{c}$). This is
the conventional wisdom in the literature of superconductivity jose1 ; jose2 ;
bard1 ; bard2 ; coo ; pde . In this letter we raise the question for a
nanostructure superconducting system whether this conventional wisdom is still
valid for an inhomogeneous SQD system with fluctuating Josephson coupling. We
will see that for inhomogeneous SQD array, our model system is in the
insulating phase even in the absence of Coulomb blockade phase. Although the
system has finite $E_{J}$, we characterize this phase as a Josephson decouple
phase (JD) because it is not yielding any superconductivity in the system. In
the present stage our model is completely the theoretical model of
inhomogeneous SQD array which predicts the JD phase. We hope that the state of
engineering of nanoscale superconducting system will find this type of system.
Our prime motivation is to predict the JD phase, after the fourty seven years
of the discovery Josephson effect. We don’t think that our model system is a
perfect model of granular superconducting system because we have built the
model to predict the JD phase only and ; abe ; shapira ; jae ; orr ; sudip ;
efe ; sim . In this study we also raise the question of the effect of on-site
Coulomb charging energy and also fluctuations of it over sites. We will see
that this effect is quite interesting. Apart from that we also study the
effect of nearest-neighbor (NN) and next-nearest-neighbor (NNN) Coulomb
interactions for inhomogeneous and homogeneous SQD.
General field-theoretical formalism for inhomogeneous SQD and homogeneous SQD
arrays:
At first we write down the model Hamiltonian of inhomogeneous SQD system with
fluctuating (with a periodicity of two lattice sites, this model is sufficient
to detect the JD phase) Josephson couplings with on-site charging energies and
intersite interactions in presence of gate voltage. The Hamiltonian is written
as
$H~{}=~{}H_{J1}~{}+~{}H_{EC0}~{}+~{}H_{EC1}~{}+~{}H_{EC2}.$ (1)
We recast the different parts of the Hamiltonian in quantum phase model as.
$H_{J1}~{}=~{}-E_{J1}\sum_{i}(1-{(-1)^{i}}{\delta}_{1})cos({\phi}_{i+1}-{\phi}_{i}),$
where ${\phi}_{i}$ and ${\phi}_{i+1}$ are quantal phase of the SQD at the
point i and i+1 respectively. Josephson couplings are fluctuating over the
sites, ${E_{J1}}(1+{\delta}_{1})$ and ${E_{J1}}(1-{\delta}_{1})$ are the
Josephson coupling strength for odd and even site respectively. We also
consider the fluctuations of on-site Coulomb charging energy over the sites.
This is represented as
$H_{EC0}~{}=~{}\frac{E_{C0}}{2}\sum_{i}(1-(-1)^{i}{\delta}_{2}){(-i\frac{\partial}{{\partial}{{\phi}_{i}}}-\frac{N}{2})^{2}}$,
where ${E_{C0}}$ is the on-site charging energy. $E_{C0}(1+{\delta}_{2})$ and
$E_{C0}(1-{\delta}_{2})$ are the on site charging energies for odd and even
sites respectively. All $\delta$’s are deviations of exchange couplings from
the homogeneous SQD. $H_{EC1}$ and $H_{EC2}$ are respectively the Hamiltonians
for nearest neighbor(NN) and next-nearest-neighbor(NNN) interations between
SQD. Now
$H_{EC1}~{}=~{}E_{Z1}\sum_{i}{n_{i}}~{}{n_{i+1}},$
and
$H_{EC2}~{}=~{}E_{Z2}\sum_{i}{n_{i}}~{}{n_{i+2}},$
where $E_{Z1}$ and $E_{Z2}$ are respectively the NN and NNN charging energies
between the dots. We see that this model is sufficient to explain JD induced
gapped phase of the system. In the phase representation,
$(-i\frac{\partial}{{\partial}{{\phi}_{i}}})$ is the operator representing the
number of Cooper pairs at the ith dot, and thus it takes only the integer
values ($n_{i}$). Hamiltonian $H_{EC0}$ accounts for the influence of gate
voltage ($eN\sim V_{g}$). $eN$ is the average dot charge induced by the gate
voltage. When the ratio $\frac{E_{J1}}{E_{C0}}\rightarrow 0$, the SQD array is
in the insulating state having a gap of the width $\sim{E_{C0}}$, since it
costs an energy $\sim E_{C0}$ to change the number of pairs at any dot. The
exceptions are the discrete points at $N~{}=~{}2n+1$, where a dot with charge
$2ne$ and $2(n+1)e$ has the same energy because the gate charge compensates
the charges of extra Cooper pair in the dot. On this degeneracy point, a small
amount of Josephson coupling leads the system to the superconducting state.
Here we would like to recast our basic Hamiltonians in the spin language.
During this process we follow Ref. ss and lar . We map this model to the spin
chain model when on-site charging energy is larger Josephson coupling. Now
$H_{J1}~{}=~{}-2~{}E_{J1}\sum_{i}(1-(-1)^{i}{{\delta}_{1}})({S_{i}}^{\dagger}{S_{i+1}}^{-}+h.c)$,
and
$H_{EC0}~{}=~{}\frac{E_{C0}}{2}\sum_{i}(1-(-1)^{i}{{\delta}_{2}}){(2{S_{i}}^{Z}-h)^{2}}.$
$H_{EC1}~{}=~{}4E_{Z1}\sum_{i}{S_{i}}^{Z}~{}{S_{i+1}}^{Z},$
$H_{EC2}~{}=~{}4E_{Z2}\sum_{i}{S_{i}}^{Z}~{}{S_{i+2}}^{Z}.$
Here $h=\frac{N-2n-1}{2}$ allows the tuning of the system around the
degeneracy point by means of gate voltage. Now we use Abelian bosonization
method to solve this problem. We recast the spinless fermion operators in
terms of field operators by this relation gia1 :
${\psi}(x)~{}=~{}~{}[e^{ik_{F}x}~{}{\psi}_{R}(x)~{}+~{}e^{-ik_{F}x}~{}{\psi}_{L}(x)]$
(2)
where ${\psi}_{R}(x)$ and ${\psi}_{L}(x)$ describe the second-quantized fields
of right- and left-moving fermions respectively and $k_{F}$ is the Fermi wave
vector. It is revealed from $H_{EC0}$ that the applied external gate voltage
on the dot systems appears as a magnetic field in the spin chain. In our
system $k_{F}$ will depend on the applied gate voltage. Therefore, one can
study the effect of gate voltage through arbitrary $k_{F}$. We would like to
express the fermionic fields in terms of bosonic field by the relation
${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))}$,
$r$ is denoting the chirality of the fermionic fields, right (1) or left
movers (-1). The operator $U_{r}$ commutes with the bosonic field. $U_{r}$ of
different species commute and $U_{r}$ of the same species anti-commute. $\phi$
field corresponds to the quantum fluctuations (bosonic) of spin and $\theta$
is the dual field of $\phi$. They are related by the relations
${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$. After continuum field theoretical
studies for arbitrary values of $k_{F}$, the model Hamiltonian becomes
$\displaystyle{H_{1}}$ $\displaystyle=$
$\displaystyle{H_{0}}+2\frac{{E_{J1}}}{2\pi{\alpha}}{{\delta}_{1}}\int
dx:cos(2\sqrt{K}{\phi}(x)-(2k_{F}-\pi)x):$
$\displaystyle+\frac{hE_{C0}}{\pi\alpha}\int dx{{\partial}_{x}}{\phi(x)}$
$\displaystyle+\frac{2hE_{C0}{{\delta}_{2}}}{\pi\alpha}\int(-1)^{x}:cos(2\sqrt{K}{\phi}(x)+2k_{F}x):~{}dx$
$\displaystyle+~{}\frac{4E_{Z1}}{{(2\pi{\alpha})}^{2}}\int:cos(4\sqrt{K}{\phi}(x)~{}-$
$\displaystyle(G-4k_{F})x-2k_{F}a):~{}dx$
$\displaystyle+~{}\frac{4E_{Z2}}{{(2\pi{\alpha})}^{2}}\int:cos(4\sqrt{K}{\phi}(x)~{}+$
$\displaystyle(G-4k_{F})x-4k_{F}a):~{}dx.$
The Bosonized Hamiltonians for homogeneous SQD can be written as
$\displaystyle{H_{2}}$ $\displaystyle=$
$\displaystyle{H_{0}}+\frac{hE_{C0}}{\pi\alpha}\int
dx{{\partial}_{x}}{\phi(x)}$ (4)
$\displaystyle+\frac{4E_{Z1}}{{(2\pi{\alpha})}^{2}}\int:cos(4\sqrt{K}{\phi}(x)$
$\displaystyle~{}-(G-4k_{F})x-2k_{F}a):~{}dx$
$\displaystyle+~{}\frac{4E_{Z2}}{{(2\pi{\alpha})}^{2}}$
$\displaystyle\int:cos(4\sqrt{K}{\phi}(x)~{}+(G-4k_{F})x-4k_{F}a):~{}dx,$
and
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle(\frac{v}{2\pi}+\frac{8E_{C0}}{{\pi}^{2}}-\frac{2{E_{J1}}^{2}}{E_{C0}})$
(6) $\displaystyle~{}\int
dx~{}[:{{({{\partial}_{x}}\theta)}^{2}}:+:{{({{\partial}_{x}}\phi)}^{2}}:~{}]$
$\displaystyle+(16E_{C0}-4\frac{{E_{J1}}^{2}}{E_{C0}})$ $\displaystyle\int
dx~{}:({\partial}_{x}{\theta}-{\partial}_{x}{\phi})({\partial}_{x}{\theta}+{\partial}_{x}{\phi}):$
Here, $H_{0}$ is the non-interacting part of the model Hamiltonian, $v$ is the
velocity of low energy excitations, one of the Luttinger liquid parameter and
the other is $K$. And $G$ is the reciprocal lattice vector.
Results and physical interpretation: Here we study the relevant physics for
single Cooper pair in alternate site for inhomogeneous and homogeneous SQD
system (here ${k_{F}}=\frac{\pi}{2}$ because the system is at half-filling).
The effective Hamiltonian for the inhomogeneous SQD reduce to
$\displaystyle{H_{1}}$ $\displaystyle=$
$\displaystyle{H_{0}}+2\frac{E_{J1}}{2\pi\alpha}{{\delta}_{1}}\int~{}dx:cos(2\sqrt{(}K)\phi(x)):$
$\displaystyle+{hE_{C0}}\int({{\partial}_{x}}\phi(x))~{}dx$
$\displaystyle+2\frac{hE_{C0}}{\pi\alpha}{{\delta}_{2}}\int~{}dx:cos(2\sqrt{(}K)\phi(x)):$
$\displaystyle-\frac{4(E_{Z1}-E_{Z2})}{{(2\pi\alpha)}^{2}}\int~{}dx:cos(4\sqrt{K}\phi(x)):.$
Our model Hamiltonian consists of three sine-Gordon couplings. The second term
of the Hamiltonian arises due to fluctuations of Josephson coupling. It yields
the gapped phase of the system. The anamolous scaling dimension of this term
is 2$K$. This phase is spontaneous, i.e., infinitesimal variation of NN
Josephson coupling around sites is sufficient to produce this state. When
${E_{c}}$ is larger than ${E_{J}}$ the system is in the gapped phase due to
the Coulomb blockade effect. If we consider the case when $E_{J}$ is much
smaller than $E_{C}$ then one should naively think that the system is in the
superconducting phase but the situation here is quite different due to the
fluctuations of Josephson coupling, its produces the gap state in the system
and blocks the superconducting phase of the system. We term this phase as
Josephson decoupling phase because it is not yielding any superconducting
phase due to the tunneling at different SQD; this phase is present even in the
absence of Coulomb blockade. This gapped state prevails until the applied gate
voltage is sufficient to break this gapped phase cbl . This prediction is
absent in all previous studies of superconductivity jose1 ; jose2 ; pde ;
bard1 ; bard2 ; coo ; ss ; lar ; lik1 ; lik2 ; baro . The third term of the
Hamiltonian arises due to constant Coulomb charging energy; it promotes the
system in different charge quantized state due to the variation of applied
gate voltage. The fourth term of the Hamiltonian is due to the fluctuations of
on-site Coulomb charging energy. It is like the staggered magnetization of the
system. It’s anamolous scaling dimension is also $2K$. Therefore, the system
is in the mixed gapped state when both terms are present. The fourth term
arises due to the NN and NNN interactions; the anamolous scaling dimension of
this term is $4K$. Therefore the physics of gapped state is mainly governed by
the second and the fourth term of the Hamiltonian.
Effective Hamiltonian for homogeneous SQD array is
$\displaystyle{H_{2}}$ $\displaystyle=$ $\displaystyle
H_{0}~{}-~{}\frac{4(E_{Z1}-E_{Z2})}{{(2\pi{\alpha})}^{2}}\int
cos(4\sqrt{K}{\phi}(x)~{})~{}dx$ (8)
$\displaystyle+\frac{hE_{C0}}{\pi\alpha}\int~{}{{\partial}_{x}}{\phi}~{}dx.$
When $E_{Z2}$ exceed some critical value, the ground state of the system is
dimerized and doubly degenerates. The dimerized ground state is the product of
spin singlet of adjacent sites hal . When $E_{Z2}$ is less than a critical
value the physics of the system is governed by the $E_{Z1}$ and the gapped
phase of the system is alike to spin-fluid phase of the system. In this model
Hamiltonian, there is no relevant sine-Gordon coupling term present due to the
variation of Josephson coupling. Therefore there is no JD phase for
homogeneous SQD. We also study our model Hamiltonian for different densities
(by varying $k_{F}$) but we are unable to find JB phase for any other fillings
for both inhomogeneous and homogeneous SQD array.
Conclusions: We have predicted the evidence for the Josephson decouple phase
for inhomogeneous SQD only at half-fillings. This is the first prediction of
Josephson decoupled phase in the literature for these type of system. There is
no evidence of Josephson decouple phase for homogeneous SQD. We have also
predicted the interesting behavior of the system due to the fluctuating on-
site Coulomb charging energy. Our prediction of Josephson decoupling phase
after the fourty seven years of Josephson effect; we hope that evidence of
this JD phase will be verified experimentally as the Josephson effect has
verified experimentally after the theoretical prediction.
## References
* (1) B. D. Josephson, Phys. Lett. 1, 251 (1962).
* (2) B. D. Josephson, Adv. Phys. 14, 419 (1965).
* (3) P. G. de Gennes Superconductivity in Metals and Alloys, W. A. Benjamin, New York (1969).
* (4) J. Bardeen, L.N. Cooper and J. R. Schriffer, Phys. Rev. 106 , 162 (1957).
* (5) J. Bardeen, L. N. Cooper and J. R. Schriffer, Phys. Rev. 108, 1175 (1957).
* (6) L. N. Cooper, Phys. Rev. 104, 1189 (1956).
* (7) S. Sarkar, Phys. Rev. B 75, 014528 (2007).
* (8) P. W. Anderson, in Lectures on the Many-Body Problems, Volume 2 (Academic, New York, 1964).
* (9) B. Abeles, Phys. Rev. B 15, (1977) 2828.
* (10) Y. Shapira and G. Deutscher, Phys. Rev. B 27, (1983) 4463.
* (11) H. Jaeger, D. Haviland, B. Orr, A. Goldman, Phys. Rev. B 40 (1989), 182.
* (12) B. G. Orr, H. M. Jaeger, A. M. Goldman and C. G. Kuper, Phys. Rev. Lett 56, (1986) 378.
* (13) S. Chakravarty $et~{}al.$, Phys. Rev. B 35, 7256.
* (14) K. B. Efetov, Sov. Phys. JETP, (1980) 2017.
* (15) E. Simanek, Sol. State. Comm 31, (1979) 419.
* (16) L. I. Glazman and A. I. Larkin, Phys. Rev. Lett. 79, 3786 (1997).
* (17) T. Giamarchi in Quantum Physics in One Dimension (Clarendon Press, Oxford 2004).
* (18) Here we discuss the effect of gate voltage on homogeneous and non-homogeneous SQD and the charge quantization physics. From an energetic point of view, when an Cooper pair added to SQD, the new electrostatic energy with added Cooper pair is smaller than the corresponding energy in the absence of Cooper pair $\frac{1}{2C}{(|Q|-|2e|)}^{2}\leq~{}\frac{Q^{2}}{2C}=>Q\geq|2e|.$ (9) Therefore we need a bias $|V|\geq\frac{2e}{2C}$ for current to flow. Hence Cooper pair transport is unfavorable when $|V|\leq\frac{2e}{2C}$, this is the Coulomb blocked phase for non-homogeneous and homogeneous SQD. When we added an extra Cooper pair to the SQD, we expect a spikes in conductance, i.e, the steps in correspondence to the biases that overcome charging energies with increasing number of Cooper pair. When one kept fixed the bias and gate voltage ($V_{g}$ ) applied to SQD (with capacitance $C_{g}$ ). The applied gate voltage will vary densities of Cooper pair in SQD. In presence of N Cooper pair in SQD, the electrostatic energy ${E_{N}}=\frac{{(2Ne)}^{2}}{2e_{g}}-2{V_{g}}N|e|$. Adding of extra Cooper pair is favorable when the above energy is minimized w.r.t number of Cooper pair, it implies $|V_{g}|=\frac{|2e|}{C_{g}}(N+1/2)$. At these gate voltage the current has a discontinuity. Therefore we define the charge quantized state when the applied gate voltage is away from the optimum gate voltage point and also for $|V|<\frac{2|e|}{2c}$.
* (19) K. K. Likharev in Dynamics of Josephson junction and circuits (Gordon and Breach 1988).
* (20) K. K. Likharev, Y. Y. Naveh and D. Averin in Physics of high $J_{c}$ Josephson junction and prospects of their RSFQ VLSI application IEEE Trans on Appl. Supercond 11, 1056.
* (21) A. Barone and G. Paterno in Dynamics of Josephson Junction and Circuits, Gordon and Breach, Philadelphia.
* (22) F. D. M. Haldane, Phys. Rev. B 25, 4925 (1982), and ibid 26, 5247 (1982).
|
arxiv-papers
| 2009-10-28T11:18:17 |
2024-09-04T02:49:06.136127
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sujit Sarkar",
"submitter": "Sujit Sarkar",
"url": "https://arxiv.org/abs/0910.5346"
}
|
0910.5354
|
# Inversion formula and Parsval theorem for complex continuous wavelet
transforms studied by entangled state representation††thanks: Project
supported by the National Natural Science Foundation of China (Grant Nos
10475056 and 10775097).
Li-yun Hu∗1,2 and Hong-yi Fan2 1College of Physics and Communication
Electronics, Jiangxi Normal University, Nanchang 330022, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China
∗Corresponding author. E-mail: hlyun2008@126.com (L-Y Hu)
###### Abstract
In a preceding Letter (Opt. Lett. 32, 554 (2007)) we have proposed complex
continuous wavelet transforms (CCWTs) and found Laguerre–Gaussian mother
wavelets family. In this work we present the inversion formula and Parsval
theorem for CCWT by virtue of the entangled state representation, which makes
the CCWT theory complete. A new orthogonal property of mother wavelet in
parameter space is revealed.
OCIS codes: 070.2590, 270.6570.
## I Introduction
Wavelet transforms (WTs) are very useful in signal analysis and detection 1 ;
2 ; 3 since it can overcome the shortcomings of nonlocality behavior of
classical Fourier analysis and thus enriches the theory of Fourier optics 4 .
The continuous WT of a signal function $f\left(x\right)\in
L^{2}\left(R\right)$ by a mother wavelet $\psi\left(x\right)$ (restricted by
the admissibility condition $\int_{-\infty}^{\infty}\psi\left(x\right)dx=0$)
is defined by
$W_{\psi}f\left(\mu,s\right)=\frac{1}{\sqrt{\mu}}\int_{-\infty}^{\infty}f\left(x\right)\psi^{\ast}\left(\frac{x-s}{\mu}\right)dx,$
(1)
where $\mu$ $\left(>0\right)$ is a scaling parameter and $s\left(\in R\right)$
is a translation parameter. The inversion of (1) is
$f\left(x\right)=\frac{1}{C_{\psi}}\int_{0}^{\infty}\frac{d\mu}{\mu^{2}}\int_{-\infty}^{\infty}W_{\psi}f\left(\mu,s\right)\psi\left(\frac{x-s}{\mu}\right)\frac{ds}{\sqrt{\mu}},$
(2)
where
$C_{\psi}=\int_{0}^{\infty}\frac{\left|\psi\left(p\right)\right|^{2}}{p}dp<\infty$
and $\psi\left(p\right)$ is the Fourier transform of $\psi\left(x\right),$ for
proving (2) we have employed the Dirac’s representation theory 5 , which has
the merit of rigour and simplicity.
In Ref.6 ; 7 , Fan and Lu have linked the one-dimensional (1D) WT with the
unitary transform (squeezing and dispacement) in quantum mechanics, i.e.,
expressing the WT as a matrix element of the single-mode squeezing-displacing
operator between the mother wavelet state vector $\left\langle\psi\right|$ and
the state vector to be transformed, such that the admissibility condition for
mother wavelets is examined in the context of quantum mechanics, in so doing a
family of the Hermite–Gaussian mother waveletes are found. Further, by
introducing the bipartite entangled state representation
$\left|\eta\right\rangle$ 8
$\left|\eta\right\rangle=\exp\left(-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right)\left|00\right\rangle,$
(3)
Fan and Lu then proposed the continuous complex wavelet transforms (CCWT) for
$g\left(\eta\right)\equiv\left\langle\eta\right|\left.g\right\rangle,\ $
$W_{\psi}g\left(\mu,\kappa\right)=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}g\left(\eta\right)\psi^{\ast}\left(\frac{\eta-\kappa}{\mu}\right),$
(4)
where $\kappa\in C.$ Correspondingly, the admissibility condition for mother
wavelets, $\int\frac{d^{2}\eta}{2\pi}\psi\left(\eta\right)=0,$ is examined in
the entangled state representations and a family of new mother wavelets (named
the Laguerre–Gaussian wavelets) are found to match the CCWT 9 , i.e., the
qualified mother wavelets $\psi\left(\eta\right)$ satisfying the admissibility
condition can be expressed as the function of $\left|\eta\right|,$
$\displaystyle\psi\left(\eta\right)$ $\displaystyle=$ $\displaystyle
e^{-\left|\eta\right|^{2}/2}\sum_{n=0}^{\infty}K_{n,n}(-1)^{n}H_{n,n}^{\ast}\left(\eta,\eta^{\ast}\right)$
(5) $\displaystyle=$ $\displaystyle
e^{-\left|\eta\right|^{2}/2}\sum_{n=0}^{\infty}n!K_{n,n}L_{n}\left(\left|\eta\right|^{2}\right),$
where $L_{n}\left(x\right)$ is the Laguerre function and
$H_{m,n}\left(x,y\right)$ is the two-variable Hermite polynomial 10 , whose
generating function is
$H_{m,n}\left(x,y\right)=\frac{\partial^{m+n}}{\partial t^{\prime n}\partial
t^{m}}\exp\left[-tt^{\prime}+tx+t^{\prime}y\right]_{t=t^{\prime}=0}.$ (6)
We emphasize that the CCWT differs from the direct product of two 1D WTs since
the squeezing transform involved in (4) is in two-mode
$\frac{1}{\mu}\psi^{\ast}\left(\frac{\eta-\kappa}{\mu}\right)=\frac{1}{\mu}\left\langle\psi\right.\left|\frac{\eta-\kappa}{\mu}\right\rangle=\left\langle\psi\right|S_{2}\left|\eta-\kappa\right\rangle,$
(7)
where $S_{2}$ is the two-mode squeezing operator
$S_{2}=\exp[(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\ln\mu]$ 11 ; 12 ,
which is in sharp contrast to the direct-product of two single-mode squeezing
(dilation) operators, and the two-mode squeezed state is simultaneously an
entangled state.
In order to complete the CCWT theory, we must ask if the corresponding
Parseval theorem exists. This is important since the inversion formula of CCWT
may appear as a lemma of this theorem. We shall solve this issue by virtue of
the merits of entangled state in quantum mechanics, to be more specific, we
shall use the property that the two-mode squeezing operator has its natural
representation in the entangled state basis (see (10) below). Noting that CCWT
involves two-mode squeezing transform, so the corresponding Parseval theorem
differs from that of the direct-product of two 1D wavelet transforms, too.
## II The quantum mechanical version of CCWT
Let us begin with putting the CCWT into the context of quantum mechanics.
Based on the idea of quantum entanglement initiated by Einstein-Podolsky-Rosen
(EPR) 13 , Fan and Klauder constructed the entangled state representation in
two-mode Fock space 8 , $\left|\eta\right\rangle$ in (3) is the common
eigenvector of two particles’ relative position $X_{1}-X_{2}$ and their
momentum $P_{1}+P_{2},$
$\left(X_{1}-X_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{1}\left|\eta\right\rangle,\text{
}\left(P_{1}+P_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{2}\left|\eta\right\rangle,$
(8)
where $X_{j}=(a_{j}+a_{j}^{{\dagger}})/\sqrt{2},$
$P_{j}=(a_{j}-a_{j}^{{\dagger}})/(\sqrt{2}i),$ ($j=1,2)$.
$\left|\eta\right\rangle$ is complete
$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=1$
($d^{2}\eta\equiv d\eta_{1}d\eta_{2},$ $\eta=\eta_{1}+i\eta_{2}),$ and
orthonormal
$\left\langle\eta\right|\left.\eta^{\prime}\right\rangle=\pi\delta\left(\eta-\eta^{\prime}\right)\delta\left(\eta^{\ast}-\eta^{\prime\ast}\right)\equiv\pi\delta^{(2)}\left(\eta-\eta^{\prime}\right)$.
Using $\left\langle\eta\right|$ and
$\psi\left(\eta\right)=\left\langle\eta\right|\left.\psi\right\rangle$ we can
recast the CCWT in (4) as
$W_{\psi}g\left(\mu,\kappa\right)=\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|g\right\rangle,$
(9)
and $U_{2}\left(\mu,\kappa\right)$ is a two-mode squeezing-translating
operator, which has its natural expression in EPR entangled state
representation,
$U_{2}\left(\mu,\kappa\right)\equiv\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left|\frac{\eta-\kappa}{\mu}\right\rangle\left\langle\eta\right|,$
(10)
when $\kappa=0,$ $U_{2}\left(\mu,0\right)=S_{2}.$
## III Parseval Theorem in the CCWT
Now let us prove the Parseval theorem for CCWT,
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}W_{\psi}g_{1}\left(\mu,\kappa\right)W_{\psi}^{\ast}g_{2}\left(\mu,\kappa\right)=C_{\psi}^{\prime}\int\frac{d^{2}\eta}{\pi}g_{2}^{\ast}\left(\eta\right)g_{1}\left(\eta\right),$
(11)
where $\kappa=\kappa_{1}+i\kappa_{2},$
$C_{\psi}^{\prime}=4\int_{0}^{\infty}\frac{d\left|\xi\right|}{\left|\xi\right|}\left|\psi\left(\xi\right)\right|^{2}.$
(12)
$\psi\left(\xi\right)$ is the Fourier transform of $\psi\left(\eta\right)$, a
mother wavelet. According to (10) and (9) the quantum mechanical version of
Parseval theorem should be
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|g_{1}\right\rangle\left\langle
g_{2}\right|U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\psi\right\rangle=C_{\psi}^{\prime}\left\langle
g_{2}\right.\left|g_{1}\right\rangle,$ (13)
where $\psi\left(\eta\right)=\left\langle\eta\right|\left.\psi\right\rangle$,
so $\psi\left(\xi\right)=$ $\left\langle\xi\right|\left.\psi\right\rangle,$
$\left|\xi\right\rangle$ is the conjugate state to $\left|\eta\right\rangle,$
$\displaystyle\left|\xi\right\rangle$ $\displaystyle=$
$\displaystyle\exp\left\\{-\frac{1}{2}\left|\xi\right|^{2}+\xi
a_{1}^{\dagger}+\xi^{\ast}a_{2}^{\dagger}-a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle$
(14) $\displaystyle=$
$\displaystyle\left(-1\right)^{a_{2}^{\dagger}a_{2}}\left|\eta\right\rangle_{\eta=\xi},\text{
\ \ \ }\xi=\xi_{1}+i\xi_{2},$
which is the common eigenstate of center-of-mass coordinate and the relative
momentum operators, i.e.,
$\left(X_{1}+X_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{1}\left|\xi\right\rangle,\text{
}\left(P_{1}-P_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{2}\left|\xi\right\rangle,$
(15)
and is complete
$\int\frac{d^{2}\xi}{\pi}\left|\xi\right\rangle\left\langle\xi\right|=1.$ (16)
The overlap between $\left\langle\xi\right|$ and $\left|\eta\right\rangle$ is
14
$\left\langle\xi\right.\left|\eta\right\rangle=\frac{1}{2}\exp[\frac{1}{2}\left(\xi^{\ast}\eta-\xi\eta^{\ast}\right)]=\frac{1}{2}\exp\left[\allowbreak
i\left(\xi_{1}\eta_{2}-\xi_{2}\eta_{1}\right)\right].$ (17)
so
$\displaystyle\psi\left(\xi\right)$ $\displaystyle=$
$\displaystyle\left\langle\xi\right|\left.\psi\right\rangle=\int\frac{d^{2}\eta}{\pi}\left\langle\xi\right|\left.\eta\right\rangle\left\langle\eta\right|\left.\psi\right\rangle$
(18) $\displaystyle=$
$\displaystyle\int\frac{d^{2}\eta}{2\pi}\exp\left[\left(\xi^{\ast}\eta-\xi\eta^{\ast}\right)/2\right]\psi\left(\eta\right).$
Eq.(11) indicates that once the state vector $\left\langle\psi\right|$
corresponding to mother wavelet is known, for any two states
$\left|g_{1}\right\rangle$ and $\left|g_{2}\right\rangle$, their overlap up to
the factor $C_{\psi}$ (determined by (12)) is just their corresponding overlap
of CCWTs in the ($\mu,\kappa$) parametric space.
Proof of Eq.(11) or (13)
We start with calculating
$U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi\right\rangle.$ Using (10) and
(17), we have
$\displaystyle U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\frac{\eta-\kappa}{\mu}\right|\left.\xi\right\rangle$
(19) $\displaystyle=$
$\displaystyle\frac{1}{\mu}\int\frac{d^{2}\eta}{2\pi}\left|\eta\right\rangle
e^{\frac{i}{\mu}\left(\xi_{2}\eta_{1}-\xi_{1}\eta_{2}+\xi_{1}\kappa_{2}-\xi_{2}\kappa_{1}\right)}$
$\displaystyle=$ $\displaystyle\frac{1}{\mu}\left|\frac{\xi}{\mu}\right\rangle
e^{\frac{i}{\mu}\left(\xi_{1}\kappa_{2}-\xi_{2}\kappa_{1}\right)},$
it follows
$\displaystyle\int\frac{d^{2}\kappa}{\pi}U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi^{\prime}\right\rangle\left\langle\xi\right|U_{2}\left(\mu,\kappa\right)$
(20) $\displaystyle=$
$\displaystyle\frac{1}{\mu^{2}}\int\frac{d^{2}\kappa}{\pi}e^{\frac{i}{\mu}\left[\left(\xi_{1}^{\prime}-\xi_{1}\right)\kappa_{2}+\left(\xi_{2}-\xi_{2}^{\prime}\right)\kappa_{1}\right]}\left|\frac{\xi^{\prime}}{\mu}\right\rangle\left\langle\frac{\xi}{\mu}\right|$
$\displaystyle=$ $\displaystyle
4\pi\left|\frac{\xi^{\prime}}{\mu}\right\rangle\left\langle\frac{\xi}{\mu}\right|\delta\left(\xi_{1}^{\prime}-\xi_{1}\right)\delta\left(\xi_{2}-\xi_{2}^{\prime}\right).$
Using (16) and (20) the left-hand side (LHS) of (13) can be reformed as
LHS of Eq.(13) (21) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa
d^{2}\xi
d^{2}\xi^{\prime}}{\pi^{3}}\left\langle\psi\right|\left.\xi\right\rangle$
$\displaystyle\times\left\langle\xi\right|U_{2}\left(\mu,\kappa\right)\left|g_{1}\right\rangle\left\langle
g_{2}\right|U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\xi^{\prime}\right\rangle\left\langle\xi^{\prime}\right|\left.\psi\right\rangle$
$\displaystyle=$ $\displaystyle
4\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\xi
d^{2}\xi^{\prime}}{\pi}\left\langle
g_{2}\right.\left|\frac{\xi^{\prime}}{\mu}\right\rangle\left\langle\frac{\xi}{\mu}\right.\left|g_{1}\right\rangle$
$\displaystyle\times\psi^{\ast}\left(\xi\right)\psi\left(\xi^{\prime}\right)\delta\left(\xi_{1}^{\prime}-\xi_{1}\right)\delta\left(\xi_{2}-\xi_{2}^{\prime}\right)$
$\displaystyle=$ $\displaystyle
4\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\xi}{\pi}\left|\psi\left(\xi\right)\right|^{2}\left\langle
g_{2}\right.\left|\frac{\xi}{\mu}\right\rangle\left\langle\frac{\xi}{\mu}\right.\left|g_{1}\right\rangle$
$\displaystyle=$
$\displaystyle\int\frac{d^{2}\xi}{\pi}\left\\{4\int_{0}^{\infty}\frac{d\mu}{\mu}\left|\psi\left(\mu\xi\right)\right|^{2}\right\\}\left\langle
g_{2}\right.\left|\xi\right\rangle\left\langle\xi\right.\left|g_{1}\right\rangle,$
where the integration value in $\\{..\\}$ is actally $\xi-$independent. Noting
that the mother wavelet $\psi\left(\eta\right)$ in Eq.(5) is just the function
of $\left|\eta\right|,$ so $\psi\left(\xi\right)$ is also the function of
$\left|\xi\right|.$ In fact, using Eqs.(5),(6) and (18), we have
$\psi\left(\xi\right)=e^{-1/2\left|\xi\right|^{2}}\sum_{n=0}^{\infty}K_{n,n}H_{n,n}\left(\left|\xi\right|,\left|\xi\right|\right),$
(22)
where we have used the integral formula
$\int\frac{d^{2}z}{\pi}e^{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(23)
So we can rewrite (21) as
$\text{LHS of
(\ref{18})}=C_{\psi}^{\prime}\int\frac{d^{2}\xi}{\pi}\left\langle
g_{2}\right.\left|\xi\right\rangle\left\langle\xi\right.\left|g_{1}\right\rangle=C_{\psi}^{\prime}\left\langle
g_{2}\right.\left|g_{1}\right\rangle,$ (24)
where
$C_{\psi}^{\prime}=4\int_{0}^{\infty}\frac{d\mu}{\mu}\left|\psi\left(\mu\xi\right)\right|^{2}=4\int_{0}^{\infty}\frac{d\left|\xi\right|}{\left|\xi\right|}\left|\psi\left(\xi\right)\right|^{2}.$
(25)
Then we have completed the proof of the Parseval theorem for CCWT in (13).
Here, we should emphasize that (13) is not only different from the product of
two 1D WTs, but also different from the usual wavelet transform in 2D.
When $\left|g_{2}\right\rangle=\left|\eta\right\rangle,$ by using (10) we see
$\left\langle\eta\right|U_{2}^{\dagger}\left(\mu,\kappa\right)\left|\psi\right\rangle=\frac{1}{\mu}\psi\left(\frac{\eta-\kappa}{\mu}\right),$
(26)
then substituting it into (13) yields
$g_{1}\left(\eta\right)=\frac{1}{C_{\psi}^{\prime}}\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi\mu}W_{\psi}g_{1}\left(\mu,\kappa\right)\psi\left(\frac{\eta-\kappa}{\mu}\right),$
(27)
which is just the inverse transform of the CCWT.
Especially, when $\left|g_{1}\right\rangle=$ $\left|g_{2}\right\rangle,$ (13)
reduces to
$\displaystyle\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|W_{\psi}g_{1}\left(\mu,\kappa\right)\right|^{2}$
$\displaystyle=$ $\displaystyle
C_{\psi}^{\prime}\int\frac{d^{2}\eta}{\pi}\left|g_{1}\left(\eta\right)\right|^{2},$
$\displaystyle\text{or
}\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|g_{1}\right\rangle\right|^{2}$
$\displaystyle=$ $\displaystyle C_{\psi}^{\prime}\left\langle
g_{1}\right.\left|g_{1}\right\rangle,$ (28)
which is named isometry of energy.
## IV New orthogonal property of mother wavelet in parameter space
On the other hand, when $\left|g_{1}\right\rangle=\left|\eta\right\rangle,$
$\left|g_{2}\right\rangle=\left|\eta^{\prime}\right\rangle$, (13) becomes
$\frac{1}{C_{\psi}^{\prime}}\int_{0}^{\infty}\frac{d\mu}{\mu^{5}}\int\frac{d^{2}\kappa}{\pi}\psi\left(\frac{\eta^{\prime}-\kappa}{\mu}\right)\psi^{\ast}\left(\frac{\eta-\kappa}{\mu}\right)=\pi\delta^{(2)}\left(\eta-\eta^{\prime}\right),$
(29)
which is a new orthogonal property of mother wavelet in parameter space
spanned by $\left(\mu,\kappa\right)$. In a similar way, we take
$\left|g_{1}\right\rangle=\left|g_{2}\right\rangle=\left|m,n\right\rangle,$ a
two-mode number state, since $\left\langle
m,n\right.\left|m,n\right\rangle=1,$ then we have
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|m,n\right\rangle\right|^{2}=C_{\psi}^{\prime},$
(30)
or take
$\left|g_{1}\right\rangle=\left|g_{2}\right\rangle=\left|z_{1},z_{2}\right\rangle,$
$\left|z\right\rangle=\exp\left(-\left|z\right|^{2}/2+za^{\dagger}\right)\left|0\right\rangle$
is the coherent state, then
$\int_{0}^{\infty}\frac{d\mu}{\mu^{3}}\int\frac{d^{2}\kappa}{\pi}\left|\left\langle\psi\right|U_{2}\left(\mu,\kappa\right)\left|z_{1},z_{2}\right\rangle\right|^{2}=C_{\psi}^{\prime}.$
(31)
This indicates that $C_{\psi}^{\prime}$ is
$\left|g_{1}\right\rangle$-independent, which coincides with the expression in
(12).
Next we examine a special example. When the mother wavelet
$\psi\left(\eta\right)$ is taken as the following form
$\psi_{M}\left(\eta\right)=\left\langle\eta\right.\left|\psi\right\rangle=e^{-1/2\left|\eta\right|^{2}}(1-\frac{1}{2}\left|\eta\right|^{2}),$
(32)
which is different from $e^{-(x^{2}+y^{2})/2}(1-x^{2})\left(1-y^{2}\right),$
the direct-product of two 1D Mexican hat wavelets (we name entangled mexican
hat wavelets (EMHWs)), using (18) we have
$\psi\left(\xi\right)=\frac{1}{2}\left|\xi\right|^{2}e^{-\frac{1}{2}\left|\xi\right|^{2}},$
(33)
which leads to
$C_{\psi}^{\prime}=\int_{0}^{\infty}\left|\xi\right|^{3}e^{-\left|\xi\right|^{2}}d\left|\xi\right|=\frac{1}{2}.$
(34)
Thus for the EMHWs (32), we see
$2\int_{0}^{\infty}\frac{d\mu}{\mu^{5}}\int\frac{d^{2}\kappa}{\pi}\psi_{M}\left(\frac{\eta^{\prime}-\kappa}{\mu}\right)\psi_{M}^{\ast}\left(\frac{\eta-\kappa}{\mu}\right)=\pi\delta^{(2)}\left(\eta-\eta^{\prime}\right).$
(35)
Eq. (35) can be checked as follows. Using (32) and the integral formula
$\displaystyle\int_{0}^{\infty}u\left(1-\frac{ux^{2}}{2}\right)\left(1-\frac{uy^{2}}{2}\right)e^{-u\frac{x^{2}+y^{2}}{2}}du$
(36) $\displaystyle=$
$\displaystyle-\frac{4(x^{4}-4x^{2}y^{2}+y^{4})}{(x^{2}+y^{2})^{4}},\text{
Re}\left(x^{2}+y^{2}\right)>0,$
we can put the left-hand side (LHS) of (35) into
LHS of (35) (37) $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}\frac{d\frac{1}{\mu^{2}}}{\mu^{2}}\int\frac{d^{2}\kappa}{\pi}e^{-\frac{x^{2}+y^{2}}{2\mu^{2}}}\left(1-\frac{x^{2}}{2\mu^{2}}\right)\left(1-\frac{y^{2}}{2\mu^{2}}\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}udu\int\frac{d^{2}\kappa}{\pi}\left(1-\frac{ux^{2}}{2}\right)\left(1-\frac{uy^{2}}{2}\right)e^{-u\frac{x^{2}+y^{2}}{2}}$
$\displaystyle=$
$\displaystyle-\int\frac{d^{2}\kappa}{\pi}\frac{4(x^{4}-4x^{2}y^{2}+y^{4})}{(x^{2}+y^{2})^{4}},$
where $x^{2}=\left|\eta^{\prime}-\kappa\right|^{2},$
$y^{2}=\left|\eta-\kappa\right|^{2}.$
When $\eta^{\prime}=\eta,$ $x^{2}=y^{2},$
$\text{LHS of
(\ref{36})}=\int\allowbreak\frac{d^{2}\kappa}{2\pi\left|\kappa-\eta\right|^{4}}=\int_{0}^{\infty}\allowbreak\int_{0}^{2\pi}\frac{drd\theta}{2\pi
r^{3}}\rightarrow\infty.$ (38)
On the other hand, when $\eta\neq\eta^{\prime}$ and noticing that
$\displaystyle x^{2}$ $\displaystyle=$
$\displaystyle\left(\eta_{1}^{\prime}-\kappa_{1}\right)^{2}+\left(\eta_{2}^{\prime}-\kappa_{2}\right)^{2},$
$\displaystyle y^{2}$ $\displaystyle=$
$\displaystyle\left(\eta_{1}-\kappa_{1}\right)^{2}+\left(\eta_{2}-\kappa_{2}\right)^{2},$
(39)
which leads to
$\mathtt{d}x^{2}\mathtt{d}y^{2}=4\left|J\right|\mathtt{d}\kappa_{1}\mathtt{d}\kappa_{2}\text{,}$
(40)
where
$J\left(x,y\right)=\left|\begin{array}[]{cc}\kappa_{1}-\eta_{1}^{\prime}&\kappa_{2}-\eta_{2}^{\prime}\\\
\kappa_{1}-\eta_{1}&\kappa_{2}-\eta_{2}\end{array}\right|$. As a result of
(39), (37) reduces to
$\text{LHS of
(\ref{36})}=-4\int_{-\infty}^{\infty}\frac{dxdy}{\pi}\frac{xy(x^{4}-4x^{2}y^{2}+y^{4})}{\left|J\right|(x^{2}+y^{2})^{4}}=0\text{,}$
(41)
where we have noticed that $J\left(x,y\right)$ is the funtion of
$\left(x^{2},y^{2}\right).$ Thus we have
$\text{LHS of
(\ref{36})}=\left\\{\begin{array}[]{cc}\infty,&\eta=\eta^{\prime},\\\
0,&\eta\neq\eta^{\prime}.\end{array}\right.=\text{RHS of (\ref{36}).}$ (42)
In sum, we have proposed the Parseval theorem corresponding to the CCWT in the
context of quantum mechanics. Our calculations are simplified greatly by using
the quantum state representations of two-mode squeezing operators. Finally, we
should emphasize that since the CCWT corresponds to two-mode squeezing
transform which differs from two single-mode squeezing operators’ direct
product, the Parseval theorem of CCWT defined in this paper differs from that
of the direct product of two 1-dimensional wavelet transforms.
Acknowledgements
This work was supported by the National Natural Science Foundation of China
(Grant Nos. 10775097), and the Research Foundation of the Education Department
of Jiangxi Province.
## References
* (1) S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001)
* (2) I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (SIAM, 1992).
* (3) M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).
* (4) J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968).
* (5) L.-Y. Hu and H.-Y. Fan, Wavelet transform in the context of quantum mechanics and new orthogonal property of mother wavelets in parameter space, J. Mod. Opt. 55, 1835-1844 (2008).
* (6) H.-Y. Fan and H.-L. Lu, General formula for finding mother wavelets by virtue of Dirac’s representation theory and the coherent state, Opt. Lett. 31, 407-409 (2006),
* (7) H.-Y. Fan and J.-F Lu , Quantum Mechanics Version of Wavelet Transform studied by virtue of IWOP technique, Commun. Theor. Phys. 41, 681-684 (2004).
* (8) H.-Y. Fan and J. R. Klauder, Eigenvectors of two particles’ relative position and total momentum Phys. Rev. A 49, 704-707 (1994).
* (9) H.-Y. Fan and H.-L. Lu, Mother wavelet for complex wavelet transform derived by EPR entangled state representation, Opt. Lett. 32, 554-556 (2007).
* (10) A. Wunsche, General Hermite and Laguerre two-dimensional polynomials, J. Phys. A-Math. and Gen. 33, 1603-1629 (2000).
* (11) R. Loudon, and P.L. Knight, Squeezed light, J. Mod. Opt. 34, 709-759 (1987).
* (12) M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University, 1997).
* (13) A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777-780 (1935).
* (14) H.-Y. Fan and H.-L. Lu, Classical optical transforms studied in the context of quantum optics via the route of developing Dirac’s symbolic method, Int. J. Mod. Phys. B, 19, 799-856 (2005).
|
arxiv-papers
| 2009-10-28T11:49:01 |
2024-09-04T02:49:06.140331
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0910.5354"
}
|
0910.5358
|
# Nonclassicality of photon-added squeezed vacuum and its decoherence in
thermal environment††thanks: Project supported by the National Natural Science
Foundation of China (Grant Nos 10775097 and 10874174) and the Research
Foundation of the Education Department of Jiangxi Province.
Li-yun Hu1,2 and Hong-yi Fan2
1College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China
Corresponding author. _E-mail address_ : hlyun2008@126.com (L-Y Hu).
###### Abstract
We study the nonclassicality of photon-added squeezed vacuum (PASV) and its
decoherence in thermal environment in terms of the sub-Poissonian statistics
and the negativity of Wigner function (WF). By converting the PASV to a
squeezed Hermite polynomial excitation state, we derive a compact expression
for the normalization factor of $m$-PASV, which is an $m$-order Legendre
polynomial of squeezing parameter $r$. We also derive the explicit expression
of WF of $m$-PASV and find the negative region of WF in phase space. We show
that there is an upper bound value of $r$ for this state to exhibit sub-
Poissonian statistics increasing as $m$ increases. Then we derive the explicit
analytical expression of time evolution of WF of $m$-PASV in the thermal
channel and discuss the loss of nonclassicality using the negativity of WF.
The threshold value of decay time is presented for the single PASV.
PACS number(s): 03.67.-a, 03.65. Ud, 42. 50.Dv
## 1 Introduction
Nonclassical states play an important role in quantum computation and quantum
imformation procession [1]. The nonclassicality of quantum states can be well-
described by some nonclassical properties, such as sub-Poissonian photon
ststistics [2], squeezing in one of the quadratures of the field [3], and
negativity of Wigner function (WF) [4]. Especially, the partial negativity of
WF implies the highly nonclassical properties of quantum states and is often
used to describe the decoherence of quantum states. Many experimental schemes
have been proposed to generate nonclassical states of optical field. Among
them, subtracting photons form and/or adding photons to quantum states have
been paid much attention because these fields exhibit an abundant of
nonclassical properties and may give access to a complete engineering of
quantum states and to fundamental quantum phenomena [5, 6, 7, 8, 10, 11, 12,
13]. For example, a single photon-subtraction squeezed vacuum (PSSV) has been
experimentally prepared with a pulsed and continuous wave squeezed vacuum [11,
12, 13]. It is very similar to quantum superpositions of coherent states with
small amplitudes [14, 15]. As another example, a single-photon addition was
experimentally performed by Zavatta et al [6], which unveils the nonclassical
features associated with the excitation of a classical coherent field by a
single light quantum. For the single PSSV, its nonclassical properties and
decoherence was investigated theoretically in two different decoherent
channels (amplitude decay and phase damping) by Biswas and Agarwal [16]. They
indicated that the WF losses its non-Gaussian nature and becomes Gaussian at
long times in amplitude decay case.
On the other hand, the combination of the photon addition and subtraction has
been successfully demonstrated in Ref.[8, 9]. In Ref.[8], photon addition and
subtraction experimentally have been employed to probe quantum commutation
rules by Parigi et al. In fact, they have implemented simple alternated
sequences of photon creation (addition) and annihilation (subtraction) on a
thermal field and observed the noncommutativity of the creation and
annihilation operators. It is interesting to notice that subtracting or adding
one photon from/to pure squeezed vacuums can generate the same output state,
i.e., squeezed single-photon state [17]. However, the resulting states
obtained by successive photon subtractions or additions are different from
each other. For instance, successive two-photon additions [$a^{{\dagger}2}$]
and successive two-photon subtractions [$a^{2}$] will result in the same state
produced by using subtraction-addition ($a^{{\dagger}}a$) and addition-
subtraction ($aa^{{\dagger}}$), (also see section 2 below) respectively. As
far as we know, the nonclassicality and decoherence of arbitrary number
photon-added squeezed vacuum states (PASV) in a dissipative channel has not
been derived analytically in the literature before.
In this paper, we shall investigate the nonclassical properties and
decoherence of single-mode PASV which is optically produced single-mode non-
Gaussian states. This work is arranged as follows. In section 2, we introduce
the single-mode PASV and discuss its nonclassicality in terms of sub-
Poissonian statistics and the negativity of its Wigner function (WF). By
converting the PASV to a squeezed Hermite polynomial excitation state, we
derive a compact expression for the normalization factor of PASV, which is an
$m$-order Legendre polynomial of the squeezing parameter $r$, where $m$ is the
number of added photons; and then derive the explicit analytical expression of
WF for any photon-added number $m$ and find the negative region of WF in phase
space. We also show that there is an upper bound value of $r$ for this state
to exhibit sub-Poissonian statistics which increases as $m$ increases. In
section 3, we derive the explicit analytical expression of time evolution of
WF of the arbitrary PASV in the thermal channel and discuss the loss of
nonclassicality in reference of the negativity of WF. The threshold value of
the decay time corresponding to the transition of the WF from partial negative
to completely positive definite is obtained at the center of the phase space,
which is independent of the squeezing parameter. We show that the WF for
single PASV has always negative value for all parameters $r$ if the decay time
$\kappa t<\frac{1}{2}\ln\frac{2\bar{n}+2}{2\bar{n}+1}\ $(see Eq.(40) below),
where $\bar{n}$ denotes the average thermal photon number in the environment
with dissipative coefficient $\kappa$. Conclusions are involved in the last
section.
## 2 Single-mode Photon added squeezed vacuum state
Various photon states have been generated by the micromaser and WFs of some
cavity fields can be measured by a scheme based on interaction between cavity
fields and atoms [18, 19]. As described in Ref. [20] when an excited atom
passes through a cavity field which is in a squeezed vacuum state then their
interaction may produce photon addition (excitation) on the squeezed vacuum
state—the excited squeezed vacuum state.
Therectically, the single-mode PASV can be obtained by repeatedly operating
the photon creation operator $a^{{\dagger}}$ on a squeezed vacuum state
$S\left(r\right)\left|0\right\rangle$, i.e.,
$\left|r,m\right\rangle\equiv
N_{r,m}a^{{\dagger}m}S\left(r\right)\left|0\right\rangle,$ (1)
where $\left|0\right\rangle$ is single mode vacuum, and $N_{r,m}\ $is the
normalization constant to be determined, $a^{{\dagger}}$ is the Bose creation
operator, and $S\left(r\right)$ is the single-mode squeezing operator
$S\left(\lambda\right)=\exp[\frac{1}{2}\left(ra^{\dagger 2}-ra^{2}\right)]$
[21, 22] with $r$ being the squeezing parameter.
### 2.1 Single-mode PASV as the squeezed Hermite polynomial excitation state
Under the transform of $S\left(\lambda\right)$ we see
$S^{{\dagger}}\left(r\right)a^{\dagger}S\left(r\right)=a^{\dagger}\cosh
r+a\sinh r,$ thus we can reform Eq.(1) as
$\displaystyle\left|r,m\right\rangle$
$\displaystyle=N_{r,m}S\left(r\right)S^{{\dagger}}\left(r\right)a^{{\dagger}m}S\left(r\right)\left|0\right\rangle$
$\displaystyle=N_{r,m}S\left(r\right)\left(a^{{\dagger}}\cosh r+a\sinh
r\right)^{m}\left|0\right\rangle.$ (2)
On the other hand, using the operator identity [23]
$\left(a\mu+\nu
a^{\dagger}\right)^{m}=\left(-i\sqrt{\frac{\mu\nu}{2}}\right)^{m}\colon
H_{m}\left(i\sqrt{\frac{\mu}{2\nu}}a+i\sqrt{\frac{\nu}{2\mu}}a^{{\dagger}}\right)\colon,$
(3)
where $H_{m}\left(x\right)$ is $m$-order single variable Hermite polynomial
whose definition is
$H_{m}\left(x\right)=\sum_{l=0}^{[m/2]}\frac{\left(-1\right)^{l}m!}{l!(m-2l)!}\left(2x\right)^{m-2l},$
we have
$\displaystyle\left|r,m\right\rangle$
$\displaystyle=\frac{\left(-i\right)^{m}}{2^{m}}N_{r,m}\sinh^{m/2}2rS\left(r\right)\colon
H_{m}\left(i\sqrt{\frac{\tanh r}{2}}a+i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\colon\left|0\right\rangle$
$\displaystyle=\frac{\left(-i\right)^{m}}{2^{m}}N_{r,m}\sinh^{m/2}2rS\left(r\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle,$ (4)
which indicates that the PASV is equivalent to a squeezed Hermite-excited
vacuum state.
Further using the generating function of $H_{m}\left(x\right)$,
$H_{m}\left(x\right)=\left.\frac{\partial^{m}}{\partial
t^{m}}\exp\left(2xt-t^{2}\right)\right|_{t=0},$ (5)
and Eq.(4), the normalization factor $N_{r,m}$ can be derived by
$\displaystyle 1$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\left\langle
0\right|H_{m}\left(-i\sqrt{\frac{\coth
r}{2}}a\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\frac{\partial^{2m}}{\partial\tau^{m}\partial
t^{m}}e^{-t^{2}-\tau^{2}}\left.\left\langle 0\right|e^{-i\sqrt{2\coth
r}a\tau}e^{i\sqrt{2\coth
r}a^{{\dagger}}t}\left|0\right\rangle\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\frac{\partial^{2m}}{\partial\tau^{m}\partial
t^{m}}\left.\exp\left(-t^{2}-\tau^{2}+2\tau t\coth
r\right)\right|_{\tau=t=0},$ (6)
where in the last step we have used the Baker-Hausdorff formula $e^{\mu
a}e^{\nu a^{\dagger}}=e^{\nu a^{\dagger}}e^{\mu a}e^{\mu\nu}$. Using the newly
found generating function of Legendre polynomial [24, 25] (see Appendix A),
$\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+\frac{2x\tau
t}{\sqrt{x^{2}-1}}\right)\right|_{t,\tau=0}=\frac{2^{m}m!}{\left(x^{2}-1\right)^{m/2}}P_{m}\left(x\right),$
(7)
we can derive the compact expression for $N_{r,m}$, i.e.,
$N_{r,m}^{-2}=m!\cosh^{m}rP_{m}\left(\cosh r\right),$ (8)
which is just the result in Ref.[20] derived by the mathematical induction
method. In particular, when $m=1,$ ($H_{1}\left(x\right)=2x$) i.e., the
single-photon added case, we see that
$\left|r,1\right\rangle=S\left(r\right)a^{{\dagger}}\left|0\right\rangle$ is
just a squeezed single-photon state; on the other hand, for the single-photon
subtracted case [17, 26], the state is
$aS\left(r\right)\left|0\right\rangle=S\left(r\right)S^{\dagger}\left(r\right)aS\left(r\right)\left|0\right\rangle=S\left(r\right)\left(a\cosh
r+a^{{\dagger}}\sinh r\right)\left|0\right\rangle\rightarrow
S\left(r\right)a^{{\dagger}}\left|0\right\rangle,$ which indicates that adding
a single-photon to the squeezed state has the same impact as annihilating a
photon from the squeezed state. While for $m\geqslant 2,$ the case is not true
(also see Fig.1). For example, successive two-photon additions
[$a^{{\dagger}2}$] and successive two-photon subtractions [$a^{2}$] will
result in the same state produced by using subtraction-addition
($a^{{\dagger}}a$) and addition-subtraction ($aa^{{\dagger}}$), respectively,
i.e., $a^{{\dagger}2}S\left(r\right)\left|0\right\rangle\rightarrow
a^{{\dagger}}aS\left(r\right)\left|0\right\rangle,a^{2}S\left(r\right)\left|0\right\rangle\rightarrow
aa^{{\dagger}}S\left(r\right)\left|0\right\rangle.$ In Ref.[27], two PSSV is
used to generate the squeezed superposition of coherent states with high
fidelities and large amplitudes.
Combining Eqs.(1) and (8) we can conveniently calculate the average photon
number $a^{{\dagger}}a$ in PASV,
$\displaystyle\left\langle r,m\right|a^{{\dagger}}a\left|r,m\right\rangle$
$\displaystyle=\left\langle r,m\right|aa^{{\dagger}}\left|r,m\right\rangle-1$
$\displaystyle=\frac{N_{r,m}^{2}}{N_{r,m+1}^{2}}-1$
$\displaystyle=\left(m+1\right)\zeta\frac{P_{m+1}\left(\zeta\right)}{P_{m}\left(\zeta\right)}-1,$
(9)
where we denote $\zeta=\cosh r$ for simplicity, and
$\displaystyle\left\langle
r,m\right|a^{{\dagger}2}a^{2}\left|r,m\right\rangle$
$\displaystyle=\left\langle
r,m\right|\left(a^{2}a^{{\dagger}2}-4aa^{{\dagger}}+2\right)\left|r,m\right\rangle$
$\displaystyle=\frac{N_{r,m}^{2}}{N_{r,m+2}^{2}}-4\frac{N_{r,m}^{2}}{N_{r,m+1}^{2}}+2$
$\displaystyle=\left(m+1\right)\zeta\left\\{\left(m+2\right)\zeta\frac{P_{m+2}\left(\zeta\right)}{P_{m}\left(\zeta\right)}-4\frac{P_{m+1}\left(\zeta\right)}{P_{m}\left(\zeta\right)}\right\\}+2,$
(10)
thus the Mandel’s $\mathcal{Q}$-parameter can be obtained by substituting
Eqs.(9) and (10) into $\mathcal{Q}\equiv\frac{\left\langle a^{\dagger
2}a^{2}\right\rangle}{\left\langle a^{{\dagger}}a\right\rangle}-\left\langle
a^{{\dagger}}a\right\rangle$. In particular, for single-photon-added case
$m=1,$ Eqs.(9) and (10) reduce to
$\displaystyle\left\langle r,1\right|a^{{\dagger}}a\left|r,1\right\rangle$
$\displaystyle=3\cosh^{2}r-2,$ (11) $\displaystyle\left\langle
r,1\right|a^{{\dagger}2}a^{2}\left|r,1\right\rangle$
$\displaystyle=\frac{3\left(3+2\tanh^{2}r\right)}{\left(\coth r-\tanh
r\right)^{2}},$ (12)
thus the $\mathcal{Q}$-parameter with $m=1$ is given by
$\mathcal{Q=}\frac{3\sinh^{2}2r}{3\cosh 2r-1}-1.$ (13)
From Eq.(13) we find that $\mathcal{Q}$ becomes negative for $m=1$ which is
satisfied for the squeezing parameter $r\lesssim 0.46$ similar to the result
of $\mathcal{Q}$ for single-photon subtracted case [16]. In order to see
clearly the variation of $\mathcal{Q}$-parameter with $r$, we show the plots
of $\mathcal{Q}$-parameter in Fig.1, from which one can clearly see that
$\mathcal{Q}$-parameter becomes negetive ($m\neq 0)$ when $r$ is less than a
certain threshold value which increases as $m$ increases; while for $m=0,$
$\mathcal{Q}$ is always positive. This implies that the nonclassicality is
enhanced by adding photon to squeezed state. We should emphasize that the WF
has negative region for all $r,$ and thus the PASV is nonclassical. In our
following work, we pay attention to the (ideal) PASV in a thermal channel.
Figure 1: (Color online) The $Q$-parameter as the function of squeezing
parameter $r$ for different $m=0,1,2,3,4,29,30.$
### 2.2 Wigner function of PASV
In order to discuss the decoherence properties of PASV in thermal environment,
in this subsection we shall derive the analytical expression of Wigner
function for PASVS. For single-mode case, the Wigner operator is defined as
[28]
$\Delta\left(q,p\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}du\left|q-\frac{u}{2}\right\rangle\left\langle
q+\frac{u}{2}\right|e^{-ipu},$ (14)
where $\left|q\right\rangle$ is the coordinate representation,
$Q\left|q\right\rangle=q\left|q\right\rangle$. Thus the Wigner function of
PASVS $\left|r,m\right\rangle$ can be calculated by
$W\left(q,p\right)=\left\langle
r,m\right|\Delta\left(q,p\right)\left|r,m\right\rangle.$ Using Eq.(4), we can
see
$W\left(q,p\right)=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\left\langle
0\right|H_{m}\left(-i\sqrt{\frac{\coth
r}{2}}a\right)S^{{\dagger}}\left(r\right)\Delta\left(q,p\right)S\left(r\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle.$ (15)
On the other hand, noticing that the single-mode squeezing operator
$S^{{\dagger}}\left(r\right)$ has its natural expression in coordinate
representation [29], i.e.,
$S^{{\dagger}}\left(r\right)=e^{r/2}\int_{-\infty}^{\infty}dq\left|qe^{r}\right\rangle\left\langle
q\right|,$ leading to
$S^{{\dagger}}\left(r\right)\left|q\right\rangle=e^{-r/2}\left|qe^{-r}\right\rangle,$
it then follows that
$\displaystyle
S^{{\dagger}}\left(r\right)\Delta\left(q,p\right)S\left(r\right)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\left(ue^{-r}\right)\left|qe^{-r}-\frac{ue^{-r}}{2}\right\rangle\left\langle
qe^{-r}+\frac{ue^{-r}}{2}\right|e^{-ipu}$
$\displaystyle=\Delta\left(qe^{-r},pe^{r}\right).$ (16)
Then substituting Eq.(16) into Eq.(15) and using Eq.(5) as well as the
coherent state representation of Wigner operator,
$\Delta\left(q,p\right)\rightarrow\Delta\left(\alpha,\alpha^{\ast}\right)=\frac{e^{2\left|\alpha\right|^{2}}}{\pi}\int\frac{d^{2}z}{\pi}\left|z\right\rangle\left\langle-z\right|e^{-2\left(z\alpha^{\ast}-z^{\ast}\alpha\right)},$
(17)
where $\alpha=(q+ip)/\sqrt{2}$ and
$\left|z\right\rangle=\exp(za^{\dagger}-z^{\ast}a)\left|0\right\rangle$ is the
coherent state [30, 31], we can put Eq.(15) into the following form,
$\displaystyle W\left(q,p\right)$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\left\langle
0\right|H_{m}\left(-i\sqrt{\frac{\coth
r}{2}}a\right)\Delta\left(qe^{-r},pe^{r}\right)H_{m}\left(i\sqrt{\frac{\coth
r}{2}}a^{{\dagger}}\right)\left|0\right\rangle$
$\displaystyle=\frac{N_{r,m}^{2}}{2^{2m}}\sinh^{m}2r\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}-\tau^{2}}\left.\left\langle
0\right|e^{-i\sqrt{2\coth
r}at}\Delta\left(qe^{-r},pe^{r}\right)e^{i\sqrt{2\coth
r}a^{{\dagger}}\tau}\left|0\right\rangle\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{r,m}^{2}e^{2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}-\tau^{2}}\left.\int\frac{d^{2}z}{\pi}e^{-\left|z\right|^{2}-\left(i\sqrt{2\coth
r}t+2\beta^{\ast}\right)z+\left(2\beta-i\sqrt{2\coth
r}\tau\right)z^{\ast}}\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{r,m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\left.\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}+2\bar{\beta}t-\tau^{2}+2\allowbreak\bar{\beta}^{\ast}\tau-2t\tau\coth
r}\right|_{\tau=t=0},$ (18)
where
$\bar{\beta}=-i\beta\sqrt{2\coth
r},\beta=(qe^{-r}+ipe^{r})/\sqrt{2}=\alpha\cosh
r-\allowbreak\alpha^{\ast}\sinh r,$ (19)
and in the last step, we have used the integration formula
$\int\frac{d^{2}z}{\pi}e^{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(20)
In order to further simplify Eq.(18), expanding the exponential item
$e^{-2t\tau\coth r}$ as sum series and using Eq.(5) we have
$\displaystyle W\left(q,p\right)$
$\displaystyle=\frac{N_{m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\sum_{l=0}^{\infty}\frac{\left(-2\coth
r\right)^{l}}{2^{2l}l!}\frac{\partial^{2l}}{\partial\left(\bar{\beta}\right)^{l}\partial\left(\bar{\beta}^{\ast}\right)^{l}}\left.\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-t^{2}+2\bar{\beta}t-\tau^{2}+2\bar{\beta}^{\ast}\tau}\right|_{\tau=t=0}$
$\displaystyle=\frac{N_{m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\sum_{l=0}^{\infty}\frac{\left(-2\coth
r\right)^{l}}{2^{2l}l!}\frac{\partial^{2l}}{\partial\left(\bar{\beta}\right)^{l}\partial\left(\bar{\beta}^{\ast}\right)^{l}}H_{m}(\bar{\beta})H_{m}(\bar{\beta}^{\ast}).$
(21)
Noticing the recurrence relations of $H_{m}(x)$,
$\frac{\mathtt{d}}{\mathtt{d}x^{l}}H_{m}(x)=\frac{2^{l}m!}{\left(m-l\right)!}H_{m-l}(x),$
(22)
then the Wigner function of $\left|r,m\right\rangle$ is given by
$\displaystyle W\left(q,p\right)$
$\displaystyle=\frac{N_{m}^{2}e^{-2\left|\beta\right|^{2}}}{2^{2m}\pi}\sinh^{m}2r\sum_{l=0}^{m}\frac{\left(m!\right)^{2}\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}(\bar{\beta})\right|^{2}$
$\displaystyle=\frac{1}{\pi}\frac{e^{-2\left|\beta\right|^{2}}\sinh^{m}r}{2^{m}P_{m}\left(\cosh
r\right)}\sum_{l=0}^{m}\frac{m!\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\left|H_{m-l}(\bar{\beta})\right|^{2},$
(23)
where $\beta$ and $\bar{\beta}$ are shown in Eq.(19). Eq.(23) seems a new
result (not reported in the literature before), related to single-variable
Hermite polynomials. In particular, when the photon-added number $m=0,1$ and
noticing that $P_{0}\left(\cosh r\right)=1$ and $P_{1}\left(\cosh
r\right)=\cosh r,$ Eq.(23) reduce to, respectively,
$\displaystyle W_{m=0}\left(q,p\right)$
$\displaystyle=\frac{1}{\pi}e^{-(q^{2}e^{-2r}+p^{2}e^{2r})},$ (24)
$\displaystyle W_{m=1}\left(q,p\right)$
$\displaystyle=\frac{1}{\pi}\left\\{2(q^{2}e^{-2r}+p^{2}e^{2r})-1\right\\}e^{-(q^{2}e^{-2r}+p^{2}e^{2r})}.$
(25)
Eq.(24) is just the WF of squeezed vacuum state, a Gaussian in phase space;
while Eq.(25) corresponds to a non-Gaussian WF in phase space due to the
presence of non-Gaussian item $2(q^{2}e^{-2r}+p^{2}e^{2r})-1$. It is clear
that the function $W_{m=1}\left(q,p\right)$ becomes negative in phase space,
when $2(q^{2}e^{-2r}+p^{2}e^{2r})<1$.
Using Eq.(23) we show the plots of WF in the phase space in Figs.2 for
different squeezing parameters $r$ and photon-added numbers $m$. One can see
clearly that there is some negative region of the WF in the phase space which
implies the nonclassicality of this state. In addition, the squeezing effect
in one of the quadratures is clear in the plots (see Figs.2a and Figs.2b),
which is another evidence of the nonclassicality of this state. The WF has its
minimum value for $m=1,3$ at the center of phase space $\left(q=p=0\right)$
(see Fig.2(a) and (d)). The case is not true for $m=2$ (see Fig2.c).
Figure 2: (Color online) The Wigner functions of PASV for (a) $m=1,r=0.3,$(b)
$m=1,r=0.8,$(c) $m=2,r=0.3$ and (d) $m=3,r=0.3.$
## 3 Decoherence of PASV in thermal environment
### 3.1 Model of Decoherence
In this section, we consider how this single-mode state (1) evolves at the
presence of thermal environment. In thermal channel, the evolution of the
density matrix for the $m$-PASV can be described by [32]
$\frac{d\rho}{dt}=\kappa\left(\bar{n}+1\right)\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho
a^{\dagger}a\right)+\kappa\bar{n}\left(2a^{\dagger}\rho
a-aa^{\dagger}\rho-\rho aa^{\dagger}\right),$ (26)
where $\kappa$ represents the dissipative coefficient and $\bar{n}$ denotes
the average thermal photon number of the environment. When $\bar{n}=0,$
Eq.(26) reduces to the master equation describing the photon-loss channel. The
corresponding time evolution density operator can be obtained as
$\rho_{r,m}\left(t\right)=e^{\kappa
t+\Gamma_{0}}\sum_{k,l=0}^{\infty}M_{k,l}\rho_{0}M_{k,l}^{\dagger},$ (27)
where $\rho_{0}=\left|r,m\right\rangle\left\langle r,m\right|$ is the initial
density matrix; $M_{k,l}$ and $M_{k,l}^{{\dagger}}$ are Hermite conjugated
operators (Kraus operator) with each other,
$M_{k,l}=e^{\left(\kappa
t+\Gamma_{0}\right)/2}\sqrt{\frac{\Gamma_{-}^{k}\Gamma_{+}^{l}e^{-2l\Gamma_{0}}}{k!l!}}e^{\Gamma_{0}a^{\dagger}a}a^{\dagger
l}a^{k},$ (28)
and $\Gamma_{+},\Gamma_{-}$ and $\Gamma_{0}$ are determined by
$T=1-e^{-2\kappa t},\text{ }\Gamma_{+}=\frac{\bar{n}T}{\bar{n}T+1},\text{
}\Gamma_{-}=\frac{\left(\bar{n}+1\right)T}{\bar{n}T+1},\text{
}\Gamma_{0}=\ln\frac{e^{-\kappa t}}{\bar{n}T+1}.$ (29)
It is not difficult to prove the $M_{k,l}$ obeys the normalization condition
$\sum_{k,l=0}^{\infty}M_{k,l}^{{\dagger}}M_{k,l}=1$ by using the technique of
integration within an ordered products of operators [33, 34].
### 3.2 Wigner function of the PASV in a thermal channel
The evolution formula of WF of the PASV can be derived as follows [35]
$W\left(\zeta,\zeta^{\ast},t\right)=\frac{2}{\left(2\bar{n}+1\right)T}\int\frac{d^{2}\alpha}{\pi}W\left(\alpha,\alpha^{\ast},0\right)e^{-2\frac{\allowbreak\left|\zeta-\alpha
e^{-\kappa t}\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}},$ (30)
where $W\left(\alpha,\alpha^{\ast},0\right)$ is the WF of the initial state.
Eq.(30) is just the evolution formula of WF in thermal channel. Thus the WF at
any time can be obtained by performing the integration when the initial WF is
known.
In a similar way to deriving Eq.(23), substituting Eq.(23) into Eq.(30) and
using the generating function of single-variable Hermite polynomials (5) and
Eq.(20), we finally obtain (see appendix B)
$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{2\sinh^{m}r}{\pi\sqrt{C}\left(2\bar{n}+1\right)T}\frac{m!e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}}{2^{m}P_{m}\left(\cosh
r\right)}e^{\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}$
$\displaystyle\times\sum_{l=0}^{m}\sum_{k=0}^{m-l}\frac{\left(-2\coth
r\right)^{l}G^{m-l-k}F^{k}}{l!k!\left[\left(m-l-k\right)!\right]^{2}}\left|H_{m-l-k}(E/\sqrt{G})\right|^{2}.$
(31)
where we have set
$\displaystyle A$ $\displaystyle=\frac{2e^{-2\kappa
t}}{\left(2\allowbreak\bar{n}+1\right)T}+2\cosh 2r,\text{ }$ (32)
$\displaystyle B$ $\displaystyle=\frac{2e^{-\kappa
t}\zeta}{\left(2\allowbreak\bar{n}+1\right)T},\text{
}C=A^{2}-4\sinh^{2}2r,\text{ }$ (33) $\displaystyle D$
$\displaystyle=\sqrt{2\coth r}i\left(B^{\ast}\sinh r-B\cosh r\right),$ (34)
$\displaystyle E$ $\displaystyle=\frac{1}{C}\left(AD-2D^{\ast}\sinh
2r\right),$ (35) $\displaystyle F$ $\displaystyle=\frac{8}{C}\left(A\cosh
2r\coth r-4\cosh^{2}r\sinh 2r\right),$ (36) $\displaystyle G$
$\displaystyle=1-\frac{16}{C}\frac{e^{-2\kappa
t}}{\left(2\allowbreak\bar{n}+1\right)T}\cosh^{2}r.$ (37)
Eq.(31) is just the analytical expression of WF for the PASV in thermal
channel. It is obvious that the WF loss its Gaussian property due to the
presence of single-variable Hermite polynomials. In particular, at the initial
time ($t=0$), noting $\sqrt{C}\left(2\bar{n}+1\right)T\rightarrow 2,$
$G\rightarrow 1,$ $F/G\rightarrow 0,$ and
$E/\sqrt{G}\rightarrow\bar{\beta}=-i\sqrt{2\coth r}\left(\zeta\cosh
r-\zeta^{\ast}\sinh r\right),$ as well as
$\frac{A}{C}\left|B\right|^{2}-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}\rightarrow-2\left|\zeta\right|^{2}\cosh
2r,\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)\rightarrow\left(\zeta^{2}+\zeta^{\ast
2}\right)\sinh 2r$, Eq.(31) just does reduce to Eq.(23), i.e., the WF of the
PASV. On the other hand, when $\kappa t\rightarrow\infty,$ noticing that
$T\rightarrow 1,B\rightarrow 0,C\rightarrow 4,D\rightarrow 0,$
$E/\sqrt{G}\rightarrow 0,G\rightarrow 1,$ $F\rightarrow\allowbreak 4\coth
r,$as well as
$H_{m}\left(0\right)=\left(-1\right)^{j}\frac{m!}{j!}\delta_{m,2j},$ then
Eq.(31) becomes
$\allowbreak\frac{1}{\pi\left(2\bar{n}+1\right)}e^{-\frac{2\left|\zeta\right|^{2}}{2\allowbreak\bar{n}+1}}$,
which is independent of photon-addition number $m$ and corresponds to the WF
of thermal state with mean thermal photon number $\bar{n}$. This indicates
that the system state reduces to a thermal state after an enough long time
interaction with the environment.
In addition, for the case of $m=0,$ single-mode squeezed vacuum, Eq.(31) just
becomes ($H_{0}(x)=1$)
$W_{m=0}\left(\zeta,\zeta^{\ast},t\right)=\mathfrak{N}e^{-\mathfrak{D}\left|\zeta\right|^{2}+\mathfrak{E(}\zeta^{\ast}{}^{2}+\zeta{}^{2})},$
(38)
where $\mathfrak{N}=\frac{2}{\pi\sqrt{C}\left(2\bar{n}+1\right)T}$ is the
normalization factor,
$\mathfrak{D}=\frac{2}{\left(2\allowbreak\bar{n}+1\right)T}-\frac{A\mathfrak{E}}{\sinh
2r},\mathfrak{E}\mathfrak{=}\frac{4e^{-2\kappa t}\sinh
2r}{\left[\left(2\allowbreak\bar{n}+1\right)T\right]^{2}C},$ Eq.(38) denotes a
Gaussian distribution function— the WF of single-mode squeezed vacuum in the
thermal channel; while for $m=1$, single–photon added case, its WF in the
thermal channel is given by ($H_{1}(x)=2x$)
$W_{m=1}\left(\zeta,\zeta^{\ast},t\right)=\frac{4\left|E\right|^{2}+F-2\coth
r}{\pi\sqrt{C}\left(2\bar{n}+1\right)T\coth
r}e^{\frac{A}{C}\left|B\right|^{2}-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}.$ (39)
In Fig.3, the WFs of the PASV with $m=1$ are depicted in phase space with
$r=0.3$ and $\bar{n}=1$ for several different $\kappa t.$ It is easy to see
that the negative region of WF gradually diminishes as the time $\kappa t$
increases. Actually, from Eq.(33) one can see that $C>0$, so at the center of
the phase space ($\alpha=\alpha^{\ast}=0$), when $F<2\coth r$ leading to the
following condition:
$\kappa t<\kappa t_{c}\equiv\frac{1}{2}\ln\frac{2\bar{n}+2}{2\bar{n}+1},$ (40)
which is independent of the squeezing parameter $r$, there always exist
negative region for WF in phase space and the WF of PASV is always positive in
the whole phase space when $\kappa t\ $exceeds the threshold value $\kappa
t_{c}$.
In Figs. 4 and 5, we plot the variation of WF in phase space for different
$\bar{n}$ and $r,$ respectively. It is found that the partial negativity of WF
decreases gradually as $\bar{n}$ (or $r$) increases for a given time. The
squeezing effect in one of the quadrature is shown in Fig.5. In addition, for
the case of large squeezing value $r$, the single-photon added squeezed state
becomes similar to a Schodinger cat state (see Fig.6). The WF becomes Gaussian
with the time evolution. In principle, by using the explicit expression of WF
in Eq.(31), we can draw the WF distribution for any photon-added case in phase
space. For instance, for $m=2,$ there are two negative regions of the WF,
which differs from the case of single PASV (see Fig.7). The absolute value of
the negative minimum of the WF decreases as $\kappa t$ increases, which leads
to the complete absence of partial negative region.
## 4 Conclusions
The nonclassical properties and decoherence of single-mode PASV in a thermal
environment have been investigated. A compact expression for the normalization
factor of PASV is derived by converting the PASV to a squeezed Hermite
polynomial excitation state. It is shown that the normalization factor is just
an $m$-order Legendre polynomial of the squeezing parameter $r$. We also
derived the explicit analytical expression of WF for any photon-added number
$m$ and found the negative region of WF in phase space. We also show that
there is an upper bound value of $r$ for this state to exhibit sub-Poissonian
statistics which increases as $m$ increases. Then we considered the effects of
decoherence to the nonclassicality of PASV when interacting with thermal
environment. For arbitrary number PASV, we derived the explicit analytical
expression of time evolution of WF and presented the loss of nonclassicality
in reference of the negativity of WF. The threshold value of the decay time
corresponding to the transition of the WF from partial negative to completely
positive definite is obtained. We find that the WF has always negative value
for all parameters $r$ if the decay time $\kappa
t<\frac{1}{2}\ln\frac{2\bar{n}+2}{2\bar{n}+1}$ for single PASV.
ACKNOWLEDGEMENTS: Work supported by the National Natural Science Foundation of
China (Grant Nos 10775097 and 10874174) and the Research Foundation of the
Education Department of Jiangxi Province.
Figure 3: (Color online) The Wigner functions of single-photon-added squeezed
vacuum states in phase space for $r=0.3,\bar{n}=1$ at $(a)$ $\kappa
t=0.05,(b)$ $\kappa t=0.1,(c)$ $\kappa t=0.2\ $and $(d)$ $\kappa t=0.5.$
Figure 4: (Color online) The Wigner functions of single-photon-added squeezed
vacuum states in phase space for $r=0.3$ and $\kappa t=0.05$ with $(a)$
$\bar{n}=0,(b)$ $\bar{n}=1,(c)$ $\bar{n}=2,(d)$ $\bar{n}=5.$ Figure 5: (Color
online) The Wigner functions of single-photon-subtracted squeezed vacuum
states in phase space for $\bar{n}=0.1$ and $\kappa t=0.05$ with $(a)$
$r=0.03,(b)$ $r=0.5,(c)$ $r=0.8,(d)$ $r=1.5.??$ Figure 6: (Color online)
Wigner function of single photon-added squeezed vacuum states in phase space
for $r=0.8,\bar{n}=0$: $(a)$ $\kappa t=0.1,(b)$ $\kappa t=0.2,(c)$ $\kappa
t=0.3,\ $and $(d)$ $\kappa t=0.7.$ Figure 7: (Color online) The Wigner
functions of photon-added squeezed vacuum states in phase space for
$r=0.7,\bar{n}=1,m=2$: $(a)$ $\kappa t=0.1,(b)$ $\kappa t=0.2.$
Appendix A: Derivation of Eq.(7)
Recalling that the newly found expression of Legendre polynomial [24, 25]
$x^{m}\sum_{{l}=0}^{\left[m/2\right]}\frac{m!}{2^{2{l}}\left({l}!\right)^{2}\left(m-2{l}\right)!}\left(1-\frac{1}{x^{2}}\right)^{{l}}=P_{m}\left(x\right),$
(A1)
which is equivalent to the well-known Legendre polynomial’s expression [36]
$P_{m}\left(x\right)=\sum_{l=0}^{[m/2]}\left(-1\right)^{l}\frac{\left(2m-2l\right)!}{2^{m}l!\left(m-l\right)!\left(m-2l\right)!}x^{m-2l},$
(A2)
though it is different in form from Eq.(A1), they actually are equal. For
instance, what we list in the following equations (the left) from Eq.(A1) are
equal to what we list from (A2) (the right)
$\displaystyle m$ $\displaystyle=0,\text{ }P_{0}\left(x\right)=1;\text{ \ }$
$\displaystyle\text{\ }m$ $\displaystyle=1,\text{ }P_{1}\left(x\right)=x;$
$\displaystyle m$ $\displaystyle=2,\text{
}P_{2}\left(x\right)=x^{2}\left[1+\frac{1}{2}\left(1-\frac{1}{x^{2}}\right)\right]=\frac{3}{2}x^{2}-\frac{1}{2};$
$\displaystyle m$ $\displaystyle=3,\text{
}P_{3}\left(x\right)=x^{3}\left[1+\frac{3}{2}\left(1-\frac{1}{x^{2}}\right)\right]=\frac{5}{2}x^{3}-\frac{3}{2}x;$
(A3)
and
$\displaystyle m$ $\displaystyle=4,\text{
}P_{4}\left(x\right)=x^{4}\left[1+3\left(1-\frac{1}{x^{2}}\right)+\frac{3}{8}\left(1-\frac{1}{x^{2}}\right)^{{2}}\right]=\frac{1}{8}\left(35x^{4}-30x^{2}+3\right);$
$\displaystyle m$ $\displaystyle=5,\text{
}P_{5}\left(x\right)=x^{5}\left[1+5\left(1-\frac{1}{x^{2}}\right)+\allowbreak\frac{15}{8}\left(1-\frac{1}{x^{2}}\right)^{{2}}\right]=\frac{1}{8}\left(63x^{5}-70x^{3}+15x\right);$
(A4)
We emphasize that the new form in Eq.(A1) cannot be directly obtained by some
series summation rearrangement technique from the orginal definition (A2).
On the other hand, noting the following relation
$\displaystyle\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.\exp\left(-t^{2}-\tau^{2}+2x\tau
t\right)\right|_{t,\tau=0}$
$\displaystyle=\sum_{n,l,k=0}^{\infty}\frac{\left(-\right)^{n+l}}{n!l!k!}\left(2x\right)^{k}\left.\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\tau^{2n+k}t^{2l+k}\right|_{t,\tau=0}$
$\displaystyle=2^{m}m!\sum_{n=0}^{\left[m/2\right]}\frac{m!}{2^{2n}\left(n!\right)^{2}\left(m-2n\right)!}x^{m-2n},$
(A5)
and then comparing Eq.(A5) with Eq.(A1) we can obtain Eq.(7).
APPENDIX B: Derivation of Wigner function Eq.(31) of PASV
In this appendix, we present the details for deriving the Wigner function
Eq.(31). Substituting Eq.(23) into Eq.(30) and noticing Eq.(19) as well as the
generating function of single-variable Hermite polynomials (5), we have
$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\bar{N}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\int\frac{d^{2}\alpha}{\pi}e^{-2\frac{\allowbreak\left|\zeta-\alpha
e^{-\kappa
t}\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}-2\left|\alpha\cosh
r-\allowbreak\alpha^{\ast}\sinh r\right|^{2}}$
$\displaystyle\times\left|H_{m-l}(-i\left(\alpha\cosh
r-\allowbreak\alpha^{\ast}\sinh r\right)\sqrt{2\coth r})\right|^{2}$
$\displaystyle=\bar{N}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}e^{-\upsilon^{2}-\tau^{2}}$
$\displaystyle\times\int\frac{d^{2}\alpha}{\pi}\left.\exp\left\\{-A\left|\alpha\right|^{2}+B_{1}\alpha+B_{2}\alpha^{\ast}+\left(\allowbreak\alpha^{2}+\alpha^{\ast}{}^{2}\right)\sinh
2r\right\\}\right|_{\tau=\upsilon=0},$ (B1)
where $A$ and $B$ are given by Eqs.(34)-(35),
$\bar{N}=\frac{2\sinh^{m}r}{\pi\left(2\bar{n}+1\right)T}\frac{m!}{2^{m}P_{m}\left(\cosh
r\right)},$ (B2)
and
$\displaystyle B_{1}$ $\displaystyle=B^{\ast}\allowbreak-i2\left(\upsilon\cosh
r+\tau\sinh r\right)\sqrt{2\coth r},$ (B3) $\displaystyle B_{2}$
$\displaystyle=B+i2\allowbreak\left(\upsilon\sinh r+\tau\cosh
r\right)\sqrt{2\coth r}.$ (B4)
Further using the following integral identity [37]:
$\displaystyle\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi
z+\eta z^{\ast}+fz^{2}+gz^{\ast 2}\right)$
$\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left[\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right],$
(B5)
whose convergent condition is Re$\left(\zeta\pm f\pm g\right)<0,\ $and
Re$\left(\frac{\zeta^{2}-4fg}{\zeta\pm f\pm g}\right)<0,$ we can put Eq.(B1)
into the following form:
$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}$
$\displaystyle\times\exp\left\\{-\upsilon^{2}-\tau^{2}+\frac{A}{C}B_{1}B_{2}+\frac{\allowbreak
1}{C}\left(B_{1}^{2}+B_{2}^{2}\right)\sinh 2r\right\\}_{\tau=\upsilon=0},$
(B6)
where $C$ and $D$ are given by (34)-(35), and
$\displaystyle B_{1}B_{2}$ $\displaystyle=B^{\ast}B+2\left(\allowbreak\upsilon
D+\tau
D^{\ast}\right)+8\left(\tau^{2}+\upsilon^{2}\right)\cosh^{2}r+8\tau\upsilon\cosh
2r\coth r,$ (B7) $\displaystyle B_{1}^{2}+B_{2}^{2}$
$\displaystyle=B^{\ast}{}^{2}+B{}^{2}-32\tau\upsilon\cosh^{2}r-8\left(\upsilon^{2}+\tau^{2}\right)\cosh
2r\coth r-\allowbreak 4\tau D-4\upsilon D^{\ast},$ (B8)
then substituting Eqs.(B7)-(B8) into Eq.(B6) yields
$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}}e^{\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}$
$\displaystyle\times\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}\exp\left\\{-G\left(\upsilon^{2}+\tau^{2}\right)+\allowbreak
2\upsilon E+2\tau E^{\ast}+\tau\upsilon F\right\\}_{\tau=\upsilon=0},$ (B9)
where $E,F,$ and $G$ are given by Eqs.(35)-(37). In order to further simplify
Eq.(B9), expanding the exponential item $e^{\tau\upsilon F}$ as sum series and
using Eqs.(5) and the formula
$\left.\frac{\partial^{m}}{\partial\upsilon^{m}}e^{-G\upsilon^{2}+\allowbreak
2\upsilon E}\right|_{\upsilon=0}=G^{m/2}H_{m}\left(E/\sqrt{G}\right),$ (B10)
we see
$\displaystyle W\left(\zeta,\zeta^{\ast},t\right)$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}+\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}}{l!\left[\left(m-l\right)!\right]^{2}}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{F^{k}}{k!}\frac{\partial^{2k}}{\partial\left(2E\right)^{k}\partial\left(2E^{\ast}\right)^{k}}\frac{\partial^{2m-2l}}{\partial\upsilon^{m-l}\partial\tau^{m-l}}\left.e^{-G\left(\upsilon^{2}+\tau^{2}\right)+\allowbreak
2\upsilon E+2\tau E^{\ast}}\right|_{\tau=\upsilon=0}$
$\displaystyle=\frac{\bar{N}}{\sqrt{C}}e^{-\frac{2\left|\zeta\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}+\frac{A}{C}\left|B\right|^{2}+\frac{\allowbreak\sinh
2r}{C}\left(B^{\ast}{}^{2}+B{}^{2}\right)}\sum_{l=0}^{m}\frac{\left(-2\coth
r\right)^{l}G^{m-l}}{l!\left[\left(m-l\right)!\right]^{2}}$
$\displaystyle\times\sum_{k=0}^{\infty}\frac{F^{k}}{k!}\frac{\partial^{2k}}{\partial\left(2E\right)^{k}\partial\left(2E^{\ast}\right)^{k}}\left|H_{m-l}\left(E/\sqrt{G}\right)\right|^{2}.$
(B11)
Then using Eq.(22) yields Eq.(31).
## References
* [1] D. Bouwmeester, A. Ekert and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, Berlin, 2000).
* [2] R. Short and L. Mandel, Phys. Rev. Lett. 51, 384 (1983).
* [3] V. V. Dodonov, J. Opt. B: Quantum Semiclassical Opt. 4, R1 (2002).
* [4] M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
* [5] J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 92, 153601 (2004).
* [6] A. Zavatta, S. Viciani, and M. Bellini, Science, 306, 660 (2004).
* [7] A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 72, 023820 (2005).
* [8] V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science, 317, 1890 (2007).
* [9] R. W. Boyd, K. W. Chan, and M. N. O’Sullivan, Science, 317, 1874 (2007).
* [10] A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 75, 052106 (2007).
* [11] J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006).
* [12] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, Ph. Grangier, Science 312, 83 (2006).
* [13] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, Opt. Express 15, 3568 (2007).
* [14] M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, Phys. Rev. A 55, 3184 (1997).
* [15] S. Glancy and H. M. de Vasconcelos, J. Opt. Soc. Am. B 25, 712 (2008).
* [16] A. Biswas and G. S. Agarwal, Phys. Rev. A 75, 032104 (2007).
* [17] H. Jeong, A. P. Lund, and T. C. Ralph, Phys. Rev. A 72, 013801 (2005).
* [18] B. T. H. Vracoe, S. Brattke, M. Weidinger, H. Walther, Nature 403, 743 (2000); S. Brattke, B.T.H. Vracoe, H. Walther, Phys. Rev. Lett. 86, 3534 (2001).
* [19] Z. M. Zhang, Chin. Phys. Lett. 20 (2003) 227; Z. M. Zhang, Chin. Phys. Lett. 21 (2004) 5.
* [20] Z. X. Zhang, H. Y. Fan, Phys. Lett. A 174 (1993) 206.
* [21] D. F. Walls andG J Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
* [22] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge: Cambidge University Press, 1997).
* [23] H.Y. Fan and Vander J. Linde, J. Phys. A 24, 2529 (1989).
* [24] L. Y. Hu and H. Y. Fan, J. Opt. Soc. Am. B, 25, 1955 (2008).
* [25] H. Y. Fan, X. G. Meng and J. S. Wang, Commun. Theor. Phys. 46, 845 (2006).
* [26] M. S. Kim, J. Phys. B: At. Mol. Opt. Phys. 41, 133001 (2008).
* [27] P. Marek, H. Jeong, and M. S. Kim, Phys. Rev. A 78, 063811 (2008).
* [28] H.-Y. Fan, H. R. Zaidi, Phys. Lett. A 124, 303 (1987).
* [29] H.-Y. Fan, Representation and Transformation Theory in Quantum Mechanics, Shanghai Scientific & Technical, Shanghai Press, 1997.
* [30] R. Glauber, Phys. Rev. 130, 2529 (1963).
* [31] R. Glauber, Phys. Rev. 131, 2766 (1963).
* [32] C. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, 2000).
* [33] H.-Y. Fan, H.-L. Lu and Y. Fan, Ann. Phys. 321, 480 (2006).
* [34] H.-Y. Fan and L. Y. Hu, Mod. Phys. Lett. B, 22, 2435 (2008).
* [35] L. Y. Hu and H.-Y. Fan, Opt. Commun. 282, 4379 (2009).
* [36] I. S. Gradshteyn and L. M. Ryzhik, Tables of Integration Series and Products (Academic Press, New York, 1980).
* [37] R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, Berlin, 2001), Appendix A.
|
arxiv-papers
| 2009-10-28T12:02:42 |
2024-09-04T02:49:06.145488
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0910.5358"
}
|
0910.5574
|
# Essential p-dimension of algebraic tori
Roland Lötscher(1) , Mark MacDonald , Aurel Meyer(2) and Zinovy
Reichstein(3)
###### Abstract.
The essential dimension is a numerical invariant of an algebraic group $G$
which may be thought of as a measure of complexity of $G$-torsors over fields.
A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for
the essential dimension of a finite $p$-group. We obtain similar formulas for
the essential $p$-dimension of a broader class of groups, which includes all
algebraic tori.
###### Key words and phrases:
Essential dimension, algebraic torus, twisted finite group, lattice
###### 2000 Mathematics Subject Classification:
20G15
(1) Roland Lötscher was partially supported by the Swiss National Science
Foundation (Schweizerischer Nationalfonds).
(2) Aurel Meyer was partially supported by a University Graduate Fellowship at
the University of British Columbia
(3) Zinovy Reichstein was partially supported by NSERC Discovery and
Accelerator Supplement grants
###### Contents
1. 1 Introduction
2. 2 Proof of Theorem 1.2
3. 3 The $p$-closure of a field
4. 4 The group $C(G)$
5. 5 Proof of Theorem 1.3(a)
6. 6 $p$-isogenies
7. 7 Proof of Theorem 1.3(b)
8. 8 An additivity theorem
9. 9 Modules and lattices
10. 10 Proof of Theorem 1.3(c)
11. 11 Tori of essential dimension $\leq 1$
12. 12 Tori split by cyclic extensions of degree dividing $p^{2}$
## 1\. Introduction
Throughout this paper $p$ will denote a prime integer, $k$ a base field of
characteristic $\neq p$ and $G$ a (not necessarily smooth) algebraic group
defined over $k$. Unless otherwise specified, all fields are assumed to
contain $k$ and all morphisms between them are assumed to be
$k$-homomorphisms.
We begin by recalling the notion of essential dimension of a functor from
[BF]. Let $\operatorname{Fields}_{k}$ be the category of field extensions
$K/k$, $\operatorname{Sets}$ be the category of sets, and
$F\colon\operatorname{Fields}_{k}\to\operatorname{Sets}$ be a covariant
functor. As usual, given a field extension $k\subset K_{0}\subset K$, we will
denote the image of $\alpha\in F(K)$ under the natural map $F(K)\to F(L)$ by
$\alpha_{L}$.
An object $\alpha\in F(K)$ is said to _descend_ to an intermediate field
$k\subseteq K_{0}\subseteq K$ if $\alpha$ is in the image of the induced map
$F(K_{0})\to F(K)$. The _essential dimension_ $\operatorname{ed}_{k}(\alpha)$
is defined as the minimum of the transcendence degrees
$\operatorname{trdeg}_{k}(K)$ taken over all fields $k\subseteq K_{0}\subseteq
K$ such that $\alpha$ descends to $K_{0}$. The essential dimension
$\operatorname{ed}_{k}(F)$ of the functor $F$ is defined as the maximal value
of $\operatorname{ed}_{k}(\alpha)$, where the maximum is taken over all fields
$K/k$ and all $\alpha\in F(K)$.
Of particular interest to us will be the Galois cohomology functor
$F_{G}\colonequals H^{1}(*,G)$, which associates to every $K/k$ the set of
isomorphism classes of $G$-torsors over $\operatorname{Spec}(K)$. The
essential dimension of this functor is usually called the essential dimension
of $G$ and is denoted by the symbol $\operatorname{ed}_{k}(G)$. Informally
speaking, this number may be thought of a measure of complexity of $G$-torsors
over fields. For example, if $k$ is an algebraically closed field of
characteristic $0$ then groups $G$ of essential dimension $0$ are precisely
the so-called special groups, i.e., algebraic groups $G/k$ with the property
that every $G$-torsor over $\operatorname{Spec}(K)$ is split, for every field
$K/k$. These groups were classified by A. Grothendieck [Gro].
For many groups the essential dimension is hard to compute, even over the
field $\mathbb{C}$ of complex numbers. The following related notion is often
more accessible. Let $F\colon\operatorname{Fields}_{k}\to\operatorname{Sets}$
be a covariant functor and $p$ be a prime integer, as above. The essential
$p$-dimension of $\alpha\in F(K)$, denoted $\operatorname{ed}_{k}(\alpha;p)$,
is defined as the minimal value of
$\operatorname{ed}_{k}(\alpha_{K^{\prime}})$, where $K^{\prime}$ ranges over
all finite field extensions of $K$ whose degree is prime to $p$. The essential
$p$-dimension of $F$, $\operatorname{ed}_{k}(F;p)$ of $F$ is once again,
defined as the maximal value of $\operatorname{ed}_{k}(\alpha;p)$, where the
maximum is taken over all fields $K/k$ and all $\alpha\in F(K)$, and once
again we will write $\operatorname{ed}_{k}(G;p)$ in place of
$\operatorname{ed}_{k}(F_{G};p)$, where $F_{G}\colonequals H^{1}(\ast,G)$ is
the Galois cohomology functor.
Note that $\operatorname{ed}_{k}(\alpha)$, $\operatorname{ed}_{k}(F)$,
$\operatorname{ed}_{k}(G)$, $\operatorname{ed}_{k}(\alpha;p)$, etc., depend on
$k$. We will write $\operatorname{ed}$ instead of $\operatorname{ed}_{k}$ if
the reference to $k$ is clear from the context. For background material on
essential dimension we refer the reader to [BR, Re, RY, BF, Me1].
We also remark that in the case of the Galois cohomology functor $F_{G}$, the
maximal value of $\operatorname{ed}_{k}(\alpha)$ and
$\operatorname{ed}_{k}(\alpha;p)$ in the above definitions is attained in the
case where $\alpha$ is a versal $G$-torsor in the sense of [GMS, Section I.5].
Since every generically free linear representation $\rho\colon
G\to\operatorname{GL}(V)$ gives rise to a versal $G$-torsor (see [GMS, Example
I.5.4]), we obtain the inequality
(1)
$\operatorname{ed}_{k}(G;p)\leq\operatorname{ed}_{k}(G)\leq\dim(V)-\dim(G)\,;$
see [Re, Therem 3.4] or [BF, Lemma 4.11]. (Recall that $\rho$ is called
generically free if there exists a $G$-invariant dense open subset $U\subset
V$ such that the scheme-theoretic stabilizer of every point of $U$ is
trivial.)
N. Karpenko and A. Merkurjev [KM] recently showed that the inequality (1) is
in fact sharp for finite constant $p$-groups.
###### Theorem 1.1.
Let $G$ be a constant $p$-group and $k$ be a field containing a primitive
$p$th root of unity. Then
$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(G)=\min\,\dim(V)\,,$
where the minimum is taken over all faithful $k$-representations
$G\hookrightarrow\operatorname{GL}(V)$.
The goal of this paper is to prove similar formulas for a broader class of
groups $G$. To state our first result, let
(2) $1\to C\to G\to Q\to 1$
be an exact sequence of algebraic groups over $k$ such that $C$ is central in
$G$ and isomorphic to $\mu_{p}^{r}$ for some $r\geq 0$. Given a character
$\chi\colon C\to\mu_{p}$, we will, following [KM], denote by
$\operatorname{Rep}^{\chi}$ the set of irreducible representations $\phi\colon
G\to\operatorname{GL}(V)$, defined over $k$, such that
$\phi(c)=\chi(c)\operatorname{Id}_{V}$ for every $c\in C$.
###### Theorem 1.2.
Assume that $k$ is a field of characteristic $\neq p$ containing a primitive
$p$th root of unity. Suppose a sequence of $k$-groups of the form (2)
satisfies the following condition:
$\gcd\\{\dim(\phi)\,|\,\phi\in\operatorname{Rep}^{\chi}\\}=\min\\{\dim(\phi)\,|\,\phi\in\operatorname{Rep}^{\chi}\\}$
for every character $\chi\colon C\to\mu_{p}$. (Here, as usual, $\gcd$ stands
for the greatest common divisor.) Then
$\operatorname{ed}_{k}(G;p)\geq\min\dim(\rho)-\dim G\,,$
where the minimum is taken over all finite-dimensional $k$-representations
$\rho$ of $G$ such that $\rho_{|\,C}$ is faithful.
Of particular interest to us will be extensions of finite $p$-groups by
algebraic tori, i.e., $k$-groups $G$ which fit into an exact sequence of the
form
(3) $1\to T\to G\to F\to 1\,,$
where $F$ is a finite $p$-group and $T$ is a torus over $k$. Note that in this
paper we will view finite groups $F$ as algebraic groups over $k$, and will
not assume they are constant, which is to say, the absolute Galois group of
$k$ may act non-trivially on the separable points of $G$. For the sake of
computing $\operatorname{ed}_{k}(G;p)$ we may assume that $k$ is a $p$-closed
field (as in Definition 3.1); see Lemma 3.3. In this situation we will show
that
(i) there is a natural choice of a split central subgroup $C\subset G$ in the
sequence (2) such that
(ii) the conditions of Theorem 1.2 are always satisfied.
(iii) Moreover, if $G$ is isomorphic to the direct product of a torus and a
finite twisted $p$-group, then a variant of (1) yields an upper bound,
matching the lower bound of Theorem 1.2.
This brings us to the main result of this paper. We will say that a
representation $\rho\colon G\to\operatorname{GL}(V)$ of an algebraic group $G$
is $p$-faithful if its kernel is finite and of order prime to $p$.
###### Theorem 1.3.
Let $G$ be an extension of a (twisted) finite $p$-group $F$ by an algebraic
torus $T$ defined over a field $k$ (of characteristic not $p$). In other
words, we have an exact sequence
$1\to T\to G\to F\to 1\,.$
Denote a $p$-closure of $k$ by $k^{(p)}$ (see Definition 3.1). Then
(a) $\operatorname{ed}_{k}(G;p)\geq\min\dim(\rho)-\dim G$, where the minimum
is taken over all $p$-faithful linear representations $\rho$ of $G_{k^{(p)}}$
over $k^{(p)}$.
Now assume that $G$ is the direct product of $T$ and $F$. Then
(b) equality holds in (a), and
(c) over $k^{(p)}$ the absolute essential dimension of $G$ and the essential
$p$-dimension coincide:
$\operatorname{ed}_{k^{(p)}}(G_{k^{(p)}})=\operatorname{ed}_{k^{(p)}}(G_{k^{(p)}};p)=\operatorname{ed}_{k}(G;p).$
If $G$ is a $p$-group, a representation $\rho$ is $p$-faithful if and only if
it is faithful. However, for an algebraic torus, “$p$-faithful” cannot be
replaced by “faithful”; see Remark 10.3.
Theorem 1.3 appears to be new even in the case where $G$ is a twisted cyclic
$p$-group, where it extends earlier work of Rost [Ro], Bayarmagnai [Ba] and
Florence [Fl]; see Corollary 9.3 and Remark 9.4.
If $G$ a direct product of a torus and an abelian $p$-group, the value of
$\operatorname{ed}_{k}(G;p)$ given by Theorem 1.3 can be rewritten in terms of
the character module $X(G)$; see Corollary 9.2. In particular, we obtain the
following formula for the essential dimension of a torus.
###### Theorem 1.4.
Let $T$ be an algebraic torus defined over a $p$-closed field $k=k^{(p)}$ of
characteristic $\neq p$. Suppose
$\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ acts on the character
lattice $X(T)$ via a finite quotient $\overline{\Gamma}$. Then
$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T;p)=\min\operatorname{rank}(L)\,,$
where the minimum is taken over all exact sequences of
$\mathbb{Z}_{(p)}\overline{\Gamma}$-lattices of the form
$(0)\to L\to P\to X(T)_{(p)}\to(0)\,,$
where $P$ is permutation and $X(T)_{(p)}$ stands for
$X(T)\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$.
In many cases Theorem 1.4 renders the value of $\operatorname{ed}_{k}(T)$
computable by known representation-theoretic methods, e.g., from [CR]. We will
give several examples of such computations in Sections 11 and 12. Another
application was recently given by Merkurjev (unpublished), who used Theorem
1.4, in combination with techniques from [Me2], to show that
$\operatorname{ed}_{k}(\operatorname{PGL}_{p^{r}};p)\geq(r-1)p^{r}+1$
for any $r\geq 1$. (For $r=2$ the above inequality is the main result of
[Me2].) This represents dramatic improvement over the best previously known
lower bounds on $\operatorname{ed}_{k}(\operatorname{PGL}_{p^{r}})$. The
question of computing $\operatorname{ed}_{k}(\operatorname{PGL}_{p^{r}})$ is a
long-standing open problem; for an overview, see [MR1, MR2].
It is natural to try to extend the formula of Theorem 1.3(b) to all $k$-groups
$G$, whose connected component $G^{0}$ is a torus. For example, the normalizer
of a maximal torus in any reductive $k$-group is of this form. For the purpose
of computing $\operatorname{ed}_{k}(G;p)$ we may assume that $k$ is $p$-closed
and $G/G^{0}$ is a $p$-group; in other words, $G$ is as in Theorem 1.3(a).
Then
(4) $\min\dim\mu-\dim(G)\leq\operatorname{ed}(G;p)\leq\min\dim\rho-\dim G\,,$
where the two minima are taken over all $p$-faithful representations $\mu$,
and $p$-generically free representations $\rho$, respectively. Here we say
that a representation $\rho$ of $G$ is $p$-generically free if the
$\ker(\rho)$ is finite of order prime to $p$, and $\rho$ descends to a
generically free representation of $G/\ker(\rho)$. The upper bound in (4)
follows from (1), in combination with Theorem 6.1; the lower bound is Theorem
1.3(a). If $G$ is a direct product of a torus and a $p$-group, then every
$p$-generically free representation is $p$-faithful (see Lemma 7.1). In this
case the lower and upper bounds of (4) coincide, yielding the exact value of
$\operatorname{ed}_{k}(G;p)$ of Theorem 1.3(b). However, if we only assume $G$
is a $p$-group extended by a torus, then faithful $G$-representations no
longer need to be generically free. We do not know how to bridge the gap
between the upper and the lower bound in (4) in this generality; however, in
all of the specific examples we have considered, the upper bound turned out to
be sharp. We thus put forward the following conjecture.
###### Conjecture 1.5.
Let $G$ be an extension of a $p$-group by a torus, defined over a field $k$ of
characteristic $\neq p$. Then
$\operatorname{ed}(G;p)=\min\dim\rho-\dim G,$
where the minimum is taken over all $p$-generically free representations
$\rho$ of $G_{k^{(p)}}$ over $k^{(p)}$.
The rest of the paper is structured as follows. Theorem 1.2 is proved in
Section 2. Section 3 is devoted to preliminary material on the $p$-closure of
a field. Theorem 1.3(a) is proved in Sections 4 and 5. In Section 6 we will
show that if $G\to Q$ is a $p$-isogeny then
$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(Q;p)$. This result playes a
key role in the proof of Theorem 1.3(b) in Section 7. At the end of Section 7
we prove a formula for the essential $p$-dimension of any finite group $G$ by
passing to a Sylow $p$-subgroup defined over $k$; see Corollory 7.2. In
Section 8 we prove the following Additivity Theorem 8.1: If $G_{1}$ and
$G_{2}$ are direct products of tori and $p$-groups, then
$\operatorname{ed}_{k}(G_{1}\times
G_{2};p)=\operatorname{ed}_{k}(G_{1};p)+\operatorname{ed}_{k}(G_{2};p)\,.$
In Section 9 we restate and amplify Theorem 1.3(b) (with $G$ abelian) in terms
of $\operatorname{Gal}(k_{\operatorname{sep}}/k)$-modules; in particular,
Theorem 1.4 stated above is a special case of Corollary 9.2 which is proved
there. In Section 10 we prove Theorem 1.3(c) by using Theorem 1.3(b),
additivity, and the lattice perspective from Section 9. The last two sections
are intended to illustrate our results by computing essential dimensions of
specific algebraic tori. In Section 11 we classify algebraic tori $T$ of
essential $p$-dimension $0$ and $1$; see Theorems 11.1 and 11.5. In Section 12
we compute the essential $p$-dimension of all tori $T$ over a $p$-closed field
$k$, which are split by a cyclic extension $l/k$ of degree dividing $p^{2}$.
## 2\. Proof of Theorem 1.2
Denote by $C^{\ast}\colonequals\operatorname{Hom}(C,\mu_{p})$ the character
group of $C$. Let $E\to\operatorname{Spec}K$ be a versal $Q$-torsor [GMS,
Example 5.4], where $K/k$ is some field extension, and let $\beta\colon
C^{\ast}\to\operatorname{Br}_{p}(K)$ denote the homomorphism that sends
$\chi\in C^{\ast}$ to the image of $E\in H^{1}(K,Q)$ in
$\operatorname{Br}_{p}(K)$ under the map
$H^{1}(K,Q)\to
H^{2}(K,C)\stackrel{{\scriptstyle\chi_{\ast}}}{{\to}}H^{2}(K,\mu_{p})=\operatorname{Br}_{p}(K)$
given by composing the connecting map with $\chi_{\ast}$. Then there exists a
basis $\chi_{1},\dotsc,\chi_{r}$ of $C^{\ast}$ such that
(5)
$\operatorname{ed}_{k}(G;p)\geq\sum_{i=1}^{r}\operatorname{ind}\beta(\chi_{i})-\dim
G,$
see [Me1, Theorem 4.8, Example 3.7]. Moreover, by [KM, Theorem 4.4, Remark
4.5]
$\operatorname{ind}\beta(\chi_{i})=\gcd\dim(\rho)\,,$
where the greatest common divisor is taken over all (finite-dimensional)
representations $\rho$ of $G$ such that $\rho_{|\,C}$ is scalar multiplication
by $\chi_{i}$. By our assumption, $\gcd$ can be replaced by $\min$. Hence, for
each $i\in\\{1,\dotsc,r\\}$ we can choose a representation $\rho_{i}$ of $G$
with
$\operatorname{ind}\beta(\chi_{i})=\dim(\rho_{i})$
such that $(\rho_{i})_{|\,C}$ is scalar multiplication by $\chi_{i}$.
Set $\rho\colonequals\rho_{1}\oplus\dotsb\oplus\rho_{r}$. The inequality (5)
can be written as
(6) $\operatorname{ed}_{k}(G;p)\geq\dim(\rho)-\dim G.$
Since $\chi_{1},\dotsc,\chi_{r}$ forms a basis of $C^{\ast}$ the restriction
of $\rho$ to $C$ is faithful. This proves the theorem. ∎
## 3\. The $p$-closure of a field
Let $K$ be a field extension of $k$ and $K_{\operatorname{alg}}$ an algebraic
closure. We will construct a field $K^{(p)}/K$ in $K_{\operatorname{alg}}$
with all finite subextensions of $K^{(p)}/K$ of degree prime to $p$ and all
finite subextensions of $K_{\operatorname{alg}}/K^{(p)}$ of degree a power of
$p$.
Fix a separable closure $K_{\operatorname{sep}}\subset K_{\operatorname{alg}}$
of $K$ and denote $\Gamma=\operatorname{Gal}(K_{\operatorname{sep}}/K)$.
Recall that $\Gamma$ is profinite and has Sylow-$p$ subgroups which enjoy
similar properties as in the finite case, see for example [RZ] or [Wi]. Let
$\Phi$ be a Sylow-$p$ subgroup of $\Gamma$ and $K_{\operatorname{sep}}^{\Phi}$
its fixed field.
###### Definition 3.1.
We call the field
$K^{(p)}=\\{a\in K_{\operatorname{alg}}|a\mbox{ is purely inseparable over
}K_{\operatorname{sep}}^{\Phi}\\}$
a $p$-closure of $K$. A field $K$ will be called $p$-closed if $K$=$K^{(p)}$.
Note that $K^{(p)}$ is unique in $K_{\operatorname{alg}}$ only up to the
choice of a Sylow-$p$ subgroup $\Phi$ in $\Gamma$. The notion of being
$p$-closed does not depend on this choice.
###### Proposition 3.2.
1. (a)
$K^{(p)}$ is a direct limit of finite extensions $K_{i}/K$ of degree prime to
$p$.
2. (b)
Every finite extension of $K^{(p)}$ is separable of degree a power of $p$; in
particular, $K^{(p)}$ is perfect.
3. (c)
The cohomological dimension of
$\Psi=\operatorname{Gal}(K_{\operatorname{alg}}/K^{(p)})$ is ${\rm
cd}_{q}(\Psi)=0$ for any prime $q\neq p$.
###### Proof.
(a) First note that $K_{\operatorname{sep}}$ is the limit of the directed set
$\\{K_{\operatorname{sep}}^{N}\\}$ over all normal subgroups $N\subset\Gamma$
of finite index. Let
$\mathcal{L}=\\{K_{\operatorname{sep}}^{N\Phi}|N\mbox{ normal with finite
index in $\Gamma$}\\}.$
This is a directed set, and since $\Phi$ is Sylow, the index of $N\Phi$ in
$\Gamma$ is prime to $p$. Therefore $\mathcal{L}$ consists of finite separable
extensions of $K$ of degree prime to $p$. Moreover,
$K_{\operatorname{sep}}^{\Phi}$ is the direct limit of fields $L$ in
$\mathcal{L}$.
If $\operatorname{char}k=0$, $K^{(p)}=K_{\operatorname{sep}}^{\Phi}$ and we
are done. Otherwise suppose $\operatorname{char}k=q\neq p$. Let
$\mathcal{E}=\\{E\subset K_{\operatorname{alg}}|E/L\mbox{ finite and purely
inseparable for some }L\in\mathcal{L}\\}.$
$\mathcal{E}$ consists of finite extensions of $K$ of degree prime to $p$,
because a purely inseparable extension has degree a power of $q$. One can
check that $\mathcal{E}$ forms a directed set.
Finally note that if $a$ is purely inseparable over
$K_{\operatorname{sep}}^{\Phi}$ with minimal polynomial $x^{q^{n}}-l$ (so that
$l\in K_{\operatorname{sep}}^{\Phi}$), then $l$ is already in some
$L\in\mathcal{L}$ since $K_{\operatorname{sep}}^{\Phi}$ is the limit of
$\mathcal{L}$. Thus $a\in E=L(a)$ which is in $\mathcal{E}$ and we conclude
that $K^{(p)}$ is the direct limit of $\mathcal{E}$.
(b) $K^{(p)}$ is the purely inseparable closure of
$K_{\operatorname{sep}}^{\Phi}$ in $K_{\operatorname{alg}}$ and
$K_{\operatorname{alg}}/K^{(p)}$ is separable, see [Win, 2.2.20]. Moreover,
$\operatorname{Gal}(K_{\operatorname{alg}}/K^{(p)})\simeq\operatorname{Gal}(K_{\operatorname{sep}}/K_{\operatorname{sep}}^{\Phi})=\Phi$
is a pro-$p$ group and so every finite extension of $K^{(p)}$ is separable of
degree a power of $p$.
(c) See [Se2, Cor. 2, I. 3]. ∎
We call a covariant functor
$\mathcal{F}\colon\operatorname{Fields}/k\to\operatorname{Sets}$ limit-
preserving if for any directed system of fields $\\{K_{i}\\}$,
$\displaystyle{\mathcal{F}(\lim_{\rightarrow}K_{i})=\lim_{\rightarrow}\mathcal{F}(K_{i})}$.
For example if $G$ is an algebraic group, the Galois cohomology functor
$H^{1}(*,G)$ is limit-preserving; see [Ma, 2.1].
###### Lemma 3.3.
Let $\mathcal{F}$ be limit-preserving and $\alpha\in\mathcal{F}(K)$ an object.
Denote the image of $\alpha$ in $\mathcal{F}(K^{(p)})$ by $\alpha_{K^{(p)}}$.
1. (a)
$\operatorname{ed}_{k}(\alpha;p)=\operatorname{ed}_{k}(\alpha_{K^{(p)}};p)=\operatorname{ed}_{k}(\alpha_{K^{(p)}})$.
2. (b)
$\operatorname{ed}_{k}(\mathcal{F};p)=\operatorname{ed}_{k^{(p)}}(\mathcal{F};p)$.
###### Proof.
(a) The inequalities
$\operatorname{ed}(\alpha;p)\geq\operatorname{ed}(\alpha_{K^{(p)}};p)=\operatorname{ed}(\alpha_{K^{(p)}})$
are clear from the definition and Proposition 3.2(b) since $K^{(p)}$ has no
finite extensions of degree prime to $p$. It remains to prove
$\operatorname{ed}(\alpha;p)\leq\operatorname{ed}(\alpha_{K^{(p)}})$. If $L/K$
is finite of degree prime to $p$,
(7) $\operatorname{ed}(\alpha;p)=\operatorname{ed}(\alpha_{L};p),$
cf. [Me1, Proposition 1.5] and its proof. For the $p$-closure $K^{(p)}$ this
is similar and uses (7) repeatedly:
Suppose there is a subfield $K_{0}\subset K^{(p)}$ and $\alpha_{K^{(p)}}$
comes from an element $\beta\in\mathcal{F}(K_{0})$, so that
$\beta_{K^{(p)}}=\alpha_{K^{(p)}}$. Write $K^{(p)}=\lim\mathcal{L}$, where
$\mathcal{L}$ is a direct system of finite prime to $p$ extensions of $K$.
Then $K_{0}=\lim\mathcal{L}_{0}$ with $\mathcal{L}_{0}=\\{L\cap
K_{0}|L\in\mathcal{L}\\}$ and by assumption on $\mathcal{F}$,
$\displaystyle{\mathcal{F}(K_{0})=\lim_{L^{\prime}\in\mathcal{L}_{0}}\mathcal{F}(L^{\prime})}$.
Thus there is a field $L^{\prime}=L\cap K_{0}$ ($L\in\mathcal{L}$) and
$\gamma\in\mathcal{F}(L^{\prime})$ such that $\gamma_{K_{0}}=\beta$. Since
$\alpha_{L}$ and $\gamma_{L}$ become equal over $K^{(p)}$, after possibly
passing to a finite extension, we may assume they are equal over $L$ which is
finite of degree prime to $p$ over $K$. Combining these constructions with (7)
we see that
$\operatorname{ed}(\alpha;p)=\operatorname{ed}(\alpha_{L};p)=\operatorname{ed}(\gamma_{L};p)\leq\operatorname{ed}(\gamma_{L})\leq\operatorname{ed}(\alpha_{K^{(p)}}).$
(b) This follows immediately from (a), taking $\alpha$ of maximal essential
$p$-dimension. ∎
###### Proposition 3.4.
Let
$\mathcal{F},\mathcal{G}\colon\operatorname{Fields}/k\to\operatorname{Sets}$
be limit-preserving functors and $\mathcal{F}\to\mathcal{G}$ a natural
transformation. If the map
$\mathcal{F}(K)\to\mathcal{G}(K)$
is bijective (resp. surjective) for any $p$-closed field extension $K/k$ then
$\operatorname{ed}(\mathcal{F};p)=\operatorname{ed}(\mathcal{G};p)\quad(\mbox{resp.
}\operatorname{ed}(\mathcal{F};p)\geq\operatorname{ed}(\mathcal{G};p)).$
###### Proof.
Assume the maps are surjective. By Proposition 3.2(a), the natural
transformation is $p$-surjective, in the terminology of [Me1], so we can apply
[Me1, Prop. 1.5] to conclude
$\operatorname{ed}(\mathcal{F};p)\geq\operatorname{ed}(\mathcal{G};p)$.
Now assume the maps are bijective. Let $\alpha$ be in $\mathcal{F}(K)$ for
some $K/k$ and $\beta$ its image in $\mathcal{G}(K)$. We claim that
$\operatorname{ed}(\alpha;p)=\operatorname{ed}(\beta;p)$. First, by Lemma 3.3
we can assume that $K$ is $p$-closed and it is enough to prove that
$\operatorname{ed}(\alpha)=\operatorname{ed}(\beta)$.
Assume that $\beta$ comes from $\beta_{0}\in\mathcal{G}(K_{0})$ for some field
$K_{0}\subset K$. Any finite prime to $p$ extension of $K_{0}$ is isomorphic
to a subfield of $K$ (cf. [Me1, Lemma 6.1]) and so also any $p$-closure of
$K_{0}$ (which has the same transcendence degree over $k$). We may therefore
assume that $K_{0}$ is $p$-closed. By assumption
$\mathcal{F}(K_{0})\rightarrow\mathcal{G}(K_{0})$ and
$\mathcal{F}(K)\rightarrow\mathcal{G}(K)$ are bijective. The unique element
$\alpha_{0}\in\mathcal{F}(K_{0})$ which maps to $\beta_{0}$ must therefore map
to $\alpha$ under the natural restriction map. This shows that
$\operatorname{ed}(\alpha)\leq\operatorname{ed}(\beta)$. The other inequality
always holds and the claim follows.
Taking $\alpha$ maximal with respect to its essential dimension, we obtain
$\operatorname{ed}(\mathcal{F};p)=\operatorname{ed}(\alpha;p)=\operatorname{ed}(\beta;p)\leq\operatorname{ed}(\mathcal{G};p)$.
∎
## 4\. The group $C(G)$
As we indicated in the Introduction, our proof of Theorem 1.3(a) will rely on
Theorem 1.2. To apply Theorem 1.2, we need to construct a split central
subgroup $C$ of $G$. In this section, we will explain how to construct this
subgroup (we will call it $C(G)$) and discuss some of its properties.
Recall that an algebraic group $G$ over a field $k$ is said to be of
multiplicative type if $G_{k_{\operatorname{sep}}}$ is diagonalizable over the
separable closure $k_{\operatorname{sep}}$ of $k$; cf., e.g., [Vo, Section
3.4]. Here, as usual, $G_{k^{\prime}}\colonequals
G\times_{\operatorname{Spec}{k}}\operatorname{Spec}(k^{\prime})$ for any field
extension $k^{\prime}/k$. Connected groups of multiplicative type are
precisely the algebraic tori.
We will use the following common conventions in working with an algebraic
group $A$ of multiplicative type over $k$.
* •
We will denote the character group of $A$ by $X(A)$.
* •
Given a field extension $l/k$, $A$ is split over $l$ if and only if the
absolute Galois group $\operatorname{Gal}(l_{\operatorname{sep}}/l)$ acts
trivially on $X(A)$.
* •
We will write $A[p]$ for the $p$-torsion subgroup $\\{a\in A\,|\,a^{p}=1\\}$
of $A$. Clearly $A[p]$ is defined over $k$.
Let $T$ be an algebraic torus. It is well known how to construct a maximal
split subtorus of $T$, see for example [Bo, 8.15] or [Wa, 7.4]. The following
definition is a variant of this.
###### Definition 4.1.
Let $A$ be an algebraic group of multiplicative type over $k$. Let $\Delta(A)$
be the $\Gamma$-invariant subgroup of $X(A)$ generated by elements of the form
$x-\gamma(x)$, as $x$ ranges over $X(A)$ and $\gamma$ ranges over $\Gamma$.
Define
$\operatorname{Split}_{k}(A)=\operatorname{Diag}(X(A)/\Delta(A))\,.$
Here $\operatorname{Diag}$ denotes the anti-equivalence between continuous
$\mathbb{Z}\Gamma$-modules and algebraic groups of multiplicative type, cf.
[Wa, 7.3].
###### Definition 4.2.
Let $G$ be an extension of a finite $p$-group by a torus, defined over a field
$k$, as in (3). Then
$C(G)\colonequals\operatorname{Split}_{k}(Z(G)[p])\,,$
where $Z(G)$ denotes the centre of $G$.
###### Lemma 4.3.
Let $A$ be an algebraic group of multiplicative type over $k$.
1. (a)
$\operatorname{Split}_{k}(A)$ is split over $k$,
2. (b)
$\operatorname{Split}_{k}(A)=A$ if and only if $A$ is split over $k$,
3. (c)
If $B$ is a $k$-subgroup of $A$ then
$\operatorname{Split}_{k}(B)\subset\operatorname{Split}_{k}(A)$.
4. (d)
For $A=A_{1}\times A_{2}$, $\operatorname{Split}_{k}(A_{1}\times
A_{2})=\operatorname{Split}_{k}(A_{1})\times\operatorname{Split}_{k}(A_{2})$,
5. (e)
If $A[p]\neq\\{1\\}$ and $A$ is split over a Galois extension $l/k$, such that
$\overline{\Gamma}=\operatorname{Gal}(l/k)$ is a $p$-group, then
$\operatorname{Split}_{k}(A)\neq\\{1\\}$.
###### Proof.
Parts (a), (b), (c) and (d) easily follow from the definition.
Proof of (e): By part (c), it suffices to show that
$\operatorname{Split}_{k}(A[p])\neq\\{1\\}$. Hence, we may assume that
$A=A[p]$ or equivalently, that $X(A)$ is a finite-dimensional
$\mathbb{F}_{p}$-vector space on which the $p$-group $\overline{\Gamma}$ acts.
Any such action is upper-triangular, relative to some $\mathbb{F}_{p}$-basis
$e_{1},\dots,e_{n}$ of $X(A)$; see, e.g., [Se1, Proposition 26, p.64]. That
is,
$\gamma(e_{i})=e_{i}+$ ($\mathbb{F}_{p}$-linear combination of
$e_{i+1},\ldots,e_{n}$)
for every $i=1,\dots,n$ and every $\gamma\in\overline{\Gamma}$. Our goal is to
show that $\Delta(A)\neq X(A)$. Indeed, every element of the form
$x-\gamma(x)$ is contained in the $\Gamma$-invariant submodule
$\operatorname{Span}(e_{2},\dots,e_{n})$. Hence, these elements cannot
generate all of $X(A)$. ∎
###### Proposition 4.4.
Suppose $G$ is an extension of a $p$-group by a torus, defined over a
$p$-closed field $k$. Suppose $N$ is a normal subgroup of $G$ defined over
$k$. Then the following conditions are equivalent:
(i) $N$ is finite of order prime to $p$,
(ii) $N\cap C(G)=\\{1\\}$,
(iii) $N\cap Z(G)[p]=\\{1\\}$,
In particular, taking $N=G$, we see that $C(G)\neq\\{1\\}$ if $G\neq\\{1\\}$.
###### Proof.
(i) $\Longrightarrow$ (ii) is obvious, since $C(G)$ is a $p$-group.
(ii) $\Longrightarrow$ (iii). Assume the contrary: $A\colonequals N\cap
Z(G)[p]\neq\\{1\\}$. By Lemma 4.3
$\\{1\\}\neq C(A)\subset N\cap C(Z(G)[p])=N\cap C(G)\,,$
contradicting (ii).
Our proof of the implication (iii) $\Longrightarrow$ (i), will rely on the
following
Claim: Let $M$ be a non-trivial normal finite $p$-subgroup of $G$ such that
the commutator $(G^{0},M)=\\{1\\}$. Then $M\cap Z(G)[p]\neq\\{1\\}$.
To prove the claim, note that $M(k_{\operatorname{sep}})$ is non-trivial and
the conjugation action of $G(k_{\operatorname{sep}})$ on
$M(k_{\operatorname{sep}})$ factors through an action of the $p$-group
$(G/G^{0})(k_{\operatorname{sep}})$. Thus each orbit has $p^{n}$ elements for
some $n\geq 0$; consequently, the number of fixed points is divisible by $p$.
The intersection $(M\cap Z(G))(k_{\operatorname{sep}})$ is precisely the fixed
point set for this action; hence, $M\cap Z(G)[p]\neq\\{1\\}$. This proves the
claim.
We now continue with the proof of the implication (iii) $\Longrightarrow$ (i).
For notational convenience, set $T\colonequals G^{0}$. Assume that
$N\triangleleft G$ and $N\cap Z(G)[p]=\\{1\\}$. Applying the claim to the
normal subgroup $M\colonequals(N\cap T)[p]$ of $G$, we see that $(N\cap
T)[p]=\\{1\\}$, i.e., $N\cap T$ is a finite group of order prime to $p$. The
exact sequence
(8) $1\to N\cap T\to N\to\overline{N}\to 1\,,$
where $\overline{N}$ is the image of $N$ in $G/T$, shows that $N$ is finite.
Now observe that for every $r\geq 1$, the commutator $(N,T[p^{r}])$ is a
$p$-subgroup of $N\cap T$. Thus $(N,T[p^{r}])=\\{1\\}$ for every $r\geq 1$. We
claim that this implies $(N,T)=\\{1\\}$ by Zariski density. If $N$ is smooth,
this is straightforward; see [Bo, Proposition 2.4, p. 59]. If $N$ is not
smooth, note that the map $c\colon N\times T\to G$ sending $(n,t)$ to the
commutator $ntn^{-1}t^{-1}$ descends to $\overline{c}\colon\overline{N}\times
T\to G$ (indeed, $N\cap T$ clearly commutes with $T$). Since $|\overline{N}|$
is a power of $p$ and $\operatorname{char}(k)\neq p$, $\overline{N}$ is smooth
over $k$, and we can pass to the separable closure $k_{\operatorname{sep}}$
and apply the usual Zariski density argument to show that the image of
$\overline{c}$ is trivial.
We thus conclude that $N\cap T$ is central in $N$. Since $\gcd(|N\cap
T|,\overline{N})=1$, by [Sch2, Corollary 5.4] the extension (8) splits, i.e.,
$N\simeq(N\cap T)\times\overline{N}$. This turns $\overline{N}$ into a
subgroup of $G$ satisfying the conditions of the claim. Therefore
$\overline{N}$ is trivial and $N=N\cap T$ is a finite group of order prime to
$p$, as claimed. ∎
For future reference, we record the following obvious consequence of the
equivalence of conditions (i) and (ii) in Proposition 4.4.
###### Corollary 4.5.
Let $k=k^{(p)}$ be a $p$-closed field and $G$ be an extension of a $p$-group
by a torus, defined over $k$, as in (3). A finite-dimensional representation
$\rho$ of $G$ defined over $k$ is $p$-faithful if and only $\rho_{|\,C(G)}$ is
faithful. ∎
## 5\. Proof of Theorem 1.3(a)
The key step in our proof will be the following proposition.
###### Proposition 5.1.
Let $k$ be a $p$-closed field, and $G$ be an extension of a $p$-group by a
torus, as in (3). Then the dimension of every irreducible representation of
$G$ over $k$ is a power of $p$.
Assuming Proposition 5.1 we can easily complete the proof of Theorem 1.3(a).
Indeed, by Proposition 3.4 we may assume that $k=k^{(p)}$ is $p$-closed. In
particular, since we are assuming that $\operatorname{char}(k)\neq p$, this
implies that $k$ contains a primitive $p$th root of unity. (Indeed, if $\zeta$
is a $p$-th root of unity in $k_{\operatorname{sep}}$ then $d=[k(\zeta):k]$ is
prime to $p$; hence, $d=1$.) Proposition 5.1 tells us that Theorem 1.2 can be
applied to the exact sequence
(9) $1\to C(G)\to G\to Q\to 1\,.$
This yields
(10) $\operatorname{ed}(G;p)\geq\min\;\dim(\rho)-\dim(G)\,,$
where the minimum is taken over all representations $\rho\colon
G\to\operatorname{GL}(V)$ such that $\rho_{|C(G)}$ is faithful. Corollary 4.4
now tells us that $\rho_{|C(G)}$ is faithful if and only if $\rho$ is
$p$-faithful, and Theorem 1.3(a) follows. ∎
The rest of this section will be devoted to the proof of Proposition 5.1. We
begin by settling it in the case where $G$ is a finite $p$-group.
###### Lemma 5.2.
Proposition 5.1 holds if $G$ is a finite $p$-group.
###### Proof.
Choose a finite Galois field extension $l/k$ such that (i) $G$ is constant
over $l$ and (ii) every irreducible linear representation of $G$ over $l$ is
absolutely irreducible. Since $k$ is assumed to be $p$-closed, $[l:k]$ is a
power of $p$.
Let $A\colonequals k[G]^{\ast}$ be the dual Hopf algebra of the coordinate
algebra of $G$. By [Ja, Section 8.6] a $G$-module structure on a $k$-vector
space $V$ is equivalent to an $A$-module structure on $V$. Now assume that $V$
is an irreducible $A$-module and let $W\subseteq V\otimes_{k}l$ be an
irreducible $A\otimes_{k}l$-submodule. Then by [Ka, Theorem 5.22] there exists
a divisor $e$ of $[l:k]$ such that
$V\otimes l\simeq e\left(\bigoplus_{i=1}^{r}{}^{\sigma_{i}}W\right)\,,$
where $\sigma_{i}\in\operatorname{Gal}(l/k)$ and $\\{{}^{\sigma_{i}}W\mid
1\leq i\leq r\\}$ are the pairwise non-isomorphic Galois conjugates of $W$. By
our assumption on $k$, $e$ and $r$ are powers of $p$ and by our choice of $l$,
$\dim_{l}W=\dim_{l}({}^{\sigma_{1}}W)=\ldots=\dim_{l}({}^{\sigma_{r}}W)$ is
also a power of $p$, since it divides the order of $G_{l}$. Hence, so is
$\dim_{k}(V)=\dim_{l}V\otimes
l=e(\dim_{l}{}^{\sigma_{1}}W+\dotsb+\dim_{l}{}^{\sigma_{r}}W)$. ∎
Our proof of Proposition 5.1 in full generality will based on leveraging Lemma
5.2 as follows.
###### Lemma 5.3.
Let $G$ be an algebraic group defined over a field $k$ and
$F_{1}\subseteq F_{2}\subseteq\dots\subset G$
be an ascending sequence of finite $k$-subgroups whose union $\cup_{n\geq
1}F_{n}$ is Zariski dense in $G$. If $\rho\colon G\to\operatorname{GL}(V)$ is
an irreducible representation of $G$ defined over $k$ then $\rho_{|\,F_{i}}$
is irreducible for sufficiently large integers $i$.
###### Proof.
For each $d=1,...,\dim(V)-1$ consider the $G$-action on the Grassmannian
$\operatorname{Gr}(d,V)$ of $d$-dimensional subspaces of $V$. Let
$X^{(d)}=\operatorname{Gr}(d,V)^{G}$ and
$X_{i}^{(d)}=\operatorname{Gr}(d,V)^{F_{i}}$ be the subvariety of
$d$-dimensional $G$\- (resp. $F_{i}$-)invariant subspaces of $V$. Then
$X_{1}^{(d)}\supseteq X_{2}^{(d)}\supseteq\ldots$ and since the union of the
groups $F_{i}$ is dense in $G$,
$X^{(d)}=\cap_{i\geq 0}X_{i}^{(d)}\,.$
By the Noetherian property of $\operatorname{Gr}(d,V)$, we have
$X^{(d)}=X_{m_{d}}^{(d)}$ for some $m_{d}\geq 0$.
Since $V$ does not have any $G$-invariant $d$-dimensional $k$-subspaces, we
know that $X^{(d)}(k)=\emptyset$. Thus, $X_{m_{d}}^{(d)}(k)=\emptyset$, i.e.,
$V$ does not have any $F_{m_{d}}$-invariant $d$-dimensional $k$-subspaces.
Setting $m\colonequals\max\\{m_{1},\dots,m_{\dim(V)-1}\\}$, we see that
$\rho_{|\,F_{m}}$ is irreducible. ∎
We now proceed with the proof of Proposition 5.1. By Lemmas 5.2 and 5.3, it
suffices to construct a sequence of finite $p$-subgroups
$F_{1}\subseteq F_{2}\subseteq\dots\subset G$
defined over $k$ whose union $\cup_{n\geq 1}F_{n}$ is Zariski dense in $G$.
In fact, it suffices to construct one $p$-subgroup $F^{\prime}\subset G$,
defined over $k$ such that $F^{\prime}$ surjects onto $F$. Indeed, once
$F^{\prime}$ is constructed, we can define $F_{i}\subset G$ as the subgroup
generated by $F^{\prime}$ and $T[p^{i}]$, for every $i\geq 0$. Since
$\cup_{n\geq 1}F_{n}$ contains both $F^{\prime}$ and $T[p^{i}]$, for every
$i\geq 0$ it is Zariski dense in $G$, as desired.
The following lemma, which establishes the existence of $F^{\prime}$, is thus
the final step in our proof of Proposition 5.1 (and hence, of Theorem 1.3(a)).
###### Lemma 5.4.
Let $1\to T\to G\xrightarrow{\pi}F\to 1$ be an extension of a $p$-group $F$ by
a torus $T$ over $k$. Then $G$ has a finite $p$-subgroup $F^{\prime}$ with
$\pi(F^{\prime})=F$.
In the case where $F$ is split and $k$ is algebraically closed this is proved
in [CGR, p. 564]; cf. also the proof of [BS, Lemme 5.11].
###### Proof.
Denote by $\widetilde{\operatorname{Ex}}^{1}(F,T)$ the group of equivalence
classes of extensions of $F$ by $T$. We claim that
$\widetilde{\operatorname{Ex}}^{1}(F,T)$ is torsion. Let
$\operatorname{Ex}^{1}(F,T)\subset\widetilde{\operatorname{Ex}}^{1}(F,T)$ be
the classes of extensions which have a scheme-theoretic section (i.e. $G(K)\to
F(K)$ is surjective for all $K/k$). There is a natural isomorphism
$\operatorname{Ex}^{1}(F,T)\simeq H^{2}(F,T)$, where the latter one denotes
Hochschild cohomology, see [DG, III. 6.2, Proposition]. By [Sch3] the usual
restriction-corestriction arguments can be applied in Hochschild cohomology
and in particular, $m\cdot H^{2}(F,T)=0$ where $m$ is the order of $F$. Now
recall that $M\mapsto\widetilde{\operatorname{Ex}}^{i}(F,M)$ and
$M\mapsto\operatorname{Ex}^{i}(F,M)$ are both derived functors of the crossed
homomorphisms $M\mapsto\operatorname{Ex}^{0}(F,M)$, where in the first case
$M$ is in the category of $F$-module sheaves and in the second, $F$-module
functors, cf. [DG, III. 6.2]. Since $F$ is finite and $T$ an affine scheme, by
[Sch1, Satz 1.2 & Satz 3.3] there is an exact sequence of $F$-module schemes
$1\to T\to M_{1}\to M_{2}\to 1$ and an exact sequence
$\operatorname{Ex}^{0}(F,M_{1})\to\operatorname{Ex}^{0}(F,M_{2})\to\widetilde{\operatorname{Ex}}^{1}(F,T)\to
H^{2}(F,M_{1})\simeq\operatorname{Ex}^{1}(F,M_{1})$. The $F$-module sequence
also induces a long exact sequence on $\operatorname{Ex}(F,*)$ and we have a
diagram
$\textstyle{\widetilde{\operatorname{Ex}}^{1}(F,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ex}^{0}(F,M_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ex}^{0}(F,M_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ex}^{1}(F,M_{1})}$$\textstyle{\operatorname{Ex}^{1}(F,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
An element in $\widetilde{\operatorname{Ex}}^{1}(F,T)$ can thus be killed
first in $\operatorname{Ex}^{1}(F,M_{1})$ so it comes from
$\operatorname{Ex}^{0}(F,M_{2})$. Then kill its image in
$\operatorname{Ex}^{1}(F,T)\simeq H^{2}(F,T)$, so it comes from
$\operatorname{Ex}^{0}(F,M_{1})$, hence is $0$ in
$\widetilde{\operatorname{Ex}}^{1}(F,T)$. In particular we see that
multiplying twice by the order $m$ of $F$,
$m^{2}\cdot\widetilde{\operatorname{Ex}}^{1}(F,T)=0$. This proves the claim.
Now let us consider the exact sequence $1\to N\to T\xrightarrow{\times
m^{2}}T\to 1$, where $N$ is the kernel of multiplication by $m^{2}$. Clearly
$N$ is finite and we have an induced exact sequence
$\widetilde{\operatorname{Ex}}^{1}(F,N)\to\widetilde{\operatorname{Ex}}^{1}(F,T)\xrightarrow{\times
m^{2}}\widetilde{\operatorname{Ex}}^{1}(F,T)$
which shows that the given extension $G$ comes from an extension $F^{\prime}$
of $F$ by $N$. Then $G$ is the pushout of $F^{\prime}$ by $N\to T$ and we can
identify $F^{\prime}$ with a subgroup of $G$. ∎
## 6\. $p$-isogenies
An isogeny of algebraic groups is a surjective morphism $G\to Q$ with finite
kernel. If the kernel is of order prime to $p$ we say that the isogeny is a
$p$-isogeny. In this section we will prove Theorem 6.1 which says that
$p$-isogenous groups have the same essential $p$-dimension. This result will
play a key role in the proof of Theorem 1.3(b) in Section 7.
###### Theorem 6.1.
Suppose $G\to Q$ is a $p$-isogeny of algebraic groups over $k$. Then
1. (a)
For any $p$-closed field $K$ containing $k$ the natural map $H^{1}(K,G)\to
H^{1}(K,Q)$ is bijective.
2. (b)
$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(Q;p)$.
###### Example 6.2.
Let $E_{6}^{sc},E_{7}^{sc}$ be simply connected simple groups of type
$E_{6},E_{7}$ respectively. In [GR, 9.4, 9.6] it is shown that if $k$ is an
algebraically closed field of characteristic $\neq 2$ and $3$ respectively,
then
$\operatorname{ed}_{k}(E_{6}^{sc};2)=3$ and
$\operatorname{ed}_{k}(E_{7}^{sc};3)=3$.
For the adjoint groups $E_{6}^{ad}=E_{6}^{sc}/\mu_{3}$,
$E_{7}^{ad}=E_{7}^{sc}/\mu_{2}$ we therefore have
$\operatorname{ed}_{k}(E_{6}^{ad};2)=3$ and
$\operatorname{ed}_{k}(E_{7}^{ad};3)=3$.
We will need two lemmas.
###### Lemma 6.3.
Let $N$ be a finite algebraic group over $k$ ($\operatorname{char}k\neq p$).
The following are equivalent:
1. (a)
$p$ does not divide the order of $N$.
2. (b)
$p$ does not divide the order of $N(k_{\operatorname{alg}})$.
If $N$ is also assumed to be abelian, denote by $N[p]$ the $p$-torsion
subgroup of $N$. The following are equivalent to the above conditions.
1. (a′)
$N[p](k_{\operatorname{alg}})=\\{1\\}$.
2. (b′)
$N[p](k^{(p)})=\\{1\\}$.
###### Proof.
(a)$\iff$(b): Let $N^{\circ}$ be the connected component of $N$ and
$N^{et}=N/N^{\circ}$ the étale quotient. Recall that the order of a finite
algebraic group $N$ over $k$ is defined as $|N|=\dim_{k}k[N]$ and
$|N|=|N^{\circ}||N^{et}|$, see for example [Ta]. If $\operatorname{char}k=0$,
$N^{\circ}$ is trivial, if $\operatorname{char}k=q\neq p$ is positive,
$|N^{\circ}|$ is a power of $q$. Hence $N$ is of order prime to $p$ if and
only if the étale algebraic group $N^{et}$ is. Since $N^{\circ}$ is connected
and finite, $N^{\circ}(k_{\operatorname{alg}})=\\{1\\}$ and so
$N(k_{\operatorname{alg}})$ is of order prime to $p$ if and only if the group
$N^{et}(k_{\operatorname{alg}})$ is. Then
$|N^{et}|=\dim_{k}k[N^{et}]=|N^{et}(k_{\operatorname{alg}})|$, cf. [Bou, V.29
Corollary].
(b)$\iff$(a′) $\Rightarrow$ (b′) are clear.
(a′) $\Leftarrow$ (b′): Suppose $N[p](k_{\operatorname{alg}})$ is nontrivial.
The Galois group $\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$
is a pro-$p$ group and acts on the $p$-group $N[p](k_{\operatorname{alg}})$.
The image of $\Gamma$ in $\operatorname{Aut}(N[p](k_{\operatorname{alg}}))$ is
again a (finite) $p$-group and the size of every $\Gamma$-orbit in
$N[p](k_{\operatorname{alg}})$ is a power of $p$. Since $\Gamma$ fixes the
identity in $N[p](k_{\operatorname{alg}})$, this is only possible if it also
fixes at least $p-1$ more elements. It follows that $N[p](k^{(p)})$ contains
at least $p$ elements, a contradiction. ∎
###### Remark 6.4.
Part (b′) could be replaced by the slightly stronger statement that
$N[p](k^{(p)}\cap k_{\operatorname{sep}})=\\{1\\}$, but we won’t need this in
the sequel.
###### Lemma 6.5.
Let $\Gamma$ be a profinite group, $G$ an (abstract) finite $\Gamma$-group and
$|\Gamma|,|G|$ coprime. Then $H^{1}(\Gamma,G)=\\{1\\}$.
The case where $\Gamma$ is finite and $G$ abelian is classical. In the
generality we stated, this lemma is also known [Se2, I.5, ex. 2].
###### Proof of Theorem 6.1.
(a) Let $N$ be the kernel of $G\to Q$ and $K=K^{(p)}$ be a $p$-closed field
over $k$. Since $K_{\operatorname{sep}}=K_{\operatorname{alg}}$ (see
Proposition 3.2(b)), the sequence of $K_{\operatorname{sep}}$-points $1\to
N(K_{\operatorname{sep}})\to G(K_{\operatorname{sep}})\to
Q(K_{\operatorname{sep}})\to 1$ is exact. By Lemma 6.3, the order of
$N(K_{\operatorname{sep}})$ is not divisible by $p$ and therefore coprime to
the order of $\Psi=\operatorname{Gal}(K_{\operatorname{sep}}/K)$. Thus
$H^{1}(K,N)=\\{1\\}$ (Lemma 6.5). Similarly, if ${}_{c}N$ is the group $N$
twisted by a cocycle $c:\Psi\to G$,
${}_{c}N(K_{\operatorname{sep}})=N(K_{\operatorname{sep}})$ is of order prime
to $p$ and $H^{1}(K,\,_{c}N)=\\{1\\}$. It follows that $H^{1}(K,G)\to
H^{1}(K,Q)$ is injective, cf. [Se2, I.5.5].
Surjectivity is a consequence of [Se2, I. Proposition 46] and the fact that
the $q$-cohomological dimension of $\Psi$ is $0$ for any divisor $q$ of
$|N(K_{\operatorname{sep}})|$ (Proposition 3.2).
This concludes the proof of part (a). Part (b) immediately follows from (a)
and Proposition 3.4. ∎
## 7\. Proof of Theorem 1.3(b)
Let $k$ be a closed field and $G=T\times F$, where $T$ is a torus and $F$ is a
finite $p$-group, defined over $k$. Our goal is to show that
(11) $\operatorname{ed}_{k}(G;p)\leq\dim(\rho)-\dim G\,,$
where $\rho$ is a $p$-faithful representation of $G$ defined over $k$.
###### Lemma 7.1.
If a representation $\rho\colon G\to\operatorname{GL}(V)$ is $p$-faithful,
then $G/\ker(\rho)\to\operatorname{GL}(V)$ is generically free. In other
words, $\rho$ is $p$-generically free.
###### Proof.
Since $\ker(\rho)$ has order prime to $p$, its image under the projection map
$G=T\times F\to F$ is trivial. Hence $\ker(\rho)\subset T$ and $T/N$ is again
a torus. So without loss of generality, we may assume $\rho$ is faithful.
Let $V_{1}\subsetneq V$ be a closed subset of $V$ such that $T$ acts freely on
$V\setminus V_{1}$. Let $n=p^{r}$ be the order of $F$ and $V_{2}$ be the
(finite) union of the fixed point sets of $1\neq g\in T[n]\times F$. Here as
usual, $T[n]$ denotes the $n$-torsion subgroup of $T$. Since $\rho$ is
faithful none of these fixed point sets are all of $V$, hence $U\colonequals
V\setminus(V_{1}\cup V_{2})$ is a dense open subset of $V$.
We claim that $\operatorname{Stab}_{G}(v)=\\{1\\}$ for every $v\in U$. Indeed,
assume $1\neq g=(t,f)\in\operatorname{Stab}_{G}(v)$. Since $v\not\in V_{2}$,
$t^{n}\neq 1$. Then $1\neq g^{n}=(t^{n},1)$ lies in both $T$ and
$\operatorname{Stab}_{G}(v)$. Since $v\not\in V_{1}$, this is a contradiction.
∎
Now suppose $\rho$ is any $p$-faithful representation of $G$. Then (1) yields
$\operatorname{ed}_{k}(G/N;p)\leq\dim(\rho)-\dim(G/\ker(\rho))=\dim(\rho)-\dim(G)\,.$
By Theorem 6.1
$\operatorname{ed}_{k}(G;p)=\operatorname{ed}(G/N;p)\leq\dim(\rho)-\dim(G)\,,$
as desired. This completes the proof of (11) and thus of Theorem 1.3(b). ∎
###### Corollary 7.2.
Let $G$ be a finite algebraic group over a $p$-closed field $k=k^{(p)}$. Then
$G$ has a Sylow-$p$ subgroup $G_{p}$ defined over $k$ and
$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(G_{p};p)=\operatorname{ed}_{k}(G_{p})=\min\dim(\rho)$
where the minimum is taken over all faithful representations of $G_{p}$ over
$k$.
###### Proof.
By assumption, $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ is a
pro-$p$ group. It acts on the set of Sylow-$p$ subgroups of
$G(k_{\operatorname{sep}})$. Since the number of such subgroups is prime to
$p$, $\Gamma$ fixes at least one of them and by Galois descent one obtains a
subgroup $G_{p}$ of $G$. By Lemma 6.3, $G_{p}$ is a Sylow-$p$ subgroup of $G$.
The first equality $\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(G_{p};p)$
is shown in [MR1, 4.1] (the reference is for smooth groups but can be
generalized to the non-smooth case as well). The minimal
$G_{p}$-representation $\rho$ from Theorem 1.3(b) is faithful and thus
$\operatorname{ed}_{k}(G_{p})\leq\dim(\rho)$, see for example [BF, Prop.
4.11]. The Corollary follows. ∎
###### Remark 7.3.
Two Sylow-$p$ subgroups of $G$ defined over $k=k^{(p)}$ do not need to be
isomorphic over $k$.
## 8\. An additivity theorem
The purpose of this section is to prove the following:
###### Theorem 8.1.
Let $G_{1}$ and $G_{2}$ be direct products of tori and $p$-groups over a field
$k$. Then $\operatorname{ed}_{k}(G_{1}\times
G_{2};p)=\operatorname{ed}_{k}(G_{1};p)+\operatorname{ed}_{k}(G_{2};p)$.
Let $G$ be an algebraic group defined over $k$ and $C$ be a $k$-subgroup of
$G$. Denote the minimal dimension of a representation $\rho$ of $G$ (defined
over $k$) such that $\rho_{|\,C}$ is faithful by $f(G,C)$.
###### Lemma 8.2.
For $i=1,2$ let $G_{i}$ be an algebraic group defined over $k$ and $C_{i}$ be
a central $k$-subgroup of $G_{i}$. Assume that $C_{i}$ is isomorphic to
$\mu_{p}^{r_{i}}$ over $k$ for some $r_{1},r_{2}\geq 0$. Then
$f(G_{1}\times G_{1};C_{1}\times C_{2})=f(G_{1};C_{1})+f(G_{2};C_{2})\,.$
Our argument is a variant of the proof of [KM, Theorem 5.1], where $G$ is
assumed to be a (constant) finite $p$-group and $C$ is the socle of $G$.
###### Proof.
For $i=1,2$ let $\pi_{i}\colon G_{1}\times G_{2}\to G_{i}$ be the natural
projection and $\epsilon_{i}\colon G_{i}\to G_{1}\times G_{2}$ be the natural
inclusion.
If $\rho_{i}$ is a $d_{i}$-dimensional $k$-representation of $G_{i}$ whose
restriction to $C_{i}$ is faithful, then clearly
$\rho_{1}\circ\pi_{1}\oplus\rho_{2}\circ\pi_{2}$ is a
$d_{1}+d_{2}$-dimensional representation of $G_{1}\times G_{2}$ whose
restriction to $C_{1}\times C_{2}$ is faithful. This shows that
$f(G_{1}\times G_{1};C_{1}\times C_{2})\leq f(G_{1};C_{1})+f(G_{2};C_{2})\,.$
To prove the opposite inequality, let $\rho\colon G_{1}\times
G_{2}\to\operatorname{GL}(V)$ be a $k$-representation such that
$\rho_{|\,C_{1}\times C_{2}}$ is faithful, and of minimal dimension
$d=f(G_{1}\times G_{1};C_{1}\times C_{2})$
with this property. Let $\rho_{1},\rho_{2},\dotsc,\rho_{n}$ denote the
irreducible decomposition factors in a decomposition series of $\rho$. Since
$C_{1}\times C_{2}$ is central in $G_{1}\times G_{2}$, each $\rho_{i}$
restricts to a multiplicative character of $C_{1}\times C_{2}$ which we will
denote by $\chi_{i}$. Moreover since $C_{1}\times
C_{2}\simeq\mu_{p}^{r_{1}+r_{2}}$ is linearly reductive $\rho_{|\,C_{1}\times
C_{2}}$ is a direct sum $\chi_{1}^{\oplus
d_{1}}\oplus\dotsb\oplus\chi_{n}^{\oplus d_{n}}$ where $d_{i}=\dim V_{i}$. It
is easy to see that the following conditions are equivalent:
(i) $\rho_{|\,C_{1}\times C_{2}}$ is faithful,
(ii) $\chi_{1},\dots,\chi_{n}$ generate $(C_{1}\times C_{2})^{*}$ as an
abelian group.
In particular we may assume that $\rho=\rho_{1}\oplus\dotsb\oplus\rho_{n}$.
Since $C_{i}$ is isomorphic to $\mu_{p}^{r_{i}}$, we will think of
$(C_{1}\times C_{2})^{*}$ as a $\mathbb{F}_{p}$-vector space of dimension
$r_{1}$ \+ $r_{2}$. Since (i) $\Leftrightarrow$ (ii) above, we know that
$\chi_{1},\dots,\chi_{n}$ span $(C_{1}\times C_{2})^{*}$. In fact, they form a
basis of $(C_{1}\times C_{2})^{*}$, i.e., $n=r_{1}+r_{2}$. Indeed, if they
were not linearly independent we would be able to drop some of the terms in
the irreducible decomposition $\rho_{1}\oplus\dots\oplus\rho_{n}$, so that the
restriction of the resulting representation to $C_{1}\times C_{2}$ would still
be faithful, contradicting the minimality of $\dim(\rho)$.
We claim that it is always possible to replace each $\rho_{j}$ by
$\rho_{j}^{\prime}$, where $\rho_{j}^{\prime}$ is either
$\rho_{j}\circ\epsilon_{1}\circ\pi_{1}$ or
$\rho_{j}\circ\epsilon_{2}\circ\pi_{2}$ such that the restriction of the
resulting representation
$\rho^{\prime}=\rho_{1}^{\prime}\oplus\dots\oplus\rho_{n}^{\prime}$ to
$C_{1}\times C_{2}$ remains faithful. Since
$\dim(\rho_{i})=\dim(\rho_{i}^{\prime})$, we see that
$\dim(\rho^{\prime})=\dim(\rho)$. Moreover, $\rho^{\prime}$ will then be of
the form $\alpha_{1}\circ\pi_{1}\oplus\alpha_{2}\circ\pi_{2}$, where
$\alpha_{i}$ is a representation of $G_{i}$ whose restriction to $C_{i}$ is
faithful. Thus, if we can prove the above claim, we will have
$\displaystyle f(G_{1}\times G_{1};C_{1}\times C_{2})$
$\displaystyle=\dim(\rho)=\dim(\rho^{\prime})=\dim(\alpha_{1})+\dim(\alpha_{2})$
$\displaystyle\geq f(G_{1},C_{1})+f(G_{2},C_{2})\,,$
as desired.
To prove the claim, we will define $\rho_{j}^{\prime}$ recursively for
$j=1,\dots,n$. Suppose $\rho_{1}^{\prime},\dots,\rho_{j-1}^{\prime}$ have
already be defined, so that the restriction of
$\rho_{1}^{\prime}\oplus\dots\oplus\rho_{j-1}^{\prime}\oplus\rho_{j}\dots\oplus\rho_{n}$
to $C_{1}\times C_{2}$ is faithful. For notational simplicity, we will assume
that $\rho_{1}=\rho_{1}^{\prime},\dots,\rho_{j-1}=\rho_{j-1}^{\prime}$. Note
that
$\chi_{j}=(\chi_{j}\circ\epsilon_{1}\circ\pi_{1})+(\chi_{j}\circ\epsilon_{2}\circ\pi_{2})\,.$
Since $\chi_{1},\dots,\chi_{n}$ form a basis $(C_{1}\times C_{2})^{*}$ as an
$\mathbb{F}_{p}$-vector space, we see that (a)
$\chi_{j}\circ\epsilon_{1}\circ\pi_{1}$ or (b)
$\chi_{j}\circ\epsilon_{2}\circ\pi_{2}$ does not lie in
$\operatorname{Span}_{\mathbb{F}_{p}}(\chi_{1},\dots,\chi_{j-1},\chi_{j+1},\dots,\chi_{n})$.
Set
$\rho_{j}^{\prime}\colonequals\begin{cases}\text{$\rho_{j}\circ\epsilon_{1}\circ\pi_{1}$
in case (a), and}\\\ \text{$\rho_{j}\circ\epsilon_{2}\circ\pi_{2}$,
otherwise.}\end{cases}$
Using the equivalence of (i) and (ii) above, we see that the restriction of
$\rho_{1}\oplus\dots\oplus\rho_{j-1}\oplus\rho_{j}^{\prime}\oplus\rho_{j+1},\dots\oplus\rho_{n}$
to $C$ is faithful. This completes the proof of the claim and thus of Lemma
8.2. ∎
###### Proof of Theorem 8.1.
We can pass to a $p$-closure $k^{(p)}$ by Lemma 3.3. Let $C(G)$ be as in
Definition 4.2. By Theorem 1.3(b)
$\operatorname{ed}(G;p)=f(G,C(G))-\dim G\,;$
cf. Corollary 4.5. Furthermore, we have $C(G_{1}\times G_{2})=C(G_{1})\times
C(G_{2})$; cf. Lemma 4.3(d). Applying Lemma 8.2 finishes the proof. ∎
## 9\. Modules and lattices
In this section we rewrite the value of $\operatorname{ed}_{k}(G;p)$ in terms
of the character module $X(G)$ for an abelian group $G$ which is an extension
of a $p$-group and a torus. Moreover we show that tori with locally isomorphic
character lattices have the same essential dimension. We need the following
preliminaries.
Let $R$ be a commutative ring (we use $R=\mathbb{Z}$ and $R=\mathbb{Z}_{(p)}$
mostly) and $A$ an $R$-algebra. An $A$-module is called an $A$-lattice if it
is finitely generated and projective as an $R$-module. For
$A=\mathbb{Z}\Gamma$ ($\Gamma$ a group) this is as usual a free abelian group
of finite rank with an action of $\Gamma$. Particular cases of
$R\Gamma$-lattices are permutation lattices $L=R[\Lambda]$ where $\Lambda$ is
a $\Gamma$-set.
For $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ the absolute Galois
group of $k$ we tacitly assume that our $R\Gamma$-lattices are continuous,
i.e. $\Gamma$ acts through a finite quotient $\overline{\Gamma}$. Under the
anti-equivalence $\operatorname{Diag}$ a $\mathbb{Z}\Gamma$-lattice
corresponds to an algebraic $k$-torus. A torus $S$ is called quasi split if it
corresponds to a permutation lattice. Equivalently $S\simeq
R_{E/k}(\operatorname{\mathbb{G}}_{m})$ where $E/k$ is étale and $R_{E/k}$
denotes Weil restriction.
Recall that $\mathbb{Z}_{(p)}$ denotes the localization of the ring
$\mathbb{Z}$ at the prime ideal $(p)$. For a $\mathbb{Z}$-module $M$ we also
write $M_{(p)}\colonequals\mathbb{Z}_{(p)}\otimes M$.
When $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ we will often pass
from $\mathbb{Z}\Gamma$-lattices to $\mathbb{Z}_{(p)}\Gamma$-lattices. This
corresponds to identifying $p$-isogeneous tori:
###### Lemma 9.1.
Let $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ and let $M,L$ be
$\mathbb{Z}\Gamma$-lattices. Then the following statements are equivalent:
1. (a)
$L_{(p)}\simeq M_{(p)}$.
2. (b)
There exists an injective map $\phi\colon L\to M$ of
$\mathbb{Z}\Gamma$-modules with cokernel $Q$ finite of order prime to $p$.
3. (c)
There exists a $p$-isogeny $\operatorname{Diag}(M)\to\operatorname{Diag}(L)$.
###### Proof.
The equivalence (b) $\Leftrightarrow$ (c) is clear from the anti-equivalence
of $\operatorname{Diag}$.
The implication (b) $\Rightarrow$ (a) follows from $Q_{(p)}=0$ and that
tensoring with $\mathbb{Z}_{(p)}$ is exact.
For the implication (a) $\Rightarrow$ (b) we use that $L$ and $M$ can be
considered as subsets of $L_{(p)}$ (resp. $M_{(p)}$). The image of $L$ under a
map $\alpha\colon L_{(p)}\to M_{(p)}$ of $\mathbb{Z}_{(p)}\Gamma$-modules
lands in $\frac{1}{m}M$ for some $m\in\mathbb{N}$ (prime to $p$) and the index
of $\alpha(L)$ in $\frac{1}{m}M$ is finite and prime to $p$ if $\alpha$ is
surjective. Since $\frac{1}{m}M\simeq M$ as $\mathbb{Z}\Gamma$-modules the
claim follows. ∎
###### Corollary 9.2.
Let $G$ be an abelian group which is an extension of a $p$-group by a torus
over $k$ and $\Gamma\colonequals\operatorname{Gal}(k_{\operatorname{sep}}/k)$
be the absolute Galois group of $k=k^{(p)}$. Let $\Gamma$ act through a finite
quotient $\overline{\Gamma}$ on $X(G)$. Then
$\operatorname{ed}_{k}(G;p)=\min\operatorname{rk}L-\dim G\,,$
where the minimum is taken over all permutation
$\mathbb{Z}\overline{\Gamma}$-lattices $L$ which admit a map of
$\mathbb{Z}\overline{\Gamma}$-modules to $X(G)$ with cokernel finite of order
prime to $p$.
If $G$ is a torus, then the minimum can also be taken over all
$\mathbb{Z}_{(p)}\overline{\Gamma}$-lattices $L$ which admit a surjective map
of $\mathbb{Z}_{(p)}\overline{\Gamma}$-modules to $X(G)_{(p)}$.
###### Proof.
Let us prove the first claim. In view of Theorem 1.3(a) it suffices to show
that the least dimension of a $p$-faithful representation of $G_{k^{(p)}}$
over $k^{(p)}$ is equal to the least rank of a permutation
$\mathbb{Z}\overline{\Gamma}$-module $L$ which admits a map to $X(G)$ with
cokernel finite of order prime to $p$.
Assume we have such a map $L\to X(G)$. Using the anti-equivalence
$\operatorname{Diag}$ we obtain a $p$-isogeny $G\to\operatorname{Diag}(L)$. We
can embed the quasi-split torus $\operatorname{Diag}(L)$ in
$\operatorname{GL}_{n}$ where $n=\operatorname{rk}L$ [Vo, Section 6.1]. This
yields a $p$-faithful representation of $G$ of dimension $\operatorname{rk}L$.
Conversely let $\rho\colon G\to\operatorname{GL}(V)$ be a $p$-faithful
representation of $G$. Since $G_{\operatorname{sep}}$ is diagonalizable, there
exist characters $\chi_{1},\dotsc,\chi_{n}\in X(G)$ such that $G$ acts on
$V_{\operatorname{sep}}$ via diagonal matrices with entries
$\chi_{1}(g),\dotsc,\chi_{n}(g)$ (for $g\in G$) with respect to a suitable
basis of $V_{\operatorname{sep}}$. Moreover $\overline{\Gamma}$ permutes the
set $\Lambda\colonequals\\{\chi_{1},\dotsc,\chi_{n}\\}$. Define a map
$\phi\colon\mathbb{Z}[\Lambda]\to X(G)$ of
$\mathbb{Z}\overline{\Gamma}$-modules by sending the basis element
$\chi_{i}\in\Lambda$ of $L\colonequals\mathbb{Z}[\Lambda]$ to itself. Then the
$p$-faithfulness of $\rho$ implies that the cokernel of $\phi$ is finite and
of order prime to $p$. Moreover $\operatorname{rk}L=|\Lambda|\leq n=\dim V$.
Now consider the case where $G$ is a torus. Assume we have a surjective map
$\alpha\colon L\to X(G)_{(p)}$ of $\mathbb{Z}_{(p)}\overline{\Gamma}$-modules
where $L=\mathbb{Z}_{(p)}[\Lambda]$ is permutation, $\Lambda$ a
$\overline{\Gamma}$-set. Then $\alpha(\Lambda)\subseteq\frac{1}{m}X(G)$ for
some $m\in\mathbb{N}$ prime to $p$ (note that $\frac{1}{m}X(G)$ can be
considered as a subset of $X(G)_{(p)}$ since $X(G)$ is torsion free). By
construction the induced map $\mathbb{Z}[\Lambda]\to\frac{1}{m}X(G)\simeq
X(G)$ becomes surjective after localization at $p$, hence its cokernel is
finite of order prime to $p$. ∎
###### Corollary 9.3.
Let $A$ be a finite (twisted) cyclic $p$-group over $k$. Let $l/k$ be a
minimal Galois splitting field of $A$, and
$\Gamma\colonequals\operatorname{Gal}(l/k)$. Then
$\operatorname{ed}(A;p)=|\Gamma|.$
###### Proof.
Since $[l:k]$ is a power of $p$, $l^{(p)}/k^{(p)}$ is a Galois extension of
the same degree and the same Galois group as $l/k$. So we can assume
$k=k^{(p)}$.
By Corollary 9.2 $\operatorname{ed}(A;p)$ is equal to the least cardinality of
a $\Gamma$-set $\Lambda$ such that there exists a map
$\phi\colon\mathbb{Z}[\Lambda]\to X(A)$ of $\mathbb{Z}\Gamma$-modules with
cokernel finite of order prime to $p$. The group $X(A)$ is a (cyclic)
$p$-group, hence $\phi$ must be surjective. Moreover $\Gamma$ acts faithfully
on $X(A)$. Surjectivity of $\phi$ implies that some element
$\lambda\in\Lambda$ maps to a generator $a$ of $X(A)$. Hence
$|\Lambda|\geq|\Gamma\lambda|\geq|\Gamma a|=|\Gamma|$. Conversely we have a
surjective homomorphism $\mathbb{Z}[\Gamma a]\to X(A)$ that sends $a$ to
itself. Hence the claim follows.
∎
###### Remark 9.4.
In the case of twisted cyclic groups of order $4$ Corollary 9.3 is due to Rost
[Ro] (see also [BF, Theorem 7.6]), and in the case of cyclic groups of order
$8$ to Bayarmagnai [Ba]. The case of constant groups of arbitrary prime power
order is due to Florence [Fl]; it is now a special case of the Karpenko-
Merkurjev Theorem 1.1.
## 10\. Proof of Theorem 1.3(c)
We will prove Theorem 1.3(c) by using the lattice point of view from Section 9
and the additivity theorem from Section 8.
Let $\overline{\Gamma}$ be a finite group. Two
$\mathbb{Z}\overline{\Gamma}$-lattices $M,N$ are said to be in the same genus
if $M_{(p)}\simeq N_{(p)}$ for all primes $p$, cf. [CR, 31A]. It is sufficient
to check this condition for divisors $p$ of the order of $\overline{\Gamma}$.
By a theorem of A.V. Roǐter [CR, Theorem 31.28] $M$ and $N$ are in the same
genus if and only if there exists a $\mathbb{Z}\overline{\Gamma}$-lattice $L$
in the genus of the free $\mathbb{Z}\overline{\Gamma}$-lattice of rank one
such that $M\oplus\mathbb{Z}\overline{\Gamma}\simeq N\oplus L$. This has the
following consequence for essential dimension:
###### Proposition 10.1.
Let $T,T^{\prime}$ be $k$-tori. If the lattices $X(T),X(T^{\prime})$ belong to
the same genus then
$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T)\text{ and
}\operatorname{ed}_{k}(T;\ell)=\operatorname{ed}_{k}(T;\ell)\text{ for all
primes }\ell.$
###### Proof.
Let $\operatorname{Gal}(k_{\operatorname{sep}}/k)$ act through a finite
quotient $\overline{\Gamma}$ on $X(T)$ and $X(T^{\prime})$. By assumption
there exists a $\mathbb{Z}\overline{\Gamma}$-lattice $L$ in the genus of
$\mathbb{Z}\overline{\Gamma}$ such that
$X(T)\oplus\mathbb{Z}\overline{\Gamma}\simeq X(T^{\prime})\oplus L$. The torus
$S=\operatorname{Diag}(\mathbb{Z}\overline{\Gamma})$ has a generically free
representation of dimension $\dim S$, hence $\operatorname{ed}_{k}(S)=0$.
Since $L$ is a direct summand of
$\mathbb{Z}\overline{\Gamma}\oplus\mathbb{Z}\overline{\Gamma}$ the torus
$S^{\prime}\colonequals\operatorname{Diag}(L)$ has
$\operatorname{ed}_{k}(S^{\prime})\leq\operatorname{ed}_{k}(S\times S)\leq 0$
as well, where the first inequality follows from [BF, Remarks 1.16 (b)].
Therefore
$\operatorname{ed}_{k}(T)\leq\operatorname{ed}_{k}(T\times
S)=\operatorname{ed}_{k}(T^{\prime}\times
S^{\prime})\leq\operatorname{ed}_{k}(T^{\prime})+\operatorname{ed}_{k}(S^{\prime})=\operatorname{ed}_{k}(T^{\prime})$
and similarly $\operatorname{ed}_{k}(T^{\prime})\leq\operatorname{ed}_{k}(T)$.
Hence $\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T^{\prime})$.
A similar argument shows that
$\operatorname{ed}_{k}(T;\ell)=\operatorname{ed}_{k}(T^{\prime};\ell)$ for any
prime $\ell$. This concludes the proof. ∎
###### Corollary 10.2.
Let $k=k^{(p)}$ be a $p$-closed field and $T$ a $k$-torus. Then
$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T;p)=\min\dim(\rho)-\dim T,$
where the minimum is taken over all $p$-faithful representations of $T$.
###### Proof.
The second equality follows from Theorem 1.3(a) and the inequality
$\operatorname{ed}_{k}(T;p)\leq\operatorname{ed}_{k}(T)$ is clear. Hence it
suffices to show $\operatorname{ed}_{k}(T)\leq\operatorname{ed}_{k}(T;p)$. Let
$\rho\colon T\to\operatorname{GL}(V)$ be a $p$-faithful representation of
minimal dimension so that $\operatorname{ed}_{k}(T;p)=\dim\rho-\dim T$. The
representation $\rho$ can be considered as a faithful representation of the
torus $T^{\prime}=T/N$ where $N\colonequals\ker\rho$ is finite of order prime
to $p$. By construction the character lattices $X(T)$ and $X(T^{\prime})$ are
isomorphic after localization at $p$. Since
$\operatorname{Gal}(k_{\operatorname{sep}}/k)$ is a (profinite) $p$-group it
follows that $X(T)$ and $X(T^{\prime})$ belong to the same genus. Hence by
Proposition 10.1 we have
$\operatorname{ed}_{k}(T^{\prime})=\operatorname{ed}_{k}(T)$. Moreover
$\operatorname{ed}_{k}(T^{\prime})\leq\dim\rho-\dim T^{\prime}$, since $\rho$
is a generically free representation of $T^{\prime}$. This finishes the proof.
∎
###### Proof of Theorem 1.3(b).
The equality
$\operatorname{ed}_{k^{(p)}}(G_{k^{(p)}};p)=\operatorname{ed}_{k}(G;p)$
follows from Lemma 3.3. Now we are assuming $G=T\times F$ for a torus $T$ and
a $p$-group $F$ over $k$, which is $p$-closed. Notice that a minimal
$p$-faithful representation of $F$ from Theorem 1.3(a) is also faithful, and
therefore $\operatorname{ed}_{k}(F;p)=\operatorname{ed}_{k}(F)$. Combining
this with Corollary 10.2 and the additivity Theorem 8.1, we see
$\operatorname{ed}(T\times
F)\leq\operatorname{ed}(T)+\operatorname{ed}(F)=\operatorname{ed}(T;p)+\operatorname{ed}(F;p)=\operatorname{ed}(T\times
F;p)\leq\operatorname{ed}(T\times F).$
This completes the proof. ∎
###### Remark 10.3.
The following example shows that “$p$-faithful” cannot be replaced by
“faithful” in the statement of Theorem 1.3(a) (and Corollary 10.2), even in
the case where $G$ is a torus.
Let $p$ be a prime number such that the ideal class group of
$\mathbb{Q}(\zeta_{p})$ is non-trivial (this applies to all but finitely many
primes, e.g. to $p=23$). This means that the subring
$R=\mathbb{Z}[\zeta_{p}]\subseteq\mathbb{Q}(\zeta_{p})$ of algebraic integers
has non-principal ideals. Let $k$ be a field which admits a Galois extension
$l$ of degree $p$ and let
$\Gamma\colonequals\operatorname{Gal}(k_{\operatorname{sep}}/k)$,
$\overline{\Gamma}\colonequals\operatorname{Gal}(l/k)\simeq\Gamma/\Gamma_{l}\simeq
C_{p}$ where $\Gamma_{l}=\operatorname{Gal}(k_{\operatorname{sep}}/l)$ and
$C_{p}$ denotes the cyclic group of order $p$.
We endow the ring $R$ with a $\mathbb{Z}\Gamma$-module structure through the
quotient map $\Gamma\to\overline{\Gamma}$ by letting a generator of
$\overline{\Gamma}$ act on $R$ via multiplication by $\zeta_{p}$. The
$k$-torus $Q\colonequals\operatorname{Diag}(R)$ is isomorphic to the Weil
restriction $R_{l/k}(\operatorname{\mathbb{G}}_{m})$ and has a $p$-dimensional
faithful representation. We will construct a $k$-torus $G$ with a $p$-isogeny
$G\to Q$, such that $G$ does not have a $p$-dimensional faithful
representation.
Let $I$ be a non-principal ideal of $R$. We may consider $I$ as a
$\mathbb{Z}\Gamma$-module and set $G\colonequals\operatorname{Diag}(I)$. We
first show that $I$ and $R$ become isomorphic as $\mathbb{Z}\Gamma$-modules
after localization at $p$. For this purpose let
$I^{\ast}=\\{x\in\mathbb{Q}(\zeta_{p})\mid xI\subseteq R\\}$ denote the
inverse fractional ideal. We have $I\oplus I^{\ast}\simeq R\oplus R$ by [CR,
Theorem 34.31]. The Krull-Schmidt Theorem [CR, Theorem 36.1] for
$\mathbb{Z}_{(p)}C_{p}$-lattices implies $I_{(p)}\simeq R_{(p)}$, hence the
claim. Therefore by Lemma 9.1 there exists a $p$-isogeny $G\to Q$, which shows
in particular that $G$ has a $p$-faithful representation of dimension $p$.
Assume that $G$ has a $p$-dimensional faithful representation. Similarly as in
the proof of Corollary 9.2 this would imply the existence of a surjective map
of $\mathbb{Z}\Gamma$-lattices $\mathbb{Z}\overline{\Gamma}\to I$. However
such a map cannot exist since $I$ is non-principal, hence non-cyclic as a
$\mathbb{Z}\Gamma$-module.
## 11\. Tori of essential dimension $\leq 1$
###### Theorem 11.1.
Let $T$ be a torus over $k$, $k^{(p)}$ a $p$-closure and
$\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$. The following are
equivalent:
1. (a)
$\operatorname{ed}_{k}(T;p)=0$.
2. (b)
$\operatorname{ed}_{k^{(p)}}(T;p)=0$.
3. (c)
$\operatorname{ed}_{k^{(p)}}(T)=0$
4. (d)
$H^{1}(K,T)=\\{1\\}$ for any $p$-closed field $K$ containing $k$.
5. (e)
$X(T)_{(p)}$ is a $\mathbb{Z}_{(p)}\Gamma$-permutation module.
6. (f)
$X(T)$ is an invertible $\mathbb{Z}\Gamma$-lattice (i.e a direct summand of a
permutation lattice).
7. (g)
There is a torus $S$ over $k^{(p)}$ and an isomorphism
$T_{k^{(p)}}\times
S\simeq\operatorname{R}_{E/k^{(p)}}(\operatorname{\mathbb{G}}_{m}),$
for some étale algebra $E$ over $k^{(p)}$.
###### Remark 11.2.
A prime $p$ for which any of these statements fails is called a torsion prime
of $T$.
###### Proof.
(a) $\Leftrightarrow$ (b) is Lemma 3.3.
(a) $\Leftrightarrow$ (d) follows from [Me1, Proposition 4.4].
(c) $\Rightarrow$ (b) is clear.
(b) $\Rightarrow$ (e): This follows from Corollary 9.2. Indeed,
$\operatorname{ed}_{k}(T;p)=0$ implies the existence of a
$\mathbb{Z}_{(p)}\Gamma$-permutation lattice $L$ together with a surjective
homomorphism $\alpha:L\to X(T)_{(p)}$ and
$\operatorname{rk}L=\operatorname{rk}X(T)_{(p)}$. It follows that $\alpha$ is
injective and $X(T)_{(p)}\simeq L$.
(e) $\Rightarrow$ (f): Let $L$ be a $\mathbb{Z}\Gamma$-permutation lattice
such that $L_{(p)}\simeq X(T)_{(p)}$. Then by [CR, Corollary 31.7] there is a
$\mathbb{Z}\Gamma$-lattice $L^{\prime}$ such that $L\oplus L\simeq X(T)\oplus
L^{\prime}$.
(g) $\Rightarrow$ (c): The torus
$R=\operatorname{R}_{E/k^{(p)}}(\operatorname{\mathbb{G}}_{m})$ has a faithful
representation of dimension $\dim R$ (over $k^{(p)}$) and hence
$\operatorname{ed}_{k^{(p)}}(R)=0$. Since $T_{k^{(p)}}$ is a direct factor of
$R$ we must have $\operatorname{ed}_{k^{(p)}}(T)\leq 0$ by [BF, Remarks 1.16
b)].
(f) $\Leftrightarrow$ (g): A permutation lattice $P$ can be written as
$P=\bigoplus_{i+1}^{m}\mathbb{Z}[\Gamma/\Gamma_{L_{i}}],$
for some (separable) extensions $L_{i}/k^{(p)}$ and
$\Gamma_{L_{i}}=\operatorname{Gal}(k_{\operatorname{alg}}/L_{i})$. Set
$E=L_{1}\times\cdots\times L_{m}$. The torus corresponding to $P$ is exactly
$\operatorname{R}_{E/k^{(p)}}(\operatorname{\mathbb{G}}_{m})$, cf. [Vo, 3.
Example 19]. ∎
###### Example 11.3.
Let $T$ be a torus over $k$ of rank $<p-1$. Then
$\operatorname{ed}_{k}(T;p)=0$. This follows from the fact that there is no
non-trivial integral representation of dimension $<p-1$ of any $p$-group, see
for example [AP, Satz]. Thus any finite quotient of
$\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$ acts trivially on
$X(T)$ and so does $\Gamma$.
###### Remark 11.4.
The equivalence of parts (d) and (f) in Theorem 11.1 can also be deduced from
[CTS, Proposition 7.4].
###### Theorem 11.5.
Let $p$ be an odd prime, $T$ an algebraic torus over $k$, and
$\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$.
1. (a)
$\operatorname{ed}(T;p)\leq 1$ iff there exists a $\Gamma$-set $\Lambda$ and
an $m\in\mathbb{Z}[\Lambda]$ fixed by $\Gamma$ such that
$X(T)_{(p)}\cong\mathbb{Z}_{(p)}[\Lambda]/\langle m\rangle$ as
$\mathbb{Z}_{(p)}\Gamma$-lattices.
2. (b)
$\operatorname{ed}(T;p)=1$ iff $m=\sum a_{\lambda}\lambda$ from part (a) is
not $0$ and for any $\lambda\in\Lambda$ fixed by $\Gamma$, $a_{\lambda}=0\mod
p$.
3. (c)
If $\operatorname{ed}(T;p)=1$ then $T_{k^{(p)}}\cong T^{\prime}\times S$ where
$\operatorname{ed}_{k^{(p)}}(S;p)=0$ and $X(T^{\prime})_{(p)}$ is an
indecomposable $\mathbb{Z}_{(p)}\Gamma$-lattice, and
$\operatorname{ed}_{k^{(p)}}(T^{\prime};p)=1$.
###### Proof.
(a) If $\operatorname{ed}(T;p)=1$, then by Corollary 9.2 there is a map of
$\mathbb{Z}\Gamma$-lattices from $\mathbb{Z}[\Lambda]$ to $X(T)$ which becomes
surjective after localization at $p$ and whose kernel is generated by one
element. Since the kernel is stable under $\Gamma$, any element of $\Gamma$
sends a generator $m$ to either itself or its negative. Since $p$ is odd, $m$
must be fixed by $\Gamma$.
The $\operatorname{ed}(T;p)=0$ case and the converse follows from Theorem 1.4
or Corollary 9.2.
(b) Assume we are in the situation of (a), and say $\lambda_{0}\in\Lambda$ is
fixed by $\Gamma$ and $a_{\lambda_{0}}$ is not $0\mod p$. Then
$X(T)_{(p)}\cong\mathbb{Z}_{(p)}[\Lambda-\\{\lambda_{0}\\}]$, so by Theorem
11.1 we have $\operatorname{ed}(T;p)=0$.
Conversely, assume $\operatorname{ed}(T;p)=0$. Then by Theorem 11.1, we have
an exact sequence $0\to\langle
m\rangle\to\mathbb{Z}_{(p)}[\Lambda]\to\mathbb{Z}_{(p)}[\Lambda^{\prime}]\to
0$ for some $\Gamma$-set $\Lambda^{\prime}$ with one fewer element than
$\Lambda$. We have
$\operatorname{Ext}^{1}_{\Gamma}(\mathbb{Z}_{(p)}[\Lambda^{\prime}],\mathbb{Z}_{(p)})=(0)$
by [CTS, Key Lemma 2.1(i)] together with the Change of Rings Theorem [CR,
8.16]; therefore this sequence splits. In other words, there exists a
$\mathbb{Z}_{(p)}\Gamma$-module homomorphism
$f\colon\mathbb{Z}_{(p)}[\Lambda]\to\mathbb{Z}_{(p)}[\Lambda]$ such that the
image of $f$ is $\langle m\rangle$ and $f(m)=m$. Then we can define
$c_{\lambda}\in\mathbb{Z}_{(p)}$ by $f(\lambda)=c_{\lambda}m$. Note that
$f(\gamma(\lambda))=f(\lambda)$ and thus
(12) $c_{\gamma(\lambda)}=c_{\lambda}$
for every $\lambda\in\Lambda$ and $\gamma\in\Gamma$. If
$m=\sum_{\lambda\in\Lambda}a_{\lambda}\lambda$, as in the statement of the
theorem, then $f(m)=m$ translates into
$\sum_{\lambda\in\Lambda}c_{\lambda}a_{\lambda}=1\,.$
Since every $\Gamma$-orbit in $\Lambda$ has a power of $p$ elements, reducing
modulo $p$, we obtain
$\sum_{\lambda\in\Lambda^{\Gamma}}c_{\lambda}a_{\lambda}=1\pmod{p}\,.$
This shows that $a_{\lambda}\neq 0$ modulo $p$, for some
$\lambda\in\Lambda^{\Gamma}$, as claimed.
(c) Decompose $X(T)_{(p)}$ uniquely into a direct sum of indecomposable
$\mathbb{Z}_{(p)}\Gamma$-lattices by the Krull-Schmidt theorem [CR, Theorem
36.1]. Since $\operatorname{ed}(T;p)=1$, and the essential $p$-dimension of
tori is additive (Thm. 8.1), all but one of these summands are permutation
$\mathbb{Z}_{(p)}\Gamma$-lattices. Now by [CR, 31.12], we can lift this
decomposition to $X(T)\cong X(T^{\prime})\oplus X(S)$, where
$\operatorname{ed}(T^{\prime};p)=1$ and $\operatorname{ed}(S;p)=0$. ∎
###### Example 11.6.
Let $E$ be an étale algebra over $k$. It can be written as
$E=L_{1}\times\cdots\times L_{m}$ with some separable field extensions
$L_{i}/k$. The kernel of the norm
$\operatorname{R}_{E/k}(\operatorname{\mathbb{G}}_{m})\to\operatorname{\mathbb{G}}_{m}$
is denoted by $\operatorname{R}_{E/k}^{(1)}(\operatorname{\mathbb{G}}_{m})$.
It is a torus with lattice
$\bigoplus_{i=1}^{m}\mathbb{Z}[\Gamma/\Gamma_{L_{i}}]\;/\;\langle
1,\cdots,1\rangle,$
where $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ and
$\Gamma_{L_{i}}=\operatorname{Gal}(k_{\operatorname{sep}}/L_{i})$. Let
$\Lambda$ be the disjoint union of the cosets $\Gamma/\Gamma_{L_{i}}$ Passing
to a $p$-closure $k^{(p)}$ of $k$, $\Gamma_{k^{(p)}}$ fixes a $\lambda$ in
$\Lambda$ iff $[L_{i}:k]$ is prime to $p$ for some $i$. We thus have
$\operatorname{ed}_{k}(\operatorname{R}_{E/k}^{(1)}(\operatorname{\mathbb{G}}_{m});p)=\left\\{\begin{array}[]{ll}1,&\mbox{$[L_{i}:k]$
is divisible by $p$ for all $i=1,...,m$}\\\ 0,&\mbox{$[L_{i}:k]$ is prime to
$p$ for some $i$.}\end{array}\right.$
## 12\. Tori split by cyclic extensions of degree dividing $p^{2}$
In this section we assume $k=k^{(p)}$ is $p$-closed. Over $k=k^{(p)}$ every
torus is split by a Galois extension of $p$-power order. We wish to compute
the essential dimension of all tori split by a Galois extension with a (small)
fixed Galois group $G$. The following theorem tells us for which $G$ this is
feasible:
###### Theorem 12.1 (A. Jones [Jo]).
For a $p$-group $G$ there are only finitely many genera of indecomposable
$\mathbb{Z}G$-lattices if and only if $G$ is cyclic of order dividing $p^{2}$.
###### Remark 12.2.
For $G=C_{2}\times C_{2}$ a classification of the (infinitely many) different
genera of $\mathbb{Z}G$-lattices has been worked out by [NA]. In contrast for
$G=C_{p^{3}}$ or $G=C_{p}\times C_{p}$ and $p$ odd (in the latter case) no
classification is known.
Hence in this section we consider tori $T$ whose minimal splitting field is
cyclic of degree dividing $p^{2}$. Its character lattice $X(T)$ is then a
$\mathbb{Z}G$-lattice where $G=\langle g|g^{p^{2}}=1\rangle$ denotes the cylic
group of order $p^{2}$. Heller and Reiner [HR], (see also [CR, 34.32])
classified all indecomposable $\mathbb{Z}G$-lattices. Our goal consists in
computing the essential dimension of $T$. By Corollary 10.2 we have
$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T;p)$, hence by the additivity
Theorem 8.1 it will be enough to find the essential $p$-dimension of the tori
corresponding to indecomposable $\mathbb{Z}G$-lattices. Recall that two
lattices are in the same genus if their $p$-localization (or equivalently
$p$-adic completion) are isomorphic. By Proposition 10.1 tori with character
lattices in the same genus have the same essential $p$-dimension, which
reduces the task to calculating the essential $p$-dimension of tori
corresponding to the $4p+1$ cases in the list [CR, 34.32].
Denote by $H=\langle\nolinebreak h|h^{p}=1\nolinebreak\rangle$ the group of
order $p$. We can consider $\mathbb{Z}H$ as a $G$-lattice with the action
$g\cdot h^{i}=h^{i+1}$. Let
$\delta_{G}=1+g+\ldots+g^{p^{2}-1}\quad\delta_{H}=1+h+\ldots+h^{p-1}$
be the “diagonals” in $\mathbb{Z}G$ and $\mathbb{Z}H$ and
$\epsilon=1+g^{p}+\ldots+g^{p^{2}-p}.$
The following $\mathbb{Z}G$-lattices represent all genera of indecomposable
$\mathbb{Z}G$-lattices (by $\langle*\rangle$ we mean the
$\mathbb{Z}G$-sublattice generated by $*$):
$\begin{array}[]{lcll}M_{1}&=&\mathbb{Z}&\\\ M_{2}&=&\mathbb{Z}H&\\\
M_{3}&=&\mathbb{Z}H/\langle\delta_{H}\rangle\\\ M_{4}&=&\mathbb{Z}G\\\
M_{5}&=&\mathbb{Z}G/\langle\delta_{G}\rangle\\\
M_{6}&=&\mathbb{Z}G\oplus\mathbb{Z}/\langle\delta_{G}-p\rangle&\\\
M_{7}&=&\mathbb{Z}G/\langle\epsilon\rangle\\\
M_{8}&=&\mathbb{Z}G/\langle\epsilon-g\epsilon\rangle&\\\
M_{9,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon-(1-h)^{r}\rangle&1\leq
r\leq p-1\\\
M_{10,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon(1-g)-(1-h)^{r+1}\rangle&1\leq
r\leq p-2\\\
M_{11,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon-(1-h)^{r},\delta_{H}\rangle&1\leq
r\leq p-2\\\
M_{12,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon(1-g)-(1-h)^{r+1},\delta_{H}\rangle&1\leq
r\leq p-2\\\ \end{array}$
In the sequel we will refer to the above list as $(\mathbb{L})$.
In $(\mathbb{L})$ we describe $\mathbb{Z}G$-lattices as quotients of
permutation lattices of minimal possible rank, whereas [CR, 34.32] describes
these lattices as certain extensions $1\to L\to M\to N\to 1$ of
$\mathbb{Z}[\zeta_{p^{2}}]$-lattices by $\mathbb{Z}H$-lattices. Therefore
these two lists look differently. Nevertheless they represent the same
$\mathbb{Z}G$-lattices. We show in the example of the lattice $M_{10,r}$ how
one can translate from one list to the other.
Let $\mathbb{Z}x$ be a $\mathbb{Z}G$-module of rank $1$ with trivial
$G$-action. We have an isomorphism
$M_{10,r}=\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon(1-g)-(1-h)^{r+1}\rangle\simeq\mathbb{Z}G\oplus\mathbb{Z}H\oplus\mathbb{Z}x/\langle\epsilon-(1-h)^{r}-x\rangle$
induced by the inclusion
$\mathbb{Z}G\oplus\mathbb{Z}H\hookrightarrow\mathbb{Z}G\oplus\mathbb{Z}H\oplus\mathbb{Z}x$.
This allows us to write $M_{10,r}$ as the pushout
$\textstyle{\mathbb{Z}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h\mapsto\epsilon}$$\scriptstyle{h\mapsto(1-h)^{r}+x}$$\textstyle{\mathbb{Z}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}H\oplus\mathbb{Z}x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{10,r}}$
Completing both lines on the right we see that $M_{10,r}$ is an extension
$0\to\mathbb{Z}H\oplus\mathbb{Z}x\to M_{10,r}\to\mathbb{Z}G/\mathbb{Z}H\to 0$
with extension class determined by the vertical map $h\mapsto(1-h)^{r}+x$ cf.
[CR, 8.12] and we identify (the $p$-adic completion of) $M_{10,r}$ with one of
the indecomposable lattices in the list [CR, 34.32].
Similarly, $M_{1},\ldots,M_{12,r}$ are representatives of the genera of
indecomposable $\mathbb{Z}G$-lattices.
###### Theorem 12.3.
Every indecomposable torus $T$ over $k$ split by $G$ has character lattice
isomorphic to one of the $\mathbb{Z}G$-lattices $M$ in the list $(\mathbb{L})$
after $p$-localization and
$\operatorname{ed}(T)=\operatorname{ed}(T;p)=\operatorname{ed}(\operatorname{Diag}(M);p)$.
Their essential dimensions are given in the tables below.
$\begin{array}[]{c|c|c}M&\operatorname{rk}M&\operatorname{ed}(T)\\\ \hline\cr
M_{1}&1&0\\\ M_{2}&p&0\\\ M_{3}&p-1&1\\\ M_{4}&p^{2}&0\\\ M_{5}&p^{2}-1&1\\\
M_{6}&p^{2}&1\end{array}\quad\quad\begin{array}[]{c|c|c}M&\operatorname{rk}M&\operatorname{ed}(T)\\\
\hline\cr M_{7}&p^{2}-p&p\\\ M_{8}&p^{2}-p+1&p-1\\\ M_{9,r}&p^{2}&p\\\
M_{10,r}&p^{2}+1&p-1\\\ M_{11,r}&p^{2}-1&p+1\\\ M_{12,r}&p^{2}&p\end{array}$
###### Proof of Proposition 12.3.
We will assume $p>2$ in the sequel. For $p=2$ the Theoerem is still true but
some easy additional arguments are needed which we leave out here.
The essential $p$-dimension of tori corresponding to $M_{1}\ldots,M_{6}$
easily follows from the discussion in section 11. Let $M$ be one of the
lattices $M_{7},\ldots,M_{12,r}$ and $T=\operatorname{Diag}M$ the
corresponding torus. We will determine the minimal rank of a permutation
$\mathbb{Z}G$-lattice $P$ admitting a homomorphism $P\to M$ which becomes
surjective after localization at $p$. Then we conclude
$\operatorname{ed}(T;p)=\operatorname{rk}P-\operatorname{rk}M$ with Corollary
9.2.
We have the bounds
(13) $\operatorname{rk}M\leq\operatorname{rk}P\leq p^{2}\;(\mbox{or
}p^{2}+p),$
where the upper bound holds since every $M$ is given as a quotient of
$\mathbb{Z}G$ (or $\mathbb{Z}G\oplus\mathbb{Z}H$). Let
$C=\operatorname{Split}_{k}(T[p])$ the finite constant group used in the proof
of Theorem 1.3. The rank of $C$ determines exactly the number of direct
summands into which $P$ decomposes. Moreover each indecomposable summand has
rank a power of $p$.
As an example, we show how to find $C$ for $M=M_{11,r}$: The relations
$g^{j}\cdot(\epsilon-(1-h)^{r});\delta_{H}$ are written out as
$\sum_{i=0}^{p-1}g^{pi+j}-\sum_{\ell=0}^{r}\binom{r}{\ell}(-1)^{\ell}h^{\ell+j},\;0\leq
j\leq p-1;\quad\sum_{i=0}^{p-1}h^{i}$
and the $k_{\operatorname{sep}}$-point of the torus are
$\begin{array}[]{ll}T(k_{\operatorname{sep}})=&{\Big{\\{}}(t_{0},\dotsc,t_{p^{2}-1},s_{0},\dotsc,s_{p-1})\mid\\\
&\prod_{i=0}^{p-1}t_{pi+j}=\prod_{\ell=0}^{r}s_{\ell+j}^{(-1)^{\ell}\binom{r}{\ell}},\;0\leq
j\leq p-1;\quad\prod_{i=0}^{p-1}s_{i}=1{\Big{\\}}}\end{array}$
and $C$ is the constant group of fixed points of the $p$-torsion $T[p]$:
$C(k)=\left\\{\left(\zeta_{p}^{i},\dotsc,\zeta_{p}^{i},\zeta_{p}^{j},\dotsc,\zeta_{p}^{j}\right)\mid\
0\leq i,j\leq p-1\right\\}\simeq\mu_{p}^{2}.$
(Note that the primitive $p$th root of unity $\zeta_{p}$ is in $k$ by our
assumption that $k$ is $p$-closed). For other lattices this is similar: $C$ is
equal to
$\operatorname{Split}_{k}(\operatorname{Diag}(P)[p])\simeq\mu_{p}^{r}$ where
$M$ is presented as a quotient $P/N$ of a permutation lattice $P$ (of minimal
rank) as in $(\mathbb{L})$ and where $r$ denotes the number of summands in a
decomposition of $P$.
$\begin{array}[]{c|c|c|c}M&\mbox{rank $C$}&\mbox{rank $M$}&\mbox{possible
$\operatorname{rk}P$}\\\ \hline\cr M_{7}&1&p^{2}-p&p^{2}\\\
M_{8}&1&p^{2}-p+1&p^{2}\\\ M_{9,r}&2&p^{2}&p^{2}+1\mbox{ or }p^{2}+p\\\
M_{10,r}&2&p^{2}+1&p^{2}+1\mbox{ or }p^{2}+p\\\
M_{11,r}&2&p^{2}-1&p^{2}+1\mbox{ or }p^{2}+p\\\ M_{12,r}&2&p^{2}&p^{2}+1\mbox{
or }p^{2}+p\end{array}$
We need to exclude the possibility $\operatorname{rk}P=p^{2}+1$ for the
lattices $M=M_{9,r},\ldots,M_{12,r}$. We can only have the value $p^{2}+1$ if
there exists a character in $M$ which is fixed under the Galois group and
nontrivial on $C$. The following Lemma 12.4 tells us, that such characters do
not exist in either case. Hence the minimal dimension of a $p$-faithful
representation of all these tori is $p^{2}+p$. ∎
###### Lemma 12.4.
For $i=9,\ldots,12$ and $r\geq 1$ every character $\chi\in M_{i,r}$ fixed
under $G$ has trivial restriction to $C$.
###### Proof.
By [Hi] the cohomology group $H^{0}(G,M_{i,r})=M_{i,r}^{G}$ of $G$-fixed
points in $M_{i,r}$ is trivial for $i=11$, has rank $1$ for $i=9,12$ and rank
$2$ for $i=10$, respectively. They are represented by $\mathbb{Z}\delta_{H}$
in $M_{9,r}$, by $\mathbb{Z}(\epsilon-(1-h)^{r})$ in $M_{12,r}$ and by
$\mathbb{Z}(\epsilon-(1-h)^{r})\oplus\mathbb{Z}\delta_{H}$ in $M_{10,r}$,
respectively. Since all these characters are trivial on
$C=\operatorname{Split}_{k}(\operatorname{Diag}(\mathbb{Z}G\oplus\mathbb{Z}H)[p]),$
the claim follows. ∎
## Acknowledgments
The authors are grateful to A. Auel, A. Merkurjev and A. Vistoli for helpful
comments and conversations.
## References
* [AP] H. Abold, W. Plesken, Ein Sylowsatz für endliche $p$-Untergruppen von ${\rm GL}(n,Z)$, Math. Ann. 232 (1978), no. 2, 183–186.
* [Ba] G. Bayarmagnai, Essential dimension of some twists of $\mu_{p^{n}}$, Proceedings of the Symposium on Algebraic Number Theory and Related Topics, 145–151, RIMS K$\hat{\rm o}$ky$\hat{\rm u}$roku Bessatsu, B4, Res. Inst. Math. Sci. (RIMS), Kyoto (2007).
* [BF] G. Berhuy, G. Favi, Essential Dimension: A Functorial Point of View (after A. Merkurjev), Doc. Math. 8:279–330 (electronic) (2003).
* [Bo] A. Borel Linear Algebraic Groups, Benjamin (1969).
* [BS] A. Borel, J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164.
* [Bou] N. Bourbaki, Algebra. II. Chapters 4–7. Translated from the French by P. M. Cohn and J. Howie. Elements of Mathematics. Springer-Verlag, Berlin, (1990).
* [BR] J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Mathematica 106:159–179.(1997).
* [CGR] V. Chernousov, Ph. Gille, Z. Reichstein, Resolving $G$-torsors by abelian base extensions, J. Algebra 296 (2006), no. 2, 561–581.
* [CTS] J.-L. Colliot-Thélène, J. J. Sansuc, Principal Homogeneous Spaces under Flasque Tori: Applications, J. Algebra 106 (1087), 148–205.
* [CR] C. W. Curtis, I. Reiner, Methods of representation theory ,vol. 1, Wiley (Interscience), 1981.
* [DG] M. Demazure, P. Gabriel, Groupes algébriques. Tome I, Masson & Cie, Paris; North-Holland Publishing Co., Amsterdam, 1970.
* [Fl] M. Florence, On the essential dimension of cyclic $p$-groups, Inventiones Mathematicae, 171 (2007), 175-189.
* [GMS] S. R. Garibaldi, A. Merkurjev, J.-P. Serre: Cohomological Invariants in Galois Cohomology, University Lecture Series, Vol. 28, American Mathematical Society, Providence, RI, (2003).
* [GR] Ph. Gille, Z. Reichstein, A lower bound on the essential dimension of a connected linear group, Comment. Math. Helv. 4, no. 1 (2009), 189–212.
* [Gro] A. Grothendieck, La torsion homologique et les sections rationnelles, Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, IHP, (1958).
* [HR] A. Heller, I. Reiner: Representations of cyclic groups in rings of integers I, Annals of Math, 76 (1962), 73-92.
* [Hi] H. Hiller, Flat Manifolds with $\mathbb{Z}/p^{2}$ Holonomy, L’Enseignement Mathématique, 31 (1985), 283–297.
* [Ja] J. C. Jantzen, Representations of Algebraic Groups. Pure and Applied Mathematics, 131. Academic Press, Orlando, Florida, (1987).
* [Jo] A. Jones, Groups with a finite number of indecomposable integral representations, Mich. Math. J, 10 (1963), 257-261.
* [Ka] G. Karpilovsky, Clifford Theory for Group Representations. Mathematics Studies, 156. North-Holland, Netherlands, (1989).
* [KM] N. Karpenko, A. Merkurjev, Essential dimension of finite p-groups, Inventiones Mathematicae, 172 (2008), 491–508.
* [Ma] B. Margaux, Passage to the limit in non-abelian Čech cohomology. J. Lie Theory 17, no. 3 (2007), 591–596.
* [Me1] A. Merkurjev, Essential dimension, in Quadratic forms – algebra, arithmetic, and geometry (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemporary Mathematics 493 (2009), 299–326.
* [Me2] A. Merkurjev Essential dimension of PGL($p^{2}$), preprint, available at http://www.math.ucla.edu/~merkurev/publicat.htm.
* [MR1] A. Meyer, Z. Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra and Number Theory, 3, no. 4 (2009), 467–487.
* [MR2] A. Meyer, Z. Reichstein, An upper bound on the essential dimension of a central simple algebra, to appear in Journal of Algebra, 10.1016/j.jalgebra.2009.09.019, preprint available at arXiv:0907.4496
* [NA] L. A. Nazarova, Unimodular representations of the four group, Dokl. Akad. Nauk SSSR, 140 (1961), 1011–1014.
* [Re] Z. Reichstein, On the Notion of Essential Dimension for Algebraic Groups, Transformation Groups, 5, 3 (2000), 265-304.
* [RY] Z. Reichstein, B. Youssin, Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-varieties, with an appendix by J. Kollar and E. Szabo, Canadian Journal of Mathematics, 52, 5 (2000), 1018–1056.
* [RZ] L. Ribes, P. Zalesskii, Profinite Groups. Springer-Verlag, Berlin, 2000.
* [Ro] M. Rost, Essential dimension of twisted $C_{4}$, available at http://www.math.uni-bielefeld.de/~rost/ed.html.
* [Sch1] H.-J. Schneider, Zerlegbare Erweiterungen affiner Gruppen J. Algebra 66, no. 2 (1980), 569–593.
* [Sch2] H.-J. Schneider, Decomposable Extensions of Affine Groups, in Lecture Notes in Mathematics 795, Springer Berlin/Heidelberg (1980), 98–115.
* [Sch3] H.-J. Schneider, Restriktion und Corestriktion für algebraische Gruppen J. Algebra, 68, no. 1 (1981), 177–189.
* [Se1] J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, 42, Springer-Verlag, 1977.
* [Se2] J.-P. Serre, Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002.
* [Ta] J. Tate, Finite flat group schemes. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 121–154, Springer, New York, 1997.
* [Vo] V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, American Mathematical Society, Providence, RI, 1998.
* [Wa] W. C. Waterhouse, Introduction to affine group schemes. Springer-Verlag, New York-Berlin, 1979.
* [Wi] J. S. Wilson, Profinite Groups. London Math. Soc. Monographs 19, Oxford University Press, New York, 1998.
* [Win] D. Winter, The structure of fields. Graduate Texts in Mathematics, no. 16. Springer-Verlag, New York-Heidelberg, 1974.
|
arxiv-papers
| 2009-10-29T08:18:35 |
2024-09-04T02:49:06.155347
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roland L\\\"otscher, Mark MacDonald, Aurel Meyer and Zinovy Reichstein",
"submitter": "Roland L\\\"otscher",
"url": "https://arxiv.org/abs/0910.5574"
}
|
0910.5793
|
# Non-Markovian entanglement dynamics between two coupled qubits in the same
environment
Wei Cui1,2, Zairong Xi1∗ and Yu Pan1,2 1Key Laboratory of Systems and Control,
Institute of Systems Science, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
2Graduate University of Chinese Academy of Sciences, Beijing 100039, People’s
Republic of China zrxi@iss.ac.cn
###### Abstract
We analyze the dynamics of the entanglement in two independent non-Markovian
channels. In particular, we focus on the entanglement dynamics as a function
of the initial states and the channel parameters like the temperature and the
ratio $r$ between $\omega_{0}$ the characteristic frequency of the quantum
system of interest, and $\omega_{c}$ the cut-off frequency of Ohmic reservoir.
We give a stationary analysis of the concurrence and find that the dynamic of
non-markovian entanglement concurrence $\mathcal{C}_{\rho}(t)$ at temperature
$k_{B}T=0$ is different from the $k_{B}T>0$ case. We find that “entanglement
sudden death” (ESD) depends on the initial state when $k_{B}T=0$, otherwise
the concurrence always disappear at finite time when $k_{B}T>0$, which means
that ESD must happen. The main result of this paper is that the non-Markovian
entanglement dynamic is fundamentally different from the Markovian one. In the
Markovian channel, entanglement decays exponentially and vanishes only
asymptotically, but in the non-Markovian channel the concurrence
$\mathcal{C}_{\rho}(t)$ oscillates, especially in the high temperature case.
Then an open-loop controller adjusted by the temperature is proposed to
control the entanglement and prolong the ESD time.
###### pacs:
03.65.Ud, 03.65.Yz, 03.67.Mn, 05.40.Ca
## 1 Introduction
Entanglement is a remarkable feature of quantum mechanics, and its
investigation is both of practical and theoretical significance. It is viewed
as a basic resource for quantum information processing (QIP) [1], like
realizing high-speed quantum computation [2] and high-security quantum
communication [3]. It is also a basic issue in understanding the nature of
nonlocality in quantum mechanics [4, 5, 6]. However, a quantum system used in
quantum information processing inevitably interacts with the surrounding
environmental system (or the thermal reservoir), which induces the quantum
world into classical world [7, 14, 21]. Thus, it is an important subject to
analyze the entanglement decay induced by the unavoidable interaction with the
environment [8, 9, 10, 11, 12]. In one-party quantum system, this process is
called decoherence [13, 14, 15, 16, 17, 18, 19]. In this paper, we will
analyze the entanglement dynamics of bipartite non-Markovian quantum system.
As well known, the system can only couple to a few environmental degrees of
freedom for short times. These will act as memory. In short time scales
environmental memory effects always appear in experiments [20]. The
characteristic time scales become comparable with the reservoir correlation
time in various cases, especially in high-speed communication. Then an exactly
analytic description of the open quantum system dynamic is needed, such as
quantum Brownian motion(QBM) [21], a two-level atom interacting with a thermal
reservoir with Lorentzian spectral density [22], and the devices based on
solid state [23] where memory effects are typically non-negligible. Due to its
fundamental importance in quantum information processing and quantum
computation, non-Markovian quantum dissipative systems have attracted much
attention in recent years [7, 24, 25, 26, 27, 28]. Recently, researches on
quantum coherence and entanglement influenced and degraded by the external
environment become more and more popular, most of the works contributed to
extend the open quantum theory beyond the Markovian approximation [29, 30,
31]. In [29], two harmonic oscillators in the quantum domain were studied and
their entanglement evolution investigated with the influence of thermal
environments. In [30], the dynamics of bipartite Gaussian states in a non-
Markovian noisy channel were analyzed. All in all, non-Markovian features of
system-reservoirs interaction have made great progress, but the theory is far
from completion, especially how the non-Markovian environmental influence the
system and what the difference is between Markovian and non-Markovian system
evolution are not clear.
In this paper we will compare the non-Markovian entanglement dynamics with the
Markovian one [32] in Ohmic reservoir with Lorentz-Drude regularization in the
following three conditions: $\omega_{0}\ll\omega_{c}$,
$\omega_{0}\approx\omega_{c}$ and $\omega_{0}\gg\omega_{c}$, where
$\omega_{0}$ is the characteristic frequency of the quantum system of interest
and $\omega_{c}$ the cut-off frequency of Ohmic reservoir. Thus,
$\omega_{c}\ll\omega_{0}$ implies that the spectrum of the reservoir does not
completely overlap with the frequency of the system oscillator and
$\omega_{0}\gg\omega_{c}$ implies the converse case. Another point of the
entanglement dynamics is the temperature. We characterize our system by low
temperature, $k_{B}T=0.03\omega_{0}$, medium temperature,
$k_{B}T=3\omega_{0}$, and the high temperature $k_{B}T=300\omega_{0}$. We give
stationary analysis of the concurrence [9] and find that the dynamics of non-
markovian entanglement concurrence $\mathcal{C}$ at temperature $k_{B}T=0$ is
fundamentally different from the $k_{B}T>0$. We find that “entanglement sudden
death” (ESD) depends on the initial state when $k_{B}T=0$, otherwise the
concurrence always disappear at finite time when $k_{B}T>0$, which means that
ESD must happen. Maniscalco S et.al studied the separability function
$S(\tau)$ in [30], where the entanglement oscillation appears for twin-beam
state in non-Markovian channels for high temperature reservoirs. The main
result of this paper is that the non-Markovian entanglement dynamics is
fundamentally different from the Markovian one. In the Markovian channel,
entanglement decays exponentially and vanishes only asymptotically, but in the
non-Markovian channel the concurrence $C_{\rho}(t)$ oscillates, especially in
the high temperature case.
The paper is organized as follows. We first introduce the open quantum system
and the non-Markovian quantum master equation for driven open quantum systems
by the noise and dissipation kernels. In Sec. III we introduce the Wootters’
concurrence and the initial “X” states. By substituting the initial states
into the master equation we get the first order coupled differential
equations, and give the stationary analysis. In Sec. IV, we numerically
analyze the Markovian and non-Markovian entanglement dynamics. Then an open-
loop controller adjusted by the temperature is proposed to control the
entanglement and prolong the ESD time. Conclusions and prospective views are
given in Sec. V.
## 2 The model
Our system consists of a pair of two-level atoms (two qubits) equally and
resonantly, coupled to a single cavity mode, with the same coupling strength.
The master equation for the reduced density matrix $\rho(t)$ which describes
its dynamics is given by [7, 18, 30, 31, 33]
$\displaystyle\frac{d\rho(t)}{dt}=\frac{\Delta(t)+\gamma(t)}{2}\sum_{j=1}^{2}\\{2\sigma_{j}^{-}\rho\sigma_{j}^{+}-\sigma_{j}^{+}\sigma_{j}^{-}\rho-\rho\sigma_{j}^{+}\sigma_{j}^{-}\\}$
$\displaystyle+\frac{\Delta(t)-\gamma(t)}{2}\sum_{j=1}^{2}\\{2\sigma_{j}^{+}\rho\sigma_{j}^{-}-\sigma_{j}^{-}\sigma_{j}^{+}\rho-\rho\sigma_{j}^{-}\sigma_{j}^{+}\\}.$
(1)
where
$\sigma^{+}=\frac{1}{2}(\sigma_{1}+i\sigma_{2}),~{}~{}\sigma^{-}=\frac{1}{2}(\sigma_{1}-i\sigma_{2})$,
with $\sigma_{1}$, $\sigma_{2}$ the Pauli matrices. The time dependent
coefficients appearing in the master equation can be written, to the second
order in the coupling strength, as follows
$\begin{array}[]{rcl}\Delta(t)&=&\int_{0}^{t}d\tau
k(\tau)\cos(\omega_{0}\tau),\\\
\gamma(t)&=&\int_{0}^{t}d\tau\mu(\tau)\sin(\omega_{0}\tau),\end{array}$ (2)
with
$\begin{array}[]{rcl}k(\tau)&=&2\int_{0}^{\infty}d\omega
J(\omega)\coth[\omega/2k_{B}T]\cos(\omega\tau),\\\
\mu(\tau)&=&2\int_{0}^{\infty}d\omega J(\omega)\sin(\omega\tau),\end{array}$
(3)
being the noise and the dissipation kernels, respectively. This master
equation (1) is valid for arbitrary temperature. The coefficient $\gamma(t)$
gives rise to a time dependent damping term, while $\Delta(t)$ the diffusive
term. The non-Markovian character is contained in the time-dependent
coefficients, which contain all the information about the short time system-
reservoir correlations [7]. In the previous equations $J(\omega)$ is the
spectral density characterizing the bath,
$J(\omega)=\frac{\pi}{2}\sum_{i}\frac{k_{i}}{m_{i}\omega_{i}}\delta(\omega-\omega_{i})$
(4)
and the index $i$ labels the different field mode of the reservoir with
frequency $\omega_{i}$. Let the Ohmic spectral density with a Lorentz-Drude
cutoff function,
$J(\omega)=\frac{2}{\pi}\omega\frac{\omega_{c}^{2}}{\omega_{c}^{2}+\omega^{2}},$
(5)
where $\omega$ is the frequency of the bath, and $\omega_{c}$ is the high-
frequency cutoff.
Then the closed analytic expressions for $\Delta(t)$ and $\gamma(t)$ are [18,
31]
$\gamma(t)=\frac{\omega_{0}r^{2}}{1+r^{2}}[1-e^{-r\omega_{0}t}\cos(\omega_{0}t)-re^{-r\omega_{0}t}\sin(\omega_{0}t)],$
(6)
$\displaystyle\Delta(t)=\omega_{0}\frac{r^{2}}{1+r^{2}}\\{\coth(\pi
r_{0})-\cot(\pi r_{c})e^{-\omega_{c}t}[r\cos(\omega_{0}t)-\sin(\omega_{0}t)]$
$\displaystyle+\frac{1}{\pi
r_{0}}\cos(\omega_{0}t)[\bar{F}(-r_{c},t)+\bar{F}(r_{c},t)-\bar{F}(ir_{0},t)-\bar{F}(-ir_{0},t)]$
$\displaystyle-\frac{1}{\pi}\sin(\omega_{0}t)[\frac{e^{-\nu_{1}t}}{2r_{0}(1+r_{0}^{2})}[(r_{0}-i)\bar{G}(-r_{0},t)+(r_{0}+i)\bar{G}(r_{0},t)]$
$\displaystyle+\frac{1}{2r_{c}}[\bar{F}(-r_{c},t)-\bar{F}(r_{c},t)]]\\},$ (7)
where $r_{0}=\omega_{0}/2\pi k_{B}T$, $r_{c}=\omega_{c}/2\pi k_{B}T$,
$r=\omega_{c}/\omega_{0}$, and
$\bar{F}(x,t)\equiv_{2}F_{1}(x,1,1+x,e^{-\nu_{1}t}),$ (8)
$\bar{G}(x,t)\equiv_{2}F_{1}(2,1+x,2+x,e^{-\nu_{1}t}).$ (9)
$\nu_{1}=2\pi k_{B}T$, and ${}_{2}F_{1}(a,b,c,z)$ is the hypergeometric
function. Note that, for time $t$ large enough, the coefficients $\Delta(t)$
and $\gamma(t)$ can be approximated by their Markovian stationary values
$\Delta_{M}=\Delta(t\rightarrow\infty)$ and
$\gamma_{M}=\gamma(t\rightarrow\infty)$. From Eqs. (6) and (7) we have
$\gamma_{M}=\frac{\omega_{0}r^{2}}{1+r^{2}},$ (10)
and
$\Delta_{M}=\omega_{0}\frac{r^{2}}{1+r^{2}}\coth(\pi r_{0}).$ (11)
Figure 1: (Color online) Dynamics of non-Markovian coefficients $\Delta(t)$
(blue solid line) and $\gamma(t)$ (red dotted line) at different temperatures:
(a)low temperature $k_{B}T=0.01$, (b)medium temperature $k_{B}T=1$, and
(c)high temperature $k_{B}(t)=100$, respectively. The other coefficients are
chosen as $r=0.1$ $\omega_{0}=1$, and $\alpha^{2}=0.01$.
Note that $\gamma(t)$ has nothing to do with the temperature [33]. In Fig.1 we
plot the time evolution of non-Markovian coefficients $\Delta(t)$ and
$\gamma(t)$ in different channel temperatures. In Fig. 1(a), the temperature
is $k_{B}T=0.01$. There are two important main points embodied in the Figure,
the first is that the coefficient $\gamma(t)$ has dominated the system
dissipation at low temperature, the other $\Delta_{M}=\gamma_{M}$ in the long
time limit. Fig. 1(b) and (c) are the evolution at the medium temperature and
high temperature respectively. The Figure shows that the larger the
temperature, the more important the coefficient $\Delta(t)$.
## 3 Concurrence and initial states
In order to describe the entanglement dynamics of the bipartite system, we use
the Wootters concurrence [9, 34]. For a system described by the density matrix
$\rho$, the concurrence $\mathcal{C}(\rho)$ is
$\mathcal{C}(\rho)=\max(0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}),$
(12)
where $\lambda_{1},\lambda_{2},\lambda_{3}$, and $\lambda_{4}$ are the
eigenvalues (with $\lambda_{1}$ the largest one) of the “spin-flipped” density
operator $\zeta$, which is defined by
$\zeta=\rho(\sigma_{y}^{A}\otimes\sigma_{y}^{B})\rho^{*}(\sigma_{y}^{A}\otimes\sigma_{y}^{B}),$
(13)
where $\rho^{*}$ denotes the complex conjugate of $\rho$ and $\sigma_{y}$ is
the Pauli matrix. $\mathcal{C}$ ranges in magnitude from 0 for a
disentanglement state to 1 for a maximally entanglement state. The concurrence
is related to the entanglement of formation $E_{f}(\rho)$ by the following
relation [34]
$E_{f}(\rho)=\varepsilon[\mathcal{C}(\rho)],$ (14)
where
$\varepsilon[\mathcal{C}(\rho)]=h[\frac{1+\sqrt{1-\mathcal{C}^{2}(\rho)}}{2}],$
(15)
and
$h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x).$ (16)
Assume that the system is initially an “X” state, which has non-zero elements
only along the main diagonal and anti-diagonal. The general structure of an
“X” density matrix is as follows
$\hat{\rho}=\left(\begin{array}[]{cccc}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho_{23}^{*}&\rho_{33}&0\\\
\rho_{14}^{*}&0&0&\rho_{44}\end{array}\right).$ (17)
Such states are general enough to include states such as the Werner states,
the maximally entangled mixed states (MEMSs) and the Bell states; and it also
arises in a wide variety of physical situations [35, 36, 37]. This particular
form of the density matrix allows us to analytically express the concurrence
as [38]
$\mathcal{C}_{\hat{\rho}}^{X}=2\max\\{0,K_{1},K_{2}\\},$ (18)
where
$\begin{array}[]{rcl}K_{1}&=&|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}},\\\
K_{2}&=&|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}}.\end{array}$ (19)
A remarkable aspect of the “X” states is that the time evolution of the master
equation (1) is maintained during the evolution. Substituting (17) into (1),
the non-markovian master equation of two-qubits system, we obtain the
following first-order coupled differential equations:
$\begin{array}[]{rcl}\dot{\rho}_{11}(t)&=&-2(\Delta(t)+\gamma(t))\rho_{11}(t)+(\Delta(t)-\gamma(t))\rho_{22}(t)+(\Delta(t)-\gamma(t))\rho_{33}(t),\\\
\dot{\rho}_{22}(t)&=&(\Delta(t)+\gamma(t))\rho_{11}(t)-2\Delta(t)\rho_{22}(t)+(\Delta(t)-\gamma(t))\rho_{44}(t),\\\
\dot{\rho}_{33}(t)&=&(\Delta(t)+\gamma(t))\rho_{11}(t)-2\Delta(t)\rho_{33}(t)+(\Delta(t)-\gamma(t))\rho_{44}(t),\\\
\dot{\rho}_{44}(t)&=&(\Delta(t)+\gamma(t))\rho_{22}(t)+(\Delta(t)+\gamma(t))\rho_{33}(t)-2(\Delta(t)-\gamma(t))\rho_{44}(t),\\\
\dot{\rho}_{23}(t)&=&-2\Delta(t)\rho_{23}(t),\\\
\dot{\rho}_{14}(t)&=&-2\Delta(t)\rho_{14}(t).\end{array}$ (20)
From Eq. (18) the concurrence $\mathcal{C}$ is dependent on the coefficients
$\Delta(t\rightarrow\infty)$ and $\gamma(t\rightarrow\infty)$ in the
asymptotic long time limit. Eqs. (10) and (11) give the stationary value of
$\gamma(t)$ and $\Delta(t)$, the Markovian limit
$\gamma_{M}\equiv\gamma(t\rightarrow\infty)=\frac{\omega_{0}r^{2}}{1+r^{2}},$
and
$\Delta_{M}\equiv\Delta(t\rightarrow\infty)=\omega_{0}\frac{r^{2}}{1+r^{2}}\coth(\frac{\omega_{0}}{2k_{B}T}).$
$\gamma_{M}$ doesn’t depend on temperature, but $\Delta_{M}$ is monotonically
increasing with respect to temperature $T$. When $T\rightarrow 0$,
$\Delta_{M}\rightarrow\frac{\omega_{0}r^{2}}{1+r^{2}}$. Noting $\coth(\pi
r_{0})\simeq 1+\frac{1}{\pi r_{0}}\simeq\frac{2k_{B}T}{\omega_{0}},$ at high
temperature
$\Delta_{M}^{HT}=2k_{B}T\frac{r^{2}}{1+r^{2}}.$ (21)
So $\Delta_{M}>\gamma_{M}$ is noticeable when temperature $k_{B}T>0$. From
Eqs. (20) we can get the stationary solution
$\begin{array}[]{rcl}\rho_{11}(t\rightarrow\infty)&=&\frac{\Delta_{M}-\gamma_{M}}{\Delta_{M}+\gamma_{M}}\rho_{33}(t\rightarrow\infty),\\\
\rho_{22}(t\rightarrow\infty)&=&\rho_{33}(t\rightarrow\infty),\\\
\rho_{33}(t\rightarrow\infty)&=&\frac{\Delta_{M}^{2}-\gamma_{M}^{2}}{4\Delta_{M}^{2}},\\\
\rho_{44}(t\rightarrow\infty)&=&\frac{\Delta_{M}+\gamma_{M}}{\Delta_{M}-\gamma_{M}}\rho_{33}(t\rightarrow\infty).\end{array}$
(22)
and
$\begin{array}[]{rcl}\rho_{23}(t\rightarrow\infty)&=&0,\\\
\rho_{14}(t\rightarrow\infty)&=&0.\end{array}$ (23)
According to Eqs. (18, 19),
$K_{1,2}(t\rightarrow\infty)=0-\frac{\Delta_{M}^{2}-\gamma_{M}^{2}}{4\Delta_{M}^{2}}<0.$
(24)
This means that entanglement must disappear in a finite time period, i.e. the
ESD must happen.
When temperature $k_{B}T=0$, $\Delta_{M}\approx\gamma_{M}$. From Eqs. (20) we
can also get the stationary solution
$\begin{array}[]{rcl}\rho_{11}(t\rightarrow\infty)&=&0,\\\
\rho_{22}(t\rightarrow\infty)&=&0,\\\ \rho_{33}(t\rightarrow\infty)&=&0,\\\
\rho_{44}(t\rightarrow\infty)&=&1.\end{array}$ (25)
and
$\begin{array}[]{rcl}\rho_{23}(t\rightarrow\infty)&=&0,\\\
\rho_{14}(t\rightarrow\infty)&=&0.\end{array}$ (26)
From Eqs. (18, 19),
$K_{1,2}(t\rightarrow\infty)=0.$ (27)
This means that entanglement maybe disappear asymptotically, or oscillates, or
other complex behaviors. In the following, we use the numerical methods to
demonstrate the concurrence evolution for a special kind of “X” state, the
$\rho_{YE}$ state.
## 4 Non-Markovian vs. Markovian entanglement dynamics
Figure 2: (Color online)Time evolution of non-Markovian concurrence as a
function of parameter “$a$” in the low temperature reservoirs.
Figure 3: (Color online)Time evolution of non-Markovian concurrence as a
function of parameter “$a$” in the medium temperature reservoirs.
Figure 4: (Color online)Time evolution of $K(t)$ for temperature
$k_{B}T=0.000001\omega_{0}$, $r=0.1$, and initial state $\hat{\rho}_{YE}$ for
the cases $a=0$ (black dotted), $a=0.3$ (cyan dash-dotted line), $a=0.5$ (red
dash line), $a=0.6$ (green dotted-dotted line), $a=0.8$ (magenta asterisk),
and $a=1.0$ (blue solid line).
Figure 5: (Color online)Comparing the non-Markovian entanglement dynamics with
the Markovian one by the time evolution of concurrence as a function of
parameter “a” in high temperature reservoirs, at $r=0.1$, $r=1$, $r=10$
respectively.
Figure 6: (Color online)Comparing the non-Markovian entanglement dynamics with
the Markovian one by the time evolution of $\mathcal{C}_{\rho}(t)$ as a
function of “$k_{B}T$” for initial state $a=0$ and $r=0.1$.
In this section, we use the formalism of the preceding section to determine
the disentanglement. As an example, let us consider an important class of
mixed states with a single parameter $a$ like the following [27, 39, 40]
$\hat{\rho}_{YE}=\frac{1}{3}\left(\begin{array}[]{cccc}a&0&0&0\\\ 0&1&1&0\\\
0&1&1&0\\\ 0&0&0&1-a\end{array}\right).$ (28)
Apparently, the concurrence of $\rho_{YE}$ is
$\mathcal{C}_{\rho}(t)=\max\\{0,K(t)\\}$, and
$K(t)=|\rho_{23}(t)|-\sqrt{\rho_{11}(t)\rho_{44}(t)}$. Initially,
$\mathcal{C}(\rho(0))=\frac{2}{3}[1-\sqrt{a(1-a)}]$. In our simulations,
$\omega_{0}=1$ are chosen as the norm unit, and we regard the temperature as a
key factor in disentanglement process, for high temperature
$k_{B}T=300\omega_{0}$, intermediate temperature $k_{B}T=3\omega_{0}$, and low
temperature $k_{B}T=0.03\omega_{0}$, respectively. Another reservoir parameter
playing a key role in the dynamics of the system is the ratio
$r=\omega_{c}/\omega_{0}$ between the reservoir cutoff frequency $\omega_{c}$
and the system oscillator frequency $\omega_{0}$. As we will see in this
section, by varying these two parameters $k_{B}T$ and
$r=\omega_{c}/\omega_{0}$, the time evolution of the open system varies
prominently from Markovian to non-Markovian.
In Fig.2, the time evolutions of the non-Markovian concurrence for various
values of the parameter $a$ in low temperature is plotted. From Fig.2 we can
see that the entanglement dynamic relies on the different values of
$r=\omega_{c}/\omega_{0}$. If the spectrum of the reservoir does not
completely overlap with the frequency of the system oscillator, $r\ll 1$, we
can see from Fig.2 that the ESD time is considerably long. As increases the
ratio $r$, the ESD time becomes shorter and shorter. With different initial
state we can see that the concurrence varies prominently. When the initial
state $a=0$, the non-Markovian entanglement decay slowly, as increasing $a$,
the entanglement decay intensely, which means that we can prepare certain
initial entanglement states and use this fact to control the system
environment in order to prolong the entanglement time.
Fig.3 is the medium temperature case. Like Fig.2, under different systems,
different entanglement initial states, corresponding to different values of
$a$, and different $r$, some decay faster, some slower. But there are some
fundamental difference between Fig.2 and Fig.3. In Section III, we get the
concurrence in the long time limit, and we affirmed that when temperature
$k_{B}T=0$, the dynamics of non-markovian entanglement concurrence
$\mathcal{C}$ is fundamentally different from the case of $k_{B}T>0$. As we
can see from Fig.3, for “$\rho_{YE}$” states, as soon as the temperature
larger than zero, the concurrence always disappear at finite time and there
were no long-lived entanglement for any value of $a$, which means that ESD
must happen. The theoretical proof is $K(t\rightarrow\infty)<0$. But when
$k_{B}T=0$, the stationary value of $K(t\rightarrow\infty)$ equals zero. So,
whether or not and when the ESD will happen are not sure in $k_{B}T=0$. In
Fig.4 we give a numerical analysis of entanglement dynamic with different
initial states and find that there exists a $\xi\in(0,1)$, for almost all
values $a>\xi$, the concurrence is completely vanished at a finite time, which
is the effect of ESD. However, for $0\leq a\leq\xi$, the entanglement of this
state decays exponentially. But when $t\rightarrow\infty$, for all initial
state, i.e. $a\in[0,1]$ the concurrence will tend to be $0$.
Fig.5 is the high temperature case. One of the remarkable phenomenon in this
figure is that the ESD time is short. In typical experimental conditions,
quantum dots are subjected to an external magnetic field $B\sim 1-10T$ [46],
the ESD time $t_{ESD}\sim(3\times 10^{-1}-3)/k_{B}T$. Another obvious
phenomenon is in high temperature the Markovian quantum system decays
exponentially and vanish only asymptotically, but in the non-Markovian system
the concurrence $C_{\rho}(t)$ oscillates, which is evidently different from
the Markovian. In this case the non-Markovian property becomes evidently. This
oscillatory phenomenon is induced by the memory effects, which allows the two
qubit entanglement to reappear after a dark period of time. This phenomenon of
revival of entanglement after finite periods of “entanglement death” appears
to be linked to the environment single qubit non-Markovian dynamics, in
particular, the $\Delta(t)<0$ at some times in some environment [31]. The
physical conditions examined here are, moreover, more similar to those
typically considered in quantum computation, where qubits are taken to be
independent and where qubits interact with non-Markovian environments typical
of solid state microdevices [41].
As we indicated above, temperature is one of the key factor in the
entanglement dynamic. Figs. 2, 3, 4, 5 are plotted in the chosen temperature,
while in Fig. 6 $k_{B}T$ ranges from $0$ to $100$. In Fig. 6 the concurrence
vs “temperature $k_{B}T$” vs $\omega_{0}t$ in $r=0.1$, and the initial state
is the “$X_{YE}$” state with $a=0$. From Fig. 6 we can compare the non-
Markovian entanglement dynamics with the Markovian one clearly. The left is
the non-Markovian one from which we can see the oscillation of the
concurrence. Moreover, at the $0$ temperature the non-Markovian effect is
faint, as the temperature rises, the non-Markovian becomes more and more
obvious, while the Markovian one decays exponentially. This phenomenon
embodies the non-Markovian effect, which is evidently different from the
Markovian property. Maniscalco S _et. al_ studied the separability function
$S(\tau)$ in [30], where entanglement oscillation appears for twin-beam state
in non-Markovian channels in high temperature reservoirs. Both of them have
the same phenomenon. Ref. [31] gave a distribution curve when
$\Delta(r,t)-\gamma(r,t)>0$ and $\Delta(r,t)-\gamma(r,t)<0$. We convince that
due to the non-Markovian memory effect, particularly $\Delta(t)<0$ in Eqs
(20), the entanglement concurrence oscillates. With $\Delta(t)-\gamma(t)>0$
the concurrence descended whilst $\Delta(t)-\gamma(t)<0$ the concurrence
ascended, which guide us to adjust the temperature to control the entanglement
evolution. In order to show this and motivate the related research we design
the open loop controller
$k_{B}T=e^{-\alpha|\Delta(t)-\gamma(t)|}k_{B}T_{0}$ (29)
where $\alpha$ is the modulation, and $k_{B}T_{0}$ is the initial temperature.
In Fig. 7, we plot the controlled entanglement evolution, where the initial
temperature is chose as $k_{B}T_{0}=30$, which oscillates and ESD occurs at
$t\approx 19$. According to Fig. 1, $\gamma(t)$ can be neglected. For
different modulation $\alpha$, different controlled entanglement evolution is
plotted, and the ESD time can be prolonged for considerable long time.
Figure 7: (Color online)Controlled entanglement evolution with different
modulation $\alpha=3$ (green dotted line), $\alpha=2$ (red dashed line),
$\alpha=1$ (blue dashed-dotted line), and initially evolution (black solid
line), respectively.
## 5 Conclusions
In this paper we have presented a procedure that allows to obtain the dynamic
of a system consisting of two identical independent qubits, each of them
locally interacting with a bosonic reservoir. A non-Markovian master equation
between two qubit systems in the same environment was obtained. We
characterize our entanglement by the temperature and the ratio $r$ between
$\omega_{0}$ the characteristic frequency of the quantum system of interest,
and $\omega_{c}$ the cut-off frequency of Ohmic reservoir. For a broad class
of initially entangled states, “X” states, by useing Wootters’ concurrence, we
analyze the long time limit phenomenon of the entanglement dynamic. We find
that the dynamic of non-markovian entanglement concurrence
$\mathcal{C}_{\rho}(t)$ at temperature $k_{B}T=0$ is fundamentally different
from $k_{B}T>0$. When $k_{B}T=0$, from our numerical analysis, we find that
“entanglement sudden death” (ESD) occurs depending on the initial state, but
if $k_{B}T>0$ the concurrence always disappear at finite time, which means
that ESD must happen. In the $k_{B}T=0$ case, we find that there exist a
$\xi\in(0,1)$, for all values $a>\xi$, the concurrence is completely vanished
in a finite time, which is the effect of ESD. However, for $0\leq a\leq\xi$,
the entanglement of this state decays exponentially. But when
$t\rightarrow\infty$, for all initial state, i.e. $a\in[0,1]$ the concurrence
will tend to be $0$. From our numerical analysis we also find that the
entanglement dynamic relies on the different values of
$r=\omega_{c}/\omega_{0}$. If $r\ll 1$, the ESD time is considerably long. As
increases the ratio $r$, the ESD time becomes shorter and shorter. Moreover,
when the initial state $a=0$, the non-Markovian entanglement decays slowly, as
increases $a$, the entanglement decays intensely. Most of all, we have shown
that the non-Markovian dynamics of entanglement, described by concurrence,
presents oscillation even revivals after entanglement disappearance, typically
for high temperature non-Markovian system. At last, we design the open loop
controller which adjust the temperature to control the entanglement and
prolong the ESD time.
## Acknowledgments
This work was supported by the National Natural Science Foundation of China
(No. 60774099, No. 60221301), the Chinese Academy of Sciences (KJCX3-SYW-S01),
and by the CAS Special Grant for Postgraduate Research, Innovation and
Practice.
## References
## References
* [1] Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
* [2] Bennett C H and DiVincenzo D P 2000 Nature(London) 404 247
* [3] Bouwmeester D, Pan J W, Weinfurter M, and Zeilinger A 1997 Nature(London) 390 575
* [4] Einstein A, Podolsky B and Rosen R 1935 Phys. Rev. 47 777
* [5] Bell J S 1964 Physics 1 195
* [6] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A and Wootters W K 1993 Phys. Rev. Lett. 70 1895
* [7] Breuer H P and Petruccione F 2002 The Theory of Open Quantum Systems (Oxford: Oxford University Press)
* [8] Weiss U 1999 Quantum Dissipative Systems (Second Edition) (Singapore: World Scientific Publishing)
* [9] Yönaç M, Yu T and Eberly J H 2007 J. Phys. B: At. Mol. Opt. Phys. 40 S45
* [10] Cui W, Xi Z R and Pan Y 2009 J. Phys. A: Math. Theor. 42 025303
* [11] Zhang J, Wu R B, Li C W and Tarn T J 2009 J. Phys. A: Math. Theor. 42 035304
* [12] Cui H P, Zou J, Li J G and Shao B 2007 J. Phys. B: At. Mol. Opt. Phys. 40 S143
* [13] Everett H 1957 Rev. Mod. Phys. 29 454-462
* [14] Zurek W H 1991 Physics Today 44 36-44
* [15] Gordon G, Erez N and Akulin V M 2007 J. Phys. B: At. Mol. Opt. Phys. 40 S95
* [16] Grace M, Brif C, Rabitz H, Walmsley I A, Kosut R L and Lidar D A 2007 J. Phys. B: At. Mol. Opt. Phys. 40 S103
* [17] Han H and Brumer P 2007 J. Phys. B: At. Mol. Opt. Phys. 40 S209
* [18] Cui W, Xi Z R and Pan Y 2008 Phys. Rev. A 77 032117
* [19] Zhang J, Li C W, Wu R B, Tarn T J, and Liu X S 2005J. Phys. A: Math. Gen. 38 6587
* [20] Rodriguez C A and Sudarshan E C G 2008 arXiv: 0803.1183v2
* [21] Zurek W H 2003 Rev. Mod. Phys. 75 715
* [22] Garraway B M 1997 Phys. Rev. A 55 2290
* [23] Chirolli L and Burkard G 2008 Advances in Physics 57 225
* [24] Hu B L, Paz J P and Zhang Y H 1992 Phys. Rev. D 45 2843
* [25] Prager T, Falcke M, Schimansky-Geier L and Zaks M A 2007 Phys. Rev. E 76 011118
* [26] Lorenz V O and Cundiff S T 2005 Phys. Rev. Lett 95 163601
* [27] Cao X and Zheng H 2008 Phys. Rev. A 77 022320
* [28] Bellomo B, Lo Franco R and Compagno G 2007 Phys. Rev. Lett. 99 160502
* [29] Liu K L and Goan H S 2007 Phys. Rev. A 76 022312
* [30] Maniscalco S, Olivares S and Paris M G A 2007 Phys Rev. A 75 062119
* [31] Maniscalco S, Piilo J, Intravaia F, Petruccione F and Messina A 2004 Phys. Rev. A 70 032113
* [32] Zhang J, Wu R B, Li C W, Tarn T J, and Wu J W 2007 Phys. Rev. A 75 022324
* [33] Intravaia F, Maniscalco S and Messina A 2003 Eur. Phys. J. B 32 97
* [34] Wootters W K 1998 Phys. Rev. Lett. 80 2245
* [35] E Hagley et al. 1997 Phys Rev. Lett. 79 1
* [36] Bose S, Fuentes-Guridi I, Knight P L and Vedral V 2001 Phys Rev. Lett. 87 050401
* [37] J S Pratt 2004 Phys Rev. Lett. 93 237205
* [38] Yu T and Eberly J H 2007 Quantum information and Computation 7 459
* [39] Yu T and Eberly J H 2004 Phys. Rev. Lett. 93 140404
* [40] Al-Qasimi A and James D F V 2008 Phys. Rev. A 77 012117
* [41] de Vega I, Alonso D and Gaspard P 2005 Phys. Rev. A 71 023812
* [42] Shresta S, Anastopoulos C, Dragulescu A and Hu B L 2005 Phys. Rev. A 71 022109
* [43] Yu T, Diósi L, Gisin N and Strunz W T 1999 Phys. Rev. A 60 91
* [44] Strunz W T and Yu T 2004 Phys. Rev. A 69 052115
* [45] Yu T 2004 Phys. Rev. A 69 062107
* [46] Hanson R et al. 2003 Phys. Rev. Lett. 91, 196802
|
arxiv-papers
| 2009-10-30T05:43:37 |
2024-09-04T02:49:06.168344
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Cui, Zairong Xi and Yu Pan",
"submitter": "Wei Cui",
"url": "https://arxiv.org/abs/0910.5793"
}
|
0910.5797
|
# Observing photonic de Broglie waves without the NOON state
Osung Kwon Department of Physics, Pohang University of Science and Technology
(POSTECH), Pohang, 790-784, Korea Young-Sik Ra Department of Physics, Pohang
University of Science and Technology (POSTECH), Pohang, 790-784, Korea Yoon-
Ho Kim yoonho72@gmail.com Department of Physics, Pohang University of Science
and Technology (POSTECH), Pohang, 790-784, Korea
###### Abstract
The photonic de Broglie wave, in which an ensemble of $N$ identical photons
with wavelength $\lambda$ reveals $\lambda/N$ interference fringes, has been
known to be a unique feature exhibited by the photon number-path entangled
state or the NOON state. Here, we report the observation of the photonic de
Broglie wave for a pair of photons, generated by spontaneous parametric down-
conversion, that are not photon number-path entangled. We also show that the
photonic de Broglie wave can even be observed for a pair of photons that are
completely separable (i.e., no entanglement in all degrees of freedom) and
distinguishable. The experimental and theoretical results suggest that the
photonic de Broglie wave is, in fact, not related to the entanglement of the
photons, rather it is related to the indistinguishable pathways established by
the measurement scheme.
###### pacs:
42.50.Dv, 42.50.-p, 42.65.Lm, 42.50.Ex
## I Introduction
The nature of multipartite quantum entanglement is often manifested in quantum
interference experiments. For example, in the case of entangled photon states
generated by spontaneous parametric down-conversion (SPDC), quantum
interference is observed in coincidence counts between two detectors, each
individually exhibiting no interference fringes rarity ; ou ; brendel .
One notable example of photonic quantum interference is the photonic de
Broglie wave in which an ensemble of $N$ identical photons with wavelength
$\lambda$ exhibits $\lambda/N$ interference fringes jacob . The photonic de
Broglie wavelength $\lambda/N$ can be observed at the $N$-photon detector
placed at an output port of an interferometer if the beam splitters that make
up the interferometer do not randomly split $N$ photons. The quantum state of
the photons in the interferometer is then the photon number-path entangled
state or the NOON state
$|\psi\rangle=(|N\rangle_{1}|0\rangle_{2}+|0\rangle_{1}|N\rangle_{2})/\sqrt{2},$
(1)
where the subscripts refer to the two interferometric paths. For this reason,
the photonic de Broglie wave has been considered to be a unique feature
exhibited by the NOON state and essential for quantum imaging and quantum
metrology boto ; dangelo ; kapale . Experimentally, photonic de Broglie waves
up to $N=4$ have been observed with corresponding NOON states fon ; eda ;
walther ; mitchell ; shige .
Note, however, that $\lambda/N$ modulations in the coincidence rate among $N$
detectors may not necessarily be of quantum origin. For instance, $\lambda/N$
modulation in coincidences among $N$ detectors reported in Ref. resch , with
each detector placed at an output port of a multi-path interferometer, is a
classical effect since the coincidence modulation is a direct result of
modulations (with different phases) observed at individual detectors. Also,
classical thermal light may exhibit sub-wavelength interference fringes in
coincidences but at the reduced visibility consistent with classical states
ferri ; xiong . Thus, the reduced-period fringe itself need not be of quantum
origin. It is, however, important to point out that $N$-th order quantum
interference, such as quantum optical $\lambda/N$ modulations due to the
photonic de Broglie wave, must exhibit high visibility (up to 100% in
principle) in the absence of any lower-order interference.
In this paper, we report an intriguing new observation of $\lambda/N$ ($N=2$)
photonic de Broglie wave interference that has no classical interpretation and
is not associated with the NOON state. We also show theoretically that
photonic de Broglie waves can even be observed for a pair of photons that are
completely separable (i.e., no entanglement in all degrees of freedom) and
distinguishable. The experimental and theoretical results suggest that the
photonic de Broglie wave is, in fact, not related to the entanglement of the
photons, rather it reflects the characteristics (i.e., the indistinguishable
pathways) of the measurement scheme.
## II Experimental observation of photonic de Broglie waves without the NOON
state
Consider the experimental setup shown in Fig. 1. A 405 nm blue diode laser,
with the full width at half maximum (FWHM) bandwidth of 0.67 nm, pumps a 3 mm
thick type-I BBO crystal to generate, via the SPDC process, a pair of energy-
time entangled photons centered at $\lambda=810$ nm. The photon pair is
coupled into the single-mode optical fiber after passing through the
interference filter with a FWHM bandwidth of 5 nm. For optimal coupling, the
pump was focused at the BBO and the focal spot was imaged to the single-mode
fiber kwon .
Figure 1: Schematic of the experiment. BS1, BS2, and BS3 are 50:50 beam
splitters. FPC is the fiber polarization controller and CC is a coincidence
counter.
The photon pair is then sent to a Mach-Zehnder interferometer (MZI), formed
with BS1 and BS2, via the different input ports $a$ and $b$. The input delay
$x_{1}$ between the photons is controlled by axially moving the output
collimator of a fiber and the interferometer path length difference $x_{2}$ is
controlled by translating one of the trombone prisms P2. A two-photon
detector, consisting of BS3 and two single-photon detectors D3 and D4, is
placed at the output mode $e$ of MZI for photonic de Broglie wave measurement
eda . Two auxiliary detectors D1 and D2 are used to adjust the input delay
$x_{1}$ by observing the Hong-Ou-Mandel (HOM) interference hong .
First, we consider the well-known photonic de Broglie wave for a biphoton NOON
state and this requires preparing the state
$|\psi\rangle=(|2\rangle_{c}|0\rangle_{d}+|0\rangle_{c}|2\rangle_{d})/\sqrt{2}$
in the MZI jacob ; eda . This can be accomplished by using HOM interference:
the photon pair arrives at BS1 (or enters the MZI) simultaneously via the
different input ports $a$ and $b$. The high-visibility HOM interference,
measured in coincidence counts between D1 and D2 as a function of $x_{1}$,
reported in Fig. 2 indicates that when the input delay is zero, i.e.,
$x_{1}=0$, the quantum state of the photons in the interferometer is indeed
the desired biphoton NOON state.
Observation of the photonic de Broglie wave for the biphoton NOON state
requires i) interfering the biphoton amplitudes $|2\rangle_{c}|0\rangle_{d}$
and $|0\rangle_{c}|2\rangle_{d}$ and ii) making a proper two-photon detection.
In the experiment, we set $x_{1}=0$ with the help of the HOM dip in Fig. 2 and
the photonic de Broglie wave corresponding to the biphoton NOON state was
observed at the two-photon detector placed at the output mode $e$ of the MZI.
The result shown in Fig. 3(a) exhibits $\lambda/2$ interference fringes as a
function of the MZI path length difference $x_{2}$.
We note that the coincidence between single photon detectors placed at modes
$e$ and $f$ also exhibits the interference fringes with the period $\lambda/2$
rarity ; ou . This $\lambda/2$ interference fringe, however, is, not related
to the photonic de Broglie wave since i) the photons are split at BS2 and ii)
it may be observed with the classical coherent state, e.g.,
$|0\rangle_{a}|\alpha\rangle_{b}$, at the input of the MZI eda ; resch .
Figure 2: The Hong-Ou-Mandel dip observed with D1 and D2. The dip visibility
is better than 98%. The arrows represent the $x_{1}$ positions at which the
biphoton interference measurements were performed with the two-photon detector
(i.e., coincidences between detectors D3 and D4).
Consider now the situation in which the photons do not enter the MZI
simultaneously. In this case, since the photons do not arrive at BS1 at the
same time, HOM interference does not occur and the quantum state of the
photons in the MZI is no longer the biphoton NOON state. The question we ask
is whether the $\lambda/2$ photonic de Broglie wave would still be observed at
the two-photon detector in mode $e$ (i.e., coincidences between detectors D3
and D4) in this case.
To probe this question, we intentionally add more time delays in mode $a$ of
the MZI so that $x_{1}\neq 0$. The arrows in Fig. 2 indicate the $x_{1}$
positions at which the biphoton interference measurements are made with the
two-photon detector in mode $e$. First, we set $x_{1}=62$ $\mu$m and $x_{2}$
is scanned for the two-photon interference measurement. At this $x_{1}$
location, there is still some Hong-Ou-Mandel interference as evidenced in Fig.
2 (i.e., the coincidence rate is still below the random coincidence rate). The
biphoton interference measured with the two-photon detector in this condition
is shown in Fig. 3(b). Interestingly, the observed interference fringes
exhibit the same $\lambda/2$ modulation with no reduction in visibility. It is
intriguing to find that the same high-visibility interference fringes with
$\lambda/2$ modulations are observed even when $x_{1}$ is completely out of
the Hong-Ou-Mandel dip region. In Fig. 3(c) and Fig. 3(d), we show the
biphoton interference observed with the two-photon detector at $x_{1}=2.8$ mm
and at $x_{1}=5.7$ mm, respectively. These data correspond to the $x_{1}$
positions marked with the arrows shown in the inset of Fig. 2.
So far, we have established experimentally that the photonic de Broglie wave
can indeed be observed without the NOON state. (Note that, differently from
Ref. resch , this is a real second-order quantum effect in the absence of any
first-order interference: the detectors D3 and D4 individually do not show any
modulations.) We now ask whether the shapes of the photonic de Broglie wave
packets would remain the same. This question is probed by measuring the the
photonic de Broglie wave packets for several different $x_{1}$ values and the
results of these measurements are shown in Fig. 4 kwon2 .
Figure 3: Biphoton interference observed at four different $x_{1}$ positions.
(a) $x_{1}=0$ $\mu$m, (b) $x_{1}=62$ $\mu$m, (c) $x_{1}=2.8$ mm. (a)
$x_{1}=5.7$ mm. The solid lines are fit to the data with the modulation
wavelength and the visibility fixed at $\lambda/2=405$ nm and $98$%,
respectively.
In Fig. 4(a), we show the typical symmetric Gaussian de Broglie wave packet
for the biphoton NOON state generated by setting $x_{1}=0\mu$m. This case
corresponds to Fig. 3(a). For non-NOON states (i.e., for $x_{1}\neq 0$), it is
found that the photonic de Broglie wave packet is modified dramatically. The
wave packet starts to become highly asymmetric (with respect to the random
coincidence rate) as soon as $x_{1}\neq 0$, see Fig. 4(b). The wave packet
envelope then takes the shape of a double-hump and a single-dip for a larger
value of $x_{1}$, see Fig. 4(c). Eventually, for a sufficiently large $x_{1}$,
small side peaks starts to appear at $x_{2}=\pm x_{1}$, see Fig. 4(d). Even
for very large values of $x_{1}$, e.g., corresponding to the positions marked
with arrows in the inset of Fig. 2, the wave packet shape remains essentially
the same as in Fig. 4(d) but the two side peaks get relocated to their new
positions, $x_{2}=\pm x_{1}$ kwon2 .
## III Photonic de Broglie wave without the NOON state
### III.1 Theoretical description
To explain the observed phenomena theoretically, we start by writing the
monochromatic laser pumped SPDC two-photon state as baek08
$|\psi\rangle_{\textrm{e}}=\int
d\omega_{s}d\omega_{i}\,\delta(\Delta_{\omega})\mathrm{sinc}(\Delta_{k}L/2)e^{i\Delta_{k}L/2}|\omega_{s},\omega_{i}\rangle,$
(2)
where the subscripts $i$, $s$, and $p$ refer to the idler, the signal, and the
pump photon, respectively. The thickness of the SPDC crystal is $L$,
$\Delta_{\omega}=\omega_{p}-\omega_{s}-\omega_{i}$, and
$\Delta_{k}=k_{p}-k_{s}-k_{i}$. Since the pump is a cw diode laser with a
rather large FWHM bandwidth, the SPDC quantum state with cw diode laser pump
should more properly be written as kwon2
$\rho=\int
d\omega_{p}\,\mathcal{S}\left(\omega_{p}\right)|\psi\rangle_{\textrm{e}}{}_{\textrm{e}}\langle\psi|,$
(3)
where the spectral power density of the pump is assumed to be Gaussian
$\mathcal{S}\left(\omega_{p}\right)\equiv\exp\left(-{(\omega_{p}-\omega_{p0})^{2}}/{2{\Delta\omega_{p}}^{2}}\right)/{\Delta\omega_{p}\sqrt{2\pi}},$
(4)
such that $\int\mathcal{S}(\omega_{p})d\omega_{p}=1$.
Figure 4: The biphoton wave packet measurements with varying input delays at
BS1. (a) $x_{1}=0$ $\mu$m, (b) $x_{1}=100$ $\mu$m, (c) $x_{1}=200$ $\mu$m, (d)
$x_{1}=500$ $\mu$m. Within the wave packets, the modulation period is
$\lambda/2$ and the visibility around $x_{2}=0$ is better than 98%. The solid
lines are the wave packet envelopes calculated using Eq. (15).
The HOM interference can be calculated by evaluating
$R_{12}=\int
dtdt^{\prime}\,tr[\rho\,E_{c}^{(-)}(t)E_{d}^{(-)}(t^{\prime})E_{d}^{(+)}(t^{\prime})E_{c}^{(+)}(t)],$
(5)
where
$\displaystyle E_{c}^{(+)}(t)$ $\displaystyle=$
$\displaystyle(iE_{a}^{(+)}(t-\tau_{1})+E_{b}^{(+)}(t))/\sqrt{2},$ (6)
$\displaystyle E_{d}^{(+)}(t)$ $\displaystyle=$
$\displaystyle(E_{a}^{(+)}(t-\tau_{1})+iE_{b}^{(+)}(t))/\sqrt{2},$ (7)
and $\tau_{1}=x_{1}/c$. The positive frequency component of the electric field
in mode $a$ is given as
$E_{a}^{(+)}(t)=\int d\omega\,a(\omega)\phi(\omega)e^{-i\omega t},$ (8)
where $a(\omega)$ is the annihilation operator for the signal photon in mode
$a$ and $E_{b}^{(+)}(t)$ for the idler photon in mode $b$ is similarly
defined. The filter transmission is assumed Gaussian
$\phi(\omega)=\exp\left(-{(\omega-\omega_{0})^{2}}/{2{\Delta\omega}^{2}}\right)/\sqrt{\Delta\omega\sqrt{\pi}},$
(9)
and $\int|\phi(\omega)|^{2}d\omega=1$. Since the natural bandwidth of SPDC,
$\textrm{sinc}(\Delta_{k}L/2)$, is much broader than the spectral filter
bandwidth $\Delta\omega$, Eq. (5) is calculated to be
$R_{12}=1-\exp(-\Delta\omega^{2}\tau_{1}^{2}/2).$ (10)
The solid line in Fig. 2 is plotted using Eq. (10) with measured spectral
filter bandwidth $\Delta\omega$.
For the photonic de Broglie wave measurement, the response of the two-photon
detector in mode $e$ must be considered and it is given as
$R_{ee}=\int
dtdt^{\prime}\,tr[\rho\,E_{e}^{(-)}(t)E_{e}^{(-)}(t^{\prime})E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)],$
(11)
where
$E_{e}^{(+)}(t)=(iE_{c}^{(+)}(t)+E_{d}^{(+)}(t-\tau_{2}))/\sqrt{2}$ (12)
and $\tau_{2}=x_{2}/c$. Equation (11) can then be re-written as
$R_{ee}=\int d\omega_{p}\mathcal{S}(\omega_{p})\int
dtdt^{\prime}\,\left|\langle
0|E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)|\psi\rangle_{\textrm{e}}\right|^{2},$
(13)
where $\langle 0|$ denotes the vacuum state. The biphoton amplitude $\langle
0|E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)|\psi\rangle_{\textrm{e}}$ contains
important information about the quantum interference and, when expanded using
the electric field operators at input modes $a$ and $b$, is calculated to be
$\begin{array}[]{ll}\langle
0|E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)|\psi\rangle_{\textrm{e}}=\\\
\frac{i}{4}\langle
0|\left[\begin{array}[]{l}E_{a}^{(+)}(t-\tau_{1}-\tau_{2})E_{b}^{(+)}(t^{\prime}-\tau_{2})+E_{a}^{(+)}(t^{\prime}-\tau_{1}-\tau_{2})E_{b}^{(+)}(t-\tau_{2})\\\
-E_{a}^{(+)}(t-\tau_{1})E_{b}^{(+)}(t^{\prime})-E_{a}^{(+)}(t^{\prime}-\tau_{1})E_{b}^{(+)}(t)\\\
-E_{a}^{(+)}(t-\tau_{1})E_{b}^{(+)}(t^{\prime}-\tau_{2})-E_{a}^{(+)}(t^{\prime}-\tau_{1})E_{b}^{(+)}(t-\tau_{2})\\\
+E_{a}^{(+)}(t-\tau_{1}-\tau_{2})E_{b}^{(+)}(t^{\prime})+E_{a}^{(+)}(t^{\prime}-\tau_{1}-\tau_{2})E_{b}^{(+)}(t)\\\
\end{array}\right]|\psi\rangle_{\textrm{e}}.\end{array}\\\ $ (14)
Note that only non-zero biphoton amplitudes are written in the above equation:
terms that contain $E_{a}^{(+)}E_{a}^{(+)}$ and $E_{b}^{(+)}E_{b}^{(+)}$ are
eventually calculated to be zero because of the nature of the input state
$|\psi\rangle_{\textrm{e}}$.
If we now consider the two-photon detector shown in Fig. 1, the normalized
coincidence rate between D3 and D4 corresponds to $R_{ee}$ and is given as
$\displaystyle R_{34}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\\{4+\exp(-(\tau_{1}-\tau_{2})^{2}\Delta\omega^{2}/2)$
(15) $\displaystyle+$
$\displaystyle\exp(-(\tau_{1}+\tau_{2})^{2}\Delta\omega^{2}/2)-2\exp(-\tau_{2}^{2}\Delta\omega^{2}/2)$
$\displaystyle-$ $\displaystyle
2\cos{(2\omega_{0}\tau_{2})}\exp(-\tau_{2}^{2}\Delta\omega_{e}^{2}/2)$
$\displaystyle\times(1+\exp(-\tau_{1}^{2}\Delta\omega^{2}/2))\\},$
where $1/{\Delta\omega_{e}^{2}}\equiv
1/\Delta\omega_{p}^{2}+1/\Delta\omega^{2}$.
Equation (15) clearly shows that the $2\omega_{0}$ or $\lambda_{0}/2$
interference fringe (corresponding to the photonic de Broglie wavelength), in
fact, is not related to the biphoton NOON state condition $\tau_{1}=0$. As
long as $\tau_{2}$ is within the effective coherence length
$\sqrt{2}/\Delta\omega_{e}$, the biphoton photonic de Broglie wave
interference can be observed regardless of the $\tau_{1}$ value.
Another interesting feature of Eq. (15) is that the shape of the biphoton de
Broglie wave packet is $\tau_{1}$ dependent while the period of interference
fringes remains the same at $2\omega_{0}$. Note also that the maximum
interference visibility is not affected by $\tau_{1}$. The theoretical result
in Eq. (15) is found to be in excellent agreement with the experimental data
in Fig. 3.
### III.2 The Feynman diagram
The interesting features of the biphoton de Broglie interference in this
experiment can be intuitively understood by analyzing the Feynman diagrams
representing the two-photon detection amplitudes.
Given the experimental setup in Fig. 1, there exist four Feynman paths in
which the photon pair exits BS2 via the output mode $e$ and these Feynman
paths are shown in Fig. 5. Since the signal, $\omega_{s}$, and idler,
$\omega_{i}$, photons must always transmit/reflect or reflect/transmit at BS3
to contribute to a final two-photon detection event, each Feynman path shown
in Fig. 5 branches off into two final Feynman amplitudes. There are, thus,
total of eight Feynman paths which lead to a detection event at the two-photon
detector in Fig. 1.
The photonic de Broglie wavelength observed in Fig. 3 is a manifestation of
quantum interference among these Feynman paths. For arbitrary $\tau_{1}$ and
$\tau_{2}$, the Feynman paths shown in Fig. 5 are clearly distinguishable (in
time). However, if $\tau_{2}=0$, all Feynman paths become indistinguishable,
regardless of $\tau_{1}$ values. This is confirmed theoretically in Eq. (15)
and experimentally in Fig. 3: high-visibility $2\omega_{0}$ or $\lambda/2$
interference fringes are observed when $\tau_{2}$ is scanned around
$\tau_{2}=0$.
In addition, it is shown in Fig. 4 that the shape of the biphoton wave packet
is dependent on the $\tau_{1}$ value. In the case that $\tau_{1}=0$, the third
and fourth Feynman paths in Fig. 5 cancel out and the wave packet envelope is
determined by the overlap between the first two Feynman paths. As shown in
Fig. 4(a), the result is a Gaussian wave packet whose width is determined by
$\Delta\omega_{e}$.
Figure 5: The Feynman paths for the photon pair. All the Feynman paths become
indistinguishable when $\tau_{2}=0$, regardless of $\tau_{1}$ values. Note
that each line (top to bottom) in Eq. (14) corresponds to each Feynman path
(left to right).
Consider now the case of $\tau_{2}=\tau_{1}$. The signal and idler photons
arrive simultaneously at BS2 for the third Feynman path in Fig. 5 and, because
of the Hong-Ou-Mandel effect, the two photons will always exit BS2 via the
same output port. If we now consider the case of $\tau_{2}=-\tau_{1}$, the
same situation occurs for the fourth Feynman path in Fig. 5. Therefore the
detection probability of the third and the fourth Feynman paths would increase
twice as big compared to $\tau_{2}\neq\pm\tau_{1}$. The net results are the
distinct side peaks observed at $\tau_{2}=\pm\tau_{1}$ in Fig. 4(d).
In general, i.e., $\tau_{1}\neq 0$, all the eight Feynman paths contribute to
quantum interference in a complex manner so an intuitive explanation becomes
difficult.
## IV Photonic de Broglie wave interference without entanglement
So far, we have shown experimentally and theoretically that the photonic de
Broglie wave is in fact not related to the photon number-path entangled or the
NOON state. The photonic de Broglie wave, instead, appears to be linked to the
underlying spectral entanglement of SPDC photons which are used for both
experimental observation and theoretical analysis baek08 . The question then
becomes whether the two input photons need to have any entanglement at all to
exhibit the photonic de Broglie wave phenomenon.
### IV.1 Photonic de Broglie wave interference for two identical photons with
no entanglement
Consider two single-photons with identical spectra and polarization, each
emitted from a separate single-photon source. It is known that HOM
interference can occur with a pair of identical single-photons san ; mos . The
biphoton NOON state resulting from HOM interference should then exhibit the
photonic de Broglie wave.
The relevant question therefore is what would happen when there is no HOM
interference between the two identical single-photons with no a priori
entanglement. Would the photonic de Broglie wave still be observed in the
absence of any entanglement between the photons?
To investigate this question, let us consider a single-photon in the pure
state at each input mode of the MZI in Fig. 1. Since the joint quantum state
of the two single-photons at the input modes of the MZI is separable, it can
be written as
$|\psi\rangle_{\textrm{s}}=\int d\omega_{a}\
\varphi(\omega_{a})|\omega_{a}\rangle\otimes\int d\omega_{b}\
\varphi(\omega_{b})|\omega_{b}\rangle,$ (16)
where the single-photon spectral amplitude is assumed to be Gaussian
$\varphi(\omega)=\exp\left(-{(\omega-\omega_{0})^{2}}/{2{\Delta\omega}^{2}}\right)/\sqrt{\Delta\omega\sqrt{\pi}},$
(17)
and $\int|\varphi(\omega)|^{2}d\omega=1$.
Given the input quantum state as in Eq. (16), the response of the MZI can now
be studied. First, the single-photon detection rates at D3 and D4 are
calculated to be constant, completely independent of $x_{1}$ and $x_{2}$. This
is because the single-photon detection probabilities due to the single-photons
in mode $a$ and in mode $b$ have the same Gaussian envelopes but are out of
phase by $180^{\circ}$. In other words, similarly to the case of entangled-
photon pairs at the input, no first-order interference can be observed.
Second, the two-photon detection rates for the photonic de Broglie wave
measurement can be calculated by evaluating
$R_{ee}^{(s)}=\int
dtdt^{\prime}\,tr[\rho^{(s)}\,E_{e}^{(-)}(t)E_{e}^{(-)}(t^{\prime})E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)],$
(18)
where $\rho^{(s)}=|\psi\rangle_{\textrm{s}}{}_{\textrm{s}}\langle\psi|$. The
above equation can then be re-written as
$R_{ee}^{(s)}=\int dtdt^{\prime}\,\left|\langle
0|E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)|\psi\rangle_{\textrm{s}}\right|^{2}.$
(19)
The biphoton amplitude $\langle
0|E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)|\psi\rangle_{\textrm{s}}$ in Eq. (19)
is evaluated to be
$\begin{array}[]{ll}\langle
0|E_{e}^{(+)}(t^{\prime})E_{e}^{(+)}(t)|\psi\rangle_{\textrm{s}}=\\\
\frac{i}{4}\langle
0|\left[\begin{array}[]{l}E_{a}^{(+)}(t-\tau_{1}-\tau_{2})E_{b}^{(+)}(t^{\prime}-\tau_{2})+E_{a}^{(+)}(t^{\prime}-\tau_{1}-\tau_{2})E_{b}^{(+)}(t-\tau_{2})\\\
-E_{a}^{(+)}(t-\tau_{1})E_{b}^{(+)}(t^{\prime})-E_{a}^{(+)}(t^{\prime}-\tau_{1})E_{b}^{(+)}(t)\\\
-E_{a}^{(+)}(t-\tau_{1})E_{b}^{(+)}(t^{\prime}-\tau_{2})-E_{a}^{(+)}(t^{\prime}-\tau_{1})E_{b}^{(+)}(t-\tau_{2})\\\
+E_{a}^{(+)}(t-\tau_{1}-\tau_{2})E_{b}^{(+)}(t^{\prime})+E_{a}^{(+)}(t^{\prime}-\tau_{1}-\tau_{2})E_{b}^{(+)}(t)\\\
\end{array}\right]|\psi\rangle_{\textrm{s}}.\end{array}\\\ $ (20)
Finally, the normalized coincidence rate on D3 and D4 in Fig. 1 is
proportional to $R_{ee}^{(s)}$ and is given as
$\displaystyle R_{34}^{(s)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\\{4+\exp(-(\tau_{1}-\tau_{2})^{2}\Delta\omega^{2}/2)$
(21) $\displaystyle+$
$\displaystyle\exp(-(\tau_{1}+\tau_{2})^{2}\Delta\omega^{2}/2)-2\exp(-\tau_{2}^{2}\Delta\omega^{2}/2)$
$\displaystyle-$ $\displaystyle
2\cos{(2\omega_{0}\tau_{2})}\exp(-\tau_{2}^{2}\Delta\omega^{2}/2)$
$\displaystyle\times(1+\exp(-\tau_{1}^{2}\Delta\omega^{2}/2))\\}.$
It is interesting to note that the result in Eq. (21) is identical to Eq. (15)
but with $\Delta\omega_{e}$ replaced by $\Delta\omega$. Effectively, this
means that SPDC pumped with a very broadband pump laser would give the
identical result as that of two separable single-photon states. The
theoretical results summarized in Fig. 6 show that the separable two-photon
state of Eq. (16) at the input of the MZI gives nearly the same result as that
of SPDC photons pumped with a laser with 2 nm FWHM bandwidth for both the NOON
state ($x_{1}=0$ $\mu$m) and non-NOON state ($x_{1}\neq 0$ $\mu$m) conditions.
Figure 6: Calculated photonic de Broglie wave packets for SPDC photons, (a)
and (b); for two identical single-photons with no entanglement, (c) and (d);
and for two distinguishable (orthogonally polarized) single-photons with no
entanglement, (e) and (f). The plots (a) and (b), (c) and (d), and (e) and (f)
are due to the theoretical results in Eq. (15), Eq. (21), and Eq. (27),
respectively. For SPDC photons, the pump bandwidth $\Delta\omega_{p}$ is
assumed to be 2 nm FWHM and the signal and the idler photons are filtered with
5 nm FWHM filters. For single-photons, they are assumed to have FWHM bandwidth
of 5 nm. Note that, since Eq. (27) is $\tau_{1}$ independent, (e) and (f) are
identical plots with different ranges.
### IV.2 Photonic de Broglie wave interference for two distinguishable
(orthogonally polarized) photons with no entanglement
In the previous section, we have seen that entanglement is in fact not
necessary for observing the photonic de Broglie wave interference of two
photons. It was however assumed that the two input single-photons were
identical. In this section, we discuss the general case in which the two input
single-photons are orthogonally polarized so that they are completely
distinguishable. Note that the experimental schematic is kept the same as in
Fig. 1: no polarization-information erasing polarizers are added to the setup.
For two orthogonally polarized single-photons, the joint quantum state is
written as
$|\psi\rangle_{\textrm{dist}}=\int d\omega_{a}\
\phi(\omega_{a})|\omega_{a}^{H}\rangle\otimes\int d\omega_{b}\
\phi(\omega_{b})|\omega_{b}^{V}\rangle,$ (22)
where the superscripts $H$ and $V$ refer to horizontal and vertical
polarization states, respectively. The counting rate at the two-photon
detector, see Fig. 1, in the output mode $e$ of BS2 is then given as
$R_{ee}^{(dist)}=\sum_{p_{1},p_{2}\in\\{H,V\\}}\int
dtdt^{\prime}\,tr[\rho^{(dist)}\,E_{e}^{p_{1}(-)}(t)E_{e}^{p_{2}(-)}(t^{\prime})E_{e}^{p_{2}(+)}(t^{\prime})E_{e}^{p_{1}(+)}(t)],$
(23)
where superscripts $p_{1}$ and $p_{2}$ denote polarizations and
$\rho^{(dist)}=|\psi\rangle_{\textrm{dist}}\ {}_{\textrm{dist}}\langle\psi|$.
Equation (23) can then be re-written as
$\displaystyle R_{ee}^{(dist)}$ $\displaystyle=$ $\displaystyle\int
dtdt^{\prime}\,\sum_{p_{1},p_{2}\in\\{H,V\\}}\left|\langle
0|E_{e}^{p_{2}(+)}(t^{\prime})E_{e}^{p_{1}(+)}(t)|\psi\rangle_{\textrm{dist}}\right|^{2}$
$\displaystyle=$ $\displaystyle\int dtdt^{\prime}\left(\left|\langle
0|E_{e}^{H(+)}(t^{\prime})E_{e}^{V(+)}(t)|\psi\rangle_{\textrm{dist}}\right|^{2}+\left|\langle
0|E_{e}^{V(+)}(t^{\prime})E_{e}^{H(+)}(t)|\psi\rangle_{\textrm{dist}}\right|^{2}\right).$
Note that terms that include electric field operators
$E_{e}^{H(+)}E_{e}^{H(+)}$ and $E_{e}^{V(+)}E_{e}^{V(+)}$ are not shown
because they eventually are calculated to be zero since the input photons are
orthogonally polarized.
The biphoton amplitudes are then expanded as
$\begin{array}[]{ll}\langle
0|E_{e}^{H(+)}(t^{\prime})E_{e}^{V(+)}(t)|\psi\rangle_{\textrm{dist}}=\\\
\frac{i}{4}\langle
0|\left[\begin{array}[]{l}E_{a}^{H(+)}(t-\tau_{1}-\tau_{2})E_{b}^{V(+)}(t^{\prime}-\tau_{2})-E_{a}^{H(+)}(t-\tau_{1})E_{b}^{V(+)}(t^{\prime})\\\
-E_{a}^{H(+)}(t-\tau_{1})E_{b}^{V(+)}(t^{\prime}-\tau_{2})+E_{a}^{H(+)}(t-\tau_{1}-\tau_{2})E_{b}^{V(+)}(t^{\prime})\\\
\end{array}\right]|\psi\rangle_{\textrm{dist}},\end{array}\\\ $ (25)
and
$\begin{array}[]{ll}\langle
0|E_{e}^{V(+)}(t^{\prime})E_{e}^{H(+)}(t)|\psi\rangle_{\textrm{dist}}=\\\
\frac{i}{4}\langle
0|\left[\begin{array}[]{l}E_{a}^{H(+)}(t^{\prime}-\tau_{1}-\tau_{2})E_{b}^{V(+)}(t-\tau_{2})-E_{a}^{H(+)}(t^{\prime}-\tau_{1})E_{b}^{V(+)}(t)\\\
-E_{a}^{H(+)}(t^{\prime}-\tau_{1})E_{b}^{V(+)}(t-\tau_{2})+E_{a}^{H(+)}(t^{\prime}-\tau_{1}-\tau_{2})E_{b}^{V(+)}(t)\\\
\end{array}\right]|\psi\rangle_{\textrm{dist }}.\end{array}\\\ $ (26)
Finally, the normalized output of the two-photon detector (i.e., coincidence
between D3 and D4) is calculated to be,
$\displaystyle R_{34}^{(dist)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\\{4-2\exp(-\tau_{2}^{2}\Delta\omega^{2}/2)-2\cos{(2\omega_{0}\tau_{2})}\exp(-\tau_{2}^{2}\Delta\omega^{2}/2)\\}.$
(27)
It is interesting to note that Eq. (27) also shows $2\omega_{0}$ modulation as
in the case of two identical single-photons, Eq. (21), and as in the case of a
pair of SPDC photons, Eq. (15). This result, therefore, reveals that photonic
de Broglie wave interference is not only unrelated to the NOON state, but it
can also be observed with completely unentangled and distinguishable photons.
Note also that Eq. (27) is completely independent of $\tau_{1}$ and Eq. (27)
can actually be obtained from Eq. (21) by letting $\tau_{1}\rightarrow\infty$.
Equation (27) is plotted in Fig. 6(e) and Fig. 6(f). The plots show very
clearly that high-visibility photonic de Broglie wave interference appear for
two orthogonally polarized single-photons. Note, however, that the shape of
the wave packet in Fig. 6(e) is quite different from Fig. 6(a) and Fig. 6(c)
but rather similar to Fig. 6(b) and Fig. 6(d). This comes from the fact that
Eq. (27) is $\tau_{1}$ independent and the other two results converge toward
Eq. (27) as $\tau_{1}$ gets bigger. This fact is also reflected in the absence
of side peaks in Fig. 6(f).
## V Conclusion
It is interesting to discuss the connection between the photonic de Broglie
wave interference and entanglement between the input photons. The
monochromatic-pumped SPDC in Eq. (2) is strongly energy-time entangled and, as
the pump bandwidth is increased, the degree of energy-time entanglement is
reduced kim05 . The experimental and theoretical results on photonic de
Broglie wave interference for broadband-pumped SPDC shown in Fig. 4(d) and in
Fig. 6(b) make it clear that the quality of the photonic de Broglie wave
interference for non-NOON states is not affected by the reduced energy-time
entanglement between the photon pair. Furthermore, Fig. 6(d) and Fig. 6(f)
show that even two unentangled and distinguishable (orthogonally polarized)
single-photons lead to essentially the same photonic de Broglie wave
interference.
These results therefore reveal that entanglement between the two photons plays
essentially no role in the manifestation of the photonic de Broglie wave
interference. Rather, it is the measurement scheme (i.e., indistinguishable
pathways established by the measurement scheme) that brings out the photonic
de Broglie wave phenomenon kim05b .
The experimental and theoretical results in this paper apply to $N=2$ photonic
de Broglie wave interference. We, however, believe that it should be possible
to extend the conclusions to the $N$ photon case.
## Acknowledgements
This work was supported, in part, by the National Research Foundation of Korea
(KRF-2006-312-C00551, 2009-0070668, and 2009-0084473) and the Ministry of
Knowledge and Economy of Korea through the Ultrafast Quantum Beam Facility
Program.
## References
* (1) J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348-1351 (1990).
* (2) Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A42, 2957-2965 (1990).
* (3) J. Brendel, E. Mohler, and W. Martienssen, “Time-resolved dual-beam two-photon interference with high visibility,” Phys. Rev. Lett. 66, 1142-1145 (1991).
* (4) J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835-4838 (1995).
* (5) A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85, 2733-2736 (2000).
* (6) M. D’Angelo, M. V. Chekhova, and Y. H. Shih, “Two-photon diffraction and Quatum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
* (7) K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
* (8) E. J. S. Fonseca, C. H. Monken, and S. Padua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868-2871 (1999).
* (9) K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the Photonic de Broglie Wavelength of Entangled Photon Pairs Generated by Spontaneous Parametric Down-Conversion,” Phys. Rev. Lett. 89, 213601 (2002).
* (10) P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161 (2004).
* (11) M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
* (12) T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the Standard Quantum Limit with Four-Entangled Photons,” Science 316, 726-729 (2007).
* (13) K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98, 223601 (2007).
* (14) F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-Resolution Ghost Image and Ghost Diffraction Experiments with Thermal Light,” Phys. Rev. Lett. 94, 183602 (2005).
* (15) J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental Observation of Classical Subwavelength Interference with a Pseudothermal Light Source,” Phys. Rev. Lett. 94, 173601 (2005).
* (16) O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Single-mode coupling efficiencies of type-II spontaneous parametric down-conversion: Collinear, noncollinear, and beamlike phase matching,” Phys. Rev. A78, 053825 (2008).
* (17) C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).
* (18) O. Kwon, Y.-S. Ra, and Y.-H. Kim, “Coherence properties of spontaneous parametric down-conversion pumped by a multi-mode cw diode laser,” Opt. Express 17, 13059-13069 (2009).
* (19) S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A77, 043807 (2008).
* (20) C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (2002).
* (21) P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded Generation of Ultrafast Single Photons in Pure Quantum States,” Phys. Rev. Lett. 100, 133601 (2008).
* (22) Y.-H. Kim and W.P. Grice, “Measurement of the spectral properties of the two- photon state generated via type II spontaneous parametric downconversion,” Opt. Lett. 30, 908-910 (2005).
* (23) Y.-H. Kim and W.P. Grice, “Quantum interference with distinguishable photons through indistinguishable pathways,” J. Opt. Soc. Am. B 22, 493 (2005).
|
arxiv-papers
| 2009-10-30T07:16:34 |
2024-09-04T02:49:06.173511
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Osung Kwon, Young-Sik Ra, Yoon-Ho Kim",
"submitter": "Osung Kwon",
"url": "https://arxiv.org/abs/0910.5797"
}
|
0910.5894
|
Noise Spectra of ac-driven quantum dots
B. H. Wu
Max Planck Institute for
the Physics of Complex Systems, Nöthnitzer Str. 38, 01087
Dresden, Germany State Key Laboratory of Functional
Materials for Informatics, Shanghai Institute of Microsystem and
Information Technology, 865 Changning Road, Shanghai 200050, China
C. Timm
Institute for Theoretical Physics, Technische
Universität Dresden, 01062 Dresden, Germany
We study the transport properties of a quantum dot driven by either
a rotating magnetic field or an ac gate voltage using the Floquet
master-equation approach. Both types of ac driving lead to
photon-assisted tunneling where quantized amounts of energy are
exchanged with the driving field. It is found that the
differential-conductance peak due to photon-assisted tunneling
does not survive in the Coulomb-blockade regime when the dot is driven
by a rotating magnetic field. Furthermore, we employ a generalized
MacDonald formula to calculate the time-averaged noise spectra of
ac-driven quantum dots. Besides the peak at zero frequency, the
noise spectra show additional peaks or dips in the presence
of an ac field. For the case of an applied ac gate voltage,
the peak or dip position is fixed at the driving frequency,
whereas the position changes with increasing amplitude for the case of a
rotating magnetic field. Additional features appear in the noise spectra if a
dc magnetic field is applied in addition to a rotating field. In all cases,
the peak or dip positions can be understood from the energy differences
of two available Floquet channels.
73.63.Kv, 73.23.Hk, 72.10.Bg, 05.60.Gg
§ INTRODUCTION
Quantum conductors based on single molecules or semiconductor
quantum dots are promising building blocks for future electronics
and model systems for the study of fundamental quantum
However, much information on quantum
conductors is beyond the reach of measurements of the
current or conductance alone. Instead, the understanding of the
transport properties calls for a study of the full counting
statistics.[2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
In the past decade, valuable information on microscopic details
of the charge transport has been obtained from measurements of the
current fluctuations or current noise.[12] Previous studies
have shown that one can extract parameters such as the average
backscattered charge,[13] the intrinsic time
scales,[14, 15] and the asymmetry of the dot-lead
coupling[16, 17] from current-noise measurements.
Most studies of the current noise have focused
on the zero-frequency noise
power $S(0)$.[18, 19, 20] The
zero-frequency noise reflects the average properties of the
tunneling. Since the finite-frequency current noise $S(\omega)$ is a
measure of the correlations between tunneling events with their time
difference conjugate to the frequency $\omega$,[21] it is
interesting to go beyond the zero-frequency limit. Aguado and
Brandes [14] have demonstrated that the noise spectra
can show dip structures at the splitting energy of an open quantum
two-level system, with their width controlled by its dissipative
In many works, the MacDonald formula[22] has
been used to study the current-noise
spectra.[23, 24, 25]
Effective in-situ manipulation of quantum conductors is a key
step for further development. Using a time-dependent field
to manipulate the dynamics of quantum dots promises to be
advantageous in situations ranging from photon-assisted inelastic
tunneling[26] to quantum pumping.[27]
When the conductor is driven by an ac field, one expects
novel features due to the interplay of intrinsic oscillation frequencies and
the external driving frequency.
Several recent studies[28, 29]
indicate that key information is hidden in the noise spectra
of the ac-driven transport. For instance, Barrett and
Stace[28] have proposed to extract the characteristic
timescales such as the inverse dephasing and relaxation rates of a solid-state
charge qubit coupled to a microwave field from the noise spectrum. Wabnig
et al.[29] have proposed
to estimate the coherence time of the spin in a quantum dot by
measuring its noise spectra under an ac magnetic field. These
results are obtained based on the rotating-wave approximation and
usually in the limit of infinite on-site Coulomb interaction.
For periodically driven systems, an appropriate theoretical tool to go
beyond the rotating-wave approximation
is the Floquet theorem.[30, 31]
Various attempts have been
made by generalizing the existing steady-state transport approaches
such as the scattering
matrix[32, 33, 34] and nonequilibrium
Green's functions[35, 36, 37]
with the help of the Floquet theorem.
However, these methods are not adequate to fully take
the Coulomb blockade in quantum dots into account, which
dominates the transport properties of small-size quantum conductors.
The quantum master equation[38] in its various
manifestations[39, 40] is able to give a good
account of the Coulomb blockade in the weak-tunneling limit. This
method has previously been generalized
using the Floquet theorem to study the current and the zero-frequency
noise power in an ac-driven
conductor.[41, 42, 38, 43, 44, 45]
In the present study, we employ the Floquet master equation in the Fock
space of an ac-driven quantum dot to study the transport properties
such as the differential conductance and the full noise spectrum in the
sequential-tunneling limit. As the ac field we consider a rotating magnetic
field as well as an ac gate voltage for comparison. We employ a
generalized MacDonald formula for the time-averaged noise spectra in the
presence of a periodic ac field. An equivalent form of the
generalized MacDonald formula has been given by Clerk
and Girvin[46] without derivation. For completeness, we
present a derivation in the appendix. Note that the authors
are concerned with a different case, namely an ac bias voltage, and do
not employ the Floquet formalism.
Our paper is organized as follows. In Sec. <ref>,
a Floquet master-equation
formalism is presented to study the transport properties of
ac-driven quantum dots. Expressions for the noise spectra are derived
based on the
full counting statistics and the generalized MacDonald formula. In
Sec. <ref>,
the transport properties of the ac-driven quantum dot are studied.
The ac field is either a rotating magnetic
field or an oscillating on-site energy due to a periodic gate voltage.
The characteristic features in the transport properties are
presented and discussed. In Sec. <ref>, a brief summary
is given.
§ FORMALISM
§.§ Model
In this paper, we study the transport properties of a single-level
quantum dot driven by an ac field. The quantum dot is coupled to
the left ($L$) and right ($R$) electron leads. The leads are assumed
to be ideal and free of interactions. The Hamiltonian of the model
system can be written as
\begin{eqnarray}
\equiv H_0 + H_T ,
\end{eqnarray}
where $H_{\mathrm{dot}}$ is the Hamiltonian of the isolated quantum
dot, which contains the effects of the ac field and the Coulomb
represents the Hamiltonian of lead $l=L,R$,
where $c_{l\k\sigma}$ ($c^\dag_{l\k\sigma}$) annihilates (creates)
an electron with spin $\sigma$, crystal momentum $\k$,
and energy $ϵ_lσ̨$ in lead $l$, and
describes the coupling between the quantum dot and the
leads, where $d^†_σ$ ($d_σ$) is the spin-$σ$
electron creation (annihilation) operator in the quantum dot.
We note that we do not assume an infinite Coulomb-interaction
strength $U→∞$, in contrast to previous studies.[44]
Instead, finite Coulomb
interaction will be included by taking the doubly
occupied state into account.
In the following, we focus on the limit of weak dot-lead coupling
and investigate the transport properties of the quantum dot using
the Floquet master-equation method. For a small quantum dot, for
which the Coulomb interaction can dominate the transport behavior,
the treatment of the Coulomb interaction must go beyond the
mean-field level. To this end, it is convenient to rewrite the dot
Hamiltonian in the electron-number basis of the Fock
space.[47, 48] In this description, the
quantum dot can either be in the empty state $|0⟩$, the singly
occupied state $|σ⟩$ with spin $σ=↑$ or
$↓$, or the doubly occupied state
$|↑↓⟩$. In the following, we denote the states
in the Fock space by Latin letters, $|a⟩=|0⟩,
|↑⟩, |↓⟩, |↑↓⟩$.
Using this orthonormal basis, the dot-lead coupling can be described
naturally with the help of Hubbard operators
\begin{eqnarray}
X_{ab}=|a\rangle\langle b|,
\end{eqnarray}
which describes the transition of the quantum dot from state
$|b⟩$ to state $|a⟩$. The second-quantized dot-electron
creation operator can thus be rewritten in terms of the Hubbard
operators as
\begin{eqnarray}
\end{eqnarray}
where $σ̅$ represents the opposite spin of $σ$ and the
factor $η_σ=±1$ for $σ=↑,↓$,
respectively, is due to the anticommutation relation of the
Fermions. In terms of these Hubbard operators, the Hamiltonian for
the dot-lead coupling and the isolated dot can be rewritten as
\begin{eqnarray}
H_\mathrm{dot}&=&\sum_{ab}H^D_{ab}|a\rangle\langle b|,\\
H_T&=&\sum_{ab,l\k\sigma}V^{ab}_{l\k\sigma} c^\dag_{l\k\sigma}
\end{eqnarray}
respectively. Here, we have made the coupling strength
$V^ab_lσ̨$ depend on the occupancy of the initial and
final states of the transition in the quantum dot. The explicit form
of the dot Hamiltonian depends on the details of the device geometry
and the external ac driving field. It will be specified in the
following sections.
\subsection{The Floquet quantum-master-equation approach}
\subsubsection{Floquet states}
Due to the presence of a time-periodic external field, the dynamics
of the quantum dot is governed by a Hamiltonian that is periodic in
time with the frequency $Ω=2π/𝒯$, i.e.,
$H_dot(t)=H_dot(t+𝒯)$, where
$𝒯$ denotes the period. The solution of the time-periodic
Hamiltonian can be simplified by the Floquet
theorem,[30, 31] which states that the solution of
the Schr\"odinger equation for the dot Hamiltonian can be obtained
from (we set $ħ=|e|=k_B=1$ in the following)
\begin{eqnarray}\label{Floq.eq1}
\left[H_\mathrm{dot}(t)-i\,\frac{\partial}{\partial
t}\right]|\alpha (t)\rangle=\varepsilon_\alpha\,|\alpha(t)\rangle,
\end{eqnarray}
where $ε_α$ is the time-independent Floquet quasienergy and
$|α(t)⟩$ is the corresponding Floquet state, which has
the same period $𝒯$,
$|α(t)⟩=|α(t+𝒯)⟩$. Here, Greek
letters are used to denote the Floquet states. Further simplification is
possible by decomposing the Floquet states into a Fourier series,
\begin{eqnarray}
|\alpha(t)\rangle&=&\sum_k e^{-ik\Omega t}\, |\alpha_k\rangle ,
\end{eqnarray}
with the reverse transformation
\begin{eqnarray}
e^{ik\Omega t}\, |\alpha(t)\rangle
\end{eqnarray}
and analogously for $H_dot(t)$.
The Fourier transform of Eq.\ (\ref{Floq.eq1}) then reads
\begin{eqnarray}
\sum_{k'} H_{\mathrm{dot},k-k'}\, |\alpha_{k'}\rangle
- k \Omega \, |\alpha_k\rangle
&=& \varepsilon_\alpha\, |\alpha_k\rangle .
\end{eqnarray}
The quasienergy $ε_α$ can evidently be restricted to the first
Brillouin zone $[0,Ω)$ of the Floquet space, while the Floquet index $k$
can assume any integer value. Equivalently, we can view
$ε_α+kΩ$ as the quasienergy in the extended zone
We also introduce the Hubbard operator in the Floquet
states to describe the transition between the Floquet states as
$X_αβ(t)=|α(t)⟩⟨β(t)|$. For the
time-dependent transport, it is more convenient to work with these
Floquet states. This is most advantageous in transformations of the following
form, which we will use in the derivation below,
\begin{eqnarray}\label{UXU}
\tilde
X_{\alpha\beta}(t')\,U_0(t',t) \nonumber \\ &=&
\end{eqnarray}
\begin{equation}
U_0(t',t) =
T_c\exp\left(-i\int^{t'}_{t}dt''\, [H_L+H_R+H_\mathrm{dot}
\end{equation}
denotes the time-evolution operator due to the Hamiltonian
in the absence of tunneling. Here, $T_c$ is the time-ordering
operator and the dot Hamiltonian is explicitly time-dependent.
\subsubsection{The Floquet quantum master equation with counting fields}
For a quantum dot coupled to external leads, the exact quantum master equation
can be written in the interaction picture as[38, 39]
\begin{eqnarray}\label{EOMrhoI}
\frac{d}{dt}\rho_I(t)=-i[H_{T,I}(t),
\rho_I(t_0)]-\int_{t_0}^tdt'\,[H_{T,I}(t), [H_{T,I}(t'),\rho_I(t')]],
\end{eqnarray}
where $A_I(t)=U^†_0(t,t_0)A(t)U_0(t,t_0)$ denotes an operator in the
interaction picture and $ρ(t)$ is the density matrix in the Fock
space of the full system.
A complete description of the electronic transport through the quantum
dot is provided by the full counting statistics.
Properties such as the noise spectrum are determined by the
counting statistics of the electrons arriving at and departing from
the leads. All information on the counting statistics is contained in
the moment-generating function $ϕ(χ_L,χ_R)=⟨exp(iχ_LN_L+iχ_RN_R)⟩$. Here, $χ_l$ represents the
counting field in the lead $l$, which counts how many electrons
have tunneled into or out of the lead.
$N_l=∑_σ̨c^†_lσ̨c_lσ̨$ is the
electron-number operator in lead $l$. We introduce the operator
\begin{eqnarray}
\mathcal{F}(\chi_L,\chi_R,t)=
\mathrm{Tr}_{\mathrm{leads}}\, e^{i\chi_LN_L+i\chi_RN_R}\rho(t) .
\end{eqnarray}
In this we follow Kaiser and
Kohler,[44] except that we introduce two counting fields.
In the limit of $χ_L→0$ and $χ_R→0$, $ℱ$
becomes the reduced density matrix of the quantum dot,
$ρ_dot=Tr_leads ρ$. Moreover, the
moment-generating function $ϕ(χ_L,χ_R, t)$ can be obtained
by tracing out the dot degrees of freedom,
$ϕ=Tr_dot ℱ$. We decompose
$ℱ$ into a Taylor series,
\begin{eqnarray}\label{TSF}
\mathcal{F}=\sum_{m=0}^\infty\sum_{n=0}^{\infty}
\frac{(i\chi_L)^{m}(i\chi_R)^{n}}{m!n!}\,\mathcal{F}^{m,n},
\end{eqnarray}
where the coefficients
\begin{eqnarray}
\mathcal{F}^{m,n}
\mathcal{F}\,\right|_{\chi_L,\chi_R\rightarrow
0}=\mathrm{Tr}_{\mathrm{leads}}\, N_L^{m}N_R^n\,\rho
\label{Fmn}
\end{eqnarray}
provide a direct
access to the moments $⟨N_L^mN_R^n⟩=Tr_dot ℱ^m,n$. In
particular, we obtain the reduced density matrix of the quantum dot,
To find the solutions
for $ℱ$, we first transform the equation of motion for the
density matrix in the interaction picture, Eq.\ (\ref{EOMrhoI}), back to
the Schr\"odinger picture,
\begin{eqnarray}\label{QME1}
\lefteqn{ \frac{d\rho(t)}{dt}+i[H_0(t),\rho(t)]=-i[H_T,
U^\dag_0(t_0,t)\rho(t_0)U_0(t_0,t)] } \nonumber \\
&& {}-\int_{t_0}^t dt'\, [H_T,
U_0^\dag(t',t) [H_T,\rho(t')] U_0(t',t)] .
\end{eqnarray}
Then, we multiply by
$e^iχ_LN_L+iχ_RN_R$ from
the left and take the trace over the lead degrees of freedom to obtain
\begin{eqnarray}\label{QME2pre}
\lefteqn{
\frac{d\mathcal{F}(\chi_L,\chi_R, t)}{dt}
+i[H_\mathrm{dot}(t), \mathcal{F}(\chi_L, \chi_R,
t)] } \nonumber \\
&& = -i\, \mathrm{Tr_{leads}}\, e^{i\chi_LN_L+i\chi_RN_R}\,
[H_T, U^\dag_0(t_0,t)\rho(t_0)U_0(t_0,t)] \nonumber \\
&& \quad {}-\int_{t_0}^t dt'\, \mathrm{Tr_{leads}}\, e^{i\chi_L
N_L+i\chi_RN_R} [H_T, U_0^\dag(t',t) [H_T,\rho(t')] U_0(t',t) ] ,
\end{eqnarray}
which is still exact.
We now assume that the full density operator is of product form at
the initial time $t_0$,
$ρ(t_0)=ρ_dot(t_0) ⊗ρ^0_leads$, where
$ρ_leads^0$ describes the leads in separate thermal equilibrium.
This assumption is reasonable since we are not interested in transient
effects coming from the initial state. Such effects have been
studied by Flindt \textit{et al.}[49]
The first term on the right-hand side of Eq.\ (\ref{QME2pre}) then vanishes.
Furthermore, we make the sequential-tunneling approximation appropriate for
weak tunneling, i.e., we treat the tunneling perturbatively to second order in
$H_T$. Since two powers of $H_T$ are already explicit in the second term on
the right-hand side of Eq.\ (\ref{QME2pre}), we can express $ρ(t')$ in terms
of the unperturbed time evolution, $ρ(t') ≈U_0^†(t,t') ρ(t)
U_0(t,t')$. This makes the master equation local in time, i.e.,
Markovian. For details, see, e.g., Ref.\ \onlinecite{PRB77195416}.
We thus do not include non-Markovian effects as studied by Flindt
\textit{et al.}[49] This is valid if the relaxation time in the
leads is short compared to the typical timescales of the dot, which in our
case include the period of the ac field. Since relaxation times in metallic
leads are of the order of femtoseconds, this is easily satisfied.
Finally, we send $t_0→-∞$ and obtain
\begin{eqnarray}\label{QME2}
\lefteqn{
\frac{d\mathcal{F}(\chi_L,\chi_R, t)}{dt}
+i[H_\mathrm{dot}(t), \mathcal{F}(\chi_L, \chi_R,
t)] } \\ \nonumber
&& =-\int_0^\infty d\tau\, \mathrm{Tr_{leads}}\, e^{i\chi_L
N_L+i\chi_RN_R}[H_T, [\tilde H_T(t-\tau, t),\rho(t)]] ,
\end{eqnarray}
where $H̃_T(t',t)=U^†_0(t',t)H_T U_0(t',t)$.
To obtain the Floquet master equation, we write Eq.\ (\ref{QME2}) in
the basis of Floquet states $|α(t)⟩$, $|β(t)⟩$.
By making use of the relation Eq.\ (\ref{UXU}) and tracing
out the lead degrees of freedom, we arrive at the equation of
motion for $ℱ$,
\begin{eqnarray}\label{EOMF}
\lefteqn{\frac{d}{dt}\mathcal{F}_{\alpha\beta}(\chi_L,\chi_R,t)=\Big\{\Big[
\mathcal{L}
+(e^{i\chi_L}-1)\mathcal{J}_{L+}+(e^{-i\chi_L}-1)\mathcal{J}_{L-}} \nonumber
\\
& & {}+(e^{i\chi_R}-1)\mathcal{J}_{R+}+(e^{-i\chi_R}-1)\mathcal{J}_{R-}\Big]
\, \mathcal{F}(\chi_L,\chi_R,t)\Big\}_{\alpha\beta},\hspace{2em}
\end{eqnarray}
where the superoperators are given by
\begin{eqnarray}
\lefteqn{ (\mathcal{LF})_{\alpha\beta}
= -i(\varepsilon_\alpha-\varepsilon_\beta)\mathcal{F}_{\alpha\beta}
d\varepsilon\,\Bigg\{\sum_{ab;mn}\sum_{\gamma\delta}\sum_{l={L,R};\sigma}} \\
\nonumber
\Gamma^{l\sigma}_{mn;ab})\,
\langle\alpha(t)|a\rangle\langle
b|\gamma(t)\rangle\langle\gamma(t-\tau)|m\rangle \langle
\nonumber
\langle\alpha(t)|a\rangle\langle b|\gamma(t)\rangle
\mathcal{F}_{\gamma\delta}(\chi_L,\chi_R,t)\langle
\delta(t-\tau)|m\rangle\langle n|\beta(t-\tau)\rangle\right.\\
\nonumber
\langle\alpha(t-\tau)|a\rangle\langle
t)\langle\delta(t)|m\rangle \langle n|\beta(t)\rangle\right.\\
\nonumber
\Gamma^{l\sigma}_{ab;mn})\,
\langle\gamma(t-\tau)|a\rangle\langle
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\mathcal{J}_{l+}\mathcal{F})_{\alpha\beta} =
\frac{1}{2\pi}\int^\infty_0d\tau\int d\varepsilon
\sum_{ab;mn}\sum_{\gamma\delta}\sum_{\sigma}
\bar{f}_l(\varepsilon)\Gamma^{l\sigma}_{ab;mn} } \nonumber \\
&&\left( e^{i\varepsilon\tau}e^{i(\varepsilon_\beta-\varepsilon_\delta)\tau}
\langle\alpha(t)|a\rangle\langle b|\gamma(t)\rangle\mathcal{F}_{\gamma\delta}(\chi_L,\chi_R,t)\langle\delta(t-\tau)|m\rangle
\langle n|\beta(t-\tau)\rangle\right. \nonumber \\
\end{eqnarray}
\begin{eqnarray}
\lefteqn{(\mathcal{J}_{l-}\mathcal{F})_{\alpha\beta} =
\frac{1}{2\pi}\int^\infty_0d\tau\int
f_l(\varepsilon)\Gamma^{l\sigma}_{mn;ab} } \nonumber \\
\langle\alpha(t)|a\rangle\langle
m\rangle\langle n|\beta(t-\tau)\rangle\right. \nonumber \\
\gamma(t-\tau)\rangle\mathcal{F}_{\gamma\delta}(\chi_L,\chi_R,t)\langle\delta(t)|m\rangle\langle
\end{eqnarray}
Here, we have defined the tunneling rate
where $ρ_l(ϵ)$ is the density of states in lead $l$.
Inserting the Taylor expansion of $ℱ$ in Eq.\
(\ref{TSF}) into its equation of motion [Eq.\ ($<ref>$)], one obtains
a hierarchy of equations for the expansion coefficients,
\begin{eqnarray}
\frac{d}{dt}\mathcal{F}^{0,0}&=&\mathcal{L}\mathcal{F}^{0,0}, \label{rhot}\\
\frac{d}{dt}\mathcal{F}^{1,0}&=&\mathcal{L}\mathcal{F}^{1,0}+(\mathcal{J}_{L+}-\mathcal{J}_{L-})\mathcal{F}^{0,0},\\
\frac{d}{dt}\mathcal{F}^{0,1}&=&\mathcal{L}\mathcal{F}^{0,1}+(\mathcal{J}_{R+}-\mathcal{J}_{R-})\mathcal{F}^{0,0},\\
\frac{d}{dt}\mathcal{F}^{2,0}&=&\mathcal{L}\mathcal{F}^{2,0}+2(\mathcal{J}_{L+}-\mathcal{J}_{L-})\mathcal{F}^{1,0}
\frac{d}{dt}\mathcal{F}^{0,2}&=&\mathcal{L}\mathcal{F}^{0,2}+2(\mathcal{J}_{R+}-\mathcal{J}_{R-})\mathcal{F}^{0,1}
\frac{d}{dt}\mathcal{F}^{1,1}&=&\mathcal{L}\mathcal{F}^{1,1}+(\mathcal{J}_{L+}-\mathcal{J}_{L-})\mathcal{F}^{0,1}
+(\mathcal{J}_{R+}-\mathcal{J}_{R-})\mathcal{F}^{1,0}, \quad\mbox{etc.}
\end{eqnarray}
As described above, these coefficients contain the full counting
statistics. The charge
current out of lead $l$ is defined as the negative of the time-derivative of
the charge in lead $l$, $I_l(t)=e dN_l/dt$. The final
expression for the current out of the left lead is given
\begin{eqnarray}\label{I2}
\langle
\, (\mathcal{J}_{L+}-\mathcal{J}_{L-})\,\mathcal{F}^{0,0} .
\end{eqnarray}
The dc component of the current then gives the time average
$I̅$. In order to find $ℱ^0,0$, we have to solve a
set of linear equations with the help of the normalization condition
of probability $Tr_dot ℱ^0,0(t) =1$.
\subsubsection{Generalized MacDonald formula for time-averaged noise spectra}
We are interested in the frequency-dependent current noise of the
quantum dot driven by an ac field. The zero-frequency current
noise for non-adiabatical driving has been investigated in
Ref.\ \onlinecite{AP16702} using the Floquet master-equation approach
in the Coulomb-blockade regime. The symmetrized current-current correlation
function is defined by
\begin{eqnarray}
S_{ll'}(t,t')=\langle \hat I_l(t)\hat I_{l'}(t')\rangle+\langle
\hat I_{l'}(t')\hat I_l(t)\rangle -2\langle \hat I_l(t)\rangle\langle
\hat I_{l'}(t')\rangle,
\label{S.tt}
\end{eqnarray}
where $Î_l(t)$ represents the current operator at the time $t$
from the lead $l$. The current-noise spectra are defined as the
Fourier transform of $S_ll'(t,t')$. Since our system is driven by
an ac field, the current noise is a double-time function. However,
the periodicity of our problem makes it possible to characterize the
spectra by averaging over one driving period.
At finite frequencies, the total current $I(t)$ measured by a
measurement device depends on both the particle and the displacement currents in
the lead-dot-lead junction. If one expresses the displacement currents by the
particle currents $I_L$ and $I_R$, one obtains the Ramo-Shockley
theorem,[50, 51, 12]
\begin{eqnarray}
I(t)=a I_L(t)-b I_R(t).
\end{eqnarray}
Here the coefficients $a$ and $b$, which satisfy
$a+b=1$, are specified by the device geometry. It is straightforward
to show that the total time-averaged noise spectrum is given by
\begin{eqnarray}
\bar S(\omega)=a^2\bar S_{LL}(\omega)+b^2\bar S_{RR}(\omega)-ab(\bar S_{LR}(\omega)+\bar
\end{eqnarray}
where $S̅_ll'(ω)$ ($l,l'=L,R$) represents the
frequency-dependent time-averaged current correlation between $I_l$
and $I_l'$,
\begin{eqnarray}
\bar S_{ll'}(\omega)=\frac{1}{\mathcal{T}}\int^\mathcal{T}_0
dt\int dt'\, e^{i\omega (t-t')}
\end{eqnarray}
In this study, we used two counting fields to derive the noise
spectra. An alternative approach is to calculate the charge fluctuation on the
dot employing the quantum regression
formula.[52, 53, 14, 3]
The two approaches are physically equivalent due to the charge
conservation condition in the transport.
The formula for the zero-frequency noise has been presented in Ref.\
\onlinecite{AP16702}. For the two-terminal device, it is adequate to
find the time-averaged zero-frequency noise from the fluctuations of
the current flowing out of a chosen lead, $S̅(0)=S̅_ll(0)$. The solution for $S̅(0)$ resulting from the Floquet
quantum master equation reads
\begin{eqnarray}
\bar S(0)=\frac{2}{\mathcal{T}}\int^\mathcal{T}_0dt\,
\delta_{lL},\delta_{lR}}_{\perp}
\end{eqnarray}
where the prefactor of $2$ is inserted to make the noise formula
consistent with Ref.\ \onlinecite{PR3361}. For Poissonian noise, we
then obtain $S̅(0)=2eI̅$. $δ_ij$ is the usual
Kronecker symbol. Following Ref.\ \onlinecite{AP16702}, the new
function $ℱ^δ_lL,δ_lR_⊥$ in the
noise expression is defined as
\begin{eqnarray}
\mathcal{F}^{\delta_{lL},\delta_{lR}}_{\perp}
\{\mathcal{F}^{\delta_{lL},\delta_{lR}}\},
\end{eqnarray}
and satisfies the equation of motion
\begin{eqnarray}
\dot{\mathcal{F}}^{\delta_{lL},\delta_{lR}}_{\perp}
\end{eqnarray}
An efficient method to find the noise spectrum is
provided by the MacDonald formula,[22] which has
been widely used in quantum
transport.[21, 24, 23, 25]
The validity of this formula requires that the current correlation
function $⟨I(t_1)I(t_2)⟩$ is only a function of the
time difference $t_1-t_2$ and that, therefore, the transport is in the
stationary regime. A direct application of the MacDonald formula to
the present time-dependent transport problem is thus not possible.
However, in the present
study the driving field is time-periodic. The discrete temporal translation
symmetry $H(t+𝒯)=H(t)$ makes it possible to generalize
the MacDonald formula for the noise spectra time-averaged
over one period as
\begin{eqnarray}
\frac{\bar S_{ll'}(\omega)}{\omega}
\frac{dt}{2i}\,\mathrm{Tr_{dot}}\left[\mathcal{S}(-i\omega)
\mathcal{F}^{\delta_{lL}+\delta_{l'L},\delta_{lR}+\delta_{l'R}}(-i\omega)
\delta_{l'R}}(i\omega)\right] ,\qquad \label{MacDonald}
\end{eqnarray}
where the composents of the superoperator $𝒮(s)$ are given by
\begin{eqnarray}
\delta_{kk'}.
\end{eqnarray}
To clarify the meaning of this definition, we note that the
superoperator $𝒮(s)$ acts on an arbitrary operator $A$
with Fourier-transformed matrix elements $A_lm;k$ in our standard
basis $|l⟩, |m⟩= |0⟩,|↑⟩,
|↓⟩, |↑↓⟩$ as
\begin{eqnarray}
[\mathcal{S}(s)\,A]_{lm;k} = \sum_{l'm'}\sum_{k'}
[\mathcal{S}(s)]_{lmk;l'm'k'}\, A_{l'm';k'} .
\end{eqnarray}
The derivation of the generalized MacDonald formula is outlined in the
appendix. In evaluating the current noise from the generalized
MacDonald formula, we encounter the Laplace transforms $ℱ^m,n(s)$
of the moments of the electron
number operators in the leads. These moments are nothing but the expansion
coefficients of $ℱ$ defined in Eq.\ (\ref{Fmn}).
Taking the trace over the dot degrees of freedom and the average
over one period makes only the matrix elements
$ℱ^m,n_ll;k=0$ of $ℱ^m,n$ that are
diagonal in the dot basis and have Floquet index $k=0$ contribute to
the final result. We arrive at the expression
\begin{eqnarray}\label{MD}
\frac{\bar
S_{ll'}(\omega)}{\omega}&=&-e^2\omega\, \mathrm{Tr_{dot}}
\end{eqnarray}
Now we require the charge moments in the left and right leads.
They can be obtained from the equation of motion for $ℱ$
[Eq.\ (\ref{EOMF})]. Suppose we switch on the counting fields at
some time $t_1$, before that time the system can be described by the
density matrix without the counting fields in the (quasi-)
stationary limit. Since we have assumed $t_0→-∞$ above, any
initial correlation have died out at time $t_1$.[49]
We set the number of electrons having tunneled into lead $l$ up to
time $t_1$ to zero, $N_l(t_1,t_1)$=0. After time $t_1$, the system
evolves under the influence of the counting fields. Then, we solve
the equations for $ℱ^m,n(t)$ by means of Laplace
transformation. For example, the solution of Eq.\ (\ref{rhot}) reads
\begin{eqnarray}
\mathcal{F}^{0,0}(s)=[\mathcal{S}(s)-\mathcal{L}]^{-1}\,\mathcal{F}^{0,0}(t_1),
\end{eqnarray}
where $ℱ^0,0(t_1)$ can be found from the stationary
master equation $ℒℱ^0,0=0$ in the absence of
counting fields. Analogously, we find expressions for the other
expansion coefficients after the Laplace transformation as
\begin{eqnarray}
\mathcal{F}^{1,0}=\left(\mathcal{S}-\mathcal{L}\right)^{-1}(\mathcal{J}_{1+}-\mathcal{J}_{1-})\mathcal{F}^{0,0},
\end{eqnarray}
\begin{eqnarray}
\mathcal{F}^{0,1}=\left(\mathcal{S}-\mathcal{L}\right)^{-1}(\mathcal{J}_{2+}-\mathcal{J}_{2-})\mathcal{F}^{0,0},
\end{eqnarray}
\begin{eqnarray}
\mathcal{F}^{2,0}=\left(\mathcal{S}-\mathcal{L}\right)^{-1}\left[2(\mathcal{J}_{1+}-\mathcal{J}_{1-})
\mathcal{F}^{1,0}+(\mathcal{J}_{1+}+\mathcal{J}_{1-})\mathcal{F}^{0,0}\right],
\end{eqnarray}
\begin{eqnarray}
\mathcal{F}^{0,2}=\left(\mathcal{S}-\mathcal{L}\right)^{-1}\left[2(\mathcal{J}_{2+}-\mathcal{J}_{2-})
\mathcal{F}^{0,1}+(\mathcal{J}_{2+}+\mathcal{J}_{2-})\mathcal{F}^{0,0}\right],
\end{eqnarray}
\begin{eqnarray}
\mathcal{F}^{1,1}=\left(\mathcal{S}-\mathcal{L}\right)^{-1}\left[(\mathcal{J}_{1+}-\mathcal{J}_{1-})
\mathcal{F}^{0,1}+(\mathcal{J}_{2+}-\mathcal{J}_{2-})\mathcal{F}^{1,0}\right],
\end{eqnarray}
where we have omitted the arguments $s$.
The solutions for these coefficients together with the generalized
MacDonald formula [Eq.\ (\ref{MacDonald})] give the desired
time-averaged current-noise spectra of the ac-driven quantum dot.
The approach
presented in this study can easily be generalized to take
more complex structures with multiple levels and
inter-level transitions into account.
\section{Results and discussion}
\label{sec:results}
In the following, we present our numerical results based on the
Floquet master-equation method and discuss the transport properties
of the single-level quantum dot with time-dependent fields.
An additional dc magnetic field along the $z$ or
$x$ direction is taken into account; it splits the energy levels of
the singly charged quantum dot due to
the Zeeman effect. In the present study, we choose the ac field to
be either a rotating magnetic field in the $xy$ plane or an ac
gate voltage. The ac gate voltage
and the rotating magnetic field will affect the quantum conductor in
quite different manners. An ac gate voltage only
changes the eigenvalues of $H_dot(t)$ periodically, which in the
adiabatic limit of large $𝒯$ become the eigenenergies.
The ac gate voltage will not induce any transition between different
eigenstates of $H_dot(t)$ (no spin flip is possible)
because the eigen\emph{states} are unaffected by the gate voltage. The
electron is trapped in one spin state. The situation
is different for a rotating magnetic field. A rotating
magnetic field does not change the eigenvalues of $H_dot(t)$ but
does change the eigenstates and thus can flip the
spin of the electron. The spin polarization of the dot
will thus evolve with the rotating magnetic field. The two types of
ac fields show drastically different
behaviors in the transport properties as we will show below.
The full Hamiltonian of the quantum dot is written as (we reiterate
that we choose $|e|=ħ=k_B=1$),
\begin{eqnarray}
H_{\mathrm{dot}}&=&\sum_\sigma(eV_G+V_{\mathrm{ac}}\cos \Omega
t +\sigma B_z)\, d^\dag_\sigma d_\sigma +U\, d^\dag_\uparrow d_\uparrow
d^\dag_\downarrow d_\downarrow \nonumber\\
&&{}+B_x\, (d^\dag_\uparrow
d_\downarrow +d^\dag_\downarrow d_\uparrow) +
d_\downarrow e^{i\Omega t}+d^\dag_\downarrow d_\uparrow
e^{-i\Omega t}),
\label{full.Hdot}
\end{eqnarray}
where $eV_G$ is the on-site energy of the quantum dot due to the dc
component of the gate voltage $V_G$, $V_ac$ is the
amplitude of the oscillating gate voltage, $U$ represents the
intra-dot Coulomb interaction, and $B_x$ and $B_z$ are half the
Zeeman energies of the singly occupied dot due to the dc magnetic
fields in the $x$ and $z$ direction, respectively. Half the Zeeman
energy of the rotating magnetic field is given by $B_ac$.
Note that while it is customary to talk about photon-assisted
processes in this context, the treatment of the electromagnetic
field in the Hamiltonian is completely classical.
We work in the sequential-tunneling regime and choose a symmetric
coupling geometry with $a=b$. We assume that the bias voltage
$V_dc$ symmetrically shifts the chemical potentials by
$μ_L,R=±eV_dc/2$. In the framework of wide-band
approximation, the tunneling rate is given by
where we have assumed the coupling strength $V_l,k,σ^ij=V$
to be a constant and have set the density of states of lead $l$ to
unity. In the present study, we have assumed the tunneling matrix to
be independent of the energy and the occupation number on the dot.
An inclusion of state-dependent tunneling is straightforward. We
assume that the electrons tunneling in and out of the dot with an
energy-independent rates $Γ=Γ^lσ_mn;ij(ϵ)$
and set $Γ=1$ as the energy unit.
\subsection{Differential conductance}
We start our discussion with the differential conductance. The
gray-scale plot Fig.\ \ref{dIdV} shows the differential conductance
$dI/dV_dc$ vs.\ the dc bias voltage $V_dc$ and the
gate voltage $V_G$ with or without an ac field. The calculations are
for the Coulomb interaction strength $U=24$ and the temperature
$k_BT=0.32$. The frequency of the ac field is $Ω=8$.
Without an ac field, Fig.\ \ref{dIdV}(a) gives the familiar diamond
structure due to the Coulomb blockade. Numerical results for the
differential conductance when the quantum dot is modulated by an ac
gate voltage are presented in Fig.\ \ref{dIdV}(b). Fig.\ \ref{dIdV}(c)
gives the results when the quantum dot is modulated by a
rotating magnetic field. Fig.\ \ref{dIdV}(d) shows the differential
conductance when the quantum dot is modulated by a rotating
magnetic field in the $xy$ plane while a dc magnetic field is applied
in the
$x$ direction, i.e., in the plane of the rotating magnetic field.
When there is an ac field, several striking features emerge in the
differential conductance: (1) At the edge of the Coulomb diamond,
the sharp differential-conductance peak for the dc transport shown
in Fig.\ \ref{dIdV}(a) is partially suppressed by the ac gate
voltage or the rotating magnetic field. Note the different gray
scales in Fig.\ 1 (a), (b), (c), and (d). This can be attributed to
the suppression of the elastic resonant peak by the photon-assisted
processes. (2) In the presence of an ac field, there are lines
parallel to the edges of the Coulomb diamond. The distance of these
lines to the peak position is approximately the frequency of the ac
field, indicating a photon-assisted tunneling process. (3) An
interesting feature of these lines can be observed inside the
Coulomb diamonds. For the ac gate voltage, the Floquet quasienergies
are spin degenerate. Therefore, the main lines in the differential
conductance plot in Fig.\ \ref{dIdV}(b) are not split. However,
satellites due to photon-assisted inelastic tunneling events appear,
in which an energy quantum of $Ω$ is absorbed from or emitted
into the driving field. We see from Fig.\ \ref{dIdV}(b) that these
additional lines remain distinct inside the Coulomb diamond. On the
other hand, when the quantum dot is driven by a rotating magnetic
field, the quasienergies are not degenerate. Therefore, the main
elastic lines are split into two at the edge of the Coulomb diamond
in Fig.\ \ref{dIdV}(c). Interestingly, the lines due to the
photon-assisted tunneling now only appear outside of the Coulomb
diamond, as can be seen in Fig.\ \ref{dIdV}(c), indicating that the
photon-assisted tunneling is forbidden inside the Coulomb diamond.
When the quantum dot is modulated by a rotating magnetic field and a
dc magnetic field is applied in the plane of the rotating field,
these lines in the Coulomb-blockade regime revive. This can be
clearly seen in Fig.\ \ref{dIdV}(d).
The disappearance of the photon-assisted tunneling inside the
Coulomb diamond for a pure rotating magnetic field can be understood
as follows. In the Coulomb diamond, the Floquet quasienergies
corresponding to the singly occupied states are far below the Fermi
energies of the two leads. A direct tunneling between the dot and
the leads is forbidden due to the Pauli principle and Coulomb
blockade. Therefore, an electron is effectively trapped in one
quantum state on the dot. According to our previous discussion, only
the spin direction of this quantum state can evolve with the
rotating magnetic field. However, its eigenvalues of
$H_dot(t)$ remain unchanged. Therefore, the electron cannot
gain extra energy from the ac magnetic field. As a consequence, we
cannot observe lines due to photon-assisted tunneling inside the
Coulomb diamond. Outside of the Coulomb diamond, the tunneling
between the dot and the leads becomes possible. When an electron is
injected from the lead into the dot, the system can absorb or emit
photons, i.e., the Floquet index $k$ can change. One could say that transport
happens via several Floquet channels. Such
photon-mediated tunneling can then give rise to the photon-assisted
differential-conductance peaks.
The situation becomes different when a dc magnetic field is applied
in the plane of the rotating magnetic field as shown in Fig.\
\ref{dIdV}(d). In that case, the eigenstates and the eigenvalues of
$H_dot(t)$ change periodically. Electrons can gain extra
energy from the ac field by absorbing or emitting a photon. In the
Coulomb diamond, electrons on the dot are able to tunnel out via the
photon-assisted tunneling and we again find the lines due to the
photon-assisted differential-conductance peaks inside the Coulomb
diamond as shown in Fig.\ \ref{dIdV}(d).
\subsection{Zero-frequency Fano factor}
In the following, we show numerical results for the time-averaged
zero-frequency noise of the quantum dot. The zero-frequency noise
has been studied by the quantum master-equation method in the
stationary [12, 16, 23] and also in the
time-dependent case.[44] Without ac field and at zero
temperature, the zero-frequency Fano factor $S(0)/2eI$ describes the
deviation of the shot noise from its Poissonian value. We choose the
parameters $T=0.32$, $U=8$, $eV_G=8$, and $Ω=4.8$. We assume
that a dc magnetic field in the $z$ direction, $B_z=1.6$, is applied
to the quantum dot. The finite value of $U$ and the Zeeman splitting
make it possible to see plateaus in the Fano factor at different
occupation numbers on the dot.[16] Fig.\ \ref{FanoVac}
shows the zero-frequency Fano factor as a function of the dc bias
with or without an ac gate voltage.
Without an ac gate voltage, $V_ac=0$, the results reproduce
the main features reported in Ref.\ \onlinecite{PRB68115105}. At
very low bias voltage $V_dc→0$, the main
contribution to the noise is the finite thermal noise while the
current as well as the shot noise are suppressed. Therefore, the
Fano factor diverges at $V_dc=0$. For low dc bias voltage,
the quantum dot operates in the Coulomb-blockade regime. With
further increasing dc bias voltage, the energy levels of the quantum
dot one by one enter the transport window defined by the dc bias.
This can be clearly identified in the Fano factor by the plateaus at
different values. The edges of the plateaus are broadened by the
finite temperature. For very large dc bias, where all the energy
levels of the quantum dot lie in the transport window, the Fano
factor approaches the well-known limit of $1/2$ for our
symmetric-coupling case.[16]
The results for the dc case
demonstrate that the plateaus of the Fano factor can
give a good account of the transport channels.[16]
In the presence of an ac field, we now consider the time-averaged Fano
factor $S̅(0)/2eI̅$.
We can see
from Fig.\ \ref{FanoVac} that for small $V_dc$
the Fano
factor becomes larger as we increase the amplitude of the ac gate
voltage. On the other hand, additional photon-assisted
transport channels are available due to the ac field, which will modify
the Fano-factor curve.
With increasing ac field, the Fano factor will thus
deviate from the plateau behavior seen in dc case due to the opening
of these photon-assisted transport channels.
When the ac gate voltage is large enough, the Floquet eigenstates
that lie outside of the transport window can contribute to the
current via photon-assisted tunneling. As a consequence, the
plateaus in the Fano-factor curve become vague. For very large bias
voltages, all the Floquet levels are well inside the transport
window. The Fano factor then will approach the same value $1/2$ as
for the time-independent transport. In Fig.\ \ref{FanoRac}, we
present our results for the zero-frequency current noise in the
presence of a rotating magnetic field. As in the case of an ac gate
voltage, the Fano factor deviates from the dc behavior with
increasing ac field. Additional plateaus can be observed in the
Fano-factor curve when we vary the dc bias voltage. Transitions between
plateaus result from additional Floquet channels becoming available.
\subsection{Frequency-dependent Fano factor}
Now we present our results for the full current-noise spectra of a
quantum dot under an ac field. The noise spectra have previously
been studied in the stationary-transport regime. An analytical
expression for the noise spectrum of a single-level quantum dot can
be found in Ref.\ \onlinecite{PRB76085325}. Unless stated otherwise,
the following calculations assume $U=24$, $V_dc=12.7$,
$T=1.6$, $Ω=8$ and $eV_G=-8$. We introduce the
frequency-dependent Fano factor $S̅(ω)/2eI̅$ to
characterize the time-averaged noise power. As discussed previously,
the ac gate voltage and the rotating magnetic field will modulate
the quantum conductor in different ways. In the following, we show
that the noise spectra are also strikingly different.
In Fig.\ \ref{SVac}, we present the results for the
frequency-dependent Fano factor $S̅(ω)/2eI̅$
as a function of the
frequency $ω$ for different amplitudes $V_ac$. No dc
magnetic field is applied. Without an ac field ($V_ac=0$), the
noise spectrum shows a peak at zero frequency and approaches a
constant value for large $ω$. The peak in the noise
spectrum is due to the elastic processes in the
transport.[29] When an ac gate voltage is applied,
additional structures in the noise spectra are expected due to
photon-assisted processes. For the present set of
parameters, one can clearly see that with increasing amplitude of
the ac gate voltage, an additional peak appears in the noise
spectrum. While the height and width of this peak vary a lot
with increasing amplitude, its peak position $ω_p$
remains almost unchanged at the external driving frequency $Ω$.
Now we turn to the rotating magnetic field in the $xy$ plane. In
Fig.\ \ref{SRac}, we plot the Fano factor as a function of the
frequency $ω$ for different amplitudes $B_ac$ of the
rotating magnetic field. Similarly to the results presented in Fig.\
\ref{SVac}, a peak is generated and the width and height of this
peak depend on the amplitude. However, the peak position is not
fixed at $Ω$ in contrast to what we have observed in Fig.\
\ref{SVac} for the ac gate voltage. Instead, its position shifts with
increasing amplitude, as shown in Fig.\ \ref{SRac}.
By comparing Fig.\ \ref{SVac} and Fig.\ \ref{SRac}, we see that the
peak position of the noise spectra behaves differently when we
increase the ac strength, depending on the type of the ac field.
Recalling that when electrons tunnel through a time-independent
quantum two level system, its current noise spectra show additional
structure at the energy difference of the two transport channels of
the system due to its internal coherent dynamics,[14]
we will show that the peak position of the noise spectra for ac
transport can be understood from the interference between two
possible Floquet transport channels. If the quantum dot is modulated
by a rotating magnetic field, the last term in the dot Hamiltonian
[Eq.\ (\ref{full.Hdot})] shows that the ac magnetic field couples
one spin state with the quasienergy $ϵ$ with a state with the
opposite spin and the quasienergy $ϵ-Ω$ (in the extended
zone scheme). The coupling strength is given by $B_ac$.
{The corresponding Floquet Hamiltonian then decomposes into $2\times
2$ blocks of the form
\begin{eqnarray}\label{hFl}
h_\mathrm{Fl} = \left(
\begin{array}{cc}
\epsilon & B_\mathrm{ac} \\
B_\mathrm{ac} & \epsilon-\Omega \\
\end{array}
\right).
\end{eqnarray}
The resulting quasienergies in the first Brillouin zone $[0,\Omega)$ are
\begin{eqnarray}
\epsilon_1 &=&
\epsilon-\frac{\Omega}{2}+\frac{\sqrt{\Omega^2+4B_\mathrm{ac}^2}}{2} , \\
\epsilon_2 &=&
\epsilon+\frac{3\Omega}{2}-\frac{\sqrt{\Omega^2+4B_\mathrm{ac}^2}}{2}
\end{eqnarray}
with the difference
\begin{eqnarray}\label{EqPeak}
\omega_p=\epsilon_2-\epsilon_1=2\Omega-\sqrt{\Omega^2+4B_\mathrm{ac}^2}
\end{eqnarray}
(these expressions hold if $B_\mathrm{ac}<\sqrt{3}\,\Omega/2$).}
If now an electron tunnels into the dot, the system ends up in a
superposition of the two Floquet states, the phases of which change
with different angular frequencies, corresponding to spin precession
with the difference frequency $ω_p$. When the electron tunnels
out again, the superposition is projected onto the spin direction of
the original electron since lead electron creation and annihilation
operators are paired with identical quantum numbers in the master
equation. This leads to interference with a typical frequency
$ω_p$, which enhances the current-current correlation function
$S_ij(t,t')$ in Eq.\ (\ref{S.tt}) for $t-t'$ being a multiple of
the period $2π/ω_p$ and thus leads to a peak in the noise
spectrum at $ω_p$. The peaks seen in Fig.\ \ref{SRac} are
indeed centered at $ω_p$ given by Eq.\ (\ref{EqPeak}).
Comparing with the stationary transport through a stationary two
level system,[14] the transport through a quantum dot
with rotating magnetic field can be understood as another type of
two level quantum system. The significant difference here is that
our two levels are defined by the Floquet channels due to a periodic
ac field and not by the true eigenenergies of an time-independent
When an ac gate voltage is applied to the quantum dot, the ac field
will not couple the different spin states. Only the eigenvalues of
$H_dot(t)$ will be modulated, see Eq.\ (\ref{full.Hdot}).
{The corresponding Floquet Hamiltonian decomposes into two infinite
blocks for the two spin directions, where each block has the
tridiagonal form}
\begin{eqnarray}
h_\mathrm{Fl} = \left(
\begin{array}{ccccc}
\ddots & & & & \\
& \epsilon+\Omega & V_\mathrm{ac}/2 & 0 & \\
& V_\mathrm{ac}/2 & \epsilon & V_\mathrm{ac}/2 & \\
& 0 & V_\mathrm{ac}/2 & \epsilon-\Omega \\
& & & & \ddots
\end{array}
\right).
\end{eqnarray}
Therefore, the electrons can tunnel through the quantum dot via
infinitely many Floquet channels with the same quasienergy in the
first Brillouin zone $[0,Ω)$ but all possible Floquet indices
$k$. The quasienergies in the extended zone scheme thus differ by
integer multiples of $Ω$. These quasienergy differences define
the peak positions in the noise spectra. In Fig.\ \ref{SVac}, a peak
at $Ω$ appears, corresponding to two channels with their
Floquet indices (photon numbers) differing by unity. One should also
expect peak structures at $nΩ$, $n>1$. However, to observe
these peak structures, one may need a stronger ac field to enable
multi-photon-assisted transport. In the inset of Fig.\ \ref{SVac}, a
small shoulder emerges at $2Ω$ for the largest amplitude,
So far in our discussion, no dc magnetic field has been considered.
For the quantum dot with an ac gate voltage and a dc magnetic field
in the $z$ direction, our results of the noise spectra for different
voltage amplitudes are displayed in Fig.\ \ref{SVacZ}. The
parameters are the same as those used in Fig.\ \ref{SVac} except
that the strength of the dc magnetic field in the $z$ direction is
$B_z = 1.6$. For the present set of parameters, the peak at $Ω$
is replaced by a dip. We observe that the appearance of the peak or
the dip depends on the detailed parameters used in our calculation.
The peak or dip position remains unchanged as we increase the
voltage amplitude. We have checked that the noise spectrum does not
depend on the direction of the dc magnetic field. This is because
the full SU(2) symmetry is preserved since we have included the time
evolution of the off-diagonal elements of the reduced density matrix
within our quantum master-equation approach.
Contrary to the quantum dot with an ac gate voltage, for the case of
a rotating magnetic field, the noise spectra do depend on the
direction of the additional dc magnetic field. When the dc magnetic
field is perpendicular to the plane of the rotating magnetic field,
the noise spectra behave much like those for a pure rotating
magnetic field. Only one peak appears at a non-zero frequency and
the peak position shifts with the amplitude of the rotating magnetic
field. Numerical results for the noise with a rotating magnetic in
the $xy$ plane and a dc magnetic field in the $z$ direction are
displayed in Fig.\ \ref{SRacZ}. The parameters are the same as in
Fig.\ \ref{SRac} and the dc magnetic field is $B_z=1.6$. It is
easy to verify that the peak position can again be determined by the
difference between two Floquet quasienergies.
When a non-zero dc magnetic field $B_x$ is applied in the plane of
the rotating magnetic field, the Floquet Hamiltonian cannot be
reduced to a $2×2$ matrix form as for the previously discussed
situation of vanishing dc magnetic field, since the $B_x$ term in
Eq.\ (\ref{full.Hdot}) mixes spin-up and spin-down states. Together
with the rotating magnetic field this couples all Floquet states
with the same quasienergy in the first Brillouin zone and different
Floquet indices. Thus the electrons can tunnel through the quantum
dot via infinitely many Floquet channels. The interference between
these Floquet channels then gives rise to much richer behavior in
the noise spectra. Numerical results for the noise spectra of a
quantum dot driven by a rotating magnetic field in the $xy$ plane
and with a dc magnetic field in the $x$ direction are presented in
Fig.\ \ref{SRacX}. The parameters used in the calculation are the
same as in Fig.\ \ref{SRac} and the dc magnetic field in $B_x=1.6$.
In Fig.\ \ref{SRacX}, more peaks are observed than for vanishing dc
magnetic field in Fig.\ \ref{SRac}. One can see that besides the
peak determined by Eq.\ (\ref{EqPeak}), there are both peaks (dips)
fixed at $Ω$ and structures at positions depending on the
ac-field amplitude $B_ac$. All peak (dip) positions
correspond to the differences of available Floquet quasienergies.
\section{Summary}
\label{sec:sum}
In this paper, the transport properties of a single-level quantum
dot modulated by either an ac gate voltage or a rotating magnetic
field have been studied within the Floquet quantum master-equation
approach in the sequential-tunneling limit. We have
employed a generalized MacDonald formula to obtain the
time-averaged current noise spectra
for both cases. Numerical results for the differential conductance
and the frequency-dependent current noise have been presented.
Besides the usual diamond structure due to the Coulomb blockade in
the differential conductance, photon-assisted tunneling can give
rise to additional lines parallel to the edges of the Coulomb
diamond. These lines cannot survive inside the Coulomb diamond in
the case of a rotating magnetic field. {This is due to the fact that
the rotating magnetic field only periodically rotates the spin
direction while the energy of the electron on the dot remains
unchanged.} The frequency-dependent noise spectra of the quantum dot
show additional peaks or dips in the presence of an ac field. The
behavior of these additional structures depends on the nature of the
ac driving field. In the case of an ac gate voltage, the position of
the finite-frequency peak is fixed at the external ac frequency,
independently of the voltage amplitude. On the other hand, in the
case of a rotating magnetic field, the peak at non-zero frequency
moves with changing amplitude of the rotating magnetic field. An
additional dc magnetic field in the plane of the rotating magnetic
field can also drastically change the noise spectra: {it leads to
the appearance of both movable and fixed peak structures in the
noise spectra}. All these peak positions are found to be determined
by the energy differences between two Floquet transport channels.
\newpage
\appendix
\section{Derivation of the generalized MacDonald formula with ac field}
\label{app:B}
A derivation of the MacDonald formula for time-independent
transport has been presented in Ref.\ \onlinecite{PRB75045340}. We
show here that the periodicity of our time-dependent Hamiltonian makes it
possible to estimate the noise spectra by generalizing the MacDonald
formula. The derivation of the generalized MacDonald formula for the
time-averaged noise spectra is outlined in the following.
We start from the Fourier-transformed current correlation function
\begin{eqnarray}\label{AB1}
S(t,\omega)=\int^\infty_{-\infty}d\tau\, e^{i\omega\tau}\left\langle\left\{
\delta I(t),\delta I(t-\tau)\right\}\right\rangle,
\end{eqnarray}
where $δI(t)=I(t)-⟨I(t)⟩$ and ${A, B}=AB+BA$. For
simplicity, we omit the lead index in the noise
expressions in this appendix.
From the definition of the current, we have
\begin{eqnarray}
\int^{t+\tau}_t dt'\, \delta I(t') &=&\int^{t+\tau}_t dt' \left[I(t')-\langle
I(t')\rangle\right] \\ \nonumber
&=&eN(t+\tau,t)-\int^{t+\tau}_t dt'\,\langle I(t')\rangle ,
\end{eqnarray}
where $N(t+τ,t)$ denotes the number of charges transferred
during the interval from $t$ to $t+τ$.
Taking the expectation value of the square of this equation, we obtain
\begin{eqnarray}\label{Eq4}
\lefteqn{
= \left\langle\int^{t+\tau}_tdt'\int^{t+\tau}_tdt''\left[\delta I(t')\delta I(t'')
\delta I(t'')\delta I(t')\right]\right\rangle } \nonumber \\
&& = \int^{t+\tau}_t dt'\int^{t+\tau}_tdt''\int^\infty_{-\infty}d\omega\,
\frac{1}{2\pi} \, S(t',\omega)\,
e^{i\omega(t'-t'')} .\hspace{15em}
\end{eqnarray}
Inserting the Fourier decomposition of the time-dependent
\begin{equation}
S(t',\omega)=S_0(\omega)+\sum_{k\neq 0}e^{-ik\Omega t'}
\end{equation}
into Eq.\ (\ref{Eq4}), we obtain
\begin{eqnarray}
\ldots&=&\int^{t+\tau}_tdt'\int^\infty_{-\infty}d\omega\,\frac{1}{2\pi}\,
\left[S_0(\omega)+\sum_k\nolimits'
e^{-ik\Omega t'}S_k(\omega)\right]e^{-i\omega t'}
\left[\frac{1}{i\omega}\,(e^{i\omega(t+\tau)} -e^{i\omega t})\right]\\
\nonumber &=&\int^\infty_{-\infty}d\omega\,\frac{1}{2\pi}\,S_0(\omega)\,
\frac{1}{\omega^2}\,(e^{-i\omega\tau}-1)(e^{i\omega\tau}-1)\\
\nonumber
e^{-ik\Omega t}(e^{-i(\omega+k\Omega)\tau}-1)(e^{i\omega\tau}-1)
\\ \nonumber
\nonumber
\frac{e^{-ik\Omega
\end{eqnarray}
where we have used the notation $∑_k'=∑_k≠0$.
Differentiation with respect to $τ$ gives
\begin{eqnarray}
\lefteqn{ \frac{d}{d\tau}\,2e^2\left\langle
\left[N(t+\tau,t)-\int^{t+\tau}_t dt'\, \langle I(t')\rangle/e
\right]^2\right\rangle
} \nonumber \\
t}\frac{1}{\omega(\omega+k\Omega)} \left(-ik\Omega e^{-ik\Omega
\tau}+i(\omega+k\Omega)e^{-i(\omega+k\Omega)\tau}-i\omega
e^{i\omega\tau}\right) . \qquad
\end{eqnarray}
We next perform a Fourier transformation
and take the time average over one period. The
second term on the right-hand side vanishes due to its periodicity.
Since the current correlation function is symmetric, $S(t,t')=S(t',t)$, it
can be shown that $S̅(ω)=1/𝒯∫^𝒯_0 dt
∫^∞_-∞ dt' S(t, t')e^iω(t-t')$ has the
property $S̅(ω)=S̅(-ω)$.
We arrive at the generalized formula for the time-averaged noise
spectrum for a periodic driving field,
\begin{eqnarray}
\lefteqn{ \frac{1}{\mathcal{T}}\int^\mathcal{T}_0dt\, 2e^2\int^\infty_{-\infty} d\tau\,
e^{i\omega \tau} \frac{\partial}{\partial \tau}\left\langle
[N(t+\tau,t)-\int^{t+\tau}_tdt'\langle I(t')\rangle/e ]^2\right\rangle } \nonumber \\
&& = \frac{1}{\mathcal{T}}\int^\mathcal{T}_0dt\, 2e^2\int^\infty_{-\infty}d\tau\,
\sin(\omega \tau)\frac{\partial}{\partial \tau}\left\langle [N(t+\tau,t)-\int^{t+\tau}_tdt'\langle I(t')\rangle/e
]^2\right\rangle \nonumber \\
&& = 2i\,\frac{\bar S(\omega)}{\omega} .
\end{eqnarray}
Noting that the integrand of the $τ$ integral is even, we
obtain the final result for the generalized MacDonald formula for the
time-averaged noise spectrum,
\begin{eqnarray}\label{MDAppend}
\frac{\bar S(\omega)}{\omega}=2e^2\,\frac{1}{\mathcal{T}}\int^\mathcal{T}_0 dt
\int^\infty_{0}d\tau\,\sin \omega\tau\: \frac{\partial}{\partial
\tau}\left\langle\left [N(t+\tau,t)-\int^{t+\tau}_tdt'\,
\langle I(t')\rangle/e\right]^2\right\rangle.
\end{eqnarray}
In comparison to the MacDonald formula for steady state
transport, an
integration of $t$ over one period is carried out to obtain the
time-averaged noise spectra. The time average can also be
expressed by an average over the initial phase of the ac field. Hence, our
expression is equivalent to the form given by Clerk and
\newpage
%Merlin.mbs v4.21 2009-07-09.
\begin{thebibliography}{10}%
\makeatletter
\providecommand \@ifxundefined [1]{%
\ifx #1\undefined \expandafter \@firstoftwo
\else \expandafter \@secondoftwo
\fi
\providecommand \@ifnum [1]{%
\ifnum #1\expandafter \@firstoftwo
\else \expandafter \@secondoftwo
\fi
\providecommand \enquote [1]{``#1''}%
\providecommand \bibnamefont [1]{#1}%
\providecommand \bibfnamefont [1]{#1}%
\providecommand \citenamefont [1]{#1}%
\providecommand\href[0]{\@sanitize\@href}%
\providecommand\@href[1]{\endgroup\@@startlink{#1}\endgroup\@@href}%
\providecommand\@@href[1]{#1\@@endlink}%
\providecommand \@sanitize [0]{\begingroup\catcode`\&12\catcode`\#12\relax}%
\@ifxundefined \pdfoutput {\@firstoftwo}{%
\@ifnum{\z@=\pdfoutput}{\@firstoftwo}{\@secondoftwo}%
\providecommand\@@startlink[1]{\leavevmode\special{html:<a href="#1">}}%
\providecommand\@@endlink[0]{\special{html:</a>}}%
\providecommand\@@startlink[1]{%
\leavevmode
\pdfstartlink
attr{/Border[0 0 1 ]/H/I/C[0 1 1]}%
\relax
\providecommand\@@endlink[0]{\pdfendlink}%
\providecommand \url [0]{\begingroup\@sanitize \@url }%
\providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix}}%
\providecommand \urlprefix [0]{URL }%
\providecommand \Eprint[0]{\href }%
\@ifxundefined \urlstyle {%
\providecommand \doi [1]{doi:\discretionary{}{}{}#1}%
\providecommand \doi [0]{doi:\discretionary{}{}{}\begingroup
\urlstyle{rm}\Url }%
\providecommand \doibase [0]{http://dx.doi.org/}%
\providecommand \Doi[1]{\href{\doibase#1}}%
\providecommand \bibAnnote [3]{%
\BibitemShut{#1}%
\begin{quotation}\noindent
\textsc{Key:}\ #2\\\textsc{Annotation:}\ #3%
\end{quotation}%
\providecommand \bibAnnoteFile [2]{%
\IfFileExists{#2}{\bibAnnote {#1} {#2} {\input{#2}}}{}%
\providecommand \typeout [0]{\immediate \write \m@ne }%
\providecommand \selectlanguage [0]{\@gobble}%
\providecommand \bibinfo [0]{\@secondoftwo}%
\providecommand \bibfield [0]{\@secondoftwo}%
\providecommand \translation [1]{[#1]}%
\providecommand \BibitemOpen[0]{}%
\providecommand \bibitemStop [0]{}%
\providecommand \bibitemNoStop [0]{.\EOS\space}%
\providecommand \EOS [0]{\spacefactor3000\relax}%
\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{J.~R.}\ \bibnamefont{Heath}}\ and\ \bibinfo
{author} {\bibfnamefont{M.~A.}\ \bibnamefont{Ratner}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Physics Today}\ }%
\textbf{\bibinfo {volume} {56}},\ \bibinfo {pages} {43} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PT5643}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{L.~S.}\ \bibnamefont{Levitov}}, \bibinfo
{author} {\bibfnamefont{H.~W.}\ \bibnamefont{Lee}},\ and\ \bibinfo {author}
{\bibfnamefont{G.~B.}\ \bibnamefont{Lesovik}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {J. Math. Phys.}\ }%
\textbf{\bibinfo {volume} {37}},\ \bibinfo {pages} {4845} (\bibinfo {year}
\bibAnnoteFile{NoStop}{JMP374845}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{C.}~\bibnamefont{Flindt}}, \bibinfo {author}
{\bibfnamefont{T.}~\bibnamefont{Novotny}},\ and\ \bibinfo {author}
{\bibfnamefont{A.-P.}\ \bibnamefont{Jauho}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {70}},\ \bibinfo {pages} {205334} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB70205334}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{M.}~\bibnamefont{Vanevic}}, \bibinfo
{author} {\bibfnamefont{Y.~V.}\ \bibnamefont{Nazarov}},\ and\ \bibinfo
{author} {\bibfnamefont{W.}~\bibnamefont{Belzig}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {99}},\ \bibinfo {pages} {076601} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL99076601}%
\BibitemOpen
\emph{\bibinfo {title} {Quantum noise in mesoscopic physics}},\ edited by\
\bibinfo {editor} {\bibfnamefont{Y.~V.}\ \bibnamefont{Nazarov}}\ (\bibinfo
{publisher} {Kluwer Academic Publishers},\ \bibinfo {address} {Dordrecht},\
\bibinfo {year} {2003})%
\bibAnnoteFile{NoStop}{Nazarov}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{J.}~\bibnamefont{Koch}}\ and\ \bibinfo
{author} {\bibfnamefont{F.}~\bibnamefont{von Oppen}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {94}},\ \bibinfo {pages} {206804} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL94206804}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{J.}~\bibnamefont{Koch}}, \bibinfo {author}
{\bibfnamefont{F.}~\bibnamefont{von Oppen}},\ and\ \bibinfo {author}
{\bibfnamefont{A.~V.}\ \bibnamefont{Andreev}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {74}},\ \bibinfo {pages} {205438} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB74205438}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{X.}~\bibnamefont{Zhong}}\ and\ \bibinfo
{author} {\bibfnamefont{J.~C.}\ \bibnamefont{Cao}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {J. Phys.: Condens. Matter}\ }%
\textbf{\bibinfo {volume} {21}},\ \bibinfo {pages} {295602} (\bibinfo {year}
\bibAnnoteFile{NoStop}{JPCM21295602}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{J.}~\bibnamefont{Bylander}}, \bibinfo
{author} {\bibfnamefont{T.}~\bibnamefont{Duty}},\ and\ \bibinfo {author}
{\bibfnamefont{P.}~\bibnamefont{Delsing}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Nature}\ }%
\textbf{\bibinfo {volume} {434}},\ \bibinfo {pages} {361} (\bibinfo {year}
\bibAnnoteFile{NoStop}{Nature434361}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{T.}~\bibnamefont{Fujisawa}}, \bibinfo
{author} {\bibfnamefont{T.}~\bibnamefont{Hayashi}}, \bibinfo {author}
{\bibfnamefont{R.}~\bibnamefont{Tomita}},\ and\ \bibinfo {author}
{\bibfnamefont{Y.}~\bibnamefont{Hirayama}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Science}\ }%
\textbf{\bibinfo {volume} {312}},\ \bibinfo {pages} {1634} (\bibinfo {year}
\bibAnnoteFile{NoStop}{Science3121634}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{S.}~\bibnamefont{Gustavsson}}, \bibinfo
{author} {\bibfnamefont{R.}~\bibnamefont{Leturcq}}, \bibinfo {author}
{\bibfnamefont{B.}~\bibnamefont{Simovi\v{c}}}, \bibinfo {author}
{\bibfnamefont{R.}~\bibnamefont{Schleser}}, \bibinfo {author}
{\bibfnamefont{T.}~\bibnamefont{Ihn}}, \bibinfo {author}
{\bibfnamefont{P.}~\bibnamefont{Studerus}}, \bibinfo {author}
{\bibfnamefont{K.}~\bibnamefont{Ensslin}}, \bibinfo {author}
{\bibfnamefont{D.~C.}\ \bibnamefont{Driscoll}},\ and\ \bibinfo {author}
{\bibfnamefont{A.~C.}\ \bibnamefont{Gossard}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {96}},\ \bibinfo {pages} {076605} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL96076605}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{Y.~M.}\ \bibnamefont{Blanter}}\ and\
\bibinfo {author} {\bibfnamefont{M.}~\bibnamefont{B\"uttiker}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rep.}\ }%
\textbf{\bibinfo {volume} {336}},\ \bibinfo {pages} {1} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PR3361}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{O.}~\bibnamefont{Zarchin}}, \bibinfo
{author} {\bibfnamefont{M.}~\bibnamefont{Zaffalon}}, \bibinfo {author}
{\bibfnamefont{M.}~\bibnamefont{Heiblum}}, \bibinfo {author}
{\bibfnamefont{D.}~\bibnamefont{Mahalu}},\ and\ \bibinfo {author}
{\bibfnamefont{V.}~\bibnamefont{Umansky}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {77}},\ \bibinfo {pages} {R241303} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB77241303}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{R.}~\bibnamefont{Aguado}}\ and\ \bibinfo
{author} {\bibfnamefont{T.}~\bibnamefont{Brandes}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {92}},\ \bibinfo {pages} {206601} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL92206601}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{F.~W.~J.}\ \bibnamefont{Hekking}}\ and\
\bibinfo {author} {\bibfnamefont{J.~P.}\ \bibnamefont{Pekola}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {96}},\ \bibinfo {pages} {056603} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL96056603}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{A.}~\bibnamefont{Thielmann}}, \bibinfo
{author} {\bibfnamefont{M.~H.}\ \bibnamefont{Hettler}}, \bibinfo {author}
{\bibfnamefont{J.}~\bibnamefont{K\"onig}},\ and\ \bibinfo {author}
{\bibfnamefont{G.}~\bibnamefont{Sch\"on}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {68}},\ \bibinfo {pages} {115105} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB68115105}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{E.~A.}\ \bibnamefont{Rothstein}}, \bibinfo
{author} {\bibfnamefont{O.}~\bibnamefont{Entin-Wohlman}},\ and\ \bibinfo
{author} {\bibfnamefont{A.}~\bibnamefont{Aharony}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {79}},\ \bibinfo {pages} {075307} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB79075307}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{S.}~\bibnamefont{Camalet}}, \bibinfo
{author} {\bibfnamefont{J.}~\bibnamefont{Lehmann}}, \bibinfo {author}
{\bibfnamefont{S.}~\bibnamefont{Kohler}},\ and\ \bibinfo {author}
{\bibfnamefont{P.}~\bibnamefont{H\"anggi}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {90}},\ \bibinfo {pages} {210602} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL90210602}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{B.~H.}\ \bibnamefont{Wu}}\ and\ \bibinfo
{author} {\bibfnamefont{J.~C.}\ \bibnamefont{Cao}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {77}},\ \bibinfo {pages} {233307} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB77233307}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{B.~H.}\ \bibnamefont{Wu}}\ and\ \bibinfo
{author} {\bibfnamefont{J.~C.}\ \bibnamefont{Cao}},\ }%
\bibinfo {note} {arXiv:0906.5266 (2009)}%
\bibAnnoteFile{NoStop}{arxiv09065266}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{M.}~\bibnamefont{Braun}}, \bibinfo {author}
{\bibfnamefont{J.}~\bibnamefont{K\"onig}},\ and\ \bibinfo {author}
{\bibfnamefont{J.}~\bibnamefont{Martinek}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {74}},\ \bibinfo {pages} {075328} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB74075328}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{D.~K.~C.}\ \bibnamefont{MacDonald}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Rep. Prog. Phys.}\ }%
\textbf{\bibinfo {volume} {12}},\ \bibinfo {pages} {56} (\bibinfo {year}
\bibAnnoteFile{NoStop}{RPP1256}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{J.~Y.}\ \bibnamefont{Luo}}, \bibinfo
{author} {\bibfnamefont{X.~Q.}\ \bibnamefont{Li}},\ and\ \bibinfo {author}
{\bibfnamefont{Y.~J.}\ \bibnamefont{Yan}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {76}},\ \bibinfo {pages} {085325} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB76085325}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{N.}~\bibnamefont{Lambert}}, \bibinfo
{author} {\bibfnamefont{R.}~\bibnamefont{Aguado}},\ and\ \bibinfo {author}
{\bibfnamefont{T.}~\bibnamefont{Brandes}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {75}},\ \bibinfo {pages} {045340} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB75045340}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{B.}~\bibnamefont{Dong}}, \bibinfo {author}
{\bibfnamefont{X.~L.}\ \bibnamefont{Lei}},\ and\ \bibinfo {author}
{\bibfnamefont{N.~J.~M.}\ \bibnamefont{Horing}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {J. Appl. Phys.}\ }%
\textbf{\bibinfo {volume} {104}},\ \bibinfo {pages} {033532} (\bibinfo {year}
\bibAnnoteFile{NoStop}{JAP104033532}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{P.~K.}\ \bibnamefont{Tien}}\ and\ \bibinfo
{author} {\bibfnamefont{J.~P.}\ \bibnamefont{Gordon}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev.}\ }%
\textbf{\bibinfo {volume} {129}},\ \bibinfo {pages} {647} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PR129647}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{P.~W.}\ \bibnamefont{Brouwer}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {58}},\ \bibinfo {pages} {R10135} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB58R10135}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{S.~D.}\ \bibnamefont{Barrett}}\ and\
\bibinfo {author} {\bibfnamefont{T.~M.}\ \bibnamefont{Stace}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {96}},\ \bibinfo {pages} {017405} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL96017405}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{J.}~\bibnamefont{Wabnig}}, \bibinfo {author}
{\bibfnamefont{B.~W.}\ \bibnamefont{Lovett}}, \bibinfo {author}
{\bibfnamefont{J.~H.}\ \bibnamefont{Jefferson}},\ and\ \bibinfo {author}
{\bibfnamefont{G.~A.~D.}\ \bibnamefont{Briggs}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {102}},\ \bibinfo {pages} {016802} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL102016802}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{G.}~\bibnamefont{Floquet}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Ann. \'Ecole Norm. Sup.}\ }%
\textbf{\bibinfo {volume} {12}},\ \bibinfo {pages} {47} (\bibinfo {year}
\bibAnnoteFile{NoStop}{Floquet}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{M.}~\bibnamefont{Grifoni}}\ and\ \bibinfo
{author} {\bibfnamefont{P.}~\bibnamefont{H\"anggi}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rep.}\ }%
\textbf{\bibinfo {volume} {304}},\ \bibinfo {pages} {229} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PR304229}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{M.}~\bibnamefont{Moskalets}}\ and\ \bibinfo
{author} {\bibfnamefont{M.}~\bibnamefont{B\"uttiker}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {66}},\ \bibinfo {pages} {205320} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB66205320}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{B.~H.}\ \bibnamefont{Wu}}\ and\ \bibinfo
{author} {\bibfnamefont{J.~C.}\ \bibnamefont{Cao}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {73}},\ \bibinfo {pages} {245312} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB73245312}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{V.}~\bibnamefont{Gudmundsson}}, \bibinfo
{author} {\bibfnamefont{G.}~\bibnamefont{Thorgilsson}}, \bibinfo {author}
{\bibfnamefont{C.}~\bibnamefont{Tang}},\ and\ \bibinfo {author}
{\bibfnamefont{V.}~\bibnamefont{Moldoveanu}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {77}},\ \bibinfo {pages} {035329} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB77035329}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{L.~E.~F.}\ \bibnamefont{{Foa Torres}}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {72}},\ \bibinfo {pages} {245339} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB72245339}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{L.}~\bibnamefont{Arrachea}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {72}},\ \bibinfo {pages} {125349} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB72125349}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{B.~H.}\ \bibnamefont{Wu}}\ and\ \bibinfo
{author} {\bibfnamefont{J.~C.}\ \bibnamefont{Cao}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {J. Phys.: Condens. Matter}\ }%
\textbf{\bibinfo {volume} {20}},\ \bibinfo {pages} {085224} (\bibinfo {year}
\bibAnnoteFile{NoStop}{JPCM20085224}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{H.-P.}\ \bibnamefont{Breuer}}\ and\ \bibinfo
{author} {\bibfnamefont{F.}~\bibnamefont{Petruccione}},\ }%
\emph{\bibinfo {title} {The Theory of Open Quantum Systems}}\ (\bibinfo
{publisher} {Oxford University Press},\ \bibinfo {address} {Oxford},\
\bibinfo {year} {2002})%
\bibAnnoteFile{NoStop}{Breuer}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{C.}~\bibnamefont{Timm}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {77}},\ \bibinfo {pages} {195416} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB77195416}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{F.}~\bibnamefont{Elste}}\ and\ \bibinfo
{author} {\bibfnamefont{C.}~\bibnamefont{Timm}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {71}},\ \bibinfo {pages} {155403} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB71155403}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{R.}~\bibnamefont{Bl\"{u}mel}}, \bibinfo
{author} {\bibfnamefont{A.}~\bibnamefont{Buchleitner}}, \bibinfo {author}
{\bibfnamefont{R.}~\bibnamefont{Graham}}, \bibinfo {author}
{\bibfnamefont{L.}~\bibnamefont{Sirko}}, \bibinfo {author}
{\bibfnamefont{U.}~\bibnamefont{Smilansky}},\ and\ \bibinfo {author}
{\bibfnamefont{H.}~\bibnamefont{Walther}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. A}\ }%
\textbf{\bibinfo {volume} {44}},\ \bibinfo {pages} {4521} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRA444521}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{H.-P.}\ \bibnamefont{Breuer}}, \bibinfo
{author} {\bibfnamefont{W.}~\bibnamefont{Huber}},\ and\ \bibinfo {author}
{\bibfnamefont{F.}~\bibnamefont{Petruccione}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. E}\ }%
\textbf{\bibinfo {volume} {61}},\ \bibinfo {pages} {4883} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRE614883}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{J.}~\bibnamefont{Lehmann}}, \bibinfo
{author} {\bibfnamefont{S.}~\bibnamefont{Kohler}}, \bibinfo {author}
{\bibfnamefont{V.}~\bibnamefont{May}},\ and\ \bibinfo {author}
{\bibfnamefont{P.}~\bibnamefont{H\"anggi}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {J. Chem. Phys.}\ }%
\textbf{\bibinfo {volume} {121}},\ \bibinfo {pages} {2278} (\bibinfo {year}
\bibAnnoteFile{NoStop}{JCP1212278}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{F.~J.}\ \bibnamefont{Kaiser}}\ and\ \bibinfo
{author} {\bibfnamefont{S.}~\bibnamefont{Kohler}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Ann. Phys. (Leipzig)}\ }%
\textbf{\bibinfo {volume} {16}},\ \bibinfo {pages} {702} (\bibinfo {year}
\bibAnnoteFile{NoStop}{AP16702}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{F.}~\bibnamefont{Cavaliere}}, \bibinfo
{author} {\bibfnamefont{M.}~\bibnamefont{Governale}},\ and\ \bibinfo {author}
{\bibfnamefont{J.}~\bibnamefont{K\"onig}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {103}},\ \bibinfo {pages} {136801} (\bibinfo {year}
\bibAnnoteFile{NoStop}{arxiv09041687}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{A.~A.}\ \bibnamefont{Clerk}}\ and\ \bibinfo
{author} {\bibfnamefont{S.~M.}\ \bibnamefont{Girvin}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {70}},\ \bibinfo {pages} {121303} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB70121303}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{S.}~\bibnamefont{Datta}},\ }%
\enquote{\bibinfo {title} {Fock space formulation for nanoscale transport},}\
\bibinfo {note} {Cond-mat/0603034 (2006)}%
\bibAnnoteFile{NoStop}{Datta_arxiv}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{M.}~\bibnamefont{Esposito}}\ and\ \bibinfo
{author} {\bibfnamefont{M.}~\bibnamefont{Galperin}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. B}\ }%
\textbf{\bibinfo {volume} {79}},\ \bibinfo {pages} {205303} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRB79205303}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{C.}~\bibnamefont{Flindt}}, \bibinfo {author}
{\bibfnamefont{T.}~\bibnamefont{Novotny}}, \bibinfo {author}
{\bibfnamefont{A.}~\bibnamefont{Braggio}}, \bibinfo {author}
{\bibfnamefont{M.}~\bibnamefont{Sassetti}},\ and\ \bibinfo {author}
{\bibfnamefont{A.-P.}\ \bibnamefont{Jauho}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Phys. Rev. Lett.}\ }%
\textbf{\bibinfo {volume} {100}},\ \bibinfo {pages} {150601} (\bibinfo {year}
\bibAnnoteFile{NoStop}{PRL100150601}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{S.}~\bibnamefont{Ramo}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {Proc. IRE}\ }%
\textbf{\bibinfo {volume} {27}},\ \bibinfo {pages} {584} (\bibinfo {year}
\bibAnnoteFile{NoStop}{Ramo1939}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{W.}~\bibnamefont{Shockley}},\ }%
\bibfield{journal}{%
\bibinfo {journal} {J. Appl. Phys.}\ }%
\textbf{\bibinfo {volume} {9}},\ \bibinfo {pages} {635} (\bibinfo {year}
\bibAnnoteFile{NoStop}{Shockley1938}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{H.~J.}\ \bibnamefont{Carmichael}},\ }%
\emph{\bibinfo {title} {Statistical Methods in Quantum Optics 1: Master
Equations and Fokker-Planck Equations}}\ (\bibinfo {publisher} {Springer},\
\bibinfo {address} {Berlin},\ \bibinfo {year} {2002})%
\bibAnnoteFile{NoStop}{MasterEquation}%
\BibitemOpen
\bibfield{author}{%
\bibinfo {author} {\bibfnamefont{C.~W.}\ \bibnamefont{Gardiner}}\ and\
\bibinfo {author} {\bibfnamefont{P.}~\bibnamefont{Zoller}},\ }%
\emph{\bibinfo {title} {Quantum Noise}},\ \bibinfo {edition} {2nd}\ ed.\
(\bibinfo {publisher} {Springer},\ \bibinfo {address} {Berlin},\ \bibinfo
{year} {2000})%
\bibAnnoteFile{NoStop}{QuantumNoise}%
\end{thebibliography}%
\newpage
\includegraphics[angle=0,width=0.8\textwidth]{dGdc_bw.eps} %{dGdc1.eps}
\includegraphics[angle=0,width=0.8\textwidth]{dGVac_bw.eps} %{dGVac1.eps}
\newpage
\includegraphics[angle=0,width=0.8\textwidth]{dGBxy_bw.eps} %{dGBxy1.eps}
\begin{figure}
\includegraphics[angle=0,width=0.8\textwidth]{dGBxyBx_bw.eps} %{dGBxyBx1.eps}
\caption{Gray-scale plots of the differential conductance
$dI/dV_\mathrm{dc}$ as a function of the dc bias voltage and the dc
gate voltage for a quantum dot (a) without any ac fields, (b)
modulated by an ac gate voltage, (c) modulated by a rotating
magnetic field, and (d) modulated by a rotating magnetic field and
with an additional dc magnetic field applied in the plane of the
rotating field. Dark regions represent low differential conductance.
The frequency of the ac field is $\Omega=8$. The amplitude of the ac
gate voltage in (b) and of the rotating magnetic field in (c), (d)
are $V_\mathrm{ac}=6.4$ and $B_\mathrm{ac}=3.2$, respectively. The
dc magnetic field in (d) is $B_x=1.6$.} \label{dIdV}
\end{figure}
\newpage
\begin{figure}
\includegraphics[width=0.8\textwidth]{FanoVac.eps}
\caption{(Color online) Fano factor of the quantum dot as a function
of dc bias voltage $V_{dc}$ in unit of $V_G$ for different
amplitudes $V_\mathrm{ac}$ of the ac gate voltage. The parameters of
the device are given in the text.}\label{FanoVac}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\textwidth]{FanoRac.eps}
\caption{(Color online) Fano factor of the quantum dot as a function
of dc bias voltage $V_{dc}$ in unit of $V_G$ for different
amplitudes $B_\mathrm{ac}$ of the rotating magnetic field. The other
parameters are the same with those of Fig.
\ref{FanoVac}.}\label{FanoRac}
\end{figure}
\begin{figure}
\includegraphics[angle=0,width=0.8\textwidth]{Vac2.eps}
\caption{(Color online) Noise spectra for different gate voltage amplitudes
$V_\mathrm{ac}$. Independently of $V_\mathrm{ac}$, the main peak position is fixed at $\Omega$.
The inset shows an enlarged view of the noise spectra
around $2\Omega$.}\label{SVac}
\end{figure}
\begin{figure}
\includegraphics[angle=0,width=0.8\textwidth]{acBxy0.eps}
\caption{(Color online) Noise spectra with a rotating magnetic field in the
$xy$ plane with various amplitudes $B_\mathrm{ac}$.
The peak position shifts with increasing $B_\mathrm{ac}$.}\label{SRac}
\end{figure}
\begin{figure}
\includegraphics[angle=0,width=0.8\textwidth]{VacBz.eps}
\caption{(Color online) Noise spectra for a quantum dot modulated by an
oscillating gate voltage with various
amplitudes $V_\mathrm{ac}$ in the presence of a dc magnetic field $B_z=1.6$
in the $z$ direction. The dip position is fixed at the driving frequency.}\label{SVacZ}
\end{figure}
\begin{figure}
\includegraphics[angle=0,width=0.8\textwidth]{acBxyZ.eps}
\caption{(Color online) Noise spectra in the presence of a rotating magnetic
field in the $xy$ plane with various amplitudes $B_\mathrm{ac}$.
A dc magnetic field $B_z=1.6$ is applied in the $z$ direction. Only one peak
appears, which shifts with $B_\mathrm{ac}$.}\label{SRacZ}
\end{figure}
\begin{figure}
\includegraphics[angle=0,width=0.8\textwidth]{acBxyX.eps}
\caption{(Color online) Noise spectra with a rotating magnetic field in the
$xy$ plane with various amplitudes $B_\mathrm{ac}$.
A dc magnetic field $B_x=1.6$ is applied in the $x$ direction.
New features appear in the noise spectra
since more Floquet channels are involved in the transport in
the presence of an in-plane magnetic field.
\end{figure}
\end{document}
|
arxiv-papers
| 2009-10-30T15:32:19 |
2024-09-04T02:49:06.181821
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B. H. Wu, and C. Timm",
"submitter": "Binhe Wu",
"url": "https://arxiv.org/abs/0910.5894"
}
|
0911.0045
|
# Disordered, stretched, and semiflexible biopolymers in two dimensions
Zicong Zhou zzhou@mail.tku.edu.tw Department of Physics, Tamkang University,
151 Ying-chuan, Tamsui 25137, Taiwan, Republic of China Béla Joós
bjoos@uottawa.ca http://www.science.uottawa.ca/~bjoos/ Ottawa Carleton
Institute for Physics, University of Ottawa Campus, Ottawa, Ontario, Canada,
K1N-6N5
###### Abstract
We study the effects of intrinsic sequence-dependent curvature for a two
dimensional semiflexible biopolymer with short-range correlation in intrinsic
curvatures. We show exactly that when not subjected to any external force,
such a system is equivalent to a system with a well-defined intrinsic
curvature and a proper renormalized persistence length. We find the exact
expression for the distribution function of the equivalent system. However, we
show that such an equivalent system does not always exist for the polymer
subjected to an external force. We find that under an external force, the
effect of sequence-disorder depends upon the averaging order, the degree of
disorder, and the experimental conditions, such as the boundary conditions.
Furthermore, a short to moderate length biopolymer may be much softer or has a
smaller apparent persistent length than what would be expected from the
“equivalent system”. Moreover, under a strong stretching force and for a long
biopolymer, the sequence-disorder is immaterial for elasticity. Finally, the
effect of sequence-disorder may depend upon the quantity considered.
Accepted to publish in Phys. Rev. E.
###### pacs:
87.15.-v, 87.10.Pq, 36.20.Ey, 87.15.A-
## I Introduction
It is known that sequence-dependent properties of biopolymers play a crucial
role in many biological processes. More specifically, sequence-disorder has
important influences on DNA packaging, transcription, replication,
recombination, and repair processes TA93 ; PH88 ; PN98 ; BDM98 ; SFCTFMWW06 ;
PT07 ; MFFA07 ; ATVDMA01 ; VAA05 ; AVADT02 ; ZJ08 . Owing to progress in
experimental techniques such as laser or magnetic tweezers and atomic force
microscopy, it is now possible to manipulate and observe single biomolecules
directly, and thus make a better comparison between theoretical predictions
and experimental observations.
In theoretical studies, a semiflexible biopolymer is often modeled as a
filament KP49 ; MS94 ; BMSS94 ; MS95 ; SABBC96 ; SFB92 ; ZJ08 ; PR00 ; ZZO00 ;
ZLJ05 ; ZJLYJ07 ; PN98 ; BDM98 ; PT07 ; AVADT02 ; ATVDMA01 ; VAA05 ; MFFA07 ;
ZZ07 . For instance, the wormlike chain (WLC) model, which views the
biopolymer as an inextensible chain with a uniform bending rigidity but with a
negligible cross section, has been used successfully to describe the entropic
elasticity of a long double-stranded DNA (dsDNA) KP49 ; MS94 ; BMSS94 ; MS95 .
However, the traditional elastic models are usually uniform and ignore the
role of sequence-disorder. Under what conditions such a simplification is
valid is therefore an intriguing question. Based on the elastic models, two
effects of sequence-disorder need to be considered. First, structural
inhomogeneity yields variations of the bending rigidity along the filament,
and results in an $s$-dependent persistence length $l_{p}(s)$ PT07 ; ZJ08 ,
where $s$ is the arc length. It has been reported that for a long biopolymer
with short-range correlation (SRC) in $l_{p}(s)$ and free of external force,
this effect can be accounted by a replacement of the $l_{p}(s)$ by an
appropriate average PT07 ; ZJ08 . However, for a short biopolymer,
inhomogeneity in $l_{p}(s)$ tends to make physical observables divergent PT07
; ZJ08 . Second, the local structure can be characterized by the intrinsic
sequence-dependent curvatures (i.e., the static curvature or the frozen-in
curvature) PH88 ; PN98 ; BDM98 ; SFCTFMWW06 ; PT07 ; MFFA07 ; ATVDMA01 ; VAA05
; AVADT02 ; ZJ08 , and this is also the focus of the present work. For a long
biopolymer with zero mean curvature, again it has been demonstrated that the
effect of intrinsic sequence-dependent curvatures can also be reduced into a
simple correction of the uniform persistence length, either free of external
force PT07 ; ZJ08 ; TTH87 ; SH95 or under moderate external force PN98 ; BDM98
; VATA03 . However, it is argued that sequence disorder is immaterial for the
elasticity of a long DNA under strong stretching force MS95 ; PN98 . In this
paper, we prove it exactly for a two dimensional biopolymer even with a
nonvanishing mean intrinsic curvature. Moreover, it is well known that the
short or intermediate-length DNAs play a more important role than the long
DNAs in biological processes, from DNA packaging, to transcription, gene
regulation and viral packaging JW01 ; RHL95 ; SLB81 . As a consequence, the
effect of intrinsic sequence-dependent curvatures for short or intermediate-
length biopolymers requires more attention. When the biopolymer is free of
external force and with SRC in intrinsic sequence-dependent curvatures, it has
been shown exactly that such a three-dimensional (3D) system is equivalent to
a system (we will refer it as the “equivalent system” henceforth) with a well-
defined (i.e., without randomness) intrinsic mean curvature and a corrected
persistence length PT07 ; ZJ08 , irrespective of its length. In this work, we
show that the same conclusion is also valid in the two-dimensional (2D) case
and present the general solution of the distribution function of the
equivalent 2D system. On the other hand, the effects of intrinsic sequence-
dependent curvatures in a short biopolymer under external force are not yet
known. In this paper, we demonstrate that under external force, the effect of
sequence-disorder for a short biopolymer is dependent on the average order and
the experimental conditions. Moreover, we find that the results are also
dependent on the boundary conditions (BC), and the short biopolymer looks
softer than what is expected from the “equivalent system”. Because
theoretically a 2D system is relatively easier to study and experiments on the
conformations of biopolymers are often conducted in a 2D environment MFFA07 ;
CSM99 , this work will focus on the 2D system.
The paper is organized as follows. In section II we set up our model. Section
III presents the exact proof of the existence of the “equivalent system” and
the general distribution function of the “equivalent system” for a force-free
biopolymer. In section IV we focus on the conformation and elasticity of the
biopolymer under constant external force. Follows a section discussing the
effects of disorder in a segment dependent curvature in the constant extension
ensemble. Finally, we end the paper with conclusions and discussions.
## II The model
The configuration of a filament with negligible cross section can be described
by the tangent vector, t$(s)$, to its contour line, where $s$ measures the
location along the filament. In two dimensions,
t$(s)=\\{\cos\phi(s),\sin\phi(s)\\}$, where the azimuthal angle $\phi$ is the
angle between the $x$-axis and t. The locus of the filament can be found by
$\displaystyle{\bf r}(s)=\\{x(s),y(s)\\}=\int_{0}^{s}{\bf t}(u)du.$ (1)
The reduced energy of the filament with intrinsic curvature but free of
external force can be written as:
$\displaystyle{\cal E}_{0}[\\{\phi(s)\\}]\equiv{E\over
k_{B}T}=\int_{0}^{L}{k\over 2}[\dot{\phi}-c(s)]^{2}ds,$ (2)
where $E$ is the energy, $\dot{\phi}\equiv d\phi/ds$, $T$ is the temperature,
$k_{B}$ is the Boltzmann constant, $L$ is the total arc length of the filament
and is a constant in the model so that the filament is inextensible,
$k=l_{p}/2$ with $l_{p}$ the 2D bare persistent length, $c(s)$ is the
intrinsic sequence-dependent curvature. Under a uniaxial applied force $f_{x}$
(along $x$-axis), the reduced energy of the filament becomes
$\displaystyle{\cal E}={\cal E}_{0}[\\{\phi(s)\\}]-f\int_{0}^{L}\cos\phi ds,$
(3)
where the reduced force is defined by $f\equiv f_{x}/k_{B}T$. When $c(s)=0$
and $l_{p}$ is a constant, it returns to the well known WLC model KP49 ; MS94
; BMSS94 ; MS95 . Note that with free boundary condition, a negative value of
$f_{x}$ only extends the polymer in the negative direction rather than the
positive direction. So it does for a long polymer since the boundary condition
becomes unimportant. Therefore, the sign of the force is meaningless in these
cases. However, it is not the case for a short polymer with a fixed initial
angle.
If both $l_{p}$ and $c(s)$ are well-defined functions of $s$, a macroscopic
quantity $B_{\phi}$ is defined as the average with Boltzmann weights over all
possible conformations,
$\displaystyle B_{\phi}={1\over{\cal Z}_{k}}\int{\cal
D}[\phi(s)]B[\\{\phi(s)\\}]\text{e}^{-{\cal E}},$ (4)
where ${\cal Z}_{k}\equiv\int{\cal D}[\phi]\text{e}^{-{\cal E}}$. Function
$B[\\{\phi(s)\\}]$ represents different physical situations. For instance, if
$B[\\{\phi(s)\\}]={\bf t}(s_{1})\cdot{\bf t}(s_{2})$, we find the
orientational correlation function (OCF); if $B[\\{\phi(s)\\}]=|{\bf
r}_{L}-{\bf r}_{0}|^{2}$, we obtain the mean end-to-end distance, where ${\bf
r}_{L}={\bf r}(L)$ and ${\bf r}_{0}={\bf r}(0)$; if
$B[\\{\phi(s)\\}]=\delta({\bf R}-\int_{0}^{L}{\bf t}ds)$, we get the
distribution function of end-to-end vector; if $B[\\{\phi(s)\\}]=\delta({\bf
r}_{L}-{\bf r}_{0})\delta[{\bf t}(L)-{\bf t}(0)]$, we find the looping
probability. $B[\\{\phi(s)\\}]$ is independent of $k$ and $c(s)$ but can be a
very complex function of $\phi(s)$ and $\dot{\phi}(s)$.
Eq. (4) uses $\phi$ as the variable of integration. However, the variable of
integration can be replaced by $\dot{\phi}(s)$, i.e., we have the following
identity (see Appendix 1 for proof)
$\displaystyle{\int{\cal D}[\phi(s)]B[\\{\phi(s)\\}]\text{e}^{-\cal
E}\over\int{\cal D}[\phi(s)]\text{e}^{-\cal E}}={\int{\cal
D}[\dot{\phi}(s)]B[\\{\phi(s)\\}]\text{e}^{-\cal E}\over\int{\cal
D}[\dot{\phi}(s)]\text{e}^{-\cal E}}.$ (5)
For a biopolymer without correlation on $c(s)$, or with SRC on $c(s)$ but in
the coarse-grained model, the distribution of $c(s)$, $W(\\{c(s)\\})$, can be
written as a Gaussian distribution with mean $\bar{c}$, and root-mean squared
deviation $1/\sqrt{\alpha}$:
$\displaystyle W(\\{c(s)\\})=\text{exp}\left[-\int{\alpha\over
2}[c(s)-\bar{c}]^{2}ds\right].$ (6)
In this case, we need to average over $c$ for all biopolymers in the system.
Note that averaging can be done in two different orders, either
$\displaystyle B\equiv\left<B[\\{\phi(s)\\}]\right>={1\over{\cal
Z}_{\alpha}}\int{\cal D}[c(s)]W(\\{c(s)\\})B_{\phi},$ (7)
where ${\cal Z}_{\alpha}\equiv\int{\cal D}[c(s)]W(\\{c(s)\\})$, or
$\displaystyle B^{\prime}$ $\displaystyle\equiv$
$\displaystyle\left<B[\\{\phi(s)\\}]\right>^{\prime}$ (8) $\displaystyle=$
$\displaystyle{1\over{\cal Z}_{k}}\int{\cal D}[\phi]\left[{1\over{\cal
Z}_{\alpha}}\int{\cal D}[c]W(\\{c\\})B[\\{\phi\\}]\text{e}^{-{\cal E}}\right]$
$\displaystyle=$ $\displaystyle{1\over{\cal Z}_{k}}\int{\cal
D}[\phi]B[\\{\phi\\}]\left[{1\over{\cal Z}_{\alpha}}\int{\cal
D}[c]W(\\{c\\})\text{e}^{-{\cal E}}\right].$
Physically, Eq. (7) corresponds to performing a conformational or thermal
average over an individual sample first, and then a disorder average over all
samples in the system, so we call this a thermal-first system. In contrast, in
Eq. (8) a disorder average over an instantaneous conformation of all samples
is first performed, followed by a conformational average over all possible
conformations, so we refer to the corresponding system as a disorder-first
system. To find a macroscopic quantity in the disorder-first system is
equivalent to averaging the sequence disorder over all samples first to create
an “equivalent” system which is then thermally averaged. So it is always
possible to construct an “equivalent system” in that case. However, in
experiments and computational simulations thermal-first averages are
performed. Therefore, this work will also focus on the thermal-first system
and show that in many cases the two systems are not equivalent.
## III Distribution Function for the Force-Free System
We first present a brief description of the force-free system. Using the
identities Eqs. (5) and
$\displaystyle\int{\cal D}[c]W(\\{c\\})\text{e}^{-{\cal E}_{0}}={{\cal
Z}_{k}^{0}{\cal Z}_{\alpha}\over{\cal Z}_{\cal H}^{0}}\text{e}^{-{\cal
H}_{0}},$ (9)
and exchanging the order of integration, Eq. (7) becomes
$\displaystyle B$ $\displaystyle=$ $\displaystyle{1\over{\cal
Z}_{k}^{0}}\int{\cal D}[\phi]B[\\{\phi(s)\\}]{\int{\cal
D}[c(s)]\text{e}^{-\cal E}W(\\{c(s)\\})\over{\cal Z}_{\alpha}}$ (10)
$\displaystyle=$ $\displaystyle{1\over{\cal Z}_{{\cal H}}^{0}}\int{\cal
D}[\phi]B[\\{\phi(s)\\}]\text{e}^{-{\cal H}_{0}},$
where
$\displaystyle{\cal H}_{0}$ $\displaystyle=$ $\displaystyle{1\over
2}\int_{0}^{L}\kappa[\dot{\phi}(s)-\bar{c}]^{2}ds,$ (11) $\displaystyle{\cal
Z}_{k}^{0}$ $\displaystyle=$ $\displaystyle\int{\cal D}[\phi]\text{e}^{-{\cal
E}_{0}},\text{ }{\cal Z}_{{\cal H}}^{0}=\int{\cal D}[\phi]\text{e}^{-{\cal
H}_{0}},$ (12)
and the effective persistent length,
$\displaystyle l^{\text{eff}}_{p}=2\kappa=2k\alpha/(k+\alpha).$ (13)
Note that Eq. (10) is valid for any $L$ and even if $\bar{c}$, $k$ and
$\alpha$ are $s$-dependent. Comparing Eqs. (4) and (10), we reach the
conclusion that a system with SRC in $c(s)$ is equivalent to a system with a
well-defined mean intrinsic curvature $\bar{c}$ and a renormalized persistence
length $l_{p}^{\text{eff}}$.
The same conclusion has been achieved for 3D biopolymers following similar
arguments ZJ08 , except that in the 2D case we have to derive Eq. (5) first
due to the convention of using $\phi$ as integral variable. This conclusion
also means that the thermal-first system is exactly the same as the disorder-
first system in the force-free case.
From the standard connection between the path integral and the Schrödinger
equation, we can find that the partition function ${\cal Z}_{\cal
H}^{0}(\phi(s),s;\phi(s_{0}),s_{0})$ ($\equiv\int{\cal
D}[\phi]\text{e}^{-{\cal H}_{0}}$) for the system with effective energy ${\cal
H}_{0}$ satisfies the following partial differential equation ZZO00 ; ZZ07 ;
HK90
$\displaystyle{\partial{\cal Z}_{\cal H}^{0}\over\partial s}=\left({1\over
2\kappa}{\partial^{2}\over\partial\phi^{2}}-\bar{c}{\partial\over\partial\phi}\right){\cal
Z}_{\cal H}^{0}.$ (14)
Fixing $\phi(s)$ at $s=s_{0}$, the boundary condition (BC) becomes
$\displaystyle{\cal Z}_{\cal
H}^{0}(\phi,s_{0};\varphi,s_{0})=\delta[\phi-\varphi],$ (15)
where $\varphi=\phi(s_{0})$. In an experiment, one usually takes $\phi(0)=0$
when $f=0$.
It is straightforward to show that the normalized function (i.e., the
distribution function with $s>s_{0}$)
$\displaystyle P(\phi,s;\varphi,s_{0})=\sqrt{1\over 2\pi
A(s,s_{0})}\text{e}^{-[\phi-\varphi-\int_{s_{0}}^{s}\bar{c}(s)ds]^{2}/2A(s,s_{0})}$
(16)
satisfies Eqs. (14) and (15), where $A(s,s_{0})=\int_{s_{0}}^{s}ds/\kappa(s)$.
Eq. (16) can also be derived directly by using a standard path integral
technique HK90 ; KLB93 (also see Appendix 2).
## IV Conformation and Elasticity of the System Under External Force
### IV.1 On the Disorder-first System
When $f\neq 0$, for the disorder-first system, the derivation leading to Eq.
(10) can be generalized easily to obtain an equivalent system with the
effective energy
$\displaystyle{\cal H}={\cal H}_{0}-f\int_{0}^{L}\cos\phi ds,$ (17)
no matter what the force may be since the force term in $\cal E$ is
independent of $c(s)$. The equivalent system has been well studied PHK05 ;
KMTS07 ; ZZ07 .
In the three dimensional case, we can also reach a similar conclusion by a
direct generalization of the proof in Ref. ZJ08 to find an equivalent system,
and an alternative proof for the three dimensional filament under weak force
can be found in Ref. PN98 .
### IV.2 General Expressions for the Thermal-first System Under Weak Force
Note that, mathematically, the conclusion of the existence of an equivalent
system in the force-free case results from the fact that both ${\cal
Z}_{k}^{0}$ and ${\cal Z}_{\alpha}$ are Gaussian integrals so are independent
of $c(s)$ or $\phi$. However, such an argument fails for the thermal-first
system with $f\neq 0$, because ${\cal Z}_{k}$ is no longer a Gaussian integral
and is dependent on $c$, so exchanging the order of integration does not
simplify the expression. In other words, there is no simple way to remove the
randomness in $c(s)$ so there is in general no “equivalent system” even under
weak force, as we will show exactly below. In this case, to first order in
$f$, as shown in Appendix 3,
$\displaystyle B$ $\displaystyle=$ $\displaystyle{1\over{\cal
Z}_{\alpha}}\int{\cal D}[\phi]B[\\{\phi\\}]\text{e}^{F}\left[\int{\cal
D}[c]{1\over{\cal Z}_{k}}W(\\{c\\})\text{e}^{-{\cal E}_{0}}\right]$ (18)
$\displaystyle\approx$ $\displaystyle B_{1}-B_{2},$ with $\displaystyle B_{1}$
$\displaystyle=$
$\displaystyle\left<B\right>_{\kappa}+f\int_{0}^{L}ds^{\prime}\left<B[\\{\phi(s)\\}]\cos\phi(s^{\prime})\right>_{\kappa},$
(19) and $\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle
f\int_{0}^{L}ds^{\prime}\text{e}^{-A^{\prime}(s^{\prime},0)/2}\left<B[\\{\phi(s)\\}]\cos[\gamma(s^{\prime})+\phi_{0}]\right>_{\kappa},$
where
$\displaystyle A^{\prime}(s,s_{0})$ $\displaystyle=$
$\displaystyle\int_{s_{0}}^{s}ds/k^{\prime}(s),\text{
}k^{\prime}(s)={k(\alpha+k)\over\alpha+2k},$ (21) $\displaystyle\gamma(s)$
$\displaystyle=$ $\displaystyle\int_{0}^{s}\dot{\gamma}(s)ds,\text{
}\dot{\gamma}(s)={k\dot{\phi}+\alpha\bar{c}\over\alpha+k},$ (22)
and $\phi_{0}=\phi(0)$ is the initial azimuthal angle. Note that $\gamma(s)$
is in general dependent on $\phi(s)$ making these expressions very complex.
We should remind that when $f\neq 0$, $\phi_{0}$ is not necessarily zero but
is dependent on the experimental conditions. In experiments, $\phi_{0}$ may be
fixed. In this case, the boundary condition at $s=0$ is given by Eq. (15).
However, experiments on stretching biopolymers usually involve attaching the
two ends of the biopolymer to beads, and it does not seem to be easy to
completely prohibit the rotation of the beads. As a consequence, it may be
difficult to fix $\phi_{0}$. In the extreme case, the polymer can rotate
freely around the origin. This can be realized by a magnetic tweezer DC07 . In
the more general case, $\phi_{0}$ may have a distribution and therefore
finally we need to average over $\phi_{0}$. It has been reported that
different boundary conditions have considerable effects on the mechanical
response of a homopolymer ZLJ05 ; DC07 . In this work, we come to the same
conclusion for a heteropolymer.
On the other hand, for the disorder-first system, we find
$\displaystyle B^{\prime}$ $\displaystyle=$ $\displaystyle{1\over{\cal
Z}_{\cal H}}\int{\cal D}[\phi]B[\\{\phi\\}]\text{e}^{-{\cal H}}\approx
B_{1}-B_{2}^{\prime},$ $\displaystyle B_{2}^{\prime}$ $\displaystyle=$
$\displaystyle
f\left<B[\\{\phi(s)\\}]\right>_{\kappa}\int_{0}^{L}ds\left<\cos\left(\phi(s)\right)\right>_{\kappa}$
(23) $\displaystyle=$ $\displaystyle
f\left<B[\\{\phi(s)\\}]\right>_{\kappa}{\cal B}_{2}^{\prime},$
$\displaystyle{\cal B}_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{L}ds\text{e}^{-A(s,0)/2}\cos\left(\phi_{0}+\int_{0}^{s}\bar{c}(s)ds\right),$
where ${\cal Z}_{{\cal H}}=\int{\cal D}[\phi]\text{e}^{-{\cal H}}$.
It is clear that in general $B_{2}\neq B_{2}^{\prime}$ since $\gamma$ is
dependent on $\phi$. It in turn leads to in general $B\neq B^{\prime}$. In
other words, in general it is impossible to find an equivalent system for the
thermal-first system under external force.
### IV.3 Elasticity of the Thermal-first System Under Weak Force
To figure out how serious the effect of the disorder in $c(s)$ or how large
the discrepancy between $B$ and $B^{\prime}$, we examine the most interesting
and also the simplest case with constant $k$, $\alpha$ and $\bar{c}=0$. It
corresponds to the WLC model and is often used to describe the entropic
elasticity of biopolymers, such as dsDNA and proteins. Experiments in 3D dsDNA
found that $k\approx 78$nm and $\kappa\approx 45$nmTTH87 ; SS90 ; BFKSDS95 ;
FBSKMSD97 . It follows that $k\approx 1.7\kappa$ for DNA. In this case,
$\gamma(s)=\kappa[\phi(s)-\phi_{0}]/\alpha$, and from Eq. (23), we can obtain
$\displaystyle B_{2}^{\prime}\approx 2\kappa
f\left(1-\text{e}^{-L/2\kappa}\right)\cos\phi_{0}\left<B[\\{\phi(s)\\}]\right>_{\kappa}.$
(24)
The extension $X$ in the thermal-first system can be found as (see Appendix 4)
$\displaystyle X$ $\displaystyle\equiv$
$\displaystyle\left<x\right>-\left<x\right>_{f=0}$ (25) $\displaystyle\approx$
$\displaystyle 2\kappa f\left[L-{k(k+3\alpha)\over k+\alpha}+{k(k+\alpha)\over
k-\alpha}\text{e}^{-L/k}\right.$ $\displaystyle\left.-{4\kappa\alpha\over
k-\alpha}\text{e}^{-L/2\kappa}\right]-{\kappa^{2}\cos(2\phi_{0})\text{e}^{-2L/\kappa}f\over
3(2k+\alpha)(3k+\alpha)}{\cal X},$ $\displaystyle{\cal X}$ $\displaystyle=$
$\displaystyle 6k^{2}\left(\text{e}^{L/k}-1\right)$
$\displaystyle+k\alpha\left(9\text{e}^{2L/\kappa}-16\text{e}^{3L/2\kappa}+12\text{e}^{L/k}-5\right)$
$\displaystyle+\alpha^{2}\left(3\text{e}^{2L/\kappa}-8\text{e}^{3L/2\kappa}+6\text{e}^{L/k}-1\right).$
On the other hand, in the disorder-first system, the extension $X^{\prime}$
is:
$\displaystyle X^{\prime}$ $\displaystyle\equiv$
$\displaystyle\left<x\right>^{\prime}-\left<x\right>^{\prime}_{f=0}$ (26)
$\displaystyle\approx$ $\displaystyle 2\kappa
f\left[L+\kappa-\kappa\left(\text{e}^{-L/2\kappa}-2\right)^{2}\right]+{\kappa^{2}f\over
3}\cos(2\phi_{0})$
$\displaystyle\cdot\left(\text{e}^{-2L/\kappa}-6\text{e}^{-L/\kappa}+8\text{e}^{-L/2\kappa}-3\right).$
From Eqs. (25) and (26), we can show that $X\rightarrow X^{\prime}$ when
$\alpha\rightarrow\infty$, as it should be since in this case the system is
free of randomness. Moreover, for a long polymer, we obtain $X\approx
X^{\prime}\approx 2\kappa Lf$, so that the averaging order is irrelevant for a
long polymer, and agrees with known results PN98 ; BDM98 ; VATA03 .
However, the discrepancy between the two systems is serious up to moderate
length at finite $\alpha$. From Eqs. (25) and (26), we find that the
extensions consist of two terms. The first term is independent of BC or
$\phi_{0}$, but the second term is dependent on cos$2\phi_{0}$ so is dependent
on the BC. As a consequence, different boundary conditions have strong effects
on the elasticity up to moderate length. Without loss in generality, we
consider two extreme cases. The first case is to fix $\phi_{0}=0$, which gives
the most stringent BC effects. The second is to let $\phi_{0}$ free so
$\left<\cos 2\phi_{0}\right>=0$, and only the BC-independent term remains. For
a better comparison, we define a ratio of the extensions,
$r_{e}=X/X^{\prime}$.
Figure 1: The ratio $r_{e}=X/X^{\prime}$ vs $L$ for the fixed BC (
$\phi_{0}=0$). From top to bottom, the parameters are: $k=1.2\kappa$,
$\alpha=6.0\kappa$(dash); $k=1.4\kappa$, $\alpha=3.5\kappa$ (dot);
$k=1.6\kappa$, $\alpha=2.67\kappa$ (dash dot); $k=1.8\kappa$,
$\alpha=2.25\kappa$ (solid); $k=2.0\kappa$, $\alpha=2.0\kappa$ (short dash).
The unit of $L$ is in $\kappa$.
Figure 2: The ratio $r_{e}=X/X^{\prime}$ vs $L$ for the free BC ($\left<\cos
2\phi_{0}\right>=0$). From top to bottom, the parameters are: $k=1.2\kappa$,
$\alpha=6.0\kappa$ (dash); $k=1.4\kappa$, $\alpha=3.5\kappa$ (dot);
$k=1.6\kappa$, $\alpha=2.67\kappa$ (dash dot); $k=1.8\kappa$,
$\alpha=2.25\kappa$ (solid); $k=2.0\kappa$, $\alpha=2.0\kappa$ (short dash).
The unit of $L$ is in $\kappa$.
Figures 1 and 2 present some typical results for $r_{e}$. From these two
figures, we can see that $r_{e}$ increases monotonously with increasing $L$,
and in general the disorder in $c(s)$ makes the biopolymer in the thermal-
first system softer, i.e. $r_{e}<1$, than the “equivalent” system, and the
effects may be still rather serious up to the length of about 20 $\kappa$.
Furthermore, we also find that the results are sensitive to BC and randomness
of $c(s)$. At first, the effect is much more serious (maybe about twice) in
the free BC (Fig. 2) than in fixing BC (Fig. 1). Second, the effect is getting
stronger with increasing randomness, i.e. with decreasing $\alpha$.
Especially, at $k\approx 1.7\kappa$ which corresponds to the experimental
value of DNA, we find that $X$ can be only about half of $X^{\prime}$ for
$L\sim\kappa$, and 20% smaller than $X^{\prime}$ up to $L\approx 10\kappa$ or
about 1500 bp. That means DNA is more likely to be in coil state or appears
softer and has a smaller apparent persistent length than that in the
“equivalent” system up to a rather long length. Finally, these results are
independent of the external force, so that a small force may produce a
considerable effect. This fact may be important in a stretching experiment for
a short polymer since it suggests that the interaction between experimental
apparatus and polymer may affect the results seriously.
### IV.4 Orientational Correlation Function (OCF) and End-to-end Distance for
the Thermal-first System Under Weak Force
Let $B[\\{\phi\\}]={\bf t}(s)\cdot{\bf
t}(s^{\prime})=\cos[\phi(s)-\phi(s^{\prime})]$, from Eqs. (18)-(22), we obtain
the orientational correlation function ($s>s^{\prime}$) for the thermal-first
system and under weak external force (see Appendix 5)
$\displaystyle\left<{\bf t}(s)\cdot{\bf
t}(s^{\prime})\right>\approx\text{e}^{-(s-s^{\prime})/2\kappa}+{\kappa\cos\phi_{0}f\over
3(3k+\alpha)(k-\alpha)}{\cal S},$ $\displaystyle{\cal S}$ $\displaystyle=$
$\displaystyle-6k(k-\alpha)\text{e}^{-2[(\alpha+2k)s+(\alpha+3k)s^{\prime}]/2k\alpha}$
(27)
$\displaystyle+8\alpha(2k+\alpha)\text{e}^{-s/2\kappa}-6k(3k+\alpha)\text{e}^{-[2\alpha
s+(k-\alpha)s^{\prime}]/2k\alpha}$
$\displaystyle+(k-\alpha)(3k+\alpha)\left[3\text{e}^{-[(\alpha+k)L+(\alpha+3k)(s-s^{\prime})]/2k\alpha}\right.$
$\displaystyle-3\text{e}^{-[L+3(s-s^{\prime})]/2\kappa}+6\text{e}^{-s^{\prime}/2\kappa}-3\text{e}^{-(L+s^{\prime}-s)/2\kappa}$
$\displaystyle\left.+2\text{e}^{-(4s-3s^{\prime})/2\kappa}+3\text{e}^{-[(\alpha+k)L+(\alpha-k)(s-s^{\prime})]/2k\alpha}\right].$
In contrast, in the disorder-first system,
$\displaystyle\left<{\bf t}(s)\cdot{\bf
t}(s^{\prime})\right>^{\prime}\approx\text{e}^{-(s-s^{\prime})/2\kappa}+{1\over
3}\kappa f\cos\phi_{0}{\cal S}^{\prime},$ $\displaystyle{\cal S}^{\prime}$
$\displaystyle=$ $\displaystyle
6\text{e}^{-s^{\prime}/2\kappa}-8\text{e}^{-s/2\kappa}+6\text{e}^{-(L+s-s^{\prime})/2\kappa}+2\text{e}^{(3s^{\prime}-4s)/2\kappa}$
(28)
$\displaystyle-3\text{e}^{-[L+3(s-s^{\prime})]/2\kappa}-3\text{e}^{-(L+s^{\prime}-s)/2\kappa}.$
The end-to-end distance can be found as
$\displaystyle R^{2}$ $\displaystyle\approx$ $\displaystyle 4\kappa
L\left[1-{2\kappa\over L}\left(1-\text{e}^{-L/2\kappa}\right)\right]$ (29)
$\displaystyle+{4\kappa^{2}\cos\phi_{0}f\over
9(k-\alpha)^{2}(k+\alpha)(2k+\alpha)(3k+\alpha)^{2}}{\cal Y},$ with
$\displaystyle{\cal Y}$ $\displaystyle=$ $\displaystyle
18(k-\alpha)^{2}(k+\alpha)(2k+\alpha)(3k+\alpha)^{2}L$
$\displaystyle-9k(k-\alpha)^{2}(3k+\alpha)^{2}(4k^{2}+11\alpha^{2}+22k\alpha)$
$\displaystyle-16\alpha(2k+\alpha)^{2}[3(k-\alpha)(k+\alpha)(3k+\alpha)L$
$\displaystyle+4k\alpha(5k+\alpha)(3k-\alpha)]\text{e}^{-L/2\kappa}$
$\displaystyle+18k\alpha(k+\alpha)^{3}(k-\alpha)^{2}\text{e}^{-(1/k+2/\alpha)L}$
$\displaystyle+18k(k+\alpha)^{3}(2k+\alpha)(3k+\alpha)^{2}\text{e}^{-L/k}$
$\displaystyle-k\alpha(k-\alpha)^{2}(2k+\alpha)(3k+\alpha)^{2}\text{e}^{-2L/\kappa}.$
On the other hand
$\displaystyle R^{\prime 2}$ $\displaystyle\approx$ $\displaystyle 4\kappa
L\left[1-{2\kappa\over L}\left(1-\text{e}^{-L/2\kappa}\right)\right]+{4\over
9}\kappa^{2}\cos\phi_{0}f{\cal Y}^{\prime}$ $\displaystyle\mbox{with}\ \ {\cal
Y}^{\prime}$ $\displaystyle=$ $\displaystyle
18L-99\kappa+16(3L+4\kappa)\text{e}^{-L/2\kappa}$
$\displaystyle+36\kappa\text{e}^{-L/\kappa}-\kappa\text{e}^{-2L/\kappa}.$
When $\alpha\rightarrow\infty$, i.e. a system without randomness, we obtain
$R^{2}=R^{\prime 2}$ as it should be. Moreover, $L\rightarrow\infty$ also
leads to $R^{2}\rightarrow R^{\prime 2}$, again supporting the conclusion that
the order in averaging is irrelevant for a long polymer PN98 ; BDM98 ; VATA03
.
Figure 3: The ratio $r_{R}=R^{2}/R^{\prime 2}$ vs $L$. $f=0.5/\kappa$ and
$\phi_{0}=0$ (the fixed BC) for all curves. From bottom to top, the parameters
are: $k=1.2\kappa$, $\alpha=6.0\kappa$ (dash); $k=1.4\kappa$,
$\alpha=3.5\kappa$ (dot); $k=1.6\kappa$, $\alpha=2.67\kappa$ (dash dot);
$k=1.8\kappa$, $\alpha=2.25\kappa$ (solid); $k=2.0\kappa$, $\alpha=2.0\kappa$
(short dash). The unit of $L$ is in $\kappa$.
Moreover, from Eqs. (29) and (IV.4), we find that the end-to-end distance can
also be divided into two terms. The first term is the force free term and is
independent of BC, but the second term is dependent on both BC and force.
Especially, for free BC ($\left<\cos 2\phi_{0}\right>=0$), the force has not
effect at all, this is quite different from the force-extension relation.
Furthermore, with finite $\left<\cos 2\phi_{0}\right>$, $R^{2}$ is also
smaller than $R^{\prime 2}$, similar to the force-extension relation. However,
the ratio $r_{R}=R^{2}/R^{\prime 2}$ is no longer a monotonic function of $L$,
but has a minimum at $L\approx 6\kappa$. Moreover, the discrepancy between $R$
and $R^{\prime}$ increases with increasing $f$ (see Eqs. (29) and (IV.4)).
Note that our results are valid only at weak force, and for DNA it means that
we require $k_{B}Tf<f_{c}\equiv k_{B}T/\kappa$. Taking the generally accepted
value $\kappa\approx$ 50 nm, we have $f_{c}\approx 0.08pN$. Fig. (3) shows
some typical results at $f=0.5/\kappa$, which corresponds to an external force
of about 0.04 pN. From Fig. (3), we can see that the discrepancy between
$R^{2}$ and $R^{\prime 2}$ is much smaller than that for extensions. This is
because the force free term dominates the value of $r_{R}$. The fact that the
disorder in $c(s)$ has quite different effects on the extension and the end-
to-end distance may be important in experiments.
### IV.5 Elasticity of the Long Biopolymer Under Large Force
Again we assume constant $k$, $\alpha$ in this part, but allow a nonvanishing
$\bar{c}(s)$. In the large force limit the filament is nearly straight, thus
t$(s)$ is nearly pointing along the direction of the force. That means
$\phi(s)\approx 0$ and the reduced energy becomes
$\displaystyle{\cal E}\approx\int_{0}^{L}{k\over
2}[\dot{\phi}-c(s)]^{2}ds+f\int_{0}^{L}{1\over 2}\phi^{2}ds,$ (31)
where we have dropped a constant term $-fL$. For a very long filament, we can
use periodic boundary conditions with negligible error, so take $q_{n}=2\pi
n/L$ with integer $n$, and expand $\phi$, $c(s)$ and $\bar{c}(s)$ as Fourier
series
$\displaystyle\phi(s)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}a_{n}\sin(q_{n}s),\text{
}\dot{\phi}(s)=\sum_{n=1}^{\infty}a_{n}q_{n}\cos(q_{n}s),$ $\displaystyle
c(s)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}c_{n}\cos(q_{n}s),\text{
}\bar{c}(s)=\sum_{n=0}^{\infty}\bar{c}_{n}\cos(q_{n}s).$ (33)
Note that to use a sine series for $\phi(s)$ is reasonable since
$\phi_{0}=\phi(L)=0$. But in general $c(L)\neq 0$ so we cannot use sine series
for it. We can also use the full Fourier series for $c(s)$ and $\bar{c}(s)$,
but it is straightforward to show that the sine part in the full Fourier
series make no contribution at all so we disregard it. Using the orthogonality
property of Fourier modes, we can reexpress the energy and extension as
$\displaystyle{\cal E}$ $\displaystyle\approx$ $\displaystyle{kc_{0}^{2}L\over
2}+{kL\over 4}\sum_{n=1}^{\infty}(a_{n}q_{n}-c_{n})^{2}+{fL\over
4}\sum_{n=1}^{\infty}a_{n}^{2},$ $\displaystyle=$
$\displaystyle{kc_{0}^{2}L\over 2}+{L\over
4}\sum_{n=1}^{\infty}\left[(kq_{n}^{2}+f)d_{n}^{2}+{kf\over
kq_{n}^{2}+f}c_{n}^{2}\right],$ with $\displaystyle d_{n}$ $\displaystyle=$
$\displaystyle a_{n}-{kq_{n}c_{n}\over kq_{n}^{2}+f},$ (35) and $\displaystyle
x$ $\displaystyle=$ $\displaystyle\int_{0}^{L}\cos\phi ds\approx
L\left(1-{1\over 4}\sum_{n=1}^{\infty}a_{n}^{2}\right)$ (36) $\displaystyle=$
$\displaystyle x_{1}+x_{2}-{L\over 2}\sum_{n=1}^{\infty}{kq_{n}c_{n}\over
kq_{n}^{2}+f}d_{n},$ where $\displaystyle x_{1}$ $\displaystyle=$
$\displaystyle L\left[1-{1\over 4}\sum_{n=1}^{\infty}d_{n}^{2}\right],$ (37)
$\displaystyle x_{2}$ $\displaystyle=$ $\displaystyle-{L\over
4}\sum_{n=1}^{\infty}{(kq_{n})^{2}\over(kq_{n}^{2}+f)^{2}}c_{n}^{2}.$ (38)
From Eqs. (4) and (LABEL:energy12)-(38), we see that in the thermal-first
system $\left<x\right>=\left<x_{1}\right>+\left<x_{2}\right>$. $x_{1}$ is
independent of $c(s)$, and we recover the well known result for the WLC model
PHK05 ; KMTS05 ; ZZ07 ,
$\displaystyle{\left<x_{1}\right>\over L}=1-{1\over 2\sqrt{fl_{p}}}.$ (39)
In contrast, $x_{2}$ is independent of $\phi(s)$ and is determined by $c(s)$.
Rewriting $W(\\{c(s)\\})$ as
$\displaystyle W(\\{c(s)\\})=\text{e}^{-{\alpha L\over
2}\left[(c_{0}-\bar{c}_{0})^{2}+{1\over
2}\sum_{n=1}^{\infty}(c_{n}-\bar{c}_{n})^{2}\right]},$ (40)
we obtain
$\displaystyle\left<x_{2}\right>$ $\displaystyle=$ $\displaystyle-{L\over
4}\sum_{n=1}^{\infty}{(kq_{n})^{2}\over(kq_{n}^{2}+f)^{2}}\left<c_{n}^{2}\right>$
(41) $\displaystyle=$ $\displaystyle-{L\over
4}\sum_{n=1}^{\infty}{(kq_{n})^{2}\over(kq_{n}^{2}+f)^{2}}\left(\bar{c}_{n}^{2}+{2\over\alpha
L}\right)$ $\displaystyle\approx$ $\displaystyle-{L\over
4}\sum_{n=1}^{\infty}{(kq_{n})^{2}\over(kq_{n}^{2}+f)^{2}}\bar{c}_{n}^{2}.$
On the other hand, replacing $k$ and $c_{n}$ by $\kappa$ and $\bar{c}_{n}$
respectively in Eqs. (LABEL:energy12)-(38), we find that in the disorder-first
system the extension becomes
$\displaystyle{\left<x\right>^{\prime}\over
L}={\left<x_{1}\right>^{\prime}+x_{2}\over L}=1-{1\over
2\sqrt{f\l_{p}^{\text{eff}}}}+{x_{2}\over L}.$ (42)
From Eqs. (36)-(42), we find that replacing $k$ by $\kappa$ one goes from the
thermal-first system to the disorder-first system. It is also interesting to
note that $\alpha$ or the width of the distribution of $c(s)$ plays no role in
the extension in both systems. Furthermore, we find that when $\bar{c}$ is a
constant, vanishing or nonvanishing, $\left<x_{2}\right>\approx 0$ since all
$\bar{c}_{n}=0$ if $n\geq 1$. On the hand hand, considering the special case
with $\bar{c}=\sigma$e-λs, in the thermal-first system we have
$\displaystyle{\left<x_{2}\right>\over L}$ $\displaystyle\approx$
$\displaystyle-{1\over
4}\sum_{n=1}^{\infty}{(kq_{n})^{2}\over(kq_{n}^{2}+f)^{2}}\bar{c}_{n}^{2}$
(43) $\displaystyle=$ $\displaystyle-{\lambda\sigma\over
2L}\sum_{n=1}^{\infty}{(kq_{n})^{2}\over(kq_{n}^{2}+f)^{2}(q_{n}^{2}+\lambda^{2})}$
$\displaystyle\approx$ $\displaystyle-{\lambda\sigma k^{2}\over
4\pi}\int_{0}^{\infty}{q^{2}dq\over(kq^{2}+f)^{2}(q^{2}+\lambda^{2})}$
$\displaystyle=$ $\displaystyle-{\lambda\sigma k^{2}\over
16\sqrt{fk}\left(\sqrt{f}+\lambda\sqrt{k}\right)^{2}}.$
A long biopolymer in general has a vanishing or small mean curvature, or in
other words in general $\sigma$ and $\lambda$ are small, so that $x_{2}$ is
negligible in either thermal-first or disorder-first systems. Since the
corrections of $\bar{c}$ are negligible in both constant and fast decay cases,
we can conclude safely that $x_{2}$ is always negligible. In the thermal-first
system, it means that sequence-disorder has no effect, which agrees with Marko
and Siggia’s argument that under a strong stretching force, disorder in
sequence is immaterial for elasticity MS95 ; PN98 . Note that the effect of
the sequence-disorder has been absorbed into $\kappa$ in the disorder-first
system, the result in this part provides another evidence of the non-
equivalence of the two systems.
## V On the Constant-extension Ensemble
Up to now, our discussions are based on a constant external force, or in the
constant-force ensemble. However, the experiments may be performed with fixed
ends, or in the constant-extension ensemble. For a long polymer, it is
believed that these two ensembles should yield the same mechanical properties.
However, it has been reported that the two ensembles are not always equivalent
for a short polymer. It is therefore interesting to ask whether sequence-
disorder has the same effects in the two ensembles. In the constant-extension
ensemble, without sequence disorder, the partition function is
$\displaystyle{\cal Z}_{e}$ $\displaystyle=$ $\displaystyle\int{\cal
D}[\phi]\delta({\bf r}_{L}-{\bf b})\text{e}^{-{\cal E}_{0}}$ (44)
$\displaystyle=$ $\displaystyle\int{\cal D}[\phi]\delta\left(\int_{0}^{L}{\bf
t}ds-{\bf b}\right)\text{e}^{-{\cal E}_{0}},$
where $\bf b$ is the constant end-to-end vector. With sequence disorder, the
disorder-first average for the ensemble becomes
$\displaystyle B^{\prime}$ $\displaystyle=$ $\displaystyle{1\over{\cal
Z}_{e}}\int{\cal D}[\phi]\left[{1\over{\cal Z}_{\alpha}}\int{\cal
D}[c]W(\\{c\\})\delta({\bf r}_{L}-{\bf b})\text{e}^{-{\cal
E}_{0}}B[\\{\phi\\}]\right]$ (45) $\displaystyle=$ $\displaystyle{1\over{\cal
Z}_{e}}\int{\cal D}[\phi]\delta({\bf r}_{L}-{\bf
b})B[\\{\phi\\}]\left[{1\over{\cal Z}_{\alpha}}\int{\cal
D}[c]W(\\{c\\})\text{e}^{-{\cal E}_{0}}\right]$ $\displaystyle=$
$\displaystyle{{\cal Z}_{k}^{0}\over{\cal Z}_{e}{\cal Z}_{\cal
H}^{0}}\int{\cal D}[\phi]\delta({\bf r}_{L}-{\bf
b})B[\\{\phi\\}]\text{e}^{-{\cal H}_{0}},$
where we have used Eq. (9) again. Now considering the special case
$B[\\{\phi\\}]=1$, which results in $B^{\prime}=1$, and
$\displaystyle{{\cal Z}_{k}^{0}\over{\cal Z}_{e}{\cal Z}_{\cal
H}^{0}}={1\over\int{\cal D}[\phi]\delta({\bf r}_{L}-{\bf b})\text{e}^{-{\cal
H}_{0}}},$ (46)
leads us to the result,
$\displaystyle B^{\prime}$ $\displaystyle=$ $\displaystyle{\int{\cal
D}[\phi]\delta({\bf r}_{L}-{\bf b})B[\\{\phi\\}]\text{e}^{-{\cal
H}_{0}}\over\int{\cal D}[\phi]\delta({\bf r}_{L}-{\bf b})\text{e}^{-{\cal
H}_{0}}},$ (47)
Therefore, we reach the conclusion that in the constant-extension ensemble for
the disorder-first system we can still find an “equivalent system” with the
effective energy ${\cal H}_{0}$.
On the other hand, the thermal-first average for the constant-extension
ensemble can be written
$\displaystyle B$ $\displaystyle=$ $\displaystyle{1\over{\cal
Z}_{\alpha}}\int{\cal D}[c]W(\\{c\\})B_{e},$ (48) $\displaystyle B_{e}$
$\displaystyle=$ $\displaystyle{1\over{\cal Z}_{e}}\int{\cal
D}[\phi]B[\\{\phi\\}]\delta({\bf r}_{L}-{\bf b})\text{e}^{-{\cal E}_{0}}.$
(49)
From Eq. (44), we find that due to the existence of the $\delta({\bf
r}_{L}-{\bf b})$ term, ${\cal Z}_{e}$ is in general dependent on $c(s)$, so
that an exchange in the order of integration cannot simplify the expression
for $B$. In other words, the existence of an “equivalent system” for the
thermal-first system in the constant-extension ensemble is still an open
question. From our experience in the constant-force ensemble, such an
“equivalent system” does not exist.
Our discussions in this section can be directly generalized to the three
dimensional system, so it completes and reassess the results of our previous
work ZJ08 .
## VI Conclusions and discussions
In summary, we present a rigorous proof that when free of external force, a 2D
semiflexible biopolymer without correlation or with SRC in intrinsic
curvatures $c(s)$ is equivalent to a system with a well-defined intrinsic
curvature and a renormalized persistence length. We obtain exact expressions
for the distribution function of the equivalent system. These conclusions can
simplify theoretical studies of semiflexible biopolymers, since the disorder
in $c(s)$ is completely removed in the equivalent system. For the system under
external force, we find that the effect of sequence-disorder is dependent on
the order in which the averaging is done or the experimental conditions. In
the disorder-first system, it is always possible to find an “equivalent
system”, no matter the external force, the length of the polymer, the
statistical ensemble or the dimension of the system. However, in the thermal-
first system, there is in general no “equivalent system” for a biopolymer up
to moderate length. Physically, this is because in the thermal-first system
the intrinsic curvatures favor defects such as kinks, buckles and loops. To
straighten these defects costs extra energy and therefore requires a larger
force. In contrast, the disorder-first system erases these extra defects
before the application of the force. We find the closed-form expression for
the force-extension relationship for the elasticity of a long biopolymer under
a strong stretching force. In the thermal-first system, we show exactly that
in this case sequence disorder is immaterial even if the biopolymer has a
nonvanishing mean intrinsic curvature. Moreover, we find that in the thermal-
first system, the results are also dependent on the boundary conditions, and
the sequence-dependent effects are much more serious in the case of free BCs
than for fixed BCs. Meanwhile, the results are dependent on the degree of
randomness and the larger the randomness, the more serious the effect. Our
results suggest that the short biopolymer may be much softer so has a smaller
apparent persistent length than what the “equivalent system” in a mechanical
experiment would predict. This fact implies that in experiments the
interaction between experimental apparatus and polymer, though may be weak,
may affect the results seriously for a short polymer. Furthermore, our results
suggest that the effects of sequence-disorder is dependent upon the quantity
measured and how it is measured. We should note that due to the existence of
an “equivalent system”, the disorder-first system is much simpler in
theoretical studies. However, it is difficult to realize the disorder-first
system in experiment.
On the other hand, we considered weak forces and large forces but not
intermediate size forces in this work, but we can expect that in this case
there will also not be an “equivalent system” even for a long polymer since
this is the case for the system under a large force. How the elasticity of a
system goes from an “equivalent system” when free of force to a disorder free
system under a large force would be an important question to address. We also
do not consider the system with LRC in $c(s)$, which deserves further
investigation.
It should also be noted that this work focussed on 2D systems. However,
whether the results apply to 3D systems is an intriguing question.
Mathematically we should reach similar conclusions since in the thermal-first
system exchanging the order of integration does not simplify the problem.
However, physically there exist some fundamental distinctions between the 3D
case and the 2D case. At first, the much stronger fluctuations in the 3D
system may dominate the intrinsic disorder so the effect may be suppressed.
This would explain why the disorder in $c(s)$ has a much smaller effect on the
end-to-end distance than on the extension. Moreover, the geometry of a polymer
with natural curvature is also very different in the 2D and 3D systems. For
example, the looped configuration in the 2D system cannot undergo an out-of-
plane buckling that would eliminate loops, but a 3D polymer will exhibit this
behavior. Furthermore, in the 2D case it is much easier to form large defects
which would be responsible for a larger decrease in extension. Therefore, the
difference in the thermal-first or disorder-first ordering may be reduced in a
3D system. But finally let us point out that the studies of the conformations
of biopolymers are often performed in a 2D environment (e.g., see MFFA07 ;
CSM99 ), so our main findings should be instructive.
## Appendix 1: Proof of Eq. (5)
Using the standard path integral methods KLB93 , for arbitrary function ${\cal
F}[\\{\phi(s)\\}]$, we can write
$\displaystyle\int{\cal D}[\phi(s)]{\cal F}[\\{\phi(s)\\}]$
$\displaystyle\propto$
$\displaystyle\lim_{N\rightarrow\infty}\prod_{j=1}^{N-1}\int d\phi_{j}{\cal
F}[\\{\phi_{j}\\}],$ (50) $\displaystyle\int{\cal D}[\dot{\phi}(s)]{\cal
F}[\\{\phi(s)\\}]$ $\displaystyle\propto$
$\displaystyle\lim_{N\rightarrow\infty}\prod_{j=1}^{N-1}\int{1\over\epsilon}d\xi_{j}{\cal
F}[\\{\phi_{j}\\}],$ (51)
where $\epsilon=L/N$, $\phi_{j}=\phi[(j-1)\epsilon]$ is the discretized
$\phi(s)$, $\xi_{j}=\phi_{j}-\phi_{j-1}$, and $\dot{\phi}$ in $B$ and $\cal E$
must be replaced by $(\phi_{j+1}-\phi_{j})/\epsilon$. The Jacobian
determinant, $J=|\partial\xi/\partial\phi|$, of $\xi^{\prime}$s with respect
to $\phi^{\prime}$s is a constant, therefore,
$\displaystyle\int{\cal D}[\dot{\phi}(s)]{\cal
F}[\\{\phi(s)\\}]\propto\lim_{N\rightarrow\infty}{J\over\epsilon^{N-1}}\prod_{j=1}^{N-1}\int
d\phi_{j}{\cal F}[\\{\phi_{j}\\}].$ (52)
Now taking ${\cal F}=B[\\{\phi(s)\\}]$e-E or e-E, from Eq. (52) we obtain Eq.
(5),
$\displaystyle{\int{\cal D}[\phi(s)]B[\\{\phi(s)\\}]\text{e}^{-\cal
E}\over\int{\cal D}[\phi(s)]\text{e}^{-\cal E}}={\int{\cal
D}[\dot{\phi}(s)]B[\\{\phi(s)\\}]\text{e}^{-\cal E}\over\int{\cal
D}[\dot{\phi}(s)]\text{e}^{-\cal E}}.$ (53)
Intuitively, both sides in the above equation are averages over all possible
configurations so they are expected to be equivalent.
## Appendix 2: A direct derivation of Eq. (16)
In this appendix, we will use the standard path integral method KLB93 to
derive Eq. (16) since it is useful. To account the more general case of
$s_{0}\neq 0$, we rewrite ${\cal H}_{0}$ as
$\displaystyle{\cal H}_{0}$ $\displaystyle=$ $\displaystyle{1\over
2}\int_{s_{0}}^{L}\kappa[\dot{\phi}(s)-\bar{c}(s)]^{2}ds.$ (54)
For large $N$, the path integral can be approximated as
$\displaystyle{\cal Z}_{\cal H}^{0}\approx C\prod_{j=1}^{N-1}\int
d\phi_{j}\text{exp}\left\\{-{\epsilon\over
2}\sum_{j=0}^{N-1}\kappa_{j}\left[{\phi_{j+1}-\phi_{j}\over\epsilon}-\bar{c}_{j}\right]^{2}\right\\}=C\prod_{j=1}^{N-1}\int
d\phi_{j}\text{exp}\left\\{-{1\over
2\epsilon}\sum_{j=0}^{N-1}\kappa_{j}\left[\phi_{j+1}-\phi_{j}-\bar{c}_{j}\epsilon\right]^{2}\right\\},$
(55)
where $C$ is a constant, $\epsilon=(L-s_{0})/N$,
$\phi_{j}=\phi[s_{0}+(j-1)\epsilon]$, $\kappa_{j}=\kappa[s_{0}+(j-1)\epsilon]$
and $\bar{c}_{j}=\bar{c}[s_{0}+(j-1)\epsilon]$ are discretized $\phi(s)$,
$\kappa(s)$ and $\bar{c}(s)$, respectively, and $\dot{\phi}$ in ${\cal H}_{0}$
is replaced by $(\phi_{j+1}-\phi_{j})/\epsilon$. Now using the identity
$\displaystyle\int_{-\infty}^{\infty}dx\text{e}^{-a(x-x_{1})^{2}-b(x_{2}-x)^{2}}=\sqrt{\pi\over
a+b}\text{exp}\left[-{1\over 1/a+1/b}(x_{1}-x_{2})^{2}\right],$ (56)
we obtain
$\displaystyle{\cal Z}_{\cal H}^{0}$ $\displaystyle\approx$ $\displaystyle
C\sqrt{2\pi\epsilon\over\kappa_{0}+\kappa_{1}}\int
d\phi_{2}d\phi_{3}\cdot\cdot\cdot d\phi_{N-1}\text{exp}\left\\{-{1\over
2\epsilon}\sum_{j=2}^{N-1}\kappa_{j}\left[\phi_{j+1}-\phi_{j}-\bar{c}_{j}\epsilon\right]^{2}\right\\}$
(57) $\displaystyle\cdot\text{exp}\left\\{-{1\over
2\epsilon/\kappa_{0}+2\epsilon/\kappa_{1}}[\phi_{2}-\phi_{0}-(\bar{c}_{0}+\bar{c}_{1})\epsilon]^{2}\right\\}$
$\displaystyle=$ $\displaystyle
C\sqrt{2\pi\epsilon\over\kappa_{0}+\kappa_{1}}\sqrt{2\pi\epsilon\over\kappa^{\prime}_{1}+\kappa_{2}}\int
d\phi_{3}d\phi_{4}\cdot\cdot\cdot d\phi_{N-1}\text{exp}\left\\{-{1\over
2\epsilon}\sum_{j=3}^{N-1}\kappa_{j}\left[\phi_{j+1}-\phi_{j}-\bar{c}_{j}\epsilon\right]^{2}\right\\}$
$\displaystyle\cdot\text{exp}\left\\{-{1\over
2\epsilon/\kappa^{\prime}_{1}+2\epsilon/\kappa_{2}}[\phi_{3}-\phi_{0}-(\bar{c}_{0}+\bar{c}_{1}+\bar{c}_{2})\epsilon]^{2}\right\\}$
$\displaystyle=$ $\displaystyle...={\cal C}\text{ exp}\left\\{-{1\over
2\sum_{j=0}^{N-1}\epsilon/\kappa_{j}}\left[\phi_{N}-\phi_{0}-\sum_{j=0}^{N-1}\bar{c}_{j}\epsilon\right]^{2}\right\\},$
where $1/\kappa^{\prime}_{1}\equiv 1/\kappa_{0}+1/\kappa_{1}$ and ${\cal C}$
is a new constant. Now let $N\rightarrow\infty$, we obtain
$\displaystyle{\cal Z}_{\cal H}^{0}={\cal C}\text{ exp}\left\\{-{1\over
2A(L,s_{0})}\left[\phi(L)-\phi(s_{0})-\int_{s_{0}}^{L}ds\bar{c}(s)\right]^{2}\right\\}.$
(58)
Normalizing the above equation we obtain ${\cal C}=1/\sqrt{2\pi A(L,s_{0})}$,
and recover Eq. (16).
## Appendix 3: Derivation of Eqs. (18)-(22)
When force is small, expanding e-E about $f=0$, we obtain
$\displaystyle{\cal Z}_{k}$ $\displaystyle=$ $\displaystyle\int{\cal
D}[\phi]\text{e}^{-\cal E}\approx{\cal Z}_{k}^{0}(1+fQ),$ (59) $\displaystyle
B$ $\displaystyle=$ $\displaystyle{1\over{\cal Z}_{\alpha}}\int{\cal
D}[\phi]B[\\{\phi\\}]\text{e}^{F}\left[\int{\cal D}[c]{1\over{\cal
Z}_{k}}W(\\{c\\})\text{e}^{-{\cal E}_{0}}\right]\approx{1\over{\cal
Z}_{\alpha}{\cal Z}_{k}^{0}}\int{\cal
D}[\phi]B[\\{\phi\\}]\text{e}^{F}\left[\int{\cal
D}[c]W(\\{c\\})(1-fQ)\text{e}^{-{\cal E}_{0}}\right]$ (60) $\displaystyle=$
$\displaystyle B_{1}-B_{2},$
where
$\displaystyle Q$ $\displaystyle=$
$\displaystyle\int_{0}^{L}ds\left<\cos\phi(s)\right>_{k}^{0},\text{ }F\equiv
f\int_{0}^{L}\cos\phi ds,$ (61) $\displaystyle B_{1}$ $\displaystyle\equiv$
$\displaystyle{1\over{\cal Z}_{\alpha}{\cal Z}_{k}^{0}}\int{\cal
D}[\phi]B[\\{\phi\\}]\text{e}^{F}\left[\int{\cal
D}[c]W(\\{c\\})\text{e}^{-{\cal E}_{0}}\right]={1\over{\cal Z}_{{\cal
H}}^{0}}\int{\cal D}[\phi]B[\\{\phi\\}]\text{e}^{-\cal H}$ (62)
$\displaystyle\approx$
$\displaystyle\left<B\right>_{\kappa}+f\int_{0}^{L}ds^{\prime}\left<B[\\{\phi(s)\\}]\cos\phi(s^{\prime})\right>_{\kappa},$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle{f\over{\cal
Z}_{\alpha}{\cal Z}_{k}^{0}}\int{\cal
D}[\phi]B[\\{\phi\\}]\text{e}^{F}R\approx{f\over{\cal Z}_{\alpha}{\cal
Z}_{k}^{0}}\int{\cal D}[\phi]B[\\{\phi\\}]{\cal R},$ (63) $\displaystyle{\cal
R}$ $\displaystyle=$ $\displaystyle\int{\cal D}[c]W(\\{c\\})Q\text{e}^{-{\cal
E}_{0}},$ (64)
and $\left<...\right>_{k}^{0}$ denotes the ensemble average with energy ${\cal
E}_{0}$
$\displaystyle\left<...\right>_{k}^{0}$ $\displaystyle\equiv$
$\displaystyle{1\over{\cal Z}_{k}^{0}}\int{\cal
D}[\phi(s)](...)\text{e}^{-{\cal E}_{0}}.$ (65)
Furthermore,
$\displaystyle{\cal R}$ $\displaystyle=$
$\displaystyle\int_{0}^{L}ds^{\prime}\left[\int{\cal
D}[c]W(\\{c\\})\text{e}^{-{\cal E}_{0}[\\{\phi(s)\\}]}{1\over{\cal
Z}_{k}^{0}}\int{\cal
D}[\phi^{\prime}]\cos[\phi^{\prime}(s^{\prime})]\text{e}^{-{\cal
E}_{0}[\\{\phi^{\prime}(s)\\}]}\right]$ (66) $\displaystyle=$
$\displaystyle\int_{0}^{L}ds^{\prime}{1\over{\cal Z}_{k}^{0}}\left[\int{\cal
D}[\phi^{\prime}]\cos[\phi^{\prime}(s^{\prime})]\int{\cal
D}[c]W(\\{c\\})\text{e}^{-{\cal E}_{0}[\\{\phi(s)\\}]-{\cal
E}_{0}[\\{\phi^{\prime}(s)\\}]}\right]$ $\displaystyle=$
$\displaystyle{G\over{\cal Z}_{k}^{0}}\text{e}^{-{\cal
H}_{0}}\int_{0}^{L}ds^{\prime}\left[\int{\cal
D}[\phi^{\prime}]\cos[\phi^{\prime}(s^{\prime})]\text{e}^{-1/2\int_{0}^{L}ds\left[k^{\prime}\left(\dot{\phi}^{\prime}(s)-\dot{\gamma}(s)\right)^{2}\right]}\right]$
$\displaystyle=$ $\displaystyle{{\cal Z}_{\alpha}{\cal Z}_{k}^{0}\over{\cal
Z}_{\cal H}^{0}}\text{e}^{-{\cal
H}_{0}}\int_{0}^{L}ds^{\prime}\left[{\int{\cal
D}[\phi^{\prime}]\cos[\phi^{\prime}(s^{\prime})]\text{e}^{-1/2\int_{0}^{L}ds\left[k^{\prime}\left(\dot{\phi}^{\prime}(s)-\dot{\gamma}(s)\right)^{2}\right]}\over\int{\cal
D}[\phi^{\prime}]\text{e}^{-1/2\int_{0}^{L}dsk^{\prime}[\dot{\phi}^{\prime}(s)-\dot{\gamma}(s)]^{2}}}\right]$
$\displaystyle=$ $\displaystyle{{\cal Z}_{\alpha}{\cal Z}_{k}^{0}\over{\cal
Z}_{\cal H}^{0}}\text{e}^{-{\cal
H}_{0}}\int_{0}^{L}ds^{\prime}\left[\text{e}^{-A^{\prime}(s^{\prime},0)/2}\cos[\gamma(s^{\prime})+\phi_{0}]\right],$
where
$\displaystyle k^{\prime}(s)={k(\alpha+k)\over\alpha+2k},\text{
}\dot{\gamma}(s)={k\dot{\phi}+\alpha\bar{c}\over\alpha+k},\text{
}\gamma(s)=\int_{0}^{s}\dot{\gamma}(s)ds,\text{ }G=\int{\cal
D}[c]\text{e}^{-1/2\int_{0}^{L}ds(2k+\alpha)c^{2}},\text{
}A^{\prime}(s,s_{0})=\int_{s_{0}}^{s}ds/k^{\prime}(s),$ (67)
and we have used an expression similar to Eq. (16) in the last two lines in
Eq. (66) to transform the path integral into simple integral. $\gamma(s)$ is
in general dependent on $\phi(s)$ and it makes the expression complex. Now the
Eq. (63) can be reduced into
$\displaystyle B_{2}$ $\displaystyle\approx$ $\displaystyle{f\over{\cal
Z}_{\cal H}^{0}}\int{\cal D}[\phi]B[\\{\phi(s)\\}]\text{e}^{-{\cal
H}_{0}}\left[\int_{0}^{L}ds^{\prime}\text{e}^{-A^{\prime}(s^{\prime},0)/2}\cos[\gamma(s^{\prime})+\phi_{0}]\right]$
(68) $\displaystyle=$ $\displaystyle
f\int_{0}^{L}ds^{\prime}\text{e}^{-A^{\prime}(s^{\prime},0)/2}\left<B[\\{\phi(s)\\}]\cos[\gamma(s^{\prime})+\phi_{0}]\right>_{\kappa}.$
where $\phi_{0}=\phi(0)$. Eqs. (62) and (68) are exactly the same as Eqs. (19)
and (LABEL:B21).
## Appendix 4: Calculations of the extension
In this case $B[\\{\phi(s)\\}]$ corresponds to cos$\phi(s)$, and so from Eqs.
(19)-(LABEL:B21), we obtain
$\displaystyle X$ $\displaystyle\equiv$
$\displaystyle\left<x\right>-\left<x\right>_{f=0}=\int_{0}^{L}ds\left<\cos\phi(s)\right>-\int_{0}^{L}ds\left<\cos\phi(s)\right>_{\kappa}$
(69) $\displaystyle=$ $\displaystyle
f\int_{0}^{L}ds\int_{0}^{L}ds^{\prime}\left[\left<\cos\phi(s)\cos\phi(s^{\prime})\right>_{\kappa}-\text{e}^{-s^{\prime}/2k^{\prime}}\left<\cos[\phi(s)]\cos[\gamma(s^{\prime})+\phi_{0}]\right>_{\kappa}\right].$
Note that Eq. (16) is valid only if $s>s_{0}$. Therefore, the integral for
$s^{\prime}$ in Eq. (74) should be divided into two parts, one is from $0$ to
$s$ and the other is from $s$ to $L$. When $s>s^{\prime}$, we have
$\displaystyle\left<\cos\phi(s)\cos\phi(s^{\prime})\right>_{\kappa}=\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}P(\phi,s;\phi^{\prime},s^{\prime})\cos\phi\cos\phi^{\prime}P(\phi^{\prime},s^{\prime};\phi_{0},0)$
(70) $\displaystyle=$
$\displaystyle\text{e}^{-(s-s^{\prime})/2\kappa}\int_{-\infty}^{\infty}d\phi^{\prime}\cos^{2}\phi^{\prime}P(\phi^{\prime},s^{\prime};\phi_{0},0)={1\over
2}\text{e}^{-(s-s^{\prime})/2\kappa}\left[1+\text{e}^{-2s^{\prime}/\kappa}\cos(2\phi_{0})\right],$
and
$\displaystyle\left<\cos\phi(s)\cos[\gamma(s^{\prime})+\phi_{0}]\right>_{\kappa}=\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}P(\phi,s;\phi^{\prime},s^{\prime})\cos\phi\cos\left[\kappa(\phi^{\prime}-\phi_{0})/\alpha+\phi_{0}\right]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
(71) $\displaystyle=$
$\displaystyle\text{e}^{-(s-s^{\prime})/2\kappa}\int_{-\infty}^{\infty}d\phi^{\prime}\cos\phi^{\prime}\cos\left[\kappa(\phi^{\prime}-\phi_{0})/\alpha+\phi_{0}\right]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
$\displaystyle=$ $\displaystyle{1\over
2}\text{e}^{-(s-s^{\prime})/2\kappa-(\alpha+\kappa)^{2}s^{\prime}/2\kappa\alpha^{2}}\left[\cos(2\phi_{0})+\text{e}^{2s^{\prime}/\alpha}\right].$
When $s<s^{\prime}$, we obtain
$\displaystyle\left<\cos\phi(s)\cos\phi(s^{\prime})\right>_{\kappa}=\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}P(\phi^{\prime},s^{\prime};\phi,s)\cos\phi\cos\phi^{\prime}P(\phi,s;\phi_{0},0)$
$\displaystyle=$
$\displaystyle\text{e}^{-(s^{\prime}-s)/2\kappa}\int_{-\infty}^{\infty}d\phi\cos^{2}\phi
P(\phi,s;\phi_{0},0)={1\over
2}\text{e}^{-(s^{\prime}-s)/2\kappa}\left[1+\text{e}^{-2s/\kappa}\cos(2\phi_{0})\right],$
$\displaystyle\left<\cos\phi(s)\cos[\gamma(s^{\prime})+\phi_{0}]\right>_{\kappa}=\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}P(\phi^{\prime},s^{\prime};\phi,s)\cos\phi\cos\left[\kappa(\phi^{\prime}-\phi_{0})/\alpha+\phi_{0}\right]P(\phi,s;\phi_{0},0)$
$\displaystyle=$
$\displaystyle\text{e}^{-\kappa(s^{\prime}-s)/2\alpha^{2}}\int_{-\infty}^{\infty}d\phi\cos\phi\cos[\kappa(\phi-\phi_{0})/\alpha+\phi_{0}]P(\phi,s;\phi_{0},0)$
$\displaystyle=$ $\displaystyle{1\over
2}\text{e}^{-\kappa(s^{\prime}-s)/2\alpha^{2}-(\alpha+\kappa)^{2}s/2\kappa\alpha^{2}}\left[\cos(2\phi_{0})+\text{e}^{2s/\alpha}\right].$
(73)
It follows
$\displaystyle X$ $\displaystyle=$ $\displaystyle 2\kappa
f\left[L-{k(k+3\alpha)\over k+\alpha}+{k(k+\alpha)\over
k-\alpha}\text{e}^{-L/k}-{4\kappa\alpha\over
k-\alpha}\text{e}^{-L/2\kappa}\right]-{\kappa^{2}\cos(2\phi_{0})\text{e}^{-2L/\kappa}f\over
3(2k+\alpha)(3k+\alpha)}{\cal X},$ (74) $\displaystyle{\cal X}$
$\displaystyle=$ $\displaystyle
6k^{2}\left(\text{e}^{L/k}-1\right)+k\alpha\left(9\text{e}^{2L/\kappa}-16\text{e}^{3L/2\kappa}+12\text{e}^{L/k}-5\right)+\alpha^{2}\left(3\text{e}^{2L/\kappa}-8\text{e}^{3L/2\kappa}+6\text{e}^{L/k}-1\right).$
In the limit $\alpha\rightarrow k$, $X$ is still finite and
$\displaystyle X(\alpha\rightarrow k)$ $\displaystyle=$ $\displaystyle 2\kappa
f\left[L-2k+(L+2k)\text{e}^{-L/k}\right]-{1\over
12}fk^{2}\cos(2\phi_{0})\left(1-2\text{e}^{-L/k}+2\text{e}^{-3L/k}-\text{e}^{-4L/k}\right).$
(75)
On the other hand, in disorder-first system,
$\displaystyle X^{\prime}$ $\displaystyle\equiv$
$\displaystyle\left<x\right>^{\prime}-\left<x\right>^{\prime}_{f=0}=\int_{0}^{L}ds\left<\cos\phi(s)\right>^{\prime}-\int_{0}^{L}\left<\cos\phi(s)\right>_{\kappa}$
(76) $\displaystyle=$ $\displaystyle
f\int_{0}^{L}ds\int_{0}^{L}ds^{\prime}\left<\cos\phi(s)\cos\phi(s^{\prime})\right>_{\kappa}-f\left[\int_{0}^{L}ds\left<\cos[\phi(s)]\right>_{\kappa}\right]^{2}$
$\displaystyle=$ $\displaystyle 2\kappa
f\left[L+\kappa-\kappa\left(\text{e}^{-L/2\kappa}-2\right)^{2}\right]+{\kappa^{2}f\over
3}\left(\text{e}^{-2L/\kappa}-6\text{e}^{-L/\kappa}+8\text{e}^{-L/2\kappa}-3\right)\cos(2\phi_{0}).$
When $\alpha\rightarrow\infty$, it is clearly that $X\rightarrow X^{\prime}$.
## Appendix 5: Calculations of the orientational correlation function and the
end-to-end Distance
In this case, $B[\\{\phi\\}]={\bf t}(s)\cdot{\bf
t}(s^{\prime})=\cos[\phi(s)-\phi(s^{\prime})]$. From Eqs. (18)-(22), we obtain
$\displaystyle\left<{\bf t}(s)\cdot{\bf t}(s^{\prime})\right>$
$\displaystyle\approx$
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\right>_{\kappa}$ (77)
$\displaystyle+f\int_{0}^{L}ds^{\prime\prime}\left[\left<\cos[\phi(s)-\phi(s^{\prime})]\cos\phi(s^{\prime\prime})\right>_{\kappa}-\text{e}^{-s^{\prime\prime}/2k^{\prime}}\left<\cos[\phi(s)-\phi(s^{\prime})]\cos[\gamma(s^{\prime\prime})+\phi_{0}]\right>_{\kappa}\right].$
The first term in above equation is simple, as
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\right>_{\kappa}=\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}P(\phi,s;\phi^{\prime},s^{\prime})\cos(\phi-\phi^{\prime})P(\phi^{\prime},s^{\prime};\phi_{0},0)=\text{e}^{-(s-s^{\prime})/2\kappa}.$
(78)
Again, due to that Eq. (16) is valid only if $s>s_{0}$, the integral for
$s^{\prime\prime}$ in Eq. (77) should be divided into several parts. If
$s>s^{\prime}>s^{\prime\prime}$, we have
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\cos[\gamma(s^{\prime\prime})+\phi_{0}]\right>_{\kappa}$
(79) $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}d\phi^{\prime\prime}P(\phi,s;\phi^{\prime},s^{\prime})P(\phi^{\prime},s^{\prime};\phi^{\prime\prime},s^{\prime\prime})\cos(\phi-\phi^{\prime})\cos[\kappa(\phi^{\prime\prime}-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime\prime},s^{\prime\prime};\phi_{0},0)$
$\displaystyle=$
$\displaystyle\text{e}^{-(s-s^{\prime})/2\kappa}\int_{-\infty}^{\infty}d\phi^{\prime\prime}\cos[\kappa(\phi^{\prime\prime}-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime\prime},s^{\prime\prime};\phi_{0},0)=\text{e}^{-(s-s^{\prime})/2\kappa-\kappa
s^{\prime\prime}/2\alpha^{2}}\cos\phi_{0}.$
If $s>s^{\prime\prime}>s^{\prime}$, we obtain
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\cos[\gamma(s^{\prime\prime})+\phi_{0}]\right>_{\kappa}$
(80) $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}d\phi^{\prime\prime}P(\phi,s;\phi^{\prime\prime},s^{\prime\prime})P(\phi^{\prime\prime},s^{\prime\prime};\phi^{\prime},s^{\prime})\cos(\phi-\phi^{\prime})\cos[\kappa(\phi^{\prime\prime}-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
$\displaystyle=$
$\displaystyle\text{e}^{-(s-s^{\prime\prime})/2\kappa}\int_{-\infty}^{\infty}d\phi^{\prime}d\phi^{\prime\prime}P(\phi^{\prime\prime},s^{\prime\prime};\phi^{\prime},s^{\prime})\cos(\phi^{\prime}-\phi^{\prime\prime})\cos[\kappa(\phi^{\prime\prime}-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
$\displaystyle=$ $\displaystyle{1\over
2}\text{e}^{-(s-s^{\prime\prime})/2\kappa-(\alpha+\kappa)^{2}(s^{\prime\prime}-s^{\prime})/2\kappa\alpha^{2}}\left(1+\text{e}^{2(s^{\prime\prime}-s^{\prime})/\alpha}\right)\int_{-\infty}^{\infty}d\phi^{\prime}\cos[\kappa(\phi^{\prime}-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
$\displaystyle=$ $\displaystyle{1\over
2}\text{e}^{-s/2\kappa+(3k+\alpha)s^{\prime}/2k\alpha-(3k+2\alpha)s^{\prime\prime}/[2\alpha(k+\alpha)]}\left(1+\text{e}^{2(s^{\prime\prime}-s^{\prime})/\alpha}\right)\cos\phi_{0}.$
If $s^{\prime\prime}>s>s^{\prime}$, we find
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\cos[\gamma(s^{\prime\prime})+\phi_{0}]\right>_{\kappa}$
(81) $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}d\phi^{\prime\prime}P(\phi^{\prime\prime},s^{\prime\prime};\phi,s)P(\phi,s;\phi^{\prime},s^{\prime})\cos(\phi-\phi^{\prime})\cos[\kappa(\phi^{\prime\prime}-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
$\displaystyle=$
$\displaystyle\text{e}^{-\kappa(s^{\prime\prime}-s)/2\alpha^{2}}\int_{-\infty}^{\infty}d\phi
d\phi^{\prime}dP(\phi,s;\phi^{\prime},s^{\prime})\cos(\phi-\phi^{\prime})\cos[\kappa(\phi-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
$\displaystyle=$ $\displaystyle{1\over
2}\text{e}^{-\kappa(s^{\prime\prime}-s)/2\alpha^{2}-(\alpha+\kappa)^{2}(s-s^{\prime})/2\kappa\alpha^{2}}\left(1+\text{e}^{2(s-s^{\prime})/\alpha}\right)\int_{-\infty}^{\infty}d\phi^{\prime}\cos[\kappa(\phi^{\prime}-\phi_{0})/\alpha+\phi_{0}]P(\phi^{\prime},s^{\prime};\phi_{0},0)$
$\displaystyle=$ $\displaystyle{1\over
2}\text{e}^{-(3k+\alpha)(s-s^{\prime})/2k\alpha-\kappa
s^{\prime\prime}/2\alpha^{2}}\left(1+\text{e}^{2(s-s^{\prime})/\alpha}\right)\cos\phi_{0}.$
Similarly, if $s>s^{\prime}>s^{\prime\prime}$, then
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\cos\left(\phi(s^{\prime\prime})\right)\right>_{\kappa}=\text{e}^{-(s-s^{\prime})/2\kappa-s^{\prime\prime}/2\kappa}\cos\phi_{0}.$
(82)
If $s>s^{\prime\prime}>s^{\prime}$, then
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\cos\left(\phi(s^{\prime\prime})\right)\right>_{\kappa}={1\over
2}\text{e}^{-(s-3s^{\prime}+3s^{\prime\prime})/2\kappa}\left(1+\text{e}^{2(s^{\prime\prime}-s^{\prime})/\kappa}\right)\cos\phi_{0}.$
(83)
If $s^{\prime\prime}>s>s^{\prime}$, then
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\cos\left(\phi(s^{\prime\prime})\right)\right>_{\kappa}={1\over
2}\text{e}^{-(3s-3s^{\prime}+s^{\prime\prime})/2\kappa}\left(1+\text{e}^{2(s-s^{\prime})/\kappa}\right)\cos\phi_{0}.$
(84)
Combining Eqs. (77)-(84), we finally obtain for $s>s^{\prime}$
$\displaystyle\left<{\bf t}(s)\cdot{\bf t}(s^{\prime})\right>$
$\displaystyle\approx$
$\displaystyle\text{e}^{-(s-s^{\prime})/2\kappa}+{\kappa\cos\phi_{0}f\over
3(3k+\alpha)(k-\alpha)}{\cal S},$ (85) $\displaystyle{\cal S}$
$\displaystyle=$
$\displaystyle-6k(k-\alpha)\text{e}^{-2[(\alpha+2k)s+(\alpha+3k)s^{\prime}]/2k\alpha}+8\alpha(2k+\alpha)\text{e}^{-s/2\kappa}-6k(3k+\alpha)\text{e}^{-[2\alpha
s+(k-\alpha)s^{\prime}]/2k\alpha}$
$\displaystyle+(k-\alpha)(3k+\alpha)\left[3\text{e}^{-[(\alpha+k)L+(\alpha+3k)(s-s^{\prime})]/2k\alpha}-3\text{e}^{-[L+3(s-s^{\prime})]/2\kappa}+6\text{e}^{-s^{\prime}/2\kappa}-3\text{e}^{-(L+s^{\prime}-s)/2\kappa}\right.$
$\displaystyle\left.+2\text{e}^{-(4s-3s^{\prime})/2\kappa}+3\text{e}^{-[(\alpha+k)L+(\alpha-k)(s-s^{\prime})]/2k\alpha}\right].$
$\left<{\bf t}(s)\cdot{\bf t}(s^{\prime})\right>$ is also finite when
$\alpha\rightarrow k$ because in this case,
$\displaystyle\left<{\bf t}(s)\cdot{\bf t}(s^{\prime})\right>$
$\displaystyle\approx$
$\displaystyle\text{e}^{-(s-s^{\prime})/2\kappa}+\kappa\cos\phi_{0}f\left[-\text{e}^{-(L+s^{\prime}-s)/k}+2\text{e}^{-s^{\prime}/k}-{13\over
6}\text{e}^{-s/k}+\text{e}^{-L/k}-{1\over
2}\text{e}^{(-3s+2s^{\prime})/k}\right.$ (86)
$\displaystyle\left.+\text{e}^{-[L+2(s-s^{\prime})]/k}+{2\over
3}\text{e}^{(-4s+3s^{\prime})/k}+\text{e}^{-[L+3(s-s^{\prime})]/k}+{s^{\prime}-s\over
k}\text{e}^{-s/k}\right].$
In contrast,
$\displaystyle\left<{\bf t}(s)\cdot{\bf t}(s^{\prime})\right>^{\prime}$
$\displaystyle\approx$
$\displaystyle\left<\cos[\phi(s)-\phi(s^{\prime})]\right>_{\kappa}+f\int_{0}^{L}ds^{\prime\prime}\left<\cos[\phi(s)-\phi(s^{\prime})]\cos\phi(s^{\prime\prime})\right>_{\kappa}$
(87)
$\displaystyle-f\left<\cos[\phi(s)-\phi(s^{\prime})]\right>_{\kappa}\int_{0}^{L}ds\left<\cos[\phi(s)]\right>_{\kappa}$
$\displaystyle=$
$\displaystyle\text{e}^{-(s-s^{\prime})/2\kappa}+f\int_{0}^{L}ds^{\prime\prime}\left<\cos[\phi(s)-\phi(s^{\prime})]\cos\phi(s^{\prime\prime})\right>_{\kappa}-2\kappa
f\cos\phi_{0}\text{e}^{-(s-s^{\prime})/2\kappa}\left(1-\text{e}^{-L/2\kappa}\right)$
$\displaystyle=$ $\displaystyle\text{e}^{-(s-s^{\prime})/2\kappa}+{1\over
3}\kappa
f\cos\phi_{0}\left(6\text{e}^{-s^{\prime}/2\kappa}-8\text{e}^{-s/2\kappa}+6\text{e}^{-(L+s-s^{\prime})/2\kappa}+2\text{e}^{(3s^{\prime}-4s)/2\kappa}\right.$
$\displaystyle\left.-3\text{e}^{-[L+3(s-s^{\prime})]/2\kappa}-3\text{e}^{-(L+s^{\prime}-s)/2\kappa}\right).$
When $\alpha\rightarrow\infty$, we obtain $\left<{\bf t}(s)\cdot{\bf
t}(s^{\prime})\right>\rightarrow\left<{\bf t}(s)\cdot{\bf
t}(s^{\prime})\right>^{\prime}$.
The end-to-end distance can be found by
$\displaystyle R^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{L}ds\int_{0}^{L}ds^{\prime}\left<{\bf t}(s)\cdot{\bf
t}^{\prime}(s^{\prime})\right>=2\int_{0}^{L}ds\int_{0}^{s}ds^{\prime}\left<\cos[\phi(s)-\phi(s^{\prime})]\right>$
(88) $\displaystyle=$ $\displaystyle 4\kappa L\left[1-{2\kappa\over
L}\left(1-\text{e}^{-{L\over
2\kappa}}\right)\right]+{4\kappa^{2}\cos\phi_{0}f\over
9(k-\alpha)^{2}(k+\alpha)(2k+\alpha)(3k+\alpha)^{3}}{\cal Y},$
$\displaystyle{\cal Y}$ $\displaystyle=$ $\displaystyle
18(k-\alpha)^{2}(k+\alpha)(2k+\alpha)(3k+\alpha)^{2}L-9k(k-\alpha)^{2}(3k+\alpha)^{2}(4k^{2}+11\alpha^{2}+22k\alpha)$
(89)
$\displaystyle-16\alpha(2k+\alpha)^{2}[3(k-\alpha)(k+\alpha)(3k+\alpha)L+4k\alpha(5k+\alpha)(3k-\alpha)]\text{e}^{-L/2\kappa}+18k\alpha(k+\alpha)^{3}(k-\alpha)^{2}\text{e}^{-(1/k+2/\alpha)L}$
$\displaystyle+18k(k+\alpha)^{3}(2k+\alpha)(3k+\alpha)^{2}\text{e}^{-L/k}-k\alpha(k-\alpha)^{2}(2k+\alpha)(3k+\alpha)^{2}\text{e}^{-2L/\kappa}.$
when $\alpha\rightarrow k$,
$\displaystyle R^{2}$ $\displaystyle=$ $\displaystyle 2kL\left[1-{k\over
L}\left(1-\text{e}^{-{L\over k}}\right)\right]+{k\cos\phi_{0}f\over
18}\left[36kL-111k^{2}+(18L^{2}+78kL+109k^{2})\text{e}^{-L/k}+3k^{3}\text{e}^{-3L/k}-k^{2}\text{e}^{-4L/k}\right].$
On the other hand
$\displaystyle R^{\prime 2}$ $\displaystyle=$ $\displaystyle 4\kappa
L\left[1-{2\kappa\over L}\left(1-\text{e}^{-{L\over
2\kappa}}\right)\right]+{4\over
9}\kappa^{2}\cos\phi_{0}f\left[18L-99\kappa+16(3L+4\kappa)\text{e}^{-L/2\kappa}+36\kappa\text{e}^{-L/\kappa}-\kappa\text{e}^{-2L/\kappa}\right].$
(91)
When $\alpha\rightarrow\infty$, we obtain $R^{2}\rightarrow R^{\prime 2}$.
## Acknowledgments
This work has been supported by the National Science Council of the Republic
of China, National Center for Theoretical Sciences at Taipei, ROC, and the
Natural Sciences and Engineering Research Council of Canada.
## References
* (1) A.A. Travers, DNA-Protein Interactions, (Chapman & Hall, Lodon, 1993).
* (2) P. J. Hagerman, Annu. Rev. Biophys. Biophys. Chem. 17, 265 (1988); D. M. Crothers, T. E. Haran, and J. G. Nadeau, J. Biol. Chem. 265, 7093 (1990).
* (3) Eran Segal, Yvonne Fondufe-Mittendorf, Lingyi Chen, AnnChristine Thåström, Yair Field, Irene K. Moore, Ji-Ping Z. Wang, and Jonathan Widom, Nature 442, 772(2006).
* (4) B. Audit, C. Thermes, C. Vaillant, Y. d’Aubenton-Carafa, J.F. Muzy, and A. Arneodo, Phys. Rev. Lett. 86, 2471 (2001).
* (5) Benjamin Audit, Cedric Vaillant, Alain Arneodo, Yves d’Aubenton-Carafa, and Claude Thermes, J. Mol. Biol. 316, 903 (2002).
* (6) C. Vaillant, B. Audit, and A. Arnéodo, Phys. Rev. Lett. 95, 068101 (2005).
* (7) J. Moukhtar, E. Fontaine, C. Faivre-Moskalenko, and A. Arneodo, Phys. Rev. Lett. 98, 178101 (2007).
* (8) Z. Zhou and B. Joós, Phys. Rev. E 77, 061906 (2008).
* (9) Y. O. Popov and A. V. Tkachenko, Phys. Rev. E. 76, 021901 (2007).
* (10) P. C. Nelson, Phys. Rev. Lett. 80, 5810 (1998).
* (11) D. Bensimon, D. Dohmi, and M. Mezard, Europhys. Lett. 42, 97 (1998).
* (12) O. Kratky and G. Porod, Recl. Trav. Chim. Pays-Bas 68, 1106 (1949).
* (13) J.F. Marko and E.D. Siggia, Science 265, 506 (1994).
* (14) C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith, Science 265, 1599 (1994).
* (15) J. F. Marko and E. D. Siggia, Macromolecules 28, 8759 (1995).
* (16) S.B. Smith, L. Finzi, and C. Bustamante, Science 258, 1122 (1992).
* (17) T.R. Strick, J.F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, Science 271, 1835 (1996).
* (18) S.V. Panyukov and Y. Rabin, Phys. Rev. E 62, 7135 (2000); ibid, Phys. Rev. E 64, 011909 (2001).
* (19) Z. Zhou, P.-Y. Lai, and B. Joós, Phys. Rev. E 71, 052801(2005).
* (20) Z. Zhou, B. Joós, P.-Y. Lai, Y.-S. Young, and J.-H. Jan, Mod. Phys. Lett. B 21, 1895-1913 (2007).
* (21) H. Zhou, Y. Zhang, Z.-C. Ou-Yang, Phys. Rev. E. 62, 1045 (2000).
* (22) Z. Zhou, Phys. Rev. E. 76, 061913 (2007).
* (23) E. N. Trifonov, R. K.-Z. Tan, and S. C. Harvey, in DNA bending and curvature, edited by W. K. Olson, M. H. Sarma, and M. Sundaralingam (Adenine Press, Schenectady, 1987).
* (24) J.A. Schellman and S.C. Harvey, Biophys. Chem. 55, 95 (1995).
* (25) C. Vaillant, B. Audit, C. Thermes, and A. Arnéodo, Phys. Rev. E. 67, 032901 (2003).
* (26) D. Shore, J. Langowski, and R.L. Baldwin, Proc. Natl Acad. Sci. USA 170, 4833 (1981).
* (27) K. Rippe, P.R. von Hippel, and J. Langowski, Trends Biochem. Sci. 20, 500 (1995).
* (28) J. Widom, Q. Rev. Biophys. 34, 269(2001).
* (29) J.A.H. Cognet, C. Pakleza, D. Cherny, E. Delain, and E.L. Cam, J. Mol. Biol. 285, 997 (1999); A. Scipioni, C. Anselmi, G. Zuccheri, B. Samori, and P. De Santis, Biophys. J., 83, 2408 (2002); M. Marilley, A. Sanchez-Sevilla J. Rocca-Serra, Mol Gen Genomics, 274, 658 (2005).
* (30) D.C. Khandekar, S.V. Lawande, and K.V. Bhagwat, Path Integrals Methods and Their Applications (World Scientific, Singapore, 1993).
* (31) H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics (World Scientific, Singapore, 1990).
* (32) A. Prasad, Y. Hori, and J. Kondev, Phys. Rev. E. 72, 041918(2005).
* (33) I.M. Kulić, H. Mohrbach, R. Thaokar, and H. Schiessel, Phys. Rev. E. 75, 011913(2007).
* (34) D. Chaudhuri, Phys. Rev. E. 75, 021803(2007).
* (35) L. Song and J.M. Schurr, Biopolymers 30, 229 (1990).
* (36) J. Bednar, P. Furrer, V. Katritch, A. Stasiak, J. Dubochet, A. Stasiak, J. Mol. Biol. 254, 579 (1995).
* (37) P. Furrer, J. Bednar, A.Z. Stasiak, V. Katritch, D. Michoud, A. Stasiak and J. Dubochet, J. Mol. Biol. 266, 711 (1997).
* (38) I.M. Kulić, H. Mohrbach, R. Thaokar, and H. Schiessel, Phys. Rev. E. 75, 011913(2007).
|
arxiv-papers
| 2009-10-31T00:49:47 |
2024-09-04T02:49:06.194790
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zicong Zhou and Bela Joos",
"submitter": "Zicong Zhou",
"url": "https://arxiv.org/abs/0911.0045"
}
|
0911.0048
|
# Validation of Models for the Flow of Granular Media
Jeffrey Picka
###### Abstract
Validation of models for powder flow requires that the models be stochastic
and that they be fit by statistical inference. Methods from spatial and
multivariate statistics can be used for model fitting and assessment. If the
quality of the fitted model is not assessed, there is a significant risk the
model will fail to represent the physics of powder flow.
## 1 Introduction
A mathematical model for powder flow must summarize what is known about the
physics of the powder and must allow useful predictions of the physical
behaviour of powders to be made. The models must be stochastic in order to
represent what is unknown about grain interactions in powder flow, and in
order to represent any sensitivity to initial conditions. Methods from
statistical inference need to be used not only to fit the models, but also to
assess the quality of the fitted models. Formal statistical methods will
complement fitting by eye and by intuition, since they will be able to
describe and compare features of realizations that the eye cannot easily see.
The methods outlined here are suited to studying two- or three-dimensional
powder flow under conditions where the powder stays in a packed or close-to-
packed state. Examples include flow in annular cells [1], flow in hoppers [2],
and flow during triaxial tests [3]. If the powder starts at rest, then the
initial arrangement of its grains in a model must be consistent with an
arrangement which could arise from the physical process of specimen
preparation. Once the powder is put into motion, the joint trajectories of the
grains in realizations of the model and in replicates of the experiment must
be sufficiently similar that a reliable prediction can be made from the model.
Methods from statistical inference are required to assess whether or not there
is evidence that the model is failing to capture the dynamics of the flow.
When the models are deterministic and produce only a single prediction for any
set of experimental conditions, no model validation procedure can demonstrate
that a model represents the dynamics of powder flow.
The word statistic is used to represent any number calculated from
observations of a physical flow process or its simulation. It need not be an
average, but could be a measure of variability around a mean or an extreme
value. The adjective statistical will be used in reference to statistical
inference, and not statistical mechanics. The goal is to undertake model
assessment by means of hypothesis tests and other tools of inference which
accommodate the natural variability of the observations, rather than trying to
eliminate variability by means of a probabilistic limiting argument.
Experiments will consist of many replicates conducted under the same
experimental conditions. For any given set of conditions, an indefinite number
of realizations can be generated from any fitted stochastic model of the
process.
## 2 The Need for Stochastic Models
Any powder flow phenomenon requires stochastic modeling. Given fixed
experimental conditions, all replicates are subject to uncontrolled variation
which cannot be eliminated by a probabilistic limiting argument.
If a powder begins at rest, then it is generally in a disordered jammed state.
This disordered state arises from a physical preparation process which is
sensitive to initial conditions and which generates unpredictable arrangements
of grains. It is possible to assume that the variability in structure between
replicates has no effect on the subsequent flow behaviour, but this would be a
dangerous assumption to make when the powder flows in a nearly-packed state.
If the physical phenomenon of interest depends strongly on the initial contact
network among the grains, then the contact network must also be stochastically
modelled. In many simulated packings and all data from physical specimens, it
is impossible to determine which grains are very close and which grains are in
contact [4]. Establishing a contact structure by means of an arbitrary
deterministic rule may result in the model failing to sample from contact
networks that are present in physical systems.
Once the powder is acted on by a force and flows, then the powder flow may be
sensitive to initial conditions. This would require stochastic modeling even
if all of the details of the grain interactions were known. Since the details
of the grain interactions are unknown, modeling requires making choices about
how to represent the many possible interactions among the grains. If
stochastic elements are introduced into the grain interactions, then these
elements can represent the ignorance of interaction mechanisms better than an
arbitrary deterministic model could.
## 3 Formulating Stochastic Models
Any stochastic model for powder flow must sample from the ensemble of jammed
initial configurations, the ensemble of viable contact networks for the
sampled configuration, and the ensemble of joint trajectories that could
emerge from the sampled contact network. Ideally, these ensembles will be
identical to those that the physical process itself samples from when the
process is repeated many times. The physical process ensembles will be defined
by a set experimental procedure, by the nature of the materials used, and by
certain macroscopic variables which govern the experimental procedure and
which are used to characterize acceptable outcomes. It will not be possible to
describe these ensembles by abstract mathematical models, as is the case in
conventional statistical mechanics.
In most cases, the model ensembles will differ from those of the physical
process. The usefulness of stochastic models will depend on their being able
to generate trajectories which cannot be distinguished from physical
trajectories by means of statistical inference. Since there will be no way to
determine the form of a stochastic model from theory, it will be necessary to
try many different models and to select the best ones by inferential methods.
If a powder begins at rest, then a stochastic model for the packed grains is
required to represent the unpredictability of the physical process which
produced the initial packing. In the best circumstances, a large number of
specimens can be prepared by that physical process and imaged by X-ray [4, 5,
6, 7], NMR [8], or confocal [9, 10] methods. From these images, initial states
can be selected at random directly from the physical ensemble for use in
stochastic models of flow.
If images of the internal structure of real specimens are not available, then
the packed states need to be generated by some type of stochastic packing
generator. There are no generators which are known to sample from the same
ensemble as a physical process. The first generators used, based on ballistic
methods [11, 12, 13, 14] or on the rearrangements of random point patterns
[15, 16, 17, 18, 19, 20], have no basis in the physics of packing formation at
all. Models based on Discrete Element Methods [DEM] [21, 22, 23, 24] are
inspired by the physics of packing formation, but all are based on
unverifiable assumptions about how grains interact. It is possible to avoid
these issues by assuming that any packing generator which can produce the
right mean volume fraction of grains is good enough, but this is a dangerous
assumption. If the generator produces initial arrangements of grains which
possess structure not found in physical specimens, then any model of an aspect
of powder flow which depends on initial grain arrangements may be useless.
Modeling of the contact network will require the use of an algorithm which
randomly chooses whether or not closely neighboring spheres are in contact.
These algorithms must avoid selecting contact networks which are mechanically
unstable.
The modeling of powder flow is generally undertaken using DEM models [24]. If
the DEM model is not stochastic, then it will produce only one predicted joint
trajectory for any one initial arrangement of grains. This joint trajectory
cannot be expected to belong to the ensemble of joint trajectories of the
physical process if any of the assumptions about grain interactions are
incorrect. Even if the single prediction is in the ensemble of trajectories
for the physical system, there is no way to tell if it is a typical trajectory
for that ensemble.
Stochastic models for powder flow can be developed from DEM models. To
represent sensitivity to initial conditions, small random perturbations of
particle positions could be introduced. To represent ignorance of the
mechanisms of grain interaction, the model could randomly choose between
several models for grain interactions at each time step. This random selection
might be based on local conditions, and may involve random selection of
friction coefficient values. A well-fitting stochastic model would sample from
an ensemble of trajectories which would approximate trajectories from the
physical ensemble to the extent that no method of statistical inference could
distinguish a sample of model trajectories from a sample of physical
trajectories.
The construction of any models for initial packings, contact networks, and
powder flows should be consistent with what is known (as opposed to what is
assumed) about the physics of powder flows. Since the models are stochastic,
it is possible that models which are much simpler than the physical process
may be useful at representing some aspect of powder flow. When such models are
proposed for reasons of computational expediency, it is necessary to show by
means of statistical inference that these simplifications have no effect on
the usefulness of the model.
Given the complexity of powder physics and the arbitrary elements of all
models for powder flow, it is not reasonable to expect that a single proposed
model will faithfully capture all of the mechanical properties of an arbitrary
powder. Instead, it would be best to focus on fitting models for one physical
phenomenon of interest and for one type of powder at a time, and then seeking
generalizations once well-fitting models are identified. Once the physical
phenomenon is chosen, it is necessary to summarize that phenomenon with a set
of response statistics. These statistics are calculated from realizations of
each fitted model, and describe whether or not the phenomenon occurred, or
describe the phenomenon. There should be as few response statistics as
possible, so as to make model fitting as simple as possible. The distributions
of the response statistics will be needed to create the intervals needed for
prediction. These distributions can only be studied through data from multiple
realizations of the model, since there is no theory available to predict their
form.
## 4 Fitting stochastic models
In any stochastic model, the parameters can be classified as physical or
calibrational. Physical parameters are values such as the acceleration of
gravity, which are assumed to be universal and are supplied by values from
other experiments. The calibrational parameters are the parameters of the
random elements of the model. Calibrational parameters can be further
subdivided into those which are associated with sensitivity to initial
conditions and those which are related to aspects of the model which reflect
ignorance of physical mechanisms. Both types of calibrational parameter must
be fitted by statistical means. There must be enough response information
available from both the model and the experiment in order to fit all of the
parameters.
The primary basis of fit for a stochastic model is the relationship between
the distribution of the response for the model and the response for the
physical experiments. The fit should be based on matching the mean response.
The presence of stochastic elements which represent the inability to model the
details of grain interactions may introduce extra variability into
realizations from the model. These stochastic elements may also change subtler
aspects of the distributions of the response from the model and from the data.
Comparing the distributions of the response for the model and the experiment
requires a sample of realizations from the model and a sample of replications
of the experiment. The number of realizations required to compare means is
relatively small, but many more are required to usefully estimate variances,
covariances, and subtler aspects of the response distributions. When small
samples are used, there is a greater risk that the model will be fit to
idiosyncrasies of the particular experimental replicates instead of being fit
to the underlying physics of the flow.
Objective fitting of the calibration parameters requires solving an
optimization problem using numerical methods. If the response were a stress-
strain curve from a triaxial test, then the mean of all curves from the
experimental replicates and the mean of all curves from realizations of the
model could be found. Calibration parameters could then be chosen to minimize
the mean square difference between the two average curves, or to minimize some
other measure of difference. The parameters could also be fit by means of
expert guessing. This method may produce a useful set of parameters, but
better solutions may be overlooked.
## 5 Assessing the fit of stochastic models
Once the model has been fit, the quality of that fit must be assessed. If the
fitted model is subject to no further objective assessment, there is a great
risk of fitting a model which fails to represent the underlying physics of
powder flow.
The subset of space occupied by the grains of a flowing powder at any point in
time can be thought of as a realization of a random set [25, 26]. These sets
are random because their structure is unpredictable between different
replicates or realizations. Their internal structure is disordered, but the
disordered pattern of grains is neither stationary nor ergodic. There is no
simple or obvious way to coordinatize the internal structure of an ensemble of
disordered patterns, and so these patterns must be summarized by sets of
descriptive statistics. These statistics may be based on single patterns
observed at a fixed time in each realization or replicate, or they could be
constructed from many patterns observed at fixed times along the joint
trajectory of the grains.
To be useful, a set of descriptive statistics has to be able to identify
common aspects of all realizations from the experiment and to identify any
systematic differences between the joint trajectories of model realizations
and experimental replicates. Finding these statistics is challenging, since
spatial statistics often lack the power to clearly distinguish outcomes from
different spatial processes [27]. Using the random set analogues of the mean
and covariance alone will also not suffice, since random set processes are
highly non-Gaussian. If model verification is based on matching one or two
spatial statistics between the model and the experiment, there is a risk that
these statistics will agree because they lack the statistical power to
distinguish the two processes, and not because the model in any way represents
the physics of powder flow.
To assess the fit of a fitted model, it is necessary to compare the
realizations and the replicates using many different statistics, all of which
summarize different aspects of the disordered patterns within each realization
or trajectory. Some of these statistics will be chosen to seek out possible
differences between model and experimental trajectories based on what is known
about the physics of powders, but others must be chosen from a large library
of descriptive statistics which compare other aspects of the trajectories.
These additional statistics are required to seek out differences which neither
the eye can see nor existing theory would suggest looking for. It will be
necessary to develop extensive libraries of statistics for this purpose, which
will include $k-$point correlation functions, point process statistics [28],
tessellation-based statistics [29, 4], statistics based on mathematical models
for physical processes applied in a non-physical context [30], statistics
found useful in the analyses of other experiments, and many other descriptors
of ordered structure which have not yet been invented.
Since the models idealize the physical processes, statistics will be found
which can identify differences between realizations of the model and
experimental replicates. These differences will be irrelevant if those
statistics do not affect the response. If it were possible to classify
beforehand exactly which statistics affected the response, then the assessment
could be based on those statistics alone. Since little is known about the
relationships between particular responses and other descriptive statistics,
these relationships also need to be established by methods from statistical
inference. The statistical analysis should be based on replicates from
physical experiments. If data from experiments are not available, then
relationships could be sought from realizations of the model. Using the model
is risky, as relationships identified from a flawed model might differ from
those found from the physical data.
The assessment of model fitness is a problem in multivariate statistical
inference [31]. While it is impossible to establish if the joint distributions
of the descriptive statistics for model and experiment are the same, three
different methods can be used to look for evidence that the distributions are
different. All methods work best when the number of observations is large and
the number of statistics is small. If there are many statistics and few
observations, then no method will produce trustworthy results.
Multivariate distributions can be compared by means of statistical tests which
seek to identify differences in ensemble means. These tests are generally
based on the assumption that both joint distributions are Gaussian, which may
not be true. If a very small number of observations are all that can be found
from the experiment and a joint distribution can be fit to statistics from
many realizations of the model, then it is also possible to test how unlikely
it is that the experimental observations came from the model distribution.
This type of test is risky, since the observations in a small experimental
sample may be unrepresentative. A third method of comparison is based on
statistical learning, also known as data mining [32]. A classification rule is
constructed with the aim of being able to distinguish replicates of the
experiment from realizations from the model. If a subset of variables can be
found which can be used to build an effective classification rule, then these
variables may identify differences between the model and the experiment.
The fit of the model may also be assessed by eye. If a simulated movie
generated by the model is visually indistinguishable from a movie of a
physical experiment, then it may be tempting to assume that the model fits.
This approach assumes that the vision of an expert can identify the mean
differences between realizations of the model and replicates of the
experiment, and can identify which of these differences affects the response.
It is possible to construct examples of spatial processes whose realizations
cannot be distinguished by eye, yet can be distinguished by statistical
methods. If an expert does claim to see differences between model and
experimental output, then the expert may not be able to express the basis of
their claim or to prove that their claim is free of any conscious or
unconscious bias.
No final and absolute decision on the quality of a fitted model can be made
from the outcome of a single experiment. The assessments are based on
statistical procedures, and so may be subject to errors. If the replicates of
the experiment do not form a representative sample from the ensemble, then
they could provide misleading evidence against an otherwise useful model. The
risk of this happening is high if the number of replications is small.
Alternatively, no evidence may be found against the model because no statistic
has yet been found which can identify important differences between the model
and the experiment. The probabilities of these errors can only be reduced
through using larger numbers of replications within experiments and through
being ingenious in the development of new statistics. The emergence of a
physical explanation for the response may only occur after comparison of
fitted models arising from repetitions of the experiment at many different
sites using the same experimental protocols.
## 6 Conclusions
The objective validation of models of powder dynamics is a problem in spatial
statistical inference. It requires that the models be stochastic, to reflect
both ignorance of the details of grain interactions and any sensitivity to
initial conditions. These models will not exactly reproduce any one observed
trajectory, but will be useful if they can produce responses with the same
distribution as an experiment and if a thorough fitness assessment reveals no
physically significant differences between trajectories from the model and
from the experiment.
Implementation of objective validation requires significant work by scientists
and statisticians. Scientists must devise the experiments, quantify the
physical property of interest with response statistics, and develop accurate
methods of imaging the full three-dimensional structure of a powder in motion.
Statisticians need to develop useful methods for comparing the joint
distributions of large numbers of descriptive statistics, and to determine how
much information is required for these methods to be effective. Statisticians
and scientists must jointly develop new descriptive spatial statistics, study
their properties, and index them in libraries that can be used for future
analyses. General methods for modeling powders can only emerge after common
elements from the analyses of many experiments are identified.
This approach to model validation and development mimics the approach taken in
the development of the first thermodynamic models. It is based on
experimentation and objective validation of model fit, and not on
extrapolation of previously existing and successful models derived for simpler
phenomena. If the models developed can be thought of as a thermodynamics of
powders, then they differ from classical models in having random state
variables, in being intended for history-dependent processes, and in
accommodating the intense multiparticle interactions which dominate dense
powder flow. If an inference-based approach to modeling is not taken, there is
a significant risk that the fitting of models could only compress data from
specific experiments, rather than the summarizing the physics of powder flow.
## References
* [1] D.S. Grebenkov, M.P. Ciamarra, M. Nicodemi, and A. Coniglio. Flow, ordering, and jamming of sheared granular suspensions. Phys. Rev. Lett., 100(7):078001, 2008.
* [2] K. To. Gravity-driven granular stead-state flows in two-dimensional hoppers and silos. Mod. Phys. Lett. B, 19(30):1751–1766, 2005.
* [3] Y.P. Cheng, M.D. Bolton, and Y. Nakata. Crushing and plastic deformation of soils simulated using DEM. Geotechnique, 54(2):131–141, 2004.
* [4] T. Aste, M. Saadatfar, and T.J. Senden. Geometrical structure of disordered sphere packings. Physical Review E, 71(6):061302, 2005.
* [5] T. Aste. Variations around disordered close packing. J.Phys. Condens. Matter, 17:S2361–S2390, 2005.
* [6] P. Richard, P. Philippe, F. Barbe, S. Bourlès, X. Thibault, and Bideau. D. Analysis by X-ray microtomography of a granular packing undergoing compression. Phys. Rev. E, 68(2):020301, 2003.
* [7] G.T. Seidler, G. Martinez, L.H. Seeley, K.H. Kim, E.A. Behne, S. Zaranek, B.D. Chapman, S.M. Heald, and D.L. Brewe. Granule-by-granule reconstruction of a sandpile from X-ray microtomography data. Phys. Rev. E, 62(6):8175–8181, 2000.
* [8] A.J. Sederman, P. Alexander, and L.F. Gladden. Structure of packed beds probed by magnetic resonance imaging. Powder Tech., 117(3):255–269, 2001.
* [9] M.M. Kohonen, D. Geromichalos, M. Scheel, C. Schier, and S. Herminghaus. On capillary bridges in wet granular materials. Physica A, 339:7–15, 2004.
* [10] M. Toiya, J. Hettinga, and W. Losert. 3d imaging of particle motion during penetrometer testing. Granular Matter, 9(5):323–329, 2007.
* [11] W.M. Visscher and M. Bolsterli. Random packing of equal and unequal spheres in two and three dimensions. Nature, 239:504–507, October 27 1972.
* [12] N.D. Aparacio and A.C.F. Cocks. On the representation of random packings of spheres for sintering simulations. Acta Metall. Mater., 43:3873–3884, 1995.
* [13] G.E. Mueller. Numerically packing spheres in cylinders. Powder Tech., 159(2):105–110, 2005.
* [14] D. Coelho, J.F. Thovert, and P.M. Adler. Geometrical and transport properties of random packings of spheres and aspherical particles. Phys. Rev. E, 55(2):1959–1978, 1997.
* [15] W.S. Jodrey and E.M. Tory. Computer simulation of isotropic, homogeneous, dense random packings of equal spheres. Powder Tech., 30:111–118, 1981.
* [16] W.S. Jodrey and E.M. Tory. Computer simulation of close random packing of equal spheres. Phys. Rev. A, 32(4):2347–2351, 1985.
* [17] M. Bargiel and J. Moscinski. C-language program for the irregular close packing of hard spheres. Comp. Phys. Comm., 64:183–192, 1991.
* [18] B.D. Lubachevsky, F.H. Stillinger, and E.N. Pinson. Disks vs. spheres: Contrasting properties of random packings. J. Stat. Phys., 64(2):510–524, 1991.
* [19] Speedy. R.J. Random jammed packings of hard discs and spheres. J. Phys. Cond. Mat., 10:4185–4194, 1998.
* [20] A.Z. Zinchenko. Algorithm for random close packing of spheres with periodic boundary conditions. J. Comp. Phys., 114(2):298–307, 1994.
* [21] P.A. Cundall and O.D.L. Strack. A discrete numerical model for granular assemblies. Geotechnique, 29:47–65, 1979.
* [22] K.Z.Y. Yen and T.K. Chaki. A dynamic simulation of particle rearrangement in powder packings with realistic interactions. J. App. Phys., 71(7):3164–3173, 1992.
* [23] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, and A.B. Yu. Discrete particle simulation of particulate systems: Theoretical developments. Chemical Engineering Science, 62:3378–3396, 2007.
* [24] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, and A.B. Yu. Discrete particle simulation of particulate systems: A review of major applications and findings. Chemical Engineering Science, 63:5728–5770, 2008.
* [25] G. Matheron. Random Sets and Integral Geometry. Wiley, New York, 1975.
* [26] I. Molchanov. Theory of Random Sets. Springer, 2005.
* [27] A.J. Baddeley and B.W. Silverman. A cautionary example on the use of 2nd-order methods for analyzing point patterns. Biometrics, 40(4):1089–1093, 1984.
* [28] D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic Geometry and Its Applications. John Wiley & Sons, 2nd edition, 1995.
* [29] J.L. Finney. Random packings and the structure of simple liquids I. The geometry of random close packing. Proc. R. Soc. A, 319:479–473, 1970.
* [30] J.D. Picka and T.D. Stewart. A conduction-based statistic for description of orientation in anisotropic sphere packings. In Powders and Grains 2005 Proceedings, pages 329–332. Balkema, 2005.
* [31] R.A. Johnson and D.W. Wichern. Applied Multivariate Statistical Analysis. Prentice-Hall, $6^{th}$ edition, 2007.
* [32] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer, 2001.
|
arxiv-papers
| 2009-10-31T01:00:10 |
2024-09-04T02:49:06.201947
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeffrey Picka",
"submitter": "Jeffrey Picka",
"url": "https://arxiv.org/abs/0911.0048"
}
|
0911.0051
|
# Objective Methods for Assessing Models for Wildfire Spread
Jeff Picka
###### Abstract
Models for wildfires must be stochastic if their ability to represent
wildfires is to be objectively assessed. The need for models to be stochastic
emerges naturally from the physics of the fire, and methods for assessing fit
are constructed to exploit information found in the time evolution of the burn
region.
## 1 Introduction
Management of wildfires requires the development of real-time forecasting
methods for the prediction of their evolution. Development of these models
requires acknowledging the difficulties associated with data acquisition, data
processing, and modeling of wildfire events. At present, many predictive
models are deterministic, and produce one forecast for every set of observed
fire conditions. When a single deterministic prediction is made, two major
questions arise: what does that single prediction represent, and how reliable
is it? There is no way to answer either of these questions unless the
deterministic model is replaced by a model which satisfies three conditions.
1. 1.
The model predicts only a single aspect of the fire rather than predicting
many diverse aspects of the fire.
2. 2.
The model must be stochastic, and make predictions in the form of estimated
probability distributions or probability maps.
3. 3.
The fit of any model must be assessed by statistical methods after it has been
calibrated.
These conditions are imposed by our lack of ability to model the fire as a
physical process, by our inability to observe the details of the internal
structure of the fire, and by the difficulties associated with simulating
sequences of regions of random shape and size.
### 1.1 Terminology
A trajectory is a sequence of regions of irregular shape and extent which
describe the cumulative region burnt by a fire over time. Even if the
prediction problem does not directly involve the burned region, it will almost
certainly involve predicting a response statistic which is defined by the
shape and size of that region (e.g. cumulative value of timber burned,
expenditure required to extinguish the fire).
A stochastic model is a collection of stochastic processes whose outcomes may
represent fire trajectories. The parameters of the stochastic model are
collections of numbers which index the stochastic processes in the model. A
fitted (stochastic) model is the one stochastic process in the model which is
chosen to best represent the fire for which a forecast needs to be made. The
fitted stochastic model can be made to produce a sequence of realizations,
each of which may represent a fire trajectory.
Since fire trajectories are unpredictable, the burn regions at any fixed time
are unpredictable in shape, size, and orientation. At a fixed time, a sample
of these shapes is a sample from a random set process [1]. The stochastic
model for the fire spread is a time series of random set processes.
## 2 Stochastic Modeling of Fires
Useful models for fires must be stochastic because of the nature of wildfires
and our ability to model complex multiscale phenomena. The stochastic aspects
of wildfires are unavoidable, and cannot be ignored in model construction,
model fitting, or model assessment.
### 2.1 Models for Multiscale Physical Phenomena
The simplest version of a multiscale physical phenomenon consists of a
collection of fundamental elements at one scale (the microscale) which act
collectively to produce a distinctive pattern of behaviour at a much larger
scale of observation (the macroscale). At the microscale, the fundamental
elements may or may not be observable. Even if they are observable, it may be
difficult to model their behaviour. It is assumed that the macroscale
phenomena are always observable.
In the ideal case, the behaviour of the fundamental elements can be used to
derive the behaviour of the macroscale phenomena. The classical example of
this is the Gibbs canonical ensemble model of an ideal gas, from which the
basic results of classical thermodynamics for an ideal gas at equilibrium can
be derived [2]. If the state of an inert gas changes very slowly, then its
physical behaviour can be modeled well by results derived from the Gibbs
model. This model works well because the fundamental units are atoms of an
inert gas, which have simple structure and which interact rarely. The
simplicity of the microscale behaviour combined with the immense number of
atoms involved results in macroscale behaviour which can be comprehensively
described by a very small number of macroscopic state variables. Microscale
simplicity also allows the derivation of models for the relationships between
state variables, which do not need to include any consideration of macroscale
fluctuations induced by microscale behaviour.
Models for the steady-state laminar flow of Newtonian fluids and the steady-
state conduction of heat through solids are also successful multiscale models
[3]. In these models, the mechanisms of energy transfer through small volumes
is simple, and macroscale flow can be modeled using partial differential
equations. In the case of heat transfer by convection or turbulent fluid flow,
rigorous definition of macroscale properties from microscale behaviour is
impossible, since there are no mathematical models that can faithfully
represent the behaviour of the fundamental elements and their interactions. In
other cases such as the modeling of the flow of powders [4], the number of
fundamental elements involved is not sufficient to eliminate fluctuations in
properties at the macroscale.
In the modeling of mass, heat, and momentum transfer, it is rare to be able to
formulate a model which faithfully captures microscale phenomena and then
derives from the microscale model a useful description of the limiting
macroscale phenomena. Any model for macroscale behaviour depends on being able
to summarize that behaviour with appropriately defined state variables, and
then being able to construct a model which describes the relationships between
the state variables. To be of use, this model must not be seriously
inconsistent with the physics at the microscale, even if the microscale
behaviour is unobserved and unmodelable. This approach can be successful if
the model developer is modeling an inherently simple multiscale phenomenon,
manages to develop a useful set of state variables, manages to find a model to
relate them, and then finds that the resulting model produces predictions
which are close enough to reality to solve the problem of interest.
### 2.2 Modeling of Wildfires
Wildfires are not amenable to modeling by simple multiscale models which have
deterministic macroscale behaviour. The combustion takes place in a vapour
which forms above pyrolyzing fuel. The heat produced by the combustion drives
the formation of more vapour, as well as slowly changing the geometry of the
combustible materials from which the vapour is produced. The combustible
material occurs in many different forms on many different length scales, and
does not have homogenous structure. The heat evolved from combustion causes
turbulent flow of air and of combusting and non-combusting fuel vapour. These
turbulent flows are of sufficient strength to affect local weather. The
modeling of any one of these phenomena is very difficult on a small scale. The
violent and complex interactions of these processes cannot be observed except
on the largest scales.
The complexity of wildfires makes it impossible to imagine modeling them at
the microscale and then establishing the limiting macroscale behaviour. The
relatively small size of the macroscale relative to the microscale together
with the complexity of interactions between the fundamental elements ensures
that, no matter how much is known about the past history of that fire, there
will be many possible future fire trajectories which are consistent with that
history. Given what can be observed about a real fire, there is an ensemble of
fire trajectories which are consistent with these observations. Given our
inability to comprehensively model the fire, nature appears to choose at
random one of the fire trajectories from the ensemble. Any model for a fire
trajectory has to sample from that ensemble in the same way that the physical
process would, and must yield a probabilistic rather than a deterministic
prediction of future fire behaviour.
Stochastic modeling of wildfire behaviour is possible, since the large-scale
behaviour of the fires is more predictable than small-scale combustion
phenomena. Any model proposed will need to be stochastic, but there will not
be any rigorous theoretical argument suggesting exactly what form this model
should take. The trajectory of each real fire will be impossible to model, and
so any model must be based on a random sample of some form of approximate
trajectory. Construction of such models will require much creative programming
and physical intuition. When constructing models, it would be unreasonable to
assume that the first successful models will be useful in all prediction
problems, or that the selection among possible approximated fire trajectories
should be undertaken via a uniform distribution. Instead, it would be best to
focus on one prediction problem at a time and then try to unify approaches
once successful models have been found.
### 2.3 The Risks Associated With Not Using Stochastic Models
In many models now in use, the macroscale behaviour of a fire is simulated by
a set of partial differential equations whose numerical solution produces a
single approximate fire trajectory. This approach to modeling has many
inherent problems which can be avoided by using stochastic models.
The most difficult problem associated with a single predictive trajectory is
that it is not clear what that trajectory represents. If it is a single
trajectory from the ensemble, then is it a typical trajectory or one with very
unusual features when compared to other trajectories in the ensemble?
Alternatively, the deterministic model may produce some form of average of
many trajectories. This implies that at a fixed time, the burn region in the
trajectory is an average of many such regions from the ensemble. Any region
produced by a geometrical averaging process will be much simpler in shape than
any single burn region from a trajectory in the ensemble.
Even if the nature of the single prediction could be determined, in both cases
the single prediction overstates the accuracy of the prediction procedure. The
deterministic prediction can be used to produce graphics which can mislead a
naive observer as to how uncertain the prediction really is, and gives no
information whatsoever as to how typical that prediction is of others from
ensemble.
Given a single trajectory from any model, it is almost certain that the
prediction will differ from the observed trajectory of the fire. If this
difference is trivial to the eye, the model may be judged to be adequate in
the modeling of that fire. If major differences are observed, it may be
tempting to conclude that the model does not fit. To make such a judgment
objectively, especially in a case where the ensemble contains a great variety
of possible trajectories, it is necessary to make the judgment of model fit
based on comparison between the observed trajectory and many simulated
trajectories from the model. This process requires the use of a stochastic
model, and is the subject of the next section.
### 2.4 Constructing a Stochastic Model
If a deterministic model samples a single trajectory from the ensemble of
physically viable trajectories, then it can be used as the basis of a
stochastic model [5]. Stochastic elements can be added which represent lack of
ability to clearly observe initial and boundary conditions, but they can also
be added to compensate for aspects of fire propagation that are not fully
understood.
## 3 Assessment of Model Fit
Any stochastic model consists of a family of stochastic processes, only one of
which is chosen to represent the fire. There is no guarantee that any method
of calibration will produce a good choice, and so the fit of the chosen model
must be assessed by statistical means.
### 3.1 Fitting a stochastic model
Fitting a model involves finding a method to choose parameter values so that
the fitted model can provide useful predictions of the evolution of a
wildfire. This choice must be based on the past history of the fire.
The parameters can be divided into two classes. Physical parameters appear in
those aspects of the model derived from physics, and take the form of physical
constants such as the acceleration of gravity. These parameters are generally
set using values established by physical experiments, often in contexts very
different from wildfires. While these parameters are not generally allowed to
be set by a model-fitting procedure, allowing these values to be fit may be
useful in model assessment.
Calibration parameters are all parameters in the model which are not set by
physical arguments. They may be physical parameters which cannot be observed
or estimated, or they may be parameters which have no clear physical meaning
but which make the model flexible enough to fit the data. They can be set by
the guesses of experts who are attempting to make the model output look right,
or by statistical means.
Development of objective fitting procedures requires that the model predict
only one specific aspect of the fire. Once the specific aspect is chosen, it
is necessary to find a small set of response statistics which fully describes
that aspect. Seeking a model which works in all predictive problems would
require finding a small set of statistics which could describe or predict all
possible responses. It is doubtful that any such universal set of predictive
statistics exists.
Given a set of response statistics, a fitted stochastic model can be found by
comparing the observed response to simulated responses. A large number of
combinations of calibration parameters could be defined, and then a large
sample of realizations generated from each stochastic process. From each
sample, a mean response would be found. The fitted model would be the
stochastic process whose mean response was closest to that of the observed
fire. A response consisting of a time series of measurements would be best for
use in this type of fitting procedure.
It is also possible to fit a model by varying the calibration parameters until
the simulated responses look right. This method needs to be compared to
objective methods since, in the hands of an expert, it may work well in some
cases. This method may miss better parameter values, may be subject to
conscious or unconscious bias, or may produce conclusions which are impossible
to explain to others.
### 3.2 Assessing the fit of a fire spread model
Once a fire spread model is fit, the adequacy of the model must be assessed by
criteria appropriate to the prediction problem. This is necessary not only
because all models are descriptive and speculative, but also because there is
a severe risk that the fitting process will only match superficial
similarities between the fire and the model realizations.
Fit assessment for a fire spread model requires comparing a sample of
simulated trajectories with the trajectory of the fire that they are modeling.
If no evidence can be found of differences which could affect the response,
then the model can be accepted as being useful. If evidence of differences are
found, then these may suggest how the model can be improved.
To assess the fit, the first stage is to summarize aspects of the fire
trajectory using descriptive statistics. These statistics may be based on the
burn region at one time, or on a sequence of burn regions in time. They may be
based on the shape of the burn region or on observable and quantifiable
properties of the fire within it.
For a descriptive statistic to summarize essential elements of the prediction
problem, it is necessary for the response to depend on the statistic. This
dependency can only be investigated by conducting statistical tests on fitted
model output and then by attempting to identify associations. It would be
better to investigate associations using fire data, but this is impossible
unless the fire can be repeated many times under the same conditions. This
classification process can reduce the number of statistics which are used to
compare the model and the data. If a statistic is found not to affect the
response, then there is no need for its values to be consistent for the model
and the data. Since the classification is based on statistical inference,
there is always a risk that a descriptive statistic will be misclassified.
It is necessary to find a large number of descriptive statistics in any model
assessment problem. Many descriptive statistics will be spatial, and will
depend on the shape and size of regions. Statistics of this kind suffer from a
lack of power, and may not be able to distinguish between spatial processes
with visibly different realizations [6]. There is also no general theory to
suggest which statistics may be useful in finding differences between the
response in the fire and the response in the fitted model.
Finding descriptive statistics is a major challenge. Given expert concerns or
past model failings, it is possible to quantify these in a similar manner to
the development of response statistics. Using these statistics alone is not
good enough, since they may not identify differences that have not been
anticipated or observed in previous modeling efforts. Realizations must be
summarized by statistics that can identify aspects of model failure that are
difficult or impossible to see, even by experts. This requires developing a
large library of statistics, all of which describe different aspects of the
burn region. Analogues of second moment statistics, such as the pair
correlation function, the nearest neighbour function, the empty space
function, the $K-$function, and the 2-point correlation function [7] are
inadequate for this purpose, since the stochastic model is not Gaussian. These
statistics are very useful, but must be supplemented by other statistics
including (possibly) statistics based on triangulations, statistics based on
models for physical properties applied out of context [8], and statistics
which have yet to be invented.
Given a set of descriptive statistics which affect the response, it is
necessary to establish if they show any evidence that the model is wrong. This
requires comparing a sample of a collection of descriptive statistic values
from the observed fire with a sample which can be generated from many
realizations of the fitted model. This comparison could be made with one
descriptive statistic at a time, but better results are expected if small
collections of statistics are compared. These comparisons are difficult to
make reliably with the data from only a single fire, but this is unavoidable
since each fire takes place under different conditions. Comparisons could be
better made between data from controlled burns undertaken under fixed
experimental conditions, but there is no guarantee that these burns would
sample from ensembles similar to those associated with real fires.
The assessment of fit will be subject to unavoidable errors since it is based
on statistical tests. Investigations of association between descriptive
statistics and the response will misclassify statistics. There will be no
guarantee that the observed trajectory is typical of trajectories from
physical process, and so differences between the observed and model
trajectories may be found for a useful model. It is also possible that no
evidence of poor fit will be found in models which are seriously wrong. This
will happen if none of the statistics used to compare the model to the data
are capable of identifying the difference. These errors are also unavoidable
in fits by expert judgment.
## 4 Conclusions
Models which faithfully represent what is known and unknown about fire spread
must be stochastic. The goal of a stochastic model is not to reproduce any one
burn model exactly, but instead to produce simulated fire trajectories which
approximate fire trajectories from the ensemble of possible fires. They need
not capture every aspect of these fires, but they need to be able to capture
the distribution of the response over the ensemble of real fires. The
assessment of fit looks for evidence that the approximate trajectories are
inconsistent with the one trajectory that is observed. A model will be useful
if this general approach to fitting and assessment allows it to be predict
outcomes successfully from many different fires.
The objective approach to model fitting presents a number of challenges which
must be addressed before it can be usefully implemented. It requires the
development of new statistics for the description of fire trajectories. It
requires the development of a stochasticized version of a model like
Prometheus [9] which can usefully be fit to data. It will be necessary to use
historical data and simulations to determine what must be observed in order to
predict fire behaviour and to assess the fit of fire models. It may require
the acquisition or development of new equipment in order to gather the
information on which predictions are based.
Given the effort required, it may seem easier to continue fitting models by
eye. This approach implicitly assumes that experts will not be influenced by
any beliefs they may have about the fitness of model and that an expert will
be able to see all possible ways in which the model could be wrong. If these
assumptions are incorrect, then mistaken assumptions about fire behaviour
could live on in flawed models for far longer than they should. Without a
trajectory-based assessment of the quality of a fitted model, it will always
be possible that a fitted model will fail to capture the physics of fire
spread.
The use of stochastic methods is also essential to making useful predictions
about fire behaviour. If a model is purely deterministic, there is no way to
determine if its single prediction is typical of those from the ensemble of
possible fires or not. Also, there will no measure of uncertainty arising from
the model that can be assigned to the prediction. This could lead to a loss of
faith in modeling, due to failures in predictions for new fires. Deterministic
predictions could also mislead non-experts into overconfidence about what is
known and can be predicted about fire behaviour. Neither outcome would be
beneficial for the development of useful models for fire spread.
## 5 Acknowledgments
This work has been partly supported by NSERC but has mostly been supported by
GEOIDE.
## References
* [1] I. Molchanov. Theory of Random Sets. Springer, 2005.
* [2] R.A. Minlos. Introduction to Mathematical Statistical Mechanics. American Mathematical Society, 2000.
* [3] R.B. Bird, W.E. Stewart, and E.N. Lightfoot. Transport Phenomena. Wiley, 1960.
* [4] J. Kakalios. Resource letter GP-1: Granular physics or non-linear dynamics in a sandbox. Am. J. Phys., 73:8–22, 2003.
* [5] T. Garcia, J. Braun, R. Bryce, and C. Tymstra. Smoothing and bootstrapping the PROMETHEUS fire growth model. Environmetrics, 19:836–848, 2008.
* [6] A.J. Baddeley and B.W. Silverman. A cautionary example on the use of 2nd-order methods for analyzing point patterns. Biometrics, 40(4):1089–1093, 1984.
* [7] D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic Geometry and Its Applications. John Wiley & Sons, 2nd edition, 1995.
* [8] J.D. Picka and T.D. Stewart. A conduction-based statistic for description of orientation in anisotropic sphere packings. In Powders and Grains 2005 Proceedings, pages 329–332. Balkema, 2005.
* [9] C. Tymstra, R.W. Bryce, B.M. Wotton, and O.B. Armitage. Development and structure of prometheus: the canadian wildland fire growth simulation model. Technical Report NOR-X-417, Natural Resources Canada, Canadian Forestry Centre Edmonton, 2009. forthcoming.
|
arxiv-papers
| 2009-10-31T01:04:40 |
2024-09-04T02:49:06.206654
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeffrey Picka",
"submitter": "Jeffrey Picka",
"url": "https://arxiv.org/abs/0911.0051"
}
|
0911.0078
|
# Compensation temperature of 3d mixed ferro-ferrimagnetic ternary alloy
Ebru Kış-Çam1 Ekrem Aydiner2 ekrem.aydiner@deu.edu.tr 1Department of Physics,
Dokuz Eylul University, 35160 Ìzmir, Turkey
2Department of Physics, Istanbul University, 34134 Istanbul, Turkey
###### Abstract
In this study, we have considered the three dimensional mixed ferro-
ferrimagnetic ternary alloy model of the type ABpC1-p where the A and X (X=B
or C) ions are alternately connected and have different Ising spins SA=3/2,
SB=1 and SC=5/2, respectively. We have investigated the dependence of the
critical and compensation temperatures of the model on concentration and
interaction parameters by using MC simulation method. We have shown that the
behavior of the critical temperature and the existence of compensation points
strongly depend on interaction and concentration parameters. In particular, we
have found that the critical temperature of the model is independent on
concentration of different types of spins at a special interaction value and
the model has one or two compensation temperature points in a certain range of
values of the concentration of the different spins.
Compensation temperature; ferro-ferrimagnetic ternary alloys; Monte Carlo
simulation.
###### pacs:
75.50.Gg; 75.10.Hk; 75.30.Kz; 05.10.Ln
††preprint: APS/123-QED
## I Introduction
Molecular-based magnetic materials have recently attracted considerable
interest and study of the magnetic properties Liu ; He ; Gmitra ; Ohkoshi1 ;
Ohkoshi2 ; Ohkoshi3 ; Ohkoshi4 ; Sato ; Pejakovic ; Ohkoshi6 ; Bobak2 ; Bobak1
; Dely ; Ohkoshi5 ; Buendia ; Dely2 ; Carling ; Bobak3 ; Zhoug . A special
class of the these materials, the so-called Prussian blue analogs, such as
(X${}_{p}^{II}$Mn${}_{1-p}^{II})_{1.5}$[CrIII(CN)6].nH2O (XII=NiII,FeII)
Ohkoshi1 ; Ohkoshi2 and
(Ni${}_{p}^{II}$Mn${}_{q}^{II}$Fe${}_{r}^{II}$)1.5[CrIII(CN)6].nH2O Ohkoshi3
which exhibit many unusual properties, for instance, occurrence of one
Ohkoshi1 or even two Ohkoshi3 compensation points, magnetic pole inversion
Ohkoshi2 ; Ohkoshi4 , the photoinduced magnetization effect Sato ; Pejakovic ,
inverted magnetic hysteresis Ohkoshi6 . These ternary alloys have
ferromagnetic-ferrimagnetic properties since they include mixed both
ferromagnetic ($J>0$) and antiferromagnetic ($J<0$) superexchange interactions
between the nearest-neighbor metal ions. The theoretical investigations of
these systems are difficult because of their structural complexity. However,
to obtain magnetic properties of the molecular-based magnetic materials, up to
now, these systems have been studied by using effective-field theory Bobak2 ,
mean field theory Bobak1 ; Dely ; Ohkoshi5 and Monte carlo simulation (MC)
methods Buendia ; Dely2 ; Carling .
In this study we consider three dimensional ferro-ferrimagnetic ABpC1-p
ternary alloy, consisting of three different Ising spins A=3/2, B=1, and
C=5/2, which corresponds to the Prussian blue analog of the type
(Ni${}_{p}^{II}$Mn${}_{1-p}^{II})_{1.5}$[CrIII(CN)6].nH2O Ohkoshi1 . In this
system, the coupling Cr-Ni is ferromagnetic and Mn-Cr is antiferromagnetic.
Our aim, in this study, is to clarify the effects of the concentration and the
interaction parameters on the magnetic behavior of the three dimensional
ternary alloy model by using MC simulation method.
## II The Model and Its Simulation
Three dimensional ferro-ferrimagnetic ABpC1-p Ising model consists of two
interpenetrating cubic sublattices as seen in Fig. 1. It can be assumed that
the A ions are located on the first cubic sublattice and the B and C ions are
randomly distributed on the second cubic sublattice with the concentration $p$
and $1-p$, respectively. Also, to construct a Hamiltonian for this system, the
ion A can be represented by spin SA, and on the other hand, ions B and C can
be represented by Ising spins SB and SC, respectively. If the interactions
between nearest neighbors can be chosen such as A ions ferromagnetically
interact with B, on the other hand, antiferromagnetically interact with C
ions, thus, spins of the Prussian blue analog of the type
(Ni${}_{p}^{II}$Mn${}_{1-p}^{II})_{1.5}$[CrIII(CN)6].nH2O can be represented
by this model where SA, SB and SC correspond to Cr, Ni and Mn, respectively.
In this study we also consider next-nearest neighbor interactions between
spins SA.
The Hamiltonian of the considered system can be written in the form
$\displaystyle
H=-\sum_{<nn>}S_{i}^{A}[J_{AB}S_{j}^{B}\varepsilon_{j}+J_{AC}S_{j}^{C}(1-\varepsilon_{j})]$
$\displaystyle-J_{AA}\sum_{<nnn>}S_{i}^{A}S_{k}^{A}$ (1)
where S${}^{A}=\pm 3/2,\pm 1/2$ for A, S${}^{B}=\pm 1,0$ for B and
S${}^{C}=\pm 5/2,\pm 3/2,\pm 1/2$ for C, on the other hand, $\varepsilon_{j}$
is a random variable which takes the value of unity if there is a spin X (SB
or SC) at the site $j$, if it not is zero. In Eq. (1), the first sum is over
the nearest-neighbor and the second one is over the next-nearest neighbor
spins. In this Hamiltonian the nearest neighbor interactions are chosen as
$J_{AB}>0$ and $J_{AC}<0$, and the next-nearest neighbor interactions are
chosen as $J_{AA}>0$.
In order to show the effects of the concentration $p$ and the interaction
parameters on the compensation and critical temperature of the three
dimensional ternary alloy model, we simulate the Hamiltonian given by Eq. (1).
To simulate this model, we employed Metropolis Monte Carlo simulation
algorithm Binder to the $L\times L\times L$ three-dimensional lattice with
periodic boundary conditions for $L=10$, $12$, $16$, $20$, $24$. One of the
cubic sublattice is fully decorated with spin SA, and spins SB and SC are
randomly distributed on the other cubic sublattice with the concentration $p$
or $1-p$, respectively. All initial spin states in the $L\times L\times L$
three-dimensional lattice are randomly assigned. Configurations are generated
by making single-spin-flip attempts, which were accepted or rejected according
to the Metropolis algorithm. To calculate the averages, data, over 20
different spin configuration, is obtained by using $50000$ Monte Carlo steps
per site after discarding $10000$ steps.
The sublattice average magnetizations per site are obtained by
$M_{A}=\frac{2}{L^{3}}\left\langle\sum_{i}^{L^{3}/2}S_{i}^{A}\right\rangle,\qquad$
(2a)
$M_{B}=\frac{2}{L^{3}}\left\langle\sum_{j=1}^{N_{B}}S_{j}^{B}\right\rangle,\qquad$
(2b)
$M_{C}=-\frac{2}{L^{3}}\left\langle\sum_{j=1}^{N_{C}}S_{j}^{C}\right\rangle$
(2c)
where $N_{B}$ denotes the number of B ions $N_{B}=pL^{3}/2$, whilst $N_{C}$
represents the number of C ions $N_{C}=(1-p)L^{3}/2$ on the same cubic
lattice. Total magnetization per site is given by
$M=\frac{1}{2}\left(M_{A}+M_{B}+M_{C}\right)\ .$ (3)
## III Results and Discussion
Figure 1: The crystallographic structure of prussian blue analog with two
interpenetrating cubic lattices.
In this section, we have given the simulation results of the ternary alloy
model ABpC1-p and we have also discussed the dependence of the critical and
compensation temperature on the concentration and other interaction parameters
in the Hamiltonian. Simulation results have been obtained for the system with
lattice size $L=10$, $12$, $16$, $20$ and $24$, however, here, we have only
presented the results of the model with lattice size $L=20$. We also note that
the critical temperature of the system for the different interaction rates and
concentrations have been obtained by using of the method of the finite-size
scaling Binder .
Figure 2: Dependence of the critical temperature on interaction ratio $R$ in
the three dimensional ternary alloy ABpC1-p for different values of $p$ when
$J_{AA}=0.0$.
In a recent study Buendia it was reported that two dimensional ternary alloy
model does not show a compensation temperature point when there is no next-
nearest neighbor interactions term in the Hamiltonian i.e., $J_{AA}=0$.
However, our simulations show that the system has a compensation point for all
$R$ (we set $R=|J_{AC}|/J_{AB}$) values in interval of $0.1\leq R\leq 2.642$
at $p=0$ when $J_{AA}=0$. This point will be considered below. Now, in order
to compare with the previous results Buendia ; Bobak3 , in Figs. 2 and 3 we
discuss the dependence of the critical temperature of the three dimensional
ternary alloy model ABpC1-p on interaction rate $R$ and concentration $p$ for
next-nearest neighbor interactions i.e., $J_{AA}=0$.
Figure 3: Dependence of the critical temperature on the concentration $p$ in
the three dimensional ternary alloy ABpC1-p for several values of interaction
ratio $R$ when $J_{AA}=0.0$. The lines show a part of the second-order
transitions separating the ferrimagnetic and paramagnetic phases.
In Fig. 2, the critical temperature of the three dimensional ternary alloy
model has been plotted as a function of $R$ for various values of $p$ when
$J_{AA}=0$. It can be seen from Fig. 2 that the critical temperature of the
system has a linear dependence on the interaction ratio $R$ and there is a
critical behavior at a special $R$ value. When $R_{c}=R=0.513$, the critical
temperature of the system has a fixed value of $T_{c}=5.47$ for all $p$
values. At $R_{c}$, the critical temperature of the system does not change
with concentration $p$. This means that neither the spin-1 ions nor spin-5/2
ions substitution to system change the critical temperature of the system at
$R_{c}$. This critical behavior has been reported in theoretical and
experimental studies Buendia ; Bobak3 ; Zhoug . The value of the $R_{c}$ for
ternary alloy ABpC1-p whose spins consist of S${}^{A}=3/2$, S${}^{B}=1$ and
S${}^{C}=5/2$ has been obtained as $R_{c}=0.4781$ in the study based on mean
field approximation Bobak3 and as $R_{c}=0.49$ in the Monte Carlo simulation
of two dimensional system Buendia . Furthermore, the experimental measurements
indicate that there are Prussian blue analogs at the $R=0.45$ have a $T_{c}$
which is almost independent of $p$ Zhoug . Fig. 2 also reveals that
concentration $p$ plays an important role for the ternary alloy model ABpC1-p
since it determine the kinds of the spins and interactions in the system. For
example, when $p=1$ and $p=0$, the system ABpC1-p fully reduces to the
ferromagnetic mixed spin-3/2 and spin-1 and ferrimagnetic mixed spin-3/2 and
spin-5/2 Ising system, respectively. As seen in Fig. 2, although $T_{c}$ of
the system is independent of $p$ at $R_{c}$, however, the total magnetization
of the system may considerably change owing to relatively small variation of
the concentration $p$. Indeed, for different values of $p$, the dependence of
critical temperature of the system on the interaction ratio $R$ is very
different above and below of $R_{c}$. This behavior can be explained by the
change of the concentration $p$ in the system. On the other hand, it can be
detected from Fig. 2 that when $R<R_{c}$, the critical temperature of the
mixed spin-3/2 and spin-5/2 system is smaller than mixed spin-3/2 and spin-1
system. On the contrary, when $R>R_{c}$, the critical temperature of the mixed
spin-3/2 and spin-5/2 system has the highest value. On the $T_{c}$ lines, the
critical temperature of the mixed spin-3/2 and spin-1 Ising system is equal to
that of the mixed spin-3/2 and spin-5/2 Ising one.
Figure 4: Magnetization of the three dimensional ternary alloy ABpC1-p vs
temperature for different values of $p$ ($J_{AA}=7.5$ and $R=1.0$).
In Fig. 3, the dependence of the critical temperature of the three dimensional
ABpC1-p system on the concentration $p$ has been shown for several values of
$R$ when $J_{AA}=0$. The lines represent part of the second-order phase
transition separating the ferrimagnetic and paramagnetic. Fig. 3 provides the
argument that the concentration $p$ determines the magnetic features of the
system mentioned above. Indeed, Fig. 3 clearly shows that the critical
temperature of the system is changed by the concentration $p$ for fixed values
of $R$. As seen from this figure that, when $R<R_{c}$, the critical
temperature of the system linearly increases with increasing of $p$, whereas,
when $R>R_{c}$, the critical temperature of the system linearly decreases with
increasing of $p$ for fixed values of $R$. However, when the values of $R$
close up $R_{c}$, the critical temperature of the system more slowly, but
linearly, change with increasing $p$, and at the critical $R_{c}$ value, the
critical temperature of the system denoted by triangle-line in Fig. 3 is
independent of the concentration $p$. On the other hand, Fig. 3 also shows
that the interaction rate $R$ plays an important role on the critical
temperature of the three dimensional ABpC1-p system. Finally we state that the
critical temperature of the model are consistent with previous result Bobak3 .
Figure 5: Magnetization of the three dimensional ternary alloy ABpC1-p vs
temperature for different values of $p$ ($J_{AA}=7.5$ and $R=2.642$).
In this study we recognize that the three dimensional ternary alloy model
ABpC1-p has one compensation behavior for $J_{AA}=0$, however, for $J_{AA}\neq
0$ it has one or multi compensation points, when other conditions are
satisfied. However, the appearance of the compensation temperature is strongly
affected by the interaction and concentration parameters. Indeed we see in the
present study that the model has not a compensation point for all values of
$p$ and $R$. The dependence of the compensation temperature behavior on
concentration and other interaction parameters has been discussed below. For
discussion, although the system has been simulated in the intervals of
$0.0\leq p<1.0$ and $0.1\leq R\leq 2.642$, the value of $J_{AA}$ used in the
present study is chosen based on previous theoretical study Buendia . The
results of simulation for $R=1.0$ and $R=2.642$ are respectively represented
in Figs. 4 and 5 for chosen parameters.
One compensation point has been found in the intervals of $0.0\leq p<0.3$ and
$0.1\leq R<2.642$ when $J_{AA}=7.5$. However, it is seen that the system has
not compensation behavior for the same values of parameters when $p\geq 0.3$.
For $R=1.0$ and several values of $p$, the compensation behavior of the system
can be seen from Fig. 4. On the other hand, as seen from Fig. 5, the
considered system has a multi compensation behavior at $R=2.642$ and $p=0.3$
for $J_{AA}=7.5$ while it has one compensation point for $0.2\leq p<0.3$.
Furthermore, our simulation data introduce that the system shows compensation
behavior at $p=0$ for all values of $R$, when $J_{AA}=0$. In addition, in the
case $J_{AA}=0$, the compensation point has been found for $R=0.25$ at
$p=0.2$, $0.25$; for $R=0.75$ at $p=0.3$; for $R=1.25$ at $p=0.3$; for $R=2.0$
at $p=0.1$, $0.3$, $0.4$; for $R=2.642$ at $p=0.1$, $0.2$, $0.3$, $0.4$.
Whereas, it has been reported in previous study that there is no compensation
point for $J_{AA}=0$ in two dimensional model Buendia .
Figure 6: Dependence of the compensation temperature $T_{comp}$ on the
interaction parameters in Hamiltonian of ternary alloy ABpC1-p for $p=0.25$.
The effect of the interaction parameters on the compensation behavior of the
three dimensional ternary model is also discussed in Fig. 6. This figure shows
dependence of the compensation temperature $T_{comp}$ of the model on
interaction parameters in Hamiltonian only for a fixed value of the
concentration parameter $p$ ($p=0.25$). In this figure, square-line represents
the behavior of the compensation point vs $J_{AA}$ for fixed values of
$J_{AB}=5$, $J_{AC}=-5$ and $p=0.25$, circle-line indicates the behavior of
the compensation point vs $J_{AB}$ for fixed values of $J_{AA}=7.5$ and
$J_{AC}=-5$ and $p=0.25$, and on the other hand, the behavior of the
compensation point vs $J_{AC}$ is plotted for fixed values of $J_{AA}=7.5$,
$J_{AB}=5$ and $p=0.25$ with triangle-line. As seen from Fig. 6 that for fixed
$p$, $J_{AB}=5$ and $J_{AC}=-5$, the compensation temperature decreases slowly
as the strength of the $J_{AA}$ increases. Similarly for fixed $p$,
$J_{AA}=7.5$ and $J_{AC}=-5$, the compensation temperature decreases slowly
with increasing of $J_{AB}$. However, for fixed $p$, $J_{AA}=7.5$ and
$J_{AB}=5$, the compensation temperature dramatically increases as $|J_{AC}|$
increases. These results indicate that the compensation temperature has a
strong dependence on the parameter $J_{AC}$ whereas its dependence on $J_{AA}$
and $J_{AB}$ is relatively weak. The characteristic behavior of the dependence
of the compensation temperature on the parameters of present model consistent
with the results of two dimensional model Buendia .
## IV Conclusion
In this study, we have considered the three dimensional ternary model ABpC1-p
whose spins consist of S${}^{A}=3/2$, S${}^{B}=1$ and S${}^{C}=5/2$. We have
investigated the dependence of the critical and compensation temperature
behavior of the considered model on concentration and interactions by using MC
simulation method. We have observed that the behavior of the critical
temperature and the existence of compensation points strongly depend on
interaction and concentration parameters. Particularly, we have found that the
critical temperature of the model is independent on concentration of different
types of spins at a critical $R_{c}$ value and the model has one or two
compensation temperature points in a certain range of values of the
concentration of the different spins. We concluded that magnetic properties of
the system ABpC1-p can be controlled by changing the relative concentration of
the different species of ions. As a result, we would like to stress that these
theoretical results can be very useful for designing molecular magnets in
experimental studies since the existence of compensation in the ternary alloy
ABpC1-p that can be setup by adjusting the proportion of compounds.
## References
* (1) W. M. Liu et al., Phy. Rev. B 65 (2002) 172416.
* (2) P. B. He and W. M. Liu, Phy. Rev. B 72 (2005) 064410.
* (3) M. Gmitra and J. Barnas, Phy. Rev. Lett. 96 (2006) 207205.
* (4) S. Ohkoshi, T. Iyoda, A. Fujishima and K. Hashimoto, Phy.Rev. B 56 (1997) 11642.
* (5) S. Ohkoshi, S. Yorozu, O. Sato, T. Iyoda, A. Fujishima and K. Hashimoto, Appl. Phys. Lett. 70 (1997) 1040.
* (6) S. Ohkoshi, Y. Abe, A. Fujishima and K. Hashimoto, Phys. Rev. Lett. 82 (1999) 1285.
* (7) S. Ohkoshi and K. Hashimoto, J. Am. Chem. Soc. 121 (1999) 10591.
* (8) O. Sato, T. Iyoda, A. Fujishima and K. Hashimoto, Science 271 (1996) 49.
* (9) D. A. Pejakovic, J. L. Manson, J. S. Miller and A. J. Eipstein, Current Appl. Phys. 1 (2001) 15.
* (10) S. Ohkoshi, T. Hozumi and K. Hashimoto, Phy. Rev. B 64 (2001) 132404.
* (11) A. Bobák, O. F. Abubrig and D. Horváth, Physica A 312 (2002) 187.
* (12) A. Bobák and J. Dely, Physica A 341, (2004) 281.
* (13) J. Dely and A. Bobák, Physica B 388 (2007) 49.
* (14) S. Ohkoshi and K. Hashimoto, Phys. Rev. B 60 (1999) 12820\.
* (15) G. M. Buendía and J. E. Villarroel, J. Magn. Magn. Mater. 310 (2007) 495.
* (16) J. Dely, A. Bobák and M. Žukovič, Phys. Lett. A 373 (2009) 3197.
* (17) S. G. Carling and P. Day, Polyhedron 20 (2001) 1525.
* (18) A. Bobák, F. O. Abubrig, T. Balcerzak, Phy. Rev. B 68 (2003) 224405.
* (19) P. Zhoug, D. Xue, H. Lou and X. Chen, Nanoletters 2 (2002) 845.
* (20) K. Binder, in: K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Springer, Berlin, 1979.
|
arxiv-papers
| 2009-10-31T13:31:09 |
2024-09-04T02:49:06.213381
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ebru Kis-Cam, Ekrem Aydiner",
"submitter": "Ekrem Aydiner Dr",
"url": "https://arxiv.org/abs/0911.0078"
}
|
0911.0125
|
# Entangled Husimi distribution and Complex Wavelet transformation††thanks:
Corresponding author. Email address: hlyun2008@126.com.
Li-yun Hu1 and Hong-yi Fan2
1College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
###### Abstract
Based on the proceding Letter [Int. J. Theor. Phys. 48, 1539 (2009)], we
expand the relation between wavelet transformation and Husimi distribution
function to the entangled case. We find that the optical complex wavelet
transformation can be used to study the entangled Husimi distribution function
in phase space theory of quantum optics. We prove that the entangled Husimi
distribution function of a two-mode quantum state $\left|\psi\right\rangle$ is
just the modulus square of the complex wavelet transform of
$e^{-\left|\eta\right|^{2}/2}$ with $\psi\left(\eta\right)$ being the mother
wavelet up to a Gaussian function.
Keywords: complex wavelet transformation, entangled Husimi distribution, IWOP
technique
## 1 Introduction
Studying distribution functions of density operator $\rho$ in phase space has
been a major topic in quantum statistical physics. Phase space technique has
proved very effective in various branches of physics. Among various phase
space distributions the Wigner function $F_{w}\left(q,p\right)$ [1, 2, 3, 4]
is the most popularly used. But the Wigner distribution function itself is not
a probability distribution due to being both positive and negative. To
overcome this inconvenience, the Husimi distribution function
$F_{h}\left(q^{\prime},p^{\prime}\right)$ is introduced [5], which is defined
in a manner that guarantees it to be nonnegative. On the other hand, the
optical wavelet transformations have been developed which can overcome some
shortcomings of the classical Fourier analysis and therefore has been widely
used in Fourier optics and information science since 1980s [6, 7, 8, 9]. In
the previous Letter [10], we have employed the optical wavelet transformation
to study the Husimi distribution function for single-mode case, and proved
that the Husimi distribution function of a quantum state
$\left|\psi\right\rangle$ is just the modulus square of the wavelet transform
of $e^{-x^{2}/2}$ with $\psi\left(x\right)$ being the mother wavelet up to a
Gaussian function, i.e.,
$\left\langle\psi\right|\Delta_{h}\left(q,p,\kappa\right)\left|\psi\right\rangle=\frac{e^{-\frac{p^{2}}{\kappa}}}{\sqrt{\pi\kappa}}\left|\int_{-\infty}^{\infty}dx\psi^{\ast}\left(\frac{x-s}{\mu}\right)e^{-x^{2}/2}\right|^{2},$
(1)
where $s=\frac{-1}{\sqrt{\kappa}}\left(\kappa q+ip\right),$
$\mu=\sqrt{\kappa},$and
$\left\langle\psi\right|\Delta_{h}\left(q,p\right)\left|\psi\right\rangle$ is
the Husimi distribution function,
$\left\langle\psi\right|\Delta_{h}\left(q,p,\kappa\right)\left|\psi\right\rangle=2\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}F_{w}\left(q^{\prime},p^{\prime}\right)\exp\left[-\kappa\left(q^{\prime}-q\right)^{2}-\frac{\left(p^{\prime}-p\right)^{2}}{\kappa}\right],$
(2)
as well as $\Delta_{h}\left(q,p,\kappa\right)$ is the Husimi operator,
$\Delta_{h}\left(q,p,\kappa\right)=\frac{2\sqrt{\kappa}}{1+\kappa}\colon\exp\left\\{\frac{-\kappa\left(q-Q\right)^{2}}{1+\kappa}-\frac{\left(p-P\right)^{2}}{1+\kappa}\right\\}\colon,$
(3)
here $\colon\colon$ denotes normal ordering; $Q=(a+a^{{\dagger}})/\sqrt{2}$
and $P=(a-a^{{\dagger}})/(\sqrt{2}\mathtt{i})$ are the coordinate and the
momentum operator, and $a_{1},a_{1}^{{\dagger}}$ the Bose annihilation and
creation operators, $[a,a^{{\dagger}}]=1,a\left|0\right\rangle=0$. Thus a
convenient approach for calculating various Husimi distribution functions of
miscellaneous quantum states is presented.
Recalling that in Ref.[11], Fan and Guo have introduced the entangled Husimi
operator $\Delta_{h}\left(\sigma,\gamma,\kappa\right)$ which is endowed with
definite physical meaning, and find that there corresponds a special two-mode
squeezed coherent state $\left|\sigma,\gamma\right\rangle_{\kappa}$
representation such that $\Delta_{h}\left(\sigma,\gamma,\kappa\right)$ $=$
$\left|\sigma,\gamma,\kappa\right\rangle\left\langle\sigma,\gamma,\kappa\right|$.
The entangled Husimi operator $\Delta_{h}\left(\sigma,\gamma,\kappa\right)$
and the entangled Husimi distribution $F_{h}\left(\sigma,\gamma,\kappa\right)$
of quantum state $\left|\psi\right\rangle$ are given by
$\Delta_{h}\left(\sigma,\gamma,\kappa\right)=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\},$
(4)
and
$F_{h}\left(\sigma,\gamma,\kappa\right)=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}F_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\},$
(5)
respectively, where
$F_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)=\left\langle\psi\right|\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\left|\psi\right\rangle$
with $\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)$ being two-mode
Wigner operator is two-mode Wigner function. Thus we are naturally led to
studying the entangled Husimi distribution function from the viewpoint of
wavelet transformation.
In this paper, we shall expand the relation between wavelet transformation and
Wigner-Husimi distribution function to the entangled case, that is to say, we
employ the complex wavelet transformation (CWT) to investigate the entangled
Husimi distribution function (EHDF) by bridging the relation between CWT and
EHDF. We prove that the entangled Husimi distribution function of a two-mode
quantum state $\left|\psi\right\rangle$ is just the modulus square of the
complex wavelet transform of $e^{-\left|\eta\right|^{2}/2}$ with
$\psi\left(\eta\right)$ being the mother wavelet up to a Gaussian function.
Thus we present a convenient approach for calculating various entangled Husimi
distribution functions of miscellaneous two-mode quantum states.
## 2 Complex wavelet transform and its quantum mechanical version
In Ref.[12], Fan and Lu have proposed the complex wavelet transform (CWT),
i.e., the CWT of a signal function $g\left(\eta\right)$ by $\psi$ is defined
by
$W_{\psi}g\left(\mu,z\right)=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}g\left(\eta\right)\psi^{\ast}\left(\frac{\eta-z}{\mu}\right),$
(6)
whose admissibility condition for mother wavelets,
$\int\frac{d^{2}\eta}{2\pi}\psi\left(\eta\right)=0,$ is examined in the
entangled state representations $\left\langle\eta\right|$ and a family of new
mother wavelets (named the Laguerre–Gaussian wavelets) are found to match the
CWT [12]. In fact, by introducing the bipartite entangled state representation
$\left\langle\eta=\eta_{1}+\mathtt{i}\eta_{2}\right|,$[13, 14]
$\left|\eta\right\rangle=\exp\left\\{-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle,$
(7)
which is the common eigenvector of relative coordinate $Q_{1}-Q_{2}$ and the
total momentum $P_{1}+P_{2}$,
$\left(Q_{1}-Q_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{1}\left|\eta\right\rangle,\text{
}\left(P_{1}+P_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{2}\left|\eta\right\rangle,$
(8)
where $Q_{j}$ and $P_{j}$ are the coordinate and the momentum operator,
related to the Bose operators
$(a_{j},a_{j}^{{\dagger}}),[a_{i},a_{j}^{{\dagger}}]=\delta_{ij}$ by
$Q_{j}=(a_{j}+a_{j}^{\dagger})/\sqrt{2}$ and
$P_{j}=(a-a^{\dagger})/(\sqrt{2}\mathtt{i})$ ($j=1,2$), we can treat (5) from
the quantum mechanically,
$W_{\psi}g\left(\mu,z\right)=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left\langle\psi\right|\left.\frac{\eta-z}{\mu}\right\rangle\left\langle\eta\right|\left.g\right\rangle=\left\langle\psi\right|U_{2}\left(\mu,z\right)\left|g\right\rangle,$
(9)
where $z=z_{1}+iz_{2}\in C,$ $0<\mu\in R,$
$g\left(\eta\right)\equiv\left\langle\eta\right|\left.g\right\rangle,\ $and
$\psi\left(\eta\right)=\left\langle\eta\right|\left.\psi\right\rangle$ are the
wavefunction of state vector $\left|g\right\rangle$ and the mother wavelet
state vector $\left|\psi\right\rangle$ in $\left\langle\eta\right|$
representation, respectively, and
$U_{2}\left(\mu,z\right)\equiv\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left|\frac{\eta-z}{\mu}\right\rangle\left\langle\eta\right|,\;\mu=e^{\lambda},$
(10)
is the two-mode squeezing-displacing operator [15, 16, 17]. Noticing that the
two-mode squeezing operator has its natural expression in
$\left\langle\eta\right|$ representation [14],
$S_{2}\left(\mu\right)=\exp\left[\left(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2}\right)\ln\mu\right]=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\left|\frac{\eta}{\mu}\right\rangle\left\langle\eta\right|,$
(11)
which is differerent from the product of two single-mode squeezing (dilation)
operators, and the two-mode squeezed state is simultaneously an entangled
state, thus we can put Eq.(10) into the following form,
$U_{2}\left(\mu,z\right)=S_{2}\left(\mu\right)\mathfrak{D}\left(z\right),$
(12)
where $\mathfrak{D}\left(z\right)$ is a two-mode displacement operator,
$\mathfrak{D}\left(z\right)\left|\eta\right\rangle=\left|\eta-z\right\rangle$
and
$\displaystyle\mathfrak{D}\left(z\right)$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}\left|\eta-z\right\rangle\left\langle\eta\right|$
$\displaystyle=\exp\left[iz_{1}\frac{P_{1}-P_{2}}{\sqrt{2}}-iz_{2}\frac{Q_{1}+Q_{2}}{\sqrt{2}}\right]$
$\displaystyle=D_{1}\left(-z/2\right)D_{2}\left(z^{\ast}/2\right).$ (13)
It the follows the quantum mechanical version of CWT is
$W_{\psi}g\left(\mu,\zeta\right)=\left\langle\psi\right|S_{2}\left(\mu\right)\mathfrak{D}\left(z\right)\left|g\right\rangle=\left\langle\psi\right|S_{2}\left(\mu\right)D_{1}\left(-z/2\right)D_{2}\left(z^{\ast}/2\right)\left|g\right\rangle.$
(14)
Eq.(14) indicates that the 2D CWT can be put into a matrix element in the
$\left\langle\eta\right|$ representation of the two-mode displacing and the
two-mode squeezing operators in Eq.(11) between the mother wavelet state
vector $\left|\psi\right\rangle$ and the state vector $\left|g\right\rangle$
to be transformed. Thus the CWT differs from the direct product of two
1-dimensional wavelet transformations.
Once the state vector $\left\langle\psi\right|$ corresponding to mother
wavelet is known, for any state $\left|g\right\rangle$ the matrix element
$\left\langle\psi\right|U_{2}\left(\mu,z\right)\left|g\right\rangle$ is just
the wavelet transform of $g(\eta)$ with respect to $\left\langle\psi\right|.$
Therefore, various quantum optical field states can then be analyzed by their
wavelet transforms.
## 3 Relation between CWT and EHDF
In the following we shall show that the entangled Husimi distribution function
(EHDF) of a quantum state $\left|\psi\right\rangle$ can be obtained by making
a complex wavelet transform of the Gaussian function
$e^{-\left|\eta\right|^{2}/2},$ i.e.,
$\left\langle\psi\right|\Delta_{h}\left(\sigma,\gamma,\kappa\right)\left|\psi\right\rangle=e^{-\frac{1}{\kappa}\left|\gamma\right|^{2}}\left|\int\frac{d^{2}\eta}{\sqrt{\kappa}\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\frac{\eta-z}{\sqrt{\kappa}}\right)\right|^{2},$
(15)
where $\mu=e^{\lambda}=\sqrt{\kappa},$ $z=z_{1}+iz_{2},$ and
$\displaystyle z_{1}$
$\displaystyle=\frac{\cosh\lambda}{1+\kappa}\left[\gamma^{\ast}-\gamma-\kappa\left(\sigma^{\ast}+\sigma\right)\right],$
(16) $\displaystyle z_{2}$
$\displaystyle=\frac{i\cosh\lambda}{1+\kappa}\left[\gamma+\gamma^{\ast}+\kappa\left(\sigma-\sigma^{\ast}\right)\right],$
(17)
and
$\left\langle\psi\right|\Delta_{h}\left(\sigma,\gamma,\kappa\right)\left|\psi\right\rangle$
is the Husimi distribution function as well as
$\Delta_{h}\left(\sigma,\gamma,\kappa\right)$ is the Husimi operator,
$\displaystyle\Delta_{h}\left(\sigma,\gamma,\kappa\right)$
$\displaystyle=\frac{4\kappa}{\left(1+\kappa\right)^{2}}\colon\exp\left\\{-\frac{\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)}{1+\kappa}\right.$
$\displaystyle-\left.\frac{\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)}{1+\kappa}\right\\}\colon$
(18)
here $\colon\colon$ denotes normal ordering of operators.
Proof of Eq.(15).
When the transformed $\left|g\right\rangle=\left|00\right\rangle$ (the two-
mode vacuum state), noticing that
$\left\langle\eta\right.\left|00\right\rangle=e^{-\left|\eta\right|^{2}/2},$
thus we can express Eq.(9) as
$\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\frac{\eta-z}{\mu}\right)=\left\langle\psi\right|U_{2}\left(\mu,z\right)\left|00\right\rangle.$
(19)
To combine the CWTs with transforms of quantum states more tightly and
clearly, using the technique of integration within an ordered product (IWOP)
[18, 19, 20, 21] of operators, we can directly perform the integral in Eq.(10)
[22]
$\displaystyle U_{2}\left(\mu,z\right)$
$\displaystyle=\frac{1}{\mu}\int\frac{d^{2}\eta}{\pi}\colon\exp\left\\{-\frac{\mu^{2}+1}{2\mu^{2}}|\eta|^{2}+\frac{\eta
z^{\ast}+z\eta^{\ast}}{2\mu^{2}}+\frac{\eta-z}{\mu}a_{1}^{\dagger}\right.$
$\displaystyle\left.-\frac{\eta^{\ast}-z^{\ast}}{\mu}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}+\eta^{\ast}a_{1}-\eta
a_{2}+a_{1}a_{2}-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}-\frac{\left|z\right|^{2}}{2\mu^{2}}\right\\}\colon$
$\displaystyle=\operatorname{sech}\lambda\exp\left[-\frac{1}{2\left(1+\mu^{2}\right)}\left|z\right|^{2}+a_{1}^{\dagger}a_{2}^{\dagger}\tanh\lambda+\frac{1}{2}\left(z^{\ast}a_{2}^{\dagger}-za_{1}^{\dagger}\right)\operatorname{sech}\lambda\right]$
$\displaystyle\times\exp\left[\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\ln\operatorname{sech}\lambda\right]\exp\left(\frac{z^{\ast}a_{1}-za_{2}}{1+\mu^{2}}-a_{1}a_{2}\tanh\lambda\right).$
(20)
where we have set $\mu=e^{\lambda}$,
$\operatorname{sech}\lambda=\frac{2\mu}{1+\mu^{2}}$,
$\tanh\lambda=\frac{\mu^{2}-1}{\mu^{2}+1}$, and we have used the operator
identity
$e^{ga^{\dagger}a}=\colon\exp\left[\left(e^{g}-1\right)a^{\dagger}a\right]\colon$.
In particular, when $z=0,$ it reduces to the usual normally ordered two-mode
squeezing operator $S_{2}\left(\mu\right)$. From Eq.(20) it then follows that
$\displaystyle U_{2}\left(\mu,z\right)\left|00\right\rangle$
$\displaystyle=\operatorname{sech}\lambda\exp\left\\{-\frac{\left(z_{1}-iz_{2}\right)\left(z_{1}+iz_{2}\right)}{2\left(1+\mu^{2}\right)}+a_{1}^{\dagger}a_{2}^{\dagger}\tanh\lambda\right.$
$\displaystyle\left.+\frac{1}{2}\left[\left(z_{1}-iz_{2}\right)a_{2}^{\dagger}-\left(z_{1}+iz_{2}\right)a_{1}^{\dagger}\right]\operatorname{sech}\lambda\right\\}\left|00\right\rangle.$
(21)
Substituting Eqs.(16), (17) and $\tanh\lambda=\frac{\kappa-1}{\kappa+1},$
$\cosh\lambda=\frac{1+\kappa}{2\sqrt{\kappa}}$ into Eq.(21) yields
$\displaystyle
e^{-\frac{1}{2\kappa}\left|\gamma\right|^{2}-\frac{\sigma\gamma^{\ast}-\gamma\sigma^{\ast}}{2\left(\kappa+1\right)}}U_{2}\left(\mu,z_{1},z_{2}\right)\left|00\right\rangle$
$\displaystyle=\frac{2\sqrt{\kappa}}{1+\kappa}\exp\left\\{-\frac{\left|\gamma\right|^{2}+\kappa\left|\sigma\right|^{2}}{2\left(\kappa+1\right)}+\frac{\kappa\sigma+\gamma}{1+\kappa}a_{1}^{\dagger}+\frac{\gamma^{\ast}-\kappa\sigma^{\ast}}{1+\kappa}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\frac{\kappa-1}{\kappa+1}\right\\}\allowbreak\left|00\right\rangle\left.\equiv\right.\left|\sigma,\gamma\right\rangle_{\kappa},$
(22)
then the CWT of Eq.(19) can be further expressed as
$e^{-\frac{1}{2\kappa}\left|\gamma\right|^{2}-\frac{\sigma\gamma^{\ast}-\gamma\sigma^{\ast}}{2\left(\kappa+1\right)}}\int\frac{d^{2}\eta}{\mu\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\frac{\eta-
z_{1}-iz_{2}}{\mu}\right)=\left\langle\psi\right.\left|\sigma,\gamma\right\rangle_{\kappa}.$
(23)
Using normally ordered form of the vacuum state projector
$\left|00\right\rangle\left\langle 00\right|=\colon
e^{-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}}\colon,$and the IWOP method as
well as Eq.(22) we have
$\displaystyle\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|$
$\displaystyle=\frac{4\kappa}{\left(1+\kappa\right)^{2}}\colon\exp\left[-\frac{\left|\gamma\right|^{2}+\kappa\left|\sigma\right|^{2}}{\kappa+1}+\frac{\kappa\sigma+\gamma}{1+\kappa}a_{1}^{\dagger}+\frac{\gamma^{\ast}-\kappa\sigma^{\ast}}{1+\kappa}a_{2}^{\dagger}\right.$
$\displaystyle\left.+\frac{\kappa\sigma^{\ast}+\gamma^{\ast}}{1+\kappa}a_{1}+\frac{\gamma-\kappa\sigma}{1+\kappa}a_{2}+\frac{\kappa-1}{\kappa+1}\left(a_{1}^{\dagger}a_{2}^{\dagger}+a_{1}a_{2}\right)-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\right]\colon$
$\displaystyle=\frac{4\kappa}{\left(1+\kappa\right)^{2}}\colon\exp\left\\{-\frac{\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)}{1+\kappa}\right.$
$\displaystyle-\left.\frac{\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)}{1+\kappa}\right\\}\colon\left.=\right.\Delta_{h}\left(\sigma,\gamma,\kappa\right).$
(24)
Now we explain why $\Delta_{h}\left(\sigma,\gamma,\kappa\right)$ is the
entangled Husimi operator. Using the formula for converting an operator $A$
into its Weyl ordering form [23]
$A=4\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\left\langle-\alpha,-\beta\right|A\left|\alpha,\beta\right\rangle\genfrac{}{}{0.0pt}{}{:}{:}\exp\\{2\left(\alpha^{\ast}a_{1}-a_{1}^{\dagger}\alpha+\beta^{\ast}a_{2}-a_{2}^{\dagger}\beta+a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\\}\genfrac{}{}{0.0pt}{}{:}{:},$
(25)
where the symbol $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$
denotes the Weyl ordering, $\left|\beta\right\rangle$ is the usual coherent
state, substituting Eq.(24) into Eq.(25) and performing the integration by
virtue of the technique of integration within a Weyl ordered product of
operators, we obtain
$\displaystyle\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|$
$\displaystyle=\frac{16\kappa}{\left(1+\kappa\right)^{2}}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\left\langle-\alpha,-\beta\right|\colon\exp\left\\{-\frac{\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)}{1+\kappa}\right.$
$\displaystyle\left.-\frac{\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)}{1+\kappa}\right\\}\colon\left|\alpha,\beta\right\rangle$
$\displaystyle\times\genfrac{}{}{0.0pt}{}{:}{:}\exp\\{2\left(\alpha^{\ast}a_{1}-a_{1}^{\dagger}\alpha+\beta^{\ast}a_{2}-a_{2}^{\dagger}\beta+a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\\}\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle=4\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{-\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)-\frac{1}{\kappa}\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\right\\}\genfrac{}{}{0.0pt}{}{:}{:},$
(26)
where we have used the integral formula
$\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}\right)=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(27)
This is the Weyl ordering form of
$\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|.$
Then according to Weyl quantization scheme [24, 25] we know the Weyl ordering
form of two-mode Wigner operator is given by
$\Delta_{w}\left(\sigma,\gamma\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\delta\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)\delta\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\delta\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(28)
thus the classical corresponding function of a Weyl ordered operator is
obtained by just replacing
$a_{1}-a_{2}^{{\dagger}}\rightarrow\sigma^{\prime},a_{1}+a_{2}^{{\dagger}}\rightarrow\gamma^{\prime},$
i.e.,
$\displaystyle
4\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{-\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)-\frac{1}{\kappa}\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\right\\}\genfrac{}{}{0.0pt}{}{:}{:}$
$\displaystyle\rightarrow
4\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\},$
(29)
and in this case the Weyl rule is expressed as
$\displaystyle\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|$
$\displaystyle=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\delta\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)\delta\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)$
$\displaystyle\times\delta\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\}$
$\displaystyle=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\}.$
(30)
In reference to Eq.(5) in which the relation between the entangled Husimi
function and the two-mode Wigner function is shown, we know that the right-
hand side of Eq. (30) should be just the entangled Husimi operator, i.e.
$\left|\sigma,\gamma\right\rangle_{\kappa\kappa}\left\langle\sigma,\gamma\right|=4\int
d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}\Delta_{w}\left(\sigma^{\prime},\gamma^{\prime}\right)\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\}=\Delta_{h}\left(\sigma,\gamma,\kappa\right),$
(31)
thus Eq. (15) is proved by combining Eqs.(31) and (23).
Motivated by the proceding Letter [10], we have futher expanded the relation
between wavelet transformation and Wigner-Husimi distribution function to the
entangled case. That is to say, we prove that the entangled Husimi
distribution function of a two-mode quantum state $\left|\psi\right\rangle$ is
just the modulus square of the complex wavelet transform of
$e^{-\left|\eta\right|^{2}/2}$ with $\psi\left(\eta\right)$ being the mother
wavelet up to a Gaussian function, i.e.,
$\left\langle\psi\right|\Delta_{h}\left(\sigma,\gamma,\kappa\right)\left|\psi\right\rangle=e^{-\frac{1}{\kappa}\left|\gamma\right|^{2}}\left|\int\frac{d^{2}\eta}{\sqrt{\kappa}\pi}e^{-\left|\eta\right|^{2}/2}\psi^{\ast}\left(\left(\eta-z\right)/\sqrt{\kappa}\right)\right|^{2}$.
Thus we have a convenient approach for calculating various entangled Husimi
distribution functions of miscellaneous quantum states. For more discussion
about the wavelet transformation in the context of quantum optics, we refer to
Refs.[26, 27].
Acknowledgement Work supported by the National Natural Science Foundation of
China under grants. 10775097 and 10874174, and the Research Foundation of the
Education Department of Jiangxi Province.
Appendix
We can check Eq.(31) by the following way.
Using the normally ordered form of the two-mode Wigner operator [11]
$\Delta_{w}\left(\sigma,\gamma\right)=\frac{1}{\pi^{2}}\colon\exp\left\\{-\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)-\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\right\\}\colon,$
(A1)
we can further perform the integration in Eq.(4) and see
$\displaystyle\Delta_{h}\left(\sigma,\gamma,\kappa\right)$
$\displaystyle=4\int\frac{d^{2}\sigma^{\prime}d^{2}\gamma^{\prime}}{\pi^{2}}\exp\left\\{-\kappa\left|\sigma^{\prime}-\sigma\right|^{2}-\frac{1}{\kappa}\left|\gamma^{\prime}-\gamma\right|^{2}\right\\}$
$\displaystyle\times\colon\exp\left\\{-\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)-\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)\right\\}\colon$
$\displaystyle=\frac{4\kappa}{\left(1+\kappa\right)^{2}}\colon\exp\left\\{-\frac{\left(a_{1}+a_{2}^{{\dagger}}-\gamma\right)\left(a_{1}^{{\dagger}}+a_{2}-\gamma^{\ast}\right)}{1+\kappa}-\frac{\kappa\left(a_{1}-a_{2}^{{\dagger}}-\sigma\right)\left(a_{1}^{{\dagger}}-a_{2}-\sigma^{\ast}\right)}{1+\kappa}\right\\}\colon$
$\displaystyle=\text{Eq.(\ref{e23})}\left.=\right.\Delta_{h}\left(\sigma,\gamma,\kappa\right),$
(A2)
which is the confirmation of Eq. (31).
## References
* [1] E. Wigner, Phys. Rev. 40, 749 (1932)
* [2] M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, Phys. Rep. 106 121 (1984); Wolfgang P Schleich, Quantum Optics in Phase Space, (Wiley-Vch, Berlin 2001)
* [3] Bužek V and Knight P L, Prog. Opt. 34 1 (1995)
* [4] Dodonov V V and Man’ko V I Theory of Nonclassical States of Light, by Taylor & Francis, New York 2003
* [5] Husimi K, Proc. Phys. Math. Soc. Jpn. 22 264 (1940)
* [6] I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).
* [7] C. K. Chui , An Introduction to Wavelets (Academic,1992).
* [8] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002)
* [9] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, (New York, 1968)
* [10] Li-yun Hu and Hong-yi Fan, Int. J. Theor. Phys. 48, 1539 (2009)
* [11] Hong-yi Fan and Qin Guo, Phys. Lett. A 358, 203 (2006)
* [12] Hong-yi Fan and Hai-liang Lu, Opt. Lett. 32, 554 (2007)
* [13] Hong-yi Fan, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 35, 1831 (1987)
* [14] Hong-yi Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994)
* [15] V. V. Dodonov, J. Opt. B: Quantum Semiclassical Opt. 4, R1 (2002)
* [16] R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987)
* [17] D. F. Walls, Nature 324, 210 (1986)
* [18] Hong-yi Fan, J Opt B: Quantum Semiclass. Opt. 5 R147 (2003)
* [19] Hu, L., Fan, H.: Int. J. Theor. Phys. (2008) DOI 10.1007/s10773-007-9533-9
* [20] H.-Y. Fan, H.-L. Lu and Y. Fan, Ann. Phys. 321, 480 (2006)
* [21] Wünsche A, J Opt B: Quantum Semiclass. Opt. 1, R11 (1999)
* [22] Hong-yi Fan and Hai-liang Lu, Int. J. Mod. Phys. B 19, 799 (2005)
* [23] C. L. Mehta, Phys. Rev. Lett. 18, 752 (1967)
* [24] Weyl H, Z.Phys. 46, 1 (1927)
* [25] Li-yun Hu and Hong-yi Fan, Phys. Rev. A 80, 022115 (2009)
* [26] Li-yun Hu and Hong-yi Fan, J. Mod. Opt. 55, 1835 (2008)
* [27] Li-yun Hu and Hong-yi Fan, arXiv:0910.5354 [quant-ph]
|
arxiv-papers
| 2009-11-01T03:33:44 |
2024-09-04T02:49:06.218232
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0911.0125"
}
|
0911.0183
|
# A Gibbs Sampling Based MAP Detection Algorithm for OFDM Over Rapidly
Varying Mobile Radio Channels
Erdal Panayırcı1, Hakan Doğan2 and H. Vincent Poor 3
1 Department of Electronics Engineering, Kadir Has University, Istanbul,
Turkey
2 Department of Electrical and Electronics Engineering, Istanbul University
1 Department of Electrical Engineering, Princeton University, Princeton, NJ
08544 USA
###### Abstract
In orthogonal frequency-division multiplexing (OFDM) systems operating over
rapidly time-varying channels, the orthogonality between subcarriers is
destroyed leading to inter-carrier interference (ICI) and resulting in an
irreducible error floor. In this paper, a new and low-complexity maximum a
posteriori probability (MAP) detection algorithm is proposed for OFDM systems
operating over rapidly time-varying multipath channels. The detection
algorithm exploits the banded structure of the frequency-domain channel matrix
whose bandwidth is a parameter to be adjusted according to the speed of the
mobile terminal. Based on this assumption, the received signal vector is
decomposed into reduced dimensional sub-observations in such a way that all
components of the observation vector contributing to the symbol to be detected
are included in the decomposed observation model. The data symbols are then
detected by the MAP algorithm by means of a Markov chain Monte Carlo (MCMC)
technique in an optimal and computationally efficient way. Computational
complexity investigation as well as simulation results indicate that this
algorithm has significant performance and complexity advantages over existing
suboptimal detection and equalization algorithms proposed earlier in the
literature.
Index Terms\- OFDM, MAP detection, Monte carlo technique, Gibbs sampling.
Intercarrier interference, fast time-varying channels.
††footnotetext: E. Panayirci is on sabbatical leave from Kadir Has
University, Istanbul, Turkey.
This research has been conducted within the NEWCOM++ Network of Excellence in
Wireless Communications and WIMAGIC Strep projects funded through the EC 7th
Framework Programs and was supported in part by the U.S. National Science
Foundation under Grant CNS-0625637.
## I Introduction
Orthogonal frequency-division multiplexing (OFDM) has been shown to be an
effective method to overcome inter-symbol interference (ISI) caused by
frequency-selective fading with a simple transceiver structure, and is
consequently used in several existing wireless local and metropolitan area
standards such as the IEEE 802.11 and IEEE 802.16 families. IEEE 802.11
wireless LAN (WLAN) technology has become very popular for providing data
services to Internet users although its overall design and feature set are not
well suited for outdoor broadband wireless access (BWA) applications[1].
Therefore, IEEE 802.16 has been developed as a new standard for BWA
applications [2]. Recently, the much-anticipated Worldwide Interoperability
for Microwave Access (WiMAX) technology was introduced to promote the 802.16
standards while introducing features to enable mobile broadband services at
vehicular speeds beyond 120 km/h.
OFDM eliminates ISI and simply uses a one-tap equalizer to compensate
multiplicative channel distortion in quasi-static channels. However, in fading
channels with very high mobility, the time variation of the channel over an
OFDM symbol period results in a loss of subchannel orthogonality which leads
to inter-carrier interference (ICI). A considerable amount of research on OFDM
receivers for quasi-static fading has been conducted, but a major hindrance to
such receivers is the lack of mobility support [3]. Since mobility support is
widely considered to be one of the key features in wireless communication
systems, and in this case ICI degrades the performance of OFDM systems, OFDM
transmission over very rapidly time varying multipath fading channels has been
considered recently in a number of recent works [4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15].
The techniques proposed in these works range from linear equalizers, based on
the zero-forcing (ZF) or the minimum mean-squared error (MMSE) criterion [4,
5, 6, 7, 8, 10, 9, 11, 12, 13], to nonlinear equalizers based on decision-
feedback or ICI cancelation [10, 9, 11, 12, 13, 14]. Also near maximum-
likelihood approaches have been proposed [16]. It has been shown that
nonlinear equalizers based on ICI cancelation generally outperform linear
approaches [12, 13, 10, 11]. However, linear equalizers still preserve their
importance mainly because they are less complex.
In [12], performances of Matched Filter (MF), Least Squares (LS), Minimum Mean
Square Error (MMSE) and MMSE with Successive Detection (SD) techniques with
optimal ordering have been investigated. However, since the number of
subcarriers is usually very large in high speed wide-band wireless standards,
even the linear MMSE equalizer considered in [12] demands very high
computational load.
The specific structure of the Doppler-induced ICI in OFDM systems operating
over highly mobile channels presents a distinctive feature of limited support
of the Doppler spread that can be exploited by the receiver. References [6, 7,
8, 10, 9] exploit the banded character of the frequency-domain channel matrix
to reach a complexity that is only linear in the number of subcarriers. In a
certain sense, the assumption of a banded frequency-domain channel matrix is a
natural extension of the time-invariant channel case, in which the frequency-
domain channel matrix is diagonal and hence banded with the smallest possible
bandwidth.
In [7], using the banded structure of the channel matrix, a simple frequency
domain equalizer has been proposed that can compensate for the loss of
subchannel orthogonality due to ICI. However, the detection performance of the
technique degrades substantially, since the data to be detected cannot fully
use the contributing observation elements. The work presented in [13] combined
[7] and [10] to derive a recursive decision feedback equalizer receiver for
ICI suppression. The iterative MMSE serial linear equalizer (SLE) of [10],
which takes the banded structure of the channel matrix into account, seems to
be one of the most promising approaches to compensate for ICI. Iterative MMSE
is then applied to estimate frequency-domain symbols. In [8], a block MMSE
equalizer for OFDM systems operating over time-varying channels is presented.
By exploiting the banded structure of the frequency-domain channel matrix, the
complexity of the resulting algorithm turns out to be smaller than that of
[10].
In this paper, a new computationally feasible, maximum a posteriori
probability (MAP)-based data symbol detection algorithm is proposed for OFDM
systems operating over highly mobile channels, as an alternative to the
existing suboptimal equalization/detection techniques summarized in the above
paragraphs. The proposed detection algorithm exploits the banded structure of
the frequency-domain channel matrix whose bandwidth is a parameter to be
adjusted according to the speed of the mobile terminal. This assumption
enables us to decompose the main received signal vector into finite numbers of
reduced-dimensional, sub-received signal vectors from which the data symbols
can be detected by the MAP algorithm in an optimal and computationally
efficient way. The decomposition is achieved in such a way that all the
components of the received vector that contribute to the symbol to be detected
are included in the decomposed observation model. Data symbols in each sub-
received signal model are then detected successively by a MAP detection
algorithm. To implement MAP symbol detection in a computationally efficient
way, we employ a Markov Chain Monte Carlo (MCMC) technique based on Gibbs
sampling, which is a powerful statistical signal processing tool to estimate a
posteriori probability (APP) values.
The resulting detection algorithm is compared with previously proposed
algorithms in terms of both bit error rate (BER) and complexity requirements.
Computational complexity investigation as well as simulation results indicate
that our algorithm has significant performance and complexity advantages over
the existing suboptimal detection and equalization algorithms.
## II System Model
Let us consider an OFDM system with $N$ subcarriers and available bandwidth
$B=1/T_{s}$ where $T_{s}$ is the sampling period. A given sampling period is
divided into $N$ subchannels by equal frequency spacing $\Delta f=B/N$. At the
transmitter, information symbols are mapped into possibly complex-valued
transmitted symbols according to the modulation format employed. The symbols
are processed by an $N-$length Inverse Fast Fourier Transform (IFFT) block
that transforms the data symbol sequence into the time domain. The time-domain
signal is extended by a guard interval containing $G$ samples whose length is
chosen to be longer than the expected delay spread to avoid ISI. The guard
interval includes a cyclically extended part of the OFDM block to avoid ICI.
Hence, the complete OFDM block duration is $P=N+G$ samples. The resulting
signal is converted to an analog signal by a digital-to-analog (D/A)
converter. After shaping with a low-pass filter (e.g. a raised-cosine filter)
with bandwidth $B$, it is transmitted through the transmit antenna with the
overall symbol duration of $PT_{s}$.
Let $h(m,l)$ represent the $l$th path (multipath component) of the time-
varying channel impulse response at time instant $t=mT_{s}$. The discrete-time
received signal can then be expressed as follows:
$y(m)=\sum_{l=0}^{L-1}h(m,l)d(m-l)+w(m),$ (1)
where the transmitted signal $d(m)$ at discrete sampling time $m$ is given by
$d(m)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}s(k)e^{j2\pi mk/N},$ (2)
$L$ is the total number of paths of the frequency selective fading channel,
and $w(m)$ is additive white Gaussian noise (AWGN) with zero mean and variance
$E\\{|w(m)|^{2}\\}=\sigma_{w}^{2}$. The sequence $s(k),k=0,1,\cdots,N-1,$ in
(2) represent either quadrature-amplitude modulation (QAM) or phase-sift-
keying (PSK) modulated data symbols with $E\\{|s(k)|^{2}\\}=1.$
At the receiver, after passing through the analog-to-digital (A/D) converter
and removing the cyclic prefix (CP), a fast Fourier transform (FFT) is used to
transform the data back into the frequency domain. Lastly, the binary data is
obtained after demodulation and channel decoding.
The fading channel coefficients $h(m,l)$ can be modeled as zero-mean complex
Gaussian random variables. Based on the wide-sense stationary uncorrelated
scattering (WSSUS) assumption, the fading channel coefficients in different
paths are uncorrelated with each other. However, these coefficients are
correlated within each individual path and have a Jakes Doppler power spectral
density having an autocorrelation function given by
$E\\{h(m,l)h^{*}(n,l)\\}=\sigma^{2}_{h_{l}}J_{0}{(2\pi f_{d}T_{s}(m-n))},$ (3)
where $\sigma^{2}_{h_{l}}$ denotes the power of the channel coefficients of
the $l$th path. $f_{d}$ is the Doppler frequency in Hertz so that the term
$f_{d}T_{s}$ represents the normalized Doppler frequency of the channel
coefficients. $J_{0}(.)$ is the zeroth order Bessel function of the first
kind.
By using (2) in (1), the received signal can be written as
$y(m)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}s(k)\sum_{l=0}^{L-1}h(m,l)e^{j\frac{2\pi
k(m-l)}{N}}+w(m),$ (4)
which upon defining the time-varying channel transfer function
$H(k,m)\triangleq{\sum}^{L-1}_{l=0}h(m,l)e^{-j2\pi lk/N},$ (5)
becomes
$y(m)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}s(k)H(k,m)e^{j2\pi mk/N}+w(m).$ (6)
The FFT output at the $k^{th}$ subcarrier, after excluding the guard interval,
can be expressed as
$\displaystyle Y(k)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}y(m)e^{-j2\pi mk/N}$ (7)
$\displaystyle=$ $\displaystyle s(k)G(k,k)+I(k)+W(k),$
where $I(k)$ is ICI caused by the time-varying nature of the channel given as
$I(k)=\sum_{i=0,i\neq k}^{N-1}s(i)G(k,i).$ (8)
$G(k,i)$ in (8) represents the average frequency domain time-varying channel
response, defined as
$G(k,i)\triangleq(1/N)\sum_{m=0}^{N-1}H(i,m)e^{j2\pi m(i-k)/N}.$ (9)
Similarly, the term $G(k,k)\triangleq\frac{1}{N}\sum_{m=0}^{N-1}H(k,m)$ in (9)
represent the portion of the average frequency domain channel response at the
$k$th subcarrier and $W(k)$ denotes discrete Fourier transform of the white
Gaussian noise $w(m)$:
$W(k)=\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}w(m)e^{-j2\pi mk/N}.$ (10)
Because of the term $I(k)$ in (7), there is an irreducible error floor even in
the training sequences since pilot symbols are also corrupted by ICI, arising
from the fact that the time-varying channel destroys the orthogonality between
subcarriers. Therefore, channel estimation should be performed either jointly
with data or before the FFT block in order to compensate for the ICI. Note
that if the channel is static or very slowly time-varying, that is
$H(k,m)\thickapprox H(k)$, then it can be easily shown that $G(k,k)=H(k)$ and
$I(k)=0$ for $k=0,1,\cdots,N-1,$ resulting in a received signal at the output
of the FFT processor corresponding to the $k$th OFDM symbol given by
$Y(k)=H(k)s(k)+W(k).$
From (6) and (7), the FFT output received signal can be expressed in vector
form as
$\bf{Y}=\bf{G}\bf{s}+\bf{W}$ (11)
where ${\bf{Y}}=[Y(0),Y(1),...,Y(N-1)]^{T}$,
${\bf{s}}=[s(0),s(1),...,s(N-1)]^{T}$ and
${\bf{W}}=[W(0),W(1),...,W(N-1)]^{T}.$ For $k,i=0,1,\cdots,N-1,$ the $(k,i)$th
element of the matrix ${\bf G}=[G(k,i)]\in\mathcal{C}^{N\times N}$
representing the time-varying channel is given by (9).
Under the assumption that the channel matrix ${\bf G}$ in (11) is perfectly
known at the receiver, the maximum likelihood (ML) detector performs an
exhaustive search over the entire set of signal vectors whose components are
selected from the signal constellation formed by the modulation scheme chosen.
Especially in IEEE 802.16 based systems, the length $N$ of each OFDM symbol is
very large; it can take values as large as $N=1024$ or even $N=2048$
especially for high mobility applications. In this case an exhaustive search
for the ML solution would be very complex since the search space has an
extremely large number of constellation points, ($|\mathcal{S}|^{N}$, where
$|\mathcal{S}|$ is the cardinality of the signal constellation). On the other
hand, all of the lower complexity linear detectors given in Table 1 are
suboptimal since they do not take into account the correlation of the
components of the transformed noise and yield noise enhancement. Recently, a
nonlinear recursive detection technique using the decision-feedback principle,
namely the MMSE-SD algorithm (VBLAST), has been proposed [12]. The performance
of VBLAST depends critically on the order in which the data vector components
are processed. To minimize error propagation effects and to improve the
detection of unreliable components, more reliable data vector components
should be detected first. Therefore, the algorithm depends on calculation of
the post-detection signal-to-interference-plus-noise ratio (SINR) based upon
MMSE detection as a measure of reliability, and so the calculation of SINR is
compulsory at each iteration. Therefore this algorithm is computationally
intensive as a number of pseudo inverse operations need to be performed.
Moreover, its complexity grows exponentially with the total number of
subcarriers.
TABLE I: Linear Detection Methods Method | Solution
---|---
Matched Filter (MF) | $\bf{\hat{s}}=\mbox{Q}\\{\bf{G^{\dagger}}\bf{Y}\\}$
Zero Forcing (ZF) | $\bf{\hat{s}}=\mbox{Q}\\{(\bf{G^{\dagger}}\bf{G})^{-1}\bf{G^{\dagger}}\bf{Y}\\}$
MMSE | $\bf{\hat{s}}=\mbox{Q}\\{(\bf{G^{\dagger}}\bf{G}+{\bf I}_{N}\sigma_{w}^{2})^{-1}\bf{G^{\dagger}}\bf{Y}\\}$
where ${\bf I}_{N}$ is the $N$-by-$N$ identity matrix.
## III MAP Detection Algorithm
The necessity of detecting large numbers of symbols in OFDM systems employed
especially in highly mobile and wide-band wireless systems represents a
significant computational burden as well as creating some convergence
problems. However, as is known [9], time-varying channels produce a nearly-
banded channel matrix ${\bf G}$ whose only main diagonal, $Q$ subdiagonals and
$Q$ superdiagonals are nonzero. The bandwidth $2Q+1$, which is defined as the
total number of non-zero diagonals in ${\bf G}$, is a parameter to be adjusted
according to the mobility-rate of the channel. The significant coefficients of
${\bf G}$ thus are confined to the $2Q+1$ central diagonals. The parameter
$Q\in\\{0,1,\cdots N/2-1\\}$ controls the target ICI response length; larger
$Q$ corresponds to a longer ICI span and, thus, increased estimation
complexity. In general, $Q$ should be chosen proportional to the width of the
Doppler spectrum of the channel.
We now present an optimal low-complexity MAP detection algorithm to detect the
data symbols ${\bf s}$ from ${\bf Y}$ taking into account the banded structure
of the channel matrix ${\bf G}$. From the observation ${\bf Y}$ in (11), the
receiver attempt to detect the OFDM output symbol ${\bf s}$, assuming that
${\bf G}$ is completely known by the receiver. The banded structure of the
channel matrix ${\bf G}$ implies that the data symbol $s(k),k=0,1,\cdots,N-1,$
contributes to a maximum of $2Q+1$ observation elements as follows:
${\bf Y}_{k}=[Y(j_{k}),Y(j_{k}+1),\cdots,Y(i_{k})]^{T},\mbox{ for}$ (12)
$j_{k}=\max\\{0,k-Q\\}\hskip 8.53581pt\mbox{and}\hskip
8.53581pti_{k}=\min\\{N-1,k+Q\\}.$
Based on this observation, the received signal in (11) can be decomposed into
$N$ reduced-dimensional sub-observations from which the data symbols can be
detected in an optimal and computationally efficient way. For a given index
$k=0,1,\cdots,N-1$ and $Q$, it can be easily shown from (11) and (12) that
${\bf Y}_{k}={\bf G}_{k}{\bf s}_{k}+{\bf W}_{k}$ (13)
where, ${\bf s}_{k}=\left[{\bf s}(j_{k})\right]$, ${\bf W}_{k}=\left[{\bf
W}(j_{k})\right]$, ${\bf G}_{k}=\left[G(i_{k},j_{k})\right]$, for
$I_{L_{k}}\triangleq\max\\{0,k-Q\\}\leq i_{k}\leq
I_{U_{k}}\triangleq\min\\{N-1,k+Q\\}$ (14)
and
$J_{L_{k}}\triangleq\max\\{0,k-2Q\\}\leq j_{k}\leq
J_{U_{k}}\triangleq\min\\{N-1,k+2Q\\}.$ (15)
Note that due to the banded structure of ${\bf G}$, some elements of the
matrices ${\bf G}_{k}$ are zero and $\mbox{dim}({\bf
G}_{k})\leq(2Q+1)\times(2(2Q+1)-1)$. The ${\bf G}_{k}$’s reach their maximum
dimension when $2Q+1\leq k\leq(N-(2Q+1))$.
For $k=0,1,\cdots,N-1,$ the MAP estimate of the data symbol $s(k)$ given ${\bf
Y}_{k}$ is
$\widehat{s}_{MAP}(k)\equiv\widehat{s}(k)=\arg\max_{s(k)\in\mathcal{S}}P(s(k)|{\bf
Y}_{k}),$ (16)
where $\mathcal{S}$ denotes the set of signal constellation points from which
$s(k)$ takes values. Based on this approach, $s(k)$ can be detected
sequentially for $k=0,1,\cdots,N-1$, incorporating the outcomes of the
previous estimates in a decision-feedback mode as follows.
$\blacksquare$ For $k=0$, determine the estimate $\widehat{s}(0)$ from (16).
$\blacksquare$ For $k=k+1$ modify the observation vector ${\bf Y}_{k}$ by
subtracting the terms coming from the contributions of the estimated data
symbols $\widehat{s}(0),\widehat{s}(1),\cdots,\widehat{s}(k-1)$ as
$\widetilde{{\bf Y}}_{k}\triangleq{\bf Y}_{k}-\sum_{l=J_{L_{k}}}^{k-1}{\bf
g}^{(l)}_{k}\widehat{s}(l)$ (17)
where ${\bf g}^{(l)}_{k}$ is the $l$th column of ${\bf G}_{k}$ and $J_{L_{k}}$
and $J_{U_{k}}$ are defined in (15).
$\blacksquare$ Determine the MAP estimate of $\widehat{s}(k)$ from
$\widetilde{{\bf Y}}_{k}=\widetilde{{\bf G}}_{k}\widetilde{{\bf s}}_{k}+{\bf
W}_{k}$ (18)
as
$\widehat{s}(k)=\arg\max_{\widetilde{s}(k)\in\mathcal{S}}P(\widetilde{s}(k)|\widetilde{{\bf
Y}}_{k}),$ (19)
where
$\widetilde{{\bf G}}_{k}\triangleq{\bf G}_{k}-\left[{\bf g}^{(0)}_{k},{\bf
g}^{(1)}_{k},\cdots,{\bf g}^{(k-1)}_{k}\right],$ (20)
and $\widetilde{{\bf s}}_{k}$ is the vector obtained by removing the first
$k-1$ elements of ${\bf s}_{k}.$
$\blacksquare$ END IF $k=N-1$.
The major problem is finding the values of $\hat{s}_{MAP}(k)$ in a
computationally efficient manner. To see this difficulty we assume that the
data symbols are independent and identically distributed binary phase shift
keying (BPSK), taking values of $+1$ and $-1$. Note that higher dimensional
signal constellations can be treated similarly with a straightforward
extension. The conditional probability of $\widetilde{s}(k)$ given the
observation vector $\bf\widetilde{Y}_{k}$ can be expressed as
$\displaystyle P(\widetilde{s}(k)=+1|\widetilde{{\bf Y}}_{k})$
$\displaystyle=$ $\displaystyle\sum_{\widetilde{{\bf
s}}_{\overline{k}}}P(\widetilde{s}(k)=+1,\widetilde{{\bf
s}}_{\overline{k}}|\widetilde{{\bf Y}}_{k})$ (21) $\displaystyle=$
$\displaystyle\sum_{\widetilde{{\bf
s}}_{\overline{k}}}P(\widetilde{s}(k)=+1|\widetilde{{\bf
s}}_{\overline{k}},\widetilde{{\bf Y}}_{k})P(\widetilde{{\bf
s}}_{\overline{k}}|\widetilde{{\bf Y}}_{k})$
where the second identity follows by applying the chain rule of probability.
The vector $\bf\widetilde{s}_{\overline{k}}$ in (21) is obtained by canceling
the component $\widetilde{s}(k)$ in $\bf\widetilde{s}_{k}$ and the summation
is over all possible values of $\bf\widetilde{s}_{\overline{k}}$. The number
of combinations that $\bf\widetilde{s}_{\overline{k}}$ takes grows
exponentially with the dimension of $\bf\widetilde{s}_{k}$ and thus becomes
prohibitive for large values of the size of this vector. Thus, we resort to
the Gibbs sampler, a Monte Carlo method to calculate the a posteriori
probabilities of the unknown symbols.
### III-A MAP Detection Based on Gibbs Sampling
The Gibbs sampler is an MCMC sampling method for numerical evaluation of
multidimensional integrals. Its popularity is gained from the facts that it is
capable of carrying out many complex Bayesian computations. In this section we
briefly explain the application of the Gibbs sampling technique to our symbol
detection problem where the observation process is given by (18). For
notational convenience we drop the index $k$ and the ”tilde” from all the
involved variables, e.g. ${\bf Y},{\bf G}$ are shorthand notations for
$\widetilde{{\bf Y}}_{k},\widetilde{{\bf G}}_{k}$, respectively. (21) can then
be expressed as
$\displaystyle P(s(k)=+1|{\bf Y})$ $\displaystyle=$ $\displaystyle\sum_{{\bf
s}_{\overline{k}}}P(s(k)=+1|{\bf s}_{\overline{k}},{\bf Y})P({\bf
s}_{\overline{k}}|{\bf Y})$ (22) $\displaystyle=$ $\displaystyle E_{{\bf
s}_{\overline{k}}|{\bf Y}}\left\\{P(s(k)=+1|{\bf s}_{\overline{k}},{\bf
Y})\right\\}.$
According to the Gibbs sampling based statistical Monte Carlo estimation
technique, an estimate of (22) can be evaluated by taking the empirical
average
$P(s(k)=+1|{\bf Y})=\frac{1}{N_{s}}\sum_{n=1}^{N_{s}}P(s(k)=+1|{\bf
s}^{(n)}_{\overline{k}},{\bf Y})$ (23)
where ${\bf s}^{(n)}_{\overline{k}}$ for $n=1,2,\cdots,N_{s}$ are samples
drawn from the conditional distribution $P({\bf s}_{\overline{k}}|{\bf Y})$.
There is a substantial body of literature concerning the Monte Carlo Gibbs
sampling technique; see, e.g., [17],[18]. One possible version of the Gibbs
sampler suitable for calculating the a posteriori probabilities in (21) may be
summarized as follows.
Let ${\bf s}=[s(0),s(1),\cdots,s(N-1)]^{T}$ be a vector of unknown data
symbols. Let ${\bf Y}$ be the observed signal. To generate random samples from
the distribution $P({\bf s}|{\bf Y})$, given the samples from the $(n-1)$th
iteration ${\bf
s}^{(n-1)}=[s^{(n-1)}(0),s^{(n-1)}(1),\cdots,s^{(n-1)}(N-1)]^{T}$, the Gibbs
algorithm iterates at the $n$th iteration as follows to generate the samples
${\bf s}^{(n)}=[s^{(n)}(0),s^{(n)}(1),\cdots,s^{(n)}(N-1)]^{T}$:
$\blacksquare$ Initialize ${\bf s}^{(0)}$ randomly;
$\blacksquare$ for $n=1,2,\cdots,N_{T}$ and for $k=0,1\cdots,N-1,$
draw sample $s^{(n)}_{k}$ from
$P\left(s(k)|s^{(n)}_{0},\cdots,s^{(n)}_{k-1},s^{(n-1)}_{k+1},\cdots,s^{(n)}_{N-1}\right).$
Note that to ensure convergence, the Gibbs iteration is usually carried out
for $N_{T}=N_{b}+N_{s}$ iterations. The first $N_{b}$ iterations of the loop
is called the burn-in period which is necessary for the Monte Carlo simulation
to reach its stationary distribution. Only the samples ${\bf
s}^{(n)}=[s^{(n)}_{0},s^{(n)}_{1},\cdots,s^{(n)}_{N-1}]^{T},n=N_{b}+1,\cdots,N_{T}$,
from the last $N_{s}$ iterations are used to calculate the expectation.
It is known that under regularity conditions [19, 18],
(i) the distribution of ${\bf s}^{(n)}$ converges to $P({\bf s}|{\bf y})$, as
$n\rightarrow\infty$.
(ii) $(1/N_{s}\sum_{n=1}^{N_{s}}P(s(k)=+1|{\bf s}^{(n)}_{\overline{k}},{\bf
Y})=\sum_{{\bf s}_{\overline{k}}}P(s(k)=+1|{\bf s}_{\overline{k}},{\bf
Y})P({\bf s}_{\overline{k}}|{\bf Y})$, as $n\rightarrow\infty$.
### III-B Implementation of the Symbol Detector
From the previous section, we recall that to compute $P(s(k)=+1|{\bf
Y}_{k})=\sum_{{\bf s}_{\overline{k}}}P(s(k)=+1,{\bf s}_{\overline{k}}|{\bf
Y}_{k})$, we need to perform the summation on the right-hand side of (22).
When ${\bf s}$ has a large dimension, the exact evaluation of this summation
may not be feasible and other more efficient techniques must be adopted. In
this section the Gibbs sampling-based Monte Carlo method summarized in the
previous section will be applied to develop a computationally efficient
algorithm for calculation of the a posteriori probabilities $P(s(k)|{\bf Y}).$
From (23), it follows that we need to evaluate $P(s(k)=+1|{\bf
s}^{(n)}_{\overline{k}},{\bf Y}),$ for $n=1,2,\cdots,N_{T}$. For this we
define
$\lambda^{(n)}_{k}\triangleq\ln\frac{P(s(k)=+1|{\bf
s}^{(n)}_{\overline{k}},{\bf Y})}{P(s(k)=-1|{\bf s}^{(n)}_{\overline{k}},{\bf
Y})},$ (24)
from which it can be easily seen that
$P(s(k)=+1|{\bf s}^{(n)}_{\overline{k}},{\bf
Y})=\frac{1}{1+\exp\left(-\lambda^{(n)}_{k}\right)}.$ (25)
$\lambda^{(n)}_{k}$ can be computed by expanding $P(s(k)=+1|{\bf
s}^{(n)}_{\overline{k}},{\bf Y})$ as
$\displaystyle P\left(s(k)=+1|{\bf s}^{(n)}_{\overline{k}},{\bf Y}\right)=$
(26) $\displaystyle\frac{p\left({\bf Y}|s(k)=+1,{\bf
s}^{(n)}_{\overline{k}})P(s(k)=+1,{\bf
s}^{(n)}_{\overline{k}}\right)}{\sum_{s(k)\in\\{+1,-1\\}}p\left({\bf
Y}|s(k)=+1,{\bf s}^{(n)}_{\overline{k}}\right)P\left(s(k)=+1,{\bf
s}^{(n)}_{\overline{k}}\right)}.$
The data symbols are assumed to be independent and equally likely, Therefore,
it follows from (26) and (24) that
$\lambda^{(n)}_{k}\triangleq\ln\frac{p({\bf Y}|s(k)=+1,{\bf
s}^{(n)}_{\overline{k}})}{p({\bf Y}|s(k)=-1,{\bf s}^{(n)}_{\overline{k}})}.$
(27)
Since $p({\bf Y}|{\bf s})\thicksim\exp(-|{\bf y}-{\bf G}{\bf s}|^{2})$, after
some algebra, (27) can be expressed as
$\lambda^{n}(s_{k})=\frac{1}{\sigma^{2}_{w}}\Re\left\\{{\bf
g}^{{\dagger}}_{k}({\bf Y}-{\bf G}_{\overline{k}}\mbox{ }{\bf
s}_{\overline{k}})\right\\},$ (28)
where $(\cdot)^{{\dagger}}$ denotes the conjugate transpose and
$\Re\\{\cdot\\}$ denotes the real part of its argument. ${\bf
G}_{\overline{k}}$ is ${\bf G}$ with its $k$th column $\bf{g}_{k}$ removed. In
summary, for $k=0,1,\cdots,N-1$, to estimate the a posteriori probabilities
$P(s(k)|{\bf Y})$ in (21), the Gibbs sampler runs over all symbols $N_{s}$
times to generate a collection of vectors $\left\\{{\bf
s}^{(n)}_{\overline{k}}\right\\}_{n=N_{b}+1}^{N_{T}}$ which are used in (23)
to compute the desired quantities.
### III-C Complexity Requirements
The computational complexity of the MAP symbol detector based on Gibbs
sampling proposed in this work is determined by the parameters $N_{s},Q,N$ and
the constellation size of the transmitted data symbols. The computation of
$\widetilde{{\bf Y}}_{k}$ in (18) for $k=0,1,\cdots,N-1$ requires a maximum of
$(4Q^{2}+2Q)N$ complex multiplications (CMs) and $(4Q^{2}+2Q)N$ complex
additions (CAs) per data block. Assuming BPSK signaling, the computation of
the a posteriori probabilities in (22) requires a maximum of
$(8Q^{2}+2Q+1)NN_{s}$ CMs and $(8Q^{2}+2Q-1)NN_{s}$ CAs and computation of the
empirical average of a posteriori probabilities of the data symbols in (23)
requires $NN_{s}$ CSs. Therefore, the whole algorithm requires of maximum
$N(4Q^{2}+2Q+(4Q^{2}+2Q)N_{s})$ CMs and $N(4Q^{2}+2Q+(4Q^{2}+2Q)N_{s})$ CAs,
leading to a total of $2N(4Q^{2}+2Q+(4Q^{2}+2Q)N_{s})$ complex operations.
Several low complexity equalization algorithms have been developed recently,
of which several are worth mentioning here to compare their computational
complexities with that of the Gibbs-based algorithm.
Ruguni et al. [8] proposed a block MMSE technique based on exploiting the
banded structure of the channel matrix ${\bf G}$. The matrix inversion was
obtained using a low-complexity decomposition such as Cholesky or the
$LDL^{{\dagger}}$ decomposition. The algorithm requires a total of
$(8Q^{2}+22Q+4)N$ complex operations. Schniter [10] proposed a linear serial
equalizer also based on exploiting the banded structure of the channel matrix.
This algorithm requires a total of $(8/3Q^{3}+2Q^{2}+5/3Q+4)N$ complex
operations. The complexity of the serial MMSE equalizer is higher than that of
the block MMSE equalizer.
In the VBLAST algorithm, matrix inversion is needed of dimension equal to the
number of OFDM subcarriers. As a result, the computational complexity of the
VBLAST receiver increases rapidly with the number of subcarriers, which makes
its real-time implementation prohibitive for large numbers of subcarriers.
As can be seen easily, the complexity of our algorithm is of the same order of
the above equalization algorithms and is lower than the VBLAST algorithm.
However, as remarked earlier, these algorithms are suboptimal as opposed to
our optimal MAP detection algorithm and perform poorly especially when the ICI
is high. It is also worth mentioning that our algorithm can be easily extended
to an iterative multiuser MAP detection scheme for OFDM systems.
## IV Simulation Results
This section presents computer simulation results of the proposed detection
methods for rapidly varying mobile radio channels. The system operates with a
5 MHz bandwidth and is divided into 512 tones ($N=512$) with a total symbol
period ($T_{s}$) of 115 $\mu$s, of which 12.8 $\mu$s constitute the CP. One
OFDM symbol thus consists of 576 samples, sixty-four of which constitute the
CP. The normalized Doppler frequencies are $f_{d1}*T_{s}=0.0307$ and
$f_{d2}*T_{s}=0.1075$, corresponding to an IEEE 802.16e mobile terminal moving
with speeds of 120 km/h and 420 km/h, respectively, for a carrier frequency of
2.4 GHz. The wireless channels between the mobile antenna and the receiver
antenna are modeled based on a realistic channel model determined by the
COST-207 project in which Typical Urban (TU) and Bad Urban (BU) channel models
are considered. For each OFDM symbol, Gibbs sampling is performed for 30
iterations, with the first 10 iterations as the burn-in period.
Figure 1: BER comparison of various detection algorithms for OFDM systems; TU
Channel (6 taps), 420 km/h Figure 2: BER comparison of various detection
algorithms for OFDM systems; TU Channel (6 taps), 120 km/h Figure 3: BER
comparison of various detection algorithms for OFDM systems; BU Channel (6
taps), 420 km/h Figure 4: BER performance of the proposed MAP algorithm based
on Gibbs Sampling for the TU channel as a function of Q values
The information symbols are BPSK modulated to yield $\mathcal{S}$ = $\\{\pm
1\\}$. Figs.1-3 compare the BER performance of the proposed Gibbs-based MAP
detection algorithm, an equalization technique based on zero forcing (ZF)
proposed in [7], linear MMSE, and the VBLAST algorithm proposed in [12], as a
function of energy per bit to noise power ratio ($E_{b}/N_{0}$) where $N_{0}$
is equal to $\sigma_{w}^{2}$.
ZF causes noise enhancement while it eliminates the ICI. Therefore, it is seen
that ZF performance is the worst. The reason for this can also be explained by
the ill-conditioned matrix $({\bf G}^{{\dagger}}{\bf G})$ to be inverted. This
problem can be solved with MMSE, which provides a good trade-off between ICI
cancellation and noise elevation by using the knowledge of the noise level. In
other words, the noise enhancement can be reduced by the insertion of the
noise power in the inverse matrix that given in Table 1.
We note that linear equalization of the received signal is suboptimal as
mentioned in Section II and hence Gibbs-based MAP detection algorithm (MAP-GS)
is proposed in Section III. It is observed that MAP-GS outperforms both ZF and
MMSE receivers while it has similar performance to VBLAST. Moreover, the exact
MAP performance is also included to benchmark the proposed algorithms. It can
be concluded from these figures that the compared algorithms have similar
performance for the speed of 120 km/h while the performance difference is
obvious for the speed of 420 km/h. Moreover, it is seen that the BER
performance of all algorithms has slightly decreased for the BU channel. In
particular, it is observed that a savings in about 2 dB is obtained at
$BER=10^{-3}$, as compared with MMSE detection for the TU channel.
It has been shown that when a proper detection technique is adopted, the time-
varying nature of the channel can be exploited as a provider of time
diversity[12]. In [12], it was demonstrated that VBLAST fully utilizes the
time diversity while suppressing the residual interference and the noise
enhancement. Similarly to VBLAST, we have seen that MAP-GS is also a useful
detection technique for time-varying channels while having lower complexity.
Therefore, in particular, it is not surprising that in simulations the
performance at 420km/h is better than that at 120km/h.
Finally, the BER performance of the proposed algorithm is presented as a
function of $Q$ in Fig. 4. The parameter $Q$ can be chosen to trade off
performance versus complexity. As a rule of thumb, we have seen that
$Q=\lfloor f_{d_{max}}/\Delta f\rfloor+1$, where $f_{d_{max}}$ is the maximum
Doppler frequency and $\Delta f$ is the subcarrier spacing, is an appropriate
choice for Rayleigh fading[10]. In this paper, $f_{d_{max}}/\Delta f$ values
are given as the normalized Doppler frequencies. It is concluded from these
curves that the selection of the $Q$ value is highly dependent on SNR values.
In particular, for $E_{b}/N_{0}=20dB$, different $Q$ values show similar
performance because the effects of ICI are not very obvious relative to the
effects of the additive Gaussian noise. The $Q$ value has a greater role for
$Eb/N_{0}$ above 25dB because ICI is dominant. We note that $Q=3$ is
sufficient for SNRs below 20dB.
## V Conclusion
Conventional detection methods such as ZF and MMSE have irreducible error
floors at high normalized Doppler frequency $f_{d}T_{s}$ since ICI corrupts
the orthogonality among subcarriers. On the other hand, more sophisticated
methods such as VBLAST require too much complexity, especially for large
numbers of subcarriers. Therefore, we have proposed a new low-complexity
maximum a posteriori probability (MAP) detection algorithm that provides
excellent performance with manageable complexity for OFDM systems via the
Gibbs sampling technique. In the simulation section, a comparison with other
previously known receiver structures has been made and it has been
demonstrated that MAP detection based on Gibbs sampling provides performance
that is close to that of the optimal MAP detection algorithm for realistic
fading conditions.
## References
* [1] Q. Ni, A. Vinel, Y. Xiao, A. Turlikov, and T. Jiang, “Investigation of bandwidth request mechanisms under point-to-multipoint mode of WiMAX networks,” _IEEE Commun. Mag._ , vol. 45, no. 5, pp. 132–138, 2007.
* [2] C. Eklund, R. Marks, K. Stanwood, and S. Wang, “IEEE standard 802.16: A technical overview of the WirelessMAN TM air interface for broadband wireless access,” _IEEE Commun. Mag._ , vol. 40, no. 6, pp. 98–107, 2002.
* [3] H. Dogan, H. Cirpan, and E. Panayirci, “Iterative channel estimation and decoding of turbo coded SFBC-OFDM systems,” _IEEE Trans. Wireless Commun._ , vol. 6, no. 8, pp. 3090–3101, 2007.
* [4] I. Barhumi, G. Leus, and M. Moonen, “Equalization for OFDM over doubly-selective channels,” _IEEE Trans. Signal Process._ , vol. 54, no. 4, pp. 1445–1458, Apr. 2006.
* [5] A. Stamoulis, S. N. Diggavi, and N. Al-Dhahir, “Intercarrier interference in MIMO OFDM,” _IEEE Trans. Signal Process._ , vol. 50, no. 10, pp. 2451–2464, Oct, 2002.
* [6] X. Huang and H.-C.Wu, “Robust and efficient intercarrier interference mitigation for OFDM systems in time-varying fading channels,” _IEEE Trans. Veh. Technol._ , vol. 56, no. 5, pp. 2517–2528, Sep. 2007.
* [7] W. G. Jeon, K. H. Chang, and Y. S. Cho, “An equalization technique for orthogonal frequency-division multiplexing systems in time-variant multipath channels,” _IEEE Trans. Commun._ , vol. 47, no. 1, pp. 27–32, Jan. 1999\.
* [8] L. Rugini, P. Banelli, and G. Leus, “Simple equalization of time-varying channels for OFDM,” _IEEE Commun. Lett._ , vol. 9, no. 7, pp. 619–621, Jul. 2005.
* [9] L. Rugini, P. Banelli, and G. Leus, “Low-complexity banded equalizers for OFDM systems in Doppler spread channels,” _EURASIP Journal on Applied Signal Processing_ , vol. 2006, no. Article ID 67404, pp. 1–13, 2006.
* [10] P. Schniter, “Low-complexity equalization of OFDM in doubly-selective channels,” _IEEE Trans. Signal Process._ , vol. 52, no. 4, pp. 1002–1011, Apr. 2004.
* [11] A. Gorokhov and J. P. Linnartz, “Robust OFDM receivers for dispersive time-varying channels: Equalization and channel acquisition,” _IEEE Trans. Commun._ , vol. 52, no. 4, pp. 572–583, Apr. 2004.
* [12] Y.-S. Choi, P. J. Voltz, and F. A. Cassara, “On channel estimation and detection for multicarrier signals in fast and selective rayleigh fading channels,” _IEEE Trans. Commun._ , vol. 49, no. 8, pp. 1375–1387, Aug. 2001\.
* [13] X. Cai and G. B. Giannakis, “Bounding performance and suppressing intercarrier interference in wireless mobile OFDM,” _IEEE Trans. Commun._ , vol. 51, no. 12, pp. 2047–2056, Dec. 2003.
* [14] S. Tomasin, A. Gorokhov, H. Yang, and J.-P. Linnartz, “Iterative interference cancellation and channel estimation for mobile OFDM,” _IEEE Trans. Wireless Commun._ , vol. 4, no. 1, pp. 238–245, Jan. 2005.
* [15] S.-J. Hwang and P. Schniter, “Efficient sequence detection of multicarrier transmissions over doubly dispersive channels,” _EURASIP J. Appl. Signal Process._ , vol. 2006, no. Article ID 93638, pp. 1–17, 2006.
* [16] S. Ohno and K. Teo, “Approximate BER expression of ML equalizer for OFDM over doubly selective channels,” in _ICASSP 2008. IEEE International Conference on_ , 31 2008-April 4 2008, pp. 3049–3052.
* [17] B. F. Boroujeny, H. Zhu and Z. Shi, “Markov chain Monte Carlo Algorithms for CDMA and MIMO communication systems,” _IEEE Trans. Signal Processing._ , vol. 54, no. 6, pp. 1896–1909, 2006.
* [18] C. Robert and G. Casella, _Monte Carlo Statistical Methods_. NewYork: Springer-Verlag, 1999.
* [19] S. Geman and D. Geman, “Stohastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” _IEEE Trans. Pattern Anal. Machine Intell._ , vol. PAMI 6, pp. 721–741, Nov 1984.
|
arxiv-papers
| 2009-11-01T18:25:31 |
2024-09-04T02:49:06.223494
|
{
"license": "Public Domain",
"authors": "Erdal Panayirci, Hakan Dogan, H. Vincent Poor",
"submitter": "Erdal Panayirci",
"url": "https://arxiv.org/abs/0911.0183"
}
|
0911.0287
|
# An additional soft X-ray component in the dim low/hard state of black hole
binaries
C. Y. Chiang1, Chris Done1, M. Still2,4, and O. Godet3
1Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK
2Mullard Space Science Laboratory, Dorking, Surrey, UK
3X-ray and Observational Astronomy Group, Department of Physics and Astronomy,
University of Leicester, LE1 7RH, UK
4NASA Ames Research Center, Moffett Field, CA 93045, USA
E-mail: chia-ying.chiang@durham.ac.uk
(Submitted to MNRAS)
###### Abstract
We test the truncated disc models using multiwavelength (optical/UV/X-ray)
data from the 2005 hard state outburst of the black hole SWIFT J1753.5-0127.
This system is both fairly bright and has fairly low interstellar absorption,
so gives one of the best datasets to study the weak, cool disc emission in
this state. We fit these data using models of an X-ray illuminated disc to
constrain the inner disc radius throughout the outburst. Close to the peak,
the observed soft X-ray component is consistent with being produced by the
inner disc, with its intrinsic emission enhanced in temperature and luminosity
by reprocessing of hard X-ray illumination in an overlap region between the
disc and corona. This disc emission provides the seed photons for Compton
scattering to produce the hard X-ray spectrum, and these hard X-rays also
illuminate the outer disc, producing the optical emission by reprocessing.
However, the situation is very different as the outburst declines. The optical
is probably cyclo-synchrotron radiation, self-generated by the flow, rather
than tracing the outer disc. Similarly, limits from reprocessing make it
unlikely that the soft X-rays are directly tracing the inner disc radius.
Instead they appear to be from a new component. This is seen more clearly in a
similarly dim low/hard state spectrum from XTE J1118+480, where the 10$\times$
lower interstellar absorption allows a correspondingly better view of the
UV/EUV emission. The very small emitting area implied by the relatively high
temperature soft X-ray component is completely inconsistent with the much
larger, cooler, UV component which is well fit by a truncated disc. We
speculate on the origin of this component, but its existence as a clearly
separate spectral component from the truncated disc in XTE J1118+480 shows
that it does not simply trace the inner disc radius, so cannot constrain the
truncated disc models.
###### keywords:
X-rays: binaries – accretion, accretion discs
††pagerange: An additional soft X-ray component in the dim low/hard state of
black hole binaries–References††pubyear: 2009
## 1 Introduction
The accretion disc in black hole binary systems (BHB) is unstable at the point
where Hydrogen goes from being predominantly neutral to ionised. This gives
rise to dramatic, transient outbursts in which the reservoir of material built
up in the outer disc during quiescence can accrete down onto the black hole.
The peak luminosity of the outburst is set by the mass in the quiescent disc
and the time taken for this to accrete. Both these depend on the size of the
disc taking part in the outburst, which is set by the binary separation. Hence
there is a link between orbital period and peak luminosity, with short period
systems showing smaller peak outburst luminosities (King & Ritter 1998;
Shahbaz et al. 1998; Lasota 2001; Done, Gierliński & Kubota 2007 hereafter
DGK07).
The peak luminosity determines the number of spectral states which are seen
during the outburst. Systems with peak luminosity below $\sim 0.1L_{Edd}$
remain in the low/hard state (LHS), where the energy output peaks at $\sim$
100 keV. Brighter systems instead show a distinct transition to a much softer
thermal dominated state (TDS, also termed a high/soft state) which peaks at
$\sim$ 1 keV, while the brightest systems can also show a very high state
(VHS, alternatively steep power law state). These dramatic changes in the
spectrum are correlated with equally dramatic changes in the rapid variability
properties (see e.g. McClintock & Remillard 2006) and the jet (Fender, Belloni
& Gallo, 2004), implying distinct changes in the nature and geometry of the
accretion flow. These can be plausibly explained if there is a hot, optically
thin, geometrically thick solution to the accretion flow equations at low
luminosities (Shapiro, Lightman & Eardley 1976; Narayan & Yi 1995).This can be
put together with a cool, optically thick, geometrically thin disc (Shakura &
Syunyaev, 1973) in the truncated disc/hot flow model where the disc
progressively replaces more of the inner hot flow as the mass accretion rate
increases. The distinct hard/soft spectral transition then marks the point at
which the cool disc extends down to the last stable orbit, replacing all the
hot flow (Esin, McClintock & Narayan 1997; DGK07).
Such models make a clear prediction that the inner radius of the cool disc
should recede as the LHS drops in luminosity. This can in principle be tested
using the colour temperature and luminosity of the cool disc component to
estimate its radius (e.g. Poutanen, Krolik & Ryde 1997; Shrader & Titarchuk
1999). However, such observations are complicated by the fact that the disc is
at low temperature at low luminosities, $\sim 0.3$ keV for a $10M_{\odot}$
black hole at $0.02L_{Edd}$, even if the disc extends down to $6R_{g}$ (where
$R_{g}=GM/c^{2}$). Such low energy X-rays cannot be seen with the 3 keV
bandpass limit of _RXTE_ , the satellite which has accumulated the most BHB
data to date (Done & Gierliński 2003; Dunn et al. 2009). CCD observations can
extend the bandpass down to lower energies, but these are still not
straightforward to interpret as the disc is not the dominant spectral
component in this state. Instead, the hard X-ray emission dominates the
spectrum, so irradiation can change the disc temperature (Gierliński, Done &
Page 2008, hereafter GDP08), while the disc luminosity can be underestimated
due to photons which are Compton upscattered into the hard X-ray spectrum
(Makishima et al., 2008). This component is still not easy to study even with
CCD detectors since low energy X-rays are absorbed by interstellar gas. Most
BHB are in the galactic plane so have gas columns of $N_{H}\geq 10^{22}$ cm-2
which effectively block emission below 1 keV.
However, there are a few black holes which have intrinsically much lower
columns which have been observed with CCD detectors, namely XTE J1118+480:
$N_{H}\sim 1.1\times 10^{20}$ cm-2, XTE J1817-330: $1.1\times 10^{21}$ cm-2
and SWIFT1753.5-0127: $2-3\times 10^{21}$ cm-2 (Cabanac et al., 2009). XTE
J1118+480 has only snapshot spectra available (Hynes et al. 2000; Esin et al.
2001; Frontera et al. 2001; 2003), but both SWIFT1753.5-0127 and XTE J1817-330
were well sampled throughout their outbursts by the _Swift_ satellite.
However, XTE J1817-330 is a long period system, so spends most of its time in
the disc dominated state, with only two, rather faint, LHS spectra at the end
of its outburst. Thus the short period LHS outburst of SWIFT1753.5-0127 is the
best candidate to study the disc evolution in the LHS. Here we analyse these
data with sophisticated models to describe the behaviour of the X-ray
irradiated disc (GDP08; Gierliński, Done & Page 2009, hereafter GDP09) to
constrain its inner radius as the outburst declines.
## 2 Data reduction
### 2.1 _Swift_
Figure 1: The top panel shows the evolution of the observed flux in the _Swift_ XRT band, while the lower panel shows the simultaneous UVOT data (in UVW1, UVW2 and B filters, where available). Figure 2: The ratio of the X-ray data (red:XRT, black:PCA, green:HEXTE) to an absorbed power law model fit to the 2-4 and 7-10 keV ’continuum dominated’ bands for spectra taken from the peak (10) through to the end of the decline (30) as marked by arrows on Fig 1. At the peak is clear that the continuum spectrum strongly declines above 70 keV, indicating that the electron temperature of the thermal Comptonisation is low enough to be seen. There is also a strong additional soft component at low energies, most probably from the accretion disc, together with a small features around the iron line and edge, indicating reflection of the continuum from the accretion disc. As the outburst declines, the electron temperature increases, so the high energy break can no longer be seen, and the soft X-ray emission drops dramatically in strength. Table 1: The table details the datasets we use from _Swift_ and _RXTE_. We refer to the combined spectrum by the last two numbers of its _Swift_ obsID. The remaining columns give the QPO frequency from fitting the _RXTE_ power spectrum, followed by the results for fitting the 0.5-200 keV X-ray data with a wabs*(diskbb+thCompml) model. We fix the absorption to $0.2\times 10^{22}$ cm-2, and assume reflection is from neutral, solar abundance material inclined at $60^{\circ}$. We tie the seed photons for Comptonisation to the disc temperature in these fits, but then also show the difference in $chi^{2}$ for removing the disc and fixing the seed photon energy for the Compton scattering to 0.1 eV Number | _Swift_ | _RXTE_ | QPO (Hz) | $kT_{disc}(keV)$ | $N_{dbb}$ | $\Gamma$ | $\Omega/2\pi$ | $N_{pl}$ | $\chi^{2}/\nu$ | $\bigtriangleup\chi^{2}$
---|---|---|---|---|---|---|---|---|---|---
00 | 00143778000 | 91094-01-01-00 | $0.64\pm 0.01$ | $0.22\pm 0.01$ | 47557 | $1.80\pm 0.02$ | $0.34^{+0.10}_{-0.13}$ | 0.58 | 712/713 | 699
03 | 00030090003 | 91094-01-01-04 | $0.83\pm 0.01$ | $0.25^{+0.00}_{-0.01}$ | 63806 | $1.86^{+0.00}_{-0.02}$ | $0.35^{+0.05}_{-0.08}$ | 1.06 | 544/548 | 1024
04 | 00030090004 | 91423-01-01-04 | $0.91\pm 0.01$ | $0.24^{+0.00}_{-0.01}$ | 66358 | $1.86\pm 0.01$ | $0.27\pm 0.06$ | 1.06 | 870/757 | 1503
07 | 00030090007 | 91094-01-02-01 | $0.75^{+0.01}_{-0.02}$ | $0.24^{+0.01}_{-0.00}$ | 69967 | $1.83^{+0.02}_{-0.01}$ | $0.36^{+0.04}_{-0.08}$ | 1.16 | 702/679 | 1112
06 | 00030090006 | 91094-01-02-01 | “ | $0.24^{+0.00}_{-0.01}$ | 83839 | $1.82^{+0.02}_{-0.01}$ | $0.32^{+0.08}_{-0.07}$ | 1.26 | 868/762 | 1674
10 | 00030090010 | 91094-01-02-00 | $0.72^{+0.00}_{-0.01}$ | $0.25^{+0.00}_{-0.01}$ | 70377 | $1.82^{+0.01}_{-0.02}$ | $0.33^{+0.03}_{-0.04}$ | 1.17 | 752/726 | 2192
08 | 00030090008 | 91094-01-02-00 | “ | $0.28\pm 0.02$ | 36514 | $1.81\pm 0.01$ | $0.28^{+0.05}_{-0.04}$ | 0.86 | 275/273 | 232
09 | 00030090009 | 91094-01-02-00 | “ | $0.24^{+0.01}_{-0.00}$ | 73119 | $1.81^{+0.01}_{-0.01}$ | $0.28\pm 0.05$ | 1.17 | 873/784 | 2955
11 | 00030090011 | 91094-01-02-02 | $0.70\pm 0.01$ | $0.25\pm 0.02$ | 59219 | $1.81\pm 0.01$ | $0.29\pm 0.06$ | 1.02 | 333/320 | 250
12 | 00030090012 | 91094-01-02-02 | “ | $0.23^{+0.01}_{-0.00}$ | 79673 | $1.81\pm 0.01$ | $0.29^{+0.05}_{-0.06}$ | 1.09 | 860/746 | 1898
13 | 00030090013 | 91094-01-02-02 | “ | $0.23^{+0.01}_{-0.00}$ | 75104 | $1.81\pm 0.01$ | $0.30\pm 0.06$ | 1.08 | 622/633 | 1134
15 | 00030090015 | 91094-01-02-03 | $0.63\pm 0.01$ | $0.23^{+0.00}_{-0.01}$ | 69620 | $1.79\pm 0.01$ | $0.31^{+0.07}_{-0.06}$ | 0.99 | 818/743 | 1231
16 | 00030090016 | 91094-01-02-03 | “ | $0.25\pm 0.01$ | 49618 | $1.79\pm 0.01$ | $0.31^{+0.06}_{-0.07}$ | 0.94 | 717/655 | 1030
18 | 00030090018 | 91423-01-03-06 | $0.47\pm 0.01$ | $0.23^{+0.01}_{-0.02}$ | 42447 | $1.73\pm 0.01$ | $0.24\pm 0.06$ | 0.64 | 396/435 | 347
19 | 00030090019 | 91423-01-04-02 | $0.45^{+0.00}_{-0.01}$ | $0.22\pm 0.01$ | 39291 | $1.72\pm 0.01$ | $0.28^{+0.07}_{-0.06}$ | 0.57 | 626/646 | 732
20 | 00030090020 | 91423-01-06-00 | $0.29\pm 0.01$ | $0.23^{+0.01}_{-0.02}$ | 17415 | $1.67^{+0.02}_{-0.01}$ | $0.20^{+0.07}_{-0.09}$ | 0.36 | 433/460 | 178
21 | 00030090021 | 91423-01-08-02 | - | $0.22\pm 0.02$ | 9122 | $1.65\pm 0.03$ | $0.19^{+0.16}_{-0.17}$ | 0.21 | 514/518 | 50
23 | 00030090023 | 91423-01-10-00 | - | $0.17\pm 0.02$ | 14876 | $1.65^{+0.01}_{-0.02}$ | $0.16^{+0.05}_{-0.09}$ | 0.18 | 635/659 | 32
24 | 00030090024 | 91423-01-11-00 | - | $0.16^{+0.02}_{-0.03}$ | 16135 | $1.67^{+0.01}_{-0.02}$ | $0.24^{+0.11}_{-0.13}$ | 0.16 | 602/631 | 19
26 | 00030090026 | 91423-01-13-00 | - | $0.17^{+0.05}_{-0.10}$ | 4629 | $1.63^{+0.02}_{-0.04}$ | $0.13^{+0.14}_{-0.13}$ | 0.11 | 559/562 | 4
30 | 00030090030 | 91423-01-16-01 | - | $0.04^{+0.05}_{-0.03}$ | 934 | $1.64^{+0.01}_{-0.02}$ | $0.09^{+0.14}_{-0.09}$ | 0.11 | 559/592 | 3
31 | 00030090031 | 91423-01-17-00 | - | $0.13^{+0.04}_{-0.12}$ | 13157 | $1.63^{+0.02}_{-0.03}$ | $0.14^{+0.16}_{-0.14}$ | 0.10 | 576/508 | 3
We use publicly available _Swift_ data from SWIFT J1753.5-0127 taken during
the period from July 2005 to July 2007. We extracted both X-ray telescope
(XRT) and UVOT data. All of our XRT data was in Windowed Timing mode, and we
extracted source counts using a circle with radius of 20 pixels. Background
was taken from an off source region using a circle of the same size. However,
the source was very bright during the peak of the outburst, and some of the
datasets were piled up as the count rate was above 100 counts $s^{-1}$. We
determined the size of the central piled up region by excluding progressively
larger radii regions until the spectra stopped softening at high energies.
This gave an exclusion region of 3 pixels for brightest observations, and 2
pixels for more moderately piled up data. The data was grouped to 20 counts
$s^{-1}$, and fit between 0.5-10 keV.
The evolution of the observed (i.e. not corrected for interstellar absorption)
flux in the X-ray (upper) and UV (lower) bands is plotted in Fig. 1. We derive
the flux by integrating the absorbed model fit over 0.5-10 keV for the X-rays.
For UV data, we find energy band for each filter and define the effective
bandpass to be between the energies determined by FWHM ($5.6\times 10^{-3}$ to
$7.45\times 10^{-3}$ keV for UVW2 filter, $4.2\times 10^{-3}$ to $5.45\times
10^{-3}$ keV for UVW1 filter, and $2.55\times 10^{-3}$ to $3.2\times 10^{-3}$
keV for B filter).
The observations fall into 3 clearly distinct time segments. The first covers
the outburst rise and decline, after which the flux remains fairly constant.
Hence in this paper we concentrate on this first data group, in order to track
the evolution of the disc during the flux decline.
### 2.2 _RXTE_
The _RXTE_ satellite also followed the 2005 outburst. We use the standard
extraction techniques with the bright source background to derive PCA spectra
from all layers of detector 2, adding 1 percent systematic error and use these
data from 3-16 keV. We also extract Standard 1 power spectra over the full
2-60 keV bandpass for intervals of 256 s, and fit these with multiple
Lorentzian components in order to determine the QPO frequency. These are given
in Table 1.
We extract HEXTE data from cluster 0 and use these over the energy range
25-200 keV. We select the subset of _Swift_ and _RXTE_ data which are taken
within one day of each other. This gives a total of 27 observations which have
quasi-simultaneous coverage of the optical/UV/X-ray/hard X-ray spectrum. Table
1 details these composite spectra.
## 3 Disc plus Comptonisation plus reflection
### 3.1 X-ray data
Figure 3: The plot shows the results of the disc temperature, inner radius and
reflection evolution obtained from the diskbb+thCompml model. The value of
inner radius were derived from the diskbb normalisation assuming an
inclination of $60^{\circ}$ and distance of 5.4kpc. As these are poorly known,
the absolute value of the radius is not well constrained. However, trends
should be robust, and clearly the derived inner radius remains constant or
mildly decreases during the outburst. This is in sharp contrast to the
expected increase from truncated disc models. Reflection (third penal) and the
QPO frequency (lower panel, see also Table 1), on the other hand, decreases
when leaving the peak, as expected if the disc is receding.
We first concentrate on the X-ray data alone, and show the different
components present in the spectrum. We fit a power law to the ’continuum
dominated’ bands at 2–4 keV and 7–10 keV, and then show a ratio of the data to
this model over the entire bandpass (assuming absorption fixed at the galactic
column of $N_{H}=0.2\times 10^{22}$ cm-2). Fig. 2 shows this at different
points along the outburst (marked by arrows in Fig. 1), from the peak to the
end of the decline. Close to the peak, the continuum clearly has a rollover at
high energies, so is most likely due to thermal Comptonisation. There is also
clearly an additional soft X-ray component, which decreases in strength as the
outburst declines. However, there are also more subtle features seen in the
high signal-to-noise PCA data, with a small excess around the iron line,
followed by a slight dip and subsequent rise to 20 keV. This is characteristic
of reflection
Thus we use a model including a disc to produce the soft X-ray component,
thermal Comptonisation of these disc seed photons to produce the hard
continuum, and reflection of the hard continuum from the surface of the disc.
We use the thcomp model to describe the Comptonisation, developed and tested
by Zdziarski, Poutanen & Johnson (2000), extended to include its reflected
continuum and self-consistent line emission by Życki, Done & Smith (1999). We
assume that the reflecting material is neutral, with solar abundances and
inclined at $60^{\circ}$. We fix the galactic absorption column at
$N_{H}=0.2\times 10^{22}$ cm-2 corresponding to an $E(B-V)=0.34$ (Cadolle Bel
et al., 2007). The results for each spectrum are detailed in Table 1. We
assess the significance of the detected disc emission by removing the disc
from the model, and fixing the seed photon temperature instead at 0.1 eV. This
results in a loss of 2 degrees of freedom, so the disc is only significant at
greater than 99 per cent confidence for $\Delta\chi^{2}>4.61$. Thus all the
spectra apart from the final 3 (26, 30 and 31) strongly require the presence
of an additional soft component. We similarly assess the significance of
reflection (driven mostly by the presence of the iron line) and find that it
is required in all the datasets apart from the final 3.
Fig. 3 shows the derived disc temperature and radius (see Table 1). The disc
radius _decreases_ as the outburst progresses (see also Cabanac et al. 2009),
in apparent conflict with the truncated disc model where the inner disc should
progressively recede during the LHS decline. Yet the amount of reflection
decreases, as expected if the disc is receding. Additionally, the low
frequency QPO seen in the variability power spectrum drops dramatically, again
as expected if the truncation radius sets the QPO frequency and is increasing
(DKG07). Thus while the behaviour of the reflected emission and QPO fit the
truncated disc models, the behaviour of the soft X-ray component does not.
### 3.2 Optical data
We show these X-ray model fits extrapolated down to the optical/UV data and
corrected for the effects of interstellar absorption. Fig. 4a shows this for
the outburst peak (10). This clearly shows that extrapolating the hard X-ray
power law spectrum down in energy overpredicts the optical/UV data. This
implies that the hard X-ray spectrum must break at UV/soft X-ray energies,
supporting our modeling of it as due to Comptonisation of seed photons from
the accretion disc. However, extrapolating the soft X-ray disc emission
underpredicts this emission, making it clear that intrinsic gravitational
energy release in the disc is insufficient to produce the observed optical/UV
data.
In general, the optical can include additional contributions from reprocessed
emission from hard X-ray illumination of the outer disc, as well as
contributions from the jet and the companion star(e.g. Russell et al. 2006).
However, SWIFT J1753.5-0127 is a short period, low mass X-ray binary, so the
companion star must faint. Instead we try to quantify the jet contribution
from observed radio flux of $\sim 2.1$ Jy at 1.7 GHz (Fender et al. 2005. We
show the extrapolation of this up through our optical/UV/X-ray bandpass
assuming a flat spectrum (black line on Fig. 4a). This is a factor $\sim 3$
below the observed optical flux, but this is the maximum possible jet
contribution as its spectrum should break where it becomes optically thin. It
seems most likely that this break is at IR frequencies (Markoff et al. 2001;
Gallo et al. 2007) so the jet is also probably negligible in the optical/UV
for these data. Thus X-ray reprocessing from hard X-ray illumination of the
outer disc seems the most likely origin for the optical/UV flux (van Paradijs,
1996).
Figure 4: The unabsorbed data and X-ray model fit (diskbb + thCompml) at the
outburst peak (top panel, spectrum 10), midway down the decline (middle panel,
spectrum 20) and in the outburst tail (lower panel, spectrum 30). The magenta
and blue components are disc emission and Comptonisation in corona,
respectively. Figure 5: The red squares show the evolution of UV excess i.e.
the observed UVW1 flux divided by the UVW1 flux predicted by the model fit to
the soft X-ray disc. This always underpredicts the observed UVW1 emission, by
an amount that increases as the source fades. The soft X-ray disc component is
not significant in the final 3 points but the trend is already clear even
without these data. The blue triangles show the ’UV deficit’ i.e. the ratio of
model to observed flux in the UVW1 filter predicted from extrapolating the
thCompml continuum down into the optical assuming the seed photons are at 0.1
eV rather than from the disc. Close to the peak of the outburst this UV
deficit is large, showing that the X-rays over-predict the UV flux.
Conversely, towards the end of the outburst, the UV data lie very close to the
extrapolation of the hard X-ray flux.
There are additional radio observations from midway down the outburst decline
(Cadolle Bel et al., 2007). Fig. 4b shows this radio emission, extrapolated as
above, onto spectrum 20, the dataset closest in time to the radio
observations. Again, the optical/UV points are not fit by either the hard
X-ray power law extrapolated down (though this time the mismatch is not so
large), or by the radio emission extrapolated up, or by the disc inferred from
the soft X-ray component, as also shown in Cadolle Bel et al. (2007).
Fig. 4c shows a spectrum from the end of the outburst (30), where there are no
radio observations. The disc emission fit to the soft X-ray component is now
very weak and cool, and its extrapolation down to the optical lies
dramatically below the observed optical/UV emission. However, the optical/UV
emission does lie remarkably close to the extrapolated hard X-ray flux in
these low-luminosity data.
We quantify this by calculating the factor by which the disc component
underpredicts the observed flux in the UVW1 filter (’UV excess’: GDP09) in
each spectrum. We also calculate a ’UV deficit’, which is the factor by which
the Comptonisation model overpredicts the flux in the UVW1 filter if its seed
photons put at 0.1 eV rather than fixed to the disc temperature. Such low
energy seed photons may be produced by cyclo-synchrotron emission by the same
thermal electrons which produce the Compton scattered emission interacting
with the tangled magnetic field in the hot flow itself (Narayan & Yi 1995; Di
Matteo, Celotti & Fabian 1997; Wardziński & Zdziarski 2000). Fig. 5 shows how
these evolutions during the outburst. This confirms the results seen in the
three individual spectra discussed above. The ’UV excess’ (red triangles)
increases strongly with time, as the soft X-ray disc component makes less and
less contribution to the UV flux. The disc is not significantly detected in
the final three spectra, but the trend in UV excess is already clearly
apparent. Conversely, the UV deficit (blue squares) decreases, and is roughly
consistent with the observed UV flux from spectrum 21 onwards.
## 4 Irradiated Disc Model
We first look at the possibility that irradiation of the disc is responsible
for all the issues highlighted in the previous section. Irradiation of the
inner disc changes its derived temperature and radius compared to that derived
from fitting purely gravitational energy release models of the disc (GDP08).
Similarly, irradiation of the outer disc can also increase the optical/UV
emission from that expected from an unilluminated disc (van Paradijs
1996;GDP09)
We use a slightly modified version of the diskir model of GDP08 so that the
disc temperature and normalisation are set by the physical variables of the
mass, distance, mass accretion rate and inner and outer radii. This has the
advantage that the outer radius is set in terms of physical units, rather than
relative to the (possibly changing) inner radius as in diskir. We parameterise
the inner and outer radii in terms of $R_{g}=GM/c^{2}$, and leave the inner
disc radius, $R_{d}$ as a free parameter in the fits. We fix the mass and
distance at the best (though poorly constrained) estimates of $12~{}M_{\odot}$
and 5.4 kpc, respectively (Zurita et al., 2008).
This model uses the diskbb parameterisation of disc temperature with radius
i.e. $T\propto r^{-3/4}$ (Mitsuda et al., 1984), so does not include a stress-
free inner boundary condition or colour temperature correction. If the disc is
truncated then the lack of stress-free inner boundary condition is probably
more appropriate, but the colour temperature correction, $f_{col}$, is still
an issue. The derived value of the inner radius is then underestimated by a
factor $f^{2}_{col}\sim 2.9$ for $f_{col}\sim 1.7$ (Kubota, Makishima &
Ebisawa, 2001). However, the main aim of our spectral fitting is to track
changes in the value of the inner disc radius, as its absolute value depends
on the poorly known system parameters. The key aspect of the truncated disc
model which we are testing is the prediction that the truncation radius
increases as the spectrum hardens during the decline. Thus we are focussing on
the relative values of the inner disc radius during the decline.
The outer radius should remain constant, as should the absorption (but see
Cabanac et al. 2009) so we constrain these by simultaneously fitting three
datasets from various points in the outburst with the modified diskir model,
together with X-ray absorption parameterised by wabs and UV reddening by
redden. Assuming standard gas to dust ratios allows us to tie
$E(B-V)=1.5\times N_{H}/10^{22}$ (GDP08). This gives a best fit at
$N_{H}=2.1\times 10^{21}cm^{-2}$ and $R_{out}=10^{5.25}R_{g}$, which are fixed
to these values in all fits hereafter.
### 4.1 Evolution of the irradiated disc model parameters
We fit this model to all the _Swift-RXTE_ datasets. Guided by previous
low/hard state data (Poutanen, Krolik & Ryde 1997; GDP08), we fix the radius
of the irradiated portion of the inner disc at $1.1\times R_{d}$. Fig. 6 shows
the evolution of the source parameters in this model. The top panel gives the
bolometric flux, which varies by an order of magnitude. This is dominated by
the hard X-ray component, whose spectral photon index (second panel) decreases
from 1.8 to 1.6 during the first section of the decline, and then stabilizes
at this value hereafter. The source did not make a transition to the high/soft
or even intermediate state, but remained in the hard state throughout the
outburst.
Figure 6: Evolution of an irradiated disc model fit to the optical/UV/X-ray
spectrum. The top panel shows the bolometric flux, taken from the unabsorbed
model integrated between 0.001-100 keV. This drops by a factor of 10 during
the outburst, close to that seen from the X-ray emission alone in Fig. 1. The
second panel shows the hard X-ray spectral index, which hardens from $1.8$ to
$1.6$. The third panel shows the mass accretion rate $\dot{m}_{d}$ required to
power the intrinsic (gravitational) emission from the disc luminosity,
$L_{d}=GM\dot{m}_{d}/2R_{d}$. This drops by nearly two orders of magnitude, so
the ratio of this to hard X-ray luminosity in the corona $L_{c}/L_{d}$
increases by a factor 10 (fourth panel). The intrinsic disc luminosity is
enhanced by irradiation of the inner disc by the hard X-ray corona. The
fraction of irradiation required drops during the decline (fifth panel) as
does the amount of (neutral) reflected emission (sixth panel). The seventh
panel shows the fraction of the bolometric flux which is required to
thermalise in the outer disc in order to make the observed optical/UV flux.
This strongly increases during the decline (see also Fig. 4). The final panel
shows the inferred inner radius of the disc. This remains remarkably constant
in this model at around $10~{}R_{g}$ (see also Fig. 2), corresponding to $\sim
30~{}R_{g}$ after a colour temperature correction. This lack of change in the
inner disc radius is in conflict with the truncated disc model predictions
that the change in spectral hardness (see second panel) is driven by changes
in the solid angle subtended by the disc to the hard X-ray source. If instead
there is a separate soft X-ray component (see Fig. 8) as required in the
similarly dim low/hard state spectra from XTE J1118+480 then these data do not
constrain the disc radius.
Figure 7: The same 3 spectra (10, 20 and 30) as in Fig. 3, fit with the
irradiated disc model and corrected (data and model) for absorption. The
magenta dashed line is the intrinsic disc emission, the red line includes
irradiation of the outer disc, and the cyan line shows the additional flux
from irradiation of the inner disc. The Comptonised emission and its reflected
spectrum are shown in blue. While the inner disc irradiation, $f{in}$, drops,
the increase in $L_{c}/L_{d}$ means that the total disc spectrum is more
distorted by irradiation at lower fluxes. Figure 8: Irradiated disc model
mass accretion rate through the disc, $\dot{m}_{d}$ versus that through the
corona, $\dot{m}_{c}$, derived from $(L_{c}/L_{d})(R_{d}/R_{c})\dot{m}_{d}$,
with $R_{c}=3.5~{}R_{g}$ fixed by requiring $\dot{m}_{c}\approx\dot{m}_{d}$
(shown by the red dashed line) for the brightest spectra. The red points show
the 3 faintest spectra where there is no significant additional soft X-ray
component. Even without these spectra the trend is clear that $\dot{m}_{c}$
gets progressively larger than $\dot{m}_{d}$ as the source declines. This is
opposite to the expected trend from a radiatively inefficient flow. are away
from the line.
The disc spectrum forms a component which connects the soft X-ray rise to the
optical emission in this model. Part of this is powered by gravitational
energy release, and the inferred mass accretion rate through the disc
decreases by more than 2 orders of magnitude during the decline (third panel).
Thus the ratio of the hard X-ray flux to this inferred intrinsic disc flux
increases by an order of magnitude (fourth panel).
The intrinsic gravitational energy of the disc is augmented by thermalisation
of the irradiating flux on the inner disc. The fraction of the hard X-ray
spectrum which illuminates the inner disc and is thermalised basically
decreases during the decline (fifth panel), as expected if the inner edge of
the disc is receding. Similarly, the amount of hard X-ray reflection also
declines (sixth panel). Conversely, the fraction of hard X-rays which
thermalise in the outer disc increases by almost an order of magnitude
(seventh panel). The final panel shows the disc inner radius. This is plainly
consistent with a more or less constant value, at $\sim 10~{}R_{g}$,
(corresponding to $\sim 30~{}R_{g}$ after a colour temperature correction)
although the error bars become large at later stages of the decline. Fig. 7a-c
show the corresponding fits to the data of Figs. 4a-c for this model.
### 4.2 Inferred mass accretion rate
We can use these derived parameters to explore how the mass accretion rate
through the disc compares to that required to power the hard X-rays. We
calculate the mass accretion rate of the corona by:
$L_{c}=\frac{\eta_{corona}}{\eta_{disc}}\frac{GM\dot{m}_{c}}{2R_{c}},$
where $\eta_{corona}/\eta_{disc}$ is the relative efficiency of converting
mass to radiation in a coronal flow which extends down to $R_{c}$ versus that
in a disc extending down to the same radius. However, the disc itself only
extends down to a radius $R_{d}$ so its luminosity from gravitational energy
release alone is $L_{d}=\frac{1}{2}GM\dot{m}_{d}/R_{d}$. Thus
$\frac{L_{c}}{L_{d}}=\frac{\eta_{corona}}{\eta_{disc}}\frac{\dot{m}_{c}}{\dot{m}_{d}}\frac{R_{d}}{R_{c}}.$
The spectral fitting parameters give $L_{c}/L_{d}$, $\dot{m}_{d}$ and $R_{d}$.
Close to the peak of the outburst, even a radiatively inefficient flow should
have $\eta_{corona}/\eta_{disc}\sim 1$. Here we also expect
$\dot{m}_{c}\sim\dot{m}_{d}$, which requires $R_{c}$ to be around 3 times
smaller than $R_{d}$ at this point. We can then calculate $\dot{m}_{c}$
assuming $\eta_{corona}/\eta_{disc}$ and $R_{c}$ remain constant. This is
plotted against $\dot{m}_{d}$ in Fig. 8. The red line is
$\dot{m}_{c}=\dot{m}_{d}$, showing clearly that these models favour a larger
mass accretion rate through the corona than through the disc on the decline.
This conclusion is strengthened if the flow is increasingly radiatively
inefficient at lower mass accretion rates, as the observed X-ray flux would
then require an even larger coronal mass accretion rate to power the same
amount of hard X-ray emission.
This conclusion is driven by the observational requirement for a soft X-ray
component in the late state decline data. This is weak, but significantly
present in all but the last three datasets (assuming that the absorption
column remains constant: Cabanac et al. 2009). It combines a low luminosity
with a fairly high (soft X-ray) temperature, which leads to the derived small
radius and low mass accretion rate. This low mass accretion rate is then much
smaller than that required to power the observed hard X-ray emission.
We illustrate this by re-coding the irradiated disc model to force
$\dot{m}_{c}=\dot{m}_{d}$ for $\eta_{corona}/\eta_{disc}=1$. This forces the
disc to have a higher mass accretion rate, so it must truncate at a larger
radius so as not to overproduce the observed soft X-ray component. Irradiation
of the inner disc should then be negligible, so we fix $f_{in}=0$ for physical
consistency, and we focus first on the X-ray data alone so we also fix
$f_{out}=0$.
We fit this to spectrum 24 (the lowest luminosity data for which the disc is
significantly detected in the soft X-ray flux), and first focus on the X-ray
data alone. The mass accretion rate through the disc is much higher ($1.2\pm
0.1\times 10^{17}$ g/s compared to $1.0\times 10^{16}$ in the diskir fits) and
the disc inner radius increases to $30~{}R_{g}$ from $\sim 5~{}R_{g}$ (i.e.
$90~{}R_{g}$ and $15~{}R_{g}$ after a colour temperature correction). This
disc contributes to the spectrum only at the softest energies of the _Swift_
XRT, and its sharp rise is rather different to the more gradual curvature seen
in the soft X-ray emission. Thus this gives a significantly worse fit
($\chi^{2}_{\nu}=626/632$ versus $600/630$, where the two extra free
parameters are $L_{c}/L_{d}$ and $f_{in}$). We can only recover the same
quality of fit with $\dot{m}_{d}=\dot{m}_{c}$ by including an additional soft
component. This allows the disc to recede back even further, to $\sim
50~{}R_{g}$ (i.e. $\sim 150~{}R_{g}$), so that the truncated disc makes no
contribution to the soft X-ray flux. Fig. 9 shows this fit including now the
optical/UV data with $f_{out}=2\times 10^{-3}$ (similar to that derived from
the original fits, see Fig. 6).
Irradiation can make such a hot, weak component if the irradiated disc area is
small. We fixed the radius of the reprocessing region at $1.1\times R_{d}$ for
the fits in Fig. 6, but as the disc recedes then the changing geometry means
that this should also drop. We allow this to be a free parameter with
$\dot{m}_{c}=\dot{m}_{d}$ and can recover as good a fit as before
($\chi^{2}=601/632$) with $\sim 1$ per cent of the bolometric flux being
reprocessed in a region with $R_{irr}=1.002R_{d}$. However, illumination of
such a tiny area of the disc surface seems unreasonable from illumination by a
central source.
Figure 9: Spectrum 24 fit by a model where $\dot{m}_{d}=\dot{m}_{c}$ and a
separate component (cyan, modelled with diskbb) to fit the soft X-ray
emission. The observed strong X-ray flux requires a large mass accretion rate
through the disc, so this has to truncate at $\sim 50~{}R_{g}$, forming the UV
peak, so as not to overproduce the soft X-ray emission.
## 5 Origin of Soft X-ray Component
### 5.1 Outburst Peak
The soft X-ray component seen at the peak of the hard state during the
outburst of SWIFT J1753.5-0127 is clearly from the disc. While the system
parameters are rather poorly known, the radii derived close to the peak are
$\sim 15~{}R_{g}$, corresponding to $\sim 45~{}R_{g}$ after a colour
temperature correction. This is consistent with the disc being recessed back
from the last stable orbit, as required for the truncated disc/hot inner flow
interpretation of the hard state. There are two independent lines of support
for this number. The first comes from the fact that the hard X-ray spectrum is
softer close to the peak, clearly consistent with the disc providing an
increasing source of seed photons to Compton cool the hot X-ray plasma. Since
most of the gravitational potential energy to power this corona is
concentrated within $20R_{g}$ then it seems most likely that the disc extends
down to similarly small radii. This means that there can also be irradiation
of the inner disc by the hard X-ray emission, leading to reflection and
thermalisation of the incident flux. We see evidence for both these processes
in the spectrum, with the extra flux from irradiation increasing the inferred
disc radius from the value of $\sim 11~{}R_{g}$ inferred from the simple
(unirradiated) disc fits to the X-ray spectrum, to $15~{}R_{g}$ (i,e, from
$30$ to $45R_{g}$ after colour temperature correction). The second is from low
frequency QPO which is at its maximum of $\sim 0.8$ Hz in these data. This
implies an outer radius of $\sim 20-30~{}R_{g}$ assuming that the QPO is
produced by Lense-Thirring (vertical) precession of the hot flow around a
$10~{}M_{\odot}$ black hole (Ingram, Done & Fragile, 2009).
The mass accretion rate required to power the disc emission is comparable with
that required to power the coronal hard X-ray emission assuming that the
corona extends down to a radius which is $\sim 5$ times smaller than that of
the disc i.e. $3.5R_{g}$ for the parameters assumed here, i.e. $\sim
9~{}R_{g}$ with a colour temperature correction. We stress again that these
system parameters are poorly known, thought this appears quite reasonable.
Much smaller radii could be potentially feasible for a hot inner flow,
irrespective of the spin of the black hole, as numerical simulations of the
MRI turbulence show that the large scale height magnetic fields can extract
energy from the infalling material beyond the last stable orbit (e.g. Krolik,
Hawley & Hirose 2005).
### 5.2 Late Stage Decline
The outburst peak spectra then form a template for comparison with the later
stages of the decline, where the bolometric flux is lower by a factor 10. The
truncated disc/hot inner flow model makes a clear prediction that these should
have a larger inner disc radius, decreasing the importance of illumination on
the inner edge of the disc, and decreasing the amount of associated
reflection. Thus illumination of the surface of the disc is not expected to be
important in distorting the derived inner disc radii in these spectra.
Similarly, as there is little (or no) overlap in radii between the hot inner
flow and disc, so the disc emission cannot be strongly suppressed by
Comptonisation as can be the case close to the transition (Kubota & Done 2004;
Makishima et al. 2008). Thus any disc component seen in these spectra should
give a fairly unbiased view of the inner radius of the flow. Yet associating
the observed soft X-ray component with this disc gives derived radii which are
as small, if not smaller than, the radii seen at the outburst peak, in clear
conflict with the truncated disc models (see Figs 4 and 6).
However, these radii are themselves in more subtle conflict with the
observations. The small disc has very small luminosity compared to the coronal
emission. This requires that the coronal mass accretion rate must be large
compared to that through the disc if the corona is powered by matter accreting
through it. Thus the corona cannot be predominantly fed by material
evaporating from the inner edge of the disc, but instead requires a completely
separate coronal flow which incorporates most of the incoming mass accretion
from the companion star. Yet it seems quite unlikely that the incoming cool
Roche lobe overflow stream would be able to form such a coronal flow at large
radii.
Instead it seems more feasible that the corona is not powered by its own mass
accretion supply, as in the truncated disc models, but is instead powered by
mass accreted through the disc, but whose energy is released in the corona
(e.g. Svensson & Zdziarski 1994). However, this model itself runs into
difficulties since the simplest idea would be for magnetic buoyancy to
transport the energy vertically. The energy is then released in a corona above
the disc, so the corona is co-spatial with it and illuminates it. This gives
rise to a reprocessed luminosity $L_{rep}=\frac{1}{2}(1-a)L_{c}$ (where $a$ is
the albedo, and the factor $\frac{1}{2}$ assumes the corona emits
isotropically) adds to the intrinsic disc luminosity (Haardt & Marashi, 1993).
For hard spectra, such as those observed here, the reflection albedo $a<0.3$,
hence the disc flux should be at least $L_{c}/3$. Yet we observe a disc flux
of $\leq L_{c}/20$ towards the end of the outburst. Thus the only way to
circumvent the reprocessing limits whilst having all the mass accrete via the
disc is if the energy is advected radially as well as vertically. Then it can
be released in a more centrally concentrated region, on size scales smaller
than the (small!) disc inner radius. The alternative of having the hard X-rays
be strongly beamed away from the disc, is ruled out by the very similar dim
low/hard state spectra seen in XTE J1118+480 (Frontera et al. 2001; 2003;
Reis, Miller & Fabian 2009), which has high inclination ($\geq 70^{\circ}$) so
that we must see a similar hard X-ray flux to that of the disc.
A potentially less arbitrary solution than using magnetic fields to transport
the energy in a dissipationless fashion is if the observed soft X-ray
component in the late stages of the decline is not from the disc. The spectra
clearly show that the observed soft X-ray emission cannot produce the
optical/UV emission without a large change in the reprocessed fraction in the
outer disc from that seen during the outburst peak. Yet the outburst remained
in the low/hard state, so there is no expected change in source geometry, so
the fraction of the total flux which illuminates the outer disc should not
change dramatically. Similarly, there is no large change in spectral shape
which might produce such a large difference in thermalisation fraction. Yet
the observed soft X-ray component is a factor 10 further below the UV emission
than at the outburst peak (Fig. 5). This could be used to argue that the real
disc is truncated, as in Fig. 7, so is larger and cooler, so that the same
amount of reprocessing can produce the optical/UV.
However, it is clear that the optical/UV emission does change character during
the decline. The rapid variability of the optical flux changes dramatically
during the decline, switching to an anti-correlation of the optical and X-ray
emission (Hynes et al., 2009). Clearly this shows that the optical is no
longer made predominantly by reprocessing of the hard X-ray flux, so it gives
only an upper limit to the irradiated disc flux.
Thus the optical emission does not trace the outer disc, so cannot constrain
our inner disc models. Instead, we use the similarly dim low/hard state
spectra from XTE J1118+480 to argue for a non-disc origin for the soft X-ray
component. This object has an absorption column which is an order of magnitude
lower than that to SWIFT J1753.5-0127. This gives a correspondingly more
sensitive view of the UV and soft X-ray emission, where it is clear that there
is a large, cool disc seen in the UV and EUV bandpass (Esin et al. 2001;
McClintock et al. 2001) which is completely inconsistent in luminosity and
temperature with the much weaker, higher temperature ’disc’ emission seen in
soft X-rays (Frontera et al. 2001; 2003; Reis, Miller & Fabian 2009). There
are plainly two components in this source, one which is consistent with a
truncated disc, making no impact on the soft X-ray emission, and another
component which produces weak soft X-ray flux. This is very similar to the
spectrum of 24 as shown in Fig. 9.
Thus for the late stage decline, our modeling with an irradiated disc is not
supported by the data. The cross-correlation of the rapid variability clearly
shows that the optical/UV cannot be made by reprocessing in the outer disc,
while XTE J1118+480 clearly shows that the soft X-rays form a separate
component to the observed UV/EUV emission from the (truncated!) disc. What
then produces the observed soft X-ray component, and what produces the
observed optical/UV emission ?
Figure 10: Two potential geometries for a soft X-ray component (highlighted in
green) from a small area, with short timescale variability dominated by
reprocessing of the hard X-rays, and additional variability at longer
timescales. With irradiation of a residual inner disc, the small area implies
a small radial extent. Conversely, with irradiation of the inner rim of a
large radius truncated disc, a small area implies a small vertical extent.
Additional long timescale variability can be generated at the disc outer or
inner edge, respectively, by clumping or turbulence.
## 6 Origin of the soft X-ray component during the decline
SWIFT J1753.5-0127 was observed by _XMM-Newton_ in March 2006 i.e. after the
end of the first intensive SWIFT campaign covering the outburst decline. These
data are analysed by Wilkinson & Uttley (2009), and have much better signal-
to-noise than those of the late decline _SWIFT_ spectra, but are very similar
in shape and intensity. The higher statistics in these data mean that the soft
component is unambiguously detected, at very high significance. The same
spectra model as used in Table 1 (diskbb+thcomp with absorption fixed at
$0.2\times 10^{22}$ cm-2)gives $kT_{disc}=0.25\pm 0.01$ keV and norm of
$1080\pm 100$.
The timing analysis of Wilkinson & Uttley (2009) showed that this soft X-ray
component variability strongly correlates with the hard X-ray variability on
short timescales i.e. is driven by reprocessing. However, on longer
timescales, there is additional variability in the soft X-ray flux, implying
that there are intrinsic fluctuations in the disc emission as well. The result
that much of the short term variability correlates with the hard X-rays rules
out a completely separate soft X-ray component such as the jet, while the
additional soft X-ray variability shows that the component is truly separate
as opposed to being produced from continuum curvature of the X-ray emission or
reflection (suggested by Hiemstra et al. 2009 as a potential origin for the
soft X-rays). These variability constraints mean that there are only two
potential origins for the soft X-ray component which arises from a small area
with relatively high temperature. Firstly, this could be emission from the
surface of a residual inner disc, forming a small ring from the last stable
orbit to some (small) outer radius. Secondly, this could be the inner face of
a truncated disc. We sketch these two possibilities in Fig. 10, and outline
them below.
### 6.1 Irradiation of a residual inner disc
Evaporation of the disc by thermal conduction is a plausible mechanism to form
a truncated disc/ hot inner flow geometry (e.g. Liu et al. 1999).
Counterintuitively, this is not most efficient at the smallest radii, as these
also have the highest coronal densities, so have the highest condensation
rates. Evaporation first erodes a gap in the disc, leaving a residual inner
disc. This gap expands radially, eventually giving a fully truncated disc if
the mass accretion rate is low enough(Mayer & Pringle, 2007). However, close
to the transition, the residual disc can remain. This geometry allows the
outer disc to carry all the mass accretion rate. This disc evaporates into a
corona, truncating it at some large radius into a coronal flow which carries
all the mass accretion rate. However, as the flow accretes to smaller radii,
the increasing density gives an increasing condensation rate and some small
fraction of the material can condense out of the hot flow. This forms a cool
ring at the innermost radii, with a mass accretion rate which is only a small
fraction of the total mass accretion rate, with the rest of the material
accreting via the coronal flow(Liu & Meyer-Hofmeister, 2001).
Reprocessing is still an issue, but if the ring is small, extending over a
very narrow range of radii, then it does not subtend a large solid angle to
the coronal X-ray emission, so could potentially produce the large observed
ratio of coronal to (inner) disc luminosity. Some reprocessing is required in
order to produce the short term variability, but there could also be intrinsic
variability in the extent of this residual disc.
### 6.2 Irradiation of the inner face of a truncated disc
The temperature of the irradiated region is required to be substantially
hotter than that of the disc itself, requiring a very small reprocessing area.
This seems very unlikely for a central source illuminating the top/bottom
surface of thin disc. However, the disc has some (small) half thickness,
$H_{d}$, so its inner rim forms a distinct, small area $\approx 4\pi
R_{d}H_{d}$ and subtends a solid angle $\approx H_{d}/R_{d}\times 4\pi$ to the
central source. The small reprocessing area found in section 4.2 above
corresponds to $H_{d}/R_{d}\sim 0.004$, predicting
$f_{in}=(1-a)H_{d}/R_{d}\approx 0.002$, where $a$ is the the reflection
albedo. This is a factor 5 smaller than $f_{in}$ derived from the data, but
potentially feasible given the large uncertainties both on the parameters and
on the modeling.
The irradiation origin then gives directly the rapid variability, while
turbulence caused by clumping instabilities on this edge could give the
required longer term additional variability.
## 7 Origin of Optical/UV emission during the decline
As noted by Motch et al. (1985) in GX339-4, the X-ray emission in the dim
low/hard state can extrapolate back quite accurately to fit the optical
spectrum (see also Corbel & Fender 2002; Nowak 2005). One way to produce this
is via a single synchrotron component from the innermost post-shock region of
the jet. This is self absorbed in the IR, strongly suppressing the emission at
lower frequencies. As the jet stretches out , each part of it produces a
synchrotron component which peaks at a lower frequency than those closer to
the center. All of these make the flat spectrum seen in the radio (e.g.
Markoff & Nowak 2005; Maitra et al. 2009). However, the non-thermal
synchrotron makes the optical and X-ray emission from a single scattering in a
single region, so it is hard to see how this can make the weak anti-
correlation between the optical and X-ray flux with lead of a few hundred
milliseconds which seems typical of this state (Motch et al. 1985; Kanbach et
al. 2001; Gandhi et al. 2008; Durant et al. 2008).
Conversely, the cross-correlation signal can be explained if the optical
emission is from the jet, while the X-ray emission is from the corona (Malzac,
Merloni & Fabian, 2004). However, the close match of the optical and X-ray
spectra is then very unexpected if these are really from different components.
Instead, if the spectrum is formed from thermal Comptonisation from self-
produced cyclo-synchrotron photons in the hot flow then there can be complex
time variability properties imprinted via propagating fluctuations through an
inhomogeneous flow (Kotov, Churazov & Gilfanov 2001; Arévalo & Uttley 2006).
Whether these can indeed explain the anti-correlation between optical and
X-ray by such spectral pivoting (Körding & Falcke, 2004) remains to be seen.
## 8 Conclusions
The combined _Swift_ and _RXTE_ observations from the black hole transient
SWIFT J1753.5-0127 give one of the best datasets to probe the evolution of the
inner edge of the accretion disc in the low/hard state. These instruments
cover the optical/UV and soft/hard X-ray bandpasses, giving a detailed picture
of the spectral evolution during the low/hard state outburst.
We fit these with a sophisticated irradiated disc model and find that this
gives a self-consistent picture around the outburst peak. Weak irradiation
increases the inferred radius of the inner disc by a factor $\sim 1.5$.
Photons from the disc are the seeds for Compton upscattering to produce the
hard X-ray emission, and this hard X-ray emission weakly illuminates the outer
disc to produce the observed optical/UV by reprocessing, as confirmed by the
optical/X-ray cross-correlation (Hynes et al., 2009).
However, we find clear evidence that the model breaks down as the source flux
declines. The optical spectra require increasingly unlikely levels of
reprocessing to explain the observed emission. A change in origin of the
optical emission is confirmed by the dramatic change in optical/X-ray cross-
correlation signal (Hynes et al., 2009). While the cross-correlation can be
explained in models where the optical is produced by the jet and the X-rays in
a corona (Malzac, Merloni & Fabian, 2004), this does not explain the excellent
match between the optical and X-ray spectra. Instead it seems more likely that
there is a single component connecting the optical and X-ray spectra. The
complex cross-correlation then remains an issue especially for a single
synchrotron component from the jet, but it may potentially be explained by
thermal Comptonisation of IR cyclo-synchrotron emission in an inhomogeneous
hot flow.
More fundamentally for the focus of this paper, the soft X-ray emission during
the decline may not be thermal emission from the disc either. If it is, its
radius does not change markedly from that seen at the outburst peak, in clear
conflict with the predictions of the truncated disc model. But it also implies
that the mass accreting through the disc is much less than the mass accretion
rate required to power the corona. Yet it seems most likely that the mass does
accrete through the outer disc, in which case the observed weak soft X-ray
disc emission is a problem. Either the mass accretes through the disc but most
of the energy is transported vertically and radially (to get around the
reprocessing limits) by magnetic fields to power a small, central hard X-ray
corona, or the soft X-rays are from an additional component, with the
truncated disc peaking in the UV.
The lower absorption to XTE J1118+480 allows us to distinguish between these
possibilities. Here, we can see a cool component peaking in the UV which is
clearly distinct from the soft X-ray emission. The UV component fits well to a
cool disc, truncated at large radii, so the soft X-ray component cannot be the
same material. This may still be associated with the inner disc, perhaps from
irradiation of its inner rim, or via a residual inner disc in the
discontinuous disc geometry predicted by evaporation models (e.g. Liu & Meyer-
Hofmeister 2001). However, it is also possible that the soft X-rays are
instead produced in a completely different way, such as ionised reflection
from grazing incidence angle illumination of the outer disc. Whatever their
origin, it is clear from XTE J1118+480 that the weak soft X-ray component seen
in the dim low/hard state does not trace the inner edge of the disc, so cannot
be used to constrain the truncated disc models.
## Acknowledgements
CYC and CD would like to thank Kim Page for help in extracting the _Swift_ XRT
data.
## References
* Arévalo & Uttley (2006) Arévalo P., Uttley P., 2006, MNRAS, 367, 801
* Cabanac et al. (2009) Cabanac C., Fender R. P., Dunn R. J. H., Körding E. G., 2009, MNRAS, 396, 1415
* Cadolle Bel et al. (2007) Cadolle Bel M., Ribó M., Rodriguez J., Chaty S., Corbel S., Goldwurm A., Frontera F., Farinelli R., D’Avanzo P., Tarana A., Ubertini P., Laurent P., Goldoni P., Mirabel I. F., 2007, ApJ, 659, 549
* Corbel & Fender (2002) Corbel S., Fender R. P., 2002, ApJ, 573, L35
* Di Matteo, Celotti & Fabian (1997) De Matteo T., Celotti A., Fabian A. C., 1997, MNRAS, 291, 805
* Done & Gierliński (2003) Done C., Gierliński M., 2003, MNRAS, 342, 1041
* Done, Gierliński & Kubota (2007) Done C., Gierliński M., Kubota A., 2007, A&ARv, 15, 1 (DGK07)
* Dunn et al. (2009) Dunn R. J. H., Fender R. P., Körding E. G., Belloni T., Cabanac C. , 2009, MNRAS, submitted
* Durant et al. (2008) Durant M., Gandhi P., Shahbaz T., Fabian A. P., Miller J., Dhillon V. S., Marsh T. R., 2008, ApJ, 682, L45
* Esin et al. (2001) Esin A. A., McClintock J. E., Drake J. J., Garcia M. R., Haswell C. A., Hynes R. I., Muno M. P., 2001, ApJ, 555, 483
* Esin, McClintock & Narayan (1997) Esin A. A., McClintock J. E., Narayan R., 1997, ApJ, 489, 865
* Fender, Belloni & Gallo (2004) Fender R. P., Belloni T. M., Gallo E., 2004, MNRAS, 355, 1105
* Fender et al. (2005) Fender R., Garrington S., Muxlow T., 2005, ATel, 558, 1
* Fender, Homan & Belloni (2009) Fender R. P., Homan J., Belloni T. M., 2009, MNRAS, 396, 1370
* Frontera et al. (2001) Frontera F., Zdziarski A. A., Amati L., Mikolajewska J., Belloni T., Del Sordo S., Haardt F., Kuulkers E., Masetti N., Orlandini M., Palazzi E., Parmar A. N., Remillard R. A., Santangelo A., Stella L., 2001, ApJ, 561, 1006
* Frontera et al. (2003) Frontera F., Amati L., Zdziarski A. A., Belloni T., Del Sordo S., Masetti N., Orlandini M., Palazzi E., 2003, ApJ, 592, 1110
* Gallo et al. (2007) Gallo E., Migliari S., Markoff S., Tomsick J. A., Bailyn C. D., Berta S., Fender R., Miller-Jones J. C. A., 2007, ApJ, 670, 600
* Gandhi et al. (2008) Gandhi P., Makishima K., Durant M., Fabian A. C. Dhillon V. S., Marsh T. R., Miller J. M., Shahbaz T., Spruit H. C., 2008, MNRAS, 390, L29
* Gierliński, Done & Page (2008) Gierliński M., Done C., Page K., 2008, MNRAS, 388, 753 (GDP08)
* Gierliński, Done & Page (2009) Gierliński M., Done C., Page K., 2009, MNRAS, 392, 1106 (GDP09)
* Haardt & Marashi (1993) Haardt F., Maraschi L., 1993, ApJ, 413, 507
* Hiemstra et al. (2009) Hiemstra B., Soleri P., Méndez M., Belloni T., Mostafa R., Wijnands R., 2009, MNRAS, 394, 2080
* Hjellming & Han (1995) Hjellming R. M., Han X., 1995, xrbi.nasa, 308
* Hynes et al. (2000) Hynes R. I., Mauche C. W., Haswell C. A., Shrader C. R., Cui W., Chaty S., 2000, MNRAS, 539, L37
* Hynes et al. (2009) Hynes R. I., O’Brien K., Mullally F., Ashcraft T., 2009, MNRAS, 399, 281
* Ingram, Done & Fragile (2009) Ingram A., Done C., Fragile P. C., 2009, MNRAS, 397, L101
* Jimenez-Garate, Raymond & Liedahl (2002) Jimenez-Garate M. A., Raymond J. C., Liedahl D. A., 2002, ApJ, 581, 1297
* Kanbach et al. (2001) Kanbach G., Straubmeier C., Spruit H. C., Belloni T., 2001, Nature, 414, 180
* King & Ritter (1998) King A. R., Ritter H., 1998, MNRAS, 293, 42
* Körding & Falcke (2004) Körding E., Falcke H., 2004, A&A, 414, 795
* Kotov, Churazov & Gilfanov (2001) Kotov O., Churazov E., Gilfanov M., 2001, MNRAS, 327, 799
* Krolik, Hawley & Hirose (2005) Krolik J. H., Hawley J. F., Hirose S., 2005, ApJ, 622, 1008
* Kubota & Done (2004) Kubota A., Done C., 2004, MNRAS, 353, 980
* Kubota, Makishima & Ebisawa (2001) Kubota A., Makishima K., Ebisawa K., ApJ, 560, 147
* Lasota (2001) Lasota J. P., 2001, NewAR, 45, 449
* Liu & Meyer-Hofmeister (2001) Liu B. F, Meyer-Hofmeister E., 2001, A&A, 372, 386
* Liu et al. (1999) Liu B. F., Yuan W., Meyer F., Meyer-Hofmeister E., Xie G. Z., 1999, ApJ, 527, L17
* Maitra et al. (2009) Maitra D., Markoff S., Brocksopp C., Noble M., Nowak M., Wilms J., 2009, MNRAS, in press, (arXiv:0904.2128)
* Makishima et al. (2008) Makishima K., Takahashi H., Yamada S., Done C., Kubota A., Dotani T., Ebisawa K., Itoh T., Kitamoto S., Negoro H., Ueda Y., Yamaoka K., 2008, PASJ, 60, 585
* Malzac, Merloni & Fabian (2004) Malzac J., Merloni A., Fabian A. C., 2004, MNRAS, 351, 253
* Markoff et al. (2001) Markoff S., Falcke H., Fender R., 2001, A&A, 372, L25
* Markoff & Nowak (2005) Markoff S., Nowak M. A., 2005, ApJ, 635, 1203
* Mayer & Pringle (2007) Mayer M., Pringle J. E., 2007, MNRAS, 376, 435
* McClintock et al. (2001) McClintock J. E., Haswell C. A., Garcia M. R., Drake J. J., Hynes R. I., Marshall H. L., Muno M. P., Chaty S., Garnavich P. M., Groot P. J., Lewin W. H. G., Mauche C. W., Miller J. M., Pooley G. G., Shrader C. R., Vrtilek S. D., 2001, ApJ., 555, 477
* McClintock & Remillard (2006) McClintock J. E., Remillard R. A., 2006, in: Compact stellar X-ray sources. W. Lewin, M. van der Klis (Eds.). Cambridge Astrophysics Series, No. 39, Cambridge University Press, 157
* Mitsuda et al. (1984) Mitsuda, K., Inoue, H., Koyama, K., Makishima, K., Matsuoka, M., Ogawara, Y., Suzuki, K., Tanaka, Y., Shibazaki, N., Hirano, T., 1984, PASJ, 36, 741
* Motch et al. (1985) Motch C., Ilovaisky S. A., Chevalier C., Angebault P., 1985, SSRv, 40, 219
* Narayan & Yi (1995) Narayan R., Yi I., 1995, ApJ, 452, 710
* Niedźwiecki (2005) Niedźwiecki A, 2005, MNRAS, 356, 913
* Nowak (2005) Nowak M., 2005, Ap&SS, 300, 159
* Poutanen, Krolik & Ryde (1997) Poutanen J., Krolik J. H., Ryde F., 1997, MNRAS, 292, L21
* Reis, Miller & Fabian (2009) Reis R. C., Miller J. M., Fabian A. C., 2009, MNRAS, 395, L52
* Russell et al. (2006) Russell D. M., Fender R. P., Hynes R. I., Brocksopp C., Homan J., Jonker P. G., Buxton M. M., 2006, MNRAS, 371, 1334
* Shapiro, Lightman & Eardley (1976) Shapiro S. L., Lightman A. P., Eardley D. M., 1976, ApJ, 204, 187
* Shahbaz et al. (1998) Shahbaz T., van der Hooft F., Casares J., Charles P. A., van Paradijs J., 1998, MNRAS, 306, 89
* Shakura & Syunyaev (1973) Shakura N. I., Syunyaev R. A., 1973, A&A, 24, 337
* Shrader & Titarchuk (1999) Shrader C.R., Titarchuk L., 1999, ApJ, 521, L121
* Svensson & Zdziarski (1994) Svensson R., Zdziarski A. A., 1994, ApJ, 436, 599
* van Paradijs (1996) van Paradijs J., 1996, ApJ, 464:L139+
* Wardziński & Zdziarski (2000) Wardziński G., Zdziarski A. A., 2000, MNRAS, 314, 183
* Wilkinson & Uttley (2009) Wilkinson T., Uttley P., 2009, MNRAS, 397, 666
* Zdziarski, Poutanen & Johnson (2000) Zdziarski A. A., Poutanen J., Johnson W. N., ApJ, 542, 703
* Zombeck (1990) Zombeck M. V., 1990, Handbook of space astronomy and astrophysics. Cambridge Univ. Press, Cambridge
* Zurita et al. (2008) Zurita C., Durant M., Torres M. A. P., Shahbaz T., Casares J., Steeghs D., 2008, ApJ, 681, 1458
* Życki, Done & Smith (1999) Życki P. T., Done C., Smith D. A., 1999, MNRAS, 305, 231
|
arxiv-papers
| 2009-11-02T10:59:20 |
2024-09-04T02:49:06.230916
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.Y. Chiang, Chris Done, M. Still, and O. Godet",
"submitter": "Chia-Ying Chiang",
"url": "https://arxiv.org/abs/0911.0287"
}
|
0911.0479
|
# Dynamical Heterogeneity in Lattice Glass Models
Richard K. Darst and David R. Reichman Department of Chemistry, Columbia
University, 3000 Broadway, New York, NY 10027, USA. Giulio Biroli Institut
de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France and CNRS, URA
2306
###### Abstract
In this paper we consider in detail the properties of dynamical heterogeneity
in lattice glass models (LGMs). LGMs are lattice models whose dynamical rules
are based on thermodynamic, as opposed to purely kinetic, considerations. We
devise a LGM that is not prone to crystallization and displays properties of a
fragile glass-forming liquid. Particle motion in this model tends to be
locally anisotropic on intermediate time scales even though the rules
governing the model are isotropic. The model demonstrates violations of the
Stokes-Einstein relation and the growth of various length scales associated
with dynamical heterogeneity. We discuss future avenues of research comparing
the predictions of lattice glass models and kinetically constrained models to
atomistic systems.
###### pacs:
blah
## I Introduction
The cause of the dramatic slowing of dynamics close to the empirically defined
glass transition is a subject of great continued interest and debate
Debenedetti and Stillinger (2001); Ediger et al. (1996). Different theoretical
proposals have been put forward aimed at describing some or all of the
phenomena commonly observed in experiments and computer simulations Garrahan
and Chandler (2002, 2003); Hedges et al. (2009); Lubchenko and Wolynes (2007);
Bouchaud and Biroli (2004); Kivelson et al. (2004); Gotze and Sjogren (1992);
Schweizer and Saltzman (2003); Dyre (2006); Langer (2006); Heuer (2008). While
these proposals are often based on completely divergent viewpoints, many of
them are able to rationalize the same observed behaviors. This fact stems from
the somewhat limited amount of information available from experiments and
simulations. Since the growth of relaxation times in glassy systems is
precipitous, it is very difficult, and in some cases impossible, to
distinguish models solely on the basis of different predictions of gross
temperature dependent relaxation behavior. In addition, computer simulations,
which are often more detailed than experiments, are limited by the range of
times scales and sizes of systems that can be studied. These difficulties have
hindered the search for a consensus on the microscopic underpinnings of
vitrification.
Despite the continued debate that revolves around the theoretical description
of supercooled liquids and glasses, little argument exists regarding the
importance of dynamical heterogeneity as a key feature of glassy behavior
Butler and Harrowell (1991); Hurley and Harrowell (1995); Kob et al. (1997);
Yamamoto and Onuki (1998); Schmidt-Rohr and Spiess (1991); Ediger (2000).
Dynamical heterogeneity refers to the fact that as a liquid is supercooled,
dynamics become starkly spatially heterogeneous, requiring the cooperative
motion of groups of particles for relaxation to occur. Dynamical heterogeneous
motion manifests in several ways, and leads to violations of the Stokes-
Einstein relation Chang and Sillescu (1997); Cicerone and Ediger (1996); Jung
et al. (2004); Xia et al. (2001); Tarjus and Kivelson (1995), cooperative
hopping motion reflected in nearly exponential tails in particle displacement
functions Berthier et al. (2005a); Stariolo and Fabricius (2006); Chaudhuri et
al. (2007); Saltzman and Schweizer (2008); Garrahan and Chandler (2002), and
growing length scales such as those associated with the recovery of Fickian
diffusion Swallen et al. (2009); Berthier (2004); Pan et al. (2005); Szamel
and Flenner (2006), growing multi-point correlation functions Lačević et al.
(2003); Dasgupta et al. (1991); Franz and Parisi (2000); Berthier et al.
(2007); Bouchaud and Biroli (2004); Franz and Montanari (2007); Biroli et al.
(2008); Berthier et al. (2005b); Biroli et al. (2006); Karmakar et al. (2009).
Indeed, the relatively recent explication of the phenomena of dynamical
heterogeneity has dramatically shifted the focus of the field and has placed
new constraints on the necessary ingredients for a successful theory of glass
formation.
Given the similarity of some aspects of dynamical heterogeneity to critical
fluctuations in standard critical phenomena, it is natural to investigate two
and three dimensional simplified coarse-grained models that encode the crucial
features of this heterogeneity. Currently, the most investigated class of
coarse-grained models are the “kinetically constrained models” (KCMs) Garrahan
and Chandler (2002, 2003); Fredrickson and Andersen (1984); Kob and Andersen
(1993); Ritort and Sollich (2003). KCMs are spin or lattice models that
generate slow, glassy relaxation via constraints on the dynamical moves that
are allowed. The slowing down of the dynamics is caused by rarefactions of
facilitating regions, also called defects. Importantly, although the dynamics
is complex the thermodynamics is trivial since the dynamical rules are such
that all configurations are equally likely. The philosophy of this viewpoint
is that thermodynamic quantities, such as the configurational entropy, are not
the fundamental underlying cause of the growing time scales in supercooled
liquids. It has been argued that the quantitative disagreement Biroli et al.
(2005) between thermodynamic features of KCMs and real experiments is of
little dynamical consequence Chandler and Garrahan (2005). In support of this
perspective is the fact that KCMs have been remarkably successful in
generating features of dynamical heterogeneity such as Stokes-Einstein
decoupling, growing dynamical length scales, and excess tails in the real-
space particle displacement function Jung et al. (2004); Pan et al. (2005);
Hedges and Garrahan (2008, 2007).
On the other hand, one may wonder if a deeper viewpoint would allow for an
understanding of the kinetic rules that govern particle motion in the
supercooled liquids. It is natural to speculate that such aspects might have
roots in the thermodynamics of configurations. Indeed, simple local Monte
Carlo “dynamics” can reproduce all features of dynamical heterogeneity seen in
Newtonian molecular dynamics simulations, and are based simply on making local
moves that are _configurationally_ allowed Berthier (2007). Lattice models
based on this concept are called “lattice glass models” (LGMs), and were first
considered by Biroli and Mézard Biroli and Mézard (2001). The rules for such
models seem at first sight like that of KCMs. For example in the simplest
versions of such models a particle may move if it is surrounded by no more
than a fixed number of nearest neighbors before and after the move McCullagh
et al. (2005); Dawson et al. (2003); Ciamarra et al. (2003); Pica Ciamarra et
al. (2003). Locally this is identical to the type of dynamical constraint that
appears in the KCMs introduced by Kob and Andersen Kob and Andersen (1993).
However, this constraint must be met globally: _all_ particles must have no
more than a fixed number of nearest neighbors. As the density of the system
increases, fewer and fewer configurations exist for which these constraints
may be satisfied. It is thus the entropy of configurations that governs the
slowing of dynamics, intimately connecting the non-trivial thermodynamic
weight of states accessible to the local dynamics. Indeed, LGMs can be solved
exactly within the Bethe approximation, or on Bethe lattices Biroli and Mézard
(2001); Ciamarra et al. (2003), and have been shown to have a glass transition
due to the vanishing of the configurational entropy. The distinction between
the KCM and LGM viewpoint is illustrated in Fig. 1.
Figure 1: Comparison and distinction of a caricature of a kinetically
constrained model with a lattice glass model. In the KCM any configuration is
allowed, but move may only be made if a particle has at least one missing
neighbor before and after the move. In the LGM, the global configuration is
defined such that all particles must have at least one missing neighbor, and
all dynamical moves must respect this rule. Note that the local environment
around the moving particle is identical in this example, while the global
configurations are distinct. Periodic boundary conditions are assumed for both
panels.
LGMs have been studied by a number of groups, but the focus has not generally
been on real-space aspects of dynamical heterogeneity. For example, Coniglio
and coworkers have developed a simple LGM that avoids crystallization and
displays many features of typical glass-forming materials, including a growing
multi-point susceptibility ($\chi_{4}(t)$) Ciamarra et al. (2003); Pica
Ciamarra et al. (2003). On the other hand, this system appears to behave as a
strong glass-former, with a stretching parameter close to one, and exhibits
essentially no Stokes-Einstein violation. Our goal in this work is to survey
in detail the dynamical behavior of a new LGM which bears similarity to the
original Biroli-Mézard model but is not prone to crystallization. The main
conclusion that we draw is that LGMs are at least as realistic as KCMs in
their description of all commonly studied features of dynamical heterogeneity.
In this regard, simple coarse-grained lattice models based on the
thermodynamic weight of states are no less viable as fundamental caricatures
of glassy liquids than are KCMs based on weights of trajectories. We conclude
our work by highlighting several key ways that LGMs and KCMs may be
distinguished. We reserve the investigation of these comparisons for a future
study. Our paper is organized as follows: Sec. II outlines the model. Sec. III
discusses both simple averaged dynamics as well as aspects of dynamical
heterogeneity. In Sec. IV we conclude with a discussion of the meaning of our
findings and the future directions to be pursued.
## II Model
Here we define the LGM that forms the basis of our simulations. The original
model of Biroli-Mézard is quite prone to crystallization Biroli and Mézard
(2001). This fact makes its use problematic for the study of glassy behavior
since crystallization always intervenes before supercooling becomes
significant. The crystallization problem persists on a square lattice for all
binary mixtures we have studied. However, we have found that certain
generalizations of the Biroli-Mézard model with three species of particles are
stable against crystallization for the densities that are sufficiently high
that glassy dynamics may be clearly observed.
Our model follows the original rules of the Biroli-Mézard model. Particles
exist on a cubic periodic lattice of side $L=15$ and each lattice site can
contain only zero or one particle. All particles, at all times, must satisfy
the condition a particle of type “$m$” must have $m$ or fewer neighbors of any
type. A neighbor is considered any particle in one of the $2d$
($d$=dimensionality) closest lattice sites along the cubic coordinate axes
111It should be noted that LGMs of the type described here involve extreme
constraints that must be globally satisfied and are thus not realistic
translations of off-lattice particle-based models. Such constraints might
indeed induce artificial behavior, especially at higher densities. It would be
most interesting to investigate “soft” versions of such models where
constraints may be locally violated at the cost of an energy penalty. In this
regard, such models would be the configurational analog of KCMs where
dynamical constraints may be broken at the cost of an energy penalty, see
Chandler, D. and J.P. Garrahan, “Dynamics on the Way to Forming Glass: Bubbles
in Space-time” arXiv:0908.0418v1; Submitted to Annual Reviews of Physical
Chemistry (2010)..
The particular three species model we employ is defined by 10% type 1
particles, 50% type 2 particles, and 40% type 3 particles. We denote this
model the “t154” model to indicate its basis in thermodynamics and to specify
the types and percentages of each particle. The composition of t154 model was
determined via trial and error by picking particle types with clashing
crystallization motifs thereby frustrating crystallization. Crystallization
was monitored by inspection of the angle resolved static structure factor,
direct inspection of configurations, and by monitoring bulk thermodynamic
quantities.
As discussed in the introduction, there appear to be strong similarities
between the rules that govern KCMs such as the Kob-Andersen model and the t154
model Kob and Andersen (1993). For example both models employ constraints with
a maximum number of neighbors, but in the Kob-Andersen model this restriction
only applies to the mobile particles, while in the t154 model applies to all
particles. Our model does not require any special dynamics methods. We employ
local canonical Monte Carlo “dynamics” via primitive translational moves
Berthier (2007). Note that for the t154 model the energy can only be zero (no
packing violations) or infinite (packing violation or overlap), thus the
acceptance criteria reduces to rejection if there is a packing violation and
acceptance otherwise. This allows us to implement an event-driven algorithm
which accelerates the simulation of lattice dynamics Bortz et al. (1975).
For thermodynamic studies we employ grand-canonical Monte Carlo with both
translational moves as well as particle insertion/deletion. Fig. 2 contains a
plot of the density of the system as a function of the chemical potential of
type 1 particles. Models which crystallize (such as original binary model of
Biroli-Mézard) have a sharp jump in this curve at the crystallization point.
Clearly, this feature is absent in the t154 model. For comparison, both curves
are displayed 222A subtle issue arises in the nature of glassy behavior
observed in the t154 model outlined in this work. LGMs could have a dynamical
percolation-like transition, as in the spiral model Biroli and Toninelli
(2008). This has been indeed found in some LGMs on the Bethe lattice Rivoire
et al. (2003) and would slow down the dynamics for reasons completely
different from the diminishing of the configurational entropy. If there is a
low-lying crystal phase then one can show that this dynamical percolation-like
transition cannot take place in finite dimension. Although we have not found a
crystal phase for the model, the existence of such a transition seems unlikely
and irrelevant for our present work. First, it can be shown that blocked
structures, if they exist, have to verify much more constraints than in the
spiral model Biroli and Toninelli (2008). Second, we have found that the
relaxation time growth of the persistence functions with increasing density in
local canonical Monte Carlo simulations are similar to those under grand-
canonical dynamics, which cannot contain any blocked structure. The union of
these two facts render the dynamical blocking scenario highly unlikely..
Figure 2: Crystallization thermodynamics in LGM. Top: The t154 model.
$\mu_{1}$ refers to the chemical potential of the type 1 particles. The
maximum density observed for the 153 lattice is .5479 (exactly 1849 out of
3375 lattice sites occupied). The three plotted quenching rates vary between a
.01 and .05 increase of $\mu_{1}$ per 10000 cycles. Bottom: A close up of the
equivalent plot for the BM model. Note the clear discontinuity upon
crystallization. Slower $\mu$-increase rates produce a sharper discontinuity.
## III Dynamical Behavior
### III.1 Simple Bulk Dynamics
In this subsection we describe the behavior of a simple 2-point observable,
namely the self-intermediate scattering function Balucani and Zoppi (1994),
defined as
$F_{s}(k,t)=\left<\frac{1}{N}\sum_{i}e^{i\mathbf{k}\cdot\left[\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right]}\right>.$
(1)
We measure $F_{s}(k,t)$ only for the type-2 particles which are present in the
greatest fraction for the three distinct species. Throughout this paper, we
report $k$-vectors using $k^{\prime}$, where $k=\frac{2\pi}{L}k^{\prime}$. We
have checked that $F_{s}(k,t)$ is qualitatively similar for the other species
of particles. The relaxation of $F_{s}(k,t)$ of the system at the wavevector
$k^{\prime}=5$ ($k=\frac{2\pi}{3}$) for various densities is shown in Fig. 3.
The bulk of the decay may be fit to a stretched exponential function,
$F_{s}(k,t)=\exp(-(t/\tau_{\alpha}(k))^{\beta(k)})$. As is customary, the
alpha-relaxation time is found by the value $F_{s}(\tau_{\alpha})=1/e$ and the
$\beta(k)$ exponent is determined by a direct fit to the terminal decay. We
find that for densities below approximately $\rho=0.48$ the value of $\beta$
saturates at the expected value $\beta=1$ characteristic of simple non-glassy
dynamics, while for the highest density simulated, $\beta=0.7$. This behavior,
over a similar range of supercooling, is reminiscent of the behavior found in
atomistic models of glass-forming liquids Kob and Andersen (1995); Wahnström
(1991). In order to better reveal the relaxation behavior, $F_{s}(t)$ is also
displayed on a log-log vs. log-time scale. In this plot, the slope of the long
time growth is related to the exponent $\beta.$ We have found that the values
of $\beta$ extracted from the slopes of the long time portion of the log-log
vs. log plot indeed coincide with that found by a direct fit to a stretched
exponential form. At the highest densities a shoulder appears in the short
time relaxation. This feature is indicative of a secondary relaxation feature
perhaps akin to beta-relaxation in realistic glass-forming liquids. It should
be noted, however, that the amplitude of this feature is very close to unity.
This is quantitatively distinct from the plateau values expected in atomistic
off-lattice models Kob and Andersen (1995); Wahnström (1991) and even LGMs
with more complicated lattice degrees of freedom Ciamarra et al. (2003); Pica
Ciamarra et al. (2003), but is similar to that encountered in simple spin
models such as variants of the Random Orthogonal Model Sarlat et al. (2009).
As is typical of fragile glass-forming systems, the t154 model exhibits
relaxation times that do not follow the (generalized) Arrhenius form Berthier
and Witten (2009). This behavior is illustrated in Fig. 4. At low densities,
plots of $\log(\tau)$ versus $\rho$ indeed follow a straight line, however in
the vicinity of $\rho\sim 0.5$ the plot of $\tau$ versus $\rho$ deviates from
this straight line and the functional density dependence of the relaxation
time becomes much more precipitous. While we have not attempted to
quantitatively characterize this density dependence, it should be noted that
the onset of increased sensitivity to changes in density occurs is the same
narrow window that marks the noticeable decrease in the values of the
stretching exponent $\beta$.
Figure 3: Decay of the self-intermediate scattering function $F_{s}(k,t)$ for
$k^{\prime}=5$ ($k=\frac{2\pi}{L}k^{\prime}$). Densities are .3, .4, .45, .48,
.50, .51, .52, .53, .535, .5375, .5400, .5425 from fastest relaxation to
slowest relaxation. These densities are used in all plots in this paper unless
otherwise indicated. Top: Plotted on a linear-log scale. Bottom: Same data as
upper panel plotted on a $\log(-\log_{10}(F_{s}(k,t)))$ vs $\log(t)$ scale.
Lowest density curves are at the top left.
Figure 4: Top: $\tau_{\alpha}$ (time at which $F_{s}(k,t)=1/e$) as a function
of density, $\rho$. Plotted for $k^{\prime}=1,2,3,4,5,6,7$, with lowest $k$ at
the top. Center: Beta stretching exponent of $F_{s}(k,t)$ (from terminal fits
$F_{s}(k,t)\sim\exp(-(t/\tau_{\alpha})^{\beta})$). Lowest $k$ curve is at the
top of the plot. Bottom: Plot of log scale $\tau_{\alpha}$ against chemical
potential $\mu$ of type 2 particles. The behavior is consistent with
$\tau_{\alpha}=5.7\exp(-21\mu_{2}/(\mu_{2}-24))$.
### III.2 Motion on the Atomic Scale
We begin our discussion of the nature of heterogeneous dynamical behavior in
the t154 LGM by observing the qualitative details of particle motion under
supercooled conditions. This will set the stage for analysis of quantitative
measures of dynamical heterogeneity in the model. For the sake of comparison,
we also investigate the analogous behavior in the Kob-Andersen model. This
comparison is useful because it suggests how models with similar local rules
but different global rules (rooted in either the purely kinetic or
thermodynamic basis of the particular model) may give rise to distinct
dynamics at the particle scale.
We start by simply observing the patterns of mobility in real space starting
from a set initial condition of the t154 model found at a given density after
equilibration. A similar analysis has been performed recently by Chaudhuri et
al. for the Kob-Andersen model, where no equilibration is required since all
initial configurations with a set density of defects are allowed Chaudhuri et
al. (2008). For a theoretical description of the dynamics of the Kob-Andersen
model, see Toninelli et al. (2004). We note that, as expected, the t154 model
exhibits regions of spatially localized particle activity against a backdrop
of transiently immobilized particles. A rather remarkable feature of the
patterns of mobility in this model is that we find evidence of string-like
motion, where a group of particles moves over a short distance, each taking
the place of the previous particle in the string Donati et al. (1998, 1999).
This motif can be seen mostly on timescales less than the $\alpha$-relaxation
time, but occasionally string-like motion may be seen to persist on longer
timescales. This behavior is demonstrated in Fig. 5.
The behavior of particle motion observed in the Kob-Andersen model is somewhat
different than that seen in the t154 model as described above. As in the t154
model, and as observed by Chaudhuri et al., motion in the Kob-Andersen model
shows similar activity regions in the vicinity of defect sites giving rise to
heterogeneous motion. However, the boundaries between active and inactive
regions at comparable timescales appear to be more distinct in the Kob-
Andersen model. Furthermore, the particle scale motion in the Kob-Andersen is
much more isotropic, exhibiting much fewer cases of directional mobility
compared with the t154 model. It would be interesting to compare the two
models by quantifying this difference via the type of directional multi-point
correlators devised by Doliwa and Heuer Doliwa and Heuer (1999). It is not
clear if the difference between the models is related to the fundamental
distinction between LGMs and KCMs or just the specifics of the particular
models considered. In particular, the t154 is a multi-component model, unlike
the Kob-Andersen model. The string-like motion on short time scales seems to
occur predominantly on the rather rough boundaries of slow clustersAppignanesi
et al. (2006). This behavior, reminiscent of the picture of dynamic
heterogeneity that put forward by Stillinger Stillinger (1988), might be
strongly influenced by compositional heterogeneity. A useful way to address
general issues related to how the initial configuration constrains subsequent
dynamics would be a systematic iso-configurational ensemble analysis comparing
LGMs and KCMs Widmer-Cooper et al. (2004). This will be the topic of a future
publication Darst et al. .
Figure 5: Examples of string-like motion apparent in the t154 model. (a) An
example of a string with all neighboring particles removed. (b) A similar
string in the context of other particles. Note that here the string is truly
isolated in space, away from other mobile particles. In these figures, type 1
particles are white, type 2 particles are blue, and type 3 are green. Sites
occupied at the initial time but vacated at the final time are shown in red.
These pictures show only the differences in position of particles between the
origin of time and the final time, not the path the particles took to achieve
that displacement. All figures are at a density of .5400, with $\Delta t$
times in (a) 251, (b) 199526. The $\alpha$-relaxation time for $k^{\prime}=5$
at this density is about $7.8\times 10^{6}$
Figure 6: Examples cluster shapes in the (a) the t154, model, density
$\rho=.5400$ and (b) the Kob-Andersen model, density $\rho=.8500$. Arrows
indicate motion between initial and final times. Time separation is 1/10th of
the $\alpha$-relaxation time. In the t154 model, we see more fractal and
disconnected clusters, while in the KA model, mobile domains tend to be
smoother clusters.
In the next few sections, we discuss how some of the most important indicators
of dynamical heterogeneity in supercooled liquids manifest in the t154 model.
The quantities that we discuss are the magnitude of violations of the Stokes-
Einstein relation, exponential tails (indicative of hopping transport) in the
van Hove function, the existence of a Fickian length scale and the development
of a dynamical length scale quantified by the multi-point function
$S_{4}(q,t)$. Unless otherwise stated, specific correlation functions and
transport coefficients are calculated with respect to type-2 particles.
### III.3 Stokes-Einstein Violation
In typical fluids a mean-field linear-response relationship asserts that the
product of the tracer particle diffusion constant and the fluid viscosity
divided by the temperature is a constant Balucani and Zoppi (1994). This
connection between diffusion and dissipation is known as the Stokes-Einstein
relationship, and empirically is known to hold even at the atomic scale in
liquids over a wide range of densities and temperatures. In supercooled
liquids, the Stokes-Einstein relation generally does not hold Chang and
Sillescu (1997); Cicerone and Ediger (1996); Jung et al. (2004); Xia et al.
(2001); Tarjus and Kivelson (1995); Berthier et al. (2005a); Stillinger and
Hodgdon (1994); Liu and Oppenheim (1996). In fact, the product of the
diffusion constant and the viscosity of a liquid may exceed that expected from
the Stokes-Einstein relation by several orders of magnitude close to the glass
transition. There are many theoretical explanations for Stokes-Einstein
violations in supercooled liquids, which essentially all invoke dynamical
heterogeneity as the fundamental factor leading to the breakdown of the simple
relationship between diffusion and viscosity. It should be noted that similar
relationships hold between the diffusion constant and the self and collective
time constants associated with the decay of density fluctuations. In this work
we focus on the relaxation time of the self-intermediate scattering function
defined above as our proxy for the fluid viscosity.
It is well known that the product $D\tau_{\alpha}$, where $\tau_{\alpha}$ is
the $\alpha$-relaxation time of the self-intermediate scattering function
shows a strong temperature/density dependence in both realistic atomic
simulations as well as in the class of KCMs that describe fragile glass-
forming liquids. No direct studies of this quantity have been made in LGMs.
The LGM of Coniglio and coworkers would appear to show essentially no Stokes-
Einstein violations because the diffusion constant and the relaxation time may
both be fit to power laws with exponents that have, within numerical accuracy,
the same magnitude Ciamarra et al. (2003); Pica Ciamarra et al. (2003). This,
however is not surprising since many of the features of the model resemble
those of a strong glass-forming system, where violations of the Stokes-
Einstein relation are, at most, weak. The features of the t154 model with
regard to non-exponential relaxation and the density dependence of the
relaxation time $\tau_{\alpha}$ indicate that this model behaves more like a
fragile glass former. Thus, we expect clear violations of the Stokes-Einstein
relation. Indeed, as shown in Fig. 7, $D\tau_{\alpha}$ increases markedly as
density is increased. Over the range densities that we can access, the
magnitude of the violation is very similar to that seen in the canonical Kob-
Andersen Lennard Jones mixture over a comparable range of changes in
relaxation time Chaudhuri et al. (2007). Interestingly, violations begin to
become pronounced at densities similar to where the relaxation times and
stretching exponents become strongly sensitive to increased density. Thus, a
consistent onset density is observed as in more realistic atomistic systems.
Figure 7: Violation of the Stokes-Einstein relation,
$D\tau_{\alpha}\sim\mathbf{constant}$, using $\tau_{\alpha}$ at
$k^{\prime}$=5. Data has been normalized to $D{\tau_{\alpha}}=1$ at the lowest
density.
### III.4 van Hove Function
It is now rather well established that an additional “quasi-universal” feature
of dynamical heterogeneity near the glass transition is contained in the shape
of the real-space van Hove function Stariolo and Fabricius (2006); Chaudhuri
et al. (2007); Saltzman and Schweizer (2008); Swallen et al. (2009). In
particular it has been argued the tails of the self van Hove function should
be approximately exponential in form. These “fat tails” imply that the rare
particles that do undergo large displacements exist in populations in excess
of what would be expected in a purely Gaussian displacement distribution.
While non-Gaussian tails should be expected of any distribution for the wings
that fall outside of limits of bounds set by the Central Limit Theorem, the
palpable exponential tails in supercooled liquids imply large non-Gaussian
effects indicative of transport that is strongly effected by heterogeneous
hopping motion.
Here, we demonstrate that such effects occur in the t154 model in a manner
similar to that seen both in experiments in colloidal and granular systems as
well as in computer simulations of atomic systems. Fig. 8 shows the self part
of the real-space van Hove function,
$G_{s}(x,t)=\left<\delta\left(x-\left|\mathbf{\hat{x}}\cdot\left(\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right)\right|\right)\right>,$
(2)
for the type two particles in the t154 model. Because we are on a lattice, we
restrict our distances along the three coordinate axes $\mathbf{\hat{x}}$
individually in our calculation. We see that for times of the order of the
$\alpha$-relaxation time, these tails are clearly visible. For very long or
short time scales, the shape of the tail deviates somewhat from the more
exponential form exhibited at intermediate times. This behavior is quite
similar to that seen in simulations of atomistic systems Chaudhuri et al.
(2007); Saltzman and Schweizer (2008), and is fully consistent with the
behavior found in KCMs Berthier et al. (2005a).
Figure 8: van Hove function for $\rho=.5375$ and various times. Distances are
measured independently along each coordinate axis. The times plotted, from
left to right, are $10^{5}$, 316227 (approx. the $\alpha$-relaxation time),
and $10^{6}$. An exponential fit to the tail of the $t=316227$ case is shown
by a dotted line.
### III.5 Fickian Length
Related to the existence of excess tails in the van Hove function is the
existence of a length scale that characterizes the anomalous transport. More
specifically, the exponential tails in the van Hove function are distinguished
from the Gaussian form of the displacement distribution obtained at relatively
short distances for fixed times. The crossover from Fickian to non-Fickian
behavior should be characterized by time scales as well as length scales over
which this crossover occurs. A non-Fickian length scale may be defined by
examining the $k$-dependent diffusion constant
$D(k)=\frac{1}{\tau_{\alpha}k^{2}}$ Berthier (2004); Pan et al. (2005); Szamel
and Flenner (2006). The wavevector that characterizes the crossover from the
expected diffusive behavior to an anomalous regime is inversely related to
such a length scale. In Fig. 9 we plot $D(k^{\prime})$. Clearly, as the
density is increased, the length scale separating the Fickian and non-Fickian
regimes increases. This behavior is consistent with that found in KCMs and
simulations of atomistic glass-forming liquids. It should be noted that
Stokes-Einstein violations, the development of exponential tails in the self
van Hove function, and a well-developed Fickian length scale are all
manifestations of related aspects of dynamically heterogeneous motion in
supercooled liquids Chaudhuri et al. (2007).
Figure 9: $k$-dependent diffusion constant $=1/\left(k^{\prime
2}\tau_{\alpha}(k)\right)$. Densities of .3000 (upper) and .5425 (lower). The
higher density curve is multiplied by a scale factor of $2.992\times 10^{5}$
for ease of comparison. A dotted flat line is included for reference of
behavior expected in the purely Fickian case.
### III.6 $\chi_{4}$ and $S_{4}$ Fluctuation Measures
The Fickian length scale is merely one length scale that arises naturally in
systems where dynamics become increasingly heterogeneous. Perhaps more
fundamental is the growth of dynamical length scales associated with multi-
point correlations of the dynamics. Supercooled liquids do not show simple
static correlations that would indicate a growing correlation length. It
should be noted that this does not exclude growing static correlations of a
more complex kind, for example point-to-set correlations Bouchaud and Biroli
(2004); Mézard and Montanari (2006); Biroli et al. (2008). Regardless,
cooperativity in dynamics may be measured via first defining a local overlap
function Garrahan and Chandler (2002); Yamamoto and Onuki (1998); Lačević et
al. (2003); Franz and Parisi (2000); Berthier et al. (2007)
$\delta
f_{k}(q,t)=\frac{1}{N}\sum_{i}e^{i\mathbf{q}\cdot\mathbf{r}_{i}(0)}\left[\cos\left(\mathbf{k}\cdot\left(\Delta\mathbf{r}_{i}(t)\right)\right)-F_{s}(k,t)\right]$
(3)
where $\Delta\mathbf{r}_{i}(t)=\mathbf{r}_{i}(0)-\mathbf{r}_{i}(t)$.
$f_{k}(q,t)$ is defined for one configuration, and the average is over all
$\mathbf{k}$ and $\mathbf{q}$ with the magnitudes $k$ and $q$. Then,
$S_{4}(q)$ is defined as
$S_{4}(q)=N\left<\left|\delta f_{k}(q,t)\right|^{2}\right>$ (4)
where this average is over the most general ensemble of configurations
Berthier et al. (2007). The $\chi_{4}$ value is defined as the limit
$S_{4}(q\to 0)$. $\chi_{4}(t)$ may be calculated strictly at $q=0$ from
$\chi_{4}(t)=N\left<\left|\delta f_{k}(q=0,t)\right|^{2}\right>$ (5)
where the average is over the entire ensemble and all $\mathbf{k}$ consistent
with the magnitude of k. Note that, as discussed in Berthier et al. (2007),
the value of $\chi_{4}(t)$ computed in this manner is a lower bound for the
extrapolation of $S_{4}(q\to 0,t)$.
The quantity $S_{4}(q,t)$ is a multi-point dynamical analog of $S(q)$. Just as
the low $q$ behavior of $S(q)$ indicates a growing (static) length scale in
systems approaching a second order phase transition, scattering from
dynamically heterogeneous regions undergoing cooperative motion will manifest
growth in the amplitude of the low $q$ region of $S_{4}^{ol}(q,t)$, indicative
of the size scale of the dynamical correlations for systems approaching the
glass transition.
The behavior of the quantity $S_{4}^{ol}(q,t)$ is shown in Fig. 10. Only
type-2 particles have been used in the calculation. As can clearly be seen,
for densities above $\rho\sim 0.5$ which constitutes the onset density of this
system, the low $q$ behavior shows a marked upturn as $q\rightarrow 0$. The
growth of $S_{4}^{ol}(q,t)$ as $q\rightarrow 0$ as density is increased
suggests a growing length scale as supercooling progresses. This non-trivial
behavior is what is found in atomistic simulated systems. Future work will be
devoted to a precise characterization of the length scale that may be
extracted from $S_{4}^{ol}(q,t)$ in the t154 model so that a comparison may be
made with recent work detailing the behavior of this length in realistic off-
lattice systems Stein and Andersen (2008); Karmakar et al. (2009).
Figure 10: Top: Top:Plot of $S_{4}(q,t)$ at $\tau_{\alpha}$ for densities
0.51, 0.52, 0.53 and 0.54. Bottom: Plot of $\chi_{4}(t)$ for the same
densities. Peak values correspond to lower bounds for of the value of
$S_{4}(q,t)$ in the upper panel at $q=0$.
## IV Conclusion
In this paper we have presented a new LGM based on the original Biroli-Mézard
model Biroli and Mézard (2001). Via the introduction of an additional species
of particle, we have demonstrated that our model is stable against
crystallization. This fact allows us to study sufficiently high density
configurations that manifest features of dynamical heterogeneity. Unlike some
previous LGMs, our model exhibits the canonical features of a fragile glass-
former. In terms of the gross features of relaxation behavior, our LGM shows
behavior similar to the standard Kob-Andersen Lennard-Jones (KALJ) mixture. In
particular, we find that the degree of violation of the Stokes-Einstein
relation and the magnitude of stretching in the decay of the self-intermediate
scattering function track the relaxation times at densities above the onset of
supercooling in a manner consistent with that seen in the KALJ system.
Features of dynamical heterogeneity such as exponential tails in the van Hove
function, the growth of a dynamical length scale as quantified by the function
$S_{4}(q,t)$, Stokes-Einstein violations and the emergence of a Fickian length
scale all occur in a manner expected from experiments and simulations of
fragile glass-forming liquids.
The similarity between the description of dynamic heterogeneity found in KCMs
and LGMs stands in stark contrast to the underlying foundations of the models
themselves. As emphasized in the introduction, KCMs are based on a constrained
dynamics for which the number of available dynamical paths leading to
relaxation becomes increasingly rare as the density increases and the number
of defects decrease. In KCMs all real-space configurations at a fixed number
of defects (excluding rare blocked configurations) are equally likely. On the
other hand LGMs are based on transitions between real-space configurations
that become increasingly scarce as the density is increased. This is not to
say that there is not a facilitated-like dynamics in LGMs. On the contrary, as
we have demonstrated in sec. III, local and sometimes anisotropic dynamics may
be generated naturally in LGMs without the explicit introduction of
facilitating defects. An important message that emerges from this study is
that the phenomenology of dynamic heterogeneity is not sufficient to
distinguish pictures or validate models based on transitions between sets of
states in configuration space from those based on sets of paths in space-time.
How then might these pictures be differentiated? While contrasting competing
models that generate seemingly similar dynamical behavior is a difficult
endeavor, several possible studies might be useful for this task. Here we
outline four avenues that could provide key information that distinguish the
purely dynamical picture from one based on transitions thermodynamic states.
a) _The mosaic length scale:_ The Random First Order Theory (RFOT) of Wolynes
and coworkers posits the existence of a static length scale which is defined
by the region over which particles are pinned by the surrounding self-
generated amorphous configuration Lubchenko and Wolynes (2007); Bouchaud and
Biroli (2004); Biroli et al. (2008). This length scale also exists in KCMs,
but it is decoupled from the relaxation dynamics of the system Jack and
Garrahan (2005). Recent atomistic computer simulations have successfully
located the mosaic length scale Biroli et al. (2008). It would be quite useful
to perform an analysis similar to that devised by Jack and Garrahan for LGMs
Jack and Garrahan (2005). Since LGMs are based on the entropy of real-space
configurations, it is expected that here the mosaic length does couple to the
glassy dynamics. Since LGMs are much simpler than atomistic off-lattice
models, the direct study of the mosaic length (and point-to-set correlations
in general) in LGMs might provide key avenues for the testing of the putative
coupling between relaxation and such length scales in simulated atomistic
systems.
b) _Correlations between configurational entropy and dynamics:_ Empirical
correlations between the configurational entropy and the $\alpha$-relaxation
time have been noted for many years, and this correlation lies at the heart of
several prominent theories. Such correlations are still widely debated, but
seem to hold at least crudely in many glass-forming systems Richert and Angell
(1998); Martinez and Angell (2001). LGMs should be expected to exhibit such
correlations, while it is known that KCMs do not exhibit such correlations.
Recently Karmakar et al. purported to show that finite-size effects of the
$\alpha$ relaxation time follow precisely the Adam-Gibbs relation between the
configurational entropy and the $\alpha$-relaxation time in the KALJ system
Karmakar et al. (2009). If true, such correlations would be a challenge to
KCMs, since it is difficult to envision how the configurational entropy would
track the $\alpha$-relaxation time for different system sizes if it were not a
crucial component of relaxation phenomena. Such correlations, however, are
subtle to measure since the Adam-Gibbs relationship is an exponential one and
the apparent correlation could depend on the somewhat indirect computational
method used in Karmakar et al. (2009) to define the configurational entropy.
It would be most useful to investigate such effects in the simpler LGMs, which
might provide a cleaner means of isolating the configurational entropy. It
should be noted that finite size effects do appear to follow an approximate
Adam-Gibbs relationship in at least one other lattice model Crisanti and
Ritort (2000). Such studies might spur more detailed investigations in
simulated atomistic systems thus allowing for a clear comparison between LGMs,
KCMs and more realistic systems.
c) _Single-particle and collective predictability ratios:_ In an important
piece of work, Jack and Berthier devised metrics that access the degree to
which single particle and collective dynamics are deterministically predicted
by a set initial configuration over a given time scale Berthier and Jack
(2007). KCMs and LGMs differ in how allowed configurations are constructed.
KCMs have explicit defects, while configurations in LGMs are determined by
global constraints, and thus do not contain explicit defects. Since the very
composition of initial conditions differ markedly in these models, one expects
that the metrics defined by Jack and Berthier would behave differently in KCMs
and LGMs. Thus, it would be very profitable to examine the density and
temperature dependence of the single particle and collective predictability
ratios in KCMs and LGMs as a possible means of distinguishing between state-
based, and dynamical constraint-based pictures Darst et al. .
d) _Evolution of the facilitation mechanism approaching the glass transition:_
Although in both KCM and LGM pictures facilitation plays an important role in
the relaxation of the system, a peculiar and different temperature and density
evolution is expected. In particular, in the KCM picture, facilitation is due
to the motion of mobility regions or defects. Dynamics slows down, and
concomitantly dynamic heterogeneity increases, because these regions become
rarer approaching the glass transition. A crucial assumption is that these
defects are conserved or at least that non-conservation is a rare event that
becomes rarer at lower temperature/high density. These assumptions impose
important constraints on the evolution of the facilitation mechanism. Thus, it
would be very interesting to examine this issue for example using the cluster
analysis developed in Candelier et al. (2009) to study the relaxation dynamics
of granular systems.
Investigation of these and other studies aimed at distinguishing the
underlying pictures that LGMs and KCMs are based on will be the subject of
future work.
###### Acknowledgements.
RKD would like to thank the John and Fannie Hertz Foundation for research
support via a Hertz Foundation Graduate Fellowship. RKD and DRR would like to
thank the NSF for financial support. We would like to thank Ludovic Berthier,
Joel Eaves, Peter Harrowell, Robert Jack, Peter Mayer and Marco Tarzia for
useful discussions.
## References
* Debenedetti and Stillinger (2001) P. Debenedetti and F. Stillinger, Nature 410, 259 (2001).
* Ediger et al. (1996) M. Ediger, C. Angell, and S. Nagel, J. Phys. Chem. 100, 13200 (1996).
* Bouchaud and Biroli (2004) J. Bouchaud and G. Biroli, J. Chem. Phys. 121, 7347 (2004).
* Dyre (2006) J. Dyre, Rev. Mod. Phys. 78, 953 (2006).
* Garrahan and Chandler (2002) J. Garrahan and D. Chandler, Phys. Rev. Lett. 89, 35704 (2002).
* Garrahan and Chandler (2003) J. Garrahan and D. Chandler, PNAS 100, 9710 (2003).
* Gotze and Sjogren (1992) W. Gotze and L. Sjogren, Rep. Prog. Phys. 55, 241 (1992).
* Hedges et al. (2009) L. Hedges, R. Jack, J. Garrahan, and D. Chandler, Science 323, 1309 (2009).
* Heuer (2008) A. Heuer, J. Phys. Cond. Mat. 20 (2008).
* Kivelson et al. (2004) D. Kivelson, S. A. Kivelson, X. Zhao, Z. Nussinov, and G. Tarjus, J. Chem. Phys. 121, 7347 (2004).
* Langer (2006) J. Langer, Phys. Rev. Lett. 97, 115704 (2006).
* Lubchenko and Wolynes (2007) V. Lubchenko and P. G. Wolynes, Ann. Rev. Phys. Chem. 58, 235 (2007).
* Schweizer and Saltzman (2003) K. Schweizer and E. Saltzman, J. Chem. Phys. 119, 1181 (2003).
* Butler and Harrowell (1991) S. Butler and P. Harrowell, J. Chem. Phys. 95, 4454 (1991).
* Hurley and Harrowell (1995) M. Hurley and P. Harrowell, Phys. Rev. E 52, 1694 (1995).
* Kob et al. (1997) W. Kob, C. Donati, S. Plimpton, P. Poole, and S. Glotzer, Phys. Rev. Lett. 79, 2827 (1997).
* Yamamoto and Onuki (1998) R. Yamamoto and A. Onuki, Phys. Rev. E 58, 3515 (1998).
* Schmidt-Rohr and Spiess (1991) K. Schmidt-Rohr and H. Spiess, Phys. Rev. Lett. 66, 3020 (1991).
* Ediger (2000) M. Ediger, Ann. Rev. Phys. Chem. 51, 99 (2000).
* Chang and Sillescu (1997) I. Chang and H. Sillescu, J. Phys. Chem. B 101, 8794 (1997).
* Cicerone and Ediger (1996) M. Cicerone and M. Ediger, J. Chem. Phys. 104, 7210 (1996).
* Jung et al. (2004) Y. Jung, J. Garrahan, and D. Chandler, Phys. Rev. E 69, 61205 (2004).
* Xia et al. (2001) X. Xia, P. Wolynes, et al., J. Phys. Chem. B 105, 6570 (2001).
* Tarjus and Kivelson (1995) G. Tarjus and D. Kivelson, J. Chem. Phys. 103, 3071 (1995).
* Berthier et al. (2005a) L. Berthier, D. Chandler, and J. Garrahan, Europhys. Lett. 69, 320 (2005a).
* Stariolo and Fabricius (2006) D. Stariolo and G. Fabricius, J. Chem. Phys. 125, 064505 (2006).
* Chaudhuri et al. (2007) P. Chaudhuri, L. Berthier, and W. Kob, Phys. Rev. Lett. 99, 60604 (2007).
* Saltzman and Schweizer (2008) E. Saltzman and K. Schweizer, Phys. Rev. E 77, 51504 (2008).
* Swallen et al. (2009) S. F. Swallen, K. Traynor, R. J. McMahon, M. D. Ediger, and T. E. Mates, J. Phys. Chem. B 113, 4600 (2009).
* Berthier (2004) L. Berthier, Phys. Rev. E 69, 20201 (2004).
* Pan et al. (2005) A. Pan, J. Garrahan, and D. Chandler, Phys. Rev. E 72, 41106 (2005).
* Szamel and Flenner (2006) G. Szamel and E. Flenner, Phys. Rev. E 73, 011504 (2006).
* Dasgupta et al. (1991) C. Dasgupta, A. Indrani, S. Ramaswamy, and M. Phani, Europhys. Lett. 15, 307 (1991).
* Lačević et al. (2003) N. Lačević, F. Starr, T. Schrøder, and S. Glotzer, J. Chem. Phys. 119, 7372 (2003).
* Franz and Parisi (2000) S. Franz and G. Parisi, J. Phys. Cond. Mat. 12, 6335 (2000).
* Berthier et al. (2007) L. Berthier, G. Biroli, J. Bouchaud, W. Kob, K. Miyazaki, and D. Reichman, J. Chem. Phys. 126, 184503 (2007).
* Franz and Montanari (2007) S. Franz and A. Montanari, J. Phys. A 40, F251 (2007).
* Biroli et al. (2008) G. Biroli, J. Bouchaud, A. Cavagna, T. Grigera, and P. Verrocchio, Nature Physics 4, 771 (2008).
* Karmakar et al. (2009) S. Karmakar, C. Dasgupta, and S. Sastry, PNAS 106, 3675 (2009).
* Berthier et al. (2005b) L. Berthier, G. Biroli, J. Bouchaud, L. Cipelletti, D. Masri, D. L’Hote, F. Ladieu, and M. Pierno, Science 310, 1797 (2005b).
* Biroli et al. (2006) G. Biroli, J. Bouchaud, K. Miyazaki, and D. Reichman, Phys. Rev. Lett. 97, 195701 (2006).
* Fredrickson and Andersen (1984) G. Fredrickson and H. Andersen, Phys. Rev. Lett. 53, 1244 (1984).
* Kob and Andersen (1993) W. Kob and H. Andersen, Phys. Rev. E 48, 4364 (1993).
* Ritort and Sollich (2003) F. Ritort and P. Sollich, Adv. in Phys. 52, 219 (2003).
* Biroli et al. (2005) G. Biroli, J. Bouchaud, and G. Tarjus, J. Chem. Phys. 123, 044510 (2005).
* Chandler and Garrahan (2005) D. Chandler and J. Garrahan, J. Chem. Phys. 123, 044511 (2005).
* Hedges and Garrahan (2008) L. Hedges and J. Garrahan, J. Phys. A: Math. and Theo. 41, 324006 (2008).
* Hedges and Garrahan (2007) L. Hedges and J. Garrahan, J. Phys. Cond. Mat. 19, 205124 (2007).
* Berthier (2007) L. Berthier, Phys. Rev. E 76, 011507 (2007).
* Biroli and Mézard (2001) G. Biroli and M. Mézard, Phys. Rev. Lett. 88, 025501 (2001).
* McCullagh et al. (2005) G. McCullagh, D. Cellai, A. Lawlor, and K. Dawson, Phys. Rev. E 71, 030102 (2005).
* Dawson et al. (2003) K. Dawson, S. Franz, and M. Sellitto, Europhys. Lett. 64, 302 (2003).
* Ciamarra et al. (2003) M. Ciamarra, M. Tarzia, A. de Candia, and A. Coniglio, Phys. Rev. E 67, 057105 (2003).
* Pica Ciamarra et al. (2003) M. Pica Ciamarra, M. Tarzia, A. de Candia, and A. Coniglio, Phys. Rev. E 68, 066111 (2003).
* Bortz et al. (1975) A. Bortz, M. Kalos, and J. Lebowitz, J. Comput. Phys 17 (1975).
* Biroli and Toninelli (2008) G. Biroli and C. Toninelli, Euro. Phys. J. B 64, 567 (2008).
* Rivoire et al. (2003) O. Rivoire, G. Biroli, O. Martin, and M. Mezard, Euro. Phys. J. B 37, 55 (2003).
* Balucani and Zoppi (1994) U. Balucani and M. Zoppi, _Dynamics of the liquid state_ (Oxford University Press, USA, New York, 1994).
* Kob and Andersen (1995) W. Kob and H. Andersen, Phys. Rev. E 51, 4626 (1995).
* Wahnström (1991) G. Wahnström, Phys. Rev. A 44, 3752 (1991).
* Sarlat et al. (2009) T. Sarlat, A. Billoire, G. Biroli, and J. Bouchaud, J. Stat. Mech p. P08014 (2009).
* Berthier and Witten (2009) L. Berthier and T. A. Witten, Europhys. Lett. 86, 10001 (2009).
* Chaudhuri et al. (2008) P. Chaudhuri, S. Sastry, and W. Kob, Phys. Rev. Lett. 101, 190601 (2008).
* Toninelli et al. (2004) C. Toninelli, G. Biroli, and D. Fisher, Phys. Rev. Lett. 92, 185504 (2004).
* Donati et al. (1998) C. Donati, J. Douglas, W. Kob, S. Plimpton, P. Poole, and S. Glotzer, Phys. Rev. Lett. 80, 2338 (1998).
* Donati et al. (1999) C. Donati, S. Glotzer, P. Poole, W. Kob, and S. Plimpton, Phys. Rev. E 60, 3107 (1999).
* Doliwa and Heuer (1999) B. Doliwa and A. Heuer, J. Phys. Cond. Mat. 11, 277 (1999).
* Appignanesi et al. (2006) G. Appignanesi, J. Rodríguez Fris, R. Montani, and W. Kob, Phys. Rev. Lett. 96, 57801 (2006).
* Stillinger (1988) F. Stillinger, J. Chem. Phys. 89, 6461 (1988).
* Widmer-Cooper et al. (2004) A. Widmer-Cooper, P. Harrowell, and H. Fynewever, Phys. Rev. Lett. 93, 135701 (2004).
* (71) R. K. Darst, D. R. Reichman, and G. Biroli, to be published.
* Stillinger and Hodgdon (1994) F. Stillinger and J. Hodgdon, Phys. Rev. E 50, 2064 (1994).
* Liu and Oppenheim (1996) C. Liu and I. Oppenheim, Phys. Rev. E 53, 799 (1996).
* Mézard and Montanari (2006) M. Mézard and A. Montanari, J. Stat. Phys. 124, 1317 (2006).
* Stein and Andersen (2008) R. Stein and H. Andersen, Phys. Rev. Lett. 101, 267802 (2008).
* Jack and Garrahan (2005) R. Jack and J. Garrahan, J. Chem. Phys. 123, 164508 (2005).
* Richert and Angell (1998) R. Richert and C. Angell, J. Chem. Phys. 108, 9016 (1998).
* Martinez and Angell (2001) L. Martinez and C. Angell, Nature 410, 663 (2001).
* Crisanti and Ritort (2000) A. Crisanti and F. Ritort, Europhys. Lett. 51, 147 (2000).
* Berthier and Jack (2007) L. Berthier and R. Jack, Phys. Rev. E 76, 41509 (2007).
* Candelier et al. (2009) R. Candelier, O. Dauchot, and G. Biroli, Phys. Rev. Lett. 102, 088001 (2009).
|
arxiv-papers
| 2009-11-03T03:29:00 |
2024-09-04T02:49:06.241832
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Richard K. Darst (1) and David R. Reichman (1) and Giulio Biroli (2)\n ((1) Columbia University, New York, USA (2) Institut de Physique Th\\'eorique,\n Gif-sur-Yvette, France)",
"submitter": "Richard Darst",
"url": "https://arxiv.org/abs/0911.0479"
}
|
0911.0558
|
# The effect of sublattice symmetry breaking on the electronic properties of a
doped graphene
Alireza Qaiumzadeh School of Physics, Institute for Research in Fundamental
Sciences (IPM), Tehran 19395-5531, Iran Institute for Advanced Studies in
Basic Sciences (IASBS), Zanjan, 45195-1159, Iran Reza Asgari 111Corresponding
author: Tel: +98 21 22280692; fax: +98 21 22280415.
E-mail address: asgari@theory.ipm.ac.ir, School of Physics, Institute for
Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran
###### Abstract
Motivated by a number of recent experimental studies, we have carried out the
microscopic calculation of the quasiparticle self-energy and spectral function
in a doped graphene when a symmetry breaking of the sublattices is occurred.
Our systematic study is based on the many-body G0W approach that is
established on the random phase approximation and on graphene’s massive Dirac
equation continuum model. We report extensive calculations of both the real
and imaginary parts of the quasiparticle self-energy in the presence of a gap
opening. We also present results for spectral function, renormalized Fermi
velocity and band gap renormalization of massive Dirac Fermions over a broad
range of electron densities. We further show that the mass generating in
graphene washes out the plasmaron peak in spectral weight.
###### pacs:
71.10.Ay, 73.63.-b, 72.10.-d, 71.55.-i
## I INTRODUCTION
Graphene is a single atomic layer of crystalline carbon on the honeycomb
lattice consists of two interpenetrating triangular sublattices A and B, has
opened up a new field for fundamental studies and applications. novoselov1 ;
novoselov2 ; novoselov3 ; novoselov4 Peculiar electronic properties of
graphene give rise possibility to come over silicon-based electronics
limitations. carbon The single-particle energy spectrum in graphene contains
two zero-energy at $K^{+}$ and $K^{-}$ points of the Brillouin zone which are
called as valleys or Dirac points. Due to the presence of the two carbon atoms
per unit cell, the quasiparticle (QP) need to be described by a two component
wave function.
The charge carriers in a pristine graphene show linear and isotropic energy
dispersion relation and massless chiral behavior for the energy scales up to 1
eV. Recently, graphene has revealed a variety of unusual transport phenomena
characteristics of two-dimensional (2D) Dirac Fermions such as an anomalous
integer quantum Hall effect at room temperature, a minimum quantum
conductivity, Klein tunneling paradox, weak and anti-localization, an absence
of Wigner crystallization phase and Shubnikov-deHaas oscillations that exhibit
a phase shift of $\pi$ due to Berry’s phase. graphene_rev1 ; graphene_rev2 ;
graphene_rev3 ; graphene_rev4 ; graphene_rev5 ; graphene_rev6 ; tomadin One
important difference between conventional electron gas and Dirac Fermion
particle is that the contribution of exchange and correlation to the chemical
potential is an increasing rather than a decreasing function of carrier-
density. This property implies that exchange and correlation increase the
effectiveness of screening, in contrast to the usual case in which exchange
and correlation weakens screening. This unusual property follows from the
difference in sublattice pseudospin chirality between the Dirac model’s
negative energy valence band states and its conduction band states.
The massless Dirac-like carriers in graphene have almost semi-ballistic
transport behavior with small resistance due to the suppression of back-
scattering process, and moreover graphene is a good thermal conductor. thermal
The mobility of carriers in graphene is quite high morozov1 ; morozov2 ;
morozov3 ; morozov4 which is much higher than the electron mobility revealed
on the semiconductor hetrostructures. eng1 ; eng2 On the other hand, by
measuring the stiffness of materials it is shown that graphene is the
strongest material in two-dimension structures. strong1 ; strong2 These
properties as well as capability to control of the type and density of charge
carriers by gate voltage or the chemical doping dop1 ; dop2 ; dop_lanzara1 ;
dop_lanzara2 make graphene an ideal candidate for superior nano-electronic
devices operating at high frequencies.
Most electronic applications are based on the presence of a gap between the
valence and conduction bands in the conventional semiconductors. The band gap
is a measure of the threshold voltage and on-off ratio of the field effect
transistors (FETs). FET1 ; FET2 Therefore, for integrating graphene into
semiconductor technology, it is crucial to induce a band gap in Dirac points
to control the transport of carriers. Consequently, band gap engineering in
graphene is a hot topic with fundamental and applied significance. gap In the
literature several routes have being proposed and applied to induce and
control a gap in graphene. One of them is using quantum confined geometries
such as quantum dots and nanoribbons. gap_ribbon1 ; gap_ribbon2 ; gap_ribbon3
; gap_ribbon4 ; gap_ribbon5 It is shown that the gap values increases by
decreasing of nanoribbon width. Another alternative way is spin-orbit coupling
whose origin is due to both intrinsic spin-orbit interactions and the Rashba
interaction. gap_spin1 ; gap_spin2 ; gap_spin3 ; gap_spin4 Another method to
generate a gap in graphene sheets is an inversion symmetry breaking of the
sublattices when the number of electrons on A and B atoms are different
gapsub1 ; gapsub2 ; gapsub3 ; gapsub4 or Kekulé kekule1 distortion, e.g.
graphene on proper substrates dop_lanzara1 ; dop_lanzara2 ; lanzara1 ;
lanzara2 ; eva ; gruneis1 ; gruneis2 ; giovannetti or adsorb of some
molecules such as water, ammonia ribeiro1 ; ribeiro2 and CrO3 zanella or an
alkali-metal sub-monolayer on graphene sheets.
Recently angle resolved photoemission spectroscopy ( ARPES) experiments on
graphene epitaxially grown on SiC and ab initio simulations reported a gap
opening in the band structure of graphene placed on proper substrates, and
suggested that interactions between the graphene sheet and the substrate leads
to symmetry breaking of the A and B sublattices and it consequences to induce
a gap in the band structure. Experimenters dop_lanzara1 ; dop_lanzara2 ;
lanzara1 ; lanzara2 ; kruczynski observed a gap of 260 meV in band structure
of the epitaxial graphene on SiC substrate due to interaction with substrate.
In addition, Zhou et al. dop_lanzara1 found a reversible metal-insulator
transition and a fine tuning of the carriers from electron to hole by
molecular doping in gapped graphene. A Density Functional Theory (DFT)
calculation confirmed a substrate induced symmetry breaking. kim . Their
results showed a gap in the band spectra of graphene about 200 meV which is in
agreement with recent experimental observation. Their calculation determined
that there is a 140 meV on-site energy difference between two sublattices. In
addition, a band gap is observed in spectra of graphene on Ni(111) substrate
gruneis1 ; gruneis2 as well as a gap about of 10 meV in suspended graphene
above a graphite substrate eva due to sublattice symmetry breaking mechanism.
Moreover, based on the ab initio calculations, it is suggested that boron
nitride substrate induced a gap of 53 meV. giovannetti Note that the gap
value calculated within DFT is in general underestimating the true band gap
value.
In this paper we consider the sublattice symmetry breaking mechanism for a gap
opening in a pristine doped graphene sheet and study the impact of gap upon
some electronic properties of QPs. To investigate the influence of gap in the
many-body properties of QP in graphene we use the random phase approximation
(RPA) and the G0W approximation. It should be noted that a detailed analysis
provided a framework for the microscopic evaluation of the QP-QP interaction
in the gapless graphene by means of the RPA was carried out by us in Ref.
[im1, ] At the beginning, we review briefly the results of the ground state
thermodynamic properties that we have already presented elsewhere. alireza
Our new results are based on the QP self-energy properties in the presence of
a gap opening in the electronic spectrum. From the self-energy we then obtain
the QP energies, renormalized Fermi velocity, spectral function which can be
compared with ARPES spectra and finally the band gap renormalization of
massive Dirac Fermions in doped graphene. We have shown that mass generating
in graphene washes out a satellite band in the spectral function in agreement
with recent experimental observations. lanzara1
This paper is organized as followed. In Section II we introduce our model
Hamiltonian and then review some ground state properties of gapped graphene.
In Section III we focus on the properties of imaginary and real parts of self-
energy for gapped graphene and then calculate QP spectral function,
renormalized Fermi velocity and band gap renormalization. Finally we conclude
in Section IV.
## II GROUND STATE THERMODYNAMIC PROPERTIES
We consider the sublattice symmetry breaking mechanism in which the densities
of particles associated to on-site energy $\mu_{a(b)}$, for A(B) sublattice
are different. The electronic structure of graphene can be reasonably good
described using a rather simple tight-binding Hamiltonian, leading to
analytical solutions for their energy dispersion and related eigenstates. The
noninteracting tight binding Hamiltonian for $\pi$ band electrons is
determined by gapsub1 ; gapsub2 ; gapsub3 ; gapsub4
$\displaystyle\hat{H_{0}}$ $\displaystyle=$ $\displaystyle
t\sum_{i}(a_{i}^{\dagger}b_{i}+{\rm
c.c.})+\mu_{a}\sum_{i}a^{\dagger}_{i}a_{i}+\mu_{b}\sum_{i}b^{\dagger}_{i}b_{i}$
(1) $\displaystyle=$ $\displaystyle t\sum_{i}(a_{i}^{\dagger}b_{i}+{\rm
c.c.})+\frac{\mu_{a}-\mu_{b}}{2}\sum_{i}(a^{\dagger}_{i}a_{i}-b^{\dagger}_{i}b_{i})+\frac{\mu_{a}+\mu_{b}}{2}\sum_{i}(a^{\dagger}_{i}a_{i}+b^{\dagger}_{i}b_{i})$
where the sums run over unit cells, $t\simeq 2.7$ eV denotes the nearest
neighbor hopping parameter and $a_{i}(b_{i})$ is Fermi annihilation operator
acts on A(B) sublattice. The second term in the noninteracting Hamiltonian
breaks the inversion symmetry and causes to a band gap with value of
$2\Delta=|\mu_{a}-\mu_{b}|$ at the Dirac points. The last term is a constant
and we left it out. The effective Hamiltonian at low excited energies lead to
a 2D massive Dirac Hamiltonian, $\mathcal{\hat{H}}_{0}=\hbar v_{\rm
F}{\vec{\sigma}}\cdot{\bf k}+\Delta\sigma_{3}$, where $\vec{\sigma}$ are Pauli
matrices and $v_{\rm F}=3ta/2\hbar\simeq 10^{6}$ m/s is the Fermi velocity
where $a\simeq 1.42$ Å is the carbon-carbon distance in honeycomb lattice. The
two eigenvalues of noninteracting Hamiltonian are given by $E_{\bf
k}=\pm\sqrt{(\hbar v_{\rm F}k)^{2}+\Delta^{2}}$ for conduction band (+) and
valance band (-) which is a fully occupied. In addition, the model Hamiltonian
can be used as an approximated model for describing a graphene antidot lattice
in the vicinity of a band gap with a small effective mass value antidot , or
moreover used as an effective Hamiltonian for the intrinsic spin-orbit
interaction in graphene where $\Delta=\Delta_{\rm SO}$ is the strength of the
spin-orbit interaction. gap_spin1 ; gap_spin2 ; gap_spin3 ; gap_spin4 If
$\mu_{a}=\mu_{b}$ the Hamiltonian reduces to massless Dirac Hamiltonian with
two chiral eigenstates having the conical band structures
$\varepsilon_{k}=\pm\hbar v_{\rm F}k$.
We consider the long-range Coulomb electron-electron interaction. We left out
the intervalley scattering and use the two component Dirac Fermion model.
Accordingly, the total interacting Hamiltonian in a continuum model at $K^{+}$
point is expressed as yafis ; Giuliani
$\hat{H}=\sum_{{\bf k},\sigma}\Psi^{\dagger}_{{\bf
k},\sigma}\mathcal{\hat{H}}_{0}\Psi_{{\bf k},\sigma}+\frac{1}{2S}\sum_{{\bf
q}\neq 0}V_{q}({\hat{n}}_{\bf q}{\hat{n}}_{-{\bf q}}-{\hat{N}}),$ (2)
where $\Psi^{\dagger}_{{\bf k},\sigma}=(\psi^{a}_{+,\sigma}({\bf
k}),\psi^{b}_{+,\sigma}({\bf k}))$ is two component pseudospinors of the
noninteracting Hamiltonian, $S$ is the sample area, ${\hat{N}}$ is the total
number operator and $V_{q}=2\pi e^{2}/\epsilon q$ is the bare Coulomb
interaction where $\epsilon$ is an average dielectric constant of the
surrounding medium. The coupling constant in graphene is
$\alpha_{gr}=g_{s}g_{v}e^{2}/\epsilon\hbar v_{\rm F}$ where $g_{s}=g_{v}=2$
being the spin and valley degeneracy, respectively. The coupling constant in
graphene depends only on the substrate dielectric constant while in the
conventional 2D electron systems is density dependent. The typical value of
dimensionless coupling constant is 1 or 2 for graphene supported on a
substrate such a SiC or SiO2.
A central quantity in the many-body techniques is the noninteracting dynamical
polarizability function $\chi^{(0)}({\bf q},i\omega,\mu)$ where $\mu$ is the
chemical potential. The problem of linear density response is set up by
considering a fluid described by the Hamiltonian, $\hat{H}$, which is subject
to an external potential. The external potential must be sufficiently weak for
low-order perturbation theory to suffice. The induced density change has a
linear relation to the external potential through the noninteracting dynamical
polarizability function. This function is recently calculated along the
imaginary frequency axis and it is given by alireza
$\displaystyle\chi^{(0)}({\bf q},i\omega,\mu\geq\Delta)$ $\displaystyle=$
$\displaystyle-\frac{g_{s}g_{v}}{2\pi\hbar^{2}v_{\rm
F}^{2}}\\{\mu-\Delta+\frac{\varepsilon_{q}^{2}}{2}(\frac{\Delta}{{y^{2}}}+\frac{x_{-}^{2}}{2y}\tan^{-1}(\frac{y}{2\Delta}))$
(3) $\displaystyle-$ $\displaystyle\frac{\varepsilon_{q}^{2}}{4y}\Re
e\left[x_{-}^{2}(\sin^{-1}\frac{z(\mu)}{x_{+}}-\sin^{-1}\frac{z(\Delta)}{x_{+}})\right]$
$\displaystyle+$ $\displaystyle\frac{\varepsilon_{q}^{2}}{4y}\Re
e\left[z(\mu)\sqrt{x_{+}^{2}-z^{2}(\mu)}-z(\Delta)\sqrt{x_{+}^{2}-z^{2}(\Delta)}\right]\\},$
where $x_{\pm}=\sqrt{1\pm
4\Delta^{2}/(\varepsilon_{q}^{2}+\hbar^{2}\omega^{2})}$,
$y=\sqrt{\varepsilon_{q}^{2}+\hbar^{2}\omega^{2}}$ and
$z(x)=(2x+i\hbar\omega)/\varepsilon_{q}$. The Fermi energy of a 2D massive
Dirac Fermion system is given by $E_{\rm F}=\mu=\sqrt{(\hbar v_{\rm F}k_{\rm
F})^{2}+\Delta^{2}}$ and the Fermi wavevector is $k_{\rm F}=\sqrt{4\pi
n/g_{s}g_{v}}$ where $n$ is the density of carriers. The noninteracting
density of states (DOS) is determined by
$D(E)=g_{s}g_{v}|E|/2\pi\hbar^{2}v_{\rm F}^{2}\Theta(E^{2}-\Delta^{2})$ which
is density dependent at the Fermi surface. It should be noticed that $D(E_{\rm
F})$ equals to $m/2\pi\hbar^{2}$ in the conventional 2D electron gas system.
Here, $\Theta(x)$ is Heaviside step function.
We now turn to present our first numerical results which are based on the
noninteracting polarization function. The static polarization function as a
function of wavevector for various gap values is shown in Fig. 1(a). The
static polarization function in gapless case is a smooth function whereas a
kink at $q=2k_{\rm F}$ occurs for gapped graphene and thus the derivatives of
$\chi^{(0)}({\bf q},0,\Delta\neq 0)$ has a singular feature. The singular
behavior is the source of several phenomena such as the Friedel oscillations
and moreover the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction which the
later is absent in gapless graphene. In Figs. 1(b) and (c) we have plotted the
dynamic polarization function as a function of frequency for wave vectors
smaller and larger than $q=2k_{\rm F}$, respectively. $\chi^{(0)}({\bf
q},i\omega)$ tends to zero like $\omega^{-1}$ at large frequency region.
The polarization function along the real $\omega$ axis can be obtained by
performing analytical continuation of Eq. (3). alireza ; pyatkovskiy1 ;
pyatkovskiy2 In Fig. 2 we have presented the real and imaginary parts of the
noninteracting polarization function as a function of frequency. Sharp cutoffs
in the imaginary part of $\chi^{(0)}({\bf q},\omega)$ are related to the rapid
swing in the real part of $\chi^{(0)}({\bf q},\omega)$. These behaviors are in
result of the fact that the real and imaginary parts of the polarization
function are related through the Kramers-Krönig relations. Importantly, the
sign change of the real part from negative to positive shows a sweep across
the electron-hole continuum. At very large gap values, the polarization
function of massive Dirac Fermions can be reduced to the polarization function
(the Lindhard’s function) of conventional two dimensional electron gas
systems, as they are determined in Figs. 1 and 2. Consequently, we settle
under situation that we can describe a range of band structures from the
Dirac’s cone (gapless graphene) to the parabolic (conventional semiconductors)
band structure behavior by tuning the gap values from zero to a large value,
respectively. We limited our calculations to the intermediate values of of
$\Delta$ and we thus expect wide range of the particular properties related to
unique behavior of the polarization function.
Figure 1: (Color online) (a): The static noninteracting polarization function
as a function of $q$ for various $\Delta$. The dimensionless noninteracting
dynamic polarization at (b): $q=0.5k_{\rm F}$ and (c): $q=2k_{\rm F}$ as a
function of $\omega$ for various $\Delta$.
Figure 2: (Color online) The gap dependence of the real and the imaginary part
of the noninteracting polarization function as a function of $\omega$ for
wavevectors (a), (c) $q=0.5k_{\rm F}$ and (b), (d) $q=2.5k_{\rm F}$
We can calculate the total ground state energy of gapped graphene within RPA.
alireza ; yafis The ground-state energies can be calculated using the
coupling constant integration technique, which has the contributions
$\varepsilon_{tot}=\varepsilon_{kin}+\varepsilon_{\rm xc}$. The kinetic energy
per particle is given by $\varepsilon_{kin}=2(E_{\rm
F}^{3}-\Delta^{3})/3\varepsilon_{\rm F}^{2}$.
As discussed previously alireza ; yafis we might subtract the vacuum energy
contribution from the total energy,
$\delta\varepsilon_{tot}=\varepsilon_{tot}(k_{\rm F})-\varepsilon_{tot}(k_{\rm
F}=0)$. Due to the number of states in the Brillouin zone must be conserved,
we do need a ultraviolet cut-off $k_{c}$, which is approximated by $\pi
k_{c}^{2}\simeq(2\pi)^{2}/A_{0}$, where $A_{0}$ is the area of the unit cell.
The dimensionless parameter $\Lambda$ is defined as $k_{c}/k_{\rm
F}\simeq(g_{s}g_{v}n^{-1}\sqrt{3}/9.09)^{1/2}\times 10^{2}$.
Figure 3: (Color online) (a) Exchange-correlation energy and (b)
compressibility as a function of $n^{-1/2}$ ( in units of 10-6 cm) for various
$\Delta$ value.
In Fig. 3, we have shown the exchange- correlation energy in units of
$\varepsilon_{\rm F}=\hbar v_{\rm F}k_{\rm F}$, as a function of $n^{-1/2}$ in
units of 10-6 cm for various $\Delta$ value. The exchange energy arises
entirely from the antisymmetry of the many-body wave function under exchange
of two electrons is positive while the correlation energy, the difference
between the ground state energy and the sum of the kinetic energy and the
exchange energy is negative. This has important implications on the
thermodynamic properties can be calculated from the derivative of the ground
state energy with respect to the density. The compressibility can be
calculated from its definition,
$\kappa^{-1}=n^{2}\partial^{2}(n\delta\varepsilon_{tot})/\partial n^{2}$. Fig.
3(b) shows the ratio between the noninteractiong value,
$\kappa_{0}=2/n\varepsilon_{\rm F}$ and the interaction value of
compressibility as a function of $n^{-1/2}$. The exchange tends to reduce the
compressibility while correlations tends to enhance it. At large $\Delta$, a
minimum structure occurs at the inverse of compressibility behavior and we
expect that at very large $\Delta$, it starts at $\kappa_{0}$ and reduces by
increasing $n^{-1/2}$ behaves like the compressibility of the conversional 2D
electron gas.
## III THE QP SELF-ENERGY AND THE SPECTRAL FUNCTION
The generation of QPs in an electron liquid leads to two effects. First it
induces a decay of a particle losing momentum via inelastic scattering which
is determined by the imaginary part of self-energy and second is the
renormalization of the dispersion relation of the carriers which is described
by the real part of self-energy. $\Re e\Sigma^{\rm ret}({\bf k},\omega)$ is
defined as the difference between the measured carrier energy $\hbar\omega$,
and the energy of free particle, $\xi_{s{\bf k}}=sE_{{\bf k}}-E_{\rm F}$. To
satisfy causality, the real and imaginary parts of self-energy are related by
a Hilbert transformation. In this section, we first derive the imaginary and
the real part of QP self-energies and then calculate some important quantities
such as a renormalized Fermi velocity, a spectral function and a band gap
renormalization in the presence of a band gap value. These quantities are
related to some important physical properties of both theoretical and
practical applications like the band structure of ARPES, the energy
dissipation rate of injected carriers and the width of the QP spectral
function. kaminski1 ; kaminski2
In the G0W approximation, the self-energy of gapped graphene is given by
($\beta=1/(k_{B}T)$) fateme :
$\displaystyle\Sigma_{s}({\bf{k}},i\omega_{n})$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\sum_{s^{\prime}}\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}F^{ss^{\prime}}({\bf k,k+q})$ $\displaystyle\times$
$\displaystyle\sum_{m=-\infty}^{+\infty}W({\bf
q},i\Omega_{m})G^{(0)}_{s^{\prime}}({\bf k+q},i\omega_{n}+i\Omega_{m}),$
where $W({\bf q},i\Omega_{m})=V_{q}/\epsilon(q,i\Omega_{m})$ is the dynamical
screened effective interaction and
$\epsilon(q,i\Omega_{m})=1-V_{q}\chi^{(0)}(q,i\Omega_{m})$ is dynamical
dielectric function in RPA. The overlap function for gapped graphene
$F^{ss^{\prime}}({\bf k,k+q})$ arises from the graphene band structure is
given by alireza
$\displaystyle F^{ss^{\prime}}({\bf
k,k+q})=\frac{1}{2}(1+ss^{\prime}\frac{\hbar^{2}v_{\rm F}^{2}{\bf k}\cdot({\bf
k}+{\bf q})+\Delta^{2}}{E_{\bf k}E_{\bf k+q}}).$ (5)
It should be noted that $F^{s=-s^{\prime}}({\bf q}=0)=0$. However, in gapless
graphene, intraband backward scattering should not be allowed, namely
$F^{s=s^{\prime}}({\bf q}=-2\textbf{k},\Delta=0)=0$, as well as
$F^{s=-s^{\prime}}({\bf q}=0,\Delta=0)=0$. In Eq. (III), $G_{s}^{(0)}({\bf
k},i\omega)=1/(i\omega-\xi_{s{\bf k}}/\hbar)$ is the noninteracting Green’s
function. Notice that in typical density of carriers in graphene namely
$n>10^{12}$cm-2, the Fermi temperature is about $T_{\rm F}=\varepsilon_{\rm
F}/k_{B}>10^{3}$K, and we therefore can eliminate temperature parameter in our
calculations. To evaluate the zero-temperature retarded self-energy we perform
the line-residue decompositions, $\Sigma^{\rm ret}_{s}({\bf
k},\omega)=\Sigma^{\rm line}_{s}({\bf k},\omega)+\Sigma^{\rm res}_{s}({\bf
k},\omega)$, where $\Sigma^{\rm line}$ is obtained by performing the analytic
continuation before summing over the Matsubara frequencies, and $\Sigma^{\rm
res}$ is the correction which must be taken into account in the total self-
energy. Giuliani At zero temperature we have
$\displaystyle\Sigma^{\rm line}_{s}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle-\sum_{s^{\prime}}\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}V_{q}F^{ss^{\prime}}({\bf k,k+q})$ $\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\Omega}{2\pi}\frac{1}{\epsilon({\bf
q},i\Omega)}\frac{1}{\omega+i\Omega-\xi_{s^{\prime}}({\bf k+q})/\hbar},$
and
$\displaystyle\Sigma^{\rm res}_{s}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\sum_{s^{\prime}}\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}\frac{V_{q}}{\epsilon({\bf q},\omega-\xi_{s^{\prime}}({\bf
k+q})/\hbar)}F^{ss^{\prime}}({\bf k,k+q})$ (7) $\displaystyle\times$
$\displaystyle[\Theta(\omega-\xi_{s^{\prime}}({\bf
k+q})/\hbar)-\Theta(-\xi_{s^{\prime}}({\bf k+q})/\hbar)].$
The line contribution of the self-energy is purely real. The imaginary part of
the self-energy has two contributions where $\Im m\Sigma_{+}^{\rm
ret}({\bf{k}},\omega)=\Im m\Sigma_{+,\rm intra}^{\rm res}({\bf{k}},\omega)+\Im
m\Sigma_{+,\rm inter}^{\rm res}({\bf{k}},\omega)$, and real part of the self
energy can be decomposed as $\Re e\Sigma_{+}^{\rm
ret}({\bf{k}},\omega)=\Sigma_{+}^{\rm line}({\bf{k}},\omega)+\Re
e\Sigma_{+,\rm inter}^{\rm res}({\bf{k}},\omega)+\Re e\Sigma_{+,\rm
intra}^{\rm res}({\bf{k}},\omega)$.
For $\omega>0$ and fixed ${\bf q}$, the RPA decay process represents
scattering of an electron from momentum ${\bm{k}}$ and energy $\omega$ to
${\bm{k}}+{\bm{q}}$ and $\xi_{s^{\prime}}({\bm{k}}+{\bm{q}})$, with all
energies in Eq. (7) measured from the Fermi energy of doped graphene. Since
the Pauli exclusion principle requires that the final state is unoccupied, it
must lie in the conduction band, i.e. $s^{\prime}=+1$. Furthermore since the
Fermi sea is initially in its ground state, the QP must lower its energy, i.e.
$\xi_{s^{\prime}}<\omega$, electrons decay by going down in energy. For
$\omega<0$, the self-energy expresses the decay of holes inside the Fermi sea,
which scatter to a final state, by exciting the Fermi sea. In this case the
final state must be occupied so both band indices are allowed for
$s^{\prime}$, and energy conservation requires that holes decay by moving up
in energy. Since photoemission measures the properties of holes produced in
the Fermi sea by photo ejection, only $\omega<0$ is relevant for this
experimental probe.
In what follows, we calculate the intraband and interband contributions of
self-energy. We have found the intraband term of residue part of self-energy
as following for various values of the frequencies,
$\displaystyle\Sigma_{+,\rm intra}^{\rm res}({\bf k},\omega>0)=C\int_{\rm
max(0,~{}k_{\rm F}-k,~{}k-\beta)}^{k+\beta}dq\int^{\rm min(\hbar\omega+E_{\rm
F},~{}\alpha_{+})}_{\rm max(E_{\rm F},~{}\alpha_{-})}dyf_{+}(y,q)$
$\displaystyle\Sigma_{+,~{}\rm intra}^{\rm res}({\bf k},\Delta-E_{\rm
F}<\hbar\omega<0)=-C\int_{\rm max(0,~{}k-k_{\rm F},~{}\beta-k)}^{k+k_{\rm
F}}dq\int^{\rm min(E_{\rm F},~{}\alpha_{+})}_{\rm max(0,~{}\hbar\omega+E_{\rm
F},~{}\alpha_{-})}dyf_{+}(y,q)$ $\displaystyle\Sigma_{+,\rm intra}^{\rm
res}({\bf k},\hbar\omega<-(\Delta+E_{\rm F}))=-C\int_{\rm max(0,~{}k-k_{\rm
F})}^{k+k_{\rm F}}dq\int^{\rm min(E_{\rm F},~{}\alpha_{+})}_{\rm
max(0,~{}\hbar\omega+E_{\rm F},~{}\alpha_{-})}dyf_{+}(y,q)~{},$ (8)
where
$f_{\pm}(y,q)=\frac{\pm(y\pm E_{k})^{2}\mp q^{2}}{\epsilon({\bf
q},\omega+E_{\rm F}\mp y)\sqrt{4k^{2}q^{2}-(y^{2}-E_{k}^{2}-q^{2})^{2}}}~{},$
$C=e^{2}/2\pi\epsilon E_{k}$, $\alpha_{\pm}=\sqrt{\hbar^{2}v^{2}_{\rm F}(k\pm
q)^{2}+\Delta^{2}}$ and
$\beta=\sqrt{\hbar^{2}\omega^{2}+\hbar^{2}v^{2}k^{2}_{\rm F}+2\hbar\omega
E_{\rm F}}$. On the other hand, the interband contribution of residue part of
the self energy is determined by
$\Sigma_{+,\rm inter}^{\rm res}({\bf k},\hbar\omega<-(\Delta+E_{\rm
F}))=-C\int_{\rm max(0,~{}k-\beta)}^{k+\beta}dq\int_{\alpha_{-}}^{\rm
min(\alpha_{+},-\hbar\omega-E_{\rm F})}dyf_{-}(y,q)~{},$ (9)
and eventually for the line contribution of self-energy we have
$\displaystyle\Sigma_{+,\rm intra}^{\rm line}({\bf{k}},\omega)$
$\displaystyle=$
$\displaystyle-\frac{e^{2}}{4\pi^{2}\epsilon}\int_{0}^{k_{c}}dq\int_{0}^{2\pi}d\phi
F^{++}({\bf
q,q+k},\Delta)\int_{-\infty}^{+\infty}d\Omega\frac{g_{+}(\phi,\Omega,q)}{\epsilon({\bf
q},i\Omega)}$ $\displaystyle\Sigma_{+,\rm inter}^{\rm line}({\bf{k}},\omega)$
$\displaystyle=$
$\displaystyle-\frac{e^{2}}{4\pi^{2}\epsilon}\int_{0}^{k_{c}}dq\int_{0}^{2\pi}d\phi
F^{+-}({\bf
q,q+k},\Delta)\int_{-\infty}^{+\infty}d\Omega\frac{g_{-}(\phi,\Omega,q)}{\epsilon({\bf
q},i\Omega)}$ (10)
where
$g_{\pm}(\phi,\Omega,q)=\frac{\hbar\omega+E_{\rm F}\mp
E_{\bf{k+q}}}{(\hbar\omega+E_{\rm F}\mp E_{{\bf
k+q}})^{2}+\hbar^{2}\Omega^{2}}$ (11)
and $\phi$ denotes an angle between ${\bf k}$ and ${\bf q}$. Note that the
real part of self-energy is $k_{c}$ dependent.
Now we are in a situation that can calculate some important physical
quantities. The QP lifetime or the single-particle relaxation time $\tau$, is
obtained by setting the frequency to the on-shell energy in imaginary part of
the self-energy, $\tau_{s}^{-1}=\Gamma_{s}({\bf k},{\xi_{s\bf
k}}/\hbar)=\frac{2}{\hbar}|\Im m\Sigma_{s}^{\rm ret}({\bf k},\xi_{s\bf
k}/\hbar)|$ where $\Gamma_{s}({\bf k},{\xi_{s\bf k}}/\hbar)$ is the quantum
level broadening of the momentum eigenstate $|s{\bf k}>$. This quantity is
identical with the Fermi’s golden rule expression for the sum of the
scattering rate of a QP and quasihole at wavevector ${\bf k}$. Giuliani From
Eqs. 8 and 9, one can conclude that total contribution of the imaginary part
of the retarded self-energy on the energy shell comes from the intraband term,
$\Im m\Sigma_{+}^{\rm ret}({\bf k},\xi_{\bf k}/\hbar)=\Im m\Sigma^{\rm
res}_{\rm intra}({\bf k},\xi_{\bf k}/\hbar)$. fateme In the case of gapless
graphene, scattering rate is a smooth function because of the absence of both
plasmon emission and interband processes. inelastic1 ; inelastic2 However,
with generating a gap and increasing the amount of it, plasmon emission cause
discontinuities in the scattering time, similar to conventional 2D electron
gas. asgari1 ; asgari2 We have thus two mechanisms for scattering of the QPs.
The excitation of electron-hole pairs which is dominant process at long
wavelength regions and the excitation of plasmon appears in a specific wave
vector. As discussed previously fateme , in clean graphene sheets the
inelastic mean free path reduces by increasing the gap whereas the mean free
path is large enough in the range of the typical gap values 10-130 meV, and
thus transport remains in the semi-ballistic regime.
The many-body interactions in graphene as a function of doping can be observed
by ARPES which plays as a central role to investigate QP properties such as
group velocity and lifetime of carriers on the Fermi surface. ARPES is a
useful complementary tool which capable of measuring the constant energy
surfaces for all partially occupied states and the fully occupied band
structure. The information of band dispersion and the Fermi surface can be
elicited from those data measured in ARPES experiments. The relation of the
Green’s function to the single-particle excitation spectrum in the interacting
fluid is expressed by its spectral function. The spectral function is related
to the retarded self-energy by the following expression Giuliani
$\displaystyle A_{s}({\bf k},\omega)=\frac{\hbar}{\pi}\frac{|\Im
m\Sigma_{s}^{\rm ret}({\bf k},\omega)|}{[\hbar\omega-\xi_{s}({\bf k})-\Re
e\delta\Sigma_{s}^{\rm ret}({\bf k,\omega})]^{2}+[\Im m\Sigma_{s}^{\rm
ret}({\bf k},\omega)]^{2}}$ (12)
where $\delta\Sigma_{s}^{\rm ret}({\bf k},\omega)=\Sigma_{s}^{\rm ret}({\bf
k},\omega)-\Sigma_{s}^{\rm ret}(k_{\rm F},0)$, and then ARPES intensity can be
described by $I({\bf k},\omega)=A({\bf k},\omega)n(\omega)$, where $n(\omega)$
is the Fermi-Dirac distribution. The spectral function is the Lorentzian
function where $\Re e\Sigma$ specifying the location of the peak of the
distribution, and $|\Im m\Sigma|$ is the linewidth. The amplitude of the the
Lorentzian function is proportional to $1/|\Im m\Sigma|$. This quantity is the
distribution of energies $\hbar\omega$, in the system when a QP with momentum
${\bf k}$, is added or removed from that. For the noninteracting system we get
$A^{(0)}({\bf k},\omega)=\delta(\omega-\xi({\bf k})/\hbar)$. The Fermi liquid
theory applies only when the spectral function at the Fermi momentum
$A^{(0)}(k=k_{\rm F},\omega)$, behaves as a delta function, and has a
broadened peak indicating damped QPs at $k\neq k_{\rm F}$.
To progress of the interband single particle excitation and plasmon effects on
the $\Im m\Sigma_{s}^{\rm ret}$, we must study the retarded self-energy on the
off-shell frequency which is $\omega\neq\xi_{s{\bf k}}/\hbar$. im1 ; im2 This
quantity gives the scattering rate of a QP with momentum ${\bf k}$ and kinetic
energy $\hbar\omega+E_{\rm F}$. The scattering rate or the linewidth raising
from electron-electron interactions is anisotropic and varies significantly
via wavevector at a constant energy. The imaginary part of self energy shows
the width of the QP spectral function.
In Fig. 4 we have shown the absolute value of the imaginary part of the self
energy in unit of $\varepsilon_{\rm F}$ for various gap values. It would be
noticed that there is an area of frequency is associated to the gap value,
$2\Delta$ in which no QP could enter in. In this case, there is a gap in the
$\Im m\Sigma$ between $\xi_{-,k=0}$ and $\xi_{+,k=0}$. We see that $\Im
m\Sigma_{+}$ vanishes as $\omega^{2}$ for $\omega$ tends to zero, a universal
properties of normal Fermi liquid. Moreover, at large frequency, $\Im
m\Sigma_{+}$ tend towards to $\omega$ linearly. Except from the Dirac point,
the conduction band $\Im m\Sigma_{+}$ peaks broaden because of the dependence
on scattering angle of $\xi({\bf k}+{\bf q})$. For low energy, only intraband
single particle excitation contributes to $\Im m\Sigma$ up to $E_{\rm F}$ and
then the interband single particle excitation contribution increases sharply
about $E_{\rm F}$. The interband contribution increases with increasing the
gap values.
To evaluate the scattering rate in interband channel, we have shown $\Im
m\Sigma_{inter(intra)}$ as a function of frequency in Fig. 5. The intraband
contribution of the imaginary part of self energy associated to scattering
rate of QP in the intraband contribution increases with increasing the gap
values while the interband contribution reduces, as we physically expected.
Moreover, by increasing of the electrons in the conduction band the interband
scattering rate reduces whereas the intraband scattering contribution
increases. The gap value suppresses the scattering rate at
$\omega=-\varepsilon_{\rm F}$.
In Fig. 6 we have plotted the real part of self-energy in unit of
$\varepsilon_{\rm F}$ as a function of the energy for various gap values.
Notice again that the real part of residue self-energy has a gap which is
associated the feature calculated in the imaginary part of self-energy. The
line part of self energy is a continues curve and then we have a jump near to
the boundary of gap values in the $\Re e\Sigma$ for gapped graphene. A kink
around $E_{\rm F}$ is associated to the interband plasmon contribution and it
is broaden due to the gap value. This feature affects noticeably in the
interacting electron density of states.
Figure 4: (Color online) The absolute value of the imaginary part of retarded
self-energy ( $+$ channel) as a function of (a) $\rm\omega$ and (b) $\rm k$,
for the various energy gaps.
Figure 5: (Color online) (a) The intraband and (b) the interband contributions
of the imaginary part of self-energy ( $+$ channel) as a function of $\omega$
for the various energy gaps at $\rm k=0.25k_{\rm F}$.
Figure 6: (Color online) The real part of retarded self-energy ( $+$ channel)
as a function of (a) $\omega$ and (b) $\rm k$, for the various energy gaps at
$\Lambda=100$. The $\Re e\Sigma_{+}$ are measured from the interaction
contribution of the chemical potential $\Re e\Sigma_{+}(k_{\rm F},\omega=0)$.
Figure 7: (Color online) The QP spectral function for $+$ channel as a
function of $\omega$ (a) and $\rm k$ (b), for the various energy gaps at
$\Lambda=100$.
As discussed before im1 ; im2 in a zero temperature and disorder free gapless
graphene, the peaks of the spectral function correspond to the nearly
solutions of Dyson’s equation in which the quasiparticle excitation energies
are obtained by $E=\xi_{+}+\Re e\delta\Sigma_{+}^{\rm ret}$. The intersection
of $\Re e\Sigma$ and the lines $E-\xi_{+}$ indicates a satellite long
wavelength plasmaron peak related to the electron-plasmon excitation due to
the long-range electron-electron Coulomb interaction and the Dyson equation
with $\Im m\Sigma=0$ corresponds to a QP peak related to the single particle
excitation. Importantly, in the presence of gap values, the plasmaron peak
suppressed. In Figs. 7(a) and (b) we have shown the energy distribution curves
(EDC) and momentum distribution curves (MDC), respectively. In the presence of
gap values, as shown in Fig. 7(a) there is only the single QP peak.
Figure 8: (Color online) The imaginary (a) and real part (b) of the $(-)$
channel of retarded self-energy as a function of $\omega$ for the various
energy gaps at $\Lambda=100$. The self-energy are measured from the
interaction contribution of the chemical potential $\Re e\Sigma_{-}(k_{\rm
F},\omega=0)$. The QP spectral function as a function of $\omega$ (c), for the
various energy gaps at $\Lambda=100$.
The valance band self-energy contributions are shown in Fig. 8. There is an
area of frequencies is associated to the gap value in which no QP could exist
in $\Im m\Sigma_{-}$ exactly the same as the conduction band. The $s=+1$ and
$-1$ peaks in $\Im m\Sigma_{s}$ in Figs. 4 and 8 separate at finite $k$
because of chirality factors which emphasize ${\bf k}$ and ${\bf q}$ in nearly
parallel directions for conduction band and ${\bf k}$ and ${\bf q}$ in nearly
opposite directions for valance band states. Consequently, at finite $\Delta$,
the QP peak of $A_{-}(k,\omega)$ which is broaden shifts toward the left in
the opposite behavior of $A_{+}(k,\omega)$. These feature have significant
effects in the interacting electron density of states.
Figure 9: (Color online) The renormalized velocity (+ channel) as a function
of the energy gap for various densities at $\alpha_{gr}=1$.
It is essential to note that the satellite band which is theoretically
predicted im1 for gapless garphene has not been seen in experiments. There
are several reasons that could wash out this feature. For example, the plasmon
damping, disorder effects, electron interactions with the buffer layer and
importantly the effect of gap at Dirac point.
One of the important information which can be extracted from ARPES spectra is
the renormalized Fermi velocity $v^{*}$. A consequence of the interaction is a
Fermi velocity renormalization from the backflow of the fluid around a moving
particle. The density of states at the Fermi energy is also changed. The QP
energy measured from the chemical potential of interacting system
$\delta\varepsilon_{sk}^{QP}$, can be calculated by solving self consistently
the Dyson equation $\delta\varepsilon_{s{\bf k}}^{QP}=\xi_{s{\bf k}}+\Re
e[\delta\Sigma_{s}^{\rm ret}({\bf{\bf
k}},\omega)]|_{\omega=\delta\varepsilon_{s{\bf k}}^{QP}/\hbar}$.Giuliani In
the isotropic systems the QP energy, depends on the magnitude of ${\bf k}$.
Expanding $\delta\varepsilon_{sk}^{QP}$ to first order in $k-k_{\rm F}$ we can
write $\delta\varepsilon_{sk}^{QP}\simeq\hbar v^{*}_{s}(k-k_{\rm F})$ which
effectively defines the renormalized velocity as $\hbar
v^{*}_{s}={d\delta\varepsilon_{sk}^{QP}}/{dk}|_{k=k_{\rm F}}$. From the Dyson
equation we can calculated the renormalized Fermi velocity as fateme ;
velocity1 ; velocity2
$\displaystyle\frac{v^{*}_{s}}{v_{s}}=\frac{\varepsilon_{\rm F}E_{\rm
F}^{-1}+(\hbar v_{s})^{-1}\partial_{k}\Re e[\delta\Sigma_{s}^{\rm ret}({\bf
k},\omega)]|_{\omega=0,k=k_{\rm F}}}{1-\hbar^{-1}\partial_{\omega}\Re
e[\delta\Sigma_{s}^{\rm ret}({\bf k},\omega)]|_{\omega=0,k=k_{\rm F}}},$ (13)
where $v_{s}=sv_{\rm F}$. It is found before fateme ; velocity1 ; velocity2 ;
gw1 ; gw2 that electron-electron interaction increases the renormalized Fermi
velocity in gapless graphene sheets which this behavior is in contrast to
conventional 2DES. asgari1 ; asgari
Fig. 9 shows the renormalized Fermi velocity in unit of the bare Fermi
velocity as a function of band gap for various carrier densities. The
renormalized Fermi velocity decreasing with increasing the gap value. $v^{*}$
is density independent after $\Delta=0.8\varepsilon_{\rm F}$ which is in good
agreement with recent experiment observation. dop_lanzara1 ; dop_lanzara2
Finally we calculated a band gap renormalization (BGR). bgr1 ; bgr2 ; bgr3 ;
bgr4 The BGR for conductance band is given by the QP self-energy at the band
edge, namely $\rm BGR=\Re e\Sigma_{+}^{\rm ret}({\bf k}=0,\omega=(\Delta-
E_{\rm F})/\hbar)$. Fig. 10 shows the BGR for the various gap values as a
function of the electron density. The BGR decreases by increasing of the
electron density and in the small energy gap values, it is less density
dependent respect to large energy gap values. In gapless case, we have
obtained a induced band gap or kink due to many-body electron-electron
interactions and it tends to a constant with increasing the electron density.
im1 ; im2 ; gw1 ; gw2 This feature is in agreement with the results obtained
within ab intio DFT calculation. gw1
Figure 10: (Color online) The band gap renormalization as a function of
electron density for various energy gap at $\alpha_{gr}=1$.
## IV SUMMERY AND CONCLUSION
We have revisited the problem of the microscopic calculation of the QP self-
energy and many-body effective velocity suppression in a gapped graphene when
the conduction band is partially occupied. We have performed a systematic
study is based on the many-body G0W approach that is established upon the
random-phase-approximation and on graphene’s massive Dirac equation continuum
model. We have carried out extensive calculations of both the real and the
imaginary part of the QP self-energy and discussed about the interband and
intraband contributions in the scattering process in the presence of gap
value. We have also presented results for the effective velocity and for the
band gap renormalization over a wide range of coupling strength. Accordingly,
we have critically examined the merits of the gap values in dynamical QP
properties.
Most feature of mass generating in graphene is the washing out of the
plasmaron peak in the spectral weight. Increasing of the gap value makes
density independent behavior of the renormalized Fermi velocity. We have shown
that the band gap renormalization in gapped graphene decreases by increasing
the carrier density at large $\Delta$. This is in contrast with the gapless
case in which many body electron-plasmon interactions induce a very small gap
in band structure. These distinct features of the massive Dirac’s Fermions are
related to mixing of the chiralities and reduce of the interband transitions
in graphene sheets.
Acknowledgment
R. A. would like to thank the Scuola Normale Superiore, Pisa, Italy for its
hospitality during the period when the final stage of this work was carried
out. A. Q. supported by IPM grant.
## References
* (1) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov 2004 Science 306 666 .
* (2) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov 2005 Nature 438 197 .
* (3) K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim 2005 Proc. Nat. Acad. Sci. 102 10451 .
* (4) Y. Zhang, Y. Tan, H. L. Stormer, P. Kim 2005 Nature 438 201 .
* (5) P. Avouris, Z. Chen, and V. Perebeinos, 2007 Nature Nanotech. 2 605 .
* (6) A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim 2009 Rev. Mod. Phys. 81 109 .
* (7) C. W. Beenakker 2008 Rev. Mod. Phys. 80 1337 .
* (8) A. K. Geim and P. Kim, 2008 Sci. Am. 298 90 .
* (9) A. K. Geim and A. H. MacDonald 2007 Physics Today 60 35 .
* (10) A. K. Geim and K. S. Novoselov 2007 Nature Mater. 6 183 .
* (11) M. I. Katsnelson, 2007 Materials Today 10 20 .
* (12) M. Polini, A. Tomadin, R. Asgari and A. H. MacDonald 2008 Phys. Rev. B 78 115426 .
* (13) A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C. N. Lau 2008 Nano Lett. 8 902 .
* (14) S.V. Morozov, K.S. Novoselov, M.I. Katsnelson, F. Schedin, D.C. Elias, J.A. Jaszczak, A.K. Geim 2008 Phys. Rev. Lett. 100 016602 .
* (15) K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stormer, and P. Kim 2008 Phys, Rev, Lett. 101 096802 .
* (16) Xu Du, Ivan Skachko, Anthoy Barker and Eva Y. Andrei 2008 Nature Nanotech. 3 491 .
* (17) K.I. Bolotin, K.J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, H.L. Stormer 2008 Solid State Commun. 146 351 .
* (18) K. Eng, R. N. McFarland, and B. E. Kane 2005 Appl. Phys. Lett. 87 052106 .
* (19) E. H. Hwang and S. Das Sarma 2007 Phys. Rev. B 75 073301 .
* (20) C. Lee, X. Wei, J. W. Kysar, J. Hone 2008 Science 321 385 .
* (21) M. Neek-Amal and R. Asgari 2009 arXiv:0903.5035
* (22) T. O. Wehling, K. S. Novoselov, S. V. Morozov, E. E. Vdovin, M. I. Katsnelson, A. K. Geim, A. I. Lichtenstein 2008 Nano Lett. 8 173 .
* (23) I. Gierz, C. Riedl, U. Starke, C. R. Ast, K. Kern 2008 Nano Lett. 8 4603 .
* (24) S. Y. Zhou, D. A. Siegel, A. V. Fedorov, and A. Lanzara 2008 Phys. Rev. Lett. 101 086402 .
* (25) D. A. Siegel, S. Y. Zhou, F. El Gabaly, A. V. Fedorov, A. K. Schmid, and A. Lanzara, 2008 Appl. Phys. Lett. 93 243119 .
* (26) Y. Lin, K. A. Jenkins, A. Valdes-Garcia, J. P. Small, D. B. Farmer, and P. Avouris 2009 Nano Lett. 9 422 .
* (27) J. Kedzierski, P. Hsu, P. Healey, P. W. Wyatt, C. L. Keast, M. Sprinkle, C. Berger, and W. A. de Heer 2008 IEEE Trans. Electron Devices 55 2078 .
* (28) K. Novoselov 2007 Nature Mater. 6 720 .
* (29) Y. W. Son, M. L. Cohen and S. G. Louie 2006 Phys. Rev. Lett. 97 216803 .
* (30) M. Y. Han, B. Ozyilmaz, Y. Zhang and P. Kim 2007 Phys. Rev. Lett. 98 206805 .
* (31) Li Yang, Cheol-Hwan Park, Young-Woo Son, M. L. Cohen, and S. G. Louie 2007 Phys. Rev. Lett. 99 186801 .
* (32) D. Finkenstadt, G. Pennington, and M. J. Mehl 2007 Phys. Rev. B 76 121405(R) .
* (33) Y.-W. Son, M. L. Cohen and S. G. Louie 2006 Nature 444 347 .
* (34) Xue-Feng Wang and T. Chakraborty 2007 Phys. Rev. B 75 033408 .
* (35) Y. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang 2007 Phys. Rev. B 75 041401(R) .
* (36) C. L. Kane and E. J. Mele 2005 Phys. Rev. Lett. 95 226801 .
* (37) H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald 2006 Phys. Rev. B 74 165310 .
* (38) G. W. Semenoff 1984 Phys. Rev. Lett. 53 2449 .
* (39) K. Ziegler, 1996 Phys. Rev. B 53 9653 .
* (40) V. P. Gusynin, S. G. Sharapov, J. P. Carbotte 2007 Int. J. Mod. Phys. B 21 4611 .
* (41) A. Bostowick, T. Ohta, J. L. McCesney, K. V. Emtsev, T. Seyller, K. Horn and E. Rotenberg 2007 New J. Phys. 9 385 .
* (42) C.-Y. Hou, C. Chamon, and C. Mudry 2007 Phys. Rev. Lett. 98 186809 .
* (43) S. Y. Zhou, G. H. Gweon, A. V. Federov, P. N. First, W. A. de Heer, D. H. Lee, F. Guinea, A. H. Castro Neto, and A. Lanzara 2007 Nature Mater. 6 770 .
* (44) S.Y. Zhou, D.A. Siegel, A.V. Fedorov, and A. Lanzara 2008 Physica E 40, 2642 .
* (45) A. Grüneis and D. V. Vyalikh 2008 Phys. Rev. B 77 193401
* (46) A. Grüneis, K. Kummer and D. V. Vyalikh 2009 New J. Phys. 11 073050 .
* (47) G. Li, A. Luican, and E. Y. Andrei 2009 Phys. Rev. Lett. 102 176804 .
* (48) G. Giovannetti, P. A. Khomyako, G. Brocks, P. J. Kelly and J. Van den Brink 2007 Phys. Rev. B 76 073103 .
* (49) M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea, and A. H. MacDonald 2008 Phys. Rev. B 77 081411(R) .
* (50) A. Qaiumzadeh and R. Asgari 2009 Phys. Rev. B 79 075414 .
* (51) R. M. Ribeiro, N. M. R. Peres, J. Coutinho and P. R. Briddon, 2008 Phys. Rev. B 78 075442 .
* (52) Eduardo V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, J. M. B. Lopes dos Santos, Johan Nilsson, F. Guinea, A. K. Geim and A. H. Castro Neto 2007 Phys. Rev. Lett. 99 216802 .
* (53) I. Zanella, S. Guerini, S. B. Fagan, J. Mendes Filho and A. G. Souza Filho 2008 Phy. Rev. B 77 073404 .
* (54) M. Mucha-Kruczyński, O. Tsyplyatyev, A. Grishin, E. McCann, Vladimir I. Fal’ko, Aaron Bostwick and Eli Rotenberg 2008 Phys. Rev. B 77 195403 .
* (55) S. Kim, J. Ihm, H. J. Choi, and Y. Son 2008 Phys. Rev. Lett. 100 176802 .
* (56) T. G. Pedersen, A. Jauho, and K. Pedersen 2009 Phy. Rev. B 79 113406 .
* (57) G. F. Giuliani and G. Vignale 2005 Quantum Theory of The Electron Liquid (Cambridge University Press, Cambridge, England) .
* (58) Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari and A. H. MacDonald 2007 Phys. Rev. Lett. 98 236601 .
* (59) P. K. Pyatkovskiy, 2009 J. Phys.: Condens. Matter 21 025506 .
* (60) B. Wunsch, T. Stauber, F. Sols and F. Guinea 2006 New J. Phys. 8 318
* (61) A. Kaminski and H. M. Fretwell 2005 New J. Phys. 7 98 .
* (62) A. Damascelli, Z. Hussain, and Z.-X. Shen 2003 Rev. Mod. Phys. 75 473 .
* (63) A. Qaiumzadeh, F. K. Joibari, and R. Asgari 2008 arXv: 0810.4681 .
* (64) E. H. Hwang, BenYu-Kaung Hu, and S. Das Sarma 2007 Phys. Rev. B 76115434 .
* (65) W. Tse, E. H. Hwang, and S. Das Sarma 2008 Appl. Phys. Lett. 93 023128 .
* (66) R. Asgari, B. Davoudi, M. Polini, G. F. Giuliani, M. P. Tosi, and G. Vignale 2005 Phys. Rev. B 71 045323 .
* (67) G. F. Giuliani and J. J. Quinn 1982 Phys. Rev. B 26 4421 .
* (68) E. H. Hwang and S. Das Sarma 2008 Phys. Rev. B 77 081412(R) .
* (69) M. Polini, R. Asgari, Y. Barlas, T. Pereg-Barnea, A. H. MacDonald 2007 Solid State Commun. 143 58 .
* (70) A. Qaiumzadeh, N. Arabchi, R. Asgari 2008 Solid State Commun. 147 172
* (71) Paolo E. Trevisanutto, Christine Giorgetti, Lucia Reining, Massimo Ladisa and Valerio Olevano 2008 Phys. Rev. Lett. 101 226405 .
* (72) C. Park, F. Giustino, C. D. Spataru, M. L. Cohen, and S. G. Louie 2009 Phys. Rev. Lett. 102 076803 .
* (73) R. Asgari and B. Tanatar 2006 Phys. Rev. B 74 075301 .
* (74) Y. Zhang and S. Das Sarma 2005 Phys. Rev. B 72 125303 .
* (75) S. Das Sarma, R. Jalabert, and S. R. Eric Yang 1990 Phys. Rev. B 41 8288 .
* (76) K. F. Berggren and B. E. Sernelius 1984 Phys. Rev. B 29 5575 .
* (77) K. F. Berggren and B. E. Sernelius 1981 Phys. Rev. B 24 1971 .
|
arxiv-papers
| 2009-11-03T12:00:51 |
2024-09-04T02:49:06.250329
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alireza Qaiumzadeh, Reza Asgari",
"submitter": "Reza Asgari",
"url": "https://arxiv.org/abs/0911.0558"
}
|
0911.0641
|
# Bene$\check{\bf S}$ condition for discontinuous exponential martingale
R. Liptser Department of Electrical Engineering Systems, Tel Aviv University,
69978 Tel Aviv, Israel liptser@eng.tau.ac.il; rliptser@gmail.com
###### Abstract.
It is known the Girsanov exponent $\mathfrak{z}_{t}$, being solution of
Doleans-Dade equation $\mathfrak{z_{t}}=1+\int_{0}^{t}\alpha(\omega,s)dB_{s}$
generated by Brownian motion $B_{t}$ and a random process $\alpha(\omega,t)$
with $\int_{0}^{t}\alpha^{2}(\omega,s)ds<\infty$ a.s., is the martingale
provided that the Bene${\rm\check{s}}$ condition
$|\alpha(\omega,t)|^{2}\leq\text{\rm
const.}\big{[}1+\sup_{s\in[0,t]}B^{2}_{s}\big{]},\ \forall\ t>0,$
holds true. In this paper, we show $B_{t}$ can be replaced by by a homogeneous
purely discontinuous square integrable martingale $M_{t}$ with independent
increments and paths from the Skorokhod space $\mathbb{D}_{[0,\infty)}$ having
positive jumps $\triangle M_{t}$ with $\mathsf{E}\sum_{s\in[0,t]}(\triangle
M_{s})^{3}<\infty$. A function $\alpha(\omega,t)$ is assumed to be nonnegative
and predictable. Under this setting $\mathfrak{z}_{t}$ is the martingale
provided that
$\alpha^{2}(\omega,t)\leq\text{\rm
const.}\big{[}1+\sup_{s\in[0,t]}M^{2}_{s-}\big{]},\ \forall\ t>0.$
The method of proof differs from the original Bene${\rm\check{s}}$ one and is
compatible for both setting with $B_{t}$ and $M_{t}$.
###### Key words and phrases:
Girsanov, exponential martingale, uniform integrability
###### 1991 Mathematics Subject Classification:
60G45, 60G46
## 1\. Introduction and main result
### 1.1. Setting of problem
A classical Girsanov’s exponent
$\mathfrak{z}_{t}=\exp\Big{(}\int_{0}^{t}\alpha(\omega,s)dB_{s}-\frac{1}{2}\int_{0}^{t}\alpha^{2}(\omega,s)ds\Big{)},$
with Brownian motion $B_{t}$ and adapted random process $\alpha(\omega,t)$
having $\int_{0}^{t}\alpha(\omega,s)ds<\infty$, forms a positive local
martingale (and supermartingale too) with $\mathsf{E}\mathfrak{z}_{t}\leq 1$.
If
$\mathsf{E}\mathfrak{z}_{t}\equiv 1,$ (1.1)
the random process $\mathfrak{z}_{t}$ is a martingale. In order (1.1) to have,
Girsanov in [2] used bounded function $\alpha(\omega,t)$ and suggested a
conjecture that (1.1) will be valid if $\alpha^{2}(\omega,t)\approx
B^{2}_{t}$. Among conditions guaranteing (1.1) (see, e.g. Novikov [10],
Kazamaki [6], the latest Krylov [7], etc), we distinguish the
Bene${\rm\check{s}}$ statement: for any $T>0$,
$``|\alpha(\omega,t)|^{2}\leq\text{\rm
const.}\big{[}1+\sup_{s\in[0,t]}B^{2}_{s}\big{]}_{t\in[0,T]}\text{''}\Rightarrow``\mathsf{E}\mathfrak{z}_{T}=1\text{''},$
(1.2)
which is derived in [1] with the help of Kazamaki [6] (see also Karatzas and
Shreve [5], ”Ustünel and Zakai [11]).
The aim of this paper is to “replace” the Brownian motion $B_{t}$ by a
homogeneous purely discontinuous square integrable martingale $M_{t}$ with
independent increments and obtain an implication similar to (1.2).
Unfortunately, a method of proof of (1.2) with $M_{t}$ instead of $B_{t}$ is
not applicable since, for example, results of Kazamaki, Novikov, and Krylov
related to martingale $M_{t}$ do not exist.
### 1.2. New approach to Bene${\bf\check{s}}$ result
We propose a new approach for the proof of (1.2) which is compatible with
$B_{t}$ and $M_{t}$. Its application to a discontinuous martingale is more
involved than to classical case with Brownian motion. So, to make our approach
at most transparent we give a sketch of the proof of (1.2) step by step.
(1) It is well known $\mathfrak{z}_{t}$ is the unique solution of Doléans-Dade
equation
$\mathfrak{z}_{t}=1+\int_{0}^{t}\mathfrak{z}_{s}\alpha(\omega,s)dB_{s}.$ (1.3)
Set $\sigma_{n}=\inf\Big{\\{}t:\Big{[}1+\sup_{s\in[0,t]}B^{2}_{s}\Big{]}\geq
n\Big{\\}}$, $B^{n}_{t}=B_{t\wedge\sigma_{n}}$ and
$\mathfrak{z}^{n}_{t}=\mathfrak{z}_{t\wedge\sigma_{n}}$. Then
$\mathfrak{z}^{n}_{t}=1+\int_{0}^{t}\mathfrak{z}^{n}_{s}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)dB_{s}.$
Evidently, $I_{\\{\sigma_{n}\geq s\\}}\alpha^{2}(\omega,s)\leq\text{const.}n.$
Consequently, $\mathsf{E}\mathfrak{z}^{n}_{t}\equiv 1.$
(2) In order to prove $\mathsf{E}\mathfrak{z}_{t}=1$ for any $t\in[0,T]$, it
suffices to show the family $\\{\mathfrak{z}^{n}_{T}\\}_{n\to\infty}$ is
uniformly integrable. With chosen $\sigma_{n}$, we have
$\mathsf{E}\mathfrak{z}^{n}_{T}=1$. Let us introduce a probability measure
$\widetilde{\mathsf{P}}^{n}_{T}\ll\mathsf{P}$ with
$d\widetilde{\mathsf{P}}^{n}_{T}=\mathfrak{z}^{n}_{T}d\mathsf{P}$ and denote
by $\widetilde{\mathsf{E}}^{n}_{T}$ the expectation symbol of
$\widetilde{\mathsf{P}}^{n}_{T}.$
Following Hitsuda [4], the uniform integrability of
$\\{\mathfrak{z}^{n}_{T}\\}_{n\to\infty}$ is verified with a convex function
$\psi(x)=x\log(x)+1-x,\ x\geq 0$ (1.4)
due to the Vallée-Poussin’s criteria since
$\lim_{x\to\infty}\frac{\psi(x)}{x}=\infty.$ Namely, we have to show that
$\sup_{n}\mathsf{E}\psi(\mathfrak{z}^{n}_{T})<\infty.$ A verification of this
condition is inconvenient. However if
$\mathsf{E}\psi(\mathfrak{z}^{n}_{T})<\infty$ a direct computation shows
$\mathsf{E}\psi(\mathfrak{z}^{n}_{T})=\widetilde{\mathsf{E}}^{n}_{T}\log\big{(}\mathfrak{z}^{n}_{T}\big{)}.$
(3) the Girsanov theorem, a random process
$(\widetilde{B}^{n}_{t})_{t\in[0,T]}$ with
$\widetilde{B}^{n}_{t}=B^{n}_{t}-\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)ds$ (1.5)
is $\widetilde{\mathsf{P}}^{n}_{T}$-martingale with the predictable quadratic
variation $\langle\widetilde{B}^{n}\rangle_{t}=\langle B^{n}\rangle_{t}\equiv
t\wedge\sigma_{n}.$ Therefore, by using (1.5), we obtain
$\log(\mathfrak{z}^{n}_{t})\stackrel{{\scriptstyle\widetilde{\mathsf{P}}^{n}_{T}}}{{=}}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)d\widetilde{B}^{n}_{s}+\frac{1}{2}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)ds.$
Since $I_{\\{\sigma_{n}\geq s\\}}\alpha^{2}(\omega,s)$ is bounded, we have
$\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)ds\leq c_{n}$ with a constant $c_{n}$ depending on
$n$. Hence, $\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)d\widetilde{B}^{n}_{s}=0.$
Therefore and in view of $I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)ds\leq\text{const.}[1+\sup_{s^{\prime}\in[0,s\wedge\sigma_{n}]}|B^{n}_{s^{\prime}}|^{2}],$
$\widetilde{\mathsf{E}}^{n}_{T}\log(\mathfrak{z}^{n}_{T})=\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)ds\\\
\leq\text{const.}\Big{[}1+\widetilde{\mathsf{E}}^{n}_{T}\sup_{s^{\prime}\in[0,T\wedge\sigma_{n}]}|B^{n}_{s^{\prime}}|^{2}\Big{]}.$
(4) Now, the proof is reduced to
$\sup_{n}\widetilde{\mathsf{E}}^{n}_{T}\sup_{s^{\prime}\in[0,T\wedge\sigma_{n}]}|B^{n}_{s^{\prime}}|^{2}<\infty.$
Denote
$V^{n}_{t}:=\widetilde{\mathsf{E}}^{n}_{T}\sup_{s^{\prime}\in[0,t\wedge\sigma_{n}]}|B^{n}_{s^{\prime}}|^{2}$.
In view of (1.5),
$\displaystyle V^{n}_{t}\leq
2\widetilde{\mathsf{E}}^{n}_{T}\Big{(}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}|\alpha(\omega,s)|ds\Big{)}^{2}+2\widetilde{\mathsf{E}}^{n}_{T}\Big{(}\sup_{s^{\prime}\in[0,t\wedge\sigma_{n}]}|\widetilde{B}^{n}_{s^{\prime}}|\Big{)}^{2}.$
By the Doob maximal inequality,
$\widetilde{\mathsf{E}}^{n}_{T}\Big{(}\sup\limits_{s^{\prime}\in[0,t\wedge\sigma_{n}]}|\widetilde{B}^{n}_{s^{\prime}}|\Big{)}^{2}\leq
4\widetilde{\mathsf{E}}^{n}_{T}|\widetilde{B}^{n}_{t\wedge\sigma_{n}}|^{2}=4\widetilde{\mathsf{E}}^{n}_{T}|(t\wedge\sigma_{n})\leq
4T,$ while by the Cauchy-Schwarz inequality
$\widetilde{\mathsf{E}}^{n}_{T}\Big{(}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}|\alpha(\omega,s)|ds\Big{)}^{2}\leq\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)ds\leq\mathbf{r}\Big{[}1+\int_{0}^{t}V^{n}_{s}\Big{]}.$
Finally, combining these estimates, we obtain an integral inequality (with the
constant $\mathbf{r}$ independent of $n$):
$V^{n}_{t}\leq\mathbf{r}\Big{[}1+\int_{0}^{t}V^{n}_{s}ds\Big{]}.$
Thus, $V^{n}_{T}\leq\mathbf{r}e^{T\mathbf{r}}$.
### 1.3. Formulation of main result
Let us explain how Brownian $B_{t}$ might be replaced by a purely
discontinuous martingale $M_{t}$. A simplest way is to replace $B_{t}$ by
$M_{t}$ in the the Doléans-Dade equation (1.3), that is,
$\mathfrak{z}_{t}=1+\int_{0}^{t}\mathfrak{z}_{s-}\alpha(\omega,s)dM_{s},$
where $\mathfrak{z}_{s-}=\lim\limits_{s^{\prime}\uparrow
s}\mathfrak{z}_{s}^{\prime}$, and adapted process $\alpha(\omega,t)$ is
replaced by its predictable version. The square integrable martingale $M_{t}$
has paths the Skorokhod space $\mathbb{D}_{[0,\infty)}.$ Denote $\langle
M\rangle_{t}$ the predictable quadratic variation of $M_{t}$ and
$M_{t-}=\lim_{t^{\prime}\uparrow t}M_{t^{\prime}}$.
We assume $\int_{0}^{t}\alpha^{2}(\omega,s)d\langle M\rangle_{s}<\infty$. A
positiveness of $\mathfrak{z}_{t}$ is warranted by assumptions
$\alpha(\omega,t)\geq 0$ and $\big{(}M_{t}-M_{t-}\big{)}I_{\\{M_{t}\neq
M_{t-}\\}}>0.$
We choose $M_{t}$ in a form of Itô’s integral
$M_{t}=\int_{0}^{t}\int_{\mathbb{R}_{+}}z\big{[}\mu(ds,dz)-\nu(ds,dz)\big{]}$
relative to, so called, martingale difference “$\mu-\nu$”, where $\mu(dt,dz)$
is the integer-valued measure $\mu=\mu(dt,dz)$ associated with a jump process
$\triangle M_{t}=M_{t}-M_{t-}$ of $M_{t}$ and $\nu(dt,dz)$ is a compensator of
$\mu(dt,dz)$. In order to have the above-mentioned properties of $M_{t}$, we
choose a deterministic compensator
$\nu(dt,dz)=K(dz)dt$
with a measure $K(dz)$ supported on $\mathbb{R}_{+}$ and
$\int_{\mathbb{R}_{+}}z^{2}K(dz)<\infty$. In particular, then,
$\langle M\rangle_{t}\equiv\mathsf{E}M^{2}_{t}\equiv\int_{0}^{t}z^{2}K(dz)ds.$
Our main result is formulated in
###### Theorem 1.1.
Assume $\int_{\mathbb{R}_{+}}z^{3}K(dz)<\infty$. Then for any $T>0$ (comp.
(1.2))
$``|\alpha(\omega,t)|^{2}\leq\text{\rm
const.}\big{[}1+\sup_{s\in[0,t]}M^{2}_{s-}\big{]}_{t\in[0,T]}\text{''}\Rightarrow``\mathsf{E}\mathfrak{z}_{T}=1\text{''},$
The method of proof is similar to one given in Section (1.2)
## 2\. The proof of Theorem 1.1
### 2.1. Preliminaries
We begin with recalling necessary notions (for more details, see e.g. [8] or
[3]). Along this paper a filtered probability space
$(\varOmega,\mathcal{F},(\mathscr{F}_{t})_{t\in[0,\infty)},\mathsf{P})$ with
“general conditions” is fixed and all random objects are defined on it.
$\mathscr{P}$ denotes predictable $\sigma$-algebra relative to
$(\mathscr{F}_{t})_{t\in[0,\infty)}$ and $\mathscr{B}_{+}$ is the Borel
$\sigma$-algebra on $\mathbb{R}_{+}$.
Henceforth, $\mathbf{r}$ denotes a generic constant taking different values at
different appearances and is independent of a number $n$ involved in the text.
We begin with know implication:
$``\alpha(\omega,t)\leq\mathbf{r}\text{''}\Rightarrow``\mathsf{E}\mathfrak{z}_{t}\equiv
1\text{''}.$ (2.1)
Set $\tau_{n}=\inf\\{t:\mathfrak{z}_{t-}\geq n\\}$ and notice
$\mathfrak{z}_{(t\wedge\tau_{n})-}\leq n$. Then
$\mathfrak{z}^{n}_{t}:=\mathfrak{z}_{t\wedge\tau_{n}}$ solves the Doleans-Dade
equation
$\mathfrak{z}^{n}_{t}=1+\int_{0}^{t}\mathfrak{z}^{n}_{s-}I_{\\{\tau_{n}\geq
s\\}}\alpha(\omega,s)dM_{s}.$ (2.2)
A boundedness of $I_{\\{\tau_{n}\geq
s\\}}\mathfrak{z}^{n}_{s-}\alpha(\omega,s)$ guarantees the process
$\mathfrak{z}^{n}_{t}$ is the square integrable martingale and
$\mathsf{E}(\mathfrak{z}^{n}_{t})^{2}=1+\mathsf{E}\int_{0}^{t}\big{(}I_{\\{\tau_{n}\geq
s\\}}\mathfrak{z}^{n}_{s-}\alpha(\omega,s)\big{)}^{2}d\langle
M\rangle_{s}ds\leq
1+\mathbf{r}\int_{0}^{t}\mathsf{E}(\mathfrak{z}^{n}_{s})^{2}ds.$ Then, a
function $V^{n}_{t}=\mathsf{E}(\mathfrak{z}^{n}_{t})^{2}$ solves the integral
inequality: $V^{n}_{t}\leq 1+\mathbf{r}\int_{0}^{t}V^{n}_{s}ds.$ So, by the
Bellman-Gronwall inequality, $V^{n}_{t}\leq e^{\mathbf{r}t}$, that is,
$\sup_{n}\mathsf{E}(\mathfrak{z}^{n}_{t})^{2}\leq e^{\mathbf{r}t}.$ Therefore,
by the Vallée-Poussin’s criteria, the family
{$\mathfrak{z}^{n}_{t})\\}_{n\to\infty}$ is uniformly integrable. So, not only
$\lim_{n\to\infty}\mathfrak{z}^{n}_{t}=\mathfrak{z}_{t}$ but also
$\mathsf{E}\mathfrak{z}_{t}=\lim_{n\to\infty}\mathsf{E}\mathfrak{z}^{n}_{t}\equiv
1.$
### 2.2. $\mathfrak{z}^{n}_{t}$ approximation of $\mathfrak{z}_{t}$. Change
of probability measure
###### Lemma 2.1.
Let $\sigma_{n}=\inf\big{\\{}t:\big{[}1+\sup_{s\in[0,t]}M^{2}_{s-}\big{]}\geq
n\big{\\}}$ and $\mathfrak{z}^{n}_{t}=\mathfrak{z}_{t\wedge\sigma_{n}}$. Then
$\mathsf{E}\mathfrak{z}^{n}_{t}\equiv 1$.
###### Proof.
We use (2.2) with $\tau_{n}$ replaced by $\sigma_{n}$:
$\mathfrak{z}^{n}_{t}=1+\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\mathfrak{z}^{n}_{s-}\alpha(\omega,s)dM_{s}.$ (2.3)
Since $\big{[}1+\sup\limits_{s\in[0,t\wedge\sigma_{n}]}M^{2}_{s-}\big{]}\leq
n$ and, then, $I_{\\{\sigma_{n}\geq
t\\}}|\alpha(\omega,t)|^{2}\leq\mathbf{r}I_{\\{\sigma_{n}\geq
t\\}}\big{[}1+\sup\limits_{s\in[0,t]}M^{2}_{s-}\big{]}\leq\mathbf{r}n$, it
remains to apply (2.1). ∎
Let $T>0$ be fixed. By Lemma 2.1 $\mathsf{E}\mathfrak{z}^{n}_{T}=1.$ As in
Section 1.2, we have to show the family
$\\{\mathfrak{z}^{n}_{T}\\}_{n\to\infty}$ is uniformly integrable. So, we
intend to to verify $\sup_{n}\mathsf{E}\psi(\mathfrak{z}^{n}_{t})<\infty$ with
the function $\psi(x)$ defined in (1.4).
Repeating arguments from Section 1.2, we introduce a probability measure
$\widetilde{\mathsf{P}}^{n}_{T}\ll\mathsf{P}$ with
$d\widetilde{\mathsf{P}}^{n}_{T}=\mathfrak{z}^{n}_{T}d\mathsf{P}$
($\widetilde{\mathsf{E}}^{n}_{T}$ denotes the expectation symbol of
$\widetilde{\mathsf{P}}^{n}_{T}$). Set $M^{n}_{t}=M_{t\wedge\sigma_{n}}$ and
rewrite (2.3) in equivalent form
$\mathfrak{z}^{n}_{t}=1+\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\mathfrak{z}^{n}_{s-}\alpha(\omega,s)dM^{n}_{s}.$ (2.4)
The random process $(M^{n}_{t})_{t\in[0,T]}$ is $\mathsf{P}$ \- square
integrable martingale. Since $\widetilde{\mathsf{P}}^{n}_{T}\ll\mathsf{P}$,
the process $(M^{n}_{t})_{t\in[0,T]}$ is
$\widetilde{\mathsf{P}}^{n}_{T}$-semimartingale obeying the unique
decomposition $M^{n}_{t}=A^{n}_{t}+\widetilde{M}^{n}_{t}$ with predictable
drift $A^{n}_{t}$ and local martingale $\widetilde{M}^{n}_{t}$ (see, e.g. [8],
Ch.4, §5, Theorem 2). Denote $\widetilde{\nu}^{n}_{T}(dt,dz)$ a compensator of
$\mu(dt,dz)$ relative to $\widetilde{\mathsf{P}}^{n}_{T}$.
###### Lemma 2.2.
1. $\widetilde{\nu}^{n}_{T}(ds,dz)=I_{\\{\sigma_{n}\geq s\\}}\big{(}1+\alpha(\omega,s)z)K(dz)ds$
2. $\widetilde{M}^{n}_{t}=M^{n}_{t}-A^{n}_{t}$ is square integrable martingale with
$\langle\widetilde{M}^{n}\rangle_{t}=\int_{0}^{t}\int_{\mathbb{R}_{+}}z^{2}I_{\\{\sigma_{n}\geq
s\\}}\big{(}1+\alpha(\omega,s)z)K(dz)ds$ (2.5)
3. $A^{n}_{t}=\int_{0}^{t}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq s\\}}\alpha(\omega,s)z^{2}K(dz)ds$
###### Proof.
1. Below, we will use a formula
$\frac{\mathfrak{z}^{n}_{s}}{\mathfrak{z}^{n}_{s-}}I_{\\{\sigma_{n}\geq
s\\}}=I_{\\{\sigma_{n}\geq s\\}}\big{(}1+\alpha(\omega,s)\triangle M^{n}_{s})$
readily derived from (2.4). Let $u(\omega,t,z)$ be bounded and
$\mathscr{P}\otimes\mathscr{B}(\mathbb{R}_{+})$-measurable function vanishing
in a vicinity of $\\{0\\}$. Write
$\displaystyle\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{T}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq
s\\}}u(\omega,s,z)\mu(dt,dz)$
$\displaystyle=\mathsf{E}\mathfrak{z}^{n}_{T}\int_{0}^{T}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq
s\\}}u(\omega,s,z)\mu(dt,dz)$
$\displaystyle=\mathsf{E}\int_{0}^{T}\int_{\mathbb{R}_{+}}\mathfrak{z}^{n}_{s}I_{\\{\sigma_{n}\geq
s\\}}u(\omega,s,z)\mu(dt,dz)$
$\displaystyle=\mathsf{E}\int_{0}^{T}\int_{\mathbb{R}_{+}}\mathfrak{z}^{n}_{s-}\frac{\mathfrak{z}^{n}_{s}}{\mathfrak{z}^{n}_{s-}}I_{\\{\sigma_{n}\geq
s\\}}u(\omega,s,z)\mu(dt,dz)$
$\displaystyle=\mathsf{E}\int_{0}^{T}\int_{\mathbb{R}_{+}}\mathfrak{z}^{n}_{s-}[1+\alpha(\omega,s)\triangle
M^{n}_{s}]I_{\\{\sigma_{n}\geq s\\}}u(\omega,s,z)\mu(dt,dz)$
$\displaystyle=\mathsf{E}\int_{0}^{T}\int_{\mathbb{R}_{+}}\mathfrak{z}^{n}_{s-}I_{\\{\sigma_{n}\geq
s\\}}\big{[}1+\alpha(\omega,s)z]u(\omega,s,z)\mu(ds,dz)$
$\displaystyle=\mathsf{E}\int_{0}^{T}\int_{\mathbb{R}_{+}}\mathfrak{z}^{n}_{s-}I_{\\{\sigma_{n}\geq
s\\}}\big{[}1+\alpha(\omega,s)z]u(\omega,s,z)K(dz)ds$
$\displaystyle=\mathsf{E}\mathfrak{z}^{n}_{T}\int_{0}^{T}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq
s\\}}\big{[}1+\alpha(\omega,s)z]u(\omega,s,z)K(dz)ds$
$\displaystyle=\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{T}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq
s\\}}\big{[}1+\alpha(\omega,s)z]u(\omega,s,z)K(dz)ds.$
The desired result follows by arbitrariness of $u(\omega,t,z)$ .
2.+3. Recall $\int_{\mathbb{R}_{+}}(z^{2}\vee z^{3})K(dz)<\infty$. Hence
$\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{T}\int_{\mathbb{R}_{+}}z^{2}\widetilde{\nu}^{n}(ds,dz)<\infty.$
Therefore
$\widehat{M}^{n}_{t}=\int_{0}^{t}\int_{\mathbb{R}_{+}}zI_{\\{\sigma_{n}\geq
s\\}}[\mu(ds,dz)-\widetilde{\nu}^{n}_{T}(ds,dz)$
is $\widetilde{\mathsf{P}}^{n}_{T}$-square integrable martingale with
predictable quadratic variation process
$\langle\widehat{M}^{n}\rangle_{t}=\int_{0}^{t}\int_{\mathbb{R}_{+}}z^{2}I_{\\{\sigma_{n}\geq
s\\}}\big{(}1+\alpha(\omega,s)z)K(dz)ds.$ (2.6)
On the other hand,
$\displaystyle
M^{n}_{t}-\widehat{M}^{n}_{t}=\int_{0}^{t}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq
s\\}}z[\widetilde{\nu}^{n}_{T}(ds,dz)-K(dz)]ds$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq
s\\}}z^{2}\alpha(\omega,s)K(dz)ds=:A^{n}_{t},$
where $A^{n}_{t}$ is well defined predictable process.
Consequently, $\widetilde{M}^{n}_{t}\equiv\widehat{M}^{n}_{t}$ in view of the
unique semimartingale decomposition with the predictable drift. ∎
### 2.3. Upper bound of $\mathsf{E}\psi(\mathfrak{z}^{n}_{T})$
Following the main idea of Section 1.2, we have to show the family
$\\{\mathfrak{z}^{n}_{T}\\}_{n\to\infty}$ is uniformly integrable. To this
end, we have to prove $\sup_{n}\mathsf{E}\psi(\mathfrak{z}^{n}_{T})<\infty$
for the function $\psi(x)=x\log(x)+1-x,\ x\geq 0.$ We show first
$\widetilde{\mathsf{E}}^{n}_{T}\log(\mathfrak{z}^{n}_{T})<\infty$
and, then, use an obvious identity
$\mathsf{E}\psi(\mathfrak{z}^{n}_{T})=\widetilde{\mathsf{E}}^{n}_{T}\log(\mathfrak{z}^{n}_{T}).$
###### Lemma 2.3.
$\widetilde{\mathsf{E}}^{n}_{T}\log(\mathfrak{z}^{n}_{T})\leq\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{T}\int_{\mathbb{R}_{+}}I_{\\{\sigma_{n}\geq
s\\}}z^{2}\alpha^{2}(\omega,s)K(dz)ds\leq\mathbf{r}\Big{[}1+\widetilde{\mathsf{E}}^{n}_{T}\sup_{s\in[0,T\wedge\sigma_{n}]}(M^{n}_{s-})^{2}\Big{]}.$
###### Proof.
It is well known (see e.g. [8], Ch. 2, §4), the Doleans-Dade equation (2.4)
obeys the unique solution
$\displaystyle\mathfrak{z}^{n}_{t}=\exp\Big{(}\int_{0}^{t}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)dM^{n}_{s}$
$\displaystyle\quad+\sum_{s\in[0,t\wedge\sigma_{n}]}\log\Big{\\{}\big{[}1+\alpha(\omega,s)\triangle
M^{n}_{s}\big{]}-\alpha(\omega,s)\triangle M^{n}_{s}\Big{\\}}\Big{)}.$
Recall $\alpha(\omega,s)\triangle M^{n}_{s}\geq 0.$ Then
$\log[1+\alpha(\omega,s)\triangle M^{n}_{s}]-\alpha(\omega,s)\triangle
M^{N}_{s}\leq 0$ and, therefore,
$\log\big{(}\mathfrak{z}^{n}_{t})\leq\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)dM^{n}_{s}.$ So, by Lemma 2.2,
$\log\big{(}\mathfrak{z}^{n}_{T})\leq\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)dA^{n}_{s}+\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)d\widetilde{M}^{n}_{s}.$
The process $\widetilde{M}^{n}_{t}$ is square integrable martingale with
$\langle\widetilde{M}^{n}\rangle_{T}$ defined in (2.5). Therefore, the Itô
integral $\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)d\widetilde{M}^{n}_{s}$ is also the square integrable
martingale with the quadratic variation
$QV:=\int_{0}^{T}\int_{\mathbb{R}_{+}}z^{2}I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)\big{[}1+\alpha(\omega,s)z]K(dz)ds.$
Let us show that QV is bounded by a constant depending on $n$. Since
$\int_{\mathbb{R}_{+}}(z^{2}\vee z^{3})K(dz)\leq\mathbf{r}$,
$QV\leq\mathbf{r}\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}[1+\alpha^{3}(\omega,s)]ds\leq\mathbf{r}T+\mathbf{r}\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}\alpha^{3}(\omega,s)ds.$
Further,
$I_{\\{\sigma_{n}\geq s\\}}\alpha^{3}(\omega,s)=[I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)]^{3/2}\leq\mathbf{r}\Big{[}1+\sup_{s^{\prime}\in[0,s\wedge\sigma_{n}]}M^{2}_{s^{\prime}-}\Big{]}^{3/2}\leq\mathbf{r}n^{3/2},$
that is, $QV\leq\mathbf{r}T[1+n^{3/2}]$.
Hence, $\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)d\widetilde{M}^{n}_{s}=0$ and
$\widetilde{\mathsf{E}}^{n}_{T}\log\big{(}\mathfrak{z}^{n}_{T})\leq\widetilde{\mathsf{E}}^{n}\int_{0}^{T}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)dA^{n}_{s}.$ So, it remains to recall the formula of
$A^{n}_{t}$ (see Lemma 2.2), and $\int_{\mathbb{R}_{+}}z^{2}K(dz)$, and
$I_{\\{\sigma_{n}\geq s\\}}\alpha^{2}(\omega,s)=I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)\leq\mathbf{r}\Big{[}1+\sup_{s^{\prime}\in[0,s\wedge\sigma_{n}]}M^{2}_{s^{\prime}-}\Big{]}.$
∎
### 2.4. Final step of the proof
Now, we are in the position to compute
$\widetilde{\mathsf{E}}^{n}_{T}\sup\limits_{s\in[0,T]}|M^{n}_{s}|^{2}.$ The
use of $M^{n}_{t}=A^{n}_{t}+\widetilde{M}^{n}_{t}$ implies
$\widetilde{\mathsf{E}}^{n}_{T}\sup\limits_{t^{\prime}\in[0,t]}|M^{n}_{t^{\prime}}|^{2}\leq
2\widetilde{\mathsf{E}}^{n}_{T}\sup\limits_{t^{\prime}\in[0,t]}|A^{n}_{t^{\prime}}|^{2}+2\widetilde{\mathsf{E}}^{n}_{T}\sup\limits_{t^{\prime}\in[0,t]}|\widetilde{M}^{n}_{t^{\prime}}|^{2}.$
In view of statement 3. of Lemma 2.2
$A^{n}_{t}=\int_{\mathbb{R}_{+}}z^{2}K(dz)\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha(\omega,s)ds$. So, by applying the Cauchy-Schwarz inequality we
obtain
$\widetilde{\mathsf{E}}^{n}_{T}\sup_{t^{\prime}\in[0,t]}|A^{n}_{t^{\prime}}|^{2}\leq\mathbf{r}\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\alpha^{2}(\omega,s)ds.$ (2.7)
Further, by the Doob maximal inequality and (2.6) we find that
$\displaystyle\widetilde{\mathsf{E}}^{n}_{T}\sup_{t^{\prime}\in[0,t]}|\widetilde{M}^{n}_{t^{\prime}}|^{2}\leq
4\widetilde{\mathsf{E}}^{n}_{T}\langle\widetilde{M}^{n}\rangle_{t}=4\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}\int_{\mathbb{R}_{+}}z^{2}I_{\\{\sigma_{n}\geq
s\\}}\big{(}1+\alpha(\omega,s)z)K(dz)ds$
$\displaystyle\leq\mathbf{r}\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\big{[}1+\alpha(\omega,s)\big{]}ds\leq\mathbf{r}\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}I_{\\{\sigma_{n}\geq
s\\}}\big{[}1+\alpha^{2}(\omega,s)\big{]}ds$ (2.8)
Now, a combination of (2.7), and (2.8) provides: for any $t\leq T$,
$\widetilde{\mathsf{E}}^{n}_{T}\sup_{t^{\prime}\in[0,t]}|M^{n}_{t^{\prime}}|^{2}\leq\mathbf{r}\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}[1+\alpha^{2}(\omega,s)]ds\leq\mathbf{r}\widetilde{\mathsf{E}}^{n}_{T}\int_{0}^{t}\Big{[}1+\sup_{s^{\prime}\in[0,s]}|M^{n}_{s^{\prime}}|^{2}\Big{]}ds.$
Hence, the function
$V^{n}_{t}:=\sup_{n}\widetilde{\mathsf{E}}^{n}_{T}\sup_{s\in[0,t]}|M^{n}_{s}|^{2}$
solves an integral inequality:
$``V^{n}_{t}\leq\mathbf{r}\Big{(}1+\int_{0}^{t}V^{n}_{s}ds\Big{)}\text{''}\Rightarrow``V^{n}_{T}\leq\mathbf{r}e^{\mathbf{r}T}\text{''}.$
∎
## References
* [1] Benes, V.E. (1971) Existence of optimal stochastic control laws _SIAM J. of Control_ , 9 , 446-475
* [2] Girsanov, I.V. (1960) On transforming a certan class of stochastic processes by absolutely continuous substitution of measures. _Theory Probab. Appl._ 5, 285-301.
* [3] Jacod J., Shiryaev A.N.: Limit theorems for stochastic processes. 2nd ed. Springer-Verlag, Berlin (2003)
* [4] Hitsuda Masuyuki. (1968) Representation of Gaussian processes equivalent to Wiener process. _Osaka J. Math._ 5, 299-312.
* [5] Karatzas, I. and Shreve, S.E. (1991): Brownian Motion and Stochastic Calculus. Springer-Verlag, New York Berlin Heidelberg.
* [6] Kazamaki, N. (1977) On a problem of Girsanov.// Tôhoku Math. J., 29 , p. 597- 600\.
* [7] Krylov, N.V. (8 May 2009) A simple proof of a result of A. Novikov. arXiv:math/020713v2 [math.PR]
* [8] Liptser, R.Sh., Shiryayev, A.N. (1989) _Theory of Martingales._ Kluwer Acad. Publ.
* [9] Liptser, R. Sh. and Shiryaev, A. N. (2000). _Statistics of Random Processes I_, 2nd ed., Springer, Berlin - New York.
* [10] Novikov, A.A. (1979) On the conditions of the uniform integrability of the continuous nonnegative martingales.// Theory of Probability and its Applications, 24, No. 4, p. 821-825.
* [11] ”Ustünel. A and Zakai, M. (2000) _Transformation of measure on Wiener space._ Springer, Berlin - New York.
|
arxiv-papers
| 2009-11-03T18:17:48 |
2024-09-04T02:49:06.257858
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Liptser",
"submitter": "R. Liptser",
"url": "https://arxiv.org/abs/0911.0641"
}
|
0911.0699
|
# The impact of feedback on the low redshift Intergalactic Medium
L. Tornatore1,2,3, S. Borgani1,2,3, M. Viel2,3 & V. Springel 4
1 Dipartimento di Astronomia dell’Università di Trieste, Via G.B. Tiepolo 11,
I-34131 Trieste, Italy
2 INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131
Trieste, Italy
3 INFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste,
Italy
4 Max-Planck Institut fuer Astrophysik, Karl-Schwarzschild Strasse 1, D-85748
Garching, Germany
###### Abstract
We analyse the evolution of the properties of the low-redshift Intergalactic
Medium (IGM) using high-resolution hydrodynamic simulations that include a
detailed chemical evolution model. We focus on the effects that two different
forms of energy feedback, strong galactic winds driven by supernova explosion
and Active Galactic Nuclei (AGN) powered by gas accretion onto super-massive
black holes (BHs), have on the thermo- and chemo-dynamical properties of of
the low redshift IGM. We find that feedback associated to winds (W) and BHs
leave distinct signatures in both the chemical and thermal history of the
baryons, especially at redshift $z<3$. BH feedback produces an amount of gas
with temperature in the range $10^{5}-10^{7}$K, the Warm Hot Intergalactic
Medium (WHIM), larger than that produced by the wind feedback. At $z=0$ the
fraction of baryons in the WHIM is about 50 per cent in the runs with BH
feedback and about 40 per cent in the runs with wind feedback. The amount of
warm baryons ($10^{4}<T<10^{5}$K) is instead at about the same level, $\sim
30$ per cent, in the runs with BH and wind feedback. Also, BH feedback
provides a stronger and more pristine enrichment of the WHIM. We find that the
metal–mass weighted age of WHIM enrichment at $z=0$ is on average a factor
$\sim 1.5$ smaller in the BH run than for the corresponding runs with galactic
winds. We present results for the enrichment in terms of mass and metallicity
distributions for the WHIM phase, both as a function of density and
temperature. Finally, we compute the evolution of the relative abundances
between different heavy elements, namely Oxygen, Carbon and Iron. While both
C/O and O/Fe evolve differently at high redshifts for different feedback
models, their values are similar at $z=0$. We also find that changing the
stellar initial mass function has a smaller effect on the evolution of the
above relative abundances than changing the feedback model. The sensitivity of
WHIM properties on the implemented feedback scheme could be important both for
discriminating between different feedback physics and for detecting the WHIM
with future far-UV and X-ray telescopes.
###### keywords:
Cosmology: theory – Methods: Numerical – Galaxies: Intergalactic Medium
## 1 Introduction
Solving the problem of the missing baryons at low redshift is one of the
important goals of observational cosmology (e.g., Persic & Salucci, 1992;
Fukugita et al., 1998; Fukugita & Peebles, 2004). Since the pioneering work by
Cen & Ostriker (1999), cosmological hydrodynamic simulations provided a
fundamental contribution in this field (see also Davé et al., 2001). Although
the baryon budget is closed by observations at high redshift,
$z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\
}2$, simulations indicate that about half of the baryons at low-$z$ should lie
in a tenuous quite elusive phase, the so-called Warm-Hot Integalactic Medium
(WHIM). The WHIM is predicted to be made of low density ($n_{\rm H}\sim
10^{-6}-10^{-4}$cm-3) and relatively high temperature ($T\sim 10^{5}-10^{7}$K)
plasma, whose main constituents are ionised Hydrogen and Helium, with traces
of heavier elements in high ionization states. Simulations have also shown
that gas at these densities traces the filamentary structures which define the
skeleton of the large-scale cosmic web, and is heated by shocks from
supersonic gravitational accretion onto the forming potential wells.
Due to its low density, the collisionally ionised WHIM should be characterised
in emission by a very low surface brightness in the UV and soft X–ray bands,
so that its detection lies beyond the capability of available detectors and
should await for instrumentation of the next generation (e.g., Yoshikawa et
al., 2004). Claims for the detection of the WHIM through soft X–ray emission
have already been presented (e.g., Zappacosta et al., 2005; Werner et al.,
2008). However, these detections are limited to special places, such as
external regions of massive galaxy clusters or large filaments connecting such
clusters, where the gas density reaches values higher than expected for the
bulk of the WHIM. A more promising approach to reveal the presence of the WHIM
lies in the detection of absorption features in the spectra of background
sources, at far ultraviolet (FUV) and soft X–ray energies, associated to
atomic transitions from highly ionized elements, using GRB (Gamma Ray Burst)
spectra (Fiore et al., 2000; Branchini et al., 2009), or in the detection of
Ly-$\alpha$ absorption from the tiny fraction of neutral hydrogen revealed
through FUV spectroscopy (see Richter et al., 2008, for a review). Although
detection of gas with $T<10^{5}$K has been obtained at FUV frequencies along a
fairly large number of sightlines (e.g., Tripp et al., 2008, and references
therein), the situation is less clear in the soft X–ray band. At these
energies, claims of WHIM detection in absorption have been presented by
different authors (e.g. Nicastro et al., 2005; Buote et al., 2009), although
they are either controversial (e.g. Kaastra et al., 2006; Rasmussen et al.,
2007) or have moderate statistical significance.
Owing to the above observational difficulties in detecting and characterizing
the physical properties of the WHIM, numerical simulations have played over
the last years the twofold role to forecast its detectability, both in
emission (e.g., Roncarelli et al., 2006; Ursino & Galeazzi, 2006) and in
absorption (e.g., Cen et al., 2001; Kravtsov et al., 2002; Viel et al., 2003;
Chen et al., 2003; Viel et al., 2005), and to predict its observational
properties, also as a function of the physical processes included (see Bertone
et al., 2008, for a review). For instance, feedback effects from star
formation and accretion onto super-massive black holes (SMBHs) are expected to
determine at the same time the process of galaxy formation and the physical
properties of the diffuse cosmic baryons. However, as of today, the mechanisms
powering feedback are still poorly understood and thus very difficult to model
in a fully self-consistent way in cosmological hydrodynamical simulations. On
the other hand, numerical simulations from different groups (e.g., Cen &
Ostriker, 2006; Davé & Oppenheimer, 2007; Kobayashi et al., 2007; Oppenheimer
& Davé, 2008; Tescari et al., 2009; Wiersma et al., 2009) confirm that the
thermo- and chemo-dynamical properties of the WHIM and, more generally, of the
Intergalactic Medium (IGM) at different redshifts are sensitive to the nature
and timing of the feedback. Indeed, metals are observed in the low density IGM
out to high redshift and it is likely that galactic winds, as observed in
galaxies in the low redshift universe, are responsible for the IGM metal
enrichment. Furthermore, SMBHs are also expected to have an impact on the gas
distribution in galaxies and to blow ejecta out to large distances, especially
during galaxy mergers.
In this paper, we will present an analysis of an extended set of cosmological
hydrodynamical simulations with the purpose of investigating the effect that
both galactic winds powered by supenova (SN) explosions and energy feedback
from accretion onto SMBHs have in determining the thermal and chemical
properties of the IGM. Although the main focus of the analysis is on the low-
redshift IGM, we will also discuss how different feedback mechanisms leave
their imprint on the evolutionary properties of diffuse baryons since $z\sim
4$. Our simulations are based on the chemo-dynamical version of the GADGET-2
code (Springel, 2005) presented by Tornatore et al. (2007), which follows the
production of different chemical species by accounting for detailed yields
from Type-Ia and Type-II SN (SN-Ia and SN-II hereafter), as well as from
intermediate and low mass stars in the thermally-pulsating asymptotic giant
branch (TP-AGB) phase, while also accounting for the mass dependent life–times
with which different stellar populations release their nucleosynthetic
products. The model of galactic winds is that introduced by Springel &
Hernquist (2003), that we consider both in its original version and in a
version based on assuming that winds are never hydrodynamically decoupled from
the surrounding medium (see also Dalla Vecchia & Schaye, 2008). As for the BH
feedback, we adopt the models originally introduced by Di Matteo et al.
(2005); Springel et al. (2005). Besides investigating the effect of different
feedback mechanisms, we will also consider the impact of changing the stellar
initial mass function (IMF) on the resulting enrichment pattern of the IGM.
Although the results of this paper have important implications for the
detectability of the WHIM in view of next-generation X–ray missions, we defer
an observationally oriented analysis of our simulations to a future work,
aimed at investigating in detail how different instrumental capabilities will
be able to characterize the WHIM properties at the level required to
discriminate between different feedback models.
The plan of the paper is as follows. In Section 2, we describe the
hydrodynamical simulations used and the feedback mechanisms adopted. Section 3
describes the main results of our analysis in terms of global gas properties
(evolution of different phases over redshift), epoch of enrichment of gas
particles, metallicity distributions at $z=0$ as a function of overdensity and
temperatures, and evolution of different chemical elements. We provide a final
discussion of our results and draw our main conclusions in Section 4. In the
following, we will assume the values of the solar metallicity as reported by
Asplund et al. (2005), with $Z/X=0.0165$ ($X$: hydrogen mass; $Z$: mass
contributed by all elements heavier than Helium).
## 2 Hydrodynamical simulations
Our simulations were carried out using a version of the parallel
hydrodynamical TreePM-SPH code GADGET-2 (Springel, 2005), which includes a
detailed implementation of the chemical enrichment as described in Tornatore
et al. (2007). The initial conditions are generated at redshift $z=99$ in a
cosmological volume with periodic boundary conditions filled with an initially
equal number of dark matter and gas particles. The matter power spectrum is
generated using CMBFAST (Seljak & Zaldarriaga, 1996) for a flat $\Lambda$CDM
model, with cosmological parameters consistent with the recent findings of
WMAP year-5 (Komatsu et al., 2009): $\Omega_{\rm 0m}=0.24,\
\Omega_{0\Lambda}=0.76,\ \Omega_{\rm 0b}=0.0413$, for the density parameters
contributed by total matter, cosmological constant and baryons, $n_{s}=0.95$
for the primordial spectral index, $H_{0}=73$ km s-1 Mpc-1 and
$\sigma_{8}=0.8$ for the normalization of the power spectrum. Besides
performing simulations within boxes of 37.5 $h^{-1}$ comoving Mpc using
$2\times 256^{3}$ gas and dark matter (DM) particles, we also run a few
simulations within a twice as large box of $75\,h^{-1}{\rm Mpc}$ on a side,
with $2\times 512^{3}$ particles so as to keep resolution constant. Such
larger boxes will allow us to keep under control any effect related to box
size. The resulting mass of the gas particles is thus $m_{\rm gas}\simeq
3.6\times 10^{7}\,h^{-1}M_{\odot}$. In these runs, the gravitational softening
is set to $\epsilon=7.5\,h^{-1}$ comoving kpc above $z=2$, while at $z<2$ it
is set to $\epsilon=2.5\,h^{-1}$ physical kpc. As for the B-spline softening
length used for the computation of the SPH forces, the lowest allowed value
was set to half of the gravitational softening. In order to address resolution
effects, we also carried out one simulation in the smaller box using $2\times
400^{3}$ particles. In this case the mass of the gas particles is $m_{\rm
gas}\simeq 9.4\times 10^{6}\,h^{-1}M_{\odot}$, with all the softening lengths
rescaled according to $m_{\rm gas}^{-1/3}$: $\epsilon=4.8\,h^{-1}$ comoving
kpc at $z>2$ and $\epsilon=1.6\,h^{-1}$ physical kpc at lower redshift. All
these simulations are evolved to $z=0$.
We point out that both a large volume and high resolution are necessary in
order to correctly describe the physical properties of the WHIM. Indeed,
shocks driven by the collapse of large-scale structures are important, while
dense environments able to trigger and power the feedback processes need to be
resolved. Our simulations of different resolutions and/or box-sizes will be
denoted by the tuple (linear size of the box in comoving $\,h^{-1}{\rm Mpc}$,
number of DM particles1/3). For instance, we will indicate with (75,512) a
simulation within a box of $75\,h^{-1}{\rm Mpc}$ and containing $512^{3}$ DM
particles (see Table 1).
Radiative cooling and heating processes are followed for a primordial mix of
hydrogen and helium and we include the dependence of cooling on metallicity by
adopting the cooling rates from the tables by Sutherland & Dopita (1993). We
also include the effect of a spatially uniform redshift–dependent Ultra Violet
Background (UVB) produced by quasars, as given by Haardt & Madau (1996), with
helium heating rates multiplied by a factor 3.3 in order to better fit
observational constraints on the temperature evolution of the IGM. This
background gives naturally a hydrogen ionization rate $\Gamma_{-12}\sim 0.8$
at $z=2-4$ (Bolton et al., 2005), which is in broad agreement with the
observations. We adopt the effective model of star formation from a multiphase
interstellar medium introduced by Springel & Hernquist (2003). In this model,
each gas particle whose density exceeds a limiting threshold value is assumed
to contain a hot and a cold phase, the latter providing the reservoir of star
formation. The two phases coexist in pressure equilibrium with their relative
fractions being computed according to the local conditions of density and
temperature. In our simulations we assume that the density threshold for a gas
particle to become multiphase and, therefore, star forming, is $n_{\rm H}=0.1$
cm-3 in terms of the number density of hydrogen atoms. Star forming gas
particles are then assumed to spawn collisionless gas particles according to
the stochastic scheme originally introduced by Katz et al. (1996). We allow
each gas particle to produce up to three generations of star particles, each
having therefore a typical mass of about one third of the initial mass of the
gas particles.
Our simulations have been carried out with the chemo-dynamical version of the
GADGET-2 code described by Tornatore et al. (2007). The included model of
chemical evolution allows us to follow the production of six different metal
species (C, O, Mg, S, Si, Fe) from Type-II and Type Ia supernovae (SNII,
SNIa), along with low and intermediate mass stars (LIMS) in the thermally-
pulsating asymptotic giant branch (TP-AGB) phase. Besides including different
contributions from SNII, SNIa and LIMS, we also include the effect of the
mass–dependent time delay with which different stellar populations release
metals. Specifically, we adopt the lifetime function given by Padovani &
Matteucci (1993). We use the stellar yields by Thielemann et al. (2003) for
SNIa, by Woosley & Weaver (1995) for SNII and by van den Hoek & Groenewegen
(1997) for LIMS. The mass-range for the SNII is $M>8M_{\rm\odot}$, while for
SNIa arising from binary systems in the mass range it is $0.8<M/M_{\odot}<8$,
with a binary fraction of 10 per cent. Finally, we use three distinct stellar
initial mass functions (IMFs): a Salpeter (1955), a Kroupa (2001) and an
Arimoto-Yoshii (1987) IMF. Our reference choice is the functional form
proposed by Kroupa (2001) which adopts a multi-slope approximation,
$\varphi(m)\propto m^{\rm-y}$ with $y=0.3$ for stellar mass
$m<0.5\,M_{\rm\odot}$, $y=1.2$ for $0.5\,M_{\rm\odot}\leq m<1\,M_{\rm\odot}$
and $y=1.7$ for $m\geq 1\,M_{\rm\odot}$. Both the IMF by Salpeter (1955) and
that by Arimoto & Yoshii (1987) have instead a single slope, with $y=1.35$ and
$y=0.95$, respectively. Our model of chemical evolution also includes stellar
mass losses, which are self–consistently computed for a given IMF and life-
time function. This means that a fraction of the mass of the star particles is
restored as diffuse gas during the evolution and distributed to the
surrounding gas particles. We refer to Tornatore et al. (2007) for a detailed
description of our implementation of chemo-dynamics in GADGET-2, to Saro et
al. (2006) and Fabjan et al. (2008) for applications to simulations of galaxy
clusters and to Tescari et al. (2009) for a description of the global IGM
properties around DLAs at $z>2$.
In the two following subsections we will summarize the main features of the
two different feedback schemes adopted in this paper.
### 2.1 Galactic outflows
The implementation of galactic outflows is extensively described by Springel &
Hernquist (2003). In this model, winds are assumed to blow from star forming
regions with a mass loading rate $\dot{M}_{\rm w}$ proportional to the star
formation rate $\dot{M}_{\rm\star}$ according to $\dot{M}_{\rm
w}=\eta\dot{M}_{\rm\star}$. Star-forming gas particles are then stochastically
selected to become part of a blowing wind, with a probability which is
proportional to their star formation rate. In its original implementation,
whenever a particle is uploaded to the wind, it is decoupled from
hydrodynamics until the density of the surrounding gas drops below a given
limiting value. This allows the wind particle to travel ‘freely’ up to few kpc
until it has left the dense star-forming phase, without directly affecting it.
As a protection against decoupling a wind particle indefinitely in case it
gets ’stuck’ in the ISM of a very massive galaxy, the maximum allowed time for
a wind particle to stay hydrodynamically decoupled is set to $t_{\rm
dec}=l_{\rm w}/v_{\rm w}$, where we fix $l_{\rm w}=10\,h^{-1}$kpc in our
reference case, while $v_{\rm w}$ is the wind speed. As for the limiting
density for hydrodynamic decoupling of gas particles, it is set to 0.5 in
units of the threshold for star formation. Unlike in Springel & Hernquist
(2003), we decide here to fix the velocity of the winds to the value $v_{\rm
w}=500\,{\rm km\,s^{-1}}$, instead of fixing the fraction of the SNII energy
powering galactic ejecta. For the efficiency of the wind mass loading, we
adopt $\eta=2$. In sum, four parameters fully specify the wind model: the wind
efficiency $\eta$, the wind speed $v_{\rm w}$, the wind free travel length
$l_{\rm w}$ (whose detailed value is unimportant) and the wind free travel
density factor.
In order to verify the effect of the hydrodynamic decoupling of wind particles
we also perform a simulation without such a decoupling, $t_{\rm dec}=0$, i.e.
keeping the particles always hydrodynamically coupled. We note that Nagamine
et al. (2007) showed that global DLAs properties are relatively insensitive to
the value of $l_{\rm w}$. On the other hand, Dalla Vecchia & Schaye (2008)
pointed out that keeping winds always hydrodynamically coupled has a
significant effect on the evolution and star formation in simulations of
isolated disk galaxies.
We stress that our wind model is not the only possible wind implementation and
others could be adopted such as those based on momentum–driven winds as
suggested by Murray et al. (2005) and Davé & Oppenheimer (2007).
### 2.2 BH feedback
We also include in our simulations the effect of feedback energy from gas
accretion onto super-massive black holes (BHs), following the scheme
originally introduced by (Springel et al. 2005, see also Di Matteo et al.
2005). In this model, BHs are represented by collisionless sink particles
initially seeded in just resolved DM haloes, which subsequently grow via gas
accretion and through mergers with other BHs during close encounters. Every
new dark matter halo, identified by a run-time friends-of-friends algorithm,
above the mass threshold $M_{\rm th}=10^{10}\,h^{-1}M_{\odot}$, is seeded with
a central BH of initial mass $10^{5}\,h^{-1}M_{\odot}$, provided the halo does
not contain any BH yet. Each BH can then grow by local gas accretion, with a
rate given by
$\dot{M}_{\rm BH}\,=\,{\rm min}\left(\dot{M}_{\rm B},\dot{M}_{\rm
Edd}\right)\,.$ (1)
Here $\dot{M}_{\rm B}$ is the accretion rate estimated with the Bondi-Hoyle-
Lyttleton formula (e.g., Bondi, 1952), while $\dot{M}_{\rm Edd}$ is the
Eddington rate. The latter is inversely proportional to the radiative
efficiency $\epsilon_{\rm r}$, which gives the radiated energy in units of the
energy associated to the accreted mass: $\epsilon_{\rm r}=L_{\rm
r}/(\dot{M}_{\rm BH}c^{2})$. Following Springel et al. (2005), we use
$\epsilon_{\rm r}=0.1$ as a reference value, which is typical for a
radiatively efficient accretion onto a Schwarzschild BH (Shakura & Sunyaev,
1973). The model then assumes that a fraction $\epsilon_{\rm f}$ of the
radiated energy is thermally coupled to the surrounding gas, so that
$\dot{E}_{\rm feed}=\epsilon_{\rm r}\epsilon_{\rm f}\dot{M}_{\rm BH}c^{2}$ is
the rate of the energy released to heat the surrounding gas. Using
$\epsilon_{\rm f}\sim 0.05$, Di Matteo et al. (2005) were able to reproduce
the observed $M_{\rm BH}-\sigma$ relation between bulge velocity dispersion
and mass of the hosted BH (see also Sijacki et al. (2008); Di Matteo et al.
(2008)). Gas particle accretion onto the BH is implemented in a stochastic
way, by assigning to each neighbouring gas particle a probability of
contributing to the accretion, which is proportional to the SPH kernel weight
computed at the particle position. In the scheme described above, this
stochastic accretion is used only to increase the dynamic mass of the BHs,
while their mass entering in the computation of the accretion rate is followed
in a continuous way, by integrating the analytic expression for $\dot{M}_{\rm
BH}$. Once the amount of energy to be thermalised is computed for each BH at a
given time-step, this energy is then distributed to the surrounding gas
particles using the SPH kernel weighting scheme.
To sum up, we run simulations with the Kroupa (2001) IMF by turning off
galactic winds (NW), with galactic winds (W), with galactic winds always
hydrodynamically coupled (CW) and with black hole feedback (BH). We summarize
in Table 1 the simulations analysed in this paper. Overall, our simulation set
allows us to address the effect of changing: (a) box size (by comparing
W37,256 with W75,512 and Way37,256 with Way75,512), (b) IMF (by comparing
W37,256 with Way37,256 and with Ws37,256), (c) resolution (by comparing
W37,256 with W37,400) and (d) nature of the energy feedback (by comparing
W37,256 with NW37,256, with CW37,256 and with BH37,256).
Run | Box size | $N_{\rm DM}^{1/3}$ | IMF | Feedback
---|---|---|---|---
W37,256 | 37.5 | 256 | Kroupa | Winds; $v_{\rm w}=500\,{\rm km\,s^{-1}}$
W75,512 | 75.0 | 512 | Kroupa | Winds; $v_{\rm w}=500\,{\rm km\,s^{-1}}$
W37,400 | 37.5 | 400 | Kroupa | Winds; $v_{\rm w}=500\,{\rm km\,s^{-1}}$
Way37,256 | 37.5 | 256 | Arimoto-Yoshii | Winds; $v_{\rm w}=500\,{\rm km\,s^{-1}}$
Ws37,256 | 37.5 | 256 | Salpeter | Winds; $v_{\rm w}=500\,{\rm km\,s^{-1}}$
NW37,256 | 37.5 | 256 | Kroupa | No feedback
CW37,256 | 37.5 | 256 | Kroupa | Coupled winds; $v_{\rm w}=500\,{\rm km\,s^{-1}}$
BH37,256 | 37.5 | 256 | Kroupa | Black Hole feedb., no winds
Table 1: Summary of the different runs. Column 1: run name; column 2: comoving
box size (units of $\,h^{-1}{\rm Mpc}$); column 3: number of DM particles;
column 4: stellar initial mass function (IMF, see text); column 5: feedback
included (see text).
## 3 Results
In Figure 1 we show the star formation rates (SFR) for the W37,256, NW37,256,
CW37,256 and BH37,256 simulations along with observational results by Hopkins
(2004). At $z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise
5.0pt\hbox{$>$}\ }3$ the NW and BH runs behave very similarly. This is not
surprising since BH feedback is not effective until a sufficiently large
number of DM haloes, massive enough to host a seed BH, is numerically
resolved. After that, gas accretion takes place in BHs with a subsequent
release of thermal energy. Once BH feedback becomes efficient, star formation
is suddenly suppressed by the expulsion of hot gas. Kinetic feedback by
galactic winds is relatively more efficient at high redshifts in depriving
relatively small haloes of the star–forming gas, while it fails in regulating
the star formation within massive haloes at later epochs. We note that
hydrodynamically coupled winds (CW) tend to produce a slightly higher SFR
around the peak of the star formation history. This is due to the fact that
coupled winds tend to thermalize their kinetic energy at smaller distances
from star forming and dense regions. The cooling time of this gas is
correspondingly shorter, which causes a larger fraction of the thermalized
wind energy to be radiated away, thus reducing the feedback efficiency. Both
runs including winds reproduce the observed behaviour of the SFR at high
redshift, $z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise
5.0pt\hbox{$>$}\ }2$, while they tend to produce too high star formation at
low redshift. On the contrary, the run with BH feedback has a too high star
formation at $z>3$, while it recovers the observed SF level at
$z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\
}2$. This illustrates once again the different role played by feedback related
to star formation and to BH accretion: while the former is efficient since
early times in regulating gas cooling within small haloes, the latter sets in
at relatively lower redshift to quench star formation within recently formed
massive haloes.
Figure 1: Star formation rate density as a function of redshift in units of
M⊙yr-1Mpc-3 for W37,256 (winds, continuous blue line), NW37,256 (no feedback,
dashed green line), CW37,256 (coupled winds, dot-dashed orange line) and
BH37,256 (black hole feedback, triple dotted-dashed line) runs. Observational
data points represented as full circles are taken from Hopkins (2004), while
the empty circles are from Bouwens et al. (2008).
### 3.1 Global gas properties
Figure 2: The $\rho_{\rm gas}-T$ phase diagrams at $z=0$ for simulations based
on the Kroupa (2001) IMF, by varying the feedback scheme, color coded
according to the metal mass fraction (see vertical bar): NW37,256 run with no
feedback (top left), W37,256 run with galactic winds (top right), CW37,256 run
with galactic winds always hydrodynamically coupled (bottom left) and BH37,256
run with BH feedback (bottom right).
We show in Figure 2 the density–temperature phase diagrams for the gas in the
four simulations based on the Kroupa (2001) IMF and including different
feedback schemes. The four panels are for the simulations without feedback
(NW37,256, top left), with feedback associated to galactic winds (W37,256, top
right), with galactic winds always hydrodynamically coupled (CW37,256, bottom
left) and with BH feedback (BH37,256, bottom right). Each phase diagram is
color-coded according to the mass in metals which is associated to the gas
particles belonging to each two-dimensional bin in the $\rho_{\rm gas}$–$T$
plane, with brighter regions corresponding to a larger metal mass. As such,
these plots convey information on the effect that different feedback schemes
have on the way in which metals are distributed in density and temperature.
Here and in the following we do not include in the baryon budget inventory the
inter–stellar medium (ISM), i.e. for gas having density larger than the
density threshold for the star formation (see Sect. 2). We note that only a
tiny fraction of the gas lies in this phase at all redshifts. Furthermore,
while our sub-grid model for star formation should provide an effective
description of the ISM, it is not obvious that this description provides a
fully realistic description of its temperature. For this reason, our gas mass
and metal budget will not include the contribution from star–forming gas.
The effect of feedback is quite strong for gas in the temperature range
$10^{5}$–$10^{7}$ K (corresponding to the WHIM phase that we will define
below; e.g. Cen & Ostriker 2006). Strong galactic winds increase the amount of
metals carried by gas with temperature in the range $10^{5}-10^{6}$K and
overdensity 111Here and in the following we denote with $\delta_{\rm b}$ the
overdensity of gas with respect to the mean cosmic baryon density
$\bar{\rho}_{\rm b}$, i.e. $\mbox{$\delta_{\rm b}$}=\rho_{\rm
b}/\bar{\rho}_{\rm b}-1$. $\mbox{$\delta_{\rm b}$}\simeq 1$–10, when compared
to the no feedback case. This is a consequence of the fact that winds are
loaded with gas particles which were star forming and, as such, have been
heavily enriched. The effect is even more dramatic in the presence of BH
feedback: even regions below the mean density are enriched with metals up to
rather high temperatures ($\sim 10^{7}$ K) when compared to the corresponding
W and CW simulations. This counter-intuitive result follows from the fact that
BHs’ release of large amount of energy in a relatively short time interval,
around $z\sim 3$. This sudden energy release turns out to be much more
efficient than winds to heat metal enriched gas to high temperatures, thus
displacing it from the haloes of star-forming regions to low-density regions.
Once brought to high entropy by BH feedback, this enriched gas is prevented
from re-accreting into collapsed haloes at lower redshift. Gas in photo-
ionization equilibrium at $T\sim 10^{4}$ K contains a comparable amount of
metals in the three runs which include feedback. In the NW case the amount of
metals present in the gas around the mean density is an order of magnitude
lower than in the runs including winds. Indeed, in the NW simulation, most of
the enriched gas remains at high density, instead of being transported away
from star-forming regions. The short cooling time of this enriched gas causes
its selective removal from the diffuse phase into the stellar phase. We note
also that relatively cold gas in very dense regions, $\rho_{\rm
gas}\simeq(10^{3}-10^{4})\bar{\rho}_{\rm b}$, is more enriched with metals in
the NW, CW and W simulations than in the BH one. This demonstrates the
efficiency of BH feedback in displacing highly enriched gas outside the core
regions of virialized haloes, while correspondingly increasing the enrichment
level of gas at lower density. As for the comparison between coupled and
decoupled winds (W and CW runs, respectively), the latter has more enriched
gas at densities approaching the threshold for the onset of star formation.
This is the consequence of the hydrodynamic coupling of winds which causes
wind particles to remain more confined in the proximity of star–forming
regions.
From a qualitative inspection of Fig. 2 we draw the following conclusions: i)
galactic winds have a large impact on the metal enrichment of the IGM at $z=0$
both for underdense-cold and mean density-hot regions of the $T-\rho_{\rm
gas}$ plane; ii) AGN feedback is more efficient than winds in enriching the
warm-hot gas at $T=10^{5}-10^{7}$ K and relatively low overdensity,
$\mbox{$\delta_{\rm b}$}\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }50$; iii) correspondingly,
very dense regions, with overdensity $\mbox{$\delta_{\rm b}$}>10^{4}$, are
less enriched with metals in the BH simulations than in the other runs.
### 3.2 Redshift evolution of the different phases
Figure 3: The fraction of mass in the warm (left panel), WHIM (middle panel)
and stellar plus hot phases (right panel, thin lines for stars and thick lines
for hot gas) as a function of redshift for the NW37,256, W37,256, CW37,256 and
BH37,256 runs (continuous blue, dashed green, dot-dashed orange and double-dot
dashed red curves, respectively).
Figure 4: Effects of resolution and box-size on the amount of gas in different
phases. The fraction of mass in the warm (left panel), WHIM (middle panel) and
stars plus hot components (right panel, thin lines for stars and thick lines
for hot gas) as a function of redshift for the W37,256, W37,400, W75,512 runs
(continuous blue, dashed green, dot-dashed orange curves, respectively).
In order to make our results directly comparable with those from previous
analyses (e.g., Davé et al., 2001; Cen & Ostriker, 2006), we define four
different phases for the baryons in our simulations: a WHIM phase, for gas
particles with temperature in the range $10^{5}-10^{7}$ K; a warm phase, for
gas particles with $T=10^{4}-10^{5}$ K; a hot phase, for gas particles with
$T>10^{7}$ K; a condensed phase, which includes all the baryonic mass
associated with star particles.
In Figure 3, we show the redshift evolution of the mass fraction in these four
different phases, comparing the effect of changing feedback scheme. The
W37,256, NW37,256, CW37,256 and BH37,256 runs are represented by the
continuous blue, dashed green, dot-dashed orange and double-dot dashed red
lines, respectively. The warm phase is reported in the left panel. Although
showing slightly different evolutions in detail, the amount of warm gas at
$z=0$ is almost the same, $\simeq 35$ per cent, for the runs with galactic
winds and with BH feedback. This gas should be partly responsible for local
Lyman-$\alpha$ forest absorption (e.g., Danforth & Shull, 2008), samples the
outskirts of galactic haloes and is weakly sensitive to the energetics and
mass-loading parameters of the energy-driven galactic wind feedback (Pierleoni
et al., 2008). The case with no feedback (NW) has instead a $\sim 10$ per cent
lower amount of warm gas at $z=0$. In the middle panel, we show the evolution
of the WHIM phase: with BH feedback the amount of WHIM at $z=0$ reaches 50 per
cent, about 10 per cent more than for the runs with winds and 15 per cent
higher than for the NW case. Similarly to the results for the warm phase,
there are small differences between the W and CW runs, also in the amount of
WHIM gas and stars. A more significant difference between W and CW runs is
found for the mass fraction in stars. The reason for this difference is in the
lower efficiency of the CW feedback model. Indeed, hydrodynamically coupled
winds deposit kinetic energy through hydrodynamical processes in relatively
higher-density environments, which are characterized by shorter cooling times,
which causes the thermalized energy to be promptly radiated away.
The evolution of the WHIM mass fraction in the BH run is initially identical
to that for the NW run at $z\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }3.5$. Indeed, at high
redshift there is still a limited number of DM haloes whose mass is large
enough to host actively accreting BHs. At lower redshift, when accretion onto
BHs is more effective, the released feedback energy efficiently heats the gas
surrounding galaxy–size haloes, thus moving gas from the warm to the WHIM
phase. This explains at the same time the lower amount of warm gas in the BH
run at intermediate redshift and the correspondingly larger amount of WHIM. As
for the effect of galactic winds, they play a role in heating circum–galactic
gas already at high redshift, soon after the onset of star formation. This
explains the larger amount of WHIM at $z\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }2.5$ in the W and CW runs
with respect to the BH run. The situation is reversed at lower redshift, when
the depths of the forming potential wells become large, making gas heating
with winds less effective with respect to BH feedback.
In the right panel we plot the evolution of the mass fractions in stars and
hot gas, which are represented by the thin and thick lines, respectively (note
the different scale in the $y-$axis). As expected, the mass in stars in the
runs with and without feedback is significantly different, largely reflecting
the behaviour of the SFR histories shown in Fig. 1: the BH run has a fraction
of stars which is five (two) times less than the NW (CW) run, as a consequence
of the quenching of star formation taking place around $z=3$, while the W run
produces a slightly smaller amount of stars when compared to the CW one.
Recently, Li & White (2009) used SDSS data to reconstruct the galaxy stellar
mass function in the local Universe. Assuming an IMF by Chabrier (2003) (very
close to the IMF by Kroupa 2001 used in the simulations shown in Figure 3),
they concluded that 3.5 per cent of the baryonic mass is locked into stars.
While clearly ruling out a model with no feedback, this observational estimate
of the stellar mass fraction is close to the predictions of the BH run,
although the star formation rate in this case is too high at
$3\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\
}z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\
}6$ and slightly too low at $z\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }0.5$. As for the W and CW
runs, they produce about two times and 2.5 times more stars than the BH run,
irrespectively, as a consequence of a star formation excess below $z\sim 1$.
These results suggest that both feedback mechanisms should be active: SN
driven galactic ejecta should regulate star formation already at high redshift
within relatively small galaxies, while BHs should quench cooling in large
haloes around the maximum of the star formation, thereby keeping gas
pressurised within such haloes down to $z=0$.
The amount of hot gas at temperatures above $10^{7}$K is instead very similar
for all the simulations and around 3 per cent. This is a consequence of the
fact that the bulk of hot gas lies within groups and clusters of galaxies,
whose gas content is marginally affected by the details of the feedback. We
note, however, that while the amount of hot gas in the BH run is larger than
in the other runs at high redshift, it drops below them at $z=1$, with an
overall trend that is opposite to that of the WHIM component. Indeed, at high
redshift BH feedback is quite efficient in heating gas to high temperature,
thus providing a sort of diffuse pre–heating. At later time, this pre–heated
gas has a harder time to fall into group–size potential wells. To quantify
this effect, we note that in the BH run about $\sim 10$ per cent of the hot
gas lies below $\mbox{$\delta_{\rm b}$}\,\sim 1$, while only
$\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\
}0.1$ per cent of hot gas lies at such low densities in the other runs.
It is worth pointing out that although at $z=0$ the total amount of hot gas is
about the same, $\simeq 3$ per cent, for all feedback models, its distribution
as a function of density has a distinct pattern in the BH run. In fact, in the
runs with no BH feedback gas is heated mostly by the process of gravitational
virialization within the potential wells of galaxy groups and clusters. As a
consequence, in those runs only about the 5 per cent of hot gas lies below
virial overdensities, $\mbox{$\delta_{\rm b}$}\sim 50$. On the other hand, a
sizeable amount of gas displaced by BH feedback lies outside virialised
haloes, with $\sim 37$ per cent of it found at
$\delta_{b}\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise
5.0pt\hbox{$<$}\ }50$. Furthermore, the heating efficiency of BH feedback
prevents hot gas from reaching densities as high as in the other runs in the
central regions of galaxy groups and clusters (see also Sijacki et al. 2007;
Bhattacharya et al. 2008; Fabjan et al. 2009). As a result, the highest
density reached by hot gas in the BH runs is lower by about a factor three
than in the NW, W and CW runs.
Interestingly, hot metals follow the fate of the gas they are associated with.
Indeed, the fractions of hot metal mass lying at different densities are quite
similar to the corresponding fractions of hot gas mass.
In order to verify the robustness of our results against box size and
numerical resolution, we compare in Figure 4 the results of the W37,256 run
with those of the W75,512 and W37,400 runs. In general, we find that there is
a very good convergence in the values of the mass fractions associated to the
WHIM, warm and star phases, at all considered redshifts. We only note that the
amount of gas in the hot phase significantly increases in the W75,512 run at
$z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\
}1.5$. This is due to the fact that a larger box can accommodate a larger
number of relatively more massive galaxy groups and clusters where a larger
amount of gas is shock–heated to a temperature above $10^{7}$K. A larger box,
with size $\sim 200\,h^{-1}{\rm Mpc}$, would certainly allow to better sample
the high end of the halo mass function and, therefore, to provide a fully
converged estimate of the mass fraction in hot gas. We note, however, that our
larger simulation box already allow us to sample the scale of galaxy groups,
which contain gas at a temperature $\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }10^{7}$ K. Indeed, the
largest halo found within the 75$\,h^{-1}{\rm Mpc}$ box has a mass of about
$3\times 10^{14}\,h^{-1}M_{\odot}$ and a X–ray emission–weighted temperature
of about $4\times 10^{7}$K. Therefore, we do not expect our baryon inventory
in simulations to be significantly affected by effects of finite box size. In
the following, we will restrict our analysis to the four (37, 256) runs, based
on the four feedback schemes (NW, W, CW, BH). For this reason, and unless
otherwise specified, we omit from now on the box-size and number of the
particles when referring to the analysed simulations.
In summary, we conclude that at $z=0$ the amount of mass in the WHIM in the
simulation including BH feedback is about 10 per cent larger than in the
simulations including galactic winds, and also displays a different redshift
evolution. Quite remarkably, this result is in quantitative agreement with
that reported by Cen & Ostriker (2006). The fact that comparable values for
the WHIM mass fraction are found with different simulation codes, based on
different hydrodynamic schemes, and using different implementations of
feedback processes highlights that this should be considered as a robust
prediction of models of cosmic structure formation. Therefore, if future
observations will falsify these predictions, this will have direct
implications on the need to include new physical processes in simulations.
### 3.3 The history of enrichment
Figure 5: Ratio between the average age of WHIM enrichment and the age of the
universe as a function of redshift ($\bar{t}_{\rm WHIM}/t_{\rm cosmic}$).
Results are shown for the W37,256, NW37,256, CW37,256 and BH37,256
simulations, represented by the continuous blue, dot-dashed green, double-dot-
dashed orange and dashed red lines, respectively.
Figure 6: Left panel: difference, in Gyr, between the average ages of
enrichment for WHIM and warm particles in the diffuse phase ($\bar{t}_{\rm
WHIM,\mbox{$\delta_{\rm b}$}\,<50}-\bar{t}_{\rm warm,\mbox{$\delta_{\rm
b}$}\,<50}$). Right panel: the same as in the left panel but for gas in the
collapsed phase ($\bar{t}_{\rm WHIM,\mbox{$\delta_{\rm b}$}\,>50}-\bar{t}_{\rm
warm,\mbox{$\delta_{\rm b}$}\,>50}$). Results are shown for the W37,256,
NW37,256, CW37,256 and BH37,256 simulations, represented by the continuous
blue, dot-dashed green, double-dot-dashed orange and dashed red lines,
respectively.
In Figure 5 we show the redshift dependence of the average age of enrichment
of the gas particles in units of the age of the universe at the same redshift.
Results are shown for warm and WHIM phases. Besides showing results for all
gas particles belonging to each of these two phases, in the following we will
also classify gas within each phase according to its overdensity.
Specifically, we will define a collapsed phase, which is made up by all gas
particles having overdensity $\mbox{$\delta_{\rm b}$}>50$ (we note that this
is the typical overdensity reached within virialised haloes), and a diffuse
phase, which contains gas at $\mbox{$\delta_{\rm b}$}<50$.
Furthermore, we define the average age of enrichment of a gas particle at
redshift $z$ as
$\bar{t}(z)\,=\,{\sum_{i}\Delta m_{Z,i}(z)t_{i}\over m_{Z,i}(z)}\,,$ (2)
where the sum is over all the time-steps performed until redshift $z$, $\Delta
m_{Z,i}$ is the mass in metals received by the particle at the $i$-th
simulation time-step, $t_{i}$ is the cosmic time at that time-step and
$m_{Z}(z)$ is the total metal mass received by a particle before $z$.
According to this definition, large values of enrichment age, at a given
redshift, correspond to a smaller look-back time for the epoch of enrichment,
while smaller values of $\bar{t}(z)$ indicate a more pristine metal
enrichment. In the limit in which all the metals were received by the
particles at the considered redshift $z$, then the enrichment age coincides
with the cosmic age at that redshift. The mean of the ages of enrichment of a
given phase is then computed by averaging eq. (2) over all particles belonging
to that phase, each particle being weighted according to its mass in metals:
$\bar{t}_{\rm
phase}(z)\,=\,{\sum_{j}\bar{t}_{j}(z)m_{Z,j}(z)\over\sum_{j}m_{Z,j}(z)}\,.$
(3)
It is worth pointing out that this way of computing the enrichment age of a
gas phase does not provide a measurement of the average time of enrichment of
the phase to which the particles belong; instead, it is an estimate of the
average time at which the metals that belong to that phase were deposited in
the gas particles currently making up that phase. Figure 5 shows the redshift
dependence of the age of enrichment: the left panel shows the enrichment epoch
of the WHIM phase in units of cosmic time at the same redshift, $\bar{t}_{\rm
phase}(z)/t_{\rm cosmic}(z)$, for the same simulations of Figure 2.
Figure 5 shows the enrichment epoch of the WHIM phase. We note that it
decreases with time in all runs, thus implying that at high redshifts the
metal mass deposition takes place at a larger fraction of the cosmic time.
Thus, the lower the value of $\bar{t}_{\rm phase}(z)/t_{\rm cosmic}(z)$, the
earlier the enrichment of the particles belonging to that phase happened. We
also point out that the rate at which the ratio $\bar{t}_{\rm WHIM}(z)/t_{\rm
cosmic}(z)$ decreases with redshift provides a measure of when the bulk of
metals are received by the particles belonging to each phase. For the NW
simulation, the WHIM enrichment is always quite recent and is due to the
combined action of star formation and structure formation: while the former
sets the time-scale at which metals are produced and assigned to the gas
surrounding star-forming regions, the latter determines the epoch at which
potential wells are sufficiently developed for gas dynamical processes to
become effective in removing gas from such regions during the hierarchical
assembly of structures. The W and CW runs exhibit a redshift dependence that
is quite similar to that of the NW run. This is not surprising since the wind
mass–load that carries the metals to the WHIM phase is also proportional to
the star formation rate, while gas–dynamical processes dominate at low
redshifts, when the star formation rate drops at a similar rate in all these
runs. A significantly different history of enrichment is instead found in the
BH run. In this case, the sudden episodes of energy release, occurring around
$2\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\
}z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\
}4$ when the mass accretion rate onto BHs peaks, are very effective in heating
and expelling a large amount of enriched gas from the galaxies. Indeed, the
faster decrease of the $\bar{t}_{\rm WHIM}(z)/t_{\rm cosmic}(z)$ ratio for the
simulation with BH feedback is a consequence of the different timing of the
enrichment episodes of the particles that then ended in the WHIM. Therefore,
while winds produce an enrichment which starts earlier and proceeds more
gradually, AGN-BH feedback heats and enriches the particles at later epochs
and over a shorter time interval. On the other hand, the lack of feedback
capable to displace enriched gas in the NW run delays the release of metals
from star–forming regions and closely links the timing of enrichment for WHIM
and warm phases to the timing of growth of large potential wells, where
stripping of enriched gas from merging haloes takes place.
Another useful diagnostic to study the enrichment timing of the IGM is
represented by the relative delays of enrichment between the WHIM and warm
phases and between the diffuse and collapsed environments. We recall that we
defined a gas particle to belong to the diffuse phase whenever it has density
contrast $\mbox{$\delta_{\rm b}$}<50$, while it belongs to the collapsed phase
for $\mbox{$\delta_{\rm b}$}>50$.
In the left panel of Figure 6 we plot $\bar{t}_{\rm WHIM,\mbox{$\delta_{\rm
b}$}\,<50}-\bar{t}_{\rm warm,\mbox{$\delta_{\rm b}$}\,<50}$, while the right
panel shows $\bar{t}_{\rm WHIM,\mbox{$\delta_{\rm b}$}\,>50}-\bar{t}_{\rm
warm,\mbox{$\delta_{\rm b}$}\,>50}$. As shown in this panel, at high redshifts
the enrichment of the diffuse WHIM phase in the NW run is on average roughly
coeval to that of diffuse warm phase, while it becomes progressively more
recent at lower redshifts (by $\sim 0.2$ Gyr at $z\sim 2$ and by $\sim 1$ Gyr
at $z=0$). As already mentioned, this evolution is purely driven by
gas–dynamical processes which bring metals from merging haloes to the diffuse
medium, whenever these haloes enter in the pressurized medium permeating large
potential wells. Since enriched gas is extracted from galaxies and groups of
galaxies at temperatures typically hotter than that of the IGM, cooler diffuse
enriched particles need more time to reach lower temperatures. This implies
that they have been stripped earlier than hotter gas, thus explaining why warm
diffuse gas has been enriched before WHIM diffuse gas.
Quite interestingly, the presence of galactic winds (W and CW runs) is
effective in improving the mixing between the warm and WHIM diffuse phases,
thus making their enrichment coeval down to lower redshifts, $z\simeq 1$. At
lower redshifts, gas stripping within large structures becomes again the
dominant process, thus recovering the same qualitative behaviour as in the NW
run. As for the run with BH feedback, its behaviour is quite close to that of
the NW run at high redshifts. At $z\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }3$ gas, displacement from
already enriched haloes to the WHIM phase becomes gradually more efficient,
thus making the enrichment of the diffuse WHIM more recent than for the
diffuse warm gas.
As for the gas in the collapsed phase ($\mbox{$\delta_{\rm b}$}>50$, right
panel), the enrichment of the warm and of the WHIM phases is almost coeval
down to $z\simeq 2$ in the NW run. The reason for this lies in the relatively
short time scale over which warm particles within haloes are heated to
$T>10^{5}$K and enriched when they approach star forming regions. At lower
redshift, the growth of progressively larger structures makes gas in a shock-
heated phase to have a progressively higher temperature, until it eventually
reaches the WHIM temperature range. At $z=0$ this gas makes up a phase with
$T>10^{5}$K, reaching overdensity of up to $\mbox{$\delta_{\rm b}$}\sim
10^{4}$, as shown in the upper left panel of Fig. 2. Differently from the warm
dense gas, the shock-heated WHIM does not lie close to star forming regions
and, as such, it did not experience recent enrichment episodes. By $z=0$, the
enrichment age of the WHIM takes place on average $\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }1$ Gyr earlier than that of
warm gas. As for the W run, we note that the effect of winds is that of
shortening the time scale that warm gas takes to reach WHIM temperatures,
thereby making even more coeval the enrichment age of the diffuse and
collapsed phases. The run with BH feedback is again very close the NW run at
$z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\
}3$. At lower redshift, BH accretion becomes efficient. The subsequent energy
feedback causes a fast removal of recently enriched gas from the very dense
warm phase, with $\mbox{$\delta_{\rm b}$}>10^{4}$, surrounding star-forming
regions. This effect is visible in Fig. 2: comparing upper left and lower
right panels one clearly notices the depletion of dense and warm gas in the BH
run. This gas is shock heated to larger temperatures, thus providing a supply
of recently enriched medium to the WHIM phase.
In general, these results show that the timing of enrichment of the WHIM does
depend on the nature of the feedback included in the simulations. The presence
of winds leaves its fingerprint in the timing of enrichment of the diffuse
component, with $\mbox{$\delta_{\rm b}$}<50$, at relatively high redshift,
$z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\
}1$. On the other hand, BH feedback has a much more evident effect at
relatively low redshift, $z\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }2$, i.e. in the regime
where it reaches the peak of efficiency in quenching star formation and in
displacing gas from star forming regions. Quite interestingly, the effect is
more pronounced for the collapsed gas phase, $\mbox{$\delta_{\rm b}$}>50$.
This suggests that studying the enrichment pattern of galaxy clusters and
groups, out to their outskirts, should represent the best diagnostic for the
role played by AGN feedback in determining the cosmic cycle of metals (e.g.,
Fabjan et al., 2009).
### 3.4 Properties of the WHIM in the local Universe
#### 3.4.1 Total and metal mass distribution as a function of density and
temperature
Figure 7: Probability distribution functions (PDF) of the total WHIM mass
(left panel) and WHIM mass in metals (right panel) at $z=0$ as a function of
gas density (in units of the mean cosmic baryon density) for the W, NW, CW, BH
simulations, represented by the continuous blue, dot-dashed green, double-dot-
dashed orange and dashed red lines, respectively.
Figure 8: Probability distribution functions of the rotal WHIM mass (left
panel) and WHIM mass in metals (right panel) at $z=0$ as a function of gas
temperature for the W, NW, CW, BH simulations, represented by the continuous
blue, dot-dashed green, double-dot-dashed orange and dashed red lines,
respectively.
While in the previous sections we analysed the evolution of different phases,
defined according to temperature criteria, we present now an analysis of the
metal and gas content of the WHIM at $z=0$ as a function of gas density and
temperature. As such, this analysis has implications for the detectability of
the WHIM in the local Universe and for connecting its properties to the
feedback mechanisms which determine its density and metallicity structure. The
results are presented in Figures 7 and 8 for the density and temperature
distributions, respectively. In the left panels of these figures we show the
probability distribution function (PDF) for the total WHIM mass, while in the
right panels we show the corresponding PDF for the metal mass in the WHIM.
These PDFs are defined so that they provide the fraction of WHIM gas mass
(metal mass) contributed by unit logarithmic intervals in gas density (Fig. 7)
and temperature (Fig. 8).
Consistently with the phase diagrams shown in Fig. 2, the left panel of Fig. 7
shows that BH feedback predicts a mass distribution for the WHIM component
which is quite different from the other simulations: it reaches smaller
underdensities, while the high–density tail is suppressed. The reason for this
is that this feedback model provides episodes of strong heating of the gas
around BHs, allowing it to reach low density regions, thus reducing the gas
content of virialised haloes. At $z=0$, we find that $\sim 50$ per cent of the
WHIM in the NW, W and CW runs has $\mbox{$\delta_{\rm b}$}<50$, thus lying
outside collapsed structures. This fraction increases to
$\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\
}80$ per cent in the run with BH feedback. Correspondingly, the peak of the
WHIM density distribution in the BH case lies at $\mbox{$\delta_{\rm
b}$}\simeq 10$, a value that increases to $\mbox{$\delta_{\rm b}$}\simeq 80$
in the other simulations. This result is in line with that found by
Bhattacharya et al. (2008) and Puchwein et al. (2008), who showed that BH
feedback is effective in removing gas from virialized haloes having masses
typical of galaxy groups. Differences between the W, CW and NW runs are
instead more prominent in the high density tail of the distribution,
$\mbox{$\delta_{\rm b}$}>10^{4}$, where they involve in any case only a tiny
fraction of the total WHIM mass. In the NW and CW runs, gas accumulates around
star forming regions, reaching densities close to the threshold density for
the onset of star formation. The effect of the decoupled winds in the W run is
that of efficiently removing this gas and bringing it to lower density
regions, with $\mbox{$\delta_{\rm b}$}\simeq 10^{3}-10^{4}$.
Similar features that describe the WHIM mass distribution are also visible in
the distribution of the mass in metals (right panel of Fig. 7). Metals are
more likely found at overdensity $\mbox{$\delta_{\rm b}$}\sim 10$ in the BH
simulation, while for the NW case this overdensity is usually between $10^{2}$
and $10^{3}$. Again, this comparison clearly shows that BH feedback is highly
efficient in displacing metal–enriched gas from high density regions, thereby
providing a more uniform enrichment of the WHIM at $z=0$. The PDFs for the CW
and W runs have a sort of double–peaked shape: one peak is at moderate
overdensity, $\mbox{$\delta_{\rm b}$}\sim 10$, that corresponds to the WHIM
lying outside virialized haloes, where winds typically deposit metal–enriched
gas; the second peak is at $\mbox{$\delta_{\rm b}$}\sim 10^{3}$, well inside
virialized haloes and close to star-forming regions. Such a double–peak
structure is more evident for the W run, where winds are more efficient in
escaping high–density regions. Finally, we note that in the BH case the metal
mass fraction at $\mbox{$\delta_{\rm b}$}>10^{2}$ is dramatically smaller (by
up to 2 orders of magnitude) than in all the other cases.
As for the temperature distribution of the WHIM, shown in the left panel of
Fig. 8, we find that winds have a negligible effect. For BH feedback, it has
the effect of reducing the mass fraction of gas above $10^{6}$K, while
correspondingly increasing the fraction of gas at lower temperature. Indeed,
BH feedback is efficient in removing gas from haloes having virial temperature
within the WHIM range. After being heated by BH feedback, this gas leaves the
halo potential wells and cools to lower temperature, $T<10^{5}$K, by adiabatic
expansion.
In general, the left panels of Figs. 7 and 8 highlight that the presence of
galactic winds has a negligible impact on the distribution of gas mass in both
density and temperature. Although the presence of BHs has a more apparent
effect on the distributions of gas mass, it is clear that the distributions of
metal mass fractions are much more sensitive to the adopted feedback model
(see right panels of Figs. 7 and 8). This is not surprising, since feedback
mostly impacts on the gas surrounding star forming regions, which has the
higher enrichment level. In the NW run, the metal mass fraction in gas at the
low-temperature boundary of the WHIM phase, $T\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }10^{5}$K, is about one
order of magnitude lower than for the hot WHIM at $T\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }10^{7}$K. The effect of
winds is that of transporting metal rich gas from high density regions, where
it reaches high temperature, to lower density regions, where is cools down by
adiabatic expansion. As expected, hydrodynamically coupled winds (CW) make it
harder for the gas to leave the densest regions, thus producing a distribution
which is intermediate between those of the NW and W runs.
#### 3.4.2 The distribution of the WHIM metallicity
Having characterized how the mass in gas and in metals is distributed as a
function of the WHIM density and temperature, we describe now how the WHIM
metallicity depends on density and temperature. This characterization of the
WHIM is more observationally oriented. Indeed, different distributions of
metallicity determine what are the best tracers (chemical elements and their
ionization states) to reveal the presence of the WHIM and characterize its
physical properties: the presence of different ionization species, that are
thought to be the best WHIM tracers, depends in fact upon local conditions of
density, temperature and ionization field (see Richter et al., 2008, for a
review).
We show in Figure 9 the distribution of the total metallicity as a function of
density within the WHIM temperature range (left panel) and as a function of
temperature (right panel). Since the metallicity distributions have always a
quite large scatter, a meaningful way of presenting the results is in terms of
the mean (continuous lines), median (dashed lines) and of the 10 and 90
percentiles (shaded area for the W run, dot–dot–dashed lines for the BH run)
of these distributions. For reasons of clarity, we do not report in these
figures the scatter for the NW and CW run.
As for the density dependence of metallicity in the W, CW and NW simulations
(upper left panel), we note that at overdensities between 10 and 100, which is
the range where the PDF of the WHIM mass distribution reaches its maximum
value (see Fig. 7), the median metallicity has a strong positive correlation
with gas density and increase from $\log(Z/Z_{\odot})\simeq-6$ to
$\log(Z/Z_{\odot})\simeq-2$. Furthermore, the overall scatter around these
median values is about two orders of magnitude, so that metallicity values as
large as $0.1Z_{\odot}$ are not unlikely. An obvious observational implication
of this large scatter in the metallicity distributions is that a fairly large
number of lines-of-sight are required, along which measuring WHIM metallicity,
in order to properly populate such a scatter. It is worth mentioning that the
large difference between the average and median metallicities in regions with
$\mbox{$\delta_{\rm b}$}\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }1$ arises from the highly
skewed distribution of metallicity in this regime: most of the underdense gas
is not enriched at all, while metals mainly lie in a small number of highly
enriched particles with $Z\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }0.01$. This is mostly true
for BH and NW runs, while in W and CW runs the distribution of metals with
particle metallicity is significantly shallower. We point out that highly
enriched particles preserve their metal content due to the intrinsic lack of
diffusivity of the SPH. Including an explicit description of metal diffusion
(e.g., Greif et al., 2009) would make such particles sharing their metal
content with the surrounding metal–poor particles.
Figure 9: Total metallicity of the WHIM (gas particles at temperatures
$10^{5}-10^{7}$ K) in solar units at $z=0$, as a function of gas density (in
units of the cosmic mean baryon density ${\left<\right.}\rho_{\rm
b}{\left.\right>}$, left panel) and temperature (right panel). In each panel,
the grey shaded area encompasses the 10 and 90 percentiles of the W run, while
the dot–dot–dashed lines show the same percentiles for BH the run. Thick
coloured lines show the average metallicities while thin dashed lines show the
median metallicities.
As for the metallicity distribution as a function of temperature (bottom
panels), we note a clear trend for metallicity to increase with temperature in
all cases, although the details of this trend have a dependence on the
feedback model. The median metallicities in the W, CW and NW runs (bottom left
panel) increase from $10^{-4}-10^{-3}Z_{\odot}$ to $\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }10^{-2}Z_{\odot}$ as the
temperature increases from $10^{6}$K to $10^{7}$K. The NW simulation predicts
a metallicity of the hottest WHIM, with $T\simeq 10^{7}$K, which is higher by
about a factor three than for the W and CW runs. This gas is located within
the virialised regions of galaxy group haloes, which have typical
overdensities of about $10^{2}-10^{3}$ (see upper left panel). In the NW run,
this gas, which remains at these overdensities for a relatively long time
before being stripped, has been more heavily enriched by the more (in fact,
exceedingly) intense star formation. Quite interestingly, the CW simulation
shows instead larger median metallicity at temperatures $10^{5.5}-10^{6}$ K,
with a shallower decline towards lower temperatures. Gas at this temperature
is a mixture of highly enriched gas in virialized haloes and of poorly
enriched gas at low density. As can be inferred from the phase diagrams shown
in Fig. 2, gas at $z=0$ in the CW run lies within the WHIM temperature range
either for $10\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise
5.0pt\hbox{$<$}\ }\mbox{$\delta_{\rm b}$}\,\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }100$ or right outside of
the star–forming regions, i.e. $10^{4}\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }\mbox{$\delta_{\rm
b}$}\,\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise
5.0pt\hbox{$<$}\ }10^{5}$, where enriched wind particles loose their kinetic
energy due to hydrodynamical interactions. As a consequence, in the CW run the
WHIM at these overdensities is more enriched than in other runs.
As for the comparison between NW and BH runs (right panels), we note that BH
feedback removes a lot of metal enriched gas from the surroundings of star
forming regions (see also Fig. 2). This explains the lower metallicity of the
densest WHIM in the BH simulation (upper left panel). Furthermore, the
suppression of star formation in the BH run also reduces the enrichment level
of the shocked gas within collapsed regions with
$10^{2}\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise
5.0pt\hbox{$<$}\ }\mbox{$\delta_{\rm b}$}\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }10^{3}$. At the same time,
BH feedback is quite effective in increasing the metallicity of gas lying
outside virialized structures, down to rather low densities,
$\mbox{$\delta_{\rm b}$}\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }10$. The reason for this
widespread enrichment is the high efficiency of BH feedback to displace highly
enriched gas from the haloes of large galaxies in the redshift range $z\simeq
2$–4, in correspondence of the peak of the BH accretion rate.
As for the dependence of the WHIM metallicity on temperature, the NW and BH
runs provides similar trends, with a lower level on enrichment in the BH run.
In general, the results of our simulations confirm that different feedback
models leave distinct signatures in the metallicity structure of the WHIM.
While these results have in principle interesting implications for the
observational characterization of WHIM, it must be emphasized that the large
scatter expected in the metallicity distributions requires that metallicity
measurements should be carried out for a fairly large number of
lines–of–sight. We defer to a future paper the presentation of an
observationally–oriented analysis of our simulations, where we discuss the
efficiency with which the metallicity structure of the WHIM can be recovered
from mock absorption spectra of background sources.
### 3.5 Evolution of the relative metal abundances
While the above analysis was aimed at describing the overall metallicity of
the different gas phases, a much richer amount of information is provided by
our chemo-dynamical simulations. Indeed, the possibility of tracing the
enrichment of different elements allows us to study differences in the timing
of enrichment of different heavy elements. In this section we explore the
redshift evolution of the ratios between the abundances of Oxygen with respect
to Iron and to Carbon, with the purpose of quantifying the signatures that IMF
and feedback mechanisms leave on the WHIM enrichment history. Atomic
transitions associated to these three elements are often used to reveal the
presence of the diffuse IGM and to measure its enrichment level at different
redshifts. For instance, CV, OVI and OVII lines are considered as the most
prominent absorption features in the far-UV and soft X-ray spectra of
background sources to reveal the presence of the WHIM in the nearby Universe
(e.g., Richter et al., 2008, for a review). At the same time, CIII, CIV and
OVI absorption lines are commonly used to trace the metal content of the IGM
at $z\raise-2.0pt\hbox{\hbox to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\
}2$ (e.g., Schaye et al., 2003; Pieri & Haehnelt, 2004). Finally emission
features associated to the OVIII line and to the Fe-L and Fe-K complexes are
used to trace the metal content of the intra-cluster and intra-group media
(e.g., Werner et al., 2008, for a review).
Since different elements are produced in different proportions by different
stellar populations, the evolution of their relative abundances is expected to
depend, for a fixed mass–dependent life-time function, on the shape of the
stellar IMF (e.g., Tornatore et al. 2007; Wiersma et al. 2009; see Borgani et
al. 2008 for a review). In general, we expect that different spatial
distributions characterize the enrichment pattern for different elements. As
an example, products of SN-II are released over a shorter time-scale than
those arising from SN-Ia. Since star particles are expected to move from their
original location where they have formed, e.g. due to merging or stripping
processes, we expect that SN-II products pollute gas particles lying very
close to star forming regions, while SN-Ia products should have a relatively
more diffuse distribution. We expect this effect to be more apparent in the
dense environment of galaxy clusters where a population of diffuse inter-
galactic stars is generated by dynamical processes. Indeed, Tornatore et al.
(2007) found from chemo-dynamical simulations of galaxy clusters that the
distribution within the intra-cluster medium (ICM) of the metals produced by
SN-II is more clumpy than that provided by SN-Ia, a prediction that has been
also confirmed by observational data (Sivanandam et al., 2009).
Figure 10: Evolution of the mean value of the [C/O] (upper panels) and [O/Fe]
(lower panels) ratios for the warm (left panels) and WHIM (right panels)
phases. Each panel shows the results for the W256,37, NW256,37, CW256,37 and
BH256,37 simulations, represented by the continuous blue, dashed green, dot-
dashed orange and double-dot dashed red lines, respectively.
In Figure 10 we plot the [C/O] and [O/Fe] ratios222Following a standard
notation, we define the relative abundance between the elements $X$ and $Y$ as
$[X/Y]=\log(Z_{X}/Z_{Y})-log(Z_{X,\odot}/Z_{Y,\odot})$. as a function of
redshifts for the four models and for both the warm (left panels) and WHIM
(right panels) phases. These three elements are produced in different
proportions by SN-II and SN-Ia: Oxygen is almost entirely produced by SN-II,
and Iron is largely provided by SN-Ia; while Carbon is produced in comparable
proportions by SN-II and SN-Ia, the main contributors to it at timescale
comparable to those of SN-Ia are AGB stars. The [C/O] ratio has been widely
investigated in a variety of environments, from dwarf galaxies (Garnett et
al., 1995), to low redshift IGM as probed by QSO absorption lines (Danforth &
Shull, 2008), to high redshift IGM at $z=2-4$ (Aguirre et al., 2008). As for
the [O/Fe] ratio, it is traced in emission with X-ray spectroscopy of
relatively poor galaxy clusters and groups (e.g., Rasmussen & Ponman, 2007)
and is used to study the relative role played by SN-II and SN-Ia in enriching
the ICM.
Figure 11: The effect of changing the IMF on the evolution of the $[C/O]$
ratio for the warm (left panel) and WHIM (right panel) phases. The solid blue
curves are for the reference run based on the IMF by Kroupa (2001) (W37,256
run), the dashed green curves are for the top–heavy IMF by Arimoto & Yoshii
(1987) (Way37,256 run) and the dot-dashed orange curves are for the IMF by
Salpeter (1955) (Ws37,256 run).
Despite the fact that different feedback prescriptions induce rather different
evolutions for the [C/O] ratio, especially for the WHIM phase, the values
attained at $z=0$ are rather similar for all the simulations. We find
[C/O]$\simeq-0.2$ and [C/O]$\simeq-0.15$ for the warm and for the WHIM phases,
respectively. These values can be compared with a similar analysis of chemo-
dynamical simulations performed by Oppenheimer & Davé (2008). At $z=0$ they
found [C/O]$\simeq-0.09$ and [C/O]$\simeq-0.05$ for the warm and warm–hot
phases, respectively, thus about 0.1 dex higher than our results (note that we
rescaled their Anders & Grevesse 1989 abundances to our used values). A
possible reason for this difference is that Oppenheimer & Davé (2008) used a
model for momentum–driven galactic ejecta. However, since they obtain rather
stable results for a variety of feedback schemes, it is not clear whether this
is a likely explanation for the difference. Another possibility lies in the
different sets of yields, mass limit for the SN-II progenitors ($10$M⊙,
instead of $8$M⊙ as in our simulations), and the mass–dependent life–time
function adopted by Oppenheimer & Davé (2008). For instance, assuming a larger
limiting mass for SN-II turns into a relatively lower amount of Oxygen
produced, which could explain the larger value of [C/O] found by Oppenheimer &
Davé (2008).
As for the evolution of the WHIM phase in the BH run, we note that [C/O] is
very close to the NW values at high redshift. This is expected, since at early
times BH accretion is quite ineffective. At these high redshifts, gas at WHIM
temperatures starts involving shock–heated gas in filaments. This diffuse gas
has been relatively more enriched by long–lived stars that had time to move
away from the highest density star-forming regions. This explains why the
relative abundance of Carbon increases with respect to Oxygen, a trend that
extends in the NW run down to $z\simeq 2$, when a nearly solar relative
abundance is attained. After this redshift, the diffuse gas whose temperature
reaches the WHIM values, has a progressively lower degree of enrichment. At
the same time, the large amount of low-redshift star formation in the absence
of feedback causes a prompt release of Oxygen from short-lived stars, thus
motivating the gentle decline of the [C/O] ratio. As for the run with BH
feedback, the WHIM starts receiving at $z\simeq 3.5$ a significant
contribution from gas expelled from haloes where gas accretion onto BHs
reaches the peak of its efficiency (see also the central panel of Fig. 3). At
lower redshift the star formation is strongly quenched in the BH run, thus
explaining the substantial flattening of [C/O] for both the warm and the WHIM
phases.
The evolution of the [C/O] ratio for the CW and W simulations is similar for
the warm and the WHIM phases, for which it rises gently by 0.1 dex and 0.2
dex, respectively, from $z=4$ to $z=0$. Differently from the BH feedback,
winds start affecting the level of star formation already at high redshift,
$z>4$. This explains the slower increase of [C/O], which extends down to low
redshift, for both the warm and the WHIM phases. While the W and CW runs
provide very similar results for the warm phase, the tendency of the decoupled
winds to transport more efficiently enriched gas outside haloes justifies the
slightly larger values of [C/O] in the WHIM phase, as found in the CW
simulation.
The values obtained at $z=2-4$ can be compared with those inferred for the IGM
by Aguirre et al. (2008) using the pixel-optical depth technique in high
resolution QSO spectra. Their findings of course depend on the strength and
shape of the assumed UV background. For a background which is made by galaxies
and quasars they find [C/O]$=-0.7\pm 0.2$ (this value refers to Anders &
Grevesse 1989 abundances and the reported errorbar corresponds to systematic
uncertainties), which is in reasonable agreement with the results of the warm
phase (left panel) that better samples the relatively low density IGM. From
the analysis made by Danforth & Shull (2008), using absorption lines in the
low redshift IGM, values of [C/O] in the range $[-1,0]$ can be inferred,
depending on which ion is used as a tracer of the metallicity. This result is
again in broad agreement with the results of our simulations.
Since Oxygen is mostly contributed by SN-II, the behaviour of the [O/Fe] ratio
has the opposite trend compared to [C/O], with an even more pronounced
redshift dependence. The decreasing trend at low redshift is due to the fact
that Fe is primarily produced by SN-Ia that have long-lived progenitors. The
values of [O/Fe] at $z=0$ are $\sim-0.15$ for the warm phase and $\sim-0.3$
for the WHIM. At $z>3$, this abundance ratio approaches the solar value. Our
findings are in good agreement with the [O/Fe] ratios of gas particles in the
ICM of isolated clusters of Tornatore et al. (2007), in particular values of
$-0.2$ are reached in the outer parts of galaxy clusters and this is close to
the value of the WHIM [O/Fe] ratio. As for the comparison with the results by
Oppenheimer & Davé (2008), we note that our predicted values of [O/Fe] are
about 0.4 dex smaller than their WHIM and warm phases once we consider the
difference abundances used. These differences highlight the need to carry out
detailed comparison between chemo-dynamical codes to distinguish the effect of
the different implementation of the chemical evolution model from the effect
of the different feedback models introduced and of the prescriptions to
distribute metals around star forming regions.
As a final analysis, we verified the effect of changing the IMF on the
resulting evolution of [C/O]. Since the slope of the IMF determines the
frequency of different SN types, we expect that a top-heavier IMF provides a
relatively larger number of SN-II and, therefore, a correspondingly lower
value of [C/O]. This is indeed confirmed by the results in Figure 11, which
show a lower value of C/O. A difference of about 0.15 dex is found for the
top–heavy IMF by Arimoto & Yoshii (1987), for both the warm and the WHIM
phase, while no sizable differences exist between the Kroupa (2001) and
Salpeter (1955) IMFs. It is interesting to note, while using a top–heavier IMF
changes the value of [C/O], its redshift dependence remains essentially
unchanged. As such, the effect of changing the IMF is different from the
effect of changing the feedback model, which instead does change the
evolutionary pattern of the relative abundances.
The main conclusion of this section is that, in spite of the dramatic
differences in the feedback schemes and star formation histories, the C/O and
O/Fe ratios conspire to reach comparable values at $z=0$ for all the
simulations, with variations of about 20–25 per cent. Larger differences, of
up to a factor $\simeq 2.5$, are instead found at higher redshift, $z=2.5$.
Much smaller differences in the evolution of the relative abundances are
instead found by changing the IMF. This implies that an observational
determination of the [C/O] evolution would provide information about the
nature of the feedback mechanism responsible for regulating star formation and
distributing metals.
## 4 Conclusions
We have presented results from an extended set of chemo-dynamical simulations
which have been carried out with the GADGET-2 code (Springel, 2005), including
the implementation of chemical evolution described by Tornatore et al. (2007).
The analysis presented here was focussed primarily on the low-redshift
properties of the inter-galactic medium (IGM) and its evolutionary properties,
by considering both gas in a warm phase (gas particles with temperature in the
range $10^{4}$–$10^{5}$ K) and in the so-called WHIM phase (gas particles with
temperature in the range $10^{5}-10^{7}$ K). The main purpose of the analysis
was to quantify the effect that different feedback mechanisms leave in the
pattern of metal enrichment. Besides a simulation not including any efficient
feedback (NW run), we performed simulations including energy feedback
resulting from accretion onto super-massive black holes (BH run; Springel et
al. 2005; Di Matteo et al. 2005), and from galactic winds powered by supernova
(SN) explosions. We considered two implementations of the latter, based either
on locally decoupling winds from hydrodynamics, so as to allow them to leave
star forming regions (W runs), or on keeping gas in the winds always
hydrodynamically coupled with the surrounding medium (CW runs).
The main results of our analysis can be summarized as follows.
* (a)
The temperature–density and metallicity–density phase diagrams at $z=0$ are
affected by the different feedback prescriptions. Including galactic winds has
the effect of enriching with metals the low–density IGM. BH feedback is even
more effective than winds in transporting metal-enriched gas from the star-
forming regions to the WHIM phase. BH feedback also causes the presence of a
non-negligible amount of relatively low–density, metal-enriched hot gas
($10^{6}-10^{7}$ K) which is not present in the simulations with galactic
winds.
* (b)
The fraction of baryonic mass associated to warm gas, with
$T=10^{4}$–$10^{5}$K, which should be associated to UV/Lyman-$\alpha$
absorption systems, varies from $\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$>$}\ }90$ per cent at high
redshift, $z>3.5$, to about 30 per cent in the local universe, with a weak
dependence on the adopted feedback model. This is in agreement with recent
observational estimates from UV spectroscopy (e.g. Danforth & Shull, 2008).
The hot phase, with $T>10^{7}$K, sums up to 3 per cent at $z=0$, again almost
independent of the feedback model. The fraction of baryons in stars is instead
the most sensitive to feedback: it ranges from about 10 per cent in the run
with no feedback (NW), to about 2 per cent in run with BH feedback.
* (c)
The WHIM phase comprises about 35 per cent of the baryon budget at $z=0$ for
the NW run, a value that increases to about 50 per cent for the run with BH
feedback. This confirms that the share of cosmic baryons in the different
phases does depend on the assumed feedback model. The redshift evolution of
the mass fraction in the WHIM also differs in the different runs, with a
stronger evolution in the run with BH feedback.
* (d)
The average age of enrichment of the warm and of the WHIM phases differs in
the different runs, by an amount which depends on the gas overdensity
$\delta_{\rm b}$. Diffuse gas ($\delta_{\rm b}$$<50$) in the warm phase at
$z=0$ is typically enriched 0.5-1 Gyr earlier than in the WHIM phase. As for
the gas within “collapsed” regions, with $\delta_{\rm b}$$>50$, the warm phase
is enriched 1 Gyr after the WHIM phase in all runs not including BH feedback.
In the BH run, enrichment of the densest WHIM as measured at $z=0$ takes place
more than $1.5$ Gyr after that of the warm medium. BHs enrich the WHIM more
promptly than W in the redshift range $2\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }z\raise-2.0pt\hbox{\hbox
to0.0pt{\hbox{$\sim$}\hss}\raise 5.0pt\hbox{$<$}\ }4$. On the other hand,
winds are more effective at higher redshift, when BH accretion is still
inefficient, and below $z\sim 2$, when star formation in the BH run has been
quenched. At all epochs, the hydrodynamically coupled winds (CW run) provide a
slightly more recent enrichment of both warm and WHIM phases when compared to
hydro-decoupled winds (W run). BH feedback provides a faster enrichment at
$z>2$, while below this redshift winds provide a more prompt WHIM enrichment.
In particular, the model with hydrodynamically coupled winds (CW) provides at
$z<3$ the most prompt enrichment of both warm and WHIM phases.
* (e)
In order to address the multi-phase nature of the WHIM, we compute the
distribution of gas and metals in the WHIM phase as a function of overdensity.
As a result of the strong heating provided by BH feedback at $z\simeq 2$–4,
the characteristic density of the WHIM in the BH run is a factor of a few
lower than in the other simulations and, correspondingly, a smaller amount of
gas is present in the dense WHIM. The same trend is also visible for the
amount of metals present in the WHIM. Typically, most of the metals lie at
over–densities between a few and 10 in the run with BH feedback, while the
typical density of the metal-enriched gas is about two orders of magnitude
higher in the run with galactic winds. The reason for this is that most of the
metals are ejected by BH heating from galaxies between $z=4$ and $z=2$. The
high entropy level reached by enriched gas heated by BH feedback makes hard
for it to be re-accreted within collapsed haloes at lower redshift. In turn,
the drop of star formation in the BH run causes a comparable suppression of
metal production at $z<2$.
* (f)
Underdense WHIM regions (voids) have a very low median metallicity, however
the metallicity of individual gas particles can reach values larger than
$0.1Z_{\odot}$ within the 90 percentile. These enriched particles have been
trasported to underdense regions both by galactic ejecta and by the action of
gas-dynamical processes. Due to the intrinsic lack of diffusion in SPH, such
particles spuriously preserve their high-metal content, rather than diffusing
metals to surrounding metal poor particles.
* (g)
The values of the [C/O] and [O/Fe] relative abundances at $z=0$ are similar
for both the WHIM and warm phases, also irrespective of the feedback model.
Their evolutionary pattern is instead sensitive to the adopted feedback model.
Using a top–heavier IMF decreases the value of the [C/O] ratio by about 0.15
dex at all redshifts.
The simulations presented in this paper have been analysed to study separately
the effects that SN-triggered winds and BH energy feedback have on the
evolution of the IGM. Clearly, in a realistic situation one expects SN and AGN
feedback to be both at work and to cooperate in determining the cosmic cycle
of baryons. However, we point out that our analysis was not aimed at
establishing a best-fit feedback model, which is able to reproduce a variety
of observational results. We aimed insted at determining the imprints that
different feedback models leave of the properties of the IGM and their
possible observational signatures. Furthermore, it is worth reminding that the
parameters that we adopted for winds and BH feedback have been fixed by
requiring each of these two feedback sources to reproduce a specific
observational constraint: the cosmic star formation rate for winds (Springel &
Hernquist, 2003) and the $M_{BH}$–$\sigma$ relation for BH feedback Di Matteo
et al. (2005). Once the two mechanisms are allowed to be both present in a
simulation, they are expected to have non–trivial interplays, which
necessarely require a re-calibration of their characteristic parameters.
The results obtained from our analysis have interesting implications for the
detectability of the WHIM and for the possibility of characterising its
thermal and chemical properties (e.g., Stocke et al., 2007). Indeed, the
possibility of detecting the WHIM through absorption lines in the spectra of
background sources does not only depend on the amount of mass in this phase,
but also on how this mass is enriched and distributed in density and
temperature. With our analysis we have demonstrated that such distributions
are rather sensitive to the adopted scheme of energy feedback. Analyses aimed
at discussing the WHIM detectability from simulations, both in emission (e.g.,
Yoshikawa et al., 2004) and in absorption (e.g., Cen et al., 2001; Viel et
al., 2005) have been so far based on approximate descriptions of the pattern
of chemical enrichment. We will present in a future paper an observationally
oriented analysis of our simulations, aimed at quantifying how the WHIM
properties can be recovered under realistic observational conditions in the
presence of different feedback schemes. There is no doubt that the possibility
of performing high-resolution spectroscopy both in the X-ray (e.g., with
micro-calorimeters onboard of large collecting area X-ray telescopes) and in
the UV band (i.e., the now operating Cosmic Origin Spectrograph onboard of
Hubble Space Telescope) will provide a leap forward in the study of the
diffuse warm-hot baryons in the local universe. Cosmological hydrodynamical
simulations, like those presented in this paper, offer the natural
interpretive framework for these future observations, which will not only
complete the census of baryons at low redshift but also characterize their
physical properties.
## Acknowledgments.
We would like to thank the anonymous referee for constructive comments that
helped improving the presentation of the results. Numerical computations have
been performed at CINECA (“Centro Interuniversitario del Nord Est per il
Calcolo Elettronico”) and CPU time has been assigned thanks to an INAF-CINECA
grant (key and standard projects), and through an agreement between CINECA and
the University of Trieste. We acknowledge useful discussions with L.
Zappacosta. This work has been partially supported by the INFN-PD51 grant, by
the ASI-AAE and ASI-COFIS Theory Grants, and by the PRIN-MIUR Grant “The
Cosmic Cycle of Baryons”.
## References
* Aguirre et al. (2008) Aguirre A., Dow-Hygelund C., Schaye J., Theuns T., 2008, ApJ, 689, 851
* Arimoto & Yoshii (1987) Arimoto N., Yoshii Y., 1987, A&A, 173, 23
* Asplund et al. (2005) Asplund M., Grevesse N., Sauval A. J., 2005, in Barnes III T. G., Bash F. N., eds, Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis Vol. 336 of Astronomical Society of the Pacific Conference Series, The Solar Chemical Composition. pp 25–+
* Bertone et al. (2008) Bertone S., Schaye J., Dolag K., 2008, Space Science Reviews, 134, 295
* Bhattacharya et al. (2008) Bhattacharya S., di Matteo T., Kosowsky A., 2008, MNRAS, 389, 34
* Bolton et al. (2005) Bolton J. S., Haehnelt M. G., Viel M., Springel V., 2005, MNRAS, 357, 1178
* Bondi (1952) Bondi H., 1952, MNRAS, 112, 195
* Borgani et al. (2008) Borgani S., Fabjan D., Tornatore L., Schindler S., Dolag K., Diaferio A., 2008, Space Science Reviews, 134, 379
* Bouwens et al. (2008) Bouwens R. J., Illingworth G. D., Franx M., Ford H., 2008, ApJ, 686, 230
* Branchini et al. (2009) Branchini E., Ursino E., Corsi A., Martizzi D., Amati L., den Herder J. W., Galeazzi M., Gendre B., Kaastra J., Moscardini L., Nicastro F., Ohashi T., Paerels F., Piro L., Roncarelli M., Takei Y., Viel M., 2009, ApJ, 697, 328
* Buote et al. (2009) Buote D. A., Zappacosta L., Fang T., Humphrey P. J., Gastaldello F., Tagliaferri G., 2009, ApJ, 695, 1351
* Cen & Ostriker (1999) Cen R., Ostriker J. P., 1999, ApJ, 514, 1
* Cen & Ostriker (2006) Cen R., Ostriker J. P., 2006, ApJ, 650, 560
* Cen et al. (2001) Cen R., Tripp T. M., Ostriker J. P., Jenkins E. B., 2001, ApJ, 559, L5
* Chabrier (2003) Chabrier G., 2003, PASP, 115, 763
* Chen et al. (2003) Chen X., Weinberg D. H., Katz N., Davé R., 2003, ApJ, 594, 42
* Dalla Vecchia & Schaye (2008) Dalla Vecchia C., Schaye J., 2008, MNRAS, 387, 1431
* Danforth & Shull (2008) Danforth C. W., Shull J. M., 2008, ApJ, 679, 194
* Davé et al. (2001) Davé R., Cen R., Ostriker J. P., Bryan G. L., Hernquist L., Katz N., Weinberg D. H., Norman M. L., O’Shea B., 2001, ApJ, 552, 473
* Davé & Oppenheimer (2007) Davé R., Oppenheimer B. D., 2007, MNRAS, 374, 427
* Di Matteo et al. (2008) Di Matteo T., Colberg J., Springel V., Hernquist L., Sijacki D., 2008, ApJ, 676, 33
* Di Matteo et al. (2005) Di Matteo T., Springel V., Hernquist L., 2005, Nat, 433, 604
* Fabjan et al. (2009) Fabjan D., Borgani S., Tornatore L., Saro A., Murante G., Dolag K., 2009, MNRAS, in press
* Fabjan et al. (2008) Fabjan D., Tornatore L., Borgani S., Saro A., Dolag K., 2008, MNRAS, 386, 1265
* Fiore et al. (2000) Fiore F., La Franca F., Vignali C., Comastri A., Matt G., Perola G. C., Cappi M., Elvis M., Nicastro F., 2000, New Astronomy, 5, 143
* Fukugita et al. (1998) Fukugita M., Hogan C. J., Peebles P. J. E., 1998, ApJ, 503, 518
* Fukugita & Peebles (2004) Fukugita M., Peebles P. J. E., 2004, ApJ, 616, 643
* Garnett et al. (1995) Garnett D. R., Skillman E. D., Dufour R. J., Peimbert M., Torres-Peimbert S., Terlevich R., Terlevich E., Shields G. A., 1995, ApJ, 443, 64
* Greif et al. (2009) Greif T. H., Glover S. C. O., Bromm V., Klessen R. S., 2009, MNRAS, 392, 1381
* Haardt & Madau (1996) Haardt F., Madau P., 1996, ApJ, 461, 20
* Hopkins (2004) Hopkins A. M., 2004, ApJ, 615, 209
* Kaastra et al. (2006) Kaastra J. S., Werner N., Herder J. W. A. d., Paerels F. B. S., de Plaa J., Rasmussen A. P., de Vries C. P., 2006, ApJ, 652, 189
* Katz et al. (1996) Katz N., Weinberg D. H., Hernquist L., 1996, ApJS, 105, 19
* Kobayashi et al. (2007) Kobayashi C., Springel V., White S. D. M., 2007, MNRAS, 376, 1465
* Komatsu et al. (2009) Komatsu E., Dunkley J., Nolta M. R., Bennett C. L., Gold B., Hinshaw G., Jarosik N., Larson D., Limon M., Page L., Spergel D. N., Halpern M., Hill R. S., Kogut A., Meyer S. S., Tucker G. S., Weiland J. L., Wollack E., Wright E. L., 2009, ApJS, 180, 330
* Kravtsov et al. (2002) Kravtsov A. V., Klypin A., Hoffman Y., 2002, ApJ, 571, 563
* Kroupa (2001) Kroupa P., 2001, MNRAS, 322, 231
* Li & White (2009) Li C., White S. D. M., 2009, ArXiv e-prints
* Murray et al. (2005) Murray N., Quataert E., Thompson T. A., 2005, ApJ, 618, 569
* Nagamine et al. (2007) Nagamine K., Wolfe A. M., Hernquist L., Springel V., 2007, ApJ, 660, 945
* Nicastro et al. (2005) Nicastro F., Mathur S., Elvis M., Drake J., Fiore F., Fang T., Fruscione A., Krongold Y., Marshall H., Williams R., 2005, ApJ, 629, 700
* Oppenheimer & Davé (2008) Oppenheimer B. D., Davé R., 2008, MNRAS, 387, 577
* Padovani & Matteucci (1993) Padovani P., Matteucci F., 1993, ApJ, 416, 26
* Persic & Salucci (1992) Persic M., Salucci P., 1992, MNRAS, 258, 14P
* Pieri & Haehnelt (2004) Pieri M. M., Haehnelt M. G., 2004, MNRAS, 347, 985
* Pierleoni et al. (2008) Pierleoni M., Branchini E., Viel M., 2008, MNRAS, 388, 282
* Puchwein et al. (2008) Puchwein E., Sijacki D., Springel V., 2008, ApJ, 687, L53
* Rasmussen et al. (2007) Rasmussen A. P., Kahn S. M., Paerels F., Herder J. W. d., Kaastra J., de Vries C., 2007, ApJ, 656, 129
* Rasmussen & Ponman (2007) Rasmussen J., Ponman T. J., 2007, MNRAS, 380, 1554
* Richter et al. (2008) Richter P., Paerels F. B. S., Kaastra J. S., 2008, Space Science Reviews, 134, 25
* Roncarelli et al. (2006) Roncarelli M., Moscardini L., Tozzi P., Borgani S., Cheng L. M., Diaferio A., Dolag K., Murante G., 2006, MNRAS, 368, 74
* Salpeter (1955) Salpeter E. E., 1955, ApJ, 121, 161
* Saro et al. (2006) Saro A., Borgani S., Tornatore L., Dolag K., Murante G., Biviano A., Calura F., Charlot S., 2006, MNRAS, 373, 397
* Schaye et al. (2003) Schaye J., Aguirre A., Kim T.-S., Theuns T., Rauch M., Sargent W. L. W., 2003, ApJ, 596, 768
* Seljak & Zaldarriaga (1996) Seljak U., Zaldarriaga M., 1996, ApJ, 469, 437
* Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
* Sijacki et al. (2008) Sijacki D., Pfrommer C., Springel V., Enßlin T. A., 2008, MNRAS, 387, 1403
* Sijacki et al. (2007) Sijacki D., Springel V., di Matteo T., Hernquist L., 2007, MNRAS, 380, 877
* Sivanandam et al. (2009) Sivanandam S., Zabludoff A. I., Zaritsky D., Gonzalez A. H., Kelson D. D., 2009, ApJ, 691, 1787
* Springel (2005) Springel V., 2005, MNRAS, 364, 1105
* Springel et al. (2005) Springel V., Di Matteo T., Hernquist L., 2005, MNRAS, 361, 776
* Springel & Hernquist (2003) Springel V., Hernquist L., 2003, MNRAS, 339, 289
* Stocke et al. (2007) Stocke J. T., Danforth C. W., Shull J. M., Penton S. V., Giroux M. L., 2007, ApJ, 671, 146
* Sutherland & Dopita (1993) Sutherland R. S., Dopita M. A., 1993, ApJS, 88, 253
* Tescari et al. (2009) Tescari E., Viel M., Tornatore L., Borgani S., 2009, ArXiv e-prints
* Thielemann et al. (2003) Thielemann F.-K., Argast D., Brachwitz F., Hix W. R., Höflich P., Liebendörfer M., Martinez-Pinedo G., Mezzacappa A., Panov I., Rauscher T., 2003, Nuclear Physics A, 718, 139
* Tornatore et al. (2007) Tornatore L., Borgani S., Dolag K., Matteucci F., 2007, MNRAS, 382, 1050
* Tripp et al. (2008) Tripp T. M., Sembach K. R., Bowen D. V., Savage B. D., Jenkins E. B., Lehner N., Richter P., 2008, ApJS, 177, 39
* Ursino & Galeazzi (2006) Ursino E., Galeazzi M., 2006, ApJ, 652, 1085
* van den Hoek & Groenewegen (1997) van den Hoek L. B., Groenewegen M. A. T., 1997, A&A Supp., 123, 305
* Viel et al. (2003) Viel M., Branchini E., Cen R., Matarrese S., Mazzotta P., Ostriker J. P., 2003, MNRAS, 341, 792
* Viel et al. (2005) Viel M., Branchini E., Cen R., Ostriker J. P., Matarrese S., Mazzotta P., Tully B., 2005, MNRAS, 360, 1110
* Werner et al. (2008) Werner N., Durret F., Ohashi T., Schindler S., Wiersma R. P. C., 2008, Space Science Reviews, 134, 337
* Wiersma et al. (2009) Wiersma R. P. C., Schaye J., Theuns T., Dalla Vecchia C., Tornatore L., 2009, ArXiv e-prints
* Woosley & Weaver (1995) Woosley S. E., Weaver T. A., 1995, ApJS, 101, 181
* Yoshikawa et al. (2004) Yoshikawa K., Dolag K., Suto Y., Sasaki S., Yamasaki N. Y., Ohashi T., Mitsuda K., Tawara Y., Fujimoto R., Furusho T., Furuzawa A., Ishida M., Ishisaki Y., Takei Y., 2004, PASJ, 56, 939
* Zappacosta et al. (2005) Zappacosta L., Maiolino R., Mannucci F., Gilli R., Schuecker P., 2005, MNRAS, 357, 929
|
arxiv-papers
| 2009-11-03T22:12:35 |
2024-09-04T02:49:06.264684
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luca Tornatore, Stefano Borgani, Matteo Viel, Volker Springel",
"submitter": "Luca Tornatore",
"url": "https://arxiv.org/abs/0911.0699"
}
|
0911.0728
|
# Thermometric Soots on Warm Jupiters
K. Zahnle NASA Ames Research Center, Moffett Field, CA 94035
Kevin.J.Zahnle@NASA.gov M. S. Marley NASA Ames Research Center, Moffett
Field, CA 94035 Mark.S.Marley@NASA.gov J. J. Fortney Department of Astronomy
and Astrophysics, University of California - Santa Cruz
###### Abstract
We use a 1D thermochemical and photochemical kinetics model to predict the
disequilibrium stratospheric chemistries of warm and hot Jupiters
($800\\!<\\!T\\!<\\!1200$ K). Thermal chemistry and vertical mixing are
generally more important than photochemistry. At 1200 K, methane is oxidized
to CO and CO2 by OH radicals from thermal decomposition of water. At $T<1000$
K, methane is reactive but stable enough to reach the stratosphere, while
water is stable enough that OH levels are suppressed by reaction with H2.
These trends raise the effective C/O ratio in the reacting gases above unity.
Reduced products such as ethylene, acetylene, and hydrogen cyanide become
abundant; further polymerization should lead to formation of PAHs (Poly-
Aromatic Hydrocarbons) and soots. Parallel shifts are seen in the sulfur
chemistry, with CS and CS2 displacing S2 and HS as the interesting
disequilibrium products. Although lower temperature is a leading factor
favoring hydrocarbons, higher rates of vertical mixing, lower metallicities,
and lower incident UV radiation also favor organic synthesis. Acetylene (the
first step toward PAHs) formation is especially favored by high eddy diffusion
coefficients $K_{zz}>10^{10}$ cm2/s. In most cases planetary compositions
inferred from transit observations will differ markedly from those inferred
from reflected or emitted light from the same planet. The peculiar properties
of HD 189733b compared to other hot Jupiters — a broadband blue haze, little
sign of Na or K, and hints of low metallicity — can be explained by an organic
haze if the planet is cool enough. Whether this interpretation applies to HD
189733b itself, many organic-rich warm Jupiters are sure to be discovered in
the near future.
planetary systems — stars: individual(HD 189733)
††slugcomment: submitted to Ap. J. Lett.
## 1 Introduction
The most easily studied transiting hot Jupiters are HD 209458b and HD189733b
(Fortney et al 2009). HD 209458b is typical of known hot Jupiters (Burrows et
al. 2008). Transit observations show a visible spectrum dominated by the 590
nm sodium resonance lines (Charbonneau et al 2002), the visibility of which
implies a clean, haze-free atmosphere (Sing et al 2008). HD 189733b is
atypical, although its details are controversial (Fortney et al 2009). In
visible light HD 189733b presents a featureless blue haze with little sign of
the expected Na and K lines (Pont et al 2008). Lecavalier des Etangs et al
(2008) suggest that these lines are obscured by enstatite grains mixed up to
microbar levels. Swain et al (2008, 2009) fit transit data between 1.5 and 2.5
$\mu$m with CO, CO2, H2O, and CH4, all with distinctly subsolar abundances
corresponding to a metallicity of $[{\rm M}/{\rm H}]\approx-0.5$. They
suggested that adding C2H2, C2H4, or possibly NH3, would improve their model’s
fit. On the other hand Fortney et al (2009) fit Désert et al’s (2009) dayside
spectrum at 4.5 $\mu$m with supersolar CO and CO2 corresponding to $0.5<[{\rm
M}/{\rm H}]<1.5$.
An organic haze offers another way to explain the appearance of HD 189733b.
Here we use a photochemical model to address the threshold for organic haze
formation in hot or warm Jupiters. In Zahnle et al (2009) we noted that
temperatures below 1200 K change the kinds of chemistry that take place. In
hot atmospheres methane is efficiently oxidized to CO. Cooler atmospheres are
functionally more reduced because H2O is more stable: organic molecules are
favored over CO, PAHs (polycyclic aromatic hydrocarbons) and soots
(disorganized agglomerations of bigger PAHs) might form and precipitate. The
PAHs and soots would be the source of the haze.
## 2 Previous Models
The possibility that organic hazes might be important in irradiated brown
dwarfs was first raised by Griffith et al (1998). However, equilibrium
chemistry does not predict the presence of organic hazes. Nor did previous
photochemical models (Liang et al 2003, Liang et al 2004). Liang et al (2004)
showed that simple hydrocarbons would not condense to form photochemical smogs
in hot solar composition atmospheres. They concluded that photochemical smogs
would not be present in the hottest hot Jupiter atmospheres that they
addressed, but they did not consider cooler atmospheres where hydrocarbons
would be more stable.
Zahnle et al (2009) developed a new 1D atmospheric chemistry model that
addresses photochemical and thermochemical disequilibrium in hot Jupiter
atmospheres. The focus of our first paper was on sulfur chemistry at
temperatures between 1200 and 2000 K. Sulfur is a relatively abundant volatile
element that equilibrium calculations suggest should be present as H2S
(Visscher et al 2006). Our disequilibrium calculations suggest that sulfur
should be present at observable altitudes as S2 or HS. Sulfur is important to
the chemistry because H2S is the most reactive primary molecule in solar
composition atmospheres at temperatures between 800 K and 2000 K. Its thermal
decomposition is the biggest source of free radicals and H in particular. We
found that HS and S2 should be abundant at the millibar level, and we
suggested that most incident light from 250 nm to 450 nm would be absorbed by
stratospheric S2 and HS. The resultant stratospheric heating can be
considerable in planets that are not hot enough for TiO and VO to evaporate in
significant quantities.
In passing we also confirmed an earlier suggestion (Fegley and Lodders 2002,
Visscher et al 2006) that CO2 should be a very good indicator of metallicity.
We showed that this result applies when kinetics are included and that the CO2
abundance and the CO2/CO ratio are insensitive to $T$ in the range
$1200<T<2000$ K.
## 3 Model
We use a standard 1D kinetics code to simulate atmospheric chemistry of hot
irradiated giant planets. Volume mixing ratios $f_{i}$ of species $i$ are
obtained by solving continuity
$N{\partial f_{i}\over\partial
t}=P_{i}-L_{i}Nf_{i}-{\partial\phi_{i}\over\partial z}$ (1)
and force (flux)
$\phi_{i}=b_{ia}f_{i}\left({m_{a}g\over kT}-{m_{i}g\over
kT}\right)-\left(b_{ia}+K_{zz}N\right){\partial f_{i}\over\partial z}$ (2)
equations. In these equations $N$ is the total number density (cm-3);
$P_{i}-L_{i}Nf_{i}$ represent chemical production and loss terms,
respectively; $\phi_{i}$ is the upward flux; $b_{ia}$, the binary diffusion
coefficient between $i$ and the background atmosphere $a$, describes true
molecular diffusion; $m_{a}$ and $m_{i}$ are the molecular masses of $a$ and
$i$; and $K_{zz}$, the eddy diffusion coefficient, parameterizes vertical
mixing as diffusion. We have implemented molecular diffusion through H2 by
setting $b_{ia}=6\times 10^{19}\left(T/1400\right)^{0.75}$ cm-1s-1 —
appropriate for CO — for all the heavy species. This is a reasonable choice
for present purposes. The physical meaning of $b_{ia}$ is the ratio of the
relative thermal velocities of two species to their collision cross section.
In the present circumstances, the relative thermal velocity is effectively
that of H2, so the only important source of variation in $b_{ia}$ is in the
different diameters of the molecules.
In practice Eqns 1 and 2 are solved as a system of second order partial
differential equations for $f_{i}(z,t)$. Steady state solutions are found by
integrating the equations through time (typically $10^{9}$ years) using a
fully implicit backward-difference method. Some aspects of the code are
briefly described in Zahnle et al (2009). Other aspects are more completely
described as applied to ancient Earth (Zahnle et al 2007) and Mars (Zahnle et
al 2008). Here we fully describe the chemical system.
In this study we solve 528 chemical reactions and 33 photolysis reactions for
58 chemical species. The 58 species are listed in Table 1. The reactions and
references are listed, and where appropriate discussed, in Tables 2 and 3. In
our scheme the hydrocarbon chemistry is truncated at C2Hm (with the exceptions
of C4H and C4H2). This means that polymerization beyond C2Hm is not included.
Thus when conditions favor polymerization carbon pools in C2Hm, because longer
carbon chains are not allowed.
New to this model are several hydrocarbon species added since Zahnle et al
(2009). Methanol (CH3OH) and the radicals CH3O and H2COH allow us to fully
describe the known hydrogenation pathways from CO to CH4. These pathways need
to be included because equilibrium chemistry predicts that CO should convert
to CH4 under warm Jupiter conditions. In practice these reactions are quite
unimportant and CO is effectively indestructible under any conditions
encountered in this study. Given that CO does not hydrogenate by gas phase
reactions to any significant extent, we see no call at this time to include
speculative pathways through the extremely unstable N2H radical that might
allow hydrogenation of N2 to NH3.
In the original preprint edition of this paper we did not include addition
reactions of hydrocarbons with OH, such as ${\rm C}_{2}{\rm H}_{4}+{\rm
OH}+{\rm M}\rightarrow{\rm adduct}+{\rm M}$. The precise nature of the adduct
is generally unknown, nor is it known in general how the adduct reacts. But as
discussed above, the C-O bond created cannot be easily undone in the
atmosphere. In the preprint we considered only the competing H abstraction
path ${\rm C}_{2}{\rm H}_{4}+{\rm OH}\rightarrow{\rm C}_{2}{\rm H}_{3}+{\rm
H}_{2}{\rm O}$. This is a reasonable approximation at high temperatures and
low pressures where the abstraction path is favored over the (3-body)
addition. But by neglecting the reaction path in which the OH directly attacks
a carbon atom, we underestimated the importance of oxidation at higher
pressures and lower temperatures, and thus the preprint overstated the
stability of organic molecules.
We have since upgraded the chemical model to explicitly include the adduct
species C2H2OH, C2H4OH, and HCNOH. Reaction rates for making the adducts are
described in the literature (R510, R512, R514). What happens to the adducts
subsequently is neither simple nor fully known. For simplicity we have limited
the options to reactions with atomic hydrogen, which is by far the most
abundant free radical in these simulations, and we have chosen simple products
that do not artificially reduce the number of free radicals (R511, R513,
R515). The specific products for R511 and R515 were picked by analogy to the
documented reaction of C2H4OH (R513). These improvements in the reaction
scheme lead to moderate changes from the original preprint in the direction
expected: in the new calculations the temperature threshold for hydrocarbon
polymerization is about 100 K cooler.
An important simplification in this study, as in its predecessors, is that we
assume isothermal atmospheres with constant eddy diffusion coefficients. This
choice facilitates presenting the results of parameter studies, as $T$ and
$K_{zz}$ prove to be the most important parameters. Published temperature-
pressure profiles for hot jupiters are themselves models. These models depend
explicitly on the assumed equilibrium chemical composition of the atmosphere;
the absence (or presence) of arbitrary amounts of hazes or clouds; time of day
or zonal circulation; etc. This study emphasizes the chemistry, and its intent
is to provide the next step in the iteration between chemical and radiative-
convective models. A “realistic” temperature profile requires at least two
parameters (an effective temperature and a mean infrared opacity at minimum),
and at least three more (an internal luminosity, visible opacity, and an
insolation geometry) for full fidelity. Application to more realistic
temperature profiles will be deferred to future work.
What $K_{zz}(z)$ should be is not well constrained. Values ranging from
$10^{3}$ at the top of the troposphere to $10^{7}$ cm2/s at the top of the
stratosphere seem to be useful for Jupiter. Because the amplitudes of upward
propagating waves grow as $p^{-0.5}$, it is often assumed that $K_{zz}\propto
p^{-0.5}$. The strong solar forcing of hot Jupiters can lead to much more
energetic motions than seen on Jupiter itself. Showman et al (2009) developed
3D models of the circulation in hot Jupiters HD 189733b and HD 209458b. They
suggested that “eddy diffusivities at 1 mbar are $\sim 10^{11}$ cm2 sec1.” For
simplicity we assume $K_{zz}$.
The background atmosphere is assumed 84% H2 and 16% He. The relative
abundances of C, N, O, and S are solar. The elements other than H and He are
scaled as a group according to metallicity. Metallicity is varied from
$-0.7\leq[{\rm M}/{\rm H}]\leq 1.7$. Surface gravity is $g=20$ m/s2. Surface
gravity is important to optical depth and therefore important to what a planet
looks like, but it has little effect on the chemistry (Zahnle et al 2009).
Incident UV radiation is 100$\times$ greater than at Earth and the solar
zenith angle is $\theta=30^{\circ}$ unless otherwise noted. When examining the
effect of photolysis we vary the UV flux from 1 to 1000 times solar.
The upper boundary is a zero flux lid at $\sim\\!1\mu$bar. For the lower
boundary we assume thermochemical equilibrium mixing ratios of H, H2O, CO,
CH4, NH3, N2, and H2S (Lodders and Fegley 2002, Visscher et al 2006). Other
species are assumed to vanish at the lower boundary. This ensures that
photochemical products flow into the deep atmosphere, where they are
presumptively recycled.
The lower boundary pressure was immaterial to our previous study because
temperatures were very high and the relevant chemical reactions very fast. But
at temperatures below 1000 K, the carbon and nitrogen chemistries become
sensitive to the choice of lower boundary conditions. Here we impose chemical
equilibrium at the lower boundary. The puzzle is where to put it. Chemical
equilibrium is not actually expected in our model. Vertical mixing at 1000 K
lifts stable molecules like NH3 faster than it reacts and, even at 100 bars,
gas phase chemistry provides no effective means of converting CO to CH4 and N2
to NH3, and is only beginning to convert excess H to H2. In practice it is
likely that CO is converted to CH4 and N2 is converted to NH3 on grains, in
particular Fe-Ni grains if present. Here we consider arbitrary lower boundary
pressures of 1 and 100 bars.
## 4 Results
The lower temperatures considered in this study result in a much richer
chemistry than we encountered in Zahnle et al (2009). Temperature, vertical
mixing, UV and EUV irradiation, metallicity, and the choice of lower boundary
conditions can all be important parameters in the 800-1200 K temperature
range. We begin with an investigation of how our results vary with a single
parameter when the other parameters are held fixed. Figures 1–4 are classic
spaghetti plots used to show how computed atmospheric compositions vary with
altitude. Figure 1 addresses temperature, Figure 2 addresses vertical mixing,
Figure 3 addresses UV irradiation, and Figure 4 addresses metallicity. Figure
1 is also used to illustrate the sensitivity to the lower boundary condition
in one particular case, and Figure 2 is also used to compare computed
disequilibrium chemical compositions to thermochemical equilibrium
compositions at 1000 K.
Spaghetti plots have their place, but it can be hard to see the entrée for the
noodles. To describe the bigger picture, we use pie charts that show computed
chemical compositions at a certain key altitudes across the general parameter
survey. The pies are arrayed in rows of constant $K_{zz}$, according to
temperature or metallicity. We use these to discuss the propensity of some
atmospheres to form soots. The discussion begins with and concentrates on
hydrocarbons, but we also provide brief overviews of nitrogen and sulfur
chemistry.
### 4.1 Parameters
Figure 1 illustrates vertical profiles of representative isothermal
atmospheres at 1200, 1000, and 800 K. These use $K_{zz}=10^{7}$ cm2/s for
vertical mixing and $[{\rm M}/{\rm H}]=0.7$ for metallicity, both comparable
to those seen in Jupiter. The overall structure of these atmospheres resembles
a poorly functioning gas stove, with methane introduced at the bottom and
flowing upward, where it is either oxidized to CO or made into bigger organic
molecules. At 1200 K, methane is quickly and cleanly oxidized, but at 800 K
methane combustion generates mostly ethylene (C2H4) and HCN, to the point
where ethylene becomes more abundant than methane. This does not mean that
C2H4 actually would be more abundant than CH4 (our model truncates
polymerization at C2Hn), but it does mean that polymerization is a major
methane reaction pathway, and it suggests that hydrocarbons bigger than C2H4
are likely to form.
The fourth panel of Figure 1 illustrates the effect of changing the lower
boundary pressure. Placing the lower boundary at 1 bar puts more methane and
less ammonia at 1 bar than there would be if the boundary were at 100 bars.
Ammonia is more abundant in the 100 bar simulations because (i) its
equilibrium abundance is higher at higher pressure and (ii) its chemical
destruction is kinetically inhibited at 1200 K and below. Hence the deeper
lower boundary puts more NH3 in the stratosphere. Similar behavior would be
seen in CH4 at $T<800$ K. But at 1000 K, putting the lower boundary at 1 bar
introduces an artificial source of methane that results in a big upward flux
that in turn results in more ethylene. As both the 1 bar and 100 bar lower
boundaries are arbitrary in this study it is not useful to take this
discussion any further here.
The chief factors behind the temperature dependence seen in Figure 1 are the
high abundances of H and H2 and the different thermal stabilities of methane
and water. Atomic H is more abundant than it would be in equilibrium.
Disequilibrium of H and H2 is closely tied to the disequilibria of H2S, HS, S,
and S2. An important reaction that does not occur at significant rates is the
3-body recombination of H to H2:
${\rm H}+{\rm H}+{\rm M}\leftrightarrow{\rm H}_{2}+{\rm M}.$ $None$
The computed H/H2 ratio seen in the kinetics models is much higher than would
be predicted by equilibrium. High disequilibrium H abundances were also a
prominent feature in the models of Liang et al (2003) and Zahnle et al (2009).
Both H2O and CH4 are attacked by H:
${\rm H}+{\rm CH}_{4}\rightarrow{\rm H}_{2}+{\rm CH}_{3}$ $None$ ${\rm H}+{\rm
H}_{2}{\rm O}\rightarrow{\rm H}_{2}+{\rm OH}.$ $None$
R6 is significantly endothermic, while R45 is nearly neutral, but both R45 and
R6 have significant activation energies. Because R6 is endothermic, the
temperature barrier is higher, so that water becomes unreactive at a higher
temperature than methane, while the reverse of R6,
${\rm H}_{2}+{\rm OH}\rightarrow{\rm H}+{\rm H}_{2}{\rm O}.$ $None$
remains fast at 1000 K. The net effect of these trends is that, at 1200 K and
above, water is very reactive and the gas is generally oxidizing, while
methane reacts so quickly that it fails to reach altitudes significantly above
the model’s lower boundary. When the temperature is below 1000 K, water
becomes stable enough that OH is scarce, while CH4 becomes stable enough to be
abundant in the atmosphere, yet unstable enough to be very reactive. As a
consequence the effective C/O ratio in the reacting gases is high and the
products are hydrocarbons.
Mismatches between the kinetics model and thermochemical equilibrium at the
lower boundary cause flows into or out of the lower atmosphere. The most
important mismatches stem from kinetic inhibition of the 3-body reactions that
hydrogenate CO to CH4 and N2 to NH3, and the 3-body reactions that
reconstitute H2 from H. The latter results in a high disequilibrium abundance
of H. Very high abundances of H are a prominent feature of all the atmospheres
in Figures 1 and 2. The overabundance of H pushes R45 farther to the right
than it would be in thermochemical equilibrium.
Figure 2 illustrates the influence of different rates of vertical transport in
1000 K atmospheres. The overall trend is that faster vertical transport lifts
CH4 to higher altitudes, and thus raises the altitudes where C2H2, C2H4, and
HCN form. Faster transport increases the efficiency of hydrocarbon formation
vis a vis CO formation, as also seen in the CO profiles, and it changes the
mix of products to favor C2H2 and HCN over C2H4. Apparently soot formation is
likeliest if $K_{zz}>10^{10}$ cm2/s.
The lower right-hand panel of Figure 2 shows what a 1000 K chemical
equilibrium atmosphere would look like. The chemical equilibrium atmosphere
can be thought of as having no vertical mixing ($K_{zz}=0$) and no
photochemistry ($I=0$), although it would not be possible to reach equilibrium
by setting $K_{zz}=0$ in the kinetics model, because there are no efficient
pathways for making NH3 from N2 or CH4 from CO (in practice we found it
difficult to obtain converged solutions with $K_{zz}<10^{5}$ cm2/s). In
general, the disequilibrium kinetic atmospheres contain a much richer mix of
chemical species. Product species are generally more abundant in the kinetics
models, but so too is ammonia. On the other hand, at the altitudes where they
would be observed, CO and CO2 tend to be present at nearly their equilibrium
abundances.
Although the disequilibrium chemistry seen in our models is primarily due to
thermal chemistry and vertical transport, photolysis can be important,
especially at high altitudes and lower temperatures. Figure 3 compares the
effects of photolysis at higher and lower levels of UV and EUV irradiation
than used in Figures 1 and 2. Figure 3 shows that UV and EUV photolysis
inhibits hydrocarbon formation. Soot’s precursors grow better in the dark. The
consequences of photolysis in warm Jupiter atmospheres are more or less
opposite to what is seen on Titan. In cold atmospheres like Titan’s, methane
photolysis leads to production of more complicated hydrocarbons. But in warm
Jupiters, methane is broken up and hydrocarbons are made by thermochemistry.
Photolysis’s role is to free OH radicals from water. Oxidation by OH inhibits
or prevents hydrocarbon polymerization. All of our atmospheres become more
oxidized, and presumably less hazy, at high altitudes due to photolysis of
H2O. In effect, the topmost atmosphere is bleached by photolysis. The
chemocline between the oxidized upper atmosphere and the reduced lower
atmosphere is especially pronounced in the 800 K atmosphere in Figure 1 where
the temperature is low enough that thermal decomposition of water is
relatively minor.
Figure 4 addresses metallicity. Enhanced metallicity favors molecules such as
N2 and CS2 that contain multiple elements other than H over molecules such as
CH4 and NH3 that have only one. Enhanced metallicity also creates a more
oxidized gas, because we have assumed a solar C/O ratio less than unity. These
two trends work together to favor CO and especially CO2 at higher
metallicities, but the two trends work to cross purposes with respect to
hydrocarbon synthesis and soot formation, because the more oxidized conditions
suppress hydrocarbon formation.
### 4.2 Hydrocarbons
Methane polymerization begins with R45, ${\rm H}+{\rm CH}_{4}\rightarrow{\rm
H}_{2}+{\rm CH}_{3}$. Detailed analysis reveals that the chief pathway for
making ${\rm C}_{2}{\rm H}_{n}$ hydrocarbons is the reaction between CH3
radicals to make ethane,
${\rm CH}_{3}+{\rm CH}_{3}+{\rm M}\rightarrow{\rm C}_{2}{\rm H}_{6}{\rm M}.$
$None$
This came as something of a surprise to us because ethane is not itself very
abundant in our models. Evidently the more abundant ${\rm C}_{2}{\rm H}_{n}$
hydrocarbons are formed by reactions such as ${\rm C}_{2}{\rm H}_{n}+{\rm
H}\rightarrow{\rm H}_{2}+{\rm C}_{2}{\rm H}_{n-1}$, which are also pushed to
the right by the high disequilibrium abundance of H.
The influence of $K_{zz}$ on carbon chemistry is viewed from a different angle
in Figure 5. Here we show how carbon is allocated at the altitude that is most
favorable to hydrocarbon production. To do this we divide carbon into 3
categories. The primary category is CH4, the source molecule. The other two
are oxidized species containing CO bonds (CO, CO2) and the more reduced
products that do not contain CO bonds (C2H2, C2H4, HCN, CS2, etc). Call these
$x_{{\rm CH}_{4}}$, $x_{{\rm CO}}$, and $x_{{\rm C}_{2}}$, respectively. Total
carbon is $\Sigma_{\rm C}=x_{{\rm CH}_{4}}+x_{{\rm CO}}+x_{{\rm C}_{2}}$. We
define the most favorable altitude $p_{f}$ for organic synthesis to be where
$x_{{\rm C}_{2}}/\Sigma_{\rm C}$ reaches its maximum value. Whilst every C-O
bond is doomed to end in the irreversible production of CO, it is possible
that much of the C in $x_{{\rm C}_{2}}$ can end in larger molecules such as
PAHs or in the irreversible production of refractory soots.
Figure 6 generalizes these results to different atmospheres as pie charts on
$T$-$K_{zz}$ and $[{\rm M}/{\rm H}]$-$K_{zz}$ grids. The relative amounts of
carbon in different species is presented at the altitude $p_{f}$. It is an
optimistic figure, in the sense that it shows the most favorable conditions
for carbon polymerization in each model. Ethylene is favored by cooler
atmospheres with strong vertical mixing, while CO and CO2 are favored by
higher temperatures, higher metallicity, and weaker vertical mixing. Acetylene
— soot’s precursor — is favored by very strong vertical mixing. The dependence
of C2H2 itself on temperature is relatively weak, but the dependence of CO on
temperature suggests that competition with oxidation makes higher temperatures
less suited to soot formation. Figure 6 also shows a clear negative dependence
on metallicity that suggests that low metallicities are more favorable to soot
formation. The origin of the effect is that metals are on balance oxidizing
because the C/O ratio in a solar gas is less than one. Therefore adding more
metals creates more oxidized conditions, other things equal.
Ethylene, acetylene, and hydrogen cyanide are first generation products of
methane. Benzene (C6H6) would be at best a third generation product (e.g.,
three acetylenes), and PAHs like coronene (C24H12) and ovalene (C32H14) are
probably two or three generations further evolved. At each step some carbon is
lost to CO. If it takes $n$ generations to make something resembling a soot,
we might expect the efficiency of soot formation to go like $x_{{\rm
C}_{2}}^{n}$. This is greater than 10% for $x_{{\rm C}_{2}}>0.8$ and $n=10$,
so we infer that atmospheres where $x_{{\rm C}_{2}}>0.8$ have potential to
develop significant organic hazes.
PAHs interact strongly with visible light (Lou Allamandola, pers. comm.).
Coronene is golden yellow, ovalene is red, and still bigger ones tend to
black. Thus PAHs by themselves may suffice to explain high altitude optical
absorption in hot Jupiters; it may not be necessary that they agglomerate into
grains, albeit grains are likely to be more refractory.
Figure 7 uses pie charts to provide an overview of the hydrocarbon chemistry
that is much easier to interpret than the spaghetti plots shown in Figures 1–4
(which are informative but daunting). Here the pie charts show carbon
speciation at two key altitudes, chosen to approximate the levels where the
optical depths would be of order $0.01$ and $1$ in clear air. Optical depths
$\tau=0.01$ and $\tau=1$ suggest what might be seen in grazing incidence
(primary eclipse) and in direct imaging (secondary eclipse), respectively. To
estimate the optical depths we have approximated the pressure-dependent
Rosseland infrared mean opacities from Freedman et al (200x), for equilibrium
chemistry for clear air of solar composition, by $\kappa\approx 0.03p^{0.6}$
cm-2/g, where $p$ is in bars. This is a very crude fit, and the assumptions on
which it is founded are of questionable relevance, yet the results may have
some value for illustrative purposes because the strongest infrared opacity
source would in all cases be water vapor. Pressure levels corresponding to
$\tau\approx 1$ and $\tau\approx 0.01$ for four different metallicities are
listed in Table 4. Perhaps the most obvious and robust conclusion to draw from
Figure 7 (and the analogous figures for nitrogen and sulfur discussed below)
is that the apparent chemical composition of a planet depends on how it is
viewed (Fortney 2005). In Figure 7, methane is much less prominent in transit
observations than it is in observations of the photosphere.
### 4.3 Nitrogen
Figure 8 provides an overview of nitrogen speciation that is analogous to
Figure 7 for carbon. Ammonia is favored by low temperatures, low metallicity
and strong vertical mixing, and N2 is favored by the opposites. HCN, which
forms most efficiently when ammonia is present and methane is polymerizing, is
most abundant on the boundary between NH3 and N2. The chief pathway for making
HCN in our models is
${\rm NH}_{2}+{\rm C}_{2}{\rm H}_{3}\rightarrow{\rm CH}_{3}+{\rm HCN}.$ $None$
This reaction is known to make an adduct; we have assumed that the C-N bond in
the adduct leads inevitably to HCN, much as we have assumed that the C-O bond
in adducts made by OH and ${\rm C}_{2}{\rm H}_{4}$ or ${\rm C}_{2}{\rm H}_{6}$
lead inevitably to CO. That a poorly characterized reaction like R337 should
play a key role is not ideal. Analogy to R289 suggests that the true path to
HCN should pass through methylamine, ${\rm CH}_{3}+{\rm NH}_{2}\rightarrow{\rm
CH}_{3}{\rm NH}_{2}$. We have not yet included this pathway in our chemical
scheme because we had no inkling that it should be important. This means that
our predictions for HCN abundances are more uncertain than those for the
hydrocarbons, and that our HCN abundances are likelier to be underestimated
than overestimated.
### 4.4 Sulfur
Sulfur species are useful because they interact strongly with blue, violet,
and UV light, which makes them potentially observable, especially in transit.
Sulfur speciation is illustrated in Figures 9 and 10. The former uses
spaghetti plots to show the vertical structure of the sulfur species in a few
selected models. The latter is an overview chart analogous to Figures 7 for
carbon and 8 for nitrogen. Equilibrium calculations predict that H2S would be
the most abundant S-containing species under nearly any condition encountered
in these atmospheres, as is seen above in Figure 2.
Zahnle et al (2009) discussed sulfur chemistry at higher temperatures
($T>1200$ K). At these temperatures S2 and HS are important at high altitudes
but neither CS nor CS2 were important. In the cooler atmospheres investigated
here both CS and CS2 become important in parallel to hydrocarbons like
ethylene, and for the same reasons: the increased availability of reactive
methane, and the decreased availability of OH from water. Consequently cooler
atmospheres with strong vertical mixing favor disequilibrium production of CS
and CS2. Diatomic sulfur (S2) is predicted to be most abundant at high
altitudes at higher temperatures, higher metallicities, and weaker vertical
mixing. These dependencies are obvious in Figure 10. It is interesting that S2
and acetylene (soot) appear to be mutually exclusive, the former indicating
relatively metal-rich placid conditions, the latter indicating strong vertical
mixing and low metallicity.
## 5 Discussion
The chemistry that we have modeled is unremarkable. Most of the important
reactions involving small organic molecules and radicals have been studied
both experimentally and theoretically. Thus the prediction that methane can
react to make ethylene rather than CO seems secure. The presumption that this
will lead to bigger hydrocarbons than ethylene seems equally secure. Whether
this process continues up to PAHs and soot is less secure.
On the surface our results appear to disagree with what Liang et al (2004)
found previously. However, there are enough differences between their
simulations and ours that the apparent disagreement is misleading. The higher
hydrocarbons that form in our models are primarily products of disequilibrium
thermochemistry, not photochemistry. They form because recombination of H to
H2 is kinetically inhibited and consequently the atomic hydrogen density is
much higher than it would be in equilibrium. Liang et al find less H than we
do because they did not include sulfur. The most important reactions governing
the H abundance are those involving sulfur, especially
${\rm H}+{\rm H}_{2}{\rm S}\leftrightarrow{\rm H}_{2}+{\rm HS}$ $None$ ${\rm
H}_{2}+{\rm HS}\leftrightarrow{\rm H}_{2}{\rm S}+{\rm H}$ $None$
and
${\rm S}+{\rm H}_{2}\leftrightarrow{\rm H}+{\rm HS}$ $None$ ${\rm H}+{\rm
HS}\leftrightarrow{\rm H}_{2}+{\rm S}.$ $None$
Reactions R156 and R157 are generally the most frequently occurring reactions
in the atmosphere by a wide margin (Zahnle et al 2009). Reactions R168 and
R169 are most important at higher altitudes (lower pressures). Photolysis has
little to do with these.
Another difference between our study and Liang et al (2003, 2004) is that
Liang et al attempt to use realistic approximations to $T(z)$ and $K_{zz}(z)$.
Constant temperature and constant $K_{zz}$ is a feature of our study. Our
emphasis here is on the survey of parameter space. Temperature and vertical
mixing are the two key parameters. The cleanest way to look at the problem is
therefore to use $T$ and $K_{zz}$ as the independent variables. More
complicated atmospheric profiles of either $T(z)$ or $K_{zz}(z)$ requires
introducing additional parameters, which introduces an additional level of
complexity that may not be justified.
For $K_{zz}$, Liang et al use $K_{zz}\propto N^{-0.6}$. Their assumption gives
$K_{zz}=2.4\times 10^{7}$ cm2/s at 1 bar, $1.5\times 10^{9}$ cm2/s at 1 mbar,
and $1.0\times 10^{11}$ cm2/s at 1 $\mu$bar (all at 1200 K). This range is
similar to our high $K_{zz}$ cases. For $T(z)$, Liang et al use three specific
profiles derived from detailed radiative transfer model results. All three are
very hot at 1 bar (the coolest is 1800 K). These temperatures are much higher
than anything we address here. In our model methane is immediately destroyed
if $T>1200$ K. We agree with Liang et al (2004) that very hot planets ought
not to have hydrocarbon hazes.
As a test we computed chemistry using the nominal $p$-$T$ profile from Fortney
et al’s (2009) most recent model of HD 189733b. Their model is hot ($T\approx
1500$ K) below 3 bars and cool ($T\approx 900$ K) above 10 mbars, with a
smooth transition in between. We note that Fortney et al include only Rayleigh
scattering, so that their computed albedo is relatively low, and thus their
$T(z)$ may not closely resemble that of a hazy planet. For the simulation we
set $K_{zz}=10^{8}$. We find very little methane above the lower boundary and
no significant polymerization. We then reduced the lower temperature to 1200
K, and then to 1100 K. At 1200 K, NH3 is abundant but hydrocarbons are not,
while at 1100 K we see rampant hydrocarbon polymerization. These results are
quite consistent with Figure 6.
In passing, we note that solar C/O is also an assumption. Snowline models of
giant planet formation can give lower C/O, while the less well known tar-line
model (Lodders 2004) predicts higher C/O. The true C/O ratio of Jupiter is not
yet known. High C/O changes the chemistry dramatically; among other things,
soots would be strongly favored at higher temperatures. The possibility that a
planet like HD 189733b might have high C/O should be in play.
## 6 Conclusions
We find that vertical mixing in hot Jupiters with temperatures below 1000 K
can provide environments suitable for turning methane into ethylene,
acetylene, and hydrogen cyanide. We speculate that such atmospheres would be
conducive to generating PAHs and perhaps soots. We also reiterate the point
that planetary compositions inferred from transit observations, which probe
high altitudes (low pressures), can differ markedly from those inferred from
reflected or emitted light from the same planet. In general, thermal chemistry
and vertical mixing are more important to disequilibrium chemistry in warm or
hot Jupiters than is photochemistry. Photochemistry plays a bigger role at low
temperatures.
The presence of abundant PAHs or soots at high altitudes could account for the
up-to-now unique properties of HD 189733b, if HD 189733b proves to be cool
enough, or its metallicity low enough, or its C/O ratio high enough, or its
vertical mixing as vigorous as Showman et al (2009) suggest. This would make
HD 189733b the first of a new class of warm Jupiters. As noted, Swain et al
(2009) reported that HD 189733b’s metallicity might be rather low. On the
other hand HD 189733 is a relatively strong source of UV (Knutson 2010), which
makes photolysis rates high, a factor that we have shown will suppress high
altitude soot formation. Whether our thermometric interpretation of HD 189733b
actually applies to HD 189733b (recent models suggest that the planet is too
hot if it has solar C/O) is not the main point: many organic-rich hazy warm
Jupiters are sure to be discovered in the near future.
## 7 Acknowledgements
KJZ thanks NASA’s Exobiology Program for support. MSM thanks NASA’s Planetary
Atmospheres Program for support
## References
* Burrows et al. (2008) Burrows, A., Budaj, J., & Hubeny, I. 2008, ApJ, 678, 1436-1457 “Theoretical Spectra and Light Curves of Close-in Extrasolar Giant Planets and Comparison with Data”
* (2) Charbonneau D, Brown TM, Noyes RW, & Gilliland RL 2002. ApJ, 568, 377-384.
* (3) Désert J.-M., Lecavelier des Etangs A., Hébrard G., Sing DK, Ehrenreich D, Ferlet R, & Vidal-Madjar A. 2009, Astrophys. J. 699, 478-485.
* (4) Fortney JJ (2005). The effect of condensates on the characterization of transiting planet atmospheres with transmission spectroscopy. Mon. Not. Rpy. Astron. Soc. 364, 649-653.
* (5) Fortney JJ, Shabram M, Showman AP, Lian Y, Freedman RS, Marley MS, and Lewis NK (2009). Transmission Spectra of Three-dimensional Hot Jupiter Model Atmospheres. Astrophys. J.
* (6) Griffith CA, Yelle RV, Marley MS 1998. “The dusty atmosphere of the brown dwarf Gliese 229B” Science 282, 2063-2067.
* (7) Knutson HA (2010) ApJ, xxx, yyy
* (8) Lecavelier des Etangs A, Pont F, Vidal-Madjar A, & Sing D. (2008) “Rayleigh scattering in the transit spectrum of HD 189733b” Astron. Astrophys. 481, L83-L86.
* (9) Liang M-C, Parkinson CD, Lee AYT, Yung YL, and Seager S, (2003) “Source of atomic hydrogen in the atmosphere of HD 209458b” Astrophys. J. 596, L247 L250.
* (10) Liang M-C, Seager S, Parkinson CD, Lee AYT, and Yung YL (2004) “On the insignificance of photochemical hydrocarbon aerosols in the atmospheres of close-in extrasolar giant planets” Astrophys. J. 605, L61 L64.
* (11) Lodders K. 2002. “Atmospheric Chemistry in Giant Planets, Brown Dwarfs, and Low-Mass Dwarf Stars. I. Carbon, Nitrogen, and Oxygen” Icarus 155, 393-424.
* (12) Lodders K. 2004. “Jupiter Formed with More Tar than Ice” Astrophys. J. 611, 587-597.
* (13) Moses JI, Fouchet T, Yelle RV, Friedson AJ, Orton GS, Read PL, Sánchez-Lavega A, Showman AP, Simon-Miller AA, Vasaveda AR (2004). In Jupiter. Eds F Bagenal, TE Dowling, WB McKinnon. Cambridge Univ. Press, pp. 129-158.
* (14) Pont F, Knutson H, Gilliland RL, Moutou C, & Charbonneau D. 2008. “Detection of atmospheric haze on an extrasolar planet: the 0.55-1.05 $\mu$m transmission spectrum of HD 189733b with the Hubble Space Telescope” Mon. Not. Roy. Astron. Soc. 385, 109-118.
* Showman et al. (2009) Showman, A. P., Fortney, J. J., Lian, Y., Marley, M. S., Freedman, R. S., Knutson, H. A., & Charbonneau, D. 2009 Astrophys. J. 699, 564-584.
* (16) Sing, D.K., Vidal-Madjar A., Désert, J.-M., Lecavelier des Etangs, A., and Ballester, G. (2008). Astrophys. J. 686, 658-666.
* (17) Swain, M. R., Vasisht, G., & Tinetti, G. 2008. Nature 452, 329-331.
* (18) Swain, M. R., Vasisht, G., Tinetti, G., Bouwman, J., Chen, P., Yung, Y., Deming, D., & Deroo, P. 2009, Astrophys. J. Lett. 690, L114-L117.
* (19) Visscher C, Lodders K, and Fegley B. (2006). Astrophys. J. 648, 1181-1195.
* (20) Zahnle KJ, Claire MW, Catling DC (2006). Geobiology 4, 271-282.
* (21) Zahnle KJ, Haberle RM, Catling DC, Kasting JF (2008). J. Geophys. Res. 113, E11004, doi:10.1029/2008JE003160.
* (22) Zahnle KJ, Marley MS, Freedman RS, Lodders K, and Fortney JJ (2009). Astrophys. J. 701, L20-L24.
Figure 1: Vertical profiles of selected H, C, N, O species in isothermal
atmospheres at three temperatures ($T=1200$, 1000, and 800 K). Fixed model
parameters are $K_{zz}=10^{7}$ cm2/s, planetary ($[{\rm M}/{\rm
H}]\\!=\\!0.7$) metallicity, $I=100$, and $g=20$ m/s2. The lower boundary
condition is thermochemical equilibrium at 100 bars. The fourth panel, to be
compared to the panel above it, raises the thermochemical equilibrium lower
boundary to 1 bar. At 1200 K methane is oxidized with little buildup of
hydrocarbons. At 800 K methane’s destruction leads to significant
disequilibrium production of organic molecules, especially ethylene (C2H4).
The horizontal gray bars denote the most favorable altitudes for hydrocarbon
formation $p_{f}$ as defined in the text. The decreasing abundance of stable
species at high altitudes is caused by molecular diffusion. Figure 2:
Vertical profiles of selected species in isothermal 1000 K atmospheres and
$[{\rm M}/{\rm H}]=0.7$ as a function of vertical mixing. Strong vertical
mixing — $K_{zz}>10^{9}$ — favors disequilibrium production of C2H4, HCN, and
especially C2H2 (acetylene, the canonical soot precursor). The horizontal gray
bars denote the most favorable altitudes for hydrocarbon formation $p_{f}$ as
defined in the text. The fourth panel shows thermochemical equilibrium
abundances at all altitudes at $T=1000$ and $[{\rm M}/{\rm H}]=0.7$. In
general, the kinetics calculations predict a much richer variety of species
than equilibrium. Figure 3: Vertical profiles of selected H, C, N, O species
in isothermal atmospheres at higher and lower levels of UV and EUV
irradiation. Fixed model parameters are $T=1000$ K, $[{\rm M}/{\rm H}]=0.7$,
and $g=20$ m/s2. At 1000 K, photolysis is most important when vertical mixing
is vigorous ($K_{zz}=10^{11}$ cm2/s). Photolysis is less important at
$K_{zz}=10^{9}$, and at $K_{zz}=10^{7}$ (not shown) is important only for NH3
and HCN. In general, UV is hostile to NH3 and small hydrocarbons other than
CH4, and through H2O photolysis creates a more oxidizing environment. These
effects suppress hydrocarbon formation and relegate hydrocarbon formation to a
deeper level in the atmosphere. Soot’s precursors, in particular, are strongly
favored by darkness. The horizontal gray bars denote the most favorable
altitudes for hydrocarbon formation $p_{f}$. For $S=1$ and $K_{zz}=10^{11}$,
this would lie above the top of the model. Figure 4: Vertical profiles of
selected H, C, N, O species in isothermal atmospheres at four metallicities.
Relative abundances of C, N, S, and O are solar. Fixed model parameters are
$K_{zz}=10^{9}$ cm2/s, $T=1000$ K, $S_{uv}=100$, and $g=20$ m/s2. The general
trend is that higher metallicities result in more strongly oxidized mixtures.
CO and especially CO2 are good indicators of metallicity. The propensity to
form soot appears to be similar for $-0.7\leq[{\rm M}/{\rm H}]\leq 0.7$, as
judged by C4H2 abundances, but higher metallicity is unfavorable. The
horizontal gray bars denote the most favorable altitudes for hydrocarbon
formation $p_{f}$. Figure 5: Methane’s fates at the most favorable altitude
$p_{f}$ for soot formation as a function of vertical mixing $K_{zz}$. Fates
are shown at $T=1000$ K (solid lines) and $T=900$ K (dashed lines). Fixed
model parameters are $[{\rm M}/{\rm H}]=0.7$, $S_{uv}=100$, $p_{0}=100$ bars,
and $g=20$ m/s2. Irreversible oxidization ($x_{{\rm CO}}$) is denoted “CO and
CO2”, interesting products ($x_{{\rm C}_{2}}$) are denoted “$\Sigma{\rm
C}_{2}{\rm H}_{n},{\rm HCN~{}etc}$”. High $K_{zz}$ favors organic molecules.
At 1000 K, soot formation seems reasonably likely for $K_{zz}\\!>\\!10^{8}$
cm2/s. The right hand axis shows that $p_{f}$ is a monotonically increasing
function of $K_{zz}$. Figure 6: Generalization of Figure 5 to other
temperatures (left, $[{\rm M}/{\rm H}]=0.7$) and metallicities (right,
$T=1000$ K). Common model parameters are $S_{uv}=100$, $p_{0}=100$ bars,
$g=20$ m/s2, and $\theta=30^{\circ}$ incidence angle for UV radiation. Each
pie-chart shows how carbon is allocated at $p_{f}$ for a specific model. To
first approximation higher values of $K_{zz}$ map to higher altitudes (lower
$p_{f}$). Left Low temperatures and high $K_{zz}$ favor organic molecules.
Very high values of $K_{zz}\geq 10^{10}$ cm2/s mix CH4 to altitudes that are
more favorable to C2H2, the traditional soot precursor. Right Low metallicity
favors hydrocarbons generally and C2H2 and HCN particularly. High metallicity
is quite oxidized. The apparent preference for CH4 at high metallicity and low
Kzz is an artifact of there being essentially no hydrocarbon synthesis and
$p_{f}$ being near the lower boundary. Figure 7: Carbon speciation at
$\tau\approx 0.01$ (top) and $\tau\approx 1$ (bottom). These approximate what
might be seen in grazing incidence in transit and in differential eclipses,
respectively. For hydrocarbons vs. temperature, fixed parameters are $[{\rm
M}/{\rm H}]=0.7$, $S_{uv}=100$, and $g=20$ m/s2. For hydrocarbons vs.
metallicity, fixed parameters are $T=1000$ K, $S_{uv}=100$, and $g=20$ m/s2.
Figure 8: Nitrogen speciation at $\tau\approx 0.01$ (top) and $\tau\approx 1$
(bottom). Cases and meanings are the same as for Figure 7. Ammonia is favored
by low temperatures, low metallicity and strong vertical mixing, and N2 is
favored by the opposites. HCN, which forms most efficiently when ammonia is
present and methane is polymerizing, is most abundant on the boundary between
NH3 and N2. Figure 9: Vertical profiles of sulfur species in isothermal
atmospheres at different temperatures and vertical mixing. S2 is favored by
high temperatures and weak vertical mixing, while CS and CS2 are favored by
lower temperatures and strong vertical mixing. H2S is the stable gas in the
deep atmosphere. CS and CS2 are found in environments that also favor C2H4. S2
tends to form at the chemical boundary between the more reduced lower
atmosphere and the more oxidized upper atmosphere. Figure 10: Sulfur
speciation at $\tau\approx 0.01$ (top) and $\tau\approx 1$ (bottom). Cases and
meanings are the same as for Figure 7. CS and CS2 are favored by cooler
atmospheres with strong vertical mixing, while S2 is favored at high altitudes
by higher temperatures, higher metallicities, and weaker vertical mixing.
Table 1
---
Chemical Species
Elements | Species
H, O | H, H2, O, OH, H2O, O2, O(1D)
CO | CO, CO2, HCO, H2CO, H2COH, CH3O, CH3OH
CHn | C, CH, CH2, 1CH2, CH3, CH4
CmHn | C2, C2H, C2H2, C2H3, C2H4, C2H5, C2H6, C2H2OH, C2H4OH, C4H, C4H2
N | N, N2, NH, NH2, NH3, CN, HCN, H2CN, HCNOH, NO, NS
S | S, HS, H2S, S2, S3, S4, S${}_{8}^{a}$, S${}_{8}^{\ast b}$, CS, CS2, OCS, HCS, H2CS, SO, HSO, SO2
$a$ – Ring (ground state)
$b$ – Linear (excited state)
Table 2
---
Chemical Reactions
| Reactantsa | | Products | Rateb | Reference
R1 | H + H + M | $\\!\\!\\!\rightarrow$ | H2 \+ M | $8.8\\!\times\\!10^{-33}\left(T/298\right)^{-0.60}$ | Ba92
| H + H | $\\!\\!\\!\rightarrow$ | H2 \+ M | $2.0\\!\times\\!10^{-10}$ |
R2 | H2 \+ M | $\\!\\!\\!\rightarrow$ | H + H +M | $1.5\\!\times\\!10^{-09}e^{-48400/T}$ | Ba92
R3 | H2 \+ O | $\\!\\!\\!\rightarrow$ | OH + H | $3.5\\!\times\\!10^{-13}\left(T/298\right)^{2.67}e^{-3160/T}$ | Ba92
R4 | H + OH | $\\!\\!\\!\rightarrow$ | H2 \+ O | $1.7\\!\times\\!10^{-13}\left(T/298\right)^{2.64}e^{-2240/T}$ | rev3
R5 | H2 \+ OH | $\\!\\!\\!\rightarrow$ | H2O + H | $1.6\\!\times\\!10^{-12}\left(T/298\right)^{1.60}e^{-1660/T}$ | Ba92
R6 | H2O + H | $\\!\\!\\!\rightarrow$ | H2 \+ OH | $6.9\\!\times\\!10^{-12}\left(T/298\right)^{1.60}e^{-9270/T}$ | Ba92
R7 | OH + OH | $\\!\\!\\!\rightarrow$ | H2O + O | $1.7\\!\times\\!10^{-12}\left(T/298\right)^{1.14}e^{-50/T}$ | Ba92
R8 | H2O + O | $\\!\\!\\!\rightarrow$ | OH + OH | $1.8\\!\times\\!10^{-11}\left(T/298\right)^{0.95}e^{-8570/T}$ | Li91
R9 | O + H + M | $\\!\\!\\!\rightarrow$ | OH + M | $4.3\\!\times\\!10^{-32}\left(T/298\right)^{-1.00}$ | Ts86
| O + H | $\\!\\!\\!\rightarrow$ | OH + M | $2.0\\!\times\\!10^{-11}$ |
R10 | OH + M | $\\!\\!\\!\rightarrow$ | O + H +M | $4.0\\!\times\\!10^{-09}e^{-50000/T}$ | Ts86
R11 | H + O | $\\!\\!\\!\rightarrow$ | OH + h$\nu$ | $9.9\\!\times\\!10^{-19}\left(T/298\right)^{-0.38}$ | Mi97
R12 | O + O + M | $\\!\\!\\!\rightarrow$ | O2 \+ M | $1.0\\!\times\\!10^{-33}\left(T/298\right)^{-1.00}$ | Wa84
| O + O | $\\!\\!\\!\rightarrow$ | O2 \+ M | $2.0\\!\times\\!10^{-11}$ |
R13 | O2 \+ M | $\\!\\!\\!\rightarrow$ | O + O +M | $1.0\\!\times\\!10^{-08}\left(T/298\right)^{-1.00}e^{-59400/T}$ | Ts86
R14 | O + OH | $\\!\\!\\!\rightarrow$ | O2 \+ H | $2.4\\!\times\\!10^{-11}e^{-353/T}$ | Ba92
R15 | H + O2 | $\\!\\!\\!\rightarrow$ | OH + O | $3.3\\!\times\\!10^{-10}e^{-8460/T}$ | Ba92
R16 | H2 \+ O2 | $\\!\\!\\!\rightarrow$ | OH + OH | $4.0\\!\times\\!10^{-11}\left(T/298\right)^{0.47}e^{-35100/T}$ | Ka05
R17 | CO + M | $\\!\\!\\!\rightarrow$ | C + O +M | $2.7\\!\times\\!10^{-03}\left(T/298\right)^{-3.52}e^{130000/T}$ | Ba92
R18 | C + O | $\\!\\!\\!\rightarrow$ | CO + M | $0$ |
| C + O | $\\!\\!\\!\rightarrow$ | CO + M | $0$ |
R19 | C + OH | $\\!\\!\\!\rightarrow$ | CO + H | $1.1\\!\times\\!10^{-10}\left(T/298\right)^{0.50}$ | Mi97
R20 | C + OH | $\\!\\!\\!\rightarrow$ | CH + O | $2.2\\!\times\\!10^{-11}\left(T/298\right)^{0.50}e^{-14800/T}$ | Mi97
R21 | C + O2 | $\\!\\!\\!\rightarrow$ | CO + O | $1.6\\!\times\\!10^{-11}$ | Ba92
R22 | H + CO | $\\!\\!\\!\rightarrow$ | C + OH | $2.8\\!\times\\!10^{-09}\left(T/298\right)^{0.50}e^{-76900/T}$ | rev21
R23 | CO + OH | $\\!\\!\\!\rightarrow$ | CO2 \+ H | $1.8\\!\times\\!10^{-14}\left(T/298\right)^{1.89}e^{583/T}$ | Li07
R24 | CO2 \+ H | $\\!\\!\\!\rightarrow$ | CO + OH | $6.1\\!\times\\!10^{-11}\left(T/298\right)^{0.64}e^{-12500/T}$ | rev23
… | … | … | … | … | …
… | … | … | … | … | …
… | … | … | … | … | …
R527 | C2H5 \+ H2CN | $\\!\\!\\!\rightarrow$ | HCN + C2H6 | $7.7\\!\times\\!10^{-12}$ |
R528 | NH2 \+ C2H5 | $\\!\\!\\!\rightarrow$ | NH + C2H6 | $3.0\\!\times\\!10^{-13}\left(T/298\right)^{1.00}e^{-4400/T}$ | Xu99
$a$ — M refers to the background atmosphere, principally H2 and He; units of
density [cm-3].
$b$ — 2-body reaction rates are in cm3s-1; 3-body rates are in cm6s-1.
Table 3
---
Photolysis Reactions
| Species | | Products | Ratea | Reference
P1 | H2O + h$\nu$ | $\\!\\!\\!\rightarrow$ | OH + H | $1.6\\!\times\\!10^{-03}$ | Sa03
P2 | CO2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | CO + O | $2.7\\!\times\\!10^{-07}$ | Ok78,Hu92
P3 | CO2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | CO + O(1D) | $3.0\\!\times\\!10^{-05}$ | Ok78,Hu92
P4 | O2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | O + O | $9.6\\!\times\\!10^{-06}$ | Sa03
P5 | O2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | O + O(1D) | $5.1\\!\times\\!10^{-04}$ | Sa03
P6 | NO + h$\nu$ | $\\!\\!\\!\rightarrow$ | N + O | $3.7\\!\times\\!10^{-06}$ | Sa03
P7 | H2S + h$\nu$ | $\\!\\!\\!\rightarrow$ | HS + H | $2.7\\!\times\\!10^{-02}$ | Ok78,Hu92
P8 | NH3 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | NH2 \+ H | $1.1\\!\times\\!10^{-02}$ | Ok78,Hu92
P9 | NH3 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | NH + H2 | $5.5\\!\times\\!10^{-03}$ | Ok78,Hu92
P10 | CH4 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | CH2 \+ H2 | $3.9\\!\times\\!10^{-04}$ | Hu92
P11 | CH4 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | CH3 \+ H | $3.9\\!\times\\!10^{-04}$ | Hu92
P12 | SO2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | SO + O | $1.7\\!\times\\!10^{-02}$ | Ok78,Hu92
P13 | SO2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | S + O2 | $6.9\\!\times\\!10^{-04}$ | Ok78,Hu92
P14 | SO2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | S + O +O | $1.7\\!\times\\!10^{-05}$ | Ok78,Hu92
P15 | SO + h$\nu$ | $\\!\\!\\!\rightarrow$ | S + O | $4.4\\!\times\\!10^{-02}$ | Ok78,Hu92
P16 | CS2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | CS + S | $3.9\\!\times\\!10^{-01}$ | Mo81,Ah92
P17 | OCS + h$\nu$ | $\\!\\!\\!\rightarrow$ | CO + S | $2.5\\!\times\\!10^{-03}$ | Sa03
P18 | S2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | S + S | $1.7\\!\times\\!10^{-01}$ | Za09
P19 | S3 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | S2 \+ S | $1.1\\!\times\\!10^{+02}$ | Za09
P20 | S4 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | S3 \+ S | $1.1\\!\times\\!10^{+01}$ | Za09
P21 | S8 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | S${}_{8}^{\ast}$ | $1.7\\!\times\\!10^{+00}$ | Ka89
P22 | S${}_{8}^{\ast}$ \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | S4 \+ S4 | $3.5\\!\times\\!10^{+00}$ | Ka89
P23 | C2H2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | C2H + H | $2.4\\!\times\\!10^{-03}$ | Ok78,Hu92
P24 | C2H4 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | C2H3 \+ H | $5.3\\!\times\\!10^{-03}$ | Ok78,Hu92
P25 | C2H6 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | C2H5 \+ H | $1.1\\!\times\\!10^{-03}$ | Ok78,Hu92
P26 | C4H2 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | C4H + H | $4.8\\!\times\\!10^{-03}$ | note
P27 | H2CO + h$\nu$ | $\\!\\!\\!\rightarrow$ | CO + H2 | $6.1\\!\times\\!10^{-03}$ | Sa03
P28 | H2CO + h$\nu$ | $\\!\\!\\!\rightarrow$ | HCO + H | $7.1\\!\times\\!10^{-03}$ | Sa03
P29 | H2CO + h$\nu$ | $\\!\\!\\!\rightarrow$ | CO + H +H | $0$ | Sa03
P30 | HCN + h$\nu$ | $\\!\\!\\!\rightarrow$ | CN + H | $2.5\\!\times\\!10^{-03}$ | Hu92
P31 | HSO + h$\nu$ | $\\!\\!\\!\rightarrow$ | HS + O | $6.1\\!\times\\!10^{-02}$ | note
P32 | HS + h$\nu$ | $\\!\\!\\!\rightarrow$ | H + S | $1.0\\!\times\\!10^{+01}$ | Za09
P33 | CH4 \+ h$\nu$ | $\\!\\!\\!\rightarrow$ | CH + H2 +H | $1.9\\!\times\\!10^{-04}$ | Ok78,Hu92
$a$ Photolysis rates at the top of the atmosphere for $S_{uv}=100$ and
$30^{\circ}$ zenith angle.
P21. Photolysis of S8 is presumed to create the linear radical
S${}_{8}^{\ast}$. This is a placeholder at high
temperatures where S8 is not expected and at low temperatures where S8
condenses.
P26. Assumes twice the cross section of C2H2.
P31. Assumes cross section of HO2.
Table 4
---
Reference Pressures
| $\tau=0.01$ | $\tau=1$
$[{\rm M}/{\rm H}]$ | $p$ [mbars] | $p$ [mbars]
0.2 | 30 | 500
1 | 10 | 200
5 | 4 | 70
50 | 1 | 16
|
arxiv-papers
| 2009-11-04T04:27:37 |
2024-09-04T02:49:06.276273
|
{
"license": "Public Domain",
"authors": "K. Zahnle, M. S. Marley, J. J. Fortney",
"submitter": "Mark S. Marley",
"url": "https://arxiv.org/abs/0911.0728"
}
|
0911.0999
|
# Studies of Quasi-Periodic Oscillations in the Black Hole Transient XTE
J1817-330
Jayashree Roy1,3, P.C. Agrawal1, B.Paul2 and K.Duorah3
1Tata Institute of Fundamental Research, Mumbai 400 005, India
2Raman Research Institute, Bangalore, 560080 , India
3Gauhati University, Guwahati 781014, India E-mail: jayashree@tifr.res.in
(Received …,; accepted …)
###### Abstract
We have used archival RXTE PCA data to investigate timing and spectral
characteristics of the transient XTE J1817-330. The data pertains to 160 PCA
pointed observations made during the outburst period 2006, January 27 to
August 2. A detailed analysis of Quasi-Periodic Oscillations (QPOs) in this
black hole X-ray binary is carried out. Power density spectra were obtained
using the light curves of the source. QPOs have been detected in the 2-8 keV
band in 10 of the observations. In 8 of these observations, QPOs are present
in the 8-14 keV and in 5 observations in the 15-25 keV band. XTE J1817-330 is
the third black hole source from which the low frequency QPOs are clearly
detected in hard X-rays. The QPO frequency lies in $\approx$ 4-9 Hz and the
rms amplitude in 1.7-13.3% range, the amplitude being higher at higher
energies. We have fitted the PDS of the observations with Lorentzian and power
law models. Energy spectra are derived for those observations in which the
QPOs are detected to investigate any dependence of the QPO characteristic on
the spectral parameters. These spectra are well fitted with a two component
model that includes the disk black body component and a power law component.
The QPO characteristics and their variations are discussed and its implication
on the origin of the QPOs are examined.
###### keywords:
stars: black hole binary: individual: XTE J1817-330 stars
††pagerange: Studies of Quasi-Periodic Oscillations in the Black Hole
Transient XTE J1817-330–LABEL:lastpage††pubyear: 2009
## 1 Introduction
Accretion powered X-ray binaries are the brightest X-ray sources in our
Galaxy. These binaries contain either an accreting neutron star or an
accreting black hole as the X-ray source. Based on estimates of the mass of
the accreting object and its X-ray characteristics, about 40 X-ray sources
have been classified as black hole binaries (Remillard and McClintock 2006a).
Of these 20 have reliable mass estimates and are, therefore, regarded as
confirmed black holes while the remaining 20 are considered to be black hole
candidates (Remillard and McClintock 2006a). A majority of the black hole
binaries (BHBs) are transients and most of them have a low mass optical
companion.
The black hole X-ray binaries have several distinctive X-ray characteristics.
During the outburst there is a strong soft component that originates in the
inner region of the hot accretion disk. The presence of a hard X-ray component
in energy spectra is another common feature of black hole binaries. The hard
X-rays arise through Compton scattering of low energy photons from the
accretion disk in a hot optically thin plasma. The resulting hard X-rays are
referred to as the thermal Compton component (McClintock and Remillard 2006).
Low frequency quasi-periodic oscillations (QPOs) in $\approx$ 0.1-40 Hz range
and high frequency QPOs $\approx$ 50-450 Hz also occur in many BHBs. Presence
of several distinct spectral states and transition from one state to another
at irregular intervals, is another distinguishing feature of BHBs. Spectral
and temporal characteristics of the sources vary from one state to another.
Remillard and McClintok (2006a) have broadly classified the spectral state for
BHB as (a) High or Soft state (HS) dominated by the thermal component, (b) Low
or Hard State (LH) marked by the hard X-ray power law and (c) Steep power law
(SPL) or very high state characterized by steep slope power law component. For
a more complete description of the spectral states McClintock and Remillard
(2006) included two more states namely an Intermediate state (IM) that occurs
when the source moves from LH to HS state and an extreme LH state which they
called as quiescent state. Gierlinski, Done and Page (2008) have included an
additional state termed as Ultra soft state (US) which is an extreme case of
high/soft state found in this source. The US state is characterized by a very
weak high energy tail with a low disk temperature and very low hardness ratio
$\leq$ 0.1 (Gierlinski, Done and Page 2008). The low frequency QPOs are
usually detected in the SPL and LH states and have been observed in 14 BHBs so
far (Remillard and McClintock 2006a). Detection of LFQPO in XTE J1817-330
indicates that LFQPOs can also be found sometime in the HS state.
An X-ray transient known as XTE J1817-330 was discovered by Remillard et al.
(2006b) on 2006 January 26, with the All Sky Monitor (ASM) on Rossi Timing
X-ray Explorer (RXTE) (Levine et al. 1996). At the time of its detection the
2-12 keV flux was 0.93 Crab. The intensity then rose to a peak value of 1.9
Crab on January 28, and then declined to 1.2 Crab by January 30. Subsequently
it decayed exponentially with a decay time of 27 days (Sala et al. 2007). From
its high/soft state at the time of the outburst, the source declined to a
low/hard state characterized by kT = 0.2 keV. A change in the intensity state
of the source occurred around $\sim$ February 9 (Shaw et al. 2006). Hard
X-rays (20-60 keV) flux increased from 46 $\pm$ 2 mCrab on February 9 to 79
$\pm$ 2 mCrab on February 14 accompanied by a decrease of soft X-ray (2-10
keV) flux from 840 $\pm$ 4 mCrab to 670 $\pm$ 4 mCrab during this period
indicating the onset of the hard state (Kuulkers et al. 2006).
Its energy spectrum was studied with the RXTE, XMM-Newton, Integral and Swift
instruments. Sala et al. (2007) measured the spectra using data from the XMM-
Newton and Integral instrument when the source was in a high/soft state during
2006 February-March. Their spectral results indicated that its energy spectrum
was typical of a BHB source with a dominant thermal disk component well
described by kT $\sim$ 0.7-0.9 keV and a thermal Compton power law component
with a photon index of $\sim$ 2-3 (Sala et al. 2007). Its spectral
characteristics were observed with X-ray telescope (XRT) on the Swift
satellite covering different stages of the outburst over 160 days. During this
period the source made transition from the high/soft state in the initial
outburst to a low hard state near the end of the outburst. The XRT spectra in
the high soft state in 0.6-10 keV are well described by a two component model
consisting of a thermal component from a optically thick and geometrically
thin accretion disk and a hard power law component. The temperature of the
inner disk producing the soft component declined from $\sim$ 0.8 keV during
the initial outburst phase to $\sim$ 0.2 keV near the end of the outburst
(Rykoff et al. 2007).
A detailed study of the spectral evolution of XTE J1817-330 at different
phases of the outburst was carried out by Gierlinski, Done and Page (2008)
using the RXTE and the Swift data. The spectra of 150 PCU2 (RXTE) observations
were modeled with the two component spectral model consisting of a disk
component and a hard thermal Compton power law component. Using the same XRT
data as used by Rykoff et al. (2007), they also found that the inner disc
temperature declined from $\sim$ 0.9 keV to $\sim$ 0.2 keV as the source
intensity declined. Based on this they claimed that the accretion disk recedes
when the source transits from the high/soft state to the low/hard state. It
may also be noted that apart from XTE J1118+480, this black hole binary has
the lowest absorption along the line of sight among all the bright black hole
candidates as obtained from the Chandra and Swift spectral data (Miller et al.
2006a;b).
Power density spectra of XTE J1817-330 obtained from the first two X-ray
observations during 06:03-17:04 UTC on 2006 February 24, revealed strong QPOs
at $\sim$8.5 Hz (Homan, Miller and Wijnands 2006). Third observation
(13:20-13:43 UTC) on the same day showed only a weak QPO at around 6.4 Hz. The
last 2 observations (14:55-17:04 UTC) again indicated the presence of strong
QPOs at 5.0 Hz.
Rupen, Dhawan and Mioduszewski (2006a), detected with VLA a radio object in
the error box of the X-ray source having a flux density of 2.1 mJy at 1.4 GHz
on 2006 January 31. The radio source faded away by 2006 February 2 (Rupen,
Dhawan and Mioduszewski 2006b). A bright optical counterpart of the X-ray
source was found at the time of the outburst with V = 11.3 magnitude and its
brightness decreased to V = 15.5 by February 10 (Torres et al. 2006). The
optical star was also detected in the near-infrared with a K magnitude of 15.0
on 2006 February 7 (D’Avanzo et al. 2006). Near-UV observations with the
Optical Monitor on the XMM-Newton showed variations in the UV flux correlated
to the hard X-ray flux variations (Sala et al. 2007).
All these characteristics strongly suggest that XTE J1817-330 is most likely a
black hole binary. This transient was repeatedly observed with the
Proportional Counter Array (PCA) on RXTE in the pointed mode during the period
2006 January 27 - August 2. We have carried out detailed timing analysis of
the PCA data to study the properties of the QPOs and in this paper we present
the results of this analysis.
## 2 Observations and Data Reduction
The data for our analysis are taken from the HEASARC data archive
(http://heasarc.gsfc.nasa.gov/). We have used data acquired from the
observations obtained during 2006, January 27 - August 2 with the 160 PCA
pointing’s. The PCA consists of five xenon proportional counter units (PCUs)
sensitive in 2-60 keV energy range with a total effective area of $\sim$6500
cm2 at $\sim$ 10 keV (Jahoda et al. 1996). We used data from only the PCU2 for
generating the light curves, energy spectra and the hardness ratio of the
source as this unit was operating in all the observations.
FTOOLS version 6.1 was employed for the analysis and the calibration data
files of epoch 3 and 4 were used for the energy response matrix. The XTE
filter file is created using task xtefilt in FTOOLS with a time step of 16
second. The binned mode data were reduced to create the light curve files for
all the observation Id’s using saextrct with a bin size of 7.8125 milli
second. Event mode data were used for constructing the power density spectra
(PDS) in the 15-25 keV energy range. These data were extracted from the light
curve files using the same binning time as used for the binned mode data using
seextrct task in FTOOLS. The spectral studies of XTE J1817-330 were carried
out by using standard 2 mode data for the PCU 2 with a binning time of 16 sec.
## 3 Data Analysis and Results
The X-ray light curve of the transient was constructed by using the 1.5-12 keV
count rates from the ASM and it is shown in Fig 1. In the ASM light curve the
source reached a peak intensity of about 1.9 Crab on 2006 January 28 and then
declined with an e-folding time of about 27 days. There is indication of a
broad peak between 2006 January 28 to 2006 February 2. After a lapse of about
120 days since the peak intensity, the source became undetectable. Similar
light curves were obtained by Rykoff et al. (2007) and Sala et al. (2007)
using the ASM daily average count rates for the entire duration of the
outburst. Sala et al. (2007) also derived variation of the hardness ratios
using the count rates in 3.0-5.0 keV / 1.5-3.0 keV and 5.0-12 keV / 3.0-5.0
keV using the ASM data from 2006 January 30 to April 30. We have computed the
hardness ratio [(6-13) keV counts / (2-6) keV counts] from the PCU 2 data and
this is shown as a function of the source intensity in Fig 2. We used the
spectral state classification of Gierlinski, Done and Page (2008) to indicate
the spectral state of the source in Figures 1 and 2. From MJD 53764 (2006
January 29) the source was in HS state for 70 days with a hardness ratio less
then 0.2. The source then moved to the US state characterized by a very low
hardness ratio ($\ll$ 0.1), and again appeared to move back to HS state. After
120 days of the HS state, the source, passed through 15 days of intermediate
state with hardness ratio in the range 0.2-0.4. Finally it reached the LH
state with a hardness ratio $>$ 0.4. Gierlinski, Page and Done (2008) had also
presented a similar plot (Figure 2) of hardness ratio obtained from 6.3-10.5
keV count rates / 3.8-6.3 count rates using the PCA (RXTE) data. There is
close resemblance between the two sets of curves even though the energy bands
are different.
### 3.1 QPO Analysis and Results
The PDS is a powerful method for probing the rapid variability in the black
hole and other accretion powered X-ray binaries. The PDS of many BHBs exhibits
narrow and broad QPOs peaks whose width and position vary with time. A search
for the QPOs in XTE J1817-330 was carried out in 2-8 keV, 8-14 keV and 15-25
keV bands. The power density spectra were constructed for all the observations
using powspec. All the power spectra were normalized such that their integral
gives the squared rms fractional variability (therefore the power spectrum is
in units of (rms)2/Hz) with the expected white noise level subtracted. The
binned mode data were used for the 2-8 keV and 8-14 keV bands and the event
mode data were used for the 15-25 keV for generating the PDS. The power law
and the Lorentzian models have been used for fitting the QPO profiles. A QPO
feature is detected in the 2-8 keV band in 10 of the observations. The
observations showing presence of the QPOs are indicated by vertical arrows in
the ASM light curve in Fig 1. It may be noted that the QPOs are detected only
when the source was bright (40-150 ASM counts sec-1). As the source intensity
declined to a level below 40 ASM counts sec-1, the QPOs disappeared. The QPO
occurrence is also indicated in the plot of hardness ratio versus source
intensity in Fig 2 by star symbol. In 6 of these 10 observations, the QPOs are
present in the 8-14 keV energy band. At the higher energy (15-25 keV), the
QPOs are detected only in 5 of the observations.
The power density spectra in the different energy ranges for three of the
observations are shown in Figures 3, 4 and 5. Prominent QPO peaks are present
in the PDS. The QPO feature at $\approx$ 5 Hz is detected prominently in the
higher energy band (15-25 keV). A first harmonic of the fundamental QPOs at
$\approx$ 10 Hz is also present in the PDS. Note that the first harmonic is
quite prominent in the 2-8 keV (Fig 3a) and the peak at $\approx$ 10 Hz in Fig
3(b) for 8-14 keV band is even stronger then the peak at $\approx$ 5 Hz. The
fundamental QPO peaks appear at about the same frequency in the plots in the
three different energy bands. The power density spectra for MJD 53790.2 are
shown in Fig 6. There is no indication of the presence of QPOs in 2-8 keV but
a broad QPO peak is clearly seen at about 9 Hz in the PDS of 8-14 keV. It is
conceivable that the 5 Hz peak is blended in the rather broad 9 Hz peak. Note
that if the 9 Hz peak is identified as the first harmonic as seen in Figs 3, 4
and 5, the peak due to fundamental QPO frequency at $\approx$ 5 Hz is
undetectable in Fig 6(a) and (b). Small but insignificant peaks can be seen in
Fig 4 at $\sim$ 3 Hz in 2-8 keV and 8-14 keV bands and at $\sim$ 3 Hz and
$\sim$ 0.8 Hz in 8-14 keV and 15-25 keV bands in Fig 5.
A summary of the characteristic of the QPOs eg., frequency, amplitude and
width for all the observations that showed the presence of the QPOs, is
presented in Table 1. It may be noticed from the table that the data of MJD
53790.2 and 53790.3, show no detectable QPOs in 2-8 keV. However the QPOs are
clearly present in the 8-14 keV at a higher frequency of $\approx$ 8-9 Hz. The
QPO peaks in the 8-14 and 15-25 keV bands are rather broad with $\delta$$\nu$
in 1.8 to 3 Hz range. This is unlike the QPOs detected in the other
observations where the peaks are narrow with $\delta$$\nu$ less then 1 Hz.
Consequently the Q values are rather low being 3 to 4. The values of the
coherence parameter (quality factor, Q$=$$\nu$/$\delta\nu$) are all greater
than 2.
We found no correlation in the amplitude of fundamental QPO and amplitude of
first harmonics. The QPO amplitudes were not found to show any correlation
with the overall source intensity or hardness ratio. No trend was observed
between the QPO amplitudes derived in the two energy bands 2-8 keV and 8-14
keV. Similarly no correlation was detected in the variation of power-law flux
with QPO fundamental frequency and rms amplitude of the QPOs.
We found from the PDS studies on 2006 February 24 that the QPOs were present
in high/soft state of the source with spectral power-law index varying in the
range 2.1-2.3. Our result are supported by the findings of Homan et al. (2006)
who discovered rapid variability in the QPO properties on the same day. The
QPOs were observed in high/soft state of the source with the spectral power-
law index from $\sim$ 2.3-2.4 (Homan et al. 2006).
To investigate whether the QPO frequency has any dependence on the source
intensity, we have plotted the QPO frequency versus the source count rate in
the three energy bands in Fig 7. These are background subtracted counts per
second taken only from the PCU2 that was working in all the observations.
There is indication of decreasing trend in the QPO frequency from $\approx$ 5
Hz to $\approx$ 4 Hz as the source intensity is increased [Fig 7(a)]. The
frequency, however, again increased as the source intensity increased by a
factor of more than two. In the 8-14 keV and 15-25 keV channels no correlation
is obvious between the frequency and the intensity in Figures 7(b), 7(c).
Hence it may be inferred that there is no clear trend of a change of the QPO
frequency with the intensity. The QPO frequency varies in an erratic manner in
a narrow band of 4.4 Hz to 5.9 Hz in the 2-8 keV channel.
### 3.2 Spectral Analysis and Results
We have analyzed the spectral data from the RXTE for those observations in
which the QPOs are detected. For a comparison the spectra are also obtained
for a few observations in which no QPOs are detected to investigate whether
there are any differences in the spectral parameters. Background subtracted
standard 2 mode data from the PCU 2 with 16 sec binning were used to construct
the spectra. The energy spectra of selected observations were fitted with a
power law model taken from XSPEC version 12 for high energy component of the
spectrum, plus a standard disk black body, diskbb model (accretion disk
consisting of multiple blackbody components) taken from XSPEC (Mitsuda et al.
1984 ; Makishima et al. 1986). It also included the photoelectric absorption
(wabs) model from (Morrison and McCammon, 1983) and the Xenon edge at
$\approx$ 4.7 keV to account for the PCA response and is not intrinsic part of
the spectrum. In the spectrum of MJD 53768 a Gaussian line model is also added
to account for the presence of an iron line at 6.4 keV. In the spectral fits a
fixed value of hydrogen column density (NH)= 1.2$\times$1021 cm-2 has been
used (Rykoff et al. 2007).
The analyzed epochs included (a) MJD 53768, 53789, 53790 and 53790.6 that show
presence of the QPOs in the entire 2-25 keV energy region, (b) MJD 53790.2 and
53790.3 in which broad QPO peaks in the 8-14 keV band are present but not in
the 2-8 keV interval (c) MJD 53778, 53780, 53786 and 53790.5 data with the QPO
detection only in the 2-8 keV band but not at $>$ 8 keV (d) MJD 53766 and
53791 data with the QPOs in the 2-14 keV interval but not at $>$ 15 keV (e)
MJD 53764 and 53775 when no QPOs are detected (f) MJD 53794 and 53797 when the
source intensity has declined and the QPOs are not present. In Table 2 we have
compiled the derived values of the temperature (Tin) of the disk black body
component, photon spectral index ($\alpha$) of the hard component as well as
the flux values of the thermal, and the power law components. Ratio of the
power law flux to the thermal flux is also computed and shown in Table 2.
Following points are to be noted from the table: (I) As expected for the black
hole binaries the thermal component is dominant in the initial part of the
outburst lasting for about first 25 days. This is obvious from the values of
flux ratio that lies in 0.06 to 0.27 range. (II) After 1/e decay of the
intensity, the thermal component declined substantially and the thermal and
the power law fluxes became comparable. (III) When the intensity declined
further (after MJD 53791) the power law flux declines and the thermal flux
dominates. Note that the power law spectral index lies in a narrow region of
2.1-2.3 for all the observations selected for the spectral studies. These
values are comparable within the errors of the photon index values estimated
by Gierlinski, Done and Page (2008) for some of the observations. The
temperature of the disk is $\approx$ 1 keV and constant during the first 30
days but there is indication of cooling of the disk as shown by kT $\approx$
0.8 with further decline of the intensity in the last two observations. A few
representative energy spectra are shown in Fig 8. The systematic errors for
all the fits are within 3%.
## 4 Discussion
The LFQPOs occur most frequently when the power law flux is the dominating
component in the energy spectrum. Some times they are also present in the high
luminosity state with the presence of a hard component. From table 1 and 2 it
will be noticed that in all the cases of the detection of LFQPOs from XTE
J1817-330, except the observations of MJD 53764 and 53775 in which no QPOs are
detected, the ratio of the power law flux to the thermal disk flux lies in
0.20 to 1.13 range consistent with its occurrence only in the states with a
significant power law component. Also note that the LFQPOs have significant
coherence (Q $=$ $\nu$/$\delta\nu$) with the Q in range of 3-11 and their rms
amplitude vary from a few percent to as high as 13%.
Variation of the QPO frequency with the source intensity is another feature
detected in some BHBs. The fundamental QPO frequency in XTE J1817-330 varies
in a narrow band of 4.4-5.9 Hz. We have investigated the variation of the QPO
frequency in the 2-8 keV band with the thermal disk component flux ( dbb ).
This is shown in Fig 9 and a trend similar to that of Fig 7(a) is seen here
indicating that the frequency is correlated with the thermal disk component.
As expected a clear 1:2 relationship of the QPO fundamental frequency and that
of the first harmonic is seen.
All the characteristics of the LFQPOs reported by us in this paper from XTE
J1817-330 are similar to those seen in the other black hole binaries and
further strengthen the black hole nature of this source (Remillard et al.
2003).
Correlation of the properties of LFQPOs with the spectral parameters of the
BHBs has been studied in detail for several sources (Muno, Remillard and
Morgan 2001; Tomsick and Kaaret 2001; Remillard et al. 2003; Belloni, Psaltis
and van der Klis 2002; Vignarca et al. 2003; Rossi, Homan and Belloni 2004).
These studies show that the LFQPO characteristics are generally well
correlated with the thermal disk and the power law components of the energy
spectra. While the QPO frequency is closely correlated with the disk flux, the
amplitude of the QPOs for the fundamental frequency is found to track the flux
of the power law component (Remillard et al. 2003). In general the QPO
amplitude is higher for the higher energy X-rays up to about 20 keV and then
it tends to decrease at the higher energy. Most of the LFQPOs detected in the
black hole binaries occur below 10 keV. In two of the BHBs namely GRS 1915+105
and XTE J1550-564 the QPOs have been reported above 20 keV (Trudolyubov,
Churazov & Gilfanov 1999; Remillard et al. 2003). In some cases the QPO
detection is claimed in a broad spectral band of 2-60 keV but since no
breakdown of QPO properties is given in the different energy intervals say
below 10 keV and above 10 keV, it is not obvious whether the QPOs have indeed
been detected in hard X-rays. The enigmatic source GRS 1915+105 is the only
black hole binary in which the QPOs in 0.8-3.0 Hz have been reported at an
energy up to 124 keV (Tomsick and Kaaret 2001). It is found that in this
object, the amplitude of the fundamental frequency QPO increases with energy
up to 29 keV and then decreases in 30-60 keV and 60-124 keV bands. Remillard
et al. (2003) investigated the QPO characteristics in the transient XTE
J1550-564 from the RXTE - PCA observations and detected LFQPOs in 2-13 keV and
13-30 keV energy channels making it only the second black hole binary in which
the LFQPOs have been detected up to 30 keV. We have detected the QPOs from XTE
J1817-330 in five of the observations up to an energy of $\sim$ 25 keV, making
it only the third BHB showing unambiguous presence of the LFQPOs at higher
energy. From Table 1 it may be noticed that the amplitude of the fundamental
QPOs is always higher in the 8-14 keV channel, being in 3.2-13.3 % range,
compared to the values of 1.7-7.0 % in the 2-8 keV channel. At still higher
energy 15-25 keV, the QPO amplitude is comparable to or slightly higher than
that in the 8-14 keV indicating that it has reached a plateau level.
A detailed study of the QPO centroid frequency, its coherence, amplitude,
phase lag and their dependence on the photon energy, is of vital importance to
understand the origin of the QPOs and pin point the emission process. The QPOs
are believed to originate in the innermost region of the accretion disk and
the most common models explain their generation to the modulation of the disk
flux by the Keplerian motion of the localized hot regions termed as ’blobs’.
Lehr, Wagoner and Wilms (2000) have developed a model to compute by Monte
Carlo simulations, the energy dependence of the QPO amplitude to probe the
site of their origin in the accretion disk. They use two components with
repeated Compton scattering to produce the high energy X-rays and assume a
radial dependence of the disk temperature. They computed the energy dependence
of the QPO amplitude for GRS 1915 + 105 and found it to be in agreement with
the observation of Morgan and Remillard (1997). Thus they are able to localize
the QPO origin in the inner disk. The amplitude of the QPOs will either
increase or decrease with energy depending on the region of the disk in which
the QPOs are produced and the temperature of the corona and its gradient. It
is reasonable to assume that the QPO frequency is related to the dynamical
time scale of the blobs and therefore, the LFQPOs observed by us in the
4.4-5.9 Hz from XTE J1817-330 will originate farther out in the accretion
disk. In the outer region, the Compton scattering corona will be relatively
cooler and the QPO amplitude will decrease with increasing energy. In our case
we have detected a marginal increase in the amplitude of the QPOs at the
higher energy in the two observations, a decrease in the amplitude in the
other two observations and no change in amplitude in one case. This suggests
that the site of QPO generation is itself dynamically varying in XTE J1817-330
due to variation in the X-ray luminosity which in turn depends on the
accretion rate.
We have detected the QPOs in the 8-14 keV band but not in the 2-8 keV for the
observations of MJD 53790.2 and 53790.3. This is similar to the detection of
the QPOs at the high energy and its absence at the low energy in some
observations from GRS 1915+105 (Chakraborti and Manickam 2000). This behavior
of GRS 1915+105 was explained by Chakrabarti and Manickam (2000) on the basis
of ”on” and ”off” (burst and quiescent) state of the source with the shock
oscillation model. Further detailed studies of the LFQPOs at the higher energy
are required to test the validity of the model.
## 5 Acknowledgment
The authors thank NASA/GSFC based High Energy Astrophysics Science Archive
Research Center for making RXTE-PCA data available on-line. We are extremely
thankful to an anonymous referees for his critical and constructive comments
and suggestions that vastly improved the content and presentation of this
paper.
## References
* (1) Belloni T., Psaltis D., van der Klis M., 2002, ApJ, 572, 392
* (2) Chakrabarti & Manickam, 2000, ApJ, 531, L41
* (3) D’Avanzo P., Goldoni P., Covino S., Campana S., Molinari E., Chincarini G., Zerbi F. M., Testa V., Tosti G., Vitali F., 2006, ATel 724
* (4) Gierlinski M., Done C., Page K., 2008, MNRAS, 388, 753
* (5) Homan J., Miller J. M., Wijnands R., 2006, ATel 752
* (6) Jahoda K., Swank J. H., Giles, A. B., Stark M. J., Strohmayer T. E., Zhang W., Morgan, E. H., 1996, SPIE, 2808, 59
* (7) Kuulkers E., Goldoni P., Shaw S. E., Brandt S., Chenevez J., Courvoisier T. J. L., Ebisawa K., Kretschmar P., Markwardt C., Mowlavi N., 2006, ATel 738
* (8) Lehr D. E., Wagoner R.V., Wilms J., 2000, astro-ph 0004211v1
* (9) Levine A. M., Bradt H., CUI W., Jernigan J. G., Morgan E.J., Remillard R.A., Shirey R. E., Smith D. A., 1996, ApJ, 469, L33
* (10) Makishima K., Maejima Y., Mitsuda K., Bradt H. V., Remillard R.A., Tuohy I. R., Hoshi R., Nakagawa M., 1986, ApJ, 308, 635
* (11) McClintock, J.E., Remillard, R.A., 2006, in Lewin, W.H.G., van der Klis, M. (Eds.), Compact Stellar X-ray Sources, Cambridge University Press, Cambridge, 157
* (12) Miller J. M., Homan J., Steeghs D., Torres M. A. P., Wijnands R., 2006a, ATel 743
* (13) Miller J. M., Homan J., Steeghs D., Wijnands R., 2006b, ATel 746
* (14) Mitsuda K., Inque H., Koyama K., Makishima K., Matsuoka M., Ogawara Y., Shibazaki N., Suzuki K., Tanaka Y., Hirano H., 1984, PASJ, 36, 741
* (15) Morrison R., McCammon D., 1983, ApJ, 270, 119
* (16) Morgan E. H., Remillard R. A., 1997, ApJ, 482, 993
* (17) Muno M., Remillard R., Morgan E., 2001, ApJ, 556, 515
* (18) Remillard R.A., Muno M.P, McClintock J.E., Orosz J., 2003, in Proc. Fourth Microquasar Workshop: New Views on Microquasars, ed. P. Durouchoux Y. Fuchs, & J. Rodriguez (Kolkata: Center for Space Physics), 57
* (19) Remillard R.A., McClintock J.E., 2006a, ARAA, 44, 49
* (20) Remillard R.A., Levine A. M., Morgan E. H., Markwardt C. B., Swank J. H., 2006b, ATel 714
* (21) Rossi S., Homan J., Belloni T., 2004, Nuclear Physics B (Proc. Suppl.), 132, 1416
* (22) Rupen M. P., Dhawan V., Mioduszewski A. J., 2006a, ATel 717
* (23) Rupen M. P., Dhawan V., Mioduszewski, A. J., 2006b, ATel 721
* (24) Rykoff E. S., Miller J. M., Steeghs D., Torres M. A. P, 2007, ApJ, 666, 1129
* (25) Sala G., Greiner J., Ajello M., Bottacini E., Haberl F., 2007, A & A, 473, 561
* (26) Shaw S. E., Zurita J., Kuulkers E., Brandt S., Chenevez J., Courvoisier T. J.L., Ebisawa K., Kretschmar P., Markwardt C., Mowlavi N., 2006, ATel 731
* (27) Tomsick & Kaaret, 2001, ApJ, 548, 401
* (28) Torres M. A. P., Steeghs D., Jonker P. G., Luhman K., McClintock J. E., Garcia M. R, 2006, ATel 733
* (29) Trudolyubov S, Churazov E., Gilfanov M., 1999, Astron. Lett., 25, 718
* (30) Vignarca F., Migliari S., Belloni T., Psaltis D., van der Klis M., 2003, A&A, 397, 729
Table 1: Summary of the characteristics of the QPO in the three energy bands.
MJD 53766 corresponds to the date 2006-01-31.
MJD | Duration | Energy | Power Law | QPO Fundamental | First Harmonic | Reduced
---|---|---|---|---|---|---
| Of Observation | Range | Index | Frequency | Width | Quality Factor | RMS | Frequency | Width | Quality Factor | RMS | Chi Sq†
| (sec) | (keV) | | (Hz) | (Hz) | | % | (Hz) | (Hz) | | % |
53766(Q1) | 3947 | 2-8 | -1.25 | $5.43_{-0.04}^{+0.04}$ | $0.81_{-0.10}^{+0.09}$ | 6.7 | 2.5 | $10.82_{-0.58}^{+0.56}$ | $1.42_{-0.98}^{+0.86}$ | 7.6 | 0.9 | 1.7
| | 8-14 | -0.56 | $5.42_{-0.20}^{+0.12}$ | $0.62_{-0.36}^{+1.59}$ | 8.7 | 3.9 | $10.86_{-0.50}^{+0.53}$ | $2.62_{-1.99}^{+2.09}$ | 4.2 | 6.8 | 0.9
53768(Q2) | 5274 | 2-8 | -1.28 | $5.39_{-0.02}^{+0.03}$ | $0.89_{-0.07}^{+0.06}$ | 6.0 | 2.6 | $10.93_{-0.16}^{+0.15}$ | $1.19_{-0.36}^{+0.27}$ | 9.2 | 1.0 | 2.4
| | 8-14 | -0.57 | $5.55_{-0.11}^{+0.13}$ | $0.74_{-0.36}^{+0.33}$ | 7.4 | 3.2 | $10.89_{-0.10}^{+0.10}$ | $2.15_{-0.33}^{+0.28}$ | 5.1 | 7.4 | 1.0
| | 15-25 | -0.32 | $5.14_{-0.23}^{+0.19}$ | $0.64_{-0.55}^{+0.58}$ | 8.1 | 3.9 | $10.83_{-0.31}^{+0.26}$ | $2.22_{-0.92}^{+0.70}$ | 4.9 | 8.1 | 1.0
53778(Q3) | 10329 | 2-8 | -1.19 | $4.41_{-0.13}^{+0.14}$ | $1.11_{-0.52}^{+0.34}$ | 3.9 | 2.0 | $7.84_{-0.22}^{+0.06}$ | $0.30_{-0.59}^{+1.19}$ | 26.1 | 1.0 | 1.7
53780(Q4) | 13549 | 2-8 | -1.09 | $4.80_{-0.08}^{+0.09}$ | $0.79_{-0.29}^{+0.24}$ | 6.1 | 2.2 | $9.83_{-0.52}^{+0.24}$ | $0.91_{-1.72}^{+0.62}$ | 10.8 | 1.3 | 1.2
| | 2-8 | -1.25 | $5.12_{-0.08}^{+0.08}$ | $1.10_{-0.20}^{+0.17}$ | 4.7 | 2.5 | $10.35_{-0.43}^{+0.54}$ | $1.82_{-1.11}^{+0.85}$ | 5.7 | 1.4 | 1.6
| | 2-8 | -1.87 | $5.05_{-0.10}^{+0.10}$ | $1.31_{-0.25}^{+0.21}$ | 3.8 | 2.5 | $10.66_{-0.31}^{+0.24}$ | $0.67_{-0.97}^{+2.31}$ | 15.8 | 1.0 | 1.5
53786(Q5) | 8413 | 2-8 | -0.38 | $5.29_{-0.13}^{+0.12}$ | $0.85_{-0.80}^{+0.35}$ | 6.2 | 2.5 | - | - | - | - | 1.3
| | 2-8 | -1.11 | $5.27_{-0.11}^{+0.11}$ | $1.53_{-0.38}^{+0.31}$ | 3.5 | 2.4 | $10.78_{-0.24}^{+0.24}$ | $1.42_{-0.66}^{+0.51}$ | 7.6 | 1.4 | 1.4
53789(Q6) | 2769 | 2-8 | -1.09 | $5.57_{-0.03}^{+0.03}$ | $0.94_{-0.06}^{+0.05}$ | 5.9 | 7.0 | $10.18_{-0.30}^{+0.32}$ | $2.92_{-1.00}^{+0.74}$ | 3.5 | 2.4 | 1.5
| | 8-14 | -2.42 | $5.57_{-0.03}^{+0.03}$ | $0.95_{-0.07}^{+0.07}$ | 5.9 | 13.2 | $10.56_{-0.46}^{+0.45}$ | $2.94_{-1.31}^{+0.93}$ | 3.6 | 5.2 | 1.2
| | 15-25 | -1.82 | $5.50_{-0.07}^{+0.07}$ | $1.01_{-0.16}^{+0.13}$ | 5.6 | 11.5 | $10.97_{-2.31}^{+2.10}$ | $2.96_{-3.29}^{+2.31}$ | 3.7 | 4.5 | 0.9
53790(Q7) | 4132 | 2-8 | -1.43 | $5.61_{-0.03}^{+0.03}$ | $1.08_{-0.08}^{+0.07}$ | 5.2 | 6.7 | $10.68_{-0.37}^{+0.53}$ | $1.89_{-1.26}^{+0.76}$ | 5.7 | 2.0 | 1.6
| | 8-14 | -2.48 | $5.60_{-0.04}^{+0.04}$ | $1.06_{-0.11}^{+0.10}$ | 5.3 | 12.3 | $12.34_{-1.53}^{+1.25}$ | $5.27_{-6.8}^{+2.67}$ | 2.3 | 6.0 | 0.9
| | 15-25 | -2.11 | $5.54_{-0.12}^{+0.13}$ | $1.38_{-0.35}^{+0.29}$ | 4.0 | 12.0 | - | - | - | - | 1.1
53790.2(Q8) | 3179 | 2-8 | -1.09 | NO QPO | - | - | - | - | - | - | - | 1.0
| | 8-14 | - | $9.06_{-0.33}^{+0.32}$ | $2.97_{-1.95}^{+1.37}$ | 3.0 | 8.8 | - | - | - | - | 1.2
53790.3(Q9) | 1967 | 2-8 | -1.12 | NO QPO | - | - | - | - | - | - | - | 1.1
| | 8-14 | - | $7.83_{-0.18}^{+0.16}$ | $2.03_{-0.90}^{+0.68}$ | 3.8 | 9.4 | - | - | - | - | 1.2
| | 15-25 | - | $7.97_{-0.37}^{+0.35}$ | $1.78_{-1.91}^{+1.28}$ | 4.5 | 9.6 | - | - | - | - | 0.9
53790.5(Q10) | 3024 | 2-8 | -0.82 | $5.88_{-0.37}^{+0.35}$ | $1.91_{-1.39}^{+0.77}$ | 3.1 | 1.7 | - | - | - | - | 1.3
53790.6(Q11) | 3686 | 2-8 | -1.16 | $4.87_{-0.03}^{+0.02}$ | $0.44_{-0.04}^{+0.04}$ | 11.0 | 5.7 | $9.60_{-0.23}^{+0.22}$ | $1.20_{-0.69}^{+0.54}$ | 8.0 | 1.9 | 1.6
| | 8-14 | -1.29 | $4.85_{-0.03}^{+0.02}$ | $0.44_{-0.05}^{+0.06}$ | 11.1 | 12.9 | $9.44_{-0.35}^{+0.37}$ | $0.43_{-2.23}^{+2.23}$ | 22.1 | 2.3 | 1.0
| | 15-25 | -0.42 | $4.91_{-0.09}^{+0.09}$ | $0.55_{-0.13}^{+0.16}$ | 9.0 | 12.6 | - | - | - | - | 1.7
53791(Q12) | 3564 | 2-8 | -1.34 | $5.03_{-0.02}^{+0.02}$ | $0.58_{-0.03}^{+0.03}$ | 8.8 | 6.4 | $9.86_{-0.10}^{+0.11}$ | $1.26_{-0.25}^{+0.20}$ | 7.8 | 2.4 | 1.6
| | 8-14 | -2.16 | $5.01_{-0.02}^{+0.02}$ | $0.59_{-0.04}^{+0.04}$ | 8.5 | 13.3 | $9.73_{-0.26}^{+0.27}$ | $1.89_{-1.26}^{+0.72}$ | 5.2 | 4.9 | 1.7
† : Reduced chi square refer to the goodness of fit of the entire spectrum.
Table 2: Summary of the spectral fit parameter for the selected observations
of XTE J1817-330. MJD 53764 corresponds to the date 2006-01-29.
MJD | DURATION OF | | power law | | | diskbb | | Reduced | Energy | Total | Flux
---|---|---|---|---|---|---|---|---|---|---|---
| OBSERVATION | | | | | | | Chi Square | Range | Flux | Ratio(1)
| | Photon Index | Normalization | Flux | Tin | Normalization | Flux | | | 3-25 keV |
| (sec) | | | (ergs/cm2/s) 10-8 | (keV) | $\times$ 103 | (ergs/cm2/s) 10-8 | | (keV) | (ergs/cm2/s 10-8) |
53764 | 3117 | 2.17($\pm 0.24$) | 0.54($\pm 0.34$) | 0.12 | 0.99($\pm 0.01$) | 3.13($\pm 0.41$) | 1.59 | 0.6 | 3-15 | 1.72 | 0.08
53766(Q1) | 3947 | 2.31($\pm 0.04$) | 2.63($\pm 0.34$) | 0.45 | 1.04($\pm 0.01$) | 2.52($\pm 0.14$) | 1.71 | 1.1 | 3-20 | 2.16 | 0.27
53768(Q2) | 5274 | 2.34($\pm 0.04$) | 2.57($\pm 0.34$) | 0.42 | 1.03($\pm 0.01$) | 2.45($\pm 0.21$) | 1.58 | 1.9 | 3-25 | 2.01 | 0.27
53775 | 3958 | 2.09($\pm 0.27$) | 0.25($\pm 0.19$) | 0.07 | 0.95($\pm 0.007$) | 2.94($\pm 0.12$) | 1.21 | 1.2 | 3-20 | 1.27 | 0.06
53778(Q3) | 10329 | 2.08($\pm 0.06$) | 0.68($\pm 0.18$) | 0.19 | 0.95($\pm 0.01$) | 2.45($\pm 0.21$) | 0.95 | 1.1 | 3-20 | 1.14 | 0.20
53780(Q4) | 13549 | 2.06($\pm 0.06$) | 0.61($\pm 0.12$) | 0.18 | 0.94($\pm 0.01$) | 2.29($\pm 0.18$) | 0.85 | 1.2 | 3-18 | 1.03 | 0.21
53786(Q5) | 8413 | 2.03($\pm 0.06$) | 0.48($\pm 0.08$) | 0.15 | 0.91($\pm 0.01$) | 2.01($\pm 0.16$) | 0.63 | 1.3 | 3-18 | 0.79 | 0.24
53789(Q6) | 2769 | 2.27($\pm 0.03$) | 2.79($\pm 0.27$) | 0.51 | 1.05($\pm 0.02$) | 0.73($\pm 0.87$) | 0.53 | 1.4 | 3-25 | 1.05 | 0.97
53790(Q7) | 4132 | 2.26($\pm 0.03$) | 2.74($\pm 0.24$) | 0.51 | 1.05($\pm 0.02$) | 0.72($\pm 0.88$) | 0.51 | 1.8 | 3-25 | 1.02 | 1.01
53790.2(Q8) | 3179 | 2.13($\pm 0.05$) | 1.80($\pm 0.26$) | 0.44 | 1.07($\pm 0.02$) | 0.59($\pm 0.74$) | 0.45 | 1.2 | 3-20 | 0.89 | 0.99
53790.3(Q9) | 1967 | 2.15($\pm 0.03$) | 2.01($\pm 0.21$) | 0.47 | 1.07($\pm 0.02$) | 0.53($\pm 0.70$) | 0.42 | 1.3 | 3-25 | 0.89 | 1.13
53790.5(Q10) | 3024 | 2.17($\pm 0.04$) | 1.58($\pm 0.20$) | 0.35 | 1.00($\pm 0.02$) | 0.94($\pm 0.13$) | 0.54 | 1.5 | 3-25 | 0.89 | 0.65
53790.6(Q11) | 3686 | 2.26($\pm 0.04$) | 2.10($\pm 0.27$) | 0.39 | 1.03($\pm 0.02$) | 1.03($\pm 0.89$) | 0.52 | 1.3 | 3-20 | 0.91 | 0.75
53791(Q12) | 3564 | 2.22($\pm 0.05$) | 2.08($\pm 0.29$) | 0.42 | 1.04($\pm 0.02$) | 0.81($\pm 0.93$) | 0.53 | 1.2 | 3-20 | 0.95 | 0.80
53794 | 3345 | 2.08($\pm 0.12$) | 0.29($\pm 0.09$) | 0.07 | 0.84($\pm 0.01$) | 2.14($\pm 0.21$) | 0.39 | 1.1 | 3-15 | 0.47 | 0.20
53797 | 5539 | 2.26($\pm 0.10$) | 0.39($\pm 0.10$) | 0.07 | 0.82($\pm 0.01$) | 2.31($\pm 0.18$) | 0.35 | 1.6 | 3-15 | 0.42 | 0.21
(1):Ratio of power law flux to diskbb flux
Figure 1: The ASM lightcurve in the 1.5 -12 keV range is shown in panel (a)
from 2006-Jan-29 (MJD 53764) to 2006-Aug-02 (MJD 53950). The count rates are
averaged over a day. The position of the observed QPOs are indicated by the
vertical arrows. The PCA light curves in the 2-6 keV and 6-13 keV energy bands
are shown in (b) and (c). The corresponding hardness ratio obtained from
counts in 6-13 keV / 2-6 keV is shown in panel (d) for the 100 pointed PCA
observations.
Figure 2: Variation of the Hardness Ratio (counts in (6-13) keV / counts in
(2-6) keV) with the count rate (s-1) in (2-13) keV from only PCU 2, is shown
during the 2006 outburst of the source. Filled $\star$ symbols represent the
positions where the QPOs are detected.
Figure 3: Power density spectra for the observation of MJD 53768. The PDS is
shown for 2-8 keV in panel (a), 8-14 keV in panel (b) and 15-25 keV in panel
(c). Arrows in the panels (a), (b) and (c) indicate the fundamental
frequencies of the QPOs at 5.39 Hz, 5.55 Hz and 5.14 Hz. The first harmonic at
$\approx$ 10 Hz is more prominent than the fundamental frequency peak in panel
(b) and (c).
Figure 4: Power density spectra for the observations of MJD 53789. The PDS is
shown for 2-8 keV in panel (a), 8-14 keV in panel (b) and 15-25 keV in panel
(c). Arrows in the panels indicate the fundamental frequencies of the QPOs at
5.57 Hz, 5.57 Hz and 5.5 Hz. The first harmonic at $\approx$ 10 Hz is weaker
compared to the fundamental frequency at $\approx$ 5 Hz in all the panels.
Figure 5: Power density spectra for observation of MJD 53790. The PDS is shown
for 2-8 keV in panel (a), 8-14 keV in panel (b) and 15-25 keV in panel (c).
Arrows in the panel (a), (b) and (c) indicate the fundamental frequencies of
QPOs at 5.61 Hz, 5.60 Hz and 5.54 Hz.
Figure 6: Power density spectrum for the MJD 53790.2 is shown in panel (a)for
2-8 keV, and (b)for 8-14 keV energy band. The arrow indicate the position of
broad QPO at 9.06 Hz in panel(b). The QPOs are not detectable in (2-8) keV but
a broad and asymmetric peak is present in panel (b) at about 9 Hz.
Figure 7: Plots of the QPO frequency versus the source intensity are shown in
the three energy bands. These are background subtracted source count rates
(counts s-1) taken only from PCU 2 in the energy intervals (a) 2-8 keV,
(b)8-14 keV and (c)15-25 keV.
Figure 8: Some representative energy spectra with the unfolded models are
shown in this figure. For details of the spectral model refer to page 5. Thick
black lines indicates the total spectrum that is a sum of all the components.
Symbol ”D” denotes the disk black body component of the models used to fit the
spectra, ”P” indicates the power law component and gaussian line model is
denoted by symbol ”G”. Fig (a) shows spectrum in the 3-25 keV for MJD 53768,
(b) for MJD 53789 and (c) for MJD 53790 in which the QPOs are detected in the
entire 3-25 keV energy range. Panel (d) shows spectrum in the 3-15 keV for MJD
$53794$ in which no QPO is detected.
Figure 9: Variations of the QPO fundamental frequency in the 2-8 keV band
(open circle) and first harmonics frequency (filled stars) with the flux of
the disk black body component is shown for XTE J1817-330 during its 2006
outburst.
|
arxiv-papers
| 2009-11-05T09:27:45 |
2024-09-04T02:49:06.287218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jayashree Roy (1 and 3), P.C. Agrawal (1), B. Paul (2) and K. Duorah\n (3), ((1) Tata Institute of Fundamental Research, (2) Raman Research\n Institute, Banglore, (3) Gauhati University, Guwahati)",
"submitter": "Jayashree Roy Ms",
"url": "https://arxiv.org/abs/0911.0999"
}
|
0911.1018
|
# Explaining LSND and MiniBooNE using altered neutrino dispersion relations
Sebastian Hollenberg, Octavian Micu111Speaker. , Heinrich Päs
Technische Universität Dortmund, Dortmund, Germany
(September 17, 2009)
###### Abstract
We investigate the possibility to explain the MiniBooNE anomaly by CPT and
Lorentz symmetry violating neutrino-antineutrino oscillations in a two
generation framework. We work with four non-zero CPT-violating parameters that
allow for resonant enhancements in neutrino-antineutrino oscillation phenomena
in vacuo which are suitably described in terms of charge conjugation
eigenstates of the system. We study the relation between the flavor, charge
conjugation and mass eigenbasis of neutrino-antineutrino oscillations and
examine the interplay between the available CPT-violating parameter space and
possible resonance structures.
The data from the MiniBooNE collaboration [1, 2] reveal a resonance-like
excess of events in the low-energy neutrino channel, but do not show a
deviation from the expected oscillation pattern in the antineutrino channel.
The LSND collaboration [3] on the other hand observes an excess in the
antineutrino channel. Also, there is a hint that the MiniBooNE results for the
resonance-like anomaly in the $\nu_{\mu}$ data actually look more like a
$\nu_{\mu}\to\bar{\nu}_{e}$ conversion than $\nu_{\mu}\to\nu_{e}$ events. In
order to understand these yet unexplained anomalies, neutrino oscillation
scenarios with altered dispersion relations have recently received attention.
The resonance structure might possibly indicate new physics and motivated
different possible explanations of this phenomenon [4, 5, 6]. Amongst other
things CPT and Lorentz symmetry violating neutrino oscillations have been
proposed [7, 8]. In the framework of a CPT- and Lorentz-violating Standard
Model extension with renormalizable operators only [9, 10, 11], mixing between
light neutrinos and antineutrinos is encountered which might provide a viable
candidate when it comes to explaining resonance features in
$\nu_{e}\rightleftharpoons\nu_{\mu}$ and
$\bar{\nu}_{e}\rightleftharpoons\bar{\nu}_{\mu}$ oscillation experiments.
Starting from the generalized Dirac equation introduced in [11], we consider a
model for only the first two neutrino and antineutrino generations, and allow
for Lorentz- and CPT-violating interactions. The off diagonal part of the
effective Hamiltonian for this case reads
$\displaystyle h_{\text{eff}}=\begin{pmatrix}-\frac{\Delta m^{2}}{4E}\cos
2\theta-\frac{c_{ee}E}{2}&\frac{\Delta m^{2}}{4E}\sin
2\theta&\frac{b_{e}E}{2}&0\\\ \frac{\Delta m^{2}}{4E}\sin 2\theta&\frac{\Delta
m^{2}}{4E}\cos 2\theta-\frac{c_{\mu\mu}E}{2}&0&\frac{b_{\mu}E}{2}\\\
\frac{b_{e}E}{2}&0&-\frac{\Delta m^{2}}{4E}\cos
2\theta-\frac{c_{ee}E}{2}&\frac{\Delta m^{2}}{4E}\sin 2\theta\\\
0&\frac{b_{\mu}E}{2}&\frac{\Delta m^{2}}{4E}\sin 2\theta&\frac{\Delta
m^{2}}{4E}\cos 2\theta-\frac{c_{\mu\mu}E}{2}\\\ \end{pmatrix},$ (1)
in the basis $(\nu_{e},\nu_{\mu},\nu_{e}^{c},\nu_{\mu}^{c})$, and where
$c_{ee}$ and $c_{\mu\mu}$ are Lorentz violating parameters; while $b_{e}$ and
$b_{\mu}$ are both Lorentz- and CPT-violating parameters. The presence of the
CPT-violating parameters induces a mixing between the neutrino and
antineutrino sectors making neutrino-antineutrino oscillations possible.
The effective Hamiltonian $h_{\text{eff}}$ can be brought to a block-diagonal
form $\tilde{h}_{\text{eff}}$ with the help of a unitary matrix $U$
$\displaystyle h_{\text{eff}}=U~{}\tilde{h}_{\text{eff}}~{}U^{\dagger}.$ (2)
It is then convenient to change from flavor basis using the unitary matrix $U$
into a new basis
$\displaystyle\left(\begin{array}[]{c}\nu_{e}\\\ \nu_{\mu}\\\ \nu_{e}^{c}\\\
\nu_{\mu}^{c}\end{array}\right)\to\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&-1&0\\\
0&1&0&-1\\\ 1&0&1&0\\\ 0&1&0&1\end{pmatrix}\left(\begin{array}[]{c}\nu_{e}\\\
\nu_{\mu}\\\ \nu_{e}^{c}\\\
\nu_{\mu}^{c}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\nu_{e}-\nu_{e}^{c}\\\
\nu_{\mu}-\nu_{\mu}^{c}\\\ \nu_{e}+\nu_{e}^{c}\\\
\nu_{\mu}+\nu_{\mu}^{c}\end{array}\right)=\left(\begin{array}[]{c}\nu_{e}^{-}\\\
\nu_{\mu}^{-}\\\ \nu_{e}^{+}\\\ \nu_{\mu}^{+}\end{array}\right),$ (19)
$\sin^{2}2\theta_{\text{eff}}$
$E[\text{eV}]$
Figure 1: Resonance structures between charge conjugation eigenstates. Shown
is the sine-squared of the effective mixing angles
$\theta_{\mathscr{C}\text{-odd}}$ (blue curve) and
$\theta_{\mathscr{C}\text{-even}}$ (red curve). We choose $b_{e}=1\times
10^{-21}$, $b_{\mu}=0.6\times 10^{-21}$, $c_{\mu\mu}=3\times 10^{-21}$,
$c_{ee}=2\times 10^{-21}$ for illustrative purposes. We take $\Delta
m^{2}=8\times 10^{-5}\text{eV}^{2}$ as well as $\sin^{2}2\theta=0.86$. In this
case the resonance energy for the $\mathscr{C}\text{-odd}$ mixing is higher as
compared to the resonance energy of the $\mathscr{C}\text{-even}$ mixing.
with the states $\nu^{-}$ and $\nu^{+}$ of the new basis being eigenstates of
the charge conjugation operator $\mathscr{C}$. In the new basis the
Hamiltonian is block diagonal, and the $\mathscr{C}$-even and the
$\mathscr{C}$-odd sectors can be diagonalized further separately.
Using the usual diagonalization procedure for each sector, the effective
mixing angles $\theta_{\mathscr{C}\text{-odd}}$ and
$\theta_{\mathscr{C}\text{-even}}$ can be calculated independently [12].
Depending on the values of the Lorentz- and CPT- violating parameters
resonances in one or in both sectors occur. The resonance energies
$E^{\mathscr{C}\text{-odd}}_{\text{res}}$ and
$E^{\mathscr{C}\text{-even}}_{\text{res}}$ can also be calculated. In general
the two resonance energies are different. As can be seen from Fig. 1, for that
particular choice of parameters both resonances exist, and the resonance
energies are different. The graph also shows that in the low energy regime the
effective mixing angles go to the standard mixing, which is the same for
neutrinos and antineutrinos. The effective mixing angles go through maximal
mixing at energies equal to the resonance energy, then as the energy increases
beyond the resonance energy, the mixing goes to zero. In the limit in which
the Lorentz- and CPT-violating parameters vanish the effective mixing angles
equal the standard mixing angle, and the resonances disappear.
The effective mixing angles and resonance energies are calculated in the
charge conjugation operator eigenbasis. The translation between the mass basis
and the flavor basis is done via a matrix $V$ which is the product of the
unitary matrix $U$ and the matrix which diagonalizes the block diagonal
Hamiltonian defined with the charge conjugation eigenbasis
$\displaystyle\left(\begin{array}[]{c}\nu_{e}\\\ \nu_{\mu}\\\ \nu_{e}^{c}\\\
\nu_{\mu}^{c}\end{array}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos\theta_{\mathscr{C}\text{-odd}}&\sin\theta_{\mathscr{C}\text{-odd}}&\cos\theta_{\mathscr{C}\text{-even}}&\sin\theta_{\mathscr{C}\text{-even}}\\\
-\sin\theta_{\mathscr{C}\text{-odd}}&\cos\theta_{\mathscr{C}\text{-odd}}&-\sin\theta_{\mathscr{C}\text{-even}}&\cos\theta_{\mathscr{C}\text{-even}}\\\
-\cos\theta_{\mathscr{C}\text{-odd}}&-\sin\theta_{\mathscr{C}\text{-odd}}&\cos\theta_{\mathscr{C}\text{-even}}&\sin\theta_{\mathscr{C}\text{-even}}\\\
\sin\theta_{\mathscr{C}\text{-odd}}&-\cos\theta_{\mathscr{C}\text{-odd}}&-\sin\theta_{\mathscr{C}\text{-even}}&\cos\theta_{\mathscr{C}\text{-even}}\end{pmatrix}\left(\begin{array}[]{c}\nu_{1}\\\
\nu_{2}\\\ \nu_{3}\\\ \nu_{4}\end{array}\right).$ (28)
The probability of oscillation is
$\displaystyle P(\beta\to\alpha)=\left|\sum_{i}V_{\beta
i}e^{-iE_{i}t}\left(V^{\dagger}\right)_{i\alpha}\right|^{2},$ (29)
where $E_{i}$ are the effective energy eigenvalues of the associated
Hamiltonian $h_{\text{eff}}$; and $\alpha$ and $\beta$ stand for the four
different neutrino species involved, i.e.
$\alpha,~{}\beta=\nu_{e},~{}\nu_{\mu},~{}\nu_{e}^{c},~{}\nu_{\mu}^{c}$. It is
important to emphasize that the CPT-violating parameters make oscillations
between neutrinos and antineutrinos from the same generation or of a different
generation possible.
We find that even a simple choice of non-zero CPT-violating coefficients
provides a workable model in which neutrino-antineutrino oscillations become
possible. The model for neutrino-antineutrino oscillations under consideration
in a CPT-violating framework gives rise to new vacuum resonances [13] which
are suitably described in terms of $\mathscr{C}$-even and $\mathscr{C}$-odd
states. Resonant mixing as defined occurs between $\mathscr{C}$-flavor
eigenstates rather than between common flavor eigenstates. Depending on the
parameter space of the CPT-violating coefficients it is possible to have none,
one (for the $\mathscr{C}$-even or $\mathscr{C}$-odd states) or two resonances
(one resonance in each sector not necessarily at the same energy). These
resonances are related to the mixing of $\mathscr{C}$-flavor eigenstates.
Another point to be made is that at least one of the neutrino-antineutrino
resonances reveals a narrower resonance width as compared to neutrino-neutrino
oscillations with altered dispersion relations in the CPT conserving case. CPT
violation distinguishes particles and antiparticles such that resonance peaks
for neutrinos and antineutrinos are not necessarily identical. Such a behavior
might be suggested by a recent analysis of the experimental neutrino data [14]
along with the hint that the signal observed at MiniBooNE looks more like a
$\nu_{\mu}\to\bar{\nu}_{e}$ conversion than $\nu_{\mu}\to\nu_{e}$ events.
Without going into further details we mention that depending on the choice of
parameters, the model predicts interesting daily and seasonal variations of
neutrino oscillation observables, which result from the Earth’s motion with
respect to a preferred frame implied by a Lorentz-violating background field.
A detailed analysis of both the direction dependence of CPT-violating
coefficients as well as the flavor oscillation probability will possibly shed
light on neutrino oscillation anomalies such as LSND and MiniBooNE.
## References
* [1] A. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], arXiv:0812.2243 [hep-ex].
* [2] A. A. Aguilar-Arevalo et al., arXiv:0904.1958 [hep-ex].
* [3] A. Aguilar et al. [LSND Collaboration], Phys. Rev. D 64, 112007 (2001) [arXiv:hep-ex/0104049].
* [4] see, e.g., H. Päs, S. Pakvasa and T. J. Weiler, Phys. Rev. D 72, 095017 (2005) [arXiv:hep-ph/0504096];
* [5] S. Hollenberg, O. Micu, H. Päs and T. J. Weiler, arXiv:0906.0150 [hep-ph].
* [6] S. Hollenberg and H. Päs, arXiv:0904.2167 [hep-ph].
* [7] V. A. Kostelecky and M. Mewes, Phys. Rev. D 70, 031902 (2004) [arXiv:hep-ph/0308300].
* [8] V. Barger, D. Marfatia and K. Whisnant, Phys. Lett. B 653, 267 (2007) [arXiv:0706.1085 [hep-ph]].
* [9] V. A. Kostelecky and R. Lehnert, Phys. Rev. D 63, 065008 (2001) [arXiv:hep-th/0012060].
* [10] D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998) [arXiv:hep-ph/9809521].
* [11] V. A. Kostelecky and M. Mewes, Phys. Rev. D 69, 016005 (2004) [arXiv:hep-ph/0309025].
* [12] S. Hollenberg, O. Micu and H. Päs, Phys. Rev. D 80 (2009) 053010 [arXiv:0906.5072 [hep-ph]].
* [13] resonant behaviour in CPT-violating neutrino oscillations was noted before, V. D. Barger, S. Pakvasa, T. J. Weiler and K. Whisnant, Phys. Rev. Lett. 85, 5055 (2000) [arXiv:hep-ph/0005197].
* [14] G. Karagiorgi, Z. Djurcic, J. M. Conrad, M. H. Shaevitz and M. Sorel, arXiv:0906.1997 [hep-ph].
|
arxiv-papers
| 2009-11-05T12:50:00 |
2024-09-04T02:49:06.293855
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sebastian Hollenberg, Octavian Micu, Heinrich P\\\"as",
"submitter": "Octavian Micu",
"url": "https://arxiv.org/abs/0911.1018"
}
|
0911.1056
|
# Detection of distant AGN by MAGIC: the transparency of the Universe to high-
energy photons
Marco Roncadelli1, Alessandro De Angelis2, Oriana Mansutti3
1INFN Sezione di Pavia via A. Bassi 6 I – 27100 Pavia Italy
2Dipartimento di Fisica Università di Udine Via delle Scienze 208 I – 33100
Udine and INAF and INFN Sezioni di Trieste Italy
3Dipartimento di Fisica Università di Udine Via delle Scienze 208 I – 33100
Udine and INFN Sezione di Trieste Italy
###### Abstract
The recent detection of blazar 3C279 by MAGIC has confirmed previous
indications by H.E.S.S. that the Universe is more transparent to very-high-
energy gamma rays than previously thought. We show that this fact can be
reconciled with standard blazar emission models provided photon oscillations
into a very light Axion-Like Particle occur in extragalactic magnetic fields.
A quantitative estimate of this effect explains the observed spectrum of
3C279. Our prediction can be tested in the near future by the satellite-borne
GLAST detector as well as by the ground-based Imaging Atmospheric Cherenkov
Telescopes H.E.S.S., MAGIC, CANGAROO III, VERITAS and by the Extensive Air
Shower arrays ARGO-YBJ and MILAGRO.
## 1 Introduction
As is well known, in the very-high-energy (VHE) band above $100\,{\rm GeV}$
the horizon of the observable Universe rapidly shrinks as the energy further
increases. This comes about because photons from distant sources scatter off
background photons permeating the Universe, thereby disappearing into
electron-positron pairs [1]. The corresponding cross section
$\sigma(\gamma\gamma\to e^{+}e^{-})$ peaks where the VHE photon energy $E$ and
the background photon energy $\epsilon$ are related by
$\epsilon\simeq(500\,{\rm GeV}/E)\,{\rm eV}$. Therefore, for observations
performed by Imaging Atmospheric Cherenkov Telescopes (IACTs) – which probe
the energy interval $100\,{\rm GeV}-100\,{\rm TeV}$ – the resulting cosmic
opacity is dominated by the interaction with ultraviolet/optical/infrared
diffuse background photons (frequency band $1.2\cdot 10^{3}\,{\rm
GHz}-1.2\cdot 10^{6}\,{\rm GHz}$, corresponding to the wavelength range
$0.25\,\mu{\rm m}-250\,\mu{\rm m}$), usually called Extragalactic Background
Light (EBL), which is produced by galaxies during the whole history of the
Universe. Neglecting evolutionary effects for simplicity, photon propagation
is controlled by the photon mean free path ${\lambda}_{\gamma}(E)$ for
$\gamma\gamma\to e^{+}e^{-}$, and so the observed photon spectrum $\Phi_{\rm
obs}(E,D)$ is related to the emitted one $\Phi_{\rm em}(E)$ by
$\Phi_{\rm obs}(E,D)=e^{-D/{\lambda}_{\gamma}(E)}\ \Phi_{\rm em}(E)~{}.$ (1)
Within the energy range in question, ${\lambda}_{\gamma}(E)$ decreases like a
power law from the Hubble radius $4.2\,{\rm Gpc}$ around $100\,{\rm GeV}$ to
$1\,{\rm Mpc}$ around $100\,{\rm TeV}$ [2]. Thus, Eq. (1) entails that the
observed flux is exponentially suppressed both at high energy and at large
distances, so that sufficiently far-away sources become hardly visible in the
VHE range and their observed spectrum should anyway be much steeper than the
emitted one.
Yet, observations have not detected the behaviour predicted by Eq. (1). A
first indication in this direction was reported by the H.E.S.S. collaboration
in connection with the discovery of the two blazars H2356-309 ($z=0.165$) and
1ES1101-232 ($z=0.186$) at $E\sim 1\,{\rm TeV}$ [3]. Stronger evidence comes
from the observation of blazar 3C279 ($z=0.536$) at $E\sim 0.5\,{\rm TeV}$ by
the MAGIC collaboration [4]. In particular, the signal from 3C279 collected by
MAGIC in the region $E<220$ GeV has more or less the same statistical
significance as the one in the range 220 GeV $<E<$ 600 GeV ($6.1\sigma$ in the
former case, $5.1\sigma$ in the latter).
A suggested way out of this difficulty relies upon the modification of the
standard Synchro-Self-Compton (SSC) emission mechanism. One option invokes
strong relativistic shocks [5]. Another rests upon photon absorption inside
the blazar [6]. While successful at substantially hardening the emission
spectrum, these attempts fail to explain why only for the most distant blazars
does such a drastic departure from the SSC emission spectrum show up.
Our proposal – usually referred to as the DARMA scenario – is quite different
[7]. Implicit in previous considerations is the hypothesis that photons
propagate in the standard way throughout cosmological distances. We suppose
instead that photons can oscillate into a new very light spin-zero particle –
named Axion-Like Parlicle (ALP) – and vice-versa in the presence of cosmic
magnetic fields, whose existence has definitely been proved by AUGER
observations [8]. Once ALPs are produced close enough to the source, they
travel unimpeded throughout the Universe and can convert back to photons
before reaching the Earth. Since ALPs do not undergo EBL absorption, the
effective photon mean free path ${\lambda}_{\gamma,{\rm eff}}(E)$ gets
increased so that the observed photons cross a distance in excess of
${\lambda}_{\gamma}(E)$. Correspondingly, Eq. (1) becomes
$\Phi_{\rm obs}(E,D)=e^{-D/{\lambda}_{\gamma,{\rm eff}}(E)}\ \Phi_{\rm
em}(E)~{},$ (2)
from which we see that even a slight increase of ${\lambda}_{\gamma,{\rm
eff}}(E)$ gives rise to a huge enhancement of the observed flux. It turns out
that the DARMA mechanism makes ${\lambda}_{\gamma,{\rm eff}}(E)$ shallower
than ${\lambda}_{\gamma}(E)$ although it remains a decreasing function of $E$.
So, the resulting observed spectrum is much harder than the one predicted by
Eq. (1), thereby ensuring agreement with observations even for a standard SSC
emission spectrum. As a bonus, we get a natural explanation for the fact that
only the most distant blazars would demand $\Phi_{\rm em}(E)$ to substantially
depart from the emission spectrum predicted by the SSC mechanism.
Our aim is to review the main features of our proposal as well as its
application to blazar 3C279.
## 2 DARMA scenario
Phenomenological as well as conceptual arguments lead to view the Standard
Model of particle physics as the low-energy manifestation of some more
fundamental and richer theory of all elementary-particle interactions
including gravity. Therefore, the lagrangian of the Standard Model is expected
to be modified by small terms describing interactions among known and new
particles. Many extensions of the Standard Model which have attracted
considerable interest over the last few years indeed predict the existence of
ALPs. They are spin-zero light bosons defined by the low-energy effective
lagrangian
${\cal L}_{\rm ALP}\ =\
\frac{1}{2}\,\partial^{\mu}\,a\,\partial_{\mu}\,a-\frac{1}{2}\,m^{2}\,a^{2}-\frac{1}{4M}\,F^{\mu\nu}\,\tilde{F}_{\mu\nu}\,a~{},$
(3)
where $F^{\mu\nu}$ is the electromagnetic field strength, $\tilde{F}_{\mu\nu}$
is its dual, $a$ denotes the ALP field whereas $m$ stands for the ALP mass.
According to the above view, it is assumed $M\gg G_{F}^{-1/2}\simeq 250\,{\rm
GeV}$. On the other hand, it is supposed that $m\ll G_{F}^{-1/2}\simeq
250\,{\rm GeV}$. The standard Axion [9] is the most well known example of ALP.
As far as generic ALPs are concerned, the parameters $M$ and $m$ are to be
regarded as independent.
So, what really characterizes ALPs is the trilinear $\gamma$-$\gamma$-$a$
vertex described by the last term in ${\cal L}_{\rm ALP}$, whereby one ALP
couples to two photons. Owing to this vertex, ALPs can be emitted by
astronomical objects of various kinds, and the present situation can be
summarized as follows. The negative result of the CAST experiment designed to
detect ALPs emitted by the Sun yields the bound $M>0.86\cdot 10^{10}\,{\rm
GeV}$ for $m<0.02\,{\rm eV}$ [10]. Moreover, theoretical considerations
concerning star cooling via ALP emission provide the generic bound
$M>10^{10}\,{\rm GeV}$, which for $m<10^{-10}\,{\rm eV}$ gets replaced by the
stronger one $M>10^{11}\,{\rm GeV}$ even if with a large uncertainty [11]. The
same $\gamma$-$\gamma$-$a$ vertex produces an off-diagonal element in the mass
matrix for the photon-ALP system in the presence of an external magnetic field
${\bf B}$. Therefore, the interaction eigenstates differ from the propagation
eigenstates and photon-ALP oscillations show up [12].
We imagine that a sizeable fraction of photons emitted by a blazar soon
convert into ALPs. They propagate unaffected by the EBL and we suppose that
before reaching the Earth a substantial fraction of ALPs is back converted
into photons. We further assume that this photon-ALP oscillation process is
triggered by cosmic magnetic fields (CMFs), whose existence has been
demonstrated very recently by AUGER observations [8]. Owing to the notorious
lack of information about their morphology, one usually supposes that CMFs
have a domain-like structure [13]. That is, ${\bf B}$ ought to be constant
over a domain of size $L_{\rm dom}$ equal to its coherence length, with ${\bf
B}$ randomly changing its direction from one domain to another but keeping
approximately the same strength. As explained elsewhere [14], it looks
plausible to assume the coherence length in the range $1-10\,{\rm Mpc}$.
Correspondingly, the inferred strength lies in the range $0.3-1.0\,{\rm nG}$
[14].
## 3 Predicted energy spectrum
Our ultimate goal consists in the evaluation of the probability
$P_{\gamma\to\gamma}(E,D)$ that a photon remains a photon after propagation
from the source to us when allowance is made for photon-ALP oscillations as
well as for photon absorption from the EBL. As a consequence, Eq. (2) gets
replaced by
$\Phi_{\rm obs}(E,D)=P_{\gamma\to\gamma}(E,D)\,\Phi_{\rm em}(E)~{}.$ (4)
We proceed as follows. We first solve exactly the beam propagation equation
arising from ${\cal L}_{\rm ALP}$ over a single domain, assuming that the EBL
is described by the “best-fit model” of Kneiske et al. [15]. Starting with an
unpolarized photon beam, we next propagate it by iterating the single-domain
solution as many times as the number of domains crossed by the beam, taking
each time a random value for the angle between ${\bf B}$ and a fixed overall
fiducial direction. We repeat such a procedure $10^{.}000$ times and finally
we average over all these realizations of the propagation process.
We find that about 13% of the photons arrive to the Earth for $E=500\,{\rm
GeV}$, representing an enhancement by a factor of about 20 with respect to the
expected flux without DARMA mechanism (the comparison is made with the above
“best-fit model”). The same calculation gives a fraction of 76% for
$E=100\,{\rm GeV}$ (to be compared to 67% without DARMA mechanism) and a
fraction of 3.4% for $E=1\,{\rm TeV}$ (to be compared to 0.0045% without DARMA
mechanism). The resulting spectrum is exhibited in Fig. 1. The solid line
represents the prediction of the DARMA scenario for $B\simeq 1\,{\rm nG}$ and
$L_{\rm dom}\simeq 1\,{\rm Mpc}$ and the gray band is the envelope of the
results obtained by independently varying ${\bf B}$ and $L_{\rm dom}$ within a
factor of 10 about such values. These conclusions hold for $m\ll
10^{-10}\,{\rm eV}$ and we have taken for definiteness $M\simeq 4\cdot
10^{11}\,{\rm GeV}$ but we have cheked that practically nothing changes for
$10^{11}\,{\rm GeV}<M<10^{13}\,{\rm GeV}$.
Our prediction can be tested in the near future by the satellite-borne GLAST
detector as well as by the ground-based IACTs H.E.S.S., MAGIC, CANGAROO III,
VERITAS and by the Extensive Air Shower arrays ARGO-YBJ and MILAGRO.
Figure 1: The two lowest lines give the fraction of photons surviving from
3C279 without the DARMA mechanism within the “best-fit model” of EBL (dashed
line) and for the minimum EBL density compatible with cosmology (dashed-dotted
line) [15]. The solid line represents the prediction of the DARMA mechanism as
explained in the text.
## References
* [1] G. G. Fazio and F. W. Stecker, Nature 226 135 (1970).
* [2] P. Coppi and F. Aharonian, Astrophys. J. 487 L9 (1997).
* [3] F. Aharonian et al. (H.E.S.S. Collaboration), Nature 440 1018 (2006).
* [4] J. Albert et al. (MAGIC Collaboration), Science 320 1752 (2008).
* [5] F. W. Stecker, M. G. Baring and E. J. Summerlin, Astrophys. J. 667 L29 (2007). F. W. Stecker and S. T. Scully, Astron. Astrophys 478 L1 (2008).
* [6] F. Aharonian, D. Khangulyan and L. Costamante, arXiv:0801.3198 (2008).
* [7] A. De Angelis, M. Roncadelli and O. Mansutti, Phys. Rev. D76 121301 (2007).
* [8] J. Abraham et al. [Pierre Auger Collaboration], Science 318 939 (2007).
* [9] J. H. Kim, Phys. Rep. 150 1 (1987). H. Y. Cheng, Phys. Rep. 158 1 (1988).
* [10] K. Zioutas et al., Phys. Rev. Lett. 94 121301 (2005). S. Andriamoje et al., JCAP 0704010 (2007).
* [11] G. G. Raffelt, Stars as Laboratories for Fundamental Physics (University of Chicago Press, Chicago, 1996).
* [12] P. Sikivie, Phys. Rev. Lett. 51 1415 (1983); (E) ibid. 52 695 (1984). L. Maiani, R. Petronzio and E. Zavattini, Phys. Lett. B175 359 (1986). G. G. Raffelt and L. Stodolsky, Phys. Rev. D37 1237 (1988).
* [13] P. P. Kronberg, Rept. Prog. Phys. 57 325 (1994). D. Grasso and H. Rubinstein, Phys. Rep. 348 163 (2001).
* [14] A. De Angelis, M. Persic and M. Roncadelli, Mod. Phys. Lett. A23 315 (2008).
* [15] T. M. Kneiske, T. Bretz, K. Mannheim and D. H. Hartmann, Astron. Astrophys. 413 807 (2004).
|
arxiv-papers
| 2009-11-05T15:23:57 |
2024-09-04T02:49:06.297855
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Roncadelli (1), Alessandro De Angelis (2), Oriana Mansutti (3)\n ((1) INFN, Sezione di Pavia, Italy, (2) Dipartimento di Fisica, Universita'\n di Udine, and INAF and INFN, Sezioni di Trieste, Italy, (3) Dipartimento di\n Fisica, Universita' di Udine, and INFN, Sezione di Trieste, Italy)",
"submitter": "Oriana Mansutti",
"url": "https://arxiv.org/abs/0911.1056"
}
|
0911.1076
|
# Evidence for an Intermediate Mass Black Hole in NGC 5408 X-1
Tod E. Strohmayer1 & Richard F. Mushotzky1
###### Abstract
We report the discovery with XMM-Newton of correlated spectral and timing
behavior in the ultraluminous X-ray source (ULX) NGC 5408 X-1. An $\approx
100$ ksec pointing with XMM/Newton obtained in January, 2008 reveals a strong
10 mHz QPO in the $>1$ keV flux, as well as flat-topped, band limited noise
breaking to a power law. The energy spectrum is again dominated by two
components, a $0.16$ keV thermal disk and a power-law with an index of
$\approx 2.5$. These new measurements, combined with results from our previous
January 2006 pointing in which we first detected QPOs, show for the first time
in a ULX a pattern of spectral and temporal correlations strongly analogous to
that seen in Galactic black hole sources, but at much higher X-ray luminosity
and longer characteristic time-scales. We find that the QPO frequency is
proportional to the inferred disk flux, while the QPO and broad-band noise
amplitude (root mean squared, rms) are inversely proportional to the disk
flux. Assuming that QPO frequency scales inversely with black hole mass at a
given power-law spectral index we derive mass estimates using the observed QPO
frequency - spectral index relations from five stellar-mass black hole systems
with dynamical mass constraints. The results from all sources are consistent
with a mass range for NGC 5408 X-1 from 1000 - 9000 $M_{\odot}$. We argue that
these are conservative limits, and a more likely range is from 2000 - 5000
$M_{\odot}$. Moreover, the recent relation from Gierlinski et al. that relates
black hole mass to the strength of variability at high frequencies (above the
break in the power spectrum) is also indicative of such a high mass for NGC
5408 X-1. Importantly, none of the above estimates appears consistent with a
black hole mass less than $\approx 1000$ $M_{\odot}$ for NGC 5408 X-1. We
argue that these new findings strongly support the conclusion that NGC 5408
X-1 harbors an intermediate mass black hole.
black hole physics - galaxies: individual: NGC 5408 - stars: oscillations -
X-rays: stars - X-rays: galaxies
11affiliationtext: Astrophysics Science Division, NASA’s Goddard Space Flight
Center, Greenbelt, MD 20771 email: tod.strohmayer, richard.mushotzky@nasa.gov
## 1 Introduction
The nature of the bright X-ray sources found in nearby galaxies, the
ultraluminous X-ray sources (ULXs), remains a major astrophysical puzzle. The
fundamental conundrum is that some of these objects have X-ray luminosities
uncomfortably high to be stellar-mass black holes (BH) without violating
standard Eddington limit arguments. Three different solutions have been
proposed for the luminosity problem. 1) The objects are intermediate-mass BHs
(Colbert & Mushotzky 1999). 2) They are stellar-mass BHs with, in some cases,
substantial beaming of their X-ray radiation (King et al. 2001), or, 3) they
are stellar-mass BHs emitting above their Eddington limit (Begelman 2006). It
is possible that some ULXs appear ultraluminous because of a combination of
all three factors (moderately higher mass, mild beaming and mild super-
Eddington emission). It may also be that these objects make up an
inhomogeneous population, comprised of both a sub-sample of intermediate-mass
BHs and moderately beamed stellar BHs (for recent reviews see Fabbiano & White
2006; Miller & Colbert 2004). Due to their extragalactic nature the study of
ULX counterparts at other wavebands has been difficult, and has so far
precluded the use of the familiar methods of dynamical astronomy to weigh
them. However, recent work has resulted in mass measurements for some Local
Group stellar BHs, including IC 10 X-1 (Prestwich et al. 2007; Silverman &
Filippenko 2008), and M33 X-7 (Orosz et al. 2007; Liu et al. 2008), but these
are not ULXs.
A substantial body of work now supports the idea that timing properties can be
used to constrain the masses of BHs. For example, McHardy et al. (2006) have
shown that when luminosity variations are taken into account, both stellar-
mass and supermassive BHs populate a “variability plane” linking their broad
band variability time-scales, luminosities and masses (see also Körding et al.
2007). Casella et al (2008) applied this relation in order to estimate the
masses of two ULXs, M82 X-1 and NGC 5408 X-1, concluding that the masses of
NGC 5408 X-1, and M82 X-1 lie in the range (in solar units) $115<M_{bh}<1300$,
and $95<M_{bh}<1260$, respectively. However, in this case the scaling was not
direct in that Casella et al. (2008) had to estimate the relevant break time-
scale from the observed QPO frequencies in M82 X-1 and NGC 5408 X-1. They did
this by applying a scaling relation between the two quantities derived only
from observations of stellar-mass BHs.
Recently, Gierlinski, Nikolajuk & Czerny (2008) have shown that the strength
of high frequency variability–parameterized as the root mean squared (rms)
amplitude integrated above the break in the power spectrum–scales
approximately linearly with BH mass. They show that a broad correlation exists
when comparing stellar and supermassive BHs, but the relation is not tight
enough to predict the mass of a stellar system by comparison, for example,
with another stellar-mass BH of known mass.
It is now firmly established that Galactic BHs accreting at high rates–often
classified as the Intermediate State (IS) or Steep Power-Law state (SPL)–show
strong correlations between their spectral and temporal parameters. In these
states a significant fraction of the X-ray luminosity is in a power-law
component whose spectral index correlates well with the frequency of a QPO
(so-called Type C QPOs) that is also commonly detected in such states (see
Sobczak et al. 2000; Vignarca et al. 2003; Kalemci et al. 2005; Remillard et
al. 2002; Casella et al. 2004). Other spectral parameters also correlate with
QPO frequency, such as the disk flux. Recent work by Shaposhnikov & Titarchuk
(2007; 2009; hereafter ST07 and ST09) has demonstrated rather convincingly
that the QPO frequency - spectral index scaling for Type C QPOs can be used as
an empirical BH mass estimator. Previous efforts have been made to use this
scaling argument to constrain the masses of ULXs (Fiorito & Titarchuk 2004;
Dewangan, Titarchuk & Griffiths 2006; Strohmayer et al. 2007, hereafter Paper
1), however, in these previous studies, there was no direct evidence to
indicate that the QPO properties seen in the ULX sources actually correlates
with the spectral parameters as in stellar-mass systems.
In this paper we present the results of new timing and spectral measurements
that show for the first time that NGC 5408 X-1 behaves very much like a
Galactic stellar-mass BH system with the exception that its characteristic
X-ray time-scales are $\sim 100$ times longer, and its luminosity is greater
by a roughly similar factor. We argue that these new findings provide strong
evidence that this system contains a few thousand solar mass BH.
## 2 New XMM-Newton Observations and Data Analysis
Paper 1 summarizes results from XMM-Newton observations obtained in 2006
January in which X-ray QPOs were detected from NGC 5408 X-1 for the first time
(we sometimes also refer to this as Observation 1). In order to look for
correlated timing and spectral behavior we sought and obtained additional XMM-
Newton observations in 2008 January. These new observations began on January
13, 2008, and continued for $\approx 116$ ksec (Observation 2). We used the
standard SAS version 8.0.0 tools to filter and extract images and event tables
for both the pn and MOS cameras. We extracted events in an 18” radius around
the source in both the pn and MOS cameras. Due to its higher count rate and
better time resolution we began our timing study using the pn data.
### 2.1 Power Spectral Timing Analysis
There was some background flaring present during our observation which broke
up the exposure into several useful intervals. We work with the four longest
intervals, labeled A through D in time order, and with exposures of 15.9,
23.0, 20.0 and 13.6 ksec, respectively. Figure 1 shows the pn lightcurve in 78
s bins for the longest interval (interval B).
We began our timing analysis by making average power spectra by combining all
good intervals. Based on the energy dependent variability behavior exhibited
in observation 1 (see Paper 1) we made power spectra in several energy bands;
0.2 - 8 keV, 1 - 8 keV, and 2 - 8 keV. We found a prominent peak at 10 mHz in
the $1-8$ keV power spectrum. Figure 2 shows the average power spectrum binned
at 0.64 mHz in which we found the peak. To assess the significance of the peak
we first rescaled the power spectrum. We did this by fitting a bending power-
law continuum to the spectrum, excluding the two highest bins in the peak to
avoid biasing the fit to higher values. We then divided the power spectrum by
the best-fitting continuum model. This procedure is important when searching
for QPOs against an intrinsic, broad-band noise component as is present in NGC
5408 X-1. We estimate the significance as the chance probability of obtaining
the highest observed peak in the rescaled power spectrum using for the noise
power distribution the $\chi^{2}$ distribution with 176 degrees of freedom
(dof). This is the expected distribution given the number of independent
frequency bins that were averaged (88). We compared the distribution of powers
in our observed power spectrum with this expected distribution and confirmed
they are consistent. This then gives a chance probability, per trial, of
$3.7\times 10^{-8}$. We searched out to 0.5 Hz and this was the 2nd power
spectrum searched, giving a total of $390\times 2=780$ independent trials,
resulting in a significance of $2.9\times 10^{-5}$, a strong detection.
To further quantify the variability we fit the same continuum model but now
including a Lorentzian to account for the QPO. This model fits well, with a
best fitting $\chi^{2}=463$ for 459 degrees of freedom, and is shown in Figure
2. Removing the QPO component from the fit results in an increase in
$\chi^{2}$ of 49. For the addition of three additional QPO parameters, this
increase in $\chi^{2}$ gives a chance probability by the F-test of $5\times
10^{-10}$, further supporting the detection.
The QPO has a centroid frequency of $10.24\pm 0.005$ mHz, and a coherence of
$\nu_{cent}/\Delta\nu_{fwhm}\approx 20$. The QPO is strong, with an average
amplitude (rms) in the $>1$ keV band of $17.4\pm 1.3\%$. We do not detect the
QPO at energies below 1 keV, with a $90\%$ upper limit on the amplitude (rms)
of $4.5\%$, demonstrating that the amplitude of the QPO is a strong function
of energy. The band-limited noise is also quite strong, with an integrated
amplitude (rms) of $41\pm 5\%$ (1 - 100 mHz). Similarly to the QPO the broad-
band noise amplitude is also stronger at higher energies. This behavior is
typical of Galactic BHs as well (Nowak et al. 1999; Belloni et al. 2005).
We next examined the intervals separately. For the longest interval (B) we
again detect the 10 mHz QPO, but we also find evidence for a second QPO at
13.4 mHz. Figure 3 (upper curve) shows the power spectrum from interval B in
the $1-8$ keV energy band. The 13.4 mHz feature is significant at a bit better
than the $3\sigma$ level, so we consider it robust. Interestingly, these two
peaks have frequencies consistent with a ratio of 4:3. A pair of QPOs with a
similar frequency ratio were also detected in observation 1 (see paper 1).
These similarities provide added confidence that the 13.4 mHz QPO is
significant. Combining the two longest intervals (B and C) above 1 keV gives a
power spectrum with a strong detection of the 10 mHz feature, as well as two
other candidate features (see Figure 3, lower curve). The higher frequency QPO
at 13.4 mHz is still present, though it is less prominent than in interval B
alone, and a third feature is suggested at 6 mHz. We modeled this power
spectrum with the same bending power-law continuum and included up to three
QPO components. We find a statistically acceptable fit with $\chi^{2}=518.8$
(497 degrees of freedom). The 10 mHz QPO is absolutely required, as removing
it from the fit increases $\chi^{2}$ by 96, better than an $8\sigma$ detection
based on the F-test. Removing the lowest and highest frequency QPO components
one at a time results in increases in $\chi^{2}$ of 17.7 and 29.2,
respectively. These correspond to F-test probabilities of $2.6\times 10^{-4}$
and $6.9\times 10^{-7}$, respectively, for the additional components. While
this provides rather strong evidence for the 13.4 mHz QPO, we regard the 6 mHz
feature as tentatively detected.
In summary, our new observations of NGC 5408 X-1 reveal a strong QPO at 10
mHz, and very strong “flat-topped” band-limited noise breaking to a power law,
with the break close to the QPO. This behavior is qualitatively similar to
results from our 2006 observations, but with the QPO and break frequencies
shifted down in frequency by about a factor of two, and with stronger rms
variability.
### 2.2 Energy Spectral Analysis
Previous spectral studies have shown that NGC 5408 X-1 has a cool thermal disk
component with $kT\approx 0.15$ keV and a power-law extending to higher
energies with a slope of $\approx 2.5$ (Kaaret et al. 2003; Soria et al. 2004;
Paper 1). We obtained a pn spectrum by extracting an 18” region around the
source. Background was obtained from a nearby circular region free of sources.
We began by fitting the present spectrum with the same model components used
to describe our 2006 data (paper 1); a relativistic disk (diskpn in XSPEC), a
power-law, and a thermal plasma (apec in XSPEC). These were modified by
successive photoelectric absorption components, one fixed at the best Galactic
value ($n_{H}=5.7\times 10^{20}$ cm-2), the other left free to account for
possible local absorption. This model provides an acceptable fit, and from a
qualitative standpoint the spectral parameters are similar to those derived
from our 2006 observations. The disk temperature of $0.16\pm 0.005$ keV is
consistent within the errors to that derived from the 2006 data. The power-law
index, at $2.47\pm 0.04$, is slightly smaller than the previous measurement,
but only different at about the 2$\sigma$ level. We searched for line emission
in the Fe band but did not detect any significant features. However, the
limits are not that constraining, with an $90\%$ confidence upper limit on the
equivalent width at 6.4 keV of $\approx 350$ eV.
The thermal plasma model parameters and flux are consistent with the 2006
data, contributing a 0.3 - 10 keV flux of $\approx 1\times 10^{-13}$ erg cm-2
s-1 in each observation. Examination of Chandra images of NGC 5408 indicate
the existence of extended emission co-spatial with NGC 5408 X-1. This emission
is unresolved in the XMM observations of NGC 5408 X-1, and we suggest that
this spectral component is likely associated with star formation in NGC 5408
and not intrinsic to the ULX. We will compare and constrast the spectral
results from the two epochs in more detail below.
## 3 Comparison of 2006 and 2008 Observations: Timing - Spectral Correlations
We can now compare the timing and spectral properties from the two epochs, and
determine whether or not NGC 5408 X-1 shows patterns of behavior similar to
that seen in Galactic BHs. We begin with a more detailed comparison of the
energy spectrum and fluxes during the two observations. For consistency we re-
extracted the spectrum from the 2006 observations to include essentially all
of the good exposure, and in each epoch the spectrum represents the total
accumulated over the entire observation. We produced count rate spectra for
each observation and Figure 4 shows the difference spectrum as a function of
energy (Observation 1 - Observation 2). One can see from the figure that the
first observation (2006) had a higher count rate, and that most of the
difference is in the $<1$ keV band, which is dominated by the disk flux. This
demonstrates that the 2006 observation had relatively more counts in the soft
band ($<1$ keV), and less in the power-law component extending to higher
energy. Table 2 provides a detailed comparison of the spectral parameters
derived from the two epochs. The unabsorbed flux (0.3 - 10 keV) in 2008 was
$3.1\times 10^{-12}$, about $12\%$ less than the 2006 observation, and the
power-law component represents a greater fraction of the total flux than in
the 2006 data. The two most important conclusions are that the disk flux was
about $20\%$ higher in Observation 1, and that the power-law component
contributes a greater fraction of the total flux in Observation 2.
We next compared power spectra accumulated during each observation. Figure 5
shows power spectra characteristic of each epoch. The upper curve is data from
our 2008 observations, and the lower curve is from 2006 (see also Paper 1,
Figure 3). These power spectra were not accumulated over the exact same energy
bands. Because the variability has a significant energy dependence, and
because a primary goal is to compare the characteristic time-scales in each
epoch, we choose to compare the spectra in the energy bands where the signal
to noise ratio of the respective QPOs is largest. For the 2006 data this
corresponds to 0.2 - 8 keV, whereas for the 2008 data we use, of the bands
searched, the 1 - 8 keV band. Table 1 compares relevant timing properties for
the two observations. Timing properties were derived from the model fits shown
in Figure 5 using the bending power-law continuum and Lorentzian components
for the QPOs. We note that the bend frequency in the $0.2-8$ keV 2006 data we
used to compare the 2006 and 2008 QPO parameters is not well constrained. A
more representative value is the bend frequency of $\approx 25$ mHz derived
from the $>2$ keV 2006 data (Paper 1). In general, the measurement of the QPO
frequency in both epochs is much more precise than the bend frequency, which
is why we emphasize it in our comparisons of the two epochs.
Closer examination of Figure 5 reveals several important conclusions
concerning the variability in NGC 5408 X-1. We see that the strongest QPO has
shifted from 20 to 10 mHz, over the same time that the disk flux dropped by
$20\%$. Additionally, the photon power-law index decreased modestly as the QPO
frequency dropped. These behaviors are entirely consistent with what has been
observed in Galactic BHs with so-called Type C QPOs (see, for example, Sobczak
et al. 2000; Vignarca et al. 2003; Kalemci et al. 2005).
Another strikingly evident feature in Figure 5 is the larger amplitude of the
variability (broad-band and QPO) in the 2008 observation (top). This quantity
is proportional to the integral of the power spectrum above the poisson level
(here a value of 2), which is clearly larger in the 2008 observations. The rms
amplitude also has a scaling with the square root of the count rate, but this
was smaller in the 2008 observations, so this effect simply makes the apparent
difference larger. We also compared the rms amplitudes in the same energy
band, and the conclusion is unchanged. Again, the higher rms variability at
the epoch of lower disk flux is strikingly similar to what is observed in
Galactic systems. Here the primary conclusion is that the variability is
mostly carried by the power-law component, and not the disk flux. So, when the
power-law becomes more dominant, then the rms amplitude increases. Finally, we
note that the modest drop in the power-law index with the decrease in QPO
frequency is also consistent with the correlations seen in Galactic systems.
In this case the relatively small change in the index accompanying a larger
change in QPO frequency suggests that the power-law index in NGC 5408 X-1 may
be near “saturation” of the scaling (see for example, Figure 5 in Vignarca et
al. 2003; ST09), that is, both the QPO frequency and index are near their
upper ranges in the correlation. This conclusion is also supported by the
behavior of the rms amplitude. The strong drop in the rms amplitude at
relatively constant power-law index is consistent with the QPO frequency being
at the higher end of the observed correlations (see, for example, the behavior
of XTE J1550-564 illustrated in Figure 5a from Sobczak et al. 2000).
## 4 Discussion and Implications
Our new observations of NGC 5408 X-1 reveal correlated variations in its
timing and spectral properties very much like a stellar-mass black hole.
Indeed, the evidence is now quite compelling that the strong low frequency QPO
detected on several occasions from NGC 5408 X-1 is a Type-C QPO analogous to
those seen in Galactic systems. It varies in frequency and amplitude with
changes in flux and spectrum in a manner entirely consistent with the behavior
in Galactic BHs such as XTE J1550-564 and GRS 1915+105. Whereas these stellar-
mass systems have characteristic QPO frequencies of a few Hz, NGC 5408 X-1
shows QPO frequencies lower by a factor $\sim 100$ while simultaneously
radiating an X-ray luminosity larger by a roughly similar factor.
Given greater confidence in the identification of and scaling properties of
the strong QPO seen in NGC 5408 X-1 we can use it to derive a mass estimate.
We essentially follow the scheme outlined by ST09. We use the QPO frequency -
power-law index correlations measured for five systems; GRS 1915+105 (Vignarca
et al. 2003), XTE J1550-564, XTE J1859+226, H 1743-322 and GX 339-4 (ST09). We
use these systems as our primary sources for mass estimates because they have
both measured power-law indices that overlap the range of observed indices in
NGC 5408 X-1, and reasonable BH mass estimates from dynamical measurements.
Since we do not yet have a correlation from NGC 5408 X-1 with many points to
scale from we carry out a simplified procedure. For our reference Galactic
systems we find the range of QPO frequencies with power-law indices between
2.4 and 2.6. We choose this index range as representative and conservative. It
is 2$\sigma$ below the lower best fit value (from 2008) and 2$\sigma$ above
the higher best-fit value (from 2006) for NGC 5408 X-1. In the case that a
range of QPO frequency exists at either end of our index range, we take the
largest range of observed frequencies. To be specific, we take the lowest
frequency measured at an index of 2.4, and the highest frequency found at an
index of 2.6. We then find the multiplicative scale factor, $f$, required to
bring the lower and upper QPO values measured for NGC 5408 X-1 (10 and 20 mHz)
into agreement with the measured frequency range for our reference systems. In
the case that a constant scale factor cannot match both the lower and higher
frequencies of the observed range, we find the scale factor that aligns the
centers of the two ranges (we call this estimate $f_{best}$). The derived mass
estimate for NGC 5408 X-1 is then the mass of the reference stellar system
times the scale factor. In one case, H 1743-322, frequency measurements exist
only near the low end of our target index range (2.4). In this case we derive
the scale factor by simply scaling to the lower QPO frequency measured in NGC
5408 X-1. The derived scale factors and mass estimates based on comparison
with these five reference systems are summarized in Table 3.
The quoted uncertainty on the best mass estimate in Table 3 reflects the
uncertainty in the masses of the reference systems (where available), that is,
assuming a “correct” scale factor. The QPO frequency and power-law index
measurements for NGC 5408 X-1 described here are rather precise, so that if
all the assumptions made with regard to the mass scaling arguments above are
accurate, then the amount of statistical error in the estimates is modest.
This leaves systematic uncertainties, that is, how accurate is the derived
scale factor? Now, some Galactic BHs do show variations in their QPO frequency
– spectral index correlations. These are primarily associated with apparent
changes in the value of the spectral index at which the correlation saturates
(see, for example, the behavior of XTE J1550-564 in Figure 7 from ST09), or
simply that the correlation is not exact, and that at a given index value,
there can be a range of measured QPO frequencies. We took into account such
variations in defining the range of QPO frequencies to scale to for each
reference system. However, the existence of a range of measured frequencies at
a given spectral index suggests a conservative way to bound the scale factor
sytematic error, by defining a minimum and maximum scale factor given the
observed QPO frequency range. We do this by scaling the lowest observed QPO
frequency in our reference systems with the highest observed QPO frequency in
NGC 5408 X-1, and vice versa. For example, for XTE J1550-564, which has
observed QPO frequencies from 2.5 - 6.5 Hz, the minimum and maximum scale
factors would be (2.5 Hz) / (0.02 Hz) = 125, and (6.5 Hz) / (0.01 Hz) = 650.
Based on these limits on the scale factor we also derive minimum and maximum
mass estimates by multiplying the maximum and minimum scale factors by the
$\pm 1\sigma$ mass limits for each reference system. Both the scale factor
ranges and minimum and maximum mass estimates are also given in Table 3. We
emphasize that we think these represent rather conservative limits.
While the derived mass ranges in Table 3 are rather large–and we emphasize
that this is not a precision mass measurement technique–there is substantial
overlap among the estimates from all the different sources. Perhaps more
interesting is that all the scaling estimates suggest a BH with a mass
comfortably greater than $1000M_{\odot}$, that is, much greater than the
current known mass range for stellar BHs. We note that the candidate BH system
4U 1630-47–which is not a dynamically confirmed BH–also has QPO frequency
measurements that overlap our target range of spectral index (see Figure 9 in
Vignarca et al. 2003). In this source the frequency range is $\approx 4-8$ Hz,
suggesting a best scale factor of 400, which falls in the range derived for
our other reference systems. While it’s mass is not known, a typical value of
$\sim 10$ $M_{\odot}$ would also suggest a mass for NGC 5408 X-1 in the range
of several thousand $M_{\odot}$, consistent with the other sources.
### 4.1 Other Concerns and Caveats
One concern with regard to QPO frequency scaling arguments has been which QPO
frequency to scale to. For example, both XTE J1550-564 and GRS 1915+105 can
sometimes show QPO frequencies below 1 Hz, approaching 0.1 Hz in the case of
XTE J1550-564. However, our observations of QPOs from NGC 5408 X-1 at
different epochs, and with different frequencies and rms amplitudes goes along
way towards alleviating this concern. In fact, the behavior of the QPO rms
amplitude in NGC 5408 X-1 is completely reversed to what one would expect for
a scaling with the lowest QPO frequencies observed in XTE J1550-564 and GRS
1915+105. Results from XTE J1550-564 show that at low QPO frequency, the rms
increases with an increase in QPO frequency (see Figure 5a in Sobczak et al.
2000), but this is exactly opposite to the behavior seen in NGC 5408 X-1. The
behavior of the QPOs in both XTE J1550-564 and GRS 1915+105 when the power-law
index and frequency are both high is a much better match to the behavior seen
in NGC 5408 X-1.
Another concern has been that while QPO frequency scalings may give one
result, a simple scaling of luminosities gives another, and that the two may
not be in agreement. Here one must try to be consistent and compare
luminosities under the same conditions. In the case of our scaling arguments
this would be to match luminosities at the appropriate power-law index and
scaled QPO frequency. An important issue here is that in many cases the
distances to BHs in the Galaxy are relatively less well known than the
distances to nearby galaxies hosting ULXs. Nevertheless, we can attempt to
compare the luminosity of NGC 5408 X-1 to some of our reference BHs. For XTE
1550-564, ST09 report X-ray spectral and QPO frequency measurements. For
power-law index and QPO frequency appropriate to our scaling the observed
X-ray flux from XTE J1550-564 (2 - 20 keV) was in the range 3 - 5 $\times
10^{-8}$ erg cm-2 s-1. The distance to this source is not very well
constrained, with estimates ranging from 2.5 - 6 kpc. Taking a flux of
$4\times 10^{-8}$ erg cm-2 s-1 as representative, we have a luminosity of
$5\times 10^{37}(d/{\rm 3kpc})^{2}$. This compares with a representative
luminosity from our 2006 observation of about $1.1\times 10^{40}$ erg cm-2 s-1
(0.2 \- 10 keV; assuming a distance of 4.8 Mpc). The luminosity ratio is then
in the range of 317 - 55, for the distance range of 2.5 - 6 kpc for XTE
J1550-564. Simply scaling up the $\pm 1\sigma$ mass range for XTE J1550-564 by
these limits gives a mass estimate for NGC 5408 X-1 of $462-3360$ $M_{\odot}$.
Interestingly, ST09 use an observed correlation in the power-law index and
bulk motion comptonization model normalization (model bmc in XSPEC) to
estimate distances as well as BH masses by scaling to a reference system.
Interestingly, they favor a distance closer to 3 kpc than 6 kpc for XTE
J1550-564, which would favor the high end of the derived mass range. For GRS
1915+105, again the distance is rather uncertain, likely being in the range
from $6-12$ kpc (Dhawan et al. 2007). For the QPO frequency range above a
representative flux range is about 2 - 5 $\times 10^{-8}$ erg cm-2 s-1 (Muno
et al. 1999), giving a range of luminosity ratio of $73-18.2$. Similar scaling
as for XTE J1550-564 would imply a mass range from $182-1314$ $M_{\odot}$.
While this range appears systematically smaller than the mass range inferred
from the QPO scaling, it still overlaps with the minimum mass estimate for GRS
1915+105.
### 4.2 Amplitude of High Frequency Variability
Recently, Gierlinski et al. (2008) have argued that the amplitude of X-ray
variability at high frequencies can be used as an estimator of BH mass. Their
hypothesis is that there is a “universal” power spectral shape for BHs at high
frequency. Here, high frequency means above the break in the power spectrum.
This universal form is roughly a power law with index of 1.5 - 2.0. They
explore the notion that this part of the power spectrum is nearly constant for
a given source, but scales with BH mass. They find that there is a roughly
linear correlation extending from the stellar-mass BHs to AGN (using type 1
Seyferts). The relation is not exact, and the scatter is such to make it
difficult to estimate the mass of a stellar BH by scaling from another
stellar-mass system, however, for order of magnitude estimates the method
seems reasonably robust. We used our power spectrum continuum models (above
the bend or break in the spectrum) to estimate the parameter $C_{M}$, which is
simply the normalization of the high frequency power-law component at 1 Hz. We
integrated our best fitting continuum models from the bend frequency to 0.5 Hz
in order to estimate $C_{M}$ (see Eqn. 2 in Gierlinski et al. 2008). We find
$\pm 1\sigma$ ranges of $-3.23<\log C_{M}<-3.13$, and $-3.66<\log C_{M}<-3.55$
for the 2008 and 2006 observations, respectively. These values correspond to
mass ranges of 1686 - 2612, and 4435 - 5714 $M_{\odot}$, respectively, for the
best fitting correlation derived by Gierlinski et al., and 3737 - 4704, and
9828 - 12661 $M_{\odot}$ using their soft-state relation derived from Cyg X-1.
While the possible mass range from this method is large, the $C_{M}$ values
for NGC 5408 X-1 fall almost midway between the stellar-mass systems and NGC
4395, a low mass AGN (see Figure 7 in Gierlinski et al. 2008), and comfortably
within the mass range consistent with intermediate mass BHs.
## 5 Summary and Conclusions
Our new XMM-Newton observations have revealed for the first time that NGC 5408
X-1 exhibits correlated X-ray timing and spectral properties quite analogous
to those exhibited by Galactic stellar-mass BHs in the “very high” or “steep
power-law” state. Its longer observed time-scales (QPO and power spectral
break frequencies) at higher luminosity compared to Galactic stellar-mass BHs
can be understood if the mass of NGC 5408 X-1 is a few thousand solar masses.
We have arrived at this conclusion by several independent arguments. Given our
present understanding of the timing properties of BHs at all mass scales, it
seems to us hard to escape the conclusion that NGC 5408 X-1 is an intermediate
mass BH.
We have again found evidence for a pair of sharp, closely spaced QPOs in NGC
5408 X-1 with a frequency ratio consistent with 4:3. We thus conclude that
this is an intrinsic feature of the system and not some artifact or
coincidence associated with having only a single observation of the source. As
we noted in Paper 1, it is possible that this results from detection of both a
Type C QPO (the stronger feature), and a Type B QPO (Casella et al. 2004). If
the source transitions from one to another, we do not have sufficient signal
to noise ratio data to “watch” such transitions occur. Rather, we simply see
average detections of each QPO.
We have argued that the observed variations in QPO frequency, accompanied by
spectral changes consistent with behavior seen in Galactic systems, allows the
mass of NGC 5408 X-1 to be estimated by scaling the observed QPO frequencies
to match those observed in stellar-mass systems of known mass. Essentially the
same method has now been used on a good number of Galactic systems and seems
empirically robust (ST09). These arguments give mass estimates for NGC 5408
X-1 in a broad range from $1-9\times 10^{3}$ $M_{\odot}$, however, we find
that none of the QPO scaling estimates are consistent with masses below
$\approx 1000$ $M_{\odot}$. Independent mass constraints based on the
amplitude of high frequency variability also appear consistent with this
range. Additional observations would help to map out the timing - spectral
correlations for NGC 5408 X-1 more clearly, and thus allow more rigorous
estimates. As noted in Paper 1, the measured disk temperature in NGC 5408 X-1
is also consistent with a few thousand $M_{\odot}$ BH if the theoretical
accretion disk scaling of $kT_{disk}\propto M^{-1/4}$ holds. Importantly, none
of these new estimates appears consistent with a mass as low as even a few
hundred solar masses. We think these new findings provide strong evidence that
NGC 5408 X-1 harbors an intermediate mass BH.
We thank the anonymous referee for a careful review of the manuscript that
helped us to improve the paper.
## References
* Begelman (2006) Begelman, M. C. 2006, ApJ, 643, 1065.
* Belloni et al. (2005) Belloni, T., Homan, J., Casella, P., van der Klis, M., Nespoli, E., Lewin, W. H. G., Miller, J. M., & Méndez, M. 2005, A&A, 440, 207.
* Casella et al. (2008) Casella, P., Ponti, G., Patruno, A., Belloni, T., Miniutti, G., & Zampieri, L. 2008, MNRAS, 387, 1707.
* Casella et al. (2004) Casella, P., Belloni, T., Homan, J., & Stella, L. 2004, A&A, 426, 587.
* Colbert & Mushotzky (1999) Colbert, E. J. M. & Mushotzky, R. F. 1999, ApJ, 519, 89.
* Dewangan et al. (2006) Dewangan, G. C., Titarchuk, L., & Griffiths, R. E. 2006, ApJ, 637, L21.
* Dhawan et al. (2007) Dhawan, V., Mirabel, I. F., Ribó, M., & Rodrigues, I. 2007, ApJ, 668, 430.
* Fabbiano & White (2006) Fabbiano, G. & White, N. E. 2006, in “Compact Stellar X-ray Sources,” ed. W. H. G. Lewin, & M. van der Klis, (Cambridge University Press: Cambridge), pg. 475.
* Filippenko & Chornock (2001) Filippenko, A. V., & Chornock, R. 2001, IAU Circ., 7644, 2.
* Fiorito & Titarchuk (2004) Fiorito, R., & Titarchuk, L. 2004, ApJ, 614, L113.
* Gierliński et al. (2008) Gierliński, M., Nikołajuk, M., & Czerny, B. 2008, MNRAS, 383, 741.
* Greiner et al. (2001) Greiner, J., Cuby, J. G., & McCaughrean, M. J. 2001, Nature, 414, 522.
* Hynes et al. (2004) R. I. Hynes, D. Steeghs, J. Casares, P. A. Charles, and K. O’Brien 2004 ApJ 609, 317.
* Kaaret et al. (2003) Kaaret, P., Corbel, S., Prestwich, A. H., & Zezas, A. 2003, Science, 299, 365.
* Kalemci et al. (2005) Kalemci, E., Tomsick, J. A., Buxton, M. M., Rothschild, R. E., Pottschmidt, K., Corbel, S., Brocksopp, C., & Kaaret, P. 2005, ApJ, 622, 508.
* King et al. (2001) King, A. R., Davies, M. B., Ward, M. J., Fabbiano, G. & Elvis, M. 2001, ApJ, 552, L109.
* Körding et al. (2007) Körding, E. G., Migliari, S., Fender, R., Belloni, T., Knigge, C., & McHardy, I. 2007, MNRAS, 380, 301.
* Liu et al. (2008) Liu, J., McClintock, J. E., Narayan, R., Davis, S. W., & Orosz, J. A. 2008, ApJ, 679, L37.
* McClintock et al. (2009) McClintock, J. E., Remillard, R. A., Rupen, M. P., Torres, M. A. P., Steeghs, D., Levine, A. M., & Orosz, J. A. 2009, ApJ, 698, 1398.
* McHardy et al. (2006) McHardy, I. M., Koerding, E., Knigge, C., Uttley, P., & Fender, R. P. 2006, Nature, 444, 730.
* Miller & Colbert (2004) Miller, M. C., & Colbert, E. J. M. 2004, International Journal of Modern Physics D, 13, 1.
* Muno et al. (1999) Muno, M. P., Morgan, E. H. & Remillard, R. A. 1999, ApJ, 527, 321.
* Muñoz-Darias et al. (2008) Muñoz-Darias, T., Casares, J., & Martínez-Pais, I. G. 2008, MNRAS, 385, 2205.
* Nowak et al. (1999) Nowak, M. A., Vaughan, B. A., Wilms, J., Dove, J. B., & Begelman, M. C. 1999, ApJ, 510, 874.
* Orosz et al. (2007) Orosz, J. A., et al. 2007, Nature, 449, 872.
* Orosz et al. (2002) Orosz, J. A., et al. 2002, ApJ, 568, 845.
* Prestwich et al. (2007) Prestwich, A. H., et al. 2007, ApJ, 669, L21.
* Remillard et al. (2002) Remillard, R, A., Muno, M. P., McClintock, J. E. & Orosz, J. A. 2002, ApJ, 580, 1030.
* Shaposhnikov & Titarchuk (2009) Shaposhnikov, N., & Titarchuk, L. 2009, ApJ, 699, 453 (ST09).
* Shaposhnikov & Titarchuk (2007) Shaposhnikov, N., & Titarchuk, L. 2007, ApJ, 663, 445.
* Silverman & Filippenko (2008) Silverman, J. M., & Filippenko, A. V. 2008, ApJ, 678, L17.
* Sobczak et al. (2000) Sobczak, G. J., McClintock, J. E., Remillard, R. A., Cui, W., Levine, A. M., Morgan, E. H., Orosz, J. A., & Bailyn, C. D. 2000, ApJ, 531, 537.
* Soria et al. (2004) Soria, R., Motch, C., Read, A. M., & Stevens, I. R. 2004, A&A, 423, 955.
* Strohmayer et al. (2007) Strohmayer, T. E., Mushotzky, R. F., Winter, L., Soria, R., Uttley, P., & Cropper, M. 2007, ApJ, 660, 580.
* Uttley & McHardy (2005) Uttley, P., & McHardy, I. M. 2005, MNRAS, 363, 586.
* Vignarca et al. (2003) Vignarca, F., Migliari, S., Belloni, T., Psaltis, D., & van der Klis, M. 2003, A&A, 397, 729.
* Zurita et al. (2002) Zurita, C., et al. 2002, MNRAS, 334, 999.
Figure 1: Lightcurve of NGC 5408 X-1 (0.2 - 15 keV band) from XMM-Newton
EPIC/pn observations showing the longest contiguous time interval used in our
power spectral analysis (Interval B). The bin size is 78.125 seconds. A
characteristic error bar is also shown.
Figure 2: Average power spectrum of NGC 5408 X-1 from the $>1$ keV EPIC/pn
data (histogram) and the best fitting model (solid). The frequency resolution
is 0.64 mHz, and each bin is an average of 44 independent power spectral
measurements. The effective exposure is $\approx 70$ ksec. A characteristic
error bar is also shown. See the text for a detailed discussion of the model,
and Table 1 for model parameters.
Figure 3: Average power spectrum of NGC 5408 X-1 from $>1$ keV EPIC/pn data.
The upper curve shows data from interval B with a frequency resolution of
0.434 mHz. Each bin is an average of 10 independent measurements. The bottom
curve is an average of the two longest intervals (B and C) at a frequency
resolution of 0.78 mHz. Each bin is an average of 32 independent measurements.
See the text for additional discussion.
Figure 4: Difference of the count rate spectra between Observation 1 and
Observation 2 (i.e. Obs1 - Obs 2). This shows that observation 1 was brighter,
and that most of the difference was associated with $<1$ keV photons, that is,
the disk component.
Figure 5: Average power spectra and best fitting models characteristic of both
observations of NGC 5408 X-1 from EPIC/pn data. The upper curve is from the
2008 data (intervals B + C), while the bottom curve is from the 2006 data. See
the text for additional details and Table 1 for a summary of the model
parameters.
Table 1: Results of Power Spectral Modeling for NGC 5408 X-111Summary of best
fit power spectral models for NGC 5408 X-1. The results from fits to three
different power spectra are shown in columns 2-4. Columns 2-4 show results
using pn data, but for different time intervals. The particular time interval
used is given with a reference in the heading to the figure the power spectrum
appears in. These fits used up to three Lorentzian components, numbered 1-3 in
order of increasing rms amplitude.
Parameters | 2008, All intervals (Fig. 2) | 2008, Interval B+C (Fig. 5) | 2006 (Fig. 5)
---|---|---|---
AaaNormalization of the bending power-law component. | $1.88\pm 0.2$ | $3.4\pm 0.6$ | $55.7\pm 5$
$\alpha_{L}$bbPower law index below the bend frequency. | $0.07\pm 0.08$ | $0.029\pm 0.1$ | $-0.42\pm 0.5$
$\alpha_{H}$ccPower law index above the bend frequency. | $1.83\pm 0.3$ | $1.7\pm 0.4$ | $1.2\pm 0.4$
$\nu_{{\rm bend}}$ (mHz)ddBend frequency, in mHz. | $10.0\pm 3$ | $6.3\pm 2$ | $2.6\pm 2$
$r_{BB}$ ($\%$)eerms amplitude of the broad-band continuum | $41\pm 5$ | $45\pm 4$ | $23\pm 4$
$\nu_{1}$ (mHz)ffCentroid frequency of the strongest QPO | $10.02\pm 0.005$ | $10.01\pm 0.002$ | $20.6\pm 0.03$
$r_{1}$ $(\%)$ggrms amplitude of the strongest QPO | $17.4\pm 1.3$ | $15.0\pm 1.2$ | $8.4\pm 1.0$
$\nu_{2}$ (mHz) | NA | $13.5\pm 0.1$ | $15.1\pm 0.2$
$r_{2}$ $(\%)$ | NA | $10.0\pm 2$ | $5.4\pm 1$
$\nu_{3}$ (mHz) | NA | $6.0\pm 0.2$ | NA
$r_{3}$ $(\%)$ | NA | $8.8\pm 2.5$ | NA
$\chi^{2}$ (dof) | 463 (459) | 518.8 (497) | 152.2 (180)
hhfootnotetext: Centroid frequency of the next strongest QPO
iifootnotetext: rms amplitude of the next strongest QPO
jjfootnotetext: Centroid frequency of the weakest QPO
kkfootnotetext: rms amplitude of the weakest QPO
llfootnotetext: Minimum $\chi^{2}$ of the fit
Table 2: Spectral Fits to XMM-Newton pn Spectra**All errors are quoted at the
90% confidence level.
Spectral parameters | 2006 | 2008
---|---|---
Model: tbabs*(diskpn \+ apec \+ pow)
nHaaHydrogen column density in units of 1020 cm-2, not including the Galactic contribution of n${}_{HGal}=5.73\times 10^{20}$ cm-2. | $7.0\pm 0.5$ | $6.5\pm 0.6$
TmaxbbDisk temperature in keV from the XSPEC disk model diskpn. The inner disk radius was fixed at 6 GM/c2. | $0.155\pm 0.004$ | $0.160\pm 0.005$
kTccPlasma temperature in keV from the XSPEC model apec. The abundances were fixed to the solar values. | $0.87\pm 0.05$ | $0.85\pm 0.04$
$\Gamma\tablenotemark{d}$ | $2.58\pm 0.04$ | $2.47\pm 0.04$
$\chi^{2}$/dof | 814/673 | 776/673
Fdisk (0.3-10, keV)eeUnabsorbed flux from the dispn component, in units of erg cm-2 s-1. | $1.5\times 10^{-12}$ | $1.1\times 10^{-12}$
Fpow (0.3-10, keV)ffUnabsorbed flux from the power-law component, in units of erg cm-2 s-1. | $2.1\times 10^{-12}$ | $1.9\times 10^{-12}$
FX (0.3-10 keV)ggUnabsorbed total flux in units of erg cm-2 s-1. | $3.7\times 10^{-12}$ | $3.1\times 10^{-12}$
Table 3: Mass Estimates for NGC 5408 X-1 from QPO Scaling**footnotemark: Source | $M_{dyn}$aaMass estimates for dynamically confirmed BHs. | $\nu_{qpo}$ rangebbRange of QPO frequencies observed when the spectral index ranges from 2.4 to 2.6. | $f_{min}$, $f_{best}$, $f_{max}$ccScale factors derived from the observed QPO range. The “best” scale factor is derived by aligning to the center of the range. The minimum and maximum values are obtained by scaling the lowest observed frequency in the reference system with the highest observed frequency in NGC 5408 X-1, and vice versa. See §4 for discussion. | $M_{min}$, $M_{best}$, $M_{max}$ddEstimated masses determined by scaling up the observed masses of the reference systems by the derived scale factors. The “best” mass estimate is obtained using the best scale factor, with uncertainties set by the uncertainties in the reference source masses. The minimum and maximum mass estimates are derived using the minimum and maximum scale factors and the $\pm 1\sigma$ mass limits. For example, the minimum mass for XTE J1550-564 is defined as $(9.5-1.1)*f_{min}=8.4\times 125=1050$ $M_{\odot}$. See §4 for additional discussion. | Refs.eeRelevant references for the mass and QPO frequency measurements: (1) Orosz et al. 2002; (2) Vignarca et al. 2003; (3) ST09; (4) Greiner, Cuby & McCaughrean 2001; (5) McClintock et al. 2009; (6) Fillipenko & Chornock 2001; (7) Zurita et al. 2002; (8) Munoz-Darias et al. 2008; (9) Hynes et al. 2004. measurements.
---|---|---|---|---|---
| $M_{\odot}$ | Hz | | $M_{\odot}$ |
XTE J1550–564 | $9.5\pm 1.1$ | 2.5 - 6.5 | 125, 300, 650 | 1050, $2850\pm 330$, 6890 | 1, 2, 3
GRS 1915+105 | $14\pm 4$ | 2.1 - 3.5 | 105, 185, 350 | 1050, $2590\pm 740$, 6300 | 2, 3, 4
H 1743–322 | $\sim 11$ | 3 - 5 | 150, 267, 500 | 1650, 2937, 5500 | 3, 5
XTE J1859+226 | 7.6 - 12.0 | 7.5 | 375, NA, 750 | 2850, NA, 9000 | 3, 6, 7
GX 339–4 | $>6$ | 5.9 - 7.8 | 295, 457, 780 | 1770, 2742, 4680 | 3, 8, 9
4U 1630–47ffThere is no dynamical mass constraint for this source. | NA | 4 - 8 | 200, 400, 800 | NA | 2
|
arxiv-papers
| 2009-11-05T16:48:20 |
2024-09-04T02:49:06.302128
|
{
"license": "Public Domain",
"authors": "Tod E. Strohmayer and Richard F. Mushotzky",
"submitter": "Tod E. Strohmayer",
"url": "https://arxiv.org/abs/0911.1076"
}
|
0911.1093
|
# A product involving the $\beta$-family in stable homotopy theory
Xiugui Liu School of Mathematical Sciences and LPMC, Nankai University,
Tianjin 300071, P. R. China xgliu@nankai.edu.cn and Wending Li School of
Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China
liwending@gmail.com
(Date: November 1, 2009)
###### Abstract.
In the stable homotopy groups $\pi_{q(p^{n}+p^{m}+1)-3}(S)$ of the sphere
spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed
an essential family $\xi_{m,n}$ for $n\geq m+2>5$. In this paper, the authors
show that the composite $\xi_{m,n}\beta_{s}\in\pi_{q(p^{n}+p^{m}+sp+s)-5}(S)$
for $2\leq s<p$ is non-trivial, where $q=2(p-1)$ and
$\beta_{s}\in\pi_{q(sp+s-1)-2}(S)$ is the known $\beta$-family. We show our
result by explicit combinatorial analysis of the (modified) May spectral
sequence.
###### Key words and phrases:
stable homotopy groups of spheres, Adams spectral sequence, May spectral
sequence, $\beta$-family
###### 2000 Mathematics Subject Classification:
55Q45, 55T15
The first author was partially supported by the National Natural Science
Foundation of China (Nos. 10501045, 10771105) and the Fund of the Personnel
Division of Nankai University.
## 1\. Introduction and statement of results
Let $A$ be the mod $p$ Steenrod algebra and $S$ the sphere spectrum localized
at an odd prime $p$. To determine the stable homotopy groups of spheres
$\pi_{\ast}(S)$ is one of the central problems in homotopy theory.
So far, several methods have been found to determine the stable homotopy
groups of spheres. For example we have the classical Adams spectral sequence
(ASS) (cf. [1]) based on the Eilenberg-MacLane spectrum $K\mathbb{Z}_{p}$,
whose $E_{2}$-term is ${\rm Ext}_{A}^{s,t}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ and
the Adams differential is given by $d_{r}:E_{r}^{s,t}\rightarrow
E_{r}^{s+r,t+r-1}$. We also have the Adams-Novikov spectral sequence based on
the Brown-Peterson spectrum $BP$(cf. [2, 3, 4]).
Throughout this paper, we fix the prime $p\geq 5$ and set $q=2(p-1)$. From
[5], we know that ${\rm Ext}_{A}^{1,1}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ has
$\mathbb{Z}_{p}$-basis consisting of $a_{0}\in{\rm
Ext}_{A}^{1,1}(\mathbb{Z}_{p},\mathbb{Z}_{p})$, $h_{i}\in{\rm
Ext}_{A}^{1,p^{i}q}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ for all $i\geq 0$ and
${\rm Ext}_{A}^{2,\ast}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ has
$\mathbb{Z}_{p}$-basis consisting of $\alpha_{2}$, $a_{0}^{2}$, $a_{0}h_{i}$
$(i>0)$, $g_{i}$ $(i\geq 0)$, $k_{i}$ $(i\geq 0)$, $b_{i}$ $(i\geq 0)$, and
$h_{i}h_{j}$ $(j\geq i+2,i\geq 0)$ whose internal degrees are $2q+1$, $2$,
$p^{i}q+1$, $q(p^{i+1}+2p^{i})$, $q(2p^{i+1}+p^{i})$, $p^{i+1}q$ and
$q(p^{i}+p^{j})$ respectively.
Let $M$ denote the Moore spectrum modulo the prime $p$ given by the
cofibration
(1.1) $S\stackrel{{\scriptstyle p}}{{\rightarrow}}S\stackrel{{\scriptstyle
i}}{{\rightarrow}}M\stackrel{{\scriptstyle j}}{{\rightarrow}}\Sigma S.$
Let $\alpha:\Sigma^{q}M\rightarrow M$ be the Adams map and $V(1)$ be its
cofibre given by the cofibration
(1.2)
$\Sigma^{q}M\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}M\stackrel{{\scriptstyle
i^{\prime}}}{{\rightarrow}}V(1)\stackrel{{\scriptstyle
j^{\prime}}}{{\rightarrow}}\Sigma^{q+1}M.$
Let $\beta:\Sigma^{(p+1)q}V(1)\rightarrow V(1)$ be the $v_{2}$-map.
Definition 1.1 We define, for $t\geq 1$, the $\beta$-family
$\beta_{t}=jj^{\prime}\beta^{t}i^{\prime}i\in\pi_{q(tp+t-1)-2}(S)$. Here the
maps $i$, $j$, $i^{\prime}$, $j^{\prime}$, and $\beta$ are given as above.
We have the following known result.
Theorem 1.1[2, Theorem 2.12] $\beta_{t}\neq 0\in\pi_{\ast}(S)$ for $p\geq 5$
and $t\geq 1$.
To determine the stable homotopy groups of spheres is very difficult. Thus not
so many families of homotopy elements in the stable homotopy groups of spheres
have been detected. See, for example, [4, 5, 6].
In [7], X. Liu obtained the following theorem, which is called the
representative theorem.
Theorem 1.2[7, Theorem 1.3] For $p\geq 5$ and $2\leq s<p$, there exists the
second Greek letter element
$\widetilde{\beta}_{s}\in{\rm
Ext}_{A}^{s,q(sp+s-1)+s-2}(\mathbb{Z}_{p},\mathbb{Z}_{p}),$
which converges to the $\beta$-family $\beta_{s}\in\pi_{q(sp+s-1)-2}(S)$ in
the ASS. Moreover, $\widetilde{\beta}_{s}$ is represented by
$a_{2}^{s-2}h_{2,0}h_{1,1}\in E_{1}^{s,q(sp+s-1)+s-2,\ast}$
in the May spectral sequence (MSS).
In [8], J. Lin detected a new family of stable homotopy groups of spheres and
showed the following theorem.
Theorem 1.3[8] For $p\geq 5$, $n\geq m+2\geq 4$. Then
$h_{0}h_{n}h_{m}\in{\rm
Ext}_{A}^{3,q(p^{n}+p^{m}+1)}(\mathbb{Z}_{p},\mathbb{Z}_{p})$
is a permanent cycle in the ASS and it converges to a family of homotopy
elements of order $p$, denoted by $\xi_{m,n}$, in the stable homotopy groups
of spheres $\pi_{q(p^{n}+p^{m}+1)-3}(S)$.
In this paper, we consider the non-triviality of the composite
$\xi_{m,n}\beta_{s}$ and obtain the following theorem.
Theorem 1.4 Let $p\geq 5$, $n\geq m+2>5$, $2\leq s<p$. Then the product
$h_{0}h_{n}h_{m}\widetilde{\beta}_{s}\neq 0\in{\rm
Ext}_{A}^{s+3,t(s)+s-2}(\mathbb{Z}_{p},\mathbb{Z}_{p})$
is a permanent cycle in the ASS and converges to a nontrivial family of
homotopy elements $\xi_{m,n}\beta_{s}\in\pi_{t(s)+s-5}(S)$, where
$t(s)=q(p^{n}+p^{m}+sp+s)$.
In this paper we make use of the ASS and the MSS to prove our theorem,
especially the MSS. The method of the proof is very elementary. By this
method, one can consider some similar problems, for example, the non-
triviality of the composite $\xi_{m,n}\gamma_{s}$, where $\gamma_{s}$ is the
known $\gamma$-family (cf. [9]).
The paper is arranged as follows: after giving some important lemmas on the
MSS in Section 2, we will prove Theorem 1.4 in Section 3.
## 2\. The ASS and some lemmas on the MSS
One of the main tools to determine the stable homotopy groups of spheres
$\pi_{\ast}(S)$ is the ASS. In 1957, Adams constructed such a machinery in the
form of a spectral sequence that making the doubly graded group ${\rm
Ext}_{A}^{\ast,\ast}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ to the $p$-primary
components of the stable homotopy groups of spheres by adapting the methods of
homological algebra. From then on, the ASS has been a powerful tool in
studying stable homotopy theory.
Let $X$ a spectrum of finite type and $Y$ a finite dimensional spectrum. Then
there is a natural spectral sequence $\\{E_{r}^{s,t},d_{r}\\}$ which is called
Adams spectral sequence and
(2.1) $E_{2}^{s,t}={\rm
Ext}_{A}^{s,t}(H^{\ast}(X;\mathbb{Z}_{p}),H^{\ast}(Y;\mathbb{Z}_{p}))\Rightarrow([Y,X]_{t-s})_{p},$
where the differential is
(2.2) $d_{r}:E_{r}^{s,t}\to E_{r}^{s+r,t+r-1}.$
If $X$ and $Y$ are sphere spectra $S$, then in the ASS
(2.3) $E_{2}^{s,t}={\rm
Ext}_{A}^{s,t}(\mathbb{Z}_{p},\mathbb{Z}_{p})\Rightarrow(\pi_{t-s}(S))_{p},$
the $p$-primary components of the group $\pi_{t-s}(S).$
There are three problems in using the ASS: the calculation of the
$E_{2}$-term, the computation of the differentials and the determination of
the nontrivial extensions from $E_{\infty}$ to $\pi_{\ast}(S)$. So, in order
to compute the stable homotopy groups of spheres with the ASS, we must compute
the $E_{2}$-term of the ASS, ${\rm
Ext}_{A}^{\ast,\ast}(\mathbb{Z}_{p},\mathbb{Z}_{p})$. The most successful tool
for computing ${\rm Ext}_{A}^{\ast,\ast}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ is
the MSS.
From [3], there is a MSS $\\{E_{r}^{s,t,\ast},d_{r}\\}$ which converges to
${\rm Ext}_{A}^{s,t}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ with $E_{1}$-term
(2.4) $E_{1}^{\ast,\ast,\ast}=E(h_{m,i}|m>0,i\geq 0)\otimes
P(b_{m,i}|m>0,i\geq 0)\otimes P(a_{n}|n\geq 0),$
where $E$ is the exterior algebra, $P$ is the polynomial algebra, and
$h_{m,i}\in E_{1}^{1,2(p^{m}-1)p^{i},2m-1},b_{m,i}\in
E_{1}^{2,2(p^{m}-1)p^{i+1},p(2m-1)},a_{n}\in E_{1}^{1,2p^{n}-1,2n+1}.$
The $r$-th May differential is
(2.5) $d_{r}:E_{r}^{s,t,u}\rightarrow E_{r}^{s+1,t,u-r},$
and if $x\in E_{r}^{s,t,\ast}$ and $y\in E_{r}^{s^{\prime},t^{\prime},\ast}$,
then $d_{r}(x\cdot y)=d_{r}(x)\cdot y+(-1)^{s}x\cdot d_{r}(y)$. From [10,
Proposition 2.5], there exists a graded commutativity in the May $E_{1}$-term
as follows:
(2.6)
$\left\\{\begin{array}[]{ll}a_{m}h_{n,j}=h_{n,j}a_{m},&h_{m,k}h_{n,j}=-h_{n,j}h_{m,k},\\\
a_{m}b_{n,j}=b_{n,j}a_{m},&h_{m,k}b_{n,j}=b_{n,j}h_{m,k},\\\
a_{m}a_{n}=a_{n}a_{m},&b_{m,n}b_{i,j}=b_{i,j}b_{m,n}.\end{array}\right.$
The first May differential $d_{1}$ is given by
(2.7)
$\left\\{\begin{array}[]{l}d_{1}(h_{i,j})=\sum\limits_{0<k<i}h_{i-k,k+j}h_{k,j},\\\
d_{1}(a_{i})=\sum\limits_{0\leq k<i}h_{i-k,k}a_{k},\\\
d_{1}(b_{i,j})=0.\end{array}\right.$
For each element $x\in E_{1}^{s,t,\mu}$, we define $\hbox{filt}~{}x=s$,
$\hbox{deg}~{}x=t$, $\hbox{M}~{}(x)=\mu$. Then we have
(2.8) $\left\\{\begin{array}[]{l}\hbox{filt}~{}h_{i,j}=\hbox{filt}~{}a_{i}=1,\
\hbox{filt}~{}b_{i,j}=2,\\\
\hbox{deg}~{}h_{i,j}=2(p^{i}-1)p^{j}=q(p^{i+j-1}+\cdots+p^{j}),\\\
\hbox{deg}~{}b_{i,j}=2(p^{i}-1)p^{j+1}=q(p^{i+j}+\cdots+p^{j+1}),\\\
\hbox{deg}~{}a_{i}=2p^{i}-1=q(p^{i-1}+\cdots+1)+1,\\\ \hbox{deg}~{}a_{0}=1,\\\
\hbox{M}~{}(h_{i,j})=\hbox{M}~{}(a_{i-1})=2i-1,\\\
\hbox{M}~{}(b_{i,j})=(2i-1)p,\end{array}\right.$
where $i\geq 1$, $j\geq 0$.
In Section 3, we will need the following lemmas on the MSS.
By the knowledge on $p$-adic expression in number theory, we have that for
each integer $t\geq 0$, it can be always expressed uniquely as
$t=q({c_{n}p^{n}+c_{n-1}p^{n-1}+\cdots+c_{1}p+c_{0}})+c_{-1},$
where $0\leq c_{i}<p$ $(0\leq i<n)$, $0<c_{n}<p$, $0\leq c_{-1}<q$.
Lemma 2.1[9, Proposition 1.1] Let $t$ as above. Let $s_{1}$ be a positive
integer with $0<s_{1}<p$. If there exists some $0\leq j\leq n$ such that
$c_{j}>s_{1}$, then in the MSS
$E_{1}^{s_{1},t,\ast}=0.$
$\Box$
Lemma 2.2 Let $t$ as above. Let $s_{1}$ be a positive integer with
$0<s_{1}<q$. If $c_{-1}>s_{1}$, then in the MSS,
$E_{1}^{s_{1},t,\ast}=0.$
Proof The proof is similar to that of [9, Proposition 1.1] and is omitted
here. $\Box$
Let $t$ as above and $s$ a given positive integer. Suppose that in the MSS a
generator $\omega\in E_{1}^{s,t,\ast}$ is of the form $w=x_{1}x_{2}\cdots
x_{m}$, where $x_{i}$ is one of $a_{k}$, $h_{l,j}$ or $b_{u,z}$, $1\leq i\leq
m$, $0\leq k\leq n+1$, $0<u+z\leq n$, $0<l+j\leq n+1$, $l>0$, $j\geq 0$,
$u>0$, $z\geq 0$. By (2.8), we can assume that for any $1\leq i\leq m$
$\hbox{deg}~{}x_{i}=q(c_{i,n}p^{n}+c_{i,n-1}p^{n-1}+\cdots+c_{i,1}p+c_{i,0})+{c}_{i,-1}$,
where $c_{i,j}=0$ or $1$ for $0\leq j\leq n$, ${c}_{i,-1}=1$ if
$x_{i}=a_{k_{i}}$, or ${c}_{i,-1}=0$. It follows that
$\hbox{deg}~{}\omega=\sum\limits_{i=1}^{m}{\hbox{deg}~{}x_{i}}=q[(\sum\limits_{i=1}^{m}{c_{i,n}})p^{n}+\cdots+(\sum\limits_{i=1}^{m}{c_{i,1}})p^{1}+\sum\limits_{i=1}^{m}{c_{i,0}}]+\sum\limits_{i=1}^{m}{c_{i,-1}}.$
For convenience, we denote $\sum\limits_{i=1}^{m}{c_{i,j}}$ by $\bar{c}_{j}$
for $j\geq-1$.
Lemma 2.3 With notation as above. If there exist three integers $-1\leq
i_{1}<i_{2}<i_{3}\leq n$ such that
$\bar{c}_{i_{1}}+\bar{c}_{i_{3}}-m>\bar{c}_{i_{2}}$, then $w$ is impossible to
exist.
Proof By (2.8) and (2.4), one easily gets the lemma. $\Box$
Lemma 2.4 With notation as above. Suppose that $m=s$, and there exist three
integers $i_{1}$, $i_{2}$ and $i_{3}$ satisfying the following conditions that
(i) $-1\leq i_{1}<i_{2}<i_{3}\leq n$;
(ii) $\overline{c}_{i_{1}}+\overline{c}_{i_{3}}-m\leq\overline{c}_{i_{2}}$;
(iii) $\overline{c}_{j}=\left\\{\begin{array}[]{cc}0&{-1\leq j<i_{1}}\\\
0&{i_{3}<j\leq n}.\end{array}\right.$
Then we have the following consequences:
(1) When $i_{1}>-1$, there are $(\overline{c}_{i_{1}}+\overline{c}_{i_{3}}-m)$
$h_{{\overline{c}_{i_{3}}-\overline{c}_{i_{1}}+1},\overline{c}_{i_{1}}}$’s
among $\omega$. Furthermore, if
$\overline{c}_{i_{1}}+\overline{c}_{i_{3}}-m>1$, then $w=0$.
(2) When $i_{1}=-1$, there are $(\overline{c}_{i_{1}}+\overline{c}_{i_{3}}-m)$
$a_{i_{3}+1}$’s among $\omega$.
Proof By (2.8) and (2.4), the desired results easily follow. $\Box$
## 3\. Proof of Theorem 1.4
In this section, we will determine two ${\rm Ext}$ groups which will be used
in the proof of Theorem 1.4. In order to do it, we first consider some May
$E_{1}$-terms $E_{1}^{u,v,\ast}$ with two given integers $u$ and $v$, and show
the following lemma.
Lemma 3.1 Let $p\geq 5$, $n\geq m+2>5$, $2\leq s<p$ and $1\leq r\leq s+3$.
Then in the MSS, we have
(3.1)
$E_{1}^{{s+3-r},{t(s)+s-r-1},\ast}=\left\\{\begin{array}[]{ll}\mathbb{Z}_{p}\\{\mathbf{g}_{1},\cdots,\mathbf{g}_{7}\\}&{r=1\
and\ s=p-1,}\\\ 0&other.\end{array}\right.$
Here, $t(s)=q(p^{n}+p^{m}+sp+s)$, and $\mathbf{g}_{1}$, $\cdots$,
$\mathbf{g}_{7}$ equal elements $a_{n}^{p-3}h_{3,0}h_{1,m}h_{n-2,2}h_{n,0}$,
$a_{n}^{p-3}h_{1,2}h_{m+1,0}h_{n-m,m}h_{n,0}$,
$a_{m+1}a_{n}^{p-4}h_{3,0}h_{n-m,m}h_{n-2,2}h_{n,0}$,
$a_{n}^{p-3}h_{3,0}h_{m-1,2}h_{n-m,m}h_{n,0}$,
$a_{n}^{p-3}h_{3,0}h_{m+1,0}h_{n-m,m}h_{n-2,2}$,
$a_{3}a_{n}^{p-4}h_{m+1,0}h_{n-m,m}h_{n-2,2}h_{n,0}$ and
$a_{m}^{p-3}h_{3,0}h_{m,0}h_{m-2,2}h_{1,n}$, respectively.
Proof We divide the proof into the following two cases.
Case 1 $s-r-1<0$. By the knowledge on $p$-adic expression in number theory and
$1\leq r\leq s+3$, we would have $s+3-r<s-r-1+q<q$. In this case
$E_{1}^{{s+3-r},{t(s)+s-r-1},\ast}=0$
by Lemma $2.2$.
Case 2 $s-r-1\geq 0$. Thus $1\leq r\leq s-1$. If $r\geq 4$, then $s+3-r<s$,
which implies that in this case $E_{1}^{{s+3-r},{t(s)+s-r-1},\ast}=0$ by Lemma
2.1. Consequently, in the rest of the proof, we always assume $r\leq 3$.
Consider $\omega=x_{1}x_{2}\cdots x_{m^{\prime}}\in
E_{1}^{s+3-r,t(s)+s-r-1,\ast}$ in the MSS, where $x_{i}$ is one of $a_{k}$,
$h_{l,j}$, $b_{u,z}$, $1\leq i\leq m^{\prime}$, $0\leq k\leq n+1$, $0<l+j\leq
n+1$, $0<u+z\leq n$, $l>0$, $j\geq 0$, $u>0$, $z\geq 0$. By $(\ref{2.8})$, we
can assume that
$\hbox{deg}~{}x_{i}=q(c_{i,n}p^{n}+c_{i,n-1}p^{n-1}+\cdots+c_{i,1}p+c_{i,0})+{c}_{i,-1}$,
where $c_{i,j}=0$ or $1$ for $0\leq j\leq n$, ${c}_{i,-1}=1$ if
$x_{i}=a_{k_{i}}$, or ${c}_{i,-1}=0$. It follows that
(3.2)
$\left\\{\begin{array}[]{l}\hbox{filt}~{}\omega=\sum\limits_{i=1}^{m^{\prime}}\hbox{filt}~{}{x_{i}}=s+3-r,\\\
\hbox{deg}~{}\omega=\sum\limits_{i=1}^{m^{\prime}}{\hbox{deg}~{}x_{i}}=q[(\sum\limits_{i=1}^{m^{\prime}}{c_{i,n}})p^{n}+(\sum\limits_{i=1}^{m^{\prime}}{c_{i,n-1}})p^{n-1}+\cdots+(\sum\limits_{i=1}^{m^{\prime}}{c_{i,m}})p^{m}\\\
{\phantom{\hbox{deg}~{}\omega+}+(\sum\limits_{i=1}^{m^{\prime}}{c_{i,m-1}})p^{m-1}+\cdots+(\sum\limits_{i=1}^{m^{\prime}}{c_{i,1}})p+(\sum\limits_{i=1}^{m^{\prime}}{c_{i,0}})]+(\sum\limits_{i=1}^{m^{\prime}}{c_{i,-1}})}\\\
{\phantom{\hbox{deg}~{}\omega}=t(s)+s-r-1.}\end{array}\right.$
Note that $\hbox{filt}~{}x_{i}=1$ or $2$ and $2\leq s<p$. From
$\sum\limits_{i=1}^{m^{\prime}}\hbox{filt}~{}x_{i}=s+3-r$, it follows that
$m^{\prime}\leq s+2<p+2.$
Using $0\leq{s,\ s-r-1}<p$ and the knowledge on the $p$-adic expression in
number theory, we have the following equations from (3.2).
(3.3)
$\left\\{\begin{array}[]{ll}\sum\limits_{i=1}^{m^{\prime}}{c}_{i,-1}=s-r-1+\lambda_{-1}q,&\lambda_{-1}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,0}+\lambda_{-1}=s+\lambda_{0}p,&\lambda_{0}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,1}+\lambda_{0}=s+\lambda_{1}p,&\lambda_{1}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,2}+\lambda_{1}=0+\lambda_{2}p,&\lambda_{2}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,3}+\lambda_{2}=0+\lambda_{3}p,&\lambda_{3}\geq
0,\\\ \cdots&\cdots\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,m-1}+\lambda_{m-2}=0+\lambda_{m-1}p,&\lambda_{m-1}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,m}+\lambda_{m-1}=1+\lambda_{m}p,&\lambda_{m}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,m+1}+\lambda_{m}=0+\lambda_{m+1}p,&\lambda_{m+1}\geq
0,\\\ \cdots&\cdots\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,n-2}+\lambda_{n-3}=0+\lambda_{n-2}p,&\lambda_{n-2}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,n-1}+\lambda_{n-2}=0+\lambda_{n-1}p,&\lambda_{n-1}\geq
0,\\\
\sum\limits_{i=1}^{m^{\prime}}c_{i,n}+\lambda_{n-1}=1.&\end{array}\right.$
By the knowledge on the $p$-adic expression and $m^{\prime}<p+2$, we have
$\lambda_{-1}=\lambda_{0}=\lambda_{1}=0$. For convenience, in the rest of the
proof we will use $\overline{c}_{j}$ to denote
$\sum\limits_{i=1}^{m^{\prime}}{c_{i,j}}$ for $-1\leq j\leq n$. From the
fourth equation of (3.3) $\overline{c}_{2}=\lambda_{2}p$, $\lambda_{2}$ may
equal $0$ or $1$.
Subcase $2.1$ $\lambda_{2}=0$.
Assertion 3.1 If $\lambda_{2}=0$, then $\lambda_{3}=\cdots=\lambda_{m-1}=0$.
Suppose $\lambda_{3}=1$. Then from the fifth equation of (3.3) we would have
$\overline{c}_{3}=p$, which implies that $m^{\prime}$ can only equal $p$ or
$p+1$. Note that $2\leq s<p$. From $\overline{c}_{3}=p$, $\overline{c}_{2}=0$
and $\overline{c}_{1}=s$, one would have
$\overline{c}_{3}+\overline{c}_{1}-m^{\prime}=p+s-m^{\prime}\geq
1>0=\overline{c}_{2}$. Thus by Lemma $2.3$, $\omega$ is impossible to exist.
Thus, $\lambda_{3}=0$. Similarly, one can show that
$\lambda_{4}=\cdots=\lambda_{m-1}=0$. Assertion 3.1 is proved.
From the $(m+2)$-th equation of ($3.3$) $\overline{c}_{m}=1+\lambda_{m}p$,
$\lambda_{m}$ may equal $0$ or $1$.
Subcase $2.1.1$ $\lambda_{m}=0$. An argument similar to that used in Assertion
$3.1$ shows that $\lambda_{m+1}=\cdots=\lambda_{n-1}=0$. Thus we have
$\overline{c}_{n}$ | $\overline{c}_{n-1}$ | $\cdots$ | $\overline{c}_{m+1}$ | $\overline{c}_{m}$ | $\overline{c}_{m-1}$ | $\cdots$ | $\overline{c}_{3}$ | $\overline{c}_{2}$ | $\overline{c}_{1}$ | $\overline{c}_{0}$ | $\overline{c}_{-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $0$ | $\cdots$ | $0$ | $1$ | $0$ | $\cdots$ | $0$ | $0$ | $s$ | $s$ | $s-r-1$
.
If $\omega$ has $h_{1,n}h_{1,m}$ as factors, one can let
$\omega=h_{1,n}h_{1,m}\omega_{1}$ by (2.6). Then
$\hbox{filt}~{}\omega_{1}=s+1-r$ and $\hbox{deg}~{}\omega_{1}=spq+sq+(s-r-1)$.
When $r>1$, $\omega_{1}$ is impossible to exist by Lemma $2.1$. So $\omega$ is
impossible to exist either. When $r=1$, $\omega_{1}$ has $(s-2)$ $a_{2}$’s
among $\omega$ if $\omega$ exists by Lemma $2.4$. Then up to sign
$\omega_{1}=a_{2}^{s-2}\omega_{2}$ with $\omega_{2}\in
E_{1}^{2,2pq+2q,\ast}=0$, which means $\omega=0$.
Similarly, $\omega$ cannot have $h_{1,n}b_{1,m-1}$, $b_{1,n-1}h_{1,m}$,
$b_{1,n-1}b_{1,m-1}$ as factors either.
Subcase 2.1.2 $\lambda_{m}=1$. In this case
$\lambda_{m+1}=\cdots=\lambda_{n-1}=1$. An argument similar to that used in
Assertion 3.1 can show that in this case $\omega$ is impossible to exist.
Subcase $2.2$ $\lambda_{2}=1$. In this case,
$(\lambda_{3},\cdots,\lambda_{m-1})$ must equal $(1,\cdots,1)$. From the
$(m+2)$-th equation of (3.3) $\overline{c}_{m}=\lambda_{m}p$, $\lambda_{m}$
may equal $0$ or $1$.
Subcase $2.2.1$ $\lambda_{m}=1$. In this case,
$(\lambda_{m+1},\cdots,\lambda_{n-1})$ must equal $(1,\cdots,1)$. Thus we have
$\overline{c}_{n}$ | $\overline{c}_{n-1}$ | $\cdots$ | $\overline{c}_{m+1}$ | $\overline{c}_{m}$ | $\overline{c}_{m-1}$ | $\cdots$ | $\overline{c}_{3}$ | $\overline{c}_{2}$ | $\overline{c}_{1}$ | $\overline{c}_{0}$ | $\overline{c}_{-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$0$ | $p-1$ | $\cdots$ | $p-1$ | $p$ | $p-1$ | $\cdots$ | $p-1$ | $p$ | $s$ | $s$ | $s-r-1$
.
If $r=2$ or $3$, then by $p\geq 5$, $2\leq s<p$ and $m^{\prime}\leq s+3-r$ one
can have
$\overline{c}_{2}+\overline{c}_{m}-m^{\prime}=p+p-m^{\prime}\geq p+p-(s+1)\geq
p>p-1=\overline{c}_{3},$
which implies that $\omega$ is impossible to exist by Lemma $2.3$.
If $r=1$, then one has $\hbox{filt}~{}\omega=s+2$. From $\overline{c}_{m}=p$,
one has $m^{\prime}\geq p$ by $c_{i,m}=0$ or $1$. Thus $m^{\prime}$ may equal
$p$ or $p+1$. If $m^{\prime}=p$, then
$\overline{c}_{2}+\overline{c}_{m}-m^{\prime}=p>p-1=\overline{c}_{3}$, which
implies that $\omega$ is impossible to exist by Lemma 2.3. Thus, in the rest
of Subcase $2.2.1$ we always assume that $r=1$ and $m^{\prime}=p+1$. Thus we
have $s=p-1$, $\hbox{filt}~{}\omega=p+1$ and $\omega=x_{1}\ \cdots\ x_{p+1}\in
E(h_{i,j}|i>0,j\geq 0)\otimes P(a_{n}|n\geq 0)$. The table above becomes
$\overline{c}_{n}$ | $\overline{c}_{n-1}$ | $\cdots$ | $\overline{c}_{m+1}$ | $\overline{c}_{m}$ | $\overline{c}_{m-1}$ | $\cdots$ | $\overline{c}_{3}$ | $\overline{c}_{2}$ | $\overline{c}_{1}$ | $\overline{c}_{0}$ | $\overline{c}_{-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$0$ | $p-1$ | $\cdots$ | $p-1$ | $p$ | $p-1$ | $\cdots$ | $p-1$ | $p$ | $p-1$ | $p-1$ | $p-3$
.
Assertion 3.2 $\omega$ has $p-1$ factors whose degrees are $q($higher terms on
$p+p^{m}+\cdots+p^{2}+$ lower terms on $p)+\epsilon$, where $\epsilon=0$ or
$1$, and two factors whose degree are $q($ higher terms on $p+p^{m})$ and
$q(p^{2}+$ lower terms on $p)+\epsilon$, respectively.
This assertion can be easily verified by (2.8) and (3.2).
Assertion 3.3 $\omega$ cannot have $h_{2,1}$ or $h_{j,m}$ ($2\leq j<n-m$) as
a factor.
Otherwise, we can let $\omega=\omega_{1}h_{2,1}$ by (2.6). Then
$\hbox{filt}~{}\omega_{1}=p$,
$\hbox{deg}~{}\omega_{1}=q[(p-1)p^{n-1}+\cdots+(p-1)p^{m+1}+pp^{m}+(p-1)p^{m-1}+\cdots+(p-1)p^{3}+(p-1)p^{2}+(p-2)p+(p-1)]+p-3$.
In this case $\omega_{1}$ is impossible to exist by Lemma $2.3$. Thus $\omega$
cannot have $h_{2,1}$ as a factor. Similarly, $\omega$ cannot have $h_{j,m}$
$(2\leq j<n-m)$ as a factor.
From Assertions 3.2 and 3.3, there must be one of $h_{1,2}h_{1,m}$,
$h_{3,0}h_{1,m}$, $a_{3}h_{1,m}$, $h_{1,2}h_{n-m,m}$, $h_{3,0}h_{n-m,m}$ or
$a_{3}h_{n-m,m}$ among $\omega$ if $\omega$ exists. By (2.6), we let
$\omega=\omega_{1}\omega_{2}$, where $\omega_{2}$ is one of the six factors
above. Then $\hbox{filt}~{}\omega_{1}=p-1$.
(i) If $\omega_{2}=h_{1,2}h_{1,m}$, then
$\hbox{deg}~{}\omega_{1}=q[(p-1)p^{n-1}+\cdots+(p-1)p^{m}+\cdots+(p-1)p+(p-1)]+p-3$.
So there must be two $h_{n,0}$’s in $\omega$ by Lemma 2.4 (1), which implies
that $\omega_{1}=0$. Then $\omega=0$.
Similarly, one can show that $\omega=0$ if $\omega_{2}=a_{3}h_{1,m}$.
(ii) If $\omega_{2}=h_{3,0}h_{1,m}$, then
$\hbox{deg}~{}\omega_{1}=q[(p-1)p^{n-1}+\cdots+(p-1)p^{m}+\cdots+(p-1)p^{2}+(p-2)p+(p-2)]+p-3$.
By Lemma 2.4, $\omega_{1}$ must equal $a_{n}^{p-3}h_{n,0}h_{n-2,2}$ up to sign
. Thus up to sign $\omega=a_{n}^{p-3}h_{3,0}h_{1,m}h_{n-2,2}h_{n,0}$, denoted
by $\mathbf{g}_{1}$.
(iii) If $\omega_{2}=h_{1,2}h_{n-m,m}$, then
$\hbox{deg}~{}\omega_{1}=q[(p-2)p^{n-1}+\cdots+(p-2)p^{m+1}+(p-1)p^{m}+(p-1)p^{m-1}+\cdots+(p-1)p^{2}+(p-1)p+(p-1)]+p-3$,
so $\omega_{1}$ has at least $p-4$ $a_{n}$’s by Lemma $2.4$. We let
$\omega_{1}=\omega_{3}a_{n}^{p-4}$ by (2.6). Thus
$\hbox{filt}~{}\omega_{3}=3$,
$\hbox{deg}~{}\omega_{3}=q(2p^{n-1}+\cdots+2p^{m+1}+3p^{m}+3p^{m-1}+\cdots+3p^{3}+3p^{2}+3p+3)+1$.
Then $\omega_{3}\in
E_{1}^{3,\hbox{deg}~{}\omega_{3},\ast}=\mathbb{Z}_{p}\\{a_{n}h_{n,0}h_{m+1,0}\\}$.
Thus up to sign $\omega=a_{n}^{p-3}h_{1,2}h_{m+1,0}h_{n-m,m}h_{n,0}$, denoted
by $\mathbf{g}_{2}$.
(iv) If $\omega_{2}=h_{3,0}h_{n-m,m}$, an argument similar to that used in
(iii) shows $\omega_{1}=a_{n}^{p-4}\omega_{3}$ with $\omega_{3}\in
E_{1}^{3,t,\ast}=\mathbb{Z}_{p}\\{a_{m+1}h_{n-2,2}h_{n,0},a_{n}h_{m-1,2}h_{n,0},a_{n}h_{m+1,0}h_{n-2,2}\\}$,
where $t=q(2p^{n-1}+\cdots+2p^{m+1}+3p^{m}+\cdots+3p^{2}+2p+2)+1$. Thus up to
sign $\omega=a_{m+1}a_{n}^{p-4}h_{3,0}h_{n-m,m}h_{n-2,2}h_{n,0}$,
$a_{n}^{p-3}h_{3,0}h_{m-1,2}h_{n-m,m}h_{n,0}$ or
$a_{n}^{p-3}h_{3,0}h_{m+1,0}h_{n-m,m}h_{n-2,2}$, denoted by $\mathbf{g}_{3}$,
$\mathbf{g}_{4}$, $\mathbf{g}_{5}$ respectively.
(v) If $\omega_{2}=a_{3}h_{n-m,m}$, by an argument similar to that used in
(iii) we have $\omega_{1}=a_{n}^{p-5}\omega_{3}$ with $\omega_{3}\in
E_{1}^{4,t^{\prime},\ast}=\mathbb{Z}_{p}\\{a_{n}h_{m+1,0}h_{n,0}h_{n-2,2}\\}$,
where $t^{\prime}=q(3p^{n-1}+\cdots+3p^{m+1}+4p^{m}+\cdots+4p^{2}+3p+3)+1$.
Thus up to sign $\omega=a_{3}a_{n}^{p-4}h_{m+1,0}h_{n-m,m}h_{n-2,2}h_{n,0}$,
denoted by $\mathbf{g}_{6}$.
Subcase $2.2.2$ $\lambda_{m}=0$. By an argument similar to that used in
Assertion $3.1$, we have $\lambda_{m+1}=\cdots=\lambda_{n-1}=0$. Thus we have
$\overline{c}_{n}$ | $\overline{c}_{n-1}$ | $\cdots$ | $\overline{c}_{m+1}$ | $\overline{c}_{m}$ | $\overline{c}_{m-1}$ | $\cdots$ | $\overline{c}_{3}$ | $\overline{c}_{2}$ | $\overline{c}_{1}$ | $\overline{c}_{0}$ | $\overline{c}_{-1}$
---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $0$ | $\cdots$ | $0$ | $0$ | $p-1$ | $\cdots$ | $p-1$ | $p$ | $s$ | $s$ | $s-r-1$
.
Obviously $\omega$ must have a factor $h_{1,n}$. One can let
$\omega=\omega_{1}h_{1,n}$ by (2.6).
If $r=2$ or $3$, it is easy to show that $\omega_{1}$ is impossible to exist
by Lemma 2.1, which implies that $\omega$ is impossible to exist either.
If $r=1$, by Lemma 2.1 it is easy to get that in this case $s$ must equal
$p-1$ and $m^{\prime}$ must equal $p+1$. By an argument similar to that used
in Subcase $2.2.1$, we get that up to sign
$\omega=a_{m}^{p-3}h_{3,0}h_{m,0}h_{m-2,2}h_{1,n}\in
E^{p+1,t(p-1)+p-3,(2m+1)p-2m-3},$ denoted by $\mathbf{g}_{7}$.
Combining Cases 1 and 2, we complete the proof of the lemma. $\Box$
By use of Lemma 3.1, we now show the non-triviality of
$h_{0}h_{n}h_{m}\widetilde{\beta}_{s}$.
Theorem 3.2 Let $p\geq 5$, $n\geq m+2>5$, $2\leq s<p$. Then the product
$h_{0}h_{n}h_{m}\widetilde{\beta}_{s}\neq 0\in{\rm
Ext}_{A}^{{s+3},{t(s)+s-2}}(\mathbb{Z}_{p},\mathbb{Z}_{p}),$
where $t(s)=q(p^{n}+p^{m}+sp+s)$.
Proof Since $h_{1,n}(n\geq 0)$ and $a_{2}^{s-2}h_{2,0}h_{1,1}$ are permanent
cycles in the MSS and converge nontrivially to $h_{n}$,
$\widetilde{\beta_{s}}\in{\rm
Ext}_{A}^{\ast,\ast}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ respectively,
$a_{2}^{s-2}h_{2,0}h_{1,1}h_{1,0}h_{1,n}h_{1,m}\in
E_{1}^{{s+3},{t(s)+s-2},\ast}$ is a permanent cycle in the MSS and converges
to $h_{0}h_{n}h_{m}\widetilde{\beta}_{s}\in{\rm
Ext}_{A}^{{s+3},{t(s)+s-2}}(\mathbb{Z}_{p},\mathbb{Z}_{p}).$
Case $1$ $s=p-1$. By (2.7), one can have that up to sign
(3.4)
$\left\\{\begin{array}[]{ll}d_{1}(\mathbf{g}_{1})=a_{n}^{p-3}h_{1,0}h_{3,0}h_{1,m}h_{n-2,2}h_{n-1,1}+\cdots&\neq
0;\\\
d_{1}(\mathbf{g}_{2})=a_{n}^{p-3}h_{1,0}h_{1,2}h_{m+1,1}h_{n-m,m}h_{n-1,1}+\cdots&\neq
0;\\\
d_{1}(\mathbf{g}_{3})=a_{n}^{p-4}a_{m+1}h_{1,0}h_{3,0}h_{n-m,m}h_{n-2,2}h_{n-1,1}+\cdots&\neq
0;\\\
d_{1}(\mathbf{g}_{4})=a_{n}^{p-3}h_{1,0}h_{3,0}h_{m-1,2}h_{n-m,m}h_{n-1,1}+\cdots&\neq
0;\\\
d_{1}(\mathbf{g}_{5})=a_{n}^{p-3}h_{1,0}h_{3,0}h_{m,1}h_{n-m,m}h_{n-2,2}+\cdots&\neq
0;\\\
d_{1}(\mathbf{g}_{6})=a_{n}^{p-4}a_{3}h_{1,0}h_{m+1,0}h_{n-m,m}h_{n-2,2}h_{n-1,1}+\cdots&\neq
0;\\\
d_{1}(\mathbf{g}_{7})=a_{m}^{p-3}h_{1,2}h_{3,0}h_{m-3,3}h_{1,n}h_{n,0}+\cdots&\neq
0.\end{array}\right.$
Obviously the first May differential of each of the seven generators contains
at least a term which is not in the first May differentials of the other
generators, which implies that $d_{1}(\mathbf{g}_{1}),\ \cdots,\
d_{1}(\mathbf{g}_{7})$ are linearly independent. Thus,
$E_{2}^{{p+1},{t(p-1)+p-3},\ast}=0.$ It follows that
$E_{r}^{{p+1},{t(p-1)+p-3},\ast}=0~{}{\rm for}~{}r\geq 2.$
Meanwhile, by (2.8) we have that $M(\mathbf{g}_{i})=(2n+1)p-2n-3$( $1\leq
i\leq 6$), $M(\mathbf{g}_{7})=(2m+1)p-2m-3$ and
$M(a_{2}^{p-3}h_{2,0}h_{1,1}h_{1,0}h_{1,n}h_{1,m})=5p-8$. Then from (2.5) one
has $a_{2}^{p-3}h_{2,0}h_{1,1}h_{1,0}h_{1,n}h_{1,m}\notin
d_{1}(E_{1}^{p+1,t(p-1)+p-3,p(2n+1)-2n-3})$ and
$a_{2}^{p-3}h_{2,0}h_{1,1}h_{1,0}h_{1,n}h_{1,m}\notin
d_{1}(E_{1}^{p+1,t(p-1)+p-3,p(2m+1)-2m-3})$. Thus we have the permanent cycle
$a_{2}^{p-3}h_{2,0}h_{1,1}h_{1,0}h_{1,n}h_{1,m}\in
E_{r}^{p+2,t(p-1)+p-3,\ast}$ cannot be hit by any May differential. It follows
that $h_{0}h_{n}h_{m}\widetilde{\beta}_{p-1}\neq 0$.
Case $2$ $2\leq s<p-1$. From Lemma 3.1 one has that in this case the May
$E_{1}$-term
$E_{1}^{{s+2},{t(s)+s-2},\ast}=0.$
Thus one has
$E_{r}^{{s+2},{t(s)+s-2},\ast}=0\ {\rm for}\ r>1.$
Consequently, the permanent cycle
$h_{1,0}h_{1,n}h_{1,m}a_{2}^{s-2}h_{2,0}h_{1,1}\in
E_{r}^{{s+3},{t(s)+s-2},\ast}$ cannot be hit by any differential in the MSS.
Then $h_{0}h_{n}h_{m}\widetilde{\beta_{s}}\neq 0\in{\rm
Ext}_{A}^{{s+3},t(s)+s-2}(\mathbb{Z}_{p},\mathbb{Z}_{p}).$
From Cases $1$ and $2$, we complete the proof of the theorem. $\Box$
Theorem 3.3 Let $p\geq 5$, $n\geq m+2>5$, $2\leq s<p$ and $2\leq r\leq s+3$.
Then
${\rm Ext}_{A}^{{s+3-r},{t(s)+s-r-1},\ast}(\mathbb{Z}_{p},\mathbb{Z}_{p})=0,$
where $t(s)=q(p^{n}+p^{m}+sp+s)$.
Proof From the case $2\leq r\leq s+3$ in Lemma 3.1, we have that in the MSS
$E_{1}^{{s+3-r},{t(s)+s-r-1},\ast}=0.$
The theorem follows easily by the MSS. $\Box$
Now we give the proof of Theorem $1.4$ .
Proof of Theorem 1.4 From Theorem 1.2, we have that $\widetilde{\beta}_{s}$
converges to $\beta$-family $\beta_{s}\in\pi_{spq+(s-1)q+s-2}(S)$ in ASS. From
Theorem $1.3$, $h_{0}h_{n}h_{m}\in{\rm
Ext}_{A}^{3,p^{n}q+p^{m}q+q}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ is a permanent
cycle in the ASS and converges to a nontrivial family of homotopy elements
$\xi_{m,n}\in\pi_{p^{n}q+p^{m}q-3}(S)$. Hence, we have that the composite
$\xi_{m,n}\beta_{s}$
is represented up to a nonzero scalar by
$h_{0}h_{n}h_{m}\widetilde{\beta}_{s}\neq 0\in{\rm
Ext}_{A}^{s+3,t(s)+s-2}(\mathbb{Z}_{p},\mathbb{Z}_{p})$
in the ASS (cf. Theorem $3.2$).
Moreover, from Theorem $3.3$, $h_{0}h_{n}h_{m}\widetilde{\beta}_{s}$ cannot be
hit by any differential in the ASS. Consequently, the corresponding homotopy
element $\xi_{m,n}\beta_{s}$ is nontrivial. This proves Theorem $1.4$. $\Box$
## References
* [1] J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press, Chicago 1974.
* [2] H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. 106 (1977), 469-516.
* [3] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Orlando: Academic Press, 1986.
* [4] C. Lee, Detection of some elements in the stable homotopy groups of spheres, Math. Z. 222 (1996), 231-246.
* [5] A. Liulevicius, The factorizations of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. 42 (1962).
* [6] R. Cohen, Odd primary infinite families in stable homotopy theory, Mem. Amer. Math. Soc. 242, 1981, viii +92.
* [7] X. Liu, Some infinite elements in the Adams spectral sequence for the sphere spectrum, J. Math. Kyoto Univ. 48 (3) (2008), 617-629.
* [8] J. Lin, A pull back theorem in the Adams spectral sequence, Acta Math. Sin. (Engl. Ser.), 24(3)(2008), 471-490.
* [9] X. Liu, A nontrivial product in the stable homotopy groups of spheres, Sci. China Ser. A 47 (6) (2004), 831-841.
* [10] X. Liu, X. Wang, A four-filtrated May spectral sequence and its applications, Acta Math. Sin. (Engl. Ser.), 24 (9) (2008), 1507-1524.
|
arxiv-papers
| 2009-11-05T18:21:38 |
2024-09-04T02:49:06.309055
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiugui Liu and Wending Li",
"submitter": "Xiugui Liu Dr",
"url": "https://arxiv.org/abs/0911.1093"
}
|
0911.1112
|
# Memento: Time Travel for the Web
Herbert Van de Sompel
Michael L. Nelson
Robert Sanderson
Los Alamos National Laboratory, NM, USA herbertv@lanl.gov Old Dominion
University, Norfolk, VA, USA mln@cs.odu.edu Los Alamos National Laboratory,
NM, USA rsanderson@lanl.gov Lyudmila L. Balakireva
Scott Ainsworth
Harihar Shankar
Los Alamos National Laboratory, NM, USA ludab@lanl.gov Old Dominion
University, Norfolk, VA, USA sainswor@cs.odu.edu Los Alamos National
Laboratory, NM, USA harihar@unm.edu
###### Abstract
The Web is ephemeral. Many resources have representations that change over
time, and many of those representations are lost forever. A lucky few manage
to reappear as archived resources that carry their own URIs. For example, some
content management systems maintain version pages that reflect a frozen prior
state of their changing resources. Archives recurrently crawl the web to
obtain the actual representation of resources, and subsequently make those
available via special-purpose archived resources. In both cases, the archival
copies have URIs that are protocol-wise disconnected from the URI of the
resource of which they represent a prior state. Indeed, the lack of temporal
capabilities in the most common Web protocol, HTTP, prevents getting to an
archived resource on the basis of the URI of its original. This turns
accessing archived resources into a significant discovery challenge for both
human and software agents, which typically involves following a multitude of
links from the original to the archival resource, or of searching archives for
the original URI. This paper proposes the protocol-based Memento solution to
address this problem, and describes a proof-of-concept experiment that
includes major servers of archival content, including Wikipedia and the
Internet Archive. The Memento solution is based on existing HTTP capabilities
applied in a novel way to add the temporal dimension. The result is a
framework in which archived resources can seamlessly be reached via the URI of
their original: protocol-based time travel for the Web.
###### category:
H.3.5 Information Storage and Retrieval Online Information Services
###### keywords:
Web Architecture, HTTP, Archiving, Content Negotiation, OAI-ORE, Time Travel
††terms: Design, Experimentation, Standardization
## 1 Introduction
“The web does not work,” my eleven year old son complained. After checking
power and network connection, I realized he meant something rather more
subtle. The URI (http://stupidfunhouse.com) he had bookmarked the year before
returned a page that didn’t look like the original at all, and definitely was
not fun. He had just discovered that the web has a terrible memory.
Let us restate the obvious: the Web is the most pervasive information
environment in the history of humanity; hundreds of millions of
people111http://www.internetworldstats.com/stats.htm access billions of
resources222http://googleblog.blogspot.com/2008/07/we-knew-web-was-big.html
using a variety of wired or wireless devices. The rapid growth of the Web was
made possible by a suite of relatively simple, yet powerful technologies
including TCP/IP, URI, HTTP, and HTML. The Web is also highly dynamic, with a
significant percentage of resources changing at different rates over time [3,
9, 18, 25]. Given the ubiquity of the Web, it is rather surprising to find how
poor its memory is regarding these continuous changes. Indeed, once a resource
has changed, accessing one of its prior versions becomes a significant
discovery challenge, no longer merely a matter of using Web protocols to
dereference its URI.
In essence, the time dimension is absent from the most common of Web
protocols, HTTP. This timelessness is even written into the W3C’s Architecture
of the World Wide Web [15], which reminds us that dereferencing a URI yields a
representation of the (current) state of the resource identified by that URI,
and highlights the impracticality of keeping prior states accessible at their
own distinct URIs:
> Resource state may evolve over time. Requiring a URI owner to publish a new
> URI for each change in resource state would lead to a significant number of
> broken references. For robustness, Web architecture promotes independence
> between an identifier and the state of the identified resource.
Nevertheless, the Web does contain a meaningful amount of records of the past.
Sites based on Content Management Systems (CMS) such as Wikimedia, the
platform used by Wikipedia, keep the current version of a page accessible at a
generic URI, while older versions remain accessible at version-specific URIs.
Special-purpose services that are concerned with persistent referencing, such
as WebCite, store a representation of the resource retrieved at the time the
service is invoked. Also, inspired by the pioneering work of the Internet
Archive, there is an ever-growing international Web Archiving [6, 22] activity
that consists of recurrently sending out crawlers to take snapshots of Web
resources, storing those in special-purpose distributed archives, and making
them accessible through tools such as the Wayback
Machine333http://www.archive.org/. Transactional archives [11] store every
materially different representation of a web server’s resources as they are
being delivered to clients. Currently, their use is primarily restricted to
applications that need to meet special legal requirements, such as keeping an
exact record of what has been delivered to users of an ecommerce or government
site, and they are therefore typically not openly accessible. Exploratory work
is ongoing regarding the establishment of a peer-to-peer web archive that
receives its content from browser caches, and that therefore can be considered
a client-side transactional archive [4]. Also personal client-side
transactional archives have been proposed [7, 30, 32] but their private
purpose excludes accessing them on the Web. Search engine caches may also
contain prior representations of resources, but they are restricted to the
most recent snapshot taken by a crawler.
Although this variety of archival solutions exists and their coverage is
growing, accessing last year’s version of a resource remains a significant
challenge. In the case of Wiki-
pedia, one has to resort to its History tab and navigate the sometimes
thousands of entries there. The situation is similar for most other version-
aware sites. For news sites, one may find the answer by searching the site’s
special purpose archive if one exists. And, as an option of last resort
unknown to many Web users, one can individually search the many Web Archives,
hoping to find a page that was archived at a time close to the desired one.
This situation is cumbersome for users who, for example, want to revisit a
bookmarked resource as it existed at the time of bookmarking. Research has
indicated that anywhere between 50% and 80% of page visits are revisits [2,
26, 31]. To an extent, this finding emphasizes the need for end-user time
travel on the Web.
The poor integration of archival content in regular Web navigation is also a
fundamental hindrance to applications that require finding, analyzing,
extracting, comparing, and otherwise leveraging historical Web information.
Examples include Zoetrope, a tool that allows interaction with and
visualization of high-resolution temporal Web data [1]; DiffIE, a Web browser
plug-in that emphasizes Web content that changed since a previous visit [32];
and time-oriented search that tracks the frequency of words and phrases in
resources over time [20]. These applications must build their own special-
purpose archives in an ad-hoc manner in order to achieve their goals.
In this paper, we present the Memento solution to allow temporal access to the
Web. Our solution is based on and is as simple as the technologies that led to
the rapid growth of the Web. It focuses on seamless access to archival content
(irrespective of its location) as part of regular Web navigation for both
human and software agents. It does not deal with the aspect of creating,
populating, and maintaining archives, but rather leverages their existence.
The remainder of the paper is structured as follows: Section 2 briefly reviews
transparent content negotiation for HTTP in order to allow a better
understanding of Section 3 which introduces the Memento solution for time
travel on the Web; Section 4 describes an experiment that provides a proof of
concept for the solution; Section 5 discusses open issues; and Section 6
provides an overview of related work; Section 7 holds our conclusion.
## 2 Content Negotiation
Transparent Content Negotiation for HTTP [14] (from here on abbreviated as
conneg) allows a client to select which representation it wants to retrieve
from a transparently negotiable resource; that is, a resource that has
multiple representations (variants) associated with it, each of which is
available from a variant resource. Currently deployed dimensions that are open
to conneg are media type, language, compression, and character set. A client
expresses preferences, possibly according to multiple dimensions, in special-
purpose HTTP Accept headers. Preferences are qualified with “quality”, or “q”,
values, that have a normalized value of 1.0 – 0.0 (an argument without a q
value is assumed to have q=1.0). For example, by using the header “Accept-
Language: en, fr;q=0.7” the client indicates that English is preferred and
French is acceptable. Based on information in these headers, a server will
either:
* •
Select an appropriate representation: There are two ways for a server to do
so. One way is to provide a “HTTP 200 OK” response with a “TCN: Choice”
header, and a Content-Location header that indicates the URI of the variant
resource that delivered the representation. The other is to provide a “HTTP
302 Found” response with a “TCN: Choice” header, and a Location header that
indicates the URI of where the client can access the variant resource.
* •
Respond with a “HTTP 406 Not Acceptable” response if the server cannot meet
the client’s preferences as stated in the request. The server then also
returns a “TCN: List” header and a list of variant resources it possesses that
are associated with the requested resource. The client can then make an
informed decision about variant selection.444The client can also force a “
HTTP 300 Multiple Choices” response
by issuing a “Negotiate: 1.0” request header. This rarely occurs in practice,
but the response is functionally equivalent to a “HTTP 406 Not Acceptable”
response.
RFC 2295 proposes a format for these lists, expressed as an Alternates
response header that can be used in both the Choice and List scenarios. Web
servers do not necessarily support all the negotiation dimensions for all of
their resources, but do indicate the supported dimensions to clients (e.g.,
“Vary: negotiate, accept-language” if the language dimension is supported).
Also, note that according to RFC 2295, variant resources do not themselves
support content negotiation555Servers must return a “HTTP 506 Variant Also
Negotiates” response if variant resources support conneg..
As an example, presume a transparently negotiable resource
http://an.example.org/paper for which the following variant resources are
available: the paper in HTML and English (paper.html.en), in PDF and English
(paper.pdf.en), and in PDF and French (paper.pdf.fr). Now presume a client
wants to access the paper and has a preference for HTML and English. The
interaction, in which the server makes a choice that fully honors the client’s
preferences, would then be (only headers relevant for conneg are shown):
GET /paper HTTP/1.1
Host: an.example.org
Accept: text/html, application/pdf;q=0.8
Accept-Language: en-US, fr;q=0.7, de;q=0.5
HTTP/1.1 200 OK
TCN: choice
Vary: negotiate, accept, accept-language
Content-Location: /paper.html.en
Content-Type: text/html
Content-Language: en
Alternates:
{"paper.html.en" 1.0 {type text/html} {language en}},
{"paper.pdf.en" 0.8 {type application/pdf} {language en}},
{"paper.pdf.fr" 0.6 {type application/pdf} {language fr}}
However, if the client prefers PDF over HTML and insists only on German
language documents (French and English have q=0.0), the interaction in which
the server cannot honor the request, and leaves the choice to the client would
be:
GET /paper HTTP/1.1
Host: an.example.org
Accept: application/pdf, text/html;q=0.8
Accept-Language: de, fr;q=0.0, en-US;q=0.0
HTTP/1.1 406 Not Acceptable
TCN: list
Vary: negotiate, accept, accept-language
Alternates: {"paper.pdf.fr" 0.8 {type application/pdf}
{language fr}}, {"paper.html.en" 0.5 {type text/html}
{language en}}, {"paper.pdf.en" 0.4
{type application/pdf} {language en}}
## 3 The Memento Solution
In this section, we introduce the two core building blocks of the Memento
solution to allow temporal navigation of the Web: HTTP content negotiation in
the datetime dimension, and an API for archives of web resources that allows
requesting an inventory of available archived resources associated with a
resource with a given URI.
### 3.1 A Memento: An Archival Resource
We introduce the term Memento to refer to an archival record of a resource.
More formally, a Memento for a resource URI-R (as it existed) at time ti is a
resource URI-Mi[URI-R@ti] for which the representation at any moment past its
creation time tc is the same as the representation that was available from
URI-R at time ti, with tc $\geq$ ti. Implicit in this definition is the notion
that, once created, a Memento always keeps the same representation.
In the remainder of this paper, the term original resource is used to refer to
a resource that itself is not a Memento of another resource, and URI-R is used
to denote its URI. URI-M is used to denote the URI of a Memento.
### 3.2 HTTP Datetime Content Negotiation
We introduce the notion of content negotiation in the datetime dimension (from
here on abbreviated as DT-conneg), allowing a client to indicate that it is
looking for past rather than current representations of a resource. This is
achieved by using a special-purpose Accept header, experimentally named
X-Accept-Datetime, which has datetimes (rather
than media type or similar) as its value:
X-Accept-Datetime: {Sun, 06 Nov 1994 08:49:37 GMT}
Generally speaking, DT-conneg works in very much the same way as existing
conneg approaches: If a client wants to retrieve a Memento of the original
resource URI-R, it issues an HTTP GET at URI-R using the X-Accept-Datetime
header to express the datetimes of the archival record(s) of URI-R in which it
is interested. The server handling this HTTP GET request tries to honor it by
delivering a representation it chooses based on the client’s datetime
preference(s), and/or by providing the client with a list of available variant
resources, each of which is a Memento of URI-R. Described in more detail
below, two distinctions exist between DT-conneg and other conneg approaches:
* •
Cases exist in which the server hosting URI-R can not itself honor the DT-
conneg request, but instead redirects to a server that can.
* •
The list of available variant resources can be too extensive to be expressed
in an Alternates header. In this case, a combination of a sizeable Alternates
header listing variants centered on the requested datetime(s), and an HTTP
Link header pointing at an extensive list of variants is used.
Before deciding on the X-Accept-Datetime header, we investigated possible
alternatives that could be used in HTTP interaction. We decided not to use the
“features” extensibility mechanism introduced by RFC 2295 because it is geared
at the fine-grained specification of variant options (e.g., paper size, color
depth) and hence is not suitable for something with the primacy of datetime.
Also, the ongoing Media Fragment work of the W3C [33] is not applicable
because it proposes expressing a segment of a resource (e.g., a region of an
image, a section of a video) as a URI fragment. It does not deal with the
notion of a resource that has changing representations over time.
### 3.3 A TimeGate: A Resource Capable of DT-conneg
We introduce the term TimeGate to refer to a transparently negotiable resource
that supports the datetime dimension. More formally, a TimeGate for an
original resource URI-R is a transparently negotiable resource URI- G[URI-R]
for which all variant resources are Mementos URI-Mi[URI-R@ti] of the resource
URI-R. Since multiple archives may host versions of URI-R, multiple TimeGates
may exist for any given resource, i.e. one per archive.
### 3.4 Time Travel: Combining DT-conneg and TimeGates
To further explain DT-conneg and TimeGates, two separate scenarios are
explored. The combination of these scenarios provides a solution for temporal
Web navigation that integrates operational web servers and archives of all
types. To allow for a better understanding, the description is restricted to
conneg in the datetime dimension only. Also, in order to keep examples simple,
requests with multiple datetime values and associated q-values are not used.
It should be noted, however, that both multi-dimensional conneg, and multiple
datetime values are possible in the proposed framework, since it builds on the
principles of RFC 2295 that provides these capabilities. Furthermore, we
assume that the server that hosts the original resource URI-R for which a
client wants to retrieve Mementos, is able to detect the existence of an
X-Accept-Datetime header.
Before describing the scenarios, let us provide some explanatory information
about the HTTP headers that are involved:
Alternates: RFC 2295 requires listing all variant resources. However, since an
extensive set of variant resources may exist in case of DT-conneg, the
Alternates listing is impractical. Therefore, Alternates only lists a limited
amount of variant resources, centered on the datetime requested by the client.
Link: To compensate for the incomplete list of variant resources in
Alternates, an HTTP Link header [23] provides a pointer to a resource (the
TimeBundle, see Section 3.5) that supports retrieving a list of all variant
resources (Mementos), and their associated metadata.
X-Archive-Interval: Indicates the entire datetime interval for which the
archival server has Mementos for URI-R.
X-Datetime-Validity: Indicates the datetime interval during which the provided
representation was valid. Certain servers, including CMS and transactional
archives, can reliably provide this information. Others, such as crawler-
driven web archives cannot.
#### 3.4.1 Web servers with archival capabilities
Some web servers handle aspects of resource archiving natively, by maintaining
explicit information about the location and datetimes of archival records of
their resources, stored internally or remotely. Many CMS, Version Control
Systems, as well as the TTApache system [8] fall under this category. But also
servers that recurrently archive into a cloud store and keep track of the URIs
of the remote archival records fit in.
When a client is looking for Mementos of an original resource URI-R hosted by
these servers, they can handle the requests internally since all the
information that is required – URIs of Mementos and their datetimes – is
available. In this case, the set-up is as follows:
* •
URI-R itself becomes a transparently negotiable resource that supports DT-
conneg to provide access to all its available Mementos. In essence, URI-R
functions as its own TimeGate URI-G. Note that typical URI-Rs for these
systems either provide access to the current version of a resource, or to a
list of all its versions (each with its own URI-M), or to a combination of
both.
* •
All Mementos URI-Mi[URI-R@ti] of URI-R become variant resources for URI-R.
Figure 1 depicts a typical, successful, DT-conneg transaction flow for this
type of server, including the HTTP headers that are used. The transactional
behavior for less trivial cases are also considered in the Memento solution
but space prevents us from discussing them here. Such cases include requesting
Mementos for datetimes that are out of the date-range for which the server has
archival records, requesting Mementos for URI-Rs that no longer exist, and the
client providing a datetime which the server is unable to parse666Details:
http://mementoweb.org/guide/http/local.
Figure 1: DT-conneg for Web servers with archival capabilities: URI-R $=$
URI-G.
#### 3.4.2 Web servers without archival capabilities
Many other servers have no local archival capabilities
whatsoever. They host resources for which only a representation of the current
state can be retrieved, and are unaware of the details regarding the existence
of Mementos of their resources in other archival servers. Naturally, such a
server cannot redirect a client that requests an archival record of one of its
URI-Rs to an appropriate Memento. However, these systems can still play a
constructive role by redirecting the client to a server that is equipped to
handle the request: an archive of web resources. In this case, the set-up is
as follows:
* •
Upon detection of the X-Accept-Datetime header in the client’s request for
URI-R, the server merely redirects (using “HTTP 302 Found”) the client to an
archival server. Note that this is not a 302 redirection that is part of a
conneg transaction, as described in Section 2. Rather it is a 302 redirection
that results from detecting the X-Accept-Datetime header.
* •
The redirection is to a TimeGate
URI-G[URI-R] that the archival server makes available for the original
resource URI-R.
* •
The archive’s URI-G is a transparently negotiable resource that supports DT-
conneg to provide access to all the Mementos that the archive has available
for URI-R.
* •
All Mementos URI-Mi[URI-R@ti] that the archive has available for URI-R become
variant resources for its URI-G.
Figure 2 depicts a typical, successful, DT-conneg transaction flow for this
type of server, and includes the HTTP headers that are involved. Again, the
transactional behavior for less trivial cases is not covered here777Details:
http://mementoweb.org/guide/http/remote. In essence, the solution is the same
as in the above case, with the exception that the TimeGates reside on an
external archival server, not on the server that hosts the original resource
URI-R. This distinction raises two important questions.
First, to which archive should a server redirect? In order to help the client,
a server should redirect to an archive that has the best archival coverage of
its resources. Servers that have an associated transactional archive should
redirect to it, servers that have explicit recurrent crawling agreements with
systems such as Archive-It888http://www.archive-it.org/ should point there,
other servers may point at their country-specific archive (such as the
Finnish, Danish, Canadian, etc. archives), and in many cases servers can point
at the Internet Archive. Note that scenarios may be envisioned in which the
redirection is subject to configuration, for example, redirection to different
archives depending on archival time-range, media type, etc. Then again, this
problem of redirecting to a specific archive could be addressed by uniformly
pointing at an aggregator service that holds crucial metadata (e.g., URI-R,
URI-G, URI-M, ti) about Mementos available in a variety of archival servers,
and that exposes cross-archive TimeGates URI-G[URI-R]. In Section 3.5, we
introduce a discovery API for archives that enables the creation of such a
TimeGate aggregator.
Second, how does the server know the URI-G of the
TimeGate for its own URI-R on an external archival server? This problem can be
addressed by introducing archive-
specific or cross-archive conventions for the syntax for URI-G of TimeGates as
a function of URI-R. This would simply formalize the status-quo as all major
web archives that use the Heritrix/Wayback solution already use such
conventions. For example, the URI to retrieve a list of all archived versions
of http://cnn.com/ is:
http://web.archive.org/web/*/http://cnn.com/
Hence, the URI that could be used as a convention for the Internet Archive’s
TimeGate for http://cnn.com/ would be:
http://web.archive.org/web/timegate/http://cnn.com/
Such a convention seems achievable in the context of the International
Internet Preservation Consortium999http://www.netpreserve.org/ that has made
archive interoperability one of its goals. However, when a TimeGate aggregator
service is introduced, URI-G syntax conventions for individual archives are
not crucial; only a convention for the aggregator’s URI-G syntax would be
essential.
Figure 2: DT-conneg for Web servers without archival capabilities: URI-R
$\neq$ URI-G.
### 3.5 Discovering Mementos: TimeBundles and TimeMaps
For discovery purposes, we introduce the notion of a resource hosted by an
archival server, via which a full overview is available of all Mementos that
the archive holds for an original resource URI-R; we name such a resource a
TimeBundle. More formally, a TimeBundle for a resource URI-R, is a resource
URI-B[URI-R] that is an aggregation of: (a) all Mementos URI-Mi[URI-R@ti]
available from an archive, (b) the archive’s TimeGate URI-G for URI-R, (c) the
original resource URI-R itself.
Given the semantics of a TimeBundle, as an aggregation of a set of resources,
all of which share a temporal relationship with URI-R, we propose to model it
as an ORE Aggregation [34]. The ORE specifications comply with the Linked Data
conventions [5], and treat an ORE Aggregation as a non-information resource
[21] described by an information resource that is accessible via an HTTP 303
redirect from the URI of the ORE Aggregation. We name the information resource
that describes the TimeBundle a TimeMap; it is a specialization of an ORE
Resource Map. The TimeMap lists the URIs of all resources that are aggregated
in the TimeBundle, as well as metadata that is available about them. We have
not formally engaged in specifying which metadata to convey in TimeMaps, but
essentials such as archival datetime, media type, and language, as well as
more specific information such as digest, number of observations, validity
time-range [4] must be considered101010An example RDF/XML TimeMap as used in
our experiment is available at http://mementoweb.org/guide/api/map1.
TimeBundles made available by archives may be leveraged in real-time client
interaction, since their URI-B is expressed as the content of the HTTP Link
header (see the HTTP headers in Figures 1 and 2). And, when an archive makes
its TimeBundles discoverable using common approaches such as SiteMaps [12],
Atom Feeds [24], or OAI-PMH [19] they become a powerful mechanism for batch
harvesting of metadata that describes an archive’s entire collection, and that
can be used for the creation of cross-archive services.
### 3.6 A TimeGate Aggregator
If various archives implement TimeBundles and associated TimeMaps, and make
them discoverable using the aforementioned techniques, then information about
Mementos hosted by different archives can be harvested into an aggregator
service. For each original resource URI-R, for which Mementos exist in the
harvested archives, such an aggregator then minimally holds the distinct URI-
Ms of each of those Mementos in the various archives, as well as their
archival datetime, media type, language etc. This information allows the
aggregator to introduce TimeGates URI-G for each of the URI-Rs for which the
harvested archives have Mementos. The variant resources for any specific
TimeGate are the Mementos for URI-R as they exist in the distributed archives.
Because the aggregator has information on Mementos across archives, its time-
granularity is finer than that of any of the individual archives. This
provides the aggregator with a better range of possibilities when redirecting
a client to a Memento in response to a request for a specific datetime. In
essence, this aggregator behaves as the archival servers discussed in Section
3.4.2, but it has a broader coverage both regarding URI-Rs and Memento
datetimes, and it does not store the Mementos itself.
Figure 3 illustrates the value such an aggregator can bring to time travel. It
shows various Mementos for the noaa.gov home page as it was around the time of
Hurricane Katrina. In order to revive how the drama unfolded, inspecting
Mementos held by different archives is required. Indeed, both the content of
the Mementos as well as their archival server changes as time progresses. Note
also that, although the Internet Archive claims to have coverage for September
9 2005, the Memento is not really available (bottom left of Figure 3; it is
not known if this is a permanent or transient error); the next available
Memento is for September 10 2005, and is available from Archive-It. In cases
like this, an aggregator could support navigation across archives and across
time.
(a) Archive-It
Thu, 08 Sep 2005 17:48:47 GMT
(b) Internet Archive
Thu, 08 Sep 2005 21:07:05 GMT
(c) Internet Archive
Fri, 09 Sep 2005 01:58:48 GMT
(d) Archive-It
Sat, 10 Sep 2005 08:11:47 GMT
Figure 3: Distributed archive coverage of www.noaa.gov. 3(a) and 3(d) come
from Archive-It and 3(b) and 3(c) come from the Internet Archive. Note that
3(c) has either a transient or permament error.
## 4 Experiment
We have performed an experiment to demonstrate the feasibility of the proposed
DT-conneg framework involving a diverse array of components that jointly
realize web time travel across various servers. The deployed environment is
depicted in Figure 4. The arrows indicate the flow of HTTP interactions shown
in Figures 1 and 2, subject to the following considerations that are directly
related to conducting a time travel experiment in a Web that is not (yet) DT-
conneg enabled.
First, as it was not realistic to try and get active development involvement
from existing archival servers within the timeframe the reported work took
place, TimeGates and TimeBundles for several archives (CMS and web archives)
were not implemented natively within those systems but rather by-proxy. This
means that they were exposed by servers under our control, which obtained the
essential information from the archives using ad-hoc techniques such as screen
scraping. While it may seem that this approach undermines the essence of the
protocol-based DT-conneg framework, it actually is a strong illustration of
its feasibility: if one can scrape the essential information from archives’
pages, it is certainly available in their databases, and hence, native
implementation should be more straightforward than by-proxy. Also, while we
rely on a by-proxy approach for certain systems, demonstrations of the
feasibility of native implementation are also available.
Second, existing web servers do not currently detect the X-Accept-Datetime
header required for time travel, and hence cannot issue the essential “HTTP
302 Found” to a TimeGate (see Section 3.4.2). These servers will currently
respond as usual, typically with an “HTTP 200 OK” or “HTTP 404 Not Found”. In
order to still be able to demonstrate the DT-conneg framework in the
experiment, the remedy is to have the time travel client detect such responses
that are unexpected from the time travel perspective, and take control by
subsequently issuing the DT-conneg request directly to a TimeGate for URI-R
exposed by an archival server (native or by-proxy). In essence, in these
cases, the client fulfills the redirecting role that the host of URI-R
normally would in the DT-conneg framework. For servers outside of our control,
there was no other option than to resort to this client approach; for servers
under our control the redirect to a TimeGate was implemented natively.
The following is a description of the components involved in the experiment:
Web servers: We equipped domains under our own control with the capability to
honor DT-conneg requests by detecting the X-Accept-Datetime header, and
redirecting to TimeGates exposed by an appropriate archival server. This was
trivially implemented using an Apache mod_rewrite
rule111111See http://mementoweb.org/tools/apache for servers we could
configure:
http://lanlsource.lanl.gov/
http://odusource.cs.odu.edu/
http://digitalpreservation.gov/
(LANL, ODU, and LoC, respectively in Figure 4). For obvious reasons, we were
not able to implement this for servers beyond our control.
Archives: Wikipedia is a prominent example of the class of servers with local
archival capabilities. TimeGates (and TimeBundles) for it were implemented by-
proxy (Wiki proxy in Figure 4). However, to demonstrate the possibility of
native implementation, a plug-in was developed that adds X-Accept-Datetime and
TimeGate capabilities to the Wikimedia platform on which Wikipedia is
based121212Plug-in at http://mementoweb.org/tools/wiki. To cover for the class
of servers that lack local archival capabilities, TimeGates (and TimeBundles)
were implemented by-proxy for the Internet Archive (IA proxy in Figure 4), the
Internet Archive’s Archive-It (AI proxy in Figure 4), the Library of Congress’
Archive-It, the Government of Canada Web Archive, and WebCite. In addition, we
developed a transactional archive platform and deployed it at LANL and ODU
(LANL TA and ODU TA in Figure 4, respectively). As the LANL and ODU web
servers respond to client requests, the representations they serve are pushed
into these archives, yielding a high-resolution archival record of their
evolving resources. It is worth noting that the described selection covers a
broad range of commonly deployed archival solutions: CMS, web-crawler based
archives, on-user-demand archives, and transactional archives.
Aggregator: Furthermore, a TimeGate aggregator (Aggr in Figure 4) was
developed that collects archival metadata from the aforementioned web
archives’ TimeBundles (some by-proxy and some native), and can hence serve as
a common target for redirection. This collecting is currently done
dynamically: as a client requests a Memento for an original resource URI-R via
the aggregator, the aggregator contacts associated TimeBundles in various
archives, merges the returned TimeMap information, and only then redirects the
client to an appropriate Memento. This experimental approach makes retrieving
Mementos via the aggregator predictably slow.
Clients: We developed a FireFox plug-in that allows setting the browser to
time travel mode, and selecting a datetime for the journey. From there
onwards, the browser adds an X-Accept-Datetime header, with the datetime value
set by the user, to every HTTP GET issued. If all targeted servers would
implement the “HTTP 302 Found” redirection upon detection of the X-Accept-
Datetime header, only archival pages would be retrieved, and all links in
those pages would be interpreted as requests for Mementos. This effectively
happens for the servers under our control (the black flows labeled [1], [2]
and [4] in Figure 4). As described above, other servers do not exhibit this
behavior (the red flows [3] and [5] in Figure 4). Implementing the remedial
behavior where the client itself takes care of the redirection turned out not
to be trivial in the Mozilla plug-in framework as it does not support
intercepting and modifying responses131313See
https://wiki.mozilla.org/Firefox/Projects/Network_Error_Pages (e.g., on 404 or
200 response codes). The result is a time travel plug-in that deals perfectly
with URI-Rs of the servers under our control but not with any others. We then
decided to develop a time travel client that runs on a server and is developed
using the Apache mod_python framework that offered the required flexibility.
The resulting gateway client handles all flows of Figure 4 correctly, and
fully demonstrates the potential of the DT-conneg framework. It is accessible
via a web form that allows entering URI-R and a datetime. Upon submitting the
time travel request, the gateway client (not the browser) fulfills the DT-
conneg requests, and once completely handled, returns the resulting Memento
page to the browser. In order to allow for continued time travel of links in
the page, they need to be rewritten to point at the gateway client. This is
merely an artifact of a server-side, not a browser-based, implementation. This
client also depicts the HTTP transactions that take place during time travel,
and allows inspecting the HTTP headers involved.
With the above components in place, an experimental environment results that
effectively demonstrates the feasibility of web time travel using the Memento
solution. Two clients, both admittedly with respective restrictions, allow
navigating the past Web in very much the same way as the current Web is
browsed; they seamlessly move across web servers and archives (CMS-style and
web archives) using the HTTP protocol, extended with DT-conneg, to try and
return a Memento that meets the client’s preference. Due to the various by-
proxy components, and the dynamic implementation of the aggregator, the
navigation can often be slow. However, the navigations that involve the
servers with full native support (flows [1] and [2] in Figure 4), those that
bypass the aggregator (flows [1], [2] and [3]), and those for which the
aggregator can respond from its limited cache (some flows [4] and [5]) perform
noticeably faster, even using the gateway client. In addition, many well
understood techniques including batch harvesting, caching, and recurrent
refreshing are available to improve the performance of the aggregator and
fundamentally improve response times.
As an illustration of our results, Figure 5 shows two navigations conducted on
November 2 2009, around 16:25:00 UTC: one in real-time, and one in time travel
mode with a datetime set to October 12 2009 16:25:00 UTC. The captions in the
figure also indicate the flow of the HTTP interactions in the experimental
environment as indicated in Figure 4. The DT-conneg framework allows a re-
navigation of both in the future. It suffices to use these respective
datetimes as the DT-conneg value, and hope that archives have records of the
resources involved141414Try it at http://mementoweb.org/demo/client1.
Figure 4: The Memento Experiment environment.
(a) http://lanlsource.lanl.gov/hello - flow 1 in Figure 4
(b) http://en.wikipedia.org/MS_Oasis_of_the_Seas - flow 3 in Figure 4
(c) http://news.bbc.co.uk/ - flow 5 in Figure 4
Figure 5: Browsing in real time (Mon, 02 Nov 2009 16:25:00 GMT) on the left
and time travel (Mon, 12 Oct 2009 16:25:00 GMT) on the right.
## 5 Discussion
In this section, we touch upon issues pertaining to the Memento solution that
require further attention. The seamless integration of archives in regular web
navigation provided by DT-conneg allows exploring novel ways to address common
problems that result from the dynamics of the Web. Some of these have
tentatively been explored in our experiment. Consider the following cases:
A URI-R vanishes, but the server that used to serve it is still operational:
In this case, the server should still issue the redirect to a TimeGate upon
detection of the DT-conneg request. This allows seamless access to a Memento
of URI-R, even if the server no longer hosts the original.
A domain vanishes: The client is looking for a current representation of a
URI-R that was hosted by the domain, but fails. The client resorts to
interaction with other archives or with a TimeBundle aggregator and arrives at
the most recent Memento of the resource.
A domain is taken over by a new custodian: The new custodian adheres to other
policies regarding which archive to redirect a DT-conneg request. The client
understands from the X-Archive-Interval returned by that archive of choice,
that it does not cover the time range in which the previous custodian operated
the domain. The client resorts to interaction with other archives or with a
TimeBundle aggregator and arrives at an appropriate Memento.
Two aspects related to the integration of the proposed Memento solution into
the existing Web infrastructure require attention. First, when issuing a
request with an X-Accept-Datetime header to a server that hosts the original
resource URI-R, all caches between the client and the server must be bypassed
in order to avoid retrieving a current representation of URI-R. In our
experiment, we enforced this behavior through a combination of two client
request headers: “Cache-Control: no-cache” to force cache revalidation and
“If-Modified-Since: Thu, 01 Jan 1970 00:00:00 GMT” to make sure that
revalidation fails. Further research is required to find an alternative for
this admittedly inelegant approach. Ideally, it should leverage existing
caching practice but extend it in such a way that caches are only bypassed in
DT-conneg when essential, but still used whenever possible (e.g., to deliver
Mementos). Second, when it comes to listing variant resources in response
headers, the DT-conneg framework cannot operate according to the letter of RFC
2295. Indeed, the RFC states: “If a response from a transparently negotiable
resource includes an Alternates header, this header MUST contain the complete
variant list bound to the negotiable resource.” This mandate is based on a
perspective expressed in the RFC that “it is expected that a typical
transparently negotiable resource will have 2 to 10 variants, depending on its
purpose.” Clearly, TimeGates as proposed in the DT-conneg framework can have
many more than 10 Mementos. We do not think this makes the conneg framework
inapplicable to the datetime dimension, but rather we believe DT-conneg
introduces a challenge that the authors of the RFC did not anticipate ten
years ago. As described, we propose a solution based on a sizeable Alternates
header combined with an HTTP Link header that leads to a complete list of
variants; other options should be explored.
An interesting characteristic of the DT-conneg framework requires more
explicit attention. When requesting a Memento for a page that contains links
to external pages, or embedded resources such as images or videos, each of
those are requested with DT-conneg from the respective servers that
host/hosted the URI-R of those resources. This is a core characteristic that
the proposed time travel framework shares with regular web navigation. It
should be noted that this is not the current behavior of pages stored in web
archives. Indeed, in order to avoid filling out an archived page with current
representations of embedded resources, web archives rewrite URIs in archived
pages to point back into the archive at archived representations of those
resources. The same happens with links in archived pages, effectively turning
the archive into an island isolated from the rest of the Web. The upside of
this approach is that archived pages are self-contained: the page and its
embedded resources were typically crawled around the same time and hence the
archived page is likely to be a faithful reconstruction of what the original
looked like at the time of the crawl. The drawback of the approach is that
navigation is restricted to the archive’s island. Navigating beyond it to
obtain an archived version of a linked resource that is not available in the
archive but might be available elsewhere on the Web, is not possible. Further
exploration is required to arrive at a strategy for web archives that would at
the same time adhere to the self-containedness principle and allow external
navigation using the DT-conneg framework when beneficial.
Another challenge pertains to selecting a Memento that best meets the client’s
conneg preferences. There are two aspects to this problem. The first relates
to the archival datetime of the Memento that an archive should return in
response to a datetime expressed by a time travel client. For certain archives
the choice is straightforward. Indeed, transactional archives and servers such
as Wikipedia know exactly during which time interval a certain Memento
functioned as the active representation of URI-R (cf. the X-Datetime-Validity
discussed in Section 3.4). Hence, they can return the Memento that was active
at the datetime specified by the client. However for resources not hosted by
such servers, it will be rare that any archive has a Memento that perfectly
matches the client’s preference. In this case, an archive (or a TimeBundle
aggregator) must make a choice. A typical approach used by existing web
archives is to choose the Memento that is the “closest” in time, regardless of
whether its archival datetime is before or after the requested datetime. But
this approach is challenged when pages have embedded resources. The more
resources required to render a page, the more variation there will be between
the requested datetime and the archival datetimes of available Mementos. As a
matter of fact, when not being sensible about the selection of Mementos, the
resulting page may never actually have existed. A second challenge relates to
multi dimensional conneg that involves the datetime dimension. Current conneg
algorithms151515See, http://httpd.apache.org/docs/2.2/content-negotiation.html
deal with variant selection in the dimensions specified in RFC 2295. These
would need to be revised to include the datetime dimension: if a client
requests an HTML Memento for a specific datetime, but only a pdf is available,
what should the archival server do? Research is required to explore both
problems.
## 6 Related Work
The goal of adding a temporal aspect to web navigation has been explored in
projects that focus on user interface enhancement. The Zoetrope project [1]
provides a rich interface for querying and interacting with a set of archived
versions of selected seed pages. The interface leverages a local archive that
is assembled by frequently polling those seed pages. The Past Web Browser [16]
provides a simpler level of interaction with changing pages, but it is
restricted to navigating existing web archives such as the Internet Archive.
And DiffIE is a plug-in for Internet Explorer that emphasizes web content that
changed since a user’s previous visit by leveraging a dedicated client cache
[32]. None of these projects propose protocol enhancements but rather use ad-
hoc techniques to achieve their goals. All could benefit from DT-conneg as a
standard mechanism for accessing prior representations of resources.
Some projects have dealt with the problem of disappeared web pages and finding
archived or replacement copies on the Web. The use of lexical signatures as
search engine query terms was proposed as a way to find content that had moved
from its original URI [28, 29]. This approach was later applied to search for
content in web archives [13, 17]. Also, when a “HTTP 404 Not Found” occurs,
the ErrorZilla FireFox plug-in161616https://addons.mozilla.org/en-
US/firefox/addon/3336 presents a user with a search page allowing her to find
disappeared pages in web archives, and the UK National Archive’s server plug-
in redirects the client to an archive of its choice. As suggested (Section 5),
in the DT-conneg framework a client could intelligently react to 404s, and
when doing so leverage available re-finding approaches.
To the best of our knowledge, very little research has explored a protocol-
based solution to augment the Web with time travel capabilities. TTApache [8]
introduced a modified version of Apache that stored archived representations
in a local transactional archive (similar to the configuration illustrated in
Figure 1). Ad-hoc RPC-style mechanisms were used to access archived
representations given the URI of their original, e.g. “page.html?02-Nov-2009”
and
“page.html?now”. This approach reveals the local scope of the problem
addressed by TTApache, as opposed to the global perspective taken by the
proposed DT-conneg framework. Indeed, the query components are issued against
a specific server, and are not maintained when a client moves to another
server as is the case with the X-Accept-Datetime header of DT-conneg. TTApache
also allowed addressing archived representations using version numbers in
query
components rather than datetimes. This capability is similar to the deprecated
“Content-Version” header field from RFC 2068 [10] and other, similar expired
proposals (e.g., [27]). Such versioning features have not found wide-spread
adoption, presumably because their address space is tied to a specific
resource or server, and not universal like the datetime of DT-conneg.
## 7 Conclusions
In Web Archiving [22], Julien Masanès expresses a vision of a global grid of
web archives realized by interconnecting existing and future ones:
> Such a grid should link Web archives so that they together form one global
> navigation space like the live Web itself. This is only possible if they are
> structured in a way close enough to the original Web and if they are openly
> accessible.
We could not agree more, and feel that our Memento solution presents a
significant step towards achieving this vision. But our approach reaches
beyond it. Indeed, the navigation space that results from our proposal is not
“like the live Web itself”, it is the Web itself, as regular navigation and
time travel are integrated. Also, it does not restrict the global archival
grid to web archives but incorporates servers (such as CMS) on the live Web
that host archival content. The Memento solution is capable of realizing this,
and does not disrupt firmly established HTTP practice. Rather, it adds to it
an orthogonal time dimension. Moreover, the Memento solution does not disrupt
existing web archives or their established operating principles, but leverages
both by tightly integrating them into the web. Time travel can be ours.
## 8 Acknowledgments
This work sponsored in part by the Library of Congress.
## References
* [1] E. Adar, M. Dontcheva, J. Fogarty, and D. S. Weld. Zoetrope: interacting with the ephemeral web. In UIST ’08: Proceedings of the 21st annual ACM symposium on User interface software and technology, pages 239–248, 2008.
* [2] E. Adar, J. Teevan, and S. T. Dumais. Resonance on the web: web dynamics and revisitation patterns. In CHI ’09: Proceedings of the 27th international conference on Human factors in computing systems, pages 1381–1390, 2009.
* [3] E. Adar, J. Teevan, S. T. Dumais, and J. L. Elsas. The web changes everything: understanding the dynamics of web content. In WSDM ’09: Proceedings of the Second ACM International Conference on Web Search and Data Mining, pages 282–291, 2009.
* [4] A. Anand, S. Bedathur, K. Berberich, R. Schenkel, and C. Tryfonopoulos. Everlast: a distributed architecture for preserving the web. In JCDL ’09: Proceedings of the 9th ACM/IEEE-CS joint conference on Digital libraries, pages 331–340, 2009.
* [5] C. Bizer, R. Cyganiak, and T. Heath. How to publish linked data on the web, 2007. http://sites.wiwiss.fu-berlin.de/bizer/pub/LinkedDataTutorial/.
* [6] A. Brown. Archiving Websites: A practical guide for information management professionals. Facet Publishing, 2006.
* [7] B. F. Cooper and H. Garcia-Molina. Infomonitor: Unobtrusively archiving a World Wide Web server. International Journal on Digital Libraries, 5(2):106–119, April 2005.
* [8] C. E. Dyreson, H. Lin, and Y. Wang. Managing versions of web documents in a transaction-time web server. In WWW ’04: Proceedings of the 13th international conference on World Wide Web, pages 422–432, 2004.
* [9] D. Fetterly, M. Manasse, M. Najork, and J. Wiener. A large-scale study of the evolution of web pages. In WWW ’03: Proceedings of the 12th international conference on World Wide Web, pages 669–678, 2003.
* [10] R. Fielding, J. Gettys, J. Mogul, H. Frystyk, and T. Berners-Lee. Hypertex transfer protocol – HTTP/1.1, Internet RFC-2068, 1997.
* [11] K. Fitch. Web site archiving: an approach to recording every materially different response produced by a Website. In 9th Australasian World Wide Web Conference, Sanctuary Cove, Queensland, Australia, July, pages 5–9, 2003.
* [12] Google, Microsoft, and Yahoo. Sitemaps XML format, 2008. http://www.sitemaps.org/protocol.php.
* [13] T. L. Harrison and M. L. Nelson. Just-in-time recovery of missing web pages. In HYPERTEXT ’06: Proceedings of the Seventeenth ACM Conference on Hypertext and Hypermedia, pages 145–156, 2006.
* [14] K. Holtman and A. Mutz. Transparent content negotiation in HTTP, Internet RFC-2295, 1998.
* [15] I. Jacobs and N. Walsh. Architecture of the world wide web, volume one. Technical Report W3C Recommendation 15 December 2004, W3C, 2004.
* [16] A. Jatowt, Y. Kawai, S. Nakamura, Y. Kidawara, and K. Tanaka. Journey to the past: proposal of a framework for past web browser. In HYPERTEXT ’06: Proceedings of the seventeenth conference on Hypertext and hypermedia, pages 135–144, 2006.
* [17] M. Klein and M. L. Nelson. Revisiting lexical signatures to (re-)discover web pages. In ECDL ’08: Proceedings of the 12th European Conference on Research and Advanced Technology for Digital Libraries, pages 371 – 382, 2008\.
* [18] W. Koehler. Web page change and persistence — a four-year longitudinal study. Journal of the American Society for Information Science and Technology, 53(2):162–171, 2002.
* [19] C. Lagoze and H. Van de Sompel. The Open Archives Initiative: building a low-barrier interoperability framework. In JCDL ’01: Proceedings of the 1st ACM/IEEE-CS Joint Conference on Digital Libraries, pages 54–62, 2001.
* [20] J. Leskovec, L. Backstrom, and J. Kleinberg. Meme-tracking and the dynamics of the news cycle. In KDD ’09: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 497–506, 2009.
* [21] R. Lewis. Dereferencing HTTP URIs. Technical Report Draft Tag Finding 04 October 2007, 2007.
* [22] J. Masanès. Web Archiving. Springer-Verlag, 2006.
* [23] M. Nottingham. Web linking, Internet Draft draft-nottinghamgm-http-link-header-06, 2009\.
* [24] M. Nottingham and R. Sayre. The Atom syndication format, Internet RFC-4287, 2005.
* [25] A. Ntoulas, J. Cho, and C. Olston. What’s new on the web?: the evolution of the web from a search engine perspective. In WWW ’04: Proceedings of the 13th international Conference on World Wide Web, pages 1–12, 2004.
* [26] H. W. E. H. Obendorf, Hartmut and M. Mayer. Web page revisitation revisited: Implications of a long-term click-stream study of browser usage. In CHI ’07: Proceedings of the 25th international conference on Human factors in computing systems, pages 597–606, 2007.
* [27] K. Ota, K. Takahashi, and K. Sekiya. Version management with meta-level links via HTTP/1.1, Internet Draft draft-ntt-http-version-00, 1996.
* [28] S.-T. Park, D. M. Pennock, C. L. Giles, and R. Krovetz. Analysis of lexical signatures for improving information persistence on the World Wide Web. ACM Transactions on Information Systems, 22(4):540–572, 2004.
* [29] T. A. Phelps and R. Wilensky. Robust hyperlinks cost just five words each. Technical Report UCB/CSD-00-1091, EECS Department, University of California, Berkeley, 2000.
* [30] H. C. Rao, Y. Chen, and M. Chen. A proxy-based personal web archiving service. SIGOPS Operating Systems Review, 35(1):61–72, 2001.
* [31] L. Tauscher and S. Greenberg. How people revisit web pages: Empirical findings and implications for the design of history systems. International Journal of Human-Computer Studies, 47(1), 1997.
* [32] J. Teevan, S. T. Dumais, D. J. Liebling, and R. L. Hughes. Changing how people view changes on the web. In UIST ’09: Proceedings of the 22nd annual ACM symposium on User interface software and technology, pages 237–246, 2009.
* [33] R. Troncy, J. Jansen, Y. Lafon, E. Mannens, S. Pfeiffer, and D. V. Deursen. Use cases and requirements for media fragments, W3C Working Draft 30 April 2009, 2009.
* [34] H. Van de Sompel, C. Lagoze, M. L. Nelson, S. Warner, R. Sanderson, and P. Johnston. Adding escience assets to the data web. In Proceedings of the Linked Data on the Web Workshop (LDOW 2009), 2009.
|
arxiv-papers
| 2009-11-05T20:52:22 |
2024-09-04T02:49:06.315030
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Herbert Van de Sompel, Michael L. Nelson, Robert Sanderson, Lyudmila\n L. Balakireva, Scott Ainsworth, Harihar Shankar",
"submitter": "Michael Nelson",
"url": "https://arxiv.org/abs/0911.1112"
}
|
0911.1339
|
# Discovery of a 115 Day Orbital Period in the Ultraluminous X-ray Source NGC
5408 X-1
Tod E. Strohmayer Astrophysics Science Division, NASA’s Goddard Space Flight
Center, Greenbelt, MD 20771; tod.strohmayer@nasa.gov
###### Abstract
We report the detection of a 115 day periodicity in SWIFT/XRT monitoring data
from the ultraluminous X-ray source (ULX) NGC 5408 X-1. Our ongoing campaign
samples its X-ray flux approximately twice weekly and has now achieved a
temporal baseline of $\approx 485$ days. Periodogram analysis reveals a
significant periodicity with a period of $115.5\pm 4$ days. The modulation is
detected with a significance of $3.2\times 10^{-4}$. The fractional modulation
amplitude decreases with increasing energy, ranging from $0.13\pm 0.02$ above
1 keV to $0.24\pm 0.02$ below 1 keV. The shape of the profile evolves as well,
becoming less sharply peaked at higher energies. The periodogram analysis is
consistent with a periodic process, however, continued monitoring is required
to confirm the coherent nature of the modulation. Spectral analysis indicates
that NGC 5408 X-1 can reach 0.3 - 10 keV luminosities of $\approx 2\times
10^{40}$ ergs s-1. We suggest that, like the 62 day period of the ULX in M82
(X41.4+60), the periodicity detected in NGC 5408 X-1 represents the orbital
period of the black hole binary containing the ULX. If this is true then the
secondary can only be a giant or supergiant star.
black hole physics - galaxies: individual: NGC 5408 - stars: oscillations -
X-rays: stars - X-rays: galaxies
## 1 Introduction
The existence of black holes in the mass range from $10^{2}-10^{4}$
$M_{\odot}$–intermediate mass black holes (IMBH)–is still not widely accepted.
It has been argued based on their extreme luminosities that some of the bright
X-ray sources found in nearby galaxies, the ultraluminous X-ray sources
(ULXs), may be IMBHs (Colbert & Mushotzky 1999), œbut it has also been
suggested that these objects may appear luminous due to beaming of their X-ray
radiation (King et al. 2001). As yet there has been no direct measurement of
the mass of a ULX.
At present the best IMBH candidates include X41.4+60, the brightest ULX in the
starburst galaxy M82 (also referred to as M82 X-1, Kaaret, Feng & Gorski 2009;
Strohmayer & Mushotzky 2003), and the ULX NGC 5408 X-1 (Strohmayer et al.
2007; Kaaret & Corbel 2009). Very recently, Farrell et al. (2009) have
reported the identification of an X-ray source (2XMM J011028.1-460421) very
near the absorption line galaxay ESO 243-49. They argue for a physical
association with this galaxy which at a redshift of $z=0.0224$ implies a
luminosity in excess of $10^{42}$ erg s-1, and suggest it is an IMBH in excess
of 500 $M_{\odot}$. If the association with ESO 243-49 is correct then this
object is the most luminous ULX currently known.
X41.4+60 is also amongst the most luminous ULXs, on occasion having a
luminosity upwards of $10^{41}$ ergs s-1 (Kaaret et al. 2009). This object
also shows a 62 day periodic modulation in its X-ray flux, which has been
proposed to be the orbital period of the binary system containing the black
hole (Kaaret, Simet & Lang 2006; Kaaret & Feng 2007). Both NGC 5408 X-1 and
X41.4+60 show quasiperiodic oscillations (QPOs) in their X-ray fluxes, and
these ULX QPOs appear at systematically lower frequencies than the analogous
QPOs observed in Galactic black hole binary systems (Strohmayer & Mushotzky
2003; Strohmayer et al. 2007). Indeed, Strohmayer & Mushotzky (2009) have
recently shown that the QPO properties in NGC 5408 X-1 correlate with the
energy spectrum and X-ray flux in a manner fully consistent with the behavior
observed for so-called Type C QPOs in Galactic systems. The Type C QPOs in
stellar mass black holes are strong (fractional rms amplitude $\sim 15\%$),
relatively coherent ($\nu_{qpo}/\Delta\nu>10$) oscillations with
characteristic frequencies of $\sim 1-10$ Hz that vary in frequency in
correlation with source flux and spectral index (see Sobczak et al 2000;
Vignarca et al. 2003; Casella, Belloni & Stella 2005). They are associated
with an energy spectral state in which approximately half or more of the flux
is carried by a power-law component with slope $\Gamma\sim 2-2.5$, commonly
referred to as the Steep Power Law state (SPL, McClintock & Remillard 2006),
or the Hard Intermediate State (HIMS, Belloni 2006). The precise origin of
these QPOs is still uncertain, but their phenomenology has been used to
estimate the masses of black holes (see Shaposhnikov & Titarchuk 2009).
The Type C QPOs in NGC 5408 X-1 identified by Strohmayer & Mushotzky (2009)
have frequencies of a few tens of mHz, a factor of 100 lower than the Galactic
black holes. Scaling the observed QPO frequencies in NGC 5408 X-1 to those
observed in Galactic systems of known mass suggests that NGC 5408 X-1 is an
IMBH with a mass greater than 1000 $M_{\odot}$ (Strohmayer & Mushotzky 2009).
Moreover, recent optical spectroscopy reported by Kaaret & Corbel (2009)
indicates that the optical counterpart to NGC 5408 X-1 is associated with
reprocessed emission from an accretion disk as well as a strongly photoionized
optical nebula. This, combined with the presence of a powerful radio nebula
also appears consistent with the presence of an IMBH (Lang et al. 2007; Soria
et al. 2006; Kaaret et al. 2003).
While it is likely that many of the ULXs are accreting black hole binary
systems, very little is known about the nature of these putative binaries. If
some of the brighter systems are indeed IMBHs accreting via Roche lobe
overflow, then their orbital periods could be as long as a few hundred days
(Portegies Zwart et al. 2004). With the exception of X41.4+60 noted above,
very few objects have been monitored with a cadence and duration that would
enable sensitive searches for X-ray periods in the range of 10s to 100s of
days. The Swift observatory uniquely provides both the X-ray imaging
sensitivity and scheduling flexibility to enable such searches. Starting with
Observing Cycle 4 Swift has been monitoring NGC 5408 X-1 several times per
week as part of an approved program. These observations have continued into
Cycle 5 and are presently ongoing. Here we present results from this campaign
which provide evidence for a 115 day periodicity in NGC 5408 X-1 which could
very likely be the orbital period of a Roche lobe overflow binary containing
an IMBH.
## 2 Data Extraction and Analysis
Monitoring of NGC 5408 X-1 with the Swift XRT began on 2008 April 9. Since
then, the source has been observed 2 - 3 times per week with a typical
exposure of 2 ksec. Aside from an $\approx 80$ day gap beginning on 2008
September 26, the observing cadence has been unbroken. Results presented here
include observations through 2009 August 5.
We began our analysis with the Level 2 cleaned XRT event files. The XRT was
operated in photon counting mode. We extracted good events using an extraction
radius of 25” around the position of NGC 5408 X-1. Most observation sequences
were made up of one to two good time intervals (GTI). In general, we combined
these to create a single count rate estimate for each observation sequence. An
important issue is to account for bad pixels and columns present on the XRT
CCD. The University of Leicester’s XRT Digest webpage (see,
http://www.swift.ac.uk/xrtdigest.shtml#analysis) provides a detailed
discussion of the problem. In order to account for the potential loss of
effective exposure due to bad pixels we constructed exposure maps for each
observation using the tool xrtexpomap. We then integrated over the extraction
region in the exposure map, weighted by the XRT point spread function (Moretti
et al. 2006). This provides an effective exposure for each observation. We
used the corrected exposures in order to estimate source counting rates for
each observation. This procedure resulted in a total of 113 count rate
estimates. The resulting light curve is shown in Figure 1. The mean rate is
0.061 s-1, and excursions from 0.01 to 0.1 s-1 are evident.
We extracted spectra from each observation using the same extraction regions
as for the count rate estimates. Background estimates were extracted from
regions of the same size nearby on the CCD. Previous observations of NGC 5408
X-1 suggest that the spectrum does not change dramatically (Kaaret et al.
2003; Strohmayer et al. 2007; Strohmayer & Mushotzky 2009), so we combined all
observations into a single, average spectrum in order to obtain an average
count rate to flux conversion. We fit the resulting spectrum with the sum of a
power-law and disk blackbody (model diskpn in XSPEC), and fixed the absorbing
column at the value measured previously (Strohmayer et al. 2007). We find a
disk temperature $kT_{max}=0.18\pm 0.02$ keV and power-law index
$\Gamma=2.6\pm 0.2$. These values are reasonably consistent with previous XMM-
Newton measurements. We find that with this spectral form an XRT count rate of
0.06 s-1 corresponds to a 0.3 - 10 keV unabsorbed flux of $4.0\times 10^{-12}$
ergs cm-2 s-1. At a distance of 4.8 Mpc this corresponds to a luminosity of
$1.1\times 10^{40}$ ergs s-1. The implied peak luminosity during these
observations was $\approx 1.9\times 10^{40}$ ergs s-1. We quote fluxes and
luminosities in the 0.3 - 10 keV range for comparison with previous
measurements, however, we note that extending the band pass down to 0.1 keV
increases these luminosity estimates by almost a factor of two.
We carried out a periodogram analysis using the methods of Scargle (1982) and
Horne & Baliunas (1986). We calculated the periodogram at 140 frequency
points, which is the number of independent frequencies estimated by the method
of Horne & Baliunas (1986), and we normalized by the total variance in the
data. The resulting power spectrum is shown in Figure 2. The highest peak in
the spectrum appears at a frequency of $1.0018\times 10^{-7}$ Hz
(corresponding to a period of 115.5 days), and has a value of 24.7. We
estimate the significance of the peak by calculating the chance probability
for obtaining this value from the $\chi^{2}$ distribution with 2 degrees of
freedom, and multiplying by the number of trials (140 frequency bins). This
yields a significance of $6.1\times 10^{-4}$, which is better than a $3\sigma$
detection. There is no evidence for significant power in any other frequency
range. The coherence $Q$, defined as the center frequency divided by the width
of the peak (measured as the FWHM), is 4.7 and is consistent with a periodic
modulation. However, the present data cover only a bit more than 4 cycles, so
further monitoring will be required to better assess the coherence of the
modulation.
We note that Kaaret & Feng (2009) have recently submitted a paper to astro-ph
(0907.5415) which reports results using some of the same data presented here.
We became aware of this work after beginning work on this manuscript. Kaaret &
Feng do not claim to detect a significant periodicity in the NGC 5408 X-1
light curve, but they do state that the highest peak they see is “near a
period of 115 days.” We believe the reason for this seeming discrepancy is
two-fold. First, we have used more data in our analysis, that is, a longer
temporal baseline, and secondly, we have included the exposure map corrections
in our light curve. Indeed, when we restrict our analysis to the time range
used by Kaaret & Feng, and ignore the exposure corrections, then we see a drop
in the peak power at 115 days to a level consistent with Kaaret & Feng. The
fact that the exposure correction increases the Fourier power is further
evidence that the 115 day modulation is a real signal.
To further explore the nature of the modulation we folded the data in several
energy bands at the measured period of 115.5 days. To do this we placed each
light curve measurement in its appropriate phase bin, and then we averaged all
measurements in a given bin. Errors were determined by averaging the
individual errors in quadrature. The resulting profiles in twelve phase bins
are shown in Figure 3. We show profiles in the full band ($E>0.2$ keV, upper
left), two soft bands ($E<1$ keV, upper right; $E<0.7$ keV, lower left), and a
hard band ($E>1$ keV, lower right), and we plot two cycles for clarity. We fit
a model to each profile that includes two Fourier components (the fundamental
and first harmonic), $I=A+B\sin 2\pi(\phi-\phi_{0})+C\sin
4\pi(\phi-\phi_{1})$. Results of these fits are summarized in Table 1. For the
hard band ($E>1$ keV) profile the harmonic term is not statistically required.
The folded profiles reveal interesting energy dependent behavior. For example,
the modulation amplitude appears to clearly decrease with increasing energy.
Defining the fractional amplitude $f_{amp}=(B^{2}+C^{2})^{1/2}/A$, we find a
significant change from $0.133\pm 0.018$ to $0.241\pm 0.017$ in going from
$E>1$ to $E<1$ keV. Indeed, recomputing the periodogram with only $E<1$ keV
photons results in an increase in the peak Fourier power to 27.33. The
detection significance then improves to $3.2\times 10^{-4}$, which includes an
additional factor of 2 for the increased number of trials. In addition to the
amplitude variations the profile appears smoother at higher energies, and the
phase of the fundamental component is significantly different above 1 keV than
below, in the sense that the harder photons appear to lag the soft photons
with a relative phase difference of $0.11\pm 0.025=12.6\pm 2.9$ days. Finally,
we computed the hardness ratio as a function of phase, where we define the
hardness ratio as the count rate for $E>1$ keV divided by the rate for $E<1$
keV. Figure 4 shows the variation of the hardness with phase, and we have also
plotted the full-band folded profile for comparison (dashed curve). The
hardness ratio shows a sharp rise to a maximum near phase 0.45, followed by a
more gradual decline. The peak of the modulation is clearly softer than the
minimum.
## 3 Discussion and Implications
A number of accreting binaries show X-ray modulations at their orbital
periods. Among black hole binaries Cyg X-1 and Cyg X-3 are well known examples
(Wen et al. 1999; Elsner 1980), and more recent detections include LMC X-3
(Boyd et al. 2001), 1E 1740.7-2942 and GRS 1758-258 (Smith et al. 2002). Many
neutron star binaries also show X-ray modulations at the binary orbital
period. The orbital modulations in accreting binaries have typically been
attributed to periodic obscuration produced by a vertically and azimuthally
structured accretion disk or scattering in an extending accretion disk corona
or stellar wind from the donor (Parmar & White 1988; Wen et al. 1999). A
recent example of an X-ray modulation at the putative orbital period in a
neutron star system is that of GX 13+1 (Corbet 2003). Thus, if ULXs are indeed
accreting black hole binaries, then it is not unexpected to detect X-ray
modulations at their orbital periods.
While their have been several claims of detection of orbital modulations in
ULXs (see, Kaaret, Simet & Lang 2006 for a brief summary), the most compelling
detection to date is that of the 62 day modulation from the M82 ULX X41.4+60
(Kaaret, Simet & Lang 2006; Kaaret & Feng 2007). The average modulation in
X41.4+60 as observed with the RXTE/PCA is roughly sinusoidal with an amplitude
of $\approx 20\%$ (Kaaret & Feng 2007). This is qualitatively consistent with
the $0.2<E<8$ keV modulation amplitude and profile that we report here for NGC
5408 X-1, and suggests that similar physical processes may produce the
observed modulation in both systems.
Superorbital periods are known in both neutron star and black hole binaries.
Well known examples include the 35 day modulation in the neutron star system
Her X-1, and the 164 day period in the putative black hole binary SS 433
(Wijers & Pringle 1999). These variations have generally been ascribed to
accretion disk precession (Ogilvie & Dubus 2001). Kaaret & Feng (2007) argued
that the 62 day period in X41.4+60 was unlikely to be a superorbital
modulation because the observed period is only consistent with the observed
superorbital periods of neutron star systems, but that the extreme luminosity
of X41.4+60 comfortably rules out an accreting neutron star (see their Figure
4 for a distribution of observed superorbital periods). Similar arguments
would appear valid for the 115 day modulation in NGC 5408 X-1. The 115 day
period is shorter than all known superorbital periods for black hole candidate
binaries, and its peak luminosity is well in excess of the Eddington limit for
a neutron star ($\approx 10^{38}$ ergs s-1).
Using X-ray timing measurements with XMM-Newton, Strohmayer et al. (2009) have
recently shown that the strong QPO observed in NGC 5408 X-1 varies in
frequency and amplitude with changes in the X-ray flux and energy spectrum in
a manner that closely mimics the correlated temporal and spectral variations
observed in stellar mass black holes in the so-called Intermediate State (also
known as the Steep Power-law State). Moreover, recent optical spectroscopy
reported by Kaaret & Corbel (2009) indicates that its optical counterpart has
a significant contribution from reprocessing of the X-ray luminosity in the
outer parts of the disk. The so-called “slim disks” that may exist at super-
Eddington accretion rates preferentially radiate more flux out along the disk
axis, and therefore have relatively less available for reprocessing
(Abramowicz et al 1988). These models thus have difficulty accounting for the
observed optical to X-ray flux ratio in NGC 5408 X-1. Further, modeling of the
observed nebular optical emission lines supports the conclusion that NGC 5408
X-1 radiates $\sim 10^{40}$ ergs s-1 in an approximately isotropic manner
(Kaaret & Corbel 2009). These observations support the notion that NGC 5408
X-1 harbors a “standard” thin, optically thick accretion disk similar to those
inferred to exist in Galactic black hole binaries.
The 115 day modulation we have found in NGC 5408 X-1 shows interesting energy
dependent effects that would appear consistent with orbital modulation. A
system which shows qualitatively similar effects to those seen in NGC 5408 X-1
is Cyg X-1. Specifically, the orbital modulation in Cyg X-1 has been modeled
in the context of absorption and scattering of X-rays in the partially ionized
wind from the companion star (Wen et al. 1999). This model predicts a decrease
in modulation amplitude with increasing energy, and a minimum in the hardness
ratio at the maximum of the modulation profile (see Wen et al. 1999, Figure
3), both of which are evident in the NGC 5408 X-1 data. While the shapes of
the modulation profiles in Cyg X-1 do not exactly match the results we find
for NGC 5408 X-1, the behavior of the $E>1$ profile (see Figure 3) appears
qualitatively similar. An important distinction would seem to be that in the
case of NGC 5408 X-1 we can directly observe the thermal disk flux whereas the
orbital modulation measurements for Cyg X-1 concern the low-hard state, when
presumably its thermal flux is weak or absent. Moreover, any disk flux appears
largely shortward of the RXTE/PCA energy band. Cyg X-1 is also a wind
accreting system, whereas wind accretion would likely be unable to account for
the high luminosity of NGC 5408 X-1.
If the 115 day period is indeed the orbital period of a Roche lobe filling
binary, then the mean density of the secondary is constrained to be
$\rho_{mean}\approx 0.2(P_{\rm days})^{-2}$ g cm-3 (Frank et al. 2002). For a
period of 115.5 days this gives $\rho_{mean}=1.50\times 10^{-5}$ g cm-3, and
the companion would have to be a giant or supergiant star. Recent observations
suggest that a substantial fraction of the optical counterpart is due to
reprocessing of the X-ray flux in an accretion disk, and while the companion
star still has not been observed directly, the recent optical measurements
suggest that it is probably a giant in the 3 - 5 $M_{\odot}$ range, and with a
spectral type of B or later (Kaaret & Corbel 2009). Binary evolution
calculations for IMBHs and massive companions appear consistent with the
notion that NGC 5408 X-1 contains a $\sim 1000M_{\odot}$ black hole with a 3 -
5 $M_{\odot}$ companion (Li 2004). Continued monitoring of NGC 5408 X-1 with
Swift is important in order to confirm the orbital nature of the modulation.
## References
* Abramowicz et al. (1988) Abramowicz, M. A., Czerny, B., Lasota, J. P., & Szuszkiewicz, E. 1988, ApJ, 332, 646.
* Belloni (2006) Belloni, T. 2006, Advances in Space Research, 38, 2801.
* Boyd et al. (2001) Boyd, P. T., Smale, A. P., & Dolan, J. F. 2001, ApJ, 555, 822.
* Casella et al. (2005) Casella, P., Belloni, T., & Stella, L. 2005, ApJ, 629, 403.
* Colbert & Mushotzky (1999) Colbert, E. J. M. & Mushotzky, R. F. 1999, ApJ, 519, 89.
* Corbet (2003) Corbet, R. H. D. 2003, ApJ, 595, 1086.
* Elsner et al. (1980) Elsner, R. F., Ghosh, P., Darbro, W., Weisskopf, M. C., Sutherland, P. G., & Grindlay, J. E. 1980, ApJ, 239, 335.
* Farrell et al. (2009) Farrell, S. A., Webb, N. A., Barret, D., Godet, O., & Rodrigues, J. M. 2009, Nature, 460, 73.
* Frank et al. (2002) Frank, J., King, A., & Raine, D. J. 2002, Accretion Power in Astrophysics, by Juhan Frank and Andrew King and Derek Raine, pp. 398. ISBN 0521620538. Cambridge, UK: Cambridge University Press, February 2002.
* Horne & Baliunas (1986) Horne, J. H., & Baliunas, S. L. 1986, ApJ, 302, 757.
* Kaaret et al. (2009) Kaaret, P., Feng, H., & Gorski, M. 2009, ApJ, 692, 653.
* Kaaret & Corbel (2009) Kaaret, P., & Corbel, S. 2009, ApJ, 697, 950.
* Kaaret & Feng (2009) Kaaret, P., & Feng, H. 2009, ApJ, 702, 1679.
* Kaaret & Feng (2007) Kaaret, P., & Feng, H. 2007, ApJ, 669, 106.
* Kaaret et al. (2006) Kaaret, P., Simet, M. G., & Lang, C. C. 2006, ApJ, 646, 174.
* Kaaret et al. (2003) Kaaret, P., Corbel, S., Prestwich, A. H., & Zezas, A. 2003, Science, 299, 365.
* King et al. (2001) King, A. R., Davies, M. B., Ward, M. J., Fabbiano, G. & Elvis, M. 2001, ApJ, 552, L109.
* Lang et al. (2007) Lang, C. C., Kaaret, P., Corbel, S., & Mercer, A. 2007, ApJ, 666, 79.
* Li (2004) Li, X.-D. 2004, ApJ, 616, L119.
* McClintock & Remillard (2006) McClintock, J. E., & Remillard, R. A. 2006, Compact stellar X-ray sources, 157.
* Moretti et al. (2006) Moretti, A., et al. 2006, Gamma-Ray Bursts in the Swift Era, 836, 676.
* Ogilvie & Dubus (2001) Ogilvie, G. I., & Dubus, G. 2001, MNRAS, 320, 485.
* Parmar & White (1988) Parmar, A. N., & White, N. E. 1988, Memorie della Societa Astronomica Italiana, 59, 147.
* Portegies Zwart et al. (2004) Portegies Zwart, S. F., Dewi, J., & Maccarone, T. 2004, MNRAS, 355, 413.
* Scargle (1982) Scargle, J. D. 1982, ApJ, 263, 835.
* Shaposhnikov & Titarchuk (2009) Shaposhnikov, N., & Titarchuk, L. 2009, ApJ, 699, 453.
* Smith et al. (2002) Smith, D. M., Heindl, W. A., & Swank, J. H. 2002, ApJ, 578, L129.
* Sobczak et al. (2000) Sobczak, G. J., McClintock, J. E., Remillard, R. A., Cui, W., Levine, A. M., Morgan, E. H., Orosz, J. A., & Bailyn, C. D. 2000, ApJ, 531, 537.
* Soria et al. (2006) Soria, R., Fender, R. P., Hannikainen, D. C., Read, A. M., & Stevens, I. R. 2006, MNRAS, 368, 1527.
* Strohmayer & Mushotzky (2009) Strohmayer, T. E. & Mushotzky, R. F., ApJ, in press.
* Strohmayer et al. (2007) Strohmayer, T. E., Mushotzky, R. F., Winter, L., Soria, R., Uttley, P., & Cropper, M. 2007, ApJ, 660, 580.
* Strohmayer & Mushotzky (2003) Strohmayer, T. E., & Mushotzky, R. F. 2003, ApJ, 586, L61.
* Vignarca et al. (2003) Vignarca, F., Migliari, S., Belloni, T., Psaltis, D., & van der Klis, M. 2003, A&A, 397, 729.
* Wen et al. (1999) Wen, L., Cui, W., Levine, A. M., & Bradt, H. V. 1999, ApJ, 525, 968.
* Wijers & Pringle (1999) Wijers, R. A. M. J., & Pringle, J. E. 1999, MNRAS, 308, 207.
Figure 1: X-ray lightcurve of NGC 5408 X-1 in the 0.2 - 8 keV band derived
from Swift/XRT measurements. A count rate of 0.06 s-1 corresponds to a 0.3 -
10 keV unabsorbed flux of $4\times 10^{-12}$ ergs cm-2 s-1. The mean count
rate of 0.061 s-1 is marked by the dashed horizontal line. Time zero
corresponds to MJD 54565.85847 (UTC).
Figure 2: Lomb-Scargle periodogram computed from the Swfit/XRT light curve of
NGC 5408 X-1. There are 140 independent frequency bins plotted with a
resolution of 9.54 nanoHz. The Nyquist frequency is 1.35 $\mu$Hz
(corresponding to a period of 8.6 days). The horizontal dashed line denotes
the $3\sigma$ detection level. The peak value of 24.7 occurs at a frequency of
$1.0018\times 10^{-7}$ Hz (a period of 115.5 days).
Figure 3: X-ray flux from NGC 5408 X-1 in different energy bands folded at a
period of 115.5 days. Profiles were computed with 12 phase bins, and two
cycles are shown for clarity. Proceeding clock-wise from the upper left panel
the energy bands are, $0.2<E<8$ keV; $0.2<E<1$ keV; $1<E<8$ keV; and
$0.2<E<0.7$ keV. The best fitting model with two Fourier components is also
plotted in each panel. Note the change in profile amplitude and shape with
energy. See Table 1 for a summary of the model fits, and the text for
additional discussion.
Figure 4: Hardness ratio as a function of phase of the 115.5 day modulation in
NGC 5408 X-1. The hardness ratio is defined as the ratio of count rates for
photons with energy $>1$ keV to that for photons with energy $<1$ keV. We also
plot the full-band (0.2 - 8 keV) folded profile for comparison (dashed curve).
Table 1: Results of Pulse Profile Modeling for NGC 5408 X-111Summary of fits to energy dependent pulse profiles for NGC 5408 X-1. The fitted model is $I=A+B\sin[2\pi(\phi-\phi_{0})]+C\sin[4\pi(\phi-\phi_{1})]$. Profile | A ($10^{-2}$) | B ($10^{-2}$) | $\phi_{0}$ | C ($10^{-2}$) | $\phi_{1}$ | $f_{amp}$aaFractional modulation amplitude defined as $(B^{2}+C^{2})^{1/2}/A$.
---|---|---|---|---|---|---
$E>0.2$ keV | $6.23\pm 0.05$ | $1.10\pm 0.08$ | $0.915\pm 0.011$ | $0.27\pm 0.07$ | $0.46\pm 0.02$ | $0.182\pm 0.013$
$E<0.7$ keV | $2.09\pm 0.03$ | $0.48\pm 0.05$ | $0.874\pm 0.014$ | $0.14\pm 0.04$ | $0.473\pm 0.024$ | $0.238\pm 0.022$
$E<1$ keV | $3.64\pm 0.04$ | $0.83\pm 0.06$ | $0.882\pm 0.011$ | $0.27\pm 0.06$ | $0.485\pm 0.016$ | $0.241\pm 0.017$
$E>1$ keV | $2.64\pm 0.03$ | $0.34\pm 0.05$ | $0.992\pm 0.023$ | $0.096\pm 0.05$ | $0.853\pm 0.040$ | $0.133\pm 0.018$
|
arxiv-papers
| 2009-11-06T19:52:43 |
2024-09-04T02:49:06.326172
|
{
"license": "Public Domain",
"authors": "Tod E. Strohmayer",
"submitter": "Tod E. Strohmayer",
"url": "https://arxiv.org/abs/0911.1339"
}
|
0911.1417
|
# On a spectral sequence for twisted cohomologies
Weiping Li Department of Mathematics, Oklahoma State University, Stillwater,
OK 74078, U.S.A wli@math.okstate.edu , Xiugui Liu School of Mathematical
Sciences and LPMC, Nankai University, Tianjin 300071, P.R.China
xgliu@nankai.edu.cn and He Wang School of Mathematical Sciences, Nankai
University, Tianjin 300071, P.R.China wanghe85@yahoo.com.cn
###### Abstract.
Let ($\Omega^{\ast}(M),d$) be the de Rham cochain complex for a smooth compact
closed manifolds $M$ of dimension $n$. For an odd-degree closed form $H$,
there are a twisted de Rham cochain complex $(\Omega^{\ast}(M),d+H_{\wedge})$
and its associated twisted de Rham cohomology $H^{*}(M,H)$. We show that there
exists a spectral sequence $\\{E^{p,q}_{r},d_{r}\\}$ derived from the
filtration $F_{p}(\Omega^{\ast}(M))=\bigoplus_{i\geq p}\Omega^{i}(M)$ of
$\Omega^{\ast}(M)$, which converges to the twisted de Rham cohomology
$H^{*}(M,H)$. We also show that the differentials in the spectral sequence can
be given in terms of cup products and specific elements of Massey products as
well, which generalizes a result of Atiyah and Segal. Some results about the
indeterminacy of differentials are also given in this paper.
###### Key words and phrases:
Spectral sequence, twisted de Rham cohomology, Massey product, differential
###### 2000 Mathematics Subject Classification:
Primary 58J52; Secondary 55T99, 81T30
The second author was partially supported by NCET and NNSFC (No. 10771105).
## 1\. Introduction
Let $M$ be a smooth compact closed manifold of dimension $n$, and
$\Omega^{\ast}(M)$ the space of smooth differential forms over $\mathbb{R}$ on
$M$. We have the de Rham cochain complex $(\Omega^{\ast}(M),d),$ where
$d:\Omega^{p}(M)\rightarrow\Omega^{p+1}(M)$ is the exterior differentiation,
and its cohomology $H^{\ast}(M)$ (the de Rham cohomology). The de Rham
cohomology with coefficients in a flat vector bundle is an extension of the de
Rham cohomology.
The twisted de Rham cohomology was first studied by Rohm and Witten [14] for
the antisymmetric field in superstring theory. By analyzing the massless
fermion states in the string sector, Rohm and Witten obtained the twisted de
Rham cochain complex $(\Omega^{\ast}(M),d+H_{3})$ for a closed 3-form $H_{3}$,
and mentioned the possible generalization to a sum of odd closed forms. A key
feature in the twisted de Rham cohomology is that the theory is not integer
graded but (like K-theory) is filtered with the grading mod $2$. This has a
close relation with the twisted K-theory and the Atiyah-Hirzebruch spectral
sequence (see [1]).
Let $H$ be $\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$, where $H_{2i+1}$ is a
closed $(2i+1)$-form. Then one can define a new operator $D=d+H$ on
$\Omega^{\ast}(M)$, where $H$ is understood as an operator acting by exterior
multiplication (for any differential form $w$, $H(w)=H\wedge w$). As in [1,
14], there is a filtration on $(\Omega^{\ast}(M),D)$:
(1.1) $K_{p}=F_{p}(\Omega^{\ast}(M))=\bigoplus\limits_{i\geq p}\Omega^{i}(M).$
This filtration gives rise to a spectral sequence
(1.2) $\\{E^{p,q}_{r},d_{r}\\}$
converging to the twisted de Rham cohomology $H^{\ast}(M,H)$ with
(1.3) $E_{2}^{p,q}\cong\left\\{\begin{array}[]{ll}H^{p}(M)&~{}~{}~{}\mbox{$q$
is even,}\\\ 0&~{}~{}~{}\mbox{$q$ is odd.}\end{array}\right.$
For convenience, we first fix some notations in this paper. The notation $[r]$
denotes the greatest integer part of $r\in{\mathbb{R}}$. In the spectral
sequence (1.2) $\\{E^{p,q}_{r},d_{r}\\}$, for any $[y_{p}]_{k}\in
E_{k}^{p,q}$, $[y_{p}]_{k+l}$ represents its class to which $[y_{p}]_{k}$
survives in $E_{k+l}^{p,q}.$ In particular, as in Proposition 3.4, for
$x_{p}\in E_{1}^{p,q}$, $[x_{p}]_{2}=[x_{p}]_{3}\in E_{2}^{p,q}=E_{3}^{p,q}$
represents the de Rham cohomology class $[x_{p}]$. $d_{r}[x_{p}]$ represents a
class in $E_{2}^{p+r,q-r+1}$ which survives to $d_{r}[x_{p}]_{r}\in
E_{r}^{p+r,q-r+1}$.
In the appendix I of [14], Rohm and Witten first gave a description of the
differentials $d_{3}$ and $d_{5}$ for the case when $D=d+H_{3}$. Atiyah and
Segal [1] showed a method about how to construct the differentials in terms of
Massey products, and gave a generalization of Rohm and Witten’s result: the
iterated Massey products with $H_{3}$ give (up to sign) all the higher
differentials of the spectral sequence for the twisted cohomology (see [1,
Proposition 6.1]). Mathai and Wu in [9, p. 5] considered the general case that
$H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$ and claimed, without proof, that
$d_{2}=d_{4}=\cdots=0$, while $d_{3}$, $d_{5}$, $\cdots$ are given by the cup
products with $H_{3}$, $H_{5}$, $\cdots$ and by the higher Massey products
with them. Motivated by the method in [1], we give an explicit description of
the differentials in the spectral sequence (1.2) in terms of Massey products.
We now describe our main results. Let $A$ denote a defining system for the
$n$-fold Massey product $\langle x_{1},x_{2},\cdots,x_{n}\rangle$ and $c(A)$
its related cocycle (see Definition 5.1). Then
(1.4) $\langle x_{1},x_{2},\cdots,x_{n}\rangle=\\{c(A)|A~{}{\rm
is~{}a~{}defining~{}system~{}for}\langle x_{1},x_{2},\cdots,x_{n}\rangle\\}$
by Definition 5.3. To obtain our desired theorems by specific elements of
Massey products, we restrict the allowable choices of defining systems for
Massey products (cf. [15]). By Theorems 4.1 and 4.3 in this paper, there are
defining systems for the two Massey products we need (see Lemma 5.5). The
notation
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}$ in
Theorem 1.1 below denotes a cohomology class in $H^{\ast}(M)$ represented by
$c(A)$, where $A$ is a defining system obtained by Theorem 4.1 (see Definition
5.6). Similarly, the notation
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{A}$ in
Theorem 1.2 below denotes a cohomology class in $H^{\ast}(M)$ represented by
$c(A)$, where $A$ is a defining system obtained by Theorem 4.3 (see Definition
5.6).
###### Theorem 1.1.
For $H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$ and $[x_{p}]_{2t+3}\in
E_{2t+3}^{p,q}$ $(t\geq 1)$, the differential of the spectral sequence (1.2)
$d_{2t+3}:E_{2t+3}^{p,q}\to E_{2t+3}^{p+2t+3,q-2t-2}$ is given by
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{t}[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3},$
and
$[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}$
is independent of the choice of the defining system $A$ obtained from Theorem
4.1.
Specializing Theorem 1.1 to the case in which $H=H_{2s+1}$ $(s\geq 2)$, we
obtain
(1.5)
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{t}[\langle\underbrace{0,\cdots,0}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}.$
Obviously, much information has been concealed in the expression above. In
particular, we give a more explicit expression of differentials for this
special case which is compatible with Theorem 1.1 (see Remark 5.14).
###### Theorem 1.2.
For $H=H_{2s+1}$ $(s\geq 1)$ only and $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$
$(t\geq 1)$, the differential of the spectral sequence (1.2)
$d_{2t+3}:E_{2t+3}^{p,q}\to E_{2t+3}^{p+2t+3,q-2t-2}$ is given by
$d_{2t+3}[x_{p}]_{2t+3}=\left\\{\begin{array}[]{ll}[H_{2s+1}\wedge
x_{p}]_{2t+3}&t=s-1,\\\
(-1)^{l-1}[\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}]_{2t+3}&t=ls-1~{}(l\geq
2),\\\ 0&\text{otherwise,}\end{array}\right.$
and
$[\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}]_{2t+3}$
is independent of the choice of the defining system $B$ obtained from Theorem
4.3.
Atiyah and Segal in [1] gave the differential expression in terms of Massey
products when $H=H_{3}$ (see [1, Proposition 6.1]). Obviously, the result of
Atiyah and Segal is a special case of Theorem 1.2.
Theorem 1.1 is essentially Theorem 5.8, and Theorem 1.2 is Theorem 5.13. Some
of the results above are known to experts in this field, but there is a lack
of mathematical proof in the literature.
This paper is organized as follows. In Section 2, we recall some backgrounds
about the twisted de Rham cohomology. In Section 3, we consider the structure
of the spectral sequence converging to the twisted de Rham cohomology, and
give the differentials $d_{i}$ ($1\leq i\leq 3$) and $d_{2k}$ $(k\geq 1)$.
With the formulas of the differentials in $E_{2t+3}^{p,q}$ in Section 4,
Theorems 1.1 and 1.2 (i.e., Theorems 5.8 and 5.13) are shown in Section 5. In
Section 6, we discuss the indeterminacy of differentials of the spectral
sequence (1.2).
## 2\. Twisted de Rham cohomology
For completeness, in this section we recall some knowledge about the twisted
de Rham cohomology. Let $M$ be a smooth compact closed manifold of dimension
$n$, and $\Omega^{\ast}(M)$ the space of smooth differential forms on $M$. We
have the de Rham cochain complex $(\Omega^{\ast}(M),d)$ with the exterior
differentiation $d:\Omega^{p}(M)\rightarrow\Omega^{p+1}(M)$, and its
cohomology $H^{\ast}(M)$ (the de Rham cohomology).
Let $H$ denote $\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$, where $H_{2i+1}$ is a
closed $(2i+1)$-form. Define a new operator $D=d+H$ on $\Omega^{\ast}(M)$,
where $H$ is understood as an operator acting by exterior multiplication (for
any differential form $w$, $H(w)=H\wedge w$, also denoted by $H_{\wedge}$). It
is easy to show that
$D^{2}=(d+H)^{2}=d^{2}+dH+Hd+H^{2}=0.$
However $D$ is not homogeneous on the space of smooth differential forms
$\Omega^{\ast}(M)=\bigoplus\limits_{i\geq 0}\Omega^{i}(M)$.
Define $\Omega^{\ast}(M)$ a new (mod 2) grading
(2.1) $\Omega^{\ast}(M)=\Omega^{o}(M)\oplus\Omega^{e}(M),$
where
(2.2) $\begin{array}[]{lclc}\Omega^{o}(M)=\bigoplus\limits_{i\geq
0\atop\rm{i\equiv
1\pmod{2}}}\Omega^{i}(M)\quad\mbox{and}\quad\Omega^{e}(M)=\bigoplus\limits_{i\geq
0\atop\rm{i\equiv 0\pmod{2}}}\Omega^{i}(M).\end{array}$
Then $D$ is homogenous for this new (mod 2) grading:
$\Omega^{e}(M)\stackrel{{\scriptstyle
D}}{{\longrightarrow}}\Omega^{o}(M)\stackrel{{\scriptstyle
D}}{{\longrightarrow}}\Omega^{e}(M).$
Define the twisted de Rham cohomology groups of $M$:
(2.3)
$H^{o}(M,H)=\frac{\mathrm{ker}[D:\Omega^{o}(M)\rightarrow\Omega^{e}(M)]}{\mathrm{im}[D:\Omega^{e}(M)\rightarrow\Omega^{o}(M)]}$
and
(2.4)
$H^{e}(M,H)=\frac{\mathrm{ker}[D:\Omega^{e}(M)\rightarrow\Omega^{o}(M)]}{\mathrm{im}[D:\Omega^{o}(M)\rightarrow\Omega^{e}(M)]}.$
###### Remark 2.1.
${\rm(i)}$ The twisted de Rham cohomology groups $H^{*}(M,H)$ $(*=o,e)$ depend
on the closed form $H$ and not just on its cohomology class. If $H$ and
$H^{{}^{\prime}}$ are cohomologous, then $H^{*}(M,H)\cong
H^{*}(M,H^{{}^{\prime}})$ $($see [1, §6]$)$.
${\rm(ii)}$ The twisted de Rham cohomology is also an important homotopy
invariant $($see [9, §1.4]$)$.
Let $E$ be a flat vector bundle over $M$ and ${\Omega}^{i}(M,E)$ be the space
of smooth differential $i$-forms on $M$ with values in $E$. A flat connection
on $E$ gives a linear map
$\nabla^{E}:{\Omega}^{i}(M,E)\to{\Omega}^{i+1}(M,E)$
such that, for any smooth function $f$ on $M$ and any
$\omega\in{\Omega}^{i}(M,E)$,
$\nabla^{E}(f\omega)=df\wedge\omega+f\cdot\nabla^{E}\omega,\ \ \
\nabla^{E}\circ\nabla^{E}=0.$
Similarly, define $\Omega^{\ast}(M,E)$ a new $\pmod{2}$ grading
(2.5) $\Omega^{\ast}(M,E)=\Omega^{o}(M,E)\oplus\Omega^{e}(M,E),$
where
(2.6) $\begin{array}[]{lclc}\Omega^{o}(M,E)=\bigoplus\limits_{i\geq
0\atop\rm{i\equiv
1\pmod{2}}}\Omega^{i}(M,E)\quad\mbox{and}\quad\Omega^{e}(M,E)=\bigoplus\limits_{i\geq
0\atop\rm{i\equiv 0\pmod{2}}}\Omega^{i}(M,E).\end{array}$
Then $D^{E}=\nabla^{E}+H_{\wedge}$ is homogenous for the new (mod 2) grading:
$\Omega^{e}(M,E)\stackrel{{\scriptstyle
D^{E}}}{{\longrightarrow}}\Omega^{o}(M,E)\stackrel{{\scriptstyle
D^{E}}}{{\longrightarrow}}\Omega^{e}(M,E).$
Define the twisted de Rham cohomology groups of $E$:
(2.7)
$H^{o}(M,E,H)=\frac{\mathrm{ker}[D^{E}:\Omega^{o}(M,E)\rightarrow\Omega^{e}(M,E)]}{\mathrm{im}[D^{E}:\Omega^{e}(M,E)\rightarrow\Omega^{o}(M,E)]}$
and
(2.8)
$H^{e}(M,E,H)=\frac{\mathrm{ker}[D^{E}:\Omega^{e}(M,E)\rightarrow\Omega^{o}(M,E)]}{\mathrm{im}[D^{E}:\Omega^{o}(M,E)\rightarrow\Omega^{e}(M,E)]}.$
Results proved in this paper are also true for the twisted de Rham cohomology
groups $H^{*}(M,E,H)$ ($*=o,e$) with twisted coefficients in $E$ without any
change.
## 3\. A spectral sequence for twisted de Rham cohomology and its
differentials $d_{i}$ ($1\leq i\leq 3$), $d_{2k}$ ($k\geq 1$)
Recall that $D=d+H$ and $H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$, where
$H_{2i+1}$ is a closed $(2i+1)$-form. Define the usual filtration on the
graded vector space $\Omega^{\ast}(M)$ to be
$K_{p}=F_{p}(\Omega^{\ast}(M))=\bigoplus\limits_{i\geq p}\Omega^{i}(M),$
and $K=K_{0}=\Omega^{\ast}(M)$. The filtration is bounded and complete,
(3.1) $K\equiv K_{0}\supset K_{1}\supset K_{2}\supset\cdots\supset
K_{n}\supset K_{n+1}=\\{0\\}.$
We have $D(K_{p})\subset K_{p}$ and $D(K_{p})\subset K_{p+1}$. The
differential $D(=d+H)$ does not preserve the grading of the de Rham complex.
However, it does preserve the filtration $\\{K_{p}\\}_{p\geq 0}.$
The filtration $\\{K_{p}\\}_{p\geq 0}$ gives an exact couple (with bidegree)
(see [13]). For each $p$, $K_{p}$ is a graded vector space with
$K_{p}=(K_{p}\cap\Omega^{o}(M))\oplus(K_{p}\cap\Omega^{e}(M))=K_{p}^{o}\oplus
K_{p}^{e},$
where $K_{p}^{o}=K_{p}\cap\Omega^{o}(M)$ and
$K_{p}^{e}=K_{p}\cap\Omega^{e}(M)$. The cochain complex $(K_{p},D)$ is induced
by $D:\Omega^{\ast}(M)\longrightarrow\Omega^{\ast}(M)$. Similar to (2.4),
there are two well-defined cohomology groups $H_{D}^{e}(K_{p})$ and
$H_{D}^{o}(K_{p})$. Note that a cochain complex with grading
$K_{p}/K_{p+1}=(K_{p}^{o}/K_{p+1}^{o})\oplus(K_{p}^{e}/K_{p+1}^{e})$
derives cohomology groups $H_{D}^{o}(K_{p}/K_{p+1})$ and
$H_{D}^{e}(K_{p}/K_{p+1})$. Since $D(K_{p})\subset K_{p+1}$, we have $D=0$ in
the cochain complex $(K_{p}/K_{p+1},D)$.
###### Lemma 3.1.
For the cochain complex $(K_{p}/K_{p+1},D)$, we have
$H_{D}^{o}(K_{p}/K_{p+1})\cong\left\\{\begin{array}[]{ll}\Omega^{p}(M)&~{}~{}~{}\mbox{$p$
is odd,}\\\ 0&~{}~{}~{}\mbox{$p$ is even.}\end{array}\right.$
and
$H_{D}^{e}(K_{p}/K_{p+1})\cong\left\\{\begin{array}[]{ll}\Omega^{p}(M)&~{}~{}~{}\mbox{$p$
is even,}\\\ 0&~{}~{}~{}\mbox{$p$ is odd.}\end{array}\right.$
###### Proof.
If $p$ is odd, then
$K_{p}\cap\Omega^{e}(M)=K_{p+1}\cap\Omega^{e}(M)~{}{\rm
and}~{}(K_{p}\cap\Omega^{e}(M))\left/(K_{p+1}\cap\Omega^{e}(M))\right.=0.$
Also
$(K_{p}\cap\Omega^{o}(M))\left/(K_{p+1}\cap\Omega^{o}(M))\right.=K^{o}_{p}/K^{o}_{p+1}\cong\Omega^{p}(M),$
and
$H_{D}^{o}(K_{p}/K_{p+1})\cong\Omega^{p}(M)~{}{\rm
and}~{}H_{D}^{e}(K_{p}/K_{p+1})=0.$
Similarly for even $p$, we have
$H_{D}^{e}(K_{p}/K_{p+1})\cong\Omega^{p}(M)~{}{\rm
and}~{}H_{D}^{o}(K_{p}/K_{p+1})=0.$
∎
By the filtration (3.1), we obtain a short exact sequence of cochain complexes
(3.2) $0\longrightarrow K_{p+1}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}K_{p}\stackrel{{\scriptstyle
j}}{{\longrightarrow}}K_{p}/K_{p+1}\longrightarrow 0,$
which gives rise to a long exact sequence of cohomology groups
(3.3)
$\begin{array}[]{cc}\cdots\longrightarrow&H_{D}^{p+q}(K_{p+1})\stackrel{{\scriptstyle
i^{\ast}}}{{\longrightarrow}}H_{D}^{p+q}(K_{p})\stackrel{{\scriptstyle
j^{\ast}}}{{\longrightarrow}}H_{D}^{p+q}(K_{p}/K_{p+1})\\\
&\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}H_{D}^{p+q+1}(K_{p+1})\stackrel{{\scriptstyle
i^{\ast}}}{{\longrightarrow}}H_{D}^{p+q+1}(K_{p})\stackrel{{\scriptstyle
j^{\ast}}}{{\longrightarrow}}\cdots.\end{array}$
Note that in the exact sequence above,
$H_{D}^{i}(K_{p})=\left\\{\begin{array}[]{ll}H_{D}^{e}(K_{p})&\mbox{$i$ is
even,}\\\ H_{D}^{o}(K_{p})&\mbox{$i$ is odd.}\end{array}\right.$
and
$H_{D}^{i}(K_{p}/K_{p+1})=\left\\{\begin{array}[]{ll}H_{D}^{e}(K_{p}/K_{p+1})&\mbox{$i$
is even,}\\\ H_{D}^{o}(K_{p}/K_{p+1})&\mbox{$i$ is odd.}\end{array}\right.$
Let
(3.4) $\begin{array}[]{cc}E_{1}^{p,q}=H_{D}^{p+q}(K_{p}/K_{p+1}),\quad
D_{1}^{p,q}=H_{D}^{p+q}(K_{p}),\\\ i_{1}=i^{\ast},\quad
j_{1}=j^{\ast},\quad\mbox{and}\quad k_{1}=\delta.\end{array}$
We get an exact couple from the long exact sequence (3.3)
(3.5)
---
$\textstyle{D_{1}^{\ast,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{D_{1}^{\ast,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{E_{1}^{\ast,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{1}}$
with $i_{1}$ of bidegree $(-1,1)$, $j_{1}$ of bidegree $(0,0)$ and $k_{1}$ of
bidegree $(1,0)$.
We have $d_{1}=j_{1}k_{1}:E_{1}^{\ast,\ast}\longrightarrow E_{1}^{\ast,\ast}$
with bidegree $(1,0)$, and $d_{1}^{2}=j_{1}k_{1}j_{1}k_{1}=0$. By (3.5), we
have the derived couple
(3.6)
---
$\textstyle{D_{2}^{\ast,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{D_{2}^{\ast,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{2}}$$\textstyle{E_{2}^{\ast,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k_{2}}$
by the following:
1. (1)
$D_{2}^{\ast,\ast}=i_{1}D_{1}^{\ast,\ast}$,
$E_{2}^{\ast,\ast}=H_{d_{1}}(E_{1}^{\ast,\ast})$.
2. (2)
$i_{2}=i_{1}|_{D_{2}^{\ast,\ast}}$, also denoted by $i_{1}$.
3. (3)
If $a_{2}=i_{1}a_{1}\in D_{2}^{\ast,\ast}$, define
$j_{2}(a_{2})=[j_{1}a_{1}]_{d_{1}}$, where $[\ \ ]_{d_{1}}$ denotes the
cohomology class in $H_{d_{1}}(E_{1}^{\ast,\ast}).$
4. (4)
For $[b]_{d_{1}}\in E_{2}^{\ast,\ast}=H_{d_{1}}(E_{1}^{\ast,\ast}),$ define
$k_{2}([b]_{d_{1}})=k_{1}b\in D_{2}^{\ast,\ast}$.
The derived couple (3.6) is also an exact couple, and $j_{2}$ and $k_{2}$ are
well-defined (see [6, 13]).
###### Proposition 3.2.
${\rm(i)}$ There exists a spectral sequence $(E^{p,q}_{r},d_{r})$ derived from
the filtration $\\{K_{n}\\}_{n\geq 0}$, where
$E_{1}^{p,q}=H^{p+q}_{D}(K_{p}/K_{p+1})$ and $d_{1}=j_{1}k_{1}$, and
$E^{p,q}_{2}=H_{d_{1}}(E^{p,q}_{1})$ and $d_{2}=j_{2}k_{2}.$ The bidegree of
$d_{r}$ is $(r,1-r)$.
${\rm(ii)}$ The spectral sequence $\\{E^{p,q}_{r},d_{r}\\}$ converges to the
twisted de Rham cohomology
(3.7) $\begin{array}[]{lclc}\bigoplus\limits_{\rm{p+q=1}}E^{p,q}_{\infty}\cong
H^{o}(M,H)\quad\mbox{and}\quad\bigoplus\limits_{\rm{p+q=0}}E^{p,q}_{\infty}\cong
H^{e}(M,H).\end{array}$
###### Proof.
Since the filtration is bounded and complete, the proof follows from the
standard algebraic topology method (see [13]). ∎
###### Remark 3.3.
1. (1)
Note that
$H_{D}^{i}(K_{p})=\left\\{\begin{array}[]{ll}H_{D}^{e}(K_{p})&\mbox{$i$ is
even,}\\\ H_{D}^{o}(K_{p})&\mbox{$i$ is odd.}\end{array}\right.$
and
$H_{D}^{i}(K_{p}/K_{p+1})=\left\\{\begin{array}[]{ll}H_{D}^{e}(K_{p}/K_{p+1})&\mbox{$i$
is even,}\\\ H_{D}^{o}(K_{p}/K_{p+1})&\mbox{$i$ is odd.}\end{array}\right.$
Then we have that $H_{D}^{i}(K_{p})$ and $H_{D}^{i}(K_{p}/K_{p+1})$ are
$2$-periodic on $i$. Consequently, the spectral sequence
$\\{E_{r}^{p,q},d_{r}\\}$ is $2$-periodic on $q$.
2. (2)
There is also a spectral sequence converging to the twisted cohomology
$H^{*}(M,E,H)$ for a flat vector bundle $E$ over $M$.
###### Proposition 3.4.
For the spectral sequence in Proposition 3.2,
${\rm(i)}$ The $E_{1}^{*,*}$-term is given by
$E_{1}^{p,q}=H_{D}^{p+q}(K_{p}/K_{p+1})\cong\left\\{\begin{array}[]{ll}\Omega^{p}(M)&~{}~{}~{}\mbox{$q$
is even,}\\\ 0&~{}~{}~{}\mbox{$q$ is odd.}\end{array}\right.$
and $d_{1}x_{p}=dx_{p}$ for any $x_{p}\in E_{1}^{p,q}$.
${\rm(ii)}$ The $E_{2}^{*,*}$-term is given by
$E_{2}^{p,q}=H_{d_{1}}(E_{1}^{p,q})\cong\left\\{\begin{array}[]{ll}H^{p}(M)&~{}~{}~{}\mbox{$q$
is even,}\\\ 0&~{}~{}~{}\mbox{$q$ is odd.}\end{array}\right.$
and $d_{2}=0$.
${\rm(iii)}$ $E_{3}^{p,q}=E_{2}^{p,q}$ and $d_{3}[x_{p}]=[H_{3}\wedge x_{p}]$
for $[x_{p}]_{3}\in E_{3}^{p,q}$.
###### Proof.
(i) By Lemma 3.1, we have the $E_{1}^{*,*}$-term as desired, and by definition
we obtain $d_{1}=j_{1}k_{1}:E_{1}^{p,q}\rightarrow E_{1}^{p+1,q}$. We only
need to consider the case when $q$ is even, otherwise $d_{1}=0.$ By (3.2) for
odd $p$ (the case when $p$ is even, is similar), we have a large commutative
diagram
(3.8)
---
$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{p+1}^{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\scriptstyle{i}$$\textstyle{K_{p}^{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\scriptstyle{j}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{p+1}^{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\scriptstyle{i}$$\textstyle{K_{p}^{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\scriptstyle{j}$$\textstyle{\Omega^{p}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{p+1}^{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\scriptstyle{i}$$\textstyle{K_{p}^{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\scriptstyle{j}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\textstyle{0}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$
where the rows are exact and the columns are cochain complexes.
Let $x_{p}\in\Omega^{p}(M)\cong H_{D}^{p+q}(K_{p}/K_{p+1})\cong E_{1}^{p,q}$
and
(3.9) $x=\sum_{i=0}^{[\frac{n-p}{2}]}x_{p+2i}$
be an (inhomogeneous) form, where $x_{p+2i}$ is a $(p+2i)$-form ($0\leq
i\leq[\frac{n-p}{2}]$). Then $x\in K_{p}^{o}$, $jx=x_{p}$ and $Dx\in
K_{p}^{e}$. Also $Dx\in K_{p+1}^{e}$. By the definition of the homomorphism
$\delta$ in (3.3), we have
(3.10) $k_{1}x_{p}=[Dx]_{D},$
where $[\ \ ]_{D}$ is the cohomology class in $H_{D}^{\ast}(K_{p+1})$. The
class $[Dx]_{D}$ is well defined and independent of the choices of $x_{p+2i}$
$(1\leq i\leq[\frac{n-p}{2}])$ (see [3, p. 116]).
Choose $x_{p+2i}=0$ ($1\leq i\leq[\frac{n-p}{2}]$). Then we have
$\displaystyle k_{1}x_{p}$ $\displaystyle=$ $\displaystyle[Dx]_{D}$
$\displaystyle=$ $\displaystyle[dx_{p}+H\wedge x_{p}]_{D}$ $\displaystyle=$
$\displaystyle[dx_{p}+\sum_{l=1}^{[\frac{n-1}{2}]}H_{2l+1}\wedge x_{p}]_{D}\in
H_{D}^{p+q+1}(K_{p+1}).$
Thus one obtains
$\displaystyle d_{1}x_{p}$ $\displaystyle=$ $\displaystyle(j_{1}k_{1})x_{p}$
$\displaystyle=$ $\displaystyle j_{1}(k_{1}(x_{p}))$ $\displaystyle=$
$\displaystyle j_{1}[dx_{p}+\sum_{l=1}^{[\frac{n-1}{2}]}H_{2l+1}\wedge
x_{p}]_{D}$ $\displaystyle=$ $\displaystyle dx_{p}.$
(ii) By the definition of the spectral sequence and (i), one obtains that
$E_{2}^{p,q}\cong H^{p}(M)$ when $q$ is even, and $E_{2}^{p,q}=0$ when $q$ is
odd. Note that $d_{2}:E^{p,q}_{2}\to E_{2}^{p+2,q-1}$. It follows that
$d_{2}=0$ by degree reasons.
(iii) Note that $[x_{p}]_{3}\in E_{3}^{p,q}$ implies $dx_{p}=0$. Choose
$x_{p+2i}=0$ for $1\leq i\leq[\frac{n-p}{2}]$, and we get
$[Dx]_{D}=[H\wedge x_{p}]_{D}=[\sum_{l=1}^{[\frac{n-1}{2}]}H_{2l+1}\wedge
x_{p}]_{D}\in H_{D}^{p+q+1}(K_{p+1}),$
where $x$ is given in the proof of (i). Note that
(3.11)
$\textstyle{H_{D}^{p+q+1}(K_{p+1})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H_{D}^{p+q+1}(K_{p+3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}^{2}}$$\scriptstyle{j_{1}}$$\textstyle{H_{D}^{p+q+1}(K_{p+3}/K_{p+4})}$$\textstyle{[Dx]_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(i_{1}^{-1})^{2}}$$\textstyle{[Dx]_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{H_{3}\wedge
x_{p}.}$
It follows that
(3.12) $\begin{array}[]{lll}d_{3}[x_{p}]_{3}&=&j_{3}k_{3}[x_{p}]_{3}\\\
&=&j_{3}(k_{1}x_{p})\\\ &=&j_{3}[Dx]_{D}\\\
&=&[j_{1}((i_{1}^{-1})^{2}[Dx]_{D})]_{3}\\\ &=&[H_{3}\wedge
x_{p}]_{3},\end{array}$
where the first, second and fourth identities follow from the definitions of
$d_{3}$, $k_{3}$ and $j_{3}$ respectively, and the third and the last
identities follow from (3.10) and (3.11). By (ii), $d_{2}=0$, so
$E_{3}^{p,q}=E_{2}^{p,q}.$ Then we have
$d_{3}[x_{p}]=[H_{3}\wedge x_{p}].$
∎
###### Corollary 3.5.
$d_{2k}=0$ for $k\geq 1.$ Therefore, for $k\geq 1$,
(3.13) $E_{2k+1}^{p,q}=E_{2k}^{p,q}.$
###### Proof.
Note that $d_{2k}:E_{2k}^{p,q}\longrightarrow E_{2k}^{p+2k,q+1-2k}.$ By
Proposition 3.4 (ii), if $q$ is odd, then $E_{2}^{p,q}=0$ which implies that
$E_{2k}^{p,q}=0.$ By degree reasons, we have $d_{2k}=0$ and
$E_{2k+1}^{p,q}=E_{2k}^{p,q}$ for $k\geq 1.$ ∎
The differential $d_{3}$ for the case in which $H=H_{3}$ is shown in [1, §6],
and the $E_{2}^{p,q}$-term is also known.
## 4\. Differentials $d_{2t+3}$ $(t\geq 1)$ in terms of cup products
In this section, we will show that the differentials $d_{2t+3}$ $(t\geq 1)$
can be given in terms of cup products.
We first consider the general case that
$H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$. For $[x_{p}]_{2t+3}\in
E_{2t+3}^{p,q}$, we let $x=\sum_{j=0}^{[\frac{n-p}{2}]}x_{p+2j}\in
F_{p}({\Omega}^{*}(M))$. Then we have
(4.1)
$\begin{array}[]{ll}Dx&=(d+\sum\limits_{i=1}^{[\frac{n-1}{2}]}H_{2i+1})(\sum\limits_{j=0}^{[\frac{n-p}{2}]}x_{p+2j})\\\
&=dx_{p}+\sum\limits_{j=0}^{[\frac{n-p}{2}]-1}(dx_{p+2j+2}+\sum\limits_{i=1}^{j+1}H_{2i+1}\wedge
x_{p+2(j-i)+2}).\end{array}$
Denote $y=Dx=\sum_{j=0}^{[\frac{n-p}{2}]}y_{p+2j+1}$, where
(4.2) $\left\\{\begin{array}[]{ll}&y_{p+1}=dx_{p},\\\
&y_{p+2j+3}=dx_{p+2j+2}+\sum\limits_{i=1}^{j+1}H_{2i+1}\wedge
x_{p+2(j-i)+2}\quad(0\leq j\leq[\frac{n-p}{2}]-1).\\\ \end{array}\right.$
###### Theorem 4.1.
For $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$ $(t\geq 1)$, there exist
$x_{p+2i}=x_{p+2i}^{(t)}$ $(1\leq i\leq t)$ such that $y_{p+2j+1}=0$ $(0\leq
j\leq t)$ and
$d_{2t+3}[x_{p}]_{2t+3}=[\sum\limits_{i=1}^{t}H_{2i+1}\wedge
x_{p+2(t-i)+2}^{(t)}+H_{2t+3}\wedge x_{p}]_{2t+3},\\\ $
where the $(p+2i)$-form $x_{p+2i}^{(t)}$ depends on $t$.
###### Proof.
The theorem is shown by mathematical induction on $t$.
When $t=1$, $[x_{p}]_{2t+3}=[x_{p}]_{5}$. $[x_{p}]_{5}\in E_{5}^{p,q}$ implies
that $dx_{p}=0$ and $d_{3}[x_{p}]=[H_{3}\wedge x_{p}]=0$ by Proposition 3.4.
Thus there exists a $(p+2)$-form $v_{1}$ such that $H_{3}\wedge
x_{p}=d(-v_{1}).$ We can choose $x_{p+2}^{(1)}=v_{1}$ to get
$y_{p+3}=dx_{p+2}^{(1)}+H_{3}\wedge x_{p}=dv_{1}+H_{3}\wedge x_{p}=0$ from
(4.2). Note that
(4.3)
$\textstyle{H_{D}^{p+q+1}(K_{p+1})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H_{D}^{p+q+1}(K_{p+5})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}^{4}}$$\scriptstyle{j_{1}}$$\textstyle{H_{D}^{p+q+1}(K_{p+5}/K_{p+6})}$$\textstyle{[Dx]_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(i_{1}^{-1})^{4}}$$\textstyle{[Dx]_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{y_{p+5},}$
we obtain
(4.4) $\begin{array}[]{ll}d_{5}[x_{p}]_{5}&=j_{5}k_{5}[x_{p}]_{5}\\\
&=j_{5}(k_{1}x_{p})\\\ &=j_{5}[Dx]_{D}\\\
&=[j_{1}(i_{1}^{-1})^{4}[Dx]_{D}]_{5}\\\ &=[y_{p+5}]_{5}.\end{array}$
The reasons for the identities in (4.4) are similar to those of (3.12). Thus
we have
$\begin{array}[]{ll}d_{5}[x_{p}]_{5}&=[dx_{p+4}+H_{3}\wedge
x_{p+2}^{(1)}+H_{5}\wedge x_{p}]_{5}\\\ &=[H_{3}\wedge
x_{p+2}^{(1)}+H_{5}\wedge x_{p}]_{5},\end{array}$
where the first identity follows from (4.4) and the definition of $y_{p+5}$ in
(4.2), and the second one follows from that $dx_{p+4}$ vanishes in
$E_{5}^{\ast,\ast}$. Hence the result holds for $t=1$.
Suppose the result holds for $t\leq m-1$. Now we show that the theorem also
holds for $t=m$.
From $[x_{p}]_{2m+3}\in E_{2m+3}^{p,q}$, we have $[x_{p}]_{2m+1}\in
E_{2m+1}^{p,q}$ and $d_{2m+1}[x_{p}]_{2m+1}=0$. By induction, there exist
$x_{p+2i}^{(m-1)}$ ($1\leq i\leq m-1$) such that
(4.5) $\left\\{\begin{array}[]{ll}&y^{(m-1)}_{p+1}(x_{p})=dx_{p}=0,\\\
&y^{(m-1)}_{p+3}(x_{p})=dx_{p+2}^{(m-1)}+H_{3}\wedge x_{p}=0,\\\
&y^{(m-1)}_{p+2i+1}(x_{p})=dx_{p+2i}^{(m-1)}+\sum_{j=1}^{i-1}H_{2j+1}\wedge
x_{p+2(i-j)}^{(m-1)}+H_{2i+1}\wedge x_{p}\\\ &\quad\quad\quad\quad\quad\
=0~{}~{}(2\leq i\leq m-1),\\\
&d_{2m+1}[x_{p}]_{2m+1}=[\sum\limits_{i=1}^{m-1}H_{2i+1}\wedge
x_{p+2(m-i)}^{(m-1)}+H_{2m+1}\wedge x_{p}]_{2m+1}=0.\end{array}\right.$
By $d_{2m}=0$ and the last equation in (4.5), there exists a $(p+2)$-form
$w_{p+2}$ such that
(4.6) $[\sum\limits_{i=1}^{m-1}H_{2i+1}\wedge
x_{p+2(m-i)}^{(m-1)}+H_{2m+1}\wedge x_{p}]_{2m-1}=d_{2m-1}[w_{p+2}]_{2m-1}.$
By induction and $[w_{p+2}]_{2m-1}\in E_{2m-1}^{p+2,q-2}$, there exist
$w^{(m-2)}_{p+2(i+1)}$ $(1\leq i\leq m-2)$ such that
(4.7) $\left\\{\begin{array}[]{ll}&y^{(m-2)}_{p+3}(w_{p+2})=dw_{p+2}=0,\\\
&y^{(m-2)}_{p+5}(w_{p+2})=dw^{(m-2)}_{p+4}+H_{3}\wedge w_{p+2}=0,\\\
&y^{(m-2)}_{p+2i+3}(w_{p+2})=dw^{(m-2)}_{p+2(i+1)}+\sum_{j=1}^{i-1}H_{2j+1}\wedge
w^{(m-2)}_{p+2(i-j+1)}+H_{2i+1}\wedge w_{p+2}\\\
&\quad\quad\quad\quad\quad\quad\ \ =0~{}~{}(2\leq i\leq m-2),\\\
&d_{2m-1}[w_{p+2}]_{2m-1}=[\sum\limits_{i=1}^{m-2}H_{2i+1}\wedge
w^{(m-2)}_{p+2(m-i)}+H_{2m-1}\wedge w_{p+2}]_{2m-1}.\end{array}\right.$
By (4.6) and the last equation in (4.7), we obtain
$[\sum\limits_{i=1}^{m-2}H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-w^{(m-2)}_{p+2(m-i)})+H_{2m-1}\wedge(x_{p+2}^{(m-1)}-w_{p+2})+H_{2m+1}\wedge
x_{p}]_{2m-1}=0.$
Note that $d_{2m-2}=0$, it follows that there exists a $(p+4)$-form $w_{p+4}$
such that
$\begin{array}[]{ll}&[\sum\limits_{i=1}^{m-2}H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-w^{(m-2)}_{p+2(m-i)})+H_{2m-1}\wedge(x_{p+2}^{(m-1)}-w_{p+2})+H_{2m+1}\wedge
x_{p}]_{2m-3}\\\ &=d_{2m-3}[w_{p+4}]_{2m-3}.\end{array}$
Keeping the same iteration process as mentioned above, we have
$\begin{array}[]{ll}&[\sum\limits_{i=1}^{2}(H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-\sum\limits_{j=1}^{m-3}w_{p+2(m-i)}^{(m-1-j)}))+\\\
&\sum\limits_{i=3}^{m-1}(H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-\sum\limits_{j=1}^{m-1-j}w_{p+2(m-i)}^{(m-1-j)}-w_{p+2(m-i)}))+H_{2m+1}\wedge
x_{p}]_{7}=0.\end{array}$
By $d_{6}=0$, it follows that there exists a $(p+2(m-2))$-form $w_{p+2(m-2)}$
such that
(4.8)
$\begin{array}[]{ll}&[\sum\limits_{i=1}^{2}(H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-\sum\limits_{j=1}^{m-3}w_{p+2(m-i)}^{(m-1-j)}))+\sum\limits_{i=3}^{m-1}(H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-\\\
&\sum\limits_{j=1}^{m-i-1}w_{p+2(m-i)}^{(m-1-j)}-w_{p+2(m-i)}))+H_{2m+1}\wedge
x_{p}]_{5}=d_{5}[w_{p+2(m-2)}]_{5}.\end{array}$
By induction and $[w_{p+2(m-2)}]_{5}\in E_{5}^{p+2(m-2),q-2(m-2)}$, there
exists $w_{p+2(m-1)}^{(1)}$ such that
(4.9)
$\left\\{\begin{array}[]{ll}&y_{p+2m-3}^{(1)}(w_{p+2(m-2)})=dw_{p+2(m-2)}=0,\\\
&y_{p+2m-1}^{(1)}(w_{p+2(m-2)})=dw_{p+2(m-1)}^{(1)}+H_{3}\wedge
w_{p+2(m-2)}=0,\\\ &d_{5}[w_{p+2(m-2)}]_{5}=[H_{3}\wedge
w_{p+2(m-1)}^{(1)}+H_{5}\wedge w_{p+2(m-2)}]_{5}.\end{array}\right.$
By (4.8), the last equation in (4.9) and $d_{4}=0$, it follows that there
exists a $(p+2(m-1))$-form $w_{p+2(m-1)}$ such that
$\begin{array}[]{ll}&[(H_{3}\wedge(x_{p+2(m-1)}^{(m-1)}-\sum\limits_{j=1}^{m-2}w_{p+2(m-1)}^{(m-1-j)}))+\sum\limits_{i=2}^{m-1}(H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-\\\
&\sum\limits_{j=1}^{m-i-1}w_{p+2(m-i)}^{(m-1-j)}-w_{p+2(m-i)}))+H_{2m+1}\wedge
x_{p}]=d_{3}[w_{p+2(m-1)}]=[H_{3}\wedge w_{p+2(m-1)}]\par\end{array}$
and $y_{p+2m-1}^{(0)}(w_{p+2(m-1)})=dw_{p+2(m-1)}=0$. Thus there exists a
$(p+2m)$-form $w_{p+2m}$ such that
(4.10)
$\sum\limits_{i=1}^{m-1}(H_{2i+1}\wedge(x_{p+2(m-i)}^{(m-1)}-\sum\limits_{j=1}^{m-i-1}w_{p+2(m-i)}^{(m-1-j)}-w_{p+2(m-i)}))+H_{2m+1}\wedge
x_{p}=dw_{p+2m}.$
By comparing (4.10) with (4.2), we choose at this time
(4.11)
$\left\\{\begin{array}[]{ll}&x_{p+2}=x_{p+2}^{(m)}=x_{p+2}^{(m-1)}-w_{p+2},\\\
&x_{p+2i}=x_{p+2i}^{(m)}=x_{p+2i}^{(m-1)}-\sum\limits_{j=1}^{i-1}w_{p+2i}^{(m-1-j)}-w_{p+2i}\quad(2\leq
i\leq m-1),\\\ &x_{p+2m}=x_{p+2m}^{(m)}=-w_{p+2m}.\end{array}\right.$
From (4.2), by a direct computation we have
(4.12) $\left\\{\begin{array}[]{ll}&y_{p+1}=y_{p+1}^{(m-1)}(x_{p})=0,\\\
&y_{p+2i-1}=y_{p+2i-1}^{(m-1)}(x_{p})-\sum\limits_{j=1}^{i-1}y^{(m-1-j)}_{p+2i-1}(w_{p+2j})=0\quad(2\leq
i\leq m),\\\ &y_{p+2m+1}=0.\end{array}\right.$
Note that
(4.13)
$\textstyle{H_{D}^{p+q+1}(K_{p+1})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
H_{D}^{p+q+1}(K_{p+2m+3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}^{2(m+1)}}$$\scriptstyle{j_{1}}$$\textstyle{H_{D}^{p+q+1}(K_{p+2m+3}/K_{p+2m+4})}$$\textstyle{[Dx]_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(i_{1}^{-1})^{2(m+1)}}$$\textstyle{[Dx]_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{y_{p+2m+3}.}$
By the similar reasons as in (3.12), the following identities hold.
(4.14)
$\begin{array}[]{ll}d_{2m+3}[x_{p}]_{2m+3}&=j_{2m+3}k_{2m+3}[x_{p}]_{2m+3}\\\
&=j_{2m+3}(k_{1}x_{p})\\\ &=j_{2m+3}[Dx]_{D}\\\
&=[j_{1}(i_{1}^{-1})^{2(m+1)}[Dx]_{D}]_{2m+3}\\\
&=[y_{p+2m+3}]_{2m+3}.\end{array}$
So we have
$\begin{array}[]{ll}d_{2m+3}[x_{p}]_{2m+3}&=[y_{p+2m+3}]_{2m+3}\\\
&=[dx_{p+2m+2}+\sum\limits_{i=1}^{m}H_{2i+1}\wedge
x_{p+2(m-i+1)}^{(m)}+H_{2m+3}\wedge x_{p}]_{2m+3}~{}{\rm by~{}(\ref{6.7})}\\\
&=[\sum\limits_{i=1}^{m}H_{2i+1}\wedge x_{p+2(m-i+1)}^{(m)}+H_{2m+3}\wedge
x_{p}]_{2m+3},\end{array}$
showing that the result also holds for $t=m$.
The proof of the theorem is finished. ∎
###### Remark 4.2.
Note that $x_{p+2i}^{(t)}$ ($1\leq i\leq t$) depends on $t$, and
$x_{p+2i}^{(t_{1})}\neq x_{p+2i}^{(t_{2})}$ on the condition that $t_{1}\neq
t_{2}$ generally. $x_{p+2i}^{(t)}$ ($1\leq i\leq t$) are related to
$x_{p+2j}^{(t-1)}$ ($1\leq j\leq t-1$, $j\leq i$).
Now we consider the special case in which $H=H_{2s+1}$ $(s\geq 1)$ only. For
this special case, we will give a more explicit result which is stronger than
Theorem 4.1.
For $x=\sum_{j=0}^{[\frac{n-p}{2}]}x_{p+2j}$, we have
$\begin{array}[]{ll}Dx&=(d+H_{2s+1})(\sum_{j=0}^{[\frac{n-p}{2}]}x_{p+2j})\\\
&=\sum_{j=0}^{s-1}dx_{p+2j}+\sum_{j=s}^{[\frac{n-p}{2}]}(dx_{p+2j}+H_{2s+1}\wedge
x_{p+2(j-s)}).\end{array}$
Denote
(4.15) $\left\\{\begin{array}[]{ll}y_{p+2j+1}=dx_{p+2j}&(0\leq j\leq s-1),\\\
y_{p+2j+3}=dx_{p+2j+2}+H_{2s+1}\wedge x_{p+2(j-s)+2}&(s-1\leq
j\leq[\frac{n-p}{2}]-1).\\\ \end{array}\right.$
Then $Dx=\sum_{j=0}^{[\frac{n-p}{2}]}y_{p+2j+1}$.
###### Theorem 4.3.
For $H=H_{2s+1}$ $(s\geq 1)$ only and $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$
$(t\geq 1)$, there exist $x_{p+2is}=x_{p+2is}^{([\frac{t}{s}])}$,
$x_{p+2(i-1)s+2j}=0$ and $x_{p+2[\frac{t}{s}]s+2k}=0$ for $1\leq
i\leq[\frac{t}{s}],$ $1\leq j\leq s-1$ and $1\leq k\leq t-[\frac{t}{s}]s$ such
that $y_{p+2u+1}=0$ $(0\leq u\leq t)$ and
$d_{2t+3}[x_{p}]_{2t+3}=\left\\{\begin{array}[]{lll}&[H_{2s+1}\wedge
x_{p}]_{2s+1}&t=s-1,\\\ &[H_{2s+1}\wedge
x_{p+2(l-1)s}^{(l-1)}]_{2t+3}&t=ls-1~{}~{}(l\geq 2),\\\
&0&otherwise,\end{array}\right.$
where the $(p+2is)$-form $x_{p+2is}^{([\frac{t}{s}])}$ depends on
$[\frac{t}{s}]$.
###### Proof.
The proof of the theorem is by mathematical induction on $s$.
When $s=1$, the result follows from Theorem 4.1.
When $s\geq 2$, we prove the result by mathematical induction on $t$. We first
show that the result holds for $t=1$. Note that $[x_{p}]_{5}\in E_{5}^{p,q}$
implies $y_{p+1}=dx_{p}=0.$ Choose $x_{p+2}=0$ and make $y_{p+3}=0$.
(i). When $s=2$, by (4.4) we have
$\begin{array}[]{ll}d_{5}[x_{p}]_{5}&=[y_{p+5}]_{5}\\\ &=[dx_{p+4}+H_{5}\wedge
x_{p}]_{5}\\\ &=[H_{5}\wedge x_{p}]_{5}.\end{array}$
(ii). When $s\geq 3$, by (4.4) we have
$d_{5}[x_{p}]_{5}=[y_{p+5}]_{5}=[dx_{p+4}]_{5}=0.$
Combining (i) and (ii), we have that the theorem holds for $t=1$.
Suppose the theorem holds for $t\leq m-1$. Now we show that the theorem also
holds for $t=m$.
Case 1. $2\leq m\leq s-1$.
By induction, the theorem holds for $1\leq t\leq m-1$. Choose $x_{p+2i}=0$
$(1\leq i\leq m)$, and from (4.15) we easily get that $y_{p+2j+1}=0$ $(0\leq
j\leq m)$. By (4.14) and (4.15), we have
$\begin{array}[]{ll}d_{2m+3}[x_{p}]_{2m+3}&=[y_{p+2m+3}]_{2m+3}\\\
&=\left\\{\begin{array}[]{lll}[dx_{p+2(m+1)}]_{2m+3}&2\leq m\leq s-2,\\\
[dx_{p+2(m+1)}+H_{2s+1}\wedge x_{p}]_{2m+3}&m=s-1,\end{array}\right.\\\
&=\left\\{\begin{array}[]{ll}0&2\leq m\leq s-2,\\\ [H_{2s+1}\wedge
x_{p}]_{2s+1}&m=s-1.\end{array}\right.\end{array}$
Case 2. $m=ls-1$ $(l\geq 2)$.
By induction, the theorem holds for $t=m-1=ls-2$. Thus, there exist
$x_{p+2is}=x_{p+2is}^{([\frac{m-1}{s}])}=x_{p+2is}^{(l-1)}$,
$x_{p+2(i-1)s+2j}=0$ and $x_{p+2(l-1)s+2k}=0$ for $1\leq i\leq l-1,$ $1\leq
j\leq s-1$ and $1\leq k\leq s-2$ such that $y_{p+2u+1}=0$ $(0\leq u\leq
ls-2)$. Choose $x_{p+2(ls-1)}=0$, and by (4.15) we get
$\begin{array}[]{ll}y_{p+2(ls-1)+1}&=dx_{p+2(ls-1)}+H_{2s+1}\wedge
x_{p+2(l-1)s-2}\\\ &=0+H_{2s+1}\wedge 0\\\ &=0.\end{array}$
Then we have
$\begin{array}[]{ll}d_{2(ls-1)+3}[x_{p}]_{2(ls-1)+3}&=[y_{p+2ls+1}]_{2(ls-1)+3}\quad{\rm
by~{}(\ref{6.5})}\\\ &=[dx_{p+2ls}+H_{2s+1}\wedge
x_{p+2(l-1)s}^{(l-1)}]_{2(ls-1)+3}\quad{\rm by~{}(\ref{6.6})}\\\
&=[H_{2s+1}\wedge x_{p+2(l-1)s}^{(l-1)}]_{2(ls-1)+3}.\end{array}$
Case 3. $m=ls$ ($l\geq 1$).
By induction, there exist
$x_{p+2is}=x_{p+2is}^{([\frac{ls-1}{s}])}=x_{p+2is}^{(l-1)}$,
$x_{p+2(i-1)s+2j}=0$ and $x_{p+2(l-1)s+2k}=0$ for $1\leq i\leq l-1$, $1\leq
j\leq s-1$ and $1\leq k\leq s-1$ such that $y_{p+2u+1}=0$ $(0\leq u\leq
ls-1)$. By the same method as in Theorem 4.1, one has that there exist
$x_{p+2is}=x_{p+2is}^{(l)}$, $x_{p+2(i-1)s+2j}=0$ and $x_{p+2(l-1)s+2k}=0$ for
$1\leq i\leq l$, $1\leq j\leq s-1$ and $1\leq k\leq s-1$ such that
$y_{p+2u+1}=0$ $(0\leq u\leq ls)$. By (4.14), (4.15) and $x_{p+2ls-2s+2}=0$,
we have
$\begin{array}[]{ll}d_{2ls+3}[x_{p}]_{2ls+3}&=[y_{p+2ls+3}]_{2ls+3}\\\
&=[dx_{p+2ls+2}+H_{2s+1}\wedge x_{p+2ls-2s+2}]_{2ls+3}\\\ &=0.\end{array}$
Case 4. $ls<m<(l+1)s-1$ ($l\geq 1$).
By induction, there exist
$x_{p+2is}=x_{p+2is}^{([\frac{m-1}{s}])}=x_{p+2is}^{(l)}$,
$x_{p+2(i-1)s+2j}=0$ and $x_{p+2ls+2k}=0$ for $1\leq i\leq l,$ $1\leq j\leq
s-1$ and $1\leq k\leq m-ls-1$ such that $y_{p+2u+1}=0$ $(0\leq u\leq m-1)$.
Choose $x_{p+2m}=0$ and make $y_{p+2m+1}=0$. By (4.14), (4.15) and
$x_{p+2m-2s+2}=0$, we have
$\begin{array}[]{ll}d_{2m+3}[x_{p}]_{2m+3}&=[y_{p+2m+3}]_{2m+3}\\\
&=[dx_{p+2m+2}+H_{2s+1}\wedge x_{p+2m-2s+2}]_{2m+3}\\\ &=0.\end{array}$
Combining Cases 1-4, we have that the result holds for $t=m$ and the proof is
completed. ∎
###### Remark 4.4.
1. (1)
Theorems 4.1 and 4.3 show that the differentials in the spectral sequence
(1.2) can be computed in terms of cup products with $H_{2i+1}$’s. The
existence of $x_{p+2i}^{(t)}$’s and $x_{p+2is}^{([\frac{t}{s}])}$’s in
Theorems 4.1 and 4.3 plays an essential role in proving Theorems 5.8 and 5.13,
respectively. Theorems 4.1 and 4.3 give a description of the differentials at
the level of $E_{2t+3}^{p,q}$ for the spectral sequence (1.2), which was
ignored in the previous studies of the twisted de Rham cohomology in [1, 9].
2. (2)
Note that Theorem 4.3 is not a corollary of Theorem 4.1, and it can not be
obtained from Theorem 4.1 directly.
## 5\. Differentials $d_{2t+3}$ ($t\geq 1$) in terms of Massey products
The Massey product is a cohomology operation of higher order introduced in
[8], which generalizes the cup product. In [10], May showed that the
differentials in the Eilenberg-Moore spectral sequence associated with the
path-loop fibration of a path connected, simply connected space are completely
determined by higher order Massey products. Kraines and Schochet [5] also
described the differentials in Eilenberg-Moore spectral sequence by Massey
products. In order to describe the differentials $d_{2t+3}$ ($t\geq 1$) in
terms of Massey products, we first recall briefly the definition of Massey
products (see [4, 10, 11, 13]). Then the main theorems in this paper will be
shown.
Because of different conventions in the literature used to define Massey
products, we present the following definitions. If $x\in\Omega^{p}(M)$, the
symbol $\bar{x}$ will denote $(-1)^{1+\hbox{deg}x}x=(-1)^{1+p}x$. We first
define the Massey triple product.
Let $x_{1}$, $x_{2}$, $x_{3}$ be closed differential forms on $M$ of degrees
$r_{1}$, $r_{2}$, $r_{3}$ with $[x_{1}][x_{2}]=0$ and $[x_{2}][x_{3}]=0,$
where [ ] denotes the de Rham cohomology class. Thus, there are differential
forms $v_{1}$ of degree $r_{1}+r_{2}-1$ and $v_{2}$ of degree $r_{2}+r_{3}-1$
such that $dv_{1}=\bar{x}_{1}\wedge x_{2}$ and $dv_{2}=\bar{x}_{2}\wedge
x_{3}$. Define the $(r_{1}+r_{2}+r_{3}-1)$-form
(5.1) $\omega=\bar{v}_{1}\wedge x_{3}+\bar{x}_{1}\wedge v_{2}.$
Then $\omega$ satisfies
$\begin{array}[]{ll}d(\omega)&=(-1)^{r_{1}+r_{2}}dv_{1}\wedge
x_{3}+(-1)^{r_{1}}\bar{x}_{1}\wedge dv_{2}\\\
&=(-1)^{r_{1}+r_{2}}\bar{x}_{1}\wedge x_{2}\wedge
x_{3}+(-1)^{r_{1}+r_{2}+1}\bar{x}_{1}\wedge x_{2}\wedge x_{3}\\\
&=0.\end{array}$
Hence a set of all the cohomology classes $[\omega]$ obtained by the above
procedure is defined to be the Massey triple product $\langle
x_{1},x_{2},x_{3}\rangle$ of $x_{1},x_{2}$ and $x_{3}$. Due to the ambiguity
of $v_{i},i=1,2$, the Massey triple product $\langle x_{1},x_{2},x_{3}\rangle$
is a representative of the quotient group
$H^{r_{1}+r_{2}+r_{3}-1}(M)/([x_{1}]H^{r_{2}+r_{3}-1}(M)+H^{r_{1}+r_{2}-1}(M)[x_{3}]).$
###### Definition 5.1.
Let $(\Omega^{\ast}(M),d)$ be de Rham complex, and $x_{1}$, $x_{2}$, $\cdots$,
$x_{n}$ closed differential forms on $M$ with $[x_{i}]\in H^{r_{i}}(M)$. A
collection of forms, $A=(a_{i,j})$, for $1\leq i\leq j\leq k$ and
$(i,j)\neq(1,n)$ is said to be a defining system for the $n$-fold Massey
product $\langle x_{1},x_{2},\cdots,x_{n}\rangle$ if
1. (1)
$a_{i,j}\in\Omega^{r_{i}+r_{i+1}+\cdots+r_{j}-j+i}(M),$
2. (2)
$a_{i,i}=x_{i}$ for $i=1,2,\cdots,k$,
3. (3)
$d(a_{i,j})=\sum\limits_{r=i}^{j-1}\bar{a}_{i,r}\wedge a_{r+1,j}.$
The $(r_{1}+\cdots+r_{n}-n+2)$-dimensional cocycle, $c(A)$, defined by
(5.2) $c(A)=\sum\limits_{r=1}^{n-1}\bar{a}_{1,r}\wedge
a_{r+1,n}\in\Omega^{r_{1}+\cdots+r_{n}-n+2}(M)$
is called the related cocycle of the defining system $A$.
###### Remark 5.2.
There is a unique matrix associated to each defining system $A$ as follows.
$\left(\begin{array}[]{ccccccc}a_{1,1}&a_{1,2}&a_{1,3}&\cdots&a_{1,n-2}&a_{1,n-1}&\\\
&a_{2,2}&a_{2,3}&\cdots&a_{2,n-2}&a_{2,n-1}&a_{2,n}\\\
&&a_{3,3}&\cdots&a_{3,n-2}&a_{3,n-1}&a_{3,n}\\\
&&&\ddots&\vdots&\vdots&\vdots\\\ &&&&a_{n-2,n-2}&a_{n-2,n-1}&a_{n-2,n}\\\
&&&&&a_{n-1,n-1}&a_{n-1,n}\\\ &&&&&&a_{n,n}\\\ \end{array}\right)_{n\times
n.}$
###### Definition 5.3.
The $n$-fold Massey product $\langle x_{1},x_{2},\cdots,x_{n}\rangle$ is said
to be defined if there is a defining system for it. If it is defined, then
$\langle x_{1},x_{2},\cdots,x_{n}\rangle$ consists of all classes $w\in
H^{r_{1}+r_{2}+\cdots+r_{n}-n+2}(M)$ for which there exists a defining system
$A$ such that $c(A)$ represents $w$.
###### Remark 5.4.
There is an inherent ambiguity in the definition of the Massey product arising
from the choices of defining systems. In general, the $n$-fold Massey product
may or may not be a coset of a subgroup, but its indeterminacy is a subset of
a matrix Massey product (see [10, §2]).
Based on Theorems 4.1 and 4.3, we have the following lemma on defining systems
for the two Massey products we consider in this paper.
###### Lemma 5.5.
(1) For $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$ ($t\geq 1$), there are defining
systems for $\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle$
obtained from Theorem 4.1.
(2) For $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$, when $t=ls-1$ ($l\geq 2$) there
are defining systems for
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle$
obtained from Theorem 4.3 .
###### Proof.
(1) From Theorem 4.1, there exist $x_{p+2j}^{(t)}$ $(1\leq j\leq t)$ such that
$y_{p+2i+1}=0$ $(0\leq i\leq t)$ and
$d_{2t+3}[x_{p}]_{2t+3}=[\sum\limits_{i=1}^{t}H_{2i+1}\wedge
x_{p+2(t-i+1)}^{(t)}+H_{2t+3}\wedge x_{p}]_{2t+3}.$ By Theorem 4.1 and (4.2),
there exists a defining system $A=(a_{i,j})$ for
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle$ as follows:
(5.3) $\left\\{\begin{array}[]{lll}&a_{t+2,t+2}=x_{p},\\\
&a_{i,i+k}=(-1)^{k}H_{2k+3}{}{}&(1\leq i\leq t+1-k,~{}0\leq k<t),\\\
&a_{i,t+2}=(-1)^{t+2-i}x_{p+2(t+2-i)}^{(t)}{}{}&(2\leq i\leq
t+1),\par\end{array}\right.$
to which the matrix associated is given by
(5.4)
$\left(\begin{array}[]{ccccccc}H_{3}&-H_{5}&H_{7}&\cdots&(-1)^{t-1}H_{2t+1}&(-1)^{t}H_{2t+3}&\\\
&H_{3}&-H_{5}&\cdots&(-1)^{t-2}H_{2t-1}&(-1)^{t-1}H_{2t+1}&(-1)^{t}x_{p+2t}^{(t)}\\\
&&H_{3}&\cdots&(-1)^{t-3}H_{2t-3}&(-1)^{t-2}H_{2t-1}&(-1)^{t-1}x_{p+2t-2}^{(t)}\\\
&&&\ddots&\vdots&\vdots&\vdots\\\ &&&&H_{3}&-H_{5}&(-1)^{2}x_{p+4}^{(t)}\\\
&&&&&H_{3}&-x_{p+2}^{(t)}\\\ &&&&&&x_{p}\\\
\end{array}\right)_{(t+2)\times(t+2).}$
The desired result follows.
(2) By Theorem 4.3, there exist $x_{p+2is}=x_{p+2is}^{(l-1)}$,
$x_{p+2(i-1)s+2j}=0$ and $x_{p+2(l-1)s+2k}=0$ for $1\leq i\leq l-1,$ $1\leq
j\leq s-1$ and $1\leq k\leq s-1$ such that $y_{p+2i+1}=0$ ($0\leq i\leq t$)
and $d_{2t+3}[x_{p}]_{2t+3}=[H_{2s+1}\wedge x_{p+2(l-1)s}^{(l-1)}]_{2t+3}.$ By
Theorem 4.3 and (4.15), there also exists a defining system $A=(a_{i,j})$ for
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle$ as
follows:
(5.5) $\left\\{\begin{array}[]{lll}&a_{i,j}=0{}{}&(1\leq i<j\leq l),\\\
&a_{i,i}=H_{2s+1}{}{}&(1\leq i\leq l),\\\ &a_{l+1,l+1}=x_{p},\\\
&a_{i,l+1}=(-1)^{l+1-i}x_{p+2(l+1-i)s}^{(l-1)}{}{}&(2\leq i\leq
l),\par\end{array}\right.$
to which the matrix associated is given by
(5.6) $\left(\begin{array}[]{ccccccc}H_{2s+1}&0&0&\cdots&0&0&\\\
&H_{2s+1}&0&\cdots&0&0&(-1)^{l-1}x_{p+2(l-1)s}^{(l-1)}\\\
&&H_{2s+1}&\cdots&0&0&(-1)^{l-2}x_{p+2(l-2)s}^{(l-1)}\\\
&&&\ddots&\vdots&\vdots&\vdots\\\ &&&&H_{2s+1}&0&(-1)^{2}x_{p+4s}^{(l-1)}\\\
&&&&&H_{2s+1}&(-1)x_{p+2s}^{(l-1)}\\\ &&&&&&x_{p}\\\
\end{array}\right)_{(l+1)\times(l+1).}$
The desired result follows.
∎
To obtain our desired theorems by specific elements of Massey products, we
restrict the allowable choices of defining systems for the two Massey products
in Lemma 5.5 (cf. [15]). By Lemma 5.5, we give the following definitions.
###### Definition 5.6.
(1) Given a class $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$ $(t\geq 1)$, a specific
element of $(t+2)$-fold Massey product
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle$, denoted by
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}$, is a
class in $H^{p+2t+3}(M)$ represent by $c(A)$, where $A$ is a defining system
obtained from Theorems 4.1. We define the $(t+2)$-fold allowable Massey
product
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{\star}$ to
be the set of all the cohomology classes $w\in H^{p+2t+3}(M)$ for which there
exists a defining system $A$ obtained from Theorem 4.1 such that $c(A)$
represents $w$.
(2) Similarly, given a class $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$ $(t\geq 1)$,
when $t=ls-1$ ($l\geq 2$) we define the specific element of $(l+1)$-fold
Massey product
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle$ and the
$(l+1)$-fold allowable Massey product
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{\star}$
by replacing Theorem 4.1 by Theorem 4.3 in (1).
###### Remark 5.7.
(1) From Definition 5.6, we can get the following
$\begin{array}[]{l}\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{\star}\subseteq\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle,\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{\star}\subseteq\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle.\end{array}$
(2) The allowable Massey product
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{\star}$ is
less ambiguous than the general Massey product
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle$. Take
$\langle H_{3},H_{3},x_{p}\rangle_{\star}$ in Definition 5.6 for example.
Suppose $H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$. By Theorem 4.1 and (4.2),
there exist $x_{p+2j}^{(1)}$ such that $y_{p+2i+1}=0\quad(0\leq i\leq 1)$ and
$d_{5}[x_{p}]_{5}=[H_{3}\wedge x_{p+2}^{(1)}+H_{5}\wedge x_{p}]_{5}.$ By Lemma
5.5, we get a defining system $A$ for $\langle H_{3},H_{3},x_{p}\rangle$ and
its related cocycle $c(A)=-H_{3}\wedge x_{p+2}^{(1)}-H_{5}\wedge x_{p}.$ Thus,
we have
(5.7) $\langle H_{3},H_{3},x_{p}\rangle_{A}=[-H_{3}\wedge
x_{p+2}^{(1)}-H_{5}\wedge x_{p}].$
Obviously, the indeterminacy of the allowable Massey product $\langle
H_{3},H_{3},x_{p}\rangle_{\star}$ is $[H_{3}]H^{p+2}(M)$. However, in the
general case, the indeterminacy of the Massey product $\langle
H_{3},H_{3},x_{p}\rangle$ is $[H_{3}]H^{p+2}(M)+H^{5}(M)[x_{p}]$.
Similarly, the allowable Massey product
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{\star}$
is less ambiguous than the general Massey product
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle$.
Now we begin to show our main theorems.
###### Theorem 5.8.
For $H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$ and $[x_{p}]_{2t+3}\in
E_{2t+3}^{p,q}$, the differential of the spectral sequence (1.2)
$d_{2t+3}:E_{2t+3}^{p,q}\to E_{2t+3}^{p+2t+3,q-2t-2}$ is given by
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{t}[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3},$
and
$[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}$
is independent of the choice of the defining system $A$ obtained from Theorem
4.1.
###### Proof.
By Lemma 5.5 (1), there exist defining systems for
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle$ given by
Theorem 4.1. For any defining system $A=(a_{i,j})$ given by Theorem 4.1, by
(5.4) we have
$c(A)=(-1)^{t}(\sum\limits_{i=1}^{t}H_{2i+1}\wedge
x_{p+2(t-i+1)}^{(t)}+H_{2t+3}\wedge x_{p}).$
By Definition 5.6, we have
(5.8)
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}=[c(A)].$
Then by Theorem 4.1, we have
$\begin{array}[]{ll}d_{2t+3}[x_{p}]_{2t+3}&=[\sum\limits_{i=1}^{t}H_{2i+1}\wedge
x_{p+2(t-i+1)}^{(t)}+H_{2t+3}\wedge x_{p}]_{2t+3}\\\
&=(-1)^{t}[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}.\end{array}$
Thus, we have
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{t}[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}.$
By the arbitrariness of $A$, we have
$[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}$
is independent of the choice of the defining system $A$ obtained from Theorem
4.1. ∎
###### Example 5.9.
For formal manifolds which are manifolds with vanishing Massey products, it is
easy to get
$E_{4}^{p,q}\cong E_{\infty}^{p,q}$
by Theorem 5.8. Note that simply connected compact Kähler manifolds are an
important class of formal manifolds (see [2]).
###### Remark 5.10.
(1) From the proof of the theorem above, we have that the specific element
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}$
represents a class in $E_{2t+3}^{\ast,\ast}$. For two different defining
systems $A_{1}$ and $A_{2}$ given by Theorem 4.1, we have
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A_{1}}\not=\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A_{2}}$
generally. However, in the spectral sequence (1.2) we have
$[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A_{1}}]_{2t+3}=[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A_{2}}]_{2t+3}.$
(2) Since the indeterminacy of
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{\star}$
does not affect our results, we will not analyze the indeterminacy of Massey
products in this paper.
(3) By Theorem 5.8,
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{t}[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}$
for $t\geq 1$ which is expressed only by $H_{3}$ and $x_{p}$. From the proof
of Theorem 5.8, we know that the expression above conceals some information,
because the other $H_{2i+1}$’s affect the result implicitly.
We have the following corollary (see [1, Proposition 6.1]).
###### Corollary 5.11.
For $H=H_{3}$ only and $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$ ($t\geq 1$), we have
that in the spectral sequence (1.2),
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{t}[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3},$
and
$[\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}$
is independent of the choice of the defining system $A$ obtained from Theorem
4.1.
###### Remark 5.12.
1. (1)
Because the definition of Massey products is different from the definition in
[1], the expression of differentials in Corollary 5.11 differs from the one in
[1, Proposition 6.1].
2. (2)
The two specific elements of
$\langle\underbrace{H_{3},\cdots,H_{3}}\limits_{t+1},x_{p}\rangle$ in Theorem
5.8 and Corollary 5.11 are completely different, and equal $[c(A_{1})]$ and
$[c(A_{2})]$ respectively, where $c(A_{i})$ $(i=1,2)$ are related cocycles of
the defining systems $A_{i}$ $(i=1,2)$ obtained from Theorem 4.1. The matrices
associated to the two defining systems are given by
$\left(\begin{array}[]{ccccccc}H_{3}&-H_{5}&H_{7}&\cdots&(-1)^{t-1}H_{2t+1}&(-1)^{t}H_{2t+3}&\\\
&H_{3}&-H_{5}&\cdots&(-1)^{t-2}H_{2t-1}&(-1)^{t-1}H_{2t+1}&(-1)^{t}x_{p+2t}^{(t)}\\\
&&H_{3}&\cdots&(-1)^{t-3}H_{2t-3}&(-1)^{t-2}H_{2t-1}&(-1)^{t-1}x_{p+2t-2}^{(t)}\\\
&&&\ddots&\vdots&\vdots&\vdots\\\ &&&&H_{3}&-H_{5}&(-1)^{2}x_{p+4}^{(t)}\\\
&&&&&H_{3}&(-1)x_{p+2}^{(t)}\\\ &&&&&&x_{p}\\\
\end{array}\right)_{(t+2)\times(t+2)}$
and
$\left(\begin{array}[]{ccccccc}H_{3}&0&0&\cdots&0&0&\\\
&H_{3}&0&\cdots&0&0&(-1)^{t}{x}_{p+2t}^{(t)}\\\
&&H_{3}&\cdots&0&0&(-1)^{t-1}{x}_{p+2t-2}^{(t)}\\\
&&&\ddots&\vdots&\vdots&\vdots\\\ &&&&H_{3}&0&(-1)^{2}{x}_{p+4}^{(t)}\\\
&&&&&H_{3}&(-1){x}_{p+2}^{(t)}\\\ &&&&&&x_{p}\\\
\end{array}\right)_{(t+2)\times(t+2),}$
respectively. Here $x_{p+2i}^{(t)}$ $(1\leq i\leq t)$ in the first matrix are
different from those in the second one.
For $H=H_{2s+1}$ $(s\geq 2)$ only (i.e., in the case $H_{i}=0$, $i\not=2s+1$)
and $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$ $(t\geq 1)$, we make use of Theorem 5.8
to get that
(5.9)
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{t}[\langle\underbrace{0,\cdots,0}\limits_{t+1},x_{p}\rangle_{A}]_{2t+3}.$
Obviously, some information has been concealed in the expression above. Now we
give another description of the differentials for this special case.
###### Theorem 5.13.
For $H=H_{2s+1}$ $(s\geq 1)$ only and $[x_{p}]_{2t+3}\in E_{2t+3}^{p,q}$
$(t\geq 1)$, the differential of the spectral sequence (1.2)
$d_{2t+3}:E_{2t+3}^{p,q}\to E_{2t+3}^{p+2t+3,q-2t-2}$ is given by
$d_{2t+3}[x_{p}]_{2t+3}=\left\\{\begin{array}[]{ll}[H_{2s+1}\wedge
x_{p}]_{2t+3}&t=s-1,\\\
(-1)^{l-1}[\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}]_{2t+3}&t=ls-1~{}(l\geq
2),\\\ 0&\text{otherwise,}\end{array}\right.$
and
$[\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}]_{2t+3}$
is independent of the choice of the defining system $B$ obtained from 4.3.
###### Proof.
When $t=s-1$, the result follows from Theorem 4.3.
When $t=ls-1$ $(l\geq 2)$, from Lemma 5.5 (2) we know that there exist
defining systems for
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle$
obtained from Theorem 4.3. For any defining system $B$ given by Theorem 4.3,
by (5.6) we get $c(B)=(-1)^{l-1}H_{2s+1}\wedge x_{p+2(l-1)s}^{(l-1)}.$ By
Definition 5.6,
(5.10)
$\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}=[c(B)].$
Then by Theorem 4.3, we have
$\begin{array}[]{ll}d_{2t+3}[x_{p}]_{2t+3}&=[H_{2s+1}\wedge
x_{p+2(l-1)s}^{(l-1)}]_{2t+3}\\\
&=(-1)^{l-1}[\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}]_{2t+3}.\end{array}$
Thus
$d_{2t+3}[x_{p}]_{2t+3}=(-1)^{l-1}[\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}]_{2t+3}.$
By the arbitrariness of $B$, we have
$[\langle\underbrace{H_{2s+1},\cdots,H_{2s+1}}\limits_{l},x_{p}\rangle_{B}]_{2t+3}$
is independent of the choice of the defining system $B$ obtained from Theorem
4.3.
For the rest cases of $t$, the results follows from Theorem 4.3.
The proof of this theorem is completed. ∎
###### Remark 5.14.
We now use the special case that $H=H_{5}$ and $d_{9}[x_{p}]_{9}$ to
illustrate the compatibility between Theorems 5.8 and 5.13 for $s=2$ and
$t=3$.
Note that in this case $H_{3}=0$ and $H_{i}=0$ for $i>5$. By Theorem 5.8, we
get the corresponding matrix associated to the defining system A for $\langle
0,0,0,0,x_{p}\rangle_{A}$ is
(5.11) $\left(\begin{array}[]{ccccccc}0&-H_{5}&0&0&\\\
&0&-H_{5}&0&-x_{p+6}^{(3)}\\\ &&0&-H_{5}&x_{p+4}^{(3)}\\\
&&&0&-x_{p+2}^{(3)}\\\ &&&&x_{p}\\\ \end{array}\right)_{5\times 5}$
and
(5.12) $\tilde{d}_{9}[x_{p}]_{9}=-[\langle 0,0,0,0,x_{p}\rangle_{A}]_{9}.$
By Theorem 5.13, in this case the matrix associated to the defining system $B$
for $\langle H_{5},H_{5},x_{p}\rangle_{B}$ is
(5.13) $\left(\begin{array}[]{ccccccc}H_{5}&0&\\\ &H_{5}&-x_{p+4}^{(1)}\\\
&&x_{p}\\\ \end{array}\right)_{3\times 3.}$
and
(5.14) $\bar{d}_{9}[x_{p}]_{9}=-[\langle H_{5},H_{5},x_{p}\rangle_{B}]_{9}.$
We claim that $\langle H_{5},H_{5},x_{p}\rangle_{\star}=\langle
0,0,0,0,x_{p}\rangle_{\star}$. For any defining system $B$ above, there is a
defining system ${\tilde{B}}$
$\left(\begin{array}[]{ccccccc}0&-H_{5}&0&0&\\\ &0&-H_{5}&0&0\\\
&&0&-H_{5}&x_{p+4}^{(1)}\\\ &&&0&0\\\ &&&&x_{p}\\\ \end{array}\right)_{5\times
5}$
for $\langle 0,0,0,0,x_{p}\rangle$ which can be obtained from Theorem 4.1 such
that
$\langle 0,0,0,0,x_{p}\rangle_{{\tilde{B}}}=\langle
H_{5},H_{5},x_{p}\rangle_{B}.$
Hence $\langle H_{5},H_{5},x_{p}\rangle_{\star}\subseteq\langle
0,0,0,0,[x_{p}]\rangle_{\star}.$ On the other hand, for any defining system
$A$ above there is also a defining system $\bar{A}$
$\left(\begin{array}[]{ccccccc}H_{5}&0&\\\ &H_{5}&-x_{p+4}^{(3)}\\\ &&x_{p}\\\
\end{array}\right)_{3\times 3.}$
for $\langle H_{5},H_{5},x_{p}\rangle$ which can be obtained from Theorem 4.3
such that
$\langle H_{5},H_{5},x_{p}\rangle_{\bar{A}}=\langle
0,0,0,0,x_{p}\rangle_{{A}}.$
Therefore $\langle 0,0,0,0,x_{p}\rangle_{\star}\subseteq\langle
H_{5},H_{5},x_{p}\rangle_{\star},$ and the claim follows.
By Theorem 5.8 and Remark 5.7, we have that
$\tilde{d}_{5}[y_{p}]_{5}=-[\langle 0,0,y_{p}\rangle_{A}]_{5}=-[-H_{5}\wedge
y_{p}]_{5}=[H_{5}\wedge y_{p}]_{5}$. By Theorem 5.13,
$\bar{d}_{5}[y_{p}]_{5}=[H_{5}\wedge y_{p}]_{5}$. By Proposition 3.4,
$\tilde{d}_{1}=\bar{d}_{1}=d$ and $\tilde{d}_{3}=\bar{d}_{3}=0$. It follows
that $\tilde{d}_{5}=\bar{d}_{5}.$
By Theorems 5.8 and 4.1, we have that $\tilde{d}_{7}[z_{p}]_{7}=[\langle
0,0,0,z_{p}\rangle_{A}]_{7}=[-H_{5}\wedge z^{(2)}_{p+2}]_{7}$, where
$z^{(2)}_{p+2}$ is an arbitrary $(p+2)$-form satisfying
$d(z^{(2)}_{p+2})=0\wedge z_{p}$. By Remark 5.10 (2), we take
$z^{(2)}_{p+2}=0$. Then we have $\tilde{d}_{7}[z_{p}]_{7}=0$, i.e.,
$\tilde{d}_{7}=0$. At the same time, we also have $\bar{d}_{7}=0$ from Theorem
5.13. Thus $\tilde{d}_{7}=\bar{d}_{7}=0$.
By $\tilde{E}_{1}^{p,q}=\bar{E}_{1}^{p,q}$, $\tilde{d}_{i}=\bar{d}_{i}$ for
$1\leq i\leq 7$ and $\langle H_{5},H_{5},x_{p}\rangle_{\star}=\langle
0,0,0,0,x_{p}\rangle_{\star}$, we can conclude that
$\tilde{d}_{9}=\bar{d}_{9}$ from (5.12) and (5.14).
## 6\. The indeterminacy of differentials in the spectral sequence (1.2)
Let $[x_{p}]_{r}\in E_{r}^{p,q}$. The indeterminacy of $[x_{p}]$ is a normal
subgroup $G$ of $H^{\ast}(M)$, which means that if there is another element
$[y_{p}]\in H^{p}(M)$ which also represents the class $[x_{p}]_{r}\in
E_{r}^{p,q}$, then $[y_{p}]-[x_{p}]\in G$.
In this section, we will show that for
$H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$ and $[x_{p}]_{2t+3}$, the
indeterminacy of the differential $d_{2t+3}[x_{p}]\in E_{2}^{p+2t+3,q-2t-2}$
is a normal subgroup of $H^{\ast}(M)$.
From the long exact sequence (3.3), we have a commutative diagram
(6.1)
---
$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H_{D}^{p+q}(K_{p+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{*}}$$\textstyle{H_{D}^{p+q}(K_{p+1}/K_{p+2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H_{D}^{p+q+1}(K_{p+2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{*}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H_{D}^{p+q}(K_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{*}}$$\textstyle{H_{D}^{p+q}(K_{p}/K_{p+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H_{D}^{p+q+1}(K_{p+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{*}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H_{D}^{p+q}(K_{p-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{*}}$$\textstyle{H_{D}^{p+q}(K_{p-1}/K_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H_{D}^{p+q+1}(K_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{*}}$$\textstyle{\cdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$
in which any sequence consisting of a vertical map $i^{*}$ followed by two
horizontal maps $j^{*}$ and $\delta$ and then a vertical map $i^{*}$ followed
again by $j^{\ast}$ and $\delta$ and iteration of this is exact. From this
diagram there is obtained a spectral sequence in which
$E_{1}^{p,q}=H_{D}^{p+q}(K_{p}/K_{p+1})$ and for $r\geq 2$, $E_{r}^{p,q}$ is
defined to be the quotient $Z_{r}^{p,q}/B_{r}^{p,q}$, where
(6.2)
$\begin{array}[]{l}Z_{r}^{p,q}=\delta^{-1}(i^{*r-1}H_{D}^{p+q+1}(K_{p+r})),\\\
B_{r}^{p,q}=j^{*}({\rm ker}[i^{*r-1}:H_{D}^{p+q}(K_{p})\rightarrow
H_{D}^{p+q}(K_{p-r+1})]).\\\ \end{array}$
We also have a sequence of inclusions
(6.3) $B_{2}^{p,q}\subset\cdots\subset B_{r}^{p,q}\subset
B_{r+1}^{p,q}\subset\cdots\subset Z_{r+1}^{p,q}\subset
Z_{r}^{p,q}\subset\cdots\subset Z_{2}^{p,q}.$
By [6, 7], the $E_{r}^{\ast,\ast}$-term defined above is the same as the one
in the spectral sequence (1.2). A similar argument about a homology spectral
sequence is given in [16, p. 472-473].
###### Theorem 6.1.
Let $H=\sum_{i=1}^{[\frac{n-1}{2}]}H_{2i+1}$ and $[x_{p}]_{r}\in E_{r}^{p,q}$
$(r\geq 3)$, then the indeterminacy of $[x_{p}]\in E_{2}^{p,q}\cong H^{p}(M)$
is the following normal subgroup of $H^{p}(M)$
$\frac{{\rm im}[\bar{\delta}:H_{D}^{p+q-1}(K_{p-r+1}/K_{p})\to
H_{D}^{p+q}(K_{p}/K_{p+1})]}{{\rm im}[d:\Omega^{p-1}(M)\to\Omega^{p}(M)]},$
where $d$ is just the exterior differentiation and $\bar{\delta}$ is the
connecting homomorphism of the long exact sequence induced by the short exact
sequence of cochain complexes
$0\longrightarrow
K_{p}/K_{p+1}\stackrel{{\scriptstyle\bar{i}}}{{\longrightarrow}}K_{p-r+1}/K_{p+1}\stackrel{{\scriptstyle\bar{j}}}{{\longrightarrow}}K_{p-r+1}/K_{p}\longrightarrow
0.$
###### Proof.
From the tower (6.3) above, we get a tower of subgroups of $E_{2}^{p,q}$
$\begin{split}B_{3}^{p,q}/B_{2}^{p,q}&\subset\cdots\subset
B_{r}^{p,q}/B_{2}^{p,q}\subset\cdots\subset Z_{r}^{p,q}/B_{2}^{p,q}\subset\\\
&\cdots\subset Z_{3}^{p,q}/B_{2}^{p,q}\subset
Z_{2}^{p,q}/B_{2}^{p,q}=E_{2}^{p,q}.\end{split}$
Note that
$E_{r}^{p,q}\cong(Z_{r}^{p,q}/B_{2}^{p,q})/(B_{r}^{p,q}/B_{2}^{p,q}).$ It
follows that the indeterminacy of $[x_{p}]$ is the normal subgroup
$B_{r}^{p,q}/B_{2}^{p,q}$ of $H^{p}(M)$.
From the short exact sequences of cochain complexes
$\begin{array}[]{c}0\longrightarrow K_{p}\stackrel{{\scriptstyle
i^{\prime}}}{{\longrightarrow}}K_{p-r+1}\stackrel{{\scriptstyle
j^{\prime}}}{{\longrightarrow}}K_{p-r+1}/K_{p}\longrightarrow 0,\\\
0\longrightarrow
K_{p}/K_{p+1}\stackrel{{\scriptstyle\bar{i}}}{{\longrightarrow}}K_{p-r+1}/K_{p+1}\stackrel{{\scriptstyle\bar{j}}}{{\longrightarrow}}K_{p-r+1}/K_{p}\longrightarrow
0,\end{array}$
we can get the following long exact sequence of cohomology groups
(6.4)
$\begin{array}[]{c}\cdots\stackrel{{\scriptstyle\delta^{\prime}}}{{\longrightarrow}}H_{D}^{s}(K_{p})\stackrel{{\scriptstyle
i^{\prime\ast}}}{{\longrightarrow}}H_{D}^{s}(K_{p-r+1})\stackrel{{\scriptstyle
j^{\prime\ast}}}{{\longrightarrow}}H_{D}^{s}(K_{p-r+1}/K_{p})\stackrel{{\scriptstyle\delta^{\prime}}}{{\longrightarrow}}\cdots,\\\
\cdots\stackrel{{\scriptstyle\bar{\delta}}}{{\longrightarrow}}H_{D}^{s}(K_{p}/K_{p+1})\stackrel{{\scriptstyle\bar{i}^{\ast}}}{{\longrightarrow}}H_{D}^{s}(K_{p-r+1}/K_{p+1})\stackrel{{\scriptstyle\bar{j}^{\ast}}}{{\longrightarrow}}H_{D}^{s}(K_{p-r+1}/K_{p})\stackrel{{\scriptstyle\bar{\delta}}}{{\longrightarrow}}\cdots,\end{array}$
where $\delta^{\prime}$ and $\bar{\delta}$ are the connecting homomorphisms.
Combining (3.3) and (6.4), we have the following commutative diagram of long
exact sequences
(6.5)
$\textstyle{H_{D}^{p+q-1}(K_{p-r+1}/K_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\delta}}$$\scriptstyle{\delta^{\prime}}$$\textstyle{H_{D}^{p+q}(K_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\scriptstyle{i^{{\prime\ast}}}$$\textstyle{H_{D}^{p+q}(K_{p-r+1})}$$\textstyle{H_{D}^{p+q}(K_{p}/K_{p+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\bar{i}^{*}}$$\textstyle{H_{D}^{p+q+1}(K_{p+1})}$$\textstyle{H_{D}^{p+q}(K_{p-r+1}/K_{p+1}).}$
Using the commutative diagram above and the fact that
$i^{{\ast}^{r-1}}={i}^{\prime\ast}$, we have
$\begin{array}[]{ll}B_{r}^{p,q}&=j^{*}({\rm
ker}[i^{*^{r-1}}:H_{D}^{p+q}(K_{p})\rightarrow H_{D}^{p+q}(K_{p-r+1})])\\\
&=j^{*}({\rm ker}[~{}i^{\prime\ast}:H_{D}^{p+q}(K_{p})\longrightarrow
H_{D}^{p+q}(K_{p-r+1})])\\\ &\cong j^{*}({\rm
im}[{\delta}^{{}^{\prime}}:H_{D}^{p+q-1}(K_{p-r+1}/K_{p})\rightarrow
H_{D}^{p+q}(K_{p})])\\\ &\cong{\rm
im}[\bar{\delta}:H_{D}^{p+q-1}(K_{p-r+1}/K_{p})\to
H_{D}^{p+q}(K_{p}/K_{p+1})].\end{array}$
When $r=2$, from (6.5) we have that
$\bar{\delta}=\delta^{\prime}j^{\ast}:H_{D}^{p+q-1}(K_{p-1}/K_{p})\to
H_{D}^{p+q}(K_{p}/K_{p+1}).$
From (3.4), it follows that $\bar{\delta}=d_{1}$. By Proposition 3.4,
$\bar{\delta}=d$. Thus we have
$\begin{array}[]{ll}B_{2}^{p,q}&\cong{\rm
im}[\bar{\delta}:H_{D}^{p+q-1}(K_{p-1}/K_{p})\to
H_{D}^{p+q}(K_{p}/K_{p+1})]\\\ &\cong{\rm
im}[d:\Omega^{p-1}(M)\to\Omega^{p}(M)].\end{array}$
The desired result follows. ∎
By Theorem 6.1, we obtain the following corollary.
###### Corollary 6.2.
In Theorem 5.8, for $d_{2t+3}[x_{p}]_{2t+3}\in E_{2t+3}^{p+2t+3,q-2t-2}$ we
have the indeterminacy of $d_{2t+3}[x_{p}]$ is a normal subgroup of
$H^{p+2t+3}(M)$
$\frac{{\rm im}[\bar{\delta}:H_{D}^{p+q}(K_{p+1}/K_{p+2t+3})\to
H_{D}^{p+q+1}(K_{p+2t+3}/K_{p+2t+4})]}{{\rm
im}[d:\Omega^{p+2t+2}(M)\to\Omega^{p+2t+3}(M)]},$
where $d$ is just the exterior differentiation and $\bar{\delta}$ is the
connecting homomorphism of the long exact sequence induced by the short exact
sequence of cochain complexes
$0\longrightarrow
K_{p+2t+3}/K_{p+2t+4}\stackrel{{\scriptstyle\bar{i}}}{{\longrightarrow}}K_{p+1}/K_{p+2t+4}\stackrel{{\scriptstyle\bar{j}}}{{\longrightarrow}}K_{p+1}/K_{p+2t+3}\longrightarrow
0.$
###### Proof.
In Theorem 6.1 $r,p$ and $q$ are replaced by $2t+3,p+2t+3$ and $q-2t-2$, then
the desired result follows. ∎
Acknowledgment The authors would like to thank Jim Stasheff for helpful
comments.
## References
* [1] M. F. Atiyah and G. B. Segal, Twisted K-theory and cohomology, Inspired by S. S. Chern, Nankai Tracts Math. P. A. Griffith (Ed.), vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 5-43, arXiv:math/0510674v1 [math.KT].
* [2] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of $K\ddot{a}hler$ manifolds, Invent. Math. 29 (3) (1975), 245–274.
* [3] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, (2002).
* [4] D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431–449.
* [5] D. Kraines and C. Schochet Differentials in the Eilenberg-Moore spectral sequence, J. Pure Appl. Algebra 2 (2) (1972), 131–148.
* [6] W. S. Massey, Exact couples in algebraic topology. I, II, Ann. of Math. (2) 56 (1952), 363–396.
* [7] W. S. Massey, Exact couples in algebraic topology. III, IV, V, Ann. of Math. (2) 57 (1953), 248–286.
* [8] W. S. Massey, Some higher order cohomology operations, 1958 Symposium internacional de topología algebraica International symposium on algebraic topology pp. 145–154 Universidad Nacional Autónoma de México and UNESCO, Mexico City.
* [9] V. Mathai and S. Wu, Analytic torsion for twisted de Rham complexes, arXiv: 0810.4204v3 [math.DG].
* [10] J. P. May, Matric Massey products, J. Algebra 12 (1969), 533–568.
* [11] J. P. May, The cohomology of augmented algebras and generalized Massey products for ${\rm DGA}$-Algebras, Trans. Amer. Math. Soc. 122 (1966), 334–340.
* [12] J. P. May, The cohomology of principal bundles, homogeneous spaces, and two-stage Postnikov systems, Bull. Amer. Math. Soc. 74 (1968), 334–339.
* [13] J. McCleary, A user’s guide to spectral sequences, second edition, Cambridge University Press, Cambridge, (2001).
* [14] R. Rohm and E. Witten, The antisymmetric tensor field in superstring theory, Ann. Physics 170 (2) (1986), 454-489.
* [15] R. T. Sharifi, Massey products and ideal class groups, J. Reine Angew. Math. 603 (2007), 1-33.
* [16] E. W. Spanier, Algebraic Topology, Springer-Verlag, New York-Berlin, (1981).
|
arxiv-papers
| 2009-11-07T13:56:58 |
2024-09-04T02:49:06.332956
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Weiping Li, Xiugui Liu and He Wang",
"submitter": "Xiugui Liu",
"url": "https://arxiv.org/abs/0911.1417"
}
|
0911.1579
|
2009 Vol. 9 No. XX, 000–000
11institutetext: Department of Astronomy, Nanjing University, Nanjing 210093,
China
Received [year] [month] [day]; accepted [year] [month] [day]
# GRB Jet Beaming Angle Statistics
Y. Gao Z.G. Dai
###### Abstract
Existing theory and models suggest that a Type I (merger) GRB should have a
larger jet beaming angle than a Type II (collapsar) GRB, but so far no
statistical evidence is available to support this suggestion. In this paper,
we obtain a sample of 37 beaming angles and calculate the probability that
this is true. A correction is also devised to account for the scarcity of Type
I GRBs in our sample. The probability is calculated to be 83% without the
correction and 71% with it.
###### keywords:
gamma rays: bursts — ISM: jets and outflows — methods: statistical
## 1 Introduction
There are two intrinsically different phenomena that give rise to gamma-ray
bursts (GRBs). One is the merging of two compact objects, such as that of a
neutron star and a black hole; the other is the core collapse of a massive
star “collapsar”), such as the birth of a Type Ib/c Supernova. According to a
classification scheme developed recently (Zhang, 2007 and Zhang et al, 2009),
a GRB obtained from the former channel is defined as a Type I GRB, while one
from the latter would be classified as a Type II GRB. This classification
scheme, though more intrinsic than the classical short/hard vs long/soft
categories, is not easy to carry out at the present stage, since many
different criteria need to be applied in order to determine the progenitor of
a GRB, few of which are decisive. One of these criteria is that Type I GRBs
are usually short/hard ( $T_{90}\leq 2\,{\rm s}$ ), while Type II GRBs are
usually long/soft ( $T_{90}>2\,{\rm s}$ ). This criteria, though supported by
many individual cases, is not decisive, as any short GRB could theoretically
have been a long GRB had it occurred at a high enough redshift. Of the
criteria that are decisive, such as whether or not a GRB emanates
gravitational radiation, most are impractical and the remainder is not
applicable to a great majority of GRBs at the present stage.
GRBs eject their energy in the form of jets. These jets have already been
modeled, and their beaming angles (also called opening angles) can be
calculated from observed physical data (Sari et al, 1999). The size of the
beaming angle, according to the models, is highly dependent on the nature of
the GRB progenitor: theoretically, Type II GRBs should have smaller beaming
angles than Type I GRBs, due to the collimation effect of the stellar wind
hugging collapsars. However, this dependency has not yet been supported
statistically, possibly due to the aggravatingly small sample of Type I GRB
jet beaming angles available, hence this paper, which specifically seeks to
find statistical evidence of this theoretical relationship.
This paper is divided into 5 sections, of which this introduction is the
first. In the second, we will list the relevant observational data and use it
to obtain the jet beaming angles of 37 GRBs. The third section will cover data
reduction and statistical analysis. The results and discussion of the analysis
will be presented in the fourth section, and the entire paper will be
summarized in the fifth section, which is the conclusion.
## 2 Collected Data
According to the literature (Sari et al, 1999), the jet beaming angle of a GRB
could be obtained by Eq.(1):
$\theta_{j}=0.057(\frac{t_{j}}{1\,{\rm
day}})^{\frac{3}{8}}(\frac{1+z}{2})^{-\frac{3}{8}}(\frac{E_{iso}(\gamma)}{10^{53}\,{\rm
erg}})^{-\frac{1}{8}}(\frac{\eta_{\gamma}}{0.2})^{\frac{1}{8}}(\frac{n}{0.1\,{\rm
cm}^{-3}})^{\frac{1}{8}},$ (1)
where $\theta_{j}$ is the jet beaming angle mentioned above, $t_{j}$ is the
time of the jet break, $z$ is the redshift at which the GRB took place,
$E_{iso}(\gamma)$ is the isotropic energy released by the burst,
$\eta_{\gamma}$ is the gamma-ray radiative efficiency of the jet and $n$ is
the density of the interstellar medium. $t_{j}$, $z$ and $E_{iso}(\gamma)$ are
observable parameters, shown in Table 1 (Appendix A). $\eta_{\gamma}$ and $n$,
however, are not observable, but thankfully do not have a very significant
effect on the value of $\theta_{j}$. Therefore, we set $\eta_{\gamma}=0.2$ and
$n=0.1\,{\rm cm}^{-3}$. The jet beaming angles obtained from these data are
presented in Appendix A and in Fig. 1 below. In Fig. 1, the bars filled in
black, stripes and blue correspond to Sample I, Sample II and unclassified jet
beaming angles (defined below) respectively.
Figure 1: Jet beaming angles: complete sample. Sample I is shown in black
filling, while Sample II is shown with striped filling. Filled in blue are
those GRBs that have not been positively identified as either Sample I or
Sample II. The horizontal axis is $\theta_{j}$ in degrees, while the vertical
axis the number of opening angles that are of the specified magnitude.
## 3 Data reduction
As shown in Appendix A, we have only managed to confirm 2 Type I bursts and 13
Type II bursts among the GRBs whose parameters we obtained.
Of the remaining GRBs, one is identified to be a short burst, while 21 of the
others are known to be long. According to experience, short GRBs are usually
of a Type I origin, while long bursts are usually of Type II. Therefore, we
incorporate the short GRBs into the Type I sample, and the long GRBs into the
Type II sample. It is hoped that, statistically, this will serve the purpose
of giving a larger sample while maintaining sample integrity.
Six bursts, namely GRB970828, GRB990510, GRB990705, GRB991216, GRB000301C and
GRB010222 cannot be identified as either Type I/II or long/short bursts, and
therefore cannot be used in the statistics. These bursts are represented by
the columns filled in blue in Fig. 1.
Thus, we obtain a “Type I + short” sample, noted as “Sample I” for the rest of
the paper, and a “Type II + long” sample, hereby noted as “Sample II”. Their
jet beaming angles are presented in Fig. 1 (black and striped filling
respectively). A primary contributing factor to the scarcity of Sample I data
is the fact that $t_{j}$ is hard to determine in short bursts, rendering Eq.
(1) unapplicable. Of the five bursts known to be of Type I origin (Zhang et
al, 2009), we obtained the opening angles for two of them (see Appendix A for
details). Of the remaining three (GRB050509B, GRB050724 and GRB061006), the
jet break time of GRB050509B and GRB061006 cannot be found, while only a lower
limit constraint could be placed on the jet break time of GRB050724. The
problem of whether lower limit constraints could be used in the sample is
discussed further in section 4. Of the short/hard bursts listed in Zhang et
al, 2009, the opening angle of only one could be ascertained. Since GRB080503,
the latest short/hard burst mentioned in Zhang et al, 2009, five bursts
(GRB080702A, GRB080905A, GRB080919, GRB081024 and GRB090510) have been
observed according to GCN reports that fall indisputably in the short/hard
category. However, no jet break time has been found for any of them.
From Fig. 1, it should be quite obvious that Sample I data have a greater
arithmetic mean in comparison to Sample II data. This is indeed the case, as
Sample I has a mean value of 10.42 degrees, while the mean of Sample II is
only 3.42 degrees. This is supportive of the statement that Type I GRBs have a
larger beaming angle than Type II GRBs. However, the fact that there are 4 of
the Sample II data that are larger than 2 of the 3 Sample II data leaves
plenty of room for argument. Therefore, statistical analysis based on the data
that lead to a quantitative result on the validity of the statement above is
required. In order to obtain such a quantitative result, statistical fitting
must be carried out. Many papers consider the Gaussian distribution under
similar circumstances (for instance: Zhang and Meszaros, 2002), and therefore
we will fit the data using the Gaussian distribution. Whether opening angles
really do follow Gaussian distributions is indeed rather problematic: this
topic will be discussed further below.
Assuming that elements taken from Samples I and II are random variables that
have Gaussian distributions ($N(\mu,\sigma^{2})$), we take their probability
densities to be
$N(\theta_{j})=\frac{1}{\sigma_{i}\sqrt{2\pi}}\exp[-\frac{1}{2}(\frac{\theta_{j}-\mu_{i}}{\sigma_{i}})^{2}],~{}~{}~{}~{}~{}(i=1,2),$
(2)
respectively. Here, $i=1$ corresponds to Sample I, while $i=2$ corresponds to
Sample II. $\mu_{i}$ and $\sigma_{i}$ are the means and standard variations of
the two samples respectively, and are hereby defined mathematically as
$\mu_{i}=\sum\limits_{k}{X_{ki}}$ and
$\sigma_{i}^{2}=\sum\limits_{k}{{(\mu_{i}-X_{ki})}^{2}}$, where $X_{ki}$ is
the ”k”th datum from Sample i.
Taking the sample mean and sample standard deviation as $\mu$ and $\sigma$
respectively for both samples ($\mu_{1}=10.42$, $\mu_{2}=3.42$,
$\sigma_{1}^{2}=46.24$, $\sigma_{2}^{2}=9.07$), we obtain best-fit curves as
shown in Figs. 2 and 3, where they have been superimposed on their respective
sample distributions for comparison. For the rest of this paper, these best-
fit distributions for Sample I and Sample II will be termed Fit I and Fit II
respectively. A Kolmogorov-Smirnov test has been preformed for Fit II,
resulting in $D_{34}=0.125$, in other words, an acceptance level of
$1-\alpha\approx 65\%$. Fit I is expected to have a very low acceptance level,
which we did not calculate but are certain that it is (possibly much) smaller
than 30%, which is one of the reasons why we decided to devise the correction
below.
Figure 2: Sample I best fit probability distribution curve superimposed on
Sample I bar graph. Figure 3: Sample II best fit probability distribution
curve superimposed on Sample II bar graph.
Next, we calculate the probability that a random variable from Fit I
($\theta_{j1}$) is larger than a random variable from Fit II ($\theta_{j2}$).
The random variable $\theta_{j1}-\theta_{j2}$ should have a distribution of
Figure 4: Distribution of random variable $\theta_{j1}-\theta_{j2}$. This is
the probability distribution of the value of the difference between a random
variable from Fit I and another from Fit II.
$N(\theta_{j1}-\theta_{j2})=\frac{1}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}\sqrt{2\pi}}\exp[-\frac{1}{2}(\frac{(\theta_{j1}-\theta_{j2})-(\mu_{1}-\mu_{2})}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}})^{2}],$
(3)
This distribution is another normal distribution with a mean of
$(\mu_{1}-\mu_{2})$ and a variance of $(\sigma_{1}^{2}+\sigma_{2}^{2})$, as
shown in Fig. 4. Thus,
$P(\theta_{j1}>\theta_{j2})=P(\theta_{j1}-\theta_{j2}>0)=\int_{0}^{+\infty}N(\theta_{j1}-\theta_{j2})d(\theta_{j1}-\theta_{j2}),$
i.e.
$P_{1}(\theta_{j1}>\theta_{j2})=\int_{0}^{+\infty}\frac{1}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}\sqrt{2\pi}}exp[-\frac{1}{2}(\frac{x-(\mu_{1}-\mu_{2})}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}})^{2}]dx.$
(4)
Calculating this by means of a Fortran program, we obtain
$P_{1}(\theta_{j1}>\theta_{j2})=0.83.$ (5)
The probability shown in Eq. (5) is the probability that a Type I jet angle is
larger than a Type II jet beaming angle, but only if Type I and II GRB beaming
angles have a distribution strictly the same as Fit I and Fit II respectively.
In the case of Fit II, 34 samples have been taken to make the fit, therefore
the difference is considered negligible. However, for Fit I, which was made
with only three samples, the effects of inconsistency must be considered.
According to Eq. (4), $P(\theta_{j1}>\theta_{j2})$ can be calculated once
$\mu_{1}$ and $\sigma_{1}^{2}$ are given as constants. But the $\mu_{1}$ and
$\sigma_{1}^{2}$ in this equation have probability distributions of their own,
which are reliant on the consistency of Fit I. They are
$t(N-1)\sim\frac{(\overline{X}-\mu_{1})\sqrt{N}}{S_{X}},$ (6)
and
$\chi^{2}(N-1)\sim\frac{(N-1)S_{X}^{2}}{\sigma_{1}^{2}},$ (7)
respectively, where $N=3$, the number of Sample I elements. Here, t and
$\chi^{2}$ correspond to a t distribution and a $\chi^{2}$ distribution
respectively, $\overline{X}$ is the sample mean, $S_{X}$ is the sample
variance, $\mu_{1}$ and $\sigma_{1}$ are the theoretical mean and standard
variation of Sample I respectively(shown here as variables).
Integrating Eq. (4) over these probability distributions, taking
$\frac{(\overline{X}-\mu_{1})\sqrt{N}}{S_{X}}$ and
$\frac{(N-1)S_{X}^{2}}{\sigma_{1}^{2}}$ to be $y$ and $z$ respectively, we
write
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P_{2}(\theta_{j1}>\theta_{j2})$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\int_{0}^{+\infty}\chi^{2}(z,2)\int_{-\infty}^{+\infty}t(y,2)\cdot
P_{1}dydz$
note that $P_{1}$, though given as a constant in Eq.(5), is dependent on
$(\mu_{1}-\mu_{2})$ and $(\sigma_{1}^{2}+\sigma_{2}^{2})$, which are not
constants for the purpose of the following calculations. Substituting the
right hand side of Eq.(4) for $P_{1}(y,z)$, and equivalent terms in $y$ and
$z$ for $\mu_{1}$ and $\sigma_{1}$, we obtain
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P_{2}(\theta_{j1}>\theta_{j2})$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\int_{0}^{+\infty}\chi^{2}(z,2)\int_{-\infty}^{+\infty}t(y,2)\cdot
P_{1}(y,z)dydz$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\int_{0}^{+\infty}\chi^{2}(z,2)\int_{-\infty}^{+\infty}t(y,2)\int_{0}^{+\infty}\frac{1}{\sqrt{g(z)+\sigma_{2}^{2}}\sqrt{2\pi}}$
$\exp[-\frac{1}{2}(\frac{x-(f(y)-\mu_{2})}{\sqrt{g(z)+\sigma_{2}^{2}}})^{2}]dxdydz,$
(8)
where $t(y,2)=\frac{1}{2\sqrt{2}}(1+\frac{y^{2}}{2})^{-\frac{3}{2}}$,
$\chi^{2}(z,2)=\frac{1}{2}exp(-\frac{z}{2})$,
$f(y)=\mu_{1}=\overline{X}-\frac{S_{X}}{\sqrt{3}}y$, and
$g(z)=\sigma_{1}^{2}=(N-1)S_{X}^{2}/z$.
Again using a Fortran program, we calculate this final result to be
$P_{2}(\theta_{j1}>\theta_{j2})=0.71.$ (9)
## 4 Discussion
Theoretically, it would have been more logical to perform a similar correction
on Sample II in conjunction with the one used on Sample I. However, this would
lead to 5-dimensional integration, which would be far more than what our PC
could manage in a short period of time (the 3-D integration for $P_{2}$ took
10 minutes at double precision, with a step length of 0.03 for $x$,$y$ and
$z$). Instead, we performed the 3-D integration correction on Samples I and II
respectively. Taking into account the first 4 digits after the decimal point,
$P_{1}$ is calculated to be 0.8272. This merely decreases to 0.8254 after the
application of the correction to Sample II, while it decreases to
$P_{2}=0.7118$ after application of the correction to Sample I. From these
numbers, we find that negligence of the Sample II correction induces an error
of only the magnitude of $10^{-3}$ (i.e. the third digit after the decimal
point), and thus find it safe to retain the first 2 digits without performing
the correction to Sample II, hence the number of digits used in the
presentation of our results (Eqs. (5) and (9)).
With an acceptance level of only 65% for even Fit II, it is indeed debatable
whether a Gaussian fit is appropriate for the Samples. However, due to the
fact that there does not exist an established fit for opening angles, a
Gaussian fit seems to be the natural choice. Further research may experiment
on other parametrical fits that may yield better acceptance levels.
It has been proposed that GRBs which would fall into the Sample I category,
but have only minimum constraints for their jet break times (such as
GRB050724) could be used in the statistics as well. In doing so, a minimum
value could be calculated for both $P_{1}$ and $P_{2}$, on the basis of a
larger sample. However, this could lead to complexities, since there are GRBs
which would fall into the Sample II category that have only minimum
constraints for their jet break times too. In this paper, we wish to avoid
such complexities, and therefore use only exact data.
The reader might want to notice that there are several problems that have not
been taken into account. Firstly, all data on $E_{iso}(\gamma)$, $z$ and
$t_{j}$ in this paper have been collected from the literature. While this
should cause no problems for $z$ and $E_{iso}(\gamma)$, it could be somewhat
problematic for $t_{j}$, since the time of the jet break is different for
different bands at which the specific observation was made. Throughout this
paper, this matter has been treated indiscriminately.Also, the data, as shown
in Table 1 (Appendix A), are presented with a certain deviation in the
literature. In other words, it is not precise. Even more prominent is the
crisis that if we take Fit I to be exactly the distribution of Type I GRB jet
beaming angles, a significant number of Type I jet beaming angles will be
smaller than zero. These problems, throughout the paper, have been treated as
insignificant details, hereby submitted to future scrutiny by the reader.
## 5 Conclusions
In this paper, we have obtained a Sample I of 3 jet beaming angles and a
Sample II of 34 jet beaming angles. These two samples are expected to be
representative of Type I and Type II GRBs respectively. After that, normal
(Gaussian) distributions have been fitted to the samples, resulting in Fit I
and Fit II. Taking these fits to be representative of Type I and Type II GRBs,
we then proceed to calculate the probability that a random variable taken from
Fit I is larger than another taken from Fit II, thereby deriving the
probability that a Type I GRB has a larger opening angle than a Type II GRB.
Taking into account the uncertainty caused by the very small Sample I, we then
devise a correction for the probability stated above, and reach a probability
that is much lower but still significantly high nevertheless.
If we take Sample I to be a sample perfectly representative of Type I GRBs,
then it could be said, with an 83% degree of certainty, that Type I GRBs have
larger jet beaming angles than Type II GRBs. However, this 83% drops to 71%
once we take into account the uncertainty caused by our small sample in Sample
I. In either case, it could be justifiably concluded, from our current
samples, that Type I GRBs generally have a larger beaming angle in comparison
to Type II GRBs.
Our results are supportive of current models and theory. Type I GRBs in
general have larger beaming angles in comparison to Type II GRBs, with a
fairly high degree of certainty, though by far not high enough to become a
decisive criterion as to whether a GRB is of Type I or Type II origin (so
far).
It is possible that with a larger Sample I, higher levels of certainty could
be reached. Therefore, further observations that yield data concerning Sample
I jet beaming angles (preferably Type I GRB jet beaming angles) are required
for progress. It is also possible that with a large enough Sample I, the
correction methods used in this paper (i.e. the correction for Sample I) will
become obsolete, however, given the current rate at which Sample I opening
angles are being derived, it seems unlikely that this would be the case in the
near future. Lastly, it would be desirable to find a parametrical fit which
could be supported theoretically, and which yields a higher level of
acceptance than the Gaussian.
###### Acknowledgements.
Many thanks to our research group at the Department of Astronomy, Nanjing
University, for valuable discussions. Thanks also to Mr. C. J. Pritchet for
discussions and for his encouragement and enlightenment to the first author.
This work was supported by the National Natural Science Foundation of China
(grant No. 10873009) and the National Basic Research Program of China (973
program) No. 2007CB815404.
## Appendix A Observed Data & Derived Jet Beaming Angles
Table 1 Data from Observations & Derived Data
GRB | $E_{iso}$ | t | z | Category | $\theta_{j}(deg)$
---|---|---|---|---|---
050709 | $0.00069^{[1]}$ | $10^{[1]}$ | $0.16^{[1]}$ | Type I[2] | 18.1957
060614 | $0.021^{[2]}$ | $1.3^{[4]}$ | $0.13^{[4]}$ | Type I[2] | 5.57885
970508 | $0.061^{[2]}$ | $25^{[4]}$ | $0.835^{[4]}$ | Type II[2] | 12.3366
980703 | $0.72^{[2]}$ | $2.49^{[4]}$ | $0.966^{[4]}$ | Type II[2] | 3.71802
990123 | $22.9^{[2]}$ | $1.8^{[4]}$ | $1.6^{[4]}$ | Type II[2] | 1.92367
990712 | $0.0672^{[5]}$ | $1.6^{[5]}$ | $0.433^{[5]}$ | Type II[2] | 4.77012
000418 | $0.75137^{[6]}$ | $25^{[6]}$ | $0.1181^{[6]}$ | Type II[2] | 8.54109
000926 | $2.71^{[2]}$ | $2.03^{[4]}$ | $2.307^{[4]}$ | Type II[2] | 2.40104
011211 | $0.67234^{[6]}$ | $1.77^{[6]}$ | $2.14^{[6]}$ | Type II[2] | 2.76815
020405 | $1^{[2]}$ | $2.74^{[4]}$ | $0.689^{[4]}$ | Type II[2] | 3.91555
020813 | $6.6^{[2]}$ | $0.46^{[4]}$ | $1.254^{[4]}$ | Type II[2] | 1.42145
000328 | $3.61^{[5]}$ | $0.8^{[5]}$ | $1.52^{[5]}$ | Type II[2] | 1.80902
030329 | $0.166^{[5]}$ | $0.5^{[5]}$ | $0.169^{[5]}$ | Type II[2] | 2.97281
041006 | $0.83^{[5]}$ | $0.16^{[5]}$ | $0.716^{[5]}$ | Type II[2] | 1.37314
050525A | $0.25^{[2]}$ | $0.16^{[4]}$ | $0.606^{[4]}$ | Type II[2] | 1.63548
051221A | $0.024^{[7]}$ | $4.1^{[4]}$ | $0.5465^{[4]}$ | short[8] | 7.50341
010921 | $0.13611^{[6]}$ | $33^{[6]}$ | $0.4509^{[6]}$ | long[9] | 13.5235
020124 | $2.15^{[5]}$ | $3^{[5]}$ | $3.198^{[5]}$ | long[9] | 2.61651
021004 | $0.55601^{[6]}$ | $7.6^{[6]}$ | $2.332^{[6]}$ | long[9] | 4.78799
030226 | $0.67^{[5]}$ | $0.84^{[5]}$ | $1.986^{[5]}$ | long[9] | 2.13395
030429 | $0.173^{[5]}$ | $1.77^{[5]}$ | $2.656^{[5]}$ | long[9] | 3.09816
050315 | $0.49^{[10]}$ | $2.6^{[4]}$ | $1.95^{[4]}$ | long[11] | 3.40529
050318 | $0.22^{[3]}$ | $0.12^{[4]}$ | $1.44^{[4]}$ | long[11] | 1.27531
050401 | $5.323^{[12]}$ | $0.06^{[12]}$ | $2.9^{[12]}$ | long[11] | 0.55384
050416A | $0.0083^{[5]}$ | $1^{[5]}$ | $0.653^{[5]}$ | long[11] | 4.92335
050820A | $9.74^{[3]}$ | $3.99^{[4]}$ | $2.61^{[4]}$ | long[11] | 2.55110
060124 | $4.1^{[3]}$ | $0.61^{[4]}$ | $2.3^{[4]}$ | long[11] | 1.45364
060206 | $0.43^{[3]}$ | $0.82^{[4]}$ | $4.05^{[4]}$ | long[11] | 1.83552
060210 | $4.15^{[13]}$ | $2.16^{[4]}$ | $3.91^{[4]}$ | long[11] | 2.00912
060526 | $0.26^{[3]}$ | $0.98^{[4]}$ | $3.21^{[4]}$ | long[11] | 2.23733
060605 | $0.25^{[14]}$ | $0.27^{[14]}$ | $3.773^{[14]}$ | long[11] | 1.32273
060814 | $0.7^{[3]}$ | $0.79^{[4]}$ | $0.84^{[4]}$ | long[11] | 2.48693
061121 | $2.25^{[3]}$ | $0.28^{[4]}$ | $1.31^{[4]}$ | long[11] | 1.33753
061126 | $1.06^{[15]}$ | $25.7^{[15]}$ | $1.1588^{[15]}$ | long[11] | 8.20785
070125 | $10.6^{[16]}$ | $3.8^{[16]}$ | $1.547^{[16]}$ | long[16] | 2.82483
071010A | $0.036^{[17]}$ | $1^{[17]}$ | $0.98^{[17]}$ | long[11] | 3.83010
080319B | $13^{[18]}$ | $0.032^{[18]}$ | $0.937^{[18]}$ | long[11] | 0.50876
970828 | $2.1982^{[6]}$ | $2.2^{[6]}$ | $0.9578^{[6]}$ | | 3.09196
990510 | $1.76349^{[6]}$ | $1.2^{[6]}$ | $1.6187^{[6]}$ | | 2.27045
990705 | $2.55952^{[6]}$ | $1^{[6]}$ | $0.8424^{[6]}$ | | 2.30918
991216 | $5.35369^{[6]}$ | $1.2^{[6]}$ | $1.02^{[6]}$ | | 2.17822
000301C | $0.43749^{[6]}$ | $7.3^{[6]}$ | $2.0335^{[6]}$ | | 5.03377
010222 | $8.57841^{[6]}$ | $0.93^{[6]}$ | $1.4768^{[6]}$ | | 1.72899
0.86[1] Fox et al. (2005); [2] Zhang et al. (2009); [3] Amati et al. (2008);
[4] Liang et al. (2008); [5] Ghirlanda et al. (2007); [6] Bloom et al. (2008);
[7] Soderberg et al. (2006); [8] Sakamoto et al. (2007); [9] Pélangeon et al.
(2008); [10] Amati L. (2007); [11] Evans et al. (2009); [12] Kamble et al.
(2009); [13] Ghirlanda et al. (2008); [14] Ferrero et al. (2009); [15] Perley
et al. (2007); [16] Chandra et al. (2008); [17] Covino et al. (2008); [18]
Racusin et al. (2008).
## References
* Amati (2007) Amati L. 2007, NCimB, 121, 1081
* Amati et al. (2008) Amati L., Guidorzi C., Frontera F. et al. 2008, MNRAS, 391, 577
* Bloom et al. (2003) Bloom J. S., Frail D. A., Kulkarni S. R. 2008, ApJ, 594, 674
* Chandra et al. (2008) Chandra P., Cenko S. B., Frail D. A. et al. 2008, ApJ, 683, 924
* Covino et al. (2008) Covino S., D´Avanzo P., Klotz A. et al. 2008, MNRAS, 388, 347
* Evans et al. (2009) Evans P. A., Beardmore A. P., Page K. L. et al. 2009, MNRAS, 397,1177
* Ferrero et al. (2009) Ferrero P., Klose S., Kann D. A. et al., 2009, A&A, 497, 729
* Fox et al. (2005) Fox D. B., Frail D. A., Price P. A. et al. 2005, Nature, 437, 845
* Frail et al. (2001) Frail D. A., Kulkarni S. R., Sari R. et al. 2001, ApJ, 522, L55
* Gao et al. (2005) Gao W. H., &Wei D. M. 2005, Chin J. Astron Astrophys, 5, 571
* Ghirlanda et al. (2007) Ghirlanda G., Nava L., Ghisellini G. , & Firmani C. 2007, A&A, 466, 127
* Ghirlanda et al. (2008) Ghirlanda G., Nava L., Ghisellini G. et al. 2008, MNRAS, 387, 319
* Kamble et al. (2008) Kamble A., Misra K., Bhattacharya D., Sagar R. 2009, MNRAS, 394, 214
* Liang et al. (2007) Liang E. W., Racusin J. L., Zhang B. et al. 2008, ApJ, 670, 565
* Pélangeon et al. (2008) Pélangeon A., Atteia J-L., Nakagawa Y. E. et al. 2008, A&A, 491, 157
* Perley et al. (2008) Perley D. A., Bloom J. S., Butler N. R. et al. 2008, ApJ, 672, 449
* Racusin et al. (2008) Racusin J. L., Karpov S. V., Sokolowski M. et al. 2008, AIPC Proceedings, 1065, 245
* Sakamoto et al. (2007) Sakamoto T., Barthelmy S. D., Barbier L. et al. 2007, ApJS, 175, 179
* Sari et al. (1999) Sari R., Piran T., Halpern J. P., 1999, ApJ, 519, L17
* Soderberg et al. (2006) Soderberg A. M., Berger E., Kasliwal M. et al. 2006, ApJ, 650, 261
* Zhang (2007) Zhang B. 2007, Chin J. Astron Astrophys, 7, 1
* Zhang et al. (2009) Zhang B., Zhang B. B., Virgili F. J. et al. 2009, ApJ, 703, 1696
* Zhang et al. (2002) Zhang B., & Meszaros P. 2002, ApJ, 571, 876
|
arxiv-papers
| 2009-11-09T03:17:34 |
2024-09-04T02:49:06.342703
|
{
"license": "Public Domain",
"authors": "Y. Gao and Z.G. Dai",
"submitter": "Xiang-Hua Li",
"url": "https://arxiv.org/abs/0911.1579"
}
|
0911.1596
|
# Relation between Optical Fresnel transformation and quantum tomography in
two-mode entangled case††thanks: Work supported by the President Foundation of
Chinese Academy of Science and the National Natural Science Foundation of
China (Grant No 10874174), and the Research Foundation of the Education
Department of Jiangxi Province.
Hong-yi Fan1 and Li-yun Hu2
1Department of Material Science and Engineering,
University of Science and Technology of China, Hefei, Anhui 230026, China
2College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China Corresponding author. Tel./fax: +86 7918120370. E-mail
address: hlyun2008@126.com.
###### Abstract
Similar in spirit to the preceding work [Opt. Commun. 282 (2009) 3734] where
the relation between optical Fresnel transformation and quantum tomography is
revealed, we study this kind of relationship in the two-mode entangled case.
We show that under the two-mode Fresnel transformation the bipartite entangled
state density $\left|\eta\right\rangle\left\langle\eta\right|$ becomes density
operator
$F_{2}\left|\eta\right\rangle\left\langle\eta\right|F_{2}^{\dagger}=\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|$,
which is just the Radon transform of the two-mode Wigner operator
$\Delta\left(\sigma,\gamma\right)$ in entangled form, i.e.,
$\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|=\pi\int d^{2}\gamma
d^{2}\sigma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\Delta\left(\sigma,\gamma\right),$
where $F_{2}$ is an two-mode Fresnel operator in quantum optics, and $s,r$ are
the complex-value expression of ${\small(}A,B,C,D).$ So the probability
distribution for the Fresnel quadrature phase is the tomography (Radon
transform of the two-mode Wigner function), correspondingly,
${}_{s,r}\left\langle\eta\right|\left.\psi\right\rangle=\left\langle\eta\right|F_{2}^{\dagger}\left|\psi\right\rangle.$
Similarly, we find
$F_{2}\left|\xi\right\rangle\left\langle\xi\right|F_{2}^{\dagger}=\left|\xi\right\rangle_{s,rs,r}\left\langle\xi\right|=\pi\int
d^{2}\sigma
d^{2}\gamma\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\Delta\left(\sigma,\gamma\right),$
where $\left|\xi\right\rangle$ is the conjugated state to
$\left|\eta\right\rangle$.
## 1 Introduction
In Ref.[1], by using the technique of integration within an ordered product
(IWOP) of operators and the coherent state representation [2, 3] we have
proved that corresponding to optical Fresnel transformation characteristic of
ray transfer matrix elements $(A,B,C,D),$ $AD-BC=1$, connecting the input
light field $f\left(x\right)$ and output light field
$g\left(x^{\prime}\right)$ by Fresnel integration [4, 5, 6]
$g\left(x^{\prime}\right)=\frac{1}{\sqrt{2\pi
iB}}\int_{-\infty}^{\infty}\exp\left[\frac{i}{2B}\left(Ax^{2}-2x^{\prime}x+Dx^{\prime
2}\right)\right]f\left(x\right)dx.$ (1)
there exists Fresnel operator $F_{1}(r,s)$ in quantum optics [7],
$\displaystyle F_{1}\left(r,s\right)$ $\displaystyle=$
$\displaystyle\sqrt{s}\int\frac{d^{2}z}{\pi}\left|sz-
rz^{\ast}\right\rangle\left\langle z\right|$ (2) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{s^{\ast}}}\exp\left(-\frac{ra^{\dagger
2}}{2s^{\ast}}\right)\colon\exp\left\\{\left(\frac{1}{s^{\ast}}-1\right)a^{\dagger}a\right\\}\colon\exp\left(\frac{r^{\ast}a^{2}}{2s^{\ast}}\right),\;\;$
such that in the coordinate $\left\langle x\right|$ representation[1, 8]
$\left\langle
x^{\prime}\right|F_{1}\left(r,s\right)\left|x\right\rangle=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax^{2}-2x^{\prime}x+Dx^{\prime
2}\right)\right],$ (3)
where $(s,r),$ $\left|s\right|^{2}-\left|r\right|^{2}=1$, are related to a
classical ray transfer matrix $\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)$ by
$s=\frac{1}{2}\left[A+D-i\left(B-C\right)\right],\;r=-\frac{1}{2}\left[A-D+i\left(B+C\right)\right],$
(4)
the unimodularity condition $AD-BC=1$ is equivalent to
$\left|s\right|^{2}-\left|r\right|^{2}=1.$ If we let
$f\left(x\right)=\left\langle x\right|\left.f\right\rangle$, then Eq.(1) is
expressed as
$g\left(x^{\prime}\right)=\int_{-\infty}^{\infty}\left\langle
x^{\prime}\right|F_{1}\left(r,s\right)\left|x\right\rangle\left\langle
x\right|\left.f\right\rangle dx=\left\langle
x^{\prime}\right|F_{1}\left(r,s\right)\left|f\right\rangle,$ (5)
which is just the quantum mechanical version of Fresnel transformation.
In a preceding paper [9], we also found that under the Fresnel transformation
the pure position density $\left|x\right\rangle\left\langle x\right|$ becomes
the tomographic density $\left|x\right\rangle_{s,rs,r}\left\langle x\right|$,
which is just the Radon transform of the Wigner operator
$\Delta\left(x,p\right),$ i.e.,
$F_{1}\left|x\right\rangle\left\langle
x\right|F_{1}^{\dagger}=\left|x\right\rangle_{s,rs,r}\left\langle
x\right|=\int_{-\infty}^{\infty}dp^{\prime}dx^{\prime}\delta\left[x-\left(Dx^{\prime}-Bp^{\prime}\right)\right]\Delta\left(x^{\prime},p^{\prime}\right).$
(6)
So the probability distribution for the Fresnel quadrature phase is the
tomography (Radon transform of Wigner function [10, 11, 12]), and the tomogram
of a state $\left|\psi\right\rangle$ is just the wave function of its Fresnel
transformed state $F_{1}^{\dagger}\left|\psi\right\rangle,$ i.e.
${}_{s,r}\left\langle x\right|\left.\psi\right\rangle=\left\langle
x\right|F_{1}^{\dagger}\left|\psi\right\rangle,$ and
$\left|x\right\rangle_{s,r}$ is expressed as
$\left|x\right\rangle_{s,r}=\frac{\pi^{-1/4}}{\sqrt{D+iB}}\exp\left\\{-\frac{A-iC}{D+iB}\frac{x^{2}}{2}+\frac{\sqrt{2}x}{D+iB}a^{\dagger}-\frac{D-iB}{D+iB}\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle.$ (7)
In this Communication we want to generalize the above conclusion to two-mode
entangled case. Firstly, we extend Eq. (3) to the two-dimensional Fresnel
transformation,
$\mathcal{K}_{2}\left(\eta^{\prime},\eta\right)=\frac{1}{2iB\pi}\exp\left[\frac{i}{2B}(A\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+D\left|\eta^{\prime}\right|^{2})\right],$
(8)
where $\eta$ is a complex number, then we construct the two-mode Fresnel
operator $F_{2}\left(r,s\right)$ such that its transformation matrix element
in the entangled state $\left|\eta\right\rangle$ representation (see below
Eq.(16)) is just the two-dimensional Fresnel transformation, i.e.,
$\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{\pi}\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle,$
then we shall prove
$F_{2}\left|\eta\right\rangle\left\langle\eta\right|F_{2}^{\dagger}=\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|=\pi\int
d^{2}\gamma
d^{2}\sigma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\Delta\left(\sigma,\gamma\right),$
(9)
i.e., we show that $\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|$
is just the Radon transform of the entangled Wigner operator
$\Delta\left(\sigma,\gamma\right)$
Our paper is arranged as follows. In section 2, we briefly review the two-mode
Fresnel operator $F_{2}\left(r,s\right)$ and then derive the 2D Fresnel
transformation in entangled state $\left|\eta\right\rangle$ representation and
introduce a new representation
$\left|\eta\right\rangle_{s,r}(=F_{2}\left(r,s\right)\left|\eta\right\rangle)$
in section 3. Section 4 is devoted to proving Eq.(9), i.e.,
$\left|\eta\right\rangle_{s,r\text{ }s,r}\left\langle\eta\right|$ as Radon
transform of entangled Wigner operator. Similar discussions are moved to the
Fresnel transformation in its ‘frequency domain’ in section 5. In the last
section, we derive the inverse Radon transformation of entangled Wigner
operator.
## 2 Two-mode Fresnel operator
Similar in spirit to the single-mode case, we introduce the two-mode Fresnel
operator $F_{2}\left(r,s\right)$ through the following 2-mode coherent state
representation [1, 13], i.e.,
$F_{2}\left(r,s\right)=s\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\left\langle
z_{1},z_{2}\right|,$ (10)
which indicates that $F_{2}\left(r,s\right)$ is a mapping of classical
symplectic transform
$\left(z_{1},z_{2}\right)\rightarrow\left(sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right)$
in phase space, where
$\left|z_{1},z_{2}\right\rangle=\exp\left\\{-\frac{1}{2}\left|z_{1}\right|^{2}-\frac{1}{2}\left|z_{2}\right|^{2}+z_{1}a_{1}^{{\dagger}}+z_{1}a_{1}^{{\dagger}}\right\\}\left|00\right\rangle$
is a usual two-mode coherent state. Concretely, the ket in (10) is
$\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\equiv\left|sz_{1}+rz_{2}^{\ast}\right\rangle_{1}\otimes\left|rz_{1}^{\ast}+sz_{2}\right\rangle_{2},\text{
}$ (11)
$s$ and $r$ are complex and satisfy the unimodularity condition
$\left|s\right|^{2}-\left|r\right|^{2}=1$. Using the IWOP technique [14, 15]
and the normal ordering of the vacuum projector
$\left|00\right\rangle\left\langle
00\right|=\colon\exp\left(-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\right)\colon,$
we perform the integral in (10) and obtain
$\displaystyle F_{2}\left(r,s\right)$
$\displaystyle=s\int\frac{1}{\pi^{2}}d^{2}z_{1}d^{2}z_{2}\colon\exp[-|s|^{2}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)-r^{\ast}sz_{1}z_{2}-rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}$
$\displaystyle+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}+z_{1}^{\ast}a_{1}+z_{2}^{\ast}a_{2}-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}]\colon$
$\displaystyle=\frac{1}{s^{\ast}}\exp\left(\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\colon\exp\left[\left(\frac{1}{s^{\ast}}-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\colon\exp\left(-\frac{r^{\ast}}{s^{\ast}}a_{1}a_{2}\right)$
$\displaystyle=\exp\left(\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\exp[\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\ln\left(s^{\ast}\right)^{-1}]\exp\left(-\frac{r^{\ast}}{s^{\ast}}a_{1}a_{2}\right).$
(12)
Thus $F_{2}\left(r,s\right)$ induces the transforms
$F_{2}\left(r,s\right)a_{1}F_{2}^{-1}\left(r,s\right)=s^{\ast}a_{1}-ra_{2}^{\dagger},\text{
\
}F_{2}\left(r,s\right)a_{2}F_{2}^{-1}\left(r,s\right)=s^{\ast}a_{2}-ra_{1}^{\dagger},$
(13)
and $F_{2}$ is actually a generalized 2-mode squeezing operator [16, 17].
$F_{2}\left(r,s\right)$ abides by the group multiplication rule. Using the
IWOP technique and (10) we obtain
$\displaystyle F_{2}\left(r,s\right)F_{2}\left(r^{\prime},s^{\prime}\right)$
$\displaystyle=ss^{\prime}\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{4}}\colon\exp\\{-|s|^{2}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)-r^{\ast}sz_{1}z_{2}$
$\displaystyle-
rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}-\frac{1}{2}[|z_{1}^{\prime}|^{2}+|z_{2}^{\prime}|^{2}+|s^{\prime}z_{1}^{\prime}+r^{\prime}z_{2}^{\prime\ast}|^{2}+|r^{\prime}z_{1}^{\prime\ast}+s^{\prime}z_{2}^{\prime}|^{2}]$
$\displaystyle+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}+z_{1}^{\prime\ast}a_{1}+z_{2}^{\prime\ast}a_{2}$
$\displaystyle+z_{1}^{\ast}\left(s^{\prime}z_{1}^{\prime}+r^{\prime}z_{2}^{\prime\ast}\right)+z_{2}^{\ast}\left(r^{\prime}z_{1}^{\prime\ast}+s^{\prime}z_{2}^{\prime}\right)-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\\}\colon$
$\displaystyle=\frac{1}{s^{\prime\prime\ast}}\exp\left(\frac{r^{\prime\prime}}{2s^{\prime\prime\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\colon\exp\left\\{\left(\frac{1}{s^{\prime\prime\ast}}-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right\\}\colon\exp\left(-\frac{r^{\prime\prime\ast}}{2s^{\prime\prime\ast}}a_{1}a_{2}\right)$
$\displaystyle=F_{2}\left(r^{\prime\prime},s^{\prime\prime}\right),$ (14)
where $\left(r^{\prime\prime},s^{\prime\prime}\right)$ are given by
$\left(\begin{array}[]{cc}s^{\prime\prime}&-r^{\prime\prime}\\\
-r^{\ast\prime\prime}&s^{\ast\prime\prime}\end{array}\right)=\left(\begin{array}[]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right)\left(\begin{array}[]{cc}s^{\prime}&-r^{\prime}\\\
-r^{\prime\ast}&s^{\prime\ast}\end{array}\right).$ (15)
Therefore, Eq.(14) is a loyal representation of the multiplication rule for
ray transfer matrices in the sense of Matrix Optics.
## 3 Two-mode Fresnel transformation in entangled state representations
By introducing the bipartite entangled state $\left|\eta\right\rangle$ [18,
19]
$\left|\eta\right\rangle=\exp\left[-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle,$
(16)
$\left|\eta=\eta_{1}+i\eta_{2}\right\rangle$ is the common eigenstate of
relative coordinate $Q_{1}-Q_{2}$ and the total momentum $P_{1}+P_{2}$, i.e.,
$\left(Q_{1}-Q_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{1}\left|\eta\right\rangle,\,\
\left(P_{1}+P_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{2}\left|\eta\right\rangle,$
(17)
where $Q_{i}=\left(a_{i}+a_{i}^{\dagger}\right)/\sqrt{2},\
P_{i}=\left(a_{i}-a_{i}^{\dagger}\right)/(\mathtt{i}\sqrt{2}),$
$\left(i=1,2.\right),$ are coordinate and momentum operators, respectively.
$\left|\eta\right\rangle$ compose a complete set
$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=1,$
then using the over-completeness relation of the coherent state and
$\left\langle
z_{1},z_{2}\right|\left.\eta\right\rangle=\exp\left[-\frac{1}{2}(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|\eta\right|^{2})+\eta
z_{1}^{\ast}-\eta^{\ast}z_{2}^{\ast}+z_{1}^{\ast}z_{2}^{\ast}\right],$ (18)
as well as
$\displaystyle\left\langle
z_{1}^{\prime},z_{2}^{\prime}\right|F_{2}\left(r,s\right)\left|z_{1},z_{2}\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\exp\left\\{-\frac{1}{2}(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{1}^{\prime}\right|^{2}+\left|z_{2}^{\prime}\right|^{2})\right.$
$\displaystyle\left.+\frac{r}{s^{\ast}}z_{1}^{\prime\ast}z_{2}^{\prime\ast}-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}+\frac{1}{s^{\ast}}\left(z_{1}^{\prime\ast}z_{1}+z_{2}^{\prime\ast}z_{2}\right)\right\\},$
(19)
we can calculate the integral kernel
$\displaystyle\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)$
$\displaystyle=\frac{1}{\pi}\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle$
$\displaystyle=\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{5}}\left\langle\eta^{\prime}\right|\left.z_{1}^{\prime},z_{2}^{\prime}\right\rangle\left\langle
z_{1}^{\prime},z_{2}^{\prime}\right|F_{2}\left(r,s\right)\left|z_{1},z_{2}\right\rangle\left\langle
z_{1},z_{2}\right.\left|\eta\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{5}}\exp\left[-(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{1}^{\prime}\right|^{2}+\left|z_{2}^{\prime}\right|^{2})-\frac{1}{2}(\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2})\right]$
$\displaystyle\times\exp\left[z_{1}^{\ast}z_{2}^{\ast}+\eta
z_{1}^{\ast}-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}+\frac{z_{1}^{\prime\ast}z_{1}+z_{2}^{\prime\ast}z_{2}}{s^{\ast}}+\frac{r}{s^{\ast}}z_{1}^{\prime\ast}z_{2}^{\prime\ast}+z_{1}^{\prime}z_{2}^{\prime}+\eta^{\prime\ast}z_{1}^{\prime}-\eta^{\prime}z_{2}^{\prime}-\eta^{\ast}z_{2}^{\ast}\right]$
$\displaystyle=\frac{1}{\left(r^{\ast}+s^{\ast}-r-s\right)\pi}\exp\left[\frac{\left(r^{\ast}-s\right)\left|\eta\right|^{2}-\left(r+s\right)\left|\eta^{\prime}\right|^{2}+\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}}{r^{\ast}+s^{\ast}-r-s}-\frac{\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2}}{2}\right].$
(20)
Using the relation between $s,r$ and $\left(A,B,C,D\right)$ in Eq.(4) we see
that Eq. (20) becomes
$\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{2iB\pi}\exp\left[\frac{i}{2B}\left(A\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+D\left|\eta^{\prime}\right|^{2}\right)\right]\equiv\mathcal{K}_{2}^{M}\left(\eta^{\prime},\eta\right),$
(21)
where the superscript $M$ only means the parameters of $\mathcal{K}_{2}^{M}$
are $\left[A,B;C,D\right]$, and the subscript $2$ implies the two-dimensional
kernel.
Operating $F_{2}\left(r,s\right)$ on $\left|\eta\right\rangle$ and using
Eqs.(12) and (18) yields
$\displaystyle F_{2}\left(r,s\right)\left|\eta\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{s^{\ast}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\exp\left[\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}+\left(\frac{1}{s^{\ast}}-1\right)\left(a_{1}^{\dagger}z_{1}+a_{2}^{\dagger}z_{2}\right)-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}\right]\left|z_{1},z_{2}\right\rangle\left\langle
z_{1},z_{2}\right|\left.\eta\right\rangle$ (22) $\displaystyle=$
$\displaystyle\frac{1}{s^{\ast}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\exp\left[-\left|z_{1}\right|^{2}+\frac{1}{s^{\ast}}\left(a_{1}^{\dagger}-r^{\ast}z_{2}\right)z_{1}+\left(\eta+z_{2}^{\ast}\right)z_{1}^{\ast}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2}\left|\eta\right|^{2}-\left|z_{2}\right|^{2}+\frac{1}{s^{\ast}}z_{2}a_{2}^{\dagger}-\eta^{\ast}z_{2}^{\ast}+\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{s^{\ast}}\int\frac{d^{2}z_{2}}{\pi}\exp\left[-\frac{s^{\ast}+r^{\ast}}{s^{\ast}}\left|z_{2}\right|^{2}+\frac{1}{s^{\ast}}\left(a_{2}^{\dagger}-\eta
r^{\ast}\right)z_{2}+\frac{1}{s^{\ast}}\left(a_{1}^{\dagger}-s^{\ast}\eta^{\ast}\right)z_{2}^{\ast}\right]$
$\displaystyle\times\exp\left[+\frac{\eta}{s^{\ast}}a_{1}^{\dagger}+\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}-\frac{1}{2}\left|\eta\right|^{2}\right]\left|00\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{s^{\ast}+r^{\ast}}\exp\left\\{-\allowbreak\frac{s^{\ast}-r^{\ast}}{2\left(s^{\ast}+r^{\ast}\right)}\left|\eta\right|^{2}+\allowbreak\frac{\eta
a_{1}^{\dagger}}{s^{\ast}+r^{\ast}}\allowbreak-\allowbreak\frac{\eta^{\ast}a_{2}^{\dagger}}{s^{\ast}+r^{\ast}}+\frac{s+r}{s^{\ast}+r^{\ast}}\allowbreak
a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle\equiv\left|\eta\right\rangle_{s,r},$
or
$\left|\eta\right\rangle_{s,r}=\frac{1}{\allowbreak
D+iB}\exp\left\\{-\frac{\allowbreak A-iC}{2\left(\allowbreak
D+iB\right)}\left|\eta\right|^{2}+\frac{\eta a_{1}^{\dagger}}{\allowbreak
D+iB}-\frac{\eta^{\ast}a_{2}^{\dagger}}{\allowbreak D+iB}+\frac{\allowbreak
D-iB}{\allowbreak
D+iB}a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle,$ (23)
where we have used the integration formula
$\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}\right)=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(24)
Noticing the completeness relation and the orthogonality of
$\left|\eta\right\rangle$ we immediately derive
$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|=1,\text{
}_{s,r}\left\langle\eta\right|\left.\eta^{\prime}\right\rangle_{s,r}=\pi\delta\left(\eta-\eta^{\prime}\right)\delta\left(\eta^{\ast}-\eta^{\prime\ast}\right),$
(25)
a generalized entangled state representation $\left|\eta\right\rangle_{s,r}$
with the completeness relation (25). From (23) we can see that
$\displaystyle a_{1}\left|\eta\right\rangle_{s,r}$ $\displaystyle=$
$\displaystyle\left(\frac{\eta}{\allowbreak D+iB}+\frac{\allowbreak
D-iB}{\allowbreak D+iB}a_{2}^{\dagger}\right)\left|\eta\right\rangle_{s,r},$
(26) $\displaystyle a_{2}\left|\eta\right\rangle_{s,r}$ $\displaystyle=$
$\displaystyle\left(-\frac{\eta^{\ast}}{\allowbreak D+iB}+\frac{\allowbreak
D-iB}{\allowbreak D+iB}a_{1}^{\dagger}\right)\left|\eta\right\rangle_{s,r},$
(27)
so we have the eigen-equations for $\left|\eta\right\rangle_{s,r}$ as follows
$\displaystyle\left[D\left(Q_{1}-Q_{2}\right)-B\left(P_{1}-P_{2}\right)\right]\left|\eta\right\rangle_{s,r}$
$\displaystyle=$
$\displaystyle\sqrt{2}\eta_{1}\left|\eta\right\rangle_{s,r},\text{ }$ (28)
$\displaystyle\left[B\left(Q_{1}+Q_{2}\right)+D\left(P_{1}+P_{2}\right)\right]\left|\eta\right\rangle_{s,r}$
$\displaystyle=$ $\displaystyle\sqrt{2}\eta_{2}\left|\eta\right\rangle_{s,r},$
(29)
We can also check Eqs.(26)-(29) by another way (see Appendix).
## 4 $\left|\eta\right\rangle_{s,r\text{ }s,r}\left\langle\eta\right|$ as the
Radon transform of entangled Wigner operator
For two-mode correlated system, it is convenient to express the Wigner
operator in the $\left|\eta\right\rangle$ representation as [20, 21, 22, 23]
$\Delta\left(\sigma,\gamma\right)=\int\frac{d^{2}\eta}{\pi^{3}}\left|\sigma-\eta\right\rangle\left\langle\sigma+\eta\right|e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma}.$
(30)
When $\sigma=\alpha-\beta^{\ast},\;\gamma=\alpha+\beta^{\ast}$, Eq. (30) is
just equal to the direct product of two single-mode Wigner operators, i.e.,
$\Delta\left(\sigma,\gamma\right)=\Delta\left(\alpha,\alpha^{\ast}\right)\otimes\Delta\left(\beta,\beta^{\ast}\right).$
Then according to the Wely correspondence rule [24]
$H\left(a_{1}^{\dagger},a_{2}^{\dagger};a_{1},a_{2}\right)=\int d^{2}\gamma
d^{2}\sigma h\left(\sigma,\gamma\right)\Delta\left(\sigma,\gamma\right),$ (31)
where $h\left(\sigma,\gamma\right)$ is the Weyl correspondence of
$H\left(a_{1}^{\dagger},a_{2}^{\dagger};a_{1},a_{2}\right),$ and
$h\left(\sigma,\gamma\right)=4\pi^{2}\mathtt{Tr}\left[H\left(a_{1}^{\dagger},a_{2}^{\dagger};a_{1},a_{2}\right)\Delta\left(\sigma,\gamma\right)\right],$
(32)
the classical Weyl correspondence of the projection operator
$\left|\eta\right\rangle_{r,sr,s}\left\langle\eta\right|$ can be calculated,
$\displaystyle
4\pi^{2}\mathtt{Tr}\left[\left|\eta\right\rangle_{r,sr,s}\left\langle\eta\right|\Delta\left(\sigma,\gamma\right)\right]$
(33) $\displaystyle=$ $\displaystyle
4\pi^{2}\int\frac{d^{2}\eta^{\prime}}{\pi^{3}}\left.{}_{r,s}\left\langle\eta\right|\left.\sigma-\eta^{\prime}\right\rangle\left\langle\sigma+\eta^{\prime}\right|\left.\eta\right\rangle_{r,s}\right.\exp(\eta^{\prime}\gamma^{\ast}-\eta^{\prime\ast}\gamma)$
$\displaystyle=$ $\displaystyle
4\pi^{2}\int\frac{d^{2}\eta^{\prime}}{\pi^{3}}\left\langle\eta\right|F_{2}^{\dagger}\left|\sigma-\eta^{\prime}\right\rangle\left\langle\sigma+\eta^{\prime}\right|F_{2}\left|\eta\right\rangle\exp(\eta^{\prime}\gamma^{\ast}-\eta^{\prime\ast}\gamma).$
Then using Eq.(20) we have
$4\pi^{2}\mathtt{Tr}\left[\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|\Delta\left(\sigma,\gamma\right)\right]=\pi\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right),$
(34)
which means the following Weyl correspondence
$\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|=\pi\int d^{2}\gamma
d^{2}\sigma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\Delta\left(\sigma,\gamma\right),$
(35)
so the projector operator
$\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|$ is just the Radon
transformation of $\Delta\left(\sigma,\gamma\right)$, $D$ and $B$ are the
Radon transformation parameter. Combining Eqs. (22)-(35) together we complete
the proof (9). Therefore, the quantum tomography in two-mode entangled case is
expressed as
$|_{s,r}\left\langle\eta\right|\left.\psi\right\rangle|^{2}=|\left\langle\eta\right|F^{\dagger}\left|\psi\right\rangle|^{2}=\pi\int
d^{2}\gamma
d^{2}\sigma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\left\langle\psi\right|\Delta\left(\sigma,\gamma\right)\left|\psi\right\rangle.$
(36)
where
$\left\langle\psi\right|\Delta\left(\sigma,\gamma\right)\left|\psi\right\rangle$
is the Wigner function. So the probability distribution for the Fresnel
quadrature phase (see Eq. (A11) in the Appendix) is the tomography (Radon
transform of the two-mode Wigner function). This is the main result of the
present paper. This new relation between quantum tomography and optical
Fresnel transform may provide experimentalists to figure out new approach for
generating tomography.
## 5 In the conjugate representation
Next we turn to the “frequency” domain, that is to say, we shall prove that
the $(A,C)$ related Radon transform of entangled Wigner operator
$\Delta\left(\sigma,\gamma\right)$ is just the pure state density operator
$\left|\xi\right\rangle_{s,rs,r}\left\langle\xi\right|,$ i.e.,
$F_{2}\left|\xi\right\rangle\left\langle\xi\right|F_{2}^{\dagger}=\left|\xi\right\rangle_{s,rs,r}\left\langle\xi\right|=\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\Delta\left(\sigma,\gamma\right)d^{2}\sigma
d^{2}\gamma,$ (37)
where
$\left|\xi\right\rangle=\exp\left[-\frac{1}{2}\left|\xi\right|^{2}+\xi
a_{1}^{\dagger}+\xi^{\ast}a_{2}^{\dagger}-a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle$
(38)
is an entangled state conjugate to $\left|\eta\right\rangle.$ By analogy with
the above procedure, we obtain the 2-dimensional Fresnel transformation in its
‘frequency domain’, i.e.,
$\displaystyle\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right)$
$\displaystyle\equiv$
$\displaystyle\frac{1}{\pi}\left\langle\xi^{\prime}\right|F_{2}\left(r,s\right)\left|\xi\right\rangle$
(39) $\displaystyle=$ $\displaystyle\int\frac{d^{2}\eta
d^{2}\sigma}{\pi^{2}}\left\langle\xi^{\prime}\right|\left.\eta^{\prime}\right\rangle\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle\left\langle\eta\right|\left.\xi\right\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{8iB\pi}\int\frac{d^{2}\sigma
d^{2}\eta}{\pi^{2}}\exp\left(\frac{\xi^{\prime\ast}\eta^{\prime}-\xi^{\prime}\eta^{\prime\ast}+\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)\mathcal{K}_{2}^{\left(\mathtt{r},s\right)}\left(\sigma,\eta\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2i\left(-C\right)\pi}\exp\left[\frac{i}{2\left(-C\right)}\left(D\left|\xi\right|^{2}+A\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right],$
where the superscript $N$ means that this transform kernel corresponds to the
parameter matrix $N=\left[D,-C,-B,A\right]$. Thus the 2D Fresnel
transformation in its ‘frequency domain’ is given by
$\Psi\left(\xi^{\prime}\right)=\int\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right)\Phi\left(\xi\right)d^{2}\xi.$
(40)
Operating $F_{2}\left(r,s\right)$ on $\left|\xi\right\rangle$ we have (also
see Appendix)
$\left|\xi\right\rangle_{s,r}=\frac{1}{\allowbreak\allowbreak
A-iC}\exp\left\\{-\frac{D+iB}{2\left(\allowbreak
A-iC\right)}\left|\eta\right|^{2}+\frac{\xi
a_{1}^{\dagger}}{A-iC}+\frac{\xi^{\ast}a_{2}^{\dagger}}{\allowbreak
A-iC}-\frac{\allowbreak A+iC}{\allowbreak
A-iC}a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle,$ (41)
or
$\left|\xi\right\rangle_{s,r}=\frac{1}{s^{\ast}-r^{\ast}}\exp\left\\{-\allowbreak\frac{s^{\ast}+r^{\ast}}{2\left(s^{\ast}-r^{\ast}\right)}\left|\xi\right|^{2}+\allowbreak\frac{\xi
a_{1}^{\dagger}}{s^{\ast}-r^{\ast}}\allowbreak+\allowbreak\frac{\xi^{\ast}a_{2}^{\dagger}}{s^{\ast}-r^{\ast}}-\frac{s-r}{s^{\ast}-r^{\ast}}\allowbreak
a_{1}^{\dagger}a_{2}^{\dagger}\right\\}\left|00\right\rangle.$ (42)
Noticing that the entangled Wigner operator in $\left\langle\xi\right|$
representation is expressed as
$\Delta\left(\sigma,\gamma\right)=\int\frac{d^{2}\xi}{\pi^{3}}\left|\gamma+\xi\right\rangle\left\langle\gamma-\xi\right|\exp(\xi^{\ast}\sigma-\sigma^{\ast}\xi),$
(43)
and using the classical correspondence of
$\left|\xi\right\rangle_{s,rs,r}\left\langle\xi\right|$ which is calculated by
$\displaystyle h(\sigma,\gamma)$ $\displaystyle=$ $\displaystyle
4\pi^{2}\mathtt{Tr}\left[\left|\xi\right\rangle_{s,r\text{
}s,r}\left\langle\xi\right|\Delta\left(\sigma,\gamma\right)\right]$
$\displaystyle=$ $\displaystyle
4\int\frac{d^{2}\xi}{\pi}\left\langle\gamma-\xi\right|F_{2}\left|\xi\right\rangle\left\langle\xi\right|F_{2}^{{\dagger}}|\gamma+\xi\rangle\exp(\xi^{\ast}\sigma-\sigma^{\ast}\xi)$
(44) $\displaystyle=$
$\displaystyle\pi\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right),$
we obtain
$\left|\xi\right\rangle_{s,r\text{
}s,r}\left\langle\xi\right|=\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\Delta\left(\sigma,\gamma\right)d^{2}\sigma
d^{2}\gamma,$ (45)
so the projector operator $\left|\xi\right\rangle_{s,r\text{
}s,r}\left\langle\xi\right|$ is another Radon transformation of the two-mode
Wigner operator, with $A$ and $C$ being the Radon transformation parameter
(‘frequency’ domain). Therefore, the quantum tomography in
${}_{s,r}\left\langle\xi\right|$ representation is expressed as the Radon
transformation of the Wigner function
$|\left\langle\xi\right|F^{\dagger}\left|\psi\right\rangle|^{2}=|_{s,r}\left\langle\xi\right|\left.\psi\right\rangle|^{2}=\pi\int
d^{2}\gamma
d^{2}\sigma\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\left\langle\psi\right|\Delta\left(\sigma,\gamma\right)\left|\psi\right\rangle,$
(46)
and ${}_{s,r}\left\langle\xi\right|=\left\langle\xi\right|F^{\dagger}.$
## 6 Inverse Radon transformation
Now we consider the inverse Radon transformation. For instance, using (35) we
see the Fourier transformation of
$\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|$ is
$\displaystyle\int
d^{2}\eta\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|\exp(-i\zeta_{1}\eta_{1}-i\zeta_{2}\eta_{2})$
(47) $\displaystyle=$ $\displaystyle\pi\int d^{2}\gamma
d^{2}\sigma\Delta\left(\sigma,\gamma\right)\exp\left[-i\zeta_{1}\left(D\sigma_{1}+B\gamma_{2}\right)-i\zeta_{2}\left(D\sigma_{2}-B\gamma_{1}\right)\right],$
the right-hand side of (47) can be regarded as a special Fourier
transformation of $\Delta\left(\sigma,\gamma\right)$, so by making its inverse
Fourier transformation, we get
$\displaystyle\Delta\left(\sigma,\gamma\right)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{4}}\int_{-\infty}^{\infty}dr_{1}\left|r_{1}\right|\int_{-\infty}^{\infty}dr_{2}\left|r_{2}\right|\int_{0}^{\pi}d\theta_{1}d\theta_{2}$
(48)
$\displaystyle\times\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle_{s,rs,r}\left\langle\eta\right|K\left(r_{1},r_{2},\theta_{1},\theta_{2}\right),$
where
$\cos\theta_{1}=\cos\theta_{2}=\frac{D}{\sqrt{B^{2}+D^{2}}},r_{1}=\zeta_{1}\sqrt{B^{2}+D^{2}},r_{2}=\zeta_{2}\sqrt{B^{2}+D^{2}}$
and
$\displaystyle K\left(r_{1},r_{2},\theta_{1},\theta_{2}\right)$
$\displaystyle\equiv$
$\displaystyle\exp\left[-ir_{1}\left(\frac{\eta_{1}}{\sqrt{B^{2}+D^{2}}}-\sigma_{1}\cos\theta_{1}-\gamma_{2}\sin\theta_{1}\right)\right]$
(49)
$\displaystyle\times\exp\left[-ir_{2}\left(\frac{\eta_{2}}{\sqrt{B^{2}+D^{2}}}-\sigma_{2}\cos\theta_{2}+\gamma_{1}\sin\theta_{2}\right)\right].$
Eq.(48) is just the inverse Radon transformation of entangled Wigner operator
in the entangled state representation. This is different from the two
independent Radon transformations’ direct product of the two independent
single-mode Wigner operators, because in (23) the
$\left|\eta\right\rangle_{s,r}$ is an entangled state. Therefore the Wigner
function of quantum state $\left|\psi\right\rangle$ can be reconstructed from
the tomographic inversion of a set of measured probability distributions
$\left|{}_{s,r}\left\langle\eta\right.\left|\psi\right\rangle\right|^{2}$,
i.e.,
$\displaystyle W_{\psi}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{4}}\int_{-\infty}^{\infty}dr_{1}\left|r_{1}\right|\int_{-\infty}^{\infty}dr_{2}\left|r_{2}\right|\int_{0}^{\pi}d\theta_{1}d\theta_{2}$
(50)
$\displaystyle\times\int_{-\infty}^{\infty}\frac{d^{2}\eta}{\pi}\left|{}_{s,r}\left\langle\eta\right.\left|\psi\right\rangle\right|^{2}K\left(r_{1},r_{2},\theta_{1},\theta_{2}\right).$
In summary, based on the preceding paper [13], we have further extended the
relation connecting optical Fresnel transformation with quantum tomography to
the entangled case. The tomography representation
${}_{s,r}\left\langle\eta\right|=\left\langle\eta\right|F_{2}^{\dagger}$ is
set up, based on which the tomogram of quantum state $\left|\psi\right\rangle$
is just the squared modulus of the wave function
${}_{s,r}\left\langle\eta\right|\left.\psi\right\rangle.$ i.e. the probability
distribution for the Fresnel quadrature phase is the tomogram (Radon transform
of the Wigner function).
Acknowledgement: Work supported by the National Natural Science Foundation of
China (Grant No 10874174) and the President Foundation of Chinese Academy of
Science, and the Research Foundation of the Education Department of Jiangxi
Province.
APPENDIX
In fact, from (13) we have
$\displaystyle F_{2}Q_{1}F_{2}^{\dagger}$
$\displaystyle=\frac{1}{2}\left(\left(A+D\right)Q_{1}-\left(B-C\right)P_{1}+\left(A-D\right)Q_{2}+\left(B+C\right)P_{2}\right),$
(A1) $\displaystyle F_{2}Q_{2}F_{2}^{\dagger}$
$\displaystyle=\frac{1}{2}\left(\left(A+D\right)Q_{2}-\left(B-C\right)P_{2}+\left(A-D\right)Q_{1}+\left(B+C\right)P_{1}\right),$
(A2)
and
$\displaystyle F_{2}P_{1}F_{2}^{\dagger}$
$\displaystyle=\frac{1}{2}\left(\left(A+D\right)P_{1}+\left(B-C\right)Q_{1}-\left(A-D\right)P_{2}+\left(B+C\right)Q_{2}\right),$
(A3) $\displaystyle F_{2}P_{2}F_{2}^{\dagger}$
$\displaystyle=\frac{1}{2}\left(\left(A+D\right)P_{2}+\left(B-C\right)Q_{2}-\left(A-D\right)P_{1}+\left(B+C\right)Q_{1}\right),$
(A4)
it then follow that
$\displaystyle F_{2}\left(Q_{1}-Q_{2}\right)F_{2}^{\dagger}$
$\displaystyle=D\left(Q_{1}-Q_{2}\right)-B\left(P_{1}-P_{2}\right),$ (A5)
$\displaystyle F_{2}\left(P_{1}+P_{2}\right)F_{2}^{\dagger}$
$\displaystyle=B\left(Q_{1}+Q_{2}\right)+D\left(P_{1}+P_{2}\right),$ (A6)
and
$\displaystyle F_{2}\left(Q_{1}+Q_{2}\right)F_{2}^{\dagger}$
$\displaystyle=A\left(Q_{1}+Q_{2}\right)+C\left(P_{1}+P_{2}\right),$ (A7)
$\displaystyle F_{2}\left(P_{1}-P_{2}\right)F_{2}^{\dagger}$
$\displaystyle=A\left(P_{1}-P_{2}\right)-C\left(Q_{1}-Q_{2}\right).$ (A8)
Noticing that
$\left[F_{2}\left(Q_{1}-Q_{2}\right)F_{2}^{\dagger},F_{2}\left(P_{1}+P_{2}\right)F_{2}^{\dagger}\right]=0$
and (17) thus the eigenvector equation of communicative operators
$D\left(Q_{1}-Q_{2}\right)-B\left(P_{1}-P_{2}\right)$ and
$B\left(Q_{1}+Q_{2}\right)+D\left(P_{1}+P_{2}\right)$ is
$\displaystyle\left[D\left(Q_{1}-Q_{2}\right)-B\left(P_{1}-P_{2}\right)\right]\left|\eta\right\rangle_{s,r}$
$\displaystyle=F_{2}\left(Q_{1}-Q_{2}\right)F_{2}^{\dagger}\left|\eta\right\rangle_{s,r}=\sqrt{2}\eta_{1}\left|\eta\right\rangle_{s,r},\text{
}$ (A9)
$\displaystyle\left[B\left(Q_{1}+Q_{2}\right)+D\left(P_{1}+P_{2}\right)\right]\left|\eta\right\rangle_{s,r}$
$\displaystyle=F_{2}\left(P_{1}+P_{2}\right)F_{2}^{\dagger}\left|\eta\right\rangle_{s,r}=\sqrt{2}\eta_{2}\left|\eta\right\rangle_{s,r},$
(A10)
thus
$\left|\eta\right\rangle_{s,r}=F_{2}\left|\eta\right\rangle=\text{Eq.(\ref{22})},$
(A11)
and we name $D\left(Q_{1}-Q_{2}\right)-B\left(P_{1}-P_{2}\right)$ or
$B\left(Q_{1}+Q_{2}\right)+D\left(P_{1}+P_{2}\right)$ the Fresnel quadrature
phase.
On the other hand, due to the communicative relation
$\left[F_{2}\left(Q_{1}+Q_{2}\right)F_{2}^{\dagger},F_{2}\left(P_{1}-P_{2}\right)F_{2}^{\dagger}\right]=0,$
and $\left|\xi\right\rangle$ (the conjugate state to
$\left|\eta\right\rangle$) is the common eigen-equation of
$\left(Q_{1}+Q_{2}\right)$ and $\left(P_{1}-P_{2}\right)$, i.e.,
$\left(Q_{1}+Q_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{1}\left|\xi\right\rangle,\text{
}\left(P_{1}-P_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{2}\left|\xi\right\rangle,$
(A12)
so the common eigenvector of $F_{2}\left(Q_{1}+Q_{2}\right)F_{2}^{\dagger}$
and $F_{2}\left(P_{1}-P_{2}\right)F_{2}^{\dagger}$ is given by
$\displaystyle\left|\xi\right\rangle_{s,r}$ $\displaystyle\equiv
F_{2}\left|\xi\right\rangle=F_{2}\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|\left.\xi\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta}{2\pi}\exp\left(\frac{\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)\left|\eta\right\rangle_{s,r}$
$\displaystyle=\text{Eq.(\ref{41})=Eq.(\ref{42}),}$ (A13)
where we have used the overlap relation of
$\left\langle\eta\right|\left.\xi\right\rangle=\frac{1}{2}\exp\left(\frac{\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)$.
The corresponding eigen-equations of $\left|\xi\right\rangle_{s,r}$ are
$\displaystyle\left[A\left(Q_{1}+Q_{2}\right)+C\left(P_{1}+P_{2}\right)\right]\left|\xi\right\rangle_{s,r}$
$\displaystyle=\sqrt{2}\xi_{1}\left|\xi\right\rangle_{s,r},$ (A14)
$\displaystyle\left[A\left(P_{1}-P_{2}\right)-C\left(Q_{1}-Q_{2}\right)\right]\left|\xi\right\rangle_{s,r}$
$\displaystyle=\sqrt{2}\xi_{2}\left|\xi\right\rangle_{s,r}.$ (A15)
## References
* [1] Hong-yi Fan and Hai-liang Lu, Phys. Lett. A 334 (2005) 132
* [2] R. J. Glauber, Phys. Rev. 131 (1963) 2766.
* [3] J. R. Klauder and B.S. Skagerstam, Coherent States, World Scientific, Singapore
* [4] D.F.V. James and G. S. Agawal, Opt. Commun. 126 (1996) 207
* [5] Kogelnik H, Appl. Opt. 4 (1965) 1562.
* [6] J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1972).
* [7] Hong-yi Fan, Commun. Theor. Phys. 40 (2003) 589
* [8] Hong-yi Fan and Hai-liang Lu, Opt. Commun. 258 (2006) 51\.
* [9] Hong-yi Fan and Li-yun Hu, Opt. Commun. 282 (2009) 3734\.
* [10] K. Vogel and H. Risken, Phys. Rev. A 40 (1989) 2847.
* [11] A. Wünsche, J. Mod. Opt. 44 (1997) 2293; A. Wünsche, Phys. Rev. A 54 (1996) 5291\.
* [12] W. Vogel and W. P. Schleich, Phys. Rev. A 44 (1991) 7642; U. Leonhardt, Measuring the Quantum State of Light, Cambridge University, New York, 1997 and references therein.
* [13] Hong-yi Fan and Jun-hua Chen, Commun. Theor. Phys. 38 (2002) 147
* [14] A. Wunsche, J. Opt. B: Quantum Semiclass. Opt. 1 (1999) R11
* [15] Hong-yi Fan, Hai-liang Lu and Yue Fan, Ann. Phys. 321 (2006) 480
* [16] D. F. Walls, Nature 324 (1986) 210.
* [17] V. Bužek, J. Mod. Opt. 37 (1990) 303; R. Loudon and P. L. Knight, J. Mod. Opt. 34 (1987) 709; V. V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1.
* [18] Hongyi Fan, Phys. Rev. A 65 (2002) 064102; Hong-yi Fan, Phys. Lett. A 294 (2002) 253
* [19] Hong-yi Fan and Fan Yue, Phys. Rev. A 54 (1996) 958
* [20] Hongyi Fan and Hai-ling Chen, Commun. Theor. Phys. 36 (2001) 651
* [21] E. Wigner, Phys. Rev. 40 (1932) 749; R. F. O’Connell and E. P. Wigner, Phys. Lett. A 83 (1981) 145.
* [22] G. S. Agawal and E. Wolf, Phys. Rev. D 2 (1970) 2161; 2 (1970) 2187; 2 (1970) 2206; M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, Phys. Rep. 106 (1984) 121
* [23] W. P. Schleich, Quantumm Optics in Phase Space, (Wiley-VCH, Berlin, 2000).
* [24] H. Weyl, Z. Phys. 46 (1927) 1
|
arxiv-papers
| 2009-11-09T07:23:08 |
2024-09-04T02:49:06.348300
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hong-yi Fan and Li-yun Hu",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0911.1596"
}
|
0911.1798
|
# Twins: The Two Shortest Period Non-Interacting Double Degenerate White Dwarf
Stars
F. Mullally11affiliation: Department of Astrophysical Sciences, Princeton
University, Princeton, NJ 08544; fergal@astro.princeton.edu , Carles
Badenes11affiliation: Department of Astrophysical Sciences, Princeton
University, Princeton, NJ 08544; fergal@astro.princeton.edu , Susan E.
Thompson22affiliation: Department of Physics and Astronomy, University of
Delaware, 217 Sharp Lab, Newark, DE 19716, USA , Robert Lupton11affiliation:
Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544; fergal@astro.princeton.edu
###### Abstract
We report on the detection of the two shortest period non-interacting white
dwarf binary systems. These systems, SDSS J143633.29+501026.8 and SDSS
J105353.89+520031.0, were identified by searching for radial velocity
variations in the individual exposures that make up the published spectra from
the Sloan Digital Sky Survey. We followed up these systems with time series
spectroscopy to measure the period and mass ratios of these systems. Although
we only place a lower bound on the companion masses, we argue that they must
also be white dwarf stars. With periods of approximately 1 hour, we estimate
that the systems will merge in less than 100 Myr, but the merger product will
likely not be massive enough to result in a Type 1a supernova.
white dwarfs — binaries: close, spectroscopic
## 1 Introduction
White dwarf stars (WDs) are the end point of stellar evolution for 98% of all
stars (Weidemann, 2000) and store the archaeological record of the Galaxy. WDs
in binaries are particularly rich systems to study. Because of their
intrinsically low luminosity the companions must also be faint, and are
frequently rare or interesting objects. As an example, WDs are ideal targets
for the direct detection of planets (Debes et al., 2005; Farihi et al., 2008;
Hogan et al., 2009; Mullally et al., 2009) and brown dwarf stars (e.g. Farihi
et al., 2005).
The Sloan Digital Sky Survey (SDSS; York et al., 2000) has increased the
number of spectroscopically identified WDs from two thousand to a few tens of
thousands. This has, in turn, allowed follow-up surveys of specific types of
WD systems, from pulsators (e.g. Mullally et al., 2005; Nitta et al., 2009),
to extremely low mass WDs (Kilic et al., 2007a) to binaries involving main-
sequence stars (e.g. Silvestri et al., 2006; Heller et al., 2009), and
binaries with neutron stars (Agüeros et al., 2009).
Binary WDs hold the solution to an enduring problem in astrophysics; the
progenitors of Type Ia Supernova (SNIa). The origin of SNIa is of great
interest given their role in galactic chemical evolution and determining the
nature of dark energy. If a WD accretes enough material that its mass
approaches the Chandrasekhar limit ($\sim$1.4 M⊙), the star can no longer be
supported by electron degeneracy pressure and explodes as a supernova. This
scenario explains the lack of observed hydrogen in the spectra of SNIa, as
well as the striking similarities in the lightcurves and spectra. However, the
source of the accreted material, remains a subject of active debate.
In the double degenerate scenario for SNIa progenitors (Iben & Tutukov, 1984;
Webbink, 1984), two WDs in a tight binary in-spiral due to the emission of
gravitational radiation. If the total mass is near the Chandrasekhar mass, the
merger results in a supernova. While theoretically appealing, it is not clear
that nature favors this method. The SPY survey (Napiwotzki et al., 2001), a
high precision radial velocity survey of over a thousand WDs, failed to find
any candidates with periods short enough to merge within the lifetime of the
Galaxy, and masses large enough to explode as SNIa (Nelemans et al., 2005;
Napiwotzki et al., 2007).
The SDSS offers an opportunity to build on the SPY survey with a much larger
sample of stars. Kleinman et al. (2004) and Eisenstein et al. (2006)
meticulously collected and classified the spectra of nearly 10,000 WD and sub-
dwarf stars, of which approximately 8,000 were single stars with hydrogen or
helium atmospheres (DAs and DBs respectively). The spectra were obtained as a
series of 3 or more 15 minute exposures usually taken consecutively (Abazajian
et al., 2009), which makes it possible to identify massive companions with
orbital periods of a few hours or less and radial velocity amplitudes
$\gtrsim$170 km.s-1. The low luminosity of the WD means that any fainter
companion must be a degenerate object (WD, brown dwarf, neutron star, etc.) or
a very late M star; any other object would be more luminous than the white
dwarf.
SWARMS (the Sloan White dwArf Radial velocity data Mining Survey, Badenes et
al., 2009) exploits these individual exposures to mine the SDSS spectroscopic
database for double degenerate white dwarf (DDWD) systems. Our survey is
complementary to the SPY survey in that it has a lower radial velocity
sensitivity, but is still sensitive to white dwarf companions for many
thousands of objects. In this paper we present two binary systems, SDSS
J143633.29+501026.8 (catalog ) and SDSS J105353.89+520031.0 (catalog ), with
periods of 1.1 and 0.96 hours respectively. These systems constitute the
shortest period non-interacting double degenerate binaries yet found, and are
significantly shorter than the 1.46 hour period of the previous record holder,
WD0957$-$666 (catalog ) (Moran et al., 1997). As there are no visible
absorption lines from the companions our mass estimates are only lower bounds,
but in each case the companion is most likely another WD.
## 2 Observations and Reductions
We identified SDSS 1436 ($g$=18.2, plate-mjd-fiber=1046-52460-594) and SDSS
1053 ($g$=18.9, 1010-52649-12) as hydrogen atmosphere (DA) WDs potentially
possessing short period companions as part of an on-going survey for DDWDs.
Although we see radial velocity variations between different exposures, no
companion is visible in the spectrum. To confirm these systems as binaries,
and to measure the orbital parameters, we observed both stars with the Dual
Imaging Spectrograph (DIS) with the 3.5m telescope at Apache Point Observatory
over 4 nights between 2009-02-05 and 2009-02-14. We used the B1200 grating
with a 1.5” slit for a dispersion of 0.62 Å per pixel and a resolution of 1.8
Å FWHM. Each exposure was 10 minutes in duration and bracketed by an exposure
of the Helium, Neon and Argon arc lamps. We took several exposures of the
spectrophotometric standard Feige 67 each night to flux calibrate our spectra.
Figure 1: De-shifted and co-added spectrum of SDSS 1436 based on observations
made at APO. The solid red line is the best fit model used to estimate the
temperature and gravity. The average S/N per pixel in this spectrum is 30, and
$\approx$7 in each individual spectra
We performed an optimal spectroscopic reduction of each spectral image and
flux standard using standard long-slit IRAF routines. To optimize the
wavelength solution we trace the arc lamp spectra with the same trace used to
extract the corresponding WD spectra. A prominent Mercury emission line at
4358 Å (courteously provided by the residents of White Sands, NM) confirmed
our wavelength offset to better than our resolution. The final flux calibrated
spectra obtained from the blue ccd spans 3790-5020 Å. We show an average
spectrum from all 4 nights for SDSS 1436 in Figure 1.
The pressure broadened lines of a DA white dwarf are well modeled as a
combination of a Gaussian core with Lorentz wings (Thompson et al., 2004). We
convert the centroid shift of Hγ to a velocity, and fit a sine curve with
constant offset to the radial velocity time-series. Radial velocity curves for
Hβ and Hδ give similar results, but the accuracy obtained by fitting Hγ alone
is sufficient for our purposes. We show the best fit folded radial velocity
curves in Figures 2 & 3 and the best fit parameters in Table 1. Given the more
sparse sampling of SDSS 1053, our period estimate is less certain than for
SDSS 1436, but our observations span nearly 2 orbits and our uncertainty
estimate is only 36s.
The residuals of the fit to SDSS 1436 show a linear trend with phase. This
trend is seen for fits to Hβ and Hδ as well. Examination of the unfolded
lightcurve confirms this trend is indeed a function of phase, not of time, and
can not be explained by some drift in our instrumental calibration. Similarly,
a third body in the system on a longer period orbit would only produce a trend
in the unfolded data. Fitting an eccentric orbit reduces the peak to peak
amplitude of the residual trend by 100 km.s-1 but does not eliminate it.
Because the eccentric orbit fit is not significantly better, and because we
have difficulty imagining a scenario in which a system that has undergone two
common envelope evolution phases could emerge with an eccentric orbit, we show
only the circular fit in Figure 2.
Figure 2: Radial velocity curve for SDSS 1436 folded at the best fit period.
The solid line is the best fit sine curve to the data. The residuals of the
fit are shown in the lower panel. The apparent linear trend in the residuals
is discussed in the text. Figure 3: Same as Figure 2, but for SDSS 1053.
### 2.1 Temperature and Gravity
The published temperature of each star from Eisenstein et al. (2006) comes
from a fit to the average of three separate exposures spanning a total of 50
minutes. As this is a significant fraction of the orbital period we were
concerned that the fit may have been biased by combining spectra with
different radial velocities. Using our best fit radial velocity curve, we
deshifted and co-added each of our 10 minute exposures to produce a high
signal-to-noise spectrum. We then fit this spectrum to the same grid of DA
models used by Eisenstein et al. (2006) (updated by Koester et al., 2009, and
kindly provided by the author). We linearly interpolated the model spectra to
produce a finer grid of $\Delta$$T_{\mathrm{eff}}$=10 K and
$\Delta\log{g}$=0.02. We fit each model to the entire spectrum from 3800-5000
Å using a least squares minimization algorithm, allowing the fit to vary by a
high order polynomial in a similar manner to Eisenstein et al. (2006). We find
best fit parameters of 17120 K, $\log{g}$=6.60 for SDSS 1436 and 16150 K, 6.35
for SDSS 1053, consistent with the estimate of Eisenstein et al. (2006) who
finds ($T_{\mathrm{eff}}$, $\log{g}$) of (16933, 6.58) and (15399, 6.28)
respectively.
For each object we combine spectra taken close to the minima and maxima of the
velocity curve. We find no evidence of spectral features in these combined
spectra and are confident that flux from the companion is not biasing our fit.
Fontaine et al. (2003) noted that temperature and gravity estimates of WDs
from independent spectra using identical reductions and identical atmosphere
models often disagree significantly more than the quoted uncertainties.
Following their approach, we adopt uncertainties of 200 K and 0.05 for our
fits, which are more conservative than the values returned by the fitting
method. We caution that these uncertainties are internal to our fitting, and
do not attempt to address limitations of the models. For example, Tremblay et
al. (2009) recently introduced an improved treatment of Stark broadening which
systematically increases the best fit gravity by 0.2 dex in this temperature
and gravity range.
Kilic et al. (2007a) independently observed and fit the spectra of both stars
(as part of a search for companions to low mass WDs) and obtained similar
results for the gravity ($\log{g}$= 6.59 and 6.40), but higher temperatures
($T_{\mathrm{eff}}$= 18339 and 18325). Given the close agreement in measured
gravity between the three measurements, the small discrepancy in temperatures
do not materially effect the stellar masses we estimate in Section 3
We simulated the effect of changing radial velocities over the course of a 10
minute exposure to estimate the effect on the best fit temperature and
gravity. Using our best fit radial velocity curve for SDSS 1436, we coadded a
series of appropriately Doppler shifted model spectra, and fit the result in a
manner similar to our data. The largest discrepancy occurs when the star is
traveling perpendicular to the line of sight, where the core of the line
appears smoothed. This blurred spectrum is preferentially fit by 500 K hotter
model with a shallower line core, but the best fit gravity, which is most
important for measuring the mass, remains unchanged.
## 3 Discussion
Comparing our best fit temperature and gravity to the WD evolution models of
Serenelli et al. (2002) we estimate masses of 0.23(01) and 0.21(01) M⊙. These
models are created by removing mass from a 1 M⊙ model at appropriate times
during red giant branch evolution, and incorporate chemical diffusion, a
nearly pure He core (with metallicity, Z=0.001) and thick H layer. The
estimated masses are close to the minimum known white dwarf mass of 0.17 M⊙
(Kilic et al., 2007a; Kawka & Vennes, 2009). Moroni & Straniero (2009)
estimates that a WD must have a mass of at least 0.33 M⊙ to have a carbon-
oxygen core. Neither object approaches this mass, and are composed almost
entirely of helium with a thin hydrogen atmosphere.
Figure 4: Distribution of known DDWD systems. The filled circles indicate the
minimum mass of the two new systems discussed in this paper. The notch on the
arrow gives the total mass assuming an inclination angle, $\iota$, of 60∘, and
the tip of the arrow shows $\iota=40^{\circ}$. Square symbols indicate
previously known double-lined (DL) binaries (where the total mass is known),
while triangles indicate single-lined (SL) systems and are only a lower bound
on the total system mass. SDSS J091709.55+463821.8 (catalog ) (open triangle)
taken from Kilic et al. (2007b), and all other systems from Nelemans et al.
(2005). The horizontal dashed line indicates the Chandrasekhar mass, while the
curved line shows the period for which the merger time is equal to the age of
the Universe.
According to the initial-final mass relation (e.g. Kalirai et al., 2008;
Williams et al., 2009), only isolated WDs with masses greater than 0.47 M⊙
have had time to evolve off the main-sequence within the lifetime of the
Universe. Although it has been argued that high metallicity progenitors can
produce lower mass WDs (Kilic et al., 2007c), that scenario is unnecessary for
these systems. Instead, the fact that these two systems are known to be
binaries suggests that growth of these lower mass WDs was truncated by a
common envelope phase and evolved through the sub-dwarf channel (Heber, 2009).
Table 1: Measured Parameters | SDSS 1436 | SDSS 1053
---|---|---
Period (hrs) | 1.15238(14) | 0.960(10)
Amplitude (km.s-1) | 388(21) | 310(14)
Seperation ($R_{\odot}$) | 0.4789(75) | 0.2924(51)
$T_{\mathrm{eff}}$(K) | 17120(200) | 16150(200)
$\log{g}$ | 6.60(05) | 6.35(05)
Mass1 (M⊙) | 0.23(01) | 0.22(01)
Mass${}_{2}/\sin{i}$ (M⊙) | 0.57(04) | 0.31(02)
Merge time (Myr) | $<$102 | $<$104
### 3.1 Nature of the Companions
Solving for the Keplerian equations of motion for SDSS 1436 gives a mass for
the companion of (0.57(04)/$\sin{\iota}$) M⊙, where $\iota$ is the inclination
of the orbit to the line of sight, consistent with a carbon-oxygen core WD.
The companion to SDSS 1053 is at least 0.31(02) M⊙. Although these are minimum
masses, and are consistent with a wide range of astrophysical objects, we
argue that the companions are most likely also WDs.
Main sequence stars can be ruled out on luminosity grounds. For SDSS 1436
(SDSS 1053), the minimum mass of the companion corresponds to a spectral type
of K8 (M1) (Habets & Heintze, 1981), which has an absolute $i$ magnitude of
7.2 (8.5) (Bilir et al., 2009; Hawley et al., 2002), considerably brighter
than the observed WD ($i=9.1~{}(9.4)$, Holberg & Bergeron, 2006). The SDSS
spectrum of either object shows no evidence of any cool companion at red
wavelengths, ruling out the possibility of a main-sequence companion. A
similar argument applies to red giant stars and other, higher luminosity
objects.
If the inclination angle, $\iota<24^{\circ}(13^{\circ})$, the companion mass
is greater than the Chandrasekhar mass and the companion must be a neutron
star (NS) or a black hole. Approximately 45 WD-NS binaries are known, and the
mass distribution of WDs in such binaries is much wider than for isolated
systems, admitting both high and low mass WDs (van Kerkwijk et al., 2005).
Agüeros et al. (2009) looked at both SDSS 1436 and SDSS 1053 during an 820 Mhz
radio survey for pulsar companions to WDs but did not detect any signal. These
observations do not exclude the possibility of a pulsar companion, not only
because the orientation of the pulsar beam may not be along the line of sight,
but also because their analysis restricted their sensitivity to orbital
periods greater than 8 hours.
Interacting WD-NS binaries are known as ultra compact X-ray binaries (UCXBs;
see Nelemans & Jonker, 2006, for a review). In these systems, the orbital
separation is so small that a WD overfills its Roche lobe and donates material
onto the surface of the neutron star via an accretion disk, emitting X-rays in
the process. The longest period UXCBs have periods of 50-55 minutes (Nelemans
& Jonker, 2006), entirely consistent with the periods of the systems under
scrutiny. If the companions were NSs, they would almost certainly be
interacting. However, the presence of hydrogen in the atmosphere of the
visible WD in both systems means that the systems are not interacting. In
double degenerate systems, mass transfers from the lower to the higher mass
star. In both systems, the higher mass star is the invisible companion. If the
visible star was losing mass, the thin hydrogen layer would be quickly
stripped, exposing the underlying helium core. Because both stars still have
their hydrogen layers we can conclude that mass transfer has not yet started.
### 3.2 Consequences of a Merger
Non-interacting DDWD companions are therefore the only possible objects
consistent with the available evidence. Non-interacting systems in short
period orbits lose orbital energy in the form of gravitational radiation and
will eventually merge. Using the equation for angular momentum loss given by
Paczyński (1967), we estimate these systems will merge in less than 102 and
104 Myr respectively. If the orbits are inclined by 45∘ to the line of sight,
the merger time decreases to $<79$ Myr
Guerrero et al. (2004) used smoothed particle hydrodynamic simulations to
predict the consequences of the merger of WDs of different masses. For the
merger of two 0.4 M⊙ helium core WDs (analogous to the minimum mass
configuration of the SDSS 1053 system), they found no thermonuclear flash, and
no mass loss. The case of a 0.4 M⊙ WD merging with a 0.6 M⊙ carbon-oxygen core
star (similar to SDSS 1436), some carbon is burned into oxygen, but there is
no thermonuclear runaway and no supernova.
Unless the unseen companions have masses $\gtrsim 1.2$ M⊙ it seems unlikely
that either system will produce a SNIa. WDs with masses greater than 1.2 M⊙ do
exist, with the most massive WD found to date being 1.33 M⊙ (Kepler et al.,
2007, assuming an oxygen-neon core). However, these stars are rare, and are
consistent with only a small range of inclinations angles for these systems.
The merger of a carbon-oxygen core WD with a helium WD most likely produces an
extreme helium star (Saio & Jeffery, 2002) or an R CrB star (Webbink, 1984;
Clayton et al., 2007). The merger of two helium core WDs (an option only for
SDSS 1053) is one evolutionary pathway to produce sdO subdwarf stars (Heber,
2009). Regardless, the merger remnant will eventually cool to become a single
WD. Liebert et al. (2005) compared the space density of high mass WDs with
that expected from a Salpeter initial mass function and a single burst stellar
population and concluded that 80% of high mass WDs were created by the merger
of lower mass stars. The merger of a carbon-oxygen WD with a He core one has
been suggested by García-Berro et al. (2007) as the origin of hot debris disks
around massive WDs.
However, analysis of the extremely low luminosity, and calcium rich, type 1b
supernova SN2008E by Perets et al. (2009) concluded that only 0.3 M⊙ of
material was ejected, and the pattern of elemental abundances (high calcium
abundances, but low sulpher) was best explained by helium fraction $>$0.5 in
the initial composition. It is conceivable that the progenitor of this
explosion involved the disruption of a helium core WD in a DDWD system.
## 4 Conclusion
We report on the detection of the two closest non-contact white dwarf binaries
known. These systems were detected by mining the spectroscopic database of the
SDSS, and followed-up with time resolved optical spectroscopy. We argue that
the companions must also be WDs: Main-sequence stars of the requisite mass are
more luminous than the primary WDs, and a neutron star or black hole in such
close proximity would have stripped off the thin outer layers of the
primaries. With periods of about an hour, these systems will merge in less
than 100 Myr and probably produce a high mass WD.
CB is supported by NASA through Chandra Postdoctoral Fellowship Award Number
PF6-70046 issued by the Chandra X-ray Observatory Center, which is operated by
the Smithsonian Astrophysical Observatory for and on behalf of NASA under
contract NAS8-03060. SET acknowledges the support of the Crystal Trust. We
wish to thank the staff of Apache Point Observatory for their assistance in
making these observations.
## References
* Abazajian et al. (2009) Abazajian, K. N., et al. 2009, ApJS, 182, 543
* Agüeros et al. (2009) Agüeros, M. A., Camilo, F., Silvestri, N. M., Kleinman, S. J., Anderson, S. F., & Liebert J. W. , 2009, ApJ, 697, 283
* Badenes et al. (2009) Badenes, C., Mullally, F., Thompson, S. E., & Lupton R. H. , 2009, ArXiv 0910.2709
* Bilir et al. (2009) Bilir, S., Karaali, S., Ak, S., Coşkunoğlu, K. B., Yaz, E., & Cabrera-Lavers, A. 2009, MNRAS, 396, 1589
* Clayton et al. (2007) Clayton, G. C., Geballe, T. R., Herwig, F., Fryer, C., & Asplund, M. 2007, ApJ, 662, 1220
* Debes et al. (2005) Debes, J. H., Sigurdsson, S., & Woodgate, B. E. 2005, AJ, 130, 1221
* Eisenstein et al. (2006) Eisenstein, D. J., et al. 2006, ApJS, 167, 40
* Farihi et al. (2008) Farihi, J., Becklin, E. E., & Zuckerman, B. 2008, ApJ, 681, 1470
* Farihi et al. (2005) Farihi, J., Zuckerman, B., & Becklin, E. E. 2005, AJ, 130, 2237
* Fontaine et al. (2003) Fontaine, G., Bergeron, P., Billères, M., & Charpinet, S. 2003, ApJ, 591, 1184
* García-Berro et al. (2007) García-Berro, E., Lorén-Aguilar, P., Pedemonte, A. G., Isern, J., Bergeron, P., Dufour, P., & Brassard P. , 2007, ApJ, 661, L179
* Guerrero et al. (2004) Guerrero, J., García-Berro, E., & Isern J. , 2004, A&A, 413, 257
* Habets & Heintze (1981) Habets, G. M. H. J., & Heintze, J. R. W. 1981, A&AS, 46, 193
* Hawley et al. (2002) Hawley, S. L., et al. 2002, AJ, 123, 3409
* Heber (2009) Heber, Ulrich 2009, Annual Review of Astronomy and Astrophysics, 47, 211
* Heller et al. (2009) Heller, R., Homeier, D., Dreizler, S., & Østensen R. , 2009, A&A, 496, 191
* Hogan et al. (2009) Hogan, E., Burleigh, M. R., & Clarke, F. J. 2009, MNRAS, 396, 2074
* Holberg & Bergeron (2006) Holberg, J. B., & Bergeron, P. 2006, AJ, 132, 1221
* Iben & Tutukov (1984) Iben Jr., I., & Tutukov, A. V. 1984, ApJS, 54, 335
* Kalirai et al. (2008) Kalirai, J. S., Hansen, B. M. S., Kelson, D. D., Reitzel, D. B., Rich, R. M., & Richer, H. B. 2008, ApJ, 676, 594
* Kawka & Vennes (2009) Kawka, A., & Vennes, S. 2009, ArXiv 0909.3249
* Kepler et al. (2007) Kepler, S. O., Kleinman, S. J., Nitta, A., Koester, D., Castanheira, B. G., Giovannini, O., Costa, A. F. M., & Althaus, L. 2007, MNRAS, 375, 1315
* Kilic et al. (2007a) Kilic, M., Allende Prieto, C., Brown, W. R., & Koester, D. 2007a, ApJ, 660, 1451
* Kilic et al. (2007b) Kilic, M., Brown, W. R., Allende Prieto, C., Pinsonneault, M. H., & Kenyon, S. J. 2007b, ApJ, 664, 1088
* Kilic et al. (2007c) Kilic, M., Stanek, K. Z., & Pinsonneault, M. H. 2007c, ApJ, 671, 761
* Kleinman et al. (2004) Kleinman S. J. Harris, H. C., et al. 2004, ApJ, 607, 426
* Koester et al. (2009) Koester, D., Kepler, S. O., Kleinman, S. J., & Nitta, A. 2009, Journal of Physics Conference Series, 172, 012006
* Liebert et al. (2005) Liebert, J., Bergeron, P., & Holberg, J. B. 2005, ApJS, 156, 47
* Moran et al. (1997) Moran, C., Marsh, T. R., & Bragaglia, A. 1997, MNRAS, 288, 538
* Moroni & Straniero (2009) Moroni, P. G. P., & Straniero, O. 2009, American Institute of Physics Conference Series, ed. ”G. Giobbi A. Tornambe G. Raimondo M. Limongi L. A. Antonelli N. Menci & E. Brocato”, 1111, 63
* Mullally et al. (2009) Mullally, F., Reach, W. T., Degennaro, S., & Burrows A. , 2009, ApJ, 694, 327
* Mullally et al. (2005) Mullally, F., Thompson, S. E., Castanheira, B. G., Winget, D. E., Kepler, S. O., Eisenstein, D. J., Kleinman, S. J., & Nitta, A. 2005, ApJ, 625, 966
* Napiwotzki et al. (2001) Napiwotzki, R., et al. 2001, AN, 322, 411
* Napiwotzki et al. (2007) ——. 2007, 15th European Workshop on White Dwarfs, ed. Napiwotzki R., & Burleigh M. R., 372, 387
* Nelemans & Jonker (2006) Nelemans, G., & Jonker, P. G. 2006, ArXiv arXiv:astro-ph/0605722
* Nelemans et al. (2005) Nelemans, G., et al. 2005, A&A, 440, 1087
* Nitta et al. (2009) Nitta, A., et al. 2009, ApJ, 690, 560
* Paczyński (1967) Paczyński, B. 1967, Acta Astronomica, 17, 287
* Perets et al. (2009) Perets, H. B., et al. 2009, ArXiv 0906.2003
* Saio & Jeffery (2002) Saio, H., & Jeffery, C. S. 2002, MNRAS, 333, 121
* Serenelli et al. (2002) Serenelli, A. M., Althaus, L. G., Rohrmann, R. D., & Benvenuto, O. G. 2002, MNRAS, 337, 1091
* Silvestri et al. (2006) Silvestri, N. M., et al. 2006, AJ, 131, 1674
* Thompson et al. (2004) Thompson, S. E., Clemens, J. C., van Kerkwijk, M. H., O’Brien, M. S., & Koester, D. 2004, ApJ, 610, 1001
* Tremblay et al. (2009) Tremblay, P.-E., Bergeron, P., & Dupuis, J. 2009, Journal of Physics Conference Series, 172, 012046
* van Kerkwijk et al. (2005) van Kerkwijk, M. H., Bassa, C. G., Jacoby, B. A., & Jonker, P. G. 2005, Binary Radio Pulsars, ed. Rasio F. A., & Stairs I. H., 328, 357
* Webbink (1984) Webbink, R. F. 1984, ApJ, 277, 355
* Weidemann (2000) Weidemann, V. 2000, A&A, 363, 647
* Williams et al. (2009) Williams, K. A., Bolte, M., & Koester, D. 2009, ApJ, 693, 355
* York et al. (2000) York, D. G., et al. 2000, AJ, 120, 1579
|
arxiv-papers
| 2009-11-09T22:17:14 |
2024-09-04T02:49:06.357108
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Mullally, Carles Badenes, Susan E. Thompson, Robert Lupton",
"submitter": "Fergal Mullally",
"url": "https://arxiv.org/abs/0911.1798"
}
|
0911.1839
|
# Monte Carlo Study of Relaxor Systems:
A Minimum Model of Pb(In1/2Nb1/2)O3
Yusuke Tomita E-mail address: ytomita@issp.u-tokyo.ac.jp Takeo Kato and Kazuma
Hirota1 Institute for Solid State PhysicsInstitute for Solid State Physics
University of Tokyo University of Tokyo Kashiwa 277-8581 Kashiwa 277-8581
Japan
1Department of Earth and Space Science Japan
1Department of Earth and Space Science Faculty of Science Faculty of Science
Osaka University Osaka University Toyonaka 560-0043 Toyonaka 560-0043 Japan
Japan
###### Abstract
We examine a simple model of Pb(In1/2Nb1/2)O3 (PIN), which includes both long-
range dipole-dipole interaction and random local anisotropy. An improved
algorithm optimized for long-range interaction has been applied to efficient
large-scale Monte Carlo simulation. We demonstrate that the phase diagram of
PIN is qualitatively reproduced by this minimum model. Some characteristic
features of relaxors such as nanoscale domain formation, slow dynamics, and
dispersive dielectric responses are also examined.
relaxors, ferroelectric phase transition, antiferroelectric phase transition,
Monte Carlo algorithm, domain formation, slow dynamics, dispersive dielectric
responses, Pb(In1/2Nb1/2)O3
Relaxors, whose first discovery dates back to over half a century ago [1],
have newly attracted much interest over the past few decades owing to their
colossal dielectric and piezoelectric responses that are appealing for
industrial applications. Despite intensive research, however, the physical
origin of the unusual properties of relaxors is not fully understood yet. [2]
This is because nanoscale intrinsic randomness in relaxor systems has to be
addressed appropriately.
As a simple example to understand the physics of relaxors, let us choose the
lead-based relaxor Pb(In1/2Nb1/2)O3 (PIN), which is the target of the present
theoretical work. The intrinsic inhomogeneities of relaxor systems originate
from the random configuration of In3+ and Nb5+ at the B-site in the perovskite
structure, while the dielectric property of such systems is governed mainly by
the replacement of Pb and O ions. This system has the advantage that the
strength of randomness can be controlled by adjusting annealing temperature.
[3] For the lowest annealing temperature, an alternate order of In and Nb
atoms stabilizes the fourfold antiferroelectric (AFE) phase. For the higher
annealing temperature, an increase in B-site randomness decreases AFE
transition temperature. Under sufficiently strong randomness in the B-site,
the ferroelectric phase accompanies relaxor properties. This behavior of PIN
suggests that the potential FE instability exists behind the AFE phase, and
that its emergence by the selective suppression of the AFE phase due to B-site
randomness is relevant to FE relaxors. Recent experimental progress in the
X-ray diffraction analysis facilitated by strong-intensity photon sources has
also enabled us to detect the potential competition between the FE and AFE
phases of PIN from the viewpoint of lattice dynamics. [4, 5]
There are several theoretical methods for elucidating the emergence of relaxor
properties. They utilize the existing effective theory dealing with a strong
randomness such as the extension of the Ginzburg-Landau-Devonshire theory, [6,
7] the spherical random-bond-random-field model, [8] and the dynamical-related
model. [9, 10] Although these approaches have provided useful description of
relaxor properties, it is unclear how their model parameters should be derived
microscopically. At present, first-principles calculation is playing a crucial
role in the microscopic description of ferroelectrics and ferroelectric
relaxors. In the pioneering work by Cohen and coworker [11, 12], the electron-
structure calculation of BaTiO3 and PbTiO3 indicated the importance of the
covalent nature among composite atoms through Ti-O hybridization. After the
success of their work, microscopic first-principles approaches have been
adopted in a number of issues. Further theoretical progress has been achieved
by a hybrid method combining the microscopic derivation of the effective
Hamiltonian and its numerical simulation. Zhong et al. have derived the
effective Hamiltonian of BaTiO3 by extracting the adiabatic potential of
important phonon modes from first-principles calculation. [13, 14] Monte Carlo
simulation for this effective Hamiltonian has enabled the successful
reproduction of the sequential finite-temperature phase transitions of BaTiO3
in good agreement with experimental results. Recently, the hybrid method has
also been applied to the study of relaxor systems. [15, 16]
Thus, the utilization of first-principles calculation is a promising method
for understanding relaxors. However, there still remain several difficulties
to overcome for the actual application of this method to the study of
relaxors. One major difficulty is in the use of the numerical solver of the
effective Hamiltonian for phonons. Even though the effective Hamiltonian
derived for relaxors is in a simple form, its numerical simulation may become
extremely difficult because it is inevitable to encounter slow dynamics
inherent to random systems. The Monte Carlo simulation of the effective
Hamiltonian needs a large number of Monte Carlo steps owing to its long
correlation time. Therefore, the development of appropriate numerical solvers
for relaxor systems seems indispensable. This situation may remind us of Monte
Carlo studies of spin-glass systems. Although the Hamiltonian of spin-glass
systems has a simple form, much effort has to be exerted for sufficient Monte
Carlo sampling to understand their peculiar slow dynamics. In the research
field of relaxors, however, the importance of developing a numerical solver
has not been addressed so far.
In this paper, we propose a simple model of ferroelectric relaxors to explain
the phase diagram of PIN. Our model includes both long-range dipole-dipole
interaction and local randomness. We apply the new effective algorithm
optimized for long-range interaction proposed by Fukui and Todo. [17] This new
algorithm enables us to simulate long-range interaction systems with the cost
of O($N$) with respect to the system size $N$, while the conventional
simulation for the same system takes the cost of O($N^{2}$). In addition to
the substantial reduction in computational time by employing this algorithm,
we adopt the exchange Monte Carlo method to attenuate the slow dynamics of
random systems. [18] We show that our simple model may qualitatively reproduce
the phase diagram of PIN. Moreover, we demonstrate that some characteristic
features of relaxors such as domain formation and dispersive dielectric
response are reproduced reasonably. Our results indicate the power of the
sophisticated Monte Carlo method as well as the possibility that relaxor
systems may be represented by a simple model.
We consider the model Hamiltonian for PIN on a 2D square lattice as
$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{i<j}\left[\frac{\mbox{\boldmath$S$}_{i}\cdot\mbox{\boldmath$S$}_{j}}{r^{3}_{ij}}-3\frac{(\mbox{\boldmath$r$}_{ij}\cdot\mbox{\boldmath$S$}_{i})(\mbox{\boldmath$r$}_{ij}\cdot\mbox{\boldmath$S$}_{j})}{r^{5}_{ij}}\right]$
(1) $\displaystyle-$
$\displaystyle\sum_{i}(\mbox{\boldmath$D$}_{i}\cdot\mbox{\boldmath$S$}_{i})^{2},$
where $\mbox{\boldmath$S$}_{i}$ is a unit vector in the $xy$-plane
representing the dipole moment on the $i$th unit cell induced by the off-
center replacement of the Pb atom. The first term of the Hamiltonian is the
dipole-dipole interaction dependent on the relative position
$\mbox{\boldmath$r$}_{ij}=\mbox{\boldmath$r$}_{j}-\mbox{\boldmath$r$}_{i}$
between the sites $i$ and $j$, while the second term describes local
anisotropy whose direction and strength are denoted as
$\mbox{\boldmath$D$}_{i}$. In order to reproduce the phase diagram of PIN, we
design our model such that the FE phase is stabilized by dipole-dipole
interaction, while the AFE phase is stabilized in the presence of an
alternative change in $\mbox{\boldmath$D$}_{i}$. Supposing that B-site
randomness affects only the local energy change through the anisotropy
parameter $\mbox{\boldmath$D$}_{i}$, we can expect that all the features of
PIN, i.e., the AFE transition in the ordered PIN, its suppression by B-site
randomness, and the appearance of the FE domain in the sufficiently disordered
PIN are reproduced.
The detailed setting of our model is given as follows. In a naive 2D square
lattice, the dipole-dipole interaction does not lead to ferroelectric
instability because the columnar antiferroelectric state is more favored in
orthogonal lattices. [19] Therefore, we modify our model slightly: We divide
the unit cells into two bipartite sublattices named P and Q, and shift only
the Q-sites in the $z$-direction by a unit length. This rearrangement of unit
cells ensures ferroelectric instability in the absence of local anisotropy.
The antiferroelectric instability is driven by an alternative arrangement of
two types of anisotropy, namely, $\mbox{\boldmath$D$}_{1}$ and
$\mbox{\boldmath$D$}_{2}$ in the P- and Q-sites, where
$\mbox{\boldmath$D$}_{1}=(D/\sqrt{2},-D/\sqrt{2})$ and
$\mbox{\boldmath$D$}_{2}=(D/\sqrt{2},D/\sqrt{2})$. The randomness of $D$ is
controlled by the probability $p$, where $\mbox{\boldmath$D$}_{1}$
($\mbox{\boldmath$D$}_{2}$) is attached to the P-(Q-)site with a probability
$(1-p)^{2}$, oppositely to $p^{2}$, and is turned off ($D$=0) with $2p(1-p)$.
The two limiting values $p=0$ and $p=1/2$ correspond to the ordered and
completely disordered PINs, respectively.
The model Hamiltonian is examined by Monte Carlo simulations under a periodic
boundary condition to minimize the strong surface effect due to long-range
interaction. Long-range interactions are summed up by the Ewald summation
technique. [20] We employ the ${\rm O}(N)$ method based on Walker’s algorithm
for efficient update [17, 21] as well as the temperature-exchange algorithm.
[18] One exchange trial between replicas was made for each of the $10$ MC
steps. The system size is taken up to $N=32\times 32$, for which $3\times
10^{6}$ MC steps for thermalization and $2\times 10^{5}$ MC steps for
measurement were needed in the severest case of $p=1/2$. The sample average is
taken over $10$ different random configurations of $\mbox{\boldmath$D$}_{i}$
for each $p$. Throughout this paper, the strength of the anisotropy is fixed
as $|\mbox{\boldmath$D$}_{i}|=1$.
Figure 1: (Color online) (a) A plot of four-fold antiferroelectric(AFE) order
parameters in the absence of randomness ($p=0$) and (b) a plot of
ferroelectric(FE) order parameters at $p=0.5$ are shown for different system
sizes $N=L\times L$. (c) Phase diagram obtained by the minimum model. Phase
boundaries in the figure are visual guides.
Figure 1(a) shows the temperature dependence of the average squared four-fold
staggered polarization $m_{\rm AF}$ for the ordered PIN ($p=0$). The pattern
of the AFE ordering, which is shown in Fig. 1(c), agrees with experimental
observations. As the system size $N=L\times L$ increases, a sharp increase in
the order parameter appears below a critical temperature, indicating the phase
transition. The transition point is determined by the crossing point of the
Binder parameters, $\langle m_{\rm AF}^{4}\rangle/\langle m_{\rm
AF}^{2}\rangle^{2}$, of $L=16$ and $32$. By a similar analysis of nonzero
values of $p$, the phase boundary of AFE in the $p$-$T$ plane is determined,
as shown by full circles in Fig. 1(c). For small values of $p$, the transition
temperature of AFE is suppressed by the B-site randomness. For sufficiently
large values of $p$, the FE domain develops at low temperatures instead of the
AFE phase. Figure 1(b) shows the average squared uniform polarization as a
function of temperature in the completely disordered case ($p=1/2$). An abrupt
increase in squared polarization below a threshold temperature for the
$N=32\times 32$ system indicates a rapid development of the FE domain. In our
simulation, however, no long-range FE ordering could be detected by finite-
size scaling because of the very slow relaxation of the Monte Carlo sampling
and the complex size dependence in the presence of strong randomness. Here, we
roughly estimate the threshold temperature at which the local FE instability
rises up by Binder-parameter analysis between $L=16$ and $32$, and plot it
using the empty circle in Fig. 1(c). At approximately $p=0.4$, the competition
between AFE and FE makes the Monte Carlo calculation severe, and the phase
boundary could not be determined. Rough interpolations for AFE and FE are
shown by the solid curves in Fig. 1(c) only as visual guides. The obtained
phase diagram is in good agreement with experimental results of PIN.
Figure 2: (Color online) Log-log plot of the normalized time autocorrelation
functions. While the autocorrelation functions decay smoothly for the no-
anisotropy system at $T=0.2$ and the random system at $T=0.3$, the
autocorrelation function shows a slow decay for the random system at $T=0.2$.
For large values of $p$, some characteristic features of relaxor systems
appear below a crossover temperature, which locates higher than the threshold
temperature for FE domain formation, as indicated roughly by the dashed line
in Fig. 1(c). The emergence of relaxor properties is indicated significantly
by the slow convergence of the Monte Carlo average. Figure 2 shows the
normalized autocorrelation of the energy $E$ for $N=32\times 32$ defined by
$\phi(t)=\frac{\langle E(0)E(t)\rangle-\langle E\rangle^{2}}{\langle
E^{2}\rangle-\langle E\rangle^{2}}$ (2)
as a function of the Monte Carlo step $t$ in the completely random case
($p=1/2$). Here, for the purpose of examining the dynamics of the system, we
turned off the exchange process between replicas. We find that the decay of
the autocorrelation function is extremely slow at $T=0.2$ in clear contrast
with that at a higher temperature $T=0.3$. This slow energy relaxation
indicates a glasslike behavior for relaxors at low temperatures. In order to
confirm that this slow relaxation comes from randomness of the B-site, we also
calculated the autocorrelation at $T=0.2$ in the absence of local anisotropy
($|\mbox{\boldmath$D$}_{i}|=0$), for which normal FE ordering is expected. As
seen in Fig. 2, the autocorrelation function follows a simple decay with no
anomalous slow relaxation. Thus, the present model is expected to exhibit
relaxor properties under sufficiently strong randomness below a crossover
temperature.
Figure 3: (Color online) Snapshots of dipole configuration for $p=0.5$ and
$T=0.1$. Each color denotes the angle of dipole moment. Reference color bars
are shown on the right. The bottom color denotes $-\pi$, and the top color
denotes $\pi$. The last graph shows the result for the same parameters but for
a model with a dipole interaction interrupted up to the next-nearest sites.
In order to visualize the glassy state realized in the relaxor phase, we show
in Fig. 3 the spatial pattern of dipole directions by taking snapshots as a
function of the MC steps for $p=0.5$ and $N=32\times 32$. We can see that
mosaic-like FE domains are formed after a finite relaxation time. The
boundaries (domain-wall regions) between the neighboring FE domains are rather
clear. One may consider that the FE domain structure is determined by a
partially ordered configuration of anisotropy, the so-called chemical
nanoregion (CNR). The present FE domain is, however, irrelevant to CNR,
because partial ordering is absent for a complete random choice of anisotropy
at each site in our model.
As has already been mentioned, the present system possesses FE instability in
the absence of local anisotropy (${\boldmath D}={\boldmath 0}$). The random
configuration of anisotropy similarly causes a local FE instability due to
effective cancellation of anisotropy, but at the same time prevents the growth
of an FE domain into a uniform FE order, as seen in Fig. 3. A small but finite
change of the snapshot in the long-time region of the Monte Carlo simulation
indicates that a very slow fluctuation of the FE domain governs the relaxor
properties in the present model. Here, we should note that the evolution of
the dipole configuration along the Monte Carlo step is not directly related to
actual real-time dynamics. The snapshots, however, provide intuitive and
fruitful insights for understanding physical processes in relaxor systems. In
our model, the long-range nature of the dipole interaction is essential for
the formation of the FE domain structure. To see this, we show a snapshot of
Monte Carlo simulation for a model with dipole interaction interrupted up to
the next-nearest site in the last graph of Fig. 3. We find that the domain
structure disappears in this modified model.
Figure 4: (Color online) Dielectric constants evaluated as functions of
temperature under the ac electric field for several frequencies in the (a)
ordered case ($p=0$) and (b) completely disordered case ($p=1/2$).
The existence of nanoscale frustrated FE domains indicates the high degeneracy
of glassy states. This degeneracy is expected to be responsible for large
dielectric responses since a significant change in the polarization of such FE
domains is possible by a small external electric field. To see this, we
measure the Monte Carlo evolution of the total polarization under a small ‘ac’
electric field, which varies periodically along the Monte Carlo step. We then
calculate the ‘ac’ dielectric constant in the linear response regime. We
should note again that the ‘ac’ dielectric constant thus obtained is not equal
to the real-time ac dielectric constant. Furthermore, we should keep in mind
that this is a nonequilibrium response since the relaxation to equilibrium
states is never realized in the glassy phase in the present simulation.
Nevertheless, the ‘ac’ dielectric response calculated here is expected to
mimic the actual ac response in a qualitative level. In Fig. 4, we show the
‘ac’ dielectric constant as a function of temperature for three frequencies in
the ordered ($p=0.0$) and completely disordered ($p=1/2$) cases. In both
cases, dielectric constant shows a peak at approximately the transition
(crossover) temperature. The marked difference is found in the low-temperature
phase. In the ordered case, dielectric constant sharply drops in the low-
temperature AFE phase, and becomes almost independent of frequency. On the
other hand, in the disordered case, it decreases gradually as temperature
decreases, and a strong frequency dependence remains. This strong dispersion
of dielectric constant at low temperatures can be explained as follows. Under
a high-frequency electric field, each dipole inside FE domains may respond to
an external field. Under a low-frequency electric field, however, those with
frustration that construct large domains start to respond. Therefore, in a
minimum model, a dipolar glass with ferroelectric ordering is realized. This
ordering causes a broadened dielectric constant and a strong dependence on
frequency in the relaxor phase.
In summary, we examined a simple theoretical model of PIN composed of dipolar
interaction and local random anisotropy. We demonstrated that efficient Monte
Carlo simulations equipped with an improved algorithm optimized for long-range
interaction may access several characteristic features of relaxors. The phase
diagram of PIN was qualitatively reproduced by appropriate inclusion of the
intrinsic competition between the AFE and FE phases. By the examination of the
Monte Carlo evolution of the dipole configuration, we demonstrated some of the
glassy behaviors inherent to relaxor systems such as the FE domain formation,
extremely slow dynamics, and strong frequency dependence of dielectric
responses.
We end this paper by mentioning the future outlook of theoretical approaches
to studying relaxors. The model we treated here includes a few artificial
assumptions. They, however, will be removed by replacing our model with the
effective Hamiltonian derived by first-principles calculation. We stress that
the smart Monte Carlo algorithm is applicable not only to the rotator model
but also to the continuous-variable model. The hybridization of the first-
principles calculation and statistical approach based on the Monte Carlo
simulation will be an effective means of elucidating the microscopic origin of
relaxors.
The computation in the present work is executed on computers at the
Supercomputer Center, Institute for Solid State Physics, University of Tokyo.
The present work is financially supported by a MEXT Grant-in-Aid for
Scientific Research (B) (19340109), by a MEXT Grant-in-Aid for Scientific
Research on Priority Areas “Novel States of Matter Induced by Frustration”
(19052002), and by the Next Generation Supercomputing Project, Nanoscience
Program, MEXT, Japan
## References
* [1] G. A. Smolenskii and A. I. Agronovskaya: Sov. Phys. Tech. Phys. A3 (1958) 1380.
* [2] For recent reviews, A. A. Bokov and Z.-G. Ye: J. Mater. Sci. 41 (2006) 31.
* [3] A. A. Bokov, I. P. Raevskii, O. I. Prokopalo, E. G. Fesenko, and V. G. Smotrakov: Ferroelectrics 54 (1984) 241.
* [4] K. Ohwada, K. Hirota, H. Terauchi, T. Fukuda, S. Tsutui, A. Q. R. Baron, J. Mizuki, H. Ohwa, and N. Yasuda: Phys. Rev. B 77 (2008) 094136.
* [5] K. Ohwada and Y. Tomita: J. Phys. Soc. Jpn. 79 (2010) 011012.
* [6] A. F. Devonshire: Adv. Phys. 3 (1954) 85.
* [7] J.-M. Liu, S. T. Lau, H. L. W. Chan, and C. L. Choy: J. Mater. Sci. 41 (2006) 163.
* [8] R. Blinc, J. Dolinšek, A. Gregorovič, B. Zalar, C. Filipič, Z. Kutnjak, A. Levstik, and R. Pirc: Phys. Rev. Lett. 83 (1999) 424.
* [9] A. J. Bell: J. Phys.: Condens. Matter 5 (1993) 8773.
* [10] B. E. Vugmeister and H. Rabitz: Phys. Rev. B 57 (1998) 7581.
* [11] R. E. Cohen: Nature 358 (1992) 136.
* [12] R. E. Cohen and H. Krakauer: Ferroelectrics 136 (1992) 65.
* [13] W. Zhong, D. Vanderbilt, and K. M. Rabe: Phys. Rev. Lett. 73 (1994) 1861.
* [14] W. Zhong, D. Vanderbilt, and K. M. Rabe: Phys. Rev. B 52 (1995) 6301.
* [15] S. Tinte, B. P. Burton, E. Cockayne, and U. V. Waghmare: Phys. Rev. Lett. 97 (2006) 137601.
* [16] B. P. Burton, E. Cockayne, S. Tinte, and U. V. Waghmare: Phys. Rev. B 77 (2008) 144114.
* [17] K. Fukui and S. Todo: J. Comp. Phys. 228 (2009) 2629.
* [18] K. Hukushima and K. Nemoto: J. Phys. Soc. Jpn. 65 (1996) 1604.
* [19] Y. Tomita: J. Phys. Soc. Jpn. 78 (2009) 114004.
* [20] J.-J. Weis: J. Phys.: Condens. Matter 15 (2003) S1471.
* [21] A. J. Walker: ACM Trans. Math. Software 3 (1977) 253.
|
arxiv-papers
| 2009-11-10T06:58:08 |
2024-09-04T02:49:06.362809
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yusuke Tomita, Takeo Kato, and Kazuma Hirota",
"submitter": "Yusuke Tomita",
"url": "https://arxiv.org/abs/0911.1839"
}
|
0911.1856
|
# Strong decays of newly observed $D_{sJ}$ states in a constituent quark model
with effective Lagrangians
Xian-Hui Zhong1,3111E-mail: zhongxh@ihep.ac.cn and Qiang Zhao2,3222E-mail:
zhaoq@ihep.ac.cn 1) Department of Physics, Hunan Normal University, and Key
Laboratory of Low-Dimensional Quantum Structures $\&$ Quantum Control of
Ministry of Education, Changsha 410081, P.R. China 2) Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, P.R. China 3)
Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, P.R.
China
###### Abstract
The strong decay properties of the newly observed states $D_{sJ}(3040)$,
$D_{sJ}(2860)$ and $D_{sJ}(2710)$ are studied in a constituent quark model
with quark-meson effective Lagrangians. We find that the $D_{sJ}(3040)$ could
be identified as the low mass physical state $|2{P_{1}}\rangle_{L}$
($J^{P}=1^{+}$) from the $D_{s}(2^{1}P_{1})$-$D_{s}(2^{3}P_{1})$ mixing. The
$D_{sJ}(2710)$ is likely to be the low-mass mixed state $|(SD)\rangle_{L}$ via
the $1^{3}D_{1}$-$2^{3}S_{1}$ mixing. In our model, the $D_{sJ}(2860)$ cannot
be assigned to any single state with a narrow width and compatible partial
widths to $DK$ and $D^{*}K$. Thus, we investigate a two-state scenario as
proposed in the literature. In our model, one resonance is likely to be the
$1^{3}D_{3}$ ($J^{P}=3^{-}$), which mainly decays into $DK$. The other
resonance seems to be the $|1{D_{2}}^{\prime}\rangle_{H}$, i.e. the high-mass
state in the $1^{1}D_{2}$-$1^{3}D_{2}$ mixing with $J^{P}=2^{-}$, of which the
$D^{*}K$ channel is its key decay mode. We also discuss implications arising
from these assignments and give predictions for their partner states such as
$|(SD)^{\prime}\rangle_{H}$, $|2{P^{\prime}_{1}}\rangle_{H}$, $2^{3}P_{0}$ and
$2^{3}P_{2}$, which could be helpful for the search for these new states in
future experiment.
###### pacs:
12.39.Fe, 12.39.Jh, 13.25.Ft, 13.25.Hw
## I Introduction
Experimental progress on the study of $D$ and $D_{s}$ states in the past few
years provides a great opportunity for theory development. Recently a new
broad resonance $D_{sJ}(3040)$ with a mass of $(3044\pm
8_{\mathrm{stat}}(^{+30}_{-5})_{\mathrm{syst}})$ MeV and a width of
$\Gamma=(239\pm 35_{\mathrm{stat}}(^{+46}_{-42})_{\mathrm{syst}})$ MeV is
reported in the $D^{*}K$ channel Aubert:2009 . Apart from the $D_{sJ}(3040)$
another two states $D_{sJ}(2710)$ and $D_{sJ}(2860)$, which were observed by
BABAR and Belle two years ago Aubert:2006mh ; jb:2007aa , are also examined.
Their branching ratio fractions between $D^{*}K$ and $DK$ are measured
Aubert:2009 ,
$\displaystyle\frac{D_{sJ}(2710)^{+}\rightarrow
D^{*}K}{D_{sJ}(2710)^{+}\rightarrow DK}=0.91\pm 0.13_{\mathrm{stat}}\pm
0.12_{\mathrm{syst}},$ (1) $\displaystyle\frac{D_{sJ}(2860)^{+}\rightarrow
D^{*}K}{D_{sJ}(2860)^{+}\rightarrow DK}=1.10\pm 0.15_{\mathrm{stat}}\pm
0.19_{\mathrm{syst}}.$ (2)
These new observations stimulate great interest in the understanding of their
nature and strong coupling properties in theory. Different theoretical
approaches for the study of the strong coupling properties of heavy-light
mesons can be found in the literature, such as the heavy quark effective field
theory approach (HQEFT) Nowak:1992um ; Deandrea:1999pa ; Deandrea:1998uz ;
Dai:1997df ; Eichten:1993ub ; Dai:1998ve ; Hiorth:2002pp ; Fajfer:2006hi ;
Fajfer:2008zz ; Kamenik:2008zza ; Bardeen:1993ae ; Bardeen:2003kt ;
Colangelo:2004vu ; Colangelo:2007ds ; Fazio , QCD sum rules Colangelo:1995ph ;
Dai:1997df ; Wang:2008tm , ${}^{3}P_{0}$ model Close:2006gr ; Close:2005se ;
Zhang:2006yj ; Wei:2006wa ; Liuxiang , and chiral quark model Pierro .
In this work, we present an analysis of these $D_{s}$ states in a constituent
quark model with effective Lagrangians for the quark-meson couplings, and try
to clarify the following issues: (i) To gain information about the structure
of the newly observed state $D_{sJ}(3040)$ according to its strong decay
properties. (ii) With the new data for the $D_{sJ}(2710)$ and $D_{sJ}(2860)$,
we reanalyze their strong decays and examine their structures again. The
quantum numbers of these two states remain controversial. The $D_{sJ}(2710)$
is identified as a state of $J^{P}=1^{-}$ in $B$ decays jb:2007aa , while it
is explained by various models as the $2^{3}S_{1}$, $1^{3}D_{1}$, the
admixtures of $2^{3}S_{1}$-$1^{3}D_{1}$, molecular structure, or tetraquark
state Zhang:2006yj ; Wei:2006wa ; Close:2006gr ; Colangelo:2007ds ;
Wang:2007nfa . There are also a lot of solutions proposed for the
$D_{sJ}(2860)$. The assignments of $1^{3}D_{3}$ or $2^{3}P_{0}$ have been
discussed in Refs. Zhang:2006yj ; Close:2006gr ; vanBeveren:2006st ;
Wei:2006wa ; Colangelo:2006rq ; Koponen:2007nr ; Zhong:2008kd . A recent
comment by Ref. eef:2009 suggests a two-state structure for the
$D_{sJ}(2860)$ in order to understand the controversial aspects arising from
its strong decays. (iii) The quark-model assignment of these states will
result in implications of their partner multiplets. We discuss some of those
relevant states, for which the experimental observations would be able to
clarify some of those theoretical and experimental issues.
By treating the light mesons (pseudoscalar and vector mesons) as effective
fields, we introduce constituent-quark-meson couplings to describe the charmed
meson strong decays into a charmed meson plus a light pseudoscalar or vector
meson in the final state. The quark-pseudoscalar-meson coupling is given by
the chiral quark model at the leading order as proposed by Manohar and Georgi
Manohar:1983md . Its application to pseudoscalar meson photoproduction in the
quark model turns out to be promising and many low-energy phenomena can be
highlighted in such a framework Li:1994cy ; Li:1997gda ; qk3 ; Zhong:2007fx ;
Zhong:2009 . In particular, the axial current conservation allows one to
extract the axial coupling in terms of the meson decay constant and a form
factor arising from the microscopic quark model wavefunctions Riska:2000gd .
With an effective quark-vector-meson coupling, one can also extract the vector
couplings in a similar way zhao:1998fn ; zhao-k1 ; zhao-kstar ; Riska:2000gd .
A natural extension of this picture is to apply this effective Lagrangian
approach to heavy-light meson strong decays involving light pseudoscalar or
vector mesons, which would be a good place to examine the validity of the
light axial and vector fields in such a transition. On the one hand, the
quark-meson coupling is the same as that defined in meson photoproduction
which is proportional to the meson decay constant. On the other hand, the
heavy-light meson in the initial and final state would provide information
about the coupling form factor and can be calculated in the quark model
framework. Thus, one can study the heavy-light meson strong decays by
combining dynamical information from meson photoproduction off nucleons.
The paper is organized as follows. In Sec. II, a brief review of the quark-
meson effective Lagrangian approach is given. The numerical results are
presented and discussed in Sec. III. Finally, a summary is given in Sec. IV.
## II framework
Figure 1: Diagrams (A) and (B) stand for the quark-meson couplings in meson-
baryon interactions and light-meson production in heavy-light meson strong
decays, respectively.
In Fig. 1, we illustrate the similarity of the quark-meson coupling in meson-
baryon and light-meson production in heavy-light meson strong decays. It
should be pointed out that since the flavor symmetries beyond the SU(3) are
badly broken, the contributions from transitions of treating the final-state
heavy-light meson as an effective field are strongly suppressed. We thus can
neglect those contributions safely in our approach. An early study of the
charmed meson strong decays can be found in Ref. Zhong:2008kd .
In the chiral quark model Manohar:1983md , the low energy quark-pseudoscalar-
meson interactions in the SU(3) flavor basis are described by the effective
Lagrangian Li:1997gda ; qk3 ; qk4 ; Zhong:2007fx ; Zhong:2009
${\cal
L}_{Pqq}=\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma^{j}_{\mu}\gamma^{j}_{5}\psi_{j}\partial^{\mu}\phi_{m}.$
(3)
where $\psi_{j}$ represents the $j$-th quark field in the hadron, and
$\phi_{m}$ is the pseudoscalar meson field.
The effective Lagrangian for quark-vector-meson interactions in the SU(3)
flavor basis is zhao:1998fn ; zhao-k1 ; zhao-kstar
${\cal
L}_{Vqq}=\sum_{j}\bar{\psi}_{j}(a\gamma^{j}_{\mu}+\frac{ib}{2m_{j}}\sigma_{\mu\nu}q^{\nu})V^{\mu}\psi_{j}\
,$ (4)
where $V^{\mu}$ represents the vector meson field with four-vector moment $q$.
Parameters $a$ and $b$ denote the vector and tensor coupling strength,
respectively.
As follows, we provide the quark-pseudoscalar and quark-vector-meson coupling
operators in a non-relativistic form Li:1997gda ; qk3 ; qk4 ; Zhong:2007fx ;
Zhong:2009 ; zhao:1998fn ; zhao-k1 ; zhao-kstar . Considering light meson
emission in a heavy-light meson strong decays, the effective quark-
pseudoscalar-meson coupling operator in the center-of-mass (c.m.) system of
the initial meson is
$\displaystyle
H_{m}=\sum_{j}\left[-\left(1+\frac{\omega_{m}}{E_{f}+M_{f}}\right)\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{q}+\frac{\omega_{m}}{2\mu_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot\textbf{p}_{j}\right]I_{j}\varphi_{m}\
.$ (5)
In a case that a light vector meson is emitted, the transition operators for
producing a transversely or longitudinally polarized vector meson are as
follows
$\displaystyle
H_{m}^{T}=\sum_{j}\left\\{i\frac{b^{\prime}}{2m_{q}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot(\mathbf{q}\times\mathbf{\epsilon})+\frac{a}{2\mu_{q}}\mathbf{p}_{j}\cdot\mathbf{\epsilon}\right\\}I_{j}\varphi_{m},$
(6)
and
$\displaystyle H_{m}^{L}=\sum_{j}\frac{aM_{v}}{|\mathbf{q}|}I_{j}\varphi_{m}\
.$ (7)
In the above three equations, q and $\omega_{m}$ are the three-vector momentum
and energy of the final-state light meson, respectively. $\textbf{p}_{j}$ is
the internal momentum operator of the $j$-th quark in the heavy-light meson
rest frame. $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ is the spin operator on
the $j$-th quark of the heavy-light system and $\mu_{q}$ is a reduced mass
given by $1/\mu_{q}=1/m_{j}+1/m^{\prime}_{j}$ with $m_{j}$ and
$m^{\prime}_{j}$ for the masses of the $j$-th quark in the initial and final
mesons, respectively. Here, the $j$-th quark is referred to the active quark
involved at the quark-meson coupling vertex. $M_{v}$ is the mass of the
emitted vector meson. The plane wave part of the emitted light meson is
$\varphi_{m}=e^{-i\textbf{q}\cdot\textbf{r}_{j}}$, and $I_{j}$ is the flavor
operator defined for the transitions in the SU(3) flavor space Zhong:2007gp ;
Li:1997gda ; qk3 ; qk4 ; Zhong:2009 ; Zhong:2007fx ; Zhong:2008kd ;
zhao:1998fn . Parameters $a$ and $b$ are the vector and tensor coupling
strengths of the quark-vector-meson couplings, respectively. Studies of vector
meson photoproduction zhao-k1 ; zhao-kstar ; JLab-Kstar ; elsa-kstar suggest
that $a=g_{\omega qq}=g_{\rho qq}\simeq-3$ and $b^{\prime}\equiv b-a\simeq 5$.
Because of vector current conservation, one has $a=g_{\rho NN}=g_{\omega
NN}/3$ Riska:2000gd ; zhao:1998fn ; zhao-kstar .
The heavy-light meson wavefunctions have been given in Ref. Zhong:2008kd , and
some of the decay amplitudes have also been deduced there. In the charmed
meson decays, the SU(4) flavor symmetry is broken. Thus, the charm quark is
treated as a spectator and the transition amplitude is proportional to the
final-state light meson decay constant associated with a form factor arising
from the convolution of the initial and final-state charmed meson
wavefunctions.
In the calculation, the standard quark model parameters are adopted. Namely,
we set $m_{u}=m_{d}=330$ MeV, $m_{s}=450$ MeV, and $m_{c}=1700$ MeV for the
constituent quark masses. The harmonic oscillator parameter $\beta$ is usually
adopted in the range of (0.4–0.5) GeV, in this work we take it as $\beta=0.45$
GeV. The decay constants for $K$ and $\eta$ mesons are $f_{K}=f_{\eta}=160$
MeV. As shown in Refs. Zhong:2007gp ; Zhong:2008kd , the flavor symmetry
breaking will lead to corrections to the quark-pseudoscalar-meson coupling
vertex, for which an additional global parameter $\delta$ is introduced. Here,
we fix its value the same as that in Refs. Zhong:2008kd ; Zhong:2007gp , i.e.
$\delta=0.557$. For the quark-vector-meson coupling strength which still
suffers relatively large uncertainties, we adopt the values extracted from
vector meson photoproduction as mentioned earlier, i.e. $a\simeq-3$ and
$b^{\prime}\simeq 5$. The masses of the mesons used in the calculations are
adopted from the PDG PDG .
Justification of the non-relativistic formulation is not obvious for the light
quark sector in the heavy-light meson transitions. This is similar to the case
of a non-relativistic quark model for baryons, where the results would rely on
the experimental data to tell how far they deviate from reality. Treating the
light meson as a chiral field somehow assumes that the light meson is produced
at short distance, and the spectators (i.e. the two spectator quarks inside a
baryon or the heavy quark in the heavy-light meson transitions) do not respond
to the internal structure of the light meson. Instead, the propagation of the
light quark pair would feel the hadronic environment from the convolution of
initial and final-state heavy-light mesons. Such an implicated assumption
means that only the processes with relatively small momentum transfers between
the light quarks inside the light meson would dominantly contribute to the
transition matrix element. This empirically supports the validity of the non-
relativistic formulation as a leading order approximation.
## III results and discussions
### III.1 $D_{sJ}(3040)$
The $D_{sJ}(3040)$ is observed in the $D^{*}K$ mode, while there is no sign of
$D_{sJ}(3040)\rightarrow DK$ in experiment Aubert:2009 . This allows its
quantum number to be $J^{P}=0^{-}$, $1^{+}$, $2^{-}$ etc. The $J^{P}=0^{-}$
state $2^{1}S_{0}$ seems not a good candidate since its predicted mass, $\sim
2.7$ GeV Pierro ; Godfrey:1985xj ; Close:2006gr ; Ebert:2009ua , is much less
than $3.04$ GeV. The predicted masses of $J^{P}=1^{+}$ and $2^{-}$ are close
to $3.04$ GeV. We hence discuss these two possibilities for the $D_{sJ}(3040)$
in this work.
First, we considered it as the $J^{P}=1^{+}$ states $2^{1}P_{1}$ and
$2^{3}P_{1}$. These two states also can decay into $D^{*}K$, $DK^{*}$,
$D^{*}_{s}\eta$, $D_{s}\phi$, $D_{0}(2400)K$, $D_{1}(2430)K$, $D_{1}(2420)K$,
$D_{2}(2460)K$, $D_{s}(2317)\eta$, $D_{s}(2460)\eta$. With a mass of 3.04 GeV,
we calculate their decay widths, which are listed in Tab. 1. From the table,
it is found that the decay width of $2^{1}P_{1}$ and $2^{3}P_{1}$ are $\sim
115$ MeV and $\sim 93$ MeV, respectively, which are too small to compare with
the data, although the decay mode, dominated by the $D^{*}K$, is consistent
with the observation Aubert:2009 . Thus, the $D_{sJ}(3040)$ may not be
considered as pure $2^{1}P_{1}$ or $2^{3}P_{1}$ state.
Figure 2: (Color online) The partial decay widths and total width of
$|2P_{1}\rangle_{L}$ with a mass of 3040 MeV as functions of mixing angle
$\phi$. The data are from BABAR Aubert:2009 .
Since the heavy-light mesons are not charge conjugation eigenstates, state
mixing between spin $\mathbf{S}=0$ and $\mathbf{S}=1$ states with the same
$J^{P}$ can occur via the spin-orbit interactions Godfrey:1986wj ;
Close:2005se ; Swanson . The physical states with $J^{P}=1^{-}$ can then be
described as
$\displaystyle|2P_{1}\rangle_{L}=+\cos(\phi)|2^{1}P_{1}\rangle+\sin(\phi)|2^{3}P_{1}\rangle,$
(8)
$\displaystyle|2{P_{1}}^{\prime}\rangle_{H}=-\sin(\phi)|2^{1}P_{1}\rangle+\cos(\phi)|2^{3}P_{1}\rangle\
,$ (9)
where the subscripts $L$ and $H$ stand for the low mass and high mass of the
physical states after the mixing.
Usually, the low mass state has a broad width while the high mass state has a
narrow width. We set the mass of $|2P_{1}\rangle_{L}$ with $3.04$ GeV, and
plot its decay width as a function of the mixing angle $\phi$, which is shown
in Fig. 2. It shows that when the mixing angle is in the range
$\phi\simeq-(40\pm 12)^{\circ}$, the total decay width, $\Gamma=(162\sim 170)$
MeV, is in the range of the experimental data (close to the lower limit of the
data) Aubert:2009 . The mixing angle predicted here is consistent with the
result $\phi\simeq-55^{\circ}$ in the heavy quark limit Godfrey:1986wj ;
Close:2005se ; Swanson . The $D^{*}K$ governs the decays of
$|2P_{1}\rangle_{L}$, while the $DK$ channel is forbidden. This is also in
agreement with the observations. These results suggest that the $D_{sJ}(3040)$
favors the $|2P_{1}\rangle_{L}$ classification.
Apart from the $D^{*}K$ mode, the $D_{1}(2430)K$, $D_{2}(2460)K$,
$D_{0}(2400)K$, $DK^{*}$, and $D_{s}^{*}\eta$ are also important in the decays
of $|2P_{1}\rangle_{L}$ as shown by Fig. 2. In particular, the partial widths
of $D_{1}(2420)K$, $D_{s}(2317)\eta$ and $D^{*}K^{*}$ turn out to be sizable.
A search for those channels would be useful for clarifying the property of the
$D_{sJ}(3040)$. With the mixing angle $\phi\simeq-55^{\circ}$, the relative
decay ratios among those decay channels are
$D^{*}K:D_{1}(2430)K:D_{2}(2460)K:D_{0}(2400)K:D_{1}(2420)K:DK^{*}:D_{s}(2317)\eta:D_{s}^{*}\eta:D^{*}K^{*}=78:17:19:13:4:8:4:11:2$.
Figure 3: (Color online) The partial decay widths and total width of
$|2P_{1}\rangle_{L}$ as functions of mass. The data are from BABAR Aubert:2009
.
Since the mass of $D_{s}(3040)$ still has a large uncertainty, it may bring
uncertainties to the theoretical predictions on the decay widths. To
investigate this effect, we plot the decay widths as a function of the mass in
Fig. 3 with the mixing angle fixed at $\phi=-50^{\circ}$. It shows that the
mass uncertainty gives rise to an uncertainty of about $\sim 70$ MeV in the
total decay width. The predicted widths are much closer to the central value
of the data with the increasing mass. The sensitivity of different decay modes
to the mass can also be seen clearly in the plot.
Table 1: The decay widths (MeV) for the $D_{sJ}(3040)$ as $1^{1}D_{2}$, $1^{3}D_{2}$, $2^{1}P_{1}$ and $2^{3}P_{1}$ candidates. | $D^{*}K$ | $DK^{*}$ | $D^{*}K^{*}$ | $D^{*}_{s}\eta$ | D(2430)K | D(2420)K | $D_{s}(2460)\eta$ | $D_{s}\phi$ | D(2400)K | D(2460)K | $D_{s}(2317)\eta$ | total
---|---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}D_{2}$ | 197 | 27 | 2 | 25 | 3 | 2 | 0.4 | 4 | 3 | 345 | 4 | 608
$1^{3}D_{2}$ | 256 | 21 | 33 | 34 | 1 | 18 | 0.01 | 0.05 | 3.4 | 512 | 1.6 | 879
$2^{1}P_{1}$ | 44 | 9 | 0.3 | 5.5 | 0.02 | 0.01 | $7.5\times 10^{-5}$ | 0.1 | 33 | 12 | 11 | 115
$2^{3}P_{1}$ | 41 | 2 | 2.5 | 7.5 | 24 | 7 | 0.5 | 0.002 | 0.09 | 9 | 0.06 | 93
Finally, we discuss the possibilities of $D_{sJ}(3040)$ as a $J^{P}=2^{-}$
candidate. There are two states, $1^{1}D_{2}$ and $1^{3}D_{2}$, with
$J^{P}=2^{-}$. If $1^{1}D_{2}$ and $1^{3}D_{2}$ have a mass of 3.04 GeV, they
can decay into the following channels, $D^{*}K$, $DK^{*}$, $D^{*}K^{*}$,
$D^{*}_{s}\eta$, $D_{s}\phi$, $D_{0}(2400)K$, $D_{1}(2430)K$, $D_{1}(2420)K$,
$D_{2}(2460)K$, $D_{s}(2317)\eta$, and $D_{s}(2460)\eta$. We calculate these
partial decay widths and list the results in Tab. 1. It shows that $D^{*}K$
and $D_{2}(2460)K$ are the two main decay channels. The total widths for both
$1^{1}D_{2}$ and $1^{3}D_{2}$ are very broad, i.e. $\Gamma\sim 608$ MeV and
$\sim 879$ MeV, respectively. They are too large to compare with the data
$\Gamma=(239\pm 35)$ MeV Aubert:2009 . Nevertheless, it shows that the
admixtures between $1^{1}D_{2}$ and $1^{3}D_{2}$ are unable to give a
reasonable explanation of the decay properties of $D_{sJ}(3040)$ as well.
Thus, the $D_{sJ}(3040)$ as a $J^{P}=2^{-}$ candidate is not favored.
In brief, the $D_{sJ}(3040)$ seems to favor a $|2P_{1}\rangle_{L}$ state with
$J^{P}=1^{+}$, which is an admixture of $2^{1}P_{1}$ and $2^{3}P_{1}$ with a
mixing angle $\phi\simeq-(40\pm 12)^{\circ}$. Our conclusion is in agreement
with that of a ${}^{3}P_{0}$ model analysis Liuxiang . The semiclassical flux
tube model Ailin and relativistic quark model Ebert:2009ua mass calculations
also support this picture.
### III.2 $D_{sJ}(2710)$
The $D_{sJ}(2710)$ was first reported by BABAR Aubert:2006mh , and its quantum
number $J^{P}=1^{-}$ was determined by Belle jb:2007aa . Recently, the decay
ratios of the $D_{sJ}(2710)$ have also been reported Aubert:2009 , which is
very useful for understanding its nature. According to the classification of
the quark model, only two states $2^{3}S_{1}$ and $1^{3}D_{1}$ with the
quantum number $J^{P}=1^{-}$ are located around the mass range $(2.7\sim 2.8)$
GeV. This state is studied by various models, e.g. as a $2^{3}S_{1}$ state
Ebert:2009ua ; Fazio , $1^{3}D_{1}$ state Wei:2006wa , or admixture of
$2^{3}S_{1}$-$1^{3}D_{1}$ Close:2006gr . It should mention that in our
previous work Zhong:2008kd an error occurred in the partial decay amplitude
of $1^{3}D_{1}\rightarrow DK$, which led to a rather small width for the
assignment of the admixture of $2^{3}S_{1}$-$1^{3}D_{1}$. Here we correct the
formulation and reanalyze the mixing scenario for the $D_{sJ}(2710)$.
Table 2: The decay widths (MeV) for the $D_{sJ}(2710)$ as $1^{3}D_{1}$ and $2^{3}S_{1}$ candidates. | $D^{0}K^{+}$ | $D^{+}K^{0}$ | $D^{*+}K^{0}$ | $D^{*0}K^{+}$ | $D_{s}\eta$ | $D^{*}_{s}\eta$ | total | $\Gamma(D^{*}K)/\Gamma(DK)$
---|---|---|---|---|---|---|---|---
$1^{3}D_{1}$ | 75 | 73.6 | 17.8 | 18.5 | 14 | 0.9 | 200 | 0.24
$2^{3}S_{1}$ | 5.4 | 5.6 | 9.0 | 9.1 | 1.7 | 0.7 | 31 | 1.65
We first assign the $D_{sJ}(2710)$ as the $2^{3}S_{1}$ and $1^{3}D_{1}$ states
and calculate its decay widths. The results are listed in Tab. 2,
respectively. For the assignment of the $2^{3}S_{1}$ state, the total decay
width and the decay branching ratio fraction between $D^{*}K$ and $DK$
channels are
$\displaystyle\Gamma\simeq 31~{}\mathrm{MeV},\ \
\frac{\Gamma(D^{*}K)}{\Gamma(DK)}\simeq 1.65.$ (10)
It shows that the predicted width $\Gamma\simeq 31$ MeV is too narrow to
compare with the data, and the predicted decay ratio $D^{*}K/DK\simeq 1.67$ is
much larger than the measurement $D^{*}K/DK\simeq 0.91\pm 0.13\pm 0.12$
Aubert:2009 . The calculations of Ref. Zhang:2006yj also tend to give a small
width $\Gamma\simeq 32$ MeV for the $2^{3}S_{1}$ configuration. The predicted
branching ratio fraction is also inconsistent with the observations
Aubert:2009 . In Ref. Close:2006gr , it is also found that the large branching
ratio fraction $D^{*}K/DK\simeq 3.55$ does not support the $D_{sJ}(2710)$ as a
pure $2^{3}S_{1}$ state.
On the other hand, if the $D_{sJ}(2710)$ is considered as a $1^{3}D_{1}$
state, the decay width and branching ratio fraction will be
$\displaystyle\Gamma\simeq 200~{}\mathrm{MeV},\ \
\frac{\Gamma(D^{*}K)}{\Gamma(DK)}\simeq 0.24.$ (11)
In this case, the branching ratio fraction $\Gamma(D^{*}K)/\Gamma(DK)\simeq
0.24$ is too small though the decay width $\Gamma\simeq 200$ MeV is roughly
consistent with the upper limit of the data jb:2007aa ; Aubert:2009 . These
results suggest that either $1^{3}D_{1}$ or $2^{3}S_{1}$ is not a good
assignment for the $D_{sJ}(2710)$.
Thus, we consider the possibilities of the $D_{sJ}(2710)$ as a mixed state of
$2^{3}S_{1}$-$1^{3}D_{1}$, for which the physical states can be expressed as
Close:2006gr
$\displaystyle|(SD)_{1}\rangle_{L}=+\cos(\phi)|2^{3}S_{1}\rangle+\sin(\phi)|1^{3}D_{1}\rangle,$
(12)
$\displaystyle|(SD)^{\prime}_{1}\rangle_{H}=-\sin(\phi)|2^{3}S_{1}\rangle+\cos(\phi)|1^{3}D_{1}\rangle\
,$ (13)
where the physical partner in the mixing is included. Assuming that the low
mass state $|(SD)_{1}\rangle_{L}$ corresponds to the $D_{sJ}(2710)$
Close:2006gr , we plot the decay properties of $|(SD)_{1}\rangle_{L}$ as
functions of the mixing angle $\phi$ in Fig. 4. It shows that with the mixing
angle $\phi\simeq(-54\pm 7)^{\circ}$, the decay width and branching ratio
fraction are
$\displaystyle\Gamma\simeq(133\pm 22)~{}\mathrm{MeV},\ \
\frac{\Gamma(D^{*}K)}{\Gamma(DK)}\simeq 0.91\mp 0.25\ ,$ (14)
which are in a good agreement with the data jb:2007aa ; Aubert:2009 .
Following this scheme, one can examine the high-mass partner
$|(SD)^{\prime}_{1}\rangle_{H}$, of which the expected mass is $\sim 2.81$ GeV
Close:2006gr . Taking into account the mass uncertainties of a region
$M\simeq(2.71\sim 2.88)$ GeV, we plot the mass-dependence of the partial and
total widths in Fig. 5. It shows that the $|(SD)^{\prime}_{1}\rangle_{H}$ also
has a broad width $\sim(120\pm 10)$ MeV, and the $DK$ channel is dominant over
others. In contrast, the partial width of $D_{s}\eta$ is also sizable, while
the $D^{*}_{s}\eta$ width is negligible. Around $M=2.81$ GeV, the predicted
branching ratio fractions are
$\displaystyle\frac{\Gamma(D_{s}\eta)}{\Gamma(DK)}\simeq
0.15,~{}\frac{\Gamma(D^{*}K)}{\Gamma(D_{s}\eta)}\simeq 0.06.$ (15)
The above mixing scheme is consistent with Ref. Close:2006gr for the low-mass
state while the predicted suppression of the $D^{*}K$ decay mode is different
from that of Ref. Close:2006gr . In Ref. Close:2006gr a very broad high-mass
state is predicted and would dominantly decay into both $DK$ and $D^{*}K$. In
our scheme, the predicted decay width for $|(SD)^{\prime}_{1}\rangle_{H}$ is
$\sim(120\pm 10)$ MeV. As a consequence, one would expect that it should
appear in the $DK$ spectrum similar to the $D_{s}(2710)$ signal. Taking into
account the still undetermined mass for $|(SD)^{\prime}_{1}\rangle_{H}$, one
possible explanation would be that the $|(SD)^{\prime}_{1}\rangle_{H}$ mass
may be larger than $M\simeq 2.88$ GeV. If so, its total width would be larger
than we estimated above and become much broader, thus, cannot be easily
identified in the present $DK$ spectrum. Interestingly, a recent study of the
$D_{s}$ spectrum suggests a larger mass for the $1^{3}D_{1}$ state
Ebert:2009ua .
It should be noted that different methods seem to lead to different
conclusions on the $D_{sJ}(2710)$ state. In Refs. Colangelo:2007ds ; Fazio ,
both the decay width and branching ratio fraction of the $D_{sJ}(2710)$ as the
$2^{3}S_{1}$ assignment can be well explained. In Ref. Ebert:2009ua , the mass
calculation also suggests that the $D_{sJ}(2710)$ is $2^{3}S_{1}$. However,
the recent study of a ${}^{3}P_{0}$ model tends to conclude that the
$D_{sJ}(2710)$ is a mixture of $2^{3}S_{1}$ and $1^{3}D_{1}$ Li:2009qu .
Therefore, additional information for $D_{sJ}(2710)\to D_{s}\eta$ and
$D_{s}^{*}\eta$, as well as a search for the $|(SD)^{\prime}_{1}\rangle_{H}$
partner in experiment would be useful for understanding the property of the
$D_{sJ}(2710)$.
Figure 4: (Color online) The partial decay widths, total width, and the decay
branching ratio fraction $\Gamma(D^{*}K)/\Gamma(D^{*}K)$ of
$|(SD)_{1}\rangle_{L}$ as functions of mass, respectively. The data are from
BABAR Aubert:2009 . Figure 5: (Color online) The partial decay widths and
total width of $|(SD)^{\prime}_{1}\rangle_{H}$ as functions of mass.
### III.3 $D_{sJ}(2860)$
The situation about the $D_{sJ}(2860)$ is still controversial and different
solutions have been proposed in the literature. In Ref. vanBeveren:2006st ,
the $D_{sJ}(2860)$ is assigned as a $J^{P}=0^{+}$ state. However, the recent
observation of $D_{sJ}(2860)\rightarrow D^{*}K$ does not support this picture.
It is also proposed to be a $J^{P}=3^{-}$ state Colangelo:2006rq ;
Zhang:2006yj ; Zhong:2008kd . However, although the decay width and decay mode
are consistent with the observation, the predicted ratio $D^{*}K/DK\simeq 0.4$
is too small to compare with the data $D^{*}K/DK\simeq 1.1$ Aubert:2009 .
Table 3: The decay widths (MeV) for the $D_{sJ}(2860)$ as $1^{3}D_{3}$, $2^{3}P_{2}$, and $1^{3}F_{2}$ candidates. | $D^{0}K^{+}$ | $D^{+}K^{0}$ | $D^{*+}K^{0}$ | $D^{*0}K^{+}$ | $D_{s}\eta$ | $D^{*}_{s}\eta$ | $DK^{*}$ | total | $\Gamma(D^{*}K)/\Gamma(DK)$
---|---|---|---|---|---|---|---|---|---
$1^{3}D_{3}$ | 12.3 | 11.8 | 5 | 4.7 | 1.7 | 0.3 | 0.2 | 36 | 0.40
$2^{3}P_{2}$ | 1.3 | 1.3 | 2.1 | 1.9 | 0.01 | 1.7 | 0.02 | 8 | 1.53
$1^{3}F_{2}$ | 21.9 | 21.4 | 0.1 | 0.1 | 5.5 | 0.02 | 0.005 | 49 | 0.005
Since the $D_{sJ}(2860)$ is observed in both $D^{*}K$ and $DK$ channels, the
allowed quantum numbers would be $1^{3}D_{3}$, $2^{3}P_{2}$ and $1^{3}F_{2}$.
We calculate the total and partial widths for these configurations and list
the results in Tab. 3.
More specifically, as the $1^{3}D_{3}$ state, the predicted width and
branching ratio fraction between the $D^{*}K$ and $DK$ channel are
$\displaystyle\Gamma\simeq
36~{}\mathrm{MeV},~{}\frac{\Gamma(D^{*}K)}{\Gamma(DK)}\simeq 0.4.$ (16)
The predicted ratio $\Gamma(D^{*}K)/\Gamma(DK)$ differs from the measurement
$D^{*}K/DK\simeq 1.1$ Aubert:2009 at the level of three standard deviations,
although the decay width is in agreement with the data. Our predictions are
consistent with those of Refs. Colangelo:2006rq ; Fazio . It should be
mentioned that the QCD-motivated relativistic quark model can not well explain
the mass of $D_{sJ}(2860)$ if it is considered as the $1^{3}D_{3}$ state
Ebert:2009ua . This could be a signal indicating the chiral symmetry in
association with the heavy quark symmetry in the heavy-light meson
transitions.
As a candidate of the $2^{3}P_{2}$ state, the decay width and branching ratio
fraction of $D_{sJ}(2860)$ are
$\displaystyle\Gamma\simeq
8~{}\mathrm{MeV},~{}\frac{\Gamma(D^{*}K)}{\Gamma(DK)}\simeq 1.53,$ (17)
where both the predicted width and ratio are inconsistent with the data. It is
interesting to mention that our predicted ratio agrees with the estimation of
Ref. eef:2009 .
If the $D_{sJ}(2860)$ is a $1^{3}F_{2}$ state, the predicted width and
branching ratio fraction are
$\displaystyle\Gamma\simeq
49~{}\mathrm{MeV},~{}\frac{\Gamma(D^{*}K)}{\Gamma(DK)}\simeq 0.005\ ,$ (18)
where the decay mode of $D^{*}K$ turns out to be negligible in comparison with
the $DK$ mode, and disagrees with the experimental observation.
Figure 6: (Color online) The partial decay widths and total width of
$2^{3}P_{0}$ as functions of mass. Figure 7: (Color online) The partial decay
widths and total width of $|1D_{2}\rangle_{H}$ with a mass of 2860 MeV as
functions of mixing angle $\phi$. The data are from BABAR Aubert:2009 ;
Aubert:2006mh .
It can be seen from the above analysis that a simple assignment of the
$D_{sJ}(2860)$ to be a pure $2^{3}P_{0}$, $1^{3}D_{3}$, $2^{3}P_{2}$ or
$1^{3}F_{2}$ cannot well explain the data. We also point out that the
$2^{3}P_{2}$ and $1^{3}F_{2}$ mixing is unable to overcome the problem either
because of the narrow width of the $2^{3}P_{2}$ state or small branching ratio
fraction $\Gamma(D^{*}K)/\Gamma(DK)\simeq 0.005$ of $1^{3}F_{2}$.
In Ref. eef:2009 , van Beveren and Rupp recently proposed an alternative
solution that there might exist two largely overlapping resonances at about
2.86 GeV, i.e. a radially excited tensor ($2^{+}$) and a scalar ($0^{+}$)
$c\bar{s}$ state. Following this two-state assumption, one would expect that
one state $D_{sJ_{1}}(2860)$ dominantly decays into $DK$, while the other one
$D_{sJ_{2}}(2860)$ dominantly decays into $D^{*}K$. Both states have a mass
around 2.86 GeV, and comparable width $\Gamma\sim 50$ MeV. This idea may shed
some light on the controversial issues. As follows, we shall investigate such
a possibility in our approach.
It shows that the decays of $2^{3}P_{0}$, $1^{3}D_{3}$ and $1^{3}F_{2}$ is
dominated by the $DK$ channel, while the decay of $1^{3}D_{2}$, $1^{1}D_{2}$,
$2^{3}P_{2}$ is dominated by the $D^{*}K$ channel. We shall identify which
states are more appropriate candidates in the two-state scenario.
First, we analyze the states dominated by $DK$ decays, i.e. $2^{3}P_{0}$,
$1^{3}D_{3}$ and $1^{3}F_{2}$. In Fig. 6 the total and partial decay widths
for the $2^{3}P_{0}$ state are revealed. It shows that the $2^{3}P_{0}$
possesses a broad decay width $\Gamma\simeq 115$ MeV at about 2.86 GeV, which
is inconsistent with the data. The $1^{3}F_{2}$ is not considered as a good
candidate of $D_{sJ_{1}}(2860)$ as well since its mass is expected to be
larger than 3.1 GeV Pierro ; Ebert:2009ua . Furthermore, our earlier analysis
suggests that the $D_{sJ}(3040)$ may favor a configuration of
$|2P_{1}\rangle_{L}$ such that the mass of the $1^{3}F_{2}$ should be larger
than the $P$ wave state $|2P_{1}\rangle_{L}$ as a consequence. In contrast, we
find that the $1^{3}D_{3}$ could be a good candidate for $D_{sJ_{1}}(2860)$
since it is dominated by the $DK$ decay mode and has a narrow width
$\Gamma\simeq 36$ MeV. The calculation results for the total and partial decay
widths have been listed in Tab. 3.
Candidates for the $D_{sJ_{2}}(2860)$ could be $1^{3}D_{2}$, $1^{1}D_{2}$, or
$2^{3}P_{2}$ which dominantly decay into $D^{*}K$. As discussed earlier in
this section and shown in Tab. 3, the $2^{3}P_{2}$ is not a good candidate
since its total width is too small to compare with the data. Nevertheless, its
expected mass should be larger than 2.86 GeV Pierro ; Ebert:2009ua .
If the $D_{sJ_{2}}(2860)$ is considered as pure $1^{3}D_{2}$ or $1^{1}D_{2}$
state, their decay widths would be $\Gamma\simeq 170$ MeV and $\Gamma\simeq
130$ MeV, respectively, which are inconsistent with the data as well. In fact,
the physical states should be the admixtures between $1^{3}D_{2}$ and
$1^{1}D_{2}$ due to the presence of the spin-orbit interactions Godfrey:1986wj
; Close:2005se ; Swanson . Thus, the mixed states can be expressed as
$\displaystyle|1D_{2}\rangle_{L}=+\cos(\phi)|1^{1}D_{2}\rangle+\sin(\phi)|1^{3}D_{2}\rangle,$
(19)
$\displaystyle|1{D_{2}}^{\prime}\rangle_{H}=-\sin(\phi)|1^{1}D_{2}\rangle+\cos(\phi)|1^{3}D_{2}\rangle\
,$ (20)
where the subscripts $L$ and $H$ denote the low-mass and high-mass state due
to the mixing. Usually, the $|1{D_{2}}^{\prime}\rangle_{H}$ have a narrow
width Godfrey:1986wj ; Close:2005se ; Swanson . We thus consider the
$|1{D_{2}}^{\prime}\rangle_{H}$ as the $D_{sJ_{2}}(2860)$ in the calculation.
In Fig. 7 the decay properties as a function of the mixing angle are plotted.
We see that around $\phi=-65^{\circ}$ or $\phi=-35^{\circ}$ the decay width is
$\Gamma\simeq 40$ MeV, which is compatible with the observation, and the decay
mode is dominated by the $D^{*}K$. With $\phi=-35^{\circ}$, the corresponding
decay branching ratio fractions are
$\displaystyle\frac{\Gamma(D^{*}K)}{\Gamma(D^{*}_{s}\eta)}\simeq
1.2,~{}\frac{\Gamma(D^{*}K)}{\Gamma(DK^{*})}\simeq 13\ ,$ (21)
which fit in the experimental data quite well. This result turns out to
support the $|1{D_{2}}^{\prime}\rangle_{H}$ to be a candidates of
$D_{sJ_{2}}(2860)$ in the two-state scenario. In the range of
$\phi=-65^{\circ}\sim-35^{\circ}$ the partial widths do not change drastically
with the mixing angle. In contrast, the suggested value is consistent with
that ($\phi=-50.7^{\circ}$) obtained in the heavy quark effective theory
Godfrey:1986wj ; Close:2005se ; Swanson ; Ebert:2009ua .
In brief, it seems likely that the abnormal property with the $D_{sJ}(2860)$
arises from two overlapping resonances with the same mass but different decay
modes. One is $1^{3}D_{3}$ and the other is $|1{D_{2}}^{\prime}\rangle_{H}$
from the $1^{3}D_{2}$ and $1^{1}D_{2}$ mixing. The $1^{3}D_{3}$ state mainly
decays into $DK$ and the $|1{D_{2}}^{\prime}\rangle_{H}$ into $D^{*}K$. With
these two largely overlapping resonances at about 2.86 GeV, we can understand
both the observed decay widths and branching ratio fractions of the
$D_{sJ}(2860)$. It shows that the $1^{3}D_{3}$ has a sizable partial width in
the $D_{s}\eta$ channel, while the $|1{D_{2}}^{\prime}\rangle_{H}\to
D^{*}_{s}\eta$ also turns out to be measurable. Further measurements of
$\Gamma(D^{*}_{s}\eta)/\Gamma(D^{*}K)$ and $\Gamma(DK)/\Gamma(D_{s}\eta)$ may
be able to distinguish the $1^{3}D_{3}$ and $|1{D_{2}}^{\prime}\rangle_{H}$
and test the two-state scenario in experiment.
Figure 8: (Color online) The partial decay widths and total width of the
$|2P^{\prime}_{1}\rangle_{H}$ state as functions of mass.
### III.4 $D_{sJ}(2|P^{\prime}_{1}\rangle_{H})$, $D_{sJ}(2^{3}P_{0})$ and
$D_{sJ}(2^{3}P_{2})$
In this subsection we discuss the implications of other states following the
consequence of the assignments for the $D_{sJ}(3040)$, $D_{sJ}(2710)$ and
$D_{sJ}(2860)$.
Since the $D_{sJ}(3040)$ seems to favor a $P$ wave with $J^{P}=1^{+}$
($|2P_{1}\rangle_{L}$), experimental evidences for the other $P$ waves,
$D_{sJ}(2|P^{\prime}_{1}\rangle_{H})$, $D_{sJ}(2^{3}P_{0})$ and
$D_{sJ}(2^{3}P_{2})$, would be important to establish the spectrum. In
particular, its high-mass partner $|2P^{\prime}_{1}\rangle_{H}$ should be
searched in experiments. Supposing that the $|2P^{\prime}_{1}\rangle_{H}$ has
a mass in the range of $(3.04\sim 3.2)$ GeV, we plot in Fig. 8 the decay
widths as functions of the mass with the mixing angle $\phi=-50^{\circ}$ fixed
by $D_{sJ}(3040)$. It shows that the $|2P^{\prime}_{1}\rangle_{H}$ width is
indeed relative narrower around $M=3.04$ GeV, although we should note that the
decay width increases fast with the increasing mass. The decay channels,
$D_{0}(2400)K$, $D_{2}(2460)K$ and $D_{s}(2317)\eta$, are predicted to be the
dominant ones, which can be investigated in experiments. In contrast, the
$D^{*}K$ channel plays a less important role in the decays.
We further study the $D_{sJ}(2^{3}P_{0})$ in detail here. The decay widths as
a function of the possible mass range $M=(2.8\sim 2.9)$ GeV are plotted in
Fig. 6. In this range the total decay width is $\Gamma\simeq(90\sim 140)$ MeV,
and increases with the increasing mass. It shows that the $DK$ channel
dominates its decays. Taking the mass of the $D_{sJ}(2^{3}P_{0})$ as
$(2.82\sim 2.84)$ GeV Close:2006gr ; Matsuki , the total width and branching
ratio fractions between $D_{s}\eta$ and $DK$ are
$\displaystyle\Gamma\simeq(101\pm
5)~{}\mathrm{MeV},~{}\frac{\Gamma(D_{s}\eta)}{\Gamma(DK)}\simeq 0.08.$ (22)
It should be pointed out that the decay properties of $2^{3}P_{0}$ are similar
to those of $|(SD)^{\prime}_{1}\rangle_{H}$ in the mass range $M<2.9$ GeV (see
Fig. 5 and Fig. 6). Both of them have comparable decay widths $\Gamma\sim 100$
MeV, and mainly decay into $DK$. To distinguish them from each other, the
measurements of their decay ratio $\Gamma(D_{s}\eta)/\Gamma(DK)$ are
important. We also note that a recent calculation suggests a larger mass of
$M\simeq 3.054$ GeV for $2^{3}P_{0}$ Ebert:2009ua . As a consequence of this
scenario, its total decay width would become much broader than we estimated
above. Thus, it may not be easily isolated in experiment.
Figure 9: (Color online) The partial decay widths and total width of
$2^{3}P_{2}$ as functions of mass.
As discussed earlier the $D_{sJ}(2860)$ does not favor the assignment of
$2^{3}P_{2}$. Thus, we investigate its decay properties and implications of
experimental measurement. We also plot its total and partial decay widths as
functions of the mass in the possible range $M=(3.04\sim 3.2)$ GeV in Fig. 9.
If $2^{3}P_{2}$ has a mass larger than 3.04 GeV, decay channels, $DK$,
$D^{*}K$, $DK^{*}$, $D^{*}K^{*}$, $D^{*}_{s}\eta$, $D_{s}\phi$, $D_{s}\eta$,
$D_{1}(2430)K$, $D_{1}(2420)K$, $D_{2}(2460)K$, $D_{s}(2460)\eta$, will open
in which $D_{1}(2430)K$ and $DK$ channels are dominant. In Fig. 9, we do not
show the results for the $D^{*}_{s}\eta$, $D_{s}\phi$ and $D_{s}(2460)\eta$
channels since they are negligibly small ($<1$ MeV). If we adopt the mass
$\sim 3.15$ GeV as predicted by Refs. Matsuki ; Ebert:2009ua ; Pierro , the
predicted width is $\Gamma\simeq 140$ MeV, and the relative decay strengths
are
$DK:D^{*}K:D_{1}(2430)K:D_{1}(2420)K:D_{2}(2460)K:DK^{*}:D^{*}K^{*}:D_{s}\eta\simeq
41:9:50:13:11:6:7:4$. It suggests that the $DK$, $D_{1}(2430)K$,
$D_{1}(2420)K$ channels may be the optimal ones for searching for the
$D_{sJ}(2^{3}P_{2})$ state in experiment.
### III.5 Sensitivity to the harmonic oscillator parameter
It should be mentioned that model-dependent feature of our model arises from
the simple treatment of harmonic oscillator potential for the heavy-light
quark system. Therefore, uncertainties with the theoretical results are
present in the choice of the quark model parameter values. The most important
parameter in our model should be the harmonic oscillator strength $\beta$,
which controls the size effect or coupling form factor from the convolution of
the heavy-light meson wavefunctions. The commonly adopted range of this
quantity is $\beta=(0.4\sim 0.5)$ GeV, and we apply $\beta=0.45$ GeV in the
above calculations.
In order to examine the sensitivity of the calculation results to $\beta$, we
plot the decay widths and ratios of $2^{3}S_{1}$, $1^{3}D_{3}$, mixed state
$|(SD)\rangle_{L}$ of $2^{3}S_{1}$-$1^{3}D_{1}$, and mixed state
$|2{P_{1}}\rangle_{L}$ of $2^{1}P_{1}$-$2^{3}P_{1}$ as a function of $\beta$
in Fig. 10. It shows that the decay widths of these excited $D_{s}$ states
exhibit some sensitivities to the parameter $\beta$. Within the range of
$\beta=(0.45\pm 0.05)$ GeV, about $30\%$ uncertainties of the decay widths
would be expected. This is a typical order of accuracy for the constituent
quark model, and can be regarded as reasonable.
The ratio $\Gamma(D^{*}K)/\Gamma(DK)$ appears to behave differently. For the
$2^{3}S_{1}$, the sensitivity of the ratio to $\beta$ is apparent. In
contrast, the ratios of $|(SD)\rangle_{L}$ and $1^{3}D_{3}$ are quite
insensitive to $\beta$. The ratio $\Gamma(D^{*}K)/\Gamma(DK)$ is not shown for
$|2{P_{1}}\rangle_{L}$ since its decay into $DK$ is forbidden.
In brief, although the harmonic oscillator parameter $\beta$ can bring some
uncertainties to the final results, within the range of $\beta=(0.4\sim 0.5)$
GeV, our major conclusions will still hold.
Figure 10: (Color online) The decay widths (upper panel) and ratios (lower
panel) of different configuration assignments as a function of $\beta$: the
solid lines are for the mixed state $|(SD)\rangle_{L}$ of
$2^{3}S_{1}$-$1^{3}D_{1}$ with a mass of 2710 MeV and mixing angle
$-55^{\circ}$; the dotted lines are for $2^{3}S_{1}$ with a mass of 2710 MeV;
the dot-dashed lines for $1^{3}D_{3}$ with a mass of 2860 MeV; and the dashed
line in the upper panel is for the mixed state $|2{P_{1}}\rangle_{L}$ of
$2^{1}P_{1}$-$2^{3}P_{1}$ with a mass of 3040 MeV and mixing angle
$-50^{\circ}$.
## IV summary
In this work we investigate the strong decays of several newly observed
charmed mesons in a constituent quark model with effective Lagrangians for the
quark-meson interactions. The decay amplitudes are extracted for light
pseudoscalar meson or vector meson productions via axial or vector current
conservation between the quark-level and hadronic level couplings. The quark-
meson couplings can then be determined by independent measurements such as
meson photoproduction and meson-baryon scatterings.
We find that the new state $D_{sJ}(3040)$ can be identified as the low mass
physical state $|2{P_{1}}\rangle_{L}$ from the
$D_{s}(2^{1}P_{1})$-$D_{s}(2^{3}P_{1})$ mixing with a mixing angle
$\phi\simeq-(40\pm 12)^{\circ}$. Further experimental search for decay modes
of $D_{1}(2430)K$, $D_{2}(2460)K$, $D_{0}(2400)K$, $DK^{*}$, and
$D_{s}^{*}\eta$ should be able to disentangle its property and test our model
predictions.
The $D_{sJ}(2710)$ seems to favor a low mass physical state $|(SD)\rangle_{L}$
from the $2^{3}S_{1}$-$1^{3}D_{1}$ mixing with a mixing angle
$\phi\simeq(-54\pm 7)^{\circ}$. Both the ratio and width are in a good
agreement with the data. The decay properties of its heavy partner
$|(SD)^{\prime}_{1}\rangle_{H}$ are also discussed. It has a broad width
$\Gamma\simeq(110\sim 140)$ MeV at the $2.8$ GeV mass region, and dominated by
the $DK$ mode. We also point out that the $|(SD)^{\prime}_{1}\rangle_{H}$
state may be searched in the $DK$ spectrum as the $D_{sJ}(2710)$ if its mass
is $\sim 2.8$ GeV. Whether the present data have contained its signal could be
a crucial criteria for various model predictions.
The $D_{sJ}(2860)$ cannot be easily explained by a single configuration of
$2^{3}P_{0}$, $2^{3}P_{2}$, $1^{3}F_{2}$ or $1^{3}D_{3}$. To overcome this
problem we follow the proposal of a two-state picture by Ref. eef:2009 and
assume that two narrow resonances may have been observed around 2.86 GeV with
a width $\Gamma\simeq(40\sim 50)$ MeV. It shows that one resonance seems to be
the $1^{3}D_{3}$, which mainly decays into $DK$. The other resonance could be
the $|1{D_{2}}^{\prime}\rangle_{H}$, which is the high-mass state from the
$1^{1}D_{2}$-$1^{3}D_{2}$ mixing, and dominantly decays into $D^{*}K$. Further
theoretical and experimental efforts are needed to disentangle the mysterious
properties about this state.
We also study the implications arising from the assignments for those observed
resonances, e.g. their partner states in the mixing. In particular, if the
$D_{sJ}(3040)$ is indeed a $P$-wave state $|2{P_{1}}\rangle_{L}$, the other
three $P$-wave states $D_{sJ}(|2P^{\prime}_{1}\rangle_{H})$,
$D_{sJ}(2^{3}P_{0})$ and $D_{sJ}(2^{3}P_{2})$ may also have measurable effects
in experiment. Their strong decay properties are predicted, which could be
useful for future experimental studies.
## Acknowledgements
This work is supported, in part, by the National Natural Science Foundation of
China (Grants 10675131 and 10775145), Chinese Academy of Sciences
(KJCX3-SYW-N2), and Ministry of Science and Technology of China
(2009CB825200).
## References
* (1) B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 80, 092003 (2009) [arXiv:0908.0806 [hep-ex]].
* (2) B. Aubert [BABAR Collaboration], Phys. Rev. Lett. 97, 222001 (2006).
* (3) J. Brodzicka et al. [Belle Collaboration], Phys. Rev. Lett. 100, 092001 (2008)
* (4) E. J. Eichten, C. T. Hill and C. Quigg, Phys. Rev. Lett. 71, 4116 (1993) [arXiv:hep-ph/9308337].
* (5) Y. B. Dai and S. L. Zhu, Phys. Rev. D 58, 074009 (1998) [arXiv:hep-ph/9802224].
* (6) A. Hiorth and J. O. Eeg, Phys. Rev. D 66, 074001 (2002) [arXiv:hep-ph/0206158].
* (7) S. Fajfer and J. F. Kamenik, Phys. Rev. D 74, 074023 (2006) [arXiv:hep-ph/0606278].
* (8) S. Fajfer and J. F. Kamenik, Int. J. Mod. Phys. A 23, 3196 (2008).
* (9) J. Kamenik and S. Fajfer, J. Phys. Conf. Ser. 110, 122015 (2008).
* (10) W. A. Bardeen and C. T. Hill, Phys. Rev. D 49, 409 (1994) [arXiv:hep-ph/9304265].
* (11) W. A. Bardeen, E. J. Eichten and C. T. Hill, Phys. Rev. D 68, 054024 (2003) [arXiv:hep-ph/0305049].
* (12) P. Colangelo, F. De Fazio and R. Ferrandes, Mod. Phys. Lett. A 19, 2083 (2004) [arXiv:hep-ph/0407137].
* (13) M. A. Nowak, M. Rho and I. Zahed, Phys. Rev. D 48, 4370 (1993) [arXiv:hep-ph/9209272].
* (14) A. Deandrea, R. Gatto, G. Nardulli and A. D. Polosa, JHEP 9902, 021 (1999) [arXiv:hep-ph/9901266].
* (15) A. Deandrea, N. Di Bartolomeo, R. Gatto, G. Nardulli and A. D. Polosa, Phys. Rev. D 58, 034004 (1998) [arXiv:hep-ph/9802308].
* (16) P. Colangelo, F. De Fazio, S. Nicotri and M. Rizzi, Phys. Rev. D 77, 014012 (2008) [arXiv:0710.3068 [hep-ph]].
* (17) F. De Fazio, arXiv:0910.0412 [hep-ph] (2009).
* (18) Y. B. Dai, C. S. Huang, M. Q. Huang, H. Y. Jin and C. Liu, Phys. Rev. D 58, 094032 (1998) [Erratum-ibid. D 59, 059901 (1999)] [arXiv:hep-ph/9705223].
* (19) P. Colangelo, F. De Fazio, G. Nardulli, N. Di Bartolomeo and R. Gatto, Phys. Rev. D 52, 6422 (1995) [arXiv:hep-ph/9506207].
* (20) Z. G. Wang, Phys. Rev. D 77, 054024 (2008) [arXiv:0801.0267 [hep-ph]].
* (21) F. E. Close, C. E. Thomas, O. Lakhina and E. S. Swanson, Phys. Lett. B 647, 159 (2007) [arXiv:hep-ph/0608139].
* (22) F. E. Close and E. S. Swanson, Phys. Rev. D 72, 094004 (2005) [arXiv:hep-ph/0505206].
* (23) Zhi-Feng Sun and Xiang Liu, Phys. Rev. D 80, 074037 (2009) [arXiv:0909.1658].
* (24) B. Zhang, X. Liu, W. Z. Deng and S. L. Zhu, Eur. Phys. J. C 50, 617 (2007) [arXiv:hep-ph/0609013].
* (25) W. Wei, X. Liu and S. L. Zhu, Phys. Rev. D 75, 014013 (2007) [arXiv:hep-ph/0612066].
* (26) M. Di Pierro and E. Eichten, Phys. Rev. D 64, 114004 (2001).
* (27) Z. G. Wang, Chin. Phys. C 32, 797 (2008) [arXiv:0708.0155 [hep-ph]].
* (28) E. Van Beveren and G. Rupp, Phys. Rev. Lett. 97, 202001 (2006) [arXiv:hep-ph/0606110].
* (29) P. Colangelo, F. De Fazio and S. Nicotri, Phys. Lett. B 642, 48 (2006) [arXiv:hep-ph/0607245].
* (30) J. Koponen, Phys. Rev. D 78, 074509 (2008) [arXiv:0708.2807 [hep-lat]].
* (31) X. H. Zhong and Q. Zhao, Phys. Rev. D 78, 014029 (2008) [arXiv:0803.2102 [hep-ph]].
* (32) E. Van Beveren and G. Rupp, arXiv:0908.1142 [hep-ph].
* (33) A. Manohar and H. Georgi, Nucl. Phys. B 234, 189 (1984).
* (34) Z. P. Li, Phys. Rev. D 50, 5639 (1994) [arXiv:hep-ph/9404269].
* (35) Z. P. Li, H. X. Ye and M. H. Lu, Phys. Rev. C 56, 1099 (1997).
* (36) Q. Zhao, J. S. Al-Khalili, Z. P. Li and R. L. Workman, Phys. Rev. C 65, 065204 (2002).
* (37) X. H. Zhong, Q. Zhao, J. He and B. Saghai, Phys. Rev. C 76, 065205 (2007).
* (38) X. H. Zhong, Q. Zhao, Phys. Rev. C 79, 045202 (2009).
* (39) D. O. Riska and G. E. Brown, Nucl. Phys. A 679, 577 (2001) [arXiv:nucl-th/0005049].
* (40) Q. Zhao, Z. P. Li and C. Bennhold, Phys. Rev. C 58, 2393 (1998); Phys. Lett. B 436, 42 (1998).
* (41) Q. Zhao, Phys. Rev. C 63, 025203 (2001).
* (42) Q. Zhao, J. S. Al-Khalili, and C. Bennhold, Phys. Rev. C 64, 052201(R) (2001).
* (43) Q. Zhao, B. Saghai and Z. P. Li, J. Phys. G 28, 1293 (2002).
* (44) X. H. Zhong and Q. Zhao, Phys. Rev. D 77, 074008 (2008) [arXiv:0711.4645 [hep-ph]].
* (45) I. Hleiqawi et al. [CLAS Collaboration], Phys. Rev. C 75, 042201 (2007) [Erratum-ibid. C 76, 039905 (2007)].
* (46) M. Nanova et al. [CBELSA/TAPS Collaboration], Eur. Phys. J. A 35, 333 (2008) [arXiv:0803.2146 [nucl-ex]].
* (47) C. Amsler et al. [Partical Data Group], Phys. Lett. B 667, 1 (2008).
* (48) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
* (49) D. Ebert, R. N. Faustov and V. O. Galkin, arXiv:0910.5612 [hep-ph].
* (50) S. Godfrey and R. Kokoski, Phys. Rev. D 43, 1679 (1991).
* (51) E. S. Swanson, Phys. Rep. 429, 243 (2006).
* (52) B. Chen, D. X. Wang and A. Zhang, Phys. Rev. D 80, 071502(R) (2009) [arXiv:0908.3261 ].
* (53) D. M. Li and B. Ma, arXiv:0911.2906 [hep-ph].
* (54) T. Matsuki, T. Morii and K. Sudoh, Eur. Phys. J.A 31,701 (2007).
|
arxiv-papers
| 2009-11-10T08:45:40 |
2024-09-04T02:49:06.368512
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xian-Hui Zhong and Qiang Zhao",
"submitter": "Xianhui Zhong",
"url": "https://arxiv.org/abs/0911.1856"
}
|
0911.1921
|
# Diagnostics of Rational Expectation Financial Bubbles with Stochastic Mean-
Reverting Termination Times
L. Lin${}^{\mbox{\,\ref{ETH}, \ref{BUAA}~{}}}$ and D.
Sornette${}^{\mbox{\,\ref{ETH}, \ref{SFI} }}$
1. 1.
Chair of Entrepreneurial Risks, Department of Management, Technology and
Economics, ETH Zurich, Kreuplatz 5, CH-8032 Zurich, Switzerland
2. 2.
School of Economics and Management, Beihang University, 100191 Beijing, China
3. 3.
Swiss Finance Institute, c/o University of Geneva, 40 blvd. Du Pont d’Arve, CH
1211 Geneva 4, Switzerland
Abstract
> We propose two rational expectation models of transient financial bubbles
> with heterogeneous arbitrageurs and positive feedbacks leading to self-
> reinforcing transient stochastic faster-than-exponential price dynamics. As
> a result of the nonlinear feedbacks, the termination of a bubble is found to
> be characterized by a finite-time singularity in the bubble price formation
> process ending at some potential critical time $\tilde{t}_{c}$, which
> follows a mean-reversing stationary dynamics. Because of the heterogeneity
> of the rational agents’ expectations, there is a synchronization problem for
> the optimal exit times determined by these arbitrageurs, which leads to the
> survival of the bubble almost all the way to its theoretical end time. The
> explicit exact analytical solutions of the two models provide nonlinear
> transformations which allow us to develop novel tests for the presence of
> bubbles in financial time series. Avoiding the difficult problem of
> parameter estimation of the stochastic differential equation describing the
> price dynamics, the derived operational procedures allow us to diagnose
> bubbles that are in the making and to forecast their termination time. The
> tests performed on three financial markets, the US S&P500 index from 1
> February 1980 to 31 October 2008, the US NASDAQ composite index from 1
> January 1980 to 31 July 2008 and the Hong Kong Hang Seng index from 1
> December 1986 to 30 November 2008, suggest the feasibility of advance bubble
> warning.
> Keywords: bubble, super-exponential regime, rational expectation, critical
> time, finite-time-singularity
## 1 Introduction
Bubbles and crashes in financial markets are of global significance because of
their effects on the lives and livelihoods of a majority of the world’s
population. While pundits and experts alike line up after the fact to claim
that a particular bubble was obvious in hindsight, the real time development
of the bubble is often characterized by either a deafening silence or a
cacophony of contradictory opinions. Here, we propose two models of financial
bubbles, from which we develop the corresponding operational procedures to
diagnose bubbles that are in the making and to forecast their termination
time. The tests performed on three financial markets, the US S&P500 index from
1 February 1980 to 31 October 2008, the US NASDAQ composite index from 1
January 1980 to 31 July 2008 and the Hong Kong Hang Seng index from 1 December
1986 to 30 November 2008, demonstrate the feasibility of advance bubble
warning on the major market regimes that were followed by crashes or extended
market downturns. The empirical results support the hypothesis that financial
bubbles result from positive feedbacks operating on the price and/or on its
momentum, leading to faster-than-exponential transients.
These results should be appreciated from the perspective of the present state-
of-art on modeling and detecting bubbles. There is no really satisfactory
theory of bubbles, which both encompasses its different possible mechanisms
and adheres to reasonable economic principles (no or limited arbitrage,
equilibrium or quasi-equilibrium with only transient deviations, bounded
rationality). Part of the reason that the literature is still uncertain on
even how to define a bubble is that an exponentially growing price can always
be argued to result from some fundamental economic factor (Gurkaynak, 2008;
Lux and Sornette, 2002). This is related to the problem that the fundamental
price is not directly observable, giving no strong anchor to understand
observed prices. Another fundamental difficulty is to go beyond equilibrium to
out-of-equilibrium set-ups (Brock, 1993; Brock and Hommes, 1999; Chiarella et
al., 2008; Hommes and Wagener, 2008).
Two conditions are in general invoked as being necessary for prices to deviate
from fundamental value. First, there must be some degree of irrationality in
the market. That is, investors’ demand for stocks must be driven by something
other than fundamentals, like overconfidence in the future. Second, even if a
market has such investors, the general argument is that rational investors
will drive prices back to fundamental value. For this not to happen, there
needs to be some limits on arbitrage. Shleifer and Vishny (1997) provide a
description for various limits of arbitrage. With respect to the equity
market, clearly the most important impediment to arbitrage are short sales
restrictions. Roughly 70% of mutual funds explicitly state (in SEC Form N-SAR)
that they are not permitted to sell short (Almazan et al., 2004). Seventy-nine
percent of equity mutual funds make no use of derivatives whatsoever (either
futures or options), suggesting further that funds do not take synthetically
short positions (Koski and Pontiff, 1999). These figures indicate that the
vast majority of funds never take short positions. Then, the argument goes
that bubbles can develop because prices reflect mainly the remaining
optimistic opinions and not the negative views of pessimistic traders who are
already out of the market, and who would take short positions, if given the
opportunity.
One important class of theories shows that there can be large movements in
asset prices due to the combined effects of heterogeneous beliefs and short-
sales constraints. The basic idea finds its root in the original CAPM
theories, in particular, the model of Lintner (1969) of asset prices with
investors having heterogeneous beliefs. Lintner and many others after him,
show that widely inflated prices can occur (Miller, 1977; Jarrow, 1980;
Harrison and Kreps, 1978; Chen et al., 2002; Scheinkman and Xiong, 2003;
Duffie et al., 2002). In these models that assume heterogeneous beliefs and
short sales restrictions, the asset prices are determined at equilibrium to
the extent that they reflect the heterogeneous beliefs about payoffs. But
short sales restrictions force the pessimistic investors out of the market,
leaving only optimistic investors and thus inflated asset price levels.
However, when short sales restrictions no longer bind investors, then prices
fall back down. This provides a possible account of the bursting of the
Internet bubble that developed in 1998-2000. Many of these models take into
account explicitly the relationship between the number of publicly tradable
shares of an asset and the propensity for speculative bubbles to form. So far,
the theoretical models based on agents with heterogeneous beliefs facing short
sales restrictions are considered among the most convincing models to explain
the burst of the Internet bubbles.
The role of “noise traders” in fostering positive feedback trading has been
emphasized by a number of models. For instance, DeLong et al. (1990)
introduced a model of market bubbles and crashes which exploits this idea of
the role of noise traders in the development of bubbles, as a possible
mechanism for why asset prices may deviate from the fundamentals over rather
long time periods. Their work was followed by a number of behavioral models
based on the idea that trend chasing by one class of agents produces momentum
in stock prices (Barberis et al., 1998; Daniel et al., 1998; Hong et al.,
2005). An influential empirical evidence on momentum strategies came from the
work of Jegadeesh and Titman (1993, 2001), which established that stock
returns exhibit momentum behavior at intermediate horizons. Strategies which
buy stocks that have performed well in the past and sell stocks that have
performed poorly in the past generate significant positive returns over 3- to
12- month holding periods. De Bondt and Thaler (1985) documented long-term
reversals in stock returns. Stocks that perform poorly in the past perform
better over the next 3 to 5 years than stocks that perform well in the past.
These findings present a serious challenge to the view that markets are semi-
strong-form efficient.
It is important to understand what mechanisms prevent arbitrageurs from
removing a bubble as soon as they see one. Abreu and Brunnermeier (2003) have
proposed that bubbles continue to grow due to a failure of synchronization of
rational traders, so that the later choose to ride rather than arbitrage
bubbles. Abreu and Brunnermeier (2003) consider a market where arbitrageurs
face synchronization risk and, as a consequence, delay usage of arbitrage
opportunities. Rational arbitrageurs are supposed to know that the market will
eventually collapse. They know that the bubble will burst as soon as a
sufficient number of (rational) traders will sell out. However, the dispersion
of rational arbitrageurs’ opinions on market timing and the consequent
uncertainty on the synchronization of their sell-off are delaying this
collapse, allowing the bubble to grow. In this framework, bubbles persist in
the short and intermediate term because short sellers face synchronization
risk, that is, uncertainty regarding the timing of the correction. As a
result, arbitrageurs who conclude that the arbitrageurs are yet unlikely to
trade against the bubble find it optimal to ride the still-growing bubble for
a while.
Bhattacharya and Yu (2008) provide a summary of recent efforts to expand on
the above concepts, in particular to address the two main questions of (i) the
cause(s) of bubbles and crashes and (ii) the possibility to diagnose them ex-
ante. Many financial economists recognize that positive feedbacks and in
particular herding is a key factor for the growth of bubbles. Herding can
result from a variety of mechanisms, such as anticipation by rational
investors of noise traders’ strategies (DeLong et al., 1990), agency costs and
monetary incentives given to competing fund managers (Dass et al., 2008)
sometimes leading to the extreme Ponzi schemes, rational imitation in the
presence of uncertainty (Roehner and Sornette, 2000) and social imitation. The
bubble models developed here build strongly on this accepted notion of
herding. We refer to Kaizoji and Sornette (2008) for an extensive review
complementing this brief survey.
The present paper takes its roots in the Johansen-Ledoit-Sornette (JLS) model
(Johansen et al., 1999, 2000) formulated in the Blanchard-Watson framework of
rational expectation bubbles (Blanchard, 1979; Blanchard and Watson, 1982).
The JLS model combined a representation of the herding behavior of noise
traders controlling a crash hazard rate with the arbitrage response of
rational traders on the asset price. One implication of the JLS model is the
transient faster-than-exponential acceleration of the price due to the
positive feedback associated with the herding behavior of noise traders. This
faster-than-exponential pattern can theoretically culminate in a finite-time
singularity, which characterizes the end of the bubble and the time at which
the crash is the most probable. Other models have explored further the
hypothesis that bubbles can be the result of positive feedbacks and that the
dynamical signature of bubbles derives from the interplay between fundamental
value investment and more technical analysis. The former can be embodied in
nonlinear extensions of the standard financial Black-Scholes model of log-
price variations (Sornette and Andersen, 2002; Ide and Sornette, 2002; Corcos
et al., 2002; Andersen and Sornette, 2004). The later requires more
significant extensions to account for the competition between inertia between
analysis and decisions, positive momentum feedbacks and negative value
investment feedbacks (Sornette and Ide, 2002). Close to our present
formulation, Sornette and Andersen (2002); Andersen and Sornette (2004)
develop a nonlinear generalization of the Black-Scholes process which can be
solved analytically. The nonlinear feedback is acting as the effect of price
on future growth, according to the view that high prices lead to a wealth
effect that drives behavioral investors to invest more aggressively.
The present paper adds to the literature by developing two related models of
transient bubbles in which their terminations occur at some potential critical
time $\tilde{t}_{c}$, which follows a mean-reversing stationary process with a
fixed unconditional mean $T_{c}$. These models provides straightforward ways
to determine the potential critical time without confronting the difficult
problem of parameter estimation of the stochastic differential equation
describing the price dynamics of price. In our models, rational arbitrageurs
can diagnose bubbles but do not know precisely when they end. These investors
are assumed to form rational expectations of the potential critical time but
not necessarily of the detailed price process itself, which form a much weaker
condition for investors’ rationality. Furthermore, we assume that rational
arbitrageurs hold consistent expectation for the potential critical time with
unbiased errors. Although rational arbitrageurs know that the bubble will
burst at its critical time, they can not make a deterministic prediction of
this time and therefore of when other arbitrageurs will sell out, because they
have little knowledge about others’ belief about the process governing the
stochastic critical time $\tilde{t}_{c}$. Our rational investors continuously
update their beliefs on the probable termination of the bubble, according to
their observation of the development of the bubble. They exit the market by
maximizing their expected payoff, based on their subjective perception of the
market bubble risk and the knowledge of the bubble dynamics. Because of the
heterogeneity of these rational agents’ expectations, there is a
synchronization problem between these arbitrageurs, which leads to the
survival of the bubble almost all the way to its theoretical end time. In this
respect, our model is reminiscent to that of Abreu and Brunnermeier (2003),
since the resilience of the bubble results from the lack of synchronization
between arbitrageurs on the decision to exit the bubble due to the
heterogeneity of their optimal exit times. Our models have the advantage of
being quantitatively testable and their concrete implementation provides
diagnostics of bubbles in real time series, as we demonstrate below.
The first model, which leads to a finite-time singularity in the price
dynamics with stochastic critical time is presented in the next section 2.
This model generalizes Sornette and Andersen’s model to allow for a mean
reversal dynamics of the bubble end. Section 3 present a second model leading
to a finite-time singularity in the momentum price dynamics with stochastic
critical time. Both models exemplify the importance of positive feedback,
which is quantified by a unique exponent $m$. A value of $m$ larger than $1$
(respectively $2$) characterize a bubble regime in the first (respectively
second) model. The two models can be solved exactly in explicit analytical
forms. These solutions provide nonlinear transformations which allow us to
develop novel tests for the presence of bubbles in financial time series.
These two classes of tests, one for each bubble model, are developed
respectively in subsections 4.1 and 4.2 and applied on three financial
markets, the US S&P500 index from 1 February 1980 to 31 October 2008, the US
NASDAQ composite index from 1 January 1980 to 31 July 2008 and the Hong Kong
Hang Seng index from 1 December 1986 to 30 November 2008. Section 5 concludes.
## 2 First bubble model: finite-time singularity in the price dynamics with
stochastic critical time
Financial bubbles are often viewed as being characterized by anomalously high
growth rates resulting from temporary over-optimistic beliefs in a ‘new
economy’ or in a ‘paradigm shift’ of the fundamental structure of productivity
gains. However, this definition is unsatisfactory because a high growth rate
associated with an exponentially growing price can always be justified by some
other fundamental valuation models which use higher discount factors and
larger dividend growth expectations, or introduce new accounting rules
incorporating for instance the benefits of real options. This definition also
leaves a large ambiguity as to when the bubble is supposed to end, or when a
crash might occur.
In contrast, we define a bubble as a transient faster-than-exponential growth
of the price, which would end in a finite-time-singularity in the absence of a
crash or change of regime. Such ‘super-exponential regime’ results from the
existence of positive nonlinear feedback mechanisms amplifying past price
increases into even faster growth rates. These positive nonlinear feedback
mechanisms may be due to a variety of causes, including derivative hedging
strategies, portfolio insurance methods or to imitative behaviors of bounded
rational arbitrageurs and of noise traders. While herding has been largely
documented to be a trait of noise traders, it is actually rational for bounded
rational agents to also enter into social imitation, as the collective
behavior may reveal information otherwise hidden to the agents. As a result of
the nonlinear positive feedbacks, the bubble price becomes less and less
coupled to the market fundamentals, and the super-exponential growth of the
price makes the market more and more unstable. In this scenario, the end of
the bubble conditional on the absence of crash occurs at a critical time at
which the market becomes maximally unstable. The end of the bubble is
therefore the time when the crash is the most probable. With or without a
crash, the end of the bubble signals the end of the transient super-
exponential growth, and the transition to a different regime, with unspecified
characteristics.
Here, we assume that sophisticated market participants are indeed aware of the
current bubble state, and that they know the price is growing towards its
final singularity which will occur at some future random critical time at
which the market may collapse with a finite probability (but not with
certainty). Our bounded rational agents are able to form unbiased rational
expectations of the critical time corresponding to the end of the bubble at
which the crash is the more probable. We assume that these sophisticated
arbitrageurs enter sequentially into the market, attracted by the potential
large gains, given their anticipation of the crash risk quantified by their
estimation of the critical end time of the bubble which is formed when they
enter the market. Because their anticipations of the bubble demise are
heterogeneous, they solve an optimal timing problem with distinct inputs,
which leads to different exit strategies. The heterogeneity in their exit
strategies is common knowledge among these arbitrageurs, and results in a lack
of coordination, ensuring the persistence of the bubble. This synchronization
problem is analogous to that identified by Abreu and Brunnermeier (2003).
However, for Abreu and Brunnermeier, the lack of synchronization stems from
the existence of heterogeneous beliefs on the start of the bubble, i.e.,
arbitrageurs have “sequential awareness” and do not know whether they have
learn the information on the mispricing early or late relative to other
rational arbitrageurs. In contrast, our model emphasizes that the lack of
synchronization results from the heterogeneous beliefs concerning the end of
the bubble. Many reports both in the academic and professional literature
state that sophisticated participants like hedge-funds correctly diagnosed the
presence of a bubble and actually “surfed” the bubbles, attracted by the
potential large gains. Many reported that the largest uncertainty was how long
it would continue its course (Gurkaynak, 2008; Sullivan, 2009)
Less sophisticated traders investing in the market have little knowledge on
the bubble duration and their action add noise which is assumed to have an
influence only on the critical time characterizing the end of the bubble,
while the super-exponential growth of the price remains robust. In the bubble
regime, a well-defined nonlinear exponent characterizes the positive feedbacks
at the origin of the bubble. The noisy character of the critical time $t_{c}$
of the end of the bubble is modeled by an Ornstein-Uhlenbeck process.
Intuitively, the price trail resembles the trace of a bug climbing erratically
along a hanging curved rope attached to a vibrating support.
Specifically, the price dynamics in the bubble regime is assumed to be
described by the following stochastic differential equation:
$\mathrm{d}p=\mu p^{m}(1+\delta(p,t))\mathrm{d}t+\sigma p^{m}\mathrm{d}W~{},$
(1)
where the exponent $m>1$ embodies the positive feedback mechanism, in which a
high price $p$ pushes even further the demand so that the return and its
volatility tend to be a nonlinear accelerating function of $p$. When $m>1$, we
will show that the price diverges in finite time. The time at which this
divergence occurs is referred to as the critical time $\tilde{t}_{c}$. As we
will see later, the term $\delta(x,t)$ is a time-varying regulator term that
governs the behavior of $\tilde{t}_{c}$, $\mu$ is the instantaneous return
rate, $\sigma$ is the volatility of the returns and $W$ is the standard Wiener
process. This model recovers the standard Black-Scholes model of the geometric
random walk with drift $\mu$ and standard deviation $\sigma$ for $m=1$ and
$\delta=0$.
Let us consider first the case where $\delta(p,t)=\sigma=0$, so that
expression (1) reduces to $\mathrm{d}p=\mu p^{m}\mathrm{d}t$, whose solution
is
$p=K(t_{c}-t)^{-\beta}~{},$ (2)
where $\beta=\frac{1}{m-1}$ , $K=(\frac{\beta}{\mu})^{\beta}$,
$t_{c}=\frac{p_{0}^{-(m-1)}}{(m-1)\mu}$ and $p_{0}$ denotes the price at the
start time of the bubble taken to be $t=0$. Since $\beta>0$ for $m>1$,
expression (2) exemplifies the existence of a finite-time (or “movable” ,
Bender and Orszag (1999)) singularity of the price that goes to infinity in
finite time as $t\to t_{c}^{-}$. This pathological behavior is the direct
consequence of the positive feedback embodied in the condition $m>1$.
Motivated by this simple analytical solution, we now consider the more general
process (1) and specify $\delta(p,t)$ in order to obtain a general process
with stochastic finite time singularities. We postulate the following specific
form for the process governing $\delta(p,t)$,
$\delta(p,t)=\alpha~{}\tilde{t}_{c}(t)+\frac{1}{2}m\mu\sigma^{2}~{}[p(t)]^{m-1}~{},$
(3)
where
$\mathrm{d}\tilde{t}_{c}=-\alpha\tilde{t}_{c}\mathrm{d}t+(\sigma/\mu)\mathrm{d}W$
(4)
follows an Ornstein-Uhlenbeck process with zero unconditional mean. The Wiener
process in (4) is the same as the one in (1). We obtain the following result.
###### Proposition 1.
Provided that $\delta(p,t)$ follows the process (3) with (4), the solution of
equation (1) can be written under a form similar to (2) as follows,
$p(t)=K(\widetilde{T_{c}}-t)^{-\beta}~{},$ (5)
with
$\beta=\frac{1}{m-1}~{},\quad
K=\left(\frac{\beta}{\mu}\right)^{\beta}~{},\quad
T_{c}=\frac{\beta}{\mu}p_{0}^{-\frac{1}{\beta}}~{},\quad\widetilde{T_{c}}=T_{c}+\tilde{t}_{c}~{}.$
(6)
The proof of Proposition (1) is given in Appendix A.
Note that
$\mathbb{E}(\widetilde{T}_{c})=\mathbb{E}(T_{c}+\tilde{t}_{c})=T_{c}~{}.$ (7)
Therefore, $T_{c}$ given in (6) is the expected time at which the bubble will
end in an explosive singularity. The Ornstein-Uhlenbeck process (4) for
$\tilde{t}_{c}(t)$ expresses that the end of the bubble cannot be known with
certainty but is instead a stochastic variable. Nonetheless, due to the
positive feedback, the price can explode in finite time, but the end of the
bubble can at best be known to follow the mean-reversing process (4). The time
$T_{c}$ can be interpreted as the consensus rational expectation formed by
sophisticated arbitrageurs of the stochastic critical time $\widetilde{T_{c}}$
as which the bubble is expected to end. But each trading day $t$ discloses a
different ‘actual’ critical time $T_{c}+t_{c}(t)$ which is vibrating around
its expected value $T_{c}$.
Our model assumes that there is no coordination mechanism that would ensure
the exchange of information among the sophisticated arbitrageurs concerning
their expectation of the end of the bubble. The arbitrageurs reveal their
private information only upon entering the market. We assume that they do so
sequentially, based on their heterogeneous beliefs on the process
$\tilde{t}_{c}$. And the process $\tilde{t}_{c}$ is an emergent property that
results from the aggregation of market beliefs rather than from the action of
single arbitrageur. Being aware of the escalating level of the bubble that has
not yet burst, each arbitrageur will ride the bubble for a while and identify
the best exit strategy according to the maximization of her risk-adjusted
return based on her belief.
Let us denote $t_{i}>0$ the time at which the $i$’s arbitrageur has entered
the market. Being aware of the form (5) of the price dynamics, at each instant
$t$, the rational arbitrageur forms a belief quantified by her hazard rate
$h_{i}(t)$, of the probability that a crash might occur in the next instant,
conditional on the fact that it has not yet happened. This allows her to
estimate the probability $1-\Pi_{i}(t)$ that the crash will not happen until
time $t$. Given the explosive form (5) of the price dynamics, we assume that
the arbitrageur forms a belief of the crash hazard rate which is of the same
form, that is,
$h_{i}(t)=\frac{\pi_{i}(t)}{1-\Pi_{i}(t)}\propto(T_{c,i}-t)^{-\beta_{i}}~{},$
(8)
where $\Pi_{i}(t)$ is the arbitrageur $i$’s conditional cumulative
distribution function of the bursting date and $\pi_{i}(t)$ represents the
associated conditional density. $T_{c,i}=T_{c}+\tilde{t}_{c,i}$ denotes the
critical time for the end of the bubble that the arbitrageur $i$ has estimated
when entering the market. We allow the exponent $\beta_{i}$ to be different
from arbitrageur to arbitrageur, so as to reflect different views on the
riskiness of the market, which can translate into distinct risk aversions: the
larger the exponent $\beta_{i}$, the more pessimistic is the view of the
arbitrageur concerning the imminence of the crash, because a larger
$\beta_{i}$ implies a faster divergence of the crash hazard rate.
The occurrence of the market collapse is posited to be triggered when a
sufficiently large number $\eta$ of arbitrageurs have exited the market,
leading to a large price movement, amplified by the herding of noise traders.
Their cumulative effect is accounted for by a postulated percent loss $\kappa$
of the crash, which is itself a random variable. Given such an environment,
the date to exit the market for a given rational arbitrageur $i$ determines
her best trade strategy, which is found as the solution of the following
optimization problem
$\max_{t}\,\,\mathbb{E}^{i}[(1-\Pi_{i}(t))\cdot\mathrm{d}p-\pi_{i}(t)\mathrm{d}t\cdot\kappa
p]~{}.$ (9)
The first term $(1-\Pi_{i}(t))\cdot\mathrm{d}p$ represents the arbitrageur’s
instantaneous benefit at $t+dt$ provided that the burst of the bubble has not
yet happened. The second term $\pi_{i}(t)\mathrm{d}t\cdot\kappa p$ is the
instantaneous cost supported by the arbitrageur when the bubble bursts. The
solution of (9) is obtained from the first-order condition
$(1-\Pi_{i}(t))\mathbb{E}^{i}(\mu
p^{m}(1+\delta(p,t)))=\pi_{i}(t)\mathbb{E}^{i}(\kappa p)~{}.$ (10)
We can thus state
###### Proposition 2.
Given a population of heterogeneous arbitrageurs, which form their expectation
of the crash hazard rate according to (8) with heterogeneous anticipated
critical times $T_{c,i}$ and exponents $\beta_{i}$ reflecting their different
views on the riskiness of the market, a given arbitrageur $i$ decides to exit
the market at the date $t_{i}^{ex}$ which is the solution of
${\mathbb{E}^{i}[dp(t^{ex}_{i})]\over\mathbb{E}^{i}[\kappa
p(t^{ex}_{i})]}=h_{i}(t^{ex}_{i})\propto(T_{i,c}-t^{ex}_{i})^{-\beta_{i}}~{}.$
(11)
Since $\mathbb{E}^{i}(p)\neq\mathbb{E}^{j}(p)$ and $h_{i}(t)\neq h_{j}(t)$, we
have $t_{i}^{ex}\neq t_{j}^{ex}$. Notwithstanding the fact that the presence
of the bubble is common knowledge among all rational arbitrageurs, the absence
of synchronization of their market exit allows the bubble to persist and run
its course up to a time close to its expected value (7).
For the price process (1) with (3), equation (11) yields
$\frac{\mathbb{E}^{i}(\mu
p^{m}(1+\alpha\tilde{t}_{c}(t^{ex}_{i})+\frac{1}{2\mu}m\sigma^{2}[p(t^{ex}_{i})]^{m-1}))}{\mathbb{E}^{i}(\kappa
p)}=h_{i}(t^{ex}_{i})\propto(T_{i,c}-t^{ex}_{i})^{-\beta_{i}}~{}.$ (12)
This synchronization problem is analogous to that identified by Abreu and
Brunnermeier (2003), with the important difference that we emphasize that the
lack of synchronization results from the heterogeneous beliefs concerning the
end of the bubble.
The corresponding observable logarithmical return for the asset price reads
$\displaystyle r_{\tau}(t)=\ln p(t+\tau)-\ln p(t)$
$\displaystyle=-\beta\ln\left(\frac{T_{c}+\tilde{t}_{c}(t+\tau)-{(t+\tau)}}{T_{c}+\tilde{t}_{c}(t)-{t}}\right)$
$\displaystyle=-\beta\ln\left(1+\frac{\Delta\tilde{t}_{c}(t)-\tau}{T_{c}+\tilde{t}_{c}(t)-t}\right)~{},$
(13)
where $\tau$ is the time interval between two observations of the price. In
the case where $T_{c}$ is large enough such that
$T_{c}+\tilde{t}_{c}(t)-t\gg\Delta\tilde{t}_{c}(t)-\tau$, expression (13) can
be approximated by its first order Taylor expansion:
$r_{\tau}(t)=\frac{\beta}{T_{c}+\tilde{t}_{c}(t)-t}(\tau-\Delta\tilde{t}_{c}(t))=\frac{1}{(m-1)(T_{c}+\tilde{t}_{c}(t)-t)}(\tau-\Delta\tilde{t}_{c}(t))$
(14)
Therefore, under the condition $m\to 1$ and
$T_{c}+\tilde{t}_{c}(t)-t\to\infty$, the logarithmical return $r_{\tau}(t)$ is
driven by the change $\Delta\tilde{t}_{c}(t)$ of the critical time on each
trading day. Although $\tilde{t}_{c}$ follows an Ornstein-Uhlenbeck process,
i.e., $\Delta\tilde{t}_{c}(t)=-\alpha\tilde{t}_{c}(t)+\varepsilon_{t}$, the
existence of correlation between successive returns will be hardly detectable
if $\alpha$ is small enough. Conversely, if $\tilde{t}_{c}$ follows a unit
root process, the logarithmical return $r_{\tau}(t)$ is only dependent on the
realization of the gaussian noise term $\varepsilon$. In this sense, only if
$T_{c}$ is not too far in the future, $m$ is sufficiently larger than $1$ and
$\tilde{t}_{c}$ is stationary, can we diagnose the existence of a bubble,
characterized by the emergence of a transient super-exponential price growth
of the form (5).
## 3 Second bubble model: finite-time singularity in the momentum price
dynamics with stochastic critical time
The price process (5) of the first bubble model might appear extreme, in the
sense that the price diverges on the approach of the critical time
$\widetilde{T_{c}}$ of the end of the bubble. However, such a divergence
cannot run its full course in our model due to the divergence of the crash
hazard rate which ensures that a crash will occur before. The critical time is
thus a ghost-like time, which is out-of-reach, and the price process (5)
describes a transient run-up that would diverge only in the hypothetical
absence of any arbitrageur. Here, we consider an alternative model in which
the price remains always finite but the faster-than-exponential growth
associated with the bubble is embodied into the price momentum, i.e., the
derivative of the logarithm of the price or logarithmic return.
Defining $y(t)=\ln p(t)$, we assume the following process for $y(t)$:
$\displaystyle\mathrm{d}y$
$\displaystyle=x(1+\gamma(x,t))\mathrm{d}t+(\sigma/\mu)x\mathrm{d}W$ (15)
$\displaystyle\mathrm{d}x$ $\displaystyle=\mu
x^{m}(1+\delta(x,t))\mathrm{d}t+\sigma x^{m}\mathrm{d}W~{},$ (16)
where the same Wiener process $W(t)$ acts on both $\mathrm{d}y$ and
$\mathrm{d}x$. The process $x(t)$ plays the role of an effective price
momentum. To see this, consider the special case $\gamma(x,t)=0$. Then,
expression (15) reduces to
$\mathrm{d}y=x\mathrm{d}t+(\sigma/\mu)x\mathrm{d}W$, which shows that
$x(t)\mathrm{d}t=\mathbb{E}[\mathrm{d}y]$ and thus $x(t)$ is the average
momentum of the price, defined as the instantaneous time derivative of the
expected logarithm of the price. The dynamics of the log-price described by
(15) with (16) is similar to previous models (Bouchaud and Cont, 1998; Farmer,
2002; Ide and Sornette, 2002; Sornette and Ide, 2002), which argued for the
presence of some inertia in the price formation process. This inertia is
related to the momentum effect (Jegadeesh and Titman, 1993, 2001; Carhart,
1997; Xue, 2003; Cooper et al., 2004). Intuitively, a price process involving
both $\mathrm{d}y$ and $\mathrm{d}x$ holds if the price variation from today
to tomorrow is based in part on decisions using analyses of the price change
between yesterday (and possibly earlier times) and today.
In the (unrealistic) deterministic limit $\gamma(x,t)=\delta(x,t)=\sigma=0$,
the two equations (15) and (16) reduce to the deterministic equation
$\frac{\mathrm{d}^{2}y}{\mathrm{d}t^{2}}=\mu\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^{m}~{},$
(17)
whose solution reads, for $m>2$,
$y(t)=A-B(T_{c}-t)^{1-\beta}~{},$ (18)
where $\beta=\frac{1}{m-1}$,
$T_{c}=(\beta/\mu)(\frac{\mathrm{d}p}{\mathrm{d}t}\big{|}_{t=t_{0}})^{-\frac{1}{\beta}}$,
$B=\frac{1}{1-\beta}(\mu/\beta)^{-\beta}$ and $A=p(T_{c})$. The condition
$m>2$ ensures that $0<1-\beta<1$. Therefore, the log-price $y(t)$ exhibits a
finite-time singularity (FTS) at $T_{c}$. But this FTS is of a different type
than in the model of the previous section: here, $y(t)$ remains finite at
$t=T_{c}$ and equal to some value $A=p(T_{c})$. The singularity is expressed
via the divergence of the momentum $x(t)=dy/dt$ which diverges at $t=T_{c}$.
As in the previous model, this FTS embodies the positive feedback mechanism,
in which a high price momentum $x$ pushes even further the demand so that the
return and its volatility tend to be nonlinear accelerating functions of $x$.
In the previous model, it is the price that provides a feedback on further
price moves, rather than the price momentum used here.
Motivated by this simple analytical solution (18), we complement the general
process (15,16) by specifying $\gamma(x,t)$ and $\delta(p,t)$ in order to
obtain solutions with stochastic finite time singularities in the momentum
with finite prices. We postulate the following specific processes
$\displaystyle\gamma(x,t)$
$\displaystyle=\alpha\tilde{t}_{c}(t)+\frac{\sigma^{2}}{2\mu}[x(t)]^{m-1}$
(19) $\displaystyle\delta(x,t)$
$\displaystyle=\alpha\tilde{t}_{c}(t)+\frac{1}{2}m\mu\sigma^{2}[x(t)]^{m-1}$
(20)
where
$\mathrm{d}\tilde{t}_{c}=-\alpha\tilde{t}_{c}\mathrm{d}t+(\sigma/\mu)\mathrm{d}W$
(21)
follows an Ornstein-Uhlenbeck process with zero unconditional mean. The Wiener
process in (21) is the same as the one in (15,16), which reflects that the
same series of news or shocks move log-price, momentum and anticipated
critical time. We obtain the following result.
###### Proposition 3.
Provided that $\gamma(x,t)$ and $\delta(p,t)$ follow the processes given
respectively by (19) and (20), then the solution of (15,16) for the log-price
$y(t)=\ln p(t)$ can be written under a form similar to expression (18) as
follows,
$y(t)=A-B(T_{c}+\tilde{t}_{c}(t)-t)^{1-\beta}$ (22)
where
$\beta=\frac{1}{m-1}~{},\quad T_{c}=\frac{\beta}{\mu}x_{0}^{1/\beta},~{}\quad
x_{0}:=x(t=0)~{},\quad B=\frac{1}{1-\beta}(\beta/\mu)^{\beta}~{},$ (23)
and $A$ is a constant.
The proof of Proposition 3 is given in appendix B.
Expression (22) describes a log-price trajectory exhibiting a FTS occurring at
an unknown future critical time $T_{c}+\tilde{t}_{c}(t)$ which itself follows
an Ornstein-Uhlenbeck walk. In the same manner as in the previous model of the
preceding section, we assume that there is no coordination mechanism that
would ensure the exchange of information among the sophisticated arbitrageurs
concerning their expectation of the time $T_{c}+\tilde{t}_{c}(t)$ of the end
of the bubble. Following step by step the same reasoning as in the previous
section, we conclude that Proposition 2 also holds for the present model.
Being aware of the escalating level of the bubble that has not yet burst, each
arbitrageur will ride the bubble for a while and identify the best exit
strategy according to the maximization of her risk-adjusted return based on
her belief. Proposition 2 determines that the exit time $t^{ex}_{i}$ for
arbitrageur $i$ is the solution of
${\mathbb{E}^{i}\left[p(t^{ex}_{i})x(t^{ex}_{i})(1+\gamma(x(t^{ex}_{i}),t^{ex}_{i}))+{\sigma^{2}\over
2\mu^{2}}p(t^{ex}_{i})[x(t^{ex}_{i})]^{2}\right]\over\mathbb{E}^{i}\left[\kappa
p(t^{ex}_{i})\right]}=h_{i}(t^{ex}_{i})\propto(T_{i,c}-t^{ex}_{i})^{-\beta_{i}}~{}.$
(24)
The observable logarithmic return for the asset price corresponding to (22)
reads
$\displaystyle r_{\tau}(t)=y(t+\tau)-y(t)$
$\displaystyle=-B[(T_{c}+\tilde{t}_{c}(t+\tau)-(t+\tau))^{1-\beta}-(T_{c}+\tilde{t}_{c}(t)-t)^{1-\beta}]$
(25)
$\displaystyle=-B(T_{c}+\tilde{t}_{c}(t)-t)^{1-\beta}[(1+\frac{\Delta\tilde{t}_{c}(t)-\tau}{T_{c}+\tilde{t}_{c}(t)-t})^{1-\beta}-1]$
(26)
For future potential critical times $T_{c}$ sufficiently far away from the
present time $t$ such that
$T_{c}+\tilde{t}_{c}(t)-t\gg\Delta\tilde{t}_{c}(t)-\tau$, expression (26) can
be simplified into
$r_{\tau}(t)=-\frac{(1-\beta)B}{(T_{c}+\tilde{t}_{c}(t)-t)^{\beta}}(\Delta\tilde{t}_{c}(t)-\tau)=\frac{\mu^{-\beta}}{[(m-1)(T_{c}+\tilde{t}_{c}(t)-t)]^{\beta}}\cdot(\tau-\Delta\tilde{t}_{c}(t))$
(27)
Eq.(27) has a structure similar to that of Eq.(14). For weak positive feedback
of the momentum on itself ($m\to 2^{+}$) and when $T_{c}$ is large enough so
that $\mu(T_{c}+\tilde{t}_{c}(t)-t)$ is slowly varying, then the logarithmical
return $r_{s}$ is essentially driven by $\Delta\tilde{t}_{c}(t)$, i.e., the
change of critical time disclosed by every trading day is the main stochastic
process. The Geometric Brownian Motion is then recovered as an approximation
in this limit when the correlation time of the Orstein-Uhlenbeck process
driving the critical goes to zero.
## 4 Empirical tests of the two bubble models
We have proposed two models in which a financial bubble is characterized by a
transient faster-than-exponential growth culminating into a finite-time
singularity at some potential critical time. Because our two models reduce to
a standard GBM in appropriate limits, the diagnostic of the presence of
bubbles according to our two models lies in the conjunction of three pieces of
evidence that characterize specific deviations from the GBM regime: (i) the
proximity of the calibrated potential critical date $T_{c}$ to the end of the
time window in which the calibration of the models are made; (ii) the
reconstructed time series of the critical time $\tilde{t}_{c}$ should be
stationary and thus reject a standard unit-root test; (iii) the critical
exponent $m$ should be significant larger than $1$, a condition for the
existence of the super-exponential regime proposed to characterize bubbles.
### 4.1 Construction of alarms from the first model
Given a financial time series of close prices at the daily scale, our purpose
is to develop a procedure using the model of section 2 to diagnose the
presence of bubbles. We use time windows of 750 trading days that we slide
with a time step of 25 days from the beginning to the end of the available
financial time series. The number of such windows is therefore equal to the
total number of trading days in the financial time series minus 750 and
divided by 25. For each window, the purpose is to decide if the model of
section 2 diagnoses an on-going bubble or not and then to compare with the
actual subsequent realization of a crash that we consider as the validation
step.
For each window $]t_{i}-750,t_{i}]$ ending at $t_{i}$, we transform the price
time series in that window into a critical time series by inverting expression
(5) for $\widetilde{T}_{c,i}(t)$:
$\widetilde{T}_{c,i}(t)=\frac{1}{K}{1\over[p(t)]^{1/\beta}}+t,\qquad
t=t_{i}-749,\cdot\cdot\cdot,t_{i}~{}.$ (28)
The critical time series $\widetilde{T}_{c,i}(t)$ is defined over the window
$i$ ending at $t_{i}$. If the model was exact and no stochastic component was
present, and in absence of estimation errors, $\widetilde{T}_{c,i}(t)$ would
be a constant equal to $T_{c}$ defined in (6). In the presence of an expected
strong stochastic component, we estimate $T_{c}$ according to (7) as the
arithmetical average of $\widetilde{T}_{c,i}(t)$
$T_{c,i}=\frac{1}{750}\sum_{t=1}^{750}\widetilde{T}_{c,i}(t)~{}.$ (29)
We can then construct $\tilde{t}_{c,i}(t)$ as
$\tilde{t}_{c,i}(t)=\widetilde{T}_{c,i}(t)-T_{c,i}~{}.$ (30)
The transformation (28) from a non-stationary possibly explosive price process
$p(t)$ into what should be a stationary time series $\widetilde{T}_{c,i}(t)$
in absence of misspecification is a key element of our methodology for bubble
detection that avoids the problems documented by Granger and Newbold (1974)
and Phillips (1986) resulting from direct calibration of price or log-price
time series.
It will not have escaped the attention of the reader that the transformation
(28) requires the knowledge of the two unknown parameters $K$ and $\beta$ that
specify the bubble process (5). We propose to determine these two parameters
by applying an optimization procedure as follows. Recall that a crucial
ingredient of the bubble model is the mean reversal nature of the potential
critical time $\tilde{t}_{c}$. This suggests to apply a unit root test on the
reconstructed time series $\tilde{t}_{c,i}(t)$ and determine the optimal
values $K_{i}^{*}$ and $\beta_{i}^{*}$ as those which make the time series
$\tilde{t}_{c,i}(t)$ as stationary as possible. We proceed in two steps. We
first search in the space of the two parameters $K$ and $\beta$ and select an
elite list of the ten best pairs $(K,\beta)$ (when they exist) which reject a
standard unit-root test of non-stationarity at the 99.5% significance level.
We implement this procedure with the t-test statistic of the Dickey-Fuller
unit-root test (without intercept). Since the Dickey-Fuller test is a lower
test, the smaller the statistics $t$, the larger is the probability to reject
the null hypothesis that $\tilde{t}_{c}$ has unit-root (is non-stationary). Of
course, only a subset of the windows will yield any solution at all, i.e., it
is quite often the case that the Dickey-Fuller unit-root test is not rejected
at the 99.5% significance level for any pair $(K,\beta)$. For those windows
for which there are selected pairs $(K,\beta)$ according to the Dickey-Fuller
test, we choose the one with the smallest variance for its corresponding time
series $\tilde{t}_{c,i}(t)$, i.e., such that the $\tilde{t}_{c,i}(t)$’s are
the closest to their mean in the variance sense. This yields the optimal
$K_{i}^{*}$ and $\beta_{i}^{*}$ that best “fit” the window $i$ in the sense
that this pair of parameters provides the closest approximation to a
stationary time series for $\tilde{t}_{c,i}(t)$ given by expression (30) for
the potential termination of the bubble.
For a given window $i$, a diagnostic for the presence of bubble is flagged and
an alarm is declared when
1. (i)
$\beta^{*}>0$ such that $m>1$ (the signature of a positive feedback in our
model) and
2. (ii)
$T_{c,i}-t_{i}<750$, i.e., the estimated termination time of the bubble is not
too distant.
Figs.1-3 depicts all the bubble alarms obtained by applying this procedure to
three major stock indices, the US S&P500 index from 1 February 1980 to 31
October 2008, the US NASDAQ composite index from 1 January 1980 to 31 July
2008 and the Hong Kong Hang Seng index from 1 December 1986 to 30 November
2008. An alarm is indicated by a vertical line positioned on the last day
$t_{i}$ of the corresponding window that passes the two criteria (i-ii). We
refine the diagnostic by presenting three alarm levels, corresponding
respectively to $T_{c,i}-t_{i}<750$, $T_{c,i}-t_{i}<500$ and
$T_{c,i}-t_{i}<250$: the closer the estimated termination of the bubble, the
stronger should be the evidence for the bubble as a faster-than-exponential
growth. Another indication is the existence of clustering of the alarms. If
indeed a bubble is developing, it should be diagnosed repeatedly by several
successive windows.
Fig.1 for the S&P500 index shows that the alarm clusters correctly identify
the upcoming of four significant market corrections or crashes. This suggests
to qualify these alarms as correct diagnostic of bubbles ending in these
corrections. The first one is the large cluster localized within 1.5 years
before the ‘Black Monday’ crash of October 1987. The second smaller cluster is
associated with a significant correction starting in July 17, 1990. The third
large cluster is also within 1.5 year before the occurrence of the turmoil
starting in August 1998, associated with the default of Russia on its debt and
the devaluation of the ruble. The detection of a bubble starting to develop
more than a year before this event suggests that this event may not have been
entirely exogenous, supporting previous evidence for this claim (Sornette,
2004). The fourth smaller cluster announced the turning point of the famous
Internet-Communication-Technology (ICT) bubble in April 2000. We note that the
strength of the bubble is quite weak for the S&P500 index. This can probably
be explained by the fact that only about 20% of its constituting firms
belonged to the ICT sector while the remaining 80% firms belonged to the “old
economy” sector. In contrast, the alarm signal is much stronger for the Nasdaq
index, as can be seen in figure 2. One can observe a fifth rather small
cluster of alarms for $T_{c,i}-t_{i}<750$ (which however disappears for
$T_{c,i}-t_{i}<500$ or smaller, suggesting a weak signal) which is dated Sept.
12, 2005 in the top panel of Fig.1. This alarm does not appear to be
associated to any nearby termination of a rising price regime. However, notice
that, in October 2007, the S&P500 peaked and then started a dramatic
accelerating downward spiral fueled by the unfolding of the global financial
crisis. This peak of October 2007 is indeed less than 750 trading days away
from the triggering of the alarm on Sept. 12, 2005. This supports other
evidence that the run-up of the S&P500 from 2003 to October 2007 was a bubble
(Sornette and Woodard, 2010). It is however a failure of the present
methodology that the alarm is short-lived and does not confirm the continuing
accelerating trend up to the peak in October 2007.
Fig. 2 paints a similar picture. A first bubble preceding the crash of October
1987 is clearly diagnosed. A very large cluster of alarms spans the period
from at least early 1997 (Phillips et al., 2007) to 2000, confirming the
diagnostic of a running ICT bubble, that ended with a crash in April 2000.
This large cluster is actually made of two sub-clusters, the first one
associated with the bubble behavior ending with the so-called Russian crisis
at the end the summer of 1998, and the second one corresponding to the well-
known ICT bubble reflecting over-optimistic expectation of a “new economy”.
This is similar to the analysis and conclusion obtained in Fig.1 for the
S&P500 index. There is small cluster of alarms ending in April 1994, which
cannot be associated with any large price movement afterwards. Finally, a
fourth cluster of alarms is dated Sept. 12, 2005 in the top panel of Fig.2. As
for the S&P500 index, this cluster of alarms does not appear to be associated
to any nearby termination of a rising price regime, but rather to the
development of an accelerating upward trending price that culminated in
October 2007 before crashing in the subsequent year.
Similar conclusions hold for the Hong Kong market, as shown in Fig.3. One can
observe the clusters of alarms associated with the successive booming phases
of the Hang Seng index followed by several corrections or crashes. One can
identify in particular the bubbles associated with the strong correction of
1992, the two Asian crises of 1994 and 1997, as well as the bubble ending in
October 2007, which is this time very clearly diagnosed for this Hong Kong
market. There is one isolated false alarm dated Feb. 1, 2001 in the top panel
of Fig.3.
### 4.2 Construction of alarms from the second model
Similarly to the procedure described in section 4.1, we transform a given
price time series in a given window $i$ of 900 successive trading days into
what should be a stationary time series of potential critical end times, if
the price series is indeed described by the bubble model of section 3.
Inverting expression (22) in Proposition 3, we get, similarly to expression
(28),
$\widetilde{T}_{c,i}(t)=t_{i}+\left(\frac{A-\ln
p(t)}{B}\right)^{\frac{1}{1-\beta}}~{},\qquad
t=t_{i}-899,\cdot\cdot\cdot,t_{i}~{}.$ (31)
The critical time series $\widetilde{T}_{c,i}(t)$ is defined within the window
$i$ ending at $t_{i}$. If the model was exact and no stochastic component was
present, and in absence of estimation errors, $\widetilde{T}_{c,i}(t)$ would
be a constant equal to $T_{c}$ defined in (23). The expected critical end time
$T_{c,i}$ of a bubble, if any, is then estimated for this window $i$ by
expression (29) (with $750$ replaced by $900$). The fluctuations around
$T_{c,i}$ are described by $\tilde{t}_{c,i}(t)$ defined by (30).
As for the first bubble model, the transformation (31) requires the
determination of parameters, here the triplet $(A,B,\beta)$. For this, we
proceed exactly as in the previous subsection, with the Dickey-Fuller unit-
root test applied to the time series $\tilde{t}_{c,i}(t)$, followed by the
selection of the best triplet $(A^{*},B^{*},\beta^{*})$ that minimize the
variance of the time series $\tilde{t}_{c,i}(t)$. The search of the additional
parameter $A$ is performed in an interval bounded from above by
$2\max_{t_{i}-899\leq t\leq t_{i}}\ln p(t)$. Then, for a given window $i$, a
diagnostic for the presence of bubble is flagged and an alarm is declared when
1. (i)
$0<\beta^{*}<1$ such that $m>2$ (the signature of a positive feedback in the
momentum price dynamics model) and
2. (ii)
$-25\leq T_{c,i}-t_{i}\leq 50$, such that the estimated termination time of
the bubble is close to the right side of the time window.
3. (iii)
We further refine the filtering by considering three levels of significance
quantified by the value of the exponent $m$: level 1 ($m>2$), level 2
($m>2.5$) and level 3 ($m>3$).
The condition $T_{c,i}-t_{i}\leq 50$ is much more stringent that its
counterpart for the first bubble model. The rational is that the price
dynamics in terms of a finite-time singularity in the price momentum
corresponds to a weaker singularity that can only be observed, in the presence
of a strong stochastic component, rather close to the potential singularity.
This explains the smaller upper bound of $50$ trading days (corresponding
approximately to two calendar months). The lower bound of $-25$ days accounts
for the fact that the analysis is performed in sliding windows with a time
step of $25$ trading days.
The results shown in Figs.4-6 complement and refine those obtained with the
bubble model tested in the previous subsection. In general, there are less
alarms when using this second model and procedure, compared with the first
bubble model and procedure of subsection 4.1. One can observe in Fig.4 for the
S&P500 index two very well-defined clusters diagnosing a bubble ending with
the crash of October 1987 and another bubble ending in October 2007. The dates
indicated in the upper panel, Oct. 17, 1987 and Oct. 9, 2007, correspond to
the right time of the last window in which an alarm is found for each of these
two clusters. The timing is thus remarkably accurate in terms of the
determination of the end of each bubble regime. It is interesting that this
second model in terms of a momentum bubble singularity is able to diagnose
unambiguously a bubble ending in October 2007, while the first model diagnosed
only an intermediate phase of this price development. This bubble can be
referred to as the “real-estate-MBS” bubble (MBS stands for morgage-backed
security, Sornette and Woodard (2010)). Using the level 1 filter for the
positive feedback exponent $m$, we observe in addition three false alarms.
Raising the condition that $m$ should be larger than $2.5$ (respectively $3$)
removes two (respectively all) of these false alarms.
Fig.5 for the Nasdaq Composite index identifies the two bubbles ending in
March 2000 (the “new economy” ICT bubble) and in Oct. 2007, without any false
alarm.
Fig.6 for the Heng Seng of Hong Kong similarly diagnoses these two bubbles as
well as those ending with the 1992 and 1994 Asian events.
## 5 Concluding remarks
We have developed two rational expectation models of financial bubbles with
heterogeneous rational arbitrageurs. Two key ingredients characterize these
models: (i) the existence of a positive feedback quantified by a nonlinear
power law dependence of price growth as a function of either price or
momentum; (ii) the stochastic mean-reversion dynamics of the termination time
of the bubble. The first model characterizes a bubble as a faster-than-
exponential accelerating stochastic price ending in a finite-time singularity
at a stochastic critical time. The second model views a bubble as a regime
characterized by an accelerating momentum ending at a finite time singularity,
also with at a stochastic critical time. This second model has the additional
feature of taking into account the existence of some inertia in the price
formation process, which is related to the momentum effect.
In these two models, the heterogeneous arbitrageurs exhibit distinct
perception for the rising risk of a crash as the bubble develops. Each
arbitrageur is assumed to know the price formation process and to determine
her exit time so as to maximize her expected gain. The resulting distribution
of exit times lead to a synchronization problem, preventing arbitraging of the
bubble and allowing it to continue its course up to close to its potential
critical time.
The explicit analytical solutions of the two models allow us to propose
nonlinear transformations of the price time series into stochastic critical
time series. The qualification of a bubble regime then boils down to
characterize the nature of the transformed stochastic critical time series,
thereby avoiding the difficult problem of parameter estimation of the
stochastic differential equation describing the price dynamics. We develop an
operational procedure that qualifies the existence of a running bubble (i) if
the critical time series is found to reject a standard unit-root test at a
high confidence level, (ii) if the exponent $m$ of the nonlinear power law
characterizing the positive feedback is sufficient large and (iii) if the
expected critical time is not too distant from the time of the analysis.
The two procedures derived from the two bubble models have been applied to
three financial markets, the US S&P500 index from 1 February 1980 to 31
October 2008, the US NASDAQ composite index from 1 January 1980 to 31 July
2008 and the Hong Kong Hang Seng index from 1 December 1986 to 30 November
2008. Specifically, we have developed criteria to flag an alarm for the
presence of a bubble, that we validate by determining if the diagnosed bubble
is followed by a crash in short order. Remarkably, we find that the major
known crashes over these periods are correctly identified with few false
alarms. The method using the second bubble model in terms of a finite-time
singularity of the price momentum seems to be more reliable with fewer false
alarms and a better detection of the two principal bubbles phases
characterizing the last 30 years or so.
These results suggest the feasibility of advance bubble warning.
Acknowledgments The authors would like to thank Stefan Riemann for useful
discussions. We acknowledge financial support from the ETH Competence Center
“Coping with Crises in Complex Socio-Economic Systems” (CCSS) through ETH
Research Grant CH1-01-08-2. This work was partly supported by the National
Natural Science Foundation of China for Creative Research Group: Modeling and
Management for Several Complex Economic System Based on Behavior (Grant No.
70521001). Lin Li also appreciates the China Scholarship Council (CSC) for
supporting his studies at ETH Zurich (No. 2008602049).
## Appendix A Proof of Proposition 1
###### Proof.
Let $n=m-1$ and $z=p^{-n}$. Employing Itö lemma for equation (1), we have
$\displaystyle\mathrm{d}z$ $\displaystyle=\frac{\partial z}{\partial
p}\mathrm{d}p+\frac{1}{2}\frac{\partial^{2}p}{\partial
p^{2}}(\mathrm{d}p)^{2}$ (32) $\displaystyle=-np^{-m}(\mu
p^{m}[1+\delta(p,t)]dt+\sigma
p^{m}dW)-\frac{1}{2}n(-n-1)p^{-m-1}\sigma^{2}p^{2m}dt$ (33)
$\displaystyle=-n\mu(1+\delta(p,t)-\frac{1}{2}(n+1)\sigma^{2}\mu
p^{m-1})dt+n\sigma dW$ (34)
Recalling that
$\delta(p,t)=\alpha\tilde{t}_{c}+\frac{1}{2}m\mu\sigma^{2}p^{m-1}$, the
previous expression can be simplified into
$\displaystyle\mathrm{d}z$
$\displaystyle=-n\mu\mathrm{d}t+n\mu(-\alpha\tilde{t}_{c}\mathrm{d}t+{\sigma\over\mu}\mathrm{d}W)$
(35) $\displaystyle=-n\mu\mathrm{d}t+n\mu\,\mathrm{d}\tilde{t}_{c}$ (36)
Integrating both sides of the above equation and with $x(t=0)=p_{0}^{-n}$, we
obtain
$p=(n\mu)^{-\frac{1}{n}}[T_{c}+\tilde{t}_{c}-t]^{-\frac{1}{n}},\qquad
T_{c}=\frac{p_{0}^{-n}}{n\mu}~{}.$ (37)
Replacing $n$ by $\frac{1}{\beta}$, this reproduces the solution (5). ∎
## Appendix B Proof of Proposition 3
###### Proof.
We now check that equation (22) is the solution of the SDEs (15) and (16). For
this, we apply Itö lemma on equation (22) by regarding $\ln p_{t}$ as a
function of $\tilde{t}_{c}$. This leads to
$\displaystyle\mathrm{d}\ln p$ $\displaystyle=\frac{\partial p}{\partial
t}\mathrm{d}t+\frac{\partial
p}{\partial\tilde{t}_{c}}\mathrm{d}\tilde{t}_{c}+\frac{1}{2}\frac{\partial^{2}p}{\partial\tilde{t}_{c}^{2}}(\mathrm{d}\tilde{t}_{c})^{2}$
(38)
$\displaystyle=(1-\beta)B(T_{c}+\tilde{t}_{c}-t)^{-\beta}\mathrm{d}t-(1-\beta)B(T_{c}+\tilde{t}_{c}-t)^{-\beta}\mathrm{d}\tilde{t}_{c}$
(39)
$\displaystyle\qquad+\frac{1}{2}\beta(1-\beta)B(T_{c}+\tilde{t}_{c}-t)^{-\beta-1}(\sigma/\mu)^{2}\mathrm{d}t$
(40)
Taking into account that $B=\frac{1}{1-\beta}(\beta/\mu)^{\beta}$ and let $Z$
represent $(\beta/\mu)^{\beta}[T_{c}+\tilde{t}_{c}-t]^{-\beta}$, the above
expression can be rewritten as
$\mathrm{d}\ln
p=Z\left[1+\alpha\tilde{t}_{c}+\frac{1}{2}(\sigma^{2}/\mu)Z^{\frac{1}{\beta}}\right]\mathrm{d}t+Z(\sigma/\mu)\mathrm{d}W~{}.$
(41)
On the other hand, it is easy to see that $Z$ is the solution of (16) in the
light of Proposition 1, which leads to $Z=x$. Furthermore, we note that
$\alpha\tilde{t}_{c}+\frac{\sigma^{2}}{2\mu}Z^{\frac{1}{\beta}}$ is nothing
but $\gamma(Z,t)$. Therefore $\ln p_{t}$ given by (22) satisfies both (15) and
(16). ∎
## References
* Abreu and Brunnermeier (2003) Abreu, D., Brunnermeier, M. K., 2003. Bubbles and crashes. Econometrica 71 (1), 173–204.
* Almazan et al. (2004) Almazan, A., Brown, K. C., Carlson, M., Chapman, D. A., 2004. Why constrain your mutual fund manager? Journal of Financial Economics 73, 289–321.
* Andersen and Sornette (2004) Andersen, J. V., Sornette, D., 2004. Fearless versus fearful speculative financial bubbles. Physica A 337 (3-4), 565–585.
* Barberis et al. (1998) Barberis, N., Shleifer, A., Vishny, R., 1998. A model of investor sentiment. Journal of Financial Economics 49 (3), 307–343.
* Bender and Orszag (1999) Bender, C., Orszag, S., 1999. Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. Springer, Heidelberg.
* Bhattacharya and Yu (2008) Bhattacharya, U., Yu, X., 2008. The causes and consequences of recent financial market bubbles: An introduction. Review of Financial Studies 21 (1), 3–10.
* Blanchard and Watson (1982) Blanchard, O., Watson, M., 1982. Bubbles, rational expectations and speculative markets. In: Wachtel, P. (Ed.), Crisis in Economic and Financial Structure: Bubbles, Bursts and Shocks.
* Blanchard (1979) Blanchard, O. J., 1979. Speculative bubbles, crashes and rational expectations. Economics Letters 3 (387-389).
* Bouchaud and Cont (1998) Bouchaud, J.-P., Cont, R., 1998. A langevin approach to stock market fluctuations and crashes. European Physical Journal B 6, 543–550.
* Brock (1993) Brock, W., 1993. Pathways to randomness in the economy: emergent nonlinearity and chaos in economics and finance. Estud. Econom. 8 (3-55).
* Brock and Hommes (1999) Brock, W., Hommes, C., 1999. Rational animal spirits. In: P.J.J. Herings, G. van der Laan, A. T. (Ed.), The Theory of Markets. North-Holland, Amsterdam, pp. 109–137.
* Carhart (1997) Carhart, M., 1997. On persistence in mutual fund performance. The Journal of Finance 52 (1), 57–82.
* Chen et al. (2002) Chen, J., Hong, H., Stein, J. C., 2002. Breadth of ownership and stock returns. Journal of Financial Economics 66, 171–205.
* Chiarella et al. (2008) Chiarella, C., Dieci, R., He, X.-Z., 2008. Heterogeneity, market mechanisms, and asset price dynamicsResearch Paper 231, Quantitative Finance Research Center, University of Technology Sydney, September 2008.
* Cooper et al. (2004) Cooper, M. J., Gutierrez, R., Hameed, A., 2004. Market states and momentum. The Journal of Finance 59 (3), 1345–1365.
* Corcos et al. (2002) Corcos, A., Eckmann, J.-P., Malaspinas, A., Malevergne, Y., Sornette, D., 2002. Imitation and contrarian behavior: hyperbolic bubbles, crashes and chaos. Quantitative Finance 2, 264–281.
* Daniel et al. (1998) Daniel, K., Hirshleifer, D., Subrahmanyam, A., 1998. Investor psychology and security market under-and overreactions. Journal of Finance, 1839–1885.
* Dass et al. (2008) Dass, N., Massa, M., Patgiri, R., 2008. Mutual funds and bubbles: The surprising role of contractual incentives. Review of Financial Studies 21 (1), 51–99.
* De Bondt and Thaler (1985) De Bondt, W. M., Thaler, R., 1985. Does the stock market overreact? Journal of Finance 40, 793–805.
* DeLong et al. (1990) DeLong, J. B., Shleifer, A., Summers, L. H., Waldmann, R. J., 1990. Noise trader risk in financial markets. Journal of Political Economy 98 (4), 703–738.
* Duffie et al. (2002) Duffie, D., Garleanu, N., Pedersen, L. H., 2002. Securities lending, shorting, and pricing. Journal of Financial Economics 66, 307–339.
* Farmer (2002) Farmer, J., 2002. Market force, ecology and evolution. Industrial and Corporate Change 11 (5), 895–953.
* Granger and Newbold (1974) Granger, C., Newbold, P., 1974. Spurious regressions in econometrics. Journal of Econometrics 2, 111–120.
* Gurkaynak (2008) Gurkaynak, R. S., 2008. Econometric tests of asset price bubbles: taking stock. Journal of Economic Surveys 22 (1), 166–186.
* Harrison and Kreps (1978) Harrison, M., Kreps, D., 1978. Speculative investor behavior in a stock market with heterogeneous expectations. Quarterly Journal of Economics 92, 323–336.
* Hommes and Wagener (2008) Hommes, C., Wagener, F., 2008. Complex evolutionary systems in behavioral finance. In: Hens, T., Schenk-Hoppe, K. (Eds.), Tinbergen Institute Discussion Paper TI 2008-054/1, CeNDEF, School of Economics, University of Amsterdam, contributed chapter to the Handbook of Financial Markets: Dynamics and Evolution. Academic Press.
* Hong et al. (2005) Hong, H., Kubik, J. D., Stein, J. C., 2005. Thy neighbor’s portfolio: Word-of-mouth effects in the holdings and trades of money managers. Journal of Finance, 2801–2824.
* Ide and Sornette (2002) Ide, K., Sornette, D., 2002. Oscillatory finite-time singularities in finance, population and rupture. Physica A 307 (1-2), 63–106.
* Jarrow (1980) Jarrow, R., 1980. Heterogeneous expectations, restrictions on short sales, and equilibrium asset prices. Journal of Finance 35, 1105–1113.
* Jegadeesh and Titman (1993) Jegadeesh, N., Titman, S., 1993. Returns to buying winners and selling losers: Implications for stock market efficiency. The Journal of Finance 48 (1), 65–91.
* Jegadeesh and Titman (2001) Jegadeesh, N., Titman, S., 2001. Profitability of momentum strategies: An evaluation of alternative explanations. Journal of Finance 54, 699–720.
* Johansen et al. (2000) Johansen, A., Ledoit, O., Sornette, D., 2000. Crashes as critical points. International Journal of Teoretical and Applied Finance 3, 219–255.
* Johansen et al. (1999) Johansen, A., Sornette, D., Ledoit, O., 1999. Predicting financial crashes using discrete scale invariance. Journal of Risk (1), 5–32.
* Kaizoji and Sornette (2008) Kaizoji, T., Sornette, D., 2008. Market bubbles and crashes. In: the Encyclopedia of Quantitative Finance ( long version at $http://arXiv.org/abs/0812.2449$ ). Wiley press.
* Koski and Pontiff (1999) Koski, J., Pontiff, J., 1999. How are derivatives used? evidence from the mutual fund industry. Journal of Finance 54 (2), 791–816.
* Lintner (1969) Lintner, J., 1969. The aggregation of investors’ diverse judgments and preferences in purely competitive security markets. Journal of Financial and Quantitative Analysis 4, 347–400.
* Lux and Sornette (2002) Lux, T., Sornette, D., 2002. On rational bubbles and fat tails. Journal of Money, Credit and Banking, Part 1 34 (2), 589–610.
* Miller (1977) Miller, E., 1977. Risk, uncertainty and divergence of opinion. Journal of Finance 32, 1151–1168.
* Phillips (1986) Phillips, P., 1986. Understanding spurious regressions in econometrics. Journal of Econometrics 31, 311–340.
* Phillips et al. (2007) Phillips, P., Wu, Y., Yu, J., 2007. Explosive Behavior and the Nasdaq Bubble in the 1990s: When Did Irrational Exuberance Escalate Asset Values? Cowles Foundation for Research in Economics, Yale University 8.
* Roehner and Sornette (2000) Roehner, B. M., Sornette, D., 2000. “Thermometers” of speculative frenzy. The European Physical Journal B-Condensed Matter 16 (4), 729–739.
* Scheinkman and Xiong (2003) Scheinkman, J., Xiong, W., 2003. Overconfidence and speculative bubbles. Journal of Political Economy 111, 1183–1219.
* Shleifer and Vishny (1997) Shleifer, A., Vishny, R., 1997. Limits of arbitrage. Journal of Finance 52, 35–55.
* Sornette (2004) Sornette, D., 2004. Why Stock Markets Crash:Critical Events in Complex Financial Systems. Princeton University Press, Princeton and Oxford.
* Sornette and Andersen (2002) Sornette, D., Andersen, J. V., 2002. A nonlinear super-exponential rational model of speculative financial bubbles. International Journal of Modern Physics C 13 (2), 171–188.
* Sornette and Ide (2002) Sornette, D., Ide, K., 2002. Theory of self-similar oscillatory finite-time singularities in finance, population and rupture. International Journal of Modern Physics C 14 (3), 267–275.
* Sornette and Woodard (2010) Sornette, D., Woodard, R., 2010. Financial bubbles, real estate bubbles, derivative bubbles, and the financial and economic crisis. Proceedings of APFA7 (Applications of Physics in Financial Analysis), “New Approaches to the Analysis of Large-Scale Business and Economic Data”, Misako Takayasu, Tsutomu Watanabe and Hideki Takayasu, eds., Springer. ($http://papers.ssrn.com/sol3/papers.cfm?abstract\\_id=1407608$).
* Sullivan (2009) Sullivan, R., Summer 2009. Taming global village risk ii: understanding and mitigating bubbles. Journal of Portfolio Management 35 (4), 131–141.
* Xue (2003) Xue, H., October 2003. Identifying factors within the apt: A new approach. AFA 2004 San Diego Meetings, Available at SSRN: $http://ssrn.com/abstract=462561orDOI:10.2139/ssrn.462561$.
Figure 1: Logarithm of the historical S&P500 stock index and corresponding
alarms shown in the three lower panels as vertical lines indicating the ends
of the windows of 750 trading days in which our procedure using the first
bubble model of section 2 flags a diagnostic for the presence of bubble. The
three lower panels corresponds to alarms for which $T_{c,i}-t_{i}<750$,
$T_{c,i}-t_{i}<500$ and $T_{c,i}-t_{i}<250$, from top to bottom. By
definition, the set of alarms of the lowest panel is included in the set of
alarms of the middle panel which is itself included in the set of alarms of
the upper panel. The exponents $m$ found for the upper panel corresponding to
$T_{c,i}-t_{i}<750$ have a mean of $2.76$ with a standard deviation of $0.33$.
Figure 2: Same as Fig.1 for the Nasdaq Composite index. The exponents $m$
found for the upper panel corresponding to $T_{c,i}-t_{i}<750$ have a mean of
$2.85$ with a standard deviation of $0.23$.
Figure 3: Same as Fig.1 for the Heng Seng index of Hong Kong. The exponents
$m$ found for the upper panel corresponding to $T_{c,i}-t_{i}<750$ have a mean
of $2.84$ with a standard deviation of $0.22$.
Figure 4: Logarithm of the historical S&P500 stock index and corresponding
alarms shown in the three lower panels as vertical lines indicating the ends
of the windows of 900 trading days in which our procedure using the second
bubble model of section 3 flags a diagnostic for the presence of bubble. The
three lower panels corresponds to alarms for which $m>2$, $m>2.5$ and $m>3$,
from top to bottom. By definition, the set of alarms of the lowest panel is
included in the set of alarms of the middle panel which is itself included in
the set of alarms of the upper panel.
Figure 5: Same as Fig.4 for the Nasdaq Composite index.
Figure 6: Same as Fig.4 for the Heng Seng index of Hong Kong.
|
arxiv-papers
| 2009-11-10T15:00:01 |
2024-09-04T02:49:06.376802
|
{
"license": "Public Domain",
"authors": "Li Lin, Didier Sornette",
"submitter": "Li Lin",
"url": "https://arxiv.org/abs/0911.1921"
}
|
0911.1962
|
# Efficient Color-Dressed Calculation of Virtual Corrections
Walter T. Giele Fermilab, Batavia, IL 60510, USA Zoltan Kunszt Institute
for Theoretical Physics, ETH, CH-8093 Zürich, Switzerland Theoretical
Physics, CERN, CH-1211 Geneva, Switzerland Jan Winter
###### Abstract
With the advent of generalized unitarity and parametric integration
techniques, the construction of a generic Next-to-Leading Order Monte Carlo
becomes feasible. Such a generator will entail the treatment of QCD color in
the amplitudes. We extend the concept of color dressing to one-loop
amplitudes, resulting in the formulation of an explicit algorithmic solution
for the calculation of arbitrary scattering processes at Next-to-Leading
order. The resulting algorithm is of exponential complexity, that is the
numerical evaluation time of the virtual corrections grows by a constant
multiplicative factor as the number of external partons is increased. To study
the properties of the method, we calculate the virtual corrections to
$n$-gluon scattering.
Fermilab-PUB-09-406-T
## 1 Introduction
Automated Leading Order (LO) generators [1, 2, 3, 4, 5] play an essential role
in experimental analyses and phenomenology in general. However, the
theoretical uncertainties associated with these generators are only understood
qualitatively. The augmentation of the LO generators with Next-to-Leading
Order (NLO) corrections will give a more quantitative understanding of the
theoretical uncertainties. This is crucial for the realization of precision
measurements at the Hadron colliders. By calculating NLO corrections using
analytic generalized unitarity methods [6, 7, 8], the one-loop amplitude is
factorized into sums over products of on-shell tree-level amplitudes. This
makes the integration of numerical generalized unitarity methods into the LO
generators attractive. One can use the LO generator as the building block for
obtaining the NLO correction, thereby negating the need for a separate
generator of all the one-loop Feynman diagrams. The generalized unitarity
approach reduces the complexity of the calculation through factorization. It
can reduce the evaluation time with increasing number of external particles
from faster than factorial growth to slower than factorial growth.
By utilizing the parametric integration method of Ref. [9] significant
progress has been made in the algorithmic implementation of generalized
unitarity based one-loop generators [10, 11] and other non-unitary methods
[12].111These methods have matured to the point where explicit NLO parton
generators for specific processes have been constructed [13, 14, 15, 16, 17].
These implementations rely on the color decomposition of the amplitude into
colorless, gauge invariant ordered amplitudes [18, 19]. At tree-level these
ordered amplitudes can be efficiently calculated by recursion relation
algorithms [20]. These algorithms are of polynomial complexity and grow
asymptotically as $n^{4}$ as the number of external partons, $n$, increases
[21]. By replacing the 4-gluon vertex by an effective 3-gluon vertex the
polynomial growth factor can be further reduced to $n^{3}$ [22, 23, 24].
At the one-loop level the ordered amplitudes generalize into primitive
amplitudes [25]. These primitive amplitudes reflect the more complicated
dipole structure of one-loop amplitudes. While the analytic structure of the
factorized one-loop amplitude in color factors and primitive amplitudes is
systematic, the subsequent calculation of the color summed virtual corrections
becomes unwieldy in the algorithmic implementation [26]. The reason for this
is the rapid growth in the number of primitive amplitudes. This rapid growth
is mainly caused by the multiple quark-pairs amplitudes. A further
complication arises from the possible presence of electro-weak particles in
the ordered amplitudes.
While in LO generators the analytic treatment of color is more manageable,
alternatives were developed for high parton multiplicity scattering amplitudes
[23, 27, 24]. Those alternatives provided a more numerical treatment of the
color, thereby facilitating the construction of tree-level Monte Carlo
programs for the automated generation of high multiplicity parton scattering
amplitudes at LO. This was accomplished by not only choosing the external
momenta and helicities, but also choosing the explicit colors of the external
partons for each scattering event considered. In doing so, the tree-level
partonic amplitude is a complex number and the absolute value squared is
simply calculated. This numerical treatment can be done in the context of
ordered amplitudes [28] by calculating the explicit color weights of each
ordered amplitude. This method was generalized to one-loop calculations in
Ref. [12]. More directly, one can reformulate the recursion relations into
color-dressed recursion relations [29, 23, 24]. These color-dressed recursion
relations integrate the now explicit color weights into the recursive formula.
The resulting algorithm is of exponential complexity and grows asymptotically
as $4^{n}$ for $n$-parton amplitudes; again, a reduction of the growth factor
to $3^{n}$ can be achieved if the 4-gluon vertex is replaced by the effective
3-gluon vertex [24].
In this paper we extend the generalized unitarity method of Ref. [10] as
implemented in Ref. [30] to incorporate the color-dressing method. The
algorithm is developed such that it can augment a dressed LO generator such as
C OMIX [5] to become a NLO generator.222The LO matrix-element generator needs
to be upgraded to allow for complex external momenta. For the numerical
examples presented in this paper, we have used our own implementation of a
color-dressed LO gluon recursion relation to calculate the virtual corrections
for $n$-gluon scattering processes.
The motivation for color dressing at the one-loop level is discussed in Sec.
2. We outline in Sec. 3 the tree-level dressed recursion relations for generic
theories expressed in terms of Feynman diagrams. We optimize the color-
sampling performance and study the phase-space integration convergence for LO
$n$-gluon scattering. The dressed formalism is extended to one-loop amplitudes
in Sec. 4. The scaling with $n$, the accuracy of the algorithm and the color-
sampling convergence of the virtual corrections to $n$-gluon scattering are
studied in some detail. We summarize our results in Sec. 5. Finally, two
appendices are added giving an explicit LO 6-quark example and details on the
color-dressed implementation of the gluon recursion relation.
## 2 Motivation for the Color-Dressed Generalized Unitarity Method
So far the numerical implementations of generalized unitarity for the
evaluation of one-loop amplitudes make use of color ordering: the ordered one-
loop amplitudes are constructed from tree-level ordered amplitudes through the
$D$-dimensional unitarity cuts. This has the advantage that the color is
factorized off the loop calculation and attached subsequently to each ordered
one-loop amplitude. For the pure gluon one-loop amplitude, this leads to a
particularly simple decomposition in terms of the adjoint generators $F$ of
${\rm SU}(N)$:
${\cal M}^{(0,1)}(1,2\ldots,n)\ \ \sim\sum_{P(2,3,\ldots,n)}{\rm
Tr}\left(F^{a_{1}}F^{a_{2}}\cdots F^{a_{n}}\right)m^{(0,1)}(1,2,\ldots,n)\ .$
(1)
The decomposition is valid for both tree-level [18] and one-loop amplitudes
[31]. Once we can calculate the colorless ordered amplitude $m(1,2,\ldots,n)$,
all other ordered amplitudes are obtained by simple permutations. All
kinematic information about the $n$-gluon amplitude is encapsulated in a
single ordered amplitude. However, we also see the drawback of this approach
as we are interested in evaluating the amplitude squared. We have to calculate
${\cal M}^{(0,1)}(1,2\ldots,n)\times\left({\cal
M}^{(0)}(1,2\ldots,n)\right)^{\dagger}$ summed over all color and spin states
of the external gluons. This immediately leads to a factorial complexity when
doing the multiplications of the full amplitudes as we have to sum over the
permutations, $P(2,3,\ldots,n)$, of the ordered amplitudes. Additionally, the
color sum has to be performed either analytically or in some numerical manner.
When including quark pairs the situation becomes even more complicated. The
reason is that the internal structure of the one-loop amplitude is not
uniquely defined by the external states, thereby affecting the color flow of
the ordered amplitudes. As a result there exist many types of ordered
amplitudes depending on the internal configuration of quark and gluon
propagators. These amplitudes are called primitive amplitudes [25] and in
general cannot be obtained from each other by simple permutations. For
example, the one-loop $q\bar{q}+n$ gluon amplitude is given by [31]
$\displaystyle{\cal M}^{(1)}(q;1,\ldots,n;\bar{q}\;\\!)\ \ \sim$ (2)
$\displaystyle\sum_{k=2}^{n}\;\sum_{P(1,\ldots,n)}\left(T^{y}T^{a_{1}}\cdots
T^{a_{k}}T^{x}\right)_{ij}\left(F^{a_{k+1}}\cdots
F^{a_{n}}\right)_{xy}m^{(1)}(q,1,\ldots,k,q,k+1,\ldots,n)\ .$
where the $T$-matrices are the fundamental generators of ${\rm SU}(N)$. While
for the full amplitude a cut line has an undetermined flavor, each primitive
amplitude has an unique flavor for all the cut lines. Therefore we can apply
generalized unitarity to the primitive amplitudes. However, from a
numerical/algorithmic point of view the evaluation of this equation becomes
tedious as can be seen for instance in the calculation of the one-loop matrix
elements for $W+5$ partons in Ref. [26].
It is clear that for an automated generator of one-loop corrections one would
like to avoid ordered/primitive amplitudes altogether. For LO matrix elements,
this can be done by applying the color-dressed recursion relations to evaluate
the (unordered) tree-level amplitudes. From these color-dressed tree-level
amplitudes we can build the one-loop color-dressed amplitudes by applying
generalized unitarity, thereby circumventing the need for primitive amplitudes
and explicit color summations. It is of interest to investigate the
feasibility of this approach. The $n$-gluon scattering process is good for
studying the behavior of the dressed algorithm. The color-ordered approach is
most effective for $n$-gluon scattering. For processes with quark-pairs, the
color-dressed approach will become even more efficient compared to the color-
ordered approach.
An additional advantage of the color-dressed algorithm is that it treats
partons and color neutral particles on the same footing. Specifically, we can
include electro-weak particles without altering the algorithm. This is in
contrast to the color-ordered algorithm, where the addition of electro-weak
particles would lead to significant modifications in the algorithmic
implementation of the method.
## 3 Dressed Recursive Techniques for Leading Order Amplitudes
In tree-level generators the Monte Carlo sampling over the external color and
helicity states has become a standard practice [23, 27, 24]. Such a color
sampling allows for the efficient evaluation of large multiplicity partonic
processes. A particular efficient implementation of the color-dressed Monte
Carlo method uses the color-flow decomposition of the multi-parton amplitudes
[23, 32, 24].
The principle of Monte Carlo sampling over the states of the external sources
generalizes to any theory expressible through Feynman rules. By explicitly
specifying the quantum numbers of the $n$ external sources, one can evaluate
the tree-level amplitude squared and differential cross section using Monte
Carlo sampling:
$\displaystyle d\,\sigma_{\rm LO}(f_{1}f_{2}\rightarrow f_{3}\cdots f_{n})\ \
=$ (3) $\displaystyle\frac{W_{\rm S}}{N_{\rm event}}\times\sum_{r=1}^{N_{\rm
event}}d\,PS^{(r)}(K_{1}K_{2}\rightarrow K_{3}\cdots K_{n})\left|\;\\!{\cal
M}^{(0)}\left({\bf f}_{1}^{(r)},{\bf f}_{2}^{(r)},\ldots,{\bf
f}_{n}^{(r)}\right)\right|^{2}\ ,$
where
${\bf
f}_{i}^{(r)}\;=\;\left\\{f_{i},h_{f_{i}},C_{f_{i}},K_{f_{i}}\right\\}^{(r)}$
(4)
denotes the flavor, spin, color and momentum four-vector of external state $i$
for event $r$.333We will use flavor to indicate the particle type, such as
e.g. gluon, up-quark, $W$-boson, etc. The constant $W_{\rm S}$ contains the
appropriate identical particle factors and Monte Carlo sampling weights. For
each event $r$, the external states are stochastically chosen such that when
summed over many events we approximate the correct differential cross section
with sufficient accuracy.
### 3.1 The Generic Recursive Formalism
To calculate the tree-level amplitude ${\cal M}^{(0)}$ in Eq. (3), we follow
the method of color-dressed recursion relations as detailed in Refs. [24, 5].
A recursion relation builds multi-particle currents from other currents. The
$m$-particle current $J_{\bf g}\left({\bf f}_{\pi}\right)$ has $m$ on-shell
particles ${\bf f}_{\pi}=\\{{\bf f}_{i}\\}_{i\in\pi}=\\{{\bf
f}_{i_{1}},\ldots,{\bf f}_{i_{m}}\\}$ where $\pi=\\{i_{1},\ldots,i_{m}\\}$ and
one off-shell particle ${\bf g}=\left\\{g,L_{g},C_{g},K_{g}\right\\}$ with
$g$, $L_{g}$, $C_{g}$ and $K_{g}$ denoting the flavor, Lorentz label, color
and four-momentum, respectively. The momentum of the off-shell particle,
$K_{g}$, is constrained by momentum conservation:
$K_{g}=-K_{\pi}=-\sum_{i\in\pi}K_{i}$.
The dressed recursion relation generates currents using the propagators and
interaction vertices of the theory. Using standard tensor notation we can
write the propagators as
$\displaystyle P^{{\bf g}_{1}{\bf g}_{2}}(Q)$ $\displaystyle=$
$\displaystyle\delta_{g_{1}g_{2}}\delta_{C_{g_{1}}C_{g_{2}}}P^{L_{g_{1}}L_{g_{2}}}(Q)\
,$ $\displaystyle P^{{\bf g}}\big{[}J({\bf f}_{\pi})\big{]}$ $\displaystyle=$
$\displaystyle\sum_{{\bf g}_{1}}P^{{\bf g}{\bf g}_{1}}(K_{\pi})J_{{\bf
g}_{1}}\big{(}{\bf f}_{\pi}\big{)}\ ,$ $\displaystyle P\big{[}J({\bf
f}_{\pi_{1}}),J({\bf f}_{\pi_{2}})\big{]}$ $\displaystyle=$
$\displaystyle\sum_{{\bf g}_{1}{\bf g}_{2}}J_{{\bf g}_{1}}\big{(}{\bf
f}_{\pi_{1}}\big{)}P^{{\bf g}_{1}{\bf g}_{2}}(K_{\pi_{1}})J_{{\bf
g}_{2}}\big{(}{\bf f}_{\pi_{2}}\big{)}\ ,$ (5)
where e.g. the gluon propagator is given by
$P^{\mu_{1}\mu_{2}}(Q)=-g^{\mu_{1}\mu_{2}}/Q^{2}$. Note that the particle sums
are taken over all quantum numbers of the off-shell particles ${\bf g}_{i}$.
Furthermore, in all expressions momentum conservation is always implicitly
understood. The on-shell tree-level $n$-particle amplitude can hence be
expressed in terms of an $(n-1)$-current,
${\cal M}^{(0)}\big{(}{\bf f}_{1},\ldots,{\bf
f}_{n}\big{)}\;=\;P^{-1}\left[J\big{(}{\bf f}_{1},\ldots,{\bf
f}_{n-1}\big{)},J\big{(}{\bf f}_{n}\big{)}\right]\ .$ (6)
We denote the interaction vertices of the theory as $V_{{\bf g}_{1}\cdots{\bf
g}_{k}}(Q_{1},\ldots,Q_{k})$. The maximal number of legs for the allowed
vertices of the theory is denoted by $V_{\rm max}$. The number of legs of the
vertex is indicated by the number of its arguments and the type of vertex is
specified by the quantum numbers of the legs. The labels ${\bf
g}_{1},\ldots,{\bf g}_{k}$ run over the values of all particles of the theory.
Symmetries and renormalizability imply that many of the vertices are set to
zero. The theory is defined by its particle content and its non-vanishing
vertices, which are generalized tensors:
$V_{{\bf g}_{1}\cdots{\bf g}_{k}}(Q_{1},\ldots,Q_{k})\;=\;V_{g_{1}\cdots
g_{k};C_{g_{1}}\cdots C_{g_{k}}}^{L_{g_{1}}\cdots
L_{g_{k}}}(Q_{1},\ldots,Q_{k})\ .$ (7)
The sum of all vertices contracted in with currents constitutes the main
building block of the recursion relation. We define it as
$D_{\bf g}\big{[}J({\bf f}_{\pi_{1}}),\ldots,J({\bf
f}_{\pi_{k}})\big{]}\;=\sum_{{\bf g}_{1}\cdots{\bf g}_{k}}V_{{\bf
gg}_{1}\cdots{\bf
g}_{k}}(K_{g}=-K_{\Pi_{k}},K_{\pi_{1}},\ldots,K_{\pi_{k}})\times J^{{\bf
g}_{1}}\big{(}{\bf f}_{\pi_{1}}\big{)}\times\cdots\times J^{{\bf
g}_{k}}\big{(}{\bf f}_{\pi_{k}}\big{)}\ ,$ (8)
where the inclusive list $\Pi_{k}$ is build up of unions of the exclusive
lists:
$\Pi_{k}=\bigcup_{i=1}^{k}\pi_{i}\ .$ (9)
Fig. 1 is a pictorial representation of Eq. (8) when using the example of QCD.
For this case, we will work out the generic vertex blob in detail in the next
subsection.
Figure 1: A graphical representation of Eq. (8) for $k=2$ and an off-shell
gluon in QCD. Because of flavor conservation only one of the two vertices can
contribute for any given partition. Figure 2: The first recursion step for
the unordered gluon current with $u,\bar{d},s,\bar{s}$ quarks and a $W^{-}$
gauge boson in the final state. There are 15 contributions corresponding to
all possible partitions of the final-state particles into two groups. Because
of flavor conservation there are only 4 non-vanishing contributions for the
“4+1” partitions (first term) and 2 non-vanishing contributions for the “3+2”
partitions (second term).
The recursion relations terminate with the one-particle currents. A one-
particle ${\bf g}$-current is defined in terms of the source
$S^{h_{f_{i}}C_{f_{i}}}_{f_{i}L_{g}}(K_{f_{i}})$. Hence, we have
$J_{\bf g}\big{(}{\bf
f}_{i}\big{)}\;=\;\delta^{gf_{i}}\delta^{C_{g}C_{f_{i}}}\,S^{h_{f_{i}}C_{f_{i}}}_{f_{i}L_{g}}(K_{f_{i}})\
.$ (10)
For example, the ${\bf g}_{1}$-gluon one-particle source with helicity
$\lambda_{1}$, color $c_{1}$ and momentum $K_{1}$ is given by $J_{\bf g}({\bf
g}_{1})=\delta^{cc_{1}}\epsilon^{\lambda_{1}}_{\mu_{1}}(K_{1})$. I.e. the
${\bf g}_{1}$-gluon source is a matrix in color space multiplied by the
helicity vector.
The $n$-particle currents are now efficiently calculated from a recursively
defined current in the following manner:
$J_{\bf g}\big{(}{\bf f}_{1},\ldots,{\bf f}_{n}\big{)}\;=\;\sum_{k=2}^{V_{\rm
max}-1}\sum_{P_{\pi_{1}\cdots\pi_{k}}(1,\ldots,n)}^{{\cal S}_{2}(n,k)}P_{\bf
g}\Big{[}D\big{[}J({\bf f}_{\pi_{1}}),\ldots,J({\bf
f}_{\pi_{k}})\big{]}\Big{]}\ ,$ (11)
where ${\cal S}_{2}(n,k)$ is the Stirling number of the second kind. The first
recursive step is graphically illustrated in Fig. (2) for the example of
$J_{\bf g}({\bf u},{\bf\bar{d}},{\bf s},{\bf\bar{s}},{\bf W^{-}})$. The sum
over $P_{\pi_{1}\cdots\pi_{k}}(1,\ldots,n)$ generates all different partitions
decomposing the set $\\{1,\ldots,n\\}$ into the non-empty subsets
$\pi_{1},\ldots,\pi_{k}$. An example for a list of different partitions is
$\displaystyle P_{\pi_{1}\pi_{2}\pi_{3}}(1,2,3,4)$ $\displaystyle=$
$\displaystyle\left\\{\pi_{1}^{(i)}\pi_{2}^{(i)}\pi_{3}^{(i)}\right\\}_{i=1}^{{\cal
S}_{2}(4,3)=6}$ (12) $\displaystyle=$ $\displaystyle\Big{\\{}\
\big{\\{}\\{1,2\\}\\{3\\}\\{4\\}\big{\\}},\,\big{\\{}\\{1,3\\}\\{2\\}\\{4\\}\big{\\}},\,\big{\\{}\\{1,4\\}\\{2\\}\\{3\\}\big{\\}},$
$\displaystyle\hphantom{\Big{\\{}\
}\big{\\{}\\{2,3\\}\\{1\\}\\{4\\}\big{\\}},\,\big{\\{}\\{2,4\\}\\{1\\}\\{3\\}\big{\\}},\,\big{\\{}\\{3,4\\}\\{1\\}\\{2\\}\big{\\}}\
\Big{\\}}\ .$
The formalism described here fully specifies an automated algorithm of
exponential complexity to calculate the LO differential cross-sections for any
theory defined in terms of Feynman rules. Owing to the characteristics of the
partitioning, the computer resources needed to calculate the $n$-particle
tree-level amplitudes asymptotically grow in proportion to ${\cal
S}_{2}(n,V_{\rm max})$. The exponential behavior arises from the large-$n$
limit of the Stirling numbers, i.e. ${\cal S}_{2}(n,V_{\rm max})\rightarrow
V_{\rm max}^{n}$ [22]. It may be possible to reduce $V_{\rm max}$ by rewriting
higher multiplicity vertices as sums of lower multiplicity vertices thereby
improving the efficiency of the recursive algorithm [23, 24]. For the case of
the Standard model this has been fully worked out in Ref. [5] and implemented
in the C OMIX LO generator.
### 3.2 Multi-Jet Scattering Amplitudes
We specify the generic recursion relations to the perturbative QCD Feynman
rules. This will give an algorithmic description of the scattering amplitudes
at LO for multi-jet production at hadron colliders.
The external sources are gluons and massless quarks. All these particles have
color and helicity as quantum numbers. Instead of the traditional color
representation in terms of fundamental generators, we choose the color-flow
representation [3, 32, 24], which is more pertinent to Monte Carlo sampling
and easily derivable from the traditional color representation by making the
following two observation: first, any internal propagating gluon has as a
color factor $\delta^{ab}={\rm Tr}\Big{(}T^{a}T^{b}\Big{)}$.444Because of this
normalization, the structure constants $f^{abc}$ are a factor of $\sqrt{2}$
larger than in the conventional definition. This color factor can be rewritten
as
${\cal M}\;=\;{\cal A}_{a}\frac{\delta^{ab}}{K^{2}}{\cal B}_{b}\;=\;{\cal
A}_{a}\frac{{\rm Tr}\left(T^{a}T^{b}\right)}{K^{2}}{\cal B}_{b}\;=\;{\cal
A}_{ij}\frac{1}{K^{2}}{\cal B}^{ji}\ .$ (13)
Second, we contract the amplitude with $T^{a_{k}}_{i_{k}j_{k}}$ for each
external gluon:
$|{\cal M}|^{2}\;=\;{\cal M}^{a}\,\delta_{ab}\left({\cal
M}^{b}\right)^{\dagger}\;=\;{\cal M}^{a}\,T^{a}_{ij}T^{b}_{ji}\left({\cal
M}^{b}\right)^{\dagger}\;=\;{\cal M}_{ij}\,{\cal M}_{ji}^{\dagger}\ .$ (14)
From these observations it follows that we can calculate the interaction
vertices in the color-flow representation by simply contracting each gluon
with $T^{a_{k}}_{i_{k}j_{k}}$ and summing over $a_{k}$. The three gluon vertex
is thus given by
$\displaystyle V_{{\bf g}_{1}{\bf g}_{2}{\bf g}_{3}}(K_{1},K_{2},K_{3})$
$\displaystyle=$ $\displaystyle
V_{i_{1}j_{1}i_{2}j_{2}i_{3}j_{3}}^{\mu_{1}\mu_{2}\mu_{3}}(K_{1},K_{2},K_{3})$
(15) $\displaystyle=$ $\displaystyle
T^{a_{1}}_{i_{1}j_{1}}T^{a_{2}}_{i_{2}j_{2}}T^{a_{3}}_{i_{3}j_{3}}\,V_{a_{1}a_{2}a_{3}}^{\mu_{1}\mu_{2}\mu_{3}}(K_{1},K_{2},K_{3})$
$\displaystyle=$ $\displaystyle
T^{a_{1}}_{i_{1}j_{1}}T^{a_{2}}_{i_{2}j_{2}}T^{a_{3}}_{i_{3}j_{3}}\,f^{a_{1}a_{2}a_{3}}\,\sqrt{2}\,\widehat{V}_{3}^{\mu_{1}\mu_{2}\mu_{3}}(K_{1},K_{2},K_{3})$
$\displaystyle=$
$\displaystyle\left(\delta^{i_{1}}_{j_{2}}\delta^{i_{2}}_{j_{3}}\delta^{i_{3}}_{j_{1}}-\delta^{i_{1}}_{j_{3}}\delta^{i_{2}}_{j_{1}}\delta^{i_{3}}_{j_{2}}\right)\widehat{V}_{3}^{\mu_{1}\mu_{2}\mu_{3}}(K_{1},K_{2},K_{3})\
,$
with
$\widehat{V}_{3}^{\mu_{1}\mu_{2}\mu_{3}}(K_{1},K_{2},K_{3})\;=\;\frac{1}{\sqrt{2}}\Big{(}(K_{1}-K_{2})^{\mu_{3}}g^{\mu_{1}\mu_{2}}+(K_{2}-K_{3})^{\mu_{1}}g^{\mu_{2}\mu_{3}}+(K_{3}-K_{1})^{\mu_{2}}g^{\mu_{3}\mu_{1}}\Big{)}\
.$ (16)
Similarly, for the four gluon vertex we find
$V_{{\bf g}_{1}{\bf g}_{2}{\bf g}_{3}{\bf
g}_{4}}\;=\;V_{i_{1}j_{1}i_{2}j_{2}i_{3}j_{3}i_{4}j_{4}}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}\;=\sum_{C(234)}\left(\delta^{i_{1}}_{j_{2}}\delta^{i_{2}}_{j_{3}}\delta^{i_{3}}_{j_{4}}\delta^{i_{4}}_{j_{1}}+\delta^{i_{1}}_{j_{4}}\delta^{i_{2}}_{j_{1}}\delta^{i_{3}}_{j_{2}}\delta^{i_{4}}_{j_{3}}\right)\widehat{V}_{4}^{\mu_{1}\mu_{3},\,\mu_{2}\mu_{4}}\
,$ (17)
with
$\widehat{V}_{4}^{\mu_{1}\mu_{2},\,\mu_{3}\mu_{4}}\;=\;2\,g^{\mu_{1}\mu_{2}}g^{\mu_{3}\mu_{4}}-g^{\mu_{1}\mu_{3}}g^{\mu_{2}\mu_{4}}-g^{\mu_{1}\mu_{4}}g^{\mu_{2}\mu_{3}}\
,$ (18)
and the sum is over the cyclic permutation of the indices $\\{2,3,4\\}$. In
the color-flow representation the quark-antiquark-gluon vertex is given by
$V_{\bf
qg\bar{q}}\;=\;V_{i,\,i_{1}j_{1},\,j}^{s\mu\bar{s}}\;=\;\left(\delta_{ij_{1}}\delta_{i_{1}j}-\frac{1}{N_{\rm
C}}\,\delta_{i_{1}j_{1}}\delta_{ij}\right)\widehat{V}^{s\mu\bar{s}}\ ,$ (19)
with
$\widehat{V}_{s\bar{s}}^{\mu}\;=\;\frac{1}{\sqrt{2}}\,\gamma_{s\bar{s}}^{\mu}\
.$ (20)
The external sources are given by
$\displaystyle J_{\bf g}({\bf g}_{1})$ $\displaystyle=$
$\displaystyle\delta^{Ii_{1}}\delta^{Jj_{1}}\varepsilon_{\mu}^{\lambda_{1}}(K_{1})\
,$ $\displaystyle J_{\bf q}({\bf q}_{1})$ $\displaystyle=$
$\displaystyle\delta^{Ii_{1}}v_{s}^{\lambda_{1}}(K_{1})\ ,$ $\displaystyle
J_{\bf\bar{q}}({\bf\bar{q}}_{1})$ $\displaystyle=$
$\displaystyle\delta^{Jj_{1}}\bar{u}_{\bar{s}}^{\lambda_{1}}(K_{1})\ ,$ (21)
where ${\bf g}=\\{g,\mu,(IJ),-K_{1}\\}$, ${\bf
g}_{1}=\\{g_{1},\lambda_{1},(i_{1}j_{1}),K_{1}\\}$, ${\bf
q}=\\{q,s,I,-K_{1}\\}$, ${\bf q}_{1}=\\{q_{1},\lambda_{1},i_{1},K_{1}\\}$,
${\bf\bar{q}}=\\{\bar{q},\bar{s},J,-K_{1}\\}$ and
${\bf\bar{q}}_{1}=\\{\bar{q}_{1},\lambda_{1},j_{1},K_{1}\\}$. The internal
propagating particles are given by
$\displaystyle P^{{\bf g}_{1}{\bf g}_{2}}(Q)$ $\displaystyle=$
$\displaystyle\delta^{i_{1}}_{j_{2}}\delta^{i_{2}}_{j_{1}}\left(\frac{-g_{\mu_{1}\mu_{2}}}{Q^{2}}\right)\
,$ $\displaystyle P^{{\bf q}_{1}{\bf q}_{2}}(Q)$ $\displaystyle=$
$\displaystyle\delta^{i_{1}}_{i_{2}}\left(Q\\!\\!\\!\\!/\,-m_{q_{1}}\right)^{-1}_{s_{1}s_{2}}\
,$ $\displaystyle P^{{\bf\bar{q}}_{1}\bar{\bf q}_{2}}(Q)$ $\displaystyle=$
$\displaystyle\delta^{j_{1}}_{j_{2}}\left(Q\\!\\!\\!\\!/\,+m_{\bar{q}_{1}}\right)^{-1}_{\bar{s}_{1}\bar{s}_{2}}\
.$ (22)
with ${\bf g}_{k}=\\{g_{k},\mu_{k},(i_{k}j_{k}),Q\\}$, ${\bf
q}_{k}=\\{q_{k},s_{k},i_{k},Q\\}$ and
${\bf\bar{q}}_{k}=\\{\bar{q}_{k},\bar{s}_{k},j_{k},Q\\}$.
We can now construct Berends–Giele recursion relations [20] using color-
dressed multi-parton currents based on Eq. (11). The result is
$\displaystyle J_{\bf q}\big{(}{\bf f}_{1},\ldots,{\bf f}_{n}\big{)}$
$\displaystyle=$ $\displaystyle\sum_{P_{\pi_{1}\pi_{2}}(1,\ldots,n)}P_{\bf
q}\Big{[}D\big{[}J({\bf f}_{\pi_{1}}),J({\bf f}_{\pi_{2}})\big{]}\Big{]}\ ,$
$\displaystyle J_{\bf g}\big{(}{\bf f}_{1},\ldots,{\bf f}_{n}\big{)}$
$\displaystyle=$ $\displaystyle\sum_{P_{\pi_{1}\pi_{2}}(1,\ldots,n)}P_{\bf
g}\Big{[}D\big{[}J({\bf f}_{\pi_{1}}),J({\bf f}_{\pi_{2}})\big{]}\Big{]}\ ,$
(23) $\displaystyle+$
$\displaystyle\sum_{P_{\pi_{1}\pi_{2}\pi_{3}}(1,\ldots,n)}P_{\bf
g}\Big{[}D\big{[}J({\bf f}_{\pi_{1}}),J({\bf f}_{\pi_{2}}),J({\bf
f}_{\pi_{3}})\big{]}\Big{]}\ ,$
where each current violating flavor conservation is defined to give zero. The
compact operator language can be expanded out to an explicit formula by adding
back in the particle attributes. For example,
$\displaystyle P_{\bf g}\Big{[}D\big{[}J({\bf f}_{\pi_{1}}),J({\bf
f}_{\pi_{2}})\big{]}\Big{]}$ $\displaystyle=$ $\displaystyle\sum_{{\bf
qg}_{1}{\bf\bar{q}}}P_{{\bf gg}_{1}}(K_{\Pi_{2}})\,V^{{\bf
qg}_{1}{\bf\bar{q}}}\,J_{\bf q}({\bf f}_{\pi_{1}})\,J_{\bf\bar{q}}({\bf
f}_{\pi_{2}})$ (24) $\displaystyle+$ $\displaystyle\sum_{{\bf g}_{1}{\bf
g}_{2}{\bf g}_{3}}P_{{\bf gg}_{1}}(K_{\Pi_{2}})\,V^{{\bf g}_{1}{\bf g}_{2}{\bf
g}_{3}}(-K_{\pi_{1}\cup\pi_{2}},K_{\pi_{1}},K_{\pi_{2}})\,J_{{\bf g}_{2}}({\bf
f}_{\pi_{1}})\,J_{{\bf g}_{3}}({\bf f}_{\pi_{2}})$ $\displaystyle=$
$\displaystyle\frac{1}{K_{\Pi_{2}}^{2}}V^{s_{1}\mu s_{2}}_{i,\,IJ,\,j}\times
J_{s_{1}}^{i}({\bf f}_{\pi_{1}})\times J_{s_{2}}^{j}({\bf f}_{\pi_{2}})$
$\displaystyle+$
$\displaystyle\frac{1}{K_{\Pi_{2}}^{2}}V^{\mu\mu_{1}\mu_{2}}_{IJi_{2}j_{2}i_{3}j_{3}}(-K_{\pi_{1}\cup\pi_{2}},K_{\pi_{1}},K_{\pi_{2}})\times
J_{\mu_{1}}^{(ij)_{2}}({\bf f}_{\pi_{1}})\times J_{\mu_{2}}^{(ij)_{3}}({\bf
f}_{\pi_{2}})\ .$
The $n$-parton tree-level matrix element is calculated using Eq. (6). We
exemplify in appendix A how to work out the 6-quark recursion steps using the
above formalism.
### 3.3 Numerical Implementation of $n$-gluon Scattering
The method of color dressing as discussed in this section relies on the
ability to perform a Monte Carlo sampling over the degrees of freedom of the
external sources. In this subsection we will study in some detail the
properties of such a sampling approach by means of the color-dressed gluonic
recursion relation. We are particularly interested in the accuracy of the
color-sampling procedure and overall speed of the implementation. The addition
of quarks and external vector bosons is a straightforward extension and will
not affect the conclusions reached in this subsection.
The explicit color-dressed gluon recursion algorithm is given in terms of
colored gluonic currents. The gluonic currents are $3\times 3$ matrices in
color space and defined as
$\displaystyle J_{\bf g}\big{(}{\bf g}_{m}\big{)}$ $\displaystyle=$
$\displaystyle\delta^{Ii_{m}}\delta^{Jj_{m}}\,\varepsilon_{\mu}^{\lambda_{m}}(K_{m})\
,$ $\displaystyle J_{\bf g}\big{(}{\bf g}_{1},\ldots,{\bf g}_{m}\big{)}$
$\displaystyle=$ $\displaystyle\sum_{P_{\pi_{1}\pi_{2}}(1,\ldots,m)}P_{\bf
g}\Big{[}D\big{[}J({\bf g}_{\pi_{1}}),J({\bf g}_{\pi_{2}})\big{]}\Big{]}$ (25)
$\displaystyle+$
$\displaystyle\sum_{P_{\pi_{1}\pi_{2}\pi_{3}}(1,\ldots,m)}P_{\bf
g}\Big{[}D\big{[}J({\bf g}_{\pi_{1}}),J({\bf g}_{\pi_{2}}),J({\bf
g}_{\pi_{3}})\big{]}\Big{]}\ .$
The color-dressed $n$-gluon amplitude is given by
${\cal M}^{(0)}\big{(}{\bf g}_{1},{\bf g}_{2},\ldots,{\bf
g}_{n}\big{)}\;=\;P^{-1}\left[J\big{(}{\bf g}_{1},{\bf g}_{2},\ldots,{\bf
g}_{n-1}\big{)},J\big{(}{\bf g}_{n}\big{)}\right]\ .$ (26)
For this specific example, we have labelled the on-shell gluons by ${\bf
g}_{i}$, the off-shell gluon is denoted by ${\bf g}$ as before. The operator
formulation of the recursive algorithm is particularly suited for an object
oriented implementation of the recursive algorithm. We have implemented the
algorithm presented above in C++. More details including the more explicit
recursion equation are shown in appendix B.
The first issue to deal with is the correctness of the implemented algorithm.
To this end we want to compare the color-dressed amplitude to existing
evaluations of the gluonic amplitudes based on ordered amplitudes. To
facilitate the comparison, we write the color-ordered expansion of the
amplitude using the color-flow representation [32]:
$\displaystyle{\cal M}^{(0)}\big{(}{\bf g}_{1},{\bf g}_{2},\ldots,{\bf
g}_{n}\big{)}$ $\displaystyle=$
$\displaystyle\sum_{P(2,\ldots,n)}{A^{(0)}}^{i_{1}\cdots i_{n}}_{j_{1}\cdots
j_{n}}(g_{1}^{\lambda_{1}},\ldots,g_{n}^{\lambda_{n}})$ (27) $\displaystyle=$
$\displaystyle T^{a_{1}}_{i_{1}j_{1}}\cdots
T^{a_{n}}_{i_{n}j_{n}}\sum_{P(2,\ldots,n)}\mbox{Tr}\big{(}F^{a_{1}}\cdots
F^{a_{n}}\big{)}\;m^{(0)}(g_{1}^{\lambda_{1}},\ldots,g_{n}^{\lambda_{n}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{P(2,\ldots,n)}\Big{(}\delta^{i_{1}}_{j_{2}}\delta^{i_{2}}_{j_{3}}\cdots\delta^{i_{n-1}}_{j_{n}}\delta^{i_{n}}_{j_{1}}+(-1)^{n}\,\delta^{i_{n}}_{j_{n-1}}\delta^{i_{n-1}}_{j_{n-2}}\cdots\delta^{i_{2}}_{j_{1}}\delta^{i_{1}}_{j_{n}}\Big{)}\;m^{(0)}(g_{1}^{\lambda_{1}},\ldots,g_{n}^{\lambda_{n}})$
$\displaystyle=$
$\displaystyle\sum_{P(2,\ldots,n)}\delta^{i_{1}}_{j_{2}}\delta^{i_{2}}_{j_{3}}\cdots\delta^{i_{n-1}}_{j_{n}}\delta^{i_{n}}_{j_{1}}\;m^{(0)}(g_{1}^{\lambda_{1}},\ldots,g_{n}^{\lambda_{n}})\
.$
The $m^{(0)}(g_{1}^{\lambda_{1}},\ldots,g_{n}^{\lambda_{n}})$ are ordered
amplitudes with the property
$m^{(0)}(1,2,\ldots,n)=(-1)^{n}\,m^{(0)}(n,\ldots,2,1)$. From the above
formulas it follows that ${A^{(0)}}^{i_{1}\cdots i_{n}}_{j_{1}\cdots
j_{n}}={A^{(0)}}^{j_{1}\cdots j_{n}}_{i_{1}\cdots i_{n}}$. By choosing the
explicit momentum, helicity and color $(ij)_{m}$ of each gluon we can compare
the numerical values of Eqs. (26) and (27). We have done the comparison up to
$2\rightarrow 12$ gluon amplitudes and found complete agreement, thereby
validating the correctness of the color-dressed algorithm.
An important consideration in calculating the color-dressed amplitudes is the
color-sampling method used in the Monte Carlo program. For a $2\rightarrow
n-2$ gluon scattering amplitude, each of the gluon color states is
stochastically chosen. The full color configuration of the event is expressed
by $\\{(ij)_{m}\\}_{m=1}^{n}$ where $i_{m}$ and $j_{m}$ each denote a color
state out of three possible ones that can be labelled $\\{1,2,3\\}$. In the
“Naive” approach one samples uniformly over all possible color states of the
gluons. The number of color configurations, $N^{\rm Naive}_{\rm col}$, and the
color-configuration weight, $W^{\rm Naive}_{\rm col}$, are given by
$N^{\rm Naive}_{\rm col}\;=\;9^{n}$ (28)
and
$W^{\rm Naive}_{\rm col}\;=\;1\ ,$ (29)
respectively. About 95% of the naive color configurations have a vanishing
color factor. This results in a rather inefficient Monte Carlo procedure when
sampling over the color states. As was noted in Ref. [24], a significant
number of the zero color-weight configurations can be removed by imposing
color conservation. This is implemented by vetoing any color configuration for
which the condition
$\exists\,c\in\\{1,2,3\\}:\sum_{m=1}^{n}\left(\delta_{i_{m},c}-\delta_{j_{m},c}\right)\neq
0$ is true. In other words, the non-vetoed color configurations can be
obtained by uniformly choosing the colors $i_{1},\ldots,i_{n}$ and
subsequently generating the colors $j_{1},\ldots,j_{n}$ through a permutation
of the list $\\{i_{1},\ldots,i_{n}\\}$. For the number of color configurations
to be sampled over, this approach, which we name “Conserved”, then yields
$N^{\rm Conserved}_{\rm
col}\;=\sum_{n_{1},n_{2},n_{3}=0}^{n}\delta_{n_{1}+n_{2}+n_{3},n}\left(\frac{n!}{n_{1}!\,n_{2}!\,n_{3}!}\right)^{2}$
(30)
where $n_{c}=\sum_{m=1}^{n}\delta_{i_{m},c}$. As this way of sampling is no
longer uniform, each generated color configuration gets an associated color
weight described by
$W^{\rm Conserved}_{\rm col}\;=\;3^{n}\,\frac{n!}{n_{1}!\,n_{2}!\,n_{3}!}\ .$
(31)
Yet, there still are non-contributing color configurations left in the
sampling set. We have to augment the selection criteria further by vetoing any
color configuration for which the condition
$\exists\,c\in\\{1,2,3\\}:[\;\forall
m\in\\{1,2,\ldots,n\\}:(i_{m}=c\rightarrow i_{m}=j_{m})\;]$ is true.555When
all colors are identical, i.e. $i_{1}=j_{1}=i_{2}=j_{2}=\cdots=i_{n}=j_{n}$,
every color factor in Eq. (27) is equal to one. We can still veto the event
because the sum over all ordered amplitudes is identical to zero at tree level
[20]. In other words, we veto a color configuration if all occurrences of a
particular color $c$ come paired: $i_{m}=j_{m}=c$. By adding this veto to the
“Conserved” generation, we obtain the “Non-Zero” Monte Carlo procedure that
has removed all color configurations with zero color weight. The number of
leftover configurations sampled over is given by
$\displaystyle N^{\textrm{Non-Zero}}_{\rm col}$ $\displaystyle=$
$\displaystyle\sum_{n_{1},n_{2},n_{3}=0}^{n}\delta_{n_{1}+n_{2}+n_{3},n}\left(\frac{n!}{n_{1}!\,n_{2}!\,n_{3}!}\right)\;\times$
(32) $\displaystyle\hskip
39.83385pt\left(\frac{n!-n_{1}!\,n_{2}!\,n_{3}!\,\big{[}1-\sum_{c}\Theta(n_{c}-1)\big{]}-\sum_{c}\Theta(n_{c}-1)\,n_{c}!\,(n-n_{c})!}{n_{1}!\,n_{2}!\,n_{3}!}\right)\
,$
where the step function $\Theta(x)=1$ for $x\geq 0$ and zero otherwise.
Scattering | Naive | Conserved | Non-Zero
---|---|---|---
$2\rightarrow 2$ | 6,561 | 639 | 378
$2\rightarrow 3$ | 59,049 | 4,653 | 3,180
$2\rightarrow 4$ | 531,441 | 35,169 | 27,240
$2\rightarrow 5$ | 4,782,969 | 272,835 | 231,672
$2\rightarrow 6$ | 43,046,721 | 2,157,759 | 1,949,178
$2\rightarrow 7$ | 387,420,489 | 17,319,837 | 16,279,212
$2\rightarrow 8$ | 3,486,784,401 | 140,668,065 | 135,526,716
Table 1: The number of color configurations sampled over when using the
different Monte Carlo color schemes.
The weight associated with each sampled color configuration has to be modified
and reads
$W^{\textrm{Non-Zero}}_{\rm
col}\;=\;\left(3^{n}-3\right)\left(\frac{n!-n_{1}!\,n_{2}!\,n_{3}!\,\big{[}1-\sum_{c}\Theta(n_{c}-1)\big{]}-\sum_{c}\theta(n_{c}-1)\,n_{c}!\,(n-n_{c})!}{n_{1}!\,n_{2}!\,n_{3}!}\right)\
.$ (33)
For up to 10-gluon scatterings, Table 1 displays the resulting number of
sampled color configurations in the column indicated “Non-Zero”. It is also
shown how this number compares to the numbers found for the “Conserved” and
“Naive” sampling scheme.
Scattering | color ordered | color dressed | color dressed
---|---|---|---
| | ($V_{\rm max}=4$) | ($V_{\rm max}=3$)
$2\to 2$ | 0.0313 | 0.117 | 0.083
$2\to 3$ | 0.169 (5.40) | 0.495 (4.24) | 0.327 (3.93)
$2\to 4$ | 0.791 (4.68) | 1.556 (3.14) | 0.822 (2.51)
$2\to 5$ | 3.706 (4.69) | 6.11 (3.93) | 2.66 (3.23)
$2\to 6$ | 17.83 (4.81) | 25.26 (4.13) | 7.55 (2.84)
$2\to 7$ | 99.79 (5.60) | 93.43 (3.70) | 24.9 (3.30)
$2\to 8$ | 557.9 (5.59) | 392.4 (4.20) | 76.1 (3.05)
$2\to 9$ | 2,979 (5.34) | 1,528 (3.89) | 228 (2.99)
$2\to 10$ | 19,506 (6.55) | 5,996 (3.92) | 693 (3.04)
$2\to 11$ | 118,635 (6.08) | 24,821 (4.14) |
$\vdots$ | | $\vdots\ \ \ \ \hphantom{{}^{(0.00)}}$ |
$2\to 15$ | | 6,248,300 ${}^{(3.98^{4})}$ |
Table 2: The time (in seconds) to evaluate 10,000 color-dressed tree-level
amplitudes for $2\to n-2$ gluon scatterings. Only color configurations with
non-zero weight are taken into account. Also indicated is the growth factor
(given in brackets) with increasing $n$. To compute the amplitudes a 2.20 GHz
Intel Core2 Duo processor was used.
Next we examine the execution time of $n$-gluon scattering amplitudes using
the “Non-Zero” color sampling. In Table 2 the CPU time needed to calculate the
color-dressed amplitudes according to Eq. (26) and Eq. (27) are compared.
The evaluation of Eq. (27) employs the ordered recursion relation [20].
Naively one would expect this evaluation to grow factorially with the number
of gluons. However this growth is considerably dampened by sampling over non-
zero color configurations only. Note that for a given event we calculate each
ordered amplitude with non-vanishing color factor independently of the other
ordered amplitudes. One can speed up the computation time by sharing the
calculated sub-currents between different orderings. This, however, is outside
the scope of this paper.
For the evaluation of Eq. (26) we use the color-dressed recursion relation of
Eq. (3.3). To study its time behavior we apply this recursion as discussed in
appendix B with and without the 4-gluon vertex. As can be seen from Table 2,
the required CPU times scales as $4^{n}$ or $3^{n}$ if the 4-gluon vertex is
neglected. This exponential scaling was derived in Ref. [22, 24]. The
derivation, following [24], uses the recursive buildup of the amplitude. To
calculate an $n$-particle amplitude using a $V$-point vertex, we have to
evaluate the $(n-1)$-particle current of Eq. (6). This current in turn is
determined by calculating all $\left({n-1\atop m}\right)$ $m$-particle sub-
currents, where $n-1\geq m\geq 2$. Each $m$-current is constructed from
smaller currents using Eq. (11) thereby employing the $V$-point vertex. All
possible partitions into $V-1$ sub-currents are given by the Stirling number
of the second kind, ${\cal S}_{2}(m,V-1)$. This leads to the following scaling
of the calculation of the $n$-particle amplitude
$T_{n}\;=\sum_{m=2}^{n-1}\left({n-1\atop m}\right){\cal
S}_{2}(m,V-1)\;=\;{\cal S}_{2}(n,V)\sim V^{n}\ .$ (34)
Consequently, the $n$-gluon amplitude using the standard 3-gluon and 4-gluon
vertex has an exponential scaling behavior $T_{n}\rightarrow 4^{n}$. This is
evident from the results shown in Table 2. As can also be seen in the table,
the scaling behaves as expected when the 4-gluon vertex is left out, i.e.
$T_{n}\rightarrow 3^{n}$. As was shown in Ref. [23, 24], the 4-gluon vertex
can be avoided and replaced by an effective 3-point vertex. This results in a
significant time gain for the evaluation of high multiplicity gluon scattering
amplitudes.
An important consideration in the usefulness of the color-sampling approach is
the convergence to the correct answer as a function of the Monte Carlo
sampling size $N_{\rm MC}$. To this end, we compare the color-sampled result
$S^{(0)}_{\rm MC}$ for the tree-level amplitude squared,
$S^{(0)}_{{\rm MC},r}\;=\;W_{\rm col}(n_{1},n_{2},n_{3})\times\left|\,{\cal
M}^{(0)}\big{(}{\bf g}^{(r)}_{1},\ldots,{\bf g}^{(r)}_{n}\big{)}\right|^{2}\
,$ (35)
to the color-summed, i.e. color-exact, result
$S^{(0)}_{{\rm col},r}\;=\sum_{i_{1},\ldots,i_{n}=1}^{3}\
\sum_{j_{1},\ldots,j_{n}=1}^{3}\left|\,{\cal M}^{(0)}\big{(}{\bf
g}^{(r)}_{1},\ldots,{\bf g}^{(r)}_{n}\big{)}\right|^{2}\ .$ (36)
We plot the ratio of the average value for the color-sampled amplitude squared
and its standard deviation over the average value of the color-summed
amplitude squared as a function of the number of evaluated Monte Carlo events:
$R\;=\;\frac{\langle S^{(0)}_{\rm MC}\rangle\,\pm\,\sigma_{\langle
S^{(0)}_{\rm MC}\rangle}}{\langle S^{(0)}_{\rm col}\rangle}\ .$ (37)
We define the ratio this way so that most of the phase-space integration
fluctuations are divided out. The average values are computed via
$\langle S^{(0)}\rangle\;=\;\frac{1}{N_{\rm MC}}\,\sum_{r=1}^{N_{\rm
MC}}S^{(0)}_{r}$ (38)
where the index $r$ numbers the different events with the only exception that
the gluon polarizations have held fixed:
$\lambda_{1},\ldots,\lambda_{n}=+-\ldots+-(+)$. The standard deviation of the
average is calculated by
$\sigma_{\langle S^{(0)}\rangle}\;=\;\frac{\sqrt{\sum_{r=1}^{N_{\rm
MC}}(S^{(0)}_{r})^{2}\,-\,N_{\rm MC}\,\langle S^{(0)}\rangle^{2}}}{N_{\rm
MC}-1}\ .$ (39)
The 4- and 6-gluon scattering results are shown in the respective top parts of
Figs. 3 and 4 for the three different sampling methods “Naive”, “Conserved”
and “Non-Zero”. The generated phase-space points were subject to the
constraints: $p_{\perp,m}>0.1\sqrt{s}$, $|\eta_{m}|<2$ and $\Delta
R_{ml}>0.7$, see also Eq. (65). As it can be seen from the two plots by
avoiding sampling over zero-weight color configurations the convergence is
greatly enhanced.
Figure 3: Top panel: comparison of Monte Carlo integrations for the various
color-sampling schemes, including the standard deviation, to the exact color-
summed result as a function of the number of evaluated phase-space points. One
obtains $1.0034\pm 0.0091$, $0.9989\pm 0.0027$ and $0.9999\pm 0.0022$ after
$10^{7}$ steps for the “Naive”, “Conserved” and “Non-Zero” sampling,
respectively. Bottom panel: number of events required to reach a given
relative accuracy on the numerical evaluation of the color-sampled amplitude.
For the definition of $R_{\rm MC}(N_{\rm MC})$ and the values of the fit
parameters determining the dashed curves, cf. the text.
Figure 4: Top panel: comparison of Monte Carlo integrations for the various
color-sampling schemes, including the standard deviation, to the exact color-
summed result as a function of the number of evaluated phase-space points. One
obtains $1.16\pm 0.32$, $0.995\pm 0.071$ and $0.913\pm 0.037$ after $10^{5}$
steps for the “Naive”, “Conserved” and “Non-Zero” sampling, respectively.
Bottom panel: number of events required to reach a given relative accuracy on
the numerical evaluation of the color-sampled amplitude. For the definition of
$R_{\rm MC}(N_{\rm MC})$, cf. the text. The fit curves in terms of
$\sigma/\mu(N_{\rm MC})$ are described by $14.0\,N_{\rm MC}^{-0.287}$,
$2.84\,N_{\rm MC}^{-0.241}$ and $3.10\,N_{\rm MC}^{-0.331}$ for the “Naive”,
“Conserved” and “Non-Zero” sampling, respectively. The “Conserved” and “Non-
Zero” approaches are slower by factors of $f=10.5$ and $f=13.3$, respectively
(see text).
Figure 5: Number of events required to reach a given relative accuracy on the
numerical evaluation of the color-sampled amplitude. For the definition of
$R_{\rm MC}(N_{\rm MC})$, cf. the text. The fit curves in terms of
$\sigma/\mu(N_{\rm MC})$ are described by $8.04\,N_{\rm MC}^{-0.530}$,
$3.25\,N_{\rm MC}^{-0.405}$ and $3.01\,N_{\rm MC}^{-0.344}$ for the “Naive”,
“Conserved” and “Non-Zero” sampling, respectively. The “Conserved” and “Non-
Zero” approaches are slower by factors of $f=9.6$ and $f=10.8$, respectively
(see text).
For $N_{\rm MC}={\cal O}(10^{5})$, we obtain sufficient accuracy in the “Non-
Zero” sampling method. To illustrate this more clearly, we show in the lower
panels of Figs. 3 and 4 as well as in Fig. 5 the number of Monte Carlo events
needed to achieve a certain relative precision in the color sampling. For
these plots, we generate $N_{\rm event}$ events, which are partitioned into
trial and sampling events via $N_{\rm event}=N_{\rm trial}\times N_{\rm MC}$.
We define as a function of $N_{\rm MC}$ the ratio
$R_{\rm MC}(N_{\rm MC})\;=\;\frac{\sum_{r=1}^{N_{\rm MC}}S^{(0)}_{{\rm
MC},r}}{\sum_{r=1}^{N_{\rm MC}}S^{(0)}_{{\rm col},r}}\;=\;\frac{\langle
S^{(0)}_{\rm MC}\rangle(N_{\rm MC})}{\langle S^{(0)}_{\rm col}\rangle(N_{\rm
MC})}$ (40)
and plot $N_{\rm MC}$ versus the relative precision $\sigma(R_{\rm
MC})/\mu(R_{\rm MC})$. The mean value
$\mu(R_{\rm MC})\;=\;\frac{1}{N_{\rm trial}}\,\sum_{k=1}^{N_{\rm
trial}}R_{{\rm MC},k}(N_{\rm MC})$ (41)
and the standard deviation
$\sigma(R_{\rm MC})\;=\;\sqrt{\frac{\sum_{k=1}^{N_{\rm trial}}\big{(}R_{{\rm
MC},k}(N_{\rm MC})\big{)}^{2}\,-\,N_{\rm trial}\,\mu^{2}(R_{\rm MC})}{N_{\rm
trial}-1}}\ .$ (42)
are computed by using a sufficiently large number of trials, i.e. $N_{\rm
trial}$ estimates of $R_{\rm MC}(N_{\rm MC})$ are calculated to obtain the
mean value and the standard deviation for $R_{\rm MC}$. For $N_{\rm
trial}>{\cal O}(100)$, we get rather smooth curves. In the 4-gluon case shown
in the lower part of Fig. 3 this gives a reasonable description for $N_{\rm
MC}<10^{5}$. The 6- and 8-gluon scatterings are more involved and require more
statistics. The trend however can be read off the respective plots in Figs. 4
and 5.
For sufficiently large $N_{\rm MC}$, the expected scaling of the relative
standard deviation $\sigma$ with the number of Monte Carlo events is
$\sigma(R_{\rm MC})\sim 1/\sqrt{N_{\rm MC}}$. As can be seen from the second
plot of Fig. 3 the scaling is as expected and we can fit to the functional
form $A\times N_{\rm MC}^{-B}$. In the 4-gluon case, we find
Naive $\displaystyle:$ $\displaystyle\ \frac{\sigma\left(R_{\rm
MC}\right)}{\mu\left(R_{\rm MC}\right)}\;=\;33.8\times N_{\rm MC}^{-0.529}$
Conserved $\displaystyle:$ $\displaystyle\ \frac{\sigma\left(R_{\rm
MC}\right)}{\mu\left(R_{\rm MC}\right)}\;=\;6.45\times N_{\rm MC}^{-0.487}$
Non-Zero $\displaystyle:$ $\displaystyle\ \frac{\sigma\left(R_{\rm
MC}\right)}{\mu\left(R_{\rm MC}\right)}\;=\;4.35\times N_{\rm MC}^{-0.484}\ .$
(43)
From these fits we can quantify the enhancements owing to the sampling
strategies. The “Conserved” sampling method improves over the “Naive” method
by a factor of $33.8/6.45=5.2$, while the improvement of the “Non-Zero” method
over the “Conserved” method yields an additional factor of $6.45/4.35=1.5$ (or
a factor of $33.8/4.35=7.8$ over the “Naive” method). The algorithm determines
the color configurations with vanishing color factor before it fully evaluates
the corresponding matrix-element weight. The differences between the various
sampling methods therefore become smaller when we measure the computer
evaluation time to reach a certain relative precision. When we express this in
numbers for the example of 4-gluon scattering, we notice that the “Conserved”
and “Non-Zero”sampling schemes are slower by factors of $f=2.42$ and $f=3.29$,
respectively. This translates into changing the fit parameter $A\to
A^{\prime}=Af^{B}$. The corresponding ratios then read $33.8/9.92=3.4$ and
$9.92/7.74=1.3$ when specifying the improvement of the “Conserved” versus the
“Naive” and the “Non-Zero” versus the “Conserved” method, respectively. We see
using improved sampling over color configurations is still highly preferred.
## 4 Dressed Generalized Unitarity for Virtual Corrections
By using the parametric integration method of Ref. [9] one can implement the
generalized unitarity method of Ref. [7] into an efficient algorithmic
solution [33]. For the evaluation of color-ordered amplitudes, the algorithm
is of polynomial complexity [21]. To calculate the dimensional regulated one-
loop amplitude we extend the parametric expressions to $D$-dimensions and
apply the cuts in several integer dimensions to determine all the parametric
coefficients [10].666If one uses an analytical implementation of the
$D$-dimensional unitarity method of Ref. [10], one can eliminate the penta-
cuts [34]. However, in numerical implementations the removal of the penta-cuts
requires performing a numerical contour integral in the complex plane [11].
The algorithm is equally applicable for the inclusion of massive quarks [37].
The power of this algorithmic solution was demonstrated in Refs. [35, 36, 30]
for pure gluonic scattering.
Given the fully specified external sources and the interaction vertices, both
real and virtual corrections can be evaluated by the recursive formulas. The
virtual corrections to the differential cross section are given by
$\displaystyle d\,\sigma^{(V)}(f_{1}f_{2}\rightarrow f_{3}\cdots
f_{n})\;=\;\frac{W_{\rm S}}{N_{\rm event}}\times\sum_{r=1}^{N_{\rm
event}}d\,PS^{(r)}(K_{1}K_{2}\rightarrow K_{3}\cdots K_{n})$ (44)
$\displaystyle 2\,\Re\left({\cal
M}^{(0)}\left({\mathbf{f}}_{1}^{(r)},\ldots,{\mathbf{f}}_{n}^{(r)}\right)^{\dagger}\times{\cal
M}^{(1)}\left({\mathbf{f}}_{1}^{(r)},\ldots,{\mathbf{f}}_{n}^{(r)}\right)\right)\
,$
where the external sources, including momenta and quantum numbers, are sampled
through a Monte Carlo procedure. The weight $W_{\rm S}$ is determined by
process dependent symmetry factors and sampling weights.
In this section we show how to use the color-dressed tree-level amplitudes
discussed in the previous section to construct the color-dressed one-loop
amplitudes. By color sampling over the external partons one can calculate the
virtual corrections using Eq. (44). The generic algorithm will be outlined and
applied to pure gluon scattering.
### 4.1 Generic Color-Dressed Generalized Unitarity
The one-loop amplitude ${\cal M}^{(1)}\left({\bf f}_{1},\ldots,{\bf
f}_{n}\right)$ is obtained by integrating the un-integrated amplitude denoted
by ${\cal A}^{(1)}\left({\bf f}_{1},\ldots,{\bf f}_{n}\mid\ell\right)$ over
the loop momentum $\ell$:
${\cal M}^{(1)}\left({\bf f}_{1},\ldots,{\bf
f}_{n}\right)\;=\int\frac{d^{D}\ell}{(2\pi)^{D}}\;{\cal A}^{(1)}\left({\bf
f}_{1},\ldots,{\bf f}_{n}\mid\ell\right)\ .$ (45)
The integrand function can be decomposed into a sum of a finite number of
rational functions of the loop momentum with loop independent coefficients
[9]. The coefficients can be calculated in terms of tree-level amplitudes.
The parametric form of the integrand is given by the triple sum of rational
functions,
${\cal A}^{(1)}\left({\bf f}_{1},\ldots,{\bf
f}_{n}\mid\ell\right)\;=\;\sum_{k=1}^{C_{\rm max}}\;\
\sum_{RP_{\pi_{1}\cdots\pi_{k}}(1,2,\ldots,n)}^{\max\left(1,\frac{1}{2}(k-1)!\right)\,{\cal
S}_{2}(n,k)}\ \sum_{g_{\Pi_{1}},\ldots,g_{\Pi_{k}}}\frac{{\cal
P}_{k}\left(\vec{C}_{g_{\Pi_{1}}\cdots
g_{\Pi_{k}}}\mid\ell\right)}{d_{g_{\Pi_{1}}}(\ell)\;d_{g_{\Pi_{2}}}(\ell)\cdots
d_{g_{\Pi_{k}}}(\ell)}\ ,$ (46)
where the sum over the propagator flavors $g_{\Pi_{1}},\ldots,g_{\Pi_{k}}$ is
required as these are not uniquely defined for unordered amplitudes.
The maximum number of denominators needed to describe the dimensional
regulated one-loop matrix element is $C_{\rm max}$. The value of $C_{\rm max}$
is given by the dimensionality of the loop momentum. For the one-loop
calculations in dimensional regularization the maximum dimension of the loop
momentum is equal to five, i.e. $C_{\rm max}=5$. The denominator terms are
defined as
$d_{f_{\Pi_{m}}}(\ell)\;=\;(\ell+K_{\Pi_{m}})^{2}-m_{f}^{2}$ (47)
with $\Pi_{m}$ given through Eq. (9). The partition sum is over
$RP_{\pi_{1}\cdots\pi_{k}}(1,2,\ldots,n)$ ($\supseteq
P_{\pi_{1}\cdots\pi_{k}}(1,2,\ldots,n)$) elements. The total number of
elements is given by $\max\left(1,\frac{1}{2}(k-1)!\right)\times{\cal
S}_{2}(n,k)$. This extended partition list now also includes non-cyclic and
non-reflective permutations over the regular partition lists
$\\{\\{\pi_{i}\\}_{i=1}^{k}\\}$; more specifically we have:
$\displaystyle RP_{\pi_{1}\pi_{2}}$ $\displaystyle=$
$\displaystyle\Big{\\{}P_{\pi_{1}\pi_{2}}\Big{\\}}$ $\displaystyle
RP_{\pi_{1}\pi_{2}\pi_{3}}$ $\displaystyle=$
$\displaystyle\Big{\\{}P_{\pi_{1}\pi_{2}\pi_{3}}\Big{\\}}$ $\displaystyle
RP_{\pi_{1}\pi_{2}\pi_{3}\pi_{4}}$ $\displaystyle=$
$\displaystyle\Big{\\{}P_{\pi_{1}\pi_{2}\pi_{3}\pi_{4}},P_{\pi_{1}\pi_{3}\pi_{4}\pi_{2}},P_{\pi_{1}\pi_{4}\pi_{2}\pi_{3}}\Big{\\}}$
$\displaystyle RP_{\pi_{1}\pi_{2}\pi_{3}\pi_{4}\pi_{5}}$ $\displaystyle=$
$\displaystyle\Big{\\{}P_{\pi_{1}\pi_{2}\pi_{3}\pi_{4}\pi_{5}},P_{\pi_{1}\pi_{3}\pi_{4}\pi_{5}\pi_{2}},P_{\pi_{1}\pi_{4}\pi_{5}\pi_{2}\pi_{3}},P_{\pi_{1}\pi_{5}\pi_{2}\pi_{3}\pi_{4}},$
(48)
$\displaystyle\phantom{\\{}P_{\pi_{1}\pi_{2}\pi_{4}\pi_{5}\pi_{3}},P_{\pi_{1}\pi_{4}\pi_{5}\pi_{3}\pi_{2}},P_{\pi_{1}\pi_{5}\pi_{3}\pi_{2}\pi_{4}},P_{\pi_{1}\pi_{3}\pi_{2}\pi_{4}\pi_{5}},$
$\displaystyle\phantom{\\{}P_{\pi_{1}\pi_{2}\pi_{5}\pi_{3}\pi_{4}},P_{\pi_{1}\pi_{5}\pi_{3}\pi_{4}\pi_{2}},P_{\pi_{1}\pi_{3}\pi_{4}\pi_{2}\pi_{5}},P_{\pi_{1}\pi_{4}\pi_{2}\pi_{5}\pi_{3}}\Big{\\}}\
.$
The polynomial dependence of the numerator functions ${\cal P}_{k}$ on the
loop momentum is specified with a vector of parametric coefficients
$\vec{C}_{g_{\Pi_{1}}\cdots g_{\Pi_{k}}}$. The explicit polynomial forms that
we are using are given in Ref. [10].
Figure 6: Graphical representation of a quadrupole-cut partitioning of the
external legs into an ordered set of four unordered subsets
${\pi_{1},\pi_{2},\pi_{3},\pi_{4}}$ of external particles. The corresponding
tree-level diagrams are connected with the propagators of particle
$g_{\Pi_{1}},g_{\Pi_{2}},g_{\Pi_{3}},g_{\Pi_{4}}$.
The dimensionality of the parameter vector $\vec{C}_{g_{\Pi_{1}}\cdots
g_{\Pi_{k}}}$ depends on the number of denominators. In the case of 5
denominators there is only one parameter, for the terms with 4 denominators we
have five parameters, etc. . The parameters are determined by putting sets of
denominators to zero and calculating the residue in terms of tree-level
amplitudes. Setting denominator factors to zero is on a par with cutting the
corresponding propagators as required by generalized $D$-dimensional
unitarity. Let $\ell_{\Pi_{1}\cdots\Pi_{c}}$ be the “on-shell” loop momentum
fulfilling the “unitarity condition”:
$d_{g_{\Pi_{1}}}(\ell_{\Pi_{1}\cdots\Pi_{c}})\;=\cdots=\;d_{g_{\Pi_{c}}}(\ell_{\Pi_{1}\cdots\Pi_{c}})\;=0\,;\qquad
c=2,\ldots,C_{\rm max}\ .$ (49)
To fulfill the unitarity conditions we allow also complex values for the
components of the loop momenta. The parametric form of the numerator functions
for $c$-cuts becomes
$\displaystyle{\cal P}_{c}\left(\vec{C}_{g_{\Pi_{1}}\cdots
g_{\Pi_{c}}}\mid\ell_{\Pi_{1}\cdots\Pi_{c}}\right)\;=\;\mbox{Res}_{g_{\Pi_{1}}\cdots
g_{\Pi_{c}}}\left({\cal A}^{(1)}\left({\bf f}_{1},\ldots,{\bf
f}_{n}\mid\ell_{\Pi_{1}\cdots\Pi_{c}}\right)\right)$ (50) $\displaystyle-$
$\displaystyle\sum_{m=c+1}^{C_{\rm max}}\ \
\sum_{PP_{\widehat{\pi}_{1},\ldots,\widehat{\pi}_{m}}(1,\ldots,n)}\delta_{\Pi_{1}\widehat{\Pi}_{1}}\cdots\delta_{\Pi_{c}\widehat{\Pi}_{c}}\
\sum_{g_{\widehat{\Pi}_{c+1}}\cdots g_{\widehat{\Pi}_{m}}}\ \frac{{\cal
P}_{m}\left(\vec{C}_{g_{\widehat{\Pi}_{1}}\cdots
g_{\widehat{\Pi}_{m}}}\mid\widehat{\ell}_{\widehat{\Pi}_{1}\cdots\widehat{\Pi}_{c}}\right)}{d_{g_{\widehat{\Pi}_{c+1}}}(\widehat{\ell}_{\widehat{\Pi}_{1}\cdots\widehat{\Pi}_{c}})\cdots
d_{g_{\widehat{\Pi}_{m}}}(\widehat{\ell}_{\widehat{\Pi}_{1}\cdots\widehat{\Pi}_{c}})}\
.$
where the sum $PP_{\widehat{\pi}_{1},\ldots,\widehat{\pi}_{m}}(1,\ldots,n)$
over all $m!$ permutations of the $m$ partitions is supplemented with the
$\delta$-functions to generate the appropriate subtraction functions. Each
individual subtraction expression has to be evaluated with the appropriate
shift of the loop-momentum,
$\widehat{\ell}_{\widehat{\Pi}_{1}\cdots\widehat{\Pi}_{c}}$. This equation
provides us with an iterative procedure starting with the highest number of
cuts. For a given number of cuts, the numerator function becomes the residue
of one-loop integrand function minus the known contributions of terms with
higher number of denominator factors. The residue of the one-loop integrand
factorizes into a product of tree-level amplitudes (see e.g. Fig. 6):
$\displaystyle\mbox{Res}_{g_{\Pi_{1}}\cdots g_{\Pi_{c}}}\left({\cal
A}^{(1)}\left({\bf f}_{1},\ldots,{\bf
f}_{n}\mid\ell_{\Pi_{1}\cdots\Pi_{c}}\right)\right)$ $\displaystyle=$
$\displaystyle{\left[d_{g_{\Pi_{1}}}(\ell)\times\cdots\times
d_{g_{\Pi_{c}}}(\ell)\times{\cal A}^{(1)}\left({\bf f}_{1},\ldots,{\bf
f}_{n}\mid\ell\right)\right]}_{\ell=\ell_{\Pi_{1}\cdots\Pi_{c}}}$ (51)
$\displaystyle=$ $\displaystyle\sum_{{\bf g}_{1}\cdots{\bf
g}_{c}}\left\\{\prod_{k=1}^{c}{\cal M}^{(0)}\big{(}\,{\bf
g}_{k}^{\dagger},\,{\bf f}_{\pi_{k}},\,{\bf g}_{k+1}\big{)}\right\\}\ ,$
where the index $k$ is cyclic (i.e. ${\bf g}_{c+1}={\bf g}_{1}$) and ${\bf
g}_{k}$ denotes the particles resulting from the cut lines.
We can determine the parametric vector $\vec{C}_{g_{\Pi_{1}}\cdots
g_{\Pi_{c}}}$ in Eq. (50) by evaluating the right hand side for a set of loop
momenta fulfilling the unitarity constraint of Eq. (49). The only physics
model input is given through the tree-level on-shell amplitudes, ${\cal
M}^{(0)}$, which we evaluate using Eqs. (6) and (11). Two of the external
lines to the on-shell tree-level amplitudes are generated by the
$D$-dimensional cut lines. These external states have in general complex,
5-dimensional momenta. This extension of the momenta does not modify the
general structure of the tree-level level recursion relations discussed in the
previous section. In this way we obtain a fully specified algorithm to
determine the parameters and thereby the parametric form on the left hand side
of Eq. (46).
It is instructive to illustrate the structure given by Eq. (46) for a simple
example. Let us consider the cut-constructible, $D=4$, part of the box terms
in 4-gluon scattering ($n=4$, $k=4$). In this case we have no pentagon terms
and the numerator functions of the box terms are parametrized by two
coefficients
$\displaystyle\sum_{RP_{1234}(1,2,3,4)}\ \ \sum_{f=\\{g,q\\}}\ \frac{{\cal
P}_{4}\left(\vec{C}_{f_{1}f_{2}f_{3}f_{4}}\mid\ell\right)}{d_{f_{1}}(\ell)d_{f_{12}}(\ell)d_{f_{123}}(\ell)d_{f_{1234}}(\ell)}$
(52) $\displaystyle=$ $\displaystyle\frac{{\cal
P}_{4}\left(\vec{C}_{g_{1}g_{2}g_{3}g_{4}}\mid\ell\right)}{d_{g_{1}}(\ell)d_{g_{12}}(\ell)d_{g_{123}}(\ell)d_{g_{1234}}(\ell)}+\frac{{\cal
P}_{4}\left(\vec{C}_{g_{1}g_{3}g_{4}g_{2}}\mid\ell\right)}{d_{g_{1}}(\ell)d_{g_{13}}(\ell)d_{g_{134}}(\ell)d_{g_{1342}}(\ell)}+\frac{{\cal
P}_{4}\left(\vec{C}_{g_{1}g_{4}g_{2}g_{3}}\mid\ell\right)}{d_{g_{1}}(\ell)d_{g_{14}}(\ell)d_{g_{142}}(\ell)d_{g_{1423}}(\ell)}$
$\displaystyle+$ $\displaystyle\frac{{\cal
P}_{4}\left(\vec{C}_{q_{1}q_{2}q_{3}q_{4}}\mid\ell\right)}{d_{q_{1}}(\ell)d_{q_{12}}(\ell)d_{q_{123}}(\ell)d_{q_{1234}}(\ell)}+\frac{{\cal
P}_{4}\left(\vec{C}_{q_{1}q_{3}q_{4}q_{2}}\mid\ell\right)}{d_{q_{1}}(\ell)d_{q_{13}}(\ell)d_{q_{134}}(\ell)d_{q_{1342}}(\ell)}+\frac{{\cal
P}_{4}\left(\vec{C}_{q_{1}q_{4}q_{2}q_{3}}\mid\ell\right)}{d_{q_{1}}(\ell)d_{q_{14}}(\ell)d_{q_{142}}(\ell)d_{q_{1423}}(\ell)}$
where
${\cal
P}_{4}\left(\vec{C}_{f_{1}f_{2}f_{3}f_{4}}\mid\ell\right)=C^{(0)}_{f_{1}f_{2}f_{3}f_{4}}+C^{(1)}_{f_{1}f_{2}f_{3}f_{4}}\times\ell{\bf\cdot}n;\
n_{\mu}=\epsilon_{\mu\mu_{1}\mu_{2}\mu_{3}}p_{1}^{\mu_{1}}p_{12}^{\mu_{2}}p_{123}^{\mu_{3}}\
.$ (53)
The parameters are calculated by using the residue formula of Eq. (51). After
the coefficients of the box functions have been obtained, one turns to
calculate the coefficients of the triangle contributions. The numerator
function for the triangle cut of the quark-loop contribution, Eq. (50),
becomes
$\displaystyle{\cal
P}_{3}\left(\vec{C}_{q_{1}q_{2}q_{34}}\mid\ell_{\Pi_{1}\Pi_{2}\Pi_{34}}\right)$
$\displaystyle=$ $\displaystyle\mbox{Res}_{q_{1}q_{2}q_{34}}\Big{(}{\cal
A}^{(1)}({\bf g}_{1},{\bf g}_{2},{\bf g}_{3},{\bf
g}_{4}\mid\ell_{\Pi_{1}\Pi_{2}\Pi_{34}})\Big{)}$ (54) $\displaystyle-$
$\displaystyle\frac{{\cal
P}_{4}\left(\vec{C}_{q_{1}q_{2}q_{3}q_{4}}\mid\ell_{\Pi_{1}\Pi_{2}\Pi_{34}}\right)}{d_{q_{123}}(\ell_{\Pi_{1}\Pi_{2}\Pi_{34}})}\;-\;\frac{{\cal
P}_{4}\left(\vec{C}_{q_{1}q_{2}q_{4}q_{3}}\mid\ell_{\Pi_{1}\Pi_{2}\Pi_{34}}\right)}{d_{q_{124}}(\ell_{\Pi_{1}\Pi_{2}\Pi_{34}})}\
,$
where the residuum of the quark loop can be calculated again using Eq. (51),
$\displaystyle\mbox{Res}_{q_{1}q_{2}q_{34}}\left({\cal A}^{(1)}({\bf
g}_{1},{\bf g}_{2},{\bf g}_{3},{\bf
g}_{4}\mid\ell_{\Pi_{1}\Pi_{2}\Pi_{34}})\right)$ $\displaystyle=$ (55)
$\displaystyle=\;\left[\,d_{q_{1}}(\ell)\times d_{q_{12}}(\ell)\times
d_{q_{1234}}(\ell)\times{\cal A}^{(1)}({\bf g}_{1},{\bf g}_{2},{\bf
g}_{3},{\bf g}_{4}\mid\ell)\,\right]_{\ell=\ell_{\Pi_{1}\Pi_{2}\Pi_{34}}}$
$\displaystyle=\sum_{{\bf q}_{1}{\bf q}_{2}{\bf q}_{3}}{\cal M}^{(0)}({\bf
q}_{1}^{\dagger},{\bf g}_{1},{\bf\bar{q}}_{2})\times{\cal M}^{(0)}({\bf
q}_{2}^{\dagger},{\bf g}_{2},{\bf\bar{q}}_{3})\times{\cal M}^{(0)}({\bf
q}_{3}^{\dagger},{\bf g}_{3},{\bf g}_{4},{\bf\bar{q}}_{1})\ .$
Finally, we can obtain the one-loop amplitude, Eq. (45), by integrating out
the parametric forms on the right hand side of Eq. (46) over the loop
momentum. In this way one finds the master-integral decomposition of the one-
loop matrix element for every specified scattering configuration point [10]:
$\displaystyle{\cal M}^{(1)}\left({\bf f}_{1},\ldots,{\bf
f}_{n}\right)\;=\int\frac{d^{D}\,\ell}{(2\pi)^{D}}\ {\cal A}^{(1)}\left({\bf
f}_{1},\ldots,{\bf f}_{n}\mid\ell\right)$ (56) $\displaystyle=$
$\displaystyle\sum_{k=1}^{C_{\rm max}}\
\sum_{RP_{\pi_{1}\cdots\pi_{k}}(1,2,\ldots,n)}\ \sum_{g_{\Pi_{1}}\cdots
g_{\Pi_{k}}}S_{F}^{(g_{\Pi_{1}}\cdots
g_{\Pi_{k}})}\times\left(\bar{C}_{g_{\Pi_{1}}\cdots g_{\Pi_{k}}}{\cal
I}_{g_{\Pi_{1}}\cdots g_{\Pi_{k}}}+\bar{\bar{C}}_{g_{\Pi_{1}}\cdots
g_{\Pi_{k}}}{\cal R}_{g_{\Pi_{1}}\cdots g_{\Pi_{k}}}\right)$
where $S_{F}^{(g_{\Pi_{1}}\cdots g_{\Pi_{k}})}$ is the loop-integral symmetry
factor (e.g. for a gluonic self-energy, the symmetry factor is $\frac{1}{2}$),
the ${\cal I}_{g_{\Pi_{1}}\cdots g_{\Pi_{k}}}$ denote the scalar master-
integral functions corresponding to the generalized cut given by the ordered
partition list $\\{\Pi_{1}\cdots\Pi_{k}\\}$ and flavors of the cut lines
$(g_{\Pi_{1}}\cdots g_{\Pi_{k}})$. The terms ${\cal R}_{g_{\Pi_{1}}\cdots
g_{\Pi_{k}}}$ are the leading terms of the higher dimensional scalar integrals
in the limit $D\to 4$,
$\displaystyle{\cal R}_{f_{\Pi_{1}}f_{\Pi_{2}}f_{\Pi_{3}}f_{\Pi_{4}}}$
$\displaystyle=$ $\displaystyle-\frac{1}{6}\ ,$ $\displaystyle{\cal
R}_{f_{\Pi_{1}}f_{\Pi_{2}}f_{\Pi_{3}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ ,$ $\displaystyle{\cal R}_{f_{\Pi_{1}}f_{\Pi_{2}}}$
$\displaystyle=$
$\displaystyle-\frac{\left(K_{\Pi_{1}}-K_{\Pi_{2}}\right)^{2}}{6}+\frac{m_{f_{\Pi_{1}}}^{2}+m_{f_{\Pi_{2}}}^{2}}{2}\
,$ $\displaystyle{\cal R}_{f_{\Pi_{1}}}$ $\displaystyle=$ $\displaystyle 0\ .$
(57)
The scalar master integrals
${\cal I}_{f_{\Pi_{1}}\cdots
f_{\Pi_{k}}}\;=\;I_{k}\left(K_{\Pi_{1}},\ldots,K_{\Pi_{k}},m_{f_{\Pi_{1}}},\ldots,m_{f_{\Pi_{k}}}\right)\
,$ (58)
can be evaluated by e.g. using the numerical package developed in Ref. [41].
In Eq. (56) the coefficients $\bar{C}$ and $\bar{\bar{C}}$ are determined by
applying Eqs. (50) and (51) using a numerical algorithm. The $\bar{\bar{C}}$
coefficients are generated due to the dimensional regularization procedure and
are associated with the higher dimensional terms in the parametric forms.
### 4.2 Numerical Results for the Virtual Corrections of $n$-gluon Scattering
We have applied the formalism of the previous sections to multi-gluon
scattering. To this end we have extended the implementation presented in Ref.
[30]. Three major changes are required to alter the generalized-unitarity
based algorithm for the evaluation of color-ordered amplitudes to a numerical
algorithm capable of calculating color-dressed one-loop amplitudes. First, in
the decomposition of the one-loop integrands (cf. Eq. (3) of Ref. [30] and Eq.
(46)), all sums over ordered cuts have to be changed into sums over
partitions, which include all configurations obtained by non-cyclic and non-
reflective permutations:
$\sum\limits_{[i_{1}|i_{k}]}\quad\to\quad\sum\limits_{{RP}_{\pi_{1}\cdots\pi_{k}}(1\ldots
n)}\quad.$ (59)
Note that $[i_{1}|i_{k}]=1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n$. Second, the
tree-level amplitudes occurring in the determination of the integrand’s
residues have to be calculated from color-dressed recursion relations. In
addition, one not only has to sum over the internal polarizations of the
gluons but also over their internal colors when computing these residues.
Third, gluon bubble coefficients need to be supplemented by a symmetry factor
of $1/2!$. The appearance of the symmetry factor is associated with the
parametrization ambiguity of the subtraction terms in the double cuts. For
example, Eq. (50) gives for one of the double cuts in 4-gluon scattering
$\displaystyle{\cal P}_{2}(\vec{C}_{g_{12}g_{34}}\mid\ell)$ $\displaystyle=$
$\displaystyle\mbox{Res}_{g_{12}g_{34}}\Big{(}{\cal A}^{(1)}({\bf g}_{1},{\bf
g}_{2},{\bf g}_{3},{\bf g}_{4}\mid\ell)\Big{)}$ (60) $\displaystyle-$
$\displaystyle\frac{{\cal
P}_{3}(\vec{C}_{g_{1}g_{2}g_{34}}\mid\ell)}{d_{g_{1}}(\ell)}\;-\;\frac{{\cal
P}_{3}(\vec{C}_{g_{2}g_{1}g_{34}}\mid\ell)}{d_{g_{2}}(\ell)}\;-\;\frac{{\cal
P}_{3}(\vec{C}_{g_{3}g_{4}g_{12}}\mid\ell)}{d_{g_{3}}(\ell)}\;-\;\frac{{\cal
P}_{3}(\vec{C}_{g_{4}g_{3}g_{12}}\mid\ell)}{d_{g_{4}}(\ell)}$ $\displaystyle-$
$\displaystyle\frac{{\cal
P}_{4}(\vec{C}_{g_{1}g_{2}g_{3}g_{4}}\mid\ell)}{d_{g_{1}}(\ell)d_{g_{123}}(\ell)}\;-\;\frac{{\cal
P}_{4}(\vec{C}_{g_{2}g_{1}g_{3}g_{4}}\mid\ell)}{d_{g_{2}}(\ell)d_{g_{213}}(\ell)}\;-\;\frac{{\cal
P}_{4}(\vec{C}_{g_{1}g_{2}g_{4}g_{3}}\mid\ell)}{d_{g_{1}}(\ell)d_{g_{124}}(\ell)}\;-\;\frac{{\cal
P}_{4}(\vec{C}_{g_{2}g_{1}g_{4}g_{3}}\mid\ell)}{d_{g_{2}}(\ell)d_{g_{214}}(\ell)}$
$\displaystyle=$ $\displaystyle\mbox{Res}_{g_{12}g_{34}}\Big{(}{\cal
A}^{(1)}({\bf g}_{1},{\bf g}_{2},{\bf g}_{3},{\bf g}_{4}\mid\ell)\Big{)}$
$\displaystyle-$ $\displaystyle\frac{{\cal
P}_{3}(\vec{C}_{g_{1}g_{2}g_{34}}\mid\ell)}{d_{g_{1}}(\ell)}\;-\;\frac{{\cal
P}_{3}(\vec{C}_{g_{1}g_{2}g_{34}}\mid-\ell+K_{1}+K_{2})}{d_{g_{1}}(-\ell+K_{1}+K_{2})}$
$\displaystyle-$ $\displaystyle\frac{{\cal
P}_{3}(\vec{C}_{g_{3}g_{4}g_{12}}\mid\ell)}{d_{g_{3}}(\ell)}\;-\;\frac{{\cal
P}_{3}(\vec{C}_{g_{3}g_{4}g_{12}}\mid-\ell+K_{3}+K_{4})}{d_{g_{3}}(-\ell+K_{3}+K_{4})}$
$\displaystyle-$ $\displaystyle\frac{{\cal
P}_{4}(\vec{C}_{g_{1}g_{2}g_{3}g_{4}}\mid\ell)}{d_{g_{1}}(\ell)d_{g_{123}}(\ell)}\;-\;\frac{{\cal
P}_{4}(\vec{C}_{g_{1}g_{2}g_{3}g_{4}}\mid-\ell+K_{1}+K_{2})}{d_{g_{1}}(-\ell+K_{1}+K_{2})d_{g_{123}}(-\ell+K_{1}+K_{2})}$
$\displaystyle-$ $\displaystyle\frac{{\cal
P}_{4}(\vec{C}_{g_{1}g_{2}g_{4}g_{3}}\mid\ell)}{d_{g_{1}}(\ell)d_{g_{124}}(\ell)}\;-\;\frac{{\cal
P}_{4}(\vec{C}_{g_{1}g_{2}g_{4}g_{3}}\mid-\ell+K_{3}+K_{4})}{d_{g_{1}}(-\ell+K_{3}+K_{4})d_{g_{124}}(-\ell+K_{3}+K_{4})}\
.$
We see that each of the four possible parametrized terms is subtracted twice
with a different choice of the loop momentum. The symmetry factor of $1/2!$
“averages” over the double subtractions.
The results of the new formalism can be tested thoroughly beyond applying the
usual consistency checks such as solving for the master-integral coefficients
with two independent sets of loop momenta. The value of the double pole (dp)
can be cross-checked against the analytic result
${\cal M}^{(1)}_{\rm dp,th}\;=\;-\,\frac{c_{\Gamma}}{\epsilon^{2}}\,n\,N_{\rm
C}\,{\cal M}^{(0)}\ .$ (61)
Moreover, for a given phase-space point, we can use the ordered algorithm of
Ref. [30] to compute the full one-loop amplitude of a certain color and
helicity (polarization) configuration. Following the color-decomposition
approach, we can analytically calculate the necessary color factors and sum up
all relevant orderings to obtain the full result. In particular, we have
employed:
$\displaystyle{\cal M}^{(1)}({\bf g}_{1},\ldots,{\bf g}_{n})\;=\sum_{P(2\cdots
n)}{A^{(1)}}^{i_{1}\cdots i_{n}}_{j_{1}\cdots
j_{n}}(g_{1}^{\lambda_{1}},\ldots,g_{n}^{\lambda_{n}})$ (62) $\displaystyle=$
$\displaystyle\sum_{P(1\cdots n-1)}\left[N_{\rm C}\,\Delta_{1\cdots
n}+\sum_{k=1}^{{\rm
int}(n/2)}\sum_{m_{1}=1}^{n-k+1}\cdots\sum_{m_{k}=m_{k-1}+1}^{n}(-1)^{k}\Delta_{m_{1}\cdots
m_{k}}\Delta_{1\cdots/m_{1}\cdots/m_{k}\cdots n}\right]m^{(1)}(12\cdots n)$
where
$\Delta_{12\cdots
n}\;=\;\delta^{i_{1}}_{j_{2}}\delta^{i_{2}}_{j_{3}}\cdots\delta^{i_{n-1}}_{j_{n}}\delta^{i_{n}}_{j_{1}}\
.$ (63)
For example,
$\displaystyle{\cal M}^{(1)}({\bf g}_{1},{\bf g}_{2},{\bf g}_{3},{\bf
g}_{4},{\bf g}_{5})$ $\displaystyle=$
$\displaystyle\sum_{P(2345)}{A^{(1)}}({\bf g}_{1},{\bf g}_{2},{\bf g}_{3},{\bf
g}_{4},{\bf g}_{5})$ (64) $\displaystyle=$
$\displaystyle\sum_{P(2345)}\Big{(}N_{c}\,\Delta_{12345}-\Delta_{1}\Delta_{2345}-\Delta_{2}\Delta_{1345}-\Delta_{3}\Delta_{1245}-\Delta_{4}\Delta_{1235}-\Delta_{5}\Delta_{1234}$
$\displaystyle\phantom{\sum_{P(2345)}}+\Delta_{12}\Delta_{345}+\Delta_{13}\Delta_{245}+\Delta_{14}\Delta_{235}+\Delta_{15}\Delta_{234}+\Delta_{23}\Delta_{145}+\Delta_{24}\Delta_{135}$
$\displaystyle\phantom{\sum_{P(2345)}}+\Delta_{25}\Delta_{134}+\Delta_{34}\Delta_{125}+\Delta_{35}\Delta_{124}+\Delta_{45}\Delta_{123}\Big{)}\,m^{(1)}(12345)\
.$
Compared to the LO color-ordered decomposition, Eq. (27), the NLO color-
ordered decomposition leads to many subleading color factors. The number of
one-loop ordered amplitudes with zero color weight is significantly smaller
than the corresponding number for tree-level ordered amplitudes. As a result,
the advantages of color dressing become more apparent at the one-loop level.
Ordered cuts.
---
# | 5-gon | box | triangle | bubble | sum | ${\rm sum}_{n}$ | total $=$ sum $\times$ | #orderings | ${\cal N}_{(ab)_{k}}=$ | ${\cal N}_{(cd)_{k}}$
$n$ | cuts | cuts | cuts | cuts | | $\overline{{\rm sum}_{n-1}\\!\\!}$ | #orderings | $=(n-1)!/2$ | $(n-2)!$ |
4 | 0 | 1 | 4 | 6 | 11 | | 33 | 3 | 2 | 3
5 | 1 | 5 | 10 | 10 | 26 | 2.36 | 312 | 12 | 6 | 7
6 | 6 | 15 | 20 | 15 | 56 | 2.15 | 3,360 | 60 | 24 | 22
7 | 21 | 35 | 35 | 21 | 112 | 2.00 | 40,320 | 360 | 120 | 40
8 | 56 | 70 | 56 | 28 | 210 | 1.88 | 529,200 | 2,520 | 720 | 144
9 | 126 | 126 | 84 | 36 | 372 | 1.77 | 7,499,520 | 20,160 | 5,040 | 756
10 | 252 | 210 | 120 | 45 | 627 | 1.69 | 113,762,880 | 181,440 | 40,320 | 2,688
11 | 462 | 330 | 165 | 55 | 1012 | 1.61 | 1,836,172,800 | 1,814,400 | 362,880 |
12 | 792 | 495 | 220 | 66 | 1573 | 1.55 | 31,394,563,200 | 19,958,400 | 3,628,800 |
Table 3: The number of cuts required for the calculation of one ordered
$n$-gluon amplitude. The column labelled “total” gives the number of cuts when
calculating all $(n-1)!/2$ ordered amplitudes needed to reconstruct the full
virtual correction. The last two columns list the number of non-zero color-
weight orderings for two special color configurations given in the text.
Unordered cuts.
---
# | pentagon | box | triangle | bubble | sum | ${\rm sum}_{n}$ | ordr total/unordr total
$n$ | cuts | cuts | cuts | cuts | $\equiv$ total | $\overline{{\rm sum}_{n-1}\\!\\!}$ | orderings | $(ab)_{k}$ | $(cd)_{k}$
4 | 0 | 3 | 6 | 3 | 12 | | 2.750 | 1.833 | 2.750
5 | 12 | 30 | 25 | 10 | 77 | 6.42 | 4.052 | 2.026 | 2.364
6 | 180 | 195 | 90 | 25 | 490 | 6.36 | 6.857 | 2.743 | 2.514
7 | 1,680 | 1,050 | 301 | 56 | 3,087 | 6.30 | 13.06 | 4.354 | 1.451
8 | 12,600 | 5,103 | 966 | 119 | 18,788 | 6.09 | 28.17 | 8.048 | 1.610
9 | 83,412 | 23,310 | 3,025 | 246 | 109,993 | 5.85 | 68.18 | 17.05 | 2.557
10 | 510,300 | 102,315 | 9,330 | 501 | 622,446 | 5.66 | 182.8 | 40.61 | 2.708
11 | 2,960,760 | 437,250 | 28,501 | 1,012 | 3,427,523 | 5.51 | 535.7 | 107.1 |
12 | 16,552,800 | 1,834,503 | 86,526 | 2,035 | 18,475,864 | 5.39 | 1699 | 308.9 |
Table 4: The number of cuts needed to calculate color-dressed $n$-gluon
amplitudes. The last three columns give ratios of total numbers of cuts
required to compute the virtual corrections in both the color-decomposition
and color-dressed approaches. The first of these columns shows the ratios for
all generic color orderings whereas the other columns show the ratios for two
specific configurations as given in the text.
For a more quantitative understanding of the one-loop amplitude decomposition,
we respectively itemize in Tables 3 and 4 how many cuts need be applied to
decompose the color-ordered and color-dressed one-loop integrands for $n$
external gluons. In both cases we separately list the numbers of pentagon,
box, triangle and bubble cuts and their sum. While for the ordered cuts these
numbers are ruled by combinatorics: ${\cal C}(n,m)=\left({n\atop m}\right)$
with $m=1,\ldots,5$; in the unordered case they are given by the Stirling
numbers777More exactly, the number of bubble cuts is given by
$2^{n-1}-1-n={\cal S}_{2}(n,2)-n$, since cuts that isolate one gluon do not
contribute. For triangle, box, and pentagon cuts, we respectively have
$(3^{n}-3\cdot 2^{n}+3)/6={\cal S}_{2}(n,3)$, $3\,{\cal S}_{2}(n,4)$ and
$12\,{\cal S}_{2}(n,5)$ where, for the determination of the latter two, the
recurrence relation ${\cal S}_{2}(n,m)={\cal S}_{2}(n-1,m-1)+m{\cal
S}_{2}(n-1,m)$ is of help. of the second kind, ${\cal S}_{2}(n,m)$, and
therefore grow more quickly with $n$ than those of the ordered cuts. This is
exemplified in the “${\rm sum}_{n}/{\rm sum}_{n-1}$” columns of the two
tables. The growth factors slowly decrease for larger $n$, approaching the
limit of $5$ for the color-dressed case. As emphasized in Table 4 the
pentagon-cut calculations dominate in this case over all other cut
evaluations. The large-$n$ growth of the total cut number is hence described
by that of ${\cal S}_{2}(n,5)$ leading to the observed large-$n$ scaling of
$5^{n}$. Using the color-decomposition approach, we have to deal with much
fewer cuts per ordering. However, the total number of ordered cuts is obtained
only after multiplying with the relevant number of orderings. When considering
all possible $(n-1)!/2$ orderings, the final numbers are given in column
“total” of Table 3. The last three columns show the number of generic
orderings and the numbers $\cal N$ of non-vanishing orderings (i.e. those
having non-zero color factors) for two color configurations
$(ab)_{k}\equiv(13)(31)(11)\ldots(11)$ and
$(cd)_{k}\equiv(22)(12)(23)(31)(11)(22)(33)(11)(22)\ldots\ $.888The first four
colors are always fixed, supplemented by the repeating sequence $(11)(22)(33)$
according to the number of gluons, i.e. for $n=5$ we have
$(cd)_{k}\equiv(22)(12)(23)(31)(11)$, while for $n=9$ we use
$(cd)_{k}\equiv(22)(12)(23)(31)(11)(22)(33)(11)(22)$. Of course, for a fair
comparison between the ordered and dressed approach, the latter two columns
are of higher interest, since zero color weights are not counted. Still, the
ratios of total numbers of ordered versus unordered cuts is always larger than
one as can be read off the last three columns of Table 4. Keeping in mind the
greater cost of evaluating dressed recursion relations, the color-
decomposition approach can be expected to outperform the dressed method as
long as these ratios remain of order ${\cal O}(1)$. This in particular is true
for simple color configurations such as $(cd)_{k}$.
The analytic knowledge of ${\cal M}^{(1)}({\bf g}_{1},\ldots,{\bf g}_{n})$
presented in Eq. (62) enables us to perform stringent tests of our algorithm
and its implementation. We consider $2\to n-2$ processes where the gluons have
possible polarization states $\lambda_{k}\in\\{+,-\\}$ and colors $(ij)_{k}$
where $i_{k},j_{k}\in\\{1,2,3\\}$ and $k=1,\ldots,n$, i.e. we make use of the
color-flow notation. Our $n$-gluon results are given in the 4-dimensional
helicity (FDH) scheme [40]. In almost all cases, we compare our new method
labelled by “drss” with the color-decomposition approach, which – since it
makes use of the ordered algorithm – we denote “ordr”. We will present all our
results for two choices of loop-momentum and spin-polarization
dimensionalities $D$ and $D_{s}$: the “4D-case” is obtained by setting
$D=D_{s}=4$ and sufficient when merely calculating the cut-constructible part
(ccp) of the one-loop amplitude. The “5D-case” specified by $D=D_{s}=5$ allows
us to determine the complete result (all) including the rational part. In NLO
calculations one identifies the momenta of the external gluons with those of
well separated jets. We therefore apply cuts on the generated $k=1,\ldots,n$
phase-space momenta ($l=3,\ldots,n$):
$|\eta_{l}|\;<\;2\,,\qquad p_{\perp,l}\;>\;0.1\,|E_{1}+E_{2}|\,,\qquad\Delta
R_{kl}\;>\;0.7\,,$ (65)
where $\eta_{l}$ and $p_{\perp,l}$ respectively denote the pseudo-rapidity and
transverse momentum of the $l$-th outgoing gluon; $\Delta R_{kl}$ describe the
pairwise geometric separations in pseudo-rapidity and azimuthal-angle space of
gluons $k$ and $l$. We perform a series of studies in the context of double-
precision computations: we investigate the accuracies with which the double
pole, single pole (sp) and finite part (fp) of the full one-loop amplitudes
are determined by our algorithm. We also examine the efficiency of calculating
virtual corrections by means of simple phase-space integrations. To begin
with, we will verify the expected exponential scaling of the computation time
for different numbers of external gluons.
| 4D-case | 5D-case
---|---|---
$n$ | ordr | drss | $\underline{\rm ordr}$ | ordr | drss | $\underline{\rm ordr}$
| $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | drss | $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | drss
4 | 0.027 | 0.026 | | 0.061 | 0.062 | | 0.43 | 0.053 | 0.052 | | 0.139 | 0.140 | | 0.38
5 | 0.159 | 0.161 | 6.04 | 0.368 | 0.364 | 5.95 | 0.44 | 0.415 | 0.412 | 7.88 | 1.026 | 1.029 | 7.37 | 0.40
6 | 1.234 | 1.235 | 7.72 | 2.152 | 2.146 | 5.87 | 0.57 | 3.887 | 3.928 | 9.45 | 7.137 | 7.124 | 6.94 | 0.55
7 | 12.07 | 12.00 | 9.75 | 13.06 | 13.08 | 6.08 | 0.92 | 41.66 | 41.61 | 10.7 | 49.62 | 49.85 | 6.98 | 0.84
8 | 131.2 | 131.3 | 10.9 | 80.22 | 80.53 | 6.15 | 1.6 | 493.2 | 498.6 | 11.9 | 348.0 | 346.9 | 6.99 | 1.4
9 | 1579 | 1563 | 12.0 | 511.6 | 507.8 | 6.34 | 3.1 | 6316 | 6296 | 12.7 | 2466 | 2470 | 7.10 | 2.6
10 | 20900 | 20480 | 13.2 | 3640 | 3629 | 7.13 | 5.7 | 88320 | 88810 | 14.0 | 21590 | 21620 | 8.75 | 4.1
Table 5: Computer times $\tau_{n}$ in seconds obtained from the 4- and 5-dimensional evaluation of $n$-gluon virtual corrections at two random phase-space points $\rm a$ and $\rm b$ using a 3.00 GHz Intel Core2 Duo processor. The results are shown for both the color-ordered and color-dressed method. All virtual corrections were evaluated twice to check for the consistency of the solutions. The $n$ gluons have colors $(ab)_{k}$ and polarizations $\kappa_{k}$ as specified in the text. Also given are the ratios $r_{n}=\tau_{n}/\tau_{n-1}$ where $\tau_{n}$ is the time to compute the correction for $n$ gluons, in particular $\tau_{n}=(\tau^{({\rm a})}_{n}+\tau^{({\rm b})}_{n})/2$. The $\tau_{n}$ ratios of the ordered versus dressed method are depicted in the respective last column of the 4- and 5-dimensional case. | 4D-case | 5D-case | 4D/5D
---|---|---|---
$n$ | ordr | drss | ordr | drss | ordr | drss
| $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | $\tau^{({\rm a})}_{n}$ | $\tau^{({\rm b})}_{n}$ | $r_{n}$ | |
4 | 0.030 | 0.030 | | 0.069 | 0.070 | | 0.059 | 0.059 | | 0.156 | 0.157 | | 0.51 | 0.44
5 | 0.180 | 0.179 | 5.98 | 0.418 | 0.413 | 5.98 | 0.464 | 0.465 | 7.87 | 1.150 | 1.148 | 7.34 | 0.39 | 0.36
6 | 1.384 | 1.383 | 7.71 | 2.419 | 2.410 | 5.81 | 4.370 | 4.340 | 9.38 | 8.036 | 7.996 | 6.98 | 0.32 | 0.30
7 | 13.53 | 13.52 | 9.78 | 14.64 | 14.65 | 6.07 | 46.65 | 46.40 | 10.7 | 56.06 | 55.99 | 6.99 | 0.29 | 0.26
8 | 147.2 | 147.5 | 10.9 | 90.48 | 91.60 | 6.22 | 550.9 | 549.5 | 11.8 | 395.2 | 391.9 | 7.02 | 0.27 | 0.23
9 | 1766 | 1764 | 12.0 | 585.9 | 585.0 | 6.43 | 7013 | 7029 | 12.8 | 2844 | 2845 | 7.23 | 0.25 | 0.21
10 | 23100 | 22830 | 13.0 | 4233 | 4208 | 7.21 | 98760 | 98360 | 14.0 | 24220 | 24410 | 8.55 | 0.23 | 0.17
Table 6: Computer times $\tau_{n}$ in seconds for the same settings as used
in Table 5, this time using a 2.66 GHz Intel Core2 Quad processor. The
rightmost part of the table depicts the ratios of 4- versus 5-dimensional
computer times for both approaches.
The scaling of the computer time can roughly be estimated by $(f\times C_{\rm
max})^{n}$. The constants $C_{\rm max}=5\,(4)$ and $1<f\leq 4$ express the
fact that the number of pentagon (box) cuts and the exponential scaling with
$n$ of the tree-level color-dressed recursion relation respectively govern the
asymptotic scaling behavior of the unordered algorithm. Although one naively
expects $f=4$, this factor is reduced by the efficient re-use of gluon
currents between different cuts. The $C_{\rm max}^{n}$ growth of the number of
cuts reflects the large-$n$ limit of the Stirling number ${\cal
S}_{2}(n,C_{\rm max})$. We show four tables summarizing our results for the
computation times $\tau_{n}$ of obtaining ${\cal M}^{(1)}({\bf
g}_{1},\ldots,{\bf g}_{n})={\cal M}_{n}^{(1)}(\lambda_{k},(ij)_{k})$ by using
two independent solutions of the unitarity constraints. The time for the re-
computation has been included in $\tau_{n}$. In real applications such a
consistency check will become unnecessary, thereby halving the evaluation time
per phase-space point. Table 5 lists the times obtained by running the 4- and
5-dimensional algorithms for the calculation of two random phase-space points
labelled “$\rm a$” and “$\rm b$”. The $n$ gluons have colors
$(ij)_{k}=(ab)_{k}$ and alternating polarizations
$\lambda_{k}=\kappa_{k}\equiv+-\ldots+-(+)$. Owing to the absence of pentagon
cuts we find that the “4D-case” calculations are faster. More importantly, the
computation time does not vary when the $n$-gluon kinematics changes. Hence,
we can calculate the ratios $r_{n}=\tau_{n}/\tau_{n-1}$ by defining
$\tau_{n}=(\tau^{({\rm a})}_{n}+\tau^{({\rm b})}_{n})/2$ and show these ratios
in the table. While for the dressed algorithm these ratios are almost stable,
they are larger and increase gradually for the method based on ordered
amplitudes. This reflects the $(n-2)!$ factorial growth of the number of non-
vanishing orderings of the color configuration $(ab)_{k}$ as given in Table 3.
For the dressed approach, we find constant ratios of $r_{n}\approx 6$ and
$r_{n}\approx 7$ in the “4D-case” and “5D-case”, respectively. This manifestly
confirms our expectation of exponential scaling. The difference between the 4-
and 5-dimensional ratios obviously arises because of the absence of pentagon
cuts in the “4D-case”. The $r_{10}$ ratios do not fit the constant trend. We
cannot exclude though that this is a consequence of the occurrence of large
structures of maps to store the vast number of color-dressed coefficients. The
increasing number of higher-cut subtractions terms may also cause deviations
from the expected scaling, which we derived from our simple arguments stated
above. Also, the conceptually easier way of storing all coefficients and
calculating the largest-$m$ cuts first is by far not the most economic in
terms of memory consumption.999It is for this reason that our calculations are
currently limited to $n=12$ in the “4D-case” and $n=10$ in the “5D-case”. For
small $n$, the lower complexity of the ordered recurrence relation facilitates
a faster calculation of the virtual corrections through ordered amplitudes.
The turnaround appears for $7<n<8$ and is just slightly above $n=7$ for the
“4D-case”. With $n\geq 8$ the dressed method becomes superior owing to the
different growth characteristics of the two approaches. This is neatly
expressed by the “ordr/drss” ratios given in Table 5.
We have cross-checked the measured computation times in a different processor
environment using exactly the same settings. The results are shown in Table 6
and consistent with those of Table 5. Instead of the “ordr/drss” ratios, here
we list ratios comparing the 4- and 5-dimensional computation for both
approaches. They stress the relative importance of the pentagon-cut
evaluations, which start to dominate the full calculation when $n$ gets large.
| 4D-case | 5D-case
---|---|---
$n$ | ordr | drss | $\underline{\rm ordr}$ | ordr | drss | $\underline{\rm ordr}$
| $\tau^{(\sigma_{k})}_{n}$ | $\tau^{(\kappa_{k})}_{n}$ | $r_{n}$ | $\tau^{(\sigma_{k})}_{n}$ | $\tau^{(\kappa_{k})}_{n}$ | $r_{n}$ | drss | $\tau^{(\sigma_{k})}_{n}$ | $\tau^{(\kappa_{k})}_{n}$ | $r_{n}$ | $\tau^{(\sigma_{k})}_{n}$ | $\tau^{(\kappa_{k})}_{n}$ | $r_{n}$ | drss
4 | 0.049 | 0.045 | | 0.074 | 0.076 | | 0.63 | 0.088 | 0.085 | | 0.153 | 0.155 | | 0.56
5 | 0.186 | 0.185 | 3.95 | 0.364 | 0.364 | 4.85 | 0.51 | 0.479 | 0.483 | 5.56 | 1.000 | 1.000 | 6.49 | 0.48
6 | 1.186 | 1.182 | 6.38 | 2.071 | 2.068 | 5.69 | 0.57 | 3.629 | 3.586 | 7.50 | 6.805 | 6.752 | 6.78 | 0.53
7 | 4.185 | 4.277 | 3.57 | 11.82 | 11.77 | 5.70 | 0.36 | 14.02 | 13.95 | 3.88 | 44.42 | 44.46 | 6.56 | 0.31
8 | 27.12 | 26.96 | 6.39 | 70.34 | 71.10 | 6.00 | 0.38 | 98.52 | 99.13 | 7.07 | 294.8 | 297.8 | 6.67 | 0.33
9 | 245.0 | 242.9 | 9.02 | 443.8 | 445.5 | 6.29 | 0.55 | 960.3 | 954.8 | 9.69 | 2080 | 2070 | 7.00 | 0.46
10 | 1442 | 1446 | 5.92 | 3265 | 3270 | 7.35 | 0.44 | 5943 | 5968 | 6.22 | 18610 | 18480 | 8.94 | 0.32
11 | | | | 28670 | 28690 | 8.78 | | | | | | | |
6 | | | | 2.044 | | 5.62 | | | | | | | |
7 | | | | 11.66 | | 5.70 | | | | | | | |
8 | | | | 68.85 | | 5.90 | | | | | | | |
9 | | | | 420.4 | | 6.11 | | | | | | | |
10 | | | | 2972 | | 7.07 | | | | | | | |
11 | | | | 26310 | | 8.85 | | | | | | | |
12 | | | | | 292000 | 11.1 | | | | | | | |
Table 7: Computer times $\tau_{n}$ in seconds obtained for the color-ordered and color-dressed evaluation of $n$-gluon virtual corrections in 4 and 5 dimensions using a 3.00 GHz Intel Core2 Duo processor. Results are shown for two different polarization choices $\sigma_{k}$ and $\kappa_{k}$. The virtual corrections were computed at the same random phase-space point with the $n$-gluon colors set to $(cd)_{k}$. The choices are specified in the text. Ratios $r_{n}=\tau_{n}/\tau_{n-1}$ are given where $\tau_{n}=(\tau^{(\sigma_{k})}_{n}+\tau^{(\kappa_{k})}_{n})/2$ is the time to evaluate the correction for $n$ gluons two times. The re-computation is used to check both solutions for their consistency. The $\tau_{n}$ ratios of the ordered versus dressed method are depicted in the respective last column of the 4- and 5-dimensional case. | 4D-case | 5D-case | 4D/5D
---|---|---|---
$n$ | ordr | drss | $\underline{\rm ordr}$ | ordr | drss | $\underline{\rm ordr}$ | ordr | drss
| $\tau_{n}$ | $r_{n}$ | $\tau_{n}$ | $r_{n}$ | drss | $\tau_{n}$ | $r_{n}$ | $\tau_{n}$ | $r_{n}$ | drss | |
4 | 0.026 | | 0.062 | | 0.42 | 0.065 | | 0.151 | | 0.43 | 0.40 | 0.41
5 | 0.222 | 8.54 | 0.394 | 6.35 | 0.56 | 0.615 | 9.46 | 1.139 | 7.54 | 0.54 | 0.36 | 0.35
6 | 1.863 | 8.39 | 2.378 | 6.04 | 0.78 | 5.544 | 9.01 | 7.970 | 7.00 | 0.70 | 0.33 | 0.30
7 | 15.06 | 8.08 | 14.58 | 6.13 | 1.03 | 50.41 | 9.09 | 56.94 | 7.14 | 0.89 | 0.30 | 0.26
8 | 129.2 | 8.58 | 93.09 | 6.38 | 1.39 | 476.7 | 9.46 | 401.5 | 7.05 | 1.19 | 0.27 | 0.23
9 | 1127 | 8.72 | 603.6 | 6.48 | 1.87 | 4483 | 9.40 | 2800 | 6.97 | 1.60 | 0.25 | 0.22
10 | 10980 | 9.74 | 3961 | 6.56 | 2.77 | 50260 | 11.2 | 25140 | 8.98 | 2.00 | 0.22 | 0.16
Table 8: Color-configuration averaged computation times $\tau_{n}$ in seconds
obtained from the 4- and 5- dimensional color-ordered and color-dressed
evaluations of $n$-gluon virtual corrections using 2.66 GHz Intel Core2 Quad
processors. Results are shown for random phase- and color-space points and
alternating gluon polarizations $\lambda_{k}=\kappa_{k}$, see text. The
respective growth factors $r_{n}=\tau_{n}/\tau_{n-1}$ are given where
$\tau_{n}$ denotes the time that is needed to calculate the $n$-gluon one-loop
amplitude two times. The re-computation is used to check the two solutions for
their consistency. Several time ratios are formed to compare the ordered with
the dressed method and the 4- with the 5-dimensional computation. These ratios
are displayed in the columns indicated accordingly.
In Table 7 we detail computation times when varying the polarizations of the
$n$ gluons while keeping their color configuration fixed. We have chosen the
two settings $\lambda_{k}=\sigma_{k}\equiv++-\ldots-$ and, as before,
$\lambda_{k}=\kappa_{k}$. In terms of colors we consider the computationally
less involved point $(ij)_{k}=(cd)_{k}$. Both amplitudes are calculated at the
same random phase-space point “$\rm c$” dissimilar from previous points “${\rm
a}$” and “${\rm b}$”. For none of our four calculational options, we notice
manifest deviations between the times $\tau^{(\lambda_{k})}_{n}$ associated
with the two polarization settings. When inspecting the “ordr/drss” ratios, we
observe that the ordered approach is advantageous in cases where only a few
orderings contribute to the result of a certain point in color space. The
fluctuations seen in the growth factors mirror the unsteady increase with $n$
in non-zero orderings depicted in the last column of Table 3. For the dressed
approach, we get similar, though somewhat smaller, growth factors compared to
the previous test. In order to validate the dressed algorithm up to $n=12$
external gluons, we introduced a few more optimizations specific to the
4-dimensional calculations.101010Some parts of the algorithm can be speed up
when pentagon cuts are completely avoided. The lower part of Table 7 shows the
computer times, which we obtained after optimization. They are consistent with
our previous findings. As mentioned before, $r_{n\geq 10}>6$ likely occur for
reasons of increasingly complex higher-cut subtractions and computer
limitations in dealing with large memory structures.
configuration: | hard colors $(ab)_{k}$ | simple colors $(cd)_{k}$ | random non-zero colors
---|---|---|---
fit values: | $a/10^{-6}$sec | $b$ | $a/10^{-6}$sec | $b$ | $a/10^{-6}$sec | $b$
4D, ordr | 1.91 | $9.75\ ^{+0.59}_{-0.56}$ | 34.5 | $5.65\ ^{+0.32}_{-0.30}$ | 4.67 | $8.57\ ^{+0.10}_{-0.09}$
5D, ordr | 2.66 | $10.99\ ^{+0.48}_{-0.46}$ | 45.6 | $6.39\ ^{+0.29}_{-0.28}$ | 7.84 | $9.46\ ^{+0.13}_{-0.12}$
4D, drss | 39.4 | $6.19\ ^{+0.09}_{-0.08}$ | 28.2 | $6.51\ ^{+0.29}_{-0.28}$ | 38.7 | $6.30\ ^{+0.04}_{-0.04}$
5D, drss | 50.8 | $7.21\ ^{+0.10}_{-0.10}$ | 62.5 | $6.92\ ^{+0.08}_{-0.09}$ | 53.3 | $7.28\ ^{+0.11}_{-0.09}$
Table 9: Parameter values $a$ and $b$ obtained from curve fitting of the
computation times $\tau_{n}$ to the functional form of $\tau_{n}=a\,b^{n}$.
The results are given for the three different $n$-gluon color assignments used
in Tables 5 (hard), 7 (simple) and 8 (random) and for all four algorithms the
4- and 5-dimensional color-ordered and color-dressed algorithm.
For the calculation of the virtual corrections, one might question whether
there exist enough points in color space that occur with many trivial
orderings. If so, the color-decomposition based method would be more efficient
on average. This is not the case for larger $n$ as shown in Table 8. For gluon
multiplicities of $n=4,\ldots,10$ and polarizations set according to
$\kappa_{k}$, we list mean computation times, growth factors, “ordr/drss” and
“4D/5D” ratios obtained for one-loop amplitude evaluations where the phase-
and color-space points have been chosen randomly. Following the method
outlined in Section 3.3, we only considered non-zero color configurations. We
averaged over many events, for $n=4,\ldots,10$ gluons, we used ${\cal
O}(10^{6}),\ldots,{\cal O}(10^{2})$ points. We observe that the pattern of the
results in Table 8 resembles that found in Tables 5 and 6 where we have
studied the more complicated color point $(ik)_{k}=(ab)_{k}$. The ratios
comparing the ordered and dressed approach are smaller with respect to those
of Tables 5 and 6. This signals that the mean number of contributing orderings
is somewhat lower than for the $(ab)_{k}$ case. We finally report dressed
growth factors that are consistent with our previous findings confirming the
approximate $6^{n}$ and $7^{n}$ growths in computational complexity of the new
method for the 4- and 5-dimensional case, respectively.
Figure 7: Computation times $\tau_{n}$ versus the number $n$ of external
gluons for the three different gluon color assignments used in Tables 5
(hard), 7 (simple) and 8 (random). The results reported in these tables are
shown for the 4- and 5-dimensional color-ordered and -dressed algorithms. The
solid and dashed curves each represent the outcomes of the fits listed in
Table 9 for both the dressed and ordered approach, respectively.
Using the results of Tables 5, 7 and 8 we have performed fits to the
functional form $\tau_{n}=a\,b^{n}$. We show the outcome of the curve fittings
in Table 9. Recall that the computation times have been obtained by using
different color assignments for the $n$ gluons. Tables 5 and 7 present results
where we have chosen $(ij)_{k}=(ab)_{k}$ and $(ij)_{k}=(cd)_{k}$ as examples
of hard and simple color configurations, respectively. We have averaged over
non-zero color settings to find the results of Table 8. Considering the
performance of the dressed algorithm, we conclude that these data are in
agreement with exponential growth for all color assignments. The errors on the
fit parameter $b$ are relatively small, only the 4-dimensional case of simple
colors is somewhat worse because we included results up to $n=12$ where parts
of the computation become less efficient as explained above. The hard- and
simple-colors case of the ordered approach show rather large errors for the
$b$-parameter signalling that the genuine scaling law is not of an exponential
kind in both cases. Interestingly, one observes an effective exponential
scaling when averaging over many non-zero color configurations. The growth
described by the $b$-parameter is however a good two units stronger for the
ordered approach than the growth seen in the color-dressed approach. To
summarize, we have plotted in Fig. 7 all computer times reported in Tables 5,
7 and 8 as a function of the number of external gluons in the range $4\leq
n\leq 12$. We have included in these plots the curves $\tau_{n}=a\,b^{n}$,
which we calculated from the respective fit parameters stated in Table 9.
Figure 8: Relative accuracies of the $1/\epsilon^{2,1,0}$ poles of $n=6$ gluon
one-loop amplitudes as determined by the double-precision color-dressed
algorithm. The gluon polarizations are given by $\lambda_{k}=+-+-+-$, colors
were chosen randomly among non-zero configurations. Vetoed events are
included, only those with unstable ortho-vectors are left out, see text for
more explanations. The mean accuracies and the number of randomly picked
phase-space points are displayed in the top row and bottom left corner of the
plot, respectively.
In the following we will discuss the quality of the semi-numerical evaluations
of ${\cal M}_{n}^{(1)}$ amplitudes for both the color-ordered and color-
dressed approaches. To this end we analyze the logarithmic relative deviations
of the double pole, single pole and finite part. Independent of the number $n$
of gluons, we define them as follows:
$\varepsilon_{\rm dp}\;=\;\log_{10}\,\frac{|{\cal M}^{(1)[1]}_{\rm
dp,num}-{\cal M}^{(1)}_{\rm dp,th}|}{|{\cal M}^{(1)}_{\rm
dp,th}|}\,,\qquad\varepsilon_{\rm s/fp}\;=\;\log_{10}\,\frac{2\,|{\cal
M}^{(1)[1]}_{\rm s/fp,num}-{\cal M}^{(1)[2]}_{\rm s/fp,num}|}{|{\cal
M}^{(1)[1]}_{\rm s/fp,num}|+|{\cal M}^{(1)[2]}_{\rm s/fp,num}|}\ ,$ (66)
where the structure of the double-poles ${\cal M}^{(1)}_{\rm dp,th}$ is known
analytically given by Eq. (61). We use two independent solutions denoted by
$[1]$ and $[2]$ to test the accuracy of the single poles and finite parts. All
results reported here were obtained by using double-precision computations. We
have run all our algorithms by choosing color configurations and phase-space
points at random. Colors are distributed according to the “Non-Zero” method
presented in Sec. 3.3. The phase-space points are accepted only if they obey
the cuts, which we have specified at the beginning of this subsection. The
gluon polarizations are always alternating set by $\lambda_{k}=\kappa_{k}$.
Figure 8 shows the $\varepsilon$ distributions in absolute normalization,
which we obtain from the 5-dimensional color-dressed calculation for the case
of $n=6$ external gluons. The number of points used to generate the plots is
given in the bottom left corner, the top rows display the means of the
double-, single-pole and finite-part distributions. Limited to double-
precision computations, we find that the numerical accuracy of our results for
${\cal M}_{n}^{(1)}$ is satisfying. With $\varepsilon$ peak positions smaller
than the respective mean values $\langle\varepsilon_{\rm d/s/fp}\rangle<-8$,
we are able to provide sufficiently accurate solutions for almost all phase-
space configurations. There is however a certain fraction of events where the
single pole and finite part cannot be determined reliably. These ${\cal
O}(100)$ events occur because in exceptional cases small denominators, such as
vanishing Gram determinants made of external momenta, cannot be completely
avoided by the generalized-unitarity algorithms. We also see accumulation
effects where larger numbers get multiplied together while determining the
subtraction of higher-cut contributions. Owing to the limited range of double-
precision calculations, such effects can lead to insufficient cancellations of
intermediate large numbers that are supposed to cancel out
eventually.111111More detailed explanations can be found in Ref. [30]. The
current implementation of the algorithm has no special treatment for these
exceptional events. One either has to come up with a more sophisticated method
treating these points separately or increase the precision with which the
corrections are calculated. Both of which is beyond the scope of this paper
and we leave it at vetoing these points. Yet, we need robust criteria that
allow us to keep track of the quality of our solutions: we first test the
orthonormal basis vectors that span the space complementary to the physical
space constructed from the external momenta associated with the particular cut
configuration under consideration. Failures in generating these basis vectors
always lead to the rejection of the event.121212We test in particular whether
the normalization of the orthonormal basis vectors deviates less than
$10^{-12}$ units from one. In the example of Fig. 8, such events occurred with
a rate of $0.6\%$ and were not included in the plot. Secondly, and more
importantly, we test the reliability of solving the systems of equations to
determine the master-integral coefficients. To this end we generate an extra
4-dimensional loop momentum during the evaluation of the bubble coefficients
establishing the cut-constructible part. Inaccuracies in solving for triangle
etc. coefficients will be also detected, since at this level all higher-cut
subtractions are necessary to obtain the correct value of the bubble
coefficients. We use the extra loop momentum to individually re-solve for the
cut-constructible bubble coefficient and compare this solution with the one
obtained in first place. We veto the event, if the deviation $\Delta_{\rm
veto}$ in the complex plane of the two solutions exceeds a certain amount. We
fix the veto cut at $\Delta_{\rm veto}=0.02$ for this publication. Having this
cross-check at hand, we gain nice control over the events populating the tail
of the accuracy distributions in Fig. 8. Applying the veto, we arrive at the
distributions presented in the top left plot of Fig. 11 where the steeper
tails clearly demonstrate the effect of the veto. Certainly, both these
shortcomings of imprecise ortho-vectors and inaccurately solved coefficients
can be lifted by switching to higher precision whenever the respective double-
precision evaluations have not passed our criteria. Accordingly, Table 10
quantifies the fractions of events, which are within the scope of the color-
dressed and color-ordered algorithms presented here. Owing to the more
complicated event structures, the fraction of rejected events increases with
$n$, where most of the events fail the bubble-coefficient test. We observe
that the loss of events is more severe for the ordered algorithm.
$n\;$ | 4D, ordr | 5D, ordr | 4D, drss | 5D, drss
---|---|---|---|---
4 | 1.0 | 1.0 | 1.0 | 1.0
5 | 0.992 | 0.991 | 0.984 | 0.984 (0.999)
6 | 0.960 | 0.960 | 0.964 | 0.972 (0.994)
7 | 0.872 | 0.873 | 0.891 | 0.892 (0.982)
8 | 0.635 | 0.642 | 0.829 | 0.825 (0.953)
9 | 0.182 (0.84) | 0.205 (0.81) | 0.532 (0.93) | 0.533 (0.903)
10 | 0.000 (0.61) | 0.000 (0.50) | 0.380 (0.86) | 0.330 (0.83)
Table 10: Fractions of $n$-gluon events that have a stable set of basis
vectors in orthogonal space and also pass the veto on inaccurate master-
integral bubble coefficients when using $\Delta_{\rm veto}=0.02$. In brackets,
fractions of $n$-gluon events that pass the test for unstable ortho-vectors.
Figure 9: Double-, single-pole and finite-part accuracy distributions (upper
part) and scatter graphs (lower part) extracted from double-precision
computations of one-loop amplitudes for $n={\tt N}=4$ gluons with
polarizations $\lambda_{k}=+-+-$ and randomly chosen non-zero color
configurations. The virtual corrections were calculated at random phase-space
points satisfying the cuts detailed in the text. Unstable solutions were
vetoed. Results from the color-dressed algorithm are compared with those of
the color-ordered method indicated by dashed curves and brighter dots in the
plots. The 5(4)-dimensional case is shown in the top left (right) and center
(bottom) part of the figure. The definitions of $\varepsilon$ and $r$ are
given in the text. All scatter graphs contain $2\times 10^{4}$ points.
Figure 10: Double-, single-pole and finite-part accuracy distributions (upper
part) and scatter graphs (lower part) extracted from double-precision
computations of one-loop amplitudes for $n={\tt N}=5$ gluons with
polarizations $\lambda_{k}=+-+-+$ and randomly chosen non-zero color
configurations. The virtual corrections were calculated at random phase-space
points satisfying the cuts detailed in the text. Unstable solutions were
vetoed. Results from the color-dressed algorithm are compared with those of
the color-ordered method indicated by dashed curves and brighter dots in the
plots. The 5(4)-dimensional case is shown in the top left (right) and center
(bottom) part of the figure. The definitions of $\varepsilon$ and $r$ are
given in the text. All scatter graphs contain $2\times 10^{4}$ points.
Figure 11: Double-, single-pole and finite-part accuracy distributions (upper
part) and scatter graphs (lower part) extracted from double-precision
computations of one-loop amplitudes for $n={\tt N}=6$ gluons with
polarizations $\lambda_{k}=+-+-+-$ and randomly chosen non-zero color
configurations. The virtual corrections were calculated at random phase-space
points satisfying the cuts detailed in the text. Unstable solutions were
vetoed. Results from the color-dressed algorithm are compared with those of
the color-ordered method indicated by dashed curves and brighter dots in the
plots. The 5(4)-dimensional case is shown in the top left (right) and center
(bottom) part of the figure. The definitions of $\varepsilon$ and $r$ are
given in the text. All scatter graphs contain $2\times 10^{4}$ points.
Figure 12: Double-, single-pole and finite-part accuracy distributions (upper
part) and scatter graphs (lower part) as obtained from double-precision
evaluations of one-loop amplitudes for $n={\tt N}=6$ gluons with polarizations
and colors set to $\lambda_{k}=++----$ and
$(ij)_{k}=(12)(21)(13)(31)(11)(22)$, respectively. The virtual corrections
were calculated at random phase-space points satisfying the cuts detailed in
the text. The veto procedure has been applied to reject unstable solutions.
The results given by the color-dressed algorithm are compared with those of
the color-ordered method indicated by dashed curves and brighter dots in the
plots. The 5(4)-dimensional case is shown in the top left (right) and center
(bottom) part of the figure. The definitions of $\varepsilon$ and $r$ are
given in the text. Each scatter graph contains $2\times 10^{4}$ points.
$94.7$($94.1$)% and $91.2$($91.0$)% of the events pass all tests in the
dressed and ordered “5(4)D-case”, respectively.
Figure 13: Double-, single-pole and finite-part accuracy distributions (upper
part) and scatter graphs (lower part) extracted from double-precision
computations of one-loop amplitudes for $n={\tt N}=7$ gluons with
polarizations $\lambda_{k}=+-+-+-+$ and randomly chosen non-zero color
configurations. The virtual corrections were calculated at random phase-space
points satisfying the cuts detailed in the text. Unstable solutions were
vetoed. Results from the color-dressed algorithm are compared with those of
the color-ordered method indicated by dashed curves and brighter dots in the
plots. The 5(4)-dimensional case is shown in the top left (right) and center
(bottom) part of the figure. The definitions of $\varepsilon$ and $r$ are
given in the text. All scatter graphs contain $2\times 10^{4}$ points.
Figure 14: Double-, single-pole and finite-part accuracy distributions (upper
part) and scatter graphs (lower part) extracted from double-precision
computations of one-loop amplitudes for $n={\tt N}=8$ gluons with
polarizations $\lambda_{k}=+-+-+-+-$ and randomly chosen non-zero color
configurations. The virtual corrections were calculated at random phase-space
points satisfying the cuts detailed in the text. Unstable solutions were
vetoed. Results from the color-dressed algorithm are compared with those of
the color-ordered method indicated by dashed curves and brighter dots in the
plots. The 5(4)-dimensional case is shown in the top left (right) and center
(bottom) part of the figure. The definitions of $\varepsilon$ and $r$ are
given in the text; the number of points contained by each scatter graph is
found in the lower left.
Figure 15: Double-, single-pole and finite-part accuracy distributions (upper
part) and scatter graphs (lower part) extracted from double-precision
computations of one-loop amplitudes for $n={\tt N}=9$ gluons with
polarizations $\lambda_{k}=+-+-+-+-+$ and randomly chosen non-zero color
configurations. The virtual corrections were calculated at random phase-space
points satisfying the cuts detailed in the text. Unstable solutions were
vetoed. Results from the color-dressed algorithm are compared with those of
the color-ordered method indicated by dashed curves and brighter dots in the
plots. The 5(4)-dimensional case is shown in the top left (right) and center
(bottom) part of the figure. The definitions of $\varepsilon$ and $r$ are
given in the text; the number of points contained by each scatter graph is
found in the lower left.
In the upper part of Figs. 9-15 we show the distributions of relative
accuracies $\varepsilon$ as occurring in the evaluation of gluon loop
corrections with $n=4,\ldots,9$ external gluons. The lower part of these
figures and Figs. 16 and 17 themselves depict scatter graphs visualizing the
relative accuracies as a function of the size of the virtual corrections for
the single pole and finite contributions only, as the double pole contribution
has no observable variance. This form of presenting the results has
information on whether certain points dominate the uncertainty of the total
correction when averaging over the phase space. The $r$-variables used in
these plots are defined by
$r\;=\;\frac{1}{2\,\pi}\;\frac{\Re\left({{\cal M}^{(0)}}^{\dagger}{\cal
M}^{(1)}\right)}{\left|\,{\cal M}^{(0)}\right|^{2}}$ (67)
and represent corrections of the order of $\alpha_{s}$. Specifically, the $r$,
$r^{\prime}$ and $r_{\rm th}$ given in the plots are obtained by employing
${\cal M}^{(1)}={\cal M}^{(1)[1]}_{\rm d/s/fp,num}$, ${\cal M}^{(1)}={\cal
M}^{(1)[2]}_{\rm s/fp,num}$ and ${\cal M}^{(1)}={\cal M}^{(1)}_{\rm dp,th}$,
respectively. In all cases we have rejected events with unreliable basis
vectors in orthogonal space. Except for the results presented in Fig. 17, we
have vetoed all events that led to unstable solutions of the bubble master-
integral coefficient using $\Delta_{\rm veto}=0.02$. The statistics concerning
these rejections is shown in Table 10.
Figure 16: Single-pole and finite-part scatter graphs extracted from the
double-precision computation of one-loop amplitudes for $n={\tt N}=10$ gluons
with polarizations $\lambda_{k}=+-+-+-+-+-$ and randomly chosen non-zero color
configurations. The virtual corrections were calculated at random phase-space
points satisfying the cuts as described in the text. Unstable solutions were
vetoed and, therefore, not included in the plots. The upper (lower) row of
plots shows the results obtained from the 5(4)-dimensional color-dressed
algorithm. For the definition of $r$, see text. The number of points contained
by each scatter graph can be found in the lower left.
Figure 17: Double-, single-pole and finite-part scatter graphs visualizing the
accuracy of double-precision evaluations of one-loop amplitudes for $n={\tt
N}=9$ and $10$ gluons of alternating polarizations. Non-zero randomly chosen
color configurations were used. Note that unstable solutions were not vetoed
and therefore included in this presentation. The virtual corrections were
calculated at random phase-space points satisfying the cuts as described in
the text. Results of the 5-dimensional algorithms either based on color
ordering (ordr) or color dressing (drss) are shown; for $n={\tt N}=9$, the
“4D-case” results are also given (ccp only). For the definition of $r$, see
text; axis labels as used in Fig. 16 are understood. The rightmost graph
contains ${\cal O}(20)$ points per $\epsilon$-pole, while the left plot of the
“5(4)D-case” has approximately $50$($120$) points per pole.
Figure 18: Finite-part versus single-pole accuracy (in double precision) as
achieved in one-loop amplitude calculations using the color-dressed approach
for various numbers $n={\tt N}$ of external gluons with polarizations
$\lambda_{k}=+-\ldots+-(+)$ and colors randomly chosen among non-zero
configurations. Note that unstable solutions have not been vetoed. The $n={\tt
N}=9$ and $n={\tt N}=10$ graphs only contain $1.6\cdot 10^{3}$ and $87$
points, respectively, whereas all other plots comprise $10^{4}$ points.
We compare in all plots of Figs. 9-15 the color-dressed with the color-ordered
approach where the results of the latter are indicated by dashed curves in the
spectra (with the $\langle\varepsilon\rangle$ given by the lower top row of
numbers) and brighter points in the scatter graphs. The $\varepsilon$ spectra
of the “5D-case” (“4D-case”) are always shown in the top left (right) parts of
the figures; the associated scatter graphs are compiled in the center (bottom)
parts. In Fig. 16 we present our results for $n=10$ gluons where for reasons
of limited statistics we solely show the scatter graphs related to the dressed
method. The veto procedure has a very strong impact on ${\cal M}_{9,10}^{(1)}$
calculations. For the purpose of direct comparisons between vetoed and non-
vetoed samples, we have added in Fig. 17 scatter plots that include vetoed
events.
In all cases we notice that the double poles are obtained very accurately with
almost no loss in precision for increasing number of gluons. The
$n$-dependence of the single-pole and finite-part precisions is not as stable
as for the double pole. We see noticeable shifts of the peak and mean
positions towards larger values when incrementing the number of external
gluons. The distribution’s tails are under good control. Because of the
introduced veto procedure, they quickly die off around $\varepsilon\approx-2$.
In rare cases worse accuracies occur, which happens more frequently for the
5-dimensional calculations. We can avoid these cases, if we extend the veto
criteria by re-solving for and testing the rational bubble coefficient as
well. For $n>9$, the limitations of double-precision computations unavoidably
lead to rather unreliable single-pole and finite-part determinations. As an
interesting fact, we observe that the color-dressed method yields throughout
results of higher precision. Moreover, the decrease in accuracy for growing
$n$ is more moderate compared to the method based on color ordering. Clearly,
on the one hand this algorithm has to be run for many orderings and may
therefore lead to an accumulation of small imprecisions. On the other hand a
rather inaccurate determination of $m^{(1)}$ may appear just for a single
ordering, in turn spoiling the overall result. Both effects make the ordered
approach less capable of delivering accurate results. Turning to the scatter
plots, we find that the most accurate but also inaccurate evaluations occur
for points distributed near the vertical line of ${\cal O}(1)$ corrections. It
is very encouraging that all top right quadrants are rather sparsely
populated, dispelling the doubts that insufficiently determined large
corrections may dominate our final results. The scatter regions of the double-
pole solutions remain almost unchanged for larger $n$, while those of the
single poles and finite parts are slightly growing gradually shifting towards
lower relative accuracies. The scatter patches of the dressed method are
displaced with respect to those of the color-decomposition approach:
advantageously, they cover regions of greater precision, in particular
populate the bottom right quadrants more densely. Due to the simplicity of the
4-gluon kinematics, the case of $n=4$ gluons stands out from the rest: the
single pole and finite part can be obtained with almost the same accuracy as
the double pole. This feature is preserved even if rational-part calculations
are included. With $5$ gluons or more it is common that all coefficients
contribute to the decomposition of the one-loop amplitude. The relative
accuracies of the single poles and finite parts therefore develop a much
different, less steeper, tail compared to the double poles. There are almost
no differences between the double- and single-pole results obtained from the
4- and 5-dimensional algorithms. This is no surprise, since the coefficients
necessary to reconstruct these poles can be determined in $4$ dimensions and
our algorithms have been set up accordingly. In the absence of rational-part
calculations it turns out that the finite parts may on average be obtained
slightly more precisely than the single poles. The tails of the $1/\epsilon$
spectra reach out to the largest $\varepsilon$-values occurring in the
evaluation of the cut-constructible part. The behavior is reversed in the
5-dimensional case owing to the addition of the rational part. For the same
reason, we note increased $\langle\varepsilon_{\rm fp}\rangle$ in the
“5D-case”, furthermore, the 5-dimensional scatter graphs show higher densities
with respect to the 4-dimensional ones at lower accuracies.
As a special case of Fig. 11 we have displayed in Fig. 12 accuracy
distributions and scatter plots for $n=6$ gluons of polarizations
$\lambda_{k}=++----$ when instead of random color-space points the fixed non-
zero color configuration $(ij)_{k}=(12)(21)(13)(31)(11)(22)$ has been
selected. We notice that all $\varepsilon$-spectra are shifted towards smaller
accuracies. Also, as illustrated by the scatter graphs, the magnitude of the
virtual corrections is bound at ${\cal O}(1)$ with the exception of the finite
piece of the cut-constructible part of the one-loop amplitudes. Interestingly,
this is corrected back by adding in the rational part.
In Ref. [30] it was shown that the finite-part accuracy of the evaluation of
ordered amplitudes is mostly correlated with that of the single poles. We have
studied this issue for the dressed algorithm in the “5D-case”. The
corresponding scatter plots also include the vetoed events and are presented
in Fig. 18. The multitude of points is distributed along the diagonal
indicating a strong correlation. As for color-ordered amplitudes the
evaluation of the rational part becomes more involved with increasing gluon
numbers. Therefore, regions of lower finite-part precision start to get
populated distorting the diagonal trend.
Finally, we want to show that the Monte Carlo sampling as defined in Eq. (44)
converges sufficiently fast for the color-dressed calculated virtual
corrections. To this end we generalize the LO discussion following Eq. (35)
with the details given in Sec. 3.3. The relevant quantity to explore in the
Monte Carlo averaging is
$S^{(0+1)}_{\rm MC}\;=\;\frac{1}{N_{\rm colpts}}\sum_{k=1}^{N_{\rm
colpts}}W_{\rm col}(n_{1},n_{2},n_{3})\times\left[\,\left|\,{\cal
M}^{(0)}_{k}\right|^{2}+\,\frac{\widehat{\alpha}_{s}}{2\,\pi}\,\Re\left({\cal
M}_{{\rm fp},k}^{(1)}\,{{\cal M}_{k}^{(0)}}^{\dagger}\right)\,\right]$ (68)
where we choose $\widehat{\alpha}_{S}=0.12$ and ${\cal M}_{{\rm fp},k}^{(1)}$
is the finite part of the virtual corrections. The sum over the $N_{\rm
colpts}$ color configurations for each phase-space point is an optional “mini-
Monte Carlo” over colors for faster convergence as a function of the number of
phase-space point evaluations. By adding the real corrections to Eq. (68) and
performing the coupling constant renormalization and mass factorization, one
obtains the gluonic contribution to the NLO multi-jet differential cross
section. Therefore, the convergence of Eq. (68) is the relevant quantity to
study.
Figure 19: Consistency test for the Monte Carlo integration of 4-gluon virtual
corrections using “Non-Zero” color sampling compared to the exact color
summing. As a function of the number $N_{\rm MC}$ of evaluated phase-space
points the $R^{(0+1)}$-ratio is plotted converging to one as it should be. The
inserted plot shows the number of phase-space evaluations needed to reach a
given relative accuracy in terms of $R^{(0+1)}_{\rm MC}(N_{\rm MC})$ while
Monte Carlo integrating; for the definitions, see text. The dashed line
depicts the fit function $\sigma/\mu=A\,N_{\rm MC}^{-B}$, see also Table 11.
By defining the $n$-gluon color-summed counterpart of $S^{(0+1)}_{\rm MC}$,
$S_{\rm col}^{(0+1)}\;=\sum_{i_{1},\ldots,i_{n}=1}^{3}\
\sum_{j_{1},\ldots,j_{n}=1}^{3}\left[\,\left|\,{\cal
M}_{k}^{(0)}\right|^{2}+\,\frac{\widehat{\alpha}_{s}}{2\,\pi}\,\Re\left({\cal
M}_{{\rm fp},k}^{(1)}\,{{\cal M}_{k}^{(0)}}^{\dagger}\right)\,\right]\ ,$ (69)
we can form the ratios
$R^{(0+1)}\;=\;\frac{\langle S^{(0+1)}_{\rm MC}\rangle\,\pm\,\sigma_{\langle
S^{(0+1)}_{\rm MC}\rangle}}{\langle S^{(0+1)}_{\rm col}\rangle}\ ,\qquad
R^{({\rm V})}\;=\;\frac{\langle S^{(0+1)}_{\rm
MC}\rangle\,\pm\,\sigma_{\langle S^{(0+1)}_{\rm MC}\rangle}}{\langle
S^{(0)}_{\rm col}\rangle}$ (70)
analogously to Eq. (37). We define the mean values and standard deviations of
the ratios similarly to Eqs. (38) and (39), respectively. Note that
$S^{(0)}_{\rm col}$ is already defined at LO by Eq. (36). As we increase the
number of Monte Carlo points, $N_{\rm MC}$, the $R^{({\rm V})}$-ratios
quantify the relative importance of the virtual corrections, while the
$R^{(0+1)}$-ratios should converge to one. For the latter, this is nicely
demonstrated in Fig. 19 for the 4-gluon virtual corrections and the “Non-zero”
sampling scheme as described in Sec. 3.3. After $15900$ events we obtain
$R^{(0+1)}=0.939\pm 0.039$, which is satisfactory for this consistency check.
As in the LO discussion we want to illustrate how many events are needed to
achieve a certain relative integration uncertainty when performing the Monte
Carlo color sampling. In analogy to Eq. (40) we can construct the ratio
$R^{(0+1)}_{\rm MC}(N_{\rm MC})\;=\;\frac{\sum_{r=1}^{N_{\rm
MC}}S^{(0+1)}_{{\rm MC},r}}{\sum_{r=1}^{N_{\rm MC}}S^{(0+1)}_{{\rm col},r}}$
(71)
as a function of $N_{\rm MC}$. Again, it is interesting to change the
normalization of the ratio and also define
$R^{({\rm V})}_{\rm MC}(N_{\rm MC})\;=\;\frac{\sum_{r=1}^{N_{\rm
MC}}S^{(0+1)}_{{\rm MC},r}}{\sum_{r=1}^{N_{\rm MC}}S^{(0)}_{{\rm col},r}}$
(72)
in order to study the impact of the virtual corrections. As before we
partition $N_{\rm event}=N_{\rm trial}\times N_{\rm MC}$ events to have a
certain number of trials to compute the corresponding mean values $\mu$ and
standard deviations $\sigma$ for $n$-gluon LO and virtual scattering according
to Eqs. (41) and (42), respectively. For the case of $R^{(0+1)}_{\rm
MC}(N_{\rm MC})$ and 4-gluon scattering, the number of Monte Carlo points
versus a given relative accuracy is shown in the inlaid plot of Fig. 19. As at
LO, the curve bends behaving as statistically determined after a certain
amount of Monte Carlo integration steps.
:) | Naive | Conserved | Non-Zero | Non-Zero, $N_{\rm colpts}=4$
---|---|---|---|---
$n$ | $B$ | $A$ | $B$ | $A$ | $A^{\prime}$ | $f$ | $B$ | $A$ | $A^{\prime}$ | $f$ | $B$ | $A$ | $A^{\prime}$ | $f$
4∗ | | | | | | | 0.479 | 3.36 | | | | | |
4 | 0.497 | 22.0 | 0.489 | 5.41 | 17.0 | 10.4 | 0.476 | 3.57 | 13.5 | 16.4 | 0.485 | 2.05 | 15.6 | 65.7
5 | 0.482 | 59.4 | 0.454 | 13.3 | 43.3 | 13.5 | 0.442 | 9.71 | 36.4 | 19.8 | 0.439 | 5.56 | 37.8 | 79.2
6 | 0.325 | 7.08 | 0.344 | 5.37 | 14.0 | 16.3 | 0.255 | 1.60 | 3.50 | 21.7 | 0.233 | 0.850 | 2.14 | 87.6
Table 11: Parameter values $B$, $A$ and $A^{\prime}$ obtained from curve
fitting of the $\sigma(R_{\rm MC})/\mu(R_{\rm MC})$ to the functional form
$A\times N_{\rm MC}^{-B}$. The results are given for the different ways of
sampling over colors in $n$-gluon scattering. The 4-gluon case marked by
“$\ast$” corresponds to the consistency check shown in Fig. 19, where
$R^{(0+1)}_{\rm MC}$ has been considered. In all other cases $R^{({\rm
V})}_{\rm MC}$ has been used, cf. Figs. 20, 21 and 22. Note that for $n=6$, we
have fitted $\sigma(R^{({\rm V})}_{\rm MC})$. The parameters
$A^{\prime}=Af^{B}$ take into account that the evaluation of a fixed number of
Monte Carlo events takes longer for the other than “Naive” color-sampling
methods. The time factors $f$ relative to the “Naive” case are also displayed.
To quantify the color-integration performances, we again perform fits to the
functional form $A\times N_{\rm MC}^{-B}$ and show the values of the fitted
parameters in Table 11 for the various cases. As argued in Sec. 3.3 for large
enough $N_{\rm MC}$, we expect a scaling of $\sigma/\mu$ that is proportional
to $1/\sqrt{N_{\rm MC}}$. The goodness of the sampling schemes is signified by
the $A$\- and $A^{\prime}$-parameters, where the latter is more important
since the time factors are included. Smaller values of these parameters
indicate a better efficiency of the sampling procedure.
:) | Naive | Conserved | Non-Zero | Non-Zero, $N_{\rm colpts}=4$
---|---|---|---|---
$n$ | $N_{\rm MC}$ | $R^{({\rm V})}$ | $N_{\rm MC}$ | $R^{({\rm V})}$ | $N_{\rm MC}$ | $R^{({\rm V})}$ | $N_{\rm MC}$ | $R^{({\rm V})}$
4 | $4\cdot\\!10^{6}$ | $0.4739\pm 0.0054$ | $4\cdot\\!10^{6}$ | $0.4750\pm 0.0017$ | $4\cdot\\!10^{6}$ | $0.4724\pm 0.0013$ | $1\cdot\\!10^{6}$ | $0.4738\pm 0.0020$
5 | 631K | $0.241\pm 0.022$ | 631K | $0.2673\pm 0.0072$ | 631K | $0.2744\pm 0.0058$ | 160K | $0.2790\pm 0.0058$
6 | 64K | $-0.10\pm 0.12$ | 64K | $-0.059\pm 0.094$ | 50.2K | $-0.076\pm 0.062$ | 16K | $-0.044\pm 0.066$
7 | 4K | $-0.87\pm 0.66$ | 4K | $-0.23\pm 0.09$ | 4K | $-0.14\pm 0.10$ | 2K | $-0.97\pm 0.65$
Table 12: Monte Carlo integration results for the $R^{({\rm V})}$ ratios as
defined in the text after $N_{\rm MC}$ phase-space point evaluations for
$n$-gluon scattering and different color-sampling schemes using color-dressed
tree-level and one-loop amplitude calculations.
Using the $R^{({\rm V})}$ and $R^{({\rm V})}_{\rm MC}(N_{\rm MC})$ ratios, we
summarize in Figs. 20-23 our Monte Carlo integration results for
$n=4,\ldots,7$ gluon processes and for the various color-sampling schemes. The
upper graphs display the averaging of $S_{\rm MC}^{(0+1)}$ normalized to the
Monte Carlo average of the color-summed LO contribution as a function of the
number of phase-space evaluations.131313As for the LO studies in Sec. 3.3, the
gluon polarizations are taken alternating and remain fixed while performing
the Monte Carlo integrations. We also indicate the estimate of the integration
uncertainty, see Eqs. (70) and (39). To compare all different test cases,
Table 12 list the final values for $R^{({\rm V})}$. In all these figures we
plot in the lower graphs the number of phase-space point evaluations needed to
reach a certain relative integration uncertainty on $R^{({\rm V})}_{\rm
MC}(N_{\rm MC})$. We show in Table 11 the results of the curve fittings
represented by the dashed lines in these plots.
As is clear from these Monte Carlo averaging tests and results, the
convergence is more than satisfactory for future applications of the color-
dressing techniques in NLO calculations. If faster sampling convergence is
required we can evaluate multiple color configurations per phase-space point.
This is shown in the graph, where we have chosen to evaluate four color
configurations at one phase-space point.
Figure 20: Upper graph: convergence of the 4-gluon virtual corrections
integration as a function of the number of evaluated phase-space points. Also
shown is the standard deviation as an estimator of the integration
uncertainty. Lower graph: convergence of the Monte Carlo integration, where
the relative integration uncertainty is shown as a function of the number of
phase-space evaluations. The dashed lines describe the fit functions
$\sigma/\mu=A\,N_{\rm MC}^{-B}$, see also Table 11. The “Naive”, “Conserved”
and “Non-Zero” color-sampling methods are explained in Sec. 3.3. The points
indicated by “Non-Zero, $N_{\rm colpts=4}$” average over 4 color
configurations per phase-space point.
Figure 21: Upper graph: convergence of the 5-gluon virtual corrections
integration as a function of the number of evaluated phase-space points. Also
shown is the standard deviation as an estimator of the integration
uncertainty. Lower graph: convergence of the Monte Carlo integration, where
the relative integration uncertainty is shown as a function of the number of
phase-space evaluations. The dashed lines describe the fit functions
$\sigma/\mu=A\,N_{\rm MC}^{-B}$, see also Table 11. The “Naive”, “Conserved”
and “Non-Zero” color-sampling methods are explained in Sec. 3.3. The points
indicated by “Non-Zero, $N_{\rm colpts=4}$” average over 4 color
configurations per phase-space point.
Figure 22: Upper graph: convergence of the 6-gluon virtual corrections
integration as a function of the number of evaluated phase-space points. Also
shown is the standard deviation as an estimator of the integration
uncertainty. Lower graph: convergence of the Monte Carlo integration, where
this time the standard deviation is shown as a function of the number of
phase-space evaluations. Note that for this case, the virtual corrections are
as large as the LO contribution so that the full result is close to zero. The
dashed lines describe the fit functions $\sigma=A\,N_{\rm MC}^{-B}$, see also
Table 11. The “Naive”, “Conserved” and “Non-Zero” color-sampling methods are
explained in Sec. 3.3. The points indicated by “Non-Zero, $N_{\rm colpts=4}$”
average over 4 color configurations per phase-space point.
Figure 23: Upper graph: convergence of the 7-gluon virtual corrections
integration as a function of the number of evaluated phase-space points. Also
shown is the standard deviation as an estimator of the integration
uncertainty. Lower graph: convergence of the Monte Carlo integration, where
for this case, only the standard deviation is shown as a function of the
number of phase-space evaluations. The “Naive”, “Conserved” and “Non-Zero”
color-sampling methods are explained in Sec. 3.3. The points indicated by
“Non-Zero, $N_{\rm colpts=4}$” average over 4 color configurations per phase-
space point.
## 5 Conclusions
In this paper we explored the possibility of color sampling within the context
of $D$-dimensional generalized unitarity. Up to now generalized unitarity has
only been used within the context of color-ordered primitive amplitudes. In
the color-ordered approach, color is treated differently from the other
quantum numbers such as spin and flavor. This makes the reconstruction of the
full one-loop amplitude rather cumbersome.
We have reformulated the $D$-dimensional generalized unitarity formalism to
include color dressing. That is, we choose the explicit color of each parton,
together with all other quantum numbers, for each Monte Carlo event. In this
way all particles, colored or colorless, are treated on an equal footing.
There is no distinction between different particles as far as the formalism
goes. Consequently, the resulting algorithm is independent of the type and
flavor of the external particles. E.g. the same algorithm calculates the
6-gluon virtual corrections, the 6-photon virtual corrections and the $W$+6
parton virtual corrections.
The use of unordered amplitudes requires the partition of the external legs
into unordered subsets. This is necessary for the calculation of the tree-
level amplitudes as well as for generating all the unitarity cuts. As a result
the complexity of the resulting algorithm is exponential. That is, the
computer time needed to calculate the virtual corrections grows with a
constant multiplicative factor when one adds external particles. In addition,
we have to sum over all color states of the internal lines. One may conclude
from these general features that the implementation of the color-dressed
$D$-dimensional generalized unitarity is less efficient in comparison with an
implementation based on ordered primitive amplitudes. As we have explicitly
demonstrated for the example of calculating the virtual corrections to
$n$-gluon scattering, this is not the case. We compared the color-sampling
approach for both the color-ordered and color-dressed case. The calculation of
the virtual corrections in the color-dressed case scales as $7^{n}$, while in
the color-ordered case the effective scaling up to 10 gluons behaves as
$9^{n}$. Moreover, the color-dressed calculation has a better accuracy in
calculating the value of the one-loop amplitude. The improved accuracy over
color-ordered evaluations increases with $n$.
As we showed for $n$-gluon scattering, the color-dressed approach becomes more
efficient than the color-ordered method for large $n$. One could argue that
the differences are small and color sampling over the ordered $n$-gluon
amplitudes will work as well. However, when including quarks and other
electro-weak particles the color-dressed approach will easily win out over the
color-ordered approach. This is because any notion of primitive amplitudes is
absent. The algorithm simply calculates the virtual correction. Moreover, the
color-dressed algorithm remains identical when including quarks and electro-
weak particles. It is this algorithmic simplicity that will enable us to
employ parallel programming to significantly improve the computer evaluation
time.
We conclude that the color-dressed formulation is competitive for calculating
one-loop virtual corrections for $n$-gluon scattering. It is expected that it
will be even more efficient in calculating virtual corrections for processes
involving quarks and electro-weak gauge bosons in addition to the gluons.
## Acknowledgments
We would like to thank Giulia Zanderighi, Kirill Melnikov, Stefan Höche and
Tanju Gleisberg for helpful discussions on the subject. Fermilab is operated
by Fermi Research Alliance, LLC, under contract DE-AC02-07CH11359 with the
United States Department of Energy.
## Appendix A The Tree-Level 6-Quark Amplitude
As an example we can take a few recursive steps in calculating the 6-quark
tree-level matrix element. We start with the definition of the tree-level
matrix element in terms of the 5-quark fermionic current
${\cal M}^{(0)}\left({\bf u},{\bf\bar{u}},{\bf d},{\bf\bar{d}},{\bf
s},{\bf\bar{s}}\right)\;=\;P^{-1}\left[J\left({\bf u},{\bf\bar{u}},{\bf
d},{\bf\bar{d}},{\bf s}\right),J\left({\bf\bar{s}}\right)\right]$ (73)
where we use the shorthand notation ${\bf u}=u_{i_{1}}^{\lambda_{1}}(K_{1})$,
${\bf\bar{u}}=\bar{u}_{j_{1}}^{-\lambda_{1}}(K_{2})$, ${\bf
d}=d_{i_{2}}^{\lambda_{2}}(K_{3})$,
${\bf\bar{d}}=\bar{d}_{j_{2}}^{-\lambda_{2}}(K_{4})$, ${\bf
s}=s_{i_{3}}^{\lambda_{3}}(K_{5})$ and
${\bf\bar{s}}={\bar{s}}_{j_{3}}^{-\lambda_{3}}(K_{6})$. The 5-quark fermionic
current decomposes into
$\displaystyle J_{\bf\bar{s}}\left({\bf u},{\bf\bar{u}},{\bf
d},{\bf\bar{d}},{\bf s}\right)$ $\displaystyle=$ $\displaystyle
P_{\bf\bar{s}}\Big{[}D\big{[}J\left({\bf d},{\bf\bar{d}},{\bf
s}\right),J\left({\bf
u},{\bf\bar{u}}\right)\big{]}\Big{]}+P_{\bf\bar{s}}\Big{[}D\big{[}J\left({\bf
u},{\bf\bar{u}},{\bf s}\right),J\left({\bf
d},{\bf\bar{d}}\right)\big{]}\Big{]}$ (74) $\displaystyle+$ $\displaystyle
P_{\bf\bar{s}}\Big{[}D\big{[}J\left({\bf s}\right),J\left({\bf
u},{\bf\bar{u}},{\bf d},{\bf\bar{d}}\right)\big{]}\Big{]}\ .$
The 3-quark fermionic current decomposes into
$J_{\bf\bar{s}}\left({\bf q},{\bf\bar{q}},{\bf
s}\right)\;=\;P_{\bf\bar{s}}\Big{[}D\big{[}J\left({\bf s}\right),J\left({\bf
q},{\bf\bar{q}}\right)\big{]}\Big{]}\ ,$ (75)
where ${\bf q}\in\\{{\bf u},{\bf d}\\}$ and
${\bf\bar{q}}\in\\{{\bf\bar{u}},{\bf\bar{d}}\\}$. The 1-quark fermionic
current is simply the source term. Finally the 4-quark gluonic current is
given by
$\displaystyle J_{\bf g}\left({\bf u},{\bf\bar{u}},{\bf
d},{\bf\bar{d}}\right)$ $\displaystyle=$ $\displaystyle P_{\bf
g}\Big{[}D\big{[}J\left({\bf u}\right),J\left({\bf d},{\bf\bar{d}},{\bf
u}\right)\big{]}\Big{]}+P_{\bf g}\Big{[}D\big{[}J\left({\bf u},{\bf
d},{\bf\bar{u}}\right),J\left({\bf\bar{u}}\right)\big{]}\Big{]}$ (76)
$\displaystyle+$ $\displaystyle P_{\bf g}\Big{[}D\big{[}J\left({\bf
d}\right),J\left({\bf u},{\bf\bar{u}},{\bf
d}\right)\big{]}\Big{]}+P_{g}\Big{[}D\big{[}J\left({\bf d},{\bf
u},{\bf\bar{u}}\right),J\left({\bf\bar{d}}\right)\big{]}\Big{]}\ ,$
and the 2-quark gluonic current is written as
$J_{\bf g}\left({\bf q},{\bf\bar{q}}\right)\;=\;P_{\bf
g}\Big{[}D\big{[}J\left({\bf
q}\right),J\left({\bf\bar{q}}\right)\big{]}\Big{]}\ .$ (77)
The above steps define the 6-quark LO amplitude recursively as would be done
by the algorithm. Note that we have ignored all flavor violating currents.
## Appendix B The Implemented Gluon Recursion Relation
Making use of the color-flow representation [32], we define the color-dressed
gluon currents as $3\times 3$ matrices of ordered gluon currents:
$J^{(IJ)}_{\mu}(g^{\lambda_{1}}_{1})\;=\;\delta^{I}_{j_{1}}\delta^{i_{1}}_{J}J_{\mu}(g^{\lambda_{1}}_{1})\
,$ (78)
where the external gluon $g_{1}$ has the polarization $\lambda_{1}$ and four-
momentum $K_{1}$, its colors are denoted by $(ij)_{1}$. The color-flow labels
of the dressed current are $(IJ)$ and $\mu$ indicates the Lorentz label. Using
this definition, the connection to the compact notation introduced in Sec. 3.1
is found as
$J_{\bf g}\big{(}{\bf
g}_{1}\big{)}\;=\;\delta^{Ii_{1}}\delta^{Jj_{1}}\,\varepsilon_{\mu}^{\lambda_{1}}(K_{1})\;\equiv\;\delta^{J}_{j_{1}}\delta^{i_{1}}_{I}J_{\mu}(g^{\lambda_{1}}_{1})\;=\;J^{(JI)}_{\mu}(g^{\lambda_{1}}_{1})\
.$ (79)
Since we only consider gluons, a plain numbering of the external particles
${\bf g}_{k}=\\{g_{k},\lambda_{k},(ij)_{k},K_{k}\\}$ is sufficient and helps
simplify the notation such that the color dressing becomes more emphasized.
Hence, in all what follows we write $J_{\bf g}({\bf
1})=\delta^{J}_{j_{1}}\delta^{i_{1}}_{I}J_{\mu}(1)=J^{(JI)}_{\mu}(1)$. Dressed
$n$-gluon currents are then described by
$J^{(IJ)}_{\mu}(1,2,\ldots,n)\;=\sum\limits_{\sigma\in
S_{n}}\delta^{I}_{j_{\sigma_{1}}}\delta^{i_{\sigma_{1}}}_{j_{\sigma_{2}}}\cdots\delta^{i_{\sigma_{n-1}}}_{j_{\sigma_{n}}}\delta^{i_{\sigma_{n}}}_{J}\
J_{\mu}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\ ,$ (80)
which follows as a consequence of the color decomposition of the tree-level
amplitude into ordered ones:
${\cal M}^{(0)}(1,2,\ldots,n,n+1)\;=\sum\limits_{\sigma\in
S_{n}}\delta^{i_{n+1}}_{j_{\sigma_{1}}}\delta^{i_{\sigma_{1}}}_{j_{\sigma_{2}}}\cdots\delta^{i_{\sigma_{n-1}}}_{j_{\sigma_{n}}}\delta^{i_{\sigma_{n}}}_{j_{n+1}}\
m^{(0)}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n},n+1)\ .$ (81)
The vectors $\sigma$ describe the elements of the permutations $S_{n}$ of the
set $\\{1,2,\ldots,n\\}$. With the color-ordered amplitudes
$m^{(0)}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n},n+1)$ expressed through
ordered $J$-currents and the definition of the dressed currents at hand, we
can re-write the last equation and formulate the tree-level amplitude in terms
of the color-dressed currents:
$\displaystyle{\cal M}^{(0)}(1,2,\ldots,n,n+1)$ $\displaystyle=$
$\displaystyle K^{2}_{\\{1,2,\ldots,n\\}}\;\times$ (82)
$\displaystyle\sum\limits_{\sigma\in
S_{n}}\delta^{I}_{j_{\sigma_{1}}}\delta^{i_{\sigma_{1}}}_{j_{\sigma_{2}}}\cdots\delta^{i_{\sigma_{n-1}}}_{j_{\sigma_{n}}}\delta^{i_{\sigma_{n}}}_{J}\
J_{\mu}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\
\delta^{J}_{j_{n+1}}\delta^{i_{n+1}}_{I}\ J^{\mu}(n+1)$ $\displaystyle=$
$\displaystyle K^{2}_{\\{1,2,\ldots,n\\}}\
J^{(IJ)}_{\mu}(1,2,\ldots,n)\;J^{(JI),\;\\!\mu}(n+1)\ .$
Owing to the simple color structure of the one-gluon current, the summation
over the color indices $(IJ)$ effectively reduces to the calculation of a
single scalar product of the ordered currents $J^{(i_{n+1}\,j_{n+1})}_{\mu}$
and $J^{(j_{n+1}\,i_{n+1}),\;\\!\mu}$. The invariant-mass prefactor $K^{2}$ is
determined by the gluon momenta via
$K^{2}_{\\{1,2,\ldots,n\\}}=(K_{1}+K_{2}+\ldots+K_{n})^{2}$. The one-gluon
current is given in Eq. (78), while the multi-gluon current is obtained
recursively. Starting from Eq. (80), one incorporates the ordered gluon
recurrence relation to evaluate $J_{\mu}(\sigma_{1},\ldots,\sigma_{n})$ and
re-groups accordingly to identify the partitioning. After some algebra, one
finds
$\displaystyle J^{IJ}_{\mu}(1,2,\ldots,n)$ $\displaystyle=$ $\displaystyle
K^{-2}_{\\{1,2,..,n\\}}\,\Bigg{[}\;\
\sum_{P_{\pi_{1}\pi_{2}}(1,\ldots,n)}\Big{(}\delta^{ILN}_{KMJ}-\delta^{INL}_{MKJ}\Big{)}\left[J^{(KL)}_{\mu}(\pi_{1}),J^{(MN)}_{\mu}(\pi_{2})\right]\
+$ (83)
$\displaystyle\sum_{P_{\pi_{1}\pi_{2}\pi_{3}}(1,\ldots,n)}\Big{(}\delta^{ILNP}_{KMOJ}+\delta^{IPNL}_{OMKJ}-\delta^{ILPN}_{KOMJ}-\delta^{INPL}_{MOKJ}\Big{)}\
\times$ $\displaystyle\hskip
59.75078pt\bigg{(}\left\\{J^{(KL)}_{\mu}(\pi_{1}),J^{(MN)}_{\mu}(\pi_{2}),J^{(OP)}_{\mu}(\pi_{3})\right\\}\;+\;\pi_{1}\leftrightarrow\pi_{2}\bigg{)}\;\
\Bigg{]}$
where we have employed the bracket notation for ordered-current operations,
which was introduced in Ref. [20]. The partition sums are explained in Sec.
3.1 and an implicit summation over the color indices $K,L,M,N,O,P$ is
understood. To efficiently compute the dressed currents, the color factors in
front of the operator brackets can be pre-calculated such that the computation
of zero color-weight contributions can be avoided. We have used the shorthand
notation
$\delta^{ik\cdots m}_{jl\cdots
n}\;=\;\delta^{i}_{j}\delta^{k}_{l}\cdots\delta^{m}_{n}\ .$ (84)
The recursion relation presented in Eq. (83) scales asymptotically as $4^{n}$,
since we kept the 4-gluon vertex as an entity in our calculation. As a
consequence we have to evaluate 3-subset partitions and the corresponding
curly brackets that merge three different dressed currents.
## References
* [1] T. Stelzer and W. F. Long, Comput. Phys. Commun. 81 (1994) 357 [arXiv:hep-ph/9401258].
* [2] M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, JHEP 0307 (2003) 001 [arXiv:hep-ph/0206293].
* [3] P. D. Draggiotis, R. H. P. Kleiss and C. G. Papadopoulos, Eur. Phys. J. C 24 (2002) 447 [arXiv:hep-ph/0202201].
* [4] E. Boos et al. [CompHEP Collaboration], Nucl. Instrum. Meth. A 534 (2004) 250 [arXiv:hep-ph/0403113].
* [5] T. Gleisberg and S. Höche, JHEP 0812 (2008) 039 [arXiv:0808.3674 [hep-ph]].
* [6] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 435 (1995) 59 [arXiv:hep-ph/9409265].
* [7] R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 725 (2005) 275 [arXiv:hep-th/0412103].
* [8] C. F. Berger, Z. Bern, L. J. Dixon, D. Forde and D. A. Kosower, Phys. Rev. D 74 (2006) 036009 [arXiv:hep-ph/0604195].
* [9] G. Ossola, C. G. Papadopoulos and R. Pittau, Nucl. Phys. B 763 (2007) 147 [arXiv:hep-ph/0609007].
* [10] W. T. Giele, Z. Kunszt and K. Melnikov, JHEP 0804, 049 (2008) [arXiv:0801.2237 [hep-ph]].
* [11] C. F. Berger et al., Phys. Rev. D 78, 036003 (2008) [arXiv:0803.4180 [hep-ph]].
* [12] A. van Hameren, C. G. Papadopoulos and R. Pittau, arXiv:0903.4665 [hep-ph].
* [13] R. Keith Ellis, K. Melnikov and G. Zanderighi, arXiv:0906.1445 [hep-ph].
* [14] K. Melnikov and M. Schulze, JHEP 0908, 049 (2009) [arXiv:0907.3090 [hep-ph]].
* [15] G. Bevilacqua, M. Czakon, C. G. Papadopoulos, R. Pittau and M. Worek, JHEP 0909 (2009) 109 [arXiv:0907.4723 [hep-ph]].
* [16] C. F. Berger et al., arXiv:0909.4949 [hep-ph].
* [17] K. Melnikov and G. Zanderighi, arXiv:0910.3671 [hep-ph].
* [18] F. A. Berends and W. Giele, Nucl. Phys. B 294, 700 (1987).
* [19] M. L. Mangano, S. J. Parke and Z. Xu, Nucl. Phys. B 298, 653 (1988).
* [20] F. A. Berends and W. T. Giele, Nucl. Phys. B 306, 759 (1988).
* [21] K. Ellis, W. Giele and Z. Kunszt, “The NLO multileg working group: Summary report”, Published in “Les Houches 2007, Physics at TeV colliders”, [arXiv:0803.0494 [hep-ph]].
* [22] M. Bruinsma, “The Caravaglios-Moretti algorithm and vanishing theorems in scalar theories”, Master Thesis, Univ. of Amsterdam, 1996.
* [23] P. Draggiotis, R. H. P. Kleiss and C. G. Papadopoulos, Phys. Lett. B 439 (1998) 157 [arXiv:hep-ph/9807207].
* [24] C. Duhr, S. Höche and F. Maltoni, JHEP 0608 (2006) 062 [arXiv:hep-ph/0607057].
* [25] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 425 (1994) 217 [arXiv:hep-ph/9403226].
* [26] R. K. Ellis, W. T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, JHEP 0901, 012 (2009) [arXiv:0810.2762 [hep-ph]].
* [27] F. Caravaglios, M. L. Mangano, M. Moretti and R. Pittau, Nucl. Phys. B 539 (1999) 215 [arXiv:hep-ph/9807570].
* [28] A. Cafarella, C. G. Papadopoulos and M. Worek, Comput. Phys. Commun. 180 (2009) 1941 [arXiv:0710.2427 [hep-ph]].
* [29] F. Caravaglios and M. Moretti, Phys. Lett. B 358, 332 (1995) [arXiv:hep-ph/9507237].
* [30] J. Winter and W. T. Giele, arXiv:0902.0094 [hep-ph].
* [31] V. Del Duca, L. J. Dixon and F. Maltoni, Nucl. Phys. B 571, 51 (2000) [arXiv:hep-ph/9910563].
* [32] F. Maltoni, K. Paul, T. Stelzer and S. Willenbrock, Phys. Rev. D 67 (2003) 014026 [arXiv:hep-ph/0209271].
* [33] R. K. Ellis, W. T. Giele and Z. Kunszt, JHEP 0803 (2008) 003 [arXiv:0708.2398 [hep-ph]].
* [34] S. D. Badger, JHEP 0901 (2009) 049 [arXiv:0806.4600v1 [hep-ph]].
* [35] W. T. Giele and G. Zanderighi, JHEP 0806, 038 (2008) [arXiv:0805.2152 [hep-ph]].
* [36] A. Lazopoulos, arXiv:0812.2998 [hep-ph].
* [37] R. K. Ellis, W. T. Giele, Z. Kunszt and K. Melnikov, arXiv:0806.3467 [hep-ph].
* [38] C. F. Berger et al., arXiv:0907.1984 [hep-ph].
* [39] F. A. Berends, W. T. Giele and H. Kuijf, Nucl. Phys. B 333, 120 (1990).
* [40] Z. Bern, A. De Freitas, L. J. Dixon and H. L. Wong, Phys. Rev. D 66, 085002 (2002) [arXiv:hep-ph/0202271].
* [41] R. K. Ellis and G. Zanderighi, JHEP 0802 (2008) 002 [arXiv:0712.1851 [hep-ph]].
* [42] T. Gleisberg and F. Krauss, Eur. Phys. J. C 53 (2008) 501 [arXiv:0709.2881 [hep-ph]].
* [43] R. Frederix, T. Gehrmann and N. Greiner, JHEP 0809 (2008) 122 [arXiv:0808.2128 [hep-ph]].
|
arxiv-papers
| 2009-11-10T18:27:35 |
2024-09-04T02:49:06.388806
|
{
"license": "Public Domain",
"authors": "Walter Giele, Zoltan Kunszt and Jan Winter",
"submitter": "Walter T. Giele",
"url": "https://arxiv.org/abs/0911.1962"
}
|
0911.2012
|
# The Grand Ensemble of Subsystems: Applications to Fitting Single-Component
Adsorption Data
Martín A. Mosquera
Universidad del Valle, Cali, Colombia
martinmo@univalle.edu.co
Abstract
In this work, it is reviewed the cell model of adsorption by using the small
system grand ensemble method. Under the common assumption that the adsorbate
phase is divided into identical and weakly-interacting subsystems, it is
suggested that the general multiparametric isotherm that arise from the theory
may be used to fit the experimental data of adsorption of gases and vapours on
microporous adorbents (type I isotherm), even though these present
heterogeneties like pore size distribution. A simplified isotherm that reduces
the number of adjustable parameters with respect to the general isotherm is
proposed. Both isotherms, due to their relative high accuracy, can be used to
estimate thermodynamical properties like isosteric and differential heats of
adsorption. Also, a simple method is presented for systems that show an
apparent variation in the coverage limit as function of temperature; for
several systems, this method reduces the fitting deviations. Finally, several
applications to fitting data of adsorption, taken from literature, of some
gases on activated carbon, molecular sieving carbon, silica gel, and pillared
clays are presented.
## 1 Introduction
Correlating adsorption data obtained from experiments or computer calculations
is necessary to save time and efforts in additional experimentation or
computing time. Because of the complexity of equilibrium adsorption
phenomenon, even today, the research of new models of adsorption is active. A
good parameter-adjustable model must fit well the experimental data and
correctly predict thermodynamic quantities; therefore, it is necessary to
analyze the model and assess its capabilities. Many real-world applications of
adsorption encompass a range of saturation up to $10^{6}$ and $10^{12}$ in
pressure [1]. Therefore, in some cases, in order to fit the adsorption data,
it may be necessary models that give experimentalist and process engineer the
possibility to set the number of adjustable parameters.
Many adsorbents used in industry and research present a certain degree of
heterogeneity in terms of a pore size distribution and certain surface
topography. Depending on each of these factors, different isotherm models are
proposed. For example, lattice gas theories that simplify the structure of the
adsorbent surface by taking into account the most important adsorption sites
of the adsorbent surface or adsorption pores have been developed, and most of
these theories are consistent with experiments and computer simulations [2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12]. A disadvantage of this kind of models is that
many gas-solid systems present unique characteristics that must be considered
in each particular model. Therefore, once a model has been developed for a
certain gas-solid system it must be tested with adsorption and thermodynamic
data.
Another commonly used method to obtain isotherms for heterogeneous systems is
the integration over a patchwise topography of adsorption sites [13]. Modified
isotherms arise by assuming an adsorption energy distribution function and an
ideal local isotherm like Langmuir. Nevertheless, although some of these
equations work well for a large number of systems, they present limitations
involved in using a particular model, e.g., some models do not provide the
correct Henry’s law limit. Additionally, this method is somewhat difficult
because the complexity of the adsorption energy distribution makes it
difficult to obtain an analytical expression for the transformed isotherm. The
Sips[14], Toth[15], and UNILAN[16] isotherms are the most widely used
isotherms of this type.
Recently, a method proposed by Reiss and Merry [17] to study the small system
grand ensemble (SSGE)[17, 18, 19, 20] has been useful to analyze the
properties of these small subsystems that all together form a homogeneous
phase. Initially, this model was proposed to study rigid spheres, but, in
principle, it can be applied to other systems [19], as shown in this work for
the adsorption case. The advantage of this SSGE method is that it can be
studied the implications of assuming interactions between a particular
subsystem and its surroundings, allowing us to analyze in detail the
properties of these small systems and their consequences for the adsorption of
gases.
In this work, it is briefly reviewed the statistical mechanical cell model of
adsorption[21, 22, 23] by using the SSGE method. The general isotherm that
arises from the cell model has been useful to fit experimental data of
adsorption of gases on zeolites [24, 25, 26, 27, 28, 29, 30, 31]. Here, an
accurate and simplified isotherm is proposed, this isotherm is similar to the
empirical isotherm proposed by Ruthven [32]. It is shown that both the
simplified and general isotherms might be applied to the adsorption of gases
on heterogeneous microporous adsorbents. To prove this hypothesis, several
experimental adsorption isotherms are correctly fitted by using the isotherm
equations studied here. From the fitting results, it is suggested that the
isotherm equations treated here can be used to fit type I experimental
adsorption isotherms and predict thermodynamic properties. Furthermore,
extrapolations to temperatures near the experimental temperature range are
possible. Because of the complexity of heterogeneous adsorbents, it is argued
that the method presented here is semi-empirical and the cells ensemble can be
considered as an abstract and idealized grand ensemble of small subsystems.
## 2 Theory
### 2.1 The General Adsorption Isotherm
Consider a single-component gas in equilibrium with an adsorbate phase.
Suppose that the adsorbate phase can be divided into $M$ identical cells that
weakly interact between each other. The total number of cells is temperature-
independent and the cell volume as well. Now, let us regard a particular cell
as a subsystem of interest and the remaining $M-1$ cells that surround this
subsystem as a bath. It is assumed that the effect of the adsorbent on the
adsorbate manifests only by means of an interaction potential and the
adsorbate presence on the adsorbent surface (or cavities) does not affect its
properties. Define the following partition functions:
$\begin{split}Q^{*}(T,&V-v_{\mathrm{s}},N-n,v_{\mathrm{s}},n)=\frac{1}{\Lambda^{dN}(N-n)!n!}\int_{v_{\mathrm{s}}}d\mathbf{R}_{1}\cdots
d\mathbf{R}_{n}\times\\\ &\int_{V-v_{\mathrm{s}}}d\mathbf{R}_{n+1}\cdots
d\mathbf{R}_{N}\exp\Big{[}-\frac{\mathscr{U}(\mathbf{R}^{N})}{k_{\mathrm{B}}T}\Big{]}\end{split}$
(1)
$Q(T,V,N)=\frac{1}{\Lambda^{dN}N!}\int_{V}d\mathbf{R}^{N}\exp\Big{[}-\frac{\mathscr{U}(\mathbf{R}^{N})}{k_{\mathrm{B}}T}\Big{]}=\frac{Z(T,V,N)}{\Lambda^{dN}N!}$
(2)
where $\Lambda$ is the deBroglie thermal wavelength, $V$ and $N$ are the
adsorbate phase volume (or area) and the total number of adsorbed molecules,
respectively, $v_{\mathrm{s}}$ and $n$ are the volume (or area) and number of
molecules of the subsystem, respectively, and $d$ is the adsorbate phase
dimension. The potential energy $\mathscr{U}$ may be decomposed into three
contributions:
$\begin{split}\mathscr{U}(\mathbf{R}^{N})=\mathscr{U}_{\mathrm{S}}(\mathbf{R}_{1},\ldots,\mathbf{R}_{n})+\mathscr{U}_{\mathrm{B}}&(\mathbf{R}_{n+1},\ldots,\mathbf{R}_{N})\\\
+\mathscr{U}_{\sigma}(\mathbf{R}^{N})\end{split}$ (3)
the term $\mathscr{U}_{\mathrm{S}}$ is the potential energy of a subsystem and
$\mathscr{U}_{\mathrm{B}}$ is the bath potential energy, these potentials are
the sum of the molecule-molecule and the molecule-solid interaction
potentials. The term $\mathscr{U}_{\sigma}$ represents the interaction between
the subsystem and the bath. Consider the usual probability density for the
canonical ensemble [23, 33]:
$\rho(\mathbf{P}^{N},\mathbf{R}^{N})=C\times\exp\Big{[}-\frac{\mathscr{H}(\mathbf{P}^{N},\mathbf{R}^{N})}{k_{\mathrm{B}}T}\Big{]}$
(4)
where $C$ is a constant. If we integrate over the subsystem’s phase space and
the bath’s phase space, we obtain the probability of finding $n$ molecules in
the subsystem volume $v_{\mathrm{s}}$ and $N-n$ molecules in the bath volume
$V-v_{\mathrm{s}}$:
$p_{n}=C\times Q^{*}(T,V-v_{\mathrm{s}},N-n,v_{\mathrm{s}},n)$ (5)
It is plausible to assume that a maximum of $n_{\mathrm{s}}$ molecules can
adsorb on the subsystem (the adsorbent has a saturation limit). Therefore, in
order to eliminate the arbitrary constant $C$, we can write:
$p_{n}=\frac{\displaystyle
p_{n}}{\displaystyle\sum_{j=0}^{n_{\mathrm{s}}}p_{j}}$ (6)
$p_{j}$ may be written as follows:
$\begin{split}p_{j}=&C\times\frac{Q^{*}(T,V-v_{\mathrm{s}},N-j,v_{\mathrm{s}},j)}{Q(T,V-v_{\mathrm{s}},N-j)Q(T,v_{\mathrm{s}},j)}\times\\\
&Q(T,V-v_{\mathrm{s}},N-j)Q(T,v_{\mathrm{s}},j)\end{split}$ (7)
By invoking the chemical potential definition we obtain:
$Q(T,V-v_{\mathrm{s}},N-j)=Q(T,V-v_{\mathrm{s}},N)\exp\Big{(}\frac{j\mu}{k_{\mathrm{B}}T}\Big{)}$
(8)
By defining
$\displaystyle A_{N-j,j}^{*}$ $\displaystyle=$ $\displaystyle-
k_{\mathrm{B}}T\ln Q^{*}(T,V-v_{\mathrm{s}},N-j,v_{\mathrm{s}},j)$ (9)
$\displaystyle A_{j}^{\mathrm{S}}$ $\displaystyle=$ $\displaystyle-
k_{\mathrm{B}}T\ln Q(T,v_{\mathrm{s}},j)$ (10) $\displaystyle
A_{N-j}^{\mathrm{B}}$ $\displaystyle=$ $\displaystyle-k_{\mathrm{B}}T\ln
Q(T,V-v_{\mathrm{s}},N-j)$ (11) $\displaystyle A_{j}^{\sigma}$
$\displaystyle=$ $\displaystyle
A_{N-j,j}^{*}-A_{N-j}^{\mathrm{B}}-A_{j}^{\mathrm{S}}$ (12)
it is obtained
$p_{j}=C\times
Q(T,V-v_{\mathrm{s}},N)\exp\Bigg{[}-\frac{(A_{j}^{\sigma}+A_{j}^{\mathrm{S}})}{k_{\mathrm{B}}T}\Bigg{]}\lambda^{j}$
(13)
where $\lambda=e^{\beta\mu}$. Hence, we have:
$p_{n}=\frac{\displaystyle\exp\Bigg{[}\frac{n\mu^{0}_{\mathrm{g}}-A_{n}^{\sigma}-A_{n}^{\mathrm{S}}}{k_{\mathrm{B}}T}\Bigg{]}a^{n}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}}\exp\Bigg{[}\frac{j\mu^{0}_{\mathrm{g}}-A_{j}^{\sigma}-A_{j}^{\mathrm{S}}}{k_{\mathrm{B}}T}\Bigg{]}a^{j}}$
(14)
where $\mu^{0}_{\mathrm{g}}$ is the gas-phase chemical potential at the
reference pressure $P^{0}$, and $a$ is the activity:
$a=\frac{f}{P^{0}}$
The term $A_{j}^{\sigma}$ may be regarded as a free energy of interaction
between the subsystem and the bath [17]. If we assume that the bath and the
subsystem weakly interact, then $A_{j}^{\sigma}$ is small. Now, we define:
$K_{j}(T)=\exp\Big{[}\frac{j\mu^{0}_{\mathrm{g}}-A_{j}^{\mathrm{S}}}{k_{\mathrm{B}}T}\Big{]}$
(15)
If the system may be divided into identical weakly-interacting [34] subsystems
then we can state that $\Xi\approx\xi^{M}$ where [22]:
$\xi=\displaystyle 1+\sum_{j=1}^{n_{\mathrm{s}}}K_{j}a^{j}$ (16)
here, $\xi$ is the subsystem’s grand canonical partition function and $\Xi$ is
the grand canonical partition function. We can calculate the saturation ($q$)
as follows:
$q=\frac{q_{\mathrm{m}}}{n_{\mathrm{s}}}\frac{\displaystyle\sum_{j=1}^{n_{\mathrm{s}}}jK_{j}a^{j}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}}K_{j}a^{j}}$ (17)
The term $K_{j}$ can be considered as an adsorption equilibrium constant.
Therefore, by similarity with chemical equilibrium constants, we can propose
the following relation:
$\ln K_{j}=\ln K_{j}^{\circ}-\frac{\Delta h_{j}}{RT}$ (18)
where $\ln K_{j}^{\circ}$ and $\Delta h_{j}$ are a change of entropy and
enthalpy, respectively, related with the adsorption of $j$ molecules on a
representative microscopic subsystem at the reference pressure $P^{0}$. Eq.
(17) was proposed by Langmuir [35] for the case in which a site can hold
several molecules and the adsorbent is a set of non-interacting sites. In
light of his proposal, the term $K_{j}$ is partitioned as follows:
$K_{j}=\frac{1}{j!}\prod_{i=1}^{j}R_{i}$ (19)
In a similar fashion, $R_{j}$ behaves as $K_{j}$:
$R_{j}=R_{j}^{\circ}\exp\Big{(}\frac{-\Delta\overline{h}_{j}}{RT}\Big{)}$ (20)
where
$\displaystyle R_{j}^{\circ}$ $\displaystyle=$
$\displaystyle\frac{jK_{j}^{\circ}}{K_{j-1}^{\circ}}$ (21)
$\displaystyle\Delta\overline{h}_{j}$ $\displaystyle=$ $\displaystyle\Delta
h_{j}-\Delta h_{j-1}$ (22)
Thus, in order to fit an individual isotherm using Eq. (17), the adjustable
parameters are $\\{R_{j}\\}$ instead of $\\{K_{j}\\}$. For the case in which
several isotherms at different temperatures have been measured, the adjustable
parameters are $\\{\ln R_{j}^{\circ}\\}$, $\\{\Delta\overline{h}_{j}\\}$, and
$q_{\mathrm{m}}$, a total of $2n_{\mathrm{s}}+1$ parameters.
The present model suggests that $\xi\rightarrow\Xi$ as the subsystem volume is
increased. Thus, in order to predict $q$, we should assume that the maximum
number of molecules $n_{\mathrm{s}}$ is large enough such that
$A_{j}^{\sigma}\rightarrow 0$ for all $j$ [17]. However, if we assume that
each $R_{j}$ is an adjustable parameter, this would give us a high number of
adjustable parameters with poor statistical confidence [31]. A solution for
this problem could be assume $n_{\mathrm{s}}$ as a small number of molecules
that adsorb on a microscopic imaginary subsystem and the parameters
$\\{R_{j}\\}$ as representative parameters of the experimental adsorption
isotherm. Thus, we are assuming that the adsorbate system can be divided into
an abstract and idealized grand ensemble of subsystems. The characteristics of
the adsorbate+adsorbent system like molecule-surface and molecule-molecule
interactions are included abstractly in these subsystems. This is an ad hoc
guess because we are not claiming that this idealized system exists, but this
assumption is a convenient picture that allows us to apply the cell model even
to heterogeneous systems despite this model was originally conceived to study
relatively homogeneous systems like zeolites.
Accordingly with Eq. (6), the probability that a subsystem with $n$ adsorbed
molecules will be found is (assuming ideal gas phase):
$p_{n}=\frac{K_{n}(P/P^{0})^{n}}{\xi}$ (23)
Eq. (17) can be written as:
$\theta=\frac{1}{n_{\mathrm{s}}}\sum_{j=1}^{n_{\mathrm{s}}}jp_{j}=\frac{1}{n_{\mathrm{s}}}\sum_{j=1}^{n_{\mathrm{s}}}f_{j}$
(24)
here, $\theta=q/q_{\mathrm{m}}$, and $f_{j}$ is a fraction of molecules that
can be found in subsystems with $j$ molecules. It is known that, at low
pressures, the leading term in Eq. (24) is $f_{1}$ (as stated by the Henry’s
law), and, as the pressure increases, the leading term is $f_{2}$. Similarly,
in a certain pressure range, each $f_{j}$ term significantly contributes to
the fractional coverage. Therefore, each parameter $K_{j}$ is estimated with
data taken in a certain range of the saturation $q$. Hence, for intervals
where there is not enough data, these parameters will give large
uncertainties.
The general isotherm (Eq. (17)) and its simplifications have been useful to
fit sets of isotherms of adsorption of gases and vapors on zeolite adsorbents
at different temperatures, where the experimental isotherms show clearly a
saturation limit, or are reported within the same range of saturation [24, 25,
26, 27, 28, 29, 30]. However, a great number of adsorption systems do not show
this behavior because experiments typically are performed within the same
pressure range; therefore, the isotherms apparently show that the saturation
limit depends on temperature (i.e., $q_{\mathrm{m}}$ tend to increase with a
decrease in temperature). Now, consider a certain isotherm at temperature
$T_{1}$, if we consider that $n_{\mathrm{s}}=3$, we have the following
adjustable parameters: $K_{1},\,K_{2},\,K_{3}$, and $q_{\mathrm{m}}(T_{1})$.
On the other hand, suppose that we have a second isotherm at temperature
$T_{2}$ and its apparent saturation limit is $q_{\mathrm{m}}(T_{2})$; if
$T_{2}<T_{1}$, then $q_{\mathrm{m}}(T_{2})>q_{\mathrm{m}}(T_{1})$. This would
suggest that, as the temperature decreases, new subsystems are created, but
this would be a physical inconsistency. To solve this problem, we introduce a
new equilibrium parameter $K_{n_{\mathrm{s}}+1}$ that take into account the
adsorption of an extra molecule on a subsystem at high pressures and low
temperatures. Because this new parameter describes the adsorption at these
conditions, it is possible that there is not enough experimental data to
estimate $\ln R_{n_{\mathrm{s}}+1}^{\circ}$ and
$\Delta\overline{h}_{n_{\mathrm{s}}+1}$, then it is plausible to assume that
$R_{n_{\mathrm{s}}+1}=R_{n_{\mathrm{s}}}$ and write
$K_{n_{\mathrm{s}}+1}=\frac{R_{n_{\mathrm{s}}}}{n_{\mathrm{s}}+1}K_{n_{\mathrm{s}}}$
(25)
Thus, we are assuming that the change of both entropy and enthalpy of
adsorption of the $(n_{\mathrm{s}}+1)$th molecule on a microscopic cell is the
same for the adsorption of the $n_{\mathrm{s}}$th molecule (i.e., $R\ln
R_{n_{\mathrm{s}}+1}^{\circ}=R\ln R_{n_{\mathrm{s}}}^{\circ}$ and
$\Delta\overline{h}_{n_{\mathrm{s}}+1}=\Delta\overline{h}_{n_{\mathrm{s}}}$).
This result is generalized as follows:
$K_{n_{\mathrm{s}}+l}=\frac{R_{n_{\mathrm{s}}}}{n_{\mathrm{s}}+l}K_{n_{\mathrm{s}}+l-1}$
(26)
Now, $\xi$ becomes:
$\xi=1+\sum_{j=1}^{n_{\mathrm{s}}+l}K_{j}a^{j}$
and the isotherm is:
$q=\frac{q_{\mathrm{m}}}{n_{\mathrm{s}}+l}\frac{\displaystyle\sum_{j=1}^{n_{\mathrm{s}}+l}jK_{j}a^{j}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}+l}K_{j}a^{j}}$ (27)
### 2.2 A Simplified Adsorption Isotherm
In order to reduce the number of adjustable parameters and reduce confidence
intervals around parameters estimates, let us re-express Eq. (17) as follows
(assuming ideal gas phase):
$q=\frac{q_{\mathrm{m}}}{n_{\mathrm{s}}}\frac{\displaystyle\sum_{j=1}^{n_{\mathrm{s}}}\frac{Z(T,v_{\mathrm{s}},j)}{(j-1)!(k_{\mathrm{B}}T/P^{0})^{j}}a^{j}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}}\frac{Z(T,v_{\mathrm{s}},j)}{j!(k_{\mathrm{B}}T/P^{0})^{j}}a^{j}}$
(28)
here $a=P/P^{0}$. It can be shown that:
$k_{\mathrm{B}}T^{2}\frac{d}{dT}\ln\Bigg{[}\frac{Z(T,v_{\mathrm{s}},j)}{(k_{\mathrm{B}}T/P^{0})^{j}}\Bigg{]}=\langle\mathscr{U}_{\mathrm{S}}(j)\rangle-
jk_{\mathrm{B}}T$ (29)
where
$\langle\mathscr{U}_{\mathrm{S}}(j)\rangle=\frac{1}{Z(T,v_{\mathrm{s}},n)}\int_{v_{\mathrm{s}}}d\mathbf{R}^{j}\mathscr{U}_{\mathrm{S}}(\mathbf{R}^{j})\exp\Big{[}-\frac{\mathscr{U}_{\mathrm{S}}(\mathbf{R}^{j})}{k_{\mathrm{B}}T}\Big{]}$
(30)
because the adsorption energy $\langle\mathscr{U}_{\mathrm{S}}(j)\rangle$ is
usually greater than $jk_{\mathrm{B}}T$, then we can assume that the latter
term is a constant $jk_{\mathrm{B}}\overline{T}$, where $\overline{T}$ is a
mean temperature. Integration of Eq. (29) gives:
$\ln\Bigg{[}\frac{Z(T,v_{\mathrm{s}},j)}{(k_{\mathrm{B}}T/P^{0})^{j}}\Bigg{]}=K_{j}^{\circ}\exp\Bigg{[}-\Bigg{(}\frac{\langle\mathscr{U}_{\mathrm{S}}(j)\rangle-
jk_{\mathrm{B}}\overline{T}}{k_{\mathrm{B}}T}\Bigg{)}\Bigg{]}$ (31)
As an special case, let us assume that the potential energy
$\mathscr{U}_{S}(j)$ can be written as follows:
$\mathscr{U}_{\mathrm{S}}(j)=\sum_{n=1}^{j}\mathscr{U}_{\mathrm{F-M}}(\mathbf{R}_{n})+\sum_{n>m}\mathscr{U}_{\mathrm{M-M}}(\mathbf{R}_{n},\mathbf{R}_{m})$
(32)
where $\mathscr{U}_{\mathrm{F-M}}$ and $\mathscr{U}_{\mathrm{F-M}}$ are the
field-molecule and the molecule-molecule interaction potentials, respectively.
Now, we obtain
$\langle\mathscr{U}_{\mathrm{S}}(j)\rangle=j\langle\mathscr{U}_{\mathrm{F-M}}\rangle+\frac{j(j-1)}{2}\langle\mathscr{U}_{\mathrm{M-M}}\rangle$
(33)
For the sake of simplicity let us assume that
$\langle\mathscr{U}_{\mathrm{F-M}}\rangle$ and
$\langle\mathscr{U}_{\mathrm{M-M}}\rangle$ do not depend on $j$ and there is a
temperature $T^{\circ}$ such that $K_{j}^{\circ}=(K^{\circ})^{j}$, where
$K^{\circ}$ is the pre-exponential term in the Henry’s constant. Under these
assumptions, it is obtained the adsorption isotherm:
$q=\frac{q_{\mathrm{m}}}{n_{\mathrm{s}}}\frac{\displaystyle\sum_{j=1}^{n_{\mathrm{s}}}\frac{e^{[jE_{\mathrm{M-F}}+j(j-1)E_{\mathrm{M-M}}/2]/RT}}{(j-1)!}(K^{\circ}a)^{j}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}}\frac{e^{[jE_{\mathrm{M-F}}+j(j-1)E_{\mathrm{M-M}}/2]/RT}}{j!}(K^{\circ}a)^{j}}$
(34)
where
$\displaystyle E_{\mathrm{M-F}}$ $\displaystyle=$
$\displaystyle-\mathcal{N}_{\mathrm{A}}\langle\mathscr{U}_{\mathrm{F-M}}\rangle$
(35) $\displaystyle E_{\mathrm{M-M}}$ $\displaystyle=$
$\displaystyle-\mathcal{N}_{\mathrm{A}}\langle\mathscr{U}_{\mathrm{M-M}}\rangle$
(36)
here, $\mathcal{N}_{A}$ is the Avogadro’s constant. If
$\langle\mathscr{U}_{\mathrm{F-M}}\rangle$ and
$\langle\mathscr{U}_{\mathrm{M-M}}\rangle$ depend on $j$, we can introduce the
equation:
$q=\frac{q_{\mathrm{m}}}{n_{\mathrm{s}}+l}\frac{\displaystyle\sum_{j=1}^{n_{\mathrm{s}}+l}\frac{\displaystyle
e^{-\Delta h_{j}/RT}}{\displaystyle(j-1)!}(K^{\circ}a)^{j}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}+l}\frac{\displaystyle e^{-\Delta
h_{j}/RT}}{\displaystyle j!}(K^{\circ}a)^{j}}$ (37)
comparing with Eq. (27) we have:
$K_{j}=\frac{(K^{\circ})^{j}}{j!}e^{-\Delta h_{j}/RT}\qquad
j=1,\ldots,n_{\mathrm{s}}$ (38)
For this simplified isotherm, from Eq. (26), we have:
$K_{n_{\mathrm{s}}+l}=\frac{\displaystyle K^{\circ}e^{-(\Delta
h_{n_{\mathrm{s}}}-\Delta h_{n_{\mathrm{s}}-1})/RT}}{\displaystyle
n_{\mathrm{s}}+l}K_{n_{\mathrm{s}}+l-1}$ (39)
Here, the fitting parameters are $\\{\Delta h_{j}\\}$, $K^{\circ}$, and
$q_{\mathrm{m}}$, a total of $n_{\mathrm{s}}+2$ adjustable parameters. We can
connect this result with the empirical isotherm proposed by Ruthven [32]
($l=0$):
$q=\frac{q_{\mathrm{m}}}{n_{\mathrm{s}}}\frac{\displaystyle\sum_{j=1}^{n_{\mathrm{s}}}\frac{A_{j}}{(j-1)!}(KP)^{j}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}}\frac{A_{j}}{j!}(KP)^{j}}\qquad A_{1}=1$ (40)
in this Eq., $K$ is the Henry’s constant, and $A_{j}$ is an empirical
parameter. The parameter $A_{j}$ is expressed as follows:
$A_{j}=\exp\Big{(}-\frac{\Delta h_{j}}{k_{\mathrm{B}}T}\Big{)}$ (41)
and
$K=\frac{K^{\circ}}{P^{0}}$ (42)
thus, the Henry’s constant in Ruthven’s Eq. is replaced by the pre-exponential
term $K^{\circ}/P^{0}$.
If the term $k_{\mathrm{B}}\overline{T}$ is not precise, Eq. (37) can be
written as
$q=\frac{q_{\mathrm{m}}}{n_{\mathrm{s}}+l}\frac{\displaystyle\sum_{j=1}^{n_{\mathrm{s}}+l}\frac{\displaystyle
e^{-\Delta
u_{j}/RT}}{\displaystyle(j-1)!}\Big{(}\frac{\tilde{K}^{\circ}a}{T^{\prime}}\Big{)}^{j}}{\displaystyle
1+\sum_{j=1}^{n_{\mathrm{s}}+l}\frac{\displaystyle e^{-\Delta
u_{j}/RT}}{\displaystyle
j!}\Big{(}\frac{\tilde{K}^{\circ}a}{T^{\prime}}\Big{)}^{j}}$ (43)
where
$T^{\prime}=\frac{T}{T^{0}}$ (44) $\Delta
u_{j}=\mathcal{N}_{\mathrm{A}}\langle\mathscr{U}_{\mathrm{S}}(j)\rangle$ (45)
here, $T^{0}$ is a reference temperature (e.g., 1 K).
### 2.3 Isosteric Heat of Adsorption
From Eq. (15) we have
$\Delta h_{j}=u_{j}-jh_{\mathrm{g}}^{0}$ (46)
where $u_{j}$ and $h_{\mathrm{g}}^{0}$ are the energy of $j$ molecules in a
subsystem and the molar enthalpy of the gas phase at $P^{0}$, respectively,
$\Delta h_{j}$ is the change of enthalpy related to the adsorption of $j$
molecules on a subsystem. It can be noticed that, in principle, $\Delta h_{j}$
is temperature-dependent, but for the sake of simplicity, we have assumed that
both $h_{\mathrm{g}}^{0}$ and $u_{j}$ are temperature-independent. The
isosteric heat of adsorption can be calculated by means of the following
formula [36]:
$q^{\mathrm{st}}=h_{\mathrm{g}}-u_{\mathrm{a}}$ (47)
where $h_{\mathrm{g}}$ and $u_{\mathrm{a}}$ are the molar enthalpy of the gas
phase and adsorbate molar internal energy respectively. For any function
$g_{j}$, define:
$\langle g_{j}\rangle_{0}=\sum_{j}p_{j}g_{j}$
It is easy to show that [37, 38]
$-q^{\mathrm{st}}=\frac{\langle j\Delta
h_{j}\rangle_{0}-n_{\mathrm{s}}\theta\langle\Delta h_{j}\rangle_{0}}{\langle
j^{2}\rangle_{0}-n_{\mathrm{s}}^{2}\theta^{2}}$ (48)
where $\theta=q/q_{\mathrm{m}}$. From the above eq, the two following limits
arise:
$\displaystyle\lim_{\theta\rightarrow 0}q^{\mathrm{st}}$ $\displaystyle=$
$\displaystyle-\Delta\overline{h}_{1}$ (49)
$\displaystyle\lim_{\theta\rightarrow 1}q^{\mathrm{st}}$ $\displaystyle=$
$\displaystyle-\Delta\overline{h}_{n_{\mathrm{s}}}$ (50)
Equivalently, if Eq. (43) is used, the differential heat of adsorption can be
calculated as follows:
$-q^{\mathrm{d}}=\frac{\langle j\Delta
u_{j}\rangle_{0}-n_{\mathrm{s}}\theta\langle\Delta u_{j}\rangle_{0}}{\langle
j^{2}\rangle_{0}-n_{\mathrm{s}}^{2}\theta^{2}}$ (51)
For the experimental isotherms studied here, both simplified isotherms (Eqs.
(37) and (43)) give the same results. Hereinafter, Eq. (37) will be used due
to its simple relationship with Eq. (48).
## 3 Results and Discussion
To fit the experimental data cited here, the unweighted least-squares
Levenberg-Marquardt algorithm was used. Details of the results of fitting Eqs.
(37) and (27) to several experimental adsorption data are shown in Tables 1
and 2; the parameters are tabulated with their corresponding marginal
confidence intervals [39, 40]. The study of these confidence intervals is
necessary to avoid over-interpretation of the parameters [41] and to estimate
uncertainties in predicted thermodynamic properties. The parameter $\ln
K^{\circ}$ (or $\ln K_{j}^{\circ}$) was used instead of $K^{\circ}$ (or
$K_{j}^{\circ}$) due to the restriction $K^{\circ}>0$ (for all $j$). To
analyze the fittings precision, the following deviation percentage parameter
was used:
$D=\frac{100\%}{N_{\mathrm{T}}}\times\sum_{j=1}^{N_{\mathrm{temp}}}\sum_{i=1}^{N_{\mathrm{P}}(j)}\Bigg{|}\frac{q(P_{i},T_{j})-q_{\mathrm{exp}}(P_{i},T_{j})}{q_{\mathrm{exp}}(P_{i},T_{j})}\Bigg{|}$
(52)
where $N_{\mathrm{temp}}$ is the number of isotherms, $N_{\mathrm{P}}(j)$ is
the number of experimental data taken at $T_{j}$ temperature, $N_{\mathrm{T}}$
is the total number of experimental data, $q(P_{i},T_{j})$ is the calculated
saturation, and $q_{\mathrm{exp}}(P_{i},T_{j})$ is the experimentally measured
saturation. It can be noticed in Table 1 that the deviation parameter $D$ in
all cases is less than $5\%$.
Figure 1: Comparison between the experimental data of $\mathrm{SF_{6}}$
adsorption on W-A(673) [42] and Eq. (17); symbols: experiment, solid line: Eq.
(37). Table 1: Estimated parameters for several systems using Eq. 37._a,b,c,d_
| $q_{\mathrm{m}}$ (or $v$) | $\ln K^{\circ}$ | $-10^{-3}\times\Delta h_{j}/R\,$ | $D$ | Temperature | Pressure
---|---|---|---|---|---|---
System | $\mathrm{mmol\,g^{-1}}$ (or $\mathrm{cm^{3}\,g^{-1}}$) | | $\mathrm{K}$ | $\%$ | range (K) | range (kPa)
$\mathrm{SF_{6}}$+W-A(673)[42] | $23.89\pm 0.44$ | $-13.67\pm 0.16$ | $3.107\pm 0.055$ | 2.97 | 266.5-297.5 | 0.1-100
$l=1$ | | | $6.112\pm 0.077$ | | |
| | | $8.08\pm 0.50$ | | |
| | | $11.62\pm 0.32$ | | |
$\mathrm{C_{3}H_{8}}$+W-A[42] | $34.6\pm 3.9$ | $-14.65\pm 0.30$ | $3.391\pm 0.096$ | 3.2 | 267-298 | 0.1-100
$l=1$ | | | $7.472\pm 0.097$ | | |
| | | $10.8\pm 0.23$ | | |
| | | $14.8\pm 0.60$ | | |
| | | $17.9\pm 0.48$ | | |
$\mathrm{HFC}$-227ea+AC[43] | $3.87\pm 0.27$ | $-16.67\pm 0.27$ | $6.066\pm 0.098$ | 2.36 | 283.15-363.15 | 0.01-100
$l=0$ | | | $11.39\pm 0.19$ | | |
| | | $16.38\pm 0.32$ | | |
| | | $20.82\pm 0.45$ | | |
| | | $24.71\pm 0.68$ | | |
$\mathrm{HFP}$+AC[43] | $3.87\pm 0.20$ | $-15.66\pm 0.20$ | $5.429\pm 0.075$ | 2.53 | 283.15-363.15 | 0.01-100
$l=0$ | | | $10.15\pm 0.15$ | | |
| | | $14.68\pm 0.23$ | | |
| | | $18.69\pm 0.32$ | | |
| | | $22.33\pm 0.45$ | | |
$\mathrm{CO_{2}}$+ZC[44] | $4.82\pm 0.39$ | $-15.71\pm 0.29$ | $5.03\pm 0.10$ | 3.84 | 273.15-353.15 | 0.05-100
$l=1$ | | | $9.39\pm 0.20$ | | |
| | | $13.45\pm 0.31$ | | |
| | | $17.00\pm 0.43$ | | |
$\mathrm{CO_{2}}$+SG[45] | $14.5\pm 1.0$ | $-15.22\pm 0.10$ | $3.068\pm 0.030$ | 1.47 | 278-328 | 50-3400
$l=4$ | | | $5.759\pm 0.052$ | | |
| | | $8.454\pm 0.094$ | | |
| | | $10.87\pm 0.12$ | | |
_a_ In all cases, $P^{0}$ is expressed in kPa.
_b_ Abbreviations: activated carbon (AC), 1,1,1,2,3,3,3-heptafluoropropane
(HFC-227a), hexafluoropropene (HFP), zeocarbon (ZC), silica gel (SG).
_c_ The parameters $\Delta h_{j}$ are tabulated in increasing order of $j$,
e.g., for the $\mathrm{SF_{6}}$+W-A(673) system, $\Delta h_{1}/R=-3.107\times
10^{3},\,\Delta h_{2}/R=-6.112\times 10^{3}$, and so on.
_d_ For $\mathrm{SF_{6}}$+W-A(673) and $\mathrm{C_{3}H_{8}}$+W-A systems, the
saturation is expressed in $\mathrm{cm^{3}\,g^{-1}}$ at 101.325 kPa and 273.15
K, and for the remaining systems it is expressed in $\mathrm{mmol\,g^{-1}}$.
Table 2: Estimated parameters for the adsorption data presented by Watson et
al. [46] using Eq. 27 (l=0)._a,b,c_
| $q_{\mathrm{m}}$ | $\ln R_{j}^{\circ}$ | $-10^{-3}\times\Delta\overline{h}_{j}/R\,$ | $D$ | Temperature | Pressure
---|---|---|---|---|---|---
System | $\mathrm{mmol\cdot g^{-1}}$ | | $\mathrm{K}$ | $\%$ | range (K) | range (kPa)
$\mathrm{CH_{4}}$+MSC | $4.412\pm 0.067$ | $-14.4\pm 1.5$ | $2.81\pm 0.41$ | 2.29 | 148-298 | 1-4000
| | $-13.88\pm 0.48$ | $1.92\pm 0.13$ | | |
$\mathrm{CO_{2}}$+MSC | $5.87\pm 0.14$ | $-13.7\pm 1.3$ | $3.26\pm 0.40$ | 1.41 | 223-323 | 25-5200
| | $-14.00\pm 0.83$ | $2.94\pm 0.25$ | | |
| | $-16.07\pm 0.82$ | $3.29\pm 0.22$ | | |
| | $-19.46\pm 0.58$ | $3.58\pm 0.17$ | | |
$\mathrm{N_{2}}$+MSC | $5.75\pm 0.12$ | $-15.6\pm 1.1$ | $2.71\pm 0.31$ | 3.14 | 115-298 | 0.01-5000
| | $-14.03\pm 0.45$ | $1.725\pm 0.099$ | | |
| | $-18.3\pm 1.0$ | $1.79\pm 0.15$ | | |
_a_ In all cases, $P^{0}$ is expressed in kPa.
_b_ Abbreviation: molecular sieving carbon (MSC).
_c_ As in 1, the parameters $\ln R_{j}^{\circ}$ and $\Delta\overline{h}_{j}$
are tabulated in increasing order of $j$, for example, for the
$\mathrm{CH_{4}}$+MSC system, $\ln R_{1}^{\circ}=-14.4,\,\ln R_{2}=-13.88$,
$\Delta\overline{h}_{1}/R=-2.81\times
10^{3},\,\Delta\overline{h}_{2}/R=-1.92\times 10^{3}$.
In Fig. 1 the fitted and experimental isotherms of $\mathrm{SF_{6}}$
adsorption on pillared clay (designated as W-A(673) [42]) are shown111For this
system, the saturation is expressed as a volume at 101.325 kPa and 273.15 K,
it was used Eq. (37). It can be noticed a good agreement between the
experimental data and Eq. (37). For this set of isotherms, the correction term
(Eq. (39)) was used ($l=1$) and it was obtained a deviation of 2.97 %. If this
correction is not applied ($l=0$) the deviation is 5.2 %. Therefore, in this
case, the correction term gives a better result. Fig. 1 suggests that Eq. (37)
is useful for isotherms with complex shapes like those reported in Ref. [42].
The experimental isotherms shown in Fig. 1 were also fitted using the
5-parameter Toth isotherm and it was obtained a deviation of 17.8%; a value
that is grater than that obtained with Eq. (37). On the other hand, Eq. (27)
could be also used to fit these experimental isotherms, but more parameters
would be necessary and this would lead to large error bars and thermodynamic
properties with large uncertainties. Bandosz et al. [42] reported additional
single-component experimental isotherms of propane and sulfur hexafluoride
adsorption on various heat-treated pillared clays; a total of six
adsorbate+adsorbent systems were studied. Given that the results are similar,
here only two analyzed systems are reported in Table 1. But, for each of the
six adsorbate+adsorbent systems the deviation was within 1 and 3 %.
Figure 2: Plot of isosteric heat of adsorption vs adsorbed volume, the system
temperature is 283.0 K; error bars correspond to 95 % confidence intervals.
Fig. 2 shows the estimated isosteric heat of adsorption as a function of
adsorbed volume and the corresponding marginal confidence intervals [39, 40]
for the $\mathrm{SF_{6}}$+W-A(673) system. This Fig. shows that the isosteric
heat of adsorption is nearly constant around 25.5 kJ/mol. Bandosz et al. [42]
used a virial type isotherm [47, 48] and the adsorption isosteric heat vs
adsorbed volume plot obtained by them presents some oscillations near zero
saturation around 25 kJ/mol; this result does not agree with that obtained
here by means of Eq. (48). These differences could be due to the method used
by Bandosz et al. [42], they fitted small subsets of 15 adsorption data by
using the virial type isotherm and used four parameters to fit each subset.
Although this method is appropriate, the error bars reported by them
correspond to standard deviations in calculated isosteric heats of adsorption.
If 95 % marginal confidence intervals are used, the error bars at low coverage
would be even larger than those reported by them and the isosteric heats
reported here would fall within such uncertainty region.
Figure 3: Isosteric heat of adsorption for $\mathrm{SF_{6}}$+WA-(673) as a
function of temperature at three different saturations.
To illustrate the dependence of the isosteric heat of adsorption in terms of
temperature, plots of $q^{\mathrm{st}}$ vs $T$ at three different saturations
are shown in Fig. 3. As expected, the isosteric heat of adsorption does not
significantly change within the experimental range of temperatures. This is a
typical behavior of the isosteric heat of adsorption that is observed in both
experiments and molecular simulations. In the case in which it is necessary to
take into account the dependency of isosteric heat of adsorption on
temperature, the well known thermochemical formula [49] to calculate chemical
equilibrium constants could be used to estimate $\ln K_{j}$ as a function of
temperature in Eq. (15); an empirical model for the heat capacity would be
necessary.
Figure 4: Comparison between the experimental data of HFC-227ea adsorption on
activated carbon [43] and Eq. (37); symbols: experiment, solid line: Eq. (37).
The fitting of the experimental isotherms of adsorption of
1,1,1,2,3,3,3-heptafluoropropane (HFC-227ea) on activated carbon is shown in
Fig. 4. As in the previous case, there is a good agreement between model and
experiment; the correction term was not necessary ($l=0$). Due to the complex
pore structure of activated carbon, it is difficult to consider a subsystem as
any specific region of the real adsorbent, for this reason it is convenient to
regard each cell as a representative subsystem. The isosteric heat of
adsorption calculated by using Eq. (48) and Toth equation are shown in Fig. 5.
It can be noticed that both curves fairly agree. The differences are
attributed to the fact that Yun et al. [43] only used two isotherms to
calculate the isosteric heat of adsorption; they employed the parameters of
each isotherm and the Clausius-Clapeyron equation to obtain the
$q^{\mathrm{st}}$ vs $q$ plot, in contrast, here the complete set of
experimental isotherms was used. A problem with the method used by Yun et al.
[43] is that the isosteric heat of adsorption is influenced by uncertainties
in adsorption data, and hence, very reliable isotherms are necessary to
determine $q^{\mathrm{st}}$.
Figure 5: Isosteric heat of HFC-227ea adsorption on activated carbon
calculated; dotted/dashed line: calculated by Yun et al. [43], solid line: Eq.
(48), dotted lines: 95 % marginal confidence intervals for Eq. (48).
a)
b)
c)
Figure 6: a) Comparison between the experimental data of $\mathrm{CO_{2}}$
adsorption on zeocarbon [44] and Eq. (17); b) distribution of molecules among
cells at 273.15 K; c) same as b) for 313.15 K.
In Fig. 6, it is shown the adsorption isotherms of $\mathrm{CO_{2}}$ on
zeocarbon [44] and the fractions of molecules distributed among cells at
273.15 K and 313.15 K. The zeocarbon synthesized by Lee et al. [44] is a
zeolite X/activated carbon mixture composed by 38.5 mass % zeolite X, 35 mass
% activated carbon, 10 mass % inert silica, and 16.5 mass % zeolite A and P.
For this system, Eq. (37) accurately fits the experimental data ($D=3.8\%$),
and the correction term ($l=1$) in this case reduces the deviation by 2 % with
respect to $l=0$. The deviation is less than that obtained by using the Toth
equation, which is one of the most used equations to fit these kind of
adsorption data. Given that the zeocarbon is a zeolite/activated carbon
composite, from these results, it can be inferred that the method used here is
empirical. Consequently, the assumption that the adsorbate phase may be
divided into identical weakly-interacting subsystems and the fictitious
representative grand ensemble of subsystems are just convenient pictures for
fitting and correlation purposes.
As it was mentioned, at low pressures the leading term is $f_{1}$ and each
$f_{j}$ significantly contributes in a certain region of the isotherm. It can
be noticed in Fig. 6a that near 100 kPa, the saturation ($q$) changes in
approximately $0.5\,\mathrm{mol\,kg^{-1}}$. Accordingly with the fitting
method used here, at the lowest experimental temperature, for which at high
pressures the saturation is approximately $q_{\mathrm{m}}$, the leading term
is $f_{n_{\mathrm{s}}+1}$, as shown in Fig. 6b. However, as temperature
increases in the high experimental pressure region, the isotherm is a
combination of several fractions $f_{j}$. For example, Fig. 6b shows that the
isotherm at 313.15 K in the high pressure region is a combination of
$f_{3},\,f_{4}$ and $f_{5}$. When temperature is further increased, the term
$f_{5}$ does not contribute to the isotherm. If the correction term is not
used and $n_{\mathrm{s}}=4$, the plot of $f_{4}$ vs $P$ is quite similar to
that shown in Fig. 6c for $f_{5}$, and thus, this term ($f_{4}$) does not
significantly contribute to the isotherm at 313.15 K. For this reason, it was
necessary to propose the correction terms shown in Eqs. (26) and (39), these
Eqs. assure that high-order fractions depend on parameters that substantially
contribute to estimate each isotherm within the experimental temperature
range. Because of the specific characteristics of each adsorbate+adsorbent
system, it is difficult to establish a priori whether the correction term is
necessary, it must be tested whether this correction reduces the fitting
standard deviation.
In Table 2, the results of the fittings of data reported in Ref. [46] are
presented. In this case, Eq. (27) was used without any correction. For the
$\mathrm{CO_{2}}$+MSC and $\mathrm{N_{2}}$+MSC systems, Eq. (27) gives better
results (in terms of deviation) than the simplified model (Eq. (37)). With Eq.
(37) the deviation is around 3.5% ($l=0$), whereas Eq. (17) gives deviations
less than 3 %. Also, the deviation obtained using Eq. (27) is less than that
obtained by using the Toth equation; this is also confirmed by F-tests. The
confidence intervals for $\ln R_{j}^{\circ}$ are larger than those obtained
for $\ln K^{\circ}$ in Table 1, this could be caused by parameter correlation
effects. To fit this set of isotherms, Watson et al. [46] used the Toth
isotherm and also obtained large confidence intervals for the $K^{\circ}$
parameter. For the $\mathrm{CH_{4}}$ system, the Toth isotherm gives better
results than Eqs. (17) and (37), this is due to an isotherm at 148 K that
present an apparent saturation limit that is less than each of the apparent
saturation limits of the isotherms at temperatures greater than 148 K. If this
isotherm at 148 K is eliminated, the results obtained by using the 5-parameter
Toth Eq. and Eq. (27) are quite similar.
The advantage of the present models is the flexibility of setting the number
of adjustable parameters; this condition is essential to fit isotherms with
complex shapes. It is known that some of the most widely used empirical
isotherms to describe type I isotherms are Sips, Toth, and Dubinin type [50,
51, 52] isotherms. These Eqs. do not reduce to the correct Henry’s law limit;
except the Toth isotherm, but this overestimates the Henry’s constant [53]. In
contrast, the isotherm models studied here have the advantage that they
present the correct Henry’s law limit and thus they can be used to estimate
this constant222The Henry’s constant estimation also depends on the accuracy
of measured adsorption data at near zero saturation.. Despite its advantages,
the model studied here cannot give site energy and pore size distribution.
However, although the model does not explicitly take into account the
adsorbent heterogeneity, the number of parameters and the degree of the
polynomial $\xi$, which are related to the subsystem size, give an idea of the
variety of adsorbent sites and molecule-molecule interactions because the
system size is related to these characteristics of the adsorbate+adsorbent
system.
## 4 Conclusions
The cell model of adsorption was reviewed and analyzed by using the SSGE
method and a simplified accurate isotherm model is proposed (Eq. (37)). It was
found that the Eqs. (27) and (37) can be applied to fit experimental data of
adsorption of gases and vapors on microporous heterogeneous adsorbents; these
isotherms give relatively accurate fittings. A simple correction that improve
the fitting results is proposed. However, in some cases this correction may
not be necessary, is must be tested. Additionally, for systems in which the
experimental temperature range is large, it is suggested that the dependence
of $\Delta h_{j}$ on temperature must be considered and a model for both gas
and adsorbate phase heat capacity could be required.
Finally, the model based on an idealized and abstract grand ensemble of
subsystems presented here is useful to correlate single-component adsorption
data and the corresponding thermodynamic properties of interest. Specifically,
the model is useful to fit type I isotherms, i.e., the reversible micropore
filling region of the experimental adsorption isotherm. The advantages of the
present model are the correlation accuracy of both adsorption and
thermodynamic data, the flexibility to set the number of adjustable parameters
and to consider variations of $\ln K_{j}$ with temperature, and the
possibility of regarding adsorptive as a real gas phase. However, the method
studied here does not consider explicitly the pore size and adsorption energy
distribution, but the size of a representative subsystem gives an idea of the
adsorbent heterogeneity because the size of the subsystem depends on this
factor.
## References
* Deitz [1971] V. R. Deitz. _Ind. Eng. Chem._ , 57:49, 1971.
* Ayappa [1999] K. G. Ayappa. _J. Chem. Phys._ , 111:4736, 1999.
* Steele [1963] W. A. Steele. _J. Phys. Chem._ , 100:2016, 1963.
* Nikitas [1996] P. Nikitas. _J. Phys. Chem._ , 100:15247, 1996.
* Kamat and Keffer [2002] M. R. Kamat and D. Keffer. _Mol. Phys._ , 100:2689, 2002.
* Nitta et al. [1984] T. Nitta, M. Kuro-oka, and T. Katayama. _J. Chem. Eng. Japan_ , 17:45, 1984.
* Aranovich et al. [2004] G. L. Aranovich, J. S. Erickson, and M. D. Donohue. _J. Chem. Phys._ , 120:5208, 2004.
* Ramirez-Pastor et al. [1999a] A. J. Ramirez-Pastor, T. P. Eggarter, V. Pereyra, and J. L. Riccardo. _Phys. Rev. B_ , 59:11027, 1999a.
* Ramirez-Pastor et al. [1999b] A. J. Ramirez-Pastor, V. D. Pereyra, and J. L. Ricardo. _Langmuir_ , 15:5707, 1999b.
* Riccardo et al. [2005] J. L. Riccardo, F. Romá, and A. J. Ramirez-Pastor. _Appl. Surf. Sci._ , 252:505, 2005.
* Romá et al. [2005] F. Romá, J. L. Ricardo, and A. J. Ramirez-Pastor. _Langmuir_ , 21:2454, 2005.
* Narkiewicz-Michałek et al. [1999] J. Narkiewicz-Michałek, P. Szabelski, W. Rudziński, and A. S. T. Chiang. _Langmuir_ , 15:6091, 1999.
* Duong [1998] D. D. Duong. _Adsorption Analysis: Equilibria and Kinetics_. Imperial College Press, London, 1998.
* Sips [1948] R. Sips. _J. Chem. Phys._ , 16:490, 1948.
* Toth [1971] J. Toth. _Acta Chim. Acad. Sci. Hung._ , 69:311, 1971.
* Sips [1950] R. Sips. _J. Chem. Phys._ , 18:1024, 1950.
* Reiss and Merry [1981] H. Reiss and G. A. Merry. _J. Phys. Chem._ , 85:3313, 1981.
* Soto-Campos et al. [1998] G. Soto-Campos, D. S. Corti, and H. Reiss. _J. Chem. Phys._ , 108:2563, 1998.
* Corti [1998] D. S. Corti. _Mol. Phys._ , 93:417, 1998.
* Heying and Corti [1998] M. Heying and D. S. Corti. _Fluid Phase Equilibr._ , 204:183, 1998.
* Hill [1953] T. L. Hill. _J. Phys. Chem._ , 57:324, 1953.
* Hill [1960] T. L. Hill. _An Introduction to Statistical Thermodynamics_. Addison–Wesley, Reading, MA, 1960.
* Hill [1956] T. L. Hill. _Statistical Mechanics_. McGraw–Hill, New York, 1956.
* Ruthven [1971] D. M. Ruthven. _Nat. Phys. Sci._ , 232:70, 1971.
* Ruthven and Loughlin [1972] D. M. Ruthven and K. F. Loughlin. _J. Chem. Soc. Farad. Trans. I_ , 68:696, 1972.
* Ayappa et al. [1999] K. G. Ayappa, C. R. Kamala, and T. A. Abinandanan. _J. Chem. Phys._ , 110:8714, 1999.
* Boddenberg et al. [1997] B. Boddenberg, G. U. Rakhmatkariev, and R. Greth. _J. Phys. Chem. B_ , 101:1634, 1997.
* Boddenberg et al. [2002] B. Boddenberg, G. U. Rakhmatkariev, S. Hufnagel, and Z. Salimov. _Phys. Chem. Chem. Phys._ , 4:4172, 2002.
* Stach et al. [1986] H. Stach, U. Lohse, H. Thamm, and W. Schirmer. _Zeolites_ , 6:76, 1986.
* Llano-Restrepo and Mosquera [2009] M. Llano-Restrepo and M. A. Mosquera. _Fluid Phase Equilibr._ , 283:73, 2009.
* Ruthven and Wong [1985] D. M. Ruthven and F. Wong. _Ind. Eng. Chem. Fund._ , 24:27, 1985.
* Ruthven [1984] D. M. Ruthven. _Principles of Adsorption and Adsorption Processes_. Wiley, New York, 1984.
* Landau and Lifshitz [1958] L. D. Landau and E. M. Lifshitz. _Statistical Physics_. Addison-Wesley, Reading, MA, 1958.
* Fowler [1966] R.H. Fowler. _Statistical Mechanics_. Cambridge University Press, London, 2 edition, 1966.
* Langmuir [1918] I. Langmuir. _J. Am. Chem. Soc._ , 40:1361, 1918.
* Gregg and Sing [1982] S. J. Gregg and K. S. W. Sing. _Adsorption, Surface Area and Porosity_. Academic Press, London, 1982.
* Nicholson and Parsonage [1982] D. Nicholson and N. G. Parsonage. _Computer Simulation and the Statistical Mechanics of Adsorption_. Academic Press, London, 1982.
* Myers et al. [1997] A. L. Myers, J. A. Calles, and G. Calleja. _Adsorption_ , 3:107, 1997.
* Bates and Watts [1988] D. M. Bates and D. G. Watts. _Nonlinear Regression Analysis and its Applications_. Wiley, New York, 1988.
* Seber and Wild [1989] G. A. F. Seber and C. J. Wild. _Nonlinear Regression_. Wiley, New York, 1989.
* Kinnlburgh [1986] D. G. Kinnlburgh. _Environ. Sci. Technol._ , 20:895, 1986.
* Bandosz et al. [1996] T. J. Bandosz, J. Jagiełło, and J. A. Schwarz. _J. Chem. Eng. Data_ , 41:880, 1996.
* Yun et al. [2000] J. Yun, D. Choi, and Y. Lee. _J. Chem. Eng. Data_ , 45:136, 2000.
* Lee et al. [2002] J. Lee, J. Kim, J. Kim J. Suh, J. Lee, and C. Lee. _J. Chem. Eng. Data_ , 47:1237, 2002.
* Berlier and Frère [1997] K. Berlier and M. Frère. _J. Chem. Eng. Data_ , 42:533, 1997.
* Watson et al. [2009] G. Watson, E. F. May, B. F. Graham, M. A. Trebble, R. D. Trengove, and K. I. Chan. _J. Chem. Eng. Data_ , 54:2701, 2009.
* Czepirsky and Jagiełło [1989] L. Czepirsky and J. Jagiełło. _Chem. Eng. Sci._ , 44:797, 1989.
* Jagiełło et al. [1995] J. Jagiełło, T. J. Bandosz, K. Putyera, and J. A. Schwarz. _J. Chem. Eng. Data_ , 40:1288, 1995.
* Prigogine and Defay [1954] I. Prigogine and R. Defay. _Chemical Thermodynamics_. Longmans Green and Co, New York, 1954.
* Dubinin and Astakhov [1947] M. M. Dubinin and L. V. Astakhov. _Proc. Acad. Sci. USSR_ , 55:331, 1947.
* Dubinin and Astakhov [1970] M. M. Dubinin and V. A. Astakhov. _Adv. Chem. Ser._ , 102:69, 1970.
* Stoeckli [1977] H. F. Stoeckli. _J. Colloid Interface Sci._ , 59:184, 1977.
* Smith et al. [2005] J. M. Smith, H. C. Van Ness, and M. M. Abbott. _Introduction to Chemical Engineering Thermodynamics_. McGraw-Hill, New York, 2005.
|
arxiv-papers
| 2009-11-11T03:39:30 |
2024-09-04T02:49:06.403043
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Martin A. Mosquera",
"submitter": "Mart\\'in Mosquera",
"url": "https://arxiv.org/abs/0911.2012"
}
|
0911.2408
|
# Highly Transitive Actions of Surface Groups
Daniel Kitroser
###### Abstract
A group action is said to be highly-transitive if it is $k$-transitive for
every $k\geq 1$. The main result of this thesis is the following:
###### Main Theorem.
The fundamental group of a closed, orientable surface of genus $>1$ admits a
faithfull, highly-transitive action on a countably infinite set. From a
topological point of view, finding a faithfull, highly-transitive action of a
surface group is equivalent to finding an embedding of the surface group into
$\text{Sym}({\mathbb{Z}})$ with a dense image. In this topological setting, we
use methods originally developed in [3] and [1] for densely embedding surface
groups in locally compact groups.
## 1 Introduction
A permutation group $G\leq\text{Sym}(X)$ is called _$k$ -transitive_ if it is
transitive on ordered $k$-tuples of distinct elements and _highly-transitive_
if it is $k$-transitive for every $k\in{\mathbb{N}}$. When $G$ is given as an
automorphism group, it can often be verified that the given action is highly
transitive. This is the case for any subgroup of $\text{Sym}(X)$ which
contains all finitely supported permutations, where $X$ is any infinite set.
Other examples include groups such as $\operatorname{Homeo}(S^{2})$ and many
of its subgroups. But for most groups the question: “what is the maximal $k$
for which the group admits a faithfull, $k$-transitive action”? is wide open.
It was shown by McDonough [4] that any non-abelian free group admits a highly
transitive action. Another proof which is more useful from our point of view
was later given by Dixon in [2], using a Baire Category type argument.
In this paper, we prove that a _surface group_ (the fundamental group of a
closed, orientable surface without boundry of genus greater than $1$) admits a
faithfull, highly-transitive action of countably infinite degree.
### 1.1 Topological interpretation
Before we continue, we introduce some notation.
###### Notation.
We denote by $S_{\infty}=\text{Sym}({\mathbb{Z}})$ the full symmetric group of
a countable set which we identify with the integer numbers. The action of
$G\leq S_{\infty}$ on ${\mathbb{Z}}$ is always from the right and the image of
an element $a\in{\mathbb{Z}}$ under $g\in G$ is denoted by $a^{g}$. If
$n\in{\mathbb{N}}$ then $[n]$ will denote the set of all integers between $-n$
and $n$.
We define a group topology on $S_{\infty}$ where a basis is given by the sets
of the form
$U(\varphi,[n])=\left\\{\psi\in S_{\infty}\ \left|\
\psi\big{|}_{[n]}=\varphi\big{|}_{[n]}\right.\right\\},\ (\varphi\in
S_{\infty},n\in{\mathbb{N}}).$
It is easy to see that if $\varphi\in S_{\infty}$ and $A\subseteq{\mathbb{Z}}$
is any finite set then $U(\varphi,A)$ is open in this topology. Notice that
this topology is the restriction to $S_{\infty}$ of the topology of pointwise
convergence on the space of all functions ${\mathbb{Z}}\to{\mathbb{Z}}$ and a
straightforward proof shows that $S_{\infty}$ is a topological group which is
completely metrizable.
The problem of finding highly-transitive subgroups of $S_{\infty}$ can now be
approached via the following proposition, derived directly from the definition
of the topology on $S_{\infty}$.
###### Proposition 1.1.
Let $G\leq S_{\infty}$. Then $G$ is highly-transitive if and only if $G$ is
dense in $S_{\infty}$.
### 1.2 The main result
In view of proposition 1.1, the main result can be reformulated as follows.
###### Theorem 1.2 (main theorem).
Let $\Gamma=\pi_{1}(\Sigma_{g})$ be the fundamental group of an orientable
surface of genus $\geq 2$. Then there exists a dense subgroup of $S_{\infty}$
which is isomorphic to $\Gamma$.
The methods used in this work to obtain dense embeddings of surface groups
into $S_{\infty}$ are analogous to those used in [3] and [1] to show that if a
locally compact group contains a dense, free subgroup of every finite rank
$>1$ then it contains a dense surface group of every genus $>1$. $S_{\infty}$
is _not_ locally-compact but as we shall see, by prooving that $S_{\infty}$
has a dense free subgroup of every finite rank $>1$ with the additional
property that certain elements of that free group generate a non-discrete
cyclic group we can apply the same methods in our setting.
## 2 Dense Embeddings of Free Groups
As a first step to proving theorem 1.2 we prove the following.
###### Theorem 2.1.
Let $r\geq 2$, let $w_{i}=w_{i}({\tau}_{1},\dots,{\tau}_{r}),\
(i\in{\mathbb{N}})$ be reduced words and assume that there exists
$j\in\\{1,\dots,r\\}$ such that every $w_{i}$ is not a conjugate of a power of
$\tau_{j}$. Then, there exist $\tau_{1},\dots,\tau_{r}\in S_{\infty}$ such
that $F=\langle{\tau_{1},\dots,\tau_{r}}\rangle$ is a dense, rank $r$ free
subgroup of $S_{\infty}$ and such that
$\langle{w_{i}({\tau}_{1},\dots,{\tau}_{r})}\rangle$ is non-discrete for every
$i\in{\mathbb{N}}$.
We prove the theorem using Baire’s Category Theorem.
###### Definition 2.2.
Let $X$ be a topological space. A subset of $X$ which is a countable
intersection of dense, open sets is called _residual_ or _generic_.
Baire’s theorem states that in a non-trivial, complete metric space, a
residual set is dense. Since they are dense and closed under countable
intersections, residual sets in a complete metric space can be thought of as
being “large”. Dixon showed in [2] that if we denote
$U=\\{({\tau}_{1},\dots,{\tau}_{r})\in S_{\infty}^{r}\ |\
\langle{\tau_{1},\dots,\tau_{r}}\rangle\text{ is dense in $S_{\infty}$}\\}$
then the set of elements in $U$ that freely generate a free group is residual
in $\overline{U}$. We use a somewhat different setting than Dixon in order to
prove the existence of the desired dense free subgroups. Fix $\sigma\in
S_{\infty}$ to be the _shift_ permutation, i.e. $\forall
a\in{\mathbb{Z}}:a^{\sigma}=a+1$. We show that for every $n\geq 1$, the set of
elements $({\tau}_{1},\dots,{\tau}_{n})\in S_{\infty}^{n}$ such that
$\langle{\sigma,\tau_{1},\dots,\tau_{n}}\rangle$ has the properties stated in
theorem 2.1 is residual is $S_{\infty}^{n}$.
###### Lemma 2.3.
Let $\gamma\in S_{\infty}$. $\langle{\gamma}\rangle\leq S_{\infty}$ is non-
discrete if and only if the orbits of $\langle{\gamma}\rangle$ are all finite
and of unbounded length.
###### Proof.
Assume first that $\langle{\gamma}\rangle$ has an infinite orbit $\Delta$ and
let $a\in\Delta$. Then, $\langle{\gamma}\rangle\cap U(1,\\{a\\})=\\{1\\}$ and
so $\langle{\gamma}\rangle$ is discrete. Now assume that the length of the
orbits of $\langle{\gamma}\rangle$ is bounded. Let $m$ be the product of the
lengths of the orbits of $\langle{\gamma}\rangle$. So, $\gamma^{m}=1$ thus,
$\langle{\gamma}\rangle$ is finite and hence discrete.
Conversely, suppose that the orbits of $\langle{\gamma}\rangle$ finite and of
unbounded length. To prove that $\langle{\gamma}\rangle$ is non-discrete it is
enough to show that every basic neighborhood of the identity contains some
non-trivial power of $\gamma$. Let $n\in{\mathbb{N}}$ and let
$\Delta=\cup_{i=1}^{k}\Delta_{i}$ be a finite union of orbits of
$\langle{\gamma}\rangle$ such that $[n]\subseteq\Delta$. By hypothesis, all
the $\Delta_{i}$’s are finite. If we set $m=\prod_{i=1}^{k}|\Delta_{i}|$ then
for every $a\in\Delta$ (and in particular for every $a\in[n]$) we have that
$a^{\gamma^{m}}=a$ and since the orbit lengths of $\langle{\gamma}\rangle$ are
unbounded, there is an orbit of $\langle{\gamma}\rangle$ which is longer then
$m$ so $\gamma^{m}$ is not the identity element. Thus, $\gamma^{m}\neq 1$ is
an element of the pointwise stablilizer $U(\text{id},[n])$. Since such
stabilizers form a basis at the identity, we are finished. ∎
###### Definition 2.4.
Let $\gamma_{1},\dots,\gamma_{n}\in S_{\infty}$ and let
$w=w(\gamma_{1},\dots,\gamma_{n})$ be any word. If $w=w_{1}w_{2}\cdots w_{n}$
where $w_{i}\in\\{\gamma_{1}^{\pm 1},\dots,\gamma_{n}^{\pm 1}\\}$ then the
_trace_ of an element $a\in{\mathbb{Z}}$ under $w$ is the ordered set
$\mathrm{tr}_{w}(a)=\\{a,a^{w_{1}},a^{w_{1}w_{2}},\dots,a^{w_{1}w_{2}\cdots
w_{n}}=a^{w}\\}.$
###### Lemma 2.5.
Fix $n\in{\mathbb{N}}$ and let $w=w(\sigma,\tau_{1},\dots,\tau_{n})$ be a
reduced word which is not conjugated to a power of $\sigma$. Then the
following sets are residual:
$\displaystyle(1)$ $\displaystyle\mathcal{F}=\ $
$\displaystyle\\{(\tau_{1},\dots,\tau_{n})\in S_{\infty}^{n}\ |\
\langle{\sigma,\tau_{1},\dots,\tau_{n}}\rangle\text{ is a free group of rank
$n$ }\\}$ $\displaystyle(2)$ $\displaystyle\mathcal{D}=\ $
$\displaystyle\\{(\tau_{1},\dots,\tau_{n})\in S_{\infty}^{n}\ |\
\langle{\sigma,\tau_{1},\dots,\tau_{n}}\rangle\text{ is dense}\\}$
$\displaystyle(3)$ $\displaystyle\mathcal{N}=\ $
$\displaystyle\\{(\tau_{1},\dots,\tau_{n})\in S_{\infty}^{n}\ |\
\langle{w(\sigma,\tau_{1},\dots,\tau_{n},)}\rangle\text{ is non-discrete}\\}$
###### Proof.
(1) Let $v$ be a reduced, non-trivial word on $n+1$ letters and consider the
set
$\mathcal{F}_{v}=\\{(\tau_{1},\dots,\tau_{n})\in S_{\infty}^{n}\ |\
v(\sigma,\tau_{1},\dots,\tau_{n})\neq 1\\}.$
If we prove that $\mathcal{F}_{v}$ is open and dense in $S_{\infty}^{n}$ for
every $v$ as above then $\mathcal{F}$ is residual since
$\mathcal{F}=\bigcap_{v\neq 1}\mathcal{F}_{v}$. Obviously, $\mathcal{F}_{v}$
is open as the inverse image of the open set $S_{\infty}\smallsetminus\\{1\\}$
under the continuous mapping $(\tau_{1},\dots,\tau_{n})\mapsto
v(\sigma,\tau_{1},\dots,\tau_{n})$. To prove that $\mathcal{F}_{v}$ is dense
let $(\varphi_{1},\dots,\varphi_{n})\in S_{\infty}^{n}$ and let
$m\in{\mathbb{N}}$. We prove that there exists
$(\tau_{1},\dots,\tau_{n})\in\mathcal{F}_{v}$ such that
$\tau_{i}\big{|}_{[m]}=\varphi_{i}\big{|}_{[m]}$ for every $1\leq i\leq n$.
First, write
$v(\sigma,\tau_{1},\dots,\tau_{n})=\sigma^{r_{1}}v_{1}\sigma^{r_{2}}v_{2}\cdots\sigma^{r_{k}}v_{k}\sigma^{r_{k+1}}$
where $r_{i}\in{\mathbb{Z}}$ and $v_{i}\in\\{\tau_{1}^{\pm
1},\dots,\tau_{n}^{\pm 1}\\}$ for every $i$. We want define the permutations
$\tau_{1},\dots,\tau_{n}$. Choose $a_{1},\dots,a_{k+1}\in{\mathbb{Z}}$ such
that the numbers $a_{1},\dots,a_{k+1},a_{1}+r_{1},\dots,a_{k+1}+r_{k+1}$ are
all distinct and define $(a_{i}+r_{i})^{v_{i}}=a_{i+1}$ (e.g. if
$v_{1}=\tau_{5}^{-1}$ define $(a_{1}+r_{1})^{\tau_{5}^{-1}}=a_{2}$ or
equivalently, define $a_{2}^{\tau_{5}}=a_{1}+r_{1}$). Note that since all the
integers involved in the definition of the $v_{i}$’s are distinct and $v$ is
reduced, the definitions of the $v_{i}$’s do not contradict each other. We now
have that $a_{1}^{v}=a_{k+1}+r_{k+1}$ and in particular, $v$ is not trivial.
In order to fulfill the condition that $\tau_{i}$ must send an element
$b\in[m]$ to $b^{\varphi_{i}}$ we choose $a_{1},\dots,a_{k+1}$ in a way that
will not contradict with this requirment. Explicitly, choose
$a_{1},\dots,a_{k+1}\in{\mathbb{Z}}\smallsetminus\left([m]\cup\bigg{(}\bigcup_{i}[m]^{\varphi_{i}}\bigg{)}\cup\bigg{(}\bigcup_{j}[m]-r_{j}\bigg{)}\cup\bigg{(}\bigcup_{i,j}[m]^{\varphi_{i}}-r_{j}\bigg{)}\right).$
Note that we can always choose $a_{1},\dots,a_{k+1}$ in the manner described
since we are only excluding a finite set of integers that we can not choose
from. Now, every $\tau_{i}$ is defined on $[m]$ and on some other elements
$\\{b_{1},\dots,b_{\ell}\\}\subseteq\\{a_{1},\dots,a_{k},a_{1}+r_{1},\dots,a_{k+1}+r_{k+1}\\}$,
sending them to $\\{c_{1},\dots,c_{\ell}\\}$ respectively. Finally, choose any
bijection
$f_{i}:{\mathbb{Z}}\smallsetminus\big{(}[m]\cup\\{b_{1},\dots,b_{\ell}\\}\big{)}\to{\mathbb{Z}}\smallsetminus\big{(}[m]^{\varphi_{i}}\cup\\{c_{1},\dots,c_{\ell}\\}\big{)}$
and define
$x^{\tau_{i}}=\begin{cases}x^{\varphi_{i}}&,x\in[m].\\\ c_{j}&,x=b_{j}\text{
for some $j$}.\\\ x^{f_{i}}&,\text{else}.\end{cases}$
So we have that every $\tau_{i}$ is a permutation which lies in the basic
neighbourhood of $\varphi_{i}$ defined by $[m]$ such that
$v(\sigma,\tau_{1},\dots,\tau_{n})$ is non-trivial.
(2) Fix some $k\in{\mathbb{N}}$ and for every two $k$-tuples
$\mathbf{x}=(x_{1},\dots,x_{k})$ and $\mathbf{y}=(y_{1},\dots,y_{k})$ of
distinct integers consider the set
$\mathcal{D}_{\mathbf{x},\mathbf{y}}=\left\\{(\tau_{1},\dots,\tau_{n})\in
S_{\infty}^{n}\ \left|\
\begin{array}[]{l}\exists\varphi\in\langle{\sigma,\tau_{1},\dots,\tau_{n}}\rangle:\\\
\forall 1\leq j\leq k:x_{j}^{\varphi}=y_{j}\end{array}\right.\right\\}.$
Notice that if
$(\tau_{1},\dots,\tau_{n})\in\bigcap_{\mathbf{x},\mathbf{y}}\mathcal{D}_{\mathbf{x},\mathbf{y}}$
where $\mathbf{x}$ and $\mathbf{y}$ range over all $k$-tuples of distinct
integers then $\langle{\sigma,\tau_{1},\dots,\tau_{n}}\rangle$ is
$k$-transitive. Thus, by proposition 1.1 we have that
$\mathcal{D}=\bigcap_{k\in{\mathbb{N}}}\bigcap_{\mathbf{x},\mathbf{y}}\mathcal{D}_{\mathbf{x},\mathbf{y}}$
and we are left to show that $\mathcal{D}_{\mathbf{x},\mathbf{y}}$ is open and
dense for every $\mathbf{x}$ and $\mathbf{y}$ as above.
First, fix $\mathbf{x}=(x_{1},\dots,x_{k}),\mathbf{y}=(y_{1},\dots,y_{k})$ as
above and let
$(\tau_{1},\dots,\tau_{n})\in\mathcal{D}_{\mathbf{x},\mathbf{y}}$. By
definition there exists
$\varphi\in\langle{\sigma,\tau_{1},\dots,\tau_{n}}\rangle$ such that
$x_{i}^{\varphi}=y_{i}$ for every $i=1,\dots,k$. Let us write
$\varphi=v(\sigma,\tau_{1},\dots,\tau_{n})$ where
$v=v(\sigma,\tau_{1},\dots,\tau_{n})$ is a word on $\\{\sigma^{\pm
1},\tau_{1}^{\pm 1},\dots,\tau_{n}^{\pm 1}\\}$ and define
$A=\bigcup_{i=1}^{k}\mathrm{tr}_{v}(x_{i})$. $A$ is finite and so the set
$\mathcal{U}=\left\\{(\psi_{1},\dots,\psi_{n})\in S_{\infty}^{n}\ \left|\
\begin{array}[]{l}\psi_{i}\big{|}_{A}=\tau_{i}\big{|}_{A},\
\psi_{i}^{-1}\big{|}_{A}=\tau_{i}^{-1}\big{|}_{A}\\\
i=1,\dots,n\end{array}\right.\right\\}$
is an open neighbourhood of $(\tau_{1},\dots,\tau_{n})$ contained in
$\mathcal{D}_{\mathbf{x},\mathbf{y}}$. Indeed, if
$(\psi_{1},\dots,\psi_{n})\in\mathcal{U}$ take
$\xi=v(\sigma,\psi_{1},\dots,\psi_{n})\in\langle{\sigma,\psi_{1},\dots,\psi_{n}}\rangle$,
then by the definition of $\mathcal{U}$, we have that $\xi$ acts the same as
$\varphi=v(\sigma,\tau_{1},\dots,\tau_{n})$ on $x_{i}$ and in particular,
$\xi$ sends $x_{i}$ to $y_{i}$. This shows that
$\mathcal{D}_{\mathbf{x},\mathbf{y}}$ is open.
Now we prove that $\mathcal{D}_{\mathbf{x},\mathbf{y}}$ is dense. Let
$(\varphi_{1},\dots,\varphi_{n})\in S_{\infty}^{n}$ and $m\in{\mathbb{N}}$.
Let $r\in{\mathbb{N}}$ be such that $x_{j}+r\notin[m]$ and
$y_{j}+r\notin[m]^{\varphi_{1}}$ for every $1\leq j\leq k$. Let
$f:{\mathbb{Z}}\smallsetminus\big{(}[m]\cup\\{x_{1}+r,\dots,x_{k}+r\\}\big{)}\to{\mathbb{Z}}\smallsetminus\big{(}[m]^{\varphi_{1}}\cup\\{y_{1}+r,\dots,y_{k}+r\\}\big{)}$
be any bijection and define
$x^{\tau_{1}}=\begin{cases}x^{\varphi_{1}}&,x\in[m]\\\
y_{j}+r&,x=x_{j}+r\text{ for some $1\leq j\leq k$}\\\
x^{f}&,\text{otherwise}\end{cases}.$
Now define $\tau_{i}=\varphi_{i}$ for every $2\leq i\leq n$ and we get that
$({\tau}_{1},\dots,{\tau}_{n})$ is an element of the basic neighbourhood of
$({\varphi}_{1},\dots,{\varphi}_{n})$ defined by $[m]$. Also, the permutation
$\xi=\sigma^{r}\tau_{1}\sigma^{-r}\in\langle{\sigma,\tau_{1},\dots,\tau_{n}}\rangle$
sends each $x_{j}$ to $y_{j}$ thus,
$({\tau}_{1},\dots,{\tau}_{n})\in\mathcal{D}_{\mathbf{x},\mathbf{y}}$.
(3) By lemma 2.3 we can equivalently write
$\mathcal{N}=\left\\{({\tau}_{1},\dots,{\tau}_{n})\in S_{\infty}^{n}\ \left|\
\begin{array}[]{l}\text{The orbits of
$\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle$ are all finite}\\\
\text{and of unbounded length.}\end{array}\right.\right\\}.$
Thus, if we define for every $t\in{\mathbb{N}}$ and $a\in{\mathbb{Z}}$:
$\displaystyle\mathcal{U}_{t}$
$\displaystyle=\\{({\tau}_{1},\dots,{\tau}_{n})\in S_{\infty}^{n}\ |\
\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle\text{ has an orbit of length
$\geq t$}\\}.$ $\displaystyle\mathcal{V}_{a}$
$\displaystyle=\\{({\tau}_{1},\dots,{\tau}_{n})\in S_{\infty}^{n}\ |\
\text{The orbit of $a$ under
$\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle$ is finite}\\}.$
we get that
$\mathcal{N}=\big{(}\bigcap_{t\in{\mathbb{N}}}\mathcal{U}_{t}\big{)}\cap\big{(}\bigcap_{a\in{\mathbb{Z}}}\mathcal{V}_{a}\big{)}$.
We now show that $\mathcal{U}_{t}$ and $\mathcal{V}_{a}$ are open and dense
for every $t\in{\mathbb{N}}$ and $a\in{\mathbb{Z}}$. Let
$({\tau}_{1},\dots,{\tau}_{n})\in\mathcal{U}_{t}$ and let $b\in{\mathbb{Z}}$
be an element belonging to an orbit of
$\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle$ of length $\geq t$. Thus,
$b,b^{w},b^{w^{2}},\dots,b^{w^{t-1}}$ are all distinct. Let
$\Delta=\bigcup_{i=1}^{k-1}\mathrm{tr}_{w^{i}}(b)$. The set
$\left\\{({\psi}_{1},\dots,{\psi}_{n})\in S_{\infty}^{n}\
\left|\begin{array}[]{l}\psi_{i}\big{|}_{\Delta}=\tau_{i}\big{|}_{\Delta},\
\psi_{i}^{-1}\big{|}_{\Delta}=\tau_{i}^{-1}\big{|}_{\Delta}\\\
i=1,\dots,n\end{array}\right.\right\\}$
is an open neighbourhood of $({\tau}_{1},\dots,{\tau}_{n})$ which is contained
in $\mathcal{U}_{t}$ hence, $\mathcal{U}_{t}$ is open.
Now, take $({\tau}_{1},\dots,{\tau}_{n})\in\mathcal{V}_{a}$ and let
$\Delta=\\{a_{1},\dots,a_{s}\\}$ be the finite orbit of
$\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle$ containing $a$. Similarly
$\left\\{({\psi}_{1},\dots,{\psi}_{n})\in S_{\infty}^{n}\
\left|\begin{array}[]{l}\psi_{i}\big{|}_{\Delta}=\tau_{i}\big{|}_{\Delta},\
\psi_{i}^{-1}\big{|}_{\Delta}=\tau_{i}^{-1}\big{|}_{\Delta}\\\
i=1,\dots,n\end{array}\right.\right\\}$
is an open neighbourhood of $({\tau}_{1},\dots,{\tau}_{n})$ which is contained
in $\mathcal{V}_{a}$.
To prove $\mathcal{U}_{t}$ is dense, let
$({\varphi}_{1},\dots,{\varphi}_{n})\in S_{\infty}^{n}$ and
$m\in{\mathbb{Z}}$. In (1) we in fact showed that we can define
$\tau_{1},\dots,\tau_{n}\in S_{\infty}$ such that for every finite set
$A\subseteq{\mathbb{Z}}$ we have that $\tau_{i}\big{|}_{A}$ acts in any way we
please and there exists some $b\in{\mathbb{Z}}$ such that $b^{w}\neq b$ where
$w=w(\sigma,\tau_{1},\dots,\tau_{n})$. By repeating the same argument we can
find $\tau_{1},\dots,\tau_{n}\in S_{\infty}$ such that
$\tau_{i}\big{|}_{[m]}=\varphi_{i}\big{|}_{[m]}$ for every $1\leq i\leq n$ and
such that there exists $b\in{\mathbb{Z}}$ such that
$b,b^{w},b^{w^{2}},\dots,b^{w^{t-1}}$ are all distinct i.e.
$\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle$ has an orbit of length
$\geq t$.
Finally, we prove that $\mathcal{V}_{a}$ is dense. Since
$\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle$ has the same orbit
structure as the cyclic group generated by any conjugate of $w$, we can assume
without loss of generality that $w$ is a cyclically reduced word that is not a
power of $\sigma$. Let $({\varphi}_{1},\dots,{\varphi}_{n})\in S_{\infty}^{n}$
and $m\in{\mathbb{Z}}$. We need to define permutations
$\tau_{1},\dots,\tau_{n}\in S_{\infty}$ such that
$({\tau}_{1},\dots,{\tau}_{n})\in\mathcal{V}_{a}$ and every $\tau_{i}$ agrees
with $\varphi_{i}$ on $[m]$. This condition can be thought of in the following
way: $\tau_{i}$ is already defined on $[m]$ for every $i$ and $\tau_{i}^{-1}$
is already defined on $[m]^{\varphi_{i}}$ for every $i$ (they act the same as
$\varphi_{i}$ and $\varphi_{i}^{-1}$ respectively) and we are left to define
$\tau_{i}$ on ${\mathbb{Z}}\smallsetminus[m]$ (and $\tau_{i}^{-1}$ on
${\mathbb{Z}}\smallsetminus[m]^{\varphi_{i}}$) in such a way that the orbit of
$a$ under $\langle{w(\sigma,\tau_{1},\dots,\tau_{n})}\rangle$ will be finite.
First we write $w(\sigma,\tau_{1},\dots,\tau_{n})=w_{1}w_{2}\cdots w_{k}$.
Now, we start by applying the positive and negative powers of $w$ to $a$
letter by letter:
$\xrightarrow{w_{\ell}}c\xrightarrow{w_{\ell+1}}\dots\xrightarrow{w_{n-1}}a_{-1}\xrightarrow{w_{n}}a=a_{0}\xrightarrow{w_{1}}a_{1}\xrightarrow{w_{2}}\dots\xrightarrow{w_{s-1}}b\xrightarrow{w_{s}}$
where $b\in{\mathbb{Z}}$ is the first element such that we need to apply to it
the permutation $w_{s}$ and $w_{s}$ is not yet defined on $b$, that is,
$w_{s}=\tau_{i}$ for some $i$ and $b\notin[m]$ or $w_{s}=\tau_{i}^{-1}$ for
some $i$ and $b\notin[m]^{\varphi_{i}}$. Note that if such an element $b$ does
not exist then, since by hypothesis at least one of the letters
$w_{1},\dots,w_{k}$ is not $\sigma$, the orbit of $a$ under
$\langle{w(\sigma,\varphi_{1},\dots,\varphi_{n})}\rangle$ is contained in
$[m]\cup\big{(}\bigcup_{i=1}^{n}[m]^{\varphi_{i}}\big{)}$, hence finite and we
can just take $\tau_{i}=\varphi_{i}$ for every $i$. We can thus assume that
such a $b$ exists. Similarly, $c\in{\mathbb{Z}}$ is the first element we reach
when applying letter by letter the negative powers of $w$ to $a$ such that we
need to apply to $c$ the permutation $w_{\ell}^{-1}$ and $w_{\ell}^{-1}$ is
not yet defined on $c$. As before, we can assume without loss of generality
that such an element $c$ exists. By hypothesis, $w$ is cyclically reduced and
so the word $w_{s}w_{s+1}\cdots w_{k}w_{1}\cdots w_{\ell}$ is reduced. Also,
by their definition, $w_{s},w_{\ell}\neq\sigma^{\pm 1}$. We wish to define
$\tau_{1},\dots,\tau_{n}$ in such a way that $b^{w_{s}w_{s+1}\cdots
w_{k}w_{1}\cdots w_{\ell}}=c$ (and of course fulfilling the condition that
$\tau_{i}$ agrees with $\varphi_{i}$ on $[m]$). By repeating the argument made
in (1), we can find two distinct elements $d_{1},d_{2}\in{\mathbb{Z}}$ that do
not lie in $[m]$ or any $[m]^{\varphi_{i}}$ such that $w_{s+1}\cdots
w_{k}w_{1}\cdots w_{\ell-1}$ sends $d_{1}$ to $d_{2}$ and define
$b^{w_{s}}=d_{1},\ d_{2}^{w_{\ell}}=c$. From this we get that $b^{w_{s}\cdots
w_{k}w_{1}\cdots w_{l}}=d_{1}^{w_{s+1}\cdots w_{k}w_{1}\cdots
w_{\ell}}=d_{2}^{w_{\ell}}=c$ thus, the orbit of $a$ is finite. Now, every
$\tau_{i}$ is defined on $[m]$ and maybe on finitely many more elements.
Again, exactly as we did in (1), we can extend the definition of $\tau_{i}$ to
${\mathbb{Z}}$ and get permutations $\tau_{1},\dots,\tau_{n}$ satisfying the
required conditions. ∎
###### proof of theorem 2.1.
Let $j\in\\{1,\dots,r\\}$ be such that every word $w_{i}$ is not a conjugate
of a power of $\tau_{j}$. If we set $\tau_{j}=\sigma$ then by lemma 2.5, the
set
$W_{i}=\left\\{(\tau_{1},\dots,\tau_{j-1},\tau_{j+1},\dots,\tau_{r})\in
S_{\infty}^{r-1}\ \left|\
\begin{array}[]{l}\langle{\tau_{1},\dots,\tau_{r}}\rangle\text{ is a dense,
rank $r$ free subgroup}\\\ \text{and
}\langle{w_{i}({\tau}_{1},\dots,{\tau}_{r})}\rangle\text{ is non-
discrete}\end{array}\right.\right\\}$
is residual and so $\bigcap_{i\in{\mathbb{N}}}W_{i}$ is residual and in
particular, not empty. ∎
## 3 Eventually Faithfull Sequences
###### Definition 3.1.
Let $G$ and $H$ be groups. A sequence $\\{f_{n}\\}_{n=1}^{\infty}$ of
homomorphisms from $G$ to $H$ is _eventually faithfull_ if for every $g\in G$
there exists $n_{0}\in{\mathbb{N}}$ such that $g\notin\text{ker}(f_{n})$ for
all $n\geq n_{0}$.
In order to prove the main theorem, we will need to produce eventually
faithfull sequences of homomorphisms from surface groups to free groups. The
following constructions apear in [3] and [1]. Let $\Gamma=\Gamma_{2r}$ be the
surface group of genus $2r$ ($r\geq 1$). We have the following presentation
for $\Gamma$
$\Gamma=\langle
a_{1},a^{\prime}_{1}\dots,a_{r},a^{\prime}_{r},b_{1},b^{\prime}_{1},\dots,b_{r},b^{\prime}_{r}\
|\
[a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}][b^{\prime}_{r},b_{r}]\cdots[b^{\prime}_{1},b_{1}]\rangle.$
Let $x=[a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}]$ and let
$h:\Gamma\to\Gamma$ be the Dehn twist around $x$, i.e.
$\displaystyle h(a_{i})=a_{i}\qquad h(b_{i})=xb_{i}x^{-1}$ $\displaystyle
h(a^{\prime}_{i})=a^{\prime}_{i}\qquad
h(b^{\prime}_{i})=xb^{\prime}_{i}x^{-1}$
Let $F$ be the free group on $2r$ free generators
$\\{\varphi_{1},\varphi^{\prime}_{1},\dots,\varphi_{r},\varphi^{\prime}_{r}\\}$
and let $k:\Gamma\to F$ be the homomorphism defined by
$\displaystyle k(a_{i})=k(b_{i})=\varphi_{i}$ $\displaystyle
k(a^{\prime}_{i})=k(b^{\prime}_{i})=\varphi^{\prime}_{i}.$
Consider the map that folds the genus $2r$ surface that has $\Gamma$ as its
fundamental group across the curve corresponding to $x$ (this curve seperates
the surface into two equal parts). The image of this map is a surface of genus
$r$ with one boundry component so it has $F$ as its fundamental group. $k$ is
the homomorphism induced on the fundamental groups by this folding map (see
figure 1). Denote $f_{n}=k\circ h^{n}$.
Figure 1:
###### Lemma 3.2 (Breuillard,Gelander,Souto,Storm [3]).
The sequence $\\{f_{n}\\}_{n=1}^{\infty}$ is eventually faithfull.
We now construct an eventually faithfull sequence of homomorphisms from an odd
genus surface group into a free group. Let $\Gamma=\Gamma_{2r+1}$ be a surface
group of genus $2r+1$ ($r\geq 1$) with the presentation
$\Gamma=\langle{a_{1},a^{\prime}_{1},\dots,a_{r},a^{\prime}_{r},b,b^{\prime},c_{1},c^{\prime}_{1},\dots,c_{r},c^{\prime}_{r}\
|\
[a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}][b^{\prime},b][c^{\prime}_{1},c_{1}]\cdots[c^{\prime}_{r},c_{r}]}\rangle$
Denote $x=[a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}]b^{\prime}$ and
let $F$ be the free group on $2r+1$ free generators
$\\{\varphi_{1},\varphi^{\prime}_{1},\dots,\varphi_{r},\varphi^{\prime}_{r},\tau\\}$.
Let $\delta:\Gamma\to\Gamma$ and $\zeta:\Gamma\to\Gamma$ be denh twists around
$x$ and $b^{\prime}$ respectively, that is
$\displaystyle\delta(a_{i})$ $\displaystyle=a_{i}$ $\displaystyle\zeta(a_{i})$
$\displaystyle=a_{i}$ $\displaystyle\delta(a^{\prime}_{i})$
$\displaystyle=a^{\prime}_{i}$ $\displaystyle\zeta(a^{\prime}_{i})$
$\displaystyle=a^{\prime}_{i}$ $\displaystyle\delta(b)$ $\displaystyle=xb$
$\displaystyle\zeta(b)$ $\displaystyle=b(b^{\prime})^{-1}$
$\displaystyle\delta(b^{\prime})$ $\displaystyle=b^{\prime}$
$\displaystyle\zeta(b^{\prime})$ $\displaystyle=b^{\prime}$
$\displaystyle\delta(c_{i})$ $\displaystyle=xc_{i}x^{-1}$
$\displaystyle\zeta(c_{i})$ $\displaystyle=c_{i}$
$\displaystyle\delta(c^{\prime}_{i})$ $\displaystyle=xc^{\prime}_{i}x^{-1}$
$\displaystyle\zeta(c^{\prime}_{i})$ $\displaystyle=c^{\prime}_{i}$
Figure 2:
Notice that $\delta$ and $\zeta$ commute.
Let $k:\Gamma\to F$ be the map induced by folding the $2r+1$ surface across
the curves $x$ and $b^{\prime}$ (these curves seperate the surface into two
surfaces of genus $r$ and two boundry components). Explicitely,
$\displaystyle k(a_{i})=\varphi_{i}$ $\displaystyle
k(a^{\prime}_{i})=\varphi^{\prime}_{i}$ $\displaystyle k(b)=1$ $\displaystyle
k(b^{\prime}_{i})=\tau$ $\displaystyle k(c_{i})=\varphi_{i}$ $\displaystyle
k(c^{\prime}_{i})=\varphi^{\prime}_{i}$
(see figure 2). Finally, denote $\rho_{n}=k\circ(\delta\circ\zeta)^{n}$.
###### Lemma 3.3 (Barlev,Gelander [1]).
The sequence $\\{\rho_{n}\\}_{n=1}^{\infty}$ is eventually faithfull.
## 4 Proof of The Main Theorem
We will need the following results.
###### Lemma 4.1.
Let $\varphi,\psi\in S_{\infty}$ be such that both $\langle{\varphi}\rangle$
and $\langle{\psi}\rangle$ are non-discrete. Then
$\langle{(\varphi,\psi)}\rangle\leq S_{\infty}^{2}$ is non-discrete.
###### Proof.
We prove that every basic neighbourhood of $(1,1)$ contains a non-trivial
element of $\langle{(\varphi,\psi)}\rangle$. Let $m\in{\mathbb{N}}$. Let
$\Delta=\bigcup_{i=1}^{\ell}\Delta_{i}$ and
$\Gamma=\bigcup_{i=1}^{k}\Gamma_{i}$ be finite unions of orbits of
$\langle{\varphi}\rangle$ and $\langle{\psi}\rangle$ respectively such that
$[m]\subseteq\Delta$ and $[m]\subseteq\Gamma$. From proposition 2.3 we have
that all the orbits of $\langle{\varphi}\rangle$ and $\langle{\psi}\rangle$
are finite and so we can define the number
$n=\prod_{i=1}^{\ell}|\Delta_{i}|\cdot\prod_{i=1}^{k}|\Gamma_{i}|.$
Notice that every element of $\Delta$ is fixed by $\varphi^{n}$ and every
element of $\Gamma$ is fixed by $\psi^{n}$ and in particular every $i\in[m]$
is fixed by $\varphi^{n}$ and $\psi^{n}$. From proposition 2.3 we also have
that the lengths of the orbits of $\langle{\varphi}\rangle$ and
$\langle{\psi}\rangle$ are unbounded and in particular
$\langle{\varphi}\rangle$ and $\langle{\psi}\rangle$ both have an orbit of
length greater then $n$ and so $\varphi^{n}$ and $\psi^{n}$ are non-trivial.
Thus $(\varphi,\psi)^{n}$ is a non-trivial element contained in the basic
neighbourhood of $(1,1)$ defined by $[m]\times[m]$. ∎
###### Lemma 4.2.
Let G be a Hausdorff topological group and let $\gamma\in G$ such that
$\langle{\gamma}\rangle$ is non-discrete. Then, for every
$n_{0}\in{\mathbb{N}}$ the set $\\{\gamma^{n}\ |\ n\geq n_{0}\\}$ is dense in
$\overline{\langle{\gamma}\rangle}$.
###### Proof.
First we notice that since $\langle{\gamma}\rangle$ is non discrete then also
$\overline{\langle{\gamma}\rangle}$ is non discrete. Let
$U\subseteq\overline{\langle{\gamma}\rangle}$ be open then we have
$\gamma^{m}\in U$ for some $m\in{\mathbb{Z}}$. If $m\geq n_{0}$ we are done so
assume $m<n_{0}$. Now, If we denote
$\displaystyle U^{\prime}$ $\displaystyle=U\gamma^{-m}$ $\displaystyle
U^{\prime\prime}$ $\displaystyle=U^{\prime}\cap(U^{\prime})^{-1}$
Then $U^{\prime\prime}$ is an open symmetric identity neighborhood and since
$\overline{\langle{\gamma}\rangle}$ is non discrete, $U^{\prime\prime}$ is not
finite. Now take
$\widetilde{U}=U^{\prime\prime}\smallsetminus\\{\gamma^{k}\ :\
|k|<n_{0}-m\\}.$
We have that $\widetilde{U}$ is open (since $G$ is Hausdorff) and non empty,
thus there exits $n\in{\mathbb{Z}}$ such that $\gamma^{n}\in\widetilde{U}$. By
the definition of $\widetilde{U}$ we have that $|n|\geq n_{0}-m$. Also, since
$\widetilde{U}\subseteq U^{\prime\prime}$ and $U^{\prime\prime}$ is symmetric
it follows that $\gamma^{n},\gamma^{-n}\in U^{\prime\prime}\subseteq
U^{\prime}$. Hence, $\gamma^{n+m},\gamma^{-n+m}\in U$ and since either
$n+m\geq n_{0}$ or $-n+m\geq n_{0}$ we are done. ∎
### 4.1 Proof of theorem 1.2 for even genus
By theorem 2.1 there exists a subgroup $F\leq S_{\infty}$ such that $F$ is
dense, free with $2r$ free generators
$\varphi_{1},\varphi^{\prime}_{1},\dots,\varphi_{r},\varphi^{\prime}_{r}\in
S_{\infty}$ and such that
$\langle{[\varphi_{1},\varphi^{\prime}_{1}]\cdots[\varphi_{r},\varphi^{\prime}_{r}]}\rangle$
is non-discrete. Denote
$\gamma=[\varphi_{1},\varphi^{\prime}_{1}]\cdots[\varphi_{r},\varphi^{\prime}_{r}]$
and $\Omega=\overline{\langle{\gamma}\rangle}$. Let $\Gamma=\Gamma_{2r}$ be a
surface group of genus $2r$ ($r\geq 1$) with the presentation
$\Gamma=\langle
a_{1},a^{\prime}_{1}\dots,a_{r},a^{\prime}_{r},b_{1},b^{\prime}_{1},\dots,b_{r},b^{\prime}_{r}\
|\
[a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}][b^{\prime}_{r},b_{r}]\cdots[b^{\prime}_{1},b_{1}]\rangle.$
Define for every $\omega\in\Omega$ a homomorphism $f_{\omega}:\Gamma\to
S_{\infty}$ by
$\displaystyle f_{\omega}(a_{i})$ $\displaystyle=\varphi_{i}$ $\displaystyle
f_{\omega}(a^{\prime}_{i})$ $\displaystyle=\varphi^{\prime}_{i}$
$\displaystyle f_{\omega}(b_{i})$ $\displaystyle=\omega\varphi_{i}\omega^{-1}$
$\displaystyle f_{\omega}(b^{\prime}_{i})$
$\displaystyle=\omega\varphi^{\prime}_{i}\omega^{-1}.$
Since every $\omega\in\Omega$ commutes with $\gamma$ this defines a
homomorphism. Indeed, for every $\omega\in\Omega$:
$\displaystyle
f_{\omega}([a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}][b^{\prime}_{r},b_{r}]\cdots[b^{\prime}_{1},b_{1}])=\underbrace{[\varphi_{1},\varphi^{\prime}_{1}]\cdots[\varphi_{r},\varphi^{\prime}_{r}]}_{\gamma}\omega\underbrace{[\varphi^{\prime}_{r},\varphi_{r}]\cdots[\varphi^{\prime}_{1},\varphi_{1}]}_{\gamma^{-1}}\omega^{-1}=$
$\displaystyle\gamma\omega\gamma^{-1}\omega^{-1}=1.$
$\Omega$ is a completely metrizable space and every element of $\Omega$
corresponds to a homomorphism $\Gamma\to S_{\infty}$ whose image contains $F$,
hence the image is dense. We are left to show that at least one of those
homorphisms is also faithfull. In fact, we show that $\chi=\\{\omega\in\Omega\
|\ \text{$f_{\omega}$ is faithfull}\\}$ is residual in $\Omega$.
For every $g\in\Gamma\smallsetminus\\{1\\}$ denote
$\chi_{g}=\\{\omega\in\Omega\ |\ f_{\omega}(g)\neq 1\\}$. Notice that
$\chi=\bigcap_{g\in\Gamma\smallsetminus\\{1\\}}\chi_{g}.$
From lemma 3.2, the sequence $\\{f_{\gamma^{n}}\\}_{n\in{\mathbb{N}}}$ is
eventually faithfull and so for every $g\in\Gamma\smallsetminus\\{1\\}$ there
exists $n_{0}\in{\mathbb{N}}$ such that $\gamma^{n}\in\chi_{g}$ for all $n\geq
n_{0}$. By lemma 4.2 we have that $\\{\gamma^{n}\ |\ n\geq n_{0}\\}$ is dense
in $\Omega$ and so, $\chi_{g}$ is dense in $\Omega$. $\chi_{g}$ is also open
as the inverse image of the the open set $S_{\infty}\smallsetminus\\{1\\}$
under the continuous map $\omega\to f_{\omega}(g)$. This shows that $\chi$ is
residual in $\Omega$. ∎
### 4.2 Proof of theorem 1.2 for odd genus
Let $\Gamma=\Gamma_{2r+1}$ be a surface group of genus $2r+1$ ($r\geq 1)$ with
the presentation
$\Gamma=\langle{a_{1},a^{\prime}_{1},\dots,a_{r},a^{\prime}_{r},b,b^{\prime},c_{1},c^{\prime}_{1},\dots,c_{r},c^{\prime}_{r}\
|\
[a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}][b^{\prime},b][c^{\prime}_{1},c_{1}]\cdots[c^{\prime}_{r},c_{r}]}\rangle$
Let $F\leq S_{\infty}$ be a dense, free subgroup with $2r+1$ free generators
$\varphi_{1},\varphi^{\prime}_{1},\dots,\varphi_{r},\varphi^{\prime}_{r},\tau\in
S_{\infty}$ such that $\langle{\tau}\rangle$ and
$\langle{[\varphi_{1},\varphi^{\prime}_{1}]\cdots[\varphi_{r},\varphi^{\prime}_{r}]\tau}\rangle$
are non-discrete (the existence of such a free subgroup is assured by theorem
2.1). Denote
$\gamma=[\varphi_{1},\varphi^{\prime}_{1}]\cdots[\varphi_{r},\varphi^{\prime}_{r}]\tau$
and $\Omega=\overline{\langle{(\gamma,\tau)}\rangle}\subseteq S_{\infty}^{2}$.
For every $(\psi,\xi)\in\Omega$ we define a homomorphism
$f_{(\psi,\xi)}:\Gamma\to S_{\infty}$ by setting
$\displaystyle f_{(\psi,\xi)}(a_{i})$ $\displaystyle=\varphi_{i}$
$\displaystyle f_{(\psi,\xi)}(b)$ $\displaystyle=\psi\xi^{-1}$ $\displaystyle
f_{(\psi,\xi)}(c_{i})$ $\displaystyle=\psi\varphi_{i}\psi^{-1}$ $\displaystyle
f_{(\psi,\xi)}(a^{\prime}_{i})$ $\displaystyle=\varphi^{\prime}_{i}$
$\displaystyle f_{(\psi,\xi)}(b^{\prime})$ $\displaystyle=\tau$ $\displaystyle
f_{(\psi,\xi)}(c^{\prime}_{i})$
$\displaystyle=\psi\varphi^{\prime}_{i}\psi^{-1}$
Since every $(\psi,\xi)\in\Omega$ commutes with $(\gamma,\tau)$ we have that
$f_{(\psi,\xi)}$ is well defined as a homomorphism because
$\displaystyle
f_{(\psi,\xi)}([a_{1},a^{\prime}_{1}]\cdots[a_{r},a^{\prime}_{r}][b^{\prime},b][c^{\prime}_{1},c_{1}]\cdots[c^{\prime}_{r},c_{r}])=$
$\displaystyle=[\varphi_{1},\varphi^{\prime}_{1}]\cdots[\varphi_{r},\varphi^{\prime}_{r}][\tau,\psi\xi^{-1}]\psi\underbrace{[\varphi^{\prime}_{1},\varphi_{1}]\cdots[\varphi^{\prime}_{r},\varphi_{r}]}_{=\tau\gamma^{-1}}\psi^{-1}=$
$\displaystyle=\gamma\psi\xi^{-1}\tau^{-1}\xi\psi^{-1}\psi\tau\gamma^{-1}\psi^{-1}=\gamma\psi\xi^{-1}\tau^{-1}\xi\tau\gamma^{-1}\psi^{-1}=$
$\displaystyle=\gamma\psi\xi^{-1}\xi\tau^{-1}\tau\gamma^{-1}\psi^{-1}=\gamma\psi\gamma^{-1}\psi^{-1}=1.$
Notice that $f_{(\gamma^{n},\tau^{n})}=\rho_{n}$ (as defined is Chapter 3) for
every $n\in{\mathbb{N}}$ and so by lemma 3.3, the sequence
$f_{(\gamma^{n},\tau^{n})}$ is eventually faithfull.
The image of every homomorphism $f_{(\gamma^{n},\tau^{n})}$ contains $F$,
hence the image is dense. Finally, we show that $\chi=\\{(\psi,\xi)\in\Omega\
|\ \text{$f_{(\psi,\xi)}$ is faithfull}\\}$ is residual in the completely
metrizable space $\Omega$ and in particular, $\chi\neq\varnothing$. For every
$g\in\Gamma\smallsetminus\\{1\\}$ we denote $\chi_{g}=\\{(\psi,\xi)\in\Omega\
|\ f_{(\psi,\xi)}(g)\neq 1\\}$. As in the previous section, $\chi_{g}$ is open
as the inverse image of an open set under a continuous map. Since
$f_{(\gamma^{n},\tau^{n})}$ is eventually faithfull there exits
$n_{0}\in{\mathbb{N}}$ such that $f_{(\gamma^{n},\tau^{n})}\in\chi_{g}$ for
all $n\geq n_{0}$. Since $\langle{\tau}\rangle$ and $\langle{\gamma}\rangle$
are non discrete we get from lemma 4.1 that $\Omega$ is non-discrete and so by
lemma 4.2 we have that $\\{(\gamma^{n},\tau^{n})\ |\ n\geq n_{0}\\}$ is dense
in $\Omega$. This shows that $\chi_{g}$ is dense, hence
$\chi=\bigcap_{g\in\Gamma\smallsetminus\\{1\\}}\chi_{g}$
is residual.
## References
* [1] J. Barlev and T. Gelander, _Compactifications and algebraic completions of limit groups_ , arXiv:0904.3771v1 (2009).
* [2] J. D. Dixon, _Most finitely generated permutation groups are free_ , Bull. London Math. Soc. 22 (1990), no. 3, 222–226.
* [3] J. Souto E. Breuillard, T. Gelander and P. Storm, _Dense embeddings of surface groups_ , Geom. Topol 10 (2006), 1373–1389.
* [4] T. P. McDonough, _A permutation representation of a free group_ , Quart. J. Math. Oxford Ser. (2) 22 (1977), no. 111, 353–356.
|
arxiv-papers
| 2009-11-12T15:15:53 |
2024-09-04T02:49:06.416429
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniel Kitroser",
"submitter": "Daniel Kitroser",
"url": "https://arxiv.org/abs/0911.2408"
}
|
0911.2489
|
# A new light boson from MAGIC observations?
Marco Roncadelli Alessandro De Angelis and Oriana Mansutti INFN, Sezione di
Pavia, via A. Bassi 6, I – 27100 Pavia, Italy Dipartimento di Fisica,
Università di Udine, Via delle Scienze 208, I – 33100 Udine, and INAF and
INFN, Sezioni di Trieste, Italy Dipartimento di Fisica, Università di Udine,
Via delle Scienze 208, I – 33100 Udine, and INFN, Sezione di Trieste, Italy
###### Abstract
Recent detection of blazar 3C279 by MAGIC has confirmed previous indications
by H.E.S.S. that the Universe is more transparent to very-high-energy gamma
rays than currently thought. This circumstance can be reconciled with
observations of nearby blazars provided that photon oscillations into a very
light Axion-Like Particle occur in extragalactic magnetic fields. The emerging
“DARMA scenario” can be tested in the near future by the satellite-borne Fermi
LAT detector as well as by the ground-based Imaging Atmospheric Cherenkov
Telescopes H.E.S.S., MAGIC, CANGAROO III, VERITAS and by the Extensive Air
Shower arrays ARGO-YBJ and MILAGRO.
## 1 MOTIVATION
Imaging Atmospheric Cherenkov Telescopes (IACTs) are providing us with an
impressive amount of information about the Universe in the energy interval
$100\,{\rm GeV}-100\,{\rm TeV}$. Observations carried out by these IACTs
concern gamma-ray sources over an extremely wide interval of distances,
ranging from the parsec scale for Galactic objects up to the Gigaparsec scale
for the fartest detected blazar 3C279. This circumstance allows not only to
infer the intrinsec properties of the sources, but also to probe the nature of
photon propagation throughout cosmic distances.
The latter fact is of paramount importance for very-high-energy (VHE) gamma-
ray astrophysics, since the horizon of the observable Universe rapidly shrinks
above $100\,{\rm GeV}$ as the energy further increases. This is due to the
fact that photons from distant sources scatter off background photons
permeating the Universe, thereby disappearing into electron-positron pairs
[1]. It turns out that the corresponding cross section $\sigma(\gamma\gamma\to
e^{+}e^{-})$ peaks where the VHE photon energy $E$ and the background photon
energy $\epsilon$ are related by $\epsilon\simeq(500\,{\rm GeV}/E)\,{\rm eV}$.
As far as observations performed by IACTs are concerned, the cosmic opacity is
dominated by the interaction with ultraviolet/optical/infrared diffuse
background photons 111Frequency band $1.2\cdot 10^{3}\,{\rm GHz}-1.2\cdot
10^{6}\,{\rm GHz}$, corresponding to the wavelength range $0.25\,\mu{\rm
m}-250\,\mu{\rm m}$., usually called Extragalactic Background Light (EBL),
which is produced by galaxies during the whole history of the Universe. Owing
to such an absorption process, photon propagation is controlled by the optical
depth ${\tau}(E,D)$, with $D$ denoting the source distance. Hence, the
observed photon flux $\Phi_{\rm obs}(E,D)$ is related to the emitted one
$\Phi_{\rm em}(E)$ by
$\Phi_{\rm obs}(E,D)=e^{-\tau(E,D)}\,\Phi_{\rm em}(E)~{}.$ (1)
Neglecting evolutionary effects on the EBL spectral energy distribution for
simplicity, the optical depth reads ${\tau}(E,D)\simeq
D/{\lambda}_{\gamma}(E)$, where ${\lambda}_{\gamma}(E)$ is the photon mean
free path for $\gamma\gamma\to e^{+}e^{-}$ referring to the present cosmic
epoch. As a consequence, Eq. (1) simplifies as
$\Phi_{\rm obs}(E,D)\simeq e^{-D/{\lambda}_{\gamma}(E)}\ \Phi_{\rm em}(E)~{}.$
(2)
The function ${\lambda}_{\gamma}(E)$ decreases like a power law from the
Hubble radius $4.3\,{\rm Gpc}$ around $100\,{\rm GeV}$ to $1\,{\rm Mpc}$
around $100\,{\rm TeV}$ [2]. Now, Eq. (2) entails that the observed flux is
exponentially suppressed both at high energy and at large distances, so that
sufficiently far-away sources become hardly visible in the VHE range and their
observed spectrum should anyway be much steeper than the emitted one.
Yet, observations carried out by IACTs have failed to detect such a behaviour.
A first indication in this respect was reported by the H.E.S.S. collaboration
in connection with the discovery of the two blazars H2356-309 ($z=0.165$) and
1ES1101-232 ($z=0.186$) at $E\sim 1\,{\rm TeV}$ [3]. Stronger evidence comes
from the observation of the blazar 3C279 ($z=0.538$) at $E\sim 0.5\,{\rm TeV}$
by the MAGIC collaboration [4]. In particular, the signal from 3C279 collected
by MAGIC in the region $E<220$ GeV has more or less the same statistical
significance as the one in the range 220 GeV $<E<$ 600 GeV ($6.1\sigma$ in the
former case, $5.1\sigma$ in the latter).
Turning the argument around and assuming standard photon propagation as
described above, the observed spectrum $\Phi_{\rm obs}(E,D)$ can only be
reproduced by an emission spectrum $\Phi_{\rm em}(E)$ much harder than for any
other blazar observed so far.
A way out of this difficulty relies upon a modification of the emission
spectrum $\Phi_{\rm em}(E)$. A possibility involves the presence of strong
relativistic shocks, which can substantially harden $\Phi_{\rm em}(E)$ [5]. A
different option invokes photon absorption inside the blazar, which has been
shown to produce again an emission spectrum $\Phi_{\rm em}(E)$ considerably
harder than previously thought [6]. While successful at increasing the
fraction of VHE emitted photons, these attempts fail to explain why only for
the most distant blazars do these mechanisms become important.
A very different solution was recently proposed by the present authors and is
usually referred to as the “DARMA scenario” [7]. Its characteristic feature is
the presence of Axion-Like Particles (ALPs) (more about this, later) and rests
upon the mechanism of photon-ALP oscillation in cosmic magnetic fields, whose
existence has definitely been proved by AUGER observations [8]. Once ALPs are
produced close enough to the source, they travel unimpeded throughout the
Universe – since they do not undergo EBL absorption – and can convert back to
photons before reaching the Earth. As a consequence, the effective photon mean
free path ${\lambda}_{\gamma,{\rm eff}}(E,D)$ gets increased so that the
observed photons cross a distance in excess of ${\lambda}_{\gamma}(E)$.
Moreover, it has been shown that the DARMA scenario works for an ALP lighter
than about $10^{-10}\,{\rm eV}$ 222Somewhat similar ideas are discussed in
[9]..
A deeper insight into the DARMA mechanism can be achieved by introducing the
probability $P_{\gamma\to\gamma}(E,D)$ that a photon remains a photon after
propagation over a distance $D$, so that we have
$\Phi_{\rm obs}(E,D)=P_{\gamma\to\gamma}(E,D)\ \Phi_{\rm em}(E)~{}.$ (3)
When only photon absorption is operative, Eq. (2) can similarly be rewritten
as
$\Phi_{\rm obs}(E,D)=P_{\gamma\to\gamma}^{(0)}(E,D)\ \Phi_{\rm em}(E)~{},$ (4)
with
$P_{\gamma\to\gamma}^{(0)}(E,D)\simeq e^{-D/{\lambda}_{\gamma}(E)}~{}.$ (5)
In the presence of photon-ALP oscillations, Eq. (5) gets replaced by
$P_{\gamma\to\gamma}(E,D)\simeq e^{-D/{\lambda}_{\gamma}(E)}\,X(E,D)$ (6)
and the above discussion entails $X(E,D)>1$. Moreover, Eq. (2) presently
becomes
$\Phi_{\rm obs}(E,D)\simeq e^{-D/{\lambda}_{\gamma,{\rm eff}}(E,D)}\ \Phi_{\rm
em}(E)~{},$ (7)
with
${\lambda}_{\gamma,{\rm eff}}(E,D)=-\frac{D}{{\rm
ln}\,P_{\gamma\to\gamma}(E,D)}~{},$ (8)
so as to guarantee consistency with Eq. (3). Next, by inserting Eq. (6) into
Eq. (8) we get
$\frac{{\lambda}_{\gamma,{\rm
eff}}(E,D)}{{\lambda}_{\gamma}(E)}\simeq\frac{D}{D-{\lambda}_{\gamma}(E)\,{\rm
ln}\,X(E,D)}$ (9)
and since $X(E,D)>1$ we find ${\lambda}_{\gamma,{\rm
eff}}(E,D)>{\lambda}_{\gamma}(E)$, which is just a formal restatement of our
previous conclusion. Still, Eq. (7) possesses the advantage to explicitly show
that even a small increase of ${\lambda}_{\gamma,{\rm eff}}(E,D)$ gives rise
to a large enhancement of the observed flux $\Phi_{\rm obs}(E,D)$. As we shall
see, the DARMA mechanism makes ${\lambda}_{\gamma,{\rm eff}}(E,D)$ shallower
than ${\lambda}_{\gamma}(E)$, although it remains a decreasing function of
$E$. So, the resulting observed spectrum is much harder than the one predicted
by Eq. (2), thereby ensuring agreement with observations even by adopting for
far-away sources the same emission spectrum characteristic of nearby ones.
Our aim is to review the main features of the DARMA scenario as well as its
application to blazar 3C279.
## 2 DARMA SCENARIO
Both phenomenological and conceptual arguments lead to view the Standard Model
of particle physics as the low-energy manifestation of some more fundamental
and richer theory of all elementary-particle interactions including gravity.
Therefore, the lagrangian of the Standard Model is expected to be modified by
small terms describing interactions among known and new particles. Many
extensions of the Standard Model which have attracted considerable interest
over the last few years indeed predict the existence of ALPs. They are spin-
zero light bosons defined by the low-energy effective lagrangian
${\cal L}_{\rm ALP}\ =\
\frac{1}{2}\,\partial^{\mu}\,a\,\partial_{\mu}\,a-\frac{m^{2}}{2}\,a^{2}-\frac{1}{4M}\,F^{\mu\nu}\,\tilde{F}_{\mu\nu}\,a~{},$
(10)
where $F^{\mu\nu}$ is the electromagnetic field strength, $\tilde{F}_{\mu\nu}$
is its dual, $a$ denotes the ALP field whereas $m$ stands for the ALP mass
333As usual, natural Lorentz-Heaviside units with $\hbar=c=1$ are employed
throughout.. According to the above view, it is assumed $M\gg
G_{F}^{-1/2}\simeq 250\,{\rm GeV}$. On the other hand, it is supposed that
$m\ll G_{F}^{-1/2}\simeq 250\,{\rm GeV}$. The standard Axion [10] is the most
well known example of ALP. As far as generic ALPs are concerned, the
parameters $M$ and $m$ are to be regarded as independent.
So, what really characterizes ALPs is the trilinear $\gamma$-$\gamma$-$a$
vertex described by the last term in ${\cal L}_{\rm ALP}$, whereby one ALP
couples to two photons. Owing to this vertex, ALPs can be emitted by
astronomical objects of various kinds, and the present situation can be
summarized as follows. The negative result of the CAST experiment designed to
detect ALPs emitted by the Sun yields the bound $M>0.86\cdot 10^{10}\,{\rm
GeV}$ for $m<0.02\,{\rm eV}$ [11]. Moreover, theoretical considerations
concerning star cooling via ALP emission provide the generic bound
$M>10^{10}\,{\rm GeV}$, which for $m<10^{-10}\,{\rm eV}$ gets replaced by the
stronger one $M>10^{11}\,{\rm GeV}$ even if with a large uncertainty [10]. The
same $\gamma$-$\gamma$-$a$ vertex produces an off-diagonal element in the mass
matrix for the photon-ALP system in the presence of an external magnetic field
${\bf B}$. Therefore, the interaction eigenstates differ from the propagation
eigenstates and photon-ALP oscillations show up [12].
We imagine that a sizeable fraction of photons emitted by a blazar convert
into ALPs because of cosmic magnetic fields (CMFs), whose existence has been
demonstrated very recently by AUGER observations [8]. These ALPs propagate
unaffected by the EBL and we suppose that a substantial fraction of them back
converts into photons before reaching the Earth ALPs. Owing to the notorious
lack of information about the morphology of CMFs, one usually supposes that
they have a domain-like structure [13]. That is, ${\bf B}$ ought to be
constant over a domain of size $L_{\rm dom}$ equal to its coherence length,
with ${\bf B}$ randomly changing its direction from one domain to another but
keeping approximately the same strength. As explained elsewhere [14], it looks
plausible to assume the coherence length in the range $1\,{\rm Mpc}-10\,{\rm
Mpc}$. Correspondingly, the inferred strength lies in the range $0.3\,{\rm
nG}-1.0\,{\rm nG}$ [14].
## 3 PREDICTED ENERGY SPECTRUM
Our ultimate goal consists in the evaluation of the probability
$P_{\gamma\to\gamma}(E,D)$ when allowance is made for photon-ALP oscillations
as well as for photon absorption from the EBL. We proceed as follows. We first
solve exactly the beam propagation equation arising from ${\cal L}_{\rm ALP}$
over a single domain, assuming that the EBL is described by the “best-fit
model” of Kneiske et al. [15]. Starting with an unpolarized photon beam, we
next propagate it by iterating the single-domain solution as many times as the
number of domains crossed by the beam, taking each time a random value for the
angle between ${\bf B}$ and a fixed overall fiducial direction. We repeat such
a procedure $10^{.}000$ times and finally we average over all these
realizations of the propagation process.
We find that about 13% of the photons arrive to the Earth for $E=500\,{\rm
GeV}$, representing an enhancement by a factor of about 20 with respect to the
expected flux without DARMA mechanism (the comparison is made with the above
“best-fit model”). The same calculation gives a fraction of 76% for
$E=100\,{\rm GeV}$ (to be compared to 67% without DARMA mechanism) and a
fraction of 3.4% for $E=1\,{\rm TeV}$ (to be compared to 0.0045% without DARMA
mechanism). The resulting spectrum is exhibited in Fig. 1. The solid line
represents the prediction of the DARMA scenario for $B\simeq 1\,{\rm nG}$ and
$L_{\rm dom}\simeq 1\,{\rm Mpc}$ and the gray band is the envelope of the
results obtained by independently varying ${\bf B}$ and $L_{\rm dom}$ within a
factor of 10 about such values. These conclusions hold for $m\ll
10^{-10}\,{\rm eV}$ and we have taken for definiteness $M\simeq 4\cdot
10^{11}\,{\rm GeV}$ but we have cheked that practically nothing changes for
$10^{11}\,{\rm GeV}<M<10^{13}\,{\rm GeV}$.
Figure 1: The two lowest lines give the fraction of photons surviving from
3C279 without the DARMA mechanism within the “best-fit model” of EBL (dashed
line) and for the minimum EBL density compatible with cosmology (dashed-dotted
line). The solid line represents the prediction of the DARMA mechanism as
explained in the text.
Our prediction can be tested in the near future by the satellite-borne Fermi
LAT detector as well as by the ground-based IACTs H.E.S.S., MAGIC, CANGAROO
III, VERITAS and by the Extensive Air Shower arrays ARGO-YBJ and MILAGRO.
## References
* [1] G. G. Fazio and F. W. Stecker, Nature 226 (1970) 135.
* [2] P. Coppi and F. Aharonian, Astrophys. J. 487 (1997) L9.
* [3] F. Aharonian et al. (H.E.S.S. Collaboration), Nature 440 (2006) 1018.
* [4] J. Albert et al. (MAGIC Collaboration), Science 320 (2008) 1752.
* [5] F. W. Stecker, M. G. Baring and E. J. Summerlin, Astrophys. J. 667 L29 (2007); F. W. Stecker and S. T. Scully, Astron. Astrophys 478 (2008) L1.
* [6] F. Aharonian, D. Khangulyan and L. Costamante, Mon. Not. R. Astron. Soc. 387 (2008) 1206.
* [7] A. De Angelis, M. Roncadelli and O. Mansutti, Phys. Rev. D76 (2007) 121301.
* [8] J. Abraham et al. (Pierre Auger Collaboration), Science 318 (2007) 939.
* [9] M. Simet, D. Hooper and P. Serpico, Phys. Rev. D77 (2008) 063001; D. Chelouche, R. Rabadan, S. S. Pavolv and F. Castejon, arXiv:0806.0411 (2008) (ApJS, in press).
* [10] G. G. Raffelt, Stars as Laboratories for Fundamental Physics (University of Chicago Press, Chicago, 1996).
* [11] K. Zioutas et al., Phys. Rev. Lett. 94 (2005) 121301; S. Andriamonje et al., JCAP 0704 (2007) 010.
* [12] P. Sikivie, Phys. Rev. Lett. 51 (1983) 1415; (E) ibid. 52 (1984) 695. L. Maiani, R. Petronzio and E. Zavattini, Phys. Lett. B175 (1986) 359; G. G. Raffelt and L. Stodolsky, Phys. Rev. D37 (1988) 1237.
* [13] P. P. Kronberg, Rept. Prog. Phys. 57 (1994) 325; D. Grasso and H. Rubinstein, Phys. Rep. 348 (2001) 163.
* [14] A. De Angelis, M. Persic and M. Roncadelli, Mod. Phys. Lett. A23 (2008) 315.
* [15] T. M. Kneiske et al., Astron. Astrophys. 413 (2004) 807.
|
arxiv-papers
| 2009-11-12T22:12:25 |
2024-09-04T02:49:06.423530
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Roncadelli (1), Alessandro De Angelis (2), Oriana Mansutti (3)\n ((1) INFN, Sezione di Pavia, Italy, (2) Dipartimento di Fisica, Universita'\n di Udine, and INAF and INFN, Sezioni di Trieste, Italy, (3)Dipartimento di\n Fisica, Universita' di Udine, and INFN, Sezione di Trieste, Italy)",
"submitter": "Oriana Mansutti",
"url": "https://arxiv.org/abs/0911.2489"
}
|
0911.2524
|
Jointly Poisson processes
Don H. Johnson and Ilan N. Goodman
Electrical & Computer Engineering Department, MS380
Rice University
Houston, Texas 77005–1892
{dhj,igoodman}@rice.edu
###### Abstract
What constitutes jointly Poisson processes remains an unresolved issue. This
report reviews the current state of the theory and indicates how the accepted
but unproven model equals that resulting from the small time-interval limit of
jointly Bernoulli processes. One intriguing consequence of these models is
that jointly Poisson processes can only be positively correlated as measured
by the correlation coefficient defined by cumulants of the probability
generating functional.
## 1 Introduction
To describe spike trains mathematically, particularly those that do not
produce deterministic sequences of spikes, point process models are usually
employed. From a mathematical viewpoint, the Poisson process is the simplest
and therefore the model that has yielded the most results. Here, events occur
randomly at a rate given by some function $\lambda(t)$ with no statistical
dependence of one event’s occurrence on the number and the timing of other
events. Unfortunately, Poisson processes cannot accurately describe spike
trains because of absolute and relative refractory effects. Here, the
occurrence of a spike influences when the next one occurs. Some spike trains
deviate even more from the Poisson model, with several spikes affecting
subsequent ones in complicated ways. Modeling these falls under the realm of
non-Poisson processes, which in many cases makes it very difficult to obtain
analytic results. Consequently, the Poisson model is used to obtain
predictions about the character of the spike train, like its information
capacity, that are understood not to be precisely accurate for any realistic
neural recording. In some cases, the Poisson process can be used to obtain
bounds on performance that can be used as well-established guideposts for
neural behavior.
When it comes to population models, in which several neurons presumably
jointly encode information, we lack even a Poisson model for all but the
simplest cases: the component point processes are either statistically
independent or conditionally independent. Data show more complicated behavior
since cross-correlation functions often show correlations among members of a
population. Consequently, what is the generalization of the single Poisson
process description to what could be termed the jointly Poisson model. Here,
we seek to describe the joint statistics for several processes, each of which
is Poisson (i.e., the marginal processes are Poisson).
## 2 Infinite Divisibility
From a probabilistic standpoint, specifying a unique joint probability
distribution that has specified marginal distributions is ill-posed, since
many joint distributions could conceivably work. The easiest way to show the
ill-posed nature of this problem is to consider the situation for Gaussian
random variables. A set of random variables $\\{{X}_{1},\dots,{X}_{M}\\}$ is
said to be jointly Gaussian if the joint probability density has the form
$p_{\mathbf{{X}}}(\mathbf{x})=\frac{1}{|2\pi\bm{\Sigma}|^{1/2}}\exp\left\\{-\frac{(\mathbf{x}-\mathbf{m})^{\prime}\bm{\Sigma}^{-1}(\mathbf{x}-\mathbf{m})}{2}\right\\},\;\mathbf{{X}}=\\{{X}_{1},\dots,{X}_{M}\\}$
Here, $\bm{\Sigma}$ is the covariance matrix, $|\cdot|$ represents the matrix
determinant, $\mathbf{m}$ is the vector of means and $\mathbf{x}^{\prime}$
represents the transpose of the vector $\mathbf{x}$. Each of the random
variables has a Gaussian marginal probability distribution. One can also find
a joint distribution _not_ of this form that also has Gaussian marginals. For
example, consider the two-dimensional case ($N=2$) when the means are zero.
Let the joint distribution be as written as above, but defined to be zero in
the first and third quadrants. To obtain a valid joint distribution, we must
multiply the above formula by two so that the total probability obtained by
integration is one. This joint distribution yields marginal distributions no
different from the jointly Gaussian case, but the random variables are _not_
jointly Gaussian because the joint distribution does not have the form written
above.
What makes the jointly Gaussian random vector special is the property of
_infinite divisibility_ : the random vector can be expressed as a sum of an
arbitrary number of statistically independent random vectors (Daley and Vere-
Jones, 1988). The probability distribution of the sum is the convolution of
the individual probability distributions. Consequently, infinite divisibility
demands that a probability distribution be expressed as the $n$-fold
convolution of a density with itself. In special cases, like the Gaussian and
the Poisson, each of the constituent random vectors has the same
distributional form (i.e., they differ only in parameter values) as do their
sum.
The characteristic function provides a more streamlined definition of what
what infinite divisibility means. The characteristic function of a random
vector $\mathbf{{X}}$ is defined to be
$\Phi_{\mathbf{{X}}}(j\mathbf{u})\mathrel{\overset{\Delta}{=}}\int
p_{\mathbf{{X}}}(\mathbf{x})e^{j\mathbf{u}^{\prime}\mathbf{x}}\,d\mathbf{x}\;.$
The characteristic function of a sum of statistically independent random
vectors is the product of the individual characteristic functions.
$\Phi_{\mathbf{{Y}}}(j\mathbf{u})=\prod_{i=1}^{n}\Phi_{\mathbf{{X}}_{i}}(j\mathbf{u})\quad\mathbf{{Y}}=\sum_{i=1}^{n}\mathbf{{X}}_{i}$
Infinite divisibility demands that
$\bigl{[}\Phi_{\mathbf{{Y}}}(j\mathbf{u})\bigr{]}^{1/n}$ also be a
characteristic function for any positive integer value of $n$. If we express a
characteristic function parametrically as $\Phi(j\mathbf{u};\bm{\theta})$,
with $\bm{\theta}$ denotes the probability distribution’s parameters, the
Gaussian case is special in that
$\bigl{[}\Phi_{\mathbf{{Y}}}(j\mathbf{u};\bm{\theta})\bigr{]}^{1/n}=\Phi_{\mathbf{{Y}}}(j\mathbf{u};\bm{\theta}/n)$.
For the jointly Gaussian case, these parameters are the mean and covariance
matrix.
$\Phi_{\mathbf{{X}}_{i}}(j\mathbf{u};\mathbf{m}_{i},\bm{\Sigma}_{i})=\exp\left\\{j\mathbf{u}^{\prime}\mathbf{m}_{i}-\mathbf{u}^{\prime}\bm{\Sigma}_{i}\mathbf{u}/2\right\\}$
Dividing these parameters by $n$ does not affect the viability of the
underlying Gaussian distribution, which makes it an infinitely divisible
random vector. The example given above of a bivariate distribution having
Gaussian marginals is not infinitely divisible as its characteristic function
does not have this property.
In the point process case, a single Poisson process is easily seen to be
infinitely divisible since the superposition of Poisson processes is also
Poisson. We must modify the just-presented mathematical formalism involving
characteristic functions because we have a random process, not a random
vector. The _probability-generating functional_ is defined as
$G[u(t)]\mathrel{\overset{\Delta}{=}}\textsf{E}\left[\exp\left\\{\int\log
u(t)\,d{N}_{t}\right\\}\right]\;,$
where the transform variable $u(t)$ is a real-valued function of time and
${N}_{t}$ is the point process’s counting function (the number of events that
have occurred prior to time $t$). It has similar properties to the moment-
generating function with one notable exception: it has no “inverse transform.”
However, the moment-generating function for the total number of counts in the
interval implicit in the integral can be found from the probability generating
function with the substitution $u(t)\rightarrow z$. Finding the probability
distribution that underlies the expected value in the above formula requires a
special series expansion. Interesting quantities, like moments can be found
from the probability-generating functional by evaluating derivatives of its
logarithm. For example, the formal derivative with respect to $u(\cdot)$ and
evaluating the result at $u(\cdot)=1$ yields the expected value.
$\displaystyle\frac{d\log G[u(t)]}{du(t)}$
$\displaystyle=\frac{1}{G[u(t)]}\left.\textsf{E}\left[\int\frac{1}{u(t)}\,d{N}_{t}\,\exp\left\\{\int\log
u(t)\,d{N}_{t}\right\\}\right]\right|_{u(t)=1}=\textsf{E}\left[\int
d{N}_{t}\right]$ $\displaystyle\frac{d\log G[u(t)]}{du(t_{0})}$
$\displaystyle=\left.\frac{1}{G[u(t)]}\textsf{E}\left[\frac{1}{u(t_{0})}d{N}_{t_{0}}\,\exp\left\\{\int\log
u(t)\,d{N}_{t}\right\\}\right]\right|_{u(t)=1}=\textsf{E}\left[d{N}_{t_{0}}\right]$
The first of these is the total variation with respect to $u(t)$ and yields
the expected number of events over the interval spanned by the integral. The
second is the derivative at the time instant $t_{0}$, which yields the
expected value of the process at that time instant.
Despite not being easily able to determine the probability distribution,
showing infinite divisibility can be seen by inspection just as with
characteristic functions. For a Poisson process, the probability-generating
functional has the special form
$G[u(t)]=\exp\left\\{\int\bigl{(}u(t)-1\bigr{)}\lambda(t)\,dt\right\\}$
To show infinite divisibility, we note that the only “parameter” of a Poisson
process is its instantaneous rate function $\lambda(t)$. As the product of
probability-generating functionals for Poisson processes yields the same form
with the total rate equaling the sum of the component rates, the Poisson
process is infinitely divisible.
What we seek here is a description of the joint probability distribution of
several marginal Poisson processes so that the vector of Poisson processes is
infinitely divisible. We exhibit here what the probability generating
functional for an infinitely divisible vector of Poisson processes must be and
show how to use this quantity to derive some of its properties. In particular,
we show that they can be constructed in a stereotypical way that elucidates
the cross-correlation behavior required of jointly Poisson processes. Somewhat
surprisingly, the range of correlation structures is quite limited, with
values for the correlation parameters tightly intertwined with each other and
with the dimensionality of the vector process. In particular, pairwise
correlation coefficients cannot be negative for any pair and must decrease as
the dimension increases.
## 3 Jointly Poisson Processes
The probability-generating functional for several point processes considered
jointly has the simple form
$G^{({M})}[\mathbf{u}(t)]\mathrel{\overset{\Delta}{=}}\textsf{E}\left[\exp\left\\{\sum_{m=1}^{{M}}\int\log
u_{m}(t)\,d{N}_{m,t}\right\\}\right]$ (1)
where the expected value is computed with respect to the joint distribution of
the point processes, which is the quantity we seek. The probability-generating
functional of component process $j$ can be found from this formula by setting
$u_{i}(t)=1$, $i\neq j$. If the processes are statistically independent, their
joint probability functional equals the product of the marginal functionals.
If the processes are added, the probability generating functional of the
result equals the joint functional evaluated at a common argument:
$G[u(t)]=G^{({M})}[u(t),u(t),\dots,u(t)]$. These properties generalize those
of moment generating functions. Furthermore, cross-_covariance_ between two
processes, $i$ and $j$ say, can be found by evaluating the second mixed
partial of the log joint probability-generating functional:
$\displaystyle\left.\frac{\partial^{2}\log G^{({M})}[\mathbf{u}(t)]}{\partial
u_{i}(t)\partial u_{j}(t)}\right|_{\mathbf{u}(t)=\mathbf{1}}$
$\displaystyle=\textsf{E}\left[\int d{N}_{i,t}\int
d{N}_{j,t}\right]-\textsf{E}\left[\int
d{N}_{i,t}\right]\cdot\textsf{E}\left[\int d{N}_{j,t}\right]$
$\displaystyle\left.\frac{\partial^{2}\log G^{({M})}[\mathbf{u}(t)]}{\partial
u_{i}(t_{i})\partial u_{j}(t_{j})}\right|_{\mathbf{u}(t)=\mathbf{1}}$
$\displaystyle=\textsf{E}\left[d{N}_{i,t_{i}}d{N}_{j,t_{j}}\right]-\textsf{E}\left[d{N}_{i,t_{i}}\right]\cdot\textsf{E}\left[d{N}_{j,t_{j}}\right]$
Again, the first expression gives the cross-covariance of counts while the
second gives the cross-covariance between the processes $i$, $j$ at the times
$t_{i},t_{j}$.
Over thirty years ago, the probability-generating functional of two marginally
Poisson processes that satisfied the infinite-divisibility condition was shown
to have the unique form (Milne, 1974)
$G^{(2)}[u_{1}(t),u_{2}(t)]=\exp\left\\{\int\bigl{(}u_{1}(t)-1\bigr{)}\nu_{1}(t)\,dt+\int\bigl{(}u_{2}(t)-1\bigr{)}\nu_{2}(t)\,dt\right.\\\
\left.+\int\\!\\!\int\bigl{(}u_{1}(s)u_{2}(t)-1\bigr{)}\nu_{c}(\alpha,\beta)\,d\alpha\,d\beta\right\\}\;.$
(2)
This joint probability-generating functional is easily interpreted. First of
all, by setting $u_{2}(t)=1$, we obtain the marginal probability-generating
functional of process 1, showing that it is a Poisson process having an
instantaneous rate of $\nu_{1}(t)+\int\nu_{c}(t,\beta)\,d\beta$. Similarly,
process 2 is also Poisson with a rate equal to
$\nu_{2}(t)+\int\nu_{c}(\alpha,t)\,d\alpha$. Also, setting $\nu_{c}(s,t)=0$
results in the product of the marginal probability-generating functionals,
corresponding to the case in which the processes are statistically
independent. Thus, the “common rate” $\nu_{c}(\alpha,\beta)$ represents a
joint rate variation that induces statistical dependence between the
processes. The simplest example is
${\nu_{c}(\alpha,\beta)=\nu_{c}(\beta)\delta(\alpha-\beta)}$, indicating an
instantaneous correlation at each moment in time. The resulting dependence
term in the probability generating functional equals
$\int\\!\\!\int\bigl{(}u_{1}(\alpha)u_{2}(\beta)-1\bigr{)}\nu_{c}(\alpha,\beta)\,d\alpha\,d\beta=\int\bigl{(}u_{1}(t)u_{2}(t)-1\bigr{)}\nu_{c}(t)\,dt\;.$
Statistically dependent Poisson processes having an infinitely divisible joint
probability distribution can be simply constructed by adding to statistically
independent Poisson processes having rates $\nu_{1}(t)$ and $\nu_{2}(t)$ what
we call the _building-block_ processes a common Poisson process having rate
$\nu_{c}(t)$ that is statistically independent of the others. This way of
constructing jointly Poisson processes amounts to the construction described
by Holgate (Holgate, 1964). An allowed variant is to delay the common process
when it is added to one but not the other building-block process. Here,
$\nu_{c}(\alpha,\beta)=\nu_{c}(\beta)\delta\bigl{(}\alpha-(\beta-
t_{0})\bigr{)}$. In this way, correlation can occur at a time lag other than
zero, but still only at a single point.
More generally, $\nu_{c}(s,t)$ depends on its arguments in different ways that
do not lead to a simple superposition of building-block Poisson processes.
Using the probability generating function, you can show that the cross-
covariance function between the two constructed processes equals the common
rate:
$\textsf{cov}\left[d{N}_{1,t_{1}},d{N}_{2,t_{2}}\right]=\nu_{c}(t_{1},t_{2})$.
One would think that many common cross-covariances could be described this
way. However, several important constraints arise.
* •
Cross-covariances must be non-negative. This condition arises because the
common rate must be non-negative so that a valid probability generating
functional results.
* •
For the constructed processes to be jointly (wide-sense) stationary, we must
have constant rates and a cross-covariance function that depends only on the
time difference. Here, the latter constraint means $\nu_{c}(s,t)=f(|s-t|)$.
Milne and Westcott (Milne and Westcott, 1972) give more general conditions for
the common rate function to be well-defined. Thus, correlation can extend
continuously over some time lag domain. Consequently, the Holgate construction
does not yield all possible jointly Poisson processes.
* •
It is not clear that the joint-rate characterization extends in its full
generality to more than pairs of Poisson processes (Milne and Westcott, 1993)
because the putative probability generating functional for the marginal
process has not been shown to correspond to a Poisson’s probability generating
functional. However, the special case of the Holgate construction technique
always works.
In sequel, we only consider jointly Poisson processes that can be constructed
in Holgate’s fashion as a superposition of building-block Poisson processes.
Calculating means and covariances from the probability generating functional
for jointly Poisson processes is very revealing.
Counts: $\displaystyle\textsf{E}\left[\int d{N}_{i,t}\right]$
$\displaystyle=\int\bigl{(}\nu_{i}(t)+\nu_{c}(t)\bigr{)}\,dt$
$\displaystyle\textsf{cov}\left[\int d{N}_{1,t},\int d{N}_{2,t}\right]$
$\displaystyle=\int\nu_{c}(t)\,dt$ Instantaneous:
$\displaystyle\textsf{E}\left[d{N}_{i,t}\right]$
$\displaystyle=\nu_{i}(t)+\nu_{c}(t)$
$\displaystyle\textsf{cov}\left[d{N}_{1,t_{1}},d{N}_{2,t_{2}}\right]$
$\displaystyle=\begin{cases}0,&t_{1}\neq t_{2}\\\
\nu_{c}(t),&t_{1}=t=t_{2}\end{cases}$
Since the variance of a Poisson process equals its mean, we find that the
second-order correlation coefficient $\rho^{(2)}(t)$ equals
Counts: $\displaystyle\rho^{(2)}(t)$
$\displaystyle=\frac{\int\nu_{c}(t)\,dt}{\sqrt{\int\bigl{(}\nu_{1}(t)+\nu_{c}(t)\bigr{)}\,dt\cdot\int\bigl{(}\nu_{2}(t)+\nu_{c}(t)\bigr{)}\,dt}}$
Instantaneous: $\displaystyle\rho^{(2)}(t)$
$\displaystyle=\begin{cases}0,&t_{1}\neq t_{2}\\\
\frac{\nu_{c}(t)}{\sqrt{(\nu_{1}(t)+\nu_{c}(t))(\nu_{2}(t)+\nu_{c}(t))}},&t_{1}=t=t_{2}\end{cases}$
Thus, the correlation coefficient between both the counts and the
instantaneous values lies in the interval $[0,1]$, with the maximal
correlation occurring in the limit of large values for the common rate.
However, note that correlation has _no_ temporal extent and for some
particular lag: given an event occurs in one process, it is correlated with
the other process at the first process’s event time and uncorrelated
(statistically independent) at all others.
We can write the probability-generating functional in terms of the rates of
the building-block processes, $\nu_{i}(t)$ and $\nu_{c}(t)$, or in terms of
the rates of the constructed processes $\lambda_{i}(t)=\nu_{i}(t)+\nu_{c}(t)$
and the correlation coefficient $\rho^{(2)}(t)$ given above.
$\displaystyle G^{(2)}[u_{1}(t),u_{2}(t)]$
$\displaystyle=\exp\left\\{\int\bigl{(}u_{1}(t)-1\bigr{)}\nu_{1}(t)\,dt+\int\bigl{(}u_{2}(t)-1\bigr{)}\nu_{2}(t)\,dt\right.$
$\displaystyle\quad\left.+\int\\!\\!\bigl{(}u_{1}(t)u_{2}(t)-1\bigr{)}\nu_{c}(t)\,dt\right\\}$
$\displaystyle G^{(2)}[u_{1}(t),u_{2}(t)]$
$\displaystyle=\exp\left\\{\int\bigl{(}u_{1}(t)-1\bigr{)}\lambda_{1}(t)\,dt+\int\bigl{(}u_{2}(t)-1\bigr{)}\lambda_{2}(t)\,dt\right.$
(3)
$\displaystyle\quad\left.+\int\\!\\!\bigl{(}u_{1}(t)-1\bigr{)}\bigl{(}u_{2}(t)-1\bigr{)}\rho^{(2)}(t)\sqrt{\lambda_{1}(t)\lambda_{2}(t)}\,dt\right\\}$
We can extend this type of analysis to three Poisson processes constructed
from six building-block processes according to the following formulas for
their rates.
$\displaystyle\lambda_{1}(t)$ $\displaystyle=\nu_{1}(t)+\nu_{4}(t)+\nu_{5}(t)$
$\displaystyle\lambda_{2}(t)$ $\displaystyle=\nu_{2}(t)+\nu_{4}(t)+\nu_{6}(t)$
$\displaystyle\lambda_{3}(t)$ $\displaystyle=\nu_{3}(t)+\nu_{5}(t)+\nu_{6}(t)$
This generates pairwise-dependent processes with no third-order dependencies.
The covariance between any pair is expressed by the building-block process
rate they share in common. Consequently,
$\rho^{(2)}_{1,2}=\frac{\nu_{4}(t)}{\sqrt{\lambda_{1}(t)\lambda_{2}(t)}}\;.$
By letting $\nu_{1}=\nu_{2}\equiv\nu^{(1)}$ and
$\nu_{4}=\nu_{5}=\nu_{6}\equiv\nu^{(2)}$, we create what we term the
_symmetric_ case, in which we have only two separately adjustable rates that
arise from the six statistically independent building-block processes. In this
case, this cross-correlation simplifies to
$\rho^{(2)}=\frac{\nu^{(2)}(t)}{\nu^{(1)}(t)+2\nu^{(2)}(t)}\leq\frac{1}{2},\;i\neq
j$ (4)
When a Poisson process having instantaneous rate $\nu^{(3)}(t)$ is added to
all three building-block processes to create third-order dependence, the
correlation coefficient becomes in the symmetric case
$\rho^{(2)}=\frac{\nu^{(2)}(t)+\nu^{(3)}(t)}{\nu^{(1)}(t)+2\nu^{(2)}(t)+\nu^{(3)}(t)},\;i\neq
j$
Now, as the common process’s rate grows, the pairwise correlation coefficient
can approach one. If we define a third-order correlation coefficient according
to
$\rho^{(3)}[d{N}_{1,t},d{N}_{2,t},d{N}_{3,t}]\mathrel{\overset{\Delta}{=}}\frac{\left.\frac{\displaystyle\partial^{3}\log
G[u_{1}(t),u_{2}(t),u_{3}(t)]}{\displaystyle\partial u_{1}(t)\partial
u_{2}(t)\partial
u_{3}(t)}\right|_{\mathbf{u}=\mathbf{1}}}{\sqrt[3]{\textsf{var}[d{N}_{1,t}]\textsf{var}[d{N}_{2,t}]\textsf{var}[d{N}_{3,t}]}}\;.$
(5)
For the symmetric Poisson example, the third-order correlation coefficient is
easily found to be
$\rho^{(3)}(t)=\frac{\nu^{(3)}(t)}{\nu^{(1)}(t)+2\nu^{(2)}(t)+\nu^{(3)}(t)}$
Combining with the expression for the second-order correlation coefficient, we
find the following bounds for the symmetric case relating the correlation
quantities.
$0\leq\rho^{(3)}\leq 2\rho^{(2)}-\rho^{(3)}\leq 1$
Note that this inequality chain indicates that
$0\leq\rho^{(3)}\leq\rho^{(2)}\leq 1$. The second-order correlation can be
bigger than $\frac{1}{2}$, but only if $\rho^{(3)}$ increases as well in a
manner defined by the inequality chain.
We need to extend this analysis to an arbitrary number of building block and
constructed processes. We can form an arbitrary number of infinitely
divisible, jointly defined Poisson processes by extending the two- and three-
process Holgate construction technique. Given ${L}$ statistically independent
Poisson processes, we create a population of $M$ statistically dependent
Poisson processes according by superimposing ${L}$ building-block processes
according to the _construction matrix_ $\mathbf{{A}}$:
$\mathbf{N}_{t}=\mathbf{{A}}\mathbf{B}_{t}$. Here, $\mathbf{N}_{t}$ and
$\mathbf{B}_{t}$ represent column vectors of constructed and building-block
Poisson processes of dimension ${M}$ and ${L}>{M}$ respectively. The entries
of the construction matrix are either $0$ or $1$. For example, the
construction matrix underlying the two- and three-process examples are
$\displaystyle{M}=2\colon$ $\displaystyle\mathbf{{A}}=\begin{bmatrix}1&0&1\\\
0&1&1\end{bmatrix}$ $\displaystyle{M}=3\colon$
$\displaystyle\mathbf{{A}}=\begin{bmatrix}1&0&0&1&1&0&1\\\ 0&1&0&1&0&1&1\\\
0&0&1&0&1&1&1\end{bmatrix}$
To introduce dependencies of all orders, ${L}\geq 2^{{M}}-1$, and we
concentrate on the case ${L}=2^{{M}}-1$ in sequel.
The probability generating functional $G^{({M})}[\mathbf{u}(t)]$ of
$\mathbf{N}_{t}$ expressed in (1) can be written in matrix form as
$G^{({M})}[\mathbf{u}(t)]=\textsf{E}\left[\exp\left\\{\int\log\mathbf{u}^{\prime}(t)\,d\mathbf{N}_{t}\right\\}\right]$
where the logarithm of a vector is defined in the Matlab sense (an element-by-
element operation). Because $\mathbf{N}_{t}=\mathbf{{A}}\mathbf{B}_{t}$, we
have
$\displaystyle G^{({M})}[\mathbf{u}(t)]$
$\displaystyle=\textsf{E}\left[\exp\left\\{\int\log\mathbf{u}^{\prime}(t)\mathbf{{A}}\,d\mathbf{B}_{t}\right\\}\right]$
$\displaystyle=\textsf{E}\left[\exp\left\\{\int\left(\mathbf{{A}}^{\prime}\log\mathbf{u}(t)\right)^{\prime}\,d\mathbf{B}_{t}\right\\}\right]$
Each component of the vector $\mathbf{{A}}^{\prime}\log\mathbf{u}(t)$
expresses which combination of components of $\mathbf{u}(t)$ are associated
with each building block process. This combination corresponds to the
constructed processes to which each building block process contributes. Since
the building block processes are statistically independent and Poisson, we
have
$G^{({M})}[\mathbf{u}(t)]=\int\left[\exp\left\\{\mathbf{{A}}^{\prime}\log\mathbf{u}(t)\right\\}-1\right]^{\prime}\bm{\nu}(t)\,dt$
Expanding the vector notation for a moment, this result can also be written as
$G^{({M})}[\mathbf{u}(t)]=\exp\left\\{\sum_{l=1}^{{L}}\int\left(\left[\prod_{m=1}^{{M}}u_{m}^{A_{m,l}}(t)\right]-1\right)\nu_{l}(t)\,dt\right\\}$
(6)
Here, $u_{m}^{A_{m,l}}(t)$ means $u_{m}(t)$ raised to the $A_{m,l}$ power. In
other words, if $A_{m,l}=1$, the term is included; if $A_{m,l}=0$ it is not.
Thus, the probability generating functional consists of a sum of terms, one
for each building block process, wherein the coefficient of each rate
$\nu_{l}(t)$ is the product of arguments corresponding to those constructed
process building block process $l$ helped to build minus one. This form is
what equation (2) describes.
However, we need to convert this result into the form of (3) so that the role
of the cumulant correlation coefficients can come to light. We can view the
cumulant moments, the mixed first partials of the logarithm of the probability
generating functional, as coefficients of the multivariate Taylor series for
$\log G^{({M})}[\mathbf{u}(t)]$ centered at the point
$\mathbf{u}(t)=\mathbf{1}$. Because the $m^{\textrm{th}}$ term in (6) contains
only multilinear combinations of $u_{m}$, second-order and higher derivatives
of these terms are zero. Consequently, the Taylor series for $\log
G^{({M})}[\mathbf{u}(t)]$ consists _only_ of multilinear terms having
$(u_{m}-1)$ as its constituents with the cumulant moments as the series
coefficients. Consequently, the jointly Poisson process can always be written
in a form generalizing (3). This coefficient equals
$\left.\frac{\partial^{k}\log G^{({M})}[\mathbf{u}(t)]}{\partial
u_{m_{1}}(t)\dots\partial
u_{m_{k}}(t)}\right|_{\mathbf{u}(t)=\mathbf{1}}=\sum_{l=1}^{{L}}\left(\prod_{m=m_{1},\ldots,m_{k}}A_{m,l}\right)\nu_{l}(t)$
(7)
Because matrix $\mathbf{{A}}$ has only binary-valued entries, the product
$\prod_{m}A_{m,l}$ equals either one or zero, bringing in the
$l^{\textrm{th}}$ building block process only if it contributes to all of the
constructed processes indexed by $m_{1},\ldots,m_{k}$. Note that the first
partial derivative expresses the rate of each constructed process:
$\lambda_{m}(t)=\sum_{l}A_{m,l}\nu_{l}(t)$.
We can normalize the Taylor series coefficient to obtain cumulant correlation
coefficients by dividing by the geometric mean of the constructed process
rates that enter into the partial derivative shown in (7).
$\displaystyle\rho^{(k)}_{m_{1},\ldots,m_{k}}(t)$
$\displaystyle\mathrel{\overset{\Delta}{=}}\frac{\left.\frac{\partial^{k}\log
G^{({M})}[\mathbf{u}(t)]}{\partial u_{m_{1}}(t)\dots\partial
u_{m_{k}}(t)}\right|_{\mathbf{u}(t)=\mathbf{1}}}{\left[\lambda_{m_{1}}(t)\cdots\lambda_{m_{k}}(t)\right]^{1/k}}$
$\displaystyle=\frac{\sum_{l=1}^{{L}}\left(\prod_{m=m_{1},\ldots,m_{k}}A_{m,l}\right)\nu_{l}(t)}{\left[\sum_{l}A_{m_{1},l}\nu_{l}(t)\cdots\sum_{l}A_{m_{k},l}\nu_{l}(t)\right]^{1/k}}$
Because the numerator expresses which building block processes are in common
with all the specified constructed processes, they and others are contained in
each term in the denominator. This property means that each cumulant
correlation coefficient is less than one and, since rates cannot be negative,
greater than or equal to zero. Similar manipulations show that
$\rho^{(k)}_{m_{1},\ldots,m_{k}}(t)\geq\rho^{(k+1)}_{m_{1},\ldots,m_{k},m_{k+1}}(t)$:
the size of the cumulant correlation coefficients cannot increase with order.
In the symmetric case, the expression for the cumulant correlation
coefficients simplifies greatly.
$\rho^{(k)}(t)=\frac{\sum_{l=k}^{M}\binom{{M}-k}{l-k}\nu^{(l)}(t)}{\sum_{l=1}^{M}\binom{{M}-1}{l-1}\nu^{(l)}(t)}$
(8)
The denominator is the rate $\lambda(t)$ of each constructed process and the
numerator is the sum of the rates of the processes that induce the dependence
of the specified order. This result makes it easier to see that the cumulant
correlation coefficients cannot increase in value with increasing order:
$0\leq\rho^{(k)}(t)\leq\rho^{(k-1)}(t)\leq 1$, $k=3,\dots,{M}$. Furthermore,
more stringent requirements can be derived by exploiting the structure
equation (8), showing that the cumulant correlation coefficients must obey the
following two relationships in the symmetric case.
$\displaystyle\sum_{k=2}^{{M}}{\rho^{(k)}(-1)^{k}\binom{{M}-1}{k-1}}\leq 1$
(9) $\displaystyle\sum_{k=m}^{{M}}{\rho^{(k)}(-1)^{k+m}\binom{{M}-m}{k-m}}\geq
0,\quad m=2,\ldots,{M}$
For example, for four jointly Poisson processes, the cumulant correlation
coefficients must satisfy the inequalities
$\begin{gathered}3\rho^{(2)}-3\rho^{(3)}+\rho^{(4)}\leq 1\\\
\rho^{(2)}-2\rho^{(3)}+\rho^{(4)}\geq 0\end{gathered}$
## 4 Relations to Jointly Bernoulli Processes
Interestingly, this form of the jointly Poisson process can be derived as the
limit of the jointly Bernoulli process when the event probability becomes
arbitrarily small. First of all, a single Poisson process is defined this way,
with the event probability equal to $\lambda(t)\Delta t$. To extend this
approach to two jointly Poisson processes, we use the Sarmanov-Lancaster model
for two jointly Bernoulli processes (Goodman, 2004). Letting ${X}_{1},{X}_{2}$
be Bernoulli random variables with event probabilities $p_{1},p_{2}$
respectively, the joint probability distribution is given by
${P}({X}_{1},{X}_{2})={P}({X}_{1}){P}({X}_{2})\left[1+\rho\frac{({X}_{1}-p_{1})({X}_{2}-p_{2})}{\sigma_{1}\sigma_{2}}\right]$
where the standard deviation $\sigma_{i}$ of each random variable equals
$\sqrt{p_{i}(1-p_{i})}$. The key to the derivation is to use the moment
generating function, defined to be the two-dimensional $z$-transform of this
joint distribution.
$\Phi(z_{1},z_{2})=\sum_{x_{1}}\sum_{x_{2}}{P}(x_{1},x_{2})z_{1}^{x_{1}}z_{2}^{x_{2}}$
Simple calculations show that for the jointly Bernoulli distribution given
above, its moment generating function is
$\displaystyle\Phi(z_{1},z_{2})$
$\displaystyle=\left[(1-p_{1})(1-p_{2})+\rho\sigma_{1}\sigma_{2}\right]+\left[p_{1}(1-p_{2})-\rho\sigma_{1}\sigma_{2}\right]z_{1}+\left[p_{2}(1-p_{1})-\rho\sigma_{1}\sigma_{2}\right]z_{2}$
$\displaystyle\quad+\left[p_{1}p_{2}+\rho\sigma_{1}\sigma_{2}\right]z_{1}z_{2}$
$\displaystyle=\bigl{(}1+p_{1}(z_{1}-1)\bigr{)}\bigl{(}1+p_{2}(z_{2}-1)\bigr{)}+(z_{1}-1)(z_{2}-1)\rho\sigma_{1}\sigma_{2}$
Letting event probabilities be proportional to the binwidth $\Delta t$, we
evaluate this expression to first order in the event probabilities. Especially
note that $\sigma_{1}\sigma_{2}\approx\sqrt{\lambda_{1}\lambda_{2}}\Delta t$
as $\Delta t\rightarrow 0$ to first order. Therefore, we have
$\displaystyle\Phi(z_{1},z_{2})\mathrel{\overset{\Delta t\rightarrow
0}{\longrightarrow}}$ $\displaystyle[1+(z_{1}-1)\lambda_{1}\Delta
t][1+(z_{2}-1)\lambda_{2}\Delta
t]+(z_{1}-1)(z_{2}-1)\rho\sqrt{\lambda_{1}\lambda_{2}}\Delta t$
$\displaystyle=$ $\displaystyle 1+\lambda_{1}\Delta
t(z_{1}-1)+\lambda_{2}\Delta
t(z_{2}-1)+\rho\sqrt{\lambda_{1}\lambda_{2}}\Delta t(z_{1}-1)(z_{2}-1)$
Evaluating the natural logarithm and using the approximation $\log(1+x)\approx
x$ for small $x$, we find that
$\log\Phi(z_{1},z_{2})\approx(z_{1}-1)\lambda_{1}\Delta
t+(z_{2}-1)\lambda_{2}\Delta
t+(z_{1}-1)(z_{2}-1)\rho\sqrt{\lambda_{1}\lambda_{2}}\Delta t$
If we sum the Bernoulli random variables in each process over a fixed time
interval, say $[0,T]$, we obtain the number of events that occur in each
process. The moment generating function of this sum is the product of the
individual joint moment generating functions, which means its logarithm equals
the sum of the logarithms of the individual functions. Since the number of
random variables increases as the binwidth decreases (equal to $T/\Delta t$)
and noting these terms are proportional to $\Delta t$, the sum becomes an
integral to yield
$\log\Phi(N_{1},N_{2})=(z_{1}-1)\int_{0}^{T}\\!\\!\\!\lambda_{1}(t)\,dt+(z_{2}-1)\int_{0}^{T}\\!\\!\\!\lambda_{2}(t)\,dt+(z_{1}-1)(z_{2}-1)\int_{0}^{T}\\!\\!\\!\rho(t)\sqrt{\lambda_{1}(t)\lambda_{2}(t)}\,dt$
If we let $\lambda_{i}(t)=\nu_{i}(t)+\nu_{c}(t)$ and substitute (4) for the
definition of the correlation coefficient, we obtain the logarithm of the
probability generating functional for two jointly Poisson processes
constructed using Holgate’s method in which $u_{i}(t)\rightarrow z_{i}$ as in
equation (3).
Generalizing this result is tedious but straightforward: jointly Bernoulli
processes converge in the limit of small event probabilities to jointly
Poisson processes interdependent on each other at the same moment. An
interesting sidelight is the normalization of the higher order dependency
terms in the Sarmanov-Lancaster expansion demanded to make the correlation
coefficient in the two models agree. In the Sarmanov-Lancaster expansion, the
$k^{\textrm{th}}$ order term has the form exemplified by
$\rho^{(k)}\frac{({X}_{1}-p_{1})\cdots({X}_{k}-p_{k})}{C_{k}}$
where $C_{k}$ is the normalization constant that depends on correlation order
and the specific choice of random variables in the term. Normally, Sarmanov-
Lancaster expansions consist of products of orthonormal functions, which in
this case would be $\prod({X}_{i}-p_{i})/\sigma_{i}$. This makes the putative
normalization constant equal to $C_{k}=\prod\sigma_{i}$. However, the higher
order correlation coefficients consequent of this definition have no
guaranteed domains as does $\rho^{(2)}$. As described above, the jointly
Poisson correlation coefficients defined via cumulants do have an orderliness.
Associating the two demands that correlation coefficient be defined as
$\rho^{(k)}\mathrel{\overset{\Delta}{=}}\frac{\textsf{E}\bigl{[}({X}_{1}-p_{1})\cdots({X}_{k}-p_{k})\bigr{]}}{\left(\prod_{i=1}^{k}\sigma_{i}^{2}\right)^{1/k}}$
The normalization $(\prod_{i=1}^{k}\sigma_{i}^{2})^{1/k}$ corresponds to the
geometric mean of the variances found in the definition (5) of correlation
coefficients for Poisson processes. In the context of the Sarmanov-Lancaster
expansion, we have
$\rho^{(k)}=\rho^{(k)}\frac{\sigma_{1}^{2}\cdots\sigma_{k}^{2}}{C_{k}\cdot\left(\prod\sigma_{i}^{2}\right)^{1/k}}\;.$
Solving for $C_{k}$, we find that
$C_{k}=\left(\sigma_{1}^{2}\cdots\sigma_{k}^{2}\right)^{\frac{k-1}{k}}\;.$
Using this normalization in the Sarmanov-Lancaster expansion now creates a
direct relationship between its parameters and those of the jointly Poisson
probability distribution. The inequality sets shown in (9) also guarantee
existence of the Sarmanov-Lancaster model (Bahadur, 1961). This change does
not affect the orthogonality so crucial in defining the Sarmanov-Lancaster
expansion, only the normality.
Because of the correspondence between jointly Bernoulli processes and jointly
Poisson processes, we can use the limit of the Sarmanov-Lancaster expansion to
represent the joint distribution of jointly Poisson processes. In particular,
we can evaluate information-theoretic quantities related to Poisson processes
using this correspondence. Since entropy and mutual information are smooth
quantities (infinitely differentiable), the small-probability limit can be
evaluated _after_ they are computed for Bernoulli processes.
## References
* Bahadur (1961) R. R. Bahadur. A representation of the joint distribution of responses to $n$ dichotomous items. In H. Solomon, editor, _Studies in Item Analysis and Prediction_ , pages 158–168. Stanford University Press, California, 1961.
* Cramér (1946) H. Cramér. _Mathematical Methods of Statistics_. Princeton University Press, 1946.
* Daley and Vere-Jones (1988) D.J. Daley and D. Vere-Jones. _An Introduction to the Theory of Point Processes_. Springer-Verlag, New York, 1988.
* Goodman (2004) I.N. Goodman. Analyzing statistical dependencies in neural populations. Master’s thesis, Dept. Electrical & Computer Engineering, Rice University, Houston, Texas, 2004.
* Holgate (1964) P. Holgate. Estimation for the bivariate Poisson distribution. _Biometrika_ , 51:241–245, 1964.
* Milne (1974) R.K. Milne. Infinitely divisible bivariate Poisson processes. _Adv. Applied Prob._ , 6:226–227, 1974.
* Milne and Westcott (1972) R.K. Milne and M. Westcott. Further results for Gauss-Poisson processes. _Adv. Applied Prob._ , 4:151–176, 1972.
* Milne and Westcott (1993) R.K. Milne and M. Westcott. Generalized multivariate Hermite distributions and related point processes. _Ann. Inst. Statist. Math._ , 45:367–381, 1993.
|
arxiv-papers
| 2009-11-13T02:03:52 |
2024-09-04T02:49:06.429257
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D.H. Johnson and I.N. Goodman",
"submitter": "Don Johnson",
"url": "https://arxiv.org/abs/0911.2524"
}
|
0911.2568
|
# Some observations on Karoubian complete strongly exceptional posets on the
projective
homogeneous varieties ††thanks: The first author is supported in part by JSPS
Grant in Aid for Scientific Research, and the second author by The National
Natural Science Foundation of China 10671142.
Kaneda Masaharu
558-8585 Sugimoto
Osaka City University
Graduate School of Science
Department of Mathematics
kaneda@sci.osaka-cu.ac.jp Ye Jiachen
Department of Mathematics
Tongji University
1239 Siping Road Shanghai 200092
P. R. China
jcye@mail.tongji.edu.cn
###### Abstract
Let ${\mathcal{P}}=G/P$ be a homogeneous projective variety with $G$ a
reductive group and $P$ a parabolic subgroup. In positive characteristic we
exhibit for $G$ of low rank a Karoubian complete strongly exceptional poset of
locally free sheaves appearing in the Frobenius direct image of the structure
sheaf of $G/P$. These sheaves are all defined over $\mathbb{Z}$, so by base
change provide a Karoubian complete strongly exceptional poset on
${\mathcal{P}}$ over $\mathbb{C}$, adding to the list of classical results by
Beilinson and Kapranov on the Grassmannians and the quadrics over
$\mathbb{C}$.
On the complex projective space $\mathbb{P}^{n}_{\mathbb{C}}$ Beilinson [Bei]
found that ${\mathcal{E}}=\coprod_{i=0}^{n}{\mathcal{O}}(-i)$ induces a
triangulated equivalence from the bounded derived category of coherent sheaves
on $\mathbb{P}^{n}_{\mathbb{C}}$ to the bounded derived category of right
modules of finite type over the endomorphism ring of ${\mathcal{E}}$. After
the discovery of similar phenomena by Kapranov [Kap83]/ [Kap88] on the
homogeneous projective varieties
${\mathcal{P}}_{\mathbb{C}}=G_{\mathbb{C}}/P_{\mathbb{C}}$ with
$G_{\mathbb{C}}=\mathrm{GL}_{n}(\mathbb{C})$ and a parabolic subgroup
$P_{\mathbb{C}}$, and on the complex quadrics, Catanese [Bö] has proposed a
conjecture on the existence of coherent sheaves ${\mathcal{E}}_{w}$ on
${\mathcal{P}}_{\mathbb{C}}$ for general complex reductive group
$G_{\mathbb{C}}$ parametrized by the coset representatives of the Weyl group
of $G_{\mathbb{C}}$ by the Weyl group of $P_{\mathbb{C}}$ such that (i)
$\mathbf{Mod}_{{\mathcal{P}}_{\mathbb{C}}}({\mathcal{E}}_{w},{\mathcal{E}}_{w})\simeq\mathbb{C}$
$\forall w$, (ii)
$\mathrm{Ext}^{i}_{{\mathcal{P}}_{\mathbb{C}}}(\coprod_{w}{\mathcal{E}}_{w},\coprod_{w}{\mathcal{E}}_{w})=0$
$\forall i>0$, (iii)
$\mathbf{Mod}_{{\mathcal{P}}_{\mathbb{C}}}({\mathcal{E}}_{x},{\mathcal{E}}_{y})\neq
0$ iff $x\leq y$ in the Chevalley-Bruhat order on $W$, and (iv) the
${\mathcal{E}}_{w}$’s generate the bounded derived category of coherent
sheaves on ${\mathcal{P}}_{\mathbb{C}}$; precisely, we will consider Karoubian
generation in this paper that the smallest triangulated subcategory of
$\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$ containing the
${\mathcal{E}}_{w}$’s and closed under taking direct summands should be the
whole of $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$. If the conjecture holds,
$\mathbf{R}\mathbf{Mod}_{{\mathcal{P}}_{\mathbb{C}}}(\coprod_{w}{\mathcal{E}}_{w},?)$
gives by Beilinson’s lemma [Bei]/[Ba] a triangulated equivalence from the
bounded derived category of coherent sheaves on ${\mathcal{P}}_{\mathbb{C}}$
to that of right modules of finite type over the endomorphism ring of
$\coprod_{w}{\mathcal{E}}_{w}$, and also supports Kontsevich’s homological
mirror conjecture [Bö].
In this paper we propose a new way of prescribing where to look for such
${\mathcal{E}}_{w}$’s. We go over to positive characteristic and exhibit for
$G$ of rank at most 2 the ${\mathcal{E}}_{w}$’s as indecomposable direct
summands of the Frobenius direct image of the structure sheaf of
${\mathcal{P}}$; in case $G$ is in type $\mathrm{G}_{2}$ we do construct those
${\mathcal{E}}_{w}$’s based on such data but are unable at present to prove
that they indeed all appear in the Frobenius direct image. To describe our
method, we have to introduce some more notations.
Let $\Bbbk$ be an algebraically closed field of positive characteristic $p$,
$G$ a reductive algebraic group over $\Bbbk$, $P$ a parabolic subgroup of $G$
and put ${\mathcal{P}}=G/P$. We assume $p>h$ the Coxeter number of $G$. Let
$F:{\mathcal{P}}\to{\mathcal{P}}$ be the absolute Frobenius endomorphism of
${\mathcal{P}}$. If $G_{1}$ is the Frobenius kernel of $G$, $F$ factors
through a natural morphism $q:{\mathcal{P}}\to G/G_{1}P$ to induce an
isomorphism $\phi$ of schemes, though not of $\Bbbk$-schemes, from $G/G_{1}P$
to ${\mathcal{P}}$. Now,
$q_{*}{\mathcal{O}}_{\mathcal{P}}\simeq{\mathcal{L}}_{G/G_{1}P}(\hat{\nabla}_{P}(\varepsilon))$
locally free ${\mathcal{O}}_{G/G_{1}P}$-module associated to $G_{1}P$-module
$\hat{\nabla}_{P}(\varepsilon)$ induced from trivial 1-dimensional $P$-module
$\varepsilon$ [Haa]. As
$F_{*}{\mathcal{O}}_{\mathcal{P}}\simeq\phi_{*}q_{*}{\mathcal{O}}_{\mathcal{P}}$,
the structure of $F_{*}{\mathcal{O}}_{\mathcal{P}}$ is controlled by that of
$\hat{\nabla}_{P}(\varepsilon)$. Let $T$ be a maximal torus of $P$. We propose
a formula to describe the $G_{1}T$-socle series on
$\hat{\nabla}_{P}(\varepsilon)$ in terms of Kazhdan-Lusztig polynomials and
examine each socle layer, which is equipped with a structure of
$G_{1}P$-module. If $W$ is the Weyl group of $G$ with length function $\ell$
and $W_{P}$ the Weyl group of $P$, $W^{P}=\\{w\in
W\mid\ell(wx)=\ell(w)+\ell(x)\ \forall x\in W_{P}\\}$ gives the set of coset
representatives of $W/W_{P}$. We find that the multiplicity space of
$G_{1}$-simple module parametrized by $w\in W^{P}$ appearing in the
$(\ell(w)+1)$-st socle layer of $\hat{\nabla}_{P}(\varepsilon)$ induce by
sheafification the desired sheaf ${\mathcal{E}}_{w}$, inverting the order on
$W^{P}$. Thus our investigation is twofold; one is to study the structure of
induced $G_{1}P$-modules, and the other is to study the sheafification of
those $G_{1}P$-modules arising from the $G_{1}T$-socle series of the induced
module.
We note that our work is also related to the tilting property of
$F_{*}{\mathcal{O}}_{\mathcal{P}}$. We observed in [HKR] that if
$F_{*}{\mathcal{O}}_{\mathcal{P}}$ is tilting, the triangulated localization
theorem holds for the endomorphism ring of ${\mathcal{O}}_{\mathcal{P}}$ over
its Frobenius twist ${\mathcal{O}}_{\mathcal{P}}^{(1)}$, which is a version of
Bezrukavnikov-Mirkovic-Rumynin localization theorem for the $\Bbbk$-algebra of
crystaline differential operators on ${\mathcal{P}}$ [BMR]. As
${\mathcal{O}}_{\mathcal{P}}$ is locally free of rank $p^{\dim{\mathcal{P}}}$
over ${\mathcal{O}}_{\mathcal{P}}^{(1)}$, the category of coherent
${\mathcal{O}}_{{\mathcal{P}}}$-modules is equivalent to the category of
coherent modules over the sheaf of small differntial operator ring
${\mathcal{M}}\\!{\it
od}_{{\mathcal{O}}_{\mathcal{P}}^{(1)}}({\mathcal{O}}_{\mathcal{P}},{\mathcal{O}}_{\mathcal{P}})$.
Since the first author presented a talk on a part of the present work at
Tongji University in 2006, a number of related works have appeared. In
particular, we have verified in [K08]/[KNS] that Kapranov’s sheaves on the
Grassmannian provide the desired sheaves in positive characteristic, while
Langer [La] has proved that $F_{*}{\mathcal{O}}_{\mathcal{Q}}$ is tilting on
the quadrics ${\mathcal{Q}}$, see also Samokhin [S07], [S1]-[S3]. One can
parametrize certain direct summands of $F_{*}{\mathcal{O}}_{\mathcal{Q}}$ by
$W^{P}$ to verify Catanese’s conjecture on the quadrics. On the projective
spaces [K09] has showed that $\hat{\nabla}_{P}(\varepsilon)$ is uniserial as
$G_{1}T$-module, and that the multiplicity spaces of the $G_{1}$-simple
isotypic components in the $G_{1}T$-socle layers in
$\hat{\nabla}_{P}(\varepsilon)$ provide the desired ${\mathcal{E}}_{w}$’s as
proposed in this work.
Our main results are stated in §1. In §§2 and 3 we show that our
${\mathcal{E}}_{w}$ possess the conjectured properties. As the verifications
are done by brute force, we omit tedious mechanical computations. In §4 we
discuss parametrization of Kapranov’s sheaves based on our observations.
We are grateful to Henning Andersen, Hashimoto Yoshitake, Tezuka Michishige
and Yagita Nobuaki for helpful dicussions. We learned of Böhning’s preprint of
[Bö] and also of [S07], [La] from Hashimoto. Thanks are also due to Adrian
Langer for explaining his work. A part of the work was done during the first
author’s visit to the second in Shanghai in the fall of 2006. He thanks Tongji
University for the hospitality and the financial support during the visit.
Some of the communications were made while the second author was visiting
Abdus Salam International Centre for Theoretical Physics in the summer of
2007, to which he thanks for the hospitality and the financial support.
$1^{\circ}$ Parabolic Humphreys-Verma modules
In this section after fixing the notations to be used throughout the
manuscript, we begin our study of induced $G_{1}P$-modules by relating them to
better-examined induced $G_{1}B$-modules, $B$ a Borel subgroup of $G$. We will
determine the $G_{1}T$-socle series on parabolic Humphreys-Verma modules for
$G$ of rank at most $2$. Based on their structural data we will define our
sheaves ${\mathcal{E}}_{w}$ on $G/P$, which verify Catanese’s conjecture.
(1.1) We will assume $G$ is a simply connected simple algebraic group over an
algebraically closed field $\Bbbk$ of characteristic $p>0$. Let $P$ be a
parabolic subgroup of $G$, $B$ a Borel subgroup of $P$, and $T$ a maximal
torus of $B$. Let $\Lambda$ be the character group of $B$, $R\subseteq\Lambda$
the root system of $G$ relative to $T$ with positive system $R^{+}$ such that
the roots of $B$ are $-R^{+}$. If $\alpha\in R$, we denote its coroot by
$\alpha^{\vee}$. Let $R^{s}$ be the set of simple roots of $R^{+}$ and
$\Lambda^{+}$ the corresponding set of dominant weights. If $\alpha\in R^{s}$,
let $\omega_{\alpha}\in\Lambda$ such that
$\langle\omega_{\alpha},\beta^{\vee}\rangle=\delta_{\alpha\beta}$
$\forall\beta\in R^{s}$. Let $W$ be the Weyl group of $G$ with distinguished
generators $s_{\alpha}$, $\alpha\in R^{s}$. For $w\in W$ and
$\lambda\in\Lambda$ set $w\bullet\lambda=w(\lambda+\rho)-\rho$ with
$\rho=\frac{1}{2}\sum_{\alpha\in R^{+}}\alpha$. We will denote the Chevalley-
Bruhat order (resp. the length function) on $W$ relative to
$\\{s_{\alpha}\mid\alpha\in R^{s}\\}$ by $\geq$ (resp. $\ell$). If
$W^{P}=\\{w\in W\mid\ell(wx)=\ell(w)+\ell(x)\ \forall x\in W_{P}\\}$, then
$W=\sqcup_{w\in W^{P}}wW_{P}$. Let $w_{0}$ (resp. $w_{P}$) be the longest
element of $W$ (resp. $W_{P}$). If we write $P=P_{I}$ for $I\subset R^{s}$,
let $R_{I}=R\cap\sum_{\alpha\in I}\mathbb{Z}\alpha$ the root system of the
standard Levi subgroup of $P$.
For an algebraic group $H$ over $\Bbbk$ let $H_{1}$ be the Frobenius kernel of
$H$ and $\mathrm{Dist}(H)$ the algebra of distributions of $H$ over $\Bbbk$.
Let $\hat{\nabla}_{P}=\mathrm{ind}_{P}^{G_{1}P}$ be the induction functor from
the category of $P$-modules to the category of $G_{1}P$-modules, which we call
the Humphreys-Verma induction. Let $\Lambda_{P}$ be the character group of
$P$. If $\nu\in\Lambda_{P}$ and if $\rho_{P}=\frac{1}{2}\sum_{\alpha\in
R^{+}\setminus R_{I}}\alpha$, there is an isomorphism of $G_{1}P$-modules
$\hat{\nabla}_{P}(\nu)\simeq\mathrm{Dist}(G_{1})\otimes_{\mathrm{Dist}(P_{1})}(\nu-2(p-1)\rho_{P})$.
Likewise we let $\nabla_{P}=\mathrm{ind}_{P}^{G}$ be the induction from the
category of $P$-modules to the category of $G$-modules. In case $P$ is $B$, we
will often suppress $B$ from the subscripts. We also let
$\nabla^{P}=\mathrm{ind}_{B}^{P}$, abbreviated as $\nabla^{\alpha}$ in case
$P=P_{\\{\alpha\\}}$ for a simple root $\alpha$. If $\lambda\in\Lambda$, we
put $\Delta^{P}(\lambda)=\mathbf{R}^{\dim
P/B}\mathrm{ind}_{B}^{P}(w_{P}\bullet\lambda)$, abbreviated as
$\Delta^{\alpha}(\lambda)$ in case $P=P_{\\{\alpha\\}}$. For an $H$-module $M$
we denote by $M^{*}$ the $\Bbbk$-linear dual of $M$ and by $M^{[1]}$ the
Frobenius twist of $M$. If $H_{1}$ acts trivially on $M$, one can untwist the
Frobenius action and obtain an $H$-module $M^{[-1]}$ such that
$(M^{[-1]})^{[1]}\simeq M$. By $\mathrm{soc}_{H}M$ we will denote the socle of
$M$ as $H$-module.
If $M$ is a $T$-module and if $\nu\in\Lambda$, $M_{\nu}$ will denote the
$\nu$-weight space of $M$. Let $\mathrm{ch}\,M=\sum_{\lambda\in\Lambda}(\dim
M_{\lambda})e^{\lambda}$ be the formal character of $M$. The simple $G$\-
(resp. $G_{1}T$-) modules are parametrized by their highest weights in
$\Lambda^{+}$ (resp. $\Lambda$), denoted $L(\lambda)$, $\lambda\in\Lambda^{+}$
(resp. $\hat{L}(\nu)$, $\nu\in\Lambda$). In order to avoid confusion, we will
let $\varepsilon$ denote trivial 1-dimensional $G$-module. For other
unexplained notations we refer to [J] except that for a category
${\mathcal{C}}$ we will denote the set of morphisms of ${\mathcal{C}}$ from
object $A$ to $B$ by ${\mathcal{C}}(A,B)$.
(1.2) If $I\subseteq R^{s}$ associated to $P$,
$\Lambda_{P}=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0\
\forall\alpha\in I\\}$. Let us begin with relating the formal character of
$\hat{\nabla}_{P}(\nu)$, $\nu\in\Lambda_{P}$, to that of
$\hat{\nabla}(\lambda)$, $\lambda\in\Lambda$. As $G_{1}T$-module
$\hat{\nabla}_{P}(\nu)$ is isomorphic to $G_{1}T$-module
$\mathrm{ind}_{P_{1}T}^{G_{1}T}(\nu)$ induced from $P_{1}T$-module $\nu$. Let
$\widehat{\mathbb{Z}[\Lambda]}$ be a completion of
$\mathbb{Z}[\Lambda]\subset\Pi_{\lambda\in\Lambda}\mathbb{Z}e^{\lambda}$ as in
[F2, 3.3] consisting of those
$(a_{\lambda}e^{\lambda})_{\lambda}\in\Pi_{\lambda\in\Lambda}\mathbb{Z}e^{\lambda}$
such that there is $\mu\in\Lambda$ for which whenever $a_{\lambda}\neq 0$,
$\lambda\leq\mu$.
###### Proposition:
Let $L$ be the standard Levi subgroup of $P$ and $U_{L}$ the unipotent radical
of the Borel subgroup $B\cap L$ of $L$. $\forall\nu\in\Lambda_{P}$, one has in
$\widehat{\mathbb{Z}[\Lambda]}$
$\mathrm{ch}\,\hat{\nabla}_{P}(\nu)=e^{\nu}\prod_{\alpha\in R^{+}\setminus
R_{I}}\frac{1-e^{-p\alpha}}{1-e^{-\alpha}}=\sum_{\begin{subarray}{c}w\in
W_{P}\\\
\gamma\in\mathbb{Z}R_{I}\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{\gamma})\mathrm{ch}\,\hat{\nabla}(w\bullet\nu+p\gamma).$
Proof: The first equality follows from the decomposition $G_{1}\simeq
P_{1}\times\prod_{\alpha\in R^{+}\setminus R_{I}}U_{\alpha,1}$ with
$U_{\alpha}$ root subgroup associated to $\alpha$. If $\Lambda_{L}$ is the
character group of $B\cap L$, one can write in
$\widehat{\mathbb{Z}[\Lambda_{L}]}$
$\displaystyle\mathrm{ch}\,\varepsilon$
$\displaystyle=\sum_{\lambda\in\Lambda_{L}}(\varepsilon:\mathrm{ind}_{(B\cap
L)_{1}T}^{L_{1}T}(\lambda))\mathrm{ch}\,\mathrm{ind}_{(B\cap
L)_{1}T}^{L_{1}T}(\lambda)\quad\text{for some
$(\varepsilon:\mathrm{ind}_{(B\cap L)_{1}T}^{L_{1}T}(\lambda))\in\mathbb{Z}$}$
$\displaystyle=\sum_{\lambda\in\Lambda_{L}}(\varepsilon:\mathrm{ind}_{B_{1}T}^{P_{1}T}(\lambda))\mathrm{ch}\,\mathrm{ind}_{B_{1}T}^{P_{1}T}(\lambda).$
Then in $\widehat{\mathbb{Z}[\Lambda]}$
$\displaystyle\mathrm{ch}\,\hat{\nabla}_{P}(\nu)$
$\displaystyle=\mathrm{ch}\,\hat{\nabla}_{P}(\nu\otimes\varepsilon)=\sum_{\lambda\in\Lambda_{L}}(\varepsilon:\mathrm{ind}_{B_{1}T}^{P_{1}T}(\lambda))\mathrm{ch}\,\hat{\nabla}_{P}(\nu\otimes\mathrm{ind}_{B_{1}T}^{P_{1}T}(\lambda))\quad\text{as
$\hat{\nabla}_{P}$ is exact}$
$\displaystyle=\sum_{\lambda\in\Lambda_{L}}(\varepsilon:\mathrm{ind}_{B_{1}T}^{P_{1}T}(\lambda))\mathrm{ch}\,\hat{\nabla}_{P}(\mathrm{ind}_{B_{1}T}^{P_{1}T}(\nu+\lambda))\quad\text{by
the tensor identity as $\nu\in\Lambda_{P}$}$
$\displaystyle=\sum_{\lambda\in\Lambda_{L}}(\varepsilon:\mathrm{ind}_{B_{1}T}^{P_{1}T}(\lambda))\mathrm{ch}\,\hat{\nabla}(\nu+\lambda)\quad\text{by
the transitivity of inductions}.$
On the other hand, Weyl’s character formula for $L$ asserts
$\displaystyle e^{0}$ $\displaystyle=\frac{\sum_{w\in
W_{P}}(-1)^{\ell(w)}e^{w\bullet_{L}0}}{\prod_{\alpha\in
R_{I}^{+}}(1-e^{-\alpha})}\quad\text{with $w\bullet_{L}0=w\rho_{L}-\rho_{L}$,
$\rho_{L}=\frac{1}{2}\sum_{\alpha\in R_{I}^{+}}\alpha$}$
$\displaystyle=\sum_{w\in
W_{P}}(-1)^{\ell(w)}e^{w\bullet_{L}0}\prod_{\alpha\in
R_{I}^{+}}\frac{1-e^{-p\alpha}}{1-e^{-\alpha}}\prod_{\alpha\in
R_{I}^{+}}\frac{1}{1-e^{-p\alpha}}$ $\displaystyle=\prod_{\alpha\in
R_{I}^{+}}\frac{1-e^{-p\alpha}}{1-e^{-\alpha}}\sum_{\begin{subarray}{c}w\in
W_{P}\\\
\gamma\in\mathbb{N}R_{I}^{+}\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{-\gamma})e^{w\bullet_{L}0-p\gamma}$
$\displaystyle=\sum_{\begin{subarray}{c}w\in W_{P}\\\
\gamma\in\mathbb{N}R_{I}^{+}\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{-\gamma})\mathrm{ch}\,\,\mathrm{ind}_{(B\cap
L)_{1}T}^{L_{1}T}(w\bullet_{L}0-p\gamma)$
$\displaystyle=\sum_{\begin{subarray}{c}w\in W_{P}\\\
\gamma\in\mathbb{Z}R_{I}\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{\gamma})\mathrm{ch}\,\,\mathrm{ind}_{B_{1}T}^{P_{1}T}(w\bullet_{L}0+p\gamma)$
$\displaystyle=\sum_{\begin{subarray}{c}w\in W_{P}\\\
\gamma\in\mathbb{Z}R_{I}\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{\gamma})\mathrm{ch}\,\,\mathrm{ind}_{B_{1}T}^{P_{1}T}(w\bullet
0+p\gamma).$
It follows that
$\displaystyle\mathrm{ch}\,\hat{\nabla}_{P}(\nu)$
$\displaystyle=\sum_{\begin{subarray}{c}w\in W_{P}\\\
\gamma\in\mathbb{Z}R_{I}\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{\gamma})\mathrm{ch}\,\hat{\nabla}(\nu+w\bullet
0+p\gamma)$ $\displaystyle=\sum_{\begin{subarray}{c}w\in W_{P}\\\
\gamma\in\mathbb{Z}R_{I}\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{\gamma})\mathrm{ch}\,\hat{\nabla}(w\bullet\nu+p\gamma)\quad\text{as
$\langle\nu,\alpha^{\vee}\rangle=0$ $\forall\alpha\in R_{I}$}.$
(1.3) Lemma: Let $\nu\in\Lambda_{P}$.
(i) There are isomorphisms of $G_{1}B$-modules
$\mathrm{soc}_{G_{1}P}\hat{\nabla}_{P}(\nu)\simeq\hat{L}(\nu)\simeq\mathrm{soc}_{G_{1}B}\hat{\nabla}_{P}(\nu)$,
and hence one may regard $\hat{\nabla}_{P}(\nu)$ as a $G_{1}B$-submodule of
$\hat{\nabla}(\nu)$.
(ii) There is an isomorphism of $G_{1}P$-modules
$\hat{\nabla}_{P}(\nu)^{*}\simeq\hat{\nabla}_{P}(2(p-1)\rho_{P}-\nu)$.
Proof: (i) One has
$G_{1}B\mathbf{Mod}(\mathrm{ind}_{P}^{G_{1}P}(\nu),\mathrm{ind}_{B}^{G_{1}B}(\nu))\simeq
B\mathbf{Mod}(\mathrm{ind}_{P}^{G_{1}P}(\nu),\nu)=\Bbbk\,\mathrm{ev}_{\nu}.$
On the other hand, $\forall\lambda\in\Lambda$,
$\displaystyle G_{1}B$
$\displaystyle\mathbf{Mod}(\hat{L}(\lambda),\mathrm{ind}_{P}^{G_{1}P}(\nu))\leq
G_{1}T\mathbf{Mod}(\hat{L}(\lambda),\mathrm{ind}_{P_{1}T}^{G_{1}T}(\nu))\simeq
P_{1}T\mathbf{Mod}(\hat{L}(\lambda),\nu)$ $\displaystyle\leq
B_{1}T\mathbf{Mod}(\hat{L}(\lambda),\nu)=G_{1}T\mathbf{Mod}(\hat{L}(\lambda),\mathrm{ind}_{B_{1}T}^{G_{1}T}(\nu))\simeq\Bbbk\delta_{\lambda\nu}.$
It follows that
$\mathrm{soc}_{G_{1}B}(\mathrm{ind}_{P}^{G_{1}P}(\nu))=\hat{L}(\nu)$, and
hence also
$\mathrm{soc}_{G_{1}P}(\mathrm{ind}_{P}^{G_{1}P}(\nu))=\hat{L}(\nu)$. Thus
$\mathrm{ind}_{P}^{G_{1}P}(\nu)\leq\mathrm{ind}_{B}^{G_{1}B}(\nu)$ as
$G_{1}B$-modules via the commutative diagram
$\textstyle{\mathrm{ind}_{P}^{G_{1}P}(\nu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\nu}}$$\textstyle{\mathrm{ind}_{B}^{G_{1}B}(\nu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\nu}}$$\textstyle{\nu.}$
(ii) follows from [J, I.8.20 and II.3.4].
(1.4) Assume now that $p>h$ the Coxeter number of $G$. We say
$\lambda\in\Lambda$ is $p$-regular iff
$\langle\lambda+\rho,\alpha^{\vee}\rangle\not\in p\mathbb{Z}$
$\forall\alpha\in R$. Lusztig’s conjecture on the irreducible characters for
$G$, equivalently for $G_{1}T$, is now a theorem for indefinitely large $p$
due to Andersen, Jantzen and Soergel [AJS] and more recently to Fiebig [F1].
In turn, Lusztig’s conjecture allows us to determine the $G_{1}T$-socle series
of $\hat{\nabla}(\lambda)$ for $p$-regular $\lambda$ [AK]. To describe it, let
$W_{p}=W\ltimes p\mathbb{Z}R$ acting on $\Lambda$ with $p\mathbb{Z}R$ by
translations. If $s_{0}$ is the reflexion with respect to the hyperplane
$\\{v\in\Lambda\otimes_{\mathbb{Z}}\mathbb{R}\mid\langle
v+\rho,\alpha_{0}^{\vee}\rangle=-p\\}$ with $\alpha_{0}^{\vee}$ the highest
coroot, $(W_{p},(s_{\alpha},s_{0}\mid\alpha\in R^{s}))$ forms a Coxeter
system. We consider also the translations by $p\Lambda$. Then
$\Lambda_{1}=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle\in[0,p[\
\forall\alpha\in R^{s}\\}$ is a fundamental domain for the action of
$p\Lambda$. Put
${\mathcal{A}}=(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})\setminus\cup_{\alpha\in
R,n\in\mathbb{Z}}\\{v\in\Lambda\otimes_{\mathbb{Z}}\mathbb{R}\mid\langle
v+\rho,\alpha^{\vee}\rangle=pn\\}$, an element of which we call an alcove. We
will denote an alcove containing $0$ by $A^{+}$. As the structure of
$G_{1}T$-socle series is uniform for $p$-regular weights in an alcove, let
$\hat{\nabla}(A)$ (resp. $\hat{L}(A)$), $A\in{\mathcal{A}}$, denote
$\hat{\nabla}(\lambda)$ (resp. $\hat{L}(\lambda)$) for $\lambda\in A$. Let
$0<\mathrm{soc}^{1}\hat{\nabla}(A)=\mathrm{soc}\hat{\nabla}(A)\leq\mathrm{soc}^{2}\hat{\nabla}(A)\leq\dots$
be the $G_{1}T$-socle series of $\hat{\nabla}(A)$ and let
$\mathrm{soc}_{i}\hat{\nabla}(A)=\mathrm{soc}^{i}\hat{\nabla}(A)/\mathrm{soc}_{i-1}\hat{\nabla}(A)$
be the $i$-th socle layer of $\hat{\nabla}(A)$. The Lusztig conjecture implies
that the Loewy length, i.e., the length of the socle series of
$\hat{\nabla}(A)$ is $\ell(w_{0})+1$, and that $\forall C\in{\mathcal{A}}$,
(1)
$Q^{C,A}=\sum_{i\in\mathbb{N}}q^{\frac{\mathrm{d}(C,A)+1-i}{2}}[\mathrm{soc}_{i}\hat{\nabla}(A):\hat{L}(C)],$
where $Q^{C,A}$ is a Kazhdan-Lusztig polynomial in indeterminate $q$ [L, 1.8],
$\mathrm{d}(C,A)$ is the distance from alcove $C$ to alcove $A$, and
$[\mathrm{soc}_{i}\hat{\nabla}(A):\hat{L}(C)]$ is the multiplicity of
$\hat{L}(C)$ as a $G_{1}T$-composition factor of
$\mathrm{soc}_{i}\hat{\nabla}(A)$. For $G$ of rank $\leq 2$ the formula (1) is
known to hold for $p\geq h$.
For $A\in{\mathcal{A}}$ let $0_{A}$ be the element of $W_{p}\bullet 0$ in $A$.
Based on (1), (1.2), and noting that
$\mathrm{soc}_{i}\hat{\nabla}_{P}(A)\leq\mathrm{soc}_{i}\hat{\nabla}(A)$
$\forall i\in\mathbb{N}$, we speculate the $G_{1}T$-socle series of
$\hat{\nabla}_{P}(A)$ with $0_{A}\in\Lambda_{P}$ to be given by
(2) $\sum_{\begin{subarray}{c}w\in W_{P}\\\ \gamma\in\sum_{\alpha\in
I}\mathbb{Z}\alpha\end{subarray}}(-1)^{\ell(w)}\dim(\mathrm{Dist}(U_{L})_{\gamma})Q^{C,w\bullet
A+{p\gamma}}=\sum_{i\in\mathbb{N}}q^{\frac{d(C,A)+1-i}{2}}[\mathrm{soc}_{i}\hat{\nabla}_{P}(A):\hat{L}(C)].$
This is (1) in case $P=B$, and specializes to (1.2) under $q\rightsquigarrow
1$. Put $w^{P}=w_{0}w_{P}$.
Proposition: Assume $G$ is of rank $\leq 2$. For $p\geq h$ each
$\hat{\nabla}_{P}(A)$ with $0_{A}\in\Lambda_{P}$ has the Loewy length
$\ell(w^{P})+1$ and the $G_{1}T$-socle series is given by the formula (2).
Proof: We may assume $P>B$. Unless $G$ is in type $\mathrm{G}_{2}$,
$\hat{\nabla}(A)$ is multiplicity-free rendering the verification mechanical.
Assume now that $G$ is in type $G_{2}$. Let $\alpha_{1}$ and $\alpha_{2}$ be
the simple roots with $\alpha_{1}$ short, and put $s_{i}=s_{\alpha_{i}}$ and
$\omega_{i}=\omega_{\alpha_{i}}$. We may translate $0_{A}$ into $\Lambda_{1}$
via $p\Lambda_{P}$. Let $P=P_{\alpha_{1}}$. We may assume $A=A^{+}$ or
$s_{0}s_{1}s_{2}s_{1}s_{0}\bullet A^{+}$. Consider the case $A=A^{+}$. We are
to test for each $C\in{\mathcal{A}}$
(3)
$\sum_{n\in\mathbb{N}}Q^{C,A^{+}-pn\alpha_{1}}-\sum_{n\in\mathbb{N}}Q^{C,s_{1}\bullet
A^{+}-pn\alpha_{1}}=\sum_{i}q^{\frac{d(C,A^{+})+1-i}{2}}[\mathrm{soc}_{i}\hat{\nabla}_{P}(A^{+}):\hat{L}(C)].$
If $Q^{C,A^{+}}$ is a monomial, we can read off the socle level of
$\hat{L}(C)$ in $\hat{\nabla}_{P}(A^{+})$ from (1.2), and (3) holds. Thus
those $C$ left to be examined are $s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}-p\omega_{2}$, $s_{1}s_{2}s_{1}s_{0}\bullet A^{+}-2p\omega_{1}$,
$s_{1}s_{0}\bullet A^{+}-p\omega_{2}$, $A^{+}+p(2\omega_{1}-2\omega_{2})$,
$A^{+}-p\omega_{2}$, $A^{+}-2p\omega_{1}$, $s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}+p(\omega_{1}-2\omega_{2})$, $s_{1}s_{2}s_{1}s_{0}\bullet A^{+}-p\rho$,
$A^{+}+p(\omega_{1}-2\omega_{2})$, $A^{+}-p\rho$, $A^{+}-3p\omega_{1}$,
$A^{+}-2p\omega_{2}$. For (3) to hold, we must have all those $\hat{L}(C)$
belonging to $\mathrm{soc}_{3}\hat{\nabla}_{P}(A^{+})$. As
$\hat{L}(s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}+p(2\omega_{1}-2\omega_{2}))=L(s_{1}s_{2}s_{1}s_{0}A^{+})\otimes(2\omega_{1}-2\omega_{2})^{[1]}\leq\mathrm{soc}_{3}\hat{\nabla}_{P}(A^{+})$
and as $\Delta^{P}(2\omega_{1}-2\omega_{2})$ is $P$-irreducible,
$G_{1}\mathbf{Mod}(L(s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}),\mathrm{soc}_{3}\hat{\nabla}_{P}(A^{+}))^{[-1]}\geq\Delta^{P}(2\omega_{1}-2\omega_{2})$.
It follows that both $\hat{L}(s_{1}s_{2}s_{1}s_{0}\bullet A^{+}-p\omega_{2})$
and $\hat{L}(s_{1}s_{2}s_{1}s_{0}\bullet A^{+}-2p\omega_{1})$ must also belong
to $\mathrm{soc}_{3}\hat{\nabla}_{P}(A^{+})$. Likewise
$\hat{L}(A^{+}+p(\omega_{1}-2\omega_{2}))$, $\hat{L}(A^{+}-p\rho)$,
$\hat{L}(A^{+}-3p\omega_{1})$. We will then be left with the following $C$:
$s_{1}s_{0}\bullet A^{+}-p\omega_{2}$, $A^{+}+p(2\omega_{1}-2\omega_{2})$,
$A^{+}-p\omega_{2}$, $A^{+}-2p\omega_{1}$, $s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}+p(\omega_{1}-2\omega_{2})$, $s_{1}s_{2}s_{1}s_{0}\bullet A^{+}-p\rho$,
$A^{+}-2p\omega_{2}$. For those $C$ if $\hat{L}(C)$ does not belong to the
third socle layer, it must lie in the 5th by (1). On the other hand, dualizing
an exact sequence of $G_{1}P$-modules
$0\to\mathrm{soc}_{5}\hat{\nabla}_{P}(A^{+})\to\hat{\nabla}_{P}(A^{+})/\mathrm{soc}^{4}\hat{\nabla}_{P}(A^{+})\to\hat{L}(s_{0}s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}-2p\omega_{2})\to 0,$
one obtains another exact sequence
$0\to\hat{L}(s_{0}s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}-2p\omega_{2})^{*}\to(\hat{\nabla}_{P}(A^{+})/\mathrm{soc}^{4}\hat{\nabla}_{P}(A^{+}))^{*}\to(\mathrm{soc}_{5}\hat{\nabla}_{P}(A^{+}))^{*}\to
0$
with
$(\hat{\nabla}_{P}(A^{+})/\mathrm{soc}^{4}\hat{\nabla}_{P}(A^{+}))^{*}\leq\hat{\nabla}_{P}(A^{+})^{*}\simeq\hat{\nabla}_{P}(3(p-1)\omega_{2})\leq\hat{\nabla}(3(p-1)\omega_{2})$
by (1.3), and hence if
$\hat{L}(C)\leq\mathrm{soc}_{5}\hat{\nabla}_{P}(A^{+})$, then
$\hat{L}(C)^{*}\leq\mathrm{soc}_{2}\hat{\nabla}(3(p-1)\omega_{2})=\mathrm{soc}_{2}\hat{\nabla}(s_{0}s_{1}s_{2}s_{1}s_{0}\bullet
A^{+}+2p\omega_{2}),$ which contradicts (1). Thus (3) holds.
Likewise the other cases.
(1.5) Assume $p>h$. For each $w\in W$ there is a unique element in $(w\bullet
0+p\Lambda)\cap\Lambda_{1}$, which we will denote by $\varepsilon_{w}$. Thus
the principal $G_{1}$-block consists of $L(\varepsilon_{w})$, $w\in W$. Put
for simplicity $L(w)=L(\varepsilon_{w})$. We know from [Y] that all $L(w)$,
$w\in W$, appear as $G_{1}$-composition factors of
$\hat{\nabla}(\varepsilon)$.
###### Corollary:
Assume $\mathrm{rk}\,G\leq 2$.
1. (i)
Each $i$-th $G_{1}T$-socle layer of $\hat{\nabla}(\varepsilon)$ admits a
decomposition as $G_{1}P$-module
$\mathrm{soc}_{i}\hat{\nabla}(\varepsilon)=\coprod_{w\in W^{P}}L(w)\otimes
G_{1}\mathbf{Mod}(L(w),\mathrm{soc}_{i}\hat{\nabla}(\varepsilon)).$
2. (ii)
Each $L(w)$, $w\in W^{P}$, appears as $G_{1}$-factor of
$\mathrm{soc}_{\ell(w)+1}\hat{\nabla}_{P}(\varepsilon)$, i.e.,
$G_{1}\mathbf{Mod}(L(w),\mathrm{soc}_{\ell(w)+1}\hat{\nabla}(\varepsilon))\neq
0$.
(1.6) Remarks: (i) In case ${\mathcal{P}}=\mathrm{GL}(E)/P$ with
$\Bbbk$-linear space $E$ of basis $e_{1},e_{2},\dots,e_{n+1}$ and with
$P=N_{\mathrm{GL}(E)}(\Bbbk e_{n+1})$, the assertions (i) and (ii) hold [K09].
(ii) Over $\mathbb{C}$ if $P^{+}_{\mathbb{C}}$ is the parabolic subgroup
opposite to $P_{\mathbb{C}}$, the $\mathrm{Dist}(G_{\mathbb{C}})$-composition
factors of Verma module
$\mathrm{Dist}(G_{\mathbb{C}})\otimes_{\mathrm{Dist}(P^{+}_{\mathbb{C}})}\varepsilon_{\mathbb{C}}$
are known to be of the form $L(w^{-1}\bullet 0)$, $w\in W^{P}$ [Hum].
(1.7) Assume $\mathrm{rk}\,G\leq 2$ and $p>h$. Put
$\mathrm{soc}_{i,w}^{1}=G_{1}\mathbf{Mod}(L(w),\mathrm{soc}_{i}\hat{\nabla}(\varepsilon))^{[-1]}$
$\forall i\in[1,\ell(w^{P})+1],w\in W^{P}$. Sheafifying
$\hat{\nabla}_{P}(\varepsilon)$ one obtains from (1.6) a filtration on
$F_{*}{\mathcal{O}}_{\mathcal{P}}\simeq\phi_{*}{\mathcal{L}}_{G/G_{1}P}(\hat{\nabla}_{P}(\varepsilon))$
of subquotients
$L(w)\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{i,w}^{1})$,
$i\in[1,\ell(w^{P})+1],w\in W^{P}$. For our second objective it is therefore
important to determine the $P$-module structure on each
$\mathrm{soc}_{\ell(w)+1,w}^{1}$, $w\in W^{P}$. In case $G$ has rank $2$, let
$\alpha_{1}$ and $\alpha_{2}$ be the simple roots with $\alpha_{1}$ short. Put
$\omega_{i}=\omega_{\alpha_{i}}$ and $s_{i}=s_{\alpha_{i}}$, $i=1,2$. Let
$P_{\alpha_{i}}$ be the standard parabolic subgroup of $G$ associated to
$\alpha_{i}$, i.e., such that $\pm\alpha_{i}$ are roots of $P_{\alpha_{i}}$.
Arguing as in [AK00]/[HKR]/[KY] we find
###### Proposition:
We have the following identifications as $P$-modules.
1. (i)
If $G=\mathrm{SL}_{2}$, then $\mathrm{soc}_{1,e}^{1}\simeq\varepsilon$ and
$\mathrm{soc}_{2,w_{0}}^{1}\simeq-\rho$.
2. (ii)
If $G=\mathrm{SL}_{3}$ and $P=P_{\alpha_{1}}$, then
$\mathrm{soc}_{1,e}^{1}\simeq\varepsilon$,
$\mathrm{soc}_{2,s_{2}}^{1}=-\omega_{2}$ and
$\mathrm{soc}_{3,w^{P}}^{1}=-2\omega_{2}$.
3. (iii)
If $G=\mathrm{SL}_{3}$ and $P=B$,
$\displaystyle\mathrm{soc}_{1,e}^{1}$ $\displaystyle\simeq\varepsilon,$
$\displaystyle\mathrm{soc}_{2,s_{1}}^{1}$ $\displaystyle\simeq-\omega_{1}$
$\displaystyle\mathrm{soc}_{2,s_{2}}^{1}$ $\displaystyle\simeq-\omega_{2},$
$\displaystyle\mathrm{soc}_{3,s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\rho)\otimes_{\Bbbk}\Delta^{\alpha_{2}}(\omega_{2}),\quad$
$\displaystyle\mathrm{soc}_{3,s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-\rho)\otimes_{\Bbbk}\Delta^{\alpha_{1}}(\omega_{1}),\quad$
$\displaystyle\mathrm{soc}_{4,w_{0}}^{1}$ $\displaystyle\simeq-\rho.$
4. (iv)
If $G=\mathrm{Sp}_{4}$ and $P=P_{\alpha_{2}}$,
$\mathrm{soc}_{1,e}^{1}\simeq\varepsilon,\quad\mathrm{soc}_{2,s_{1}}^{1}\simeq-\omega_{1},\quad\mathrm{soc}_{3,s_{2}s_{1}}^{1}\simeq-2\omega_{1},\quad\mathrm{soc}_{4,w^{P}}^{1}\simeq-3\omega_{1}.$
5. (v)
If $G=\mathrm{Sp}_{4}$ and $P=P_{\alpha_{1}}$,
$\mathrm{soc}_{1,e}^{1}\simeq\varepsilon,\quad\mathrm{soc}_{2,s_{2}}^{1}\simeq-\omega_{2},\quad\mathrm{soc}_{3,s_{1}s_{2}}^{1}\simeq\Delta^{\alpha_{1}}(\omega_{1}-2\omega_{2}),\quad\mathrm{soc}_{4,w^{P}}^{1}\simeq-2\omega_{2}.$
6. (vi)
If $G=\mathrm{Sp}_{4}$ and $P=B$,
$\displaystyle\mathrm{soc}_{1,e}^{1}$ $\displaystyle\simeq\varepsilon,$
$\displaystyle\mathrm{soc}_{2,s_{1}}^{1}$ $\displaystyle\simeq-\omega_{1},$
$\displaystyle\mathrm{soc}_{2,s_{2}}^{1}$ $\displaystyle\simeq-\omega_{2},$
$\displaystyle\mathrm{soc}_{3,s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\omega_{2})\otimes_{\Bbbk}\ker(\Delta(\omega_{1})\twoheadrightarrow\omega_{1}),$
$\displaystyle\mathrm{soc}_{3,s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-\rho)\otimes\Delta^{\alpha_{1}}(\omega_{1}),$
$\displaystyle\mathrm{soc}_{4,s_{1}s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-\rho)\otimes_{\Bbbk}(\Delta(\omega_{2})/(-\omega_{2})),$
$\displaystyle\mathrm{soc}_{4,s_{2}s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\rho)\otimes_{\Bbbk}(\Delta(\omega_{1})/(-\omega_{1})),$
$\displaystyle\mathrm{soc}_{5,w_{0}}^{1}$ $\displaystyle\simeq-\rho.$
7. (vii)
If $G$ is of type $\mathrm{G}_{2}$ and $P=P_{\alpha_{2}}$,
$\displaystyle\mathrm{soc}_{1,e}^{1}$
$\displaystyle\simeq\varepsilon,\qquad\mathrm{soc}_{2,s_{1}}^{1}\simeq-\omega_{1},\qquad\mathrm{soc}_{3,s_{2}s_{1}}^{1}\simeq-2\omega_{1},$
$\displaystyle\mathrm{soc}_{4,s_{1}s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-3\omega_{1})\otimes(\Delta(\omega_{1})/\mathrm{Dist}(P_{\alpha_{2}})(\Delta(\omega_{1})_{-\alpha_{1}})),$
$\displaystyle\mathrm{soc}_{5,s_{2}s_{1}s_{2}s_{1}}^{1}$
$\displaystyle\simeq-3\omega_{1},\qquad\mathrm{soc}_{6,w^{P}}^{1}\simeq-4\omega_{1}.$
8. (viii)
If $G$ is of type $\mathrm{G}_{2}$ and $P=P_{\alpha_{1}}$,
$\displaystyle\mathrm{soc}_{1,e}^{1}$
$\displaystyle\simeq\varepsilon,\qquad\mathrm{soc}_{2,s_{2}}^{1}\simeq-\omega_{2},$
$\displaystyle\mathrm{soc}_{3,s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\omega_{2})\otimes\ker(\Delta(\omega_{1})\twoheadrightarrow\Delta^{\alpha_{1}}(\omega_{1})),\qquad\mathrm{soc}_{4,s_{2}s_{1}s_{2}}^{1}\simeq\Delta^{\alpha_{1}}(\omega_{1}-2\omega_{2}),$
$\displaystyle\mathrm{soc}_{5,s_{1}s_{2}s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-2\omega_{2})\otimes(\Delta(\omega_{1})/\Delta^{\alpha_{1}}(\omega_{1}-\omega_{2})),\qquad\mathrm{soc}_{6,w^{P}}^{1}\simeq-2\omega_{2}.$
(1.8) If $G$ is in type $\mathrm{G}_{2}$ and $P=B$, we are unfortunately not
able to detemine the $B$-module structure on $\mathrm{soc}_{i,w}^{1}$ at
present. To speculate, we make use of the following
###### Lemma:
Let $G$ be an arbitrary simply connencted reducive algebraic group over
$\Bbbk$. Assume $p\geq 2(h-1)$ and that Lusztig’s conjecture on irreducible
$G_{1}T$-modules hold. Let $\mathrm{rad}_{G,i}\nabla(p\rho)$,
$i\in\mathbb{N}$, be the $i$-th layer of the radical series on $\nabla(p\rho)$
as $G$-module, and for each $\lambda\in\Lambda_{1}$ let
$L(\lambda)\otimes(\mathrm{rad}_{G,i,\lambda}^{1}\nabla(p\rho))^{[1]}$ (resp.
$L(\lambda)\otimes(\mathrm{soc}_{i}^{1}\hat{\nabla}(\varepsilon))^{[1]}$)
denote the $L(\lambda)$-isotypic component of
$\mathrm{rad}_{G,i}\nabla(p\rho)$ (resp.
$\mathrm{soc}_{i}\hat{\nabla}(\varepsilon)$ as $G_{1}T$-module). Then there is
for each $i$ a surjective homomorphism of $B$-modules
$(-\rho)\otimes_{\Bbbk}\mathrm{rad}_{G,i,\lambda}^{1}\nabla(p\rho)\twoheadrightarrow\mathrm{soc}_{\ell(w_{0})+1-i,\lambda}^{1}\hat{\nabla}(\varepsilon).$
Proof: By the hypothesis on $p$ one has
$\langle\nu^{1}+\rho,\alpha^{\vee}\rangle\leq p$ for any weight $\nu$ of
$\nabla(p\rho)$ and any $\alpha\in R^{+}$, where $\nu=\nu^{0}+p\nu^{1}$ with
$\nu^{0}\in\Lambda_{1}$ and $\nu^{1}\in\Lambda$. Then by [AK, 8.1.2] the
radical series as $G$-module and as $G_{1}T$-module on $\nabla(p\rho)$
coincide: $\forall i\in\mathbb{N}$,
(1) $\mathrm{rad}_{G,i}\nabla(p\rho)=\mathrm{rad}_{G_{1}T,i}\nabla(p\rho).$
On the other hand, as observed in the proof of [AK, 8.2], the natural
homomorphism $\nabla(p\rho)\to\hat{\nabla}(p\rho)$ of $G_{1}B$-modules is
surjective, which therefore induces by (1) a commutative diagram of
$G_{1}T$-modules
$\textstyle{\mathrm{rad}_{G,i}\nabla(p\rho)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{rad}_{G_{1}T,i}\hat{\nabla}(p\rho)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{soc}_{\ell(w_{0})+1-i}\hat{\nabla}(p\rho)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{p\rho\otimes_{\Bbbk}\mathrm{soc}_{\ell(w_{0})+1-i}\hat{\nabla}(\varepsilon).}$
As $\mathrm{rad}_{G_{1}T,i}\hat{\nabla}(p\rho)$ is equipped with a structure
of $G_{1}B$-module, the induced map $\mathrm{rad}_{G,i}\nabla(p\rho)\to
p\rho\otimes_{\Bbbk}\mathrm{soc}_{\ell(w_{0})+1-i}\hat{\nabla}(\varepsilon)$
above is a surjective homomorphism of $G_{1}B$-modules, and hence the
assertion.
(1.9) Assuming the Jantzen conjecture on the filtration of Weyl modules [A83],
the radical series of $\Delta(p\rho)$ as $G$-module is available from [A87].
Based also on the character data (1.4), we are thus led to speculate on the
$B$-module structure of $\mathrm{soc}_{\ell(w)+1,w}^{1}$, $w\in W$, in type
$\mathrm{G}_{2}$ to be as follows:
$\displaystyle\mathrm{soc}_{1,e}^{1}$ $\displaystyle\simeq\varepsilon,$
$\displaystyle\mathrm{soc}_{2,s_{1}}^{1}$ $\displaystyle\simeq-\omega_{1},$
$\displaystyle\mathrm{soc}_{2,s_{2}}^{1}$ $\displaystyle\simeq-\omega_{2},$
$\displaystyle\mathrm{soc}_{3,s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-\rho)\otimes\Delta^{\alpha_{1}}(\omega_{1}),$
$\displaystyle\mathrm{soc}_{3,s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\omega_{2})\otimes\ker(\Delta(\omega_{1})\twoheadrightarrow\omega_{1}),$
$\displaystyle\mathrm{soc}_{4,s_{2}s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\rho)\otimes(\Delta(\omega_{1})/\mathrm{Dist}(P_{\alpha_{2}})v_{3}),$
$\displaystyle\mathrm{soc}_{4,s_{1}s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{1}})(\Bbbk(v_{2}+v_{1})+\Delta(\omega_{2})_{-\alpha_{2}})\\},$
$\displaystyle\mathrm{soc}_{5,s_{1}s_{2}s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(\Bbbk(v_{4}+v_{3})+\Delta(\omega_{2})_{-3\omega_{1}+\omega_{2}})\\},$
$\displaystyle\mathrm{soc}_{5,s_{2}s_{1}s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-\rho)\otimes(\Delta(\omega_{1})/\Delta^{\alpha_{1}}(\omega_{1}-\omega_{2})),$
$\displaystyle\mathrm{soc}_{6,s_{2}s_{1}s_{2}s_{1}s_{2}}^{1}$
$\displaystyle\simeq(-\rho)\otimes(\Delta(\omega_{1})/(-\omega_{1})),$
$\displaystyle\mathrm{soc}_{6,s_{1}s_{2}s_{1}s_{2}s_{1}}^{1}$
$\displaystyle\simeq(-\rho)\otimes\\{\varepsilon\oplus(\Delta(\omega_{2})/(-\omega_{2}))\\},$
$\displaystyle\mathrm{soc}_{7,w_{0}}^{1}$ $\displaystyle\simeq-\rho,$
where $v_{i}\in\Delta(\omega_{i})_{\alpha_{1}}\setminus 0$, $i\in\\{1,2\\}$,
and $v_{i+2}=\displaystyle
F_{1}^{(2)}v_{i}\in\Delta(\omega_{i})_{-\alpha_{1}}$, $i\in\\{1,2\\}$, with
$F_{1}$ a root vector belonging to $-\alpha_{1}$ in the Chevalley basis of the
Lie algebra of $G$.
We remark, in particular, that the Jantzen conjecture implies in type
$\mathrm{G}_{2}$ that $\mathrm{soc}^{1}_{6,s_{1}s_{2}s_{1}s_{2}s_{1}}$ should
be decomposable, contrary to the cases for $\mathrm{SL}_{2}$,
$\mathrm{SL}_{3}$, $\mathrm{Sp}_{4}$.
(1.10) We now set
${\mathcal{E}}_{w}={\mathcal{L}}_{\mathcal{P}}(\mathrm{soc}_{\ell(w)+1,w}^{1})$,
$w\in W^{P}$, or rather we actually employ for
$({\mathcal{E}}_{w})_{P/P}\otimes_{{\mathcal{O}}_{{\mathcal{P}},P/P}}\Bbbk$
the $P$-module on the right hand side of $\mathrm{soc}_{\ell(w)+1,w}^{1}$
given in (1.7) and (1.9), and forget about the characteristic restriction. In
type $\mathrm{G}_{2}$ when $P=B$ we take
${\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}s_{1}}={\mathcal{L}}_{{\mathcal{P}}}((-\rho)\otimes(\Delta(\omega_{2})/(-\omega_{2})))$
instead.
###### Theorem:
Let $p>0$ be arbitrary unless otherwise specified.
1. (i)
The ${\mathcal{E}}_{w}$, $w\in W^{P}$, Karoubian generate
$\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$.
2. (ii)
Unless $G$ is in type $\mathrm{G}_{2}$ with $P=P_{\alpha_{1}}$ or $B$,
$\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}_{w},{\mathcal{E}}_{w})\simeq\Bbbk$
$\forall w\in W^{P}$,
$\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}_{x},{\mathcal{E}}_{y})\neq 0$ iff
$x>y$ $\forall x,y\in W^{P}$, and
$\mathrm{Ext}_{\mathcal{P}}^{i}(\coprod_{w\in
W^{P}}{\mathcal{E}}_{w},\coprod_{w\in W^{P}}{\mathcal{E}}_{w})\neq 0$ $\forall
i>0$.
3. (iii)
If $G$ is in type $\mathrm{G}_{2}$ and $P=P_{\alpha_{1}}$, assume $p\geq 3$.
Then the same hold true for the ${\mathcal{E}}_{w}$, $w\in W^{P}$, as in (ii).
4. (iv)
If $G$ is in type $\mathrm{G}_{2}$ and $P=B$, assume $p\geq 7$. Then the same
hold true for the ${\mathcal{E}}_{w}$, $w\in W^{P}=W$, as in (ii).
The arguments for the theorem will be given in §§2 and 3.
(1.11) Remarks: (i) In (iii) above the restriction on $p$ is sharp. If $p=2$,
we have, see §3, $\forall i\in\mathbb{N}$,
$\mathrm{Ext}^{i}_{\mathcal{P}}({\mathcal{E}}_{s_{2}},{\mathcal{E}}_{s_{1}s_{2}})\simeq\begin{cases}\Bbbk&\text{if
$i=0,1$}\\\ 0&\text{else}.\end{cases}$
(ii) In (iv) if $p=2$, we have $\forall i\in\mathbb{N}$,
$\mathrm{Ext}_{\mathcal{P}}^{i}({\mathcal{E}}_{s_{2}},{\mathcal{E}}_{s_{1}s_{2}})\simeq\begin{cases}\Bbbk&\text{if
$i=0,1$},\\\ 0&\text{else}.\end{cases}$
Also if $p=3$,
$\mathbf{Mod}_{\mathcal{P}}({\mathcal{E}}_{s_{2}s_{1}},{\mathcal{E}}_{s_{1}s_{2}s_{1}})\neq
0$.
(iii) If we employ the order reversing involution $w\mapsto w_{0}ww_{P}$ on
$W^{P}$, Theorem 1.10 verifies for groups of rank $\leq 2$ Catanese’s
conjecture over $\mathbb{C}$ by base change. Our present parametrization
appears more natural particularly in the case of quadrics, see §4.
$2^{\circ}$ Karoubian completeness
Let $\\{{\mathcal{E}}_{w}|w\in W^{P}\\}$ be the coherent sheaves on
${\mathcal{P}}=G/P$ defined in (1.7) and (1.9). In this section we will show
that they are Karoubian complete for
$\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$, i.e., they Karoubian generate the
bounded derived category $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$ of
coherent sheaves on ${\mathcal{P}}$.
Write $P=P_{I}$, $I\subset R^{s}$, and recall $2\rho_{P}=\sum_{\alpha\in
R^{+}\setminus R_{I}}\alpha$. As ${\mathcal{L}}_{\mathcal{P}}(2\rho_{P})$ is
ample on ${\mathcal{P}}$, to show that the ${\mathcal{E}}_{w}$, $w\in W^{P}$,
Karoubian generate $\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$, it is enough
by a result attributed to Kontsevich by Positselskii [BMR] to verify that all
${\mathcal{L}}_{\mathcal{P}}(-2n\rho_{P})$, $n\in\mathbb{N}^{+}$, are
Karoubian generated by the ${\mathcal{E}}_{w}$’s; we actually show that all
${\mathcal{L}}_{\mathcal{P}}(\lambda)$, $\lambda\in\Lambda_{P}$, are Karoubian
generated by the ${\mathcal{E}}_{w}$’s. This has been done in the case of
$G\in\\{\mathrm{SL}_{2},\mathrm{SL}_{3}\\}$ with $P=B$ in [HKR] and in the
case of $G=\mathrm{Sp}_{4}$ with $P=B$ in [KY]; precisely, in [HKR] and in
[KY] it was assumed that $p\geq h$, whose arguments carry over, however,
verbatim to arbitrary characteristic thanks to the new definition of the
${\mathcal{E}}_{w}$’s. Our argument for $G$ in type $\mathrm{G}_{2}$ and $P=B$
is essentially the same.
If $P>B$ in cases $G=\mathrm{Sp}_{4}$ or in type $\mathrm{G}_{2}$, we will
also make use of a projection formula [Bö, 3.3.2]: put ${\mathcal{B}}=G/B$. If
$\bar{\pi}:{\mathcal{B}}\to{\mathcal{P}}$ is the natural morphism,
(1)
$\mathrm{id}_{\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})}\simeq(\mathbf{R}\bar{\pi}_{*}{\mathcal{O}}_{\mathcal{B}})\otimes^{\mathbf{L}}_{\mathcal{P}}\
?\simeq(\mathbf{R}\bar{\pi}_{*})\circ\bar{\pi}^{*}:\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})\to\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}}).$
Recall from [J, I.5.19] that $\forall m,n\in\mathbb{Z}$,
(2)
$(\mathbf{R}\pi_{*}){\mathcal{L}}_{\mathcal{B}}(m\omega_{1}+n\omega_{2})\simeq\begin{cases}{\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{1}}(m\omega_{1}+n\omega_{2}))\simeq(\mathbf{R}\bar{\pi}_{*}){\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{1}}(m\omega_{1}+n\omega_{2}))\\\
\hskip 142.26378pt\text{if $P=P_{\alpha_{1}}$ and $m\geq 0$}\\\
{\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{2}}(m\omega_{1}+n\omega_{2}))\simeq(\mathbf{R}\bar{\pi}_{*}){\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{2}}(m\omega_{1}+n\omega_{2}))\\\
\hskip 142.26378pt\text{if $P=P_{\alpha_{2}}$ and $n\geq 0$},\end{cases}$
and from [J, I.5.17] that $\forall M\in P\mathbf{Mod}$,
(3)
$\bar{\pi}^{*}{\mathcal{L}}_{\mathcal{P}}(M)\simeq{\mathcal{L}}_{\mathcal{B}}(M).$
In the following we will often describe the setup in which a module $M$ admits
a filtration whose subquotients are $M_{1}$, $M_{2}$,…, $M_{r}$ from the
bottom to the top by a diagram
$M=\begin{tabular}[]{|c|}\hline\cr$M_{r}$\\\ \hline\cr$\vdots$\\\
\hline\cr$M_{2}$\\\ \hline\cr$M_{1}$\\\ \hline\cr\end{tabular}.$
If $M_{i}$ is a direct sum of $M_{i1},\dots,M_{is}$, we will insert
$M_{i1}\mid M_{i2}\mid\dots\mid M_{is}$ in place of $M_{i}$.
Let us illustrate our argument in case $G$ is in type $\mathrm{G}_{2}$ and
$P=P_{\alpha_{2}}$. Let $\hat{\mathcal{E}}=\langle{\mathcal{E}}_{w}\mid w\in
W^{P}\rangle$ denote the triangulated subcategory of
$\mathrm{D}^{b}(\mathrm{coh}{\mathcal{P}})$ Karoubian generated by
${\mathcal{E}}_{w}$, $w\in W^{P}$. One has $\Lambda_{P}=\mathbb{Z}\omega_{1}$.
Note first that in
${\mathcal{E}}_{s_{1}s_{2}s_{1}}={\mathcal{L}}_{{\mathcal{P}}}((-3\omega_{1})\otimes(\Delta(\omega_{1})/\mathrm{Dist}(P)(v_{3})))$
one has
$\mathrm{Dist}(P)(v_{3})=\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\ \hline\cr\end{tabular},$
and hence
$\displaystyle(-3\omega_{1})\otimes(\Delta(\omega_{1})/\mathrm{Dist}(P)(v_{3}))$
$\displaystyle\simeq(-3\omega_{1})\otimes\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr$\varepsilon$\\\
\hline\cr\end{tabular}=\begin{tabular}[]{|c|}\hline\cr$-2\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-4\omega_{1}+\omega_{2})$\\\
\hline\cr$-3\omega_{1}$\\\ \hline\cr\end{tabular}.$
It follows, by the presence of
${\mathcal{E}}_{s_{2}s_{1}s_{2}s_{1}}={\mathcal{L}}_{\mathcal{P}}(-3\omega_{1})$
and ${\mathcal{E}}_{s_{2}s_{1}}={\mathcal{L}}_{\mathcal{P}}(-2\omega_{1})$ in
$\hat{\mathcal{E}}$, that
${\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{2}}(-4\omega_{1}+\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{P}}(\Delta^{\alpha_{2}}(-4\omega_{1}+\omega_{2}))\in\hat{\mathcal{E}}$.
Then, as
$\displaystyle\hat{\mathcal{E}}$
$\displaystyle\ni\Delta(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{P}}(-2\omega_{1})$
$\displaystyle\simeq{\mathcal{L}}_{\mathcal{P}}(\Delta(\omega_{1})\otimes_{\Bbbk}(-2\omega_{1}))\quad\text{by
the tensor identity}$
$\displaystyle\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr$\varepsilon$\\\
\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr\end{tabular})\otimes(-2\omega_{1}))\quad\text{using the $P$-module
structure on $\Delta(\omega_{1})$}$
$\displaystyle\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$-\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-3\omega_{1}+\omega_{2})$\\\
\hline\cr$-2\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-4\omega_{1}+\omega_{2})$\\\
\hline\cr$-3\omega_{1}$\\\ \hline\cr\end{tabular}),$
one obtains
${\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{2}}(-3\omega_{1}+\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{P}}(\Delta^{\alpha_{2}}(-3\omega_{1}+\omega_{2}))\in\hat{\mathcal{E}}$.
In turn, as
$\displaystyle\hat{\mathcal{E}}$
$\displaystyle\ni\Delta(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{P}}(-\omega_{1})\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr$\varepsilon$\\\
\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr\end{tabular})\otimes(-\omega_{1}))\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\varepsilon$\\\
\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-3\omega_{1}+\omega_{2})$\\\
\hline\cr$-2\omega_{1}$\\\ \hline\cr\end{tabular}),$
${\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{2}}(-2\omega_{1}+\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{P}}(\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2}))\in\hat{\mathcal{E}}$.
Then, as
$\displaystyle\hat{\mathcal{E}}$
$\displaystyle\ni\Delta(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{P}}(-3\omega_{1})\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr$\varepsilon$\\\
\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\ \hline\cr\end{tabular})\otimes(-3\omega_{1}))$
$\displaystyle\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$-2\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-4\omega_{1}+\omega_{2})$\\\
\hline\cr$-3\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-5\omega_{1}+\omega_{2})$\\\
\hline\cr$-4\omega_{1}$\\\ \hline\cr\end{tabular}),$
${\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{2}}(-5\omega_{1}+\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{P}}(\Delta^{\alpha_{2}}(-5\omega_{1}+\omega_{2}))\in\hat{\mathcal{E}}$.
It follows, as
$\displaystyle\hat{\mathcal{E}}$
$\displaystyle\ni\Delta(\omega_{2})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{P}}(-2\omega_{1})\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{2}}(\omega_{2})$\\\
\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr\begin{tabular}[]{c|c}$\Delta^{\alpha_{2}}(-3\omega_{1}+2\omega_{2})$&$\varepsilon$\end{tabular}\\\
\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-3\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular})\otimes(-2\omega_{1}))$
$\displaystyle\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-3\omega_{1}+\omega_{2})$\\\
\hline\cr\begin{tabular}[]{c|c}$\Delta^{\alpha_{2}}(-5\omega_{1}+2\omega_{2})$&$-2\omega_{1}$\end{tabular}\\\
\hline\cr$\Delta^{\alpha_{2}}(-4\omega_{1}+\omega_{2})$\\\
\hline\cr$-3\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-5\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular}),$
that
${\mathcal{L}}_{\mathcal{P}}(\Delta^{\alpha_{2}}(-5\omega_{1}+2\omega_{2}))\in\hat{\mathcal{E}}$.
If $p\geq 3$,
$\Delta^{\alpha_{2}}(-5\omega_{1}+2\omega_{2})\simeq\nabla^{\alpha_{2}}(-5\omega_{1}+2\omega_{2})$.
If $p=2$, as
$\Delta^{\alpha_{2}}(-5\omega_{1}+2\omega_{2})=\begin{tabular}[]{|c|}\hline\cr$L^{\alpha_{2}}(-5\omega_{1}+2\omega_{2})$\\\
\hline\cr$-2\omega_{1}$\\\ \hline\cr\end{tabular}$ with
$L^{\alpha_{2}}(-5\omega_{1}+2\omega_{2})$ simple $P_{\alpha_{2}}$-module of
highest weight $-5\omega_{1}+2\omega_{2}$, and as
${\mathcal{L}}_{\mathcal{P}}(-2\omega_{1})\in\hat{\mathcal{E}}$, so does
${\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{2}}(-5\omega_{1}+2\omega_{2}))$.
Thus,
${\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{2}}(-5\omega_{1}+2\omega_{2}))\in\hat{\mathcal{E}}$
regardless of the characteristic.
Also, as
$\hat{\mathcal{E}}\ni\Delta(\omega_{1})\otimes_{\Bbbk}{\mathcal{O}}_{\mathcal{P}}\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr$\varepsilon$\\\
\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\ \hline\cr\end{tabular}),$
one obtains
${\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\nabla^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular})\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular})\in\hat{\mathcal{E}}.$
Likewise, as
$\displaystyle\hat{\mathcal{E}}$
$\displaystyle\ni\Delta(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}(-4\omega_{1})\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr$\varepsilon$\\\
\hline\cr$\Delta^{\alpha_{2}}(-2\omega_{1}+\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr\end{tabular})\otimes(-4\omega_{1}))\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$-3\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-5\omega_{1}+\omega_{2})$\\\
\hline\cr$-4\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-6\omega_{1}+\omega_{2})$\\\
\hline\cr$-5\omega_{1}$\\\ \hline\cr\end{tabular}),$
one has
${\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\nabla^{\alpha_{2}}(-6\omega_{1}+\omega_{2})$\\\
\hline\cr$-5\omega_{1}$\\\
\hline\cr\end{tabular})\simeq{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{2}}(-6\omega_{1}+\omega_{2})$\\\
\hline\cr$-5\omega_{1}$\\\ \hline\cr\end{tabular})\in\hat{\mathcal{E}}.$
Let $\tilde{\mathcal{E}}$ be the triangulated subcategory of
$\mathrm{D}^{b}(\mathrm{coh}{\mathcal{B}})$ Karoubian generated by
${\mathcal{L}}_{\mathcal{B}}(-r\omega_{1})$, $r\in[0,4]$,
${\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{2}}(-s\omega_{1}+\omega_{2}))$
and ${\mathcal{L}}_{\mathcal{B}}(-s\omega_{1}+\omega_{2})$, $s\in[2,5]$,
${\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{2}}(-5\omega_{1}+2\omega_{2}))$,
${\mathcal{L}}_{\mathcal{B}}(-5\omega_{1}+2\omega_{2})$,
${\mathcal{L}}_{\mathcal{B}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\nabla^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular})$, and
${\mathcal{L}}_{\mathcal{B}}(\begin{tabular}[]{|c|}\hline\cr$\nabla^{\alpha_{2}}(-6\omega_{1}+\omega_{2})$\\\
\hline\cr$-5\omega_{1}$\\\ \hline\cr\end{tabular})$. By (1), (2) and (3) it
suffices to show that all
${\mathcal{L}}_{\mathcal{B}}(n\omega_{1})\in\tilde{\mathcal{E}}$,
$n\in\mathbb{Z}$. As all
${\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{2}}(-s\omega_{1}+\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{B}}(\begin{tabular}[]{|c|}\hline\cr$-s\omega_{1}+\omega_{2}$\\\
\hline\cr$(-s+3)\omega_{1}-\omega_{2}$\\\ \hline\cr\end{tabular})$ and
${\mathcal{L}}_{\mathcal{B}}(-s\omega_{1}+\omega_{2})$, $s\in[2,5]$, belong to
$\tilde{\mathcal{E}}$, one obtains
${\mathcal{L}}_{\mathcal{B}}((-s+3)\omega_{1}-\omega_{2})\in\tilde{\mathcal{E}}$.
Likewise, as
${\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{2}}(-5\omega_{1}+2\omega_{2}))\in\tilde{\mathcal{E}}$,
${\mathcal{L}}_{\mathcal{B}}(\omega_{1}-2\omega_{2})\in\tilde{\mathcal{E}}$.
Then, as
$\nabla(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{B}}(-\omega_{2})\in\tilde{\mathcal{E}}$,
${\mathcal{L}}_{\mathcal{B}}(2\omega_{1}-2\omega_{2})\in\tilde{\mathcal{E}}$.
Likewise, as
$\nabla(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{B}}(-\rho)\in\tilde{\mathcal{E}}$,
${\mathcal{L}}_{\mathcal{B}}(-2\omega_{2})\in\tilde{\mathcal{E}}$. As
$\nabla(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{B}}(-3\omega_{1}+\omega_{2})\in\tilde{\mathcal{E}}$,
${\mathcal{L}}_{\mathcal{B}}(-4\omega_{1}+2\omega_{2})\in\tilde{\mathcal{E}}$.
Then, as
$\nabla(\omega_{2})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{B}}(-\omega_{1})\in\tilde{\mathcal{E}}$,
${\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{2}}(-\omega_{1}+\omega_{2}))\in\tilde{\mathcal{E}}$.
Then, as
${\mathcal{L}}_{\mathcal{B}}(\begin{tabular}[]{|c|}\hline\cr$\omega_{1}$\\\
\hline\cr$\nabla^{\alpha_{2}}(-\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular})\in\tilde{\mathcal{E}}$ by definition,
${\mathcal{L}}_{\mathcal{B}}(\omega_{1})\in\tilde{\mathcal{E}}$. Likewise, as
$\nabla(\omega_{1})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{B}}(-4\omega_{1}+\omega_{2})\in\tilde{\mathcal{E}}$,
${\mathcal{L}}_{\mathcal{B}}(-6\omega_{1}+2\omega_{2})\in\tilde{\mathcal{E}}$.
Then, as
$\nabla(\omega_{2})\otimes_{\Bbbk}{\mathcal{L}}_{\mathcal{B}}(-3\omega_{1})\in\tilde{\mathcal{E}}$,
${\mathcal{L}}_{\mathcal{B}}(\nabla^{\alpha_{2}}(-6\omega_{1}+\omega_{2}))\in\tilde{\mathcal{E}}$.
Then, as
${\mathcal{L}}_{\mathcal{B}}(\begin{tabular}[]{|c|}\hline\cr$\nabla^{\alpha_{2}}(-6\omega_{1}+\omega_{2})$\\\
\hline\cr$-5\omega_{1}$\\\ \hline\cr\end{tabular})\in\tilde{\mathcal{E}}$ by
definition, ${\mathcal{L}}_{\mathcal{B}}(-5\omega_{1})\in\tilde{\mathcal{E}}$.
Thus ${\mathcal{L}}_{\mathcal{B}}(k\omega_{1})\in\tilde{\mathcal{E}}$ $\forall
k\in[-5,1]$. As $\dim\nabla(\omega_{1})=7$, one now obtains all
${\mathcal{L}}_{\mathcal{B}}(n\omega_{1})\in\tilde{\mathcal{E}}$,
$n\in\mathbb{Z}$, from the exact sequence
$0\to{\mathcal{L}}_{\mathcal{B}}(-7\omega_{1})\otimes_{\Bbbk}\wedge^{7}\nabla(\omega_{1})\to{\mathcal{L}}_{\mathcal{B}}(-6\omega_{1})\otimes_{\Bbbk}\wedge^{6}\nabla(\omega_{1})\to\\\
{\mathcal{L}}_{\mathcal{B}}(-5\omega_{1})\otimes_{\Bbbk}\wedge^{5}\nabla(\omega_{1})\to\dots\to{\mathcal{L}}_{\mathcal{B}}(-\omega_{1})\otimes_{\Bbbk}\wedge^{1}\nabla(\omega_{1})\to{\mathcal{O}}_{\mathcal{B}}\to
0.$
This finishes a verification in the case of $P_{\alpha_{2}}$ in type
$\mathrm{G}_{2}$.
The other cases are handled entirely similarly.
$3^{\circ}$ Extensions
In this section we will compute the extensions among our ${\mathcal{E}}_{w}$’s
given in (1.7) and (1.9) to verify (1.10).
The cases for $G=\mathrm{SL}_{2},\mathrm{SL}_{3},\mathrm{Sp}_{4}$ for $P$ a
Borel subgroup have been done in [HKR] and [KY] if $p\geq h$ the Coxeter
number of $G$; the new presentations in (1.7) somewhat ease the computations
in [KY], and moreover, adopting those ${\mathcal{E}}_{w}$’s, $w\in W$, we can
get rid of the restrictions on the characteristic.
Let us next explain the characteristic restrictions in type $\mathrm{G}_{2}$
stated in (1.11). Let $P=P_{\alpha_{1}}$. One has
$\displaystyle\mathrm{Ext}^{\bullet}_{\mathcal{P}}$
$\displaystyle({\mathcal{E}}_{s_{2}},{\mathcal{E}}_{s_{1}s_{2}})\simeq\mathrm{Ext}^{\bullet}_{\mathcal{P}}({\mathcal{L}}_{\mathcal{P}}(-\omega_{2}),{\mathcal{L}}_{\mathcal{P}}((-\omega_{2})\otimes\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{1}}(2\omega_{1}-\omega_{2})$\\\
\hline\cr$\Delta^{\alpha_{1}}(\omega_{1}-\omega_{2})$\\\
\hline\cr\end{tabular}))$
$\displaystyle\simeq\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{1}}(2\omega_{1}-\omega_{2})$\\\
\hline\cr$\nabla^{\alpha_{1}}(\omega_{1}-\omega_{2})$\\\
\hline\cr\end{tabular}))\simeq\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\Delta^{\alpha_{1}}(2\omega_{1}-\omega_{2})))$
$\displaystyle\simeq\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$L^{\alpha_{1}}(2\omega_{1}-\omega_{2})$\\\
\hline\cr$\varepsilon$\\\ \hline\cr\end{tabular}))\quad\text{as $p=2$}$
with $L^{\alpha_{1}}(2\omega_{1}-\omega_{2})$ simple $P_{\alpha_{1}}$-module
of highest weight $2\omega_{1}-\omega_{2}$. On the other hand, in
characteristic $2$
$0=\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\nabla^{\alpha_{1}}(2\omega_{1}-\omega_{2})))=\mathrm{H}^{\bullet}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\begin{tabular}[]{|c|}\hline\cr$\varepsilon$\\\
\hline\cr$L^{\alpha_{1}}(2\omega_{1}-\omega_{2})$\\\
\hline\cr\end{tabular})),$
and hence
$\mathrm{H}^{i}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(L^{\alpha_{1}}(2\omega_{1}-\omega_{2})))\simeq\mathrm{H}^{i-1}({\mathcal{P}},{\mathcal{L}}_{\mathcal{P}}(\varepsilon))\simeq\delta_{i-1,0}\Bbbk=\delta_{i1}\Bbbk$,
and (1.11.i) follows. Likewise (1.11.ii).
Now, to compute all
$\mathrm{Ext}^{\bullet}_{\mathcal{B}}({\mathcal{E}}_{x},{\mathcal{E}}_{y})$,
the case of $G$ in type $\mathrm{G}_{2}$ with $P=B$ is by far the hardest. As
$|W|=12$, there are 144 of them. Put ${\mathcal{B}}=G/B$. Let us exhibit the
computation of
$\mathrm{Ext}^{\bullet}_{\mathcal{B}}({\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}s_{1}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$,
which is most complicated among them. In view of the mal-behaviour in
characteristic $2$ and $3$ as noted above, and in order for irreducible
$G$-module $\nabla(\omega_{1})$ not to appear as a composition factor of
$\nabla(\rho)$, and for other reasons, we will assume $p\geq 7$. A basic idea
is to exploit multiple guises of the $B$-module structure defining the
${\mathcal{E}}_{w}$. For example, the $B$-module defining
${\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}s_{1}}$ is
$(-\rho)\otimes(\Delta(\omega_{2})/(-\omega_{2}))$ with
$\Delta(\omega_{2})/(-\omega_{2})$ having a filtration of $B$-modules
$\omega_{2}$ $(-\omega_{2})\otimes\Delta^{\alpha_{1}}(3\omega_{1})$
$(-\omega_{2})\otimes\Delta^{\alpha_{1}}(2\omega_{1})$ $\varepsilon$
$(-2\omega_{2})\otimes\Delta^{\alpha_{1}}(3\omega_{1})$ . But
$\Delta(\omega_{2})$ also admits a filtration by $P_{\alpha_{1}}$-modules
$\mathrm{Dist}(P_{\alpha_{1}})(\Delta(\omega_{2})_{-\alpha_{2}})<\mathrm{Dist}(P_{\alpha_{1}})(\Bbbk
v_{2}+\Delta(\omega_{2})_{-\alpha_{2}})<\mathrm{Dist}(P_{\alpha_{1}})(\Delta(\omega_{2})_{\leq\alpha_{1}})$
with $v_{2}$ as in (1.9) and where
$\Delta(\omega_{2})_{\leq\alpha_{1}}=\sum_{\nu\leq\alpha_{1}}\Delta(\omega_{2})_{\nu}$
with
$\mathrm{Dist}(P_{\alpha_{1}})(\Delta(\omega_{2})_{-\alpha_{2}})=\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{1}}(3\omega_{1}-2\omega_{2})$\\\
\hline\cr$-\omega_{2}$\\\ \hline\cr\end{tabular}$. Moreover,
$\Delta(\omega_{2})/\mathrm{Dist}(P_{\alpha_{1}})(\Delta(\omega_{2})_{-\alpha_{2}})=\begin{tabular}[]{|c|}\hline\cr$\omega_{1}\otimes\ker(\Delta(\omega_{1})\twoheadrightarrow\omega_{1})$\\\
\hline\cr$\Delta^{\alpha_{2}}(\omega_{2})\otimes(-3\omega_{1}+\omega_{2})$\\\
\hline\cr$-2\omega_{1}+\omega_{2}$\\\ \hline\cr\end{tabular}$. Thus
(1)
$\mathrm{Ext}_{\mathcal{B}}^{\bullet}({\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}s_{1}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})\simeq\mathrm{Ext}_{\mathcal{B}}^{\bullet}(\begin{tabular}[]{|c|}\hline\cr${\mathcal{E}}_{s_{1}s_{2}}$\\\
\hline\cr${\mathcal{L}}(\Delta^{\alpha_{2}}(\omega_{2})\otimes(-4\omega_{1}))$\\\
\hline\cr${\mathcal{L}}(-3\omega_{1})$\\\
\hline\cr${\mathcal{L}}(\Delta^{\alpha_{1}}(3\omega_{1}-2\omega_{2})\otimes(-\rho))$\\\
\hline\cr\end{tabular},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}}).$
By definition
${\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}}={\mathcal{L}}_{\mathcal{B}}((-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(\Bbbk(v_{4}+v_{3})+\Delta(\omega_{2})_{-3\omega_{1}+\omega_{2}})\\})$.
We note under the assumption $p\geq 7$ that
$\mathrm{Dist}(P_{\alpha_{2}})(\Bbbk(v_{4}+v_{3})+\Delta(\omega_{2})_{-3\omega_{1}+\omega_{2}})=\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})$,
which admits a $P_{\alpha_{2}}$-filtration
$(-2\omega_{1})\otimes\Delta^{\alpha_{2}}(\omega_{2})$ $-\omega_{1}$
$(-3\omega_{1})\otimes\Delta^{\alpha_{2}}(\omega_{2})$ and also a
$B$-filtration $-2\omega_{1}+\omega_{2}$
$(-\rho)\otimes\Delta^{\alpha_{1}}(2\omega_{1})$ $-\omega_{2}$ .
We claim that
$\mathrm{Ext}_{\mathcal{B}}^{\bullet}({\mathcal{E}}_{s_{1}s_{2}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$,
$\mathrm{Ext}_{\mathcal{B}}^{\bullet}({\mathcal{L}}(\Delta^{\alpha_{2}}(\omega_{2})\otimes(-4\omega_{1})),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$
and
$\mathrm{Ext}_{\mathcal{B}}^{\bullet}({\mathcal{L}}(-3\omega_{1}),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}}))$
all vanish. First,
$\mathrm{Ext}_{\mathcal{B}}^{\bullet}({\mathcal{E}}_{s_{1}s_{2}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})\simeq\mathrm{Ext}_{\mathcal{B}}^{\bullet}({\mathcal{L}}((-\omega_{2})\otimes\ker(\Delta(\omega_{1})\twoheadrightarrow\omega_{1})),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$
with $\forall i\in\mathbb{N}$,
$\displaystyle\mathrm{Ext}_{\mathcal{B}}^{i}$
$\displaystyle({\mathcal{L}}(-\omega_{2}+\omega_{1}),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}((\omega_{2}-\omega_{1}-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}((-2\omega_{1})\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}((-2\omega_{1})\otimes\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})))$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}((-2\omega_{1})\otimes\begin{tabular}[]{|c|}\hline\cr$(-2\omega_{1})\otimes\Delta^{\alpha_{2}}(\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr$(-3\omega_{1})\otimes\Delta^{\alpha_{2}}(\omega_{2})$\\\
\hline\cr\end{tabular}))\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}(\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{2}}(-4\omega_{1}+\omega_{2})$\\\
\hline\cr$-3\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-5\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular}))$ $\displaystyle=0\quad\text{as $p\geq 5$}$
while
$\mathrm{Ext}_{\mathcal{B}}^{i}({\mathcal{L}}((-\omega_{2})\otimes\Delta(\omega_{1})),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})\simeq\mathrm{Ext}_{\mathcal{B}}^{i}({\mathcal{E}}_{s_{2}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})\otimes\Delta(\omega_{1})^{*}$
with
$\displaystyle\mathrm{Ext}_{\mathcal{B}}^{i}$
$\displaystyle({\mathcal{E}}_{s_{2}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\mathrm{Ext}_{\mathcal{B}}^{i}({\mathcal{L}}(-\omega_{2}),{\mathcal{L}}((-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}((-\omega_{1})\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}((-\omega_{1})\otimes\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})))$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}((-\omega_{1})\otimes\begin{tabular}[]{|c|}\hline\cr$(-2\omega_{1})\otimes\Delta^{\alpha_{2}}(\omega_{2})$\\\
\hline\cr$-\omega_{1}$\\\
\hline\cr$(-3\omega_{1})\otimes\Delta^{\alpha_{2}}(\omega_{2})$\\\
\hline\cr\end{tabular}))\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}(\begin{tabular}[]{|c|}\hline\cr$\Delta^{\alpha_{2}}(-3\omega_{1}+\omega_{2})$\\\
\hline\cr$-2\omega_{1}$\\\
\hline\cr$\Delta^{\alpha_{2}}(-4\omega_{1}+\omega_{2})$\\\
\hline\cr\end{tabular}))$ $\displaystyle=0\quad\text{by
\cite[cite]{[\@@bibref{}{J}{}{}, II.6.18]}}.$
and hence
$\mathrm{Ext}_{\mathcal{B}}^{\bullet}({\mathcal{E}}_{s_{1}s_{2}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})=0$.
Next,
$\displaystyle\mathrm{Ext}^{\bullet}_{\mathcal{B}}$
$\displaystyle({\mathcal{L}}(\Delta^{\alpha_{2}}(\omega_{2})\otimes(-4\omega_{1})),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})\simeq\mathrm{Ext}^{\bullet}_{\mathcal{B}}({\mathcal{L}}(\Delta^{\alpha_{2}}(\omega_{2})\otimes(-4\omega_{1})),$
$\displaystyle\hskip
56.9055pt{\mathcal{L}}((-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{\bullet}({\mathcal{B}},{\mathcal{L}}((-\omega_{2})\otimes
3\omega_{1}\otimes\nabla^{\alpha_{2}}(-3\omega_{1}+\omega_{2})\otimes$
$\displaystyle\hskip
56.9055pt\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))=0$
as
$3\omega_{1}\otimes\nabla^{\alpha_{2}}(-3\omega_{1}+\omega_{2})\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}$
is equipped with a structure of $P_{\alpha_{2}}$-module. Likewise,
$\displaystyle\mathrm{Ext}_{\mathcal{B}}^{i}$
$\displaystyle({\mathcal{L}}(-3\omega_{1}),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\mathrm{Ext}_{\mathcal{B}}^{i}({\mathcal{L}}(-3\omega_{1}),{\mathcal{L}}((-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}((-\omega_{2})\otimes
2\omega_{1}\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))=0.$
It now follows $\forall i\in\mathbb{N}$ that
$\displaystyle\mathrm{Ext}_{\mathcal{B}}^{i}$
$\displaystyle({\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}s_{1}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})\simeq\mathrm{Ext}_{\mathcal{B}}^{i}({\mathcal{L}}(\Delta^{\alpha_{1}}(3\omega_{1}-2\omega_{2})\otimes(-\rho)),{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\mathrm{Ext}_{\mathcal{B}}^{i}({\mathcal{L}}(\Delta^{\alpha_{1}}(3\omega_{1}-2\omega_{2})\otimes(-\rho)),$
$\displaystyle\hskip
56.9055pt{\mathcal{L}}((-\rho)\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1}-\omega_{2})\otimes\\{(\Delta(\omega_{2})\oplus\Delta(\omega_{1}))/\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})\\}))$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1}-\omega_{2})\otimes\mathrm{Dist}(P_{\alpha_{2}})(v_{4}+v_{3})))$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1}-\omega_{2})\otimes\begin{tabular}[]{|c|}\hline\cr$-2\omega_{1}+\omega_{2}$\\\
\hline\cr$(-\rho)\otimes\Delta^{\alpha_{1}}(2\omega_{1})$\\\
\hline\cr$-\omega_{2}$\\\ \hline\cr\end{tabular}))$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}(\begin{tabular}[]{|c|}\hline\cr$\nabla^{\alpha_{1}}(3\omega_{1})\otimes(-2\omega_{1})$\\\
\hline\cr$\nabla^{\alpha_{1}}(3\omega_{1}-\omega_{2})\otimes(-\omega_{1})\otimes\nabla^{\alpha_{1}}(2\omega_{1}-\omega_{2})$\\\
\hline\cr$\nabla^{\alpha_{1}}(3\omega_{1}-2\omega_{2})$\\\
\hline\cr\end{tabular}))$
with
$\displaystyle\mathrm{H}^{i+1}({\mathcal{B}},$
$\displaystyle{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1})\otimes(-2\omega_{1})))\simeq\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1})\otimes(-\omega_{2})))\quad\text{by
the Serre duality }$
$\displaystyle\simeq\mathrm{H}^{i}({\mathcal{B}},{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1}-\omega_{2})))$
$\displaystyle=0=\mathrm{H}^{\bullet}({\mathcal{B}},{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1}-\omega_{2})\otimes(-\omega_{1})\otimes\nabla^{\alpha_{1}}(2\omega_{1}-\omega_{2}))).$
One thus obtains that
$\displaystyle\mathrm{Ext}_{\mathcal{B}}^{i}({\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}s_{1}},{\mathcal{E}}_{s_{1}s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\mathrm{H}^{i+1}({\mathcal{B}},{\mathcal{L}}(\nabla^{\alpha_{1}}(3\omega_{1}-2\omega_{2})))$
$\displaystyle\simeq\delta_{i+1,1}\Bbbk=\delta_{i0}\Bbbk\quad\text{by
\cite[cite]{[\@@bibref{}{J}{}{}, II.6.18]}}.$
$4^{\circ}$ Kapranov’s sheaves
In [K08]/[KNS] we showed that Kapranov’s sheaves on the Grassmannians [Kap83]
constitute a tilting sheaf in positive characteristic if the characteristic is
large enough, and their parametrization by $W^{P}$ verifies Catanese’s
conjecture. In this section We will briefly discuss Kapranov’s sheaves on the
flag variety of $\mathrm{GL}_{3}$ and on the quadrics for future study.
(5.1) Let us first consider the flag variety ${\mathcal{B}}=\mathrm{GL}(E)/B$
with $E$ of dimension 3. If $p\geq 3$, as well as in characteristic 0,
Kapranov’s sheaves
${\mathcal{E}}_{e}={\mathcal{L}}_{\mathcal{B}}(\nabla^{{\alpha_{2}}}(-\omega_{1})\otimes(-\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{B}}(-\rho)$,
${\mathcal{E}}_{s_{1}}={\mathcal{L}}_{\mathcal{B}}(\nabla^{{\alpha_{2}}}(-\omega_{1}+\omega_{2})\otimes(-\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{B}}(\nabla^{{\alpha_{2}}}(\omega_{2})\otimes(-\rho))$,
${\mathcal{E}}_{s_{2}s_{1}}={\mathcal{L}}_{\mathcal{B}}(-\omega_{2})$,
${\mathcal{E}}_{s_{2}}={\mathcal{L}}_{\mathcal{B}}(\nabla^{{\alpha_{2}}}(-\omega_{1}))\simeq{\mathcal{L}}_{\mathcal{B}}(-\omega_{1})$,
${\mathcal{E}}_{s_{1}s_{2}}={\mathcal{L}}_{\mathcal{B}}(\nabla^{{\alpha_{2}}}(-\omega_{1}+\omega_{2}))\simeq{\mathcal{L}}_{\mathcal{B}}(\nabla^{{\alpha_{2}}}(\omega_{2})\otimes(-\omega_{1}))$,
and ${\mathcal{E}}_{s_{2}s_{1}s_{2}}={\mathcal{O}}_{\mathcal{B}}$ from
[Kap88]/[Bö, 2.2.3] form a complete strongly exceptional sequence on the flag
variety. The nonvanishing
$\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{x},{\mathcal{E}}_{y})$, $x\neq y$,
are, however, given by
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{e},{\mathcal{E}}_{s_{1}})$
$\displaystyle\simeq\nabla(\omega_{2}),\quad$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{e},{\mathcal{E}}_{s_{2}})$
$\displaystyle\simeq\nabla(\omega_{2}),$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{e},{\mathcal{E}}_{s_{1}s_{2}})$
$\displaystyle\simeq\begin{tabular}[]{|c|}\hline\cr$\nabla(2\omega_{2})$\\\
\hline\cr$\nabla(\omega_{1})$\\\ \hline\cr\end{tabular},\quad$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{e},{\mathcal{E}}_{s_{2}s_{1}})$
$\displaystyle\simeq\nabla(\omega_{1}),$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{e},{\mathcal{E}}_{s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\nabla(\rho),\quad$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{1}},{\mathcal{E}}_{s_{2}})$
$\displaystyle\simeq\Bbbk,$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{1}},{\mathcal{E}}_{s_{2}s_{1}})$
$\displaystyle\simeq\nabla(\omega_{2}),\quad$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{1}},{\mathcal{E}}_{s_{1}s_{2}})$
$\displaystyle\simeq\nabla(\omega_{2})^{\oplus 2},$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{1}},{\mathcal{E}}_{s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\begin{tabular}[]{|c|}\hline\cr$\nabla(2\omega_{2})$\\\
\hline\cr$\nabla(\omega_{1})$\\\ \hline\cr\end{tabular},\qquad$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{2}},{\mathcal{E}}_{s_{1}s_{2}})$
$\displaystyle\simeq\nabla(\omega_{2}),$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{2}},{\mathcal{E}}_{s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\nabla(\omega_{1}),\quad$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{1}s_{2}},{\mathcal{E}}_{s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\nabla(\omega_{2}),$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{2}s_{1}},{\mathcal{E}}_{s_{1}s_{2}})$
$\displaystyle\simeq\Bbbk,\quad$
$\displaystyle\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{s_{2}s_{1}},{\mathcal{E}}_{s_{2}s_{1}s_{2}})$
$\displaystyle\simeq\nabla(\omega_{2}).$
In particular, $|\\{w\in
W\setminus\\{s_{1}s_{2}\\}\mid\mathbf{Mod}_{\mathcal{B}}({\mathcal{E}}_{w},{\mathcal{E}}_{s_{1}s_{2}})\neq
0\\}|=4$, and hence there is no reindexing of these ${\mathcal{E}}_{w}$ by $W$
such that the Catanese conjecture hold, contrary to our construction (1.8),
(1.10).
(5.2) Let us next consider the quadric ${\mathcal{Q}}={\mathcal{Q}}_{n}$ of
dimension $n\geq 3$ in odd characteristic. Let $E$ be a $\Bbbk$-linear space
of dimension $n+2$. In case $n$ is odd, write $n=2m+1$. Let
$e_{1},\dots,e_{m+1},e_{0},e_{-m-1},\dots,e_{-1}$ be a basis of $E$ and define
a quadratic form $q$ on $E$ by
$q(\sum_{k=-m-1}^{m+1}x_{k}e_{k})=x_{1}x_{-1}+\dots+x_{m+1}x_{-m-1}+x_{0}^{2}$.
If $n$ is even, write $n=2m$. Let $e_{1},\dots,e_{m+1},e_{-m-1},\dots,e_{-1}$
be a basis of $E$ equipped with quadratic form
$q(\sum_{k=1}^{m+1}(x_{k}e_{k}+x_{-k}e_{-k})=x_{1}x_{-1}+\dots+x_{m+1}x_{-m-1}$.
In either case let $G=\mathrm{SO}(E)$, $P=\mathrm{N}_{G}(\Bbbk e_{-1})$. Then
a closed immersion $i:G/P\to\mathbb{P}(E)$ via $gP\mapsto[ge_{-1}]$ identifies
$G/P$ with ${\mathcal{Q}}$.
If $n=2m+1$,
$T=\\{\mathrm{diag}(\zeta_{1},\dots,\zeta_{m+1},1,\zeta_{m+1}^{-1},\dots,\zeta_{1}^{-1})\mid\zeta_{i}\in\Bbbk^{\times}\
\forall i\\}$ forms a maximal torus of $G$. Take as simple roots
$\alpha_{1}=\varepsilon_{1}-\varepsilon_{2},\dots,\alpha_{m}=\varepsilon_{m}-\varepsilon_{m+1},\alpha_{m+1}=\varepsilon_{m+1}$
with
$\varepsilon_{k}:\mathrm{diag}(\zeta_{1},\dots,\zeta_{m+1},\zeta_{0},\zeta_{-m-1},\dots,\zeta_{-1})\mapsto\zeta_{k}$.
Then the fundamental weights are given by
$\omega_{1}=\varepsilon_{1},\omega_{2}=\varepsilon_{1}+\varepsilon_{2},\dots,\omega_{m}=\varepsilon_{1}+\dots+\varepsilon_{m},\omega_{m+1}=\frac{1}{2}(\varepsilon_{1}+\dots+\varepsilon_{m+1})$,
and $W^{P}=\\{e,s_{1},s_{2}s_{1},\dots,s_{m+1}s_{m}\dots
s_{2}s_{1},s_{m}s_{m+1}s_{m}\dots s_{2}s_{1},s_{m-1}s_{m}s_{m+1}s_{m}\dots
s_{2}s_{1},\dots,\linebreak s_{2}\dots s_{m-1}s_{m}s_{m+1}s_{m}\dots
s_{2}s_{1},s_{1}s_{2}\dots s_{m-1}s_{m}s_{m+1}s_{m}\dots s_{2}s_{1}\\}$.
Define ${\mathcal{E}}_{e}={\mathcal{O}}_{\mathcal{Q}}$,
${\mathcal{E}}_{s_{1}}=\linebreak{\mathcal{O}}_{\mathcal{Q}}(-1)$,
${\mathcal{E}}_{s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-2)$, …,
${\mathcal{E}}_{s_{m}\dots s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-m)$,
${\mathcal{E}}_{s_{m+1}s_{m}\dots
s_{2}s_{1}}=\linebreak{\mathcal{L}}(\nabla^{P}(\omega_{m+1}))(-m-1)$,
${\mathcal{E}}_{s_{m}s_{m+1}s_{m}\dots
s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-m-1)$, …, ${\mathcal{E}}_{s_{2}\dots
s_{m}s_{m+1}s_{m}\dots
s_{2}s_{1}}=\linebreak{\mathcal{O}}_{\mathcal{Q}}(-n+2)$, and
${\mathcal{E}}_{s_{1}\dots s_{m}s_{m+1}s_{m}\dots
s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-n+1)$.
If $n=2m$,
$T=\\{\mathrm{diag}(\zeta_{1},\dots,\zeta_{m+1},\zeta_{m+1}^{-1},\dots,\zeta_{1}^{-1})\mid\zeta_{i}\in\Bbbk^{\times}\
\forall i\\}$ forms a maximal torus of $G$. Take as simple roots
$\alpha_{1}=\varepsilon_{1}-\varepsilon_{2},\dots,\alpha_{m-1}=\varepsilon_{m-1}-\varepsilon_{m},\alpha_{m}=\varepsilon_{m}-\varepsilon_{m+1},\alpha_{m+1}=\varepsilon_{m}+\varepsilon_{m+1}$
with
$\varepsilon_{k}:\mathrm{diag}(\zeta_{1},\dots,\zeta_{m+1},\zeta_{-m-1},\dots,\zeta_{-1})\mapsto\zeta_{k}$.
Then the fundamental weights are
$\omega_{1}=\varepsilon_{1},\omega_{2}=\varepsilon_{1}+\varepsilon_{2},\dots,\omega_{m-1}=\varepsilon_{1}+\dots+\varepsilon_{m-1},\omega_{m}=\frac{1}{2}(\varepsilon_{1}+\dots+\varepsilon_{m}-\varepsilon_{m+1}),\omega_{m+1}=\frac{1}{2}(\varepsilon_{1}+\dots+\varepsilon_{m}+\varepsilon_{m+1})$,
and $W^{P}=\\{e,s_{1},s_{2}s_{1},\dots,s_{m}s_{m-1}\dots
s_{1},s_{m+1}s_{m-1}\dots s_{2}s_{1},\linebreak s_{m}s_{m+1}s_{m-1}\dots
s_{1},s_{m-1}s_{m}s_{m+1}s_{m-1}\dots
s_{1},s_{m-2}s_{m-1}s_{m}s_{m+1}s_{m-1}\dots s_{1},\dots,\linebreak
s_{1}s_{2}\dots s_{m-1}s_{m}s_{m+1}s_{m-1}\dots s_{1}\\}$. Define
${\mathcal{E}}_{e}={\mathcal{O}}_{\mathcal{Q}}$,
${\mathcal{E}}_{s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-1)$,
${\mathcal{E}}_{s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-2)$, …,
${\mathcal{E}}_{s_{m-1}\dots s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-(m-1))$,
${\mathcal{E}}_{s_{m}\dots
s_{2}s_{1}}={\mathcal{L}}_{\mathcal{Q}}(\nabla^{P}(\omega_{m+1}))(-m)$,
${\mathcal{E}}_{s_{m+1}s_{m-1}\dots
s_{2}s_{1}}=\linebreak{\mathcal{L}}(\nabla^{P}(\omega_{m}))(-m)$,
${\mathcal{E}}_{s_{m}s_{m+1}s_{m-1}\dots
s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-m)$, …, ${\mathcal{E}}_{s_{2}\dots
s_{m}s_{m+1}s_{m-1}\dots s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-n+2)$, and
${\mathcal{E}}_{s_{1}\dots s_{m}s_{m+1}s_{m}\dots
s_{2}s_{1}}={\mathcal{O}}_{\mathcal{Q}}(-n+1)$.
In either case Langer [La] shows that the ${\mathcal{E}}_{w}$, $w\in W^{P}$,
verify Catanese’s conjecture, and that for $p\geq n+1$ the Coxeter number of
$G$ all ${\mathcal{E}}_{w}$’s appear as direct summands of
$F_{*}{\mathcal{O}}_{\mathcal{Q}}$. More precisely, let
$A=\mathrm{S}_{\Bbbk}(E^{*})/(q)$ and put $\bar{A}=A/(a^{p}\mid a\in E^{*})$.
Let $\bar{A}_{j}$ be the $j$-th homogeneous part of $\bar{A}$. If $n=2m+1$,
$F_{*}{\mathcal{O}}_{\mathcal{Q}}\simeq\\{\coprod_{i\in[0,n[\setminus\\{m+1\\}}{\mathcal{O}}_{Q}(-i)\otimes_{\Bbbk}\bar{A}_{ip}\\}\oplus\\{{\mathcal{O}}_{\mathcal{Q}}(-m-1)\otimes_{\Bbbk}\bar{A}_{mp-n}\\}\oplus\\\
{\mathcal{L}}(\nabla^{P}(\omega_{m+1}))^{\oplus r}(-m-1)$
with $r=\frac{\dim\bar{A}_{(m+1)p}-\dim\bar{A}_{mp-n}}{\dim L(\omega_{m+1})}$,
while for $n=2m$
$F_{*}{\mathcal{O}}_{\mathcal{Q}}\simeq\\{\coprod_{i\in[0,n[\setminus\\{m\\}}{\mathcal{O}}_{Q}(-i)\otimes_{\Bbbk}\bar{A}_{ip}\\}\oplus\\{{\mathcal{O}}_{\mathcal{Q}}(-m)\otimes_{\Bbbk}\bar{A}_{mp-n}\\}\oplus\\\
\\{{\mathcal{L}}(\nabla^{P}(\omega_{m})\oplus\nabla^{P}(\omega_{m+1}))\\}^{\oplus
s}(-m)$
with $s=\frac{\dim\bar{A}_{mp}-\dim\bar{A}_{mp-n}}{\dim\\{L(\omega_{m})\oplus
L(\omega_{m+1})\\}}$. Note that our parametrization of the ${\mathcal{E}}_{w}$
is different from that of Böhning’s, but is consistent with (1.10) in case
$n=3$.
## References
* [A83] Andersen, H.H., Filtrations of cohomology modules for Chevalley groups, Ann. Sci. ENS. 16 (1983), 495–528
* [A87] Andersen, H.H., Jantzen’s Filtrations of Weyl Modules, Math. Z. 194 (1987), 127–142
* [AJS] Andersen, H.H., Jantzen, J.C. and Soergel, W., Representations of quantum groups at a $p$-th root of unity and of semisimple groups in characteristic $p$ : independence of $p$, Astérisque 220, 1994 (SMF)
* [AK] Andersen, H.H. and Kaneda M., Loewy series of modules for the first Frobenius kernel in a reductive algebraic group, Proc. LMS (3) 59 (1989), 74–98
* [AK00] Andersen, H.H. and Kaneda M., On the $D$-affinity of the flag variety in type $B_{2}$, Manuscripta Math. 103 (2000), no. 3, 393–399
* [Ba] Baer, D., Tilting sheaves in representation theory of algebras, Manus. Math. 60 (1988), 323-347
* [Bei] Beilinson, A. A., Coherent sheaves on $\mathbb{P}^{n}$ and problems of linear algebra, Func. Anal. Appl. 12 (1979), 214-216.
* [BMR] Bezrukavnikov, R.,Mirkovic, I.and Rumynin, D.,Localzation of modules fora semisimple Lie algebra in prime characteristic, to appear in Ann. Math..
* [Bö] Böhning, C., Derived categories of coherent sheaves on rational homogeneous manifolds, Doc. Math. 11 (2006), 261-331
* [F1] Fiebig, P., Sheaves on affine Schubert varieties, modular representations and Lusztig’s conjecture, arXive0711:0871v2
* [F2] Fiebig, P., Lusztig’s conjecture as a moment graph problem, arXive0712:3909
* [Haa] Haastert, B., Über Differentialoperatoren und $\mathbb{D}$-Moduln in positiver Charakteristik, Manusc. Math. 58 (1987), 385–415
* [H] Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York 1977
* [HKR] Hashimoto Y., Kaneda M. and Rumynin, D., On localization of $\bar{D}$-modules, in “Representations of Algebraic Groups, Quantum Groups, and Lie Algebras,” Contemp. Math. 413 (2006), 43-62
* [Hum] Humphreys, J. E., Representations of Semisimple Lie Algebras in the BGG Category ${\mathcal{O}}$, GSM 94, AMS 2008
* [J] Jantzen, J. C., Representations of Algebraic Groups, 2003 (AMS)
* [KY] Kaneda M. and Ye J.-C., Equivariant localization of $\bar{D}$-modules on the flag variety of the symplectic group of degree $4$ , J. Alg. 309 (2007), 236–281
* [K08] Kaneda M., Kapranov’s tilting sheaf on the Grassmannian in positive characteristic, Alg. Repr. Th. 11 (2008), 347-354
* [K09] Kaneda M., The structure of Humphreys-Verma modules for projective spaces, J. Alg. 322 (2009), 237-244
* [KNS] Kaneda M., Naito S. and Sagaki D., Kapranov’s tilting sheaf on the Grassmannian revisited, pp.188-197 in Proceedings of the 10-th Meeting of the Representation Theory of Algebraic and Quantum Groups 2007
* [Kap83] Kapranov, M. M., The derived category of coherent sheaves on Grassmannians, Functional Anal. i Prilozhen, 17 (1983), 78-79
* [Kap88] Kapranov, M. M., On the derived category of coherent sheaves on some homogeneous spaces, Inv. Math., 92 (1988), 479-508
* [La] Langer, A., D-affinity and Frobenius morphism on quadrics, IMRN (2008), rnm 145
* [L] Lusztig, G., Hecke algebras and Jantzen’s generic decomposition patterns, Adv. Math. 37 (1980), 121-164
* [S07] Samokhin, A., On the D-affinity of quadrics in positive characteristic, C. R. Acad. Sci. Paris, Ser. I 344 (2007), 377-382
* [S1] Samokhin, A., On the D-affinity of flag varieties in positive characteristic, arXiv:0906.1555v2
* [S2] Samokhin, A., Tilting bundles via the Frobenius morphism, arXiv:0904.1235v2
* [S3] Samokhin, A., A vanishing theorem for small differential operators in positive characteristic, arXiv:0707.0913v2
* [Y] Ye J.-c., Filtrations of principal indecomposable modules of Frobenius kernels of reductive groups, Math. Z. 189 (1985), 515-527
|
arxiv-papers
| 2009-11-13T12:26:03 |
2024-09-04T02:49:06.436467
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Masaharu Kaneda, Jiachen Ye",
"submitter": "Ye Jiachen",
"url": "https://arxiv.org/abs/0911.2568"
}
|
0911.2619
|
# Active Flows in Diagnostic of Troubleshooting on Backbone Links
Andrei M. Sukhov
Dmitry I. Sidelnikov
Alexey Galtsev
Samara State Aerospace University Moskovskoe sh., 34 Samara, 443086, Russia
amskh@yandex.ru Institute of Organic Chemistry of RAS Leninsky pros., 47
Moscow, 119991, Russia sid@free.net Samara State Aerospace University
Moskovskoe sh., 34 Samara, 443086, Russia galaleksey@gmail.com Alexey P.
Platonov
Mikhail V. Strizhov
Russian Institute for Public Networks Kurchatova sq. 1 Moscow, 123182,
Russia plat@ripn.net Samara State Aerospace University Moskovskoe sh., 34
Samara, 443086, Russia strizhov@ip4tv.ru
(10 November 2009)
###### Abstract
This paper aims to identify the operational region of a link in terms of its
utilization and alert operators at the point where the link becomes overloaded
and requires a capacity upgrade. The number of active flows is considered the
real network state and is proposed to use a proxy for utilization. The
Gaussian approximation gives the expression for the confidence interval on an
operational region. The easy rule has been formulated to display the network
defects by means of measurements of router loading and number of active flows.
Mean flow performance is considered as the basic universal index characterized
quality of network services provided to single user.
###### category:
C.2.3 Computer-communication networks Network Operations
###### keywords:
network monitoring
###### category:
C.4 Performance of systems Measurement techniques
###### keywords:
Network States on Flow Level, Mean Flow Performance, Test for Network Quality
††titlenote: The research is supported partially by grant N06-07-89074 of
RFBR††titlenote: corresponding author
## 1 Introduction
Typically, the following four values are used for the estimation of the
network quality:
* •
Link or router loading (or available bandwidth for end-to-end connection)
* •
Round trip time
* •
Packet loss rate
* •
IP packet delay variation (network jitter)
The available bandwidth, round trip time, packet loss rate and IP packet delay
variation describe the quality of connectivity between two remote points or
end-to-end connection. The link utilization is applied to the monitoring of a
single hop between two routers [9, 16].
Network operators need to know when their backbone or peering links must be
upgraded. Boundary values of network parameters may serve as an indicator e.g.
as the current values of the network parameters reach a defined limit, the
links have to be upgraded. The problem with this method is that there is no
standardized set of network parameters to monitor. Each provider has its own
set of technical specifications aimed at avoiding overload. Big providers,
like Sprint [9], rely on the results of their own research. Usually, network
operators monitor peak and average link utilization levels and upgrade their
links when the utilization level is in the range 30%-60%.
The main focus of this paper is to use flow-based analysis [5] to monitor the
backbone link and identify when an upgrade is needed. Previous work by Chuck
Fraleigh et al [9] addressed a similar provisioning problem to reduce the per
packet end-to-end delay. Dina Papagiannaki et al [16], at Infocom 03,
introduced a methodology on the basis of SNMP statistics to predict when and
where link additions/upgrades should take place in an IP backbone network.
For resource management in IP networks, several recent proposals [6] advocate
for a flow oriented architecture. A flow is classically identified by the
usual 5tuple composed of the source and destination addresses together with
the source and destination port numbers and protocol type. The basic
principles of flow-aware networking rely on the fact that flows are the
elementary entities associated with user transactions. In order to provide end
users with an acceptable quality, it is essential to share the bandwidth of
the network by taking into account flows [4, 11, 13, 15]. To efficiently
implement flow aware resource management techniques, however, it is essential
to estimate the number of flows simultaneously active on a link in an
operational network [2, 8].
Traffic accounting mechanisms based on flows should be considered as passive
measurement mechanisms. Information is gathered by flows are useful for many
purposes:
* •
Understanding the behaviour of existing networks
* •
Planning for network development and expansion
* •
Quantifying network performance
* •
Verifying the quality of network service
* •
Attribution of network usage to users
Unfortunately, today there is no common view on how to estimate the connection
quality, and how to find bottlenecks in networks.
Barakat et al [3] propose a model that relies on flow-level information to
compute the total (aggregate) rate of data observed on an IP backbone link.
For modelling purposes, the traffic is viewed as the superposition (i.e.
multiplexing) of a large number of flows that arrive at random times and that
stay active for random periods.
Our paper presents a technique for estimating the network behaviour based on
the utilization curve which is representing correlation between link
utilization and the number of active flows in it. We implicitly argue that the
number of active flows may be considered as the real network state [2, 17] and
is consequently, a better indication of utilization or desired operating
point.
Our objective is to gather knowledge and plot the curve for network quality
and related terms for utilization such as: length of operational region, mean
flow performance, confidence interval, points of overload, etc. Traffic
measurement and analysis is extended to consider the network quality for a
large network or for a high-speed backbone [10, 14]. This will help discover
possible bottlenecks on external links of enterprize network or backbones [9].
We locate the threshold point at which the addition of new flows does not
increase the link utilization.
In order to prove our hypothesis we took measurements from the border gateway
routers of Russian Internet Service Provider FREEnet. FREEnet (The Network For
Research, Education and Engineering), an academic and research network, was
founded on July 20, 1991, by the N.D.Zelinsky Institute of Organic Chemistry
(IOC RAS) at the Center of Computer Assistance for Chemical Research (CACR).
It assists and fulfills the networking needs of the research institutes of
Russian Academy of Sciences, universities, colleges, and other research and
academic institutions of Russia. FREEnet provides IPv4/IPv6 connectivity,
worldwide multicast IP connectivity, DNS services, stratum 1 time service,
mail relaying, collocation and hosting services, assorted information
services.
Total capacity of upstream links is presently 2.1 Gbps, 1.3 Gbps links
interconnect FREEnet with its peers (excluding FREEnet members). There are 6
regional branches that operate independently but they are cooperating as peers
in accordance with FREEnet Charter principles. Tens of thousands researchers
from hundreds institutions countrywide enjoy FREEnet services.
Earlier, in order to prove our hypothesis we took measurements from the border
gateway routers of Russian ISP “SamaraTelecom” (ST) and from HEAnet -
Ireland’s National Research and Education Network.
In the course of experiment average flow performance in investigated networks
considerably differs. So flow performance in provincial Russian networks more
then one order less similar indicator HEAnet. These results are completely not
surprising; essentially they reflect quality of network services. At once
there was an idea to apply the model found us for the comparative analysis of
large networks or date-centers. Thus it is possible to use unique parameter -
mean flow performance.
This paper describes an apllication of simple flow-based model [2, 3] to
identify the quality of the connection and the instance when a backbone
upgrade is required. We present our findings under the following headings:
* •
Section 2 \- using queueing theory for flow-based analysis of a backbone
* •
Section 3 discusses three states of network on flow level
* •
Section 4 \- a test for network quality
* •
Section 5 \- results from experiments conducted in the real ISP’s
* •
Section 6 describes techniques of comparison of a communication quality in the
large networks.
## 2 The Flow Based Model
In this paper we present traffic as a stationary process, using the results
from the papers of Barakat et al [3] and Ben Fredj et al [4]. They proposed a
traffic model for uncongested backbone links that is simple enough to be used
in network operations and engineering. The model of Barakat et al relies on
Poisson shot-noise. With only 3 parameters ($\lambda$, arrival rate of flows,
${\mathbb{E}}[S_{n}]$, average size of a flow, and
${\mathbb{E}}[S^{2}_{n}/D_{n}]$, average value for the ratio of the square of
a flow size and its duration), the model provides approximations for the
average of the total rate (the throughput) on a backbone link and for its
variations at short timescales. The model is designed to be general so that it
can be easily used without any constraints from the definition of flows, or on
the application or the transport protocol. In summary, this model allows us to
completely characterize the data rate on a backbone link based on the
following inputs:
* •
Session arrivals in any period where the traffic intensity is approximately
constant are accurately modelled by a homogeneous Poisson process of finite
rate $\lambda$. The measurements of Barakat et al [3] showed that the arrival
rate $\lambda$ remains pretty constant for at least a 30 minute interval. In
general, this assumption can be relaxed to more general processes such as MAPs
(Markov Arrival Processes) [1], or non homogeneous Poisson processes, but we
will keep working with it for simplicity of the analysis.
* •
The distribution of flow sizes ${S_{n}}$ and flow durations ${D_{n}}$. In this
paper we denote $T_{n}$ as the arrival time of the $n$-th flow, $S_{n}$ as its
size (e.g., in bits), and $D_{n}$ as its duration (e.g., in seconds).
Sequences ${S_{n}}$ and ${D_{n}}$ also form independently of each other and
are identically distributed sequences.
* •
The flow rate function (shot) is $X_{n}(t)$. A flow is called active at time
$t$ when $T_{n}\leq t\leq T_{n}+D_{n}$. Define as $X_{n}(t-T_{n})$ the rate of
the $n$-th flow at time $t$ (e.g., in bits/s), with $X_{n}(t-T_{n})$ equal to
zero for $t<T_{n}$ and for $t>T_{n}+D_{n}$.
Define $B(t)$ as the total rate of data (e.g., in bits/s) on the modeled link
at time $t$. It is determined by adding the rates of the different flows. We
can then write
$B(t)=\sum_{n\in{\mathbb{Z}}}X_{n}(t-T_{n})$ (1)
The process from Eq. (1) can describe the number of active flows $N$ found at
time $t$ in an $M/G/\infty$ queue [12], if $X_{n}(t-T_{n})=1$ at
$t\in[T_{n},T_{n}+D_{n}]$.
The model presented by Barakat et al [3] can compute the average and the
variation of traffic on the backbone. In summary:
* •
The average total rate of the traffic is given by the two parameters $\lambda$
and ${\mathbb{E}}[S_{n}]$:
${\mathbb{E}}[B(t)]=\lambda{\mathbb{E}}[S_{n}]$ (2)
* •
The variance of the total rate ${\mathbb{V}}[B(t)]$ (i.e., burstiness of the
traffic) is given by the two parameters $\lambda$ and
${\mathbb{E}}[S^{2}_{n}/D_{n}]$:
${\mathbb{V}}[B(t)]=\lambda{\mathbb{E}}[S^{2}_{n}/D_{n}]$ (3)
It should be mentioned that Eq. (2) is true only for the ideal case of a
backbone link of unrestricted capacity, that can be applied to underloaded
links. The main drawback of the ratio (2) is its lack of definite usage
limits, due to the fact that variables $\lambda,{\mathbb{E}}[S_{n}]$
describing the system are in no way connected with its current state. The
average flow size ${\mathbb{E}}[S_{n}]$ does not depend on a specific system,
it is an universal value determined by the current distribution of file sizes
found in the Internet.
The arrival rate of flows $\lambda$ describes the user’ s behaviour and doesn’
t depend on the network state and utilization. The cumulative number of flows
that arrive at a link will remain linear even if the network has problems and
doesn’ t satisfy all the incoming demands.
In order to describe the real network state with arbitrary load we should use
Little’ s law:
$N=\lambda{\mathbb{E}}[D_{n}]$ (4)
Here ${\mathbb{E}}[D_{n}]$ is the mean duration of flow and $N$ is the mean
number of active flows. Formula (4) is true [12] for any flow duration and
thus for an arbitrary flow size distribution and rate limit. This formula
describes the network state more precisely than Eq. (2) as the average number
of active flows on the bandwidth unit increases with the utilization. In other
words, the average duration of flow ${\mathbb{E}}[D_{n}]$ enables us to judge
the real network state in contrast to its average value ${\mathbb{E}}[S_{n}]$.
## 3 Network States on Flow Level
In order to analyze the connection quality at the backbone area or the link to
the provider we are going to investigate a graphical dependence between the
link utilization $U$ and the number of active flows $N$ in it [2]. These
variables are easy measurable quantities in despite of average values
${\mathbb{E}}[D_{n}]$ and ${\mathbb{E}}[S_{n}]$. The separate network state is
pictured by single point on coordinate plane with axes $N$ and $U$. The curve
depicting average values has been shown in Fig. 1.
Figure 1: Link utilization vs the number of active flows
On the curve shown in Fig. 1, three parts can be identified, corresponding to
the different network states. The first part of the curve describes the
network state close to the ideal. If the investigated link has unrestricted
capacity there should be a stable linear relationship between number of lows
$N$ and link utilization $U$. The curve describing the network behavior beyond
a certain point will be convex. The linear part of the curve corresponding to
the ideal network behavior is defined as the operational region. The
operational region ends at the threshold point which should be found
experimentally. The dislocation of this point depends on many factors, such
as, transport layer protocol, network topology, the amount of buffering at the
link, etc.
The second part of the curve corresponds to the moderately loaded network,
when the diversion from the ideal network state becomes obvious. There is an
increase in the average duration of a flow compared to the working area, and
therefore, a larger number of active flows on the bandwidth unit
characteristic of this network state.
The third part of the curve corresponds to the totally disabled network with
considerable packet loss evident. We propose some simple preliminary models
for an overloaded link, accounting for user impatience and reattempt
behaviour. In a real network, if demand exceeds capacity, the number of flows
in progress does not increase indefinitely. As perflow throughput decreases,
some flows or sessions will be interrupted, due either to user impatience or
to aborts by TCP or higher layer protocols.
In the end of this section estimation of confidence interval is given for the
operational region of our curve. Since the total rate is the result of
multiplexing of $N(t)$ flows of independent rates, the Central Limit Theorem
[12] tells us that the distribution of $B(t)$ tends to Gaussian at high loads,
which is typical of backbone links. As it is mentioned in Section 2, the
variance of the total rate requires two parameters: the arrival rate of flows
$\lambda$ and the expectation of the ratio between the square of the size of a
flow duration ${\mathbb{E}}[S^{2}_{n}/D_{n}]$ (see Eq. 3). It tells us that
the total rate should lie between
${\mathbb{E}}[B]-A(\varepsilon)\sqrt{{\mathbb{V}}(B)}$ and
${\mathbb{E}}[B]+A(\varepsilon)\sqrt{{\mathbb{V}}(B)}$ in order to provide a
required quality of service. When we talked about the required quality of
service we implied the accordance of network behavior to Eq. (2), here
$A(\varepsilon)$ is the $\varepsilon$-quantile of the centered and normalized
total rate $B(t)$.
Taking into account Eqs. (2-4) and theorems about average values the
confidence interval of the bandwidth $B$ on a operational region of our curve
is
$B=b(N\pm\alpha A(\varepsilon)\sqrt{N}),$ (5)
here $b=E[S_{n}/D_{n}]$ is the average flow performance which characterizesis
the speed of user communication, coefficient $\alpha$ could be found
experimentally.
## 4 Testbed Setup
In order to prove our hypothesis we took measurements on border gateway
routers from FREEnet, HEAnet - Ireland’s National Research and Education
Network, and also from the Russian ISP “SamaraTelecom” (ST). All networks have
several internal and external links. ST’s basic load lies on one channel to
the Internet, whereas HEAnet and FREEnet relies on a number of connections.
Measurements from Gigabit links were taken for FREEnet and 155Mbps, 622Mbps
are for HEAnet and ST. The utilization of these links varies widely from 5% to
60% with a clearly identifiable busy period.
A passive monitoring system based on Cisco’s NetFlow [7] technology was used
to collect link utilization values and active flow numbers in real-time. In
Moscow and Samara we measured on a Cisco 7206 router with NetFlow switched on.
At HEAnet a Cisco 12008 was utilized. A detailed description of Cisco NetFlow
can be found in the Cisco documentation [7].
This is achieved using the following commands on the Cisco 7206:
* •
sh ip cache flow \- gives information about the number of active and inactive
flows, about the parameters of the flows in the real time.
* •
show interface summary \- gives information about the current link
utilization.
On a GSR Router these commands look like:
* •
enable
* •
attach slot-number
* •
show ip cache flow
The FREEnet data was obtained using scripts running every 30 minutes from
middle of January to the end of March 2008. The data sets from two routers of
FREEnet have been collected for further analysis. The full loading of first
router varies in limits of hundreds megabits per second (100-220 Mbps) and
tens megabits for second router (see Fig. 5). During the tests we fixed
information about any network events that could influence on connection
quality. The ST data were recorded at 30-minute intervals, twenty-four hours a
day for a week to discover network behavior with different loading levels. The
HEAnet data was obtained using scripts running every 5 minutes for a period of
72 hours. It is quite easy to write a script, which will collect the data from
the router to the management server.
## 5 Experimental Validation and Diagnostic of Troubleshooting
The network state at every instant describes by point on two-dimensional plot
where abscise shows the number of active flow $N$ and ordinate shows router
loading $B$. Basic tasks of experimental validation of our model consist in
* •
construction of curve of average values and its comparison with theoretical
prediction showed on Fig. 1
* •
calculation of variance for flow performances and verification of parabolic
form of confidence interval from Eq. (5)
* •
examination on normal distribution for flow performances
In order to do experimental test of our model the data set should be divided
into several intervals depending on number of active flows. Inside each
interval the average values and their variance were calculated for flow
performances as well as for other parameters characterizing network states.
Table 1: Parameters for active flows of FREEnet, Data Set 1, 2008 n | $N$ | $B$, | $\sigma(B)$, | $b$, | $\sigma(b)$,
---|---|---|---|---|---
| | Mbps | Mbps | bps | bps
1 | 17489 | 113.1 | 23.1 | 6784 | 1386
2 | 23260 | 126.0 | 21.4 | 5682 | 965
3 | 27007 | 152.0 | 39.2 | 5628 | 1452
4 | 34902 | 156.7 | 26.9 | 4990 | 770
5 | 45104 | 163.9 | 33.9 | 3634 | 752
6 | 55019 | 176.3 | 33.2 | 3205 | 604
7 | 64778 | 215.4 | 42.2 | 3325 | 652
The earlier tests have been conducted on the boundary router of the ST and
HEAnet, and they didn’t allow to verify the model with high precision. The new
data from FREEnet contains thousands of points describing network states.
FREEnet Data Set 1 has been divided into seven intervals according to the
number of active flows (15000 - 20000, 20000 - 25000, 25000 - 30000, 30000 -
40000, 50000 - 60000, $\geq$60000). Inside each interval basic parameters
characterizing active flows have been calculated and the result is represented
in the Table 1. Here $N$ is the average number of active flows for the
interval denoted by $n$. $B$ describes the average router loading in Megabit
per second (Mbps), and $b$ is average flow performance measured in bit per
second (bps). $\sigma(b),\sigma(B)$ are the standard deviation for flow
performances $b$ and router loading $B$ correspondingly.
Figure 2: FREEnet router loading vs the number of active flows, DataSet1
The results of the measurement for FREEnet Data Set 1 are pictured on Fig. 2.
Basic curve is constructed as the line of average values, it describes network
states on flow level. The error bar restricts the confidential interval for
network states. Comparison with theoretical prediction from Section 3 leads to
the conclusion that the area of network exploitation lies inside the
operational region.
Figure 3: The operational region of network, DataSet1
In order to restrict the operational region the straight portion of curve from
Fig. 2 should be marked as it is shown on Fig. 3. The slope angle of this
straight line is found as the average flow performance $b$. Only three initial
points from investigated data set may be placed in the limits of operational
region. The angle of inclination gives the average flow performance equal to
5700 bps for FREEnet. If the number of flows exceeds 30000 then network gets
moderately loading that leads to a reduction of the flow performance. The
router loading does not increase uniformly with the number of requests, and
the connection quality becomes almost twice as bad (see Table 1).
Table 2: Parameters for active flows of FREEnet, Data Set 2, 2008 n | $N$ | $B$, | $\sigma(B)$, | $b$, | $\sigma(b)$,
---|---|---|---|---|---
| | Mbps | Mbps | bps | bps
1 | 5446 | 15.42 | 2.25 | 2843 | 413
2 | 6531 | 17.11 | 2.45 | 2364 | 377
3 | 7508 | 17.74 | 2.35 | 2364 | 313
4 | 8370 | 18.92 | 2.24 | 2261 | 268
5 | 9443 | 20.67 | 3.81 | 2190 | 404
6 | 15495 | 28.05 | 5.40 | 1811 | 349
In Table 2 the Data Set 2 from the second router of FREEnet with lower loading
is presented. Investigated region divides into six intervals according to the
number of active flows (5000 - 6000, 6000 - 7000, 7000 - 8000, 9000 - 10000,
$\geq$10000). Operational region for second router of FREEnet is restricted by
10 000 active flows as it is shown in Fig. 4. Only last intervals should be
excluded from straight portion of utilization curve.
Figure 4: The operational region of network, DataSet2
Fig. 5 illustrates the network states and the form of confidence interval for
operation region with normal quantile function $A(\varepsilon)=0.05$.
A correlation coefficient indicates the strength and direction of a linear
relationship between two random variables. In order to verify the parabolic
form of confidence interval the correlation coefficient between variables
$\sigma_{i}(B)$ and $\sqrt{N_{i}}$ should be calculated for both DataSet of
FREEnet (see Tables 1 and 2). Comparing second and forth columns of mentioned
Tables the values of correlation coefficient are equal 0.70 for DataSet1 and
0.93 for DataSet2. These magnitudes allow us to say about high correlation
between theoretical model and its experimental examination.
Figure 5: Confidence interval for operation region, DataSet2 from FREEnet
A significant question concerns the numerical value for the the numerical
coefficient $\alpha_{1},\alpha_{2}$ from Eq. (5). The function
$A(\varepsilon)$ can be computed using the Gaussian approximation, which gives
for example $A(0.05)=1.96$. Our data from Tables 1, 2 allow calculating their
magnitudes:
$\alpha_{1}\approx 13,\alpha_{2}\approx 4.5$ (6)
Table 3: Statistical tests, Data Set 1, 2008 n | $\chi^{2}$ for | Gaussian | Correlation
---|---|---|---
| $\alpha=0.95$ | test | coefficient
1 | not enough data | |
2 | not enough data | |
3 | not enough data | |
4 | 3.49 (9.49) | $+$ | 0.70
5 | not enough data | |
6 | 3.45 (7.81) | $+$ |
7 | 0.50 (9.49) | $+$ |
Significant assumption underlies a theoretical model that distribution of flow
performances $b$ may be considered as Gaussian distribution. The number of
testing network states inside many intervals from Tables 1, 2 allow to check a
given set for similarity to the normal distribution. Here we use the Pearson
$\chi^{2}$ test and the results of this test could be found in Tables 3 and 4.
Column 2 shows the value of $\chi^{2}$ for $\alpha=0.95$, in round brackets it
is shown Table values for $\chi^{2}$. It should be noted that all investigated
intervals with sufficient number of states discover the normal type of
distribution.
Table 4: Statistical tests, Data Set 2, 2008 n | $\chi^{2}$ for | Gaussian | Correlation
---|---|---|---
| $\alpha=0.95$ | test | coefficient
1 | 9.15 (12.6) | $+$ |
2 | 10.0 (11.1) | $+$ |
3 | 3.24 (14.1) | $+$ | 0.93
4 | 10.0 (14.1) | $+$ |
5 | not enough data | |
6 | 1.94 (14.1) | $+$ |
In conclusion of this Section it should be noted that expression (5) allows
formulating the easy rule how to display the network defects. If two
consistent measurements running every 5(30) minutes give the deviation of real
network states $B_{i},N_{i}$ from the confidence interval with $A(0.05)$ then
network problem has been detected. Confidence interval is described by flow
performance $b$ and the values $\alpha,\sigma(B)$ which may be found only as
result of data processing. Unfortunately, the corresponding software is not
completed yet.
Figure 6: Detection of anomaly network state
This rule received an apt illustration during FREEnet network testing. A
network incident has been detected: a wide links to large FTP server has been
temporally turned down. These anomaly network states $B_{i},N_{i}$ departure
the confidence interval corresponding to standard behavior of investigated
network and form separate cluster as it has shown on Fig. 6.
So our model receives the experimental confirmation and the diagnostic method
based on introduction of confidence interval may be applied to network
monitoring.
## 6 Parameters of Comparative Studies
During experiments it has been noticed that the average flow performances in
investigated networks are considerable different.
For different Russian networks this parameter varies from hundreds bps to tens
Kbps, for regional network it exceeds 2Kbps very seldom. For comparison, the
mean flow performance in HEAnet differs on almost one order and equals 15000
bps that allows them to provide high speed video applications through public
network. In other words the investigated Russian network needs expanding
bandwidth of trunk links, especially those ones that connect remote regions.
It is problem number one for Russian Networks for Science and Education.
The idea evolved to apply our model for comparative analysis of large networks
or data-center connectivity. The equation (5) allows us to compare the length
of operation regions, mean flow performances, the width of confidence
intervals, etc.
It should be noted that the curve of mean values from Fig. 2 describes the
behavior of investigated network as a whole. Mean flow performance $b$ allows
to judge about the quality of network services provided to single user. The
calculation of active flow number has a significant feature; the flow is
considered active for a long time after transmission of the last packet.
Therefore the real speed of network services exceeds the value $b$ almost by
one order.
From the end user’s point of view the mean flow performance is the basic
universal index. Principally, this value should be considered as the basic
index characterizing the quality of network services.
There is a dependence of access to high-speed Internet services and mean flow
performance. For example, streaming video began possible to look at our
university as soon as the given indicator has exceeded value in 10 Kbps. It is
very much a rough estimate, additional researches are necessary to find
boundary values of mean flow performance, necessary for introduction of this
or that high-speed service. This problem in networks of cellular operators
where standards of data transmission GPRS, EDGE and even 3G are not capable to
give yet high speeds to an unlimited circle of clients is especially actual.
## 7 Conclusion
In this paper some methods are described that allow us to evaluate connection
quality in large networks based on flow technology, and give an indication as
to when capacity needs to be increased. At the moment we are working on
developing utilities, which will make it possible to construct the dependence
of link loading on the number of active flows automatically, and calculate the
length of the operation region as well as coefficients for confidence
interval.
The special attention has been given how by means of the constructed model to
make a technique of comparison of the big networks. It has been established
that as object for comparison the unique parameter - mean flow performance can
act. The high-grade comparative analysis should contain the conformity table
between mean flow performance and possibility of start high-speed Internet
services that demands the further experiments.
This paper has demonstrated that it is possible to easily determine the
confidence interval, operation region and the overload point of a network
connection, utilizing low cost commodity hardware and simple software. This
allows us to identify the anomaly network states and moment when a backbone
upgrade is required. Further experiments are necessary in order to develop
software utilities for this purpose. Thus, providing analytical
generalizations, we established common terminology for processes, taking place
in networks.
## References
* [1] E. Altman, K. Avratchenkov, C. Barakat, A stochastic model for TCP/IP with stationary random losses, ACM SIGCOMM, September 2000
* [2] F. Afanasiev, A. Petrov, and A. Sukhov, A Flow-based analysis of Internet traffic, Russian Edition of Network Computing, 5(98) (2003) 92-95 (arXiv:cs/0306037)
* [3] C. Barakat, P. Thiran, G. Iannaccone, C. Diot, P. Owezarski P., A flow-based model for Internet backbone traffic, IEEE Transactions on Signal Processing - Special Issue on Signal Processing in Networking, vol. 51, no. 8 (2003) 2111-2124
* [4] S. Ben Fredj, T. Bonald T., A. Proutiere, G. Regnie, J. Roberts, Statistical Bandwidth Sharing: A Study of Congestion at Flow Level, ACM SIGCOMM, August 2001
* [5] N. Brownlee, C. Mills, G. Ruth, Traffic Flow Measurement: Architecture (RFC 2722), October 1999
* [6] Y. Chabchoub, C. Fricker, F. Guillemin, and P. Robert, A Study of Flow Statistics of IP Traffic with Application to Sampling, Lecture Notes in Computer Science, 4516 (2007) 678-689
* [7] Cisco IOS NetFlow site, Cisco Systems, http://www.cisco.com/go/netflow/
* [8] L. Deri, nProbe: an Open Source NetFlow Probe for Gigabit Networks, TERENA 2003
* [9] C. Fraleigh, F. Tobagi, C. Diot, Provisioning IP Backbone Networks to Support Latency Sensitive Traffic, INFOCOM, Volume: 1 (2003) 375-385
* [10] R. Lippmann, J. Haines, D. Fried, J. Korba and K. Das, The 1999 DARPA off-line intrusion detection evaluation, Computer Networks, 34(4) (2000) 579-595
* [11] Y. Jiang, P. Emstad, A. Nevin, V. Nicola, and M. Fidler, Measurement based admission control for a flow-aware network, in Next Generation Internet Networks, (2005) 318 325.
* [12] L. Kleinrock, Queueing Systems, Wiley, NY, 1975, Vol. I: Theory
* [13] A. Kortebi, L. Muscariello, S. Oueslati, and J. Roberts, Minimizing the overhead in implementing flow-aware networking, in ANCS 05: Proceedings of the 2005 symposium on Architecture for networking and communications systems. New York, NY, USA: ACM Press (2005) 153 162
* [14] NSS Group, Intrusion Detection Systems Group Test (Edition 4), NSS Group, 2004
* [15] S. Oueslati and J. Roberts, A new direction for quality of service: Flow aware networking, in Next Generation Internet Networks (2005) 226-232
* [16] K. Papagiannaki, N. Taft, Z.-L Zhang, C. Diot, Long-Term Forecasting of Internet Backbone Traffic: Observations and Initial Models, INFOCOM 2003
* [17] W. Yang W., J. Gong, W. Ding, X. Wu, Network Traffic Emulation for IDS Evaluation, IFIP International Conference on Network and Parallel Computing, ISBN: 978-0-7695-2943-1 (2007) 608-612
|
arxiv-papers
| 2009-11-13T14:31:58 |
2024-09-04T02:49:06.446149
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.M. Sukhov, D.I. Sidelnikov, A. Galtsev, A.P. Platonov, M.V. Strizhov",
"submitter": "Andrei Sukhov M",
"url": "https://arxiv.org/abs/0911.2619"
}
|
0911.2648
|
# The Transition from Heavy Fermion to Mixed Valence in Ce1-xYxAl3:
A Quantitative Comparison with the Anderson Impurity Model
E. A. Goremychkin Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439 ISIS Pulsed Neutron and Muon Source, Rutherford Appleton
Laboratory, OX11 0QX, UK R. Osborn ROsborn@anl.gov Materials Science
Division, Argonne National Laboratory, Argonne, IL 60439 I. L. Sashin Joint
Institute for Nuclear Research, Dubna, Moscow Region, 141980 Russia P.
Riseborough Department of Physics, Temple University, Philadelphia, PA 19122,
USA B. D. Rainford Department of Physics, University of Southampton, S017
1BJ, UK D. T. Adroja ISIS Pulsed Neutron and Muon Source, Rutherford
Appleton Laboratory, OX11 0QX, UK J. M. Lawrence Department of Physics,
University of California, Irvine, CA 92697, USA
###### Abstract
We present a neutron scattering investigation of Ce1-xYxAl3 as a function of
chemical pressure, which induces a transition from heavy fermion behavior in
CeAl3 (T${}_{\mathrm{K}}=5$ K) to a mixed valence state at $x=0.5$
(T${}_{\mathrm{K}}=150$ K). The crossover can be modeled accurately on an
absolute intensity scale by an increase in the k-f hybridization, Vkf, within
the Anderson Impurity Model. Surprisingly, the principal effect of the
increasing Vkf is not to broaden the low-energy components of the dynamic
magnetic susceptibility but to transfer spectral weight to high energy.
###### pacs:
71.27.+a,71.70.Ch,75.40.Gb,78.70.Nx
The Anderson Impurity Model (AIM) has been invoked to describe the
thermodynamic, transport, and spectroscopic properties of strongly correlated
electron systems for nearly fifty years Anderson (1961). It is believed to
contain the essential physics of materials in which a narrow band of quasi-
localized electrons are hybridized with a broader band of itinerant electrons,
and provides a mechanism describing the formation and screening of local
moments in metallic systems. The AIM has been most widely applied to describe
fluctuating moments in rare earth $f$-electron systems, ranging from strongly
hybridized mixed valent systems to more weakly hybridized heavy fermion
systems Fulde (1988). Although it is a single-impurity model, it has been
successfully applied to concentrated rare earth systems, except at the lowest
temperatures where lattice coherence cannot be ignored.
In spite of its widespread use in heavy fermion and mixed valence physics,
there have been no direct quantitative comparisons of ab initio calculations
of the dynamic magnetic susceptibility, $\mathrm{Im}\,\chi(\omega)$, with
experiment covering the crossover between the two regimes. In recent years, it
has become possible to perform detailed calculations of
$\mathrm{Im}\,\chi(\omega)$ using the Non-Crossing Approximation (NCA) in the
presence of crystal field splittings of the 4$f$-electron ground multiplet
Bickers et al. (1987); Kuramoto (1983), but these theoretical advances have
never been tested against inelastic neutron scattering data, which directly
measure $\mathrm{Im}\,\chi(\omega)$. The purpose of this letter is to present
the first detailed quantitative comparison of AIM/NCA calculations with
neutron scattering data as a function of the hybridization strength, Vkf,
covering the transition from heavy fermion to mixed valence behavior. There is
excellent agreement on an absolute intensity scale between theory and
experiment, although the relative strength of the transverse and longitudinal
susceptibility requires an adjustment for anisotropic hybridization. The low-
frequency broadening of the magnetic response is typically proportional to the
Kondo temperature, TK, so it is surprising that increasing the hybridization
does not significantly increase the width of the low-energy contributions to
$\mathrm{Im}\,\chi(\omega)$. Instead, the transition is predominantly
characterized by a transfer of spectral weight from the two low-energy
components, respectively a narrow quasielastic response and a crystal field
excitation, to a broad high-energy tail.
Figure 1: The unit cell volume, the static susceptibility measured at 8K and
the magnetization in a field of 12T measured at 2K of Ce1-yLayAl3 and
Ce1-xYxAl3. The solid circles are measurements and the open squares are
predictions of a localized crystal field model. The lines are guides to the
eye.
Figure 2: S(Q,$\omega$) of Ce1-xYxAl3 as a function of $x$ measured at an
average scattering angle of (a) $60^{\circ}$ and (b) $15^{\circ}$, with an
incident energy of (a) 3.12 meV and (b) 35 meV. The solid lines are fits to a
quasielastic Lorentzian lineshape (dashed line) and an elastic peak (dotted
line) convolved with the instrumental resolution (FWHM $\sim 90\mu$eV). The
inset shows the fitted values of the static susceptibility, $\chi_{0}$, and
the half-width of the quasielastic peak, $\Gamma$.
The variation in hybridization is achieved through the application of chemical
pressure on the canonical heavy fermion system CeAl3Andres et al. (1975).
Doping lanthanum onto the cerium sites expands the lattice, progressively
localizing the 4$f$ electrons while doping with yttrium compresses the
lattice, inducing a transition from the heavy fermion behavior of pure CeAl3,
characterized by a Kondo temperature of T${}_{\mathrm{K}}=5$K, to a mixed
valence state in Ce0.5Y0.5Al3, where T${}_{\mathrm{K}}=150$K. Comparison with
the cell volumes of pure CeAl3 under pressure shows that the chemical pressure
in Ce1-xYxAl3 at $x=0.5$ is equivalent to 27 kbar. The bulk measurements in
Figure 1 illustrate how chemical pressure, and the consequent increase in
hybridization, reduces the static susceptibility and high-field magnetization
from the values predicted in a pure crystal field model Goremychkin et al.
(1999).
Polycrystalline samples of Ce1-xYxAl3, with $x$ = 0.0, 0.05, 0.1, 0.2, 0.3,
0.4, and 0.5, Ce1-yLayAl3 with $y$ = 0.2, 0.3, 0.4, 0.5, and 0.95, along with
non-magnetic samples of LaAl3 and La0.5Y0.5Al3, were made by arc-melting
stoichiometric quantities of the constituent elements, with a total mass of
approximately 30 g for each sample, followed by annealing in a vacuum at 900∘C
for four to five weeks. Powder neutron diffraction confirmed that each sample
was single-phase. The inelastic neutron scattering experiments were performed
on the IN6 spectrometer at the Institut Laue Langevin, using an incident
neutron energy of 3.12 meV, and the LRMECS spectrometer at the Intense Pulsed
Neutron Source, in Argonne National Laboratory, using incident energies of 35
and 80 meV. All spectra were normalized on an absolute intensity scale using a
standard vanadium plate and corrected for self-shielding, absorption, and form
factor. Our own comparison with inelastic scattering standards has shown that
it is possible to achieve accuracies of 1% or better, a precision that is
confirmed here in our comparisons of data taken at different incident neutron
energies.
Figure 2 shows corrected data from IN6 and LRMECS at a temperature of 8 K
covering the low and intermediate energy range. In Figure 2(a), the magnetic
response is fitted to a single quasielastic Lorentzian lineshape convolved
with the instrumental resolution.
$\displaystyle S(Q,\omega)\propto
F^{2}(Q)\left[n(\omega)+1\right]\frac{\chi_{0}}{2\pi}\frac{\omega\Gamma}{\left(\omega^{2}+\Gamma^{2}\right)}$
(1)
where $F(Q)$ is the Ce3+ magnetic form factor, $n(\omega)$ is the Bose
population factor, $\chi_{0}$ is the static susceptibility, and $\Gamma$ the
half width.
With increasing $x$, there is only a modest increase in the linewidth $\Gamma$
from 1.06 meV at $x=0$ to 1.26 meV at $x=0.3$, but a dramatic reduction in the
spectral weight, $\chi_{0}$, becoming negligibly small at $x=0.5$. The energy
range of Fig. 2(b) covers the crystal field excitation, which is observed at
6.7 meV in CeAl3. With increasing hybridization, the crystal field peak falls
sharply in intensity, with a slight shift to higher energy, and is difficult
to resolve at $x\geq 0.3$.
Figure 3: S(Q,$\omega$) of Ce0.6Y0.4Al3 (open circles), Ce0.6La0.4Al3 (solid
circles), and LaAl3 (diamonds) measured at an average scattering angle of
$15^{\circ}$ with an incident energy of 35 meV (left axis) and 80 meV (right
axis) at 8 K, normalized on an absolute scale. The shaded area shows the
estimated magnetic scattering. The inset shows the magnetic scattering
integrated from 40 to 70 meV as a function of La and Y doping.
At all values of $x$, there is an additional component to the magnetic
response in the form of a high-energy tail, extending to greater than 70 meV,
that cannot be accounted for by the low-energy quasielastic Lorentzian or the
crystal field excitation. It is also just visible at $y=0.3$, but it has
negligible intensity at $y=0.4$. At these energies, multiple scattering
produces a significant phonon contribution to the measured inelastic
scattering and must be subtracted before comparing to theory. Figure 3
illustrates this with data at $x=0.4$, which shows significant additional
magnetic intensity when compared to the non-magnetic LaAl3 data. The inset to
Figure 3 shows that the high-energy magnetic component, integrated from 40 to
70 meV, falls with increasing cell volume, i.e., with decreasing
hybridization, becoming negligible in Ce0.6La0.4Al3.
Figure 4: Theoretical calculations of S($\omega$) determined by the Anderson
Impurity Model using the Non-Crossing Approximation at a temperature of 8 K.
Qualitatively, the observations are consistent with AIM/NCA calculations,
which follow the method of Kuramoto Kuramoto (1983) and Bickers et al. Bickers
et al. (1987) (Figure 4). We assume a Gaussian conduction band of half-width
$W=3$ eV, as in Ref. Riseborough (2003) with the ground state of the Ce3+ 4$f$
electrons at $E_{f}=-2$ eV. The 14-fold degeneracy of the $f$-states is lifted
by the free-atom spin-orbit coupling and the crystal field potential, which,
in hexagonal point group symmetry, is diagonal in $J_{z}$. The $J=\frac{5}{2}$
multiplet is therefore split into three Kramers doublets,
$\Gamma_{7}=\left|\pm\frac{1}{2}\right>$,
$\Gamma_{8}=\left|\pm\frac{5}{2}\right>$, and
$\Gamma_{9}=\left|\pm\frac{3}{2}\right>$. In an earlier publication, we showed
that the $\Gamma_{9}$ doublet is the ground state, with the other two doublets
nearly degenerate at an energy of approximately 7 meV Goremychkin et al.
(1999); in the calculation, the $\Gamma_{8}$ and $\Gamma_{7}$ doublets were
assumed to be at 6 meV and 7.5 meV, respectively. The hybridization matrix
element was assumed to be isotropic and its value was treated as a free
parameter in order to simulate the effect of increasing $x$.
For low values of the hybridization, the AIM/NCA calculations show a sharp
quasielastic response, well-defined crystal field excitation, and a broad high
energy tail. The high-energy tail in the spectrum is a consequence of the
dynamic screening of the local moments by the conduction electrons Anderson
(1967a). The screening of the magnetic moments can be considered as an
iterated orthogonality catastrophe Anderson (1967b), which occurs since the
conduction electrons experience a sequence of local spin-dependent potentials
caused by successive flips of the local spin. This process results in a power-
law decay of the spectrum at high energies, similar to that found in the x-ray
edge problem Mahan (1967).
As the hybridization increases, the calculated quasielastic and crystal field
contributions gradually decrease in intensity, but their spectral weight is
transferred into the high energy tail. This agrees with the experimental
trends with increasing $x$. Since the theoretical predictions cannot be
described by simple analytic functions, we have put this comparison on a
quantitative footing by directly comparing integrals over energy ranges
corresponding to the three components in the scattering with both the
experimental and calculated intensities placed on an absolute scale with no
adjustable parameters. The quasielastic component is represented by an
integral from -3 to 2 meV (using the fitted profile to exclude the elastic
nuclear scattering), the crystal field component from 5 to 15 meV, and the
high energy component from 40 to 70 meV. For the latter two components,
similar integrals were performed on LaAl3 and La0.5Y0.5Al3, and subtracted
after interpolating for the correct yttrium concentration. The integrals in
the high-energy tail are not affected by this interpolation since the phonon
scattering at these energies is dominated by the aluminum contribution.
Figure 5: The magnetic cross section of Ce1-xYxAl3 as a function of $x$
integrated over three energy regions, -3 to 2 meV (triangles), 5 to 15 meV
(circles) and 40 to 70 meV (squares), and corrected for the form factor. The
solid lines are the same integrals determined from the theoretical
calculations using the AIM/NCA model as a function of Vkf, using a linear
mapping of Vkf to $x$. The dashed line is the same as the solid line shifted
to illustrate the effect of increasing Vkf by $\sim 20$ meV at each value of
$x$.
Figure 5(a) shows that two contributions to $\mathrm{Im}\,\chi(\omega)$, the
narrow quasielastic and high energy components, are in excellent agreement
with the AIM/NCA calculations with Vkf = 443 meV at $x=0$ and 477 meV at
$x=0.5$. It appears that there is a nearly linear relation between Vkf and
$x$, although there is no a priori reason for this to be so. The important
result is that both the low energy and high energy regions of scattering have
absolute cross sections that are accurately predicted at each value of Vkf
over the entire transition from heavy fermion to mixed valence states.
On the other hand, Figure 5(b) shows that the measured intensities in the
intermediate energy region, which, at lower values of $x$, includes the
crystal field excitation, are much lower than predicted if we use the same
values of Vkf. Reasonable agreement between theory and experiment can only be
achieved in this energy range if we increase the value of Vkf by approximately
20 meV at each value of $x$. To explain this apparent discrepancy, it is
important to know that, because the crystal field potential is diagonal in
$J_{z}$, the quasielastic response is purely longitudinal, whereas the crystal
field excitation is purely transverse in character. The anomaly in the crystal
field intensities therefore implies that the k-f hybridization in CeAl3 is
anisotropic.
In earlier investigations, we used the Anisotropic Kondo Model (AKM) to
explain the anomalous spin dynamics in Ce1-xLaxAl3 Goremychkin et al. (2000,
2002). However, the present results provide the first direct evidence of the
anisotropy in Vkf and its impact on the dynamic magnetic susceptibility. There
is some evidence that anisotropic hybridization may also be important in other
strongly correlated electron systems, such as CeCu6-xAuxLöhneysen et al.
(2007), where the magnetic fluctuations appear two-dimensional, and URu2Si2
Elgazzar et al. (2009), where we have proposed that the AKM plays an important
role in generating hidden order Goremychkin et al. (2002).
In conclusion, a quantitative comparison of inelastic neutron scattering data
with NCA calculations of the Anderson Impurity Model have provided, for the
first time, a consistent description of the evolution of the spin dynamics
from heavy fermion to mixed valence behavior. Because of the non-analytic
character of the Kondo coupling, a 10% increase in the k-f hybridization
produces an increase in the energy scale of spin fluctuations of more than an
order-of-magnitude. However, the most significant result is that the
transition between the two regimes does not primarily result from a general
broadening of the dynamic magnetic susceptibility but by a significant shift
of spectral weight from from the narrow components that characterize well-
localized $f$-electron states to a broad magnetic response that is present at
all hybridization strengths, including heavy fermion compounds such as CeAl3.
This research was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, Division of Materials Sciences and Engineering under Award #
DE-AC02-06CH11357.
## References
* Anderson (1961) P. W. Anderson, Phys Rev 124, 41 (1961).
* Fulde (1988) P. Fulde, J Phys F 18, 601 (1988).
* Bickers et al. (1987) N. E. Bickers, D. Cox, and J. W. Wilkins, Phys. Rev. B 36, 2036 (1987).
* Kuramoto (1983) Y. Kuramoto, Z Phys B 53, 37 (1983).
* Andres et al. (1975) K. Andres, J. E. Graebner, and H. R. Ott, Phys Rev Lett 35, 1779 (1975).
* Goremychkin et al. (1999) E. Goremychkin, R. Osborn, and I. Sashin, J Appl Phys 85, 6046 (1999).
* Riseborough (2003) P. Riseborough, Phys Rev B 67, 045102 (2003).
* Anderson (1967a) P. W. Anderson, Phys Rev 164, 352 (1967a).
* Anderson (1967b) P. W. Anderson, Phys Rev Lett 18, 1049 (1967b).
* Mahan (1967) G. D. Mahan, Phys Rev 163, 612 (1967).
* Goremychkin et al. (2000) E. Goremychkin, R. Osborn, B. Rainford, and A. Murani, Phys Rev Lett 84, 2211 (2000).
* Goremychkin et al. (2002) E. Goremychkin, R. Osborn, B. Rainford, T. Costi, A. Murani, C. Scott, and P. King, Phys Rev Lett 89, 147201 (2002).
* Löhneysen et al. (2007) H. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev Mod Phys 79, 1015 (2007).
* Elgazzar et al. (2009) S. Elgazzar, J. Rusz, M. Amft, P. M. Oppeneer, and J. A. Mydosh, Nat Mater 8, 337 (2009).
|
arxiv-papers
| 2009-11-13T17:08:31 |
2024-09-04T02:49:06.451777
|
{
"license": "Public Domain",
"authors": "E. A. Goremychkin, R. Osborn, I. L. Sashin, P. Riseborough, B. D.\n Rainford, D. T. Adroja, and J. M. Lawrence",
"submitter": "Ray Osborn",
"url": "https://arxiv.org/abs/0911.2648"
}
|
0911.2660
|
# Maximum GCD Among Pairs of Random Integers
R. W. R. Darling $\&$ E. E. Pyle
Mathematics Research Group, National Security Agency
9800 Savage Road, Fort George G. Meade, Maryland 20755-6515
###### Abstract
ABSTRACT: Fix $\alpha>0$, and sample $N$ integers uniformly at random from
$\left\\{1,2,\ldots,\left\lfloor e^{\alpha N}\right\rfloor\right\\}$. Given
$\eta>0$, the probability that the maximum of the pairwise GCDs lies between
$N^{2-\eta}$ and $N^{2+\eta}$ converges to 1 as $N\to\infty$. More precise
estimates are obtained. This is a Birthday Problem: two of the random integers
are likely to share some prime factor of order $\left.N^{2}\right/\log[N]$.
The proof generalizes to any arithmetical semigroup where a suitable form of
the Prime Number Theorem is valid.
## 1\. Main Result
Whereas the distribution of the sizes of the prime divisors of a random
integer is a well studied subject — see portions of Billingsley (1999) — the
authors are unaware of any published results on the pairwise Greatest Common
Divisors (GCD) among a large collection of random integers. Theorem 1.1
establishes probabilistic upper and lower bounds for the maximum of these
pairwise GCDs.
### 1.1 Theorem
Suppose $\alpha>0$, and $T_{1},\ldots,T_{N}$ is a random sample, drawn with
replacement, from the integers $\left\\{n\in\mathbb{N}:n\leq e^{\alpha
N}\right\\}$. Let $\Gamma_{j,k}$ denote the Greatest Common Divisor of
$T_{j}$ and $T_{k}$. For any $\eta>0$,
$\lim_{N\to\infty}\mathbb{P}\left[N^{2-\eta}<\underset{1\leq j<k\leq
N}{\max}\left\\{\Gamma_{j,k}\right\\}<N^{2+\eta}\right]=1.$ (1)
Indeed there are more precise estimates: for all $s\in(0,1)$, and $b>0$, the
right side of (2) is finite, and
$\mathbb{P}\left[\underset{1\leq j<k\leq
N}{\max}\left\\{\Gamma_{j,k}\right\\}\geq
N^{2/s}b^{1/s}\right]\leq\frac{1}{2b}\prod_{p\in\mathcal{P}}\left(1+\frac{p^{s}-1}{p^{2}-p^{s}}\right),$
(2)
where $\mathcal{P}$ denotes the rational primes; while if $\Lambda_{j,k}$
denotes the largest common prime factor of $T_{j}$ and $T_{k}$, then for all
$\theta>0$,
$\lim_{N\to\infty}\mathbb{P}\left[\underset{1\leq k<j\leq
N}{\max}\left\\{\Lambda_{j,k}\right\\}<\frac{N^{2}}{\log\left[N^{\theta}\right]}\right]\leq
e^{-\theta/8}.$ (3)
Supplement: There is an upper bound, similar to (2), for the radical (i.e the
largest square-free divisor) $\text{rad}\left[\Gamma_{j,k}\right]$ of the GCD:
$\mathbb{P}\left[\underset{1\leq j<k\leq
N}{\max}\left\\{\text{rad}\left[\Gamma_{j,k}\right]\right\\}\geq
N^{2/s}b^{1/s}\right]\leq\frac{1}{2b}\prod_{p\in\mathcal{P}}\left(1-p^{-2}+p^{s-2}\right).$
(4)
The proof, which is omitted, uses methods similar to those of Proposition 2.2,
based upon a Bernoulli model for occurrence of prime divisors, instead of a
Geometric model for prime divisor multiplicities. For example, when $s=0.999$,
the product on the right side of (4) is approximately 12.44; for the right
side of (2), it is approximately 17.64.
### 1.2 Overview of the Proof of Theorem 1.1
Let $Z_{i}^{k}$ be a Bernoulli random variable, which takes the value 1 when
prime $p_{i}$ divides $T_{k}$. As a first step towards the proof, imagine
proving a comparable result in the case where $\left\\{Z_{i}^{k},1\leq k\leq
N,i\geq 1\right\\}$ were independent, and
$\mathbb{P}\left[Z_{i}^{k}=1\right]=1\left/p_{i}\right.$. The harder parts of
the proof arise in dealing with the reality that, for fixed $k$,
$\left\\{Z_{i}^{k},i\geq 1\right\\}$ are negatively associated, and change
with $N$. Convergence of the series
$\sum_{p\in\mathcal{P}}p^{-2}\log[p]<\infty$
ensures that the parameter $\alpha$, which governs the range of integers being
sampled, appears neither in (1), (2), nor (3). However the proof for the lower
bound depends crucially on an exponential (in $N$) rate of growth in the
range, in order to moderate the dependence among $\left\\{Z_{i}^{k},i\geq
1\right\\}$ for fixed $k$.
Consider primes as labels on a set of urns; the random variable $T_{j}$
contributes a ball to the urn labelled $p$ if prime $p$ divides $T_{j}$. The
lower bound comes from showing that, with asymptotic probability at least
$1-e^{-\theta/8}$ , some urn with a label
$p>N^{2}/\log\left[N^{\theta}\right]$ contains more than one ball; in that
case prime $p$ is a common divisor of two distinct members of the list
$T_{1},\ldots,T_{N}$. The upper bound comes from an exponential moment
inequality.
If $T_{1},\ldots,T_{N}$ were sampled uniformly without replacement from the
integers from 1 to $N^{2}$, the lower bound (3) would fail; see the analysis
in Billingsley (1999) of the distribution of the largest prime divisor of a
random integer. In the case of sampling from integers from 1 to $N^{r}$, where
$r\geq 3$, the upper bound (2) remains valid, but we do not know whether the
lower bound (3) holds or not.
### 1.3 Generalizations to Arithmetical Semigroups
Although details will not be given, the techniques used to prove Theorem 1.1
will be valid in the more general context of a commutative semigroup $G$ with
identity element 1, containing a countably infinite subset
$\mathcal{P}:=\left\\{p_{1},p_{2},\ldots\right\\}$ called the primes of $G$,
such that every element $a\neq 1$ of $G$ has a unique factorization of the
form
$a=\prod_{i\geq
1}p_{i}^{e_{i}},\left(e_{1},e_{2},\ldots\right)\in\mathbb{Z}_{+}^{\infty}$
where all but finitely many $\left(e_{i}\right)$ are zero. Assume in addition
that $G$ is an arithmetical semigroup in the sense of Knopfmacher (1990),
meaning that there exists a real-valued norm $|\cdot|$ on $G$ such that:
$\bullet$ $|1|=1$, $\left|p_{i}\right|>1$ for all $p_{i}\in\mathcal{P}$.
$\bullet$ $|ab|=|a||b|$ for all $a,b\in G$.
$\bullet$ The set $\pi_{G}[x]:=\left\\{i\geq 1:\left|p_{i}\right|\leq
e^{x}\right\\}$ is finite, for each real $x>0$.
The only analytic condition needed is an abstract form of the Prime Number
Theorem (see Knopfmacher (1990), Chapter 6):
$\lim_{x\to\infty}xe^{-x}\left|\pi_{G}[x]\right|=1,$
used in the proof of Proposition 4.1.This in turn will imply convergence of
series such as:
$\sum_{p\in\mathcal{P}}\log\left[1+|p|^{s-2}\right],s<1,$
which appear (in an exponentiated form) in the bound (2). For example,
Landau’s Prime Ideal Theorem provides such a result in the case where $G$ is
the set of integral ideals in an algebraic number field, $\mathcal{P}$ is the
set of prime ideals, and $|a|$ is the norm of $a$. Knopfmacher (1990) also
studies a more general setting where, for some $\delta>0$,
$\lim_{x\to\infty}xe^{-\delta x}\left|\pi_{G}[x]\right|=\delta.$
The authors have not attempted to modify Theorem 1.1 to fit this case.
## 2\. Pairwise Minima in a Geometric Probability Model
### 2.1 Geometric Random Vectors
Let $\mathcal{P}:=\left\\{p_{1},p_{2},\ldots\right\\}$ denote the rational
primes $\\{2,3,5,\ldots\\}$ in increasing order. Let $\mathcal{I}$ denote the
set of non-negative integer vectors $\left(e_{1},e_{2},\ldots\right)$ for
which $\sum e_{i}<\infty$. Let $X_{1},X_{2},\ldots$ be (possibly dependent)
positive integer random variables, whose joint law has the property that, for
every $k\in\mathbb{N}$. and every
$\left(e_{1},e_{2},\ldots\right)\in\mathcal{I}$ for which $e_{k}=0$,
$\mathbb{P}\left[X_{k}\geq m|\underset{i\neq
k}{\cap}\left\\{X_{i}=e_{i}\right\\}\right]\leq\left(\frac{1}{p_{k}}\right){}^{m}.$
(5)
Let $\zeta$ denote the random vector:
$\zeta:=\left(X_{1},X_{2},\ldots\right)\in\mathbb{N}^{\mathbb{N}}.$ (6)
Consider the finite-dimensional projections of $X_{1},X_{2},\ldots$ as a
general model for prime multiplicities in the prime factorization of a random
integer, without specifying exactly how that integer will be sampled. Let
$\zeta^{(1)},\zeta^{(2)},\ldots,\zeta^{(N)}$ be independent random vectors,
all having the same law as $\zeta$ in (6). Write $\zeta^{(k)}$ as
$\left(X_{1}^{k},X_{2}^{k},\ldots\right)$. Then
$L_{j,k}:=\sum_{i}\min\left\\{X_{i}^{k},X_{i}^{j}\right\\}\log\left[p_{i}\right]$
is a model for the log of the GCD of two such random integers. We shall now
derive an upper bound for
$\Delta_{N}:=\underset{1\leq k<j\leq N}{\max}\left\\{L_{j,k}\right\\},$
which models the log maximum of the pairwise GCD among a set of $N$ "large,
random" integers.
### 2.2 Proposition
Assume the joint law of the components of $\zeta$ satisfies (5).
(i) For every $s\in(0,1)$, the following expectation is finite:
$\mathbb{E}\left[e^{sL_{k,j}}\right]<\prod_{i}\left(1+\frac{p_{i}^{s}-1}{p_{i}^{2}-p_{i}^{s}}\right)=:C_{s}<\infty,s<1.$
(7)
(ii) For any $s\in(0,1)$, and $b>\left.C_{s}\right/2$, for $C_{s}$ as in
(7), there is an upper bound:
$\mathbb{P}\left[\Delta_{N}\geq\log\left[N^{2/s}\right]+s^{-1}\log[b]\right]\leq\frac{C_{s}}{2b}<1.$
(8)
Proof: Consider first the case where $X_{1},X_{2},\ldots$ are independent
Geometric random variables, and
$\mathbb{P}\left[X_{k}\geq
m\right]=\left(\frac{1}{p_{k}}\right){}^{m},m=1,2,\ldots$
It is elementary to check that, for $s\in(0,1)$, and any $p\in\mathcal{P}$, if
$X^{\prime\prime},X^{\prime}$ are independent Geometric random variables with
$\mathbb{P}\left[X^{\prime\prime}\geq
m\right]=p^{-m}=\mathbb{P}\left[X^{\prime}\geq m\right],m=1,2,\ldots,$
then their minimum is also a Geometric random variable, which satisfies
$\mathbb{E}\left[p^{s\min\left\\{X^{\prime\prime},X^{\prime}\right\\}}\right]=1+\frac{p^{s}-1}{p^{2}-p^{s}}<1+p^{s-2}.$
It follows from the independence assumption that
$\mathbb{E}\left[e^{sL_{k,j}}\right]=\mathbb{E}\left[\prod_{i}p_{i}^{s\min\left\\{X_{i}^{k},X_{i}^{j}\right\\}}\right]=\prod_{i}\left(1+\frac{p_{i}^{s}-1}{p_{i}^{2}-p_{i}^{s}}\right)=C_{s}.$
This verifies the assertion (7). Markov’s inequality shows that, for any
$s\in(0,1)$
$C_{s}\geq e^{st}\mathbb{P}\left[L_{k,j}\geq t\right].$
Furthermore
$\mathbb{P}\left[\underset{1\leq k<j\leq N}{\max}\left\\{L_{k,j}\right\\}\geq
t\right]=\mathbb{P}\left[\underset{1\leq k<j\leq N}{\cup}\left\\{L_{k,j}\geq
t\right\\}\right]$
$\leq\sum_{1\leq k<j\leq N}\mathbb{P}\left[L_{k,j}\geq
t\right]=\frac{N(N-1)}{2}\mathbb{P}\left[L_{k,j}\geq t\right].$
It follows that, for $s\in(0,1)$, $b>0$, and
$t:=s^{-1}\log\left[bN^{2}\right]$
$\mathbb{P}\left[\Delta_{N}\geq\log\left[N^{2/s}\right]+s^{-1}\log[b]\right]\leq\frac{N^{2}}{2}e^{-st}C_{s}=\frac{C_{s}}{2b}.$
It remains to consider the case where $X_{1},X_{2},\ldots$ satisfies (5),
without the independence assumption. Choose a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ on which independent Geometric random
variables $X_{1}^{\prime},X_{2}^{\prime},\ldots$ and
$X_{1}^{\prime\prime},X_{2}^{\prime\prime},\ldots$ are defined, such that for
all $i\geq 1$,
$\mathbb{P}\left[X_{i}^{\prime\prime}\geq
m\right]=p_{i}^{-m}=\mathbb{P}\left[X_{i}^{\prime}\geq m\right],m=1,2,\ldots.$
We propose to construct $\zeta^{(1)}=\left(X_{1}^{1},X_{2}^{1},\ldots\right)$
and $\zeta^{(2)}=\left(X_{1}^{2},X_{2}^{2},\ldots\right)$ by induction, on
this probability space $(\Omega,\mathcal{F},\mathbb{P})$, so that for each
$n\geq 1$, $\left\\{\left(X_{i}^{1},X_{i}^{2}\right)1\leq i\leq n\right\\}$
have the correct joint law, and
$X_{i}^{1}\leq X_{i}^{\prime};X_{i}^{2}\leq X_{i}^{\prime\prime},\text{
}i=1,2,\ldots.$
Once this is achieved, monotonicity implies
$\mathbb{E}\left[e^{sL_{1,2}}\right]\leq\mathbb{E}\left[\prod_{i}p_{i}^{s\min\left\\{X_{i}^{\prime},X_{i}^{\prime\prime}\right\\}}\right],$
so the desired result will follow from the previous one for independent
Geometric random variables.
Since $\zeta^{(1)}$ and $\zeta^{(2)}$are independent, it suffices to construct
$\zeta^{(1)}$ in terms of $X_{1}^{\prime},X_{2}^{\prime},\ldots$ so that
$X_{i}^{1}\leq X_{i}^{\prime}$ for all $i$. Let $\left(U_{i,j},i\geq 1,j\geq
0\right)$ be independent Uniform$(0,1)$ random variables. Suppose either
$i=1$, or else some values
$X_{1}^{1}=e_{1},X_{2}^{1}=e_{2},\ldots,X_{i-1}^{1}=e_{i-1}$ have already been
determined. By assumption, there exists parameters
$q_{i,k}:=\mathbb{P}\left[X_{i}\geq
k|\underset{j<i}{\cap}\left\\{X_{j}=e_{j}\right\\}\right]\leq\left(\frac{1}{p_{i}}\right){}^{k},k=1,2,\ldots.$
Use these to construct $X_{i}^{\prime}$ and $X_{i}^{1}$ as follows:
$X_{i}^{\prime}:=\min\left\\{k:U_{i,0}U_{i,1}\ldots
U_{i,k}>\left(\frac{1}{p_{i}}\right){}^{k}\right\\};$
$X_{i}^{1}:=\min\left\\{k:U_{i,0}U_{i,1}\ldots U_{i,k}>q_{i,k}\right\\}\leq
X_{i}^{\prime}.$
This completes the construction and the proof, giving the result (8).
$\square$
## 3\. Lower Bound for Largest Collision
### 3.1 Random Vectors with Independent Components
Let $\mathcal{P}:=\left\\{p_{1},p_{2},\ldots\right\\}$ denote the rational
primes $\\{2,3,5,\ldots\\}$ in increasing order, and let
$a_{j}:=\left(\log\left[p_{j}\right]\right){}^{1/2}$. Instead of the Geometric
model (5), switch to a Bernoulli model in which $Z_{1},Z_{2},\ldots$ are
independent Bernoulli random variables, with
$\mathbb{P}\left[Z_{j}=1\right]:=\frac{1}{p_{j}}.$ (9)
Let $\xi$ denote the random vector
$\xi:=\left(a_{1}Z_{1},a_{2}Z_{2},\ldots\right)\in[0,\infty)^{\mathbb{N}}.$
(10)
under this new assumption, and let $\xi^{(1)},\xi^{(2)},\ldots,\xi^{(N)}$ be
independent random vectors, all having the same law as $\xi$. Note that
$\xi^{(1)}\cdot\xi^{(2)}$ is not a suitable model for the GCD of two random
integers, because the independence assumption (9) is not realistic. However it
is a useful context to develop the techniques which will establish the lower
bound in Theorem 1.1.
Write $\xi^{(k)}=\left(a_{1}Z_{1}^{k},a_{2}Z_{2}^{k},\ldots\right)$. We seek a
lower bound on the log of the largest prime $p_{i}$ at which a "collision"
occurs, meaning that $Z_{i}^{j}=1=Z_{i}^{k}$ for some $j,k$:
$\Delta_{N}^{\prime}:=\underset{1\leq k<j\leq
N}{\max}\left\\{\underset{i}{\max}\left\\{Z_{i}^{j}Z_{i}^{k}\log\left[p_{i}\right]\right\\}\leq\underset{1\leq
k<j\leq N}{\max}\left\\{\xi^{(k)}\cdot\xi^{(j)}\right\\}.\right.$
### 3.2 Proposition
Given $\delta\in(0,\infty)$, define $\varphi_{N}:=\varphi_{N}[\delta]$
implicitly by the identity
$\underset{\varphi_{N}}{\overset{2\varphi_{N}}{\int}}\frac{N^{2}dx}{2x^{2}\log[x]}=\delta.$
(11)
Under the assumption of independence of the components of the random vector
(10),
$\lim_{N\to\infty}\mathbb{P}\left[\Delta_{N}^{\prime}\geq\log\left[\varphi_{N}[\delta]\right]\right]\geq
1-e^{-\delta}.$ (12)
Remark: From the integration bounds:
$\frac{1}{2\varphi_{N}\log\left[2\varphi_{N}\right]}=\frac{1}{\log\left[2\varphi_{N}\right]}\underset{\varphi_{N}}{\overset{2\varphi_{N}}{\int}}\frac{dx}{x^{2}}<\frac{2\delta}{N^{2}}<\frac{1}{\log\left[\varphi_{N}\right]}\underset{\varphi_{N}}{\overset{2\varphi_{N}}{\int}}\frac{dx}{x^{2}}=\frac{1}{2\varphi_{N}\log\left[\varphi_{N}\right]}.$
it follows that $\varphi_{N}$, defined in (11), satisfies
$\varphi_{N}\log\left[\varphi_{N}\right]/N^{2}\to 0.25/\delta.$ Hence for all
sufficiently large $N$, $\varphi_{N}<\left.N^{2}\right/2$, and
$\varphi_{N}>\frac{N^{2}}{4\delta\log\left[2\varphi_{N}\right]}>\frac{N^{2}}{8\delta\log[N]}.$
(13)
The proof uses the following technical Lemma, which the reader may treat as a
warm-up exercise for the more difficult Proposition 4.1.
### 3.3 Lemma
Let $\mathcal{P}_{N}$ denote the set of primes $p$ such that
$\varphi_{N}<p\leq 2\varphi_{N}$. Let
$\left\\{Z_{p}^{k},p\in\mathcal{P}_{N},1\leq k\leq N\right\\}$ be independent
Bernoulli random variables, where $\mathbb{P}\left[Z_{p}^{k}=1\right]=1/p$.
Take $D_{p}:=Z_{p}^{1}+\ldots+Z_{p}^{N}$. Then
$\lim_{N\to\infty}\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cup}\left\\{D_{p}\geq
2\right\\}\right]=1-e^{-\delta}.$ (14)
Proof: Binomial probabilities give:
$\mathbb{P}\left[D_{p}\leq
1\right]=\left(1-\frac{1}{p}\right)^{N}+\frac{N}{p}\left(1-\frac{1}{p}\right)^{N-1}=\text{
}\left(1-\frac{1}{p}\right)^{N}\left(1+\text{ }\frac{N}{p-1}\right)$
$=\text{
}\left(1-\frac{N}{p}+\frac{N(N-1)}{2p^{2}}-\ldots\right)\left(1+\text{
}\frac{N}{p-1}\right)=1-\frac{N^{2}}{2p^{2}}+O\left(\frac{N}{\varphi_{N}^{2}}\right)+O\left(\left(\frac{N}{\varphi_{N}}\right){}^{3}\right).$
Independence of $\left\\{Z_{p}^{k},p\in\mathcal{P}_{N},1\leq k\leq N\right\\}$
implies independence of $\left\\{D_{p},p\in\mathcal{P}_{N}\right\\}$, so
$\log\left[\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cap}\left\\{D_{p}\leq
1\right\\}\right]\right]=\text{
}\sum_{p\in\mathcal{P}_{N}}\log\left[\mathbb{P}\left[D_{p}\leq
1\right]\right]$
$=\sum_{p\in\mathcal{P}_{N}}\log\left[1-\frac{N^{2}}{2p^{2}}\right]+O\left(\frac{N\left|\mathcal{P}_{N}\right|}{\varphi_{N}^{2}}\right)+O\left(\frac{N^{3}\left|\mathcal{P}_{N}\right|}{\varphi_{N}^{3}}\right).$
Using the estimates
$\varphi_{N}\log\left[\varphi_{N}\right]=O\left(N^{2}\right)$,
$\left|\mathcal{P}_{N}\right|=O\left(\varphi_{N}/\log\left[\varphi_{N}\right]\right)$,
and $p\left/\varphi_{N}\right.\leq 2$, the last expression becomes
$=-\sum_{p\in\mathcal{P}_{N}}\frac{N^{2}}{2p^{2}}+O\left(\frac{N^{2}}{\varphi_{N}^{2}}\right)+O\left(\frac{N}{\varphi_{N}\log\left[\varphi_{N}\right]}\right)+O\left(\frac{N^{3}}{\varphi_{N}^{2}\log\left[\varphi_{N}\right]}\right).$
All terms but the first vanish in the limit, while the Prime Number Theorem
ensures that
$\lim_{N\to\infty}\sum_{p\in\mathcal{P}_{N}}\frac{N^{2}}{2p^{2}}=\delta.$
Therefore
$\lim_{N\to\infty}\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cap}\left\\{D_{p}\leq
1\right\\}\right]=\text{ }e^{-\delta}.$
Thus the limit (14) follows. $\square$
#### 3.3.1 Proof of Proposition
According to our model, if $D_{p}\geq 2$ for some $p=p_{i}\in\mathcal{P}_{N}$,
then there are indices $1\leq k<j\leq N$ for which $Z_{i}^{j}=1=Z_{i}^{k}$.
Since $\log\left[p_{i}\right]\geq\log\left[\varphi_{N}[\delta]\right]$,
$\lim_{N\to\infty}\mathbb{P}\left[\Delta_{N}^{\prime}\geq\log\left[\varphi_{N}[\delta]\right]\right]\geq\lim_{N\to\infty}\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cup}\left\\{D_{p}\geq
2\right\\}\right]=1-e^{-\delta}.$
This verifies (12). $\square$
## 4\. Application: Pairwise GCDs of Many Uniform Random Integers
We shall now prove an analogue of Lemma 3.3 which applies to random integers,
dropping the independence assumption for the components of the random vector
(10).
### 4.1 Proposition
Suppose $\alpha>0$, and $T_{1},\ldots,T_{N}$ is a random sample, drawn with
replacement, from the integers $\left\\{n\in\mathbb{N}:n\leq e^{\alpha
N}\right\\}$. Given $\delta\in(0,\infty)$, define
$\varphi_{N}:=\varphi_{N}[\delta]$ implicitly by the identity (11) . Let
$\mathcal{P}_{N}$ denote the set of primes $p$ such that $\varphi_{N}<p\leq
2\varphi_{N}$; for $p\in\mathcal{P}_{N}$ let $D_{p}$ denote the number of
elements of $\left\\{T_{1},\ldots,T_{N}\right\\}$ which are divisible by $p$ .
Then
$\lim_{N\to\infty}\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cup}\left\\{D_{p}\geq
2\right\\}\right]=1-e^{-\delta}.$ (15)
Proof: As noted above, the Prime Number Theorem ensures that
$\lim_{N\to\infty}\sum_{p\in\mathcal{P}_{N}}\frac{N^{2}}{2p^{2}}=\delta.$
More generally, the alternating series for the exponential function ensures
that there is an even integer $d\geq 1$ such that, given $\epsilon\in(0,1)$,
for all sufficiently large $N$,
$1-e^{-\delta/(1+\epsilon)}<\sum_{r=1}^{d}(-1)^{r+1}I_{r}<1-e^{-\delta/(1-\epsilon)}$
where, for $\left\\{p_{1},\ldots,p_{r}\right\\}\subset\mathcal{P}_{N}$
$I_{r}{}{F39E}:=\sum_{p_{1}<\ldots<p_{r}}\frac{N^{2r}}{2^{r}\left(p_{1}\ldots
p_{r}\right){}^{2}},r=1,2,\ldots,d.$
Because $\varphi_{N}/N^{2}\to 0$, it follows that, for every
$\left\\{p_{1},\ldots,p_{d}\right\\}\subset\mathcal{P}_{N}$,
$\frac{p_{1}\ldots p_{d}}{e^{\alpha
N}}<\frac{\left(\varphi_{N}\right){}^{d}}{e^{\alpha N}}<e^{2d\log[N]-\alpha
N}\to 0.$
Suppose that, for this constant value of $d$, we fix some
$\left\\{p_{1},\ldots,p_{d}\right\\}\subset\mathcal{P}_{N}$; instead of
sampling $T_{1},\ldots,T_{N}$ uniformly from integers up to $e^{\alpha N}$,
sample $T_{1}^{\prime},\ldots,T_{N}^{\prime}$ uniformly from integers up to
$p_{1}\ldots p_{d}\left\lfloor e^{\alpha N}/\left(p_{1}\ldots
p_{d}\right)\right\rfloor.$
¿From symmetry considerations, the Bernoulli random variables
$B_{1}^{\prime},\ldots,B_{d}^{\prime}$ are independent, with parameters
$1\left/p_{1}\right.,\ldots,1\left/p_{d}\right.$, respectively where
$B_{i}^{\prime}$ is the indicator of the event that $p_{i}$ divides
$T_{1}^{\prime}$. By elementary reasoning,
$\mathbb{P}\left[D_{p}\geq
2\right]=\frac{N^{2}}{2p^{2}}+O\left(\left(N\left/\varphi_{N}\right.\right){}^{3}\right);$
$\mathbb{P}\left[D_{p_{1}}\geq 2,\ldots,D_{p_{r}}\geq
2\right]=\frac{N^{2r}}{2^{r}\left(p_{1}\ldots
p_{r}\right){}^{2}}+O\left(\left(N\left/\varphi_{N}\right.\right){}^{2r+1}\right),r=1,2,\ldots,d.$
If we were to sample $T_{1},\ldots,T_{N}$ instead of
$T_{1}^{\prime},\ldots,T_{N}^{\prime}$, the most that such a probability could
change is
$\mathbb{P}\left[\underset{i=1}{\overset{N}{\cup}}\left\\{T_{i}\neq
T_{i}^{\prime}\right\\}\right]\leq\frac{Np_{1}\ldots p_{d}}{e^{\alpha
N}}<e^{(2d+1)\log[N]-\alpha N}.$
The same estimate holds for any choice of
$\left\\{p_{1},\ldots,p_{d}\right\\}\subset\mathcal{P}_{N}$. By the inclusion-
exclusion formula, taken to the first $d$ terms,
$\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cup}\left\\{D_{p}\geq
2\right\\}\right]\geq\sum_{p\in\mathcal{P}_{N}}\mathbb{P}\left[D_{p}\geq
2\right]-\sum_{p_{1}<p_{2}}\mathbb{P}\left[D_{p_{1}}\geq 2,D_{p_{2}}\geq
2\right]+\ldots-\sum_{p_{1}<\ldots<p_{d}}\mathbb{P}\left[D_{p_{1}}\geq
2,\ldots,D_{p_{d}}\geq 2\right]$
$=\sum_{r=1}^{d}(-1)^{r+1}I_{r}+O\left(\left(N\left/\varphi_{N}\right.\right){}^{3}\right)+\left(\begin{array}[]{c}N\\\
d\end{array}\right)e^{(2d+1)\log[N]-\alpha N}.$
So under this simplified model, the reasoning above combines to show that, for
all sufficiently large $N$,
$1-e^{-\delta/(1+\epsilon)}<\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cup}\left\\{D_{p}\geq
2\right\\}\right]<1-e^{-\delta/(1-\epsilon)}.$
Since $\epsilon$ can be made arbitrarily small, this verifies the result.
$\square$
### 4.2 Proof of Theorem 1.1
Suppose $\alpha>0$, and $T_{1},\ldots,T_{N}$ is a random sample, drawn with
replacement, from the integers $\left\\{n\in\mathbb{N}:n\leq e^{\alpha
N}\right\\}$. Let $\Lambda_{j,k}$ denote the largest common prime factor of
$T_{j}$ and $T_{k}$. Take
$\Delta_{N}^{\prime}:=\underset{1\leq k<j\leq
N}{\max}\left\\{\log\left[\Lambda_{j,k}\right]\right\\}.$
In the language of Proposition 4.1, if $D_{p}\geq 2$ for some
$p\in\mathcal{P}_{N}$, then there are indices $1\leq k<j\leq N$ for which
$\Lambda_{j,k}\text{ }>\varphi_{N}$. So inequality (13) and Proposition 4.1
imply that, for any $\theta=8\delta>0$
$\lim_{N\to\infty}\mathbb{P}\left[\Delta_{N}^{\prime}\geq
2\log[N]-\log\left[\log\left[N^{\theta}\right]\right]\right]\geq\lim_{N\to\infty}\mathbb{P}\left[\Delta_{N}^{\prime}\geq\log\left[\varphi_{N}[\theta/8]\right]\right]$
$\geq\lim_{N\to\infty}\mathbb{P}\left[\underset{p\in\mathcal{P}_{N}}{\cup}\left\\{D_{p}\geq
2\right\\}\right]=1-e^{-\theta/8}.$
This is precisely the lower bound (3). For any $\eta>0$, the lower bound in
(1) follows from:
$\lim_{N\to\infty}\mathbb{P}\left[\Delta_{N}^{\prime}>(2-\eta)\log[N]\right]=1.$
Let $\Gamma_{j,k}\geq\Lambda_{j,k}$ denote the Greatest Common Divisor of
$T_{j}$ and $T_{k}$. To obtain the upper bound (2) on $\Gamma_{j,k}$, it
suffices by Proposition 2.2 to check that condition (5) is valid, when $X_{i}$
denotes the multiplicity to which prime $p_{i}$ divides $T_{1}$. Take any
positive integer $r\geq 1$, any prime $p_{k}$ coprime to $r$, and any $m\geq
1$. The conditional probability that $p_{k}^{m}$ divides $T_{1}$, given that
$r$ divides $T_{1}$, is
$\frac{\left\lfloor e^{\alpha
N}/\left(rp_{k}^{m}\right)\right\rfloor}{\left\lfloor\left.e^{\alpha
N}\right/r\right\rfloor}\leq\left(\frac{1}{p_{k}}\right){}^{m}.$
So condition (5) holds. Thus (8) holds, which is equivalent to (2).
Finally we derive the upper bound in (1), for an arbitrary $\eta>0$. Fix
$\epsilon\in(0,1)$ and $\eta>0$. Select $s\in(0,1)$ to satisfy $2/s=2+\eta/2$.
Then choose $b=\left.C_{s}^{\prime}\right/\epsilon$. According to (8),
$\mathbb{P}\left[\Delta_{N}\geq(2+\eta/2)\log[N]+s^{-1}\log[b]\right]\leq\epsilon/2.$
For any $N$ sufficiently large so that $(\eta/2)\log[N]>s^{-1}\log[b]$,
$\mathbb{P}\left[\Delta_{N}\geq(2+\eta)\log[N]\right]\leq\epsilon/2.$
This yields the desired bound (1). $\square$
## References
* [1] Patrick Billingsley, Convergence of Probability Measures, Wiley, 1999.
* [2] John Knopfmacher, Abstract Analytic Number Theory, Dover, New York, 1990.
|
arxiv-papers
| 2009-11-13T20:58:51 |
2024-09-04T02:49:06.456417
|
{
"license": "Public Domain",
"authors": "R. W. R. Darling, E. E. Pyle",
"submitter": "R W R Darling Ph. D.",
"url": "https://arxiv.org/abs/0911.2660"
}
|
0911.2765
|
School of Mathematical Sciences, University of Nottingham, Nottingham NG72RD,
England
# Multifractality and Freezing Phenomena in Random Energy Landscapes: an
Introduction 111Lectures at International Summer School ”Fundamental Problems
in Statistical Physics XII” held on August 31 - September 11, 2009 at Leuven,
Belgium
Yan V Fyodorov222e-mail: yan.fyodorov@nottingham.ac.uk
###### Abstract
We start our lectures with introducing and discussing the general notion of
multifractality spectrum for random measures on lattices, and how it can be
probed using moments of that measure. Then we show that the Boltzmann-Gibbs
probability distributions generated by logarithmically correlated random
potentials provide a simple yet nontrivial example of disorder-induced
multifractal measures. The typical values of the multifractality exponents can
be extracted from calculating the free energy of the associated Statistical
Mechanics problem. To succeed in such a calculation we introduce and discuss
in some detail two analytically tractable models for logarithmically
correlated potentials. The first model uses a special definition of distances
between points in space and is based on the idea of multiplicative cascades
which originated in theory of turbulent motion. It is essentially equivalent
to statistical mechanics of directed polymers on disordered trees studied long
ago by B. Derrida and H. Spohn in [12]. In this way we introduce the notion of
the freezing transition which is identified with an abrupt change in the
multifractality spectrum. Second model which allows for explicit analytical
evaluation of the free energy is the infinite-dimensional version of the
problem which can be solved by employing the replica trick. In particular, the
latter version allows one to identify the freezing phenomenon with a mechanism
of the replica symmetry breaking (RSB) and to elucidate its physical meaning.
The corresponding 1-step RSB solution turns out to be marginally stable
everywhere in the low-temperature phase. We finish with a short discussion of
recent developments and extensions of models with logarithmic correlations, in
particular in the context of extreme value statistics. The first appendix
summarizes the standard elementary information about Gaussian integrals and
related subjects, and introduces the notion of the Gaussian Free Field
characterized by logarithmic correlations. Three other appendices provide the
detailed exposition of a few technical details underlying the replica analysis
of the model discussed in the lectures.
KEYWORDS: Multifractality; Freezing; Random Energy Model; Replica Symmetry
Breaking; Gaussian Free Field.
## 1 Introduction
Investigations of multifractal measures of diverse origin is for several
decades a very active field of research in various branches of applied
mathematical sciences like chaos theory, geophysics, oceanology, climate
studies, and finance, and in such areas of physics as turbulence and
statistical mechanics [1], and theory of quantum disordered systems [2]. The
main characteristics of multifractal patterns of data is to possess high
variability over a wide range of space or time scales, associated with huge
fluctuations in intensity which can be visually detected (see fig. 1). Another
common feature is presence of certain long-ranged powerlaw-type correlations
in data values.
Figure 1: Multifractal probability density for a model of quantum particle at
the critical point of Quantum Hall Effect, see [2]. Courtesy of F. Evers, A.
Mirlin and A. Mildenberger, unpublished.
To set the notations, consider a certain (e.g. hypercubic) lattice of linear
extent $L$ in $N-$dimensional space, with $M\sim L^{N}$ standing for the total
number of sites in the lattice. The measures of interest are usually defined
via weights $p_{i}$ associated with every lattice site $i=1,2,\ldots,M$ and
appropriately normalized to the total weight equal to unity as sketched below:
$0\leq p_{i}\leq 1,\quad\sum_{i=1}^{M}p_{i}=1$${\bf p}_{i}$ Figure 2: A square
lattice with weights attributed to the lattice sites.
One can imagine a few different spatial arrangements of weights $p_{i}$ across
the lattice sites. In the case of simply extended measures the weights are of
similar magnitude at each lattice site, the normalisation condition then
implying the scaling $p_{i}\sim M^{-1}$ in the large-$M$ limit. As a
generalisation of the above example one can imagine the non-zero weights
$p_{i}$ supported evenly on a fractal subset of lattice sites of effective
dimension $0\leq N_{ef}<N$. In the limiting case of $N_{ef}=0$ we then deal
with localised measures characterized by the weights $p_{i}$ essentially
different from zero only inside one or few blobs of finite total volume. In
such a situation weights stay finite even when $M\to\infty$, that is
$p_{i}=O(M^{0})$. Finally, in the most interesting case of multifractal
measures the weights scale differently at different sites: $p_{i}\sim
M^{-\alpha_{i}}$ 333Usually one defines exponents via the relation $p_{i}\sim
L^{-N\alpha_{i}}$ i.e. by the reference to linear scale $L$ instead of the
total number of sites $M\sim L^{N}$. We however find it more convenient to get
rid of trivial spatial dimension factor $N$, and concentrate only on essential
parameter behaviour. The full set of exponents $0\leq\alpha_{i}<\infty$ can be
conveniently characterized by the density
$\rho(\alpha)=\sum_{i=1}^{M}\,\delta(\alpha-\alpha_{i})$ whose scaling
behaviour in the large-$M$ limit is expected to be nontrivial:
$\rho(\alpha)\sim M^{f(\alpha)}$, with the convex function $f(\alpha)$ known
in this context as the multifractality spectrum or singularity spectrum, see
Fig. 3. In view of the identity $\int_{0}^{\infty}\rho(\alpha)\,d\alpha\equiv
M$ we see that at the point of maximum $\alpha=\alpha_{0}$ we must have
$f(\alpha_{0})=1$. Note also that the total number
$m(\alpha)=\int_{0}^{\alpha}\rho(\alpha)\,d\alpha$ of sites of the lattice
characterized by the scaling exponents $\alpha_{i}<\alpha(<\alpha_{0})$
satisfies for $M\gg 1$ the inequality $m(\alpha)\sim M^{f(\alpha)}\geq 1$,
hence $f(\alpha)\geq 0$ for $\alpha<\alpha_{0}$. Modifying this argument one
can show $f(\alpha)\geq 0$ also for $\alpha>\alpha_{0}$. The condition
$f(\alpha)=0$ defines generically the minimal $\alpha_{-}$ and maximal
$\alpha_{+}$ threshold values of the exponents which can be observed in a
given typical pattern. Note that the constraint $p_{i}\leq 1$ implies
$\alpha_{-}\geq 0$.
$1$$0$$f(\alpha)$$\alpha_{-}$$\alpha_{+}$$\alpha_{0}$$\alpha$ Figure 3: Shape
of a typical multifractality spectrum.
An alternative, frequently more practical way of describing multifractality is
via the set of exponents $\tau_{q}$ characterizing the large-$M$ behaviour of
the so-called inverse participation ratios (IPR’s) $P_{q}$ which are simply
the moments of the corresponding measure:
$P_{q}=\sum_{i=1}^{M}\,p_{i}^{q}=\int\,M^{-q\alpha}\rho(\alpha)\,d\alpha\,.$
(1)
Substituting in the above definition the relation $\rho(\alpha)\sim
M^{f(\alpha)}$ one can evaluate the integral in the large-$M$ limit by the the
steepest descent (also known as Laplace) method, see Appendix A. One then
finds the relation between $\tau_{q}$ and $f(\alpha)$ given by the Legendre
transform:
$P_{q}\sim M^{-\tau_{q}},\quad\tau_{q}=q\alpha-f(\alpha)\quad\mbox{where}\quad
q=\frac{df}{d\alpha}\,\,.$ (2)
In particular, at the point of maximum $q=0$ and as from the very definition
$\tau_{0}=-1$ we immediately see that
$f(\alpha_{0})\equiv\max_{\alpha}\\{f(\alpha)\\}=1$, cf. Fig. 3.
The above description is valid for multifractal measures of any nature. In
recent years important insights were obtained for disorder-generated
multifractality, see [2] and [3] for a comprehensive discussion in the context
of Anderson localisation transitions, and [4, 5] for examples related to
Statistical Mechanics in disordered media which are closer to the context of
the present lectures. One of the specific features of multifractality in the
presence of disorder is a possibility of existence of two different sets of
exponents, $\tau_{q}$ versus $\tilde{\tau}_{q}$, governing the scaling
behaviour of typical $P_{q}$ versus disorder averaged IPR’s, $<P_{q}>\sim
M^{-\tilde{\tau}_{q}}$. So by definition
$\tau_{q}=-\frac{\left\langle\ln{P_{q}}\right\rangle}{\ln{M}},\quad\tilde{\tau_{q}}=-\frac{\ln{\left\langle
P_{q}\right\rangle}}{\ln{M}},\quad$ (3)
Here and henceforth the brackets stand for the averaging over different
realisations of the disorder. The first type of averaging featuring in the
above equation is traditionally called in the literature ”quenched” , and
second one is known as ”annealed”. It is known that the ”quenched” values
correspond to values of exponents which one finds in a ”typical” realisation
of disorder. The possibility of ”annealed” average to produce results
different from typical is related to a possibility of disorder-averaged
moments to be dominated by exponentially rare configurations in some parameter
range. A related aspect of the problem is that the ”annealed” multifractality
spectrum recovered from the multifractal exponents $\tilde{\tau}_{q}$ via the
Legendre transform (1) can be negative: $\tilde{f}(\alpha)<0$, see fig. 4.
$1$$0$$\tilde{f}(\alpha)$$\alpha_{-}$$\alpha_{+}$$\alpha_{0}$$\alpha$ Figure
4: Shape of an ”annealed” multifractality spectrum with negative parts
(dotted) extracted from the disorder-averaged moments and reflecting
exponentially rare events, see the text.
Indeed, those values reflect events which are exponentially rare [6] and need
exponentially many realisations of disorder to be observed experimentally or
numerically. On the other hand, as was noted in [2], when dealing with typical
multifractality spectrum $f^{typ}(\alpha)$ by exploiting the relation (1) one
has to specify the limits of integration over $\alpha$ to be precisely
$\alpha_{-}\leq\alpha\leq\alpha_{+}$. IPR moments are then given by
$P^{typ}_{q}=\int_{\alpha_{-}}^{\alpha_{+}}\,M^{-q\alpha+f^{typ}(\alpha)}d\alpha\sim
M^{-\tau^{typ}_{q}}\,,$ (4)
and calculating the above integral by the steepest descent method reveals that
typical (that is quenched) exponents $\tau_{q}=\tau_{q}^{typ}$ are related to
$f^{typ}(\alpha)$ by Legendre transform only in the range
$\frac{df}{d\alpha}|_{\alpha_{+}}=q_{min}\leq q\leq
q_{max}=\frac{df}{d\alpha}|_{\alpha_{-}}$, whereas outside that interval the
exponents behave linearly in $q$, that is $\tau^{typ}_{q}=q\alpha_{\pm}$, see
fig. 5. We will not dwell on the differences ”quenched” vs. ”annealed”
exponents further and direct the interested reader to the recent works [5] and
[3] for more detail and further references 444Note that unfortunately the
definitions of the termination of the multifractality spectrum used in [5] and
in [3] are essentially different. The work [5] uses the definitions set up in
the comprehensive review [2] which could be consulted in case of confusion..
In the present set of lectures we will concentrate exclusively on calculating
typical (=”quenched”) values of IPR exponents for some class of models.
$\tau_{q}=\tau^{typ}_{q}$$q_{min}$$-1$$1$$q_{max}$$q$ Figure 5: $q$-dependence
of typical (”quenched”) multifractality exponents $\tau_{q}$. Dotted lines
show linear behaviour, see the text.
Introduced through the moments involving summation over all the lattice sites,
cf. (1), the multifractality by itself says nothing about more delicate
questions, for example about spatial correlations between weights at two
different sites of the lattice with coordinates, say, ${\bf x}_{1}$ and ${\bf
x}_{2}$, separated by a given distance $|{\bf x}_{1}-{\bf x}_{2}|$. The most
natural assumption which is satisfied by vast majority of multifractal
measures of actual experimental interest is the power-law decay of
correlations implied by full statistical spatial self-similarity of the random
measure:
$\left\langle p^{q}({\bf x}_{1})p^{s}({\bf x}_{2})\right\rangle\propto
L^{-N\,y(q,s)}\delta^{-N\,z(q,s)},\quad\delta=|{\bf x}_{1}-{\bf x}_{2}|\,.$
(5)
As statistical homogeneity of the random measure implies for local averages
$\left\langle p^{q}({\bf
x}_{1})\right\rangle=\frac{1}{M}\sum_{i=1}^{M}\,p_{i}^{q}\sim
L^{-N-N\tau_{q}}$ the equation (5) after setting $\delta\sim L$ yields the
relation for exponents:
$y(q,s)+z(q,s)-2=\tau_{q}+\tau_{s}$ (6)
which follows from assuming the decoupling $\left\langle p^{q}({\bf
x}_{1})p^{s}({\bf x}_{2})\right\rangle\approx\left\langle p^{q}({\bf
x}_{1})\right\rangle\left\langle p^{s}({\bf x}_{2})\right\rangle$ at large
separations $\delta=|{\bf x}_{1}-{\bf x}_{2}|\sim L\to\infty$. On the other
hand, for sites separated by a single lattice spacing $\delta=1$ we must have
$\left\langle p^{q}({\bf x}_{1})p^{s}({\bf
x}_{2})\right\rangle\approx\left\langle p^{q+s}({\bf x}_{1})\right\rangle\sim
L^{-N-N\tau_{q+s}}$, which after comparing with (5) allows one to relate the
exponents governing the spatial correlations to the multifractality exponents
as [7]
$y(q,s)=1+\tau_{q+s},\quad z(q,s)=1+\tau_{q}+\tau_{s}-\tau_{q+s}$ (7)
Further, it turns out to be instructive to exploit (5) for evaluating the
following correlation function:
$\left\langle\ln{p({\bf x}_{1})}\ln{p({\bf
x}_{2})}\right\rangle=\frac{\partial^{2}}{\partial q\partial s}\left\langle
p^{q}({\bf x}_{1})p^{s}({\bf x}_{2})\right\rangle|_{q=s=0}$ (8)
Remembering $\tau_{0}=-1$, and $\left\langle\ln{p({\bf
x}_{1})}\right\rangle=-N\ln{L}\frac{\partial\tau_{q}}{\partial q}|_{q=0}$ we
obtain after straightforward manipulations the following fundamental relation
$\left\langle\ln{p({\bf x}_{1})}\ln{p({\bf
x}_{2})}\right\rangle-\left\langle\ln{p({\bf
x}_{1})}\right\rangle\left\langle\ln{p({\bf
x}_{2})}\right\rangle=-g^{2}\,\ln{\frac{|x_{1}-x_{2}|}{L}},\quad
g^{2}=N\frac{\partial^{2}\tau_{q+s}}{\partial q\partial s}|_{q=s=0}>0$ (9)
valid for arbitrary self-similar multifractal field. In other words, we have
demonstrated that multifractality plus statistical selfsimilarity and
homogeneity of the random weights necessarily imply that the logarithms of
such weights must be correlated logarithmically in space.
Inverting such an argument suggests that possibly the simplest way to generate
random multifractal weights in the lattice is by constructing quantities
$\ln{p_{i}}$ at every lattice site $i$ as Gaussian-distributed random
variables correlated in precisely the way prescribed by (9). The resulting
model has a very natural interpretation in terms of the equilibrium
statistical mechanics. Indeed, consider a single classical particle subject to
a random Gaussian potential $V({\bf x})$. It is the standard fact of theory of
random processes[8] that if such a particle moves under the influence of the
thermal white noise according to the Langevin equation
$\dot{\bf x}=-\frac{\partial}{\partial{\bf x}}V\left({\bf x}\right)+\xi({\bf
x},t),\quad\overline{\xi({\bf x_{1}},t_{1})\xi({\bf
x_{2}},t_{2})}=2T\delta(t_{1}-t_{2})$
then the probability $P({\bf x},t)$ to find such a particle at a point ${\bf
x}$ of the sample of finite size $L$ will converge to the equilibrium Gibbs-
Boltzmann measure
$P({\bf x},t\to\infty)\to p_{\beta}({\bf x})=\frac{1}{Z(\beta)}\exp{-\beta
V({\bf x})}\,$
characterized by the inverse temperature $\beta={1}/{T}$. The normalization
$\int_{|{\bf x}|\leq L}p_{\beta}({\bf x})d{\bf x}\,=1$ implies the value of
the partition function to be given by
$Z(\beta)=\int_{|{\bf x}|\leq L}\exp{-\beta V({\bf x})}\,d{\bf x}\,.$ (10)
As obviously $\ln{p_{\beta}({\bf x})}=const-\beta V({\bf x})$ the weights
$p_{\beta}({\bf x})$ according to our discussion will be multifractal if the
potential $V({\bf x})$ is chosen logarithmically correlated in space:
$\left\langle V\left({\bf x}_{1}\right)\,V\left({\bf
x}_{2}\right)\right\rangle=-\,g^{2}\ln{\left[\frac{({\bf x}_{1}-{\bf
x}_{2})^{2}+a^{2}}{L^{2}}\right]},\quad a\ll L,\quad{\bf
x}\in\mathbb{R}^{N}\,,$ (11)
where we assumed $|{\bf x}|<L$, and the parameter $a$ stands for a small-scale
cutoff.
According to the general discussion, the multifractal structure of the Gibbs-
Boltzmann measure can be extracted from the knowledge of moments
$\quad P_{q}=\int_{|{\bf x}|\leq L}p^{q}_{\beta}({\bf x})\,d{\bf
x}=\frac{Z(\beta q)}{\left[Z(\beta)\right]^{q}}\sim
L^{-N\tau_{q}}\quad\mbox{as}\quad L\to\infty\,.$ (12)
Identifying $M\sim(L/a)^{N}$ , the Eqs.(12) and (10) imply the following
expression for the typical exponents $\tau_{q}$ in terms of the appropriately
normalized free energy of the system
$\quad\tau_{q}=|q|\beta{\cal F}(|q|\beta)-q\beta{\cal F}(\beta),\quad{\cal
F}(\beta)=-\lim_{M\to\infty}\frac{\left\langle\ln{Z(\beta)}\right\rangle}{\beta\ln{M}}\,.$
(13)
As shown in the Appendix A, the most natural random field with logarithmic
correlations corresponds to the so-called Gaussian Free Field (GFF) in two
spatial dimensions $N=2$, as well as its one-dimensional subsets. It is one of
the fundamental objects in physics and various issues of its statistics
attracted a lot of interest recently in conformal field theory, Schramm-
Loewner evolution, and two-dimensional quantum gravity, see e.g. some
discussion in [9]. Technically the problem of extracting the multifractality
exponents $\tau_{q}$ for the GFF amounts to ability to calculate efficiently
the disorder average of the free energy (13). Such task is in general
considered to be one of the most difficult problems in the statistical
mechanics of systems with quenched disorder and we will not be able to perform
such calculation explicitly in $N=2$ GFF case 555Actually, in recent years
some sophisticated probabilistic methods were developed which allowed to
address somewhat similar questions for GFF, see e.g. [10] and the references
therein. That development however goes beyond the remit of the present
lectures.. Instead, we are going to outline such calculation for two
particular choices of the models with logarithmically correlated potentials
where such calculation is indeed feasible. The first model uses a special
definition of distances between points in space and is based on the idea of
multiplicative cascades which originated in the theory of turbulence, see e.g.
discussion and further references in [11]. In fact, the model is essentially
equivalent to statistical mechanics of directed polymers on disordered trees
studied long ago in the seminal paper by B. Derrida and H. Spohn [12]. Our
second model will use standard Euclidean distances but exploits high
dimensionality of the embedding space: $N\to\infty$. Although the details of
the two models and the corresponding methods of solution may look rather
different, there is a general consensus that they address essentially the same
physics: the so-called freezing transition common to all disordered systems
with logarithmic correlations. And indeed we shall see that the resulting
multifractality spectrum will be identical. In the final section we will give
a short account of recent works on different aspects of logarithmically
correlated potentials.
## 2 Statistical mechanics for logarithmically correlated potentials
generated by multiplicative cascades
The construction we are going to describe below can be easily carried out in
any spatial dimension, but for simplicity we consider the one-dimensional case
of an interval of length $L$ with the left end at the origin. With each point
$0\leq{\bf X}\leq L$ of such an interval we can associate an infinite binary
string generated by expansion
${\bf
X}=L\left(\frac{x_{1}}{2}+\frac{x_{2}}{2^{2}}+\ldots+\frac{x_{n}}{2^{n}}+\ldots\right)=(x_{1}x_{2}x_{3}\ldots
x_{n}\ldots)$ (14)
where each $x_{n}$ is either $0$ or $1$. For some numbers the binary string is
not unique but by choosing the expansion with infinite number of zeroes to the
right it can always be made unique ( e.g. we use for $L/2$ the string
$(100\ldots)$ rather than $(0111\ldots)$). Then for any two points ${\bf X}$
and ${\bf Y}$ in the interval we can introduce the distance function defined
as $d({\bf X},{\bf Y})=\frac{L}{2^{n+1}}$ where $n$ is the maximal number of
first binary digits shared by ${\bf X}$ and ${\bf Y}$. For example, if ${\bf
X}=(0*****\ldots)$ and ${\bf Y}=(1*****\ldots)$ then $n=0$, hence $d({\bf
X},{\bf Y})=\frac{L}{2}$ (which is obviously the maximal possible distance
between the points in the interval), if ${\bf X}=(00****\ldots)$ and ${\bf
Y}=(01****\ldots)$ then $n=1$, hence $d({\bf X},{\bf Y})=\frac{L}{2^{2}}$,
etc. One can check that such a function $d({\bf X},{\bf Y})$ indeed satisfies
all the axioms for the distances: (i) $d({\bf X},{\bf Y})\geq 0,\forall{\bf
X}\neq{\bf Y}$, and $d({\bf X},{\bf Y})=0$ implies ${\bf X}={\bf Y}$ (ii)
$d({\bf X},{\bf Y})=d({\bf Y},{\bf X})$ and the triangle inequality (iii)
$d({\bf X},{\bf Y})+d({\bf Y},{\bf Z})\geq d({\bf X},{\bf Z})$ for any triple
${\bf X},{\bf Y},{\bf Z}$.
Now, let us associate with every point ${\bf X}$ an infinite set of random
i.i.d. variables $\phi_{k}({\bf X}),\,\,k=0,1,2,\ldots,\infty$ with zero mean
and variances chosen to satisfy:
$\left\langle\phi_{k}\left({\bf X}\right)\phi_{l}\left({\bf
Y}\right)\right\rangle=2g^{2}\ln{2}\,\delta_{l,k}\delta_{(x_{1}x_{2}x_{3}\ldots
x_{k}),(y_{1}y_{2}y_{3}\ldots y_{k})}$ (15)
where we used the Kronecker symbol: $\delta_{A,B}=1$ for $A=B$ and zero
otherwise, for any two objects $A$ and $B$ of arbitrary nature. Finally, with
any point ${\bf X}$ of the interval we associate a random potential
$V\left({\bf X}\right)$ according to the rule
$V\left({\bf X}\right)=\phi_{0}\left({\bf X}\right)+\phi_{1}\left({\bf
X}\right)+\ldots=\sum_{k=0}^{\infty}\phi_{k}\left({\bf X}\right)\,.$ (16)
This construction implies for any ${\bf X}\neq{\bf Y}$:
$\left\langle V\left({\bf X}\right)V\left({\bf
Y}\right)\right\rangle=\sum_{k=0}^{\infty}\left\langle\phi^{2}_{k}\left({\bf
X}\right)\right\rangle=2g^{2}\ln{2}\,(n+1)\,,$ (17)
where we assumed that the two points ${\bf X}$ and ${\bf Y}$ share precisely
$n$ first digits in the binary expansion. This implies that they are separated
by the distance $d({\bf X},{\bf Y})=\frac{L}{2^{n+1}}$, hence the above
formula takes the form
$\left\langle V\left({\bf X}\right)V\left({\bf
Y}\right)\right\rangle=-2g^{2}\ln{\frac{d({\bf X},{\bf Y})}{L}},\quad{\bf
X}\neq{\bf Y}.$ (18)
We see then that with respect to the chosen distance the constructed random
potential is logarithmically correlated in space. When dealing with
logarithmically correlated potentials one has to ensure the proper
regularization at small distances, as the logarithm obviously diverges for
${\bf X}\to{\bf Y}$. Various regularization schemes are possible, and in the
present situation one of the most natural is to replace continuous space of
the interval with a discrete lattice structure. In the particular case under
consideration we introduce a ”lattice” of $2^{K}=M$ sites, each site located
at one of the points ${\bf X}_{N}=\frac{N}{2^{K}},\quad
N=0,1,2,\ldots,2^{K}-1$. We can visualise this construction via the tree
diagram, associating the random fields $\phi_{l}({\bf X})$ to every branch of
the tree as sketched in Fig. 6 for $K=3$:
$\phi_{0}$$\phi_{1}(0)$$\phi_{1}(1)$$\phi_{2}(00)$$\phi_{2}(01)$$\phi_{2}(10)$$\phi_{2}(11)$$\phi_{3}(000)$$...$$...$$...$$\phi_{3}(111)$$0$$\frac{1}{8}$$\frac{2}{8}$$\frac{3}{8}$$\frac{4}{8}$$\frac{5}{8}$$\frac{6}{8}$$\frac{7}{8}$${\sf{K=3}}$
Figure 6: Lattice of $8$ sites and the corresponding tree diagram associating
random fields to every branch of the tree.
Now we can define the distances in the same fashion as before, but since the
maximal number of common digits can be at most $K$ we get for the variance of
the random potential a finite value (cf. (11)):
$\left\langle V^{2}\left({\bf
X}\right)\right\rangle=2g^{2}\ln{2}(K+1)\equiv-2g^{2}\ln{\frac{a}{L}}\,,$ (19)
where we have introduced the effective lattice cutoff given by $a=L/2^{K+1}$.
For this regularized lattice version we can now introduce the well-defined
Boltzmann-Gibbs weights
$p_{\beta}({\bf X}_{N})=\frac{1}{Z_{K}(\beta)}\exp{-\beta V({\bf
X}_{N})},\quad Z_{K}(\beta)=\sum_{N=0}^{2^{K}-1}\,\exp{-\beta V({\bf X}_{N})}$
(20)
and try to calculate the associated free energy $\langle\ln
Z_{K}(\beta)\rangle$, hence to extract the multifractality exponents
$\tau_{q}$, see (13). The value of the potential $V({\bf X}_{N})$ associated
with each lattice site ${\bf X}_{N}$ is obviously obtained by adding all the
random fields $\phi({\bf X})$ along the unique path connecting the site to the
top level of the tree diagram. This implies the essentially multiplicative
nature of the cascade model for the weight factors $exp\\{-\beta V({\bf
X}_{N})\\}$. The most efficient way to organize calculations amounts to
exploiting such a multiplicative structure combined with the hierarchical
organization of the model which is obvious from the tree diagram decomposition
as shown in Fig.7 below. The described structure implies that
$Z_{K}(\beta)=e^{-\beta\phi_{0}}\left[Z_{K-1}^{(L)}(\beta)+Z_{K-1}^{(R)}(\beta)\right]$
(21)
where $Z_{K-1}^{(L/R)}(\beta)$ corresponds to the left/right-hand subtree of
the tree in Fig.7 which is of the depth $K-1$ as reflected in the lower index.
$\phi_{0}$$Z_{K-1}^{(L)}(\beta)$$Z_{K-1}^{(R)}(\beta)$$...$$...$$...$$\underbrace{\quad\,\,\mbox{Left}\quad}$$\underbrace{\quad\,\,\mbox{Right}\quad}$
Figure 7: The tree diagram decomposition leading to recursive relations for
the partition function.
Note that the fields $\phi({\bf X})$ entering $Z_{K-1}^{(L)}(\beta)$ are
statistically independent of those entering $Z_{K-1}^{(R)}(\beta)$. To make
the direct use of the structure of the equation (21) it is expedient to
introduce the generating function
$G_{K}(p)=\left\langle e^{-pZ_{K}(\beta)}\right\rangle,\quad p\geq 0\,,$ (22)
which is simply the Laplace transform of the probability density of the
partition function. Denoting the probability density of the distribution for
the variable $\phi_{0}$ with ${\cal P}(\phi_{0})$ and exploiting that the
variables $Z_{K-1}^{(L)}(\beta)$ and $Z_{K-1}^{(R)}(\beta)$ are independent of
each other and identically distributed (i.i.d.) we arrive at the relation:
$G_{K}(p)=\int{\cal P}(\phi_{0})\left\langle
e^{-pe^{-\beta\phi_{0}}[Z^{(L)}_{K-1}(\beta)+Z^{(R)}_{K-1}(\beta)]}\right\rangle\,d\phi_{0}\equiv\int{\cal
P}(\phi)G^{2}_{K-1}\left(pe^{-\beta\phi}\right)\,d\phi\,.$ (23)
Precisely in the same way we can relate $G_{K-1}$ to $G_{K-2}$, etc. in a kind
of recursive procedure which starts with the obvious initial condition
$G_{0}(p)=e^{-p}$. Finally, it turns out that the subsequent analysis becomes
more transparent if one introduces a new variable
$x=-\frac{1}{\beta}\ln{p}\in(-\infty,\infty)$. We arrive therefore at the
recursion relations
$G_{l}(x)=\int{\cal P}(\phi)G^{2}_{l-1}\left(x+\phi\right)\,d\phi,\quad
l=1,2,\ldots K\,\,\mbox{and}\,\,G_{0}(x)=e^{-e^{-\beta x}}\,,$ (24)
where we have replaced $G_{l}\left(p=e^{-\beta\,x}\right)\to G_{l}(x)$, with
some abuse of notations.
Note: If from the very beginning we had considered a tree with arbitrary
constant branching $s>1$ instead of the binary tree with $s=2$ the above
recursion would be simply replaced by
$G_{l}(x)=\int{\cal P}(\phi)G^{s}_{l-1}\left(x+\phi\right)\,d\phi,\quad
G_{0}(p)=e^{-e^{-\beta x}}\,.$ (25)
where $\left\langle V^{2}\left({\bf X}\right)\right\rangle=2g^{2}(K+1)\ln{s}$
is the variance of the underlying logarithmically correlated potential, cf.
(19), and $M=s^{K}$ is the total number of points in the lattice.
To understand better the nature of the solution of the above equations in the
thermodynamic limit $K\to\infty$ it is instructive to consider the following
limiting case for the branching parameter: $s=1+\delta,\,\,\delta\ll 1$ . This
implies scaling the variable $\phi$ in such a way that $<\phi^{2}>\equiv
2g^{2}\ln{s}\approx 2g^{2}\delta$.
To be specific, one may just wish to use the Gaussian distribution ${\cal
P}(\phi)=\frac{1}{\sqrt{2\pi\delta}g}\exp{-\frac{\phi^{2}}{4g^{2}\delta}}$.
Then the right-hand side of (25) takes the form
$\int\frac{1}{\sqrt{2\pi\delta}g}e^{-\frac{\phi^{2}}{4g^{2}\delta}}\,G^{1+\delta}_{l-1}\left(x+\phi\right)\,d\phi\equiv\int_{-\infty}^{\infty}e^{-\frac{y^{2}}{2}}\,G^{1+\delta}_{l-1}\left(x+gy\sqrt{2\delta}\right)\,\frac{dy}{\sqrt{2\pi}}$
which after straightforwardly expanding in powers of $\delta$ reduces (25) to
$G_{l}(x)=G_{l-1}\left(x\right)+\delta\left[G_{l-1}\left(x\right)\ln{G_{l-1}\left(x\right)}+g^{2}\frac{d^{2}}{dx^{2}}G_{l-1}\left(x\right)\right]+O(\delta^{2})\,.$
(26)
Thus in such an approximation the function $G_{l}(x)$ experiences only small
change in one step of iteration: $G_{l}(x)-G_{l-1}(x)\propto\delta$.
Introducing to this end the variable $t=l\delta$ and consider it to be
continuous in the interval $t\in[0,t_{max}=K\delta\approx\ln{M}]$ we can
replace $G_{l}(x)\to G(x,t)$ and approximately write to the leading order
$G_{l}(x)-G_{l-1}(x)\approx\delta\frac{\partial}{\partial t}G(x,t)$. In this
approximation the relation (26) is replaced by a partial differential equation
on the function $G(x,t)$:
$\frac{\partial G}{\partial t}=g^{2}\frac{\partial^{2}G}{\partial
x^{2}}+G\ln{G},\quad G(x,0)=e^{-e^{-\beta x}}\,.$ (27)
We also note that (i) by its very definition the function $G(x,t)$ satisfies
the following conditions:
$0\leq G(x,t)\leq 1,\quad G(x\to-\infty,t)=0,\quad G(x\to\infty,t)=1\,$ (28)
and (ii) the values $G(x,t)=0$ and $G(x,t)=1$ solves the equation (27). All
these observations are typical for the partial differential equations having
the so-called travelling waves solutions of the form
$G(x,t)=W[x-m(t)],\quad\frac{d}{dt}m(t)\equiv c(t)\to
c\,t\quad\mbox{when}\quad t\to\infty,\,\,c=const>0$ (29)
where the constant $c$ plays the role of the asymptotic velocity of the front
propagation, see Fig.8.
$1$$0$$G(x,t)$$x$$\rightarrow ct$ Figure 8: Sketch of a typical front of the
travelling wave solution.
Substituting such a form to (28), denoting $\tau=x-m(t)$ (so that e.g.
$\frac{\partial G}{\partial t}=-\frac{dW}{d\tau}\frac{d}{dt}m(t)$) we see that
the partial differential equation in the limit $t\to\infty$ implies an
ordinary differential equation for $W(\tau)$ which can be written as:
$g^{2}\ddot{W}+c\dot{W}+\frac{d}{dW}U(W)=0,\quad\mbox{where}\quad
U(W)=\frac{W^{2}}{2}\left(\ln{W}-\frac{1}{2}\right)$ (30)
where we have introduced the notations $\dot{W}\equiv\frac{dW}{d\tau}$ and
$\ddot{W}\equiv\frac{d^{2}W}{d\tau^{2}}$. Obviously, we can interpret the
latter equation as the Newtonian equation describing the motion of a classical
particle of mass $g^{2}$ on the interval of the fictitious ”coordinate”
$W\in[0,1]$ in fictitious ”time” $\tau$ subject to the dissipative force
(”friction”) $c\dot{W}$ plus the potential force generated by the potential
$U(W)$ sketched in Fig. 9:
$1$$0$$U(W)$$W$stable equilibriumunstableequilibrium Figure 9: Sketch of the
potential driving the motion of a fictitious overdamped Newtonian particle,
see the text.
By inspection we see that the position $W=0$ corresponds to the maximum of the
potential, hence it is unstable equilibrium, and $W=1$ is the stable
equilibrium (minimum of the potential). As by its physical meaning $W\leq 1$
the motion of such particle must be overdamped, that is it should approach the
stable equilibrium $W=1$ in a monotonic way as $\tau\to\infty$ (i.e. the
damping should be strong enough to avoid oscillations around the stable
equilibrium which would bring $W$ out of the physical interval.) To this end,
let us consider in more detail the motion in the vicinity of the stable
equilibrium by expanding: $W=1-v,\,\,v(\tau)\ll 1$ so that to the linear order
$W\ln{W}\approx-v$ and (30) is reduced to the linear second-order differential
equation $g^{2}\ddot{v}+c\dot{v}+v=0$ whose general solution is given by
$v(\tau)=Ae^{\lambda_{+}\tau}+Be^{\lambda_{-}\tau},\quad\lambda_{\pm}=-\frac{1}{2g^{2}}(c\pm\sqrt{c^{2}-4g^{2}})$
(31)
To have a non-oscillatory (”overdamped”) asymptotic behaviour for
$\tau\to\infty$ is only possible for $c\geq 2g$, so that
$v(\tau\to\infty)\approx\left\\{\begin{array}[]{cc}Be^{-\frac{\tau}{2g^{2}}(c-\sqrt{c^{2}-4g^{2}})},&c>2g\\\
B\tau e^{-\frac{\tau}{g}},&c=2g\end{array}\right.$ (32)
To determine the value of $c$ it is natural to recall that according to the
definition (22) $G_{K}(x)=\left\langle\exp\\{-e^{-\beta
x}Z_{K}(\beta)\\}\right\rangle$ so that naively expanding for $x\to\infty$
gives $G_{K}(x\to\infty)\approx 1-e^{-\beta x}\left\langle
Z_{K}(\beta)\right\rangle+\ldots$. Using the Gaussian distribution of the
random potential chosen for the present model one finds
$\left\langle Z_{K}(\beta)\right\rangle=M\left\langle e^{-\beta V({\bf
X})}\right\rangle=M\exp\left\\{\frac{\beta^{2}}{2}<V^{2}({\bf
X}>\right\\}\approx Me^{\beta^{2}g^{2}\ln{M}}=M^{1+\beta^{2}g^{2}}\,,$ (33)
which implies
$G_{K}(x\to\infty)\approx
1-e^{-\beta\tau_{max}},\,\,\mbox{with}\,\,\tau_{max}=x-ct_{\max},\,\,t_{\max}\equiv\ln{M}\,\mbox{and}\,c\equiv
c(\beta)=\frac{1}{\beta}+\beta g^{2}\,.$ (34)
The above formula for the velocity $c=c(\beta)$ ensures the consistency
between the asymptotic behaviour in (34) and in (32) as for such a choice
holds the relation $\beta\equiv\frac{1}{2g^{2}}(c-\sqrt{c^{2}-4g^{2}})$.
Moreover, the choice is also compatible with the condition for overdamped
motion as $c-2g=(\beta g-1)^{2}\geq 0$. However there is some subtlety in that
formula which is most apparent if we follow the function $c(\beta)$ starting
from the high-temperature regime $\beta\ll g^{-1}$. We see that the wavefront
velocity $c(\beta)$ decreases with increasing $\beta$ (decreasing temperature)
down to the minimal value $c(\beta=g^{-1})=2g$, and then for $T<T_{c}=g$
starts increasing again, as schematically shown below.
$g^{-1}$$0$$2g$$c(\beta)$$\beta$
Temperature dependence of the front velocity. The dotted branch is unphysical
and should be replaced with the constant value $c=2g$.
A rigorous mathematical analysis of the travelling wave equations by
Bramson[13] revealed that such conclusion is however not quite correct.
Namely, Bramson proved that for the initial conditions of the type (27) the
actual velocity of the travelling wave front is indeed given by
$c(\beta)=\frac{1}{\beta}+\beta g^{2}$ for $\beta<\beta_{c}=g^{-1}$, but
sticks to the minimal value $c_{min}=2g$ everywhere in the low-temperature
regime $\beta>\beta_{c}$. Such a picture implies, in particular the asymptotic
form $W(\tau\to\infty)\approx 1-e^{-\beta_{c}\tau}$, or equivalently the
asymptotics
$G(x,t)\approx 1-e^{-\beta_{c}(x-c_{min}t)},\quad T<T_{c}=g$ (35)
so that the profile of the function $G(x,t)$ turns out to be temperature
independent (”frozen”) everywhere in the lower-temperature phase. Such
behaviour certainly signals of a kind of strong non-analyticity, as e.g. it
invalidates the expansion of the exponent in $\left\langle\exp\\{-e^{-\beta
x}Z_{K}(\beta)\\}\right\rangle$ which underlay our ”naive” analysis. It is
therefore appropriate to call such a drastic change of the behaviour a phase
transition, which is known in the literature as the freezing transition.
Qualitatively, the same picture holds generically for an arbitrary branching
$s>1$, that is for the solution of recursive equation (25). Namely, an
extension of the above analysis [12] shows that in the thermodynamic limit
$K\to\infty$ the solution takes the form
$G_{K}(x)=W[x-c_{\beta}K],\quad
c_{\beta}=\left\\{\begin{array}[]{cc}\frac{1}{\beta}\log{\left[s\int{\cal
P}(\phi)\,e^{-\beta\phi}\,d\phi\right]},&\beta<\beta_{c}\\\
\frac{1}{\beta_{c}}\log{\left[s\int{\cal
P}(\phi)\,e^{-\beta_{c}\phi}\,d\phi\right]},&\beta>\beta_{c}\end{array}\right.$
(36)
where $\beta_{c}$ is the point at which the function $c_{\beta}$ from the
upper line in (36) has its minimum:
$\frac{d}{d\beta}c_{\beta}|_{\beta=\beta_{c}}=0$. Such a knowledge allows one
to calculate our main object of interest, the mean free energy
$-\beta\overline{F}(\beta)=\lim_{K\to\infty}\frac{1}{K}\left\langle\ln{Z_{K}(\beta)}\right\rangle$.
To this end it is convenient to use the following integral representation for
the logarithm:
$\ln{Z}=\int_{0}^{\infty}\left[e^{-p}-e^{-pZ}\right]\frac{dp}{p}\,.$ (37)
Remembering the definition of the generating function: $G_{K}(p)=\left\langle
e^{-pZ_{K}(\beta)}\right\rangle$ and $G_{0}(p)=e^{-p}$ and also the relation
$p=e^{\beta x}$ we after averaging of (37) arrive at the important identity:
$\left\langle\ln{Z_{K}(\beta)}\right\rangle=\beta\int_{-\infty}^{\infty}\left[G_{0}(x)-G_{K}(x)\right]\,dx\,.$
(38)
Inspecting the travelling wave form of the solution (36) we observe that in
the limit $K\gg\frac{1}{\beta c_{\beta}}$ the difference $G_{0}(x)-G_{K}(x)$
(sketched in Fig.10) is approximately equal to unity inside the interval
$x\in[\frac{1}{\beta},K\,c_{\beta}]$, and is negligibly small outside:
$1$$0$$G_{0}(x)-G_{K}(x)$$x$$\beta^{-1}$$C_{\beta}K$ Figure 10:
This immediately produces the simple result for the limiting free energy:
$-\beta\overline{F}(\beta)=\lim_{K\to\infty}\frac{1}{K}\left\langle\ln{Z_{K}(\beta)}\right\rangle=\beta\,c_{\beta}$
(39)
with $c_{\beta}$ given by (36). Remembering $M\approx s^{K}$ and using the
relation (13) for the typical multifractality exponents, we find
$\tau_{q>0}=\frac{1}{\ln{s}}\,\,\beta\,q\,[c_{\beta}-c_{\beta q}]\,.$ (40)
In particular, for the earlier considered case of the Gaussian distribution
${\cal
P}(\phi)=\frac{1}{\sqrt{2\pi\delta}g}\exp{-\frac{\phi^{2}}{4g^{2}\delta}}$ we
find
$-\beta\overline{F}(\beta)={\ln{s}}\left\\{\begin{array}[]{cc}1+\frac{\beta^{2}}{\beta_{c}^{2}},&\beta<\beta_{c}=1/g\\\
2\frac{\beta}{\beta_{c}},&\beta>\beta_{c}=1/g\end{array}\right..$ (41)
We see that the only control parameter for the model is
$\gamma=\beta^{2}/\beta_{c}^{2}$. After a simple calculation using (40) we
recover the multifractality exponents for this case, which we are going to
present only in the range $q>1$:
$\tau_{q>1}=\left\\{\begin{array}[]{c}(q-1)(1-\gamma q),\quad
0\leq\gamma<\frac{1}{q^{2}}\\\
q(1-\sqrt{\gamma})^{2},\quad\frac{1}{q^{2}}<\gamma<1\\\
0,\,\,\,\quad\gamma>1\end{array}\right.\,.$ (42)
The phenomenon of vanishing of the exponents $\tau_{q>1}$ in the low-
temperature phase $\gamma=\beta^{2}g^{2}>1$ is one of the manifestations of
freezing. It is qualitatively interpreted in terms of the Boltzmann measure
being essentially localised on a few sites for low enough temperature or
strong enough disorder. The typical multifractality spectrum corresponding to
the above exponents is obtained according to the Legendre transform (2) which
gives
$f(\alpha)=\left\\{\begin{array}[]{c}1-\frac{1}{4\gamma}\left[\alpha-(1+\gamma)\right]^{2}\quad\mbox{for}\quad\gamma<1\\\
-\frac{1}{4\gamma}\left[\alpha^{2}-4\sqrt{\gamma}\alpha\right]\quad\mbox{for}\quad\gamma>1\end{array}\right.\,,$
(43)
where the expression in the first line formally assumes the range of exponents
$\alpha_{-}=(1-\sqrt{\gamma})^{2}\leq\alpha\leq 1+\gamma=\alpha_{0}$, whereas
in second line $0\leq\alpha\leq 2\sqrt{\gamma}=\alpha_{0}$. The upper bound
$\alpha_{0}$ here corresponds to the point of maximum of $f(\alpha)$ and is
related to the formal restriction $q>1$ in (42). In fact however it is not
difficult to find $\tau_{q}$ for any $q$ and show that the expressions (43)
are valid in a wider range $\alpha\in[\alpha_{-},\alpha_{+}]$ where the
boundary $\alpha_{+}$ is the largest root of $f(\alpha)=0$.
$1$$0$$f(\alpha)$$(1-\sqrt{\gamma})^{2}$$(1+\sqrt{\gamma})^{2}$$\alpha$$\gamma<1$${\sf{Fig.\,\,11a}}$multifractality
spectrumin the high-temperature
phase$1$$0$$f(\alpha)$$4\sqrt{\gamma}$$\alpha$$\gamma>1$${\sf{Fig.\,\,11b}}$multifractality
spectrumbelow freezing temperature
Exploiting the relation (4) for the typical multifractality spectrum one has
to specify the limits of integration over $\alpha$ to be precisely
$\alpha_{-}\leq\alpha\leq\alpha_{+}$. Substituting there (43) and calculating
the integral by the steepest descent method reproduces the values (42) of the
quenched exponents, that is $\tau^{typ}\equiv\tau_{q}$. Such a calculation
confirms that the change of behaviour of the exponent $\tau_{q}^{typ}$ to
linear in $q$ for $\gamma>1/q^{2}$ is induced by the dominance of the boundary
point $\alpha_{-}$ in the integration over $\alpha$, in agreement with general
discussion after (4).
Thinking in terms of the multifractality spectrum it is also easy to see that
the freezing phenomenon at $\gamma>1$ is related to $\alpha_{-}=0$, when the
leftmost end of the curve $f(\alpha)$ hits the vertical axis precisely at zero
level: $f(0)=0$, see Fig.11b.
## 3 Statistical mechanics for logarithmically correlated potentials in
Euclidean spaces of high dimensionality
As was discussed in the Introduction, we consider the Gibbs partition function
of a classical particle confined to a spherical box of some finite radius $L$.
We denote the corresponding domain as $\\{D_{L}:\,|{\bf x}|\leq L\\}$. As
before our main goal is to calculate the ensemble average of the free energy
$F=-\frac{1}{\beta}\,\ln{Z_{\beta}},\quad Z=\int_{D_{L}}\exp{-\beta V({\bf
x})}\,d{\bf x}\,,$ (44)
where $\beta=1/T$ stands for the inverse temperature and $d{\bf x}$ is the
standard volume element in $N-$dimensional Euclidean space. The average of the
logarithm of the partition function is one of the central problems in the
whole physics of disordered systems, and is usually performed with the help of
the so-called replica trick, i.e. the formal identity
$\left\langle\ln{Z_{\beta}}\right\rangle=\lim_{n\to
0}\frac{1}{n}\ln{\left\langle Z_{\beta}^{n}\right\rangle},\quad
Z_{\beta}^{n}=\int_{D_{L}}e^{-\beta\sum_{a=1}^{n}V({\bf
x_{a}})}\prod_{a=1}^{n}d{\bf x}_{a}\,.$ (45)
The random Gaussian-distributed potential $V({\bf x})$ is characterized by
zero mean and the covariance specified by the pair correlation function (11).
Performing the averaging over the Gaussian disorder in Eq.(45) according to
the formula (89), we in the standard way arrive at the following expression:
$\left\langle Z_{\beta}^{n}\right\rangle=e^{\gamma
n\ln{\frac{L}{a}}}\int_{D_{L}}e^{-\gamma\sum_{a<b}\ln{\left[\frac{({\bf
x}_{1}-{\bf x}_{2})^{2}+a^{2}}{L^{2}}\right]}}\prod_{a=1}^{n}d{\bf x}_{a}\,,$
(46)
where we recall the definition of the main control parameter of the problem:
$\gamma=\beta^{2}g^{2}$. To achieve further progress one has to suggest an
efficient way of working with the resulting multidimensional non-Gaussian
integral. To this end one may notice that the integrand in Eq.(46) in fact
possesses a high degree of invariance: it depends on $N-$component vectors
${\bf x_{a}}$ only via $n(n+1)/2$ scalar products $q_{ab}={\bf x_{a}}{\bf
x_{b}},\,\,a\leq b$, and is therefore invariant with respect to an arbitrary
simultaneous $O(N)$ rotation of all vectors ${\bf x}_{a}$. Moreover, our
choice of the integration domain respects this invariance. To this end,
introduce $N\times n$ rectangular matrix $X=({\bf x}_{1},...,{\bf x}_{n})$
such that the $N-$ component vector ${\bf x}_{i}$ forms $i-$th column of such
a matrix. Then the matrix $Q=X^{T}X$ is $n\times n$ positive definite, whose
entries are precisely the scalar products $q_{ab}={\bf x_{a}}{\bf
x_{b}},\,\,a\leq b$. An efficient method of dealing with integrals possessing
such type of invariance is based on the fundamental identity
$\int_{|{\bf x}_{1}|<L}...\int_{|{\bf x}_{n}|<L}{\cal
I}\left(X^{T}X\right)\,d{\bf x}_{1}\ldots d{\bf x}_{n}={\cal
C}_{N,n}\int_{D_{L}^{(Q)}}{\cal
I}(Q)\,\left[\det{Q}\right]^{\frac{N-n-1}{2}}\,dQ\,,$ (47)
where $\quad{\cal
C}_{N,n}=\frac{\pi^{\frac{n}{2}\left(N-\frac{n-1}{2}\right)}}{\prod_{k=0}^{n-1}\Gamma\left(\frac{N-k}{2}\right)}$
and we assumed $N\geq n+1$. The integration domain in the right-hand side is
simply $D_{L}^{(Q)}=\\{Q\geq 0,\,q_{aa}\leq L^{2},\,a=1,\ldots n\\}$, the
volume element is $dQ=\prod_{a\leq b}dq_{ab}$. The above formula seem to
appear originally in [14] but has not been much in use before it was
independently rediscovered in the context of theory of random matrices in
[15]. In [16] it was exploited in the present context. Since the relation
turns out to be quite useful in a few applications we present in the Appendix
B its derivation taken from [15] with the purpose of making the notes self-
contained.
Applying such a transformation gives in our case:
$\left\langle Z_{\beta}^{n}\right\rangle={\cal C}_{N,n}e^{\gamma
n\ln{\frac{L}{a}}}\int_{D_{L}^{(Q)}}e^{-\gamma\sum_{a<b}\ln{\left[\frac{q_{aa}+q_{bb}-2q_{ab}+a^{2}}{L^{2}}\right]}}\left[\det{Q}\right]^{\frac{N-n-1}{2}}\,dQ\,,$
(48)
So far all our manipulations were exact for any spatial dimension, provided
$N\geq n+1$. For any finite $N<\infty$ no further simplifications seem
possible, any ways to proceed to analysis of (48) are presently unknown and
yet to be found.
The situation is better if we agree to consider the dimension $N$ as one more
control parameter and let it to be large: $N\gg 1$. After appropriate
rescaling of the coupling constant $g\to g\sqrt{N}$ (i.e. $\gamma\to N\gamma$)
and also rescaling the integration variables $Q\to\frac{a^{2}}{2}Q$ we can
rewrite the exact expression for the averaged replicated partition function in
the following form
$\left\langle Z_{\beta}^{n}\right\rangle={\cal
C}_{N,n}\left(\frac{a^{2}}{2}\right)^{Nn/2}e^{N\gamma
n^{2}\ln{\frac{L}{a}}}\int_{D_{Q}}\left(\mbox{det}Q\right)^{-(n+1)/2}e^{-N\Phi_{n}(Q)}\,dQ$
(49)
where
$\Phi_{n}(Q)=-\frac{1}{2}\ln{(\det{Q})}+\gamma\sum_{a<b}\ln{\left[\frac{1}{2}(q_{aa}+q_{bb})-q_{ab}+1\right]}$
(50)
and $N$ is assumed to satisfy the constraint $N>n$. The final integration
domain $D_{Q}$ is: $D_{Q}=\\{Q\geq
0,\,q_{aa}\leq\,R^{2}=2L^{2}/a^{2},\,a=1,\ldots n\\}$. The form of the
integrand in Eq.(49) is precisely one required for the possibility of
evaluating the replicated partition function in the limit $N\to\infty$ by the
multidimensional Laplace (also known as the ”steepest descent” or ”saddle-
point”) method. The effective free energy relevant for extracting the
multifractality is then calculated by replica trick as (see (13) and (45))
$\beta{\cal
F}(\beta)=-\lim_{M\to\infty}\frac{\left\langle\ln{Z_{\beta}}\right\rangle}{\ln{M}}=\lim_{L\to\infty}\frac{1}{\ln{L}}\lim_{n\to
0}\frac{1}{n}\Phi_{n}(Q)$ (51)
where we have replaced $\ln{M}\approx N\ln{L}$, and the entries of the matrix
$Q$ should be chosen to satisfy the extremal conditions:
$\frac{\partial\Phi_{n}(Q)}{\partial q_{ab}}=0$ for $a\leq b$. This yields, in
general, the system of $n(n+1)/2$ equations:
$-\left[Q^{-1}\right]_{aa}+\gamma\sum_{b(\neq
a)}^{n}\left[\frac{1}{2}(q_{aa}+q_{bb})-q_{ab}+1\right]^{-1}=0,\quad
a=1,2,\ldots,n$ (52)
and
$-\left[Q^{-1}\right]_{ab}-\gamma\left[\frac{1}{2}(q_{aa}+q_{bb})-q_{ab}+1\right]^{-1}=0,\quad
a\neq b$ (53)
One should also ensure that the solutions to these equations respects the
constraint $q_{aa}\leq R^{2}$ for all $a=1,\ldots,n$ imposed by the presence
of the boundaries of the integration domain $D_{Q}$, and also the fact of $Q$
being positive definite. However, the above equations obviously imply
$\left[Q^{-1}\right]_{aa}=-\sum_{b(\neq
a)}\left[Q^{-1}\right]_{ab},\quad\forall a=1,2,\ldots,n\,.$ (54)
The above condition ensures that the matrix $Q^{-1}$ must have at least one
zero eigenvalue (which corresponds to the uniform eigenvector with all
components equal) which is obviously inconsistent with constraints on $Q$. We
interpret such a failure as manifestation of the fact that the functional
$\Phi_{n}(Q)$ cannot achieve its extremum inside the domain $Q>0,q_{aa}\leq
R^{2}$. This means that such an extremum should be looked for at the boundary
of the domain: $q_{aa}=R^{2},\,\forall a=1,2,\ldots,n$. In turn, it means that
when searching for such an extremum we only vary $\Phi_{n}(Q)$ with respect
the off-diagonal entries, and therefore only have to satisfy the equation
(53).
Our procedure of investigating the equations (52,53) in the replica limit
$n\to 0$ will follow the standard pattern suggested by developments in spin
glass theory[18]. We first seek for the so-called ”replica symmetric”
solution, and then investigate its stability depending on $\gamma$. When the
replica symmetric solution is found inadequate, it should be replaced by the
hierarchical (”Parisi”, or ”ultrametric”) ansatz for the matrix elements
$q_{ab}$, with various levels of replica symmetry breaking.
### 3.1 Analysis of the model within the Replica Symmetric Ansatz.
The Replica Symmetric Ansatz amounts to searching for a solution to (52),(53)
within subspace of $n\times n$ symmetric positive definite matrices $Q$ such
that $q_{aa}=q_{d}=R^{2}$, for any $a=1,\ldots n$, and $q_{a<b}=q_{0}$,
subject to the constraints $0<q_{0}\leq R^{2}$ to ensure positive
definiteness. Inverting such a $Q$ yields the matrix $Q^{-1}$ of the same
structure, with the diagonal entries all equal and given by
$p_{d}=\frac{R^{2}+q_{0}(n-2)}{(R^{2}-q_{0})(R^{2}+q_{0}(n-1))}$ (55)
and all off-diagonal entries given by
$p_{0}=-\frac{q_{0}}{(R^{2}-q_{0})(R^{2}+q_{0}(n-1))}$ (56)
Note, that
$p_{d}-p_{0}=\frac{1}{R^{2}-q_{0}}$ (57)
In the replica limit $n\to 0$ the equations (53) and (56) give in this way the
equation for $q_{0}$:
$\frac{q_{0}}{(R^{2}-q_{0})^{2}}-\frac{\gamma}{\left(R^{2}-q_{0}+1\right)}=0$
(58)
It is convenient to define the variable $d_{0}=R^{2}-q_{0}$ satisfying $0\leq
d_{0}\leq R^{2}$ and reduce (58) to the simple quadratic equation
$(\gamma+1)d_{0}^{2}-d_{0}(R^{2}-1)-R^{2}=0$. Choosing the solution with
$d_{0}>0$ and remembering that we are actually interested in the large$-L$
limit $R^{2}=2L^{2}/a^{2}\gg 1$ we find
$d_{0}=\frac{1}{2(1+\gamma)}[R^{2}-1+\sqrt{(R^{2}-1)^{2}+4R^{2}(1+\gamma)}]\approx\frac{R^{2}}{\gamma+1},$
(59)
Now we should calculate the value of the functional $\Phi_{n}(Q)$ for the
replica symmetric solution. It is easy to show that
$\det{Q}=(R^{2}-q_{0})^{n-1}[R^{2}+(n-1)q_{0}]$, so that in the limit $n\to 0$
we easily find from (50)
$\lim_{n\to
0}\frac{1}{n}\Phi_{n}(Q)=-\frac{1}{2}\ln{d_{0}}-\frac{\gamma}{2}\ln{(d_{0}+1)}\approx-(1+\gamma)\ln{L}+O(\ln{a})$
(60)
where we again considered the limit $L\gg a$. This shows that the effective
free energy (51) is given by
$\beta{\cal F}(\beta)=\lim_{L\to\infty}\frac{1}{\ln{L}}\lim_{n\to
0}\frac{1}{n}\Phi_{n}(Q)=1+\gamma\equiv 1+\beta^{2}g^{2}\,.$ (61)
This coincides precisely with the high-temperature ($T>T_{c}=g$, i.e.
$\gamma<1$) result for the ”cascade model” of the previous section, cf. (41),
which is valid before the freezing mechanism becomes operative. Our next goal
is to understand how the freezing emerges and is maintained for $T<T_{c}$.
### 3.2 Analysis within the Parisi scheme of the replica symmetry breaking.
The standard way of revealing the breakdown of the replica-symmetric solution
is to perform a stability analysis following the pattern of the famous de
Almeida-Thouless paper [19] in the theory of spin glasses, i.e. magnetic
systems with random interactions. Such analysis can be straightforwardly done
for the present type of system, see Appendix D of the present lectures, and
shows that for a given value of $R$ the replica symmetric solution becomes
unstable for the temperatures $T<T_{c}=g\frac{R^{2}-1}{R^{2}+1}$. Therefore at
low temperatures stable solution will have to be one with a broken symmetry in
the replica space. To derive the corresponding expression for the free energy
of our model we will follow a particular heuristic scheme of the replica
symmetry breaking proposed originally by Parisi in the theory of spin glasses,
see e.g. [18], or more recently [20] 666In recent years the use of the scheme
was justified by alternative rigorous mathematical procedures. For the model
under consideration the corresponding equations were re-derived recently by a
rigorous methods in [21] without any recourse to the powerful but ill-defined
replica trick.. To make the present set of lectures self-contained we describe
in full detail the structure of the matrix $Q$, the ensuing Parisi function
$x(q)$ and the main steps of the derivation in Appendix C in full detail777The
Appendix is taken verbatim from [16], but we used this opportunity to correct
the important formula (121) which appeared in [16] in a distorted form.. Here
we just sketch those objects schematically for the convenience of the reader:
$n$$m_{1}$$m_{2}$$m_{3}$$q_{0}$$q_{0}$$q_{1}$$q_{1}$$q_{1}$$q_{1}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$$q_{2}$${\sf{Fig.\,\,12a}}$Schematic
hierarchical structure of${\mbox{the matrix }\,Q\,\,\mbox{in Parisi
parametrisation}}$$n$$0$$x(q)$$q$$q_{d}$$q_{k}$$...$$q_{0}$$q_{1}$$q_{1}$$m_{1}$$m_{1}$$m_{k}$$1$${\sf{Fig.\,\,12b}}$Step-
wise Parisi function for finite integer $n$
We are actually interested in the replica limit $n\to 0$. According to the
Parisi prescription explained in detail in the Appendix C, in such a limit
$x(q)$ becomes non-decreasing function of the variable $q$ and the system can
be fully described in terms of such an object. The function depends non-
trivially on its argument in the interval $q_{0}\leq q\leq q_{k}$, with
$q_{0}\geq 0$ and $q_{k}\leq q_{d}$. Outside that interval the function stays
constant:
$x(q<q_{0})=0,\quad\mbox{and}\quad x(q>q_{k})=1.$ (62)
In general, the function $x(q)$ also depends on the increasing sequence of $k$
positive parameters $m_{i}$ satisfying the following inequalities
$0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{k}\leq m_{k+1}=1\,.$ (63)
If the number of levels of the Parisi hierarchy $K$ tends to infinity we may
think of the function $x(q)$ as continuous in the interval $q_{0}\leq q\leq
q_{k}$, with possible jumps at the end of the interval: $q=q_{0}$ and
$q=q_{k}$.
As is shown in the Appendix C, in the replica limit the following identity
must hold for any differentiable function $g(q)$:
$\lim_{n\to
0}\frac{1}{n}Tr\left[g(Q)\right]=g\left(q_{d}-q_{k}\right)+\int_{0}^{q_{k}}g^{\prime}\left(\int_{q}^{q_{d}}x(\tilde{q})\,d\tilde{q}\right)\,dq\,.$
(64)
In particular, for the first term entering the replica functional Eq.(50)
application of the rule Eq.(64) gives
$\lim_{n\to
0}\frac{1}{n}\left[Tr\ln{(Q)}\right]=\ln{(q_{d}-q_{k})}+\int_{0}^{q_{k}}\frac{1}{\int_{q}^{q_{d}}x(\tilde{q})\,d\tilde{q}}\,dq\,.$
(65)
The last term in Eq.(50) is also easily dealt with in the Parisi scheme (see
Appendix C), where it can be written as
$-\gamma\lim_{n\to
0}\sum_{l=0}^{k}(m_{l+1}-m_{l})\ln{(q_{d}-q_{l}+1)}=-\gamma\int_{0}^{q_{d}}\ln{(q_{d}-q+1)}x^{\prime}(q)\,dq,$
(66)
by using explicitly the derivative of the generalized function Eq. (115).
Using integration by parts and taking into account the properties Eq.(62) we
finally arrive at the required free energy functional for the phase with
broken replica symmetry
$\displaystyle\lim_{n\to
0}\frac{1}{n}\Phi_{n}(Q)=-\frac{1}{2}\left[\ln{\left(q_{d}-q_{k}\right)}+\int_{0}^{q_{k}}\frac{1}{q_{d}-q_{k}+\int_{q}^{q_{k}}x(\tilde{q})\,d\tilde{q}}\,dq\right]$
$\displaystyle-\frac{\gamma}{2}\left(\ln{(q_{d}-q_{k}+1)}+\int_{q_{0}}^{q_{k}}\frac{1}{q_{d}-q+1}\,x(q)\,dq\right)\equiv\phi\\{x(q)\\}$
(67)
The functional $\phi\\{x(q)\\}$ should be now extremized with respect to the
non-negative non-decreasing continuous function $x(q)$, whereas as we know the
variable $q_{d}$ must be fixed to its boundary value $q_{d}=R^{2}$. To this
end we find it convenient to introduce two parameters
$d_{min}=R^{2}-q_{k},\,d_{max}=R^{2}-q_{0}$ satisfying $0\leq d_{min}\leq
d_{max}\leq R^{2}$ and also to use $t=R^{2}-q$ as the new integration variable
simultaneously replacing (with some abuse of notations) $x(q=R^{2}-t)\to
x(t)$. Such a renamed function $x(t)$ is now non-increasing in the interval
$t\in[d_{min},d_{max}]$, and satisfies $x(t<d_{min})=1,\,x(t>d_{max})=0$.
As the result, the above functional assumes a somewhat simpler form:
$-2\phi\\{x(t)\\}=\ln{\left(d_{min}\right)}+\int_{d_{min}}^{R^{2}}\frac{\,dt}{d_{min}+\int_{d_{min}}^{t}x(\tilde{t})\,d\tilde{t}}+\gamma\ln{(d_{min}+1)}+\gamma\int_{d_{min}}^{d_{max}}\frac{x(t)dt}{t+1}$
(68)
Varying the functional Eq.(68) with respect to such a function $x(t)$ gives
after due manipulations with integrals the expression
$-2\delta\phi\\{x(t)\\}=\int_{d_{min}}^{d_{max}}\,S(t)\,\delta x(t)=0,\quad
S(t)=\gamma\frac{1}{t+1}-\int_{t}^{R^{2}}\frac{d\tilde{t}}{\left[d_{min}+\int_{d_{min}}^{\tilde{t}}x(\tau)\,d\tau\right]^{2}}\,,$
(69)
Requiring the variation to vanish therefore amounts to the condition
$S(t)=0,\,\,\forall t\in[d_{min},d_{max}]$. As this obviously implies
$\frac{d}{dt}S(t)=0$ we can differentiate Eq.(69) once, and immediately get
the equation
$d_{min}+\int_{d_{min}}^{t}x(\tau)\,d\tau=\frac{t+1}{\sqrt{\gamma}},\quad\forall
t\in[d_{min},d_{max}]\,\Rightarrow x(t)=\frac{1}{\sqrt{\gamma}}$ (70)
What remains to be determined are the values for parameters $d_{min}$ and
$d_{max}$. To this end, we substitute the value $t=d_{min}$ into the first of
relations Eq.(70) which shows that
$d_{min}=\frac{1}{\sqrt{\gamma}-1}\,.$ (71)
Next, we use the condition $S(d_{max})=0$, which in view of (69) and $x(t)=0$
for $t\in[d_{max},R^{2}]$ gives the relation
$\gamma\frac{1}{d_{max}+1}=\int_{d_{max}}^{R^{2}}\frac{d\tilde{t}}{\left[d_{min}+\int_{d_{min}}^{d_{max}}x(\tau)\,d\tau\right]^{2}}\,.$
Substituting here the expressions (70,71) yields after a simple algebra the
$\gamma-$independent result:
$d_{max}=\frac{R^{2}-1}{2},$ (72)
completing the solution. According to the general procedure the solution makes
sense as long as $d_{min}\leq d_{max}$, and using $\sqrt{\gamma}=g/T$ it is
easy to check that the condition can be rewritten as $T\leq
T_{c}=g\frac{R^{2}-1}{R^{2}+1}$ which defines the low-temperature phase of the
model for finite $R$, with the same $T_{c}$ as follows from the stability
analysis (Appendix D). In the thermodynamic limit $\lim_{R\to\infty}T_{c}=g$,
that is $\gamma_{c}=1$ as expected.
The value of the functional at the extremum can be easily calculated by
substituting $x(t)=\gamma^{-1/2}$ for $t\in[d_{min},d_{max}]$ and $x(t)=0$ for
$t\in[d_{max},R^{2}]$ into (68) and using (71) and (72). This gives after some
algebra
$\displaystyle-\phi\\{x(t)\\}=\sqrt{\gamma}\ln{\frac{R^{2}+1}{2}}+\frac{\sqrt{\gamma}}{2}-\frac{1}{2}(\sqrt{\gamma}-1)^{2}\ln{(\sqrt{\gamma}-1)}+\frac{1}{2}(\gamma-2\sqrt{\gamma})\ln{\sqrt{\gamma}}$
(73)
which finally implies in the thermodynamic limit $L\to\infty$ for the
effective free energy the value
$\beta{\cal F}(\beta)=\lim_{L\to\infty}\frac{1}{\ln{L}}\lim_{n\to
0}\frac{1}{n}\Phi_{n}(Q)=2\sqrt{\gamma}=2\beta g\,.$ (74)
In particular it shows that the free energy value in the low-temperature phase
is frozen i.e. given by the temperature-independent constant ${\cal
F}(\beta)=2g$. This fully corroborates the picture obtained in the framework
of logarithmic cascades of the previous section, see (41).
Before finishing this section it makes sense to discuss in more detail the
picture associated with the freezing transition which manifests itself via the
spontaneous breakdown of replica symmetry. The general interpretation of the
freezing below $T_{c}$ is that the partition function becomes dominated by a
finite number of sites where the random potential is particularly low, and
where the particle ends up spending most of its time [22]. For a more
quantitative description of the particle localization, useful in the
following, it is natural to employ the overlap function defined as the mean
probability for two independent particles placed in the same random potential
to end up at a given distance to each other. Denoting the scaled Euclidean
distance (squared) between the two points in the sample as ${\cal D}$, and
employing the Boltzmann-Gibbs equilibrium measure $p_{\beta}({\bf
x})=\frac{1}{Z(\beta)}\exp{-\beta V({\bf x})}$ the above probability in
thermodynamic equilibrium should be given by
$\pi({\cal D})=\left\langle\int_{|{\bf x}_{1}|<L}\,d{\bf
x}_{1}\,p_{\beta}({\bf x}_{1})\int_{|{\bf x}_{2}|<L}d{\bf
x}_{2}\,p_{\beta}({\bf x}_{2})\,\delta\left({\cal D}-\frac{1}{2}|{\bf
x}_{1}-{\bf x}_{2}|^{2}\right)\right\rangle_{V}\,$ (75)
where again $\delta$ denotes the Dirac’s $\delta$-function. The disorder
averaging in (75) can be calculated following the same standard steps of the
replica approach as the free energy itself ( see Appendix A of [17]). With the
function $\pi({\cal D})$ in hand we can ask, in particular what is the
probability for the particle in logarithmically correlated potential to end up
at ${\cal D}=O(a^{2})$, i.e. at a distance of order of the small cutoff scale.
The answer turns out to be zero in the high-temperature phase $T>T_{c}$,
confirming the particle delocalization over the sample. In contrast, in the
low-temperature phase $T<T_{c}$ the probability is finite:
$\pi\left(O(a^{2})\right)=1-T/T_{c}$, since both particles can be trapped by
one and the same, or nearby favorable, deep minima. At a formal level such a
behaviour is directly related to the shape of the function $x(t)$ which in our
case turned out to be rather simple and consisting of three flat regions (see
Fig. 13b):
$x(0<t<d_{min})=1,\quad x(d_{min}<t<d_{max})=\gamma^{-1/2},\quad
x(d_{max}<t<R^{2})=0\,.$ (76)
This essentially means that from the very beginning we could restrict
ourselves to the first non-trivial level $k=1$ of the Parisi hierarchical
scheme, see Eq.(110, 111) instead of assuming the most general Parisi scheme
for $Q$ at the outset of our procedure. Such a simplified form (see Fig. 13a)
of $Q$ below the transition is typical for the random energy models and is
known in the literature as 1-step RSB scheme, see e.g. [20]. The equilibrium
values of the parameters $q_{0},q_{1}$ and $m_{1}\equiv m$ found from directly
extremizing the corresponding functional $\frac{1}{n}\Phi_{n}(Q)|_{n\to 0}$
(or equivalently from solving the equations (53)) are given by
$q_{0}=\frac{R^{2}+1}{2},\,q_{1}=R^{2}-\frac{1}{\sqrt{\gamma}-1}\,,\,\,\mbox{
and}\quad m=\frac{1}{\sqrt{\gamma}}\,.$ (77)
These values fully agree with those of the function $x(t)$ obtained from the
general Parisi Ansatz. Finally, in the Appendix D we discuss stability of the
1-step RSB solution for the logarithmic potential, and find it is actually
marginally stable everywhere in the low-temperature phase. The latter feature
is usually associated with the infinite-step Parisi Ansatz, see e.g. [20].
This is another manifestation of the fact that the logarithmic case is very
special and shares both features of the full-scale infinite and 1-step replica
symmetry breaking.
$n$$m$$1$$R^{2}$$R^{2}$$R^{2}$$R^{2}$$R^{2}$$R^{2}$$R^{2}$$R^{2}$$R^{2}$$q_{0}$$q_{0}$$q_{1}$$q_{1}$$q_{1}$$q_{1}$$q_{1}$$q_{1}$${\sf{Fig.\,\,13a}}$$\mbox{Structure
of the matrix }Q\,\,\mbox{in}$1-step Replica Symmetry
Breaking$1$$0$$x(t)$$t$$R^{2}$$d_{max}$$d_{min}$$\frac{1}{\sqrt{\gamma}}$${\sf{Fig.\,\,13b}}$Parisi
function for the logarithmic model with 1-step RSB
## 4 Summary, Historical background and Recent Extensions
In this set of lectures we have addressed in some detail the spatial
structures of the Boltzmann-Gibbs measure describing a single particle that
thermally equilibrated under a random potential with logarithmic correlations.
We have been able to calculate the multifractality spectrum of the measure and
revealed the associated freezing transition by analysing the ensemble-averaged
free energy in two special cases by two complementary methods. The first model
introduced logarithmic correlations via employing the hierarchical
”multiplicative cascades” construction and associated definition of the
distance function. This way allowed us to perform the analysis of the freezing
transition in the framework of a certain travelling wave equation satisfied by
an appropriately defined generating function of partition function moments. In
the second case the spatial dimension $N$ of the system was assumed to be
large which helped to employ the replica trick combined with the steepest
descent method, and to relate freezing to the phenomenon of spontaneous
replica symmetry breaking. In both cases the resulting free energy appears to
be given by essentially the same expression.
As the present day understanding of freezing and related phenomena has already
a history of almost thirty years, it is certainly useful to be aware of a
broader context of the problem under consideration. To this end it is
appropriate to mention that an extreme ”toy model” case of the problem in hand
is represented by the famous Random Energy Model (REM) by Derrida where the
freezing phenomenon was discovered and investigated for the first time [23,
24]. The REM in some loose sense can be looked at as a limiting ”zero-
dimensional” $N=0$ case of the model we studied elsewhere in this set of
lectures. It amounts essentially to replacing the logarithmically correlated
random potential by a collection of $M$ uncorrelated Gaussian variables with
the variances chosen to be scaled with $M$ in the same way as in the
logarithmic case: $<V_{i}^{2}>=2g^{2}\ln{M}$. REM is simple enough to allow
explicit calculation of the free energy by direct counting of degrees of
freedom, and the result essentially coincides with (41). A very informative
account of the REM problem can be found in the fifth chapter of [20].
Understanding quantitatively the generic statistical-mechanical behaviour of
disordered systems for finite $N$ is notoriously difficult, and even the
simplest cases like our single-particle model still present considerable
challenges. To this end we first need to mention a general attempt of
investigating such model for finite dimensions $N<\infty$ in the thermodynamic
limit $L\to\infty$ undertaken in an insightful paper by Carpentier and Le
Doussal [25]. The approach of Carpentier and Le Doussal was based on applying
a kind of real-space renormalisation group treatment to the free energy
distribution. The authors concluded that for finite spatial dimensions the
model with logarithmically correlated potetial is really distinguished among
others of similar kind. Namely, if correlations of the random potential grow
faster than logarithm with the distance, then in the thermodynamic limit the
corresponding Boltzmann-Gibbs measure turn out to be always localised at any
temperature $T<\infty$. At the same time, if the correlations decay to zero
for large separations (such potentials are natural to call ”short-ranged”)
than the Boltzmann-Gibbs measure turns out to be always trivially extended at
any positive temperature $T>0$. And only for the marginal situation of
logarithmic correlations the true REM-like freezing transition indeed happens
at some finite $T=T_{c}>0$, at any dimension $N\geq 1$. Indeed, for that case
the renormalisation group yielded after some clever albeit not fully
controlled approximations a kind of travelling wave equation for the
generating function, akin to (27). Fortunately, the logarithmic growth is not
at all an academic oddity. The paper of Carpentier and Le Doussal can be
warmly recommended for describing the present model in a broad physical
context and elucidating its relation to quite a few other interesting and
important physical systems, as e.g. quantum Dirac particle in a random
magnetic field [26], and directed polymers on trees with disorder [12]. The
latter works played the fundamental role in advancing the understanding of the
freezing transition. Our presentation in the Section 2 is actually based on an
adaptation of material from [26] and [12], with the pedagogic example of
branching tending to unity inspired by [27].
Another line of research which deserves mentioning was pursued recently in
[17] where it was revealed that the picture of potentials with short-ranged,
long-ranged, and logarithmic correlations presented in [25] is still
incomplete, and misses a rich class of possible behaviour that survives in the
thermodynamic limit $L\to\infty$. Namely, given any increasing function
$\Phi(y)$ for $0<y<1$, it was suggested to consider Gaussian random potentials
whose two-point correlation functions (covariances) take the following scaling
form
$\left\langle V\left({\bf x}_{1}\right)\,V\left({\bf
x}_{2}\right)\right\rangle=-2\ln{L}\,\,\Phi\left(\frac{\ln{\left[({\bf
x}_{1}-{\bf x}_{2})^{2}+a^{2}\right]}}{2\ln{L}}\right),\quad a\ll L,\quad{\bf
x}\in\mathbb{R}^{N}\,$ (78)
which generalizes our (11). Actually, the above expression gives back (11) for
the special case $\Phi(y)=g^{2}(y-1)$. As shown in [17] the potential with the
covariance (78) can be constructed by a superimposing several logarithmically
correlated potentials of the type (11) with different cutoff scales $a_{i}$,
and allow those cutoff scales to depend on the system size $L$ in a power-law
way: $a_{i}\sim L^{\nu_{i}},\,0<\nu_{i}<1$ .
The equilibrium statistical mechanics of such system in the limit $N\to\infty$
and $L\to\infty$ turns out to be precisely equivalent to that of the
celebrated Derrida’s Generalized Random Energy Model (GREM) see [28] and
references therein. Namely, the system experiences a kind of freezing
transition at the critical temperature $T_{c}=\sqrt{\Phi^{\prime}(1)}$. Below
this temperature the equilibrium free energy turns out to be in the
thermodynamic limit $L\to\infty$
$-{\cal
F}(T)=T\nu_{*}(T)+\frac{\left[\Phi(\nu_{*})-\Phi(0)\right]}{T}+2\int_{\nu_{*}}^{1}\sqrt{\Phi^{\prime}(y)}\,dy\,,\quad
0\leq T\leq T_{c}\,,$ (79)
where the parameter $\nu_{*}$ is related to the temperature $T$ via the
equation $T^{2}=\Phi^{\prime}(\nu_{*})$. For $T>T_{c}$ the free energy is
instead given by
$\displaystyle-{\cal F}(T)=T+\frac{\left[\Phi(1)-\Phi(0)\right]}{T}\,.$ (80)
Using the two-point probability defined in (75) these expressions for the free
energy can be given a clear interpretation as describing a continuous sequence
of ”freezing transitions” which start at $T_{c}$ and continue at all lower
temperatures, with freezing happening on smaller and smaller spatial scales
with decreasing temperature [17]. This is related also to the nature of the
replica symmetry breaking, which requires for its description the full
infinite sequence $K\to\infty$ of hierarchy levels in the Parisi scheme of
Appendix $C$. Such a rich picture results in a more complicated
multifractality spectrum $f(\alpha)$ which in contrast to (43) is in general
non-parabolic. However, it is appropriate to mention that the Boltzmann-Gibbs
probability measures generated by the random potentials described in (78) are
rather peculiar, as for any non-linear function $\Phi(y)$ they do not satisfy
the standard spatial self-similarity property (5). Instead, it is easy to
check that the exponents $y(q,s)$ and $z(q,s)$ governing the spatial decay of
correlations between weights in (5) will be non-trivial functions of the
variable $\frac{\ln{|{\bf x}_{1}-{\bf x}_{2}|}}{\ln{L}}$ rather than simple
constants. In this way, the exponents governing the decay of correlations for
two points separated by the distance, say, $|{\bf x}_{1}-{\bf x}_{2}|\sim
L^{1/2}$ will be different from those separated by, say, $|{\bf x}_{1}-{\bf
x}_{2}|\sim L^{1/3}$. Though such behaviour is certainly not prohibited by
first principles, it remains to be seen whether random multifractal measures
with such peculiar spatial structure could appear in interesting applications
in physics or other sciences.
Although our lectures were centered around the notion of the multifractality
spectrum, there is a different, and in essence deeper aspect of the freezing
transition which attracted considerable research interest recently: the issue
of the extreme value statistics [29, 25, 30, 31]. This goes beyond the
calculation of the ensemble-averaged value for the free energy
$F=-T\ln{Z}(\beta)$, but aims to describe precise form of the fluctuations
around that mean value. Technically it amounts to our ability to calculate the
shape of the generating function $G(p)$ defined in (22) in much finer detail
(note that in the context of calculating typical multifractality exponents
actual form of that function appeared to large extent irrelevant). As
$lim_{T\to 0}F=\min_{{\bf x}}V({\bf x})$ it is obvious that at low enough
temperatures the free energy fluctuations are dominated by the distribution of
the deepest minimum of the random potential in a given sample. Classifying
possible types of extreme value statistics for strongly correlated random
variables is an open problem in probability theory with many important
applications in natural sciences and beyond, see [25] and the references
therein. In particular, it was argued in [25] that logarithmically correlated
potentials represent a new universality class for extreme value statistics,
and recent works [30, 31] on extremes of the two-dimensional Gaussian free
field (see definition of this important object in Appendix A1 below) along
various curves further substantiated that claim. Another aspect of the problem
which certainly deserves to be mentioned here are intriguing but so far poorly
explored connections to two-dimensional quantum gravity models as noticed in
[34], discussed in [25], and most recently in [31]. Some speculations about
relevance of the REM-type models in the string theory context can be found in
[35].
Finally, let us mention that there exists a completely different source of
interest in multifractal random processes $\&$ measures with logarithmic
correlations motivated by growing applications in financial mathematics, see
e.g. [32], [33] for the background information and further references.
Although the questions addressed there are formally rather different, one can
recognize a common mathematical structure. It is therefore natural to expect a
fruitful merger of the two lines of research in the nearest future.
Acknowledgements. My understanding of some aspects of the subject of the
present lectures was informed by discussions on various occasions with Jean-
Philippe Bouchaud, Pierre Le Doussal and Alexander Mirlin. I am grateful to
them as well as to Hans-Juergen Sommers and Alberto Rosso for collaboration at
various stages, and to Ferdinand Evers for kindly providing picture Fig. 1 for
the present notes.
## Appendix A Elementary facts about Gaussian integrals and processes, the
steepest descent method, and the Gaussian free field
The fundamental role in applications is played by the standard Gaussian
integral
$\int_{-\infty}^{\infty}e^{-\frac{a}{2}y^{2}+b\,y}\,\frac{dy}{\sqrt{2\pi}}=\frac{1}{\sqrt{a}}\,e^{\frac{b^{2}}{2a}},\quad
Re{(a)}>0,\,\forall b$ (81)
Suppose now we are interested in finding the asymptotic behaviour for large
values of a parameter $N$ of the following integral
$\int_{y_{1}}^{y_{2}}e^{-NF(y)}\phi(x)\,dy,\quad N\gg 1$ (82)
where $F(y)$ and $\phi(y)$ are some given infinitely differentiable functions.
It is clear that if the function $F(y)$ is monotonically increasing/decreasing
in the interval $y\in[y_{1},y_{2}]$, then the integral will be dominated by
the vicinity of the left/right end of the interval, and for getting the
leading asymptotics it is therefore enough to expand $F(y)$ around the
corresponding point up to the linear term only. For example for
$F^{\prime}(y)>0,\forall y\in[a,b]$, we write $F(y)\approx
F(y_{1})+F^{\prime}(y_{1})(y-y_{1})+\ldots$ which gives
$\int_{y_{1}}^{y_{2}}e^{-NF(y)}\phi(y)\,dy\approx\frac{1}{NF^{\prime}(y_{1})}\,e^{-NF(y_{1})}\phi(y_{1})+O(N^{-2})$
(83)
where we assumed that generically $\phi(y_{1})\neq 0$ (otherwise one has also
to expand $\phi(y)$ around $y_{1}$, which will change the result slightly).
Similarly, if the function $F(y)$ has a single maximum in some point $y_{0}$
inside the interval, then subdividing the integration domain into two
subintervals $y\in[y_{1},y_{0}]$ and $y\in[y_{0},y_{1}]$ we can apply the
above consideration to each of the new intervals. For example, if
$F(y_{1})<F(y_{2})$ we have the same asymptotics as above in (83), whereas for
$F(y_{1})>F(y_{2})$ we have
$\int_{y_{1}}^{y_{2}}e^{-NF(y)}\phi(y)\,dy\approx\frac{1}{NF^{\prime}(y_{2})}\,e^{-NF(y_{2})}\phi(y_{2})[1+O(1/N)]$
(84)
Finally, the most interesting case arises if $F(y)$ has a single minimum in
some point $y_{0}\in[y_{1},y_{2}]$, that is $F^{\prime}(y_{0})=0$ and
$F^{\prime\prime}(y_{0})>0$. In such a case the integral will be obviously
dominated by the vicinity of the point of minimum, around which we can
therefore expand as $F(y)\approx
F(y_{1})+\frac{F^{\prime\prime}(y_{1})}{2}(y-y_{1})^{2}+\ldots$. Substituting
this approximation back to the integral and again assuming that generically
$\phi(y_{0})\neq 0$ we find after application of (81) with
$a=F^{\prime\prime}(y_{0})$, the asymptotics
$\int_{y_{1}}^{y_{2}}e^{-NF(y)}\phi(y)\,dy\approx\sqrt{\frac{2\pi}{NF^{\prime\prime}(y_{0})}}\,e^{-NF(y_{0})}\phi(y_{0})[1+O(1/N)]$
(85)
These formulae represent the essence of the steepest descent (a.k.a. the
Laplace) method of asymptotic evaluations of integrals.
All the formulae can be naturally extended to the multivariable case. The
multivariable generalisation of the Gaussian integral is given by
$\int\ldots\int
e^{-\frac{1}{2}\sum_{ij}A_{ij}y_{i}y_{j}+\sum_{i}b_{i}y_{i}}\frac{dy_{1}\ldots
dy_{n}}{(2\pi)^{n/2}}=\frac{1}{\sqrt{\det{A}}}\,\,e^{\frac{1}{2}\sum_{ij}[A^{-1}]_{ij}b_{i}b_{j}}$
(86)
where $n\times n$ matrix $A$ is assumed to be real symmetric
$A_{ij}=A_{ji},\,\forall i,j$ and positive definite, i.e. all its eigenvalues
$\lambda_{i}$ are positive. Then the inverse matrix $A^{-1}$ is well-defined
and the determinant $\det{A}=\prod_{i=1}^{n}\lambda_{i}\neq 0$. In fact
introducing the scalar product for two vectors as $({\bf y},{\bf
x})=\sum_{i}y_{i}x_{i}$ the quadratic form in the exponential can be written
as $\sum_{ij}A_{ij}y_{i}y_{j}\equiv({\bf y},\,A{\bf y})$. The matrix is
positive definite iff $({\bf y},\,A{\bf y})>0,\,\forall{\bf y}$ 888In fact the
domain of validity of the formula (86) is broader, and allows the matrix $A$
to have complex eigenvalues with positive real parts.
The analogue of (85) has the form
$\int\ldots\int e^{-NF(y_{1},,\ldots y_{n})}\phi(y_{1},\ldots
y_{n})\,dy_{1}\ldots
dy_{n}\approx\sqrt{\frac{(2\pi)^{n}}{N^{n}\det{\delta_{2}F|_{min}}}}\,e^{-NF(y_{1},,\ldots
y_{n})}\phi(y_{1},,\ldots y_{n})|_{min}$ (87)
where we assumed that the function $F(y_{1},,\ldots y_{n})$ has a single
minimum at some point, and $\delta_{2}F|_{min}$ is the $n\times n$ Hessian
matrix $(\delta_{2}F)_{ij}=\frac{\partial^{2}}{\partial y_{i}\partial
y_{j}}F(y_{1},,\ldots y_{n})$ evaluated at the point of minimum of $F$.
Let us clarify the probabilistic meaning of the integral (86). Suppose that
$n$ real variables $v_{1},\ldots,v_{n}$ are Gaussian-distributed, that is
their normalized joint probability density of the vector ${\bf
v}=(v_{1},\ldots,v_{n})$ is given by ${\cal
P}(v_{1},\ldots,v_{n})=e^{-\frac{1}{2}({\bf v},\,A{\bf
v})}\sqrt{\frac{\det{A}}{(2\pi)^{n}}}$ with some positive definite matrix
$A_{ij}$. Denoting the averaging over such a distribution with the angular
brackets $\langle\ldots\rangle$ we can rewrite (86) for any given vector ${\bf
b}=(b_{1},\ldots,b_{n})$ as
$\left\langle e^{({\bf b},\,{\bf v})}\right\rangle=e^{\frac{1}{2}({\bf
b},\,A^{-1}{\bf b})}\quad\Rightarrow\quad\langle
v_{i}\rangle=0\,\,\mbox{and}\,\,\left\langle
v_{i}v_{j}\right\rangle=[A^{-1}]_{ij},\,\,\forall i,j$ (88)
where the identities for the mean value and the pair correlation functions
(a.k.a. covariances) immediately follow after expanding in the Taylor series
with respect to $b_{i}$.
### A.1 Gaussian random fields: ”massive” vs ”free”.
The last expression is the basis for discussing properties of random processes
(i.e. random functions $V(x)$ of a single real variable $x$) which are a
particular case of random fields representing random functions $V({\bf x})$ of
$N-$dimensional vector ${\bf x}=(x_{1},\ldots,x_{N})$. The field $V({\bf x})$
is called Gaussian if for any choice of the number $n=1,2,\ldots,\infty$ of
points ${\bf x}_{1},{\bf x}_{2},\ldots,{\bf x}_{n}$ in the space the joint
probability density ${\cal P}(v_{1},\ldots,v_{n})$ of $n$ values of the field
in those points, that is $v_{1}=V({\bf x}_{1}),v_{2}=V({\bf
x}_{2}),\ldots,v_{n}=V({\bf x}_{n})$ are given by a Gaussian distribution with
some matrix $A_{ij}$. Such random field is uniquely determined by the two-
point correlation function (the covariance) $\langle V({\bf x}_{1})\,V({\bf
x}_{2})\rangle=f({\bf x}_{1},{\bf x}_{2})$ in terms of which the analogue of
(88) reads
$\left\langle\exp{\left[{\int b({\bf x})V({\bf x})\,d^{N}{\bf
x}}\right]}\right\rangle=\exp{\left[\frac{1}{2}\int\int f({\bf x}_{1},{\bf
x}_{2})b({\bf x}_{1})\,b({\bf x}_{2})\,\,d^{N}{\bf x}_{1}d^{N}{\bf
x}_{2}\right]}$ (89)
for any suitable function $b({\bf x})$. If we define the scalar product of any
two functions $a({\bf x})$ and $b({\bf x})$ in the standard way as $({\bf
a},{\bf b})=\int a({\bf x})b({\bf x})\,d^{N}{\bf x}$, we see that the
quadratic form in the exponential of the right-hand side is $({\bf
a},\,\hat{F}{\bf b})$, where the linear integral operator $\hat{F}$ is defined
via the kernel $f({\bf x}_{1},{\bf x}_{2})$. If one then defines the inverse
operator as $\hat{A}=\hat{F}^{-1}$ , the joint probability density of the
random field $V({\bf x})$ can be symbolically written using the scalar product
as
${\cal P}\left[V({\bf x})\right]=\frac{1}{{\cal
N}}\,\exp{\left[-\frac{1}{2}(V,\hat{A}V)\right]},$ (90)
where ${\cal N}$ is the suitable normalisation constant.
To illustrate the latter approach, we briefly describe the paradigmatic
example of the massive Gaussian field in $N$ dimensions which is of importance
for us here, and is also central for the modern theory of phase transitions.
The probability of a given configuration $V({\bf x})$ of such field is given
by (90) with the quadratic form defined by
$(V,\,\hat{A}V)=\int\left(m^{2}V^{2}({\bf x})+\kappa^{2}\,[\nabla V({\bf
x})]^{2}\right)\,d^{N}{\bf x}\equiv\int V({\bf
x})\left[m^{2}-\kappa^{2}\,\Delta\right]\,V({\bf x})\,d^{N}{\bf x}$ (91)
where the ”mass” $m$ and the ”stiffness” $\kappa$ of the field are two
parameters, $\nabla$ is the gradient operator and $\Delta$ is the Laplacian:
$\Delta=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}$. Second form
follows from the first one after applying the integration by parts and
assuming that the random field $V({\bf x})$ vanishes at infinity. In such an
example the role of the operator $\hat{A}$ is obviously played by the second-
order differential operator $\hat{A}=m^{2}-\kappa^{2}\,\Delta$. Such operators
are called ”local” as their action on any function involves only values of
that function and its derivatives in the same point of the space. Knowing
$\hat{A}$ explicitly allows one to find the two-point correlation function
$\langle V({\bf x})\,V({\bf y})\rangle=f({\bf x},{\bf y})$ as the kernel of
the operator inverse to $A$ , hence satisfying the differential equation
$\left[m^{2}-\kappa^{2}\,\Delta\right]\,f({\bf x},{\bf y})=\delta({\bf x}-{\bf
y}),$ (92)
where the Laplacian is assumed to act on the first argument, and $\delta({\bf
x}-{\bf y})=\int e^{i{\bf q}({\bf x}-{\bf y})}\frac{d^{N}{\bf q}}{(2\pi)^{N}}$
stands for the appropriate Dirac delta-function. By applying the Fourier
transform to the equation immediately gives the two-point correlation function
as
$\langle V({\bf x})\,V({\bf y})\rangle=\int\frac{e^{i{\bf q}({\bf x}-{\bf
y})}}{(m^{2}+\kappa^{2}{\bf q}^{2})}\frac{d^{N}{\bf q}}{(2\pi)^{N}}$ (93)
To calculate the above integral it is convenient to use the identity
$(m^{2}+\kappa^{2}{\bf
q}^{2})^{-1}=\int_{0}^{\infty}e^{-t(m^{2}+\kappa^{2}{\bf q}^{2})}\,dt$ and
change the order of integration, which gives
$\langle V({\bf x})\,V({\bf y})\rangle=\int_{0}^{\infty}e^{-tm^{2}}\,dt\int
e^{i{\bf q}({\bf x}-{\bf y})-\kappa^{2}t{\bf q}^{2}}\frac{d^{N}{\bf
q}}{(2\pi)^{N}}$ (94)
$=\left(\frac{1}{4\pi\kappa^{2}}\right)^{N/2}\int_{0}^{\infty}e^{-tm^{2}-\frac{1}{4\kappa^{2}t}({\bf
x}-{\bf
y})^{2}}\,\frac{dt}{t^{N/2}}=\frac{1}{\left(2\pi\right)^{N/2}}\frac{m^{N/2-1}}{\kappa^{N/2+1}}\frac{K_{N/2-1}\left(\frac{m}{\kappa}|{\bf
x}-{\bf y}|\right)}{|{\bf x}-{\bf y}|^{N/2-1}}$
where we have used (88) with $A_{ij}\to 2\kappa^{2}t\delta_{ij},{\bf
b}\to({\bf x}-{\bf y})$ to evaluate the Gaussian integral in the first line,
and $K_{\nu}(z)$ is the so-called Macdonald function, see the formula 3.471.9
of [36]. In particular, for $N=2$ and $m\to 0$ we have from the expansion
3.471.9 of [36]
$\langle V({\bf x})\,V({\bf
y})\rangle=\frac{1}{2\pi\kappa^{2}}K_{0}\left(\frac{m}{\kappa}|{\bf x}-{\bf
y}|\right)\approx-\frac{1}{2\pi\kappa^{2}}\ln{\left[\frac{|{\bf x}-{\bf
y}|}{2\kappa/m}\right]},\quad|{\bf x}-{\bf y}|\ll\frac{\kappa}{m}.$ (95)
We conclude that the limit of 2D massless Gaussian field provides us with a
random field with logarithmic correlations.
The massless Gaussian field is also known in the modern literature as the
Gaussian Free Field (GFF) and considered to be an object of fundamental
importance. It can be defined on any domain ${\bf D}$ of $N-$dimensional space
using the following construction. Consider an eigenproblem for the Laplace
operator $-\Delta$ acting on functions in ${\bf D}$, and denote ${\bf
e}_{j}({\bf x}),$ $j=1,2,\ldots,\infty$ its eigenfunctions corresponding to
the Dirichlet boundary conditions ( i.e. vanishing at the boundary
$\partial{\bf D}$) and let $\lambda_{j}>0$ be the corresponding eigenvalues.
Then the functions $\tilde{{\bf e}}_{j}({\bf
x})=\frac{1}{\sqrt{\lambda_{j}}}{\bf e}_{j}({\bf x})$ form an orthonormal
basis of the Hilbert space H with respect to the so-called Dirichlet scalar
(or ”inner”) product
$\left(f,g\right)=\int_{{\bf D}}\left(\nabla f\cdot\nabla g\right)d^{N}{\bf
x}=-\int_{{\bf D}}\left(f\cdot\Delta g\right)d^{N}{\bf x}$ (96)
for functions $f({\bf x})$ on ${\bf D}$ vanishing at the boundary
$\partial{\bf D}$. Introduce now a set $\zeta_{j},\,\,j=1,2,\ldots,\infty$ of
standard Gaussian independent, identically distributed real variables with
mean zero and unit variance each:
$\langle\zeta_{j}\rangle=0,\,\langle\zeta_{j}^{2}\rangle=1$. Then the GFF
$V({\bf x})$ on the domain ${\bf x}\in{\bf D}$ is defined as the formal sum
$V({\bf x})=\sum_{j=1}^{\infty}\zeta_{j}\,\tilde{{\bf e}}_{j}({\bf x}),$ (97)
from which it immediately follows that it is a Gaussian field with the
covariance given by
$\left\langle V({\bf x}_{1})V({\bf
x}_{2})\right\rangle=\sum_{j=1}^{\infty}\,\frac{1}{\lambda_{j}}{\bf
e}_{j}({\bf x}_{1}){\bf e}_{j}({\bf x}_{2})=-\left(\Delta^{-1}\right)({\bf
x}_{1},{\bf x}_{2})$ (98)
which is nothing else but the Green function $G({\bf x}_{1},{\bf x}_{2})$ of
the Laplace operator on the domain ${\bf D}$. Note however that mathematically
$V({\bf x})$ is rather subtle (e.g. the sum in (97) does not converge
pointwise and fails in general to be an element of the Hilbert space H ).
Because of this and other subtleties an extra mathematical care is needed to
define the object fully rigorously, see references in [9]. The physicists
however work with such an object without further ado, and we finish this
section by two simple but important examples. In the first example we deal
with the GFF on a one-dimensional domain, the interval ${\bf D}=[0,1]$. The
Laplacian in one dimension is simply $\Delta=-\frac{d^{2}}{dx^{2}}$ and the
eigenfunctions/eigenvalues of the Dirichlet problem are given by
$e_{n}(x)=\sqrt{2}\sin{n\pi x},\lambda_{n}=\pi^{2}n^{2}$ so that the GFF in
this particular case is given by a random Fourier series
$V(x)=\sum_{n=1}^{\infty}\zeta_{n}\frac{\sqrt{2}}{\pi n}\sin{n\pi x}$ (compare
with the periodic $1/f$ noise in the end of this Appendix). The corresponding
Green function can be easily found to be given by
$G(x_{1},x_{2})=x_{1}(1-x_{2})$ for $x_{2}>x_{1}$ and
$G(x_{1},x_{2})=x_{2}(1-x_{1})$ for $x_{2}<x_{1}$. One immediately recognizes
that the one-dimensional version of the GFF for such a domain coincides with
the version of the Brownian motion called Brownian bridge, which is
conditioned to return to the origin after a given time.
Our second example is much more relevant in the context of the present
lectures and deals with GFF defined on the two-dimensional disk: ${\bf
D}=|z|<L$ where we use the complex coordinate $z=x+iy$. The Green function for
the Dirichlet problem on such a domain is well known and is given by
$G(z_{1},z_{2})=-\frac{1}{2\pi}\ln{\frac{L|z_{1}-z_{2}|}{L^{2}-z_{1}z_{2}}}$.
In particular, for any two points $|z_{1,2}|\ll L$ (i.e. well inside the disk)
the Green function reduces to expression equivalent to the full-plane formula
(95) which is the basis for models with logarithmic correlations.
Using the full-plane logarithmic GFF it is easy to construct various one-
dimensional Gaussian random processes with logarithmic correlations. In
particular, sampling the values of such GFF along a circle of unit radius with
coordinates $z=e^{it},\,t\in[0,2\pi)$ we get a Gaussian process with the
covariance $\langle
V(t_{1})V(t_{2})\rangle=-\frac{1}{2\pi}\ln{|e^{it_{1}}-e^{it_{2}}|}$. Such a
process can be shown to be equivalent to a random Fourier series of the form
$V(t)=\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\left[v_{n}e^{int}+\overline{v}_{n}e^{-int}\right]$,
where $v_{n},\overline{v}_{n}$ are independent, identically distributed
complex Gaussian variables with mean zero and variance $\langle
v_{n}\overline{v}_{n}\rangle=1$ (compare with the earlier Brownian bridge
example). As the mean-square value (the ”spectral power”) of the coefficient
in front of a given Fourier harmonic with index $n$ in this case decays like
$1/n$ such signals are known in many applications as $1/f$ noises.
## Appendix B Proof of the identity (47)
We start with identically rewriting the left-hand side of (47) as
$\int{\cal I}\left(X^{T}X\right)\,dX=\lim_{\epsilon\to 0^{+}}\int{\cal
I}(Q){\cal J}_{\epsilon}(Q)\,dQ\,$ (99)
where
$\quad{\cal J}_{\epsilon}(Q)=\int
e^{-\frac{\epsilon}{2}Tr[X^{T}X]}\delta\left(Q-X^{T}X\right)\,dX$ (100)
and $\delta(x)$ stands for the appropriate Dirac $\delta-$distribution in the
matrix space. As usual $\delta-$function can be expressed via the Fourier
transform $\delta(x)=\int e^{-ifx}\frac{df}{2\pi}$ its matrix analogue can be
defined via the following Fourier representation:
$\delta\left(Q-X^{T}X\right)=\int
e^{-\frac{i}{2}Tr[(Q-X^{T}X)F_{n}]}\,dF_{n},\quad
dF_{n}=\prod_{i}\frac{d[F_{n}]_{ii}}{4\pi}\prod_{i<j}\frac{d[F_{n}]_{ij}}{2\pi}$
(101)
with the integration going over $n\times n$ real symmetric matrices:
$[F_{n}]_{ij}=[F_{n}]_{ji}$. Substituting such a representation into the
expression for ${\cal J}_{\epsilon}(Q)$ and changing the order of integration
over $dF_{n}$ and $dX$ one may notice that the integral over $X$ is
essentially a product of $N$ identical Gaussian multivariable integrals (86)
where the role of $A$ is played by the matrix $\epsilon{\bf 1}_{n}-iF_{n}$.
The integrals are well-defined due to $\epsilon>0$. Applying (86) we arrive at
${\cal J}_{\epsilon}(Q)=(2\pi)^{\frac{Nn}{2}}{\cal J}_{n,N,\epsilon}(Q)$,
where
${\cal J}_{n,N,\epsilon}(Q)=\int
e^{-\frac{i}{2}Tr[QF_{n}]}\frac{1}{[\det\left(\epsilon{\bf
1}_{n}-iF_{n}\right)]^{N/2}}\,dF_{n},$ (102)
and we have indicated explicitly the dependence on $n$ and $N$ for the sake of
future reference. Notice that the integrand is invariant with respect to the
rotations $F\to\hat{O}\hat{F}\hat{O}^{-1}$ where $O$ are orthogonal matrices
satisfying $O^{T}O=1$. As $Q$ is real symmetric matrix, it can be brought to
the diagonal form by an orthogonal transformation. Hence the result of the
integration can depend only on the eigenvalues $q_{1},q_{2},...,q_{n}$ of
$\hat{Q}$. Thus, it is enough to take $\hat{Q}$ to be diagonal from the very
beginning. Now we separate the first eigenvalue from the rest:
$\hat{Q}=\mbox{diag}(q_{1},q_{2},...,q_{n})\equiv\mbox{diag}(q_{1},\hat{Q}_{n-1})$
and accordingly decompose the matrix $F_{n}$ as
$F_{n}=\left(\begin{array}[]{cc}f_{11}&{\bf f}\\\ {\bf
f}^{T}&F_{n-1}\end{array}\right)\quad,\quad
dF_{n}=\frac{df_{11}}{4\pi}\frac{d{\bf f}}{(2\pi)^{n-1}}dF_{n-1}$ (103)
where ${\bf f}=\left(f_{12},f_{13},....,f_{1n}\right)$ is a $n-1$ component
vector.
Next step is to use the well-known property of the determinants composed of
four blocks:
$\det{\left(\epsilon{\bf 1}_{n}-iF_{n}\right)}=\det{\left(\epsilon{\bf
1}_{n-1}-iF_{n-1}\right)}\left(\epsilon-if_{11}+{\bf f}\left[\epsilon{\bf
1}_{n-1}-iF_{n-1}\right]^{-1}{\bf f}^{T}\right)$
which gives:
$\displaystyle{\cal J}_{n,N,\epsilon}(\hat{Q})$ $\displaystyle=$
$\displaystyle\int
d\hat{F}_{n-1}e^{-\frac{i}{2}\mbox{Tr}\left(\hat{F}_{n-1}\hat{Q}_{n-1}\right)}\left[\det{\left(\epsilon{\bf
1}_{n-1}-iF_{n-1}\right)}\right]^{-N/2}$ $\displaystyle\times$
$\displaystyle\int\frac{d{\bf
f}}{(2\pi)^{n-1}}\int_{-\infty}^{\infty}\frac{df_{11}}{4\pi}e^{-\frac{i}{2}f_{11}q_{1}}\frac{1}{\left(\epsilon-
if_{11}+{\bf f}\left[\epsilon{\bf 1}_{n-1}-iF_{n-1}\right]^{-1}{\bf
f}^{T}\right)^{N/2}}$
The last integral over $f_{11}$ can be explicitly evaluated by using the
formula 3.382.7 of [36]:
$\int e^{-ifp}\frac{1}{(\beta-
if)^{\nu}}\frac{df}{2\pi}=\frac{p^{\nu-1}}{\Gamma{(\nu)}}e^{-\beta
p}\theta(p),\quad Re(\nu,\beta)>0$ (105)
where $\Gamma(\nu)$ is the Euler Gamma-function, and $\theta(x)=1$ for $x>0$
and zero otherwise. Taking into account $\epsilon>0$, the result of the
integration over $f_{11}$ gives
$\frac{1}{2\Gamma(N/2)}\theta(q_{1})\left(\frac{q_{1}}{2}\right)^{N/2-1}\exp\left\\{-\frac{1}{2}q_{1}\left(\epsilon+{\bf
f}\left[\epsilon{\bf 1}_{n-1}-iF_{n-1}\right]^{-1}{\bf
f}^{T}\right)\right\\}\,.$ (106)
Now the integration over the vector $d{\bf f}$ becomes the standard Gaussian
and can be performed using (86) yielding the factor:
$\left(\frac{1}{2\pi
q_{1}}\right)^{\frac{n-1}{2}}\mbox{det}^{1/2}\left(\epsilon{\bf
1}_{n-1}-iF_{n-1}\right)$
Collecting all the factors we arrive at the recursive relation
${\cal
J}_{n,N,\epsilon}(\hat{Q})=\frac{\pi^{-\frac{n-1}{2}}}{2^{n}\Gamma(N/2)}\left(\frac{q_{1}}{2}\right)^{\frac{N-n-1}{2}}\theta(q_{1})e^{-\frac{1}{2}\epsilon
q_{1}}{\cal J}_{n-1,N-1,\epsilon}(\hat{Q}_{n-1})$ (107)
This relation can be iterated further, and assuming $N>n$ we arrive at the
last step to (105) which gives
${\cal
J}_{1,N-n+1,\epsilon}(q_{n})=\frac{1}{2\Gamma(\frac{N-n+1}{2})}\left(\frac{q_{n}}{2}\right)^{\frac{N-n-1}{2}}\theta(q_{n})e^{-\frac{1}{2}\epsilon
q_{n}}$ (108)
and serves as an ”initial condition” for our iteration scheme. This
immediately yields the result:
${\cal
J}_{n,N,\epsilon}(Q)=\frac{1}{2^{\frac{Nn}{2}}\pi^{\frac{n(n-1)}{4}}}\frac{1}{\prod_{j=0}^{n-1}\Gamma\left(\frac{N-j}{2}\right)}\,\mbox{det}^{\frac{N-n-1}{2}}\left[Q\right]e^{-\frac{1}{2}\epsilon\mbox{Tr}\,Q}\,\,\prod_{j=1}^{n}\theta(q_{j})$
(109)
for $N\geq n+1$. As ${\cal J}_{\epsilon}(Q)=(2\pi)^{\frac{Nn}{2}}{\cal
J}_{n,N,\epsilon}(Q)$, in the limit $\epsilon\to 0$ the above relation yields
precisely the required identity (47).
## Appendix C Parisi matrix, its eigenvalues and evaluation of traces in the
replica limit.
We start with describing the well known structure of the $n\times n$ matrix
$Q$ in the Parisi parametrisation, see Fig.12a. At the beginning we set $n$
diagonal entries $q_{\alpha\alpha}$ all to the same value
$q_{\alpha\alpha}=0$. This value will be maintained at every but last step of
the recursion. The off-diagonal part of the matrix $Q$ in the Parisi scheme is
built recursively as follows. At the first step we single out from the
$n\times n$ matrix $Q$ the chain consisting of $n/m_{1}$ blocks of the size
$m_{1}\leq n$, each situated on the main diagonal. All off-diagonal entries
$q_{\alpha\beta},\,\alpha\neq\beta$ inside those blocks are filled in with the
same value $q_{\alpha\beta}=q_{1}\leq 0$, whereas all the remaining
$n^{2}(1-1/m_{1})$ entries of the matrix $Q$ are set to the value $0<q_{0}\leq
q_{1}$. The latter entries remain from now on intact to the end of the
procedure, whereas some entries inside the diagonal $m_{1}\times m_{1}$ blocks
will be subject to a further modification. At the next step of iteration in
each of those diagonal blocks of the size $m_{1}$ we single out the chain of
$m_{2}/m_{1}$ smaller blocks of the size $m_{2}\leq m_{1}$, each situated on
the main diagonal. All off-diagonal entries
$q_{\alpha\beta},\,\alpha\neq\beta$ inside those sub-blocks are filled in with
the same value $q_{\alpha\beta}=q_{2}\geq q_{1}$, whereas all the remaining
entries of the matrix $Q$ hold their old values. At the next step only some
entries inside diagonal blocks of the size $m_{2}$ will be modified., etc.
Iterating this procedure step by step one obtains after $k$ steps a
hierarchically built structure characterized by the sequence of integers
$n=m_{0}\geq m_{1}\geq m_{2}\geq\ldots\geq m_{k}\geq m_{k+1}=1$ (110)
and the values placed in the diagonal blocks of the $Q$ matrix satisfying:
$0<q_{0}\leq q_{1}\leq q_{2}\leq\ldots\leq q_{k}$ (111)
Finally, we complete the procedure by filling in the $n$ diagonal entries
$q_{\alpha\alpha}$ of the matrix $Q$ with one and the same value
$q_{\alpha\alpha}=q_{d}\geq q_{k}$.
For the subsequent analysis we need the eigenvalues of the Parisi matrix $Q$.
Those can be found easily together with the corresponding eigenvectors built
according to a recursive procedure which uses the sequence Eq.(110). It is
convenient to visualize eigenvectors as being ”strings” of $n$ boxes numbered
from $1$ to $n$, with $l^{th}$ component being a content of the box number
$l$.
At the first step $i=1$ we choose the eigenvector to have all $n$ boxes filled
with the same content equal to unity. The corresponding eigenvalue is non-
degenerate and equal to
$\lambda_{1}=q_{d}+q_{k}(m_{k}-1)+q_{k-1}(m_{k-1}-m_{k})+\ldots+q_{1}(m_{1}-m_{2})+q_{0}(m_{0}-m_{1})$
(112)
Now, at the subsequent steps $i=2,3,\ldots,k+2$ one builds eigenvectors by the
following procedure. The string of $n$ boxes of an eigenvector belonging to
$i^{th}$ family are subdivided into $n/m_{i-1}$ substrings of the length
$m_{i-1}$, and numbered accordingly by the index $j=1,2,\ldots,n/m_{i-1}$. All
$m_{i-1}$ boxes of the first substring $j=1$ are filled invariably with all
components equal to $1$. Next we fill $m_{i-1}$ boxes in one (and only one) of
the remaining $\frac{n}{m_{i-1}}-1$ substrings with all components equal to
$-1$. In doing so we however impose a constraint that the substrings with the
indices $j$ given by $j=1+l\frac{m_{i-2}}{m_{i-1}}$ should be excluded from
the procedure, with $l$ being any integer satisfying $1\leq
l\leq\frac{n}{m_{i-2}}-1$. After the choice of a particular substring is made,
we fill all $n-2m_{i-1}$ boxes of the remaining substrings with identically
zero components. It is easy to see that all $d_{i}=n/m_{i-1}-n/m_{i-2}$
different eigenvectors of $i^{th}$ family built in such a way correspond to
one and the same $d_{i}-$degenerate eigenvalue
$\lambda_{i}=q_{d}+q_{k}(m_{k}-1)+q_{k-1}(m_{k-1}-m_{k})+\ldots+q_{i-1}(m_{i-1}-m_{i})-q_{i-2}(m_{i-1})$
(113)
In this way we find all $n$ possible eigenvalues, the last being equal to
$\lambda_{k+2}=q_{d}-q_{k}\,m_{k+1}\equiv q_{d}-q_{k}.$ (114)
The completeness of the procedure follows from the fact that sum of all the
degeneracies $d_{i}$ is equal to
$1+\left(\frac{n}{m_{1}}-1\right)+\left(\frac{n}{m_{2}}-\frac{n}{m_{1}}\right)+\ldots+\left(\frac{n}{m_{k+1}}-\frac{n}{m_{k}}\right)=n$
Note that all the found eigenvalues are positive due to inequalities Eq.(111)
between various $q_{i}$, which is required by the positive definiteness of the
matrix $Q$. Note also that all eigenvectors built in this way are obviously
linearly independent, although the eigenvectors belonging to the same family
are not orthogonal. The latter fact however does not have any bearing for our
considerations.
To facilitate the subsequent treatment it is convenient to introduce the
following (generalized) function of the variable $q$, see Fig.12b:
$x(q)=n+\sum_{l=0}^{k}(m_{l+1}-m_{l})\,\theta(q-q_{l})$ (115)
where we use the notation $\theta(z)$ for the Heaviside step function:
$\theta(z)=1$ for $z>0$ and zero otherwise. In view of the inequalities
Eq.(110,111) the function $x(q)$ is piecewise-constant non-increasing, and
changes between $n$ and $1$ as follows:
$x(q<q_{0})=m_{0}\equiv
n,\,\,x(q_{0}<q<q_{1})=m_{1},\,\ldots,\,x(q_{k-1}<q<q_{k})=m_{k},\,x(q>q_{k})=m_{k+1}\equiv
1$ (116)
Comparison of this form with Eq.(115) makes evident the validity of a useful
inversion formula:
$\frac{1}{x(q)}=\frac{1}{n}+\sum_{l=0}^{k}\left(\frac{1}{m_{l+1}}-\frac{1}{m_{l}}\right)\,\theta(q-q_{l})$
(117)
which will be exploited by us shortly.
As observed by Crisanti and Sommers[37] one can represent the eigenvalues
Eq.(113) of the Parisi matrix in a compact form via the following remarkable
identities:
$\lambda_{1}=\int_{0}^{q_{d}}x(q)\,dq=nq_{0}+\int_{q_{0}}^{q_{d}}x(q)\,dq,\quad\lambda_{i+2}=\int_{q_{i}}^{q_{d}}x(q)\,dq,\quad
i=0,1,\ldots,k$ (118)
As a consequence, these relations imply for any analytic function $g(x)$ the
identity
$\frac{1}{n}Tr\left[g(Q)\right]=\frac{1}{n}\sum_{i=1}^{k+2}g(\lambda_{i})\,d_{i}=\frac{1}{n}g\left(nq_{0}+\int_{q_{0}}^{q_{d}}x(q)\,dq\right)+\sum_{l=0}^{k}\left(\frac{1}{m_{l+1}}-\frac{1}{m_{l}}\right)g\left(\int_{q_{l}}^{q_{d}}x(q)\,dq\right)$
(119)
Next one observes that taking the derivative of the generalized function from
Eq.(117) produces
$\frac{d}{dq}\left[\frac{1}{x(q)}\right]=\sum_{l=0}^{k}\left(\frac{1}{m_{l+1}}-\frac{1}{m_{l}}\right)\,\delta(q-q_{l}).$
(120)
This fact allows one to rewrite the sum in Eq.(119) in terms of an integral,
yielding
$\frac{1}{n}Tr\left[g(Q)\right]=\frac{1}{n}g\left(nq_{0}+\int_{q_{0}}^{q_{d}}x(q)\,dq\right)+\int_{q_{0}-0}^{q_{k}+0}g\left(\int_{q}^{q_{d}}x(\tilde{q})\,d\tilde{q}\right)\,\frac{d}{dq}\left[\frac{1}{x(q)}\right]\,dq,$
where the short-hand notation $q\pm 0$ designates the limit from below/above.
Further performing integration by parts, and using
$x(q>q_{k})=1,\,x(q<q_{0})=n$, we finally arrive at
$\frac{1}{n}Tr\left[g(Q)\right]=\frac{1}{n}\left[g\left(nq_{0}+\int_{q_{0}}^{q_{d}}x(q)\,dq\right)-g\left(\int_{q_{0}}^{q_{d}}x(q)\,dq\right)\right]+\int_{q_{0}}^{q_{k}}g^{\prime}\left(\int_{q}^{q_{d}}x(\tilde{q})\,d\tilde{q}\right)dq+g(q_{d}-q_{k}).$
(121)
We are actually interested in the replica limit $n\to 0$. According to the
Parisi prescription in such a limit the inequality Eq.(110) should be
reversed:
$n=0\leq m_{1}\leq m_{2}\leq\ldots\leq m_{k}\leq m_{k+1}=1$ (122)
and the function $x(q)$ is now transformed to a non-decreasing function of the
variable $q$ in the interval $q_{0}\leq q\leq q_{k}$, and satisfying outside
that interval the following properties
$x(q<q_{0})=0,\quad\mbox{and}\quad x(q>q_{k})=1.$ (123)
In general,such a function also depends on the increasing sequence of $k$
parameters $m_{l}$ described in Eq.(63) .
The form of Eq.(121) makes it easy to perform the limit $n\to 0$ explicitly,
and to obtain after exploitation of Eq.(62) an important identity Eq.(64)
helping to evaluate the traces in the replica limit. Finally, let us mention
the existence of an efficient method of the ”replica Fourier transform”
allowing one to diagonalise (and otherwise work) with much more general types
of hierarchical matrices, see [20] for more details.
## Appendix D Stability analysis of the saddle-point solution
Our starting point is the functional $\Phi_{n}(Q)$ from (50) whose extrema we
look for in the space of positive definite matrices $Q$ constrained to have
the diagonal entries $q_{aa}=R^{2}$. The independent variables are all off-
diagonal entries $q_{(ab)}$ where $(ab)$ stands for $n(n-1)/2$ ”ordered” pairs
with $a<b$, and the stationary values are found from the equations (83). The
stability matrix in this space is given by
$A_{(ab),(cd)}=\frac{\partial^{2}}{\partial q_{(ab)}\partial
q_{(cd)}}\Phi_{n}(Q)$ which should be evaluated at the saddle-point solution.
In a general situation we should distinguish three types of entries of that
matrix: the diagonal entries
$A_{(ab),(ab)}=\left[\left(Q^{-1}\right)_{aa}\left(Q^{-1}\right)_{bb}+\left(Q^{-1}\right)^{2}_{ab}\right]-\gamma\frac{1}{(R^{2}-q_{ab}+1)^{2}}\,,$
(124)
the entries for the case when the ordered pairs $(ab)$ and $(cd)$ share one
common replica, that is
$A_{(ab),(ac)}=\left[\left(Q^{-1}\right)_{aa}\left(Q^{-1}\right)_{bc}+\left(Q^{-1}\right)_{bc}\left(Q^{-1}\right)_{ab}\right],\,\,b<c$
(125)
and a similar expression for $A_{(ab),(cb)}\,,a<c$, and finally the entries
for the ordered pairs $(ab)$ and $(cd)$ which do not share any common replica:
$A_{(ab),(cd)}=\left[\left(Q^{-1}\right)_{ac}\left(Q^{-1}\right)_{bd}+\left(Q^{-1}\right)_{ad}\left(Q^{-1}\right)_{bc}\right],\,\,$
(126)
If we are interested in investigating stability of the replica-symmetric
solution, we should substitute to the above equations $q_{ab}=q_{0},\forall
a\neq b$ as well as $\left(Q^{-1}\right)_{aa}=p_{d},\forall a$ and
$\left(Q^{-1}\right)_{ab}=p_{0},\forall a<b$, with $p_{d}$ and $p_{0}$ taken
from (55,56). This gives for the entries of the stability matrix
$A_{(ab),(ab)}\equiv
A_{1}=p_{d}^{2}+p_{0}^{2}-\gamma\frac{1}{(R^{2}-q_{0}+1)^{2}}\,,\quad
A_{(ab),(ac)}\equiv A_{2}=p_{0}p_{d}+p_{0}^{2}$ (127)
and $\quad A_{(ab),(cd)}\equiv A_{3}=2p_{0}^{2}$. As discovered by De Almeida
and Thouless [19] the eigenvalues/eigenvectors of such $n(n-1)/2\times
n(n-1)/2$ matrix can be found explicitly. There are three families of
eigenvectors. The first family consists of a single ”replica-symmetric”
eigenvector ${\bf e}_{1}$ with all components $\left[{\bf
e}_{1}\right]_{(ab)}=1$. The corresponding eigenvalue is equal to the sum of
all entries in one row of $A$ that is
$\lambda_{1}=A_{1}+2(n-2)A_{2}+\frac{(n-2)(n-3)}{2}A_{3}$. Next family
consists of $d_{2}=n-1$ eigenvectors ${\bf e}_{2}^{c},\,c=1,\dots n-1$ with
one replica index $c$ singled out. For example, suppose that $c=1$, then
$\left[{\bf e}^{1}_{2}\right]_{(ab)}=\frac{n-2}{2}$ if $a=1$ or $b=1$, and
$\left[{\bf e}^{1}_{2}\right]_{(ab)}=-1$ otherwise (note that such eigenvector
is orthogonal to ${\bf e}_{1}$). The corresponding eigenvalue shared by all
the eigenvectors in the family is $\lambda_{2}=A_{1}+(n-4)A_{2}-(n-3)A_{3}$.
Finally , third family consists of $d_{3}=\frac{n(n-3)}{2}$ eigenvectors ${\bf
e}^{(cd)}_{3}$ with an ordered pair of replica indices $c<d$ singled out. For
example, if $(cd)=(12)$ then components of the corresponding eigenvector are
$\left[{\bf e}^{(12)}_{3}\right]_{(12)}=\xi$ , $\left[{\bf
e}^{(11)}_{3}\right]_{(ab)}=\psi$ if $a=1,2$ or $b=1,2$, and otherwise
$\left[{\bf e}^{(11)}_{3}\right]_{(ab)}=\rho$ where the values of
$\xi,\psi,\rho$ should be chosen to make ${\bf e}^{(11)}_{3}$ orthogonal to
${\bf e}^{(1)}_{2}$ and ${\bf e}_{1}$. The eigenvalue shared by the third
family turns out to be given by an $n-$independent expression
$\lambda_{3}=A_{1}-2A_{2}+A_{3}$. Since $1+d_{2}+d_{3}=n(n-1)/2$ no more
eigenvalues are possible.
It is well known in general (and can be easily checked for our model) that it
is third family which gives rise to ”dangerous” fluctuations breaking down the
replica symmetry of the saddle-point solution in the limit $n\to 0$ below some
critical temperature $T_{c}$ at which $\lambda_{3}$ vanishes. Substituting the
expressions (127) into $\lambda_{3}$ and using the relation (57) we find that
the condition $\lambda_{3}=0$ is equivalent to
$\frac{1}{R^{2}-q_{0}}=\frac{\gamma^{1/2}}{R^{2}-q_{0}+1}\,.$ (128)
Finally, using for the combination $R^{2}-q_{0}\equiv d_{0}$ the equation (59)
we find after simple algebra that the critical value of the parameter
$\sqrt{\gamma}=\beta g$ is given by
$\sqrt{\gamma}_{c}=\frac{R^{2}+1}{R^{2}-1}$ as was quoted in the text.
Let us now turn to the stability issue for the one-step RSB solution which we
claimed to be the correct choice below the critical temperature. According to
Fig.12a the one-step solution is characterized by the matrices $Q$ with all
diagonal entries still equal to $R^{2}$, and two different values of the off-
diagonal entries $q_{1}>q_{0}$. The size of blocks containing $q_{1}$ is equal
to $m$. Such more complicated structure of $Q$ generates more types of
different elements in the stability matrix $A$, and although its
eigenvalues/eigenvectors can be still successfully found [16], actual analysis
becomes long. Referring the interested reader to Appendix B3 of [16] for a
detailed exposition, we give below a very brief summary of the the outcome of
the procedure. Actually the stability analysis of [16] was performed for
models with general random potential characterized by the covariance
$\left\langle V\left({\bf x}_{1}\right)\,V\left({\bf
x}_{2}\right)\right\rangle=\,Nf\left(({\bf x}_{1}-{\bf x}_{2})^{2}/2N\right)$.
It was found that the matrix $A$ in that general case has nine different
eigenvector families, and the stability is controlled by two of them with
eigenvalues given by
$\Lambda^{*}_{0}=\left[{1\over R^{2}-q_{1}+m(q_{1}-q_{0})}-{1\over
T}\sqrt{f^{\prime\prime}(R^{2}-q_{0})}\right]\left[{1\over
R^{2}-q_{1}+m(q_{1}-q_{0})}+{1\over
T}\sqrt{f^{\prime\prime}(R^{2}-q_{0})}\right]$ (129)
and
$\Lambda^{*}_{K}=\left[{1\over R^{2}-q_{1}}-{1\over
T}\sqrt{f^{\prime\prime}(R^{2}-q_{1})}\right]\left[{1\over
R^{2}-q_{1}}+{1\over T}\sqrt{f^{\prime\prime}(R^{2}-q_{1})}\right]\,,$ (130)
assuming $f^{\prime\prime}(x)>0$. If both of the above eigenvalues are non-
negative, and the second derivative $f^{\prime\prime}(x)$ is monotonically
decreasing with $x$ then all the remaining seven eigenvalues are strictly
positive and the system is stable.
It is easy to understand that the logarithmic case (11) considered in this
paper is recovered, after all due rescalings in the limit $N\gg 1$, by the
choice $f(x)=-g^{2}\ln{(x+1)}$ so that
$\frac{1}{T}\sqrt{f^{\prime\prime}(x)}=\frac{\sqrt{\gamma}}{x+1}$. Using now
the equilibrium values for the parameters $q_{1},q_{0},m$ given in (77) one
finds after a straightforward algebra that the first brackets in (129),(130)
identically vanish leaving us with $\Lambda^{*}_{0}=\Lambda^{*}_{K}=0$
everywhere in the low-temperature phase. This implies indeed that the
corresponding one-step RSB solution is marginally stable.
## References
* [1] Paladin G and Vulpiani A Phys. Rep. 156 (1987) 147
* [2] Evers F and Mirlin AD, Rev. Mod. Phys. 80 (2008) 1355
* [3] Foster MS, Ryu S, and Ludwig AWW Phys. Rev. B 80 (2009) 075101
* [4] Monthus C and Garel T Phys. Rev. E 75 (2007), Art. No. 051122
* [5] Fyodorov YV J. Stat. Mech. (2009) P07022 [e-preprint arXiv:0903.2502]
* [6] Mandelbrot B Physica A 163 (1990), 306; Chabra AB and Sreenivasan KR Phys. Rev. A 43 (1990) 1114
* [7] see e.g. Eq.(2.30) of [ME].
* [8] Van Kampen NG Stochastic Processes in Physics and Chemistry (3rd ed., North-Holland, 2007)
* [9] Duplantier B and Sheffield S (2008), e-preprint arXiv:0808.1560 [math.PR]
* [10] Bolthausen E, Deuschel J-D and Giacomin G (2001) Ann. Probab.29 (2001) 1670; Daviaud O, Ann. Prob. 34 (2006), 962.
* [11] Schmitt FG Eur. Phys. J. B 34 (2003) 85 ; Schmitt FG and Chainais F Eur. Phys. J. B 58, (2007) 149
* [12] Derrida B, Spohn H J.Stat.Phys. 51 (1988) 817
* [13] Bramson MMem. Am. Math. Soc. 44 (1983) 285
* [14] Percus JK , Comm. Pure and Appl. Math. 40 (1986) 449
* [15] Fyodorov YV , Nuclear Phys. B 621 (2002), 643
* [16] Fyodorov YV and Sommers H-J Nucl. Phys. B [FS] 764 (2007), 128
* [17] Fyodorov YV and Bouchaud JP J. Phys.A: Math.Theor 41 (2008) 324009
* [18] Mezard M, Parisi G and Virasoro MA , ”Spin glass theory and beyond” (World Scientific, Singapore, 1987)
* [19] de Almeida JRL and Thouless DJ J.Phys.A 11 (1978) 983
* [20] De Dominicis C , Giardina I ”Random Fields and Spin Glasses” (Cambridge University Press, 2006)
* [21] Klimovsky A (2009), ”Parisi landscapes in high-dimensional Euclidean spaces”, talk at the workhop ”Mathematical Models from Physics and Biology”, April 2009, Bonn, Germany
* [22] Monthus C, Bouchaud J-P 1996 J. Phys. A: Math. Gen. 29 3847; Ben Arous G, Cerny J, 2006, Dynamics of trap models, arXiv:math.PR/0603344.
* [23] Derrida B Phys. Rev. B 24 (1981) 2613
* [24] Gardner E and Derrida B J.Phys.A 22 (1989) 1975
* [25] Carpentier D and Le Doussal P Phys. Rev. E 63 (2001), 026110
* [26] Chamon C, Mudry C and Wen X-G Phys. Rev. Lett. 77 (1996) 4194; Castillo H E, Chamon C C, Fradkin E, Goldbart P M, and Mudry C Phys. Rev. B 56 (1997) 10668
* [27] Saakian DB , Phys. Rev. E 65 (2002) 067104
* [28] Bovier A 2006, Statistical Mechanics of Disordered Systems: a Mathematical Perspective (Cambridge University Press)
* [29] Bouchaud J-P and Mézard M J. Phys. A: Math. Gen. 30 (1997) 7997
* [30] Fyodorov YV and Bouchaud JP J. Phys.A: Math.Theor 41 (2008) 372001
* [31] Fyodorov YV, Le Doussal P, and Rosso A, J. Stat. Mech. (2009) P10005 [e-preprint arXiv:0907.2359 [cond-mat.dis-nn]]
* [32] Muzy J-F, Delour J, and Bacry E Eur. Phys. J. B 17 (2000) 537 Bacry E, Delour J, and Muzy J-F Phys. Rev. E 64 (2001) 026103; Bacry E, Muzy J-F Comm. Math. Phys. 236 (2003), 449
* [33] Ostrovsky D, J. Stat. Phys. 127 (2007), 935 and Comm. Math. Phys. 288 (2009) 287
* [34] Kogan II, Mudry C, and Tsvelik AM Phys. Rev. Lett. 77 (1996) 707
* [35] Saakian DB J. Stat. Mech. (2009) P07003
* [36] Gradshteyn I S, and Ryzhik I M , Table of Integrals, Series, and Products, 6th ed. Academic Press, 2000
* [37] Crisanti A, Sommers H-J Z. Phys. B 87(1992) 341
|
arxiv-papers
| 2009-11-14T11:58:28 |
2024-09-04T02:49:06.464578
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yan V Fyodorov",
"submitter": "Yan V. Fyodorov",
"url": "https://arxiv.org/abs/0911.2765"
}
|
0911.2831
|
# Charged rotating dilaton black branes in AdS universe
A. Sheykhi1,2111sheykhi@mail.uk.ac.ir and S.H. Hendi3222hendi@mail.yu.ac.ir 1
Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran
2 Research Institute for Astrophysics and Astronomy of Maragha (RIAAM),
Maragha, Iran
3 Department of Physics, College of Science, Yasouj University, Yasouj 75914,
Iran
###### Abstract
We present the metric for the $(n+1)$-dimensional charged rotating dilaton
black branes with cylindrical or toroidal horizons in the background of anti-
de Sitter spacetime. We find the suitable counterterm which removes the
divergences of the action in the presence of the dilaton potential in all
higher dimensions. We plot the Penrose diagrams of the spacetime and reveal
that the spacetime geometry crucially modifies in the presence of the dilaton
field. The conserved and thermodynamic quantities of the black branes are also
computed.
## I Introduction
The motivation idea for studying higher dimensional black holes with a
negative cosmological constant arises from the correspondence between the
gravitating fields in an anti-de Sitter (AdS) spacetime and conformal field
theory living on the boundary of the AdS spacetime Witt1 . It was argued that
the thermodynamics of black holes in AdS spaces can be identified with that of
a certain dual CFT in the high temperature limit Witt2 . Having the AdS/CFT
correspondence idea at hand, one can gain some insights into thermodynamic
properties and phase structures of strong ’t Hooft coupling CFTs by studying
thermodynamics of AdS black holes. According to the AdS/CFT correspondence,
the rotating black holes in AdS space are dual to certain CFTs in a rotating
space Haw , while charged ones are dual to CFTs with chemical potential Cham1
; Cham2 . The most general higher dimensional uncharged rotating black holes
in AdS space have been recently found Haw ; Gib . As far as we know, rotating
black holes for the Maxwell field minimally coupled to Einstein gravity in
higher dimensions, do not exist in a closed form and one has to rely on
perturbative or numerical methods to construct them in the background of
asymptotically flat kunz1 ; Aliev2 and AdS kunz2 spacetimes. There has also
been recent interest in constructing the analogous charged rotating solutions
in the framework of gauged supergravity in various dimensions Cvetic0 ;
Cvetic1 ; Cvetic2 .
There has been a renewed interest in studying scalar coupled solutions of
general relativity ever since new black hole solutions have been found in the
context of string theory. The low energy effective action of string theory
contains two massless scalars namely dilaton and axion. The dilaton field
couples in a nontrivial way to other fields such as gauge fields and results
into interesting solutions for the background spacetime CDB1 ; CDB2 ; Hor2 ;
Hor3 . These solutions CDB1 ; CDB2 ; Hor2 ; Hor3 , however, are all
asymptotically flat. The presence of Liouville-type dilaton potential, which
is regarded as the generalization of the cosmological constant, changes the
asymptotic behavior of the solutions to be neither asymptotically flat nor
(A)dS. While in the presence of one or two Liouville-type potential, black
holes/branes which are neither asymptotically flat nor (A)dS have been
explored in MW ; CHM ; Cai ; Clem ; Deh1 ; Deh2 ; Sheykhi0 ; Sheykhi1 ; DPH ;
DHSR , magnetic dilaton solutions coupled to nonlinear electrodynamics have
also been investigated DSH . Although these kind of solutions may shed some
light on the possible extensions of AdS/CFT correspondence, they are
physically less interesting due to their unusual asymptotic behavior.
Recently, the dilaton potential leading to (A)dS-like solutions of dilaton
gravity has been found Gao1 ; Gao2 ; Gao3 . It was shown that the cosmological
constant is coupled to the dilaton in a very nontrivial way. With an
appropriate combination of three Liouville-type dilaton potentials, a class of
static dilaton black hole solutions in (A)dS spaces has been obtained by using
a coordinates transformation which recast the solution in the schwarzschild
coordinates system Gao1 ; Gao2 . Such potential may arise from the
compactification of a higher dimensional supergravity model Gid which
originates from the low energy limit of a background string theory. More
recently, one of us has constructed a class of magnetic rotating solutions in
four-dimensional Einstein-Maxwell-dilaton gravity with Liouville-type
potential in the background of AdS spaces Sheykhi2 . Although these solutions
are not black holes and represent spacetimes with conic singularities,
asymptotically AdS charged rotating black string solutions in four-dimensional
Einstein-Maxwell-dilaton gravity has also been constructed Sheykhi3 . So far,
exact higher dimensional charged rotating dilaton black hole/brane solutions
for an arbitrary dilaton-electromagnetic coupling constant in the background
of AdS spacetime have not been constructed. In this paper we intend to
construct exact, charged rotating dilaton black branes with cylindrical or
toroidal horizons in higher dimensional AdS spacetimes and investigate their
properties.
This paper is outlined as follows. Section II is devoted to a brief review of
the field equations and the general formalism of calculating the conserved
quantities. We shall also present the suitable counterterm which removes the
divergences of the action in the presence of the dilaton potential. In section
III, we construct the $(n+1)$-dimensional charged rotating dilaton black
branes with a complete set of rotation parameters and investigate their
properties. We also obtain the conserved and thermodynamical quantities of the
$(n+1)$-dimensional black brane solutions. We finish with conclusion in the
last section.
## II Basic Equations and counterterm method
We consider the $(n+1)$-dimensional theory in which gravity is coupled to
dilaton and Maxwell field with an action
$\displaystyle I_{G}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{n+1}x\sqrt{-g}\left({R}\text{
}-\frac{4}{n-1}(\nabla\Phi)^{2}-V(\Phi)-e^{-4\alpha\Phi/(n-1)}F_{\mu\nu}F^{\mu\nu}\right)$
(1)
$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n}x\sqrt{-\gamma}\Theta(\gamma),$
where ${R}$ is the scalar curvature, $\Phi$ is the dilaton field,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor, and $A_{\mu}$ is the electromagnetic potential.
$\alpha$ is an arbitrary constant governing the strength of the coupling
between the dilaton and the Maxwell field. The last term in Eq. (1) is the
Gibbons-Hawking surface term. It is required for the variational principle to
be well-defined. The factor $\Theta$ represents the trace of the extrinsic
curvature for the boundary ${\partial\mathcal{M}}$ and $\gamma$ is the induced
metric on the boundary. While $\alpha=0$ corresponds to the usual Einstein-
Maxwell-scalar theory, $\alpha=1$ indicates the dilaton-electromagnetic
coupling that appears in the low energy string action in Einstein’s frame. For
arbitrary value of $\alpha$ in (A)dS spaces the form of the dilaton potential
in arbitrary dimensions is chosen as Gao2
$\displaystyle{V}({\Phi})$ $\displaystyle=$
$\displaystyle\frac{2\Lambda}{n(n-2+\alpha^{2})^{2}}\Big{\\{}-\alpha^{2}\left[(n+1)^{2}-(n+1)\alpha^{2}-6(n+1)+\alpha^{2}+9\right]e^{{-4(n-2){\Phi}}/{[(n-1)\alpha]}}$
(2)
$\displaystyle+(n-2)^{2}(n-\alpha^{2})e^{{4\alpha{\Phi}}/({n-1})}+4\alpha^{2}(n-1)(n-2)e^{{-2{\Phi}(n-2-\alpha^{2})}/{[(n-1)\alpha}]}\Big{\\}}.$
Here $\Lambda$ is the cosmological constant. For later convenience we redefine
$\Lambda=-n(n-1)/2l^{2}$, where $l$ is the AdS radius of spacetime. It is
clear the cosmological constant is coupled to the dilaton in a very nontrivial
way. This type of the dilaton potential was introduced for the first time by
Gao and Zhang Gao2 . They derived, by applying a coordinates transformation
which recast the solution in the Schwarzchild coordinates system, the static
dilaton black hole solutions in the background of (A)dS universe. For this
purpose, they required the existence of the (A)dS dilaton black hole solutions
and extracted successfully the form of the dilaton potential leading to (A)dS-
like solutions. They also argued that this type of derived potential can be
obtained when a higher dimensional theory is compactified to four dimension,
including various supergravity models Gid . In the absence of the dilaton
field ($\Phi=0=\alpha$), the potential (2) reduces to ${V}({\Phi})=2\Lambda$,
and the action (1) recovers the action of Einstein-Maxwell gravity with
cosmological constant. The equations of motion can be obtained by varying the
action (1) with respect to the gravitational field $g_{\mu\nu}$, the dilaton
field $\Phi$ and the gauge field $A_{\mu}$ which yields the following field
equations
$\mathcal{R}_{\mu\nu}=\frac{4}{n-1}\left(\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{4}g_{\mu\nu}V(\Phi)\right)+2e^{{-4\alpha\Phi}/{(n-1)}}\left(F_{\mu\eta}F_{\nu}^{\text{
}\eta}-\frac{1}{2(n-1)}g_{\mu\nu}F_{\lambda\eta}F^{\lambda\eta}\right),$ (3)
$\nabla^{2}\Phi=\frac{n-1}{8}\frac{\partial
V}{\partial\Phi}-\frac{\alpha}{2}e^{-{4\alpha\Phi}/{(n-1})}F_{\lambda\eta}F^{\lambda\eta},$
(4)
$\partial_{\mu}\left(\sqrt{-g}e^{{-4\alpha\Phi}/({n-1})}F^{\mu\nu}\right)=0.$
(5)
The conserved charges of the spacetime can be calculated through the use of
the substraction method of Brown and York BY . Such a procedure causes the
resulting physical quantities to depend on the choice of reference background.
For asymptotically AdS solutions, the way that one can calculate these
quantities and obtain finite values for them is through the use of the
counterterm method inspired by AdS/CFT correspondence Witt1 . In this paper we
deal with the spacetimes with zero curvature boundary, $R_{abcd}(\gamma)=0$,
and therefore the counterterm for the stress energy tensor should be
proportional to $\gamma^{ab}$. We find the suitable counterterm which removes
the divergences in the form
$I_{ct}=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n}x\sqrt{-\gamma}\left(-\frac{(n-1)(n-2)}{2l}+\frac{\sqrt{-n(n-1)V(\Phi)}}{2}\right).$
(6)
In the absence of the dilaton field, ${V}({\Phi})=2\Lambda=-n(n-1)/l^{2}$, and
Eq. (6) reduces to
$I_{ct}=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n}x\sqrt{-\gamma}\left(\frac{n-1}{l}\right),$
(7)
which is the counterterm of the asymptotically AdS spaces. Having the total
finite action $I=I_{G}+I_{\mathrm{ct}}$ at hand, we can use the quasilocal
definition to construct a divergence free stress-energy tensor BY . Thus we
write down the finite stress-energy tensor in $(n+1)$-dimensional Einstein-
dilaton gravity with three Liouville-type dilaton potentials (2) in the
following form
$T^{ab}=\frac{1}{8\pi}\left[\Theta^{ab}-\Theta\gamma^{ab}+\left(-\frac{(n-1)(n-2)}{2l}+\frac{\sqrt{-n(n-1)V(\Phi)}}{2}\right)\gamma^{ab}\right].$
(8)
The first two terms in Eq. (8) are the variation of the action (1) with
respect to $\gamma_{ab}$, and the last two terms are the variation of the
boundary counterterm (6) with respect to $\gamma_{ab}$. To compute the
conserved charges of the spacetime, one should choose a spacelike surface
$\mathcal{B}$ in $\partial\mathcal{M}$ with metric $\sigma_{ij}$, and write
the boundary metric in ADM (Arnowitt-Deser-Misner) form:
$\gamma_{ab}dx^{a}dx^{a}=-N^{2}dt^{2}+\sigma_{ij}\left(d\varphi^{i}+V^{i}dt\right)\left(d\varphi^{j}+V^{j}dt\right),$
where the coordinates $\varphi^{i}$ are the angular variables parameterizing
the hypersurface of constant $r$ around the origin, and $N$ and $V^{i}$ are
the lapse and shift functions respectively. When there is a Killing vector
field $\mathcal{\xi}$ on the boundary, then the quasilocal conserved
quantities associated with the stress tensors of Eq. (8) can be written as
$Q(\mathcal{\xi)}=\int_{\mathcal{B}}d^{n-1}x\sqrt{\sigma}T_{ab}n^{a}\mathcal{\xi}^{b},$
(9)
where $\sigma$ is the determinant of the metric $\sigma_{ij}$, $\mathcal{\xi}$
and $n^{a}$ are, respectively, the Killing vector field and the unit normal
vector on the boundary $\mathcal{B}$. For boundaries with timelike
($\xi=\partial/\partial t$) and rotational
($\varsigma=\partial/\partial\varphi$) Killing vector fields, one obtains the
quasilocal mass and angular momentum
$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{n-1}x\sqrt{\sigma}T_{ab}n^{a}\xi^{b},$ (10)
$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{n-1}x\sqrt{\sigma}T_{ab}n^{a}\varsigma^{b}.$
(11)
provided the surface $\mathcal{B}$ contains the orbits of $\varsigma$. These
quantities are, respectively, the conserved mass and angular momenta of the
system enclosed by the boundary $\mathcal{B}$. Note that they will both depend
on the location of the boundary $\mathcal{B}$ in the spacetime, although each
is independent of the particular choice of foliation $\mathcal{B}$ within the
surface $\partial\mathcal{M}$.
## III Charged rotating dilaton black brane
Our aim here is to construct the $(n+1)$-dimensional rotating solutions of the
field equations (3)-(5) with $k$ rotation parameters and investigate their
properties. The rotation group in $(n+1)$-dimensions is $SO(n)$ and therefore
the number of independent rotation parameters for a localized object is equal
to the number of Casimir operators, which is $[n/2]\equiv k$, where $[x]$ is
the integer part of $x$. Inspired by awad , we take the metric of
$(n+1)$-dimensional rotating solution with cylindrical or toroidal horizons
and $k$ rotation parameters in the form
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-U(r)\left(\Xi
dt-{{\sum_{i=1}^{k}}}a_{i}d\phi_{i}\right)^{2}+\frac{r^{2}}{l^{4}}R^{2}(r){{\sum_{i=1}^{k}}}\left(a_{i}dt-\Xi
l^{2}d\phi_{i}\right)^{2}$
$\displaystyle-\frac{r^{2}}{l^{2}}R^{2}(r){\sum_{i<j}^{k}}(a_{i}d\phi_{j}-a_{j}d\phi_{i})^{2}+\frac{dr^{2}}{W(r)}+\frac{r^{2}}{l^{2}}R^{2}(r)dX^{2},$
$\displaystyle\Xi^{2}$ $\displaystyle=$ $\displaystyle
1+\sum_{i=1}^{k}\frac{a_{i}^{2}}{l^{2}},$ (12)
where $a_{i}$’s are $k$ rotation parameters. The functions $U(r)$, $W(r)$ and
$R(r)$ should be determined and $l$ has the dimension of length which is
related to the cosmological constant $\Lambda$ for the case of Liouville-type
potential with constant $\Phi$. The angular coordinates are in the range
$0\leq\phi_{i}\leq 2\pi$ and $dX^{2}$ is the Euclidean metric on the
$(n-k-1)$-dimensional submanifold with volume $\Sigma_{n-k-1}$. The Maxwell
equation (5) can be integrated immediately to give
$\displaystyle F_{tr}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{U(r)}{W(r)}}\frac{q\Xi
e^{{4\alpha\Phi}/{(n-1)}}}{\left(rR\right)^{n-1}},$ $\displaystyle
F_{\phi_{i}r}$ $\displaystyle=$ $\displaystyle-\frac{a_{i}}{\Xi}F_{tr}.$ (13)
where $q$, an integration constant, is the charge parameter of the black
brane. Using metric (12) and the Maxwell fields (III), one can show that
equations (3)-(4) have solutions of the form
$\displaystyle
U(r)=-\left(\frac{c}{r}\right)^{n-2}\left[1-\left(\frac{b}{r}\right)^{n-2}\right]^{1-\gamma\left(n-2\right)}-\frac{2\Lambda
r^{2}}{n(n-1)}\left[1-\left(\frac{b}{r}\right)^{n-2}\right]^{\gamma},$ (14)
$\displaystyle
W(r)=\Bigg{\\{}-\left(\frac{c}{r}\right)^{n-2}\left[1-\left(\frac{b}{r}\right)^{n-2}\right]^{1-\gamma\left(n-2\right)}-\frac{2\Lambda
r^{2}}{n(n-1)}\left[1-\left(\frac{b}{r}\right)^{n-2}\right]^{\gamma}\Bigg{\\}}$
$\displaystyle\times\left[1-\left(\frac{b}{r}\right)^{n-2}\right]^{\gamma(n-3)},$
(15)
$\displaystyle\Phi(r)=\frac{n-1}{4}\sqrt{\gamma(2+2\gamma-n\gamma)}\ln\left[1-\left(\frac{b}{r}\right)^{n-2}\right],$
(16) $\displaystyle
R(r)=\left[1-\left(\frac{b}{r}\right)^{n-2}\right]^{\gamma/2},$ (17)
Here $b$ and $c$ are integration constants and the constant $\gamma$ is
$\gamma=\frac{2\alpha^{2}}{(n-2)(n-2+\alpha^{2})}.$ (18)
The charge parameter $q$ is related to $b$ and $c$ by
$q^{2}=\frac{(n-1)(n-2)^{2}}{2(n-2+\alpha^{2})}b^{n-2}c^{n-2}.$ (19)
When ($\alpha=0=\gamma$), the above solution recovers the asymptotically AdS
charged rotating black branes presented in Deh3 ; awad . In the particular
case $n=3$ these solutions reduce to the asymptotically AdS charged rotating
dilaton black strings Sheykhi3 . For $n=3$ and $\alpha=0$ they reduce to
charged rotating black string solutions presented in Lem0 . Inserting
solutions (14)-(17) into the Maxwell fields (III), they can be simplified as
$\displaystyle F_{tr}$ $\displaystyle=$ $\displaystyle\frac{q\Xi}{r^{n-1}},$
$\displaystyle F_{\phi_{i}r}$ $\displaystyle=$
$\displaystyle-\frac{a_{i}}{\Xi}F_{tr}.$ (20)
As one can see from Eq. (III), in the background of AdS universe, the dilaton
field does not exert any direct influence on the matter fields $F_{tr}$ and
$F_{\phi_{i}r}$’s $(i=1,...,k)$, however, the dilaton field modifies the
geometry of the spacetime as it participates in the field equations. This is
in contrast to the solutions presented in Sheykhi0 . The solutions of Ref.
Sheykhi0 are neither asymptotically flat nor (A)dS and the gauge field
crucially depends on the scalar dilaton field. The gauge potential $A_{\mu}$
corresponding to the electromagnetic tensor (III) can be obtained as
$A_{\mu}=-\frac{q}{(n-2)r^{n-2}}\left(\Xi\delta_{\mu}^{t}-a_{i}\delta_{\mu}^{i}\right)\hskip
39.83368pt{\text{(no sum on i)}}.$ (21)
Figure 1: Penrose diagram for black brane with two horizon ($\alpha=0$)
located at $r=b$ and $r=r_{+}$. The dotted curves represent
$r=\mathrm{const}.$
Figure 2: Penrose diagram for extreme black brane with $\alpha=0$ and one
horizon located at $r=r_{+}=r_{\mathrm{h}}$. The dotted curves represent
$r=\mathrm{const}.$
Figure 3: Penrose diagram for black brane with $\alpha\neq 0$ and one horizon
located at $r_{+}$. The dotted curves represent $r=\mathrm{const}.$
This spacetime is asymptotically AdS, since the functions $W(r)$ and $U(r)$
behave as $-2\Lambda\left(r^{2}/[n(n-1)]\right)$ as $r\rightarrow\infty$.
Indeed, for large values of $r$, in four dimensions $(n=3)$ the functions
$W(r)$ and $U(r)$ behave as $-2\Lambda\left(r^{2}/6+p_{1}r+p_{2}\right)$,
while in higher dimensions ($n\geq 4$) they behave like $-2\Lambda
r^{2}/[n(n-1)]$. Here $p_{1}$ and $p_{2}$ are functions of $\alpha$. This
implies that the falloff rate of the solutions in four dimension is much
slower than in higher dimensions. The Kretschmann invariant
$R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$ and the Ricci scalar $R$
diverge at $r=0$ and therefore there is an essential singularity located at
$r=0$. For all $\alpha$, the surface $r=r_{+}$ is an event horizon (the
positive root of Eq. $W(r=r_{+})=0$). The surface $r=b$ is a curvature
singularity except for the case $\alpha=0$ when it is a nonsingular inner
horizon. This is consistent with the idea that the inner horizon is unstable
in the Einstein-Maxwell theory. Therefore, our solutions describe black branes
only in the case $b<r_{+}$ Hor3 . For $\alpha=0$ the metric (III) is real in
the range $0\leq r<\infty$, while for $\alpha>0$, it is real only in the range
$b\leq r<\infty$. Thus, in order to have a real metric, we restrict the
spacetime to the region $r\geq b$. We plot Penrose diagrams of spacetime in
Figs. 1-3. From these figures we find out that the casual structure is
asymptotically well behaved. It is notable to mention that in contrast to the
solutions of Ref. DPH , here we have a spacelike singularity with one horizon
and the solutions are asymptotically AdS, at all times, in the presence of
dilaton field. It is also worthwhile to note that the dilaton field $\Phi(r)$
and the electromagnetic fields $F_{tr}$ and $F_{\phi_{i}r}$’s become zero as
$r\rightarrow\infty$. As in the case of rotating black hole solutions of the
Einstein gravity, the above metric has both Killing and event horizons. The
Killing horizon is a null surface whose null generators are tangent to a
Killing field. It is easy to see that the Killing vector
$\chi=\partial_{t}+{{{\sum_{i=1}^{k}}}}\Omega_{i}\partial_{\phi_{i}},$ (22)
is the null generator of the event horizon, where $k$ denote the number of
rotation parameters Deh4 . We can obtain the temperature and angular velocity
of the horizon by analytic continuation of the metric. The analytical
continuation of the Lorentzian metric by $t\rightarrow i\tau$ and
$a\rightarrow ia$ yields the Euclidean section, whose regularity at $r=r_{+}$
requires that we should identify $\tau\sim\tau+\beta_{+}$ and
$\phi_{i}\sim\phi_{i}+\beta_{+}\Omega_{i}$, where $\beta_{+}$ and $\Omega_{i}$
’s are the inverse Hawking temperature and the $i$th component of angular
velocity of the horizon, respectively. The Hawking temperature of the black
brane on the horizon $r_{+}$ can be calculated using the relation
$T_{+}=\beta_{+}^{-1}=\left(\frac{U^{\text{
}^{\prime}}}{4\pi\Xi\sqrt{U/W}}\right)_{r=r_{+}}.$ (23)
where a prime denotes derivative with respect to $r$. It is a matter of
calculation to show that
$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi\Xi
r_{+}}\Bigg{\\{}n+(n-1)[\gamma(n-2)-2]\left(\frac{b}{r_{+}}\right)^{n-2}\Bigg{\\}}\left(\frac{c}{r_{+}}\right)^{n-2}\left[1-\left(\frac{b}{r_{+}}\right)^{n-2}\right]^{-\gamma(n-1)/2}$
(24) $\displaystyle\Omega_{i}$ $\displaystyle=$ $\displaystyle\frac{a_{i}}{\Xi
l^{2}}.$ (25)
where we have used equation $W(r=r_{+})=0$ for omitting $\Lambda$. It is easy
to check that for $\alpha\geq\sqrt{n}$, the temperature is always positive,
while for $\alpha<\sqrt{n}$ the temperature is positive definite provided we
have
$\displaystyle r_{+}>b\left(\frac{-n}{(n-1)[\gamma(n-2)-2]}\right)^{-1/(n-2)}$
(26)
The entropy of the black brane typically satisfies the so called area law of
the entropy which states that the entropy of the black hole is a quarter of
the event horizon area Beck . This near universal law applies to almost all
kinds of black holes, including dilaton black holes/branes, in Einstein
gravity hunt . Denoting the volume of the hypersurface boundary at constant
$t$ and $r$ by $V_{n-1}=(2\pi)^{k}\Sigma_{n-k-1}$, we can show that the
entropy per unit volume $V_{n-1}$ of the black brane is
${S}=\frac{\Xi
r_{+}^{n-1}}{4l^{n-2}}\left[1-\left(\frac{b}{r_{+}}\right)^{n-2}\right]^{\gamma(n-1)/2}.$
(27)
The mass per unit volume $V_{n-1}$ of the black brane can be calculated
through the use of Eq. (10). We find
$\displaystyle{M}=\frac{(3\Xi^{2}-1)c}{16\pi
l}+\frac{\alpha^{2}(\alpha^{2}-1)b^{3}}{24\pi l^{3}(\alpha^{2}+1)^{3}}\hskip
19.91684pt\mathrm{for}\ \ n=3,$ (28)
$\displaystyle{M}=\frac{(n\Xi^{2}-1)c^{n-2}}{16\pi l^{n-2}}\hskip
88.2037pt\mathrm{for}\ \ n\geq 4.$ (29)
Let us note that the mass expression in four dimension differs from higher
dimensions. This is due to the fact that, for large values of $r$, the falloff
rate of the solutions in four and higher dimensions are different. Thus, the
mass in four dimension depends on the dilaton coupling constant $\alpha$ while
in higher dimensions it is independent of $\alpha$ and coincides with the mass
of charged rotating black branes in Einstein gravity Deh3 . The angular
momentum per unit volume $V_{n-1}$ of the black brane can be calculated
through the use of Eqs. (11). We obtain
$J_{i}=\frac{n\Xi c^{n-2}a_{i}}{16\pi l^{n-2}}.$ (30)
For $a_{i}=0$ ($\Xi=1$), the angular momentum per unit volume vanishes, and
therefore $a_{i}$’s are the rotational parameters of the spacetime. Next, we
calculate the electric charge of the solutions. To determine the electric
field we should consider the projections of the electromagnetic field tensor
on special hypersurfaces. The normal to such hypersurfaces is
$u^{0}=\frac{1}{N},\text{ \ }u^{r}=0,\text{ \ }u^{i}=-\frac{V^{i}}{N},$ (31)
where $N$ and $V^{i}$ are the lapse function and shift vector. Then the
electric field is
$E^{\mu}=g^{\mu\rho}e^{\frac{-4\alpha\phi}{n-1}}F_{\rho\nu}u^{\nu}$, and the
electric charge per unit volume $V_{n-1}$ can be found by calculating the flux
of the electric field at infinity, yielding
${Q}=\frac{\Xi q}{4\pi l^{n-2}}.$ (32)
The electric potential $U$, measured at infinity with respect to the horizon,
is defined by Cham2 ; Cal
$U=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(33)
where $\chi$ is the null generators of the event horizon given by Eq. (22). It
is a matter of calculation to show that
$U=\frac{q}{(n-2)\Xi{r_{+}^{n-2}}}.$ (34)
## IV Conclusions
It is well-known that in the presence of Liouville-type dilaton potential,
which is regarded as the generalization of the cosmological constant, the
asymptotic behavior of the solutions change to be neither asymptotically flat
nor (A)dS. As a matter of fact, with the exception of a pure cosmological
constant, no dilaton (A)dS solution exists with the presence of only one or
two Liouville-type potential MW . In this paper, with an appropriate
combination of three Liouville-type dilaton potentials, we obtained a new
class of charged rotating black brane solutions in $(n+1)$-dimensional
Einstein-Maxwell-dilaton gravity with cylindrical or toroidal horizons in the
background of AdS spaces and investigated their properties. We found a
suitable counterterm which removes the divergences of the action in the
presence of dilaton potential in all higher dimensions. We ploted the Penrose
diagrams associated with these spacetimes. These diagrams show that for
$\alpha=0$, the solutions can be interpreted as black brane with two event
horizons, an extreme black brane or a naked singularity provided the
parameters of the solutions are chosen suitably. In this case we encounter a
timelike singularity which is located at $r=0$. We found out that for
$\alpha>0$, the solutions represent black brane with one horizon. In this case
we have a spacelike singularity at $r=b$. We also computed the conserved and
thermodynamic quantities of the black branes by using the conterterm method
inspired by AdS/CFT correspondence. Interestingly enough, we found that the
mass expression in four dimension depends on the dilaton coupling constant
$\alpha$, while in higher dimensions it is independent of $\alpha$. This can
be understood easily, since in four dimension the functions $W(r)$ and $U(r)$
behave as $-2\Lambda\left(r^{2}/6+p_{1}r+p_{2}\right)$ for large $r$, while in
higher dimensions ($n\geq 4$) they behave like $-2\Lambda r^{2}/[n(n-1)]$.
###### Acknowledgements.
We thank the anonymous referee for constructive comments. We are also grateful
to Prof. M.H. Dehghani for helpful discussions. This work has been supported
by Research Institute for Astronomy and Astrophysics of Maragha.
## References
* (1) E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998);
J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).
* (2) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998).
* (3) S. W. Hawking, C. J. Hunter and M. Taylor, Phys. Rev. D 59, 064005 (1999).
* (4) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Phys. Rev. D 60, 064018 (1999);
R. G. Cai and K. S. Soh, Mod. Phys. Lett. A 14, 1895 (1999).
* (5) M. Cvetic and S. S. Gubser, J. High Energy Phys. 04, 024 (1999).
* (6) G. W. Gibbons, H. Lu, D. N. Page and C. N. Pope, Phys. Rev. Lett. 93, 171102 (2004);
G. W. Gibbons, H. Lu, D. N. Page and C. N. Pope, J. Geom. Phys. 53, 49 (2005).
* (7) J. Kunz, F. Navarro-Lerida and J. Viebahn, Phys. Lett. B 639, 362 (2006).
* (8) A. N. Aliev, Phys. Rev. D 74, 024011 (2006).
* (9) J. Kunz, F. Navarro-Lerida and E. Radu, Phys. Lett. B 649, 463 (2007);
A. N. Aliev, Phys. Rev. D 75, 084041 (2007);
H. C. Kim and R. G. Cai, Phys. Rev. D 77, 024045 (2008);
Y. Brihaye and T. Delsate, arXiv:0806.1583.
* (10) M. Cvetic and D. Youm, Phys. Rev. D 54, 2612 (1996);
D. Youm, Phys. Rep. 316, 1 (1999).
* (11) M. Cvetic and D. Youm, Nucl. Phys. B 477, 449 (1996).
* (12) M. Cvetic and D. Youm, Nucl. Phys. B 476, 118 (1996);
Z. W. Chong, M. Cvetic, H. Lu, and C. N. Pope, Phys. Rev. D 72, 041901 (2005).
* (13) G. W. Gibbons and K. Maeda, Nucl. Phys. B 298, 741 (1988);
T. Koikawa and M. Yoshimura, Phys. Lett. B 189, 29 (1987);
D. Brill and J. Horowitz, Phys. Lett. B 262, 437 (1991).
* (14) D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D 43, 3140 (1991);
R. Gregory and J. A. Harvey, Phys. Rev. D 47, 2411 (1993).
* (15) G. T. Horowitz and A. Strominger, Nucl. Phys. B 360, 197 (1991).
* (16) J. H. Horn and G. Horowitz, Phys. Rev. D 46, 1340 (1992).
* (17) S. J. Poletti and D. L. Wiltshire, Phys. Rev. D 50, 7260 (1994) ;
S. J. Poletti, J. Twamley and D. L. Wiltshire, Phys. Rev. D 51, 5720 (1995).
* (18) K. C. K. Chan, J. H. Horne and R. B. Mann, Nucl. Phys. B 447, 441 (1995).
* (19) R. G. Cai, J. Y. Ji and K. S. Soh, Phys. Rev D 57, 6547 (1998);
R. G. Cai and Y. Z. Zhang, _ibid._ 64, 104015 (2001).
* (20) G. Clement, D. Gal’tsov and C. Leygnac, Phys. Rev D 67, 024012 (2003);
G. Clement and C. Leygnac, Phys. Rev. D 70, 084018 (2004).
* (21) M. H Dehghani, Phys. Rev. D 71, 064010 (2005).
* (22) M. H. Dehghani and N. Farhangkhah, Phys. Rev. D 71, 044008 (2005).
* (23) A. Sheykhi, M. H. Dehghani and N. Riazi, Phys. Rev. D 75, 044020 (2007);
A. Sheykhi, M. H. Dehghani, N. Riazi and J. Pakravan Phys. Rev. D 74, 084016
(2006);
A. Sheykhi and N. Riazi, Phys. Rev. D 75, 024021 (2007);
R.G. Cai, Y.Z. Zhang, Phys. Rev. D 54 4891 (1996).
* (24) A. Sheykhi, Phys. Rev. D 76, 124025 (2007);
A. Sheykhi, Phys. Lett. B 662, 7 (2008).
* (25) M. H. Dehghani, J. Pakravan and S. H. Hendi, Phys. Rev. D 74, 104014 (2006);
S. H. Hendi, J. Math. Phys. 49, 082501 (2008).
* (26) M. H. Dehghani, S. H. Hendi, A. Sheykhi and H. Rastegar Sedehi. JCAP 0702, 020 (2007).
* (27) M. H. Dehghani, A. Sheykhi and S. H. Hendi, Phys. Lett. B 659, 476 (2008).
* (28) C. J. Gao and S. N. Zhang, Phys. Rev. D 70, 124019 (2004).
* (29) C. J. Gao and S. N. Zhang, Phys. Lett. B 605, 185 (2005).
* (30) C. J. Gao and S. N. Zhang, Phys. Lett. B 612 127 (2005).
* (31) S. B. Giddings, Phys. Rev. D 68, 026006 (2003);
E. Radu and D. H. Tchrakian, Class. Quant. Grav. 22, 879 (2005).
* (32) A. Sheykhi, Phys. Lett. B 672,101 (2009).
* (33) A. Sheykhi, Phys. Rev. D 78, 064055 (2008).
* (34) J. Brown and J. York, Phys. Rev. D 47, 1407 (1993).
* (35) A. M. Awad, Class. Quant. Grav. 20, 2827 (2003).
* (36) M. H. Dehghani, Phys. Rev. D 66, 044006 (2002);
M. H. Dehghani and A. Khodam-Mohammadi, _ibid._ 67, 084006 (2003).
* (37) J. P.S. Lemos and V. T. Zanchin, Phys. Rev. D 54, 3840 (1996);
J. P. S. Lemos, Class. Quant. Grav. 12, 1081 (1995);
J. P. S. Lemos, Phys. Lett. B 353, 46 (1995).
* (38) M. H. Dehghani and R. B. Mann, Phys.Rev. D 73, 104003 (2006).
* (39) J. D. Beckenstein, Phys. Rev. D 7, 2333 (1973);
S. W. Hawking, Nature (London) 248, 30 (1974);
G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).
* (40) C. J. Hunter, Phys. Rev. D 59, 024009 (1999);
S. W. Hawking, C. J Hunter and D. N. Page, Phys. Rev. D 59, 044033 (1999);
R. B. Mann Phys. Rev. D 60, 104047 (1999).
* (41) M. M. Caldarelli, G. Cognola and D. Klemm, Class. Quant. Grav. 17, 399 (2000).
|
arxiv-papers
| 2009-11-15T04:03:33 |
2024-09-04T02:49:06.477686
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi and S.H. Hendi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0911.2831"
}
|
0911.2949
|
11institutetext: NAOC / Yunnan Observatory, Chinese Academy of Sciences,
Kunming 650011, China. sujie@ynao.ac.cn, ly@ynao.ac.cn
22institutetext: Graduate School of the Chinese Academy of Sciences, Beijing
100049, China.
Received 2009 Sep. 18; accepted 2009 Nov. 12
# On the DB gap of white dwarf evolution: effects of hydrogen mass fraction
and convective overshooting
Jie Su 11 2 2 Yan Li 11
###### Abstract
We investigate the spectral evolution of white dwarfs by considering the
effects of hydrogen mass in the atmosphere and convective overshooting above
the convection zone. Our numerical results show that white dwarfs with $M_{\rm
H}\sim 10^{-16}~{}M_{\odot}$ show DA spectral type between $46,000\lesssim
T_{\rm eff}\lesssim 26,000~{}{\rm K}$ and DO or DB spectral type may appears
on either side of this temperature range. White dwarfs with $M_{\rm H}\sim
10^{-15}~{}M_{\odot}$ appear as DA stars until they cool to $T_{\rm eff}\sim
31,000~{}{\rm K}$, from then on they will evolve into DB white dwarfs as a
result of convective mixing. If $M_{\rm H}$ in the white dwarfs more than
$10^{-14}~{}M_{\odot}$, the convective mixing will not occur when $T_{\rm
eff}>20,000~{}{\rm K}$, thus these white dwarfs always appear as DA stars.
White dwarfs within the temperature range $46,000\lesssim T_{\rm eff}\lesssim
31,000~{}{\rm K}$ always show DA spectral type, which coincides with the DB
gap. We notice the importance of the convective overshooting and suggest that
the overshooting length should be proportional to the thickness of the
convection zone to better fit the observations.
###### keywords:
convection — stars: evolution — stars: white dwarfs
## 1 Introduction
White dwarfs can be classified into several spectral types in terms of the
spectral characteristics. The current classification system was introduced by
Sion et al. ([1983]) and has been modified several times. In this system, the
spectral type of a white dwarf is denoted by a letter D plus another letter
indicating its spectral characteristics. Sometimes, a suffix is added to
indicate some other features (polarization, magnetic field, pulse, etc.).
Table 1 lists a spectral classification scheme from McCook & Sion ([1999]).
Table 1: White dwarf spectral types Spectral Type | Characteristics
---|---
DA | | Only Balmer lines; no He I or metals present
DB | | He I lines; no H or metals present
DC | | Continuous spectrum, no lines deeper than $5\%$ in any part of the electromagnetic spectrum
DO | | He II strong; He I or H present
DZ | | Metal lines only; no H or He lines
DQ | | Carbon features, either atomic or molecular in any part of the electromagnetic spectrum
P | (suffix) | Magnetic white dwarfs with detectable polarization
H | (suffix) | Magnetic white dwarfs without detectable polarization
X | (suffix) | Peculiar or unclassifiable spectrum
E | (suffix) | Emission lines are present
? | (suffix) | Uncertain assigned classification; a colon (:) may also be used
V | (suffix) | Optional symbol to denote variability
Most of white dwarfs are of DA type which have hydrogen-dominated atmospheres.
They are found at all effective temperatures from $170,000~{}{\rm K}$ down to
about $4,500~{}{\rm K}$ (Kurtz et al. [2008]). DA white dwarfs occupy the vast
majority (about $75\%$) of all known white dwarfs.
About $25\%$ of the observed white dwarfs have helium-dominated atmospheres
which can be divided further into two spectral types. The DO white dwarfs are
found between approximately $100,000$ to $45,000~{}{\rm K}$, their outer
atmospheres being dominated by singly ionized helium (He II). The DB white
dwarfs are found between approximately $30,000$ to $12,000~{}{\rm K}$, their
outer atmospheres being dominated by neutral helium (He I).
There is an interesting fact that few white dwarfs with helium-dominated
atmospheres (DO or DB type) are found in the effective temperature range of
$45,000\lesssim T_{\rm eff}\lesssim 30,000~{}{\rm K}$. This is the so-called
DB gap. The reason for this phenomenon is not clear. On one side, there is
strong gravitational field in the white dwarf, which may cause a stratified
atmosphere. The so-called gravitational settling effect lets the light element
(hydrogen) floating to the stellar surface and the heavy element (helium)
sinking to the bottom of the stellar envelope. The typical mass fractions of
hydrogen and helium in the white dwarfs are $M_{\rm H}/M_{\rm tot}\lesssim
10^{-4}$ and $M_{\rm He}/M_{\rm tot}\lesssim 10^{-2}$, which are the mass
threshold for residual nuclear burning (Tremblay & Bergeron [2008]). If there
is no mixing between hydrogen and helium, we will expect all white dwarfs to
be DA stars. On the other side, however, many observational data show that
$M_{\rm H}$ may be significantly lower than the typical value. The existence
of a large number of non-DA white dwarfs indicates that some physical
mechanisms are competing with the gravitational settling and make the spectral
type of some DA white dwarfs changed. The DB gap is suspected to be due to the
competition between the gravitational settling and convective mixing.
Fontaine & Wesemael ([1987]) proposed that when a white dwarf starts cooling
from the hot PG $1159$ type star, hydrogen is mixed within the outer helium
envelope and the white dwarf shows the DO spectral type. After that, hydrogen
gradually floats to the stellar surface in the strong gravitational field.
When the DO star cools to $T_{\rm eff}\sim 45,000~{}{\rm K}$, hydrogen has
accumulated enough at the surface, and the white dwarf is turned into a DA
star. They supposed further that as soon as the DA star cools down to $T_{\rm
eff}\sim 30,000~{}{\rm K}$, the He I/II ionization zone in the stellar
envelope becomes convective. The convective motion may penetrate into the top
hydrogen layer and mix the hydrogen atmosphere into the helium envelope, the
less abundant hydrogen being overwhelmed by the more abundant helium and
becoming undetectable. So the DA star then appears as a DB white dwarf.
Shibahashi ([2005], [2007]) proposed a different scenario for the DB gap.
During the early stage of a white dwarf’s evolution, the convection in the He
II/III ionization zone mixes the hydrogen layer and results in a helium-
dominated atmosphere, and the white dwarf appears as a DO star. When the star
cools down to around $T_{\rm eff}\sim 45,000~{}{\rm K}$, the He II/III
ionization zone becomes deep enough that convection disappears in the DO
star’s atmosphere, so hydrogen floating to the surface and then the white
dwarf being transformed into a DA star. When the white dwarf cools further
down to $T_{\rm eff}\sim 30,000~{}{\rm K}$, the He I/II ionization zone
generates a convection zone again, as Fontaine & Wesemael’s proposal, a
similar mixing between H and He occurs and the white dwarf appears as a DB
star.
The difference between the two scenarios of the DB gap is the time scale of
the gravitational settling. In Fontaine & Wesemael’s assumption, the settling
process happens slowly as the cooling of the white dwarf. But in Shibahashi’s
assumption, it happens quickly as soon as the convection is turned off.
However, the recent data of the Sloan Digital Sky Survey (SDSS) suggest that
several DB white dwarfs do appear in the DB gap (Eisenstein et al. [2006]).
These facts imply that the formation mechanism of the DB gap are not clear and
more works should be done.
In the present paper, we calculate a series of white dwarf evolutionary models
to investigate the spectral evolution of white dwarfs caused by the convective
mixing. The details of model calculations and input physics are presented in
Section 2. In Section 3 we discuss the results of our numerical models.
Conclusions are summarized in Section 4.
## 2 model details and input physics
We have used a modified version of the White Dwarf Evolution Code (WDEC),
which was originally described by Martin Schwarzschild to simulate the
evolution of the white dwarf. Some details of the WDEC has been described in
Lamb & Van Horn ([1975]) and Wood ([1990]). Here, we only present some
summaries of the input physics in our models.
The equation of state (EOS) used in the present calculations is composed of
two parts which apply to different regions. The first part of the EOS is used
for the degenerate, completely ionized interior of the white dwarf. In this
region, we use the EOS tables provided by Lamb ([1974]). For a given chemical
composition the needed values are obtained by two-dimension, four-point
Aitken-Lagrange interpolation in terms of variables $\lg P$ and $\lg T$. For a
specific C/O mixture, they are obtained by interpolation between the carbon
and oxygen tables using the additive volume technique of Fontaine et al.
([1977]). The second part of the EOS is used for the partial ionized envelope
where non-ideal effect is important. We use the Saumon et al. ([1995]) EOS for
hydrogen and helium mixtures. The new EOS include some new physical treatments
of partial ionization caused by pressure and temperature. Mixtures of hydrogen
and helium are also obtained by the additive volume technique.
The total opacity ($\kappa$) is given by
$\dfrac{1}{\kappa}=\dfrac{1}{\kappa_{\rm r}}+\dfrac{1}{\kappa_{\rm c}}~{},$
(1)
where $\kappa_{\rm r}$ is the radiative opacity and $\kappa_{\rm c}$ is the
conductive opacity. We use the OPAL radiative opacities in our calculations.
The new tables include some new physical factors, e.g. the L-S coupling effect
of iron atoms. The conductive opacities consist of two parts which are from
Itoh et al. ([1983], [1984]) and Hubbard & Lampe ([1969]). In the actual
calculations, we use Itoh et al. opacities only in $\lg\rho\geq 1.8$ region
and Hubbard & Lampe opacities in $\lg\rho\leq 1.5$ region, in the range of
$1.5<\lg\rho<1.8$ linear interpolation being performed.
The rates of neutrino energy loss used in our calculations are provided by
Itoh and his collaborators. The rates of neutrino energy loss due to pair,
photo, plasma and bremsstrahlung processes are from Itoh et al. ([1989]), and
the rate of recombination neutrino is from Kohyama et al. ([1993]).
The high surface gravity of the white dwarf leads to gravitational segregation
of the elements in the stellar envelope, and thus models of white dwarf must
include compositionally stratified envelopes. In our calculations, we adopt
approximations of the equilibrium diffusion profiles introduced by Wood
([1990]) (see Figure 1). Our calculations do not include the impact of
convective mixing on the H/He profile. Although this may not be correct in
details, we still expect it to be a reasonable approximation, because the
abundance of hydrogen in the mixing region is several orders of magnitude less
than the helium abundance.
Figure 1: Approximations to the diffusive equilibrium profiles.
We use the standard mixing-length theory of Böhm-Vitense ([1958]) to deal with
the convection. We set the mixing-length $l$ to be equal to one local pressure
scale height, i.e.,
$l=H_{\rm P}=-\cfrac{dr}{d\ln P}=\cfrac{P}{\rho g}~{}.$ (2)
Boundaries of the convection zones are determined by the Schwarzschild
criterion. And we set the integration step to be equal to $H_{\rm P}/8$ in our
calculations.
## 3 evolutionary results
We have computed a series of white dwarf evolutionary models with mass
$M=0.6~{}M_{\odot}$, which is the typical mass of white dwarf. In order for a
DA white dwarf to change its surface chemical composition as a result of the
convective mixing, it ought to have a very thin hydrogen layer, so the
hydrogen mass of the model is supposed to vary between $10^{-16}$ to
$10^{-14}~{}M_{\odot}$. The helium mass in the computed model envelope is
fixed to be $5.0\times 10^{-5}~{}M_{\odot}$. We assume that the heavy elements
have sunk during the early phase of the cooling process to the white dwarf’s
interior due to the so-called gravitational settling effect. As a result the
envelope of the white dwarf is only composed of hydrogen and helium, and the
metallicity in the envelope is assumed to be $Z=0$. All models are evolved
from $T_{\rm eff}\sim 90,000~{}{\rm K}$ down to $T_{\rm eff}=10,000~{}{\rm
K}$. In this section we present the results of our calculations.
### 3.1 Importance of the convective overshooting
We have examined the development of convection zone in our white dwarf models.
Figure 2 shows the convection zone of a DA model with $M_{\rm
H}=10^{-15}~{}M_{\odot}$ evolving with the decreasing effective temperature.
The horizontal axis is the effective temperature which denotes the
evolutionary sequence, and the vertical axis denotes the location within the
envelope of the model. We use $1-r/R$ as the vertical axis scale in Figure 2
(a) and use $\lg(1-M_{r}/M)$ in Figure 2 (b). The solid line corresponds to
the location of the model’s photosphere ($\tau=1$) and the dashed lines
correspond to the Schwarzschild boundaries of the convection zone.
(a) (b)
Figure 2: Location of the convection zone.
It can be found that convection occurs completely within the helium layer
which is located under the photosphere which can be regarded as the innermost
point visible to us in the white dwarf. Just above the He convection zone
there is a thin layer with a mean molecular weight gradient (the so-called
$\mu$-barrier). In the case of very thin hydrogen envelope ($M_{\rm
H}<10^{-15}~{}M_{\odot}$), Vauclair & Reisse ([1977]) have argued that the
$\mu$-barrier provides extra buoyancy to restrict the upper boundary of the He
convection zone just below it. However, at the upper boundary which is
determined by the convective stability criterion (at the $\mu$-barrier, the
Ledoux criterion $\nabla_{\rm rad}\geq\nabla_{\rm ad}+d\ln\mu/d\ln P$ is
adopted), the convective motion does not stop but moves upward further due to
inertia. In this sense, the fluid parcels with kinetic energy may penetrate
into the $\mu$-barrier until their velocity drop to zero. We perform a rough
calculation, supposing that the velocities of the fluid parcels is equal to
the mean flow velocity ($\bar{v}$) in the convection zone. When the fluid
parcels enter the $\mu$-barrier, a force $F$ (the resultant force of gravity
and buoyancy, neglecting the viscous force) acts on them and reduces the
velocity. The kinetic energy of the fluid parcels $E_{\rm
K}\sim\bar{v}^{2}/2$. When the fluid parcels deplete all the kinetic energy,
they can move beyond the convective boundary a length $L\sim E_{\rm
K}/F\sim\bar{v}^{2}/2F$. As shown in Fig. 3, the thickness of the
$\mu$-barrier $d_{\rm\mu}$ is around $3\times 10^{3}~{}{\rm cm}$ and
$L>d_{\rm\mu}$ when $T_{\rm eff}<40,000~{}{\rm K}$, and the maximum value of
$L$ is about $3\times 10^{4}~{}{\rm cm}$ which is one order of magnitude
greater than $d_{\rm\mu}$. Therefore, it seems reasonable to believe that the
convective overshooting can go cross the $\mu$-barrier. Once He penetrates
into the $\mu$-barrier, an effective mixing will flatten the composition
gradient and weaken the $\mu$-barrier. As a result, the upper boundary of the
He convection zone will thus be able to extend outward. We expect that the
convective motion can extend upward to the stellar surface, or at least, to
the photosphere in order to mix the upper hydrogen layer and to ensure that
helium can be observed. Thus, the role of convective overshooting appears to
be decisive. We suppose that the convective overshooting is able to reach the
photosphere. This suggestion let us set the distance from the photosphere down
to the top of the Schwarzschild boundary as the minimum length of the
overshooting region ($l_{\rm ovs}$). The geometrical length between the two
Schwarzschild boundaries is regarded as the length of the convection zone
($l_{\rm con}$). These two lengths vary with the evolution of the effective
temperature as shown in Figure 4. We also compare $l_{\rm ovs}$ with $l_{\rm
con}$, the ratio given in Figure 5.
Figure 3: $L$ and $d_{\rm\mu}$ versus $T_{\rm eff}$. Figure 4: The evolution
of the length of overshooting region and that of the convection zone. $l_{\rm
ovs}$ is the length of the overshooting region and $l_{\rm con}$ the length of
the convection zone. Figure 5: The ratio of $l_{\rm ovs}$ to $l_{\rm con}$
varies with $T_{\rm eff}$.
As shown in the Figure 4, $l_{\rm ovs}$ keeps almost a constant (about
$2~{}{\rm km}$), while $l_{\rm con}$ increases slowly to a few hundred meters
during a long evolutionary time scale. That is to say, the convective motion
must overshoot to a distance which is several times thicker than the
convection zone itself. Figure 6 is similar to Figure 4 but $l_{\rm ovs}$ and
$l_{\rm con}$ are expressed in unit of the local pressure scale height
($H_{\rm P}$). It is shown that $l_{\rm ovs}$ is no more than $4H_{\rm P}$.
Figure 6: Similar to Figure 4, but the two lengths are expressed in unit of
$H_{\rm P}$.
We discuss from another side of view the extension of convective overshooting
that concerns the masses in the convection and overshooting regions. We denote
$M_{\rm ovs}$ the mass within the overshooting region and $M_{\rm con}$ the
mass of the convection zone. Figure 7 shows $\lg M_{\rm ovs}$ and $\lg M_{\rm
con}$ vary with the effective temperature, respectively. Figure 8 shows the
variation of the ratio of $M_{\rm ovs}$ to $M_{\rm con}$. It can be seen that
$M_{\rm ovs}/M_{\rm con}$ decreases rapidly with $T_{\rm eff}$ and $M_{\rm
ovs}$ accounts only for a small fraction of $M_{\rm con}$ for the models with
relatively low $T_{\rm eff}$. The convection zone is thickening in the
evolutionary process and causes a rapid increase in $M_{\rm con}$. At $T_{\rm
eff}\sim 38,000~{}{\rm K}$, $M_{\rm con}$ is greater than $M_{\rm ovs}$. It is
interesting to note, that although $l_{\rm ovs}$ is considerably large than
$l_{\rm con}$, the matter in the overshooting region has a low density
comparing with the dense, turbulent convection zone. For example, when the
white dwarf model cools down to $T_{\rm eff}\sim 30,000~{}{\rm K}$, the ratio
of $l_{\rm ovs}$ to $l_{\rm con}$ is less than $3.8$ and $M_{\rm ovs}$ is only
about $1/3$ of $M_{\rm con}$. Therefore, we may reasonably believe that the
convective motion can extend to the photosphere by force of the convective
overshooting.
Figure 7: Masses of overshooting region and convection zone vary with $T_{\rm
eff}$. $M_{\rm ovs}$ is the mass of the overshooting region and $M_{\rm con}$
is the mass of the convection zone. Figure 8: The ratio of $M_{\rm ovs}$ to
$M_{\rm con}$ varies with $T_{\rm eff}$.
### 3.2 Determination of the transition temperature
If the convection zone in the helium envelope of a DA white dwarf can extend
upward to the photosphere due to the convective overshooting, the hydrogen
atmosphere will be mixed into the convective helium layer and the white dwarf
will have the opportunity to change its apparent chemical composition, in
other words, to transform into a DB star. In this section, we will discuss
when such transformation occurs.
We assume that the convective mixing region includes the overshooting region
and the convection zone, in which hydrogen and helium are homogeneously mixed.
The mass of the mixing zone is denoted as $M_{\rm mix}$, and the mass of
hydrogen in the mixing zone is denoted as $M_{\rm Hmix}$. The remainder is
helium whose mass is equal to $M_{\rm mix}-M_{\rm Hmix}$. When the mass of
helium exceeds the mass of hydrogen in the mixing zone, i.e.
$M_{\rm mix}-M_{\rm Hmix}>M_{\rm Hmix}~{},$ (3)
that is,
$M_{\rm mix}>2M_{\rm Hmix}~{},$ (4)
hydrogen will be overwhelmed by helium and we assume that the transformation
of the spectral type will occur. We use $M_{\rm mix}=2M_{\rm Hmix}$ as a
critical condition for a DA star evolving into a DB one. The effective
temperature at this critical point is called the transition temperature.
### 3.3 Discussions of the calculation results
We have computed a series model of a DA white dwarf with $M_{\rm H}=1.0\times
10^{-15}~{}M_{\odot}$. The convection zone varies with the effective
temperature as shown in Figure 9, in which we use $\lg(1-M_{r}/M)$ to indicate
the location of the convection zone. The convective motion appears just below
the H/He interface ($\lg(1-M_{r}/M)\approx-15$) and extends into the stellar
interior when the white dwarf model cools down. When the convective mixing
zone becomes thick enough, Eq. (4) is satisfied (indicated by the thick dashed
line). It can be found that the transition temperature ($T_{\rm tr}$) of this
model is about $31,000~{}{\rm K}$. If the convective overshooting can reach to
the photosphere at this temperature, we will observe that the white dwarf
evolves into a DB star. According to the discussions in Section 3.1, the
required length of the convective overshooting is about $3H_{\rm P}$.
Figure 9: A schematic representation of the spectral evolution of a white
dwarf model with $M_{\rm H}=1.0\times 10^{-15}~{}M_{\odot}$.
Other series of white dwarf evolution models we have computed show that the
thicker the hydrogen layer is, the lower the transition temperature will be.
Table 2 lists the value of $T_{\rm tr}$ of our models with different $M_{\rm
H}$. It can be noticed that the transition temperature of the model with
$M_{\rm H}=1.0\times 10^{-14}~{}M_{\odot}$ is below $20,000~{}{\rm K}$. It is
expected therefore that models with the hydrogen layers heavier than
$10^{-14}~{}M_{\odot}$ must have $T_{\rm tr}$ lower than $18,000~{}{\rm K}$.
So a DA white dwarf with $M_{\rm H}>10^{-15}~{}M_{\odot}$ may have the
opportunity to change its spectral type when it cools much below $T_{\rm
eff}\sim 30,000~{}{\rm K}$. Moreover, greater $M_{\rm H}$ also requires
stronger convective overshooting to bring helium to the stellar surface.
Table 2: Transition temperatures for different white dwarf models $M_{\rm H}~{}(M_{\odot})$ | $T_{\rm tr}~{}(~{}{\rm K})$ | $M_{\rm H}~{}(M_{\odot})$ | $T_{\rm tr}~{}(~{}{\rm K})$
---|---|---|---
$l=H_{\rm P}$ | $l=2H_{\rm P}$ | $l=H_{\rm P}$ | $l=2H_{\rm P}$
$1.0\times 10^{-15}$ | $31,084$ | $31,084$ | $1.8\times 10^{-15}$ | $25,191$ | $26,222$
$1.1\times 10^{-15}$ | $29,349$ | $29,769$ | $2.0\times 10^{-15}$ | $24,001$ | $26,209$
$1.2\times 10^{-15}$ | $27,920$ | $29,006$ | $3.0\times 10^{-15}$ | $22,112$ | $24,004$
$1.4\times 10^{-15}$ | $27,509$ | $28,456$ | $4.0\times 10^{-15}$ | $20,785$ | $22,799$
$1.5\times 10^{-15}$ | $27,257$ | $28,240$ | $5.0\times 10^{-15}$ | $19,966$ | $22,477$
$1.6\times 10^{-15}$ | $26,360$ | $27,466$ | $1.0\times 10^{-14}$ | $18,460$ | $19,802$
Furthermore, a more efficient convection can change the deepening of the inner
boundary of the He convection during the white dwarf’s evolution, and thus
change the transition temperature. In the MLT (mixing-length theory), the
mixing-length $l$ represents the efficiency of convective heat transfer. We
considered a series of models, in which the mixing-length is set to $2H_{\rm
P}$, which is $2$ times larger than before. Our calculations show that the
deepening of the convection zone do occur at a higher effective temperature.
As shown in Fig. 10, the convection zone of a model with $M_{\rm
H}=10^{-15}~{}M_{\odot}$ deepening at $T_{\rm eff}\sim 30,000~{}{\rm K}$
(compared with Figure 2). For this reason Eq. (4) will be satisfied earlier
and it will lead to a change of $T_{\rm tr}$ (see Table 2). However, the
variation of $T_{\rm tr}$ is relatively small, especially around
$30,000~{}{\rm K}$. We therefore believe that the efficiency of the convection
will not significantly affect the results.
(a) (b)
Figure 10: Similar to Figure 2, but $l=2H_{\rm P}$.
The above discussions imply that DB white dwarfs are likely born from DA white
dwarfs because of the convective mixing. Is it possible that a DO white dwarf
can evolve into a DA star? Our calculations show that if a DA white dwarf with
a sufficiently thin hydrogen layer of $M_{\rm H}\sim 10^{-16}~{}M_{\odot}$,
this transformation is possible. As shown in Figure 11, during the early time
of the evolution, the model’s $T_{\rm eff}$ is very high and hydrogen is
completely ionized in its atmosphere. The photosphere lies deeply in the
helium layer and thus helium is visible, resulting in the white dwarf
appearing as a DO star. When the white dwarf model cools down to $T_{\rm
eff}\sim 63,000~{}{\rm K}$, the photosphere rises to the hydrogen layer, but
at the same time convection appears in the He II/III ionization zone. Because
of the thin hydrogen layer, the convective overshooting can possibly reach to
the photosphere and Eq. (4) can easily be satisfied. The convective motion in
the helium layer will dilute the hydrogen layer, so the white dwarf is still a
DO star.
Figure 11: A schematic representation of the formation of a DO white dwarf
from a DA star with $M_{\rm H}\sim 10^{-16}~{}M_{\odot}$.
When the white dwarf cools down below $60,000~{}{\rm K}$, as shown in Figure
12, the location of the photosphere will quickly move up toward the stellar
surface. The extension of the convective motion can not go so far to reach the
photosphere. Therefore the convective mixing of helium is invisible and
hydrogen will re-accumulate in the atmosphere, making the DO white dwarf
transforming into a DA star. This result is similar to the Shibahashi’s
assumption (Shibahashi [2005], [2007]). According to the observational data,
there are no DO white dwarfs below $T_{\rm eff}\sim 45,000~{}{\rm K}$. This
fact allows us to adjust the overshooting length to let our model change its
spectral type at $T_{\rm eff}\sim 45,000~{}{\rm K}$. We can assume reasonably
that the overshooting length is proportional to the dimension of the
convection zone. In practice, we choose the overshooting length to be the
length of convection zone (being approximately equal to $H_{\rm P}$) plus
$0.375H_{\rm P}$. Our numerical result shows that at an effective temperature
($T_{\rm eff}\sim 26,000~{}{\rm K}$) the convection zone has developed thick
enough that the convective overshooting can reach again above the photosphere.
Then the hydrogen layer will be mixed with the helium layer and the white
dwarf will become a DB star.
Figure 12: A schematic representation of the spectral evolution of a whte
dwarf model with $M_{\rm H}\sim 10^{-16}~{}M_{\odot}$. The dash-dotted line
corresponds to the boundary of the overshooting region.
There is another possibility for DO white dwarfs transforming into DA stars in
the literature. It is probably that the progenitors of DO white dwarfs have
undergone a born-again phase and burnt most of the hydrogen envelope (Althaus
et al. [2005]). The mass loss due to stellar wind during the hot white dwarf
stage is likely to throw away the superficial hydrogen and prevents the
gravitational settling (Unglaub & Bues [2000]). When the white dwarfs cool
down to $T_{\rm eff}\sim 45,000~{}{\rm K}$, hydrogen previously left in the
internal layer is able to float to the stellar surface and DA white dwarfs are
formed.
## 4 Conclusions
From the above investigations we have found that the DB gap could be explained
as a consequence of the convective mixing in white dwarfs. DA white dwarfs
with $M_{\rm H}/M_{\odot}\sim 10^{-16}$ have opportunities to transform into
DO white dwarfs at $T_{\rm eff}\gtrsim 46,000~{}{\rm K}$ or DB white dwarfs at
$T_{\rm eff}\lesssim 26,000~{}{\rm K}$, respectively. DA white dwarfs with
$M_{\rm H}/M_{\odot}\sim 10^{-15}$ will can transform into DB stars below
$T_{\rm eff}\sim 31,000~{}{\rm K}$. White dwarfs with $M_{\rm H}$ greater than
$10^{-14}~{}M_{\odot}$ always appear as DA stars at $T_{\rm eff}\gtrsim
18,000~{}{\rm K}$. It is obvious that in the effective temperature range
between $46,000\lesssim T_{\rm eff}\lesssim 31,000~{}{\rm K}$ almost all of
white dwarfs have the DA spectral type, as shown in Figure 13. This scenario
substantially coincides with the observation. We can also estimate that the
hydrogen mass is $M_{\rm H}/M_{\odot}\sim 10^{-16}$ for the DO white dwarfs
and is $M_{\rm H}/M_{\odot}\sim 10^{-15}$ for the hot DB white dwarfs ($T_{\rm
eff}>20,000~{}{\rm K}$).
Figure 13: A schematic representation of the DB gap. $q_{H}=M_{H}/M_{\odot}$.
Based on our numerical results, the convective overshooting plays a crucial
role in the formation of the so-called DB gap, through the convective mixing
effect. It allows the convective motion penetrating into the hydrogen layer
and makes helium in the deep stellar interior being observable on the stellar
photosphere. The overshooting length is an important parameter of the model.
According to our results, the overshooting length should be proportional to
the thickness of the convection zone, which gives better agreement between the
model results and observations.
The hydrogen mass $M_{\rm H}$ is another important parameter, which is used as
a criterion for deciding when helium is dominant in the atmosphere of the
white dwarf. It determine decisively the critical effective temperature for
the white dwarf changing its spectral type.
###### Acknowledgements.
We thank Q. S. Zhang for many valuable discussions. This work is supported by
the National Key Fundamental Research Project through grant 2007CB815406.
## References
* [2005] Althaus, L. G., Serenelli, A. M., Panei, J. A., Córsico, A. H., García-Berro, E., Scóccola, C. G. 2005, A&A, 435, 631
* [1958] Böhm-Vitense, E. 1958, ZsAp, 46, 108
* [2006] Eisenstein, D. J., et al. 2006, AJ, 132, 676
* [1977] Fontaine, G., Graboske, H. C., Jr., Van Horn, H. M. 1977, ApJS, 35, 293
* [1987] Fontaine G., Wesemael F., 1987, In: Philip A. G. D., Hayes D. S., Liebert J., eds, IAU Colloq. 95, Second Conference on Faint Blue Stars. Davis Press, Schenectady, NY, p. 319
* [1969] Hubbard, W. B., Lampe, M., 1969, ApJS, 18, 297
* [1983] Itoh, N., Mitake, S., Iyetomi, H., Ichimaru, S. 1983, ApJ, 273, 774
* [1989] Itoh, N., Adachi, T., Nakagawa, M., Kohyama, Y., Munakata, H. 1989, ApJ, 339, 354
* [1993] Kohyama, Y., Itoh, N., Obama, A., Mutoh, H. 1993, ApJ, 415, 267
* [2008] Kurtz, D. W., Shibahashi, H., Dhillon, V. S., Marsh, T. R., Littlefair, S. P. 2008, MNRAS, 389, 1771
* [1974] Lamb, D. Q. 1974, PhD thesis, The University of Rochester
* [1975] Lamb, D. Q., Van Horn, H. M. 1975, ApJ, 200, 306
* [1999] McCook, G. P., Sion, E. M. 1999, ApJS, 121, 1
* [1984] Mitake, S., Ichimaru, S., Itoh, N. 1984, ApJ, 277, 375
* [1995] Saumon, D., Chabrier, G., Van Horn, H. M. 1995, ApJS, 99, 713
* [2005] Shibahashi H., 2005, In: Alecian G., Richard O., Vauclair S., eds, EAS Publ. Ser. Vol. 17, Element Stratification in Stars: 40 Years of Atomic Diffusion. EDP Sciences, Paris, p. 143
* [2007] Shibahashi H., 2007, AIPC, 948, 35
* [1983] Sion, E. M., Greenstein, J. L., Landstreet, J. D., Liebert, J., Shipman, H. L., Wegner, G. A. 1983, ApJ, 269, 253
* [2008] Tremblay, P. -E., Bergeron, P. 2008, ApJ, 672, 1144
* [2000] Unglaub, K., Bues, I. 2000, A&A, 359, 1042
* [1977] Vauclair, G., Reisse, C. 1977, A&A, 61, 415
* [1990] Wood, M. A. 1990, PhD thesis, The University of Texas at Austin
|
arxiv-papers
| 2009-11-16T07:09:03 |
2024-09-04T02:49:06.485028
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jie Su and Yan Li",
"submitter": "Jie Su",
"url": "https://arxiv.org/abs/0911.2949"
}
|
0911.2971
|
# Quantum mechanics and geodesic deviation in the brane world
S. M. M. Rasouli A. F. Bahrehbakhsh S. Jalalzadeh s-jalalzadeh@sbu.ac.ir M.
Farhoudi Department of Physics, Shahid Beheshti University, G. C., Evin,
Tehran 19839, Iran
###### Abstract
We investigate the induced geodesic deviation equations in the brane world
models, in which all the matter forces except gravity are confined on the
3-brane. Also, the Newtonian limit of induced geodesic deviation equation is
studied. We show that in the first Randall-Sundrum model the Bohr-Sommerfeld
quantization rule is as a result of consistency between the geodesic and
geodesic deviation equations. This indicates that the path of test particle is
made up of integral multiples of a fundamental Compton-type unit of length
$h/mc$.
###### pacs:
04.50.-h, 03.65.Ta, 04.20.Cv
## I
The mission to formulate a consistent quantum theory of gravity has maintained
physicists busy since the first attempt by Rosenfeld in 1930. In spite of much
work, no definitive progress has been made. Nowadays, there are many
interesting attempts to quantize gravity. In this paper we take an opposite
direction: we will show that quantum objects can be constructed from
gravitationalgeometrical effects. Actually, the idea of geometrization of
quantum mechanics has been considered in different approaches. For example,
one can increase the number of dimensions of spacetime in Kaluza-Klein (KK)
models of gravity 1 , Weylian spacetime 2 , scalar-tensor theories of gravity
3 or other possible extensions of Einstein general relativity. Recently, it
has been shown that the existence of non-compact extra dimensions leads to
quantum effects in the classically induced 4-dimensional (4D) physical
entities 4 . In 5 , to construct semi-classical quantum gravity from geometric
properties of brane, the authors have used the Induced Matter Theory (IMT)
which is an extension of the KK theory. In this approach, not only the gauge
fields are unified with gravity (geometry) but also the matter fields are
unified with gravity and have geometrical origin, constructed from extrinsic
curvature 6 . The origin of quantum effects in fact is the fluctuation of
matter fields around 4D spacetime.
In this paper we discuss the existence of quantum effects in the most famous
model of brane gravity. In this model and its extensions, the presence of non-
compact extra dimension is not in fact for the unification shame, but for the
explanation of hierarchy problem without using supersymmetry RStwo99 .
The idea that our familiar 4D spacetime is a hypersurface (brane) embedded in
a 5D bulk has been experiencing a phenomenal interest during the last decade.
The behavior of geodesics and the Newtonian limit of linearized gravity for
the Randall-Sundrum (RS) and an alternative brane background have been
investigated extensively MVV00 . Also, Ref. Y00 has looked into the geodesic
motions of a test particle in the bulk spacetime in RS scenario. The induced
4D geodesic equation on the brane, to which we assume that the matter fields
except gravity is confined, is given by JS05
$\displaystyle\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}=0\,,$
(1)
where $\tau$ is the proper time defined on the brane and
$\Gamma^{\mu}_{\alpha\beta}$ are 4D Christoffel symbols derived from the
induced metric. (Here and throughout we shall use $A,B=0,1,2,3,5$ to denote 5D
coordinates, $\mu,\nu=0,1,2,3$ to denote the standard 4D ones and
$\bar{A}=1,2,..,5,\ \bar{\mu}=1,2,3$ denotes spacelike counterparts).
Note that in Eq. (1) the effect of the existence of bulk space is hidden in
the induced metric which one can obtain via induced Einstein field equations
SMS00 . To obtain the induced geodesic (1), we usually start from the geodesic
equation of a test particle in the bulk space and then reduce it to the 4D
hypersurface. One can use the same procedure to acquire induced geodesic
deviation (GD) equation. For example in Kaluza-Klein theory authors of KMMH00
used the same method to obtain GD on this kind of compact models. Hence, we
start with the GD equation associated to the bulk space, namely
$\displaystyle\frac{{}^{(5)}D^{2}\xi^{A}}{DS^{2}}={\cal
R}^{A}_{\,\,\,BCD}\frac{dx^{B}}{dS}\frac{dx^{C}}{dS}\xi^{D},$ (2)
where ${\cal R}^{A}_{\,\,\,BCD}$ is the Reimann tensor for the bulk space,
$\xi^{A}$ is an infinitesimal GD vector, $D/DS$ denotes the pull–back of
covariant derivatives and $S$ is an affine parameter for the bulk space. To
induce Eq. (2) on the brane we need induced components of the Reimann tensor
of the bulk space on the brane, _i.e_. Gauss-Codazzi equations. In the
Gaussian normal frame, explicit calculation directly gives
$\displaystyle{\cal
R}^{\mu}_{\,\,\,\alpha\beta\gamma}=R^{\mu}_{\,\,\,\alpha\beta\gamma}+K_{\alpha\beta}K^{\mu}_{\,\,\,\gamma}-K_{\alpha\gamma}K^{\mu}_{\,\,\,\beta},$
(3)
and
$\displaystyle{\cal R}^{\mu}_{\,\,\,4\alpha
4}=K^{\mu}_{\,\,\,\alpha,4}-K^{\sigma}_{\,\,\,\alpha}K_{\sigma}^{\,\,\,\mu},$
(4)
where $R^{\mu}_{\,\,\,\alpha\beta\gamma}$ is 4D Reimann tensor and
$K_{\mu\nu}$ denotes the extrinsic curvature. Inserting Eqs. (3) and (4) into
the Eq. (2) gives
$\displaystyle\begin{array}[]{cc}\frac{D^{2}\xi^{\mu}}{DS^{2}}=\\\ \\\
\left(R^{\mu}_{\,\,\,\alpha\beta\gamma}+K_{\alpha\beta}K^{\mu}_{\,\,\,\gamma}-K_{\alpha\gamma}K^{\mu}_{\,\,\,\beta}\right)\frac{dx^{\alpha}}{dS}\frac{dx^{\beta}}{dS}\xi^{\gamma}+\\\
\\\
\epsilon\left(K^{\mu}_{\,\,\,\alpha,4}-K_{\alpha\sigma}K^{\sigma\mu}\right)\left[\frac{dx^{4}}{dS}\frac{dx^{\alpha}}{dS}\xi^{4}-(\frac{dx^{4}}{dS})^{2}\xi^{\alpha}\right].\end{array}$
(10)
Now the derivatives with respect to the 5D line element $dS$, should be
replaced by the derivatives with respect to the 4D Affine parameter. To attend
to this aim, we rewrite Eq. (10) with a general parameter $\lambda$, which
parameterizes 4D motion as
$\displaystyle\frac{{}^{(5)}D^{2}\xi^{\mu}}{DS^{2}}=\left(\frac{d\lambda}{dS}\right)^{2}\frac{{}^{(5)}D^{2}\xi^{\mu}}{D\lambda^{2}}+\frac{d\lambda}{dS}\frac{d}{d\lambda}(\frac{d\lambda}{dS})\frac{{}^{(5)}D\xi^{\mu}}{D\lambda}\,,$
(11)
where the relation between $5$ and $4$–dimensional covariant differentiations
is given by
$\displaystyle\begin{array}[]{cc}\frac{{}^{(5)}D\xi^{\mu}}{D\lambda}=\frac{d\xi^{\mu}}{d\lambda}+^{(5)}\Gamma^{\mu}_{AB}\frac{dx^{A}}{d\lambda}\xi^{B}=\frac{D\xi^{\mu}}{D\lambda}\\\
\\\ -\epsilon
K^{\mu}_{\,\,\,\alpha}\frac{dx^{\alpha}}{d\lambda}\xi^{4}-\epsilon
K^{\mu}_{\,\,\,\alpha}\frac{dx^{4}}{d\lambda}\xi^{\alpha},\end{array}$ (15)
so that in the second equality, 5D Christoffel symbols have been replaced by
the 4D counterparts using their relations obtained in Ref. JS05 . Now, from
Eqs. (10), (11) and (15) we obtain
$\displaystyle\begin{array}[]{cc}\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}=R^{\mu}_{\,\,\,\alpha\beta\gamma}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}\xi^{\gamma}+\\\
\\\
\left(K_{\alpha\beta}K^{\mu}_{\,\,\,\beta}-K_{\alpha\gamma}K^{\mu}_{\,\,\,\beta}\right)\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}\xi^{\gamma}+\\\
\\\
\epsilon\left(K^{\mu}_{\,\,\,\alpha,4}-K^{\rho}_{\,\,\,\alpha}K^{\mu}_{\,\,\,\rho}\right)\left[\frac{dx^{\alpha}}{d\lambda}\frac{dx^{4}}{d\lambda}\xi^{4}-(\frac{dx^{4}}{d\lambda})^{2}\xi^{\alpha}\right]-\\\
\\\ \left[\frac{D\xi^{\mu}}{D\lambda}-\epsilon
K^{\mu}_{\,\,\,\alpha}\left(\frac{dx^{\alpha}}{d\lambda}\xi^{4}+\frac{dx^{4}}{d\lambda}\xi^{\alpha}\right)\right]\left(\frac{d\lambda}{dS}\right)^{-1}\frac{d}{d\lambda}(\frac{d\lambda}{dS}).\end{array}$
(23)
The above induced GD equation can be used in various brane models. For
example, in the Induced Matter Theory (IMT) 4 ; WP92 , the test particles are
not in general, confined to the specific fixed brane J07 . In this case, since
the extra component of velocity of the test particle, $u^{4}=dx^{4}/d\lambda$,
does not vanish, all the extra terms in the right hand side of Eq. (23) will
be present. Another important point in the IMT is choice of $\lambda$, the
parameterization of the path. Usually, in the literature has been assumed that
the line element of the brane, which is defined here as the proper time
“$d\tau$”, is logical and convenient. However, the non-integrability property
of induced physical quantities on the brane dictates that the parameterization
of the path is, in general, deferent from the 4D proper time J07 . On the
other hand, in the brane phenomenological models where matter field has been
confined on the fixed brane, the 4D proper time defined on the brane is
required as a suitable parameterization of motion. In this paper, we would
like to study GD in the brane models based on the Horava and Witten theory
HW96 , hence, we will assume that all the matter fields, except gravity, are
confined on the fixed brane. Therefore, in Eq. (23), $d\lambda$ will be
substituted by $d\tau$, the proper time defined on the brane. Furthermore, we
assume that the velocity of test particles along the extra dimension vanishes.
Imposing the above assumptions on Eq. (23), we obtain
$\displaystyle\begin{array}[]{cc}\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=R^{\mu}_{\,\,\,\alpha\beta\gamma}u^{\alpha}u^{\beta}\xi^{\gamma}+\\\
\\\
\left(K_{\alpha\beta}K^{\mu}_{\,\,\,\gamma}-K_{\alpha\gamma}K^{\mu}_{\,\,\,\beta}\right)u^{\alpha}u^{\beta}\xi^{\gamma},\end{array}$
(27)
where $u^{\alpha}=\frac{dx^{\alpha}}{d\tau}$ denotes $4$-velocity of the test
particles defined on the brane.
In general relativity, the Newtonian limit of GD equation leads us to the form
of the field Equations MVV00 . Hence we derive and analyze the Newtonian limit
of Eq. (23). We elaborate tensor equation (2) in the local rest frame for one
of the two test particles $A_{1}$ and $A_{2}$ with coordinates $x^{A}(s,\eta)$
and $x^{A}(s,\eta+\delta\eta)$, respectively. In this frame ${\cal
G}_{AB}=\eta_{AB}$ and $dS=dt$. This means that $A_{1}$ promotes its clock to
the master clock indicating coordinates time. Also, ${{}^{(5)}D}/{DS}=d/dt$,
$x^{A}=(t,0,0,0,0)$ and $u^{A}=(1,0,..,0)$. We are left with
$\displaystyle\frac{d^{2}\xi^{\bar{A}}}{dt^{2}}={\cal
R}^{\bar{A}}_{\,\,\,00\bar{B}}\xi^{\bar{B}}\hskip
28.45274pt(\bar{A}=1,2,3,4).$ (28)
At this point $A_{1}$ recalls that according to the classical mechanics both
he and $A_{2}$ move in a stationary gravitational field:
${\bf{\ddot{r}}}_{A_{1}}={\bf{F}}({\bf{r}}_{A_{1}})$ and
${\bf{\ddot{r}}}_{A_{2}}={\bf{F}}({\bf{r}}_{A_{2}})$. Setting
$\xi^{B}={\bf{r}}^{B}_{A_{2}}-{\bf{r}}^{B}_{A_{1}}$ gives
$\displaystyle\begin{array}[]{cc}\frac{d^{2}\xi^{\bar{A}}}{dt^{2}}=F^{\bar{A}}({\bf{r}}_{A}+{\xi})-F^{\bar{A}}({\bf{r}}_{A})\simeq\\\
\\\
F^{\bar{A}}_{\,\,\,,\bar{B}}\xi^{\bar{B}}=-\Phi^{,{\bar{A}}}_{\,\,\,,{\bar{B}}}\xi^{\bar{B}},\end{array}$
(32)
where $\Phi$ is the gravitational potential in the bulk space. Comparing Eqs.
(28) and (32) gives
$\displaystyle{\cal
R}^{\bar{A}}_{\,\,\,00\bar{B}}=-\Phi^{,\bar{A}}_{\,\,\,,\bar{B}}.$ (33)
Now, using this equation and recalling Eqs. (3) and (4) we find
$\displaystyle\begin{array}[]{cc}R^{\bar{\mu}}_{\,\,\,00\bar{\mu}}+K_{00}K-K_{0\bar{\mu}}K^{\bar{\mu}}_{\,\,\,0}-K_{,5}-\\\
\\\
K^{\bar{\mu}}_{\,\,\,\bar{\nu}}K_{\bar{\mu}}^{\,\,\,\bar{\nu}}=-\Phi^{,\bar{A}}_{\,\,\,,\bar{A}}\hskip
14.22636pt(\bar{\mu}=1,2,3).\end{array}$ (37)
The classical field equations in the bulk space is
$\displaystyle\Phi_{,\bar{A}\bar{A}}=-\Lambda+(-\sigma+k^{2}_{5}\rho)\delta(x^{5})\,,$
(38)
where according to the spirit of brane models, we have assumed existence of
the bulk cosmological constant $\Lambda$, tension of brane $\sigma$ and the
matter density $\rho$. Consequently, we obtain
$\displaystyle\begin{array}[]{cc}R^{\bar{\mu}}_{\,\,\,00\bar{\mu}}+K_{00}K-K_{0\bar{\mu}}K^{\bar{\mu}}_{\,\,\,0}-K_{,5}-K^{\bar{\mu}}_{\,\,\,\bar{\nu}}K_{\bar{\mu}}^{\,\,\,\bar{\nu}}=\\\
\\\ -\Lambda+(-\sigma+k^{2}_{5}\rho)\delta(x^{4}).\end{array}$ (42)
Integration along normal direction gives the Newtonian limit of the Israel
junction condition as
$\displaystyle\left[K\right]=-k^{2}_{5}\rho+\sigma\,,$ (43)
where $[X]:=\lim_{x^{4}\rightarrow 0^{+}}X-\lim_{x^{4}\rightarrow 0^{-}}X$.
Also, if we impose the $Z_{2}$ symmetry then we obtain
$\displaystyle K^{+}=\frac{1}{2}(k^{2}_{5}\rho-\sigma),$ (44)
which is the Newtonian version of Israel junction condition obtained in HW96 .
Now, we obtain the GD equation in the RS brane world scenario. In the RS
scenario, it has been proposed a 5D bulk space, which is described by the
metric RStwo99 ; RSone99
$\displaystyle dS^{2}=e^{-2k|y|}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}\,,$ (45)
where $y=r\phi$ signifies the extra spacelike dimension with compactification
radius $r$, $k=\sqrt{-\Lambda/12M^{3}}$ and $\Lambda$ is the bulk cosmological
constant, and $M$ is fundamental 5D Plank scale. The factor
$e^{-2k\left|y\right|}$ is called warp factor and the geometry of the extra
dimension is orbifolded by $S^{1}/Z_{2}$. In the RSI scenario it can be shown
that even if Higgs or any other mass parameter in the 5D Lagrangian is of the
order of Planck scale, $m_{0}\simeq 10^{16}$ TeV, on the visible brane, it
gets warped by a factor of the form
$\displaystyle m=m_{0}e^{-kr\pi}.$ (46)
Thus by assuming $kr=11.84$, one gets $m\simeq 1$ TeV. Using RSI metric (45)
we obtain
$\displaystyle K_{\mu\nu}=k\frac{|y|}{y}e^{-2k|y|}\eta_{\mu\nu}.$ (47)
The constant slices at $y=0$ and $y=r\pi$ are known as the hidden and visible
branes respectively, which the observable universe being identified with
latter. Therefore, the GD equation (2) on the visible brane becomes
$\displaystyle\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\ddot{\xi}^{\mu}=k^{2}e^{-2\pi
kr}(\eta_{\alpha\beta}\eta^{\mu}_{\ \gamma}-\eta_{\alpha\gamma}\eta^{\mu}_{\
\beta})u^{\alpha}u^{\beta}\xi^{\gamma},$ (48)
where a dot denotes derivative with respect to the brane proper time. On the
other hand, solving the geodesic equation (1) on this brane model gives the
constant 4-velocity of test particle as $u^{\mu}=const\,,$ which shows that
the initially parallel geodesics will always remain parallel as a property of
$4D$ Minkowski spacetime. The solution of equation (48) for massive test
particles is
$\displaystyle\xi^{\mu}=f^{\mu}e^{ike^{-\pi kr}\tau},$ (49)
where $f^{\mu}$ is the integration constant. Equation (49) implies that the
distance between two geodesics oscillate contrary to the geodesic equation.
The consistency of this solution with geodesic equation then impose the
following restriction
$\displaystyle cke^{-\pi kr}\tau=n\pi,\ n=0,1,2,...,$ (50)
where $c$ is the speed of light which is not considered, here, to be unity.
Also it is well known that
$\displaystyle\int p_{\mu}dx^{\mu}=\int mu_{\mu}dx^{\mu}=\int
m\left(\frac{ds}{d\tau}\right)^{2}d\tau=mc^{2}\tau,$ (51)
where $p_{\mu}$ is the induced 4-momentum of the test particle and $m$ is the
rest mass. Comparing Eqs. (50) and (51) gives
$\displaystyle\int p_{\mu}dx^{\mu}=n\pi\frac{mce^{\pi kr}}{k}.$ (52)
Replacing $m$ from Eq. (46) into the above equation, and if we set $k\sim
1/l_{Pl}$, Eq. (52) reduces to
$\displaystyle\int p_{\mu}dx^{\mu}=nh.$ (53)
Which is similar to the old quantum theory quantization condition but is less
stringent, for the old quantum conditions were the integration being taken for
a closed curve. On the other hand, Eq. (50) leads to
$\displaystyle\tau=n\frac{h}{mc^{2}},$ (54)
indicating that the proper time of the test particle is made up of integral
multiples of a fundamental unit of length $h/mc^{2}$. This result suggests is
that the world-line of the test particle is to be considered as made up of
these units of length, nothing less being observable directly or indirectly in
experiment. Note that according to the flint it could be concluded from (54)
that the smallest interval of time and distance then are given by
$\displaystyle\begin{array}[]{cc}\delta
t=\frac{h}{mc^{2}}\frac{1}{\sqrt{1-\beta^{2}}},\\\ \\\ \delta
l=\frac{h}{mc}\frac{\beta}{\sqrt{1-\beta^{2}}},\end{array}$ (58)
where $\beta=\frac{v}{c}$ and the following uncertainty relations
$\displaystyle\Delta p_{\mu}\Delta x_{\mu}\sim\frac{2h}{n-1}.$ (59)
Note that in relations (58) both of them are depend upon the velocity of the
test particle. For velocities approaching the velocity of light they become
very large which means that it is impossible to measure intervals of time and
length in association with such rapidly moving particles. Hence it seems that
the deduction from the existence of a least proper time is that any accurate
measurements on a particle moving with such velocity would be impossible. Also
in equation (59) the worst case is for $n=1$, but this is no practical
significance for it corresponds to an observation of one fundamental unit of
length which is recorded as corresponding to zero proper time. In this
uncertainty relation for a large amount of $n$, the right hand side of (59)
vanish, i.e. this equation naturally contains classical limit. Since the
minimum length and time intervals that can be measured are given by (58) then
the maximum uncertainly on 3-momentum and energy becomes
$\displaystyle\begin{array}[]{cc}\delta
p\sim\frac{2mc}{n-1}\frac{\sqrt{1-\beta^{2}}}{\beta},\\\ \\\ \delta
E\sim\frac{2mc^{2}}{n-1}{\sqrt{1-\beta^{2}}}.\end{array}$ (63)
The conclusion is that the above uncertainties vanish when the velocity of
test particle reach the velocity of light, while the corresponding uncertainty
on time and length tends to infinity, but their product remains finite. we
have obtained the above uncertainty relations for massive test particles. Note
that the existence of minimum spatial and causal structures also will be
appearer in seeking for quantum gravity such as the loop quantum gravity loop
or string theory string . The modification of special relativity in which a
minimum length, which may be the Planck length, joins the speed of light as an
invariant is done in Ref. smolin . We now discuss about light quanta or
massless particles. In this case we have $u_{\mu}u^{\mu}=0$ and therefor
equation (48) becomes
$\displaystyle\frac{D^{2}\xi}{D\tau^{2}}=-k^{2}e^{-2\pi
kr}u_{\gamma}u^{\mu}\xi^{\gamma}.$ (64)
If we assume a solution like $\xi^{\mu}=f^{\mu}(\tau)$, then by inserting into
the above equation and by considering null conditionality for 4-velocity we
obtain $d^{2}f^{\mu}/d\tau^{2}=0$ and consequently
$\displaystyle\xi^{\mu}=A^{\mu}\tau+B^{\mu},$ (65)
where $A^{\mu}$ and $B^{\mu}$ are constants of integration. This result shows
that the extension of the massive test particle case to the photons is not
correct. The above solution shows classically propagating massless particles
in parallel or cross propagating photons. Note that the case of massless
particles can be drive in this approach and the Wesson suggestions 4 can not
lead us to this result. In fact difference behavior of photons are proceed
from confinement of gauge fields on the brane. Also As we know, the concepts
of time in general relativity and quantum theory differ intensely from each
other. Time in quantum theory is an external parameter, whereas in general
relativity time is dynamical one. Consequently, a consistent theory of quantum
gravity should exhibit a new concept of time. In general relativity spacetime
is dynamical and therefore there is no absolute time. Spacetime influences
material clocks in order to allow them to show proper time. The clocks, in
turn, react on the metric and change the geometry Zeh . In this sense, the
metric itself is a clock. A quantization of the metric can thus be interpreted
as a quantization of the concept of time. In this paper we showed that the
consistency of geodesic and geodesic deviation equations on the RS brane
dictates the quantization of proper time or clock rate. Note that this
quantity cannot be dealt with as operators in ordinary quantum theories. The
advantage of this model is that it makes General Relativity compatible with de
Broglie ideas, allows a geometric interpretation of de Broglie waves without
any generalization of Riemannian spacetime. In this direction the problem
needs more survey.
## References
* (1) Gonseth F. and Juvet, G., C. R. Acad. Sci., 185, 535, (1927) 535; 732(E); Fisher J. W., _Proc. R. Soc. London_ , _Ser_. _A_ , 123 (1929) 489.
Fisher, J.W., Proc. R. Soc. London, Ser. A, 123, 489, (1929).
* (2) Santamato E., _Phys. Rev. D_ 29 216, (1984); Wheeler J. T., _Phys. Rev. D_ , 41 431 (1990).
* (3) F. Shojai, A. Shojai, M. Golshani, _Mod.Phys.Lett. A_ , 13 (1998) 2915.
* (4) P. S. Wesson, Space–Time–Matter, (World Scientific, Singapore) (1999); _Gen. Rel. Grav_. 36, 451 (2004); Class. Quant. Grav. 19, 2825 (2002).
* (5) P. Moyassari and S. Jalalzadeh, _Gen. Rel. Grav_. 39, 1467 (2007).
* (6) S Jalalzadeh1, B Vakili1, F Ahmadi1 and H R Sepangi, _Class. Quantum Grav_. 23 (2006) 6015.
* (7) L. Randall and R. Sundrum, _Phys. Rev. Lett_. 83, 3370 (1999).
* (8) W. Muck, K. S. Viswanathan and I. Volovich, _Phys. Rev. D_ , 62, 105019 (2000).
* (9) Youm D., Phys. Rev. D, 62, 084002 (2000).
* (10) S. Jalalzadeh and H. R. Sepangi, _Class. Quant. Grav._ 22, 2035 (2005).
* (11) Shiromizu T., Maeda K. I. and Sasaki M., _Phys. Rev. D_ , 62, 024012 (2000).
* (12) Kerner R., Martin J., Mignemi S. and van Holten J. W., _Phys. Rev. D_ , 63, 027502 (2000).
* (13) Wesson P. S. and Ponce de Leon J., _J. Math. Phys_. 33, 3883 (1992).
* (14) Jalalzadeh S., _Gen. Rel. Grav_. 39, 387 (2007).
* (15) Horava P. and Witten E., _Nucl. Phys. B_ , 460, 506 (1996).
* (16) Randall L. and Sundrum R., _Phys. Rev. Lett_. 83, 4690 (1999).
* (17) Flint H. T. and Richardson O. W., _Proc. Roy. Soc._ __A, 117 , 637 (1927).
* (18) Rovelli C., _Living Rev. Rel_. 1 1 (1998).
* (19) Polchinski J., _String Theory, Vol. I: An Introduction to the Bosonic String_ (Cambridge University Press, Cambridge) 1998.
* (20) Magueijo J. and Smolin L., _Phys. Rev. Lett_. 88 (2002) 190403.
* (21) Zeh, H. D. _The physical basis of the direction of time_ , 4th edition. (Springer, Berlin) 2001.
|
arxiv-papers
| 2009-11-16T07:49:53 |
2024-09-04T02:49:06.491284
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. M. M. Rasouli, A. F. Bahrehbakhsh, S. Jalalzadeh and M. Farhoudi",
"submitter": "Shahram Jalalzadeh",
"url": "https://arxiv.org/abs/0911.2971"
}
|
0911.3038
|
# On a curious property of 3435.
Daan van Berkel
###### Abstract
Folklore tells us that there are no uninteresting natural numbers. But some
natural numbers are more interesting then others. In this article we will
explain why $3435$ is one of the more interesting natural numbers around.
We will show that $3435$ is a _Munchausen number_ in base 10, and we will
explain what we mean by that. We will further show that for every base there
are finitely many Munchausen numbers in that base.
Folklore tells us that there are no uninteresting natural numbers. The
argument hinges on the following observation: _Every subset of the natural
numbers is either empty, or has a smallest element_.
The argument usually goes something like this. If there would be any
uninteresting natural numbers, the set $\mathcal{U}$ of all these
uninteresting natural numbers would have a smallest element, say
$u\in\mathcal{U}$. But $u$ in it self has a very remarkable property. $u$ is
the smallest uninteresting natural number, which is very interesting indeed.
So $\mathcal{U}$, the set of all the uninteresting natural numbers, can not
have a smallest element, therefore $\mathcal{U}$ must be empty. In other
words, all natural numbers are interesting.
Having established this result, exhibiting an interesting property of a
specific natural number is often left as an excercise for the reader. Take for
example the integer $3435$. At first it does not seem that remarkable, until
one stumbles upon the following identity.
$3435=3^{3}+4^{4}+3^{3}+5^{5}$
This coincidence is even more remarkable when one discovers that there is only
one other natural number which shares this property with $3435$, namely
$1=1^{1}$.
In this article we will establish the claim made and generalize the result.
## Munchausen Number
Through out the article we will use the following notation. $b\in\mathbb{N}$
will denotate a base and therefore the inequility $b\geq 2$ will hold
throughout the article. For every natural number $n\in\mathbb{N}$, the _base
$b$ representation of $n$_ will be denoted by
$[c_{m-1},c_{m-2},\ldots,c_{0}]_{b}$, so $0\leq c_{i}<b$ for all
$i\in\\{0,1,\ldots,m-1\\}$ and $n=\sum_{i=0}^{m-1}c_{i}b^{i}$. Furtheremore,
we define a function
$\theta_{b}:\mathbb{N}\rightarrow\mathbb{N}:n\mapsto\sum_{i=0}^{m-1}c_{i}^{c_{i}}$,
where $n=[c_{m-1},c_{m-2},\ldots,c_{0}]_{b}$. We will further adopt the
convention that $0^{0}=1$, in accordance with $1^{0}=1$, $2^{0}=1$ etcetera.
* An integer $n\in\mathbb{N}$ is called a _Munchausen number in base $b$_ if and only if $n=\theta_{b}(n)$. $\circ$
So by the equality in the introduction we know that $3435$ is a Munchausen
number in base $10$.
* A related concept to Munchausen number is that of Narcissistic number. (See for example [1], [2] and [3].)
The reason for picking the name Munchausen number stems from the visual of
raising oneself, a feat demonstrated by the famous Baron von Munchausen ([4]).
Andrew Baxter remarked that the Baron is a narcissistic man indeed, so I think
the name is aptly chosen. $\triangleleft$
The following two lemmas will be used to proof the main result of this
article: for every base $b\in\mathbb{N}$ there are only finitely many
Munchausen numbers in base $b$.
* For all $n\in\mathbb{N}$: $\theta_{b}(n)\leq(\log_{b}(n)+1)(b-1)^{b-1}$. $\diamond$
* Notice that the function $x\mapsto x^{x}$ is strictly increasing if $x\geq\frac{1}{e}$. This can be seen from the derivative of $x^{x}$ which is $x^{x}(\log(x)+1)$. This last expression is clearly positive for $x>\frac{1}{e}$. Together with the definition of $0^{0}=1$, we see that $x^{x}$ is increasing for all the nonnegative integers.
For all $n\in\mathbb{N}$ with $n=[c_{m-1},c_{m-2},\ldots,c_{0}]_{b}$ we have
the ineqalities $0\leq c_{i}\leq b-1$ for all $i$ within $0\leq i<m$.
So
$\theta_{b}(n)=\sum_{i=0}^{m-1}c_{i}^{c_{i}}\leq\sum_{i=0}^{m-1}(b-1)^{b-1}=m\times(b-1)^{b-1}$.
Now, the number of digits in the base $b$ represantation of $n$ equals
$\left\lfloor\log_{b}(n)+1\right\rfloor$. In other words
$m:=\left\lfloor\log_{b}(n)+1\right\rfloor\leq\log_{b}(n)+1$.
So $\theta_{b}(n)\leq(\log_{b}(n)+1)(b-1)^{b-1}$ $\square$
* If $n\in\mathbb{N}$ and $n>2b^{b}$ then $\frac{n}{\log_{b}(n)+1}>(b-1)^{b-1}$. $\diamond$
* Let $n\in\mathbb{N}$ such that $n>2b^{b}$. Notice that $x\mapsto\frac{x}{\log_{b}(x)}$ is strictly increasing if $x>e$. To see this notice that the derivative of $\frac{x}{\log_{b}{x}}$ is $\log(b)\frac{\log(x)-1}{\log^{2}(x)}$ which is positive for $x>e$. Furthermore $\log_{b}(2)+1\leq 2\leq b=b\log_{b}(b)$.
Now, because $n>2b^{b}>e$, from the following chain of ineqalities:
$\frac{n}{\log_{b}(n)+1}>\frac{2b^{b}}{b\log_{b}(b)+\log_{b}(2)+1}\geq\frac{2b^{b}}{2b\log_{b}(b)}=b^{b-1}>(b-1)^{b-1}$
we can deduce that $\frac{n}{\log_{b}(n)+1}>(b-1)^{b-1}$ $\square$
With both lemma’s in place we can present without further ado the main result
of this article.
* For every base $b\in\mathbb{N}$ with $b\geq 2$: there are only finitely many Munchausen numbers in base $b$. $\diamond$
* By the preceding lemma’s we have, for all $n\in\mathbb{N}$ with $n>2b^{b}$: $n>(\log_{b}(n)+1)(b-1)^{b-1}\geq\theta_{b}(n)$.
So, in order for $n$ to equal $\theta_{b}(n)$, $n$ must be less then or equal
to $2b^{b}$. This proves that there are only finitely many Munchausen numbers
in base $b$. $\square$
## Exhaustive Search
The proposition in the preceding section tells use that for every base
$b\in\mathbb{N}$, Munchausen numbers in that base only occur within the
interval $[1,2b^{b}]$. This makes it possible to exhaustively search for
Munchausen numbers in each base.
Figure 1 lists all the Munchausen numbers in the bases 2 through 10. So for
example in base $4$, $29$ and $55$ are the only non-trivial Munchausen
numbers. Furthermore, the base $4$ representation of $29$ and $55$ have a
striking resemblance. For $29=[1,3,1]_{4}=1^{1}+3^{3}+1^{1}$ and
$55=[3,1,3]_{4}=3^{3}+1^{1}+3^{3}$.
Figure 1: Munchausen numbers in base 2 through 10. Base | Munchausen Numbers | Representation
---|---|---
2 | 1, 2 | $[1]_{2}$, $[1,0]_{2}$
3 | 1, 5, 8 | $[1]_{3}$, $[1,2]_{3}$, $[2,2]_{3}$
4 | 1, 29, 55 | $[1]_{4}$, $[1,3,1]_{4}$, $[3,1,3]_{4}$
5 | 1 | $[1]_{5}$
6 | 1, 3164, 3416 | $[1]_{6}$, $[2,2,3,5,2]_{6}$, $[2,3,4,5,2]_{6}$
7 | 1, 3665 | $[1]_{7}$, $[1,3,4,5,4]_{7}$
8 | 1 | $[1]_{8}$
9 | 1, 28, 96446, 923362 | $[1]_{9}$, $[3,1]_{9}$, $[1,5,6,2,6,2]_{9}$, $[1,6,5,6,5,4,7]_{9}$
10 | 1, 3435 | $[1]_{10}$, $[3,4,3,5]_{10}$
The sequence of Munchausen numbers is listed as sequence A166623 at the OEIS.
(See [5]. For the related sequence of Narcissistic numbers see [6])
The code in listing 1 is used to produce the numbers in figure 1. There are
two utility functions. These are munchausen and next. munchausen calculates
$\theta_{b}(n)$ given a base $b$ representation of $n$. next returns the base
$b$ representation of $n+1$ given a base $b$ representation of $n$.
I would like to conclude this article with a question my wife asked me while I
was writing this: “But what about $20082009$?”
Listing 1: GAP code finding Munchausen numbers
⬇
next := function(coefficients, b)
local i;
coefficients[1] := coefficients[1] + 1;
i := 1;
while coefficients[i] = b do
coefficients[i] := 0;
i := i + 1;
if (i <= Length(coefficients)) then
coefficients[i] := coefficients[i] + 1;
else
Add(coefficients, 1);
fi;
od;
return coefficients;
end;
munchausen := function(coefficients)
local sum, coefficient;
sum := 0;
for coefficient in coefficients do
sum := sum + coefficient^coefficient;
od;
return sum;
end;
for b in [2..10] do
max := 2*b^b;
n := 1; coefficients := [1];
while n <= max do
sum := munchausen(coefficients);
if (n = sum) then
Print(n, ”\n”);
fi;
n := n + 1;
coefficients := next(coefficients, b);
od;
od;
## References
* [1] Clifford A. Pickover. Wonders of Numbers. Oxford University Press, 2001.
* [2] Wikipedia. Narcissistic Number. http://en.wikipedia.org/wiki/Narcissistic_number.
* [3] Wolfram Math World. Narcissistic Number. http://mathworld.wolfram.com/NarcissisticNumber.html.
* [4] Wikipedia. Baron Munchhausen. http://en.wikipedia.org/wiki/Baron_Munchhausen.
* [5] The On-Line Encyclopedia of Integer Sequences. A166623. http://www.research.att.com/~njas/sequences/A166623.
* [6] The On-Line Encyclopedia of Integer Sequences. A005188. http://www.research.att.com/~njas/sequences/A005188.
|
arxiv-papers
| 2009-11-16T14:24:57 |
2024-09-04T02:49:06.497219
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daan van Berkel",
"submitter": "Daan van Berkel",
"url": "https://arxiv.org/abs/0911.3038"
}
|
0911.3239
|
# Hawking radiation and black hole spectroscopy in Hořava-Lifshitz gravity
Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail: bibhas@bose.res.in
###### Abstract
Hawking radiation from the black hole in Hořava-Lifshitz gravity is discussed
by a reformulation of the tunneling method given in [17]. Using a density
matrix technique the radiation spectrum is derived which is identical to that
of a perfect black body. The temperature obtained here is proportional to the
surface gravity of the black hole as occurs in usual Einstein gravity. The
entropy is also derived by using the first law of black hole thermodynamics.
Finally, the spectrum of entropy/area is obtained. The latter result is also
discussed from the viewpoint of quasi-normal modes. Both methods lead to an
equispaced entropy spectrum, although the value of the spacing is not the
same. On the other hand, since the entropy is not proportional to the horizon
area of the black hole, the area spectrum is not equidistant, a finding which
also holds for the Einstein-Gauss-Bonnet theory.
## 1 Introduction
Inspired by condensed matter models of dynamical critical system, Hořava
proposed a new four dimensional theory of gravity [1], popularly known as
Hořava-Lifshitz gravity. Since then a lot of attention has been given in
several directions [2, 3] of this Hořava-Lifshitz gravity theory.
Recently, it has also been shown that a static spherically symmetric black
hole solution exists in this theory for the Lifshitz point $z=3$ [4, 5] and
for $z=4$ [6] 111For other black hole solutions of Hořava-Lifshitz gravity
with different conditions see [7, 8]. A study of the thermodynamic properties
of this black hole has also been done [5, 9, 10, 6, 11]. Surprisingly,
however, there does not exist any detailed study of the Hawking effect [12],
except for some sporadic attempts [13, 14]. The motivation of this paper is to
fill such a gap. We feel this study to be important since the Hawking
radiation is crucial to give the black holes one of its thermodynamic
properties making it consistent with the rest of physics.
The Hawking effect can be discussed by Hawking’s original approach [12] or
anomaly method [15, 16]. Here we will discuss the Hawking effect by a
physically intuitive picture - a reformulation [17, 18] of the standard
tunneling formalism 222For more references on tunneling mechanism see [22]
[19, 20, 21]. The advantage of this approach [17, 18] is that, in contrast to
[19, 20, 27, 28, 29], the spectrum is directly obtained instead of just the
temperature.
In this paper, we will study the propagation of scalar fields on the
background of Hořava-Lifshitz black hole spacetime for $z=3$. Following the
reformulation of the tunneling mechanism [17, 18, 23] the explicit expressions
of the left and right moving modes in the semi-classical limit (i.e.
$\hbar\rightarrow 0$) as seen by an asymptotic observer will be given.
Exploiting a density matrix technique the radiation spectrum will be derived.
We find that the distribution function exactly matches with the black body
radiation with a temperature proportional to the surface gravity of the black
hole. This is a new result in the context of black holes in Hořava-Lifshitz
gravity.
Also, we will calculate the thermodynamic entities of the black hole. Using
the first law of thermodynamics the expression for the entropy will be
derived. In this case the entropy is not proportional to the area of the event
horizon as happens in Einstein gravity; rather it has an extra additive
logarithmic term involving the area.
Another purpose of this paper is to study the nature of entropy/area spectrum
of the Hořava-Lifshitz black hole. Here we will use two methods: tunneling
method [23] and quasi-normal mode (QNM) method [24, 25, 26]. In both
approaches the entropy (and not area) spectrum is seen to be equispaced.We
also observe that, although there is a discrepancy in the value of the entropy
spacing in the tunneling and quasi normal mode approaches, the order of
magnitude is same. The probable reason for such discrepancy is also discussed
here.
The organization of the paper is as follows. In section-2 a short discussion
on the black hole solution of Hořava-Lifshitz gravity and the expressions of
the modes of the scalar particle as seen by an asymptotic observer are
presented. The radiation spectrum is calculated in the next section. Explicit
forms of the temperature and the entropy of the black hole are derived in
section-4. Section-5 is devoted for the discussions on the entropy/area
spectrum. Finally, we give the concluding remarks.
## 2 Tunneling in black hole in Hořava-Lifshitz gravity
The spherically symmetric static black hole solution at the Lifshitz point
$z=3$ in Hořava-Lifshitz theory was obtained as [5],
$\displaystyle
ds^{2}_{H}=-N^{2}(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{a}$ (1)
where $d\Omega^{2}_{a}$ is the line element for a two dimensional Einstein
space with constant scalar curvature $2a$. Without loss of generality, one can
take $a=0,\pm 1$ respectively. The metric coefficients are given by [5],
$\displaystyle f(r)=a+x^{2}-\alpha
x^{(2\lambda\pm\sqrt{6\lambda-2})/(\lambda-1)},$ (2) $\displaystyle
N(r)=x^{-(1+3\lambda\pm 2\sqrt{6\lambda-2})/(\lambda-1)}\sqrt{f(r)}$ (3)
where $\alpha$ is the integration constant and $\lambda$ is the coupling
constant, susceptible to quantum corrections [1]. Here $x=r\sqrt{-\Lambda}$
with $\Lambda$ is the cosmological constant. Now, for the above metric
coefficients to be real, $\lambda$ must be greater than $\frac{1}{3}$. Hence
in this case, as explained in [4], the cosmological constant $\Lambda$ is
negative.
Henceforth we will consider the $\lambda=1$ case, which is of particular
interest. The metric coefficients can then be obtained from (2) and (3) after
taking the limit $\lambda\rightarrow 1$. Here it must be noted that for the
positive sign of (2,3), this limit does not exist, while for the other sign,
it exists. Therefore, for the negative sign, it leads to the following
solutions:
$\displaystyle N^{2}(r)=f(r)=a+x^{2}-\alpha\sqrt{x};\,\,\,\
x=r\sqrt{-\Lambda}~{}.$ (4)
This solution is asymptotically $AdS_{4}$ and has a singularity at $x=0$ if
$\alpha\neq 0$. This singularity could be covered by the black hole horizon at
$x_{+}$, the largest root of the equation $f(x_{+})=0$. For $a=1$, equation
(4) reduces to that obtained in [4]. The ADM mass of this black hole is given
by [5],
$\displaystyle
M=\frac{p^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{a}}{16}\alpha^{2}~{}.$ (5)
Since $\Lambda$ is negative [4] and $\alpha$ is a real constant, the black
hole mass $M$ is always positive and real.
Now to find the modes of the particle as seen by the observer situated outside
the event horizon, consider the massless Klein-Gordon (KG) equation
$g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0$ under the background (1).
Proceeding in a similar way as presented in [17, 18] we obtain that the modes
which are travelling in the “in” and “out” sectors of the black hole horizon
are connected through the expressions
$\displaystyle\phi^{(R)}_{in}=e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}$
(6) $\displaystyle\phi^{(L)}_{in}=\phi^{(L)}_{out},$ (7)
where $\omega$ is the effective energy of the emitted particle as measured at
infinity and $\kappa$ is the surface gravity defined by
$\displaystyle\kappa=\frac{1}{2}\frac{df(r)}{dr}\Big{|}_{r=r_{+}}~{}.$ (8)
Here $L$ ($R$) refers to the ingoing (outgoing) mode. $r_{+}$ is the event
horizon of the black hole (1). In the set (6,7), the left hand side modes of
the equality represent the pair produced inside the black hole while those on
the right hand side of equality represent the modes of that pair if one
observes from outside the black hole. Since the physical quantities for the
black hole are measured from the outside observer, we will always use this set
of transformations in the subsequent analysis.
Now, since the left moving mode travels towards the center of the black hole,
its probability to go inside, as measured by an external observer, is
$\displaystyle P^{(L)}=|\phi^{(L)}_{in}|^{2}=|\phi^{(L)}_{out}|^{2}=1$ (9)
where we have used (7) to recast $\phi^{(L)}_{in}$ in terms of
$\phi^{(L)}_{out}$ since measurements are done by an outside observer. This
shows that the left moving (ingoing) mode is trapped inside the black hole, as
expected. On the other hand the right moving mode, i.e. $\phi^{(R)}_{in}$,
tunnels through the event horizon. So proceeding in the similar way we obtain
the tunneling probability as seen by an external observer as
$P^{(R)}=|\phi^{(R)}_{in}|^{2}=|e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}|^{2}=e^{-\frac{2\pi\omega}{\hbar\kappa}}$,
i.e. there is a finite probability for the outgoing mode to cross the horizon.
## 3 Radiation spectrum
In this section we will derive the emission spectrum from the black hole in
the tunneling approach with the help of density matrix technique developed in
[18]. Now to find this spectrum, we first consider $n$ number of non-
interacting virtual pairs that are created inside the black hole. Each of
these pairs is represented by the modes defined in the left side of the
equality of (6, 7). Then the physical state of the system, observed from
outside, is given by,
$\displaystyle|\Psi>=N\sum_{n}|n^{(L)}_{\textrm{in}}>\otimes|n^{(R)}_{\textrm{in}}>=N\sum_{n}e^{-\frac{\pi
n\omega}{\hbar{\kappa}}}|n^{(L)}_{\textrm{out}}>\otimes|n^{(R)}_{\textrm{out}}>$
(10)
where use has been made of the transformations (6) and (7). Here
$|n^{(L)}_{\textrm{out}}>$ corresponds to $n$ number of left going modes and
so on while $N$ is a normalization constant which can be determined by using
the normalization condition $<\Psi|\Psi>=1$. This immediately yields,
$N=\Big{(}\displaystyle\sum_{n}e^{-\frac{2\pi
n\omega}{\hbar{\kappa}}}\Big{)}^{-\frac{1}{2}}$. The sum will be calculated
for both bosons and fermions. For bosons $n=0,1,2,3,....$ whereas for fermions
$n=0,1$. With these values of $n$ we obtain the normalization constant as
[18],
$\displaystyle
N_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar{\kappa}}}\Big{)}^{\frac{1}{2}};\,\,\
N_{(\textrm{fermion})}=\Big{(}1+e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}^{-\frac{1}{2}}$
(11)
From here on our analysis will be only for bosons since for fermions the
analysis is identical. For bosons the density matrix operator of the system is
given by,
$\displaystyle{\hat{\rho}}_{(\textrm{boson})}$ $\displaystyle=$
$\displaystyle|\Psi>_{(\textrm{boson})}<\Psi|_{(\textrm{boson})}$ (12)
$\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}\sum_{n,m}e^{-\frac{\pi
n\omega}{\hbar\kappa}}e^{-\frac{\pi
m\omega}{\hbar\kappa}}|n^{(L)}_{\textrm{out}}>\otimes|n^{(R)}_{\textrm{out}}><m^{(R)}_{\textrm{out}}|\otimes<m^{(L)}_{\textrm{out}}|$
on exploiting (10) with the normalization (11). Now tracing out the ingoing
(left) modes we obtain the density matrix for the outgoing modes,
$\displaystyle{\hat{\rho}}^{(R)}_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}\sum_{n}e^{-\frac{2\pi
n\omega}{\hbar\kappa}}|n^{(R)}_{\textrm{out}}><n^{(R)}_{\textrm{out}}|$ (13)
Therefore the average number of particles detected at asymptotic infinity is
given by,
$\displaystyle<n>_{(\textrm{boson})}={\textrm{trace}}({\hat{n}}{\hat{\rho}}^{(R)}_{(\textrm{boson})})$
$\displaystyle=$
$\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar\kappa}}\Big{)}\sum_{n}ne^{-\frac{2\pi
n\omega}{\hbar\kappa}}$ (14) $\displaystyle=$
$\displaystyle\frac{1}{e^{\frac{2\pi\omega}{\hbar\kappa}}-1}$
where the trace is taken over all $|n^{(R)}_{\textrm{out}}>$ eigenstates. This
is the Bose distribution. Similar analysis for fermions leads to the Fermi
distribution:
$\displaystyle<n>_{(\textrm{fermion})}=\frac{1}{e^{\frac{2\pi\omega}{\hbar\kappa}}+1}$
(15)
Note that both these distributions correspond to a black body spectrum with a
temperature given by the Hawking expression,
$\displaystyle T_{H}=\frac{\hbar\kappa}{2\pi}$ (16)
Correspondingly, the Hawking flux can be obtained by integrating the above
distribution functions over all $\omega$’s. For fermions it is given by,
$\displaystyle{\textrm{Flux}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\omega~{}d\omega}{e^{\frac{2\pi\omega}{\hbar
K}}+1}=\frac{\hbar^{2}\kappa^{2}}{48\pi}$ (17)
Similarly, the Hawking flux for bosons can be calculated, leading to the same
answer.
## 4 Black hole thermodynamics
In the previous section, the emission spectrum of the particle from the black
hole has been derived. This is a perfectly black body spectrum with the
temperature given by (16). Now to find out the explicit form of the
temperature, we will derive the surface gravity for this black. Use of the
definition (8) and explicit form of metric coefficient (4) yield the value of
the surface gravity as,
$\displaystyle\kappa=\frac{1}{2}\Big{(}2x_{+}\sqrt{-\Lambda}-\frac{\alpha}{2}\sqrt{\frac{-\Lambda}{x_{+}}}\Big{)}=\frac{3x_{+}^{2}-a}{4x_{+}}\sqrt{-\Lambda}$
(18)
where, in the last step, we have substituted the value of $\alpha$ from the
equation $f(x_{+})=0$. Therefore substituting this in (16) the Hawking
temperature of the black hole is given by
$\displaystyle T_{H}=\frac{\hbar(3x_{+}^{2}-a)}{8\pi x_{+}}\sqrt{-\Lambda}$
(19)
which has been obtained earlier in [5, 9, 10] by different methods. It is
noted that there exits an extremal limit (i.e. temperature vanishes) at
$x_{+}=\sqrt{\frac{a}{3}}$, in which
$\alpha=4\Big{(}\frac{a}{3}\Big{)}^{3/4}$.
The next step is to find the entropy of the black hole. As shown in [10], the
first law of thermodynamics $dM=T_{H}dS_{BH}$ holds in this case. We shall use
this to find the entropy. Using (5) and (19) in the first law of
thermodynamics and integrating, we obtain,
$\displaystyle S_{BH}$ $\displaystyle=$ $\displaystyle\frac{\pi
p^{2}\mu^{2}\Omega_{a}}{4\hbar}(x_{+}^{2}+2a\ln x_{+})+S_{0}$ (20)
$\displaystyle=$
$\displaystyle\frac{c^{3}}{4G}\Big{(}A-\frac{a\Omega_{a}}{\Lambda}\ln\frac{A}{A_{0}}\Big{)}~{}.$
Here in the last step, the horizon area $A=\Omega_{a}r_{+}^{2}$ has been used.
$A_{0}$ is the integration constant of dimension of length square. Note that,
the entropy is not just proportional to area, as usually happens in Einstein
gravity, rather it has an additive term proportional to logarithmic of area.
## 5 Entropy and area spectrum
In this section we will derive the spectrum for the entropy as well as the
area of the black hole following two methods: tunneling method [23] and QNM
method [24, 25, 26]. Then a comparison of the results obtained in both methods
will be done.
### 5.1 Tunneling method
It has already been mentioned that the pair production occurs inside the
horizon. The relevant modes are given by left side of the equality of (6, 7).
It has also been shown in the previous section that the left mode is trapped
inside the black hole while the right mode can tunnel through the horizon
which is observed at asymptotic infinity. Therefore, the average value of
$\omega$ will be computed as
$\displaystyle<\omega>=\frac{\displaystyle{\int_{0}^{\infty}\left(\phi^{(R)}_{in}\right)^{*}\omega\phi^{(R)}_{in}d\omega}}{\displaystyle{\int_{0}^{\infty}\left(\phi^{(R)}_{in}\right)^{*}\phi^{(R)}_{in}d\omega}}~{}.$
(21)
It should be stressed that the above definition is unique since the pair
production occurs inside the black hole and it is the right moving mode that
eventually escapes (tunnels) through the horizon.
Since the observer is located outside the event horizon, it is essential to
recast the “in” expressions into their corresponding “out” expressions using
the map (6) and then perform the integrations. This yields,
$\displaystyle<\omega>$ $\displaystyle=$
$\displaystyle\frac{\displaystyle{\int_{0}^{\infty}e^{-\frac{\pi\omega}{\hbar\kappa}}\left(\phi^{(R)}_{out}\right)^{*}\omega
e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}d\omega}}{\displaystyle{\int_{0}^{\infty}e^{-\frac{\pi\omega}{\hbar\kappa}}\left(\phi^{(R)}_{out}\right)^{*}e^{-\frac{\pi\omega}{\hbar\kappa}}\phi^{(R)}_{out}d\omega}}=\frac{\displaystyle{\int_{0}^{\infty}\omega
e^{-\beta\omega}d\omega}}{\displaystyle{\int_{0}^{\infty}e^{-\beta\omega}d\omega}}=\beta^{-1}$
(22)
where $\beta$ is the inverse Hawking temperature
$\displaystyle\beta=\frac{2\pi}{\hbar\kappa}=\frac{1}{T_{H}}~{}.$ (23)
In a similar way one can compute the average squared energy of the particle
detected by the asymptotic observer, $<\omega^{2}>=\frac{2}{\beta^{2}}$. Hence
the uncertainty in the detected energy $\omega$ is given by,
$\displaystyle\left(\Delta\omega\right)=\sqrt{<\\!\\!\omega^{2}\\!\\!>-<\\!\\!\omega\\!\\!>^{2}}\,=\,\beta^{-1}=T_{H}$
(24)
which is nothing but the Hawking temperature $T_{H}$. This uncertainty can be
seen as the lack of information in energy of the black hole due to the
particle emission. Now since, as stated earlier, ‘$\omega$’ is the effective
energy of the emitted particle as measured by the outside observer and in the
context of information theory, entropy is the lack of information, then
substituting equation (24) in the first law of black hole mechanics
$\displaystyle T_{H}(\Delta S_{BH})=\Delta\omega$ (25)
one obtains
$\displaystyle\Delta S_{BH}=1~{}.$ (26)
This shows that the entropy of the black hole is quantized in units of the
identity. Also, the entropy spectrum is equispaced and is given by
$\displaystyle S_{n}=n$ (27)
where $n$ is an integer. Since the analysis is semi-classical, the above
result is valid only for large $n$. Similar nature of entropy spectrum was
obtained in Einstein and Einstein-Gauss-Bonnet theory [37, 24, 25, 31, 32, 26,
33, 34, 35, 23].
A couple of comments are in order here. First, the entropy quantum is
universal in the sense that it is independent of the black hole parameters.
This universality was also derived in the context of the new interpretation of
quasi-normal moles of black holes [26, 33] for the case of Einstein gravity.
Second, in the Einstein gravity, the same value was also obtained earlier by
Hod by considering the Heisenberg uncertainty principle and Schwinger-type
charge emission process [36].
### 5.2 Quasi-normal mode (QNM) method
In the above analysis, the entropy spectrum is derived by the tunneling
mechanism. In the following, we will derive this by the well known method
prescribed in [26]. Here the frequency of the quasi-normal modes (QNM) plays
an important role.
According to this method, a black hole behaves like a damped harmonic
oscillator whose frequency is given by
$f=(f_{R}^{2}+f_{I}^{2})^{\frac{1}{2}}$, where $f_{R}$ and $f_{I}$ are the
real and imaginary parts of the frequency of the QNM. In the large $n$ ($n$ is
an integer) limit $f_{I}>>f_{R}$. Consequently one has to use $f_{I}$ rather
than $f_{R}$ in the adiabatic quantity [25],
$\displaystyle I_{adiab}=\int\frac{dW}{\Delta f(W)},\,\,\,\ \Delta
f=f_{n+1}-f_{n}$ (28)
where $W$ is the energy of the QNM.
Here we shall calculate the entropy spectrum by using the adiabatic invariant
quantity (28). Since in the large $n$ limit, the imaginary part of the
frequency of the QNM is relevant, our next task is to find this. It will be
derived by the method prescribed in [38]. For simplicity, we consider a
massless scalar field satisfying the wave equation
$\nabla^{\mu}\nabla_{\mu}\phi=0$ in the space-time (1) where the metric
coefficients are given by (4). We look for a solution to this wave equation of
the form,
$\displaystyle\phi=\frac{1}{r}F(r)Y_{lm}(\theta,\phi)e^{\frac{iEt}{\hbar}}$
(29)
with Re($E$)$>0$. Substituting this in the wave equation and performing some
simple algebra we obtain the following “Schr$\ddot{o}$dinger like equation”,
$\displaystyle\Big{[}-\frac{d^{2}}{dr^{*2}}+V(r)\Big{]}F(r)=\frac{E^{2}}{\hbar^{2}}F(r)$
(30)
where the effective potential $V(r)$ is given by,
$\displaystyle
V(r)=f(r)\Big{[}\frac{l(l+1)}{r^{2}}+\frac{f^{\prime}(r)}{r}\Big{]}$ (31)
and the tortoise coordinate $r^{*}$ is defined as,
$\displaystyle r^{*}=\int\frac{dr}{f(r)}.$ (32)
In principle (30) can be solved with a particular set of boundary conditions.
But unfortunately, this equation cannot be solved exactly. Therefore to solve
this one has to take the help of some approximate method. Here, we shall use
the approximation method prescribed in [38]. Note that the effective potential
(31) vanishes at the horizon ($r^{*}\rightarrow-\infty$) and diverges at
spatial infinity ($r^{*}\rightarrow\infty$). Therefore, the QNMs are defined
to be those for which one has purely ingoing plane wave at the horizon and no
wave at spatial infinity, i.e.
$\displaystyle
F(r)|_{QNM}\sim\left\\{\begin{array}[]{ll}e^{\frac{iEr^{*}}{\hbar}}&\textrm{at
$r^{*}\rightarrow-\infty$}\\\ 0&\textrm{ at
$r^{*}\rightarrow\infty$}\end{array}\right.$ (35)
Now we will solve equation (30) in the near horizon limit and then impose the
above boundary conditions to find the frequency of QNM.
Expansion of the metric coefficient around the event horizon yields,
$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
f^{\prime}(r_{+})(r-r_{+})+{\cal{O}}[(r-r_{+})^{2}]$ (36) $\displaystyle=$
$\displaystyle 2{\kappa}(r-r_{+})+{\cal{O}}[(r-r_{+})^{2}]~{}.$
Here in the last step (8) has been used. Substituting this in the definition
of ‘$r^{*}$’ (32) and performing the integration we obtain,
$\displaystyle r^{*}$ $\displaystyle=$
$\displaystyle\int\frac{dr}{2{\kappa}(r-r_{+})+{\cal{O}}[(r-r_{+})^{2}]}$ (37)
$\displaystyle\simeq$
$\displaystyle\frac{1}{2\kappa}\ln(r-r_{+})+{\cal{O}}[r]$
Keeping the first term of ‘$f(r)$’ (36) only and substituting in (31) yields,
$\displaystyle V(r)\simeq
2\kappa(r-r_{+})\Big{[}\frac{l(l+1)}{r^{2}}+\frac{2\kappa}{r}\Big{]}$ (38)
Now substituting $\epsilon=r-r_{+}$ in the above and Taylor expanding around
$\epsilon=0$ we obtain the near horizon form of the effective potential:
$\displaystyle V(\epsilon)\simeq
2\kappa\epsilon\Big{[}\frac{l(l+1)}{r_{+}^{2}}(1-\frac{2\epsilon}{r_{+}})+\frac{2\kappa}{r_{+}}(1-\frac{\epsilon}{r_{+}})\Big{]}~{}.$
(39)
Therefore, keeping only the first term in (37) and then substituting (39) in
(30), we obtain the near horizon “Schrodinger like equation”:
$\displaystyle-4\kappa^{2}\epsilon^{2}\frac{d^{2}F}{d\epsilon^{2}}-4\kappa^{2}\epsilon\frac{dF}{d\epsilon}+2\kappa\epsilon\Big{[}\frac{l(l+1)}{r_{+}^{2}}(1-\frac{2\epsilon}{r_{+}})+\frac{2\kappa}{r_{+}}(1-\frac{\epsilon}{r_{+}})\Big{]}F=\frac{E^{2}}{\hbar^{2}}F$
(40)
Solution of the above equation yields,
$\displaystyle
F\sim\epsilon^{\frac{iE}{2\hbar\kappa}}U\Big{[}\frac{1}{4}\Big{(}2-\frac{i(\frac{\sqrt{\kappa}(2r_{+}\kappa+l+l^{2})}{\sqrt{r_{+}}\sqrt{r_{+}\kappa+l+l^{2}}}-\frac{2E}{\hbar})}{\kappa}\Big{)},1+\frac{iE}{\hbar\kappa},\frac{2i\epsilon\sqrt{r_{+}\kappa+l+l^{2}}}{r{{}_{+}}{{}^{3/2}}\sqrt{\kappa}}\Big{]}$
(41)
where ‘$U[.....]$’ is the confluent hypergeometric function. In the limit
$\epsilon<<1$, the above solution reduces to the form
$\displaystyle F$ $\displaystyle\sim$
$\displaystyle{\textrm{(constant)}}~{}\epsilon^{-\frac{iE}{2\hbar\kappa}}\frac{\Gamma(\frac{iE}{\hbar\kappa})}{\Gamma(\frac{1}{2}-\frac{i(\frac{\sqrt{\kappa}(2r_{+}\kappa+l+l^{2})}{\sqrt{r_{+}}\sqrt{r_{+}\kappa+l+l^{2}}}-\frac{2E}{\hbar})}{4\kappa})}$
(42) $\displaystyle+$
$\displaystyle{\textrm{(constant)}}\epsilon^{\frac{iE}{2\hbar\kappa}}\frac{\Gamma(-\frac{iE}{\hbar\kappa})}{\Gamma(\frac{1}{2}-\frac{i(\frac{\sqrt{\kappa}(2r_{+}\kappa+l+l^{2})}{\sqrt{r_{+}}\sqrt{r_{+}\kappa+l+l^{2}}}+\frac{2E}{\hbar})}{4\kappa})}~{}.$
In the above expression we did not mention the explicit form of the “constant”
since in this analysis it is not necessary. The second term of (42) represents
the ingoing wave. Now since there is no outgoing wave in the QNM (35) at
$r^{*}\rightarrow-\infty$, the first term should vanish. This will happen at
poles of the gamma function of the denominator of first term. The poles of
this gamma function will ultimately determine the imaginary part of the
frequency of the QNMs. The poles are given by
$\displaystyle E_{n}=\Big{[}1+i(2n+1)\Big{]}\hbar\kappa$ (43)
for the $l=0$ mode and $n=1,2,3,...$. At these poles the gamma function on the
numerator does not vanish. So, these will give the imaginary part of the
frequency of the QNM. Hence, in this case the imaginary part of the frequency
of the QNMs is
$\displaystyle{\textrm{Im}}~{}f_{n}=(2n+1)\kappa=(2n+1)\frac{2\pi
T_{H}}{\hbar},$ (44)
where $f=\frac{E}{\hbar}$. Of course, one can check this value by the
perturbation method as done in [39] for another black hole solution in Hořava-
Lifshitz gravity.
Now the energy of the black hole is given by the ADM mass ‘$M$’ (5) and since
from (44) ${\textrm{Im}}~{}\Delta f={\textrm{Im}}(f_{n+1}-f_{n})=\frac{4\pi
T_{H}}{\hbar}$, the adiabatic invariant quantity (28) in this case yields,
$\displaystyle I_{adiab}=\frac{\hbar}{4\pi}\int\frac{dM}{T_{H}}~{}.$ (45)
Use of first law of thermodynamics, $T_{H}dS_{BH}=dM$, then leads to,
$\displaystyle I_{adiab}=\frac{\hbar}{4\pi}\int
dS_{BH}=\frac{\hbar}{4\pi}S_{BH}~{}.$ (46)
Finally, the Bohr-Sommerfield quantization rule
$\displaystyle I_{adiab}=n\hbar,$ (47)
gives the spacing of the entropy spectrum:
$\displaystyle S_{n}=4\pi n~{}.$ (48)
Although the exact value (27, 48) of the equi-spacing in the two methods does
not coincide, their order of magnitude is same. This discrepancy may be due to
the following reason. It has been shown in [36] that, if one calculates the
entropy spectrum following [37], by incorporating both the uncertainty
relation and the Schwinger mechanism, then the spacing between two adjacent
levels is different from the calculation [37] where the latter effect is not
considered. In this connection, one must note that the tunneling mechanism has
a similarity with Schwinger mechanism [19, 40] and also, as we have explained
earlier, the uncertainty relation is there. On the contrary, the QNM method
incorporates only the uncertainty relation through Bohr-Sommerfield
quantization rule (47). Hence it not surprising that we obtained a different
spacing in the entropy spectrum in both methods.
Finally, from the expression for the entropy of the black hole given by (20),
we observe that it is not proportional to the area. Therefore, in this case,
the area spacing is not equidistant. This is contrary to Einstein gravity but
agrees with other examples like Einstein-Gauss-Bonnet gravity.
## 6 Conclusions
Some aspects of the quantum nature of black holes were studied for the black
hole solution recently found in Hořava-Lifshitz gravity. We mainly
concentrated on the Hawking effect and black hole spectroscopy. The analysis
was done by our reformulated tunneling approach [17, 18]. The advantage of
this reformulated approach is that the emission spectrum was directly obtained
instead of just the temperature, as happens in the conventional formulations
[19, 20]. Also, as an application, the nature of spectroscopy of the black
hole was discussed, following [23].
In the semi-classical limit, the analysis showed that the emission spectrum
was perfectly black body with a temperature proportional to the surface
gravity. This reproduced the familiar form which occurs in known theories,
e.g. Einstein and Einstein-Gauss-Bonnet gravities. Using the first law of
thermodynamics the entropy was also calculated. The standard Bekenstein-
Hawking area law was violated since there was an additional term proportional
to logarithmic of area.
Also, we discussed about the spectrum of entropy/area of the black hole in two
distinct ways - the tunneling and QNM approaches. Both revealed that the
entropy spectrum was equispaced in the large quantum number limit as usually
happens for Einstein gravity and Einstein-Gauss-Bonnet gravity. On the other
hand, since the entropy was not proportional to the area, the area spectrum
was not equispaced. Consequently, it has a similarity with the Einstein-Gauss-
Bonnet theory, rather than the usual Einstein gravity. We hope that the
several new results and insights gained from our analysis would help in
providing a better understanding of the black holes in Hořava-Lifshitz
gravity.
Acknowledgement :
I thank Prof. Rabin Banerjee for illuminating discussions and a careful
reading of the manuscript.
## References
* [1] P. Horava, Phys. Rev. D 79, 084008 (2009) [arXiv:0901.3775 [hep-th]].
* [2] P. Horava, JHEP 0903, 020 (2009) [arXiv:0812.4287 [hep-th]].
P. Horava, Phys. Rev. Lett. 102, 161301 (2009) [arXiv:0902.3657 [hep-th]].
A. Volovich and C. Wen, JHEP 0905, 087 (2009) [arXiv:0903.2455 [hep-th]].
G. Calcagni, JHEP 0909, 112 (2009) [arXiv:0904.0829 [hep-th]].
E. Kiritsis and G. Kofinas, Nucl. Phys. B 821, 467 (2009) [arXiv:0904.1334
[hep-th]].
T. Takahashi and J. Soda, Phys. Rev. Lett. 102, 231301 (2009) [arXiv:0904.0554
[hep-th]].
J. Kluson, JHEP 0907, 079 (2009) [arXiv:0904.1343 [hep-th]].
A. Wang and Y. Wu, JCAP 0907, 012 (2009) [arXiv:0905.4117 [hep-th]].
D. Blas, O. Pujolas and S. Sibiryakov, JHEP 0910, 029 (2009) [arXiv:0906.3046
[hep-th]].
J. Z. Tang and B. Chen, [arXiv:0909.4127 [hep-th]].
* [3] Y. S. Myung, Phys. Lett. B 679, 491 (2009) [arXiv:0907.5256 [hep-th]].
Y. S. Myung, Phys. Lett. B 681, 81 (2009) [arXiv:0909.2075 [hep-th]].
* [4] H. Lu, J. Mei and C. N. Pope, Phys. Rev. Lett. 103, 091301 (2009) [arXiv:0904.1595 [hep-th]].
* [5] R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 80, 024003 (2009) [arXiv:0904.3670 [hep-th]].
* [6] R. G. Cai, Y. Liu and Y. W. Sun, JHEP 0906, 010 (2009) [arXiv:0904.4104 [hep-th]].
* [7] E. O. Colgain and H. Yavartanoo, JHEP 0908, 021 (2009) [arXiv:0904.4357 [hep-th]].
* [8] T. Harada, U. Miyamoto and N. Tsukamoto, [arXiv:0911.1187 [gr-qc]].
* [9] R. G. Cai, L. M. Cao and N. Ohta, Phys. Lett. B 679, 504 (2009) [arXiv:0905.0751 [hep-th]].
* [10] R. G. Cai and N. Ohta, arXiv:0910.2307 [hep-th].
* [11] Y. S. Myung and Y. W. Kim, arXiv:0905.0179 [hep-th].
Y. S. Myung, Phys. Lett. B 678, 127 (2009) [arXiv:0905.0957 [hep-th]].
R. B. Mann, JHEP 0906, 075 (2009) [arXiv:0905.1136 [hep-th]].
* [12] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)].
* [13] J. J. Peng and S. Q. Wu, [arXiv:0906.5121 [hep-th]].
* [14] D. Y. Chen, H. Yang and X. T. Zu, Phys. Lett. B 681, 463 (2009) [arXiv:0910.4821 [gr-qc]].
* [15] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [arXiv:gr-qc/0502074].
S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. Lett. 96, 151302 (2006)
[arXiv:hep-th/0602146].
* [16] R. Banerjee and S. Kulkarni, Phys. Rev. D 77, 024018 (2008) [arXiv:0707.2449 [hep-th]].
R. Banerjee and S. Kulkarni, Phys. Lett. B 659, 827 (2008) [arXiv:0709.3916
[hep-th]].
* [17] R. Banerjee and B. R. Majhi, Phys. Rev. D 79, 064024 (2009) [arXiv:0812.0497 [hep-th]].
* [18] R. Banerjee and B. R. Majhi, Phys. Lett. B 675, 243 (2009) [arXiv:0903.0250 [hep-th]].
* [19] K. Srinivasan and T. Padmanabhan, Phys. Rev. D 60, 024007 (1999) [arXiv:gr-qc/9812028].
* [20] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042 (2000) [arXiv:hep-th/9907001].
* [21] M. Arzano, A. J. M. Medved and E. C. Vagenas, JHEP 0509, 037 (2005) [arXiv:hep-th/0505266];
R. Banerjee and B. R. Majhi, Phys. Lett. B 662, 62 (2008) [arXiv:0801.0200
[hep-th]];
* [22] B. R. Majhi and S. Samanta, arXiv:0901.2258 [hep-th].
* [23] R. Banerjee, B. R. Majhi and E. C. Vagenas, [arXiv:0907.4271 [hep-th]].
* [24] S. Hod, Phys. Rev. Lett. 81, 4293 (1998) [arXiv:gr-qc/9812002].
* [25] G. Kunstatter, Phys. Rev. Lett. 90, 161301 (2003) [arXiv:gr-qc/0212014].
* [26] M. Maggiore, Phys. Rev. Lett. 100, 141301 (2008) [arXiv:0711.3145 [gr-qc]].
* [27] M. Angheben, M. Nadalini, L. Vanzo and S. Zerbini, JHEP 0505, 014 (2005) [arXiv:hep-th/0503081];
E. T. Akhmedov, V. Akhmedova and D. Singleton, Phys. Lett. B 642, 124 (2006)
[arXiv:hep-th/0608098];
R. Banerjee and B. R. Majhi, JHEP 0806, 095 (2008) [arXiv:0805.2220 [hep-th]];
R. Banerjee and B. R. Majhi, Phys. Lett. B 674, 218 (2009) [arXiv:0808.3688
[hep-th]];
* [28] R. Kerner and R. B. Mann, Class. Quant. Grav. 25, 095014 (2008) [arXiv:0710.0612 [hep-th]];
R. Kerner and R. B. Mann, Phys. Lett. B 665, 277 (2008) [arXiv:0803.2246 [hep-
th]];
R. Di Criscienzo and L. Vanzo, Europhys. Lett. 82, 60001 (2008)
[arXiv:0803.0435 [hep-th]];
B. R. Majhi, Phys. Rev. D 79, 044005 (2009) [arXiv:0809.1508 [hep-th]].
* [29] V. Akhmedova, T. Pilling, A. de Gill and D. Singleton, Phys. Lett. B 666, 269 (2008) [arXiv:0804.2289 [hep-th]].
E. T. Akhmedov, T. Pilling and D. Singleton, Int. J. Mod. Phys. D 17, 2453
(2008) [arXiv:0805.2653 [gr-qc]].
* [30] R. Banerjee, B. R. Majhi and S. Samanta, Phys. Rev. D 77, 124035 (2008) [arXiv:0801.3583 [hep-th]].
* [31] M. R. Setare, Phys. Rev. D 69, 044016 (2004) [arXiv:hep-th/0312061];
M. R. Setare, Gen. Rel. Grav. 37, 1411 (2005) [arXiv:hep-th/0401063].
* [32] M. R. Setare and E. C. Vagenas, Mod. Phys. Lett. A 20, 1923 (2005) [arXiv:hep-th/0401187];
M. R. Setare, Class. Quant. Grav. 21, 1453 (2004) [arXiv:hep-th/0311221].
* [33] E. C. Vagenas, JHEP 0811, 073 (2008) [arXiv:0804.3264 [gr-qc]];
A. J. M. Medved, Class. Quant. Grav. 25, 205014 (2008) [arXiv:0804.4346 [gr-
qc]].
* [34] D. Kothawala, T. Padmanabhan and S. Sarkar, Phys. Rev. D 78, 104018 (2008) [arXiv:0807.1481 [gr-qc]].
* [35] S. W. Wei, R. Li, Y. X. Liu and J. R. Ren, [arXiv:0901.0587 [hep-th]].
* [36] S. Hod, Phys. Rev. D 59, 024014 (1999) [arXiv:gr-qc/9906004];
* [37] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
* [38] T. R. Choudhury and T. Padmanabhan, Phys. Rev. D 69, 064033 (2004) [arXiv:gr-qc/0311064].
T. Padmanabhan, Class. Quant. Grav. 21, L1 (2004) [arXiv:gr-qc/0310027].
A. J. M. Medved, D. Martin and M. Visser, Class. Quant. Grav. 21, 2393 (2004)
[arXiv:gr-qc/0310097].
* [39] R. A. Konoplya, Phys. Lett. B 679, 499 (2009) [arXiv:0905.1523 [hep-th]].
* [40] S. P. Kim, J. Korean Phys. Soc. 53, 1095 (2008) [arXiv:0709.4313 [hep-th]].
|
arxiv-papers
| 2009-11-17T08:59:33 |
2024-09-04T02:49:06.504047
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/0911.3239"
}
|
0911.3245
|
# Effect of Holstein phonons on the optical conductivity of gapped graphene
Kh. Jahanbani Institute for Advanced Studies in Basic Sciences (IASBS),
Zanjan, 45195-1159, Iran School of Physics, Institute for Fundamental
Sciences, (IPM) 19395-5531 Tehran, Iran Reza Asgari 111Corresponding author:
Tel: +98 21 22280692; fax: +98 21 22280415.
E-mail address: asgari@theory.ipm.ac.ir, School of Physics, Institute for
Fundamental Sciences, (IPM) 19395-5531 Tehran, Iran
###### Abstract
We study the optical conductivity of a doped graphene when a sublattice
symmetry breaking is occurred in the presence of the electron-phonon
interaction. Our study is based on the Kubo formula that is established upon
the retarded self-energy. We report new features of both the real and
imaginary parts of the quasiparticle self-energy in the presence of a gap
opening. We find an analytical expression for the renormalized Fermi velocity
of massive Dirac Fermions over broad ranges of electron densities, gap values
and the electron-phonon coupling constants. Finally we conclude that the
inclusion of the renormalized Fermi energy and the band gap effects are indeed
crucial to get reasonable feature for the optical conductivity.
###### pacs:
78.67.-n, 71.10.Ay, 73.25.+i, 72.80.-r
## 1 Introduction
There is a considerable interest in understanding the effects on properties of
particle due to the interactions with environment, for instance the coupling
of electrons to lattice vibrations or electron-phonon coupling. The electron-
phonon coupling plays an essential role in the theory of high temperature
superconductivity and they exist in other material such as nanotubes, C60
molecules and other fullerenes rmp . Also it is important to consider the
electron-phonon coupling in transport properties.
Graphene, a single layer of carbon atoms, Geim is disputable the first true
two-dimensional lattices. Graphene is thermodynamically stable and there is
indeed ripple structures on graphene sheets. Lattice displacements due to the
ripple structures are symmetric with respect to their close carbon atoms and
couple to the carrier densities. The electrons moving through the sheet are
coupled to the out-of-plane phonons and therefore the electron-phonon coupling
plays an important role in the transport properties akturk ; basko ; park .
The coupling of electrons to out-of-plane optical phonons can be modeled by a
Holstein type coupling Holstein . In this model the coupling of electrons to
dispersionless optical phonons is essentially local. The electron-phonon
coupling has been carefully examined and has been shown to give rise to Kohn
anomalies in the phonon dispersion at edge points in the Brillouin zone where
the phonons can be studied by Raman spectroscopy piscanec1 ; piscanec2 ;
pisana . An alternative strategy for the electron-phonon coupling measurement
is based on the analysis of the $G$-peak linewidths and its broadening.
The optical conductivity is one of the most useful tools to investigate the
basic properties of materials. Both the excitation spectrum of materials such
gaps, phonons and interband transitions and the scattering mechanisms leave
their distinct traces in transport. It was shown that the infrared
conductivity of graphene is basically independent of the frequency peres2 ;
peresijmp ; peresprb78 ; gusynin and experimentally confirmed this manner li
; nair . The effect of electron-phonon interaction in gapless graphene has
been discussed by several authors Stauber ; calandra ; tse ; peres ; stauperes
directed towards understanding this effect on the optical conductivity.
The energy spectrum of the Dirac electrons in a graphene layer that
epitaxially grown on a SiC substrate has been measured by Zhou et al. Zhou
and they observed an energy gap of about 200 meV opened up in the electronic
spectrum. They attributed the opening up of the gap is due to the breaking of
the $A$ and $B$ sublattices symmetry novoselov . The optical response of a
gapped graphene is of important for an understanding of optoelectronic
devices. Moreover, the optical spectroscopy can be used for measurements of
the magnitude of the energy gap.
In this paper we consider the sublattice symmetry breaking mechanism for a gap
opening in a pristine doped graphene sheet and study the impact of the
electron-phonon coupling on the electronic conductivity of the electron-doped
gapped graphene using Kubo formula at zero-temperature. We show that the
renormalized velocity is suppressed due to the electron-phonon interaction.
There is a shift in the chemical potential and we show that the interacting
chemical potential is less than the noninteracting one due to the electron-
phonon coupling. The optical conductivity is affected by Pauli blocking below
twice value of the renormalized interacting chemical potential and gap values.
## 2 Model Hamiltonian and theory
We consider the simplest form of Hamiltonian that describes the interaction of
electron with an optical phonon mode, called the Holstein model. The honeycomb
lattice can be consider in terms of two triangular sublattices $A$ and $B$. We
consider electrons in $\pi$-orbital of carbon atoms by using the tight-binding
Hamiltonian in addition to the effect of the electron-phonon coupling due to
localized Holstein phonons and a gap opening procedure due to sublattice
symmetry breaking alireza . The total Hamiltonian in momentum space can be
expressed as
$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum_{k,\sigma}[\phi(k)a_{{\bf k},\sigma}^{\dagger}b_{{\bf
k},\sigma}+h.c]$ (1) $\displaystyle+$ $\displaystyle
D\sum_{p,k,\sigma}\chi_{0}[a^{\dagger}_{{\bf p},\sigma}a_{{\bf
p+k},\sigma}+b^{\dagger}_{{\bf p},\sigma}b_{{\bf p+k},\sigma}](c_{\bf
k}+c_{\bf-k}^{\dagger})$ $\displaystyle+$
$\displaystyle\sum_{k}\omega_{0}c_{\bf k}^{\dagger}c_{\bf
k}+\Delta\sum_{k}[a_{{\bf k},\sigma}^{\dagger}a_{{\bf k},_{\sigma}}-b_{{\bf
k},\sigma}^{\dagger}b_{{\bf k},\sigma}]$ $\displaystyle-$
$\displaystyle\mu_{0}\sum_{k}[a_{{\bf k},\sigma}^{\dagger}a_{{\bf
k},\sigma}+b_{{\bf k},\sigma}^{\dagger}b_{{\bf k},\sigma}]$
where $a_{{\bf k},\sigma}$ or $b_{{\bf k},\sigma}$ is the fermion annihilation
operator in $k-$space on sublattice $A$ or $B$, respectively and $t$ is the
nearest neighbor hopping parameter castro . The band gap, $2\Delta$ has a
nonzero value as a result of breaks the symmetry between sublattices, A and B.
We consider that the noninteracting chemical potential, $\mu_{0}$ be larger
than the gap value representing the electron-doped system. The electron-phonon
coupling is determined by $D$ and furthermore $\omega_{n}$ denotes the
fermionic Matsubara frequency. Moreover, $c_{\bf k}$ is the annihilation
phonon operator. $\omega_{0}$ is the frequency of the out of plane vibrations
of the optical phonon and $\chi_{0}=\sqrt{\frac{\hbar}{2MN\omega_{0}}}$ with M
is ion’s mass and N denotes the number of unit cells.
$\phi(k)=\sum_{\delta}\exp^{-i{\bf\delta}\cdot{\bf k}}$ with $\delta$ being
the vectors connecting the three nearest neighbors on the honeycomb lattice
Mahan . $\phi(k)$ reduces to $\hbar v_{\rm F}k/t$ in the Dirac cone
approximation castro .
The matrix element of noninteracting Green’s function with the gap of the
electronic spectrum is determined by following expression
$\displaystyle G_{\alpha\beta}^{0}(k,i\omega_{n})=\frac{1}{2}\sum_{\lambda=\pm
1}\left(\delta_{\alpha,\beta}+\frac{\lambda\Upsilon_{\alpha,\beta}}{\xi_{{}_{\Delta}}(k)}\right)\frac{1}{i\hbar\omega_{n}+\mu_{0}-\lambda\xi_{{}_{\Delta}(k)}},$
in which $\alpha,\beta=A,B$ and we have defined parameters
$\Upsilon_{AA}=-\Upsilon_{BB}=\Delta$,
$\Upsilon_{AB}=\Upsilon^{*}_{BA}=-t\phi(k)$. The quasiparticle excitation
energy is $\xi_{{}_{\Delta}}(k)=\sqrt{t^{2}|\phi(k)|^{2}+\Delta^{2}}$. Note
that at zero-temperature $\mu_{0}=\xi_{{}_{\Delta}}(k_{\rm F})$ with $k_{\rm
F}$ is the Fermi momentum of charge carriers.
An exact evaluation of the self-energy is only possible in some special cases.
The matrix elements of the self-energy calculated to the lowest order in the
electron-phonon interaction and is defined as
$\Sigma_{\alpha,\beta}(i\omega_{n},p)=-k_{B}T\sum_{k,\nu}D^{2}\chi_{0}^{2}D^{(0)}(k,i\nu)G^{(0)}_{\alpha,\beta}(p-q,i\omega_{n}-i\nu)$
(2)
where $G^{(0)}_{\alpha,\beta}$ and $D^{(0)}$ are the zero-order electron and
phonon Green’s functions, respectively Mahan ; Jonson . In Holstein phonons,
$\chi_{0}$ and $D^{(0)}(k,i\nu)$ are momentum independent and thus the phonon
propagator is simplified by
$D^{(0)}(k,i\nu)=-2\omega_{0}/(\nu^{2}+\omega_{0}^{2}).$ (3)
We restrict our calculations to the lowest order self-energy that is
sufficient if Migdal’s theorem, states that vertex corrections in the
electron-phonon interaction can be neglected if the typical phonon frequencies
are sufficiently smaller than the electronic energy scale, is valid.
Therefore, we can neglect the vertex corrections since the self-energy is
$k$-independent. Using the contour integration, we can perform the summation
over the bosonic frequency in the expression of the self-energy and finally
the self-energy yields as
$\displaystyle\Sigma_{AA}(i\omega_{n})$ $\displaystyle=$
$\displaystyle\frac{D^{2}\chi_{0}^{2}}{2}\sum_{k,\lambda=\pm
1}\left(1+\frac{\lambda\Delta}{\xi_{{}_{\Delta}(k)}}\right)\times$ (4)
$\displaystyle\left\\{\frac{N_{0}+n_{F}(\lambda\xi_{{}_{\Delta}(k)}-\mu_{0})}{i\hbar\omega_{n}+\hbar\omega_{0}-\lambda\xi_{{}_{\Delta}(k)}+\mu_{0}}\\!+\\!\frac{N_{0}+1-n_{F}(\lambda\xi_{{}_{\Delta}(k)}-\mu_{0})}{i\hbar\omega_{n}-\hbar\omega_{0}-\lambda\xi_{{}_{\Delta}(k)}+\mu_{0}}\right\\},$
where $N_{0}=1/(e^{\hbar\omega_{0}/k_{B}T}-1)$ and $n_{F}(x)$ denotes the
Fermi-Dirac distribution function. To calculate $\Sigma_{BB}(i\omega_{n})$,
the gap value $\Delta$ might be replaced by $-\Delta$ in Eq. 4. It should be
noted that $\Sigma_{AB}(i\omega_{n})=\Sigma_{BA}(i\omega_{n})=0$ in the Dirac
cone approximation. The explicit expression of the self-energy will be
computed in the following.
### 2.1 Finite doping with a gap opening
We consider the low excited electron energy where the noninteracting electron
spectrum energy is given by $\sqrt{(\hbar v_{\rm F}k)^{2}+\Delta^{2}}$ alireza
. To evaluate the zero-temperature retarded self-energy evaluated at the Fermi
surface, we integrate Eq. 4 over $k$ and then decompose the results into
$\Sigma^{j}(\omega)=\Sigma^{j}_{0}(\omega)+\Delta\Sigma^{j}(\omega)$ where
$\displaystyle\Re e\Sigma^{j}_{0}(\omega)=\frac{\hbar
A_{c}}{2\pi}(g\omega_{0})^{2}\\{-\frac{\omega_{j}}{v_{F}^{2}}\ln|\frac{\Delta^{2}+\hbar^{2}v_{F}^{2}k_{c}^{2}}{(\hbar\omega+\mu_{0})^{2}-(\hbar\omega_{0}+\Delta)^{2}}|+\frac{\omega_{0}}{v_{F}^{2}}\ln|\frac{\omega_{+}+\omega_{0}}{\omega_{-}-\omega_{0}}|\\}$
(5) $\displaystyle\Im m\Sigma^{j}_{0}(\omega)$ $\displaystyle=$
$\displaystyle-\pi\frac{\hbar
A_{c}}{2\pi}(g\omega_{0})^{2}\\{-\frac{\omega_{j}+\omega_{0}}{v_{F}^{2}}\Theta(-\omega_{+}-\omega_{0})\Theta(\hbar\omega+\mu_{0}+\hbar\omega_{0}+\sqrt{\hbar^{2}v_{F}^{2}k_{c}^{2}+\Delta^{2}})$
$\displaystyle+$
$\displaystyle\frac{\omega_{j}-\omega_{0}}{v_{F}^{2}}\Theta(\omega_{-}-\omega_{0})\Theta(-\hbar\omega-\mu_{0}+\hbar\omega_{0}+\sqrt{\hbar^{2}v_{F}^{2}k_{c}^{2}+\Delta^{2}})\\}$
here $\omega_{j}=\omega+(\mu_{0}+j\Delta)/\hbar$ with $j=+1(-1)$ refers to
sublattice $A$($B$). $k_{c}$ is the ultraviolet cut-off momentum Stauber and
finally the coupling constant $g=\sqrt{N}D\chi_{0}/\omega_{0}$ being the order
of unity. The area of the unit cell is $A_{c}=a^{2}3\sqrt{3}/2$ with
$a=1.42$Å. The extra terms take the following form as
$\displaystyle\Re e\Delta\Sigma^{j}(\omega)=\frac{\hbar
A_{c}}{2\pi}(g\omega_{0})^{2}\\{-\frac{\omega_{j}}{v_{F}^{2}}\ln|\frac{(\omega+\omega_{0})(\omega_{-}-\omega_{0})}{(\omega-\omega_{0})(\omega_{-}+\omega_{0})}|-\frac{\omega_{0}}{v_{F}^{2}}\ln|\frac{\omega^{2}-\omega_{0}^{2}}{(\omega_{-})^{2}-\omega_{0}^{2}}|\\}$
(6) $\displaystyle\Im m\Delta\Sigma^{j}(\omega)=-\pi\frac{\hbar
A_{c}}{2\pi}(g\omega_{0})^{2}\\{$
$\displaystyle\\!\\!\\!\\!\\!\\!\frac{\omega_{j}+\omega_{0}}{v_{F}^{2}}~{}\Theta(\omega_{-}+\omega_{0})\Theta(-\omega-\omega_{0})$
(7) $\displaystyle-$
$\displaystyle\frac{\omega_{j}-\omega_{0}}{v_{F}^{2}}\Theta(-\omega+\omega_{0})~{}\Theta(\omega_{-}-\omega_{0})\\}$
If $\Delta=0$, the self-energy reduces to massless Dirac graphene which
addressed in Ref Stauber . Therefore, we have generalized the retarded self-
energy expression to gapped graphene. Once the retarded self-energy is
obtained, the quasiparticle properties of system due to the interaction of the
electron-phonon can be calculated. The renormalized electronic spectrum is
given by the Dyson equation as $E_{\bf k}=\xi_{\Delta}(k)+\Re e\Sigma(E_{\bf
k})$. Notice that according to the Dyson equation, we might distinguish the
noninteracting chemical potential from the chemical potential of the
interacting system due to the fact that $\Re e\Sigma(\omega)$ is not vanished
for doped graphene when $\omega$ tends to zero. We thus have
$\mu=\mu_{0}+\Re e\Sigma(\omega)|_{\omega=0}~{}.$ (8)
The renormalized velocity, on the other hand, is given by
$\frac{v^{\star}}{v_{\rm F}}=\frac{\hbar v_{\rm F}k/\xi_{\Delta}(k)+(v_{\rm
F}\hbar)^{-1}\partial_{k}\Re
e\Sigma(k,\omega)}{1-\hbar^{-1}\partial_{\omega}\Re
e\Sigma(k,\omega)}|_{\omega=0,k=k_{F}}$ (9)
within the Dyson scheme Mahan . The self-energy is independent of the
momentum, accordingly its $k$-derivative is zero. Consequently, the
renormalized velocity is obtained analytically
$\displaystyle\frac{v_{\rm F}}{v^{*}(1+(\Delta/\hbar v_{\rm F}k_{\rm
F})^{1/2})}$ $\displaystyle=$ $\displaystyle
1+\left(\frac{g{\omega}_{0}}{v_{\rm
F}{k}_{c}}\right)^{2}\\{\ln|\frac{({\Delta}^{2}+({\hbar v_{\rm
F}k_{c}})^{2})}{(({\mu_{0}}+{\hbar\omega}_{0})^{2}-{\Delta}^{2})}|$ (10)
$\displaystyle-$
$\displaystyle({\mu_{0}}+j{\Delta}+\hbar{\omega}_{0})\frac{2({\mu_{0}}+\hbar{\omega}_{0})}{({\mu_{0}}+\hbar{\omega}_{0})^{2}-{\Delta}^{2}}+2\frac{{\mu_{0}}+j{\Delta}}{\hbar{\omega}_{0}}\\}.$
### 2.2 Optical Conductivity
The optical conductivity can be calculated from the Kubo formalism. To this
end, we need to obtain the current operator which is a composition of the
paramagnetic and diamagnetic terms, i.e.
$j_{\alpha}=j^{P}_{\alpha}+j^{D}_{\alpha\beta}{\emph{A}}_{\beta}$. We do need
to modify the hopping parameter in the presence of an electromagnetic field
Stauber and then expand it up to the second order in the vector potential
$\overrightarrow{\emph{A}}(t)$. The current operator expressions do not change
in the presence of the gap value and therefore by assuming that the electric
field is in the direction of $x$-axis, we have
$j_{x}^{P}=-i\xi\sum_{\sigma,k}[(\phi(k)-3)a_{\sigma}^{\dagger}(k)b_{\sigma}(k)-(\phi^{\star}(k)-3)a_{\sigma}(k)b_{\sigma}^{\dagger}(k)]$
(11)
where $\xi=tea/\hbar$ and then the Kubo formula for conductivity is given by
$\sigma_{xx}(\omega)=\frac{<j_{x}^{D}>}{iA_{s}(\omega+i\eta)}+\frac{\Lambda_{xx}(\omega+i\eta)}{i\hbar
A_{s}(\omega+i\eta)}$ (12)
where $A_{s}$ is the area of sample and
$\Lambda_{xx}(i\omega_{n})=\int_{0}^{\hbar/k_{B}T}d\tau
e^{i\omega_{n}\tau}<T_{\tau}j_{x}^{P}(\tau)j_{x}^{P}(0)>$ Mahan . We have
ignored vertex corrections in the Kubo formula since we worked in nearly
highly electron doped graphene for which the Dirac cone approximation is
applicable. It was shown that the vertex corrections is essential for the low
density carriers of the DC conductivity of graphene. cappelluti After a
lengthy but straightforward algebra, we find
$\displaystyle\Im m\Lambda_{xx}(\omega)\\!$ $\displaystyle=$
$\displaystyle\\!\\!\xi^{2}\frac{A_{s}}{8\pi}\\!\int_{0}^{k_{c}}\\!kdk\int_{-\infty}^{\infty}\frac{d\epsilon}{2\pi}(n_{F}(\epsilon+\omega)-n_{F}(\epsilon))$
(13) $\displaystyle\times$
$\displaystyle\left\\{(2t^{2}|\phi(k)|^{4})A_{AB}(k,\epsilon)A_{AB}(k,\epsilon+\omega)\right.$
$\displaystyle+$
$\displaystyle(9-|\phi(k)|^{2})~{}[A_{AA}(k,\epsilon)A_{BB}(k,\epsilon+\omega)$
$\displaystyle+$ $\displaystyle
A_{BB}(k,\epsilon)A_{AA}(k,\epsilon+\omega)]\\}$
where the spectral functions are the imaginary part of Green’s function which
take the following forms:
$\displaystyle A_{\alpha,\beta}=-2\Im
m\\{\frac{\Phi_{\alpha,\beta}}{{(\Omega_{+}-\Sigma_{BB}(i\omega_{n}))(\Omega_{-}-\Sigma_{AA}(i\omega_{n}))-t^{2}|\phi(k)|^{2}}}\\}.$
Here
$\displaystyle\Omega_{\pm}=i\hbar\omega_{n}+\mu\pm\Delta,~{}~{}~{}~{}~{}~{}~{}~{}~{}\Phi_{AA(BB)}=\Omega_{+(-)}-\Sigma_{BB(AA)}(i\omega_{n})$
and $\Phi_{AB}=\Phi_{BA}=1$. The integral over $k$ in Eq. 13 can be performed
analytically and accordingly one dimensional integral will be needed to be
calculated numerically. Note that the interacting chemical potential is used
instead of the noninteracting one because of the nonzero value of
$\Sigma_{j}(0)$ . It should be noted that by setting $\Delta=0$, the optical
conductivity results are different with the results given in Ref. Stauber due
to the fact that we have implemented the interacting Fermi energy in the
formalism.
## 3 Numerical Results
We have considered the system with the phonon energy being
$\hbar\omega_{0}=0.2~{}eV$ peres . Although the order of coupling constant is
unity, we consider a larger value to seek its effect better. We have found
that the value of the quasiparticle properties for sublattices $A$ and $B$ are
different at most about $0.8\%$ due to the gap opening. We will then present
only the results of the sublattice $A$.
In Fig. 1, we have shown the results of the real and imaginary parts of the
retarded self-energy for the electron-doped system, ($\mu>\Delta$) at
$n=5\times 10^{12}$cm-2. $\Im m\Sigma(\omega)$ vanishes in
$|\omega|<\omega_{0}$ at which point it jumps up to a finite value because
only then can a quasiparticle decay by boson emission. It drops towards the
zero for $\omega<-\omega_{0}$ and then increases linearly showing a marginal
type physics which happens in the Coulomb electron interactions in undoped
graphene polini . Notice that $\Im m\Sigma$ is not symmetric with respect to
change of the sign of frequency. In addition, $\Im m\Sigma(\omega)$ vanishes
when $|\hbar\omega+\hbar\omega_{0}+\mu_{0}|<\Delta$ due to the effect of the
gap opening and $\Im m\Sigma$ tends to zero at $-\hbar\omega_{0}-\mu_{0}$ for
$\Delta=0$. These behaviors can be determined explicitly from expressions
given by Eqs. 2.1 and 7.
In Fig. 1b we can see logarithmic type singularities dogan at
$\omega=\pm\omega_{0}$ and $\omega=-\omega_{0}-(\mu_{0}-\Delta)/\hbar$ for the
results of $\Re e\Sigma$. The extra singular behavior is due to the gap
effect. It should be noted that the singularity at $\omega=\pm\omega_{0}$
would be washed out if a momentum dependence of phonon spectra is used. In
addition, there is a cancelation of the logarithmic singularity at
$\omega=-\omega_{0}-(\mu_{0}+\Delta)/\hbar$. The logarithmic singularity can
be determined to the argument of the logarithm in Eqs. 5 and 6. We have
obtained an expression for the interacting density of states too through the
spectral function. The singularities manner lead to kink structures in the
interacting electronic density of states. In the results, there are three kink
structures in the interacting density of states where one of them is
associated to the gap. The kink structures would affect to physical quantities
and transport properties through the interacting electronic density of states.
The renormalized velocity as function of the densities, gap values and the
coupling constants are shown in Fig. 2. The renormalized velocity is
suppressed due to the electron-phonon interaction and the gap values too. We
have found a nonmonotonic behavior of $v^{*}$ with respect to the electron
density when the gap value increases and results are shown in Fig. 2b. At
small gap values, $v^{*}$ decreases with increasing density however it changes
behavior at large gap values and behaves like conventional two-dimensional
electron systems. Therefore, we expect that the electron-phonon interaction
renormalized the electronic quantities at the Fermi surface by a factor
$v^{*}/v_{\rm F}$.
The optical conductivity scaled by $\sigma_{0}=e^{2}/4\hbar$ as a function of
energy for different values of (a) the coupling constants and (b) the gap
values are shown in Fig. 3. First of all, $\sigma$ tends to a minimum value at
$\omega_{0}$. Moreover it basically increases around $\omega>\omega_{0}$ due
to the contribution of the Holstein phonon sideband. In the case of
noninteracting electron-phonon system, $\sigma$ has a sharp structure, step
function manner, at $2\mu_{0}$ due to interband transitions and the
conductivity increases by a factor of two, $\sigma=2\sigma_{0}$ at
$\omega=2\Delta$ and finally at higher frequencies decreases and approaches to
$\sigma_{0}$ gusynin . By switching interaction on, the chemical potential
becomes weaker and consequently the position of the sharp structure changes to
$2\mu$ which is smaller than $2\mu_{0}$. This behavior is clearly shown in the
Fig. 3 which did not consider in results discussed in Ref. Stauber . At $g=0$,
the conductivity is larger than $\sigma_{0}$ about $2\mu$ and then tends to
$\sigma_{0}$ in gapped graphene. However, $\sigma$ always remains smaller than
$\sigma_{0}$ in gapless graphene. The gap dependence on the optical
conductivity is shown in Fig. 3b. First, the gap opening makes the chemical
potential bigger therefore the sharp structure in the $\sigma$ tends to larger
$\omega$ values. Second, the scattering mechanism increases by increasing the
electron densities and then the optical conductivity changes and becomes
smaller.
Another point of interest for experiments is the density dependence ( in units
of 1012 cm-2) of the optical conductivity ( Fig. 4) as a function of frequency
at $2\Delta=0.2$ eV. Note that the noninteracting chemical potential values
associated to the electron densities used in Fig. 4 are
$\mu_{0}=0.154,0.279,0.382$ and $0.831$ eV, respectively with giving
$\Delta=0.1$ eV. The optical conductivity increases by increasing the electron
density around $\omega_{0}$ however $\sigma$ decreases faster by increasing
the density at high frequency. The sharp structure of the optical conductance
tends to higher frequency by increasing the electron density. The sharp
position occurs at $2\mu$ which is always smaller than $2\mu_{0}$ for the same
system.
## 4 Conclusion
we have calculated the optical conductivity of gapped graphene, including the
effect of the lowest order self-energy diagram due to the electron-phonon
interaction by Holstein Hamiltonian. We have reported an extra logarithmic
singular behavior associated to gap value in the real part of the self-energy.
We have found the density, gap value and the electron-phonon coupling
dependence of the renormalized velocity and the interacting chemical
potential. The optical conductivity is affected by these physical quantities
and Pauli blocking below twice value of the renormalized chemical potential
and the gap values. We conclude that the inclusion of the renormalized Fermi
energy and the band gap affects are indeed crucial to get reasonable feature
for the optical conductivity. The gap dependence of the optical conductivity
would be verified by experiments.
## 5 acknowledgments
R. A thank S.G. Sharapov for stimulating discussion. We are grateful A.
Qaiumzadeh for useful comments. We thank Centro de Ciencias de Benasque, Spain
where this work was completed.
Note added- In final stage of preparing this manuscript, we became aware of a
related work for gapless graphene carbotte .
## References
* (1) O. Gunnarsson, Rev. Mod. Phys. 69 (1997) 575 and references therein.
* (2) K. S. Novoselov, A. K. Geim, S. V. Morozov, et.al., Science, 306 (2004) 666 .
* (3) Akin Akturk and Neil Goldman, Journal of Applied physics, 103 (2008) 053702 .
* (4) D. M. Basko and I. L. Aleiner, Phys. Rev. B 77 (2008) 041409 (R) .
* (5) Cheol-Hwang Park, Feliciaon Giustino, Marivin L. Cohen and Steven G. Louie, Nano Letters 8 (2008) 4229 .
* (6) T. Holstein, Ann. Phys. (N.Y.) 8, 325 (1959); 8 (1959) 343 .
* (7) S. Piscanec et al., Phys. Rev. Lett. 93 (2004) 185503 .
* (8) S. Piscanec et al., Phys. Rev. B 75 (2007) 035427 .
* (9) S. Pisana et al. Nature Mater. 6 (2007) 198 .
* (10) N. M. R. Peres, F. Guinea and A. H. Castro Neto, Phys. Rev. B 73(2006) 125411
* (11) N. M. R. Peres, and T. Stauber, Int. J. Mod. Phys. B, 16 (2008) 2529 .
* (12) T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B, 78 (2008) 085432 .
* (13) V. P. Gusynin, S. G. Sharapov and J. P. Carbotte, Phys. Rev. Lett. 96(2006) 256802 .
* (14) Z. Q. Li, et.al., Nat. Phys., 4(2008) 532 .
* (15) R. R. Nair et. al., Science, 320 (2008) 1308 .
* (16) T. Stauber, and N. M. R. Peres, J. Phys.: Condens. Matter 20 (2008) 055002.
* (17) M. Calandra and F. Mauri, Phys. Rev. B 76 (2007) 205411 .
* (18) W.-K. Tse and Das Sarma, Phys. Rev. Lett 99 (2007) 236802 .
* (19) N. M. R. Peres, T. Stauber and A. H. Castro Neto, EPL 84 (2008) 38002 .
* (20) T. Stauber, and N. M. R. Peres, Phys. Rev. B, 78(2008) 085418 .
* (21) S.Y. Zhou et. al., Nature Mater., 76 (2007) 770 .
* (22) K. S. Novoselov, et.al., nature 438(2005) 197 .
* (23) E. Cappelluti and L. Benfatto, Phys. Rev. B 79 (2009) 035419 .
* (24) A. Qaiumzadeh and R. Asgari, Phys. Rev. B 79 (2009) 075414 .
* (25) A. H. Castro Neto et al., Rev. Mod. Phys., 81 (2009) 109 .
* (26) Gerald D. Mahan, Many-Particle physics, (Plenum Peress, New York 1990) 2nd edition.
* (27) M. Jonson, and G. D. Mahan, Phys. Rev. B, 21 (1980) 4223 .
* (28) F. Dogan and F. Marsiglio, Phys. Rev. B 68 (2003) 165102 .
* (29) M. Polini et al., Phys. Rev. B 77 (2008) 081411(R) .
* (30) J. P. Carbotte, E. J. Nicol and S. G. Sharapov, arXiv:0908.2608 .
Figure 1: (Color online) Imaginary (a) and real (b) parts of the self-energy
as a function of energy evaluated at Fermi energy for different gap values at
the coupling constant $g=3.0$ and density $n=5.0\times 10^{12}$ cm-2.
Figure 2: (Color online) Renormalized electron velocity for (a) the different
values of coupling constants at $n=5.0\times 10^{12}$ cm-2, (b) the different
value of density (in units of $10^{12}$ cm-2) at $g=3.0$.
Figure 3: (Color online) Optical conductivity as a function of energy for (a)
the different values of coupling constant at $\Delta=0.1~{}eV$ and (b) the
different value of $\Delta$ at $g=3.0$. We consider $n=1\times
10^{13}~{}cm^{-2}$.
Figure 4: (Color online) Optical conductivity as a function of energy for
different values of density ( in unites of $10^{12}$ cm-2) at $g=3.0$ and
$\Delta=0.1~{}eV$
|
arxiv-papers
| 2009-11-17T09:43:55 |
2024-09-04T02:49:06.509432
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kh. Jahanbani, Reza Asgari",
"submitter": "Reza Asgari",
"url": "https://arxiv.org/abs/0911.3245"
}
|
0911.3295
|
010005 2009 J. A. Bertolotto 010005
Using Monte Carlo simulations and finite-size scaling analysis, the critical
behavior of attractive rigid rods of length $k$ ($k$-mers) on square lattices
has been studied. An ordered state, with the majority of $k$-mers being
horizontally or vertically aligned, was found. This ordered phase is separated
from the disordered state by a continuous transition occurring at a critical
density $\theta_{c}$, which increases linearly with the magnitude of the
lateral interactions.
# The effect of the lateral interactions on the critical behavior of long
straight rigid rods on two-dimensional lattices
P. Longone [inst1] D. H. Linares [inst1] A. J. Ramirez-Pastor[inst1] E-mail:
antorami@unsl.edu.ar
(9 July 2009; 6 November 2009)
††volume: 1
99 inst1 Departamento de Física, Instituto de Física Aplicada, Universidad
Nacional de San Luis-CONICET, Chacabuco 917, D5700BWS San Luis, Argentina.
## 1 Introduction
The study of systems of hard non-spherical colloidal particles has, for many
years, been attracting a great deal of interest and the activity in this field
is still growing [1–14]. An early seminal contribution to this subject was
made by Onsager [1] with his paper on the isotropic-nematic (I-N) phase
transition in liquid crystals. The Onsager’s theory predicted that very long
and thin rods interacting with only excluded volume interaction can lead to
long-range orientational (nematic) order. Thus, at low densities, the
molecules are typically far from each other and the resulting state is an
isotropic gas. However, at large densities, it is more favorable for the
molecules to align spontaneously (there are many more ways of placing nearly
aligned rods than randomly oriented ones), and a nematic phase is present at
equilibrium.
Interestingly, a number of papers have appeared recently, in which the I-N
transition was studied in two dimensions [10–14]. In Ref. [10], the authors
gathered strong numerical evidence to suggest that a system of square
geometry, with two allowed orientations, shows nematic order at intermediate
densities for $k\geq 7$ and provided a qualitative description of a second
phase transition (from a nematic order to a non-nematic state) occurring at a
density close to $1$. However, the authors were not able to determine the
critical quantities (critical point and critical exponents) characterizing the
I-N phase transition occurring in the system. This problem was resolved in
Refs. [11, 12], where an accurate determination of the critical exponents,
along with the behavior of Binder cumulants, showed that the transition from
the low-density disordered phase to the intermediate-density ordered phase
belongs to the 2D Ising universality class for square lattices and the three-
state Potts universality class for honeycomb and triangular lattices. Later,
the I-N phase transition was analyzed by combining Monte Carlo (MC)
simulations and theoretical analysis [13, 14]. The study in Refs. [13, 14]
allowed (1) to obtain $\theta_{c}$ as a function of $k$ for square, triangular
and honeycomb lattices, being $\theta_{c}(k)\propto k^{-1}$ (this dependence
was already noted in Ref. [10]); and (2) to determine the minimum value of $k$
($k_{min}$), which allows the formation of a nematic phase on triangular
($k_{min}=7$) and honeycomb ($k_{min}=11$) lattices.
In a recent paper, Fischer and Vink [15] indicated that the transition studied
in Refs. [10–14] corresponds to a liquid-gas transition, rather than I-N. This
interpretation is consistent with the 2D-Ising critical behavior observed for
monodisperse rigid rods on square lattices [11]. This point will be discussed
in more detail in Sec. III.
In contrast to the systems studied in Refs. [10–14], many rod-like biological
polymers are formed by monomers reversibly self-assembling into chains of
arbitrary length. Consequently, these systems exhibit a broad equilibrium
distribution of filament lengths. A model of self-assembled rigid rods has
been recently considered by Tavares et al. [16]. The authors focused on a
system composed of monomers with two attractive (sticky) poles that polymerize
reversibly into polydisperse chains and, at the same time, undergo a
continuous I-N phase transition. The obtained results revealed that nematic
ordering enhances bonding. In addition, the average rod length was described
quantitatively in both phases, while the location of the ordering transition,
which was found to be continuous, was predicted semiquantitatively by the
theory.
Beyond the differences between lattice geometry and the characteristics of the
rods (self-assembled or not), one fundamental feature is preserved in all the
studies mentioned above. This is the assumption that only excluded volume
interactions between the rods are considered (except in Ref. [16], where
monomers with two attractive bonding sites polymerize into polydisperse rods).
Moreover, one often encounters phrases in the literature, such as “This theory
[Onsager’s theory] shows that repulsive interactions [excluding volume
interactions] alone can lead to long-range orientational nematic order,
disproving the notion that attractive interactions are a prerequisite” [17],
which could be ambiguous with respect to the role that attractive lateral
interactions between the rods should play in reinforcing (or not) the nematic
order.
In this context, it is of interest and of value to inquire how the existence
of lateral interactions between the rods influences the phase transition
occurring in the system. The objective of this paper is to provide a thorough
analysis in this direction. For this purpose, an exhaustive study of the phase
transition occurring in a system of attractive rigid rods deposited on square
lattices was performed. The results revealed that $(i)$ the orientational
order survives in the presence of attractive lateral interactions; $(ii)$ the
critical density shifts to higher values as the magnitude of the lateral
interactions is increased; and $(iii)$ the continuous transition becomes first
order for interaction strength $w>w_{c}$ (in absolute values).
The outline of the paper is as follows. In Sec. II we describe the lattice-gas
model and the simulation scheme. In Sec. III we present the MC results.
Finally, the general conclusions are given in Sec. IV.
## 2 Lattice-gas model and Monte Carlo simulation scheme
We address the general case of adsorbates assumed to be linear rigid particles
containing $k$ identical units ($k$-mers), with each one occupying a lattice
site. Small adsorbates would correspond to the monomer limit ($k=1$). The
distance between $k$-mer units is assumed to be equal to the lattice constant;
hence exactly $k$ sites are occupied by a $k$-mer when adsorbed (see Fig. 1).
The surface is represented as an array of $M=L\times L$ adsorptive sites in a
square lattice arrangement, where $L$ denotes the linear size of the array. In
order to describe the system of $N$ $k$-mers adsorbed on $M$ sites at a given
temperature $T$, let us introduce the occupation variable $c_{i}$ which can
take the values $c_{i}=0$ if the corresponding site is empty and $c_{i}=1$ if
the site is occupied. On the other hand, molecules adsorb or desorb as one
unit, neglecting any possible dissociation. Under these considerations, the
Hamiltonian of the system is given by
$H=w\sum_{\langle i,j\rangle}c_{i}c_{j}-N(k-1)w+\epsilon_{o}\sum_{i}c_{i}$ (1)
where $w$ is the nearest-neighbor (NN) interaction constant which is assumed
to be attractive (negative), $\langle i,j\rangle$ represents pairs of NN sites
and $\epsilon_{o}$ is the energy of adsorption of one given surface site. The
term $N(k-1)w$ is subtracted in eq. (1) since the summation over all the pairs
of NN sites overestimates the total energy by including $N(k-1)$ bonds
belonging to the $N$ adsorbed $k$-mers. Because the surface was assumed to be
homogeneous, the interaction energy between the adsorbed dimer and the atoms
of the substrate $\epsilon_{o}$ was neglected for the sake of simplicity.
Figure 1: Linear tetramers adsorbed on square lattices. Full and empty circles
represent tetramer units and empty sites, respectively.
In order to characterize the phase transition, we use the order parameter
defined in Ref. [11], which in this case can be written as
$\delta=\frac{\left|{n}_{h}-{n}_{v}\right|}{{n}_{h}+{n}_{v}}$ (2)
where $n_{h}(n_{v})$ is the number of rods aligned along the horizontal
(vertical) direction. When the system is disordered $(\theta<\theta_{c})$, all
orientations are equivalents and $\delta$ is zero. As the density is increased
above $\theta_{c}$, the $k$-mers align along one direction and $\delta$ is
different from zero. Thus, $\delta$ appears as a proper order parameter to
elucidate the phase transition.
The problem has been studied by grand canonical MC simulations using a typical
adsorption-desorption algorithm. The procedure is as follows. Once the value
of the chemical potential $\mu$ is set, a linear $k$-uple of nearest-neighbor
sites is chosen at random and an attempt is made to change its occupancy state
with probability $W={\rm min}\left\\{1,\exp\left(-\Delta
H/k_{B}T\right)\right\\}$, where $\Delta H=H_{f}-H_{i}$ is the difference
between the Hamiltonians of the final and initial states and $k_{B}$ is the
Boltzmann constant. In addition, displacement (diffusional relaxation) of
adparticles to nearest-neighbor positions, by either jumps along the $k$-mer
axis or reptation by rotation around the $k$-mer end, must be allowed in order
to reach equilibrium in a reasonable time. A MC step (MCs) is achieved when
$M$ $k$-uples of sites have been tested to change its occupancy state.
Typically, the equilibrium state can be well reproduced after discarding the
first $r^{\prime}=10^{7}$ MCs. Then, the next $r=2\times 10^{7}$ MCs are used
to compute averages.
Figure 2: Adsorption isotherms (coverage versus chemical potential) for
$k=10$, $L=100$ different $w/k_{B}T$’s as indicated. Inset: Adsorption phase
diagram of attractive $10$-mers on square lattices.
In our MC simulations, we varied the chemical potential $\mu$ and monitored
the density $\theta$ and the order parameter $\delta$, which can be calculated
as simple averages. The reduced fourth-order cumulant $U_{L}$ introduced by
Binder [18] was calculated as:
$U_{L}=1-\frac{\langle\delta^{4}\rangle}{3\langle\delta^{2}\rangle^{2}},$ (3)
where $\langle\cdots\rangle$ means the average over the MC simulation runs.
All calculations were carried out using the BACO parallel cluster (composed by
60 PCs each with a 3.0 GHz Pentium-4 processor and 90 PCs each with a 2.4 GHz
Core 2 Quad processor) located at Instituto de Física Aplicada, Universidad
Nacional de San Luis-CONICET, San Luis, Argentina.
## 3 Results
The calculations were developed for linear $10$-mers ($k=10$). With this value
of $k$ and for non-interacting rods, it is expected the existence of a nematic
phase at intermediate densities [10]. The surface was represented as an array
of adsorptive sites in a square lattice arrangement with conventional periodic
boundary conditions. The effect of finite size was investigated by examining
lattices with $L=50,100,150,200$.
In order to understand the basic phenomenology, we consider, in the first
place, the behavior of the adsorption isotherms in presence of attractive
lateral interactions between the $k$-mers.
Fig. 2 shows typical adsorption isotherms (coverage versus $\mu/k_{B}T$) for
linear $10$-mers with different values of the lateral interaction (the solid
circles represent the Langmuir case, $w/k_{B}T=0$).
The isotherms shift to lower values of chemical potential, and their slopes
increase as the ratio $w/k_{B}T$ increases (in absolute value). For
interaction strength above the critical value ($w>w_{c}$, in absolute values)
the system undergoes a first-order phase transition, which is observed in the
clear discontinuity in the adsorption isotherms111In this situation, which has
been observed experimentally in numerous systems, the only phase which one
expects is a lattice-gas phase at low coverage, separated by a two-phase
coexistence region from a “lattice-fluid” phase at higher coverage. This
condensation of a two-dimensional gas to a two-dimensional liquid is similar
to that of a lattice-gas of attractive monomers. However, the symmetry
particle-vacancy (valid for monoatomic particles) is broken for $k$-mers and
the isotherms are asymmetric with respect to $\theta=0.5$.. In the case
studied, this critical value is approximately $w_{c}/k_{B}T\approx-0.80$ (or
$k_{B}T_{c}/w\approx-1.25$). The behavior of the adsorption isotherms also
allows us to calculate the phase diagram of the adsorbed monolayer in
“temperature-coverage” coordinates. In fact, once obtained the real value of
the chemical potential (or critical chemical potential $\mu_{c}$) in the two-
phase region, the corresponding critical densities can be easily calculated.
By repeating this procedure for different temperatures ranging between $0$ and
$T_{c}$, the coexistence curve can be built [20]. A typical phase diagram,
obtained in this case for attractive $10$-mers, is shown in the inset of Fig.
2.
On the basis of the study in Fig. 2, our next objective is to obtain evidence
for the existence of nematic order in the range $-0.80\leq w/k_{B}T<0$ of
attractive interactions. For this purpose, the behavior of the order parameter
$\delta$ as a function of coverage was analyzed for $k=10$, $L=100$ and
different values of the lateral interaction. The results are shown in Fig. 3,
revealing that $(i)$ the orientational order survives in the presence of
attractive lateral interactions and $(ii)$ the critical density shifts to
higher values as the magnitude of the lateral interactions is increased.
Figure 3: Surface coverage dependence of the nematic order parameter for
$k=10$, $L=100$ different $w/k_{B}T$’s as indicated.
In order to corroborate the results obtained in the last figure, we now study
the dependence of $\theta_{c}$ on $w/k_{B}T$. In the case of the standard
theory of FSS [18, 19], when the phase transition is temperature driven, the
technique allows for various efficient routes to estimate $T_{c}$ from MC
data. One of these methods, which will be used in this case, is from the
temperature dependence of $U_{L}(T)$, which is independent of the system size
for $T=T_{c}$. In other words, $T_{c}$ is found from the intersection of the
curve $U_{L}(T)$ for different values of $L$, since $U_{L}(T_{c})=$const. In
our study, we modified the conventional FSS analysis by replacing temperature
by density [11]. Under this condition, the critical density has been estimated
from the plots of the reduced four-order cumulants $U_{L}(\theta)$ plotted
versus $\theta$ for several lattice sizes. As an example, Fig. 4 shows the
results for $w/k_{B}T=-0.125$. In this case, the value obtained was
$\theta_{c}=0.542(2)$. In the inset, the data are plotted over a wider range
of temperatures, exhibiting the typical behavior of the cumulants in the
presence of a continuous phase transition.
Figure 4: Curves of $U_{L}(\theta)$ vs $\theta$ for $k=10$, $w/k_{B}T=-0.125$
and square lattices of different sizes. From their intersections one obtained
$\theta_{c}$. In the inset, the data are plotted over a wider range of
densities. Figure 5: Temperature-coverage phase diagram corresponding to
attractive $k$-mers with $k=10$. The inset in the upper-left (lower-right)
corner shows a typical configuration in the nematic (isotropic) phase.
The procedure of Fig. 4 was repeated for $-0.80\leq w/k_{B}T<0$, showing that
the values of $\theta_{c}$ increase linearly with the magnitude of the lateral
couplings (see solid squares in Fig. 5). The critical line (dotted line in the
figure) was obtained from the linear fit of the numerical data. As it is
possible to observe, the range of coverage at which the transition occurs
diminishes as $w/k_{B}T$ is increased (in absolute value). This finding
indicates that the presence of attractive lateral interactions between the
rods does not favor the formation of nematic order in the adlayer. The
phenomenon can be understood from the behavior of the second virial
coefficient, which will initially decrease on introducing attractive $w$. This
decrease implies that the isotherms shift to lower values of chemical
potential, and consequently, the critical point shifts to higher densities.
We did not assume any particular universality class for the transitions
analyzed here in order to calculate their critical densities, since the
analysis relied on the order parameter cumulant’s properties. However, the
fixed value of the cumulants, $U^{*}=0.617(9)$, is consistent with the
extremely precise transfer matrix calculation of $U^{*}=0.6106901(5)$ [21] for
the 2D Ising model. This finding may be taken as an indication that the phase
transition belongs to the 2D Ising universality class.
With respect to the behavior of the system for $w/k_{B}T<-0.80$, the adsorbed
layer “jumps” from a low-coverage phase to a high-coverage phase. This effect,
which has been discussed in Fig. 2, is represented in Fig. 5 by the dashed
coexistence line. The low-coverage phase is an isotropic state, similar to
that observed for $w/k_{B}T>-0.80$ and low density (see inset in the lower-
right corner of Fig. 5). On the other hand, the high-coverage phase is also an
isotropic state, but characterized by the presence of local orientational
order (domains of parallel $k$-mers). A typical configuration in this regime
is shown in Fig. 6).
Figure 6: Typical configuration of the adlayer in the high-coverage phase and
$w/k_{B}T<-0.80$.
Finally, it is worth pointing out that: $(1)$ the behavior of the order
parameter in Fig. 3 clearly indicates that the transition from the low-density
disordered phase to the intermediate-density ordered phase is an isotropic to
nematic phase transition (when all the words have the usual meaning). In this
case, the transition under study belongs to the 2D Ising universality class.
It can also be thought of as an unmixing or liquid-gas transition [15]. For
this reason we have called gas and liquid to the phases reported in Fig. 5;
and $(2)$ even though it has not been rigorously proved yet, a second phase
transition for non-interacting rods at high densities has been theoretically
predicted [10] and numerically confirmed [13]. This result has not been
confirmed for the case of attractive rods. An exhaustive study on this subject
will be the object of future work.
## 4 Conclusions
We have addressed the critical properties of attractive rigid rods on square
lattices with two allowed orientations, and shown the dependence of the
critical density on the magnitude of the lateral interactions $w/k_{B}T$. The
results were obtained by using MC simulations and FSS theory.
Several conclusions can be drawn from the present work. On the one hand, we
found that even though the presence of attractive lateral interactions between
the rods does not favor the formation of nematic order in the adlayer, the
orientational order survives in a range that goes from $w/k_{B}T=0$ up to
$w_{c}/k_{B}T\approx-0.80$ ($w_{c}/k_{B}T$ represents the critical value at
which occurs a typical transition of condensation in the adlayer). In this
region of $w/k_{B}T$, the critical density increases linearly with the
magnitude of the lateral couplings. On the other hand, the evaluation of the
fixed point value of the cumulants $U^{*}=0.617(9)$ indicates that, as in the
case of non-interacting rods, the observed phase transition belongs to the
universality class of the two-dimensional Ising model.
With respect to the behavior of the system for $w/k_{B}T<-0.80$, the
continuous transition becomes first order. Thus, the adsorbed layer jumps from
a low-coverage phase, similar to that observed for $w/k_{B}T>-0.80$ and low
density, to an isotropic phase at high coverage, characterized by the presence
of local orientational order (domains of parallel $k$-mers)
Future efforts will be directed to $(1)$ extend the study to repulsive lateral
interaction between the $k$-mers; $(2)$ obtain the whole phase diagram in the
space (temperature-coverage-rod’s size); $(3)$ develop an exhaustive study on
critical exponents and universality and $(4)$ characterize the second phase
transition from a nematic order to a non-nematic state occurring at high
density.
###### Acknowledgements.
This work was supported in part by CONICET (Argentina) under project number
PIP 112-200801-01332; Universidad Nacional de San Luis (Argentina) under
project 322000 and the National Agency of Scientific and Technological
Promotion (Argentina) under project 33328 PICT 2005.
## References
* [1] L Onsager, The effects of shape on the interaction of colloidal particles, Ann. N. Y. Acad. Sci. 51, 627 (1949).
* [2] P J Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10, 51 (1942); P J Flory, Principles of Polymers Chemistry, Cornell University Press, Ithaca, NY (1953).
* [3] M L Huggins, Some properties of solutions of long-chain compounds, J. Phys. Chem. 46, 151 (1942); M L Huggins, Thermodynamic properties of solutions of long-chain compounds, Ann. N. Y. Acad. Sci. 43, 1 (1942); M L Huggins, Theory of solutions of high polymers, J. Am. Chem. Soc. 64, 1712 (1942).
* [4] J P Straley, Liquid crystals in two dimensions, Phys. Rev. A 4, 675 (1971).
* [5] J Vieillard-Baron, Phase transitions of the classical hard-ellipse system, J. Chem. Phys. 56, 4729 (1972).
* [6] D Frenkel, R Eppenga, Evidence for algebraic orientational order in a two-dimensional hard-core nematic, Phys. Rev. A 31, 1776 (1985).
* [7] K J Strandburg, Two-dimensional melting, Rev. Mod. Phys. 60, 161 (1988).
* [8] A J Phares, F J Wunderlich, Thermodynamics and molecular freedom of dimers on plane triangular lattices, J. Math. Phys. 27, 1099 (1986).
* [9] A J Phares, F J Wunderlich, J D Curley, D W Grumbine Jr, Structural ordering of interacting dimers on a square lattice, J. Phys. A: Math. Gen. 26, 6847 (1993).
* [10] A Ghosh, D Dhar, On the orientational ordering of long rods on a lattice, Eur. Phys. Lett. 78, 20003 (2007).
* [11] D A Matoz-Fernandez, D H Linares, A J Ramirez-Pastor, Determination of the critical exponents for the isotropic-nematic phase transition in a system of long rods on two-dimensional lattices: Universality of the transition, Europhys. Lett. 82, 50007 (2008).
* [12] D A Matoz-Fernandez, D H Linares, A J Ramirez-Pastor, Critical behavior of long linear $k$-mers on honeycomb lattices, Physica A 387, 6513 (2008).
* [13] D H Linares, F Romá, A J Ramirez-Pastor, Entropy-driven phase transition in a system of long rods on a square lattice, J. Stat. Mech. P03013 (2008).
* [14] D A Matoz-Fernandez, D H Linares, A J Ramirez-Pastor, Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations, J. Chem. Phys. 128, 214902 (2008).
* [15] T Fischer, R L C Vink, Restricted orientation “liquid crystal” in two dimensions: Isotropic-nematic transition or liquid-gas one(?), Europhys. Lett. 85, 56003 (2009).
* [16] J M Tavares, B Holder, M M Telo da Gama, Structure and phase diagram of self-assembled rigid rods: Equilibrium polydispersity and nematic ordering in two dimensions, Phys. Rev. E 79, 021505 (2009).
* [17] H H Wensink, Columnar versus smectic order in systems of charged colloidal rods, J. Chem. Phys. 126, 194901 (2007).
* [18] K Binder, Applications of the Monte Carlo Method in Statistical Physics. Topics in current Physics, Springer, Berlin (1984).
* [19] V Privman, Finite Size Scaling and Numerical Simulation of Statistical Systems, World Scientific, Singapore (1990).
* [20] T L Hill, An Introduction to Statistical Thermodynamics, Addison Wesley Publishing Company, Reading, MA (1960).
* [21] G Kamieniarz, H W J Blöte, Universal ratio of magnetization moments in two-dimensional Ising models, J. Phys. A: Math. Gen. 26, 201 (1993).
|
arxiv-papers
| 2009-11-17T12:59:32 |
2024-09-04T02:49:06.515243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Longone, D. H. Linares, A. J. Ramirez-Pastor",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/0911.3295"
}
|
0911.3297
|
# Nucleon spin structure and pQCD frontier on the move
Roman S. Pasechnik roman.pasechnik@fysast.uu.se High Energy Physics,
Department of Physics and Astronomy, Uppsala University Box 516, SE-75120
Uppsala, Sweden Dmitry V. Shirkov Oleg V. Teryaev Bogoliubov Lab, JINR,
Dubna 141980, Russia Olga P. Solovtsova Vyacheslav L. Khandramai Gomel
State Technical University, Gomel 246746, Belarus
###### Abstract
The interplay between higher orders of the perturbative QCD (pQCD) expansion
and higher-twist contributions in the analysis of recent Jefferson Lab data on
the lowest moment of the spin-dependent proton $\Gamma_{1}^{p}(Q^{2})$ at
$0.05<Q^{2}<3\,{\rm GeV}^{2}$ is studied. We demonstrate that the values of
the higher-twist coefficients $\mu^{p,n}_{2k}\,$ extracted from the data by
using the singularity-free analytic perturbation theory provide a better
convergence of the higher-twist series than with the standard perturbative
QCD. From the high-precision proton data, we extract the value of the singlet
axial charge $a_{0}(1\,{\rm GeV}^{2})=0.33\pm 0.05$. We observe a slow $Q^{2}$
dependence of fitted values of the twist coefficient $\mu_{4}$ and $a_{0}$
when going to lower energy scales, which can be explained by the
renormalization group evolution of $\mu_{4}(Q^{2})$ and $a_{0}(Q^{2})$. As the
main result, a good quantitative description of all the Jefferson Lab data
sets down to $Q\simeq 350$ MeV is achieved.
###### pacs:
11.10.Hi, 11.55.Hx, 11.55.Fv, 12.38.Bx, 12.38.Cy
††preprint:
## I Introduction
The spin structure of the nucleon remains the essential problem of
nonperturbative QCD and hadronic physics. One of its most significant
manifestations is the so-called spin crisis or spin puzzle related to the
surprisingly small fraction of proton polarization carried by quarks
Anselmino:1994gn ; Leader08 . This problem attracted attention to the
peculiarities of the underlying QCD description of the nucleon spin, in
particular, to the role of the gluonic anomaly (see Anselmino:1994gn ;
Efremov:1989sn and references therein). The natural physical interpretation
of these effects was the gluon (circular) polarization, while the experimental
indications of its smallness may also point to a possible manifestation of the
anomaly via the strangeness polarization OT09 . The key point is its
consideration as a kind of heavy-quarks polarization Polyakov:1998rb due to
the multiscale OT09 picture of the nucleon exploring the fact that strange
quark mass is much (as the squared ratios matter) smaller than the nucleon one
and, in turn, larger than higher-twist parameters.
Higher-twist parameters (known also as the color polarizabilities) are
important ingredients of the nucleon spin structure. Their extraction from
experimental studies is relatively complicated as they are most pronounced at
low momentum transfer $Q$. Although in this region very accurate Jefferson Lab
(JLab) data are now available, higher-twist contributions are shadowed by
Landau singularities of QCD coupling. As was shown in Ref. Bjour , this
problem may be solved by the use of singularity-free couplings which allowed a
quite accurate extraction of higher twist (HT) and a fairly good description
of data down to rather low $Q$. The object of investigation in Bjour was the
difference of the lowest moments $\Gamma^{p,n}_{1}$ of proton and neutron
structure functions $g_{1}$, which corresponds to the renowned Bjorken sum
rule (BSR) Bj66 . At finite $Q^{2}$ the moments $\Gamma^{p,n}_{1}$ are
modified by higher order radiative corrections and higher-twist power
corrections, as dictated by the operator product expansion (OPE). Such
generalized ($Q^{2}$-dependent) BSR became a convenient and renowned target
ground for testing different possibilities of combining both the perturbative
and nonperturbative QCD contributions in the low-energy domain (see, for
example, Refs. Kodaira79 ; SofTer ).
The global higher twist analysis of the data on the spin-dependent proton
structure function $g^{p}_{1}$ at relatively large $1<Q^{2}<30\,\text{{\rm
GeV}}^{2}$, was performed in Ref. Osipenko:2004xg . While the $1/Q^{2}$ term
in the OPE works at relatively high scales $Q^{2}\gtrsim 1\,\text{{\rm
GeV}}^{2}$, higher-twist power corrections $1/Q^{4},\,1/Q^{6},$ etc., start to
play a significant role at lower scales, where the influence of the ghost
singularities in the coefficient functions within the standard perturbation
theory (PT) becomes more noticeable. It affects the results of extraction of
the higher twists from the precise experimental data leading to unstable OPE
series and huge error bars Bjour . It seems natural that the weakening or
elimination of the unphysical singularities of the QCD coupling would allow
shifting the perturbative QCD (pQCD) frontier to a lower energy scale and
getting more exact information about the nonperturbative part of the process
described by the higher-twist series.
As was shown in Ref. Bjour , the situation becomes better if one uses for a
running coupling a more precise iterative solution of the renormalisation
group (RG) equation in the form of the so-called denominator representation
denom06 instead of the Particle Data Group loop $1/L$ expansion pdg08 ,
especially at the two-loop level. In this investigation, to avoid completely
the unphysical singularities at $Q=\Lambda_{QCD}\sim 400\,\text{{\rm MeV}}$ we
deal with the ghost-free analytic perturbation theory (APT) apt96-7 (for a
review on APT concepts and algorithms, see also Ref. Sh-revs ), which recently
proved to be an intriguing candidate for a quantitative description of light
quarkonia spectra within the Bethe-Salpeter approach BSAPT , and the so-called
glueball-freezing model proposed recently by Yu. A. Simonov in Ref. Simonov
(below, SGF model) to avoid the renormalon renormalon ambiguity in QCD. Other
versions of frozen $\alpha_{s}$ models were developed earlier in Ref. IR-freez
. As it will seen below that APT and SGF approaches predict very close
couplings at $Q\gtrsim\Lambda_{QCD}$, whereas they have different infrared-
stable points at $Q=0$. Consequently, as it was shown in Ref. Bjour , these
models lead to very close perturbative parts of the Bjorken sum
$\Gamma_{1,pert}^{p-n}$. The higher-twist contributions turned out to be very
close, too. Here, we would like to discuss this point in more detail.
In the current paper we study the interplay between higher orders of the pQCD
expansion and higher-twist contributions using the recent JLab data on the
lowest moments of the spin-dependent proton and neutron structure functions
$\Gamma_{1}^{p,n}(Q^{2})$ and $\Gamma_{1}^{p-n}(Q^{2})$ in the range
$0.05<Q^{2}<3\,{\rm GeV}^{2}$ Deur:2008rf . Thus, we extend and generalize the
analysis started in Ref. Bjour by considering also the singlet channel
involving the $\Gamma_{1}^{p,n}(Q^{2})$ for the proton (providing the most
accurate data) and the neutron structure functions separately. This allows, in
particular, determining the singlet axial charge $a_{0}$ coming into both
$\Gamma_{1}^{p,n}(Q^{2})$ moments, which in the quark-parton model is
identified with the total spin carried by quarks in the proton. For this
purpose, we perform the global analysis of the JLab precise low-energy data on
$\Gamma_{1}^{p}(Q^{2})$ JLab08data using the advantages of the APT and SGF
model, and extract the singlet axial charge $a_{0}$, as well as the
coefficient $\mu^{p,n}_{4}$ of the $1/Q^{2}$ subleading twist-4 term, which
contains information on quark-gluon correlations in nucleons.
The paper is organized as follows. In Sec. 2, the lowest moments analysis for
the polarized structure functions $g^{p,n}_{1}$ in the framework of the
conventional PT approach is performed. In Sec. 3, we dwell briefly on the APT,
its ideas and the results of its application to $\Gamma^{p,n}_{1}(Q^{2})$. In
Sec. 4, we apply the formalism to the analysis of the low-energy data on the
first moments $\Gamma^{p,n}_{1}(Q^{2})$ and compare the results with the
results of other researchers concerning the singlet axial constant $a_{0}$ and
gluon polarization $\Delta g$ at low $Q^{2}\lesssim 1\,\text{{\rm GeV}}^{2}$.
Section 5 contains discussion and some concluding remarks.
## II Spin sum rules in conventional PT
### II.1 First moments of spin structure functions $g_{1}^{p,n}$
The lowest moments of spin-dependent proton and neutron structure functions
$g^{p,n}_{1}$ are defined as follows:
$\displaystyle\Gamma_{1}^{p,n}(Q^{2})=\int^{1}_{0}dx\,g^{p,n}_{1}(x,Q^{2})\,,$
(1)
with $x=Q^{2}/2M\nu$, the energy transfer $\nu$, and the nucleon mass $M.$ The
upper limit includes the proton/neutron elastic contribution at $x=1$. This
contribution becomes essential if the OPE is used to study the evolution of
the integral in the moderate and low momentum transfer region $Q^{2}\lesssim
1\,\text{{\rm GeV}}^{2}$ Ji . It is of special interest to analyze data with
the elastic contribution excluded, since the low-$Q^{2}$ behavior of
“inelastic” contributions to their nonsinglet combination
$\Gamma^{p-n}_{1}(Q^{2})$, i.e. BSR, is constrained by the Gerasimov-Drell-
Hearn (GDH) sum rule GDH , and one may investigate its continuation to a low
scale SofTer . So below we study inelastic contributions
$\Gamma^{p,n}_{inel,1}(Q^{2})$ using the corresponding low-energy JLab data
JLab08data . Note that the influence of the “elastic” contribution is
noticeable starting from the higher-twist $\sim\mu_{6}$ term which is natural
due to a decrease of the elastic contribution with growing $Q^{2}$ Bjour .
At large $Q^{2}$ the moments $\Gamma_{1}^{p,n}(Q^{2})$ are given by the OPE
series in powers of $1/Q^{2}$ with the expansion coefficients related to
nucleon matrix elements of operators of a definite twist (defined as the
dimension minus the spin of the operator), and coefficient functions in the
form of pQCD series in $\alpha_{s}^{n}$ (see, e.g., Ref. kataev ). In the
limit $Q^{2}\gg M^{2}$ the moments are dominated by the leading twist
contribution, $\mu_{2}^{p,n}(Q^{2})$, which is given in terms of matrix
elements of the twist-2 axial vector current,
$\bar{\psi}\gamma^{\mu}\gamma_{5}\psi$. This can be decomposed into flavor
singlet and nonsinglet contributions. The total expression for the
perturbative part of $\Gamma^{p,n}_{1}(Q^{2})$ including the HT contributions
reads
$\displaystyle\Gamma^{p,n}_{1}(Q^{2})=\frac{1}{12}\left[\biggl{(}\pm
a_{3}+\frac{1}{3}a_{8}\biggr{)}E_{NS}(Q^{2})+\frac{4}{3}a^{inv}_{0}\,E_{S}(Q^{2})\right]+\sum_{i=2}^{\infty}\frac{\mu^{p,n}_{2i}(Q^{2})}{Q^{2i-2}},$
(2)
where $E_{S}$ and $E_{NS}$ are the singlet and nonsinglet Wilson coefficients,
respectively, calculated as series in powers of $\alpha_{s}$ Larin:1997qq .
These coefficient functions for $n_{f}=3$ active flavors in the $\overline{\rm
MS}$ scheme are
$\displaystyle E_{NS}(Q^{2})$ $\displaystyle=$ $\displaystyle
1-\frac{\alpha_{s}}{\pi}-3.558\left(\frac{\alpha_{s}}{\pi}\right)^{2}-20.215\left(\frac{\alpha_{s}}{\pi}\right)^{3}-O(\alpha_{s}^{4})\,,$
(3) $\displaystyle E_{S}(Q^{2})$ $\displaystyle=$ $\displaystyle
1-\frac{\alpha_{s}}{\pi}-1.096\left(\frac{\alpha_{s}}{\pi}\right)^{2}-O(\alpha_{s}^{3})\,.$
(4)
The triplet and octet axial charges $a_{3}\equiv g_{A}=1.267\pm 0.004$ pdg08
and $a_{8}=0.585\pm 0.025$ Goto:1999by , respectively, are extracted from weak
decay matrix elements and are known from $\beta$-decay measurements. As for
the singlet axial charge $a_{0}$, it is convenient to work with its RG
invariant definition in the $\overline{\rm MS}$ scheme
$a_{0}^{inv}=a_{0}(Q^{2}=\infty)$, in which all the $Q^{2}$ dependence is
factorized into the definition of the Wilson coefficient $E_{S}(Q^{2})$.
In contrast to the proton and neutron spin sum rules (SSRs), the singlet and
octet contributions are canceled out, giving rise to more fundamental BSR
$\displaystyle\Gamma^{p-n}_{1}(Q^{2})=\frac{g_{A}}{6}E_{NS}(Q^{2})+\sum_{i=2}^{\infty}\frac{\mu^{p-n}_{2i}(Q^{2})}{Q^{2i-2}},$
(5)
which is analyzed here along with the proton SSR in more detail than in Ref.
Bjour . The first nonleading twist term Shuryak can be expressed chen06
$\displaystyle\mu_{4}^{p-n}\approx\frac{4\,M^{2}}{9}f_{2}^{p-n},$
in terms of the color polarizability $f_{2}$.
The RG $Q^{2}$ evolution of the axial singlet charge $a_{0}(Q^{2})$ and
nonsinglet higher-twist $\mu^{p-n}_{4}(Q^{2})$ is Shuryak
$\displaystyle a_{0}(Q^{2})$ $\displaystyle=$ $\displaystyle
a_{0}(Q_{0}^{2})\exp\left\\{\frac{\gamma_{2}}{(4\pi)^{2}\beta_{0}}[\alpha_{s}(Q^{2})-\alpha_{s}(Q_{0}^{2})]\right\\},\quad\gamma_{2}=16n_{f},$
(6) $\displaystyle\mu_{4}^{p-n}(Q^{2})$ $\displaystyle=$
$\displaystyle\mu_{4}^{p-n}(Q_{0}^{2})\left[\frac{\alpha_{s}(Q^{2})}{\alpha_{s}(Q_{0}^{2})}\right]^{\gamma_{0}/8\pi\beta_{0}},\quad\beta_{0}=\frac{33-2n_{f}}{12\pi},\quad\gamma_{0}=\frac{16}{3}C_{F}\,.$
(7)
In the NLO we may write
$\displaystyle a_{0}(Q^{2})$ $\displaystyle\simeq$ $\displaystyle
a_{0}(Q_{0}^{2})\left[1+\Delta_{1}(Q^{2})+{\cal O}(\alpha_{s}^{2})\right],$
(8) $\displaystyle\Delta_{1}(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{\gamma_{2}}{(4\pi)^{2}\beta_{0}}[\alpha_{s}(Q^{2})-\alpha_{s}(Q_{0}^{2})],\quad\frac{\gamma_{2}}{(4\pi)^{2}\beta_{0}}=\frac{4}{3\pi}.$
As a first step of our analysis, in Eq. (2) we will neglect the weak
dependence of $\mu^{p,n}_{2i}$ on $\log Q^{2}$. Note that the evolution of the
higher-twist terms $\mu_{6,8,\,...}$ in Eq. (2) is still unknown. As a next
step we discuss the possible influence of the $\mu_{4}(Q^{2})$ evolution on
our results. The $Q^{2}$ evolution of the proton higher-twist term
$\mu^{p}_{4}(Q^{2})$ is assumed to be the same as the evolution of the
nonsinglet twist $\mu^{p-n}_{4}(Q^{2})$. This may be justified by the relative
smallness of the singlet higher-twist term.
Table 1: Current NLO fit results for the axial singlet charge $a_{0}$. Reference | LSS LSS06 | DSSV DSSV | AAC AAC08 | HERMES HERMES06 | COMPASS COMPASS06
---|---|---|---|---|---
$Q_{0}^{2},\,\text{{\rm GeV}}^{2}$ | 1.0 | 10.0 | 4.0 | 5.0 | 3.0
$a_{0}$ | $0.24\pm 0.07$ | 0.24 | $0.25\pm 0.05$ | $0.32\pm 0.04$ | $0.35\pm 0.06$
Let us discuss current results for the nucleon spin structure and higher
twists. In Table 1, we list the fit results for the axial singlet charge
$a_{0}$ from the literature including all global NLO PT analyses and the
recent results obtained directly from deuteron data on $\Gamma_{1}^{d}$ by
COMPASS COMPASS06 and HERMES HERMES06 . The global fit results for $a_{0}$
are somewhat lower than that from the deuteron data. It was mentioned in the
most recent review Leader08 that the reason for such a discrepancy is not
completely understood. Further, we analyze this issue in more detail.
Table 2: Current NLO fit results for the highest-twist term $\mu_{4}/M^{2}$. The uncertainties are statistical only. Target | Proton proton | Neutron neutron | p – n Bj04-tw | p – n Bj08-tw | p – n Bjour
---|---|---|---|---|---
$Q^{2},\,\text{{\rm GeV}}^{2}$ | 0.6 – 10.0 | 0.5 – 10.0 | 0.5 – 10.0 | 0.66 – 10.0 | 0.12 – 3.0
$\mu_{4}/M^{2}$ | $-0.065\pm 0.012$ | $0.019\pm 0.002$ | $-0.06\pm 0.02$ | $-0.04\pm 0.01$ | $-0.048\pm 0.002$
A detailed higher-twist analysis based on the combined SLAC and JLab data [on
proton, neutron $\Gamma_{1}^{p,n}(Q^{2})$ JLab-old-data and nonsinglet
$\Gamma_{1}^{p-n}(Q^{2})$ moments Bj08-tw ] was performed in Refs. proton ;
neutron ; Bj04-tw ; Bj08-tw . In Table 2, we show the current results for the
twist-4 coefficient $\mu_{4}/M^{2}$ at $Q^{2}=1\,\text{{\rm GeV}}^{2}$
extracted from $\Gamma_{1}^{p,n}$ data. As we have seen from our previous
analysis Bjour , a satisfactory description of the low-energy JLab data on the
Bjorken sum rule down to $Q_{min}\sim\Lambda_{QCD}\simeq 350\,\text{{\rm
MeV}}\,$ can be achieved by using APT and taking into account only three
higher-twist terms $\mu^{p-n}_{4,6,8}$. Including only the twist-4 term
$\mu^{p-n}_{4}/M^{2}$, this method allowed us to get its value with noticeably
higher accuracy than in the standard PT approach, shifting the applicability
of the pQCD expansion down to $Q^{2}_{min}=0.47\,\text{{\rm GeV}}^{2}$. The
higher-twist analysis of the most recent precise JLab experimental data on the
proton spin sum rule JLab08data has not been carried out yet in the
literature. This gives us a reasonable motivation for a detailed data analysis
and studying the higher-twist effects at low-energy scale both in the standard
PT, APT and “infrared-frozen” $\alpha_{s}$ approaches.
### II.2 The running coupling
The infrared behavior of the strong coupling is crucial for the extraction of
the nonperturbative information from the low-energy data. Within the pQCD, the
$\alpha_{s}$ coupling can be found by a solution of the RG equation
$\displaystyle\frac{d\alpha_{s}}{dL}=-\beta_{0}\alpha_{s}^{2}(1+b_{1}\alpha_{s}+b_{2}\alpha_{s}^{2}+\,...)\,,$
where $L=\ln(Q^{2}/\Lambda^{2})$ and $b_{k}=\beta_{k}/\beta_{0}$. The standard
PT running coupling $\alpha_{s}$ is usually taken in the form [see, for
example, Eq. (6) in the recent review bethke09 or Eq. (9.5) in the PDG review
pdg08 ] expanded in a series over $\ln L/L\,$, i.e.
$\displaystyle\alpha_{s}^{(3)}(L)=\frac{1}{\beta_{0}L}-\frac{b_{1}}{\beta_{0}^{2}}\frac{\ln
L}{L^{2}}+\frac{1}{\beta_{0}^{3}L^{3}}\left[b_{1}^{2}(\ln^{2}L-\ln
L-1)+b_{2}\right].$ (9)
Here, the $1/L^{2}$ term corresponds to the 2-loop contribution and the
$1/L^{3}$ term is usually referred to as “the 3-loop one.” Actually, the
pieces of genuine 2-loop contribution proportional to $b_{1}\,$ are entangled
with the higher-loop ones. This defect is absent in the more compact
denominator representation denom06 , which at 2, 3-loop levels has the
following forms:
$\displaystyle\frac{1}{\alpha_{s}^{(2),D}(L)}=\beta_{0}\,L+b_{1}\ln\left(L+\frac{b_{1}}{\beta_{0}}\right),~{}~{}\frac{1}{\alpha_{s}^{(3),D}(L)}=\beta_{0}\,L+b_{1}\ln\left(L+\frac{b_{1}}{\beta_{0}}\,\ln
L\right)+\frac{b_{1}^{2}-b_{2}}{\beta_{0}\,L},$ (10)
which, being generic for the PDG expression (9), are closer to the
corresponding iterative RG solutions and, hence, more precise. Advantages of
formulas (10) in the higher-twist analysis of the Bjorken sum rule were
demonstrated in our previous work Bjour .
Figure 1: The NLO running coupling $\alpha_{s}$ in different approaches.
In Fig. 1, we compare the behavior of the two-loop running coupling
$\alpha_{s}$ at low $Q^{2}$ scales in different approaches. The long-dashed
line is the exact two-loop PT result, the dotted line is the denominator
representation (10) (referred to as “Denom” below), and the short-dashed line
is the PDG expression (9). As one can see from this figure, the NLO Denom
coupling is much closer to the corresponding numerical RG solution than the
$1/L$-expanded PDG expression.
In Fig. 1, we also show two models of the infrared-stable running coupling.
One of them is the Simonov “glueball-freezing model” (SGF-model) Simonov ,
represented by the dash-dotted line, with the $1/L$-type loop expansion for
the “infrared-frozen” coupling similar to PDG
$\displaystyle\phantom{AAAAAA}\alpha_{B}(Q^{2})=\alpha_{s}^{(2)}(\bar{L})\,,\quad\bar{L}=\ln\left(\frac{Q^{2}+M_{0}^{2}}{\Lambda^{2}}\right),$
(11)
where the two-loop $\alpha_{s}^{(2)}$ is taken in the form of the first two
terms in Eq. (9) with logarithm modified by a “glueball mass” $M_{0}\sim
1\,\text{{\rm GeV}}$. Note, the usual PT expansion in powers of $\alpha_{B}$
in the coefficient functions (3) and (4) is adopted. The solid line
corresponds to the second model of the infrared-stable coupling – the APT
running coupling, which will be discussed in detail below in the next section.
As one can see from Fig. 1, the SGF and APT couplings are very similar in the
low-energy domain $\Lambda_{QCD}<Q\lesssim 1$ GeV though their infrared limits
are different. Also, a comparison of APT and PT couplings over a wide range of
$Q^{2}$, $1\leq Q^{2}\leq 10^{4}$ GeV2, can be found in Ref. APT-GLS .
Note, we extract values of $\Lambda_{QCD}$ corresponding to different models
of the running coupling, by evolution from the world experimental data on
$\alpha_{s}(M_{Z}^{2})$ as a normalization point in each particular order of
PT.
### II.3 Stability and duality
In the following, when calculating the observables in any particular order of
perturbation theory, we will employ the prescription for the coefficient
functions in the infrared region, where the order of the power $\alpha_{s}$
series in the coefficient functions is matched with the loop order in
$\alpha_{s}$ itself. For example, for the nonsinglet coefficient function in
the Bjorken sum rule, we write consequently (for details, see Ref. HERMES06 )
$\displaystyle E^{LO}_{NS}=1,\quad
E^{NLO}_{NS}=1-\frac{\alpha^{NLO}_{s}}{\pi},\quad
E^{N^{2}LO}_{NS}=1-\frac{\alpha^{N^{2}LO}_{s}}{\pi}-3.558\Big{(}\frac{\alpha_{s}^{N^{2}LO}}{\pi}\Big{)}^{2},\,\ldots~{}~{}~{}$
(12)
We see that the leading singular behavior in the coefficient function
$\sim\ln^{n}L/L^{m}$ when $L\to 0$ comes from the highest power of
$\alpha_{s}$. So in the infrared domain the influence of singularities gets
stronger in higher orders of perturbation theory that may affect the data
analysis below $1\,\text{{\rm GeV}}^{2}$. This fact explains our observation
made in Ref. Bjour , where we showed that the higher PT orders yield a worse
description of the BSR data in comparison with the leading order. We observe a
similar picture for the precise JLab data on $\Gamma_{1}^{p}(Q^{2})$
JLab08data probably implying the asymptotic character of the series in powers
of $\alpha_{s}$ (see Fig. 2).
Figure 2: Best fits of JLab and SLAC data on BSR $\Gamma_{1}^{p-n}(Q^{2})$ (left panel) and proton SSR $\Gamma_{1}^{p}(Q^{2})$ (right panel) calculated at various loop orders. Table 3: Dependence of the best fit results of BSR $\Gamma^{p-n}_{1}(Q^{2})$ and proton SSR $\Gamma^{p}_{1}(Q^{2})$ data (elastic contribution excluded) on the order of perturbation theory [NLO and NNLO Denom couplings (10) are used]. The corresponding fit curves are shown in Fig. 2. The minimal borders of fitting domains in $Q^{2}$ are settled from the ad hoc restriction $\chi^{2}\leqslant 1$ and monotonous behavior of the resulting fitted curves. Target | Method | $\;Q^{2}_{min},\,\text{{\rm GeV}}^{2}\;$ | $\quad a_{0}^{inv}\quad$ | $\quad\mu_{4}/M^{2}\quad$ | $\quad\mu_{6}/M^{4}\quad$ | $\quad\mu_{8}/M^{6}\quad$
---|---|---|---|---|---|---
| LO | 0.121 | $0.29(2)$ | $-0.089(3)$ | $0.016(1)$ | $-0.0010(1)$
proton | NLO | 0.17 | $0.38(2)$ | $-0.070(5)$ | $0.010(2)$ | $0.0004(3)$
| NNLO | 0.38 | $0.37(5)$ | $-0.034(19)$ | $-0.025(20)$ | $0.017(6)$
| LO | 0.17 | – | $-0.126(5)$ | $0.037(3)$ | $-0.004(1)$
p – n | NLO | 0.17 | – | $-0.076(5)$ | $0.019(3)$ | $-0.001(1)$
| NNLO | 0.38 | – | $-0.026(11)$ | $-0.035(15)$ | $0.026(5)$
The corresponding fit results for HT terms, extracted in different orders of
PT, are listed in Table 3. We see that with raising the loop order the values
of $\mu^{p}_{4,8}$ terms increase, whereas $\mu^{p}_{6}$ decreases, yielding a
“swap” between the higher orders of PT and HT terms. Such a “swap” between PT
and HT terms (decreasing HT term by including more terms of PT and using
resummation of PT series) was previously observed in Refs. Kotikov:1992ht ;
Parente:1994bf . A similar situation holds when fitting
$\Gamma^{p}_{1}(Q^{2})$ data over the fixed range $0.8\,\text{{\rm
GeV}}<Q<2.0\,\text{{\rm GeV}}\,$, where it is sufficient to take into account
only one twist term $\mu_{4}$.
Figure 3: Best fits of JLab and SLAC data on BSR $\Gamma_{1}^{p-n}(Q^{2})$
(left panel) and proton SSR $\Gamma_{1}^{p}(Q^{2})$ (right panel) calculated
in various loop orders with fixed $Q_{min}=0.8$ GeV.
In Fig. 3, we show fits of BSR data (left panel) and proton SSR data (right
panel) in different orders of perturbation theory taking only into account the
$\mu_{4}$ term. One can see there that the higher-loop contributions are
effectively “absorbed” into the value of $\mu_{4}$ which decreases in
magnitude with increasing loop order while all the fitting curves are very
close to each other. This observation reveals a kind of “duality” between the
perturbative $\alpha_{s}$ series and nonperturbative $1/Q^{2}$ series. A
similar phenomenon was observed before for the structure function $F_{3}$ in
Refs. Kataev:1997nc ; SidKat .
This also means the appearance of a new aspect of quark hadron duality, the
latter being the necessary ingredient of all the QCD applications in the low-
energy domain. Usually, it is assumed Shifman:1978bx that the perturbative
effects are less important there than the power ones due to a nontrivial
structure in the QCD vacuum.
In our case, the PT corrections essentially enter into the game, so that the
pQCD higher order terms are relevant in the domain where the concepts of
traditional hadronic physics are usually applied.
Table 4: Dependence of the best $(3+1)$-parametric fit results of
$\Gamma^{p}_{1}(Q^{2})$ data (elastic contribution excluded) on
$\Lambda_{n_{f}=3}$ in NLO Denom PT.
$\;\Lambda_{QCD},\,\text{{\rm MeV}}\;$ | $\quad Q^{2}_{min},\,\text{{\rm GeV}}^{2}\quad$ | $\quad a_{0}^{inv}\quad$ | $\quad\mu_{4}/M^{2}\quad$ | $\quad\mu_{6}/M^{4}\quad$ | $\quad\mu_{8}/M^{6}\quad$
---|---|---|---|---|---
300 | 0.14 | $0.40(2)$ | $-0.077(3)$ | $0.014(1)$ | $-0.0005(2)$
400 | 0.24 | $0.39(3)$ | $-0.064(8)$ | $0.006(5)$ | $0.002(1)$
500 | 0.35 | $0.34(4)$ | $-0.028(13)$ | $-0.033(11)$ | $0.019(3)$
The interplay between partonic and hadronic degrees of freedom in the
description of GDH SR and BSR may also be observed in the surprising
similarity between the results of “resonance” Burkert:1992tg and “parton”
SofTer approaches.
One may ask to what extent these results are affected by the unphysical
singularities when approaching $Q\sim\Lambda_{QCD}$ in the PT series for
$\Gamma^{p,n}_{1,PT}$. Their influence becomes essential at $Q<1\,\text{{\rm
GeV}}$ where the HT terms start to play an important role. The minimal border
of the fitting domain $Q_{min}$ is tightly connected with the value of
$\Lambda_{QCD}$; i.e. it is a scale, below which the influence of the ghost
singularities becomes too strong and destroys the fit. To see how the
$Q^{2}_{min}$ scale and fit results for the $\mu$ terms change with varying
$\Lambda_{QCD}$, we have performed three different NLO fits with
$\Lambda_{QCD}=300,\,400,\,500\,\text{{\rm MeV}}$ (see Table 4). It turns out
that the term $\mu_{4}$ is quite sensitive to the Landau singularity position,
and its value noticeably increases with increasing $\Lambda_{QCD}$. The APT
and “soft-frozen” models are free of such a problem, thus providing a reliable
tool of investigating the behavior of HT terms extracted directly from the
low-energy data Bjour . This provides a motivation for the analysis performed
in the next section.
## III Moments $\Gamma_{1}^{p,n}(Q^{2})$ in Analytic Perturbation Theory
The moments of the structure functions are analytic functions in the complex
$Q^{2}$ plane with a cut along the negative real axis, as was demonstrated in
Ref. W78 (see also Ref. Ashok_suri ). On the other hand, the standard PT
approach does not support these analytic properties. The influence of
requiring these properties to hold in the DIS description was studied
previously by Igor Solovtsov and coauthors in Refs. APT-GLS ; MSS . Here we
continue this investigation by applying the APT method, which gives the
possibility of combining the RG resummation with correct analytic properties
of the QCD corrections, to the low-energy data on nucleon spin sum rules
$\Gamma_{1}^{p,n}(Q^{2})$.
In the framework of the analytic approach we can write the expression for
$\Gamma_{1}^{p,n}(Q^{2})$ in the form
$\displaystyle\Gamma^{p,n}_{1,APT}(Q^{2})=\frac{1}{12}\left[\biggl{(}\pm
a_{3}+\frac{1}{3}a_{8}\biggr{)}E^{APT}_{NS}(Q^{2})+\frac{4}{3}a^{inv}_{0}\,E^{APT}_{S}(Q^{2})\right]+\sum_{i=2}^{\infty}\frac{\mu^{APT;\,p,n}_{2i}(Q^{2})}{Q^{2i-2}}\,,$
(13)
which is analogous to one in the standard PT (2). The corresponding NNLO APT
modification of the singlet and nonsinglet coefficient functions is
$\displaystyle E^{APT}_{NS}(Q^{2})$ $\displaystyle=$ $\displaystyle
1-0.318\,{\cal A}^{(3)}_{1}(Q^{2})-0.361\,{\cal A}^{(3)}_{2}(Q^{2})-\,...\,,$
(14) $\displaystyle E^{APT}_{S}(Q^{2})$ $\displaystyle=$ $\displaystyle
1-0.318\,{\cal A}^{(3)}_{1}(Q^{2})-0.111\,{\cal A}^{(3)}_{2}(Q^{2})-\,...\,,$
(15)
where ${\cal A}^{(3)}_{k}$ is the analyticized $k$th power of 3-loop PT
coupling in the Euclidean domain
$\displaystyle\mathcal{A}^{(n)}_{k}(Q^{2})=\frac{1}{\pi}\int^{+\infty}_{0}\frac{\mathrm{Im}([\alpha_{s}^{(n)}(-\sigma,n_{f})]^{k})\,d\sigma}{\sigma+Q^{2}},\qquad
n=3\,.$ (16)
In the one-loop case, the APT Euclidean functions are simple enough apt96-7 :
$\displaystyle{\cal
A}_{1}^{(1)}(Q^{2})=\frac{1}{\beta_{0}}\left[\frac{1}{L}+\frac{\Lambda^{2}}{\Lambda^{2}-Q^{2}}\right]\,,\quad
L=\ln\left(\frac{Q^{2}}{\Lambda^{2}}\right),$ (17) $\displaystyle{\cal
A}_{2}^{(1)}(l)=\frac{1}{\beta_{0}^{2}}\left[\frac{1}{L^{2}}-\frac{Q^{2}\,\Lambda^{2}}{(Q^{2}-\Lambda^{2})^{2}}\right],\;{\cal
A}_{k+1}^{(1)}=-\,\frac{1}{k\,\beta_{0}}\,\frac{d\,{\cal A}_{k}^{(1)}}{dL}\,,$
i.e. the higher functions ${\cal A}_{k}$ are related to the lower ones
recursively by differentiating. Analogous two- and three-loop level
expressions involve the special Lambert function and are more intricate, and
they can be found in Refs. Magr:00 ; K-Magr:01 . It should be stressed that
the APT couplings are stable with respect to different loop orders at low-
energy scales $Q^{2}\lesssim 1\,\text{{\rm GeV}}^{2}$ Sh-revs . This feature
is absent in the standard PT approach, as reflected in Fig. 2.
Meanwhile, even for the three-loop APT case, there exists a possibility to
employ the effective log approach proposed by Igor Solovtsov and one of the
authors in Ref. SolSh99 . In the present context, in the region
$\,Q<5\,\,\text{{\rm GeV}}$ one may use simple model one-loop expressions (17)
with some effective logarithm $L^{*}\,$:
$\displaystyle{\cal A}_{1,2,3}^{(3)}(L)\to{\cal A}_{1,2,3}^{mod}=\,{\cal
A}_{1,2,3}^{(1)}(L^{*})\,,\quad L^{*}\simeq
2\,\ln(Q/\Lambda^{(1)}_{eff}),\quad\Lambda^{(1)}_{eff}\simeq
0.50\,\Lambda^{(3)}.$ (18)
Thus, instead of the exact three-loop expressions for the APT functions, in
Eq. (15) one can use the one-loop expressions (17) with the effective
$\Lambda$ parameter $\Lambda_{mod}=\Lambda^{(1)}_{eff}\,$ whose value is given
by the last relation (18). This model was successfully applied for higher-
twist analysis of low-energy data on BSR in our previous work Bjour , and also
in the $\Upsilon$ decay analysis in Ref. ShZ05 .
The maximal errors of the model (18) for the first and the second functions
are $\delta\mathcal{A}^{mod}_{1}/\mathcal{A}^{mod}_{1}\simeq 4\%$ and
$\delta\mathcal{A}^{mod}_{2}/\mathcal{A}^{mod}_{2}\simeq 8\%$ at
$Q\sim\Lambda_{n_{f}=3}\,,$ which seem to be sufficiently accurate. Indeed, as
far as ${\cal A}_{1}(Q=400\,\text{{\rm MeV}})=0.532\,$ and ${\cal
A}_{2}(400\,\text{{\rm MeV}})=0.118\,,$ the total error in $\Gamma^{p}_{\rm
1,APT}\,$ is mainly determined by the first term, being of the order
$\delta\Gamma_{1}^{p}/\Gamma_{1}^{p}\simeq\delta\mathcal{A}^{mod}_{1}/\pi\sim
1\,\%\,,$ i.e., less than the data uncertainty.
Figure 4: Evolution of $a_{0}(Q^{2})$ normalized at $Q_{0}^{2}=1\,\text{{\rm
GeV}}^{2}$.
Figure 5: Evolution of $\mu^{p-n}_{4}(Q^{2})$ normalized at
$Q_{0}^{2}=1\,\text{{\rm GeV}}^{2}$.
In order to take into account the one-loop $Q^{2}$ evolution of the axial
singlet charge $a_{0}(Q^{2})$, we use expression (II.1) substituting the one-
loop analytic coupling ${\cal A}_{1}^{(1)}(L)$. The contribution of the
$\sim{\cal A}_{1}$ term to $a_{0}(Q^{2})$ at, for example,
$Q^{2}=0.1\,\text{{\rm GeV}}^{2}$ with normalization point at
$Q_{0}^{2}=1\,\text{{\rm GeV}}^{2}$ is $\Delta_{1}(0.1\,\text{{\rm
GeV}}^{2})\simeq 0.11$; i.e. the evolution contributes about 10% when one
shifts the pQCD border down to $\Lambda_{QCD}$ (see Fig. 5).
For the evolution of the twist-4 term $\mu_{4}(Q^{2})$ (7), we have to
“analyticize” the fractional power $(\alpha_{s})^{\nu}$. For this purpose we
apply the fractional APT approach developed in Ref. Bakulev . At the one-loop
level in the Euclidean domain we have
$\displaystyle{\cal
A}_{\nu}^{(1)}(L)=\frac{1}{L^{\nu}}-\frac{F(e^{-L},1-\nu)}{\Gamma(\nu)}.$ (19)
Here $F(z,\nu)$ is the Lerch transcendent function. In this case, the
evolution of the nonsinglet twist-4 term in BSR reads
$\displaystyle\mu_{4,APT}^{p-n}(Q^{2})=\mu_{4,APT}^{p-n}(Q_{0}^{2})\,\frac{{\cal
A}_{\nu}^{(1)}(Q^{2})}{{\cal
A}_{\nu}^{(1)}(Q_{0}^{2})},\qquad\nu=\frac{32}{81}.$ (20)
The corresponding evolution is shown in Fig. 5. As follows from this figure,
the evolution from $1\,\text{{\rm GeV}}$ to $\Lambda_{QCD}$ increases the
absolute value of $\mu_{4,APT}^{p-n}$ by about 20 %.
## IV Numerical results
### IV.1 Nonsinglet case: the Bjorken sum rule
Figure 6: Best 1,2,3-parametric fits of the JLab and SLAC data on Bjorken SR calculated with different models of running coupling. Table 5: Combined fit results of BSR for the HT terms in APT, the SGF model and the standard PT approach. Method | $Q_{min}^{2},\,\text{{\rm GeV}}^{2}$ | $\quad\mu_{4}/M^{2}\quad$ | $\quad\mu_{6}/M^{4}\quad$ | $\quad\mu_{8}/M^{6}\quad$
---|---|---|---|---
| 0.50 | $-0.043(3)$ | 0 | 0
NLO PDG | 0.30 | $-0.074(3)$ | $~{}0.026(7)$ | 0
| 0.27 | $-0.049(4)$ | $-0.010(3)$ | 0.010(1)
| 0.47 | $-0.049(3)$ | 0 | 0
NLO Denom | 0.17 | $-0.069(4)$ | 0.014(1) | 0
| 0.17 | $-0.065(7)$ | 0.011(3) | 0.0003(7)
| 0.47 | $-0.061(3)$ | 0 | 0
NLO SGF | 0.19 | $-0.073(3)$ | 0.010(3) | 0
| 0.10 | $-0.077(4)$ | 0.014(5) | $-0.0008(3)$
| 0.47 | $-0.055(3)$ | 0 | 0
NNLO APT | 0.17 | $-0.062(4)$ | 0.008(2) | 0
no evolution | 0.10 | $-0.068(4)$ | 0.010(3) | $-0.0007(3)$
| 0.47 | $-0.051(3)$ | 0 | 0
NNLO APT | 0.17 | $-0.056(4)$ | 0.0087(4) | 0
with evolution | 0.10 | $-0.058(4)$ | 0.0114(6) | $-0.0005(8)$
In Fig. 6, we show best fits of the combined data set for the BSR function
$\Gamma_{1}^{p-n}(Q^{2})$ with NLO Denom (solid lines) and PDG (dashed lines)
couplings and NNLO APT (dash-dotted lines) at fixed $\Lambda_{QCD}$ value
corresponding to the world average. We also show here the pQCD part of the BSR
at different values of $\Lambda_{QCD}=300,\,400,\,500$ MeV calculated within
APT (short-dashed lines) and the SGF model Simonov at different values of the
glueball mass $M_{0}=1.2,\,1.0,\,0.8\,\text{{\rm GeV}}$ (with $\Lambda=360$
MeV) (dotted lines).
The corresponding numerical results are given in Table 5. As we have seen
before in Fig. 1, the behavior of SGF and APT couplings is very similar in the
low-energy domain $\Lambda_{QCD}<Q\lesssim 1$ GeV. As a result, the
corresponding perturbative parts of BSR in Fig. 6 and results for higher-twist
terms in Table 5 turn out to be close, too. Our fits in APT and the SGF model
give the HT values indicating a better convergence of the OPE series due to
decreasing magnitudes and alternating signs of consecutive terms, in contrast
to the usual PT fit results.
As is seen from Table 5, there is some sensitivity of fitted values of
$\mu_{4}$ with respect to $Q_{min}$ variations; namely, it increases in
magnitude when one incorporates into the fit the data points at lower
energies. This property of the fit may be treated as the slow (logarithmic)
evolution $\mu_{4}(Q^{2})$ with $Q^{2}$ which becomes more noticeable at
broader fitting ranges in $Q^{2}$, as discussed above. So for completeness we
included in Table 5 APT fits for $\mu_{4}(Q_{0}^{2})$ taking into account
their RG evolution with $Q_{0}=1\,\text{{\rm GeV}}$ as a normalization point.
We see that the fit results become more stable with respect to $Q_{min}$
variations.
However, there is still a problem with how to treat the evolution of higher-
twist terms $\mu_{6,8,..}(Q^{2})$ which again may turn out to be important
when one goes to lower $Q^{2}$, since the fit becomes more sensitive to very
small variations of $\mu_{6,8,..}$ with $Q^{2}$.
Note that the APT functions $\mathcal{A}_{k}$ contain the $(Q^{2})^{-k}\,$
power contributions which effectively change the fitted values of $\mu$ terms.
In particular, subtracting an extra $(Q^{2})^{-1}$ term induced by the APT
series
$\displaystyle\Gamma^{p-n}_{1,APT}(Q^{2})\simeq\frac{g_{A}}{6}+f\biggl{(}\frac{1}{\ln(Q^{2}/{\Lambda^{(1)}_{eff}}^{2})}\biggr{)}+\varkappa\frac{{\Lambda^{(1)}_{eff}}^{2}}{Q^{2}}+{\cal
O}\left(\frac{1}{Q^{4}}\right)$
with $\varkappa=0.43$ and using the value $\mu_{4,APT}^{p-n}/M^{2}=-0.058$
(with evolution) from Table 5, we finally get
$\displaystyle\frac{\mu_{4,APT}^{p-n}+\varkappa{\Lambda^{(1)}_{eff}}^{2}}{M^{2}}\simeq\frac{\mu^{p-n}_{4}(1\,\text{{\rm
GeV}}^{2})}{M^{2}}\simeq-0.042\,,\quad\Lambda^{(1)}_{eff}\sim 0.18\,\text{{\rm
GeV}}\,$ (21)
that nicely correlates with the result in Ref. Bj08-tw :
$\mu^{p-n}_{4}/M^{2}\simeq-0.045.$ This demonstrates the concert of the APT
analysis with the usual PT one for the BSR data at $Q^{2}\geq 1\;\text{{\rm
GeV}}^{2}$.
We do not take into account RG evolution in $\mu_{4}$ for the standard PT
calculations since the only effect of that would be the enhancement of the
Landau singularities by extra divergencies at $\Lambda_{QCD}$ (see Fig. 5),
whereas at higher $Q^{2}\sim 1\,\text{{\rm GeV}}^{2}$ the evolution is
negligible with respect to other uncertainties. In ghost-free models, however,
the evolution gives a noticeable effect at low $Q\sim\Lambda_{QCD}$. Note that
our previous result in Ref. Bjour , obtained without taking into account the
RG evolution, turned out to be slightly larger than (21)
$\mu^{p-n}_{4}/M^{2}\simeq-0.048$ which is very close to the corresponding
value obtained with the most precise Denom PT coupling and is shown in Table
5.
### IV.2 Singlet case: spin sum rules $\Gamma_{1}^{p,n}$ and nucleon spin
structure
Turn now to the three-loop APT part of the proton moment
$\Gamma^{p}_{1,APT}(Q^{2})$. Its value is quite stable with respect to small
variations of $\Lambda$, in contrast to the huge instability of
$\Gamma^{p}_{1,PT}$: it changes now by about $2\%-3\%$ within the interval
$\Lambda^{(3)}=300-500\,\text{{\rm MeV}}\,$. The same was previously observed
for the Bjorken function $\Gamma^{p-n}_{1,APT}(Q^{2})$ in Ref. Bjour . Because
of this fact the low-$Q^{2}$ data on $\Gamma^{p}_{1}(Q^{2})$ cannot be used
for determination of $\Lambda$ in the APT approach.
Extending the analysis of Ref. MSS to lower $Q^{2}$ scales, we estimated the
relative size of APT contributions to $\Gamma_{1}^{p}(Q^{2})$. It turned out
that the third term $\sim\mathcal{A}_{3}$ contributes no more than $5\%$ to
the sum, thus supporting the practical convergence of the APT series.
Table 6: Sensitivity of the best APT fit results of proton $\Gamma^{p}_{1}(Q^{2})$ data (elastic contribution excluded) to $\Lambda_{n_{f}=3}$ variations. The minimal fitting border is $Q_{min}^{2}=0.12\,\text{{\rm GeV}}^{2}$. $\;\Lambda_{QCD},\,\text{{\rm MeV}}\;$ | $\qquad a^{inv}_{0}\qquad$ | $\quad\mu_{4}/M^{2}\quad$ | $\quad\mu_{6}/M^{4}\quad$ | $\quad\mu_{8}/M^{6}\quad$
---|---|---|---|---
300 | 0.43(3) | $-0.082(4)$ | 0.015(9) | $-0.0009(5)$
400 | 0.45(3) | $-0.081(4)$ | 0.015(9) | $-0.0009(5)$
500 | 0.47(3) | $-0.080(4)$ | 0.014(9) | $-0.0009(5)$
To see how the numerical fit results are sensitive to $\Lambda_{(n_{f}=3)}$ in
APT, we fulfilled four different fits of the proton $\Gamma^{p}_{1}(Q^{2})$
data with $\Lambda_{QCD}=300,\,400,\,500\,\text{{\rm MeV}}$ as we did before
in the standard PT. The results of these fits are shown in Table 6. Comparing
these results with the data from Table 4, we see that the corresponding
results in the standard PT are much more sensitive to $\Lambda$ variations
than ones in APT.
Figure 7: Best (1,2,3+1)-parametric fits of the JLab and SLAC data on
$\Gamma^{p}_{1}$ (elastic contribution excluded).
In Fig. 7, we show best fits of the combined data set for the function
$\Gamma_{1}^{p}(Q^{2})$ (the data uncertainties are statistical only) in the
standard PT (PDG and Denom versions) and the APT approaches. We have also
shown the perturbative parts of $\Gamma_{1}^{p}(Q^{2})$ calculated in APT and
the SGF model. They are close to each other down to $Q\sim\Lambda$, similar to
the BSR analysis in the previous subsection. A similar observation was made in
the analysis of the small $x$ spin averaged structure functions in Ref.
Kotikov .
In Table 7, we present the combined fit results of the proton
$\Gamma^{p}_{1}(Q^{2})$ data (elastic contribution excluded) in APT, the SGF
model and conventional PT in PDG and denominator forms. One can see there is
noticeable sensitivity of the extracted $a^{inv}_{0}$ and $\mu_{4}$ with
respect to the minimal fitting scale $Q^{2}_{min}$ variations, which may be
(at least, partially) compensated by their RG $\log Q^{2}$ evolution, similar
to the BSR case. For completeness we included in Table 7 APT fits for
$a^{inv}_{0}(Q_{0}^{2})$ and $\mu_{4}(Q_{0}^{2})$, taking into account their
RG evolution.
Table 7: Combined fit results of the proton $\Gamma^{p}_{1}(Q^{2})$ data (elastic contribution excluded). APT fit results $a_{0}$ and $\mu^{APT}_{4,6,8}$ (at the scale $Q_{0}^{2}=1\,\text{{\rm GeV}}^{2}$) are given without and with taking into account the RG $Q^{2}$ evolution of $a_{0}(Q^{2})$ and $\mu^{APT}_{4}(Q^{2})$. Method | $Q_{min}^{2},\,\text{{\rm GeV}}^{2}$ | $\qquad a_{0}\qquad$ | $\quad\mu_{4}/M^{2}\quad$ | $\quad\mu_{6}/M^{4}\quad$ | $\quad\mu_{8}/M^{6}\quad$
---|---|---|---|---|---
| 0.59 | 0.33(3) | $-0.050(4)$ | 0 | 0
NLO PDG | 0.35 | 0.43(5) | $-0.087(9)$ | 0.024(5) | 0
| 0.29 | 0.37(5) | $-0.060(15)$ | -0.001(8) | 0.006(5)
| 0.59 | 0.35(3) | $-0.058(4)$ | 0 | 0
NLO Denom | 0.20 | 0.38(3) | $-0.076(4)$ | 0.013(1) | 0
| 0.17 | 0.38(4) | $-0.070(8)$ | 0.010(4) | 0.0004(5)
| 0.47 | 0.32(4) | $-0.056(4)$ | 0 | 0
NLO SGF | 0.17 | 0.36(3) | $-0.071(4)$ | 0.0082(9) | 0
$M_{0}=1\,\text{{\rm GeV}}$ | 0.10 | 0.40(4) | $-0.080(4)$ | 0.0134(9) | $-0.0007(6)$
| 0.47 | 0.35(4) | $-0.054(4)$ | 0 | 0
NNLO APT | 0.17 | 0.39(3) | $-0.069(4)$ | 0.0081(8) | 0
no evolution | 0.10 | 0.43(3) | $-0.078(4)$ | 0.0132(9) | $-0.0007(5)$
| 0.47 | 0.33(4) | $-0.051(4)$ | 0 | 0
NNLO APT | 0.17 | 0.31(3) | $-0.059(4)$ | 0.0098(8) | 0
with evolution | 0.10 | 0.32(4) | $-0.065(4)$ | 0.0146(9) | $-0.0006(5)$
As we already mentioned, the evolution of the $\mu^{p}_{4}(Q^{2})$ is taken to
be the same as for the nonsinglet term $\mu^{p-n}_{4}(Q^{2})$, allowing one to
keep only one fitting parameter $\mu^{p}_{4}(Q_{0}^{2})$ instead of two in the
general case. We also tested that the singlet anomalous dimension instead of
the nonsinglet one [resulting in the same $Q^{2}$ evolution of
$\mu^{p}_{4}(Q^{2})$ as that of $\mu^{p+n}_{4}(Q^{2})$] leads to close fit
results within error bars.
Figure 8: Behavior of $\chi^{2}/D.o.f.$ and $\mu^{p}_{4}$ from the proton data
fits (with only one $1/Q^{2}$ term) as functions of $a_{0}$ at different
values of $Q_{min}^{2}$ (the numbers at the curves) in the APT (left panels)
and PT (right panels) cases.
Figure 8 demonstrates the characteristic values of the proton data fits
$\chi^{2}/D.o.f.$ (upper row) and the twist-4 coefficient $\mu_{4}$ (lower
row) as functions of $a_{0}$ at different values of $Q_{min}^{2}$ (numbers at
the curves). One can see that at lower $Q^{2}$ ($Q_{min}^{2}<1~{}\text{{\rm
GeV}}^{2}$) the APT description (left panels) turns out to be more precise and
stable than that in the standard PT (right panels). Though we have taken the
fitted values of $a_{0}$ and higher twists $\mu_{2i}$ in the minima of each
$\chi^{2}/D.o.f.$ curve as best fits, the naive constraint
$\chi^{2}/D.o.f.\leq 1$ (dotted horizontal lines mark $1$) provides a quite
wide spread in the allowable values of the fit parameters. However, it would
be reasonable to take the spread between different minima as an optimistic
error bar of our analysis. This gives us the following result: $a_{0}=0.33\pm
0.05$, which is consistent with the recent analysis by COMPASS COMPASS06 and
HERMES HERMES06 (see Table 1).
In Fig. 9, we show the best fit results for the less precise neutron
$\Gamma^{n}_{1}(Q^{2})$ data. Again, the APT fit gives the HT values
demonstrating a better convergence of the OPE series, in contrast to the usual
PT fit results. Fits with APT and more precise Denom PT couplings lead to a
much smaller value of $\mu^{n}_{4}$ and more stable fitting curves than that
with the PDG coupling. Also the axial singlet charge $a_{0}$ extracted within
APT from the neutron data turns out to be very close to the one extracted from
more precise proton data (see Table 7).
Figure 9: Best (2+1)-parametric fits of the JLab and SLAC data on
$\Gamma^{n}_{1}$ calculated with NLO Denom (solid line) and PDG (dashed line)
couplings and NNLO APT (dash-dotted line).
To obtain the genuine value of the twist-4 term $\mu_{4}^{p}$, we act in a
similar way as for the BSR case in the previous subsection, namely,
subtracting an extra $(Q^{2})^{-1}$ term induced by the APT series
$\displaystyle E^{APT}_{NS}(Q^{2})$ $\displaystyle=$ $\displaystyle
E_{NS}(\alpha_{s}=\alpha_{s}^{LO}(Q^{2}))+\varkappa^{NS}_{4}\,\frac{{\Lambda^{(1)}_{eff}}^{2}}{Q^{2}}+{\cal
O}\left(\frac{1}{Q^{4}}\right),$ $\displaystyle E^{APT}_{S}(Q^{2})$
$\displaystyle=$ $\displaystyle
E_{S}(\alpha_{s}=\alpha_{s}^{LO}(Q^{2}))+\varkappa^{S}_{4}\,\frac{{\Lambda^{(1)}_{eff}}^{2}}{Q^{2}}+{\cal
O}\left(\frac{1}{Q^{4}}\right)$ (22)
with $\Lambda^{(1)}_{eff}\sim 0.18\,\text{{\rm
GeV}},\,\varkappa^{NS}_{4}=2.035$, and $\varkappa^{S}_{4}=0.661$, and using
the fit result in APT (with evolution) $\mu_{4}^{p,APT}/M^{2}=-0.065$ from
Table 7, we obtain
$\displaystyle\frac{\mu^{p}_{4}(1\,\text{{\rm
GeV}}^{2})}{M^{2}}\simeq\frac{1}{M^{2}}\left(\mu_{4}^{p,APT}+\frac{1}{12}\biggl{(}a_{3}+\frac{1}{3}a_{8}\biggr{)}\varkappa^{NS}_{4}{\Lambda^{(1)}_{eff}}^{2}+\frac{1}{9}a^{inv}_{0}\,\varkappa^{S}_{4}{\Lambda^{(1)}_{eff}}^{2}\right)\simeq-0.055\,.$
(23)
Analogously, for a neutron we have $\mu^{n}_{4}/M^{2}\simeq-0.010.$
Subtracting it from the proton value (23), we get for the nonsinglet twist-4
term $\mu^{p-n}_{4}/M^{2}\simeq-0.045\,,$ which is close to the result in Ref.
Bj04-tw , showing up the consistence of the APT analysis with the usual PT one
for the proton and neutron SSR $\Gamma^{p,n}_{1}$ data at $Q^{2}\geq
1\;\text{{\rm GeV}}^{2}$. Our result (23) is also consistent with the previous
extraction at higher energies in Ref. proton within the error bars (see also
Table 2).
It is worth noting that the best APT fit allows one to describe low-energy
JLab data on $\Gamma^{p,n}_{1}$ at scales down to $Q\sim 350\,\text{{\rm
MeV}}$ with only the first three terms of the OPE series, unlike the usual PT
case, where such fits happened to be impossible (due to the ghost issue) even
for an increasing number of HT terms. This means that the lower bound of the
pQCD applicability (supported by power HT terms) now may be shifted down to
$Q\sim\Lambda_{\rm QCD}\simeq 350$ MeV.
However, it seems to be difficult to get a description in the region
$Q<\Lambda_{\rm QCD}$. This is not surprising, because the expansion in
positive powers of $Q^{2}$ and its matching SofTer with the HT expansion are
relevant here. In this respect, the $\Lambda_{\rm QCD}$ scale appears as a
natural border between “higher-twist” and “chiral” nonperturbative physics.
Figure 10: Scale dependence of the gluon polarization $\Delta g$, obtained for
different versions of perturbation theory – in APT (solid line), in
conventional PT (dashed line), and in the SGF model (dash-dotted line).
Finally, in Fig. 10, we show the scale dependence of the gluon polarization
$\Delta g$ obtained in APT, PT, and the SGF model. In conventional PT the
value of $\Delta g$ is small at the lower scale $Q^{2}\sim 0.3\,\text{{\rm
GeV}}^{2}$ (see Ref. Wakamatsu:2007ar ). However, as one can see from Fig. 10,
one may evolve $\Delta g$ starting from higher scales $Q^{2}>1\,\text{{\rm
GeV}}^{2}$ down to the deep infrared region and observe that the smallness of
$\Delta g$ is a consequence of the Landau singularities in $\alpha_{s}$.
Applying different ghost-free models we see that $\Delta g$ is much higher at
$Q^{2}\lesssim 0.5\,\text{{\rm GeV}}^{2}$ than one predicted in the standard
PT.
## V Conclusion and Outlook
The singlet axial charge $a_{0}$ is the essential element of the nucleon spin
structure which is related to the average total quark polarization in the
nucleon. In this paper, we systematically extracted this quantity from very
accurate JLab data on the first moments of spin structure functions
$g^{p,n}_{1}$.
These data were obtained at low $Q^{2}$ region $0.05<Q^{2}<3\,{\rm GeV}^{2}$,
and therefore, a special attention was paid to the QCD coupling in this
domain. We demonstrated that the denominator form (10) of the QCD coupling
$\alpha_{s}$ is more suitable at the low $Q^{2}$ (see Figs. 1 and 2). In
particular, at the two-loop level it happens to be quite close to the exact
numerical solution of the corresponding two-loop RG equation for $Q\gtrsim
0.5\,\text{{\rm GeV}}$.
The performed analysis includes even lower $Q\sim\Lambda_{QCD}$ and involves
the QCD coupling which is free of Landau singularities. For this purpose we
used the APT apt96-7 and the soft glueball-freezing model Simonov for the
infrared-finite QCD coupling $\alpha_{s}$. It was shown that the singularity-
free APT and SGF QCD couplings are very close in the domain $Q\gtrsim
400\,\text{{\rm MeV}}$.
One can argue that large order perturbative and nonperturbative contributions
are mixed up, and the duality between them is expected (see Ref. ZN09 ). We
tested a separation of perturbative and nonperturbative physics and performed
a systematic comparison of the extracted values of the higher-twist terms in
different versions of perturbation theory. A kind of duality between higher
orders of PT and HT terms is observed so that higher order terms absorb part
of the HT contributions moving the pQCD frontier between the PT and HT
contribution to lower $Q$ values in both nonsinglet and singlet channels (see
Fig. 3). As expected, the value of $a_{0}$ changes substantially when coming
from LO to NLO, whereas it is quite stable in higher-loop approximations.
The perturbative contribution to the proton spin sum rule $\Gamma^{p}_{1}$ and
to the Bjorken sum rule $\Gamma^{p-n}_{1}$ in the APT approach and the SGF
model is less than 5 % for $Q>\Lambda$. This explains the similarity of the
extracted higher-twist parameters for these two modifications of QCD
couplings.
In the APT approach the convergence of both the higher orders and HT series is
much better. In both the nonsinglet and singlet case, while the twist-4 term
happened to be larger in magnitude in the APT than in the conventional PT, the
subsequent terms are essentially smaller and quickly decreasing (as the APT
absorbs some part of nonperturbative dynamics described by HT). This is the
main reason for the shift of the pQCD frontier to lower $Q$ values. A
satisfactory description of the proton SSR and BSR data down to
$Q\sim\Lambda_{QCD}\simeq 350\,\text{{\rm MeV}}\,$ was achieved by taking the
higher-twist and (analytic) higher order perturbative contributions into
account simultaneously (see Figs. 6 and 7). The best accuracy for the
extracted values of $a_{0}$ and higher-twist contributions $\mu_{2i}$ is
achieved for the most precise proton SSR data while the analysis of the data
on the neutron SSR shows the compatibility with the analysis of the BSR which
is free from the singlet contribution.
For the first time we considered the QCD evolution at low $Q^{2}$ of both the
leading twist $a_{0}$ and the higher-twist $\mu_{4}$ terms using the
(fractional) analytic perturbation theory Bakulev and also the related
evolution of the average gluon polarization $\Delta g$. Account of this
evolution, which is most important at low $Q^{2}$, improves the stability of
the extracted parameters whose $Q^{2}$ dependence diminishes (see Table 7). As
a result, we extract the value of the singlet axial charge
$a_{0}(1\,\text{{\rm GeV}}^{2})=0.33\pm 0.05$. This value is very close to the
corresponding COMPASS $0.35\pm 0.06$ COMPASS06 and HERMES $0.35\pm 0.06$
HERMES06 results.
The RG evolution of $a_{0}$ is related to the evolution of the average gluon
polarization $\Delta g$ Anselmino:1994gn ; Leader08 . The results of the
evolution of $\Delta g$ in the analytic perturbation theory and in the
standard PT was compared (see Fig. 10). The decrease of $\Delta g$ at low
$Q^{2}$ in APT is not so dramatic as in the standard PT case Wakamatsu:2007ar
.
In a sense, it could be natural if the main reason for the significant shift
of the pQCD frontier to lower $Q^{2}$ scales was the disappearance of
unphysical singularities in perturbative series. Note that the data at very
low $Q\sim\Lambda_{QCD}$ are usually dropped from the analysis of $a_{0}$ and
the higher-twist term in the standard PT analysis because of Landau
singularities. At the same time, the compatibility of our results for $a_{0}$,
extracted from the low energy JLab data with previous results COMPASS06 ;
HERMES06 demonstrates the universality of the nucleon spin structure at large
and low $Q^{2}$ scales. It will be very interesting to explore the interplay
between perturbative and nonperturbative physics against other low energy
experimental data.
## ACKNOWLEDGMENTS
This work was partially supported by RFBR Grants No. 07-02-91557, No.
08-01-00686, No. 08-02-00896-a, and No 09-02-66732, the JINR-Belorussian Grant
(Contract No. F08D-001), and RF Scientific School Grant No. 1027.2008.2. We
are thankful to A.P. Bakulev, J.P. Chen, G. Dodge, A.E. Dorokhov, S.B.
Gerasimov, G. Ingelman, A.L. Kataev, S.V. Mikhailov, A.V. Sidorov, D.B.
Stamenov, and N.G. Stefanis for valuable discussions.
## References
* (1) M. Anselmino, A. Efremov and E. Leader, Phys. Rept. 261, 1 (1995) [Erratum-ibid. 281, 399 (1997)].
* (2) S. E. Kuhn, J. P. Chen and E. Leader, Prog. Part. Nucl. Phys. 63, 1 (2009).
* (3) A. V. Efremov, J. Soffer and O. V. Teryaev, Nucl. Phys. B346, 97 (1990).
* (4) O. V. Teryaev, To appear in Proceedings of XIII Workshop of High Energy Spin Physics, DSPIN’09, Dubna, Russia September 1 - 5, 2009.
* (5) M. V. Polyakov, A. Schafer and O. V. Teryaev, Phys. Rev. D 60, 051502 (1999) [arXiv:hep-ph/9812393].
* (6) R. S. Pasechnik, D. V. Shirkov and O. V. Teryaev, Phys. Rev. D 78, 071902 (2008).
* (7) J. D. Bjorken, Phys. Rev. 148, 1467 (1966); Phys. Rev. D 1, 1376 (1970).
* (8) J. Kodaira et al., Nucl. Phys. B159, 99 (1979);
J. Kodaira, Nucl. Phys. B165, 129 (1980);
S. A. Larin, F. V. Tkachov and J. A. Vermaseren, Phys. Rev. Lett. 66, 862
(1991);
S. A. Larin and J. A. Vermaseren, Phys. Lett. B 259, 345 (1991);
M. Anselmino, B. L. Ioffe and E. Leader, Sov. J. Nucl. Phys. 49, 136 (1989).
* (9) J. Soffer and O. Teryaev, Phys. Rev. Lett. 70, 3373 (1993); Phys. Rev. D 70, 116004 (2004).
* (10) M. Osipenko et al., Phys. Lett. B 609, 259 (2005) [arXiv:hep-ph/0404195].
* (11) A. Deur, V. Burkert, J. P. Chen and W. Korsch, Phys. Lett. B 665, 349 (2008).
* (12) D. V. Shirkov, Nucl. Phys. Proc. Suppl. 162, 33 (2006).
* (13) C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008).
* (14) D. V. Shirkov and I. L. Solovtsov, JINR Rapid Comm. 2 [76-96], 5 (1996) [arXiv:hep-ph/9604363]; Phys. Rev. Lett. 79, 1209 (1997);
K. A. Milton and I. L. Solovtsov, Phys. Rev. D 55, 5295 (1997).
* (15) D. V. Shirkov and I. L. Solovtsov, Theor. Math. Phys. 150, 132 (2007).
* (16) M. Baldicchi, A. V. Nesterenko, G. M. Prosperi, D. V. Shirkov and C. Simolo, Phys. Rev. Lett. 99, 242001 (2007).
* (17) Yu. A. Simonov, Phys. Atom. Nucl. 65, 135 (2002); 66, 764 (2003); J. Nonlin. Math. Phys. 12, S625 (2005).
* (18) M. Beneke, Phys. Rept. 317, 1 (1999).
* (19) G. Curci, M. Greco and Y. Srivastava, Phys. Rev. Lett. 43, 834 (1979); Nucl. Phys. B159, 451 (1979);
C. Berger et al. [PLUTO Collaboration], Phys. Lett. B 100, 351 (1981);
J. M. Cornwall, Phys. Rev. D 26, 1453 (1982);
N. G. Stefanis, Phys. Rev. D 40, 2305 (1989) [Erratum-ibid. D 44, 1616
(1991)];
N. N. Nikolaev and B. M. Zakharov, Z. Phys. C 49, 607 (1991); C 53, 331
(1992);
Y. L. Dokshitzer, V. A. Khoze and S. I. Troian, Phys. Rev. D 53, 89 (1996).
* (20) K. V. Dharmawardane et al., Phys. Lett. B 641 11 (2006);
P. E. Bosted et al., Phys. Rev. C 75, 035203 (2007);
Y. Prok et al. [CLAS Collaboration], Phys. Lett. B 672, 12 (2009).
* (21) X. D. Ji and J. Osborne, J. Phys. G 27, 127 (2001).
* (22) S. B. Gerasimov, Yad. Fiz. 2, 598 (1965) [Sov. J. Nucl.Phys. 2, 430 (1966)];
S. D. Drell and A. C. Hearn, Phys. Rev. Lett. 16, 908 (1966).
* (23) A. L. Kataev, Phys. Rev. D 50, 5469 (1994); Mod. Phys. Lett. A 20, 2007 (2005).
* (24) S. A. Larin, T. van Ritbergen and J. A. M. Vermaseren, Phys. Lett. B 404, 153 (1997).
* (25) Y. Goto et al. [Asymmetry Analysis collaboration], Phys. Rev. D 62, 034017 (2000).
* (26) E. V. Shuryak and A. I. Vainshtein, Nucl. Phys. B199, 451 (1982); B201, 141 (1982).
* (27) J. P. Chen, nucl-ex/0611024;
J. P. Chen, A. Deur and Z. E. Meziani, Mod. Phys. Lett. A 20, 2745 (2005).
* (28) V. Y. Alexakhin et al. [COMPASS Collaboration], Phys. Lett. B 647, 8 (2007).
* (29) A. Airapetian et al. [HERMES Collaboration], Phys. Rev. D 75, 012007 (2007).
* (30) E. Leader, A. V. Sidorov and D. B. Stamenov, Phys. Rev. D 75, 074027 (2007).
* (31) D. de Florian, R. Sassot, M. Stratmann and W. Vogelsang, Phys. Rev. Lett. 101, 072001 (2008).
* (32) M. Hirai and S. Kumano [Asymmetry Analysis Collaboration], Nucl. Phys. B813, 106 (2009).
* (33) R. Fatemi et al. [CLAS Collaboration], Phys. Rev. Lett. 91, 222002 (2003);
M. Amarian et al., Phys. Rev. Lett. 89, 242301 (2002);
M. Amarian et al. [Jefferson Lab E94-010 Collaboration], Phys. Rev. Lett. 92,
022301 (2004).
* (34) G. Cvetic, A. Y. Illarionov, B. A. Kniehl and A. V. Kotikov, Phys. Lett. B 679, 350 (2009) [arXiv:0906.1925 [hep-ph]].
* (35) A. Deur et al., Phys. Rev. D 78, 032001 (2008).
* (36) A. Deur, arXiv:nucl-ex/0508022.
* (37) Z. E. Meziani et al., Phys. Lett. B 613, 148 (2005) [arXiv:hep-ph/0404066].
* (38) A. Deur et al., Phys. Rev. Lett. 93, 212001 (2004) [arXiv:hep-ex/0407007].
* (39) S. Bethke, arXiv:0908.1135 [hep-ph].
* (40) K. A. Milton, I. L. Solovtsov and O. P. Solovtsova, Phys. Rev. D 60, 016001 (1999).
* (41) A. V. Kotikov, G. Parente and J. Sanchez Guillen, Z. Phys. C 58, 465 (1993).
* (42) G. Parente, A. V. Kotikov and V. G. Krivokhizhin, Phys. Lett. B 333, 190 (1994) [arXiv:hep-ph/9405290].
* (43) A. L. Kataev, A. V. Kotikov, G. Parente and A. V. Sidorov, Phys. Lett. B 417, 374 (1998) [arXiv:hep-ph/9706534].
* (44) A. L. Kataev, G. Parente and A. V. Sidorov, Nucl. Phys. B573, 405 (2000).
* (45) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147, 385 (1979).
* (46) V. D. Burkert and B. L. Ioffe, Phys. Lett. B 296, 223 (1992).
* (47) W. Wetzel, Nucl. Phys. B 139, 170 (1978).
* (48) Ashok suri, Phys. Rev. D 4, 570 (1971).
* (49) K. A. Milton, I. L. Solovtsov and O. P. Solovtsova, Phys. Lett. B 439, 421 (1998).
* (50) B. A. Magradze, JINR Comm. E2-2000-222, Oct 2000. 19 pp. [hep-ph/0010070].
* (51) D. S. Kourashev and B. A. Magradze, Theor. Math. Phys., 135, 531 (2003) [hep-ph/0104142].
* (52) I. L. Solovtsov and D. V. Shirkov, Theor. Math. Phys. 120, 1220 (1999).
* (53) D. V. Shirkov and A. V. Zayakin, Phys. Atom. Nucl. 70: 775-783, (2007).
* (54) A. P. Bakulev, S. V. Mikhailov and N. G. Stefanis, Phys. Rev. D 72, 074014 (2005) [Erratum-ibid. D 72, 119908 (2005)]; 75, 056005 (2007) [Erratum-ibid. D 77, 079901 (2008)];
N. G. Stefanis, arXiv:0902.4805 [hep-ph].
* (55) M. Wakamatsu and Y. Nakakoji, Phys. Rev. D 77, 074011 (2008).
* (56) S. Narison and V. I. Zakharov, Phys. Lett. B 679, 355 (2009) [arXiv:0906.4312 [hep-ph]].
|
arxiv-papers
| 2009-11-17T13:05:39 |
2024-09-04T02:49:06.520943
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roman S. Pasechnik, Dmitry V. Shirkov, Oleg V. Teryaev, Olga P.\n Solovtsova, Vyacheslav L. Khandramai",
"submitter": "Olga Solovtsova",
"url": "https://arxiv.org/abs/0911.3297"
}
|
0911.3378
|
# Effective potential and warp factor dynamics
Michael R. Douglas1,&
1Simons Center for Geometry and Physics
Stony Brook University
Stony Brook, NY 11794 USA
&I.H.E.S., Le Bois-Marie, Bures-sur-Yvette, 91440 France
douglas@max2.physics.sunysb.edu
###### Abstract:
We define an effective potential describing all massless and massive modes in
the supergravity limit of string/M theory compactification which is valid off-
shell, i.e. without imposing the equations of motion. If we neglect the warp
factor, it is unbounded below, as is the case for the action in Euclidean
quantum gravity. By study of the constraint which determines the warp factor,
we solve this problem, obtaining a physically satisfying and tractable
description of the dynamics of the warp factor.
## 1 Introduction
If our universe is described by string/M theory, there exist six or seven
extra dimensions of space, not yet detected by experiment. This is possible
because the extra dimensions can take the form of a small, compact manifold
$X$. A basic question in string/M theory is to know what types of manifold are
allowed, and to find general relations between their geometry and physics. In
the present state of the art, this is generally done by solving the equations
of motion for the supergravity theory which describes the low energy limit,
and then taking into account various stringy and quantum effects.
The first works on string compactification assumed that some supersymmetry is
preserved at the compactification scale, for good physical and mathematical
reasons. Well known physical arguments suggest low energy supersymmetry;
supersymmetry favors stability; and supersymmetry places strong constraints on
the local geometry of $X$. In the best studied cases, $X$ is a complex Kähler
manifold, so powerful techniques of algebraic geometry are available for the
analysis.
More recently, there has been a shift towards techniques which do not assume
supersymmetry. After all, supersymmetry is broken in our universe, and we
don’t know at what scale it is broken. String/M theory suggests many other
solutions to the hierarchy problem, such as large extra dimensions [4] or
warped hierarchies [29]. Or, the hierarchy might simply be a chance property,
following from a lucky tuning of parameters in a small subset of vacua. Given
a measure factor (a probability distribution over vacua), presumably emerging
from early cosmology, it might turn out that this class of compactifications
outweighs those with low energy supersymmetry [14, 33].
Another reason to be interested in compactifications without supersymmetry is
to get models of inflation, because the required positive vacuum energy breaks
supersymmetry. Indeed this is a reason to study not just vacua, meaning long-
lived compactifications to maximally symmetric space-times, but the dynamics
on the larger configuration space which contains the vacua.
At present the main technique for addressing any of these questions is to
start from a class of compactifications, say with a specific choice of
topology for the compact manifold $X$, and derive a four (or $d$) dimensional
effective field theory which describes theories within this class. One begins
by identifying various “pseudo-moduli fields” such as metric moduli, brane
positions and the like, which parameterize the vacua within this class. One
then derives an effective potential, which is a function of the these fields.
This is usually done by combining various exact results for related
supersymmetric compactifications, brane world-volume theories, and
supersymmetric quantum field theories, in a first approximation by adding
them, and then considering corrections.
Having derived an effective potential, one then looks for its local minima.
The first issue is that, since the potential always goes to zero at large
volume and weak coupling [13, 22], one must find effects that produce a
barrier to this runaway. Having done this, the easiest way to argue that local
minima exist at finite moduli is to show that, to a good approximation, the
effective potential in the regime under study is a sum of many different,
loosely correlated, contributions from different sources: fluxes, branes,
quantum effects, curvature, and so on. By approximately balancing two or three
of these contributions and then tuning their precise strengths, one can obtain
potentials with local minima.
One important conclusion from this work is that there is no particular
obstacle within string theory to realizing de Sitter space-time [25], and thus
the small observed dark energy could be a cosmological constant. Indeed, given
that one can obtain local minima with a small negative vacuum energy, compared
to the individual contributions, it would seem that only an incredible
conspiracy between the different effects would prevent one from obtaining
similar vacua with small positive vacuum energy. In relatively simple and
controllable models such as the explicit KKLT realizations of [10], one could
in principle get enough control to rule out such conspiracies. In practice,
one brings in some physical intuition at this point, using arguments such as
parametric separation of energy scales of different effects (so they cannot
always cancel) or spatial separation of effects from different regions in $X$.
Combining these arguments with the lack of any proposed mechanism for the
supposed conspiracy, the conclusion seems well enough justified, though better
arguments would certainly be welcome.
This general approach and many examples are reviewed in [15]. It is very
effective in determining general structure and relations between parameters
such as masses and coupling constants; for example the general differences
between IIb, IIa, large volume and heterotic compactifications are all readily
understandable in these terms. One can also show that certain classes of
models cannot contain vacua, or cannot realize slow-roll inflation [24],
because of runaways to large volume or weak coupling. Other reasons to develop
this approach are that it could in principle be generalized to nongeometric
compactifications, and finally it is the best we can hope to do for the vast
majority of compactifications.
Once one grants the validity of the effective action, the question of the
existence of vacua with positive vacuum energy, and even general
nonsupersymmetric vacua, becomes in principle straightforward to answer. It
also leads to a simple picture of dynamics. The best studied example is the
dynamics of inflation, which can be realized by slowly rolling down a nearly
flat potential. More recently, models have been suggested which take advantage
of other structure, such as in the kinetic terms [3]. Although one expects the
effective action picture to break down at high energies, in the best case at
the (higher dimensional) Planck scale, this can still be well above the energy
scale of slow-roll inflation.
One problem with the effective potential approach is that string theory and
even simple Kaluza-Klein reduction involve an infinite set of fields, while
the usual truncation to the pseudo-moduli fields is somewhat ad hoc. Although
there is a simple argument that one can solve the equations of motion for
massive fields in terms of light fields, this ignores the possibility of
tachyonic modes, which will destabilize vacua. A candidate vacuum for present-
day physics must be tachyon-free, while configurations with tachyons are
important for cosmology. In dynamical situations in which fields undergo large
variations, of course the splitting of fields into “massless” and “massive”
can change drastically.
There are simple arguments that massive modes are under control, but these
tend to use supersymmetry. For example, one often builds up an effective
potential by combining sectors which individually respect different
supersymmetries. Another simple argument is that, even after supersymmetry
breaking, the effective potential is approximately a sum of squares, up to
corrections of order $F/M_{Planck}^{2}$. Such arguments seem believable given
a hierarchy between the supersymmetry breaking scale and the masses of the new
modes, but are not convincing otherwise.
A second level of analysis in which one can see the massive modes is to solve
the full $9+1$ or $10+1$-dimensional supergravity field equations, or the beta
function conditions for a conformal world-sheet theory, or perhaps even the
full string field theory equations of motion. These various approaches include
more and more modes at the cost of eventual intractability. Of course, one
does not need an exact solution for all the modes; even the analysis of
linearized fluctuations of massive modes around a solution would go a long way
towards answering the stability question.
A problem with these approaches is that they are classical (from the point of
view of space-time), while the existing constructions of vacua usually rely on
quantum corrections for stability and supersymmetry breaking. To try to
address this problem, one can note that, despite being formulated in higher
dimensions or with string fields, we can still think of these approaches as
each based on an effective potential, essentially defined as the higher
dimensional part of the action, but now taken as a function of all of the
massless and massive modes. The higher dimensional equations of motion
correspond to solving the condition $\partial V/\partial\phi^{i}=0$, while the
linearized stability analysis simply corresponds to computing the mass matrix
$\partial^{2}V/\partial\phi^{i}\partial\phi^{j}$. To the extent that one can
make this idea precise, one can then try to deal with quantum corrections by
the same prescription which was used before, namely to simply add them in, or
make other corresponding adjustments of the effective potential. We could
refer to this entire class of approaches as “semiclassical,” to be contrasted
with a fully nonperturbative approach such as gauge-gravity duality, which
unfortunately is not known to exist for theories with positive vacuum energy.
Since in the supergravity limit, a vacuum is a solution of well understood
higher dimensional equations of motion, the problem of constructing an
effective potential which takes massive modes into account is in principle
just one of isolating the appropriate terms in the original action. Following
up on works such as [34, 21, 9], in [23] Giddings and Maharana proposed an
effective potential, based on reinterpreting the constraint in the Einstein
equations. They went on to show that it worked in examples such as that of
[21].
While we did not know about this proposal during the course of our work, and
only found out about it after having distributed our work as a preprint, in
general we followed a similar approach, and obtained an effective potential
consistent with that of [23], but with many differences in our discussion. We
show explicitly that a critical point of the potential solves all the
equations of motion, and we have eliminated the implicit assumption that
space-time is Minkowski made at various points in [23]. Another difference is
that the two proposals use different conventions for the warp factor. The
conventions followed here have the great advantage that the constraint becomes
linear, enabling us to go on and better understand the physics in §3 and §4.
On further consideration of the resulting effective potential, one realizes
that there is a significant problem with its interpretation, which we now
explain.
### 1.1 Questions addressed in this work
In this work we develop the analysis of compactifications starting from the
equations of motion of general relativity coupled to the type of matter
sectors which appear in maximal supergravity theories, including possible
stringy corrections. We believe that the same ideas would apply to any of the
semiclassical approaches, if we understood the relevant configuration spaces.
Some issues which we try to shed light on include:
* •
The definition of the four (or $d$) dimensional effective potential. While it
is in principle clear how to define this for a compactified field theory (not
including gravity), this is not the case once gravity is included. Indeed,
there are well-known difficulties in making a global definition of energy in
general relativity [37]. The usual response in this context (as in [15]) is to
work in approximately asymptotically flat backgrounds, and appeal to the
standard definition for this case. This is obviously not satisfactory when
doing cosmology.
One particularly interesting contribution to the effective potential is minus
the integral of the scalar curvature of the compactification manifold,
$V_{eff}=-\frac{1}{2}\int\sqrt{g}R^{(k)}.$ (1)
This term can be obtained by the general procedure we just discussed, of
regarding the higher dimensional part of the action as an effective potential,
as we will review in section §2 (see Eq. (12)). At least naively, it is
responsible for anticorrelating the curvature of the compactification manifold
with that of space-time, as in the well-known ${AdS}_{p}\times S^{q}$
solutions. While there are additional terms in the Einstein equation and this
is oversimplified, at first sight it seems reasonable to think of this as one
of the many terms in the effective potential.
However, a serious problem with this interpretation of Eq. (1) is that the
integral of the scalar curvature can be made arbitrarily large and positive,
by making a rapidly varying conformal transformation of the metric. Thus, Eq.
(1) is unbounded below, and does not even have local minima. This problem is
closely related to the fact that the action in Euclidean gravity is unbounded
below, and to the “wrong sign kinetic term” for the conformal factor which may
be familiar from minisuperspace treatments of gravity. As such, one would
expect that it is solved (at least classically) by imposing the appropriate
constraints. In particular, in the canonical formulation, the Hamiltonian
constraint must be imposed at each point in space, and determines the
conformal factor in terms of the other fields. At least in asymptotically flat
space-times, the resulting energy would be non-negative, by the positive
energy theorem [31, 38].
However, it is not so clear how the Hamiltonian constraint solves this
problem, because it directly determines only the conformal factor on
$D-1$-dimensional space, leaving the conformal factor on $X$ free to vary. It
corresponds to the “warp factor” constraint in the existing analyses, which is
a clue to the physical interpretation.
Is the potential Eq. (1) bounded below, and if so why? If it is, what
determines the correct conformal factor with which to evaluate it? Can we
solve for this variable to simplify the potential?
* •
The nature of the warp factor constraint. In doing Kaluza-Klein reduction, the
$d$-dimensional part of the Einstein equation (in fact, the Hamiltonian
constraint) turns into a partial differential equation for the warp factor. By
analysis of this equation, in [11, 27] it was shown that one cannot realize de
Sitter space-time in pure supergravity, i.e. without stringy corrections or
singularities. Later it was argued [21, 12] that this could be evaded using
orientifolds and/or “$0$-form flux” (the Romans mass term in IIa
supergravity), and this is no longer considered a major obstacle.
As the warp factor plays such an important role in the physics, perhaps even
solving the hierarchy problem, it would be nice to have a better conceptual
understanding of it. Are there other conditions for this constraint to have a
solution? What if the various sources to it are large, or widely separated on
$X$ ? What if the sources evolve in time?
* •
The possibility of compactification on negatively curved manifolds, and the
nature of flux or other effects needed to stabilize negative curvature.
According to Eq. (1), negative scalar curvature makes a positive contribution
to the vacuum energy, which could make de Sitter space easier to realize. This
point has been particularly emphasized by Silverstein, who has proposed a
variety of constructions of this type [30, 32]. There are many other works on
the subject, including [8].
Now, the no-go theorems of [11, 27] do not make any assumption about the
curvature, and would hold in this case as well. Thus, even with negative
curvature, one still needs to call on stringy effects to get de Sitter
compactifications. Is negative curvature actually relevant for this, and if so
why? Perhaps consistent compactifications of this type are always dual to
others of more familiar types?
The possibility of compactification on negatively curved manifolds raises
another point, which is that there are far more of these than manifolds of
zero or positive curvature. This is illustrated by the familiar example of
Riemann surfaces, for which the curvature is proportional to the Euler
characteristic $\chi=2-2g$, and is true in higher dimensions as well. If such
manifolds could be used to get compactifications with negative or small
positive vacuum energy, it seems almost inevitable that they would dominate
the landscape.
In this work, we will give an explicit expression for the effective potential
in supergravity compactifications, Eq. (47), mostly answering the first two
questions, and making a start on the third.
Besides understanding solutions, another important application of an effective
potential is to study time dependence. In particular, slow-roll inflation can
be described as gradient descent. With this potential, this leads to equations
similar to the Ricci flow equations, as we will discuss elsewhere. We might
also hope that this will shed light on the deeper questions regarding the
existence of the effective potential raised by Banks in [5], or on the old
problem of the conformal factor in Euclidean quantum gravity.
## 2 Effective potential
### 2.1 General discussion
We consider $D$-dimensional solutions of general relativity coupled to matter,
with an action such as
$S=\int\sqrt{-g}\left(R^{(D)}-\frac{1}{2}\sum_{p}|F^{(p)}|^{2}\right).$ (2)
We take metric signature $-+++\ldots$, and define units so that the
fundamental Planck scale $M_{Pl,D}=1$.
The equations of motion are the Einstein equation,
$R_{MN}-\frac{1}{2}g_{MN}R=T_{MN}$ (3)
coupled to $p$-form gauge field strengths, with equation of motion $d*F=0$ and
stress-energy
$T_{MN}=pF_{MI_{1}\cdots I_{p-1}}F_{N}^{I_{1}\cdots
I_{p-1}}-\frac{1}{2}g_{MN}F^{2}.$ (4)
We choose a nonstandard normalization so that the case $p=0$ can be treated
uniformly.
At least in the absence of Chern-Simons terms, one can choose to use an action
in terms of either $F^{(p)}$ or its dual $F^{(D-p)}$. In the following, we
will use this freedom to consider all background fluxes as magnetic fluxes
(so, $F_{0\ldots}=0$), just to simplify the discussion.
We consider compactification on a $k=D-d$-dimensional compact manifold $M$ to
$d$-dimensional maximally symmetric space-time (Minkowski, AdS, dS). Whenever
there is any ambiguity, we superscript the metric and curvatures with the
dimension of space-time $D,k$ or $d$. We superscript forms with their rank –
since forms can be pulled back, generally there is no ambiguity.
We make a Kaluza-Klein warped metric ansatz,
$ds^{2}=\eta_{\mu\nu}e^{2A(y)}dx^{\mu}dx^{\nu}+g_{ij}(y)dy^{i}dy^{j}.$ (5)
Here $\eta_{\mu\nu}$ is a maximally symmetric metric in $d$ dimensions, which
could be dS, AdS or Minkowski, and $g_{ij}$ is a metric on $M$.
Our goal is to write a $d$-dimensional effective action,111 We absorb the
customary $1/16\pi$ into our definition of $G_{N}$.
$S^{(d)}=\int\sqrt{-g^{(d)}}\left(\frac{1}{G_{N}}R^{(d)}-\frac{1}{2}G_{ab}(\phi)\partial\phi^{a}\partial\phi^{b}-2V_{eff}(\phi)+\ldots\right),$
(6)
whose equations of motion agree with the KK reduction of the $D$-dimensional
Einstein equations. By $\ldots$ we denote gauge field actions and other terms
we will not treat in detail here. Thus, the $d$-dimensional Einstein equation
is
$R^{(d)}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{(d)}=G_{N}T^{(d)}_{\mu\nu}$ (7)
with
$T^{(d)}_{\mu\nu}=-\frac{1}{2}G_{ab}(\phi)\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}-\frac{1}{2}g_{\mu\nu}\left(\frac{1}{2}G_{ab}(\phi)\partial\phi^{a}\partial\phi^{b}+2V_{eff}(\phi)\right);$
(8)
in particular $T_{00}=V_{eff}+\ldots$.
In principle, we now want to rewrite the $D$-dimensional fields using a mode
decomposition, as appears in [17] and many other works. We would then
substitute these expressions into the $k$-dimensional part of the action, do
the integral over $X$ and reinterpret the results as terms in Eq. (6). Any
term with no space-time derivatives would become part of the effective
potential.
The main difficulty in doing explicit mode expansions is to diagonalize the
various Laplacians on $X$ which appear as kinetic terms. In a mathematical
sense this step is well understood, and is not directly relevant for our
purposes. Since we do not need to diagonalize the metric $G_{ab}(\phi)$ on
field space, any complete basis for functions on $X$ would suffice. Thus, we
can regard the $D$-dimensional fields as generating functions for the totality
of massive modes in the $d$-dimensional compactified theory.
### 2.2 Einstein equations
The subtleties we are concerned with appear elsewhere, and can be seen by
reviewing the standard discussion of compactification based on the
$D$-dimensional Einstein equations. Since these look rather different in $d$
and $k$ dimensions, we consider the two components separately. The
$d$-dimensional components can be written
$R^{(d)}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{(d)}=T^{(d)}_{\mu\nu}+\frac{1}{2}g_{\mu\nu}R^{(k)}$
(9)
and we would like to interpret the right-hand side as an effective
$d$-dimensional stress tensor. Since the $d$ dimensions have maximal symmetry,
we lose nothing by taking the trace,
$\displaystyle-\frac{d-2}{2}R^{(d)}$ $\displaystyle=$ $\displaystyle
T^{(d)}+\frac{d}{2}R^{(k)}$ (10) $\displaystyle=$
$\displaystyle-d\cdot\Lambda$ (11)
The right hand side of this equation defines an effective cosmological
constant,
$\Lambda=T_{00}-\frac{1}{2}R^{(k)},$ (12)
with a contribution from the curvature of $X$. To get a $d$-dimensional
equation, we then integrate over $X$, producing Eq. (1). More precisely, all
three terms depend on the volume of $X$, which will lead to the dependence on
Newton’s constant $1/G_{N}$ in the result.
For standard (classical) sources of energy, except for negative potential
energy (which is not present in the string/M theory case), $T_{00}\geq 0$, so
naively one might expect de Sitter to be easy to realize. On the other hand,
the scalar curvature also contributes, so one needs to decide which effect is
more important.
The $k$-dimensional curvature can be determined by using the $D$-dimensional
trace of the Einstein equation,
$\left(1-\frac{D}{2}\right)R=T^{(d)}+T^{(k)},$ (13)
to solve for $R$ in Eq. (3), obtaining
$R^{(k)}_{ij}=T^{(k)}_{ij}-\frac{1}{D-2}g_{ij}\left(T^{(d)}+T^{(k)}\right).$
(14)
Taking the trace, one sees that the sign of $R^{(k)}$ is correlated to that of
$T^{(k)}$, while the $T^{(d)}=-dT_{00}$ contribution now favors positive
internal curvature. Thus, the effects of $T^{(d)}$ in the two equations
conflict with each other.
Solving for $R^{(k)}$ and substituting back into Eq. (12), we get
$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle
T_{00}-\frac{1}{2}\left(\frac{d-2}{D-2}T^{(k)}-\frac{k}{D-2}T^{(d)}\right)$
(15) $\displaystyle=$
$\displaystyle-\frac{(d-2)(k-2)}{2(D-2)}T_{00}-\frac{d-2}{2(D-2)}T^{(k)}.$
(16)
Evidently the total effect of $T_{00}$ always favors AdS, with the only hope
for dS being to have $T^{(k)}$ very negative compared to $T_{00}$, more
precisely
$T^{(k)}<-(k-2)T_{00}.$ (17)
Now, taking magnetic flux $F_{0\ldots}=0$ in Eq. (4), we have
$T^{(k)}=(p-\frac{k}{2})F^{2}=(2p-k)T_{00}.$ (18)
Thus, this condition will never hold for any $p\geq 1$, or sums of such
stress-tensors.
Thus, de Sitter space-time can only be realized if the matter stress tensor
violates the strong energy condition [37], which is
$0\leq R_{00}=T_{00}-\frac{1}{D-2}g_{00}T.$ (19)
Terms which violate this condition include positive potential energy, certain
string/M theory corrections, and the $p=0$ flux which appears in massive IIa
supergravity. Conceptually, this is the statement that the strong energy
condition is preserved under restriction; if it holds on $M\times X$ then it
will hold on $M$ [19, 20]. This no-go theorem was independently rediscovered
by deWit-Hari Dass-Smit [11] and Maldacena-Nunez [27].
Coming back to Eq. (12), while it might seem that we have justified Eq. (1),
there is an issue. We needed to use the $D$-dimensional Einstein equation Eq.
(3) in order to determine $R^{(k)}$. On the other hand, the effective
potential was supposed to be defined independently of solving the $d$ or
$D$-dimensional Einstein equation, so it is not clear that Eq. (1) can be
interpreted in this way.
Certainly, we cannot interpret Eq. (16) as an effective potential. This is
clear from the nontrivial factor in the relation between $T_{00}$ and
$\Lambda$, which even has the wrong sign. This factor arose because we solved
various equations of motion, including Eq. (3), and substituted the solutions.
While the final results are correct, identifying terms in partial results with
terms in an effective action is usually not.
We could correctly identify the curvature term in Eq. (12) with a term Eq. (1)
in the $d$-dimensional effective potential, if we could do this without
solving any equations of motion. However, we cannot simply relax Eq. (3), as
part of it (the $G_{00}$ component) is in fact a constraint. As we mentioned
in the introduction, if we do not enforce this constraint, the Einstein action
and thus the potential Eq. (1) is unbounded below and has no local minima. To
see this, evaluate Eq. (1) (using the results in the appendix) for the
conformally transformed metrics
${\tilde{g}}_{ij}=e^{2B}g_{ij},$ (20)
obtaining
$V_{eff}=\frac{1}{2}\int\sqrt{g}e^{(k-2)B}\left(-R^{(k)}[g]-(k-2)(k-1)(\nabla
B)^{2}\right),$ (21)
with the wrong sign for the derivative term.
Now it is true that the potential in gravity and supergravity is often
unbounded below, but only to one or a few directions in field space. On the
other hand, with a negative definite spatial derivative term, every short
wavelength perturbation of the conformal factor is tachyonic, so the theory
would be completely unstable. Somehow, this problem must be fixed by
incorporating the constraints of the $D$-dimensional theory.
Having seen the problem, it is not hard to come up with a strategy for dealing
with it. The constraints have two parts, a “zero mode” part and a “non-zero
mode” part. The former become constraints in the $d$-dimensional theory, while
the latter need to be solved in the process of dimensional reduction.
### 2.3 Incorporating the warp factor
We now study the dependence of the action on the warp factor, and consider the
general warped ansatz
$ds^{2}\equiv{\tilde{g}}_{AB}dx^{A}dx^{B}=e^{2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2B(y)}g_{ij}dy^{i}dy^{j}.$
(22)
We denote the metric with $A=B=0$ as $g_{AB}$, and the metric Eq. (22) as
${\tilde{g}}_{AB}$.
As $B$ is redundant with $g_{ij}$, for now we will take $B=0$, until we are
ready to discuss this part of the problem. Using the results in the appendix,
the $k$-dimensional part of the Einstein equations becomes
$R^{(k)}_{ij}-\frac{1}{2}g_{ij}R^{(k)}=T^{(k)}_{ij}+d\nabla_{i}\nabla_{j}A+d\nabla_{i}A\nabla_{j}A+g_{ij}\left(\frac{1}{2}e^{-2A}R^{(d)}-d\nabla^{2}A-\frac{d(d+1)}{2}(\nabla
A)^{2}.\right)$ (23)
The $d$-dimensional trace Eq. (10) becomes
$-\frac{d-2}{2}e^{-2A}R^{(d)}+d(d-1)\nabla^{2}A+\frac{d^{2}(d-1)}{2}(\nabla
A)^{2}=T^{(d)}+\frac{d}{2}R^{(k)}.$ (24)
with
$T^{(d)}=-\frac{d}{2}\sum_{p}|F^{(p)}|^{2}+T^{(d)}_{string}$ (25)
At this point we have added the term $T^{(d)}_{string}$ which represents the
non-classical sources present in superstring theory. We will not use its
detailed form, only the fact that it allows violating the inequality
$T^{(d)}\leq 0$.
Note that, except for $R^{(d)}$, every term in Eq. (24) comes with the same
weight $e^{\alpha A}$ with $\alpha=0$.222 See for example equations (2.12) and
(2.13) of [11]. This dependence has only two sources: the overall
$\sqrt{g^{(d)}}$, which sits in front of every term, and explicit factors of
the $d$-dimensional metric $g_{\mu\nu}$ or its inverse. However, since every
term is a scalar in space-time, and does not contain $d$-dimensional
derivatives, the $d$-dimensional metric cannot appear. This also includes
sources in $T^{(d)}_{string}$ which are space-time scalars. While the argument
we just gave does not cover Chern-Simons terms or electric field strengths,
they also have the same weight, as we argue in §2.7.
Eq. (24) can be dramatically simplified by the change of variable
$u\equiv e^{dA/2},$ (26)
and becomes
$-\frac{d-2}{2}R^{(d)}u^{1-4/d}=-{2(d-1)}\nabla^{2}u+\left(\frac{d}{2}R^{(k)}+T^{(d)}\right)u.$
(27)
This change of variables is used to great advantage in related mathematical
work, on the Yamabe problem [26] and in Perelman’s entropy functional [28].
Readers who have looked at this, or at Tseytlin’s discussion of an entropy
functional for sigma models [35] will recognize many ingredients of the
following discussion, for example Eq. (60).
### 2.4 Effective potential
We seek a functional of $u$ and $g_{ij}$ (and matter fields) whose variation
produces the two Einstein equations. Let us try direct substitution of the
ansatz into the $D$-dimensional action. We continue to take $B=0$, then
$\displaystyle S_{eff,R}$ $\displaystyle=$
$\displaystyle\int\sqrt{{\tilde{g}}}\left({\tilde{R}}^{(D)}-\frac{1}{2}|F^{(p)}|^{2}\right)$
$\displaystyle=$
$\displaystyle\int\sqrt{g}\left(u^{2-4/d}R^{(d)}+u^{2}R^{(k)}+\frac{4(d-1)}{d}(\nabla
u)^{2}-\frac{u^{2}}{2}|F^{(p)}|^{2}\right).$
The similarity to the string action with the dilaton is not coincidental, and
is because one can also obtain the dilaton by dimensional reduction.
Varying this with respect to $g_{ij}^{(k)}$, we get
$\displaystyle
u^{2}\left(R^{(k)}_{ij}-\frac{1}{2}g_{ij}R^{(k)}\right)-\nabla_{i}\nabla_{j}u^{2}+g_{ij}\nabla^{2}u^{2}$
$\displaystyle=$ $\displaystyle
u^{2}T^{(k)}_{ij}-\frac{4(d-1)}{d}\nabla_{i}u\nabla_{j}u$
$\displaystyle-\frac{1}{2}g_{ij}\left(-\frac{4(d-1)}{d}(\nabla
u)^{2}-R^{(d)}u^{2-4/d}\right)$
which can be checked to be equivalent to Eq. (23) with the substitution Eq.
(26). Here $T^{(k)}_{ij}$ is defined to be the variation of the matter action
with respect to $g_{ij}^{(k)}$.
Since $e^{2A}=u^{4/d}$, the variation $\delta g^{\mu\nu}=g^{\mu\nu}$ which
leads to Eq. (24), should be equivalent to varying with respect to $u$ and
multiplying by $-d/4$, and it is.
If we now try to identify the effective potential by direct comparison of Eq.
(2) and Eq. (6), we find
$\int\sqrt{g}\frac{u^{2}}{2}\left(-R^{(k)}+\frac{1}{2}\sum_{p}|F^{(p)}|^{2}-\frac{2}{d}T^{(d)}_{string}\right)-\frac{2(d-1)}{d}(\nabla
u)^{2}$ (30)
which is close but not quite right, because the $R^{(d)}$ term is missing from
the equations of motion. However, we cannot simply add it back, as we are
trying to define an effective potential which is purely a functional of the
$k$-dimensional fields.
Rather, we implement the strategy described at the end of §2.2. The equation
$0=\delta S/\delta u$ is a constraint equation, so we need to enforce it in
the definition of $V_{eff}$, except for a zero mode part. The zero mode part
is exactly the part sourced by the $R^{(d)}$ term, and thus we can do this by
replacing $R^{(d)}$ with an undetermined constant $C$. If the $d$-dimensional
equations equate this to $R^{(d)}$, we will get the correct $k$-dimensional
equations of motion.
On the other hand, from the point of view of the $d$-dimensional equations Eq.
(7), we are adding an extra term to $V_{eff}$. To reconcile the various
equations, we need to subtract the same term without the $u$ dependence.
While this may seem a bit ad hoc, there is another way to justify it. The term
in Eq. (2.4) which becomes the $d$-dimensional Einstein term is
$\int\sqrt{g}u^{2-4/d}\,R^{(d)}.$ (31)
Thus we identify the $d$-dimensional Newton’s constant as
$\frac{1}{G_{N}}=M_{Planck,d}^{d-2}=\int\sqrt{g}e^{(d-2)A}=\int\sqrt{g}u^{2-4/d},$
(32)
so we can interpret the constant $C$ as a Lagrange multiplier which enforces
this definition. This leads to
$V_{eff}=\frac{1}{2}\int\sqrt{g}\left[u^{2}\left(-R^{(k)}+\frac{1}{2}\sum_{p}|F^{(p)}|^{2}-\frac{2}{d}T^{(d)}_{string}\right)-\frac{4(d-1)}{d}(\nabla
u)^{2}\right]+\frac{1}{2}C\left(\frac{1}{G_{N}}-\int\sqrt{g}u^{2-4/d}\right)$
(33)
With this definition, the $\delta V_{eff}/\delta u=0$ constraint becomes
$-\frac{d-2}{2}Cu^{1-4/d}=-{2(d-1)}\nabla^{2}u+\left(T^{(d)}+\frac{d}{2}R^{(k)}\right)u,$
(34)
Given $C$, this is a sensible constraint, but we need to explain how to choose
$C$, and how this eventually implies Eq. (27). Before explaining this, let us
look at how we would solve Eq. (34).
### 2.5 Solving the constraint
To begin, let us set $C=0$, and instead add a term $\lambda u$ on the left
hand side. We get a Schrödinger-type equation on $X$,333 This equation (with
$T^{(d)}=0$) appears in [28, 35], although not with the interpretation of $u$
as a warp factor. It is also the $d\rightarrow\infty$ limit of Eq. (34), a
relation used in the math literature. Also, on replacing $d$ with $2-k$, Eq.
(34) with $T^{(d)}=0$ becomes the Yamabe equation [26].
$\lambda u=-\nabla^{2}u+\frac{d}{2(d-1)}U\cdot u.$ (35)
with a “local potential”
$U\equiv\frac{1}{2}R^{(k)}+\frac{1}{d}T^{(d)}.$ (36)
As is very familiar, such an equation has solutions $u_{i}$ for a discrete
spectrum of eigenvalues $\lambda_{i}$, and the $u_{i}$ form a complete basis
for functions on $X$.
This is the relevant equation for Minkowski space-time with $R^{(d)}=0$, so
let us discuss this case first. Since $u>0$ by its definition Eq. (26), the
only acceptable solution is the ground state $u_{0}$. However, the eigenvalue
$\lambda_{0}$ typically will not be zero. For this to be true, the integral of
the potential (against $u$) must be zero, which is the “warp factor
constraint” of [6, 21]. For compactification with $X$ Ricci flat and nonzero
flux, one can only satisfy this by adding sources with $T^{(d)}_{string}>0$,
such as orientifolds or certain $\alpha^{\prime}$ corrections. The same will
be true if we try to realize Minkowski space-time using an $X$ with negative
total scalar curvature $\int\sqrt{g}R^{(k)}<0$.
Keeping in mind our Schrödinger equation intuition, we return to the actual
constraint equation Eq. (34). Integrating over $X$, we derive the warp factor
constraint,
$-\frac{d-2}{2}C\int\sqrt{g}u^{1-4/d}=\int\sqrt{g}u\left(T^{(d)}+\frac{d}{2}R^{(k)}\right).$
(37)
While this is still a necessary condition, once we allow $C\neq 0$ it can
always be solved, and relates $C$ to the scale of $u$.
This is not yet the constraint of the no-go theorems [11, 27], which (as we
discuss in §5.1) uses the $k$-dimensional Einstein equation to control the
sign of the right-hand-side. If one knows that the local potential Eq. (36)
has a definite sign, clearly $C$ (and ultimately $R^{(d)}$) must have the
opposite sign. However, the local potential $U$ need not have a definite sign,
in which case one needs to know the warp factor $u$ to find the sign of the
integral in Eq. (37). Even when it has a definite sign, we need this
information to estimate $C$.
Rather fortuitously, for $d=4$, the equation Eq. (34) is linear inhomogeneous,
so we can easily get more information. Given the normalized eigenfunctions
$u_{i}$, satisfying Eq. (35) with eigenvalues $\lambda_{i}$ and
$\int\sqrt{g}u_{i}u_{j}=\delta_{ij},$ (38)
we can write the solution as
$u(y)=-\frac{1}{6}C\sum_{i}u_{i}(y)\frac{1}{\lambda_{i}}\int_{X}\sqrt{g}\,u_{i}.$
(39)
If there are any modes with $\lambda_{i}=0$ (normally there will not be), they
can be left out of the sum, and enter the solution with undetermined
coefficients.
To summarize, although we cannot solve the constraint Eq. (27) before knowing
the $d=4$-dimensional curvature $R^{(d)}$, we can solve it up to an overall
coefficient. The equation Eq. (34) obtained by varying Eq. (33) is equivalent
to Eq. (27); we just need to relate the coefficient $C$ to $R^{(d)}$.
The same idea can be applied in $d\neq 4$, and by rescaling $u$ one can set
$C=1$ in Eq. (34), restoring it with the relation $u\propto C^{d/4}$. Of
course, one would have to solve a nonlinear equation. At first sight, the
cases $d>4$ would appear similar to $d=4$, as the nonlinearity is mild. On the
other hand, for $d=3$ the source term blows up as $u\rightarrow 0$, which
looks significant; however we leave the analysis of this for subsequent work.
### 2.6 Interpreting the constraint
To summarize the discussion so far, we seek a $k$-dimensional functional whose
variation leads to the Einstein equations for maximally symmetric
compactifications. By direct substitution of the ansatz, we obtain Eq. (2.4)
which has this property, but it is not purely $k$-dimensional, since it
depends on the $d$-dimensional curvature $R^{(d)}$.
Usually (this is a convention), the $d$-dimensional effective action is
defined in Einstein frame, i.e. with fixed Newton’s constant $G_{N}$. But this
is easy to obtain from Eq. (2.4), because $R^{(d)}$ and $G_{N}$ are conjugate
variables. Thus the effective potential is the Legendre transform of Eq.
(2.4),
$V_{eff}=\frac{1}{2G_{N}}R^{(d)}-\frac{1}{2}S_{eff,R}\,\bigg{|}_{\delta
V_{eff}/\delta R^{(d)}=0}\\\ $ (40)
with $R^{(d)}$ relabeled $C$, and the factor $\frac{1}{2}$ introduced to agree
with the conventions of Eq. (6). This reproduces Eq. (33).
The condition $\delta V_{eff}/\delta R^{(d)}=0$ defining the Newton constant
is
$\frac{1}{G_{N}}=M_{Planck,d}^{d-2}=\int\sqrt{g}e^{(d-2)A}=\int\sqrt{g}u^{2-4/d}.$
(41)
This is an independent condition on $u$ and can also be used to determine the
overall normalization of the warp factor. Thus, we choose the coefficient $C$
in order to satisfy this condition.
Substituting the solution of the constraint Eq. (34) into the effective
potential $V_{eff}$ defined in Eq. (33), and using the definition Eq. (41), we
get
$V_{eff}=\frac{d-2}{2d}\frac{C}{G_{N}}.$ (42)
Thus, if we use this effective potential in Eq. (6), and impose the Einstein
equation following from this action, we obtain Eq. (10) with $C=R^{(d)}$, and
have reproduced the discussion of §2. We stress that we do not need to know
$R^{(d)}$ to compute it, rather this is done by solving Eq. (34), and then
imposing Eq. (41). Nor does one need to impose the $k$-dimensional Einstein
equations; it is defined for general metric and field configurations.
Substituting Eq. (39) into Eq. (41) (for $d=4$), we find that
$\frac{1}{C}=-\frac{1}{6}G_{N}\sum_{i}\frac{1}{\lambda_{i}}\bigg{|}\int\sqrt{g}u_{i}\bigg{|}^{2}.$
(43)
and
$\frac{1}{G_{N}^{2}V_{eff}}=-\frac{2}{3}\sum_{i}\frac{1}{\lambda_{i}}\bigg{|}\int\sqrt{g}u_{i}\bigg{|}^{2},$
(44)
an explicit expression for $V_{eff}$ in terms of $k$-dimensional quantities.
### 2.7 Incorporating the dilaton and other fields
Essentially the same results apply to very general matter theories coupled to
Einstein gravity; in particular supergravity, because every term in the
effective potential will have the same dependence on the warp factor. A
general argument to this effect is as follows. Since the effective potential
is a scalar, whose integral makes a contribution to the $d$-dimensional action
proportional to the space-time volume, the same factor of the $d$-dimensional
space-time volume element must appear in every term. Then, using the
definition made in Eq. (5), the dependence on the warp factor is the same as
the dependence on this volume element.
For example, the NS sector of the type II string actions in string frame leads
to
$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\int\sqrt{g^{(k)}}e^{-2\Phi}\left[u^{2}\left(-R^{(k)}-4(\nabla\Phi)^{2}+\frac{1}{2}|H^{(3)}|^{2}-T^{(d)}_{string}\right)-\frac{4(d-1)}{d}(\nabla
u)^{2}\right]$ (45)
$\displaystyle+C\left(\frac{1}{G_{N}}-\int\sqrt{g}e^{-2\Phi}u^{2-4/d}\right).$
In this form, it is not manifest that the dilaton contribution to the
potential is bounded below. One could make this manifest either by going to
Einstein frame in $k$ dimensions, or (for IIa theory) going to an M theory
description in $k+1$ dimensions. Our later arguments that the potential is
bounded below will assume that this has been done.
Two cases in which the warp factor dependence may not be immediately obvious
are contributions from electric flux, and contributions from the Chern-Simons
terms. The easy way to argue in the first case is to use duality to reexpress
the action in terms of magnetic flux. Thus, a $p$-form electric flux behaves
in expressions such as Eq. (18) like a $D-p$-form magnetic flux.
A more direct argument to this effect uses the fact that a background electric
flux compatible with maximal space-time symmetry must have $p\geq d$, and
transforms like the $d$-dimensional volume form multiplied by a $p-d$-form on
$X$. Then, it is quantized in units of the $d$-dimensional volume form [7].
Taking into account the space-time metric in the term $|F^{(p)}|^{2}$, this
term is independent of the warp factor, so again the entire dependence comes
from the volume form $\sqrt{g^{(d)}}$.
As for the Chern-Simons terms, these will only contribute to the effective
potential in the presence of a background electric field; for example in M
theory we can have
$\int C^{(3)}\wedge G^{(4)}\wedge G^{(4)}$ (46)
and a background $G^{(4)}=Gdx^{0}dx^{1}dx^{2}dx^{3}$ in $d=4$. Again, the
quantization condition on the electric field will force the same warp factor
dependence.
## 3 Physical discussion
Let us recap the final expression for the effective potential. We have
eliminated all dependence on $d$-dimensional physics except the number $d$,
which we now set to $d=4$. The metric, curvature, fluxes and warp factor $u$
are all defined on $X$, as is the additional “stringy source”
$T^{(d)}_{string}$.
$V_{eff}=\frac{1}{2}\int\sqrt{g}\left[u^{2}\left(-R+\frac{1}{2}\sum_{p}|F^{(p)}|^{2}-\frac{1}{2}T^{(d)}_{string}\right)-3(\nabla
u)^{2}\right]+\frac{1}{2}C\left(\frac{1}{G_{N}}-\int\sqrt{g}\,u\right).$ (47)
To use it, one enforces the warp factor constraint $\delta V_{eff}/\delta
u=0$, which in $d=4$ is linear,
$-\frac{1}{6}C=\left(-\nabla^{2}+\frac{2}{3}U\right)u;\qquad
U=\frac{1}{2}R-\frac{1}{4}\sum_{p}|F^{(p)}|^{2}+\frac{1}{4}T^{(d)}_{string}$
(48)
thus determining $C$ and $u$ up to an overall normalization, and the warped
volume constraint
$\frac{1}{G_{N}}=\int\sqrt{g}u,$
thus determining the normalization. Due to the simple form of Eq. (47), its
final value on substituting $u$ is simply $C/4G_{N}$, as in Eq. (42).
### 3.1 Physical regimes
With the main result in hand, let us discuss some of its physics in $d=4$.
First, we consider a constant warp factor,
$u=\frac{1}{G_{N}\mbox{Vol}X},$ (49)
so Eq. (47) reduces to
$V_{eff,unwarped}=-\frac{\int\sqrt{g}\,U}{(G_{N}\mbox{Vol}X)^{2}}.$ (50)
In this case, the original intuition leading to Eq. (1) is correct. This will
be exact if $U$ is constant.
Before we continue, there are two general conventions we could take for
$G_{N}$. When discussing fixing of moduli, such as $\mbox{Vol}X$, we want to
exhibit the dependence of different terms of the potential on the moduli, so
we should choose a fixed $G_{N}$ in fundamental units. In this case we have a
universal $1/(\mbox{Vol}X)^{2}$ (in $d=4$) prefactor for the potential, as in
[22].
As the simplest possible example, taking $X=S^{k}$, with $\mbox{Vol}X=L^{k}$,
and turning on flux $F^{(p)}$ with $p=k$, we have
$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\sqrt{g}u^{2}(-R+\frac{1}{2}||F||^{2})$ (51)
$\displaystyle=$
$\displaystyle\frac{1}{2G_{N}^{2}L^{2k}}\left(-R\,L^{k-2}+\frac{F^{2}}{2L^{k}}\right)$
(52)
where $R>0$ is the curvature of a diameter $1$ sphere. This potential has a
unique negative minimum. In this approximation, one recovers many of the
effective potentials found in the string compactification literature.
Once the volume is fixed, one is better off taking $G_{N}\propto
1/\mbox{Vol}X$, in which case $u\sim 1$. This is the convention we will follow
below.
As we now explain, there appear to be two general regimes, distinguished by
whether the warp factor is slowly varying or not. Since the new physics of the
warp factor has to do with its variation, the first might also be called “weak
warping,” or some more euphonious version of this.
### 3.2 Slowly varying warp factor
If the variation of $U$ is small, we can try to ignore the derivatives in Eq.
(48), to find
$u=-\frac{C}{4U}=-\frac{G_{N}V_{eff}}{U}.$ (53)
One necessary condition for this to make sense is that $U$ must have a
definite sign. Treating the derivatives as a perturbation, and solving to
first order, another condition for this approximation to be good is
$\bigg{|}\nabla^{2}\left(\frac{1}{U}\right)\bigg{|}<<1.$ (54)
While these conditions can be satisfied, for example in AdS compactifications,
they are quite restrictive. Localized sources like branes and orientifold
planes will violate them. Even without these, if $U$ varies on a scale $L$ (in
fundamental units), say set by the size of $X$, then we require
$L^{k+2}\cdot|G_{N}^{2}V_{eff}|>>1.$ (55)
Thus one requires large vacuum energy, large volume or both.
In this approximation, $V_{eff}$ is always smaller in magnitude than Eq. (50).
To see this, we begin by substituting Eq. (53) into Eq. (47), to get an
explicit expression for $V_{eff}$ in terms of $U$,
$\frac{1}{G_{N}^{2}V_{eff}}=\pm\left\|{\frac{1}{U}}\right\|_{L^{2-4/d}},$ (56)
where the sign is opposite to that of $U$, and we use the mathematical
notation
$\left\|{f}\right\|_{L^{\alpha}}=\left(\int\sqrt{g}\,|f|^{\alpha}\right)^{1/\alpha}.$
(57)
Then, we use the Cauchy-Schwarz inequality
$\left\|{f}\right\|\,\left\|{g}\right\|\geq(f,g)^{2},$ (58)
applied to $f=U^{1-2/d}$ and $g=U^{2/d-1}$.
Thus, in this regime warping increases the effective potential for AdS, and
would decrease it for dS.
### 3.3 The ground state approximation
When the condition Eq. (54) is not satisfied, we need to take the derivatives
into account. Let us consider the idea that, to a good approximation, the warp
factor will be proportional to the ground state wave function $u_{0}$. When it
is, we can drop the other terms in Eq. (39), to find
$u\sim cu_{0};\qquad\frac{1}{c}\equiv G_{N}\int\sqrt{g}u_{0};\qquad
C\sim-6G_{N}c^{2}\lambda_{0}$ (59)
and
$V_{eff}=-\frac{3}{2}c^{2}\lambda_{0}.$ (60)
Here $u_{0}\sim 1/\sqrt{\mbox{Vol}X}$ (since it is normalized), so
$c\sim\sqrt{\mbox{Vol}X}$ and $u\sim 1$.
Of course, a constant warp factor is the ground state for constant $U$, so
there will some nearby regime in which this is a good approximation. Here are
various arguments that this regime is large:
* •
Out of all the source terms $\int\sqrt{g}u_{i}$ in Eq. (39), only $i=0$ has a
positive definite integrand. For the other eigenmodes, the integral will have
cancellations, typically making it much smaller than $1/G_{N}$.
* •
Since the other eigenfunctions always have nodes and are negative in some part
of $X$, if they come in with significant coefficients, the solution is in
danger of violating the consistency condition that $u>0$ at each point.
* •
Normally, the $1/\lambda_{i}$ factor will be maximized for $i=0$. This is
clear if the spectrum is positive. If $\lambda_{0}<0$, there are good reasons
(which we discuss below) to expect that the other $\lambda_{i}>0$, in which
case all but a few will have $\lambda_{i}>>|\lambda_{0}|$.
In §3.5, we will further discuss the validity of this approximation. To
oversimplify a bit, it will be good in the opposite regime from Eq. (55),
$L^{k+2}\cdot|G_{N}^{2}V_{eff}|\lesssim 1.$ (61)
In this approximation, warping significantly increases the effective
potential:
$V_{eff}=-\frac{3}{2}c^{2}\lambda_{0}>V_{eff,unwarped}=-\frac{\int\sqrt{g}\,U}{(G_{N}\mbox{Vol}X)^{2}}.$
(62)
This follows from the variational bound for the wave function $u={\rm
constant}$,
$-\lambda_{0}\geq-\frac{\int\sqrt{g}\,U}{\mbox{Vol}X},$ (63)
the Cauchy-Schwarz inequality
$\frac{1}{c^{2}}=G_{N}^{2}(\int\sqrt{g}u_{0})^{2})\leq G_{N}^{2}\mbox{Vol}X,$
(64)
and finally $3/2>1$.
### 3.4 Basic picture
The two approximations we discussed do not cover all the possibilities, and
one might study others, or develop a mixed picture by using each in different
regions and patching them together. However the general picture is similar
enough in both that we leave this for subsequent work, and instead try to
outline the physical picture.
As is familiar from quantum mechanics, the ground state wave function will
have most of its support in potential wells. This is also true (although less
so) for Eq. (53) with $U>0$.
Here, the local potential is minus the energy density $T_{00}$, so the warp
factor will be concentrated in regions of positive energy. Conversely, regions
with negative energy will have this energy warped (or redshifted) away.
Most sources, such as fluxes, make positive contributions to the energy
density. This also includes negative curvature. On the other hand, positive
curvature, and some stringy sources like orientifolds, make negative
contributions to the energy density.
Physically, the warp factor relates energy scales on $X$ to those in space-
time. In the language of AdS/CFT (see for example [36]), one says that $u$ is
large in the “UV” and small in the “IR.” The usual AdS solutions are supported
by $X$ with positive curvature. This contributes a negative energy density
which outweighs the flux energy, and sends $u\rightarrow 0$, consistent with
the picture we just gave. Conversely, the parts of $X$ with positive energy
density, either from flux or from negative curvature, are the UV.
Thus, the basic physics is that, while the vacuum energy density is a sum of
effects from different regions of the manifold, each given by minus the local
potential $U$, we must take into account the local warp factor in adding
energy densities, which we do with the weighing $u^{2}\cdot U$. This favors
the UV region, with positive energy, and thus raises $V_{eff}$.
This gives us a physical answer to the puzzle raised in the introduction, that
varying the conformal factor might send the effective potential to arbitrarily
negative values. Such variations produce positive scalar curvature, so
decrease the warp factor, and redshift away the negative energy. In §4 we will
try to demonstrate this from the equations.
In an actual vacuum, the local potential $U$ is determined by the
$k$-dimensional Einstein equation in a way we will discuss below. If one can
find solutions in which it is negative in some regions and positive in others,
since the warp factor weighs the negative regions more heavily, taking it into
account should make it easier to find de Sitter solutions. On the other hand,
the no-go theorems show that negative $U$ is not so easy to obtain.
### 3.5 Going through $\Lambda=0$
What happens if one varies parameters in $k$ dimensions (say pseudo-moduli or
fluxes), so that $V_{eff}$ crosses zero? While in conventional field theory,
there is nothing special about zero energy, in gravity there definitely is.
Can we see any sign of this in $k$ dimensions?
As we recalled in the introduction, the constructions of de Sitter vacua using
the effective potential, implicitly or explicitly rely on precisely such a
continuation of parameters. For example, one style of argument [25] is to
combine sources of vacuum energy which are computable in a supersymmetric AdS
vacuum, with a single additional supersymmetry breaking energy which “uplifts”
the vacuum to dS. This might be justified by continuing the parameters from a
better controlled nonsupersymmetric AdS vacuum, or conversely some flaw in the
reasoning might appear at this step. Thus, it is crucial to understand this
point.
At first sight, the series solution Eq. (39) for the warp factor looks
singular as an eigenvalue $\lambda$ passes through zero. We can get a simple
model for this by considering the family of local potentials obtained by a
simple shift $a$ of the energy,
$U_{a}\equiv U+a.$ (65)
In massive IIa theory, the $(F^{(0)})^{2}$ source has precisely this effect
(with $a<0$).
Clearly the eigenfunctions $u_{i}$ are independent of $a$, while the
eigenvalues simply become $\lambda_{i}+a$. Thus the series solution becomes
$u(y)=-\frac{C}{6}\sum_{i}u_{i}(y)\frac{1}{\lambda_{i}+a}\int\sqrt{g}u_{i}.$
(66)
As we approach the resonance, it should be a good approximation to keep only
the resonant term in the sum. Now, there are two cases:
* •
An excited state, i.e. $i>0$, passes through zero, $\lambda_{i}+a\sim 0$.
Assuming that the matrix element $\int\sqrt{g}u_{i}\neq 0$, which (in the
absence of some special consideration such as symmetry) will almost surely be
true, we will have $u\sim u_{i}$ near the resonance. But since the excited
state wave functions $u_{i}$ all have nodes, this is inconsistent with
positivity of $u$. Before we reach this point, we will find $u=0$ somewhere on
$X$, so our assumptions must break down.
* •
On the other hand, if the ground state energy passes through zero, we are
simply doing the same truncation to the ground state that we discussed in
§3.3. All of the approximate results given there, such as Eq. (59) and Eq.
(60), have sensible continuations through $\lambda_{0}=0$.
Thus, we see no obstacle to continuing $V_{eff}$ through zero to a de Sitter
vacuum with $\lambda_{0}<0$, up to the point where $\lambda_{1}$ crosses zero.
Somewhere before that point, at a vacuum energy
$\Lambda\sim M_{Planck,4}^{4}(\lambda_{1}-\lambda_{0}),$ (67)
there will be some sort of transition. It is not clear whether this can be
described using supergravity. If it can, perhaps the factorized form Eq. (5)
of the metric breaks down. If not, then assuming $u$ is analytic (as will be
true for $U$ analytic) and vanishes at a point $y_{0}$, the metric near this
point will look like (in $d=4$)
$ds^{2}=\alpha|y-y_{0}|\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}+\ldots,$ (68)
in other words an event horizon. The precise meaning of this depends on the
metric on $X$, and since the $k$-dimensional Einstein equation has source
terms which blow up as $u\rightarrow 0$, we need to look at details of the
example to work this out.
One could get good estimates for this gap using general results on Schrödinger
operators. For example, if the wavefunction is localized to a potential well
$U\sim m^{2}x^{2}/2$, there will be excited states with $\Delta\lambda\sim m$.
Again, the best way to do this probably depends on details of the example at
hand.
A simplified picture is to grant that
$\lambda_{1}-\lambda_{0}\sim L^{-2},$ (69)
which leads to the bound Eq. (61).
### 3.6 Cosmological application
In subsequent work we will try to develop a picture of early cosmological
dynamics based on these results. The basic idea is that the equations of
motion for slow roll inflation,
$3H\frac{\partial\phi^{a}}{\partial t}=-G^{ab}\frac{\partial
V_{eff}}{\partial\phi^{b}},$ (70)
can be computed using Eq. (47), to get a flow on the space of metrics and
other fields on $X$ similar to Ricci flow. This is somewhat like Perelman’s
treatment of Ricci flow as a gradient flow [28].
As is manifest from its definition, the effective potential $V_{eff}$
decreases under the flow; thus one expects that the local potential $U$
becomes more positive, and the eigenvalues $\lambda_{i}$ each increase, in a
way we can roughly model by taking Eq. (65) with $a\sim t$. In this situation
the consistency condition we just discussed might be a significant constraint
on the compactifications which lead to a sensible cosmology.
## 4 Stability
As we saw in §3, we expect the problem raised in the introduction, that the
effective potential is unbounded below under varying the conformal factor, to
be solved by the warp factor. Such variations produce positive scalar
curvature, so decrease the warp factor, and redshift away the negative energy.
In this section we look at how one might show this from the equations. In
fact, at this point we have a precise mathematical
###### Conjecture 1
Consider a conformal class of metrics $\tilde{g}=e^{2B}g$ on a $k$-dimensional
manifold $X$; then the functional Eq. (47) evaluated at its critical point
$\delta V/\delta u=0$, considered as a function on the space of all conformal
factors $B$ with fixed warped volume and volume
$\int\sqrt{\tilde{g}}u^{2-4/d}=C_{1};\qquad\int\sqrt{\tilde{g}}=C_{2};$ (71)
and all $F$, is bounded below.
To be precise, the volumes $C_{1}$ and $C_{2}$ should be defined in the
$k$-dimensional Einstein frame.444 By “Einstein frame,” one means conventions
in which $V_{eff}\sim-\int\sqrt{g}R$ without any field-dependent prefactor, of
the type which appears (for example) in Eq. (45). Such a prefactor can be
removed by a field-dependent conformal transformation.
Although this is not the same claim as the positive energy theorem (there are
AdS vacua), clearly it is related, and perhaps it can proven using some
variation of the arguments used there [31, 38]. Here we make some nonrigorous
but suggestive arguments.
### 4.1 Linearized stability
This was already checked for various explicit solutions in the early Kaluza-
Klein literature. Let us check it at short distances, for which one can simply
take a flat background metric. Thus, we allow a general conformal factor $B$
as in Eq. (22), turning Eq. (2.4) into
$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\int-\sqrt{g}e^{dA+(k-2)B}\left(R^{(k)}-2(k-1)\nabla^{2}B-(k-2)(k-1)(\nabla
B)^{2}+d(d-1)(\nabla A)^{2}\right)$ $\displaystyle=$
$\displaystyle-\int\sqrt{g}e^{dA+(k-2)B}\left(R^{(k)}+2d(k-1)\nabla A\nabla
B+(k-2)(k-1)(\nabla B)^{2}+d(d-1)(\nabla A)^{2}\right)$
While every term individually is negative definite, the determinant of the
$2\times 2$ matrix of kinetic terms is
$\det=(k-1)(k-2)d(d-1)-d^{2}(k-1)^{2}=-d(k-1)(D-2)<0$ (73)
so it has signature $(+1,-1)$. As we expect, the conformal factor is the only
mode with a wrong sign kinetic term. Solving the constraint and substituting
back in, can in principle produce a positive definite effective potential. We
check that this is true at the linearized level for short wavelength
fluctuations of $B$; the linearization of Eq. (24) is
$p^{2}A=-\nabla^{2}A=\frac{k-1}{d-1}\nabla^{2}B$
and the prefactor of the kinetic term becomes
$2d(k-1)\frac{k-1}{d-1}-(k-2)(k-1)-d(d-1)\left(\frac{k-1}{d-1}\right)^{2}=\frac{D-2}{(k-1)(d-1)}.$
(74)
### 4.2 WKB analysis
Next we do a WKB analysis of the effect of short length variations of the
conformal factor. We insert the ansatz
$u\propto e^{dA/2\hbar}$ (75)
into Eq. (48), and take the formal $\hbar\rightarrow 0$ limit, obtaining
$\frac{d}{4}g^{ij}(\nabla_{i}A)\nabla_{j}(dA+2(k-2)\hbar
B)=\frac{\hbar^{2}}{6}U+{\cal O}(\hbar^{2}),$ (76)
where the extra term on the left hand side comes from the connection. Using
the standard formulas, the curvature contribution to $U$ will go as
$U\sim\frac{(k-1)(k-2)}{2}e^{-2B}(\nabla B)^{2},$ (77)
so by taking $B\sim 1/\hbar$ we keep the growing term in the curvature, while
$C$ and the background value of $U$ can be neglected. Thus we are effectively
in the “ground state” regime of §3.3. This also makes the $e^{-2B}$ factor
scale in the same way as the metric $g^{ij}\sim e^{-2A}$.
The WKB estimate for this contribution to the effective potential is then
$V_{eff}\sim-2\int\sqrt{g}e^{dA}U$ (78)
While the integrand looks exponentially small, we should be careful as the
conformal factor can give us exponentially large $\sqrt{g}$. In a small
region, we can approximate $A$ and $B$ by their linear parts $A\sim a\cdot x$
and $B\sim b\cdot x$; then
$da(da+2(k-2)b)=\frac{(k-1)(k-2)}{3}b^{2},$ (79)
while $\sqrt{g}\sim e^{kB}$. We choose $b>0$, while out of the two solutions
for $a$, we choose the one with $a<0$ by the WKB ansatz. One can then show
that the overall exponent $da+(k-2)b$ is negative for all $k>2$.
While the computation looks similar to the linearized analysis, it is not the
same because it takes into account the exponentials in the integrand, and is
thus nonlinear.
### 4.3 Radially symmetric ansatz
Another nonlinear test at short distances can be done by considering a
radially symmetric conformal factor, say
$e^{2B}=(a^{2}+r^{2})^{\gamma},$ (80)
For $-2<\gamma<-1$, the conformal factor becomes large at $r\sim a$, producing
a region of large volume and large positive curvature. On the other hand, it
becomes small for $r>>a$, so one can patch such a region (or “bubble”) into a
general manifold $X$. This type of construction is used in the study of the
Yamabe problem [26].
While we omit the details here, one can show that the effective potential Eq.
(47) remains bounded below in these metrics as well.
To summarize this section, a variety of simple modifications to the conformal
factor have the full nonlinear energy bounded below. Since this would seem to
be necessary for string/M theory compactification on $X$ to be well defined,
we conjecture that it is always true.
From a physics point of view, it would be even more interesting if this worked
for some $X$ and failed for others, as this could give us a consistency
condition and cut down on the plethora of vacua. The only evidence for this we
see so far is that the case $d=3$ might be special, since the source in Eq.
(34) has a negative power of $u$ in this case.
While one might go on to conjecture that $V$ is bounded below for all metrics
on a given manifold $X$, there are some mathematical reasons to doubt this
[2]. Physically it is not required, as long as $V$ has local minima, for which
the tunnelling rates to lower minima are very small (as in [25]).
When $V$ is bounded below, the bound can be regarded as a topological
invariant of $X$, directly analogous to the Yamabe invariant [26]. Of course,
it might be the same invariant, as turned out to be the case for Perelman’s
entropy [2].
## 5 Solutions
So far, we avoided using the $k$-dimensional Einstein equation Eq. (2.4), so
that we could derive an effective potential which is valid off-shell. In this
section we generally assume we are expanding around a solution.
### 5.1 Solving the $k$-dimensional equations
We now bring in the Einstein equation, reinterpreted as above:
$\displaystyle
u^{2}\left(R^{(k)}_{ij}-\frac{1}{2}g_{ij}R^{(k)}\right)-\nabla_{i}\nabla_{j}u^{2}+g_{ij}\nabla^{2}u^{2}$
$\displaystyle=$ $\displaystyle
u^{2}T^{(k)}_{ij}-\frac{4(d-1)}{d}\nabla_{i}u\nabla_{j}u$
$\displaystyle-\frac{1}{2}g_{ij}\left(-\frac{4(d-1)}{d}(\nabla
u)^{2}-Cu^{2-4/d}\right)$
Note that $T^{(d)}_{string}$ does not appear in this formula, however since
normally it depends on the metric on $X$, terms from it will appear in
$T^{(k)}_{ij}$.
Its trace is
$\frac{k-2}{2}R^{(k)}=2(k-1)u^{-1}\nabla^{2}u+\frac{2(D-2)}{d}(u^{-1}\nabla
u)^{2}-\frac{k}{2}u^{-4/d}C-T^{(k)}.$ (82)
Finally, the flux equations of motion are
$d*(u^{2}F)=dF=0.$ (83)
It might seem tempting to eliminate $C$ using the constraint Eq. (24), to get
an equation in terms of $k$-dimensional quantities, equivalent to Eq. (14).
However this is not a good idea, because we would lose the knowledge that $C$
is constant on $X$.
Rather, we can use Eq. (24) to eliminate either $R^{(k)}$ or $-\nabla^{2}u$.
If we do the first, we get
$-(D-2)\nabla^{2}u^{2}+\left(dT^{(k)}-(k-2)T^{(d)}\right)u^{2}=(D-2)Cu^{2-4/d}.$
(84)
Integrating this equation over the manifold gives
$(D-2)C=-G_{N}\int\sqrt{g}\left(dT^{(k)}-(k-2)T^{(d)}\right)u^{2},$ (85)
which is the constraint appearing in the no-go theorems of [11, 27]. Writing
it in terms of $u^{2}$, it becomes a Schrödinger equation with a nonlinear
source.
If we do the other elimination, we get
$R^{(k)}=-\frac{4(d-1)}{d}(u^{-1}\nabla
u)^{2}+u^{-4/d}C+\frac{2(d-1)}{D-2}T^{(k)}-\frac{2(k-1)}{D-2}T^{(d)}.$ (86)
Both are interesting equations. The combinations of flux stress-tensors which
appear are
$\alpha T^{(k)}+\beta
T^{(d)}=\frac{1}{2}\sum_{p}((2p-k)\alpha-d\beta)|F^{(p)}|^{2}.$ (87)
In the original trace equation Eq. (82), we have
$-T^{(k)}-\frac{k}{2}T^{(d)}=\frac{1}{4}\sum_{p}(-4p+k(d+2))|F^{(p)}|^{2},$
(88)
which for $k>4$ is always positive. For Eq. (84), we have
$dT^{(k)}-(k-2)T^{(d)}=d\sum_{p}(p-1)|F^{(p)}|^{2}.$ (89)
As in the no-go theorems, this is non-negative except for $p=0$. Finally, in
Eq. (86) we have
$2(d-1)T^{(k)}-2(k-1)T^{(d)}=\sum_{p}((d-1)2p+k-d)|F^{(p)}|^{2}$ (90)
Again, for $k>d$ this is always positive.
To summarize, Eq. (82) and Eq. (86) make the point that flux favors positive
scalar curvature on $X$, while stringy corrections are needed to get negative
scalar curvature. We can now go on to consider how the curvature is
distributed in the internal dimensions, how this impacts the warp factor, and
whether the consistency condition $\lambda_{1}>0$ of the previous section is
significant or not. We will look at this in examples in future work.
### 5.2 Conformal factor
Next, we introduce the conformal factor. Fixing all the other fields, we would
like to find the conformal factor which minimizes $V_{eff}$. Granting that
this is bounded below, there should be a minimum satisfying $\delta
V_{eff}/\delta B=0$ (of course, there could be other critical points). We need
to fix the volume modulus to have a minimum; rather than do this physically we
simply impose $\mbox{Vol}(X)=\int\sqrt{g}v^{2k/(k-2)}$ with a Lagrange
multiplier $D$.
Taking $v=e^{(k-2)B/2}$, and $F=0$, we can rewrite Eq. (47) as
$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\sqrt{g^{(k)}}\bigg{[}-u^{2}v^{2}R-\frac{4(k-1)}{k-2}(\nabla
v)\nabla(u^{2}v)-\frac{4(d-1)}{d}v^{2}(\nabla u)^{2}$ (93)
$\displaystyle\qquad\qquad+\frac{u^{2}}{2}\sum_{p}v^{2(k-2p)/(k-2)}|F^{(p)}|^{2}\bigg{]}$
$\displaystyle+C\left(\frac{1}{G_{N}}-\int\sqrt{g}v^{2+4/(k-2)}u^{2-4/d}\right)+D\left(\mbox{Vol}(X)-\int\sqrt{g}v^{2+4/(k-2)}\right).$
The various $p$-form flux energies come with different powers of the conformal
factor $v$, given by the $L$ scaling of §3, while Chern-Simons terms do not
depend on the conformal factor. We will not quote these in the equations
below, but they can be easily added.
We expect that $V_{eff}$ is minimized at the simultaneous critical point
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\delta V_{eff}}{\delta
u}=-uv^{2}R+\frac{4(k-1)}{k-2}\,uv\,\nabla^{2}v+\frac{4(d-1)}{d}\nabla(v^{2}\nabla
u)-\frac{d-2}{d}Cv^{2+4/(k-2)}u^{1-4/d}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\delta V_{eff}}{\delta
v}=-u^{2}vR+\frac{2(k-1)}{k-2}\nabla^{2}(u^{2}v)+\frac{2(k-1)}{k-2}\nabla(u^{2}\nabla
v)$ $\displaystyle\qquad\qquad-\frac{4(d-1)}{d}v(\nabla
u)^{2}-\frac{k}{k-2}v^{1+4/(k-2)}\left(Cu^{2-4/d}+D\right).$
We can simplify the $u$ constraint a bit by the change of variables $u=w/v$,
to get
$\displaystyle 0$ $\displaystyle=$ $\displaystyle-
wvR-\left(\frac{4(d-1)}{d}-\frac{4(k-1)}{k-2}\right)\,w\,\nabla^{2}v+\frac{4(d-1)}{d}v\nabla^{2}w-\frac{d-2}{d}Cv^{2+4/(k-2)}(w/v)^{1-4/d}$
(94) $\displaystyle 0$ $\displaystyle=$
$\displaystyle-w^{2}R+\frac{2(k-1)}{k-2}\,v\nabla
v^{-1}\nabla(w^{2})-\frac{4(d-1)}{d}(\nabla w-w\nabla\log
v)^{2}-\frac{k}{k-2}v^{2+4/(k-2)}\left(C(w/v)^{2-4/d}+D\right).$
Except for the source term, the constraint is now linear in $v$ and $w$
separately.
### 5.3 Comparison with supersymmetric ansatzes
Any supersymmetric ansatz should be using the conformal factor which minimizes
the potential, so let us sketch how this works in two well-studied examples.
In [6], K. and M. Becker constructed M theory compactifications on Calabi-Yau
fourfolds, thus $d=3$ and $k=8$. They take $B=-A/2$ (see their Eqs. (2.11) and
(2.30)), so $v=u^{-1}$. Their construction involves both electric and magnetic
four-form flux, so in the magnetic notation uses $p=4$ and $p=7$. Stringy
ingredients include the Chern-Simons term, and an $R^{4}$ anomaly term.
In [21], Giddings, Kachru and Polchinski developed supersymmetric type IIb
compactification on Calabi-Yau threefolds with flux. Here $d=4$ and $k=6$. We
will consider the special case in which the dilaton is constant. They take
$B=-A$, so again $v=u^{-1}$. This construction uses $p=3$ and $p=5$ flux, a
Chern-Simons term $F\wedge H\wedge C^{(4)}$, and orientifold sources to
$T^{(d)}_{string}$.
The two constructions are similar enough that we can make a unified
discussion. Both are based on Calabi-Yau manifolds, so $R=0$. Both lead to
Minkowski space-time, so we can set $C=0$. We note in passing that the ground
state approximation of §3.3 is always exact in this case.
Both constructions use conformally invariant flux with $2p=k$ (call this
$F_{a}$), and an additional $p>k/2$ magnetic flux (call this $F_{b}$) which is
determined by the warp factor, $F_{b}=*^{(D)}\epsilon^{(d)}\wedge d(u^{2})$,
where $*^{(D)}$ is the $D$-dimensional Hodge star in the warped metric.555This
follows from the cancellation between warped tension and potential energy for
supersymmetric space-time filling M2 or D3 branes. Substituting the ansatz, in
both this becomes $F_{b}=*^{(k)}d(v^{2})$.
In both cases, the function $w$ of Eq. (94) is set to a constant (say $1$), so
this equation becomes linear. For Minkowski space-time, $C=0$ and one has an
integral constraint on $T^{(d)}$, which in some sense is a supersymmetric
partner to a topological constraint on $F_{a}^{2}-T^{(d)}_{string}$ (the M2 or
D3 tadpole condition).
Let us derive the equations of motion by re-expressing the effective potential
Eq. (93) in terms of $v$ and $w$,
$V_{eff}=\frac{1}{2}\int\sqrt{g^{(k)}}\bigg{[}-\frac{4(k-1)}{k-2}(\nabla
v)\nabla(w^{2}/v)-\frac{4(d-1)}{d}v^{2}(\nabla(w/v))^{2}+\frac{w^{2}}{2v^{2}}(F_{a}^{2}+v^{-2}F_{b}^{2})\bigg{]}$
(95)
There is also a term $\frac{w^{2}}{2v^{2}}T^{(d)}_{string}$, but since this
always comes with $F_{a}^{2}$ and has the same dependence on $v$ and $w$, we
leave it out until the end.
After varying $V_{eff}$, we set $w=1$, to find
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\delta V_{eff}}{\delta
w}=\alpha\frac{1}{v}\nabla^{2}v+\frac{1}{2v^{2}}(F_{a}^{2}+v^{-2}F_{b}^{2})$
(96) $\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\delta
V_{eff}}{\delta v}=-\alpha(v^{-1}\nabla^{2}v-v^{-2}(\nabla
v)^{2})-\frac{1}{2v^{2}}(F_{a}^{2}+2v^{-2}F_{b}^{2})$ (97)
where
$\alpha\equiv\left(\frac{4(k-1)}{k-2}-\frac{4(d-1)}{d}\right)>0.$
Subtracting these equations, we find
$\alpha(\nabla v)^{2}=\frac{1}{2v^{2}}F_{b}^{2},$ (98)
which is satisfied by $F_{b}=\sqrt{\alpha/2}\nabla v^{2}$. Substituting in Eq.
(96), and restoring $T^{(d)}_{string}$, we find
$-\alpha\nabla^{2}v^{2}=F_{a}^{2}-T^{(d)}_{string},$
which is the warp factor equation in both cases.
Two points worth observing are the supersymmetry relation between the warp
factor and $F_{b}$, and how the positive curvature required by Eq. (82) is
provided by the variation of the conformal factor $v$.
It would be very interesting to find a derivation of the effective
superpotential along the lines of §2, starting with the $D$-dimensional
supergravity Lagrangian and an unbroken supersymmetry, and producing a
functional of the $k$-dimensional fields.
Acknowledgements
This work grew out of ongoing discussions with Renata Kallosh on
compactification using negatively curved manifolds, to whom we give special
thanks. Also influential were many discussions over the years with Tom Banks
on problems with the effective potential in gravity and string theory. We also
thank Michael Anderson, Xiuxiong Chen, Gary Gibbons, Marcus Khuri, Luca
Mazzucato, John Morgan, Joe Polchinski, Dennis Sullivan, Peter van
Nieuwenhuizen and Edward Witten for valuable discussions. Finally, we thank
Steve Giddings for pointing out [23] and subsequent discussions.
This research was supported in part by DOE grant DE-FG02-92ER40697.
## Appendix A Conventions
Supergravity considerations will be accurate when all geometric length scales,
such as the diameter and volume of $X$ and curvature lengths, are greater than
$1$ in fundamental units (we ignore the string coupling), so we take
$M_{Planck,D}=1$ to make this condition evident. We furthermore take the
coordinates on $X$ to range over order $1$, so a geometric length scale $L$
goes as $L^{2}\sim g_{ij}$. Then $\mbox{Vol}X\sim\sqrt{g}\sim L^{k}$,
derivatives $\nabla\sim 1$, and the scalar curvature $R\sim 1/L^{2}$. The
integrals of the $p$-form fluxes over homology cycles $\Sigma^{(p)}$ are
quantized as
$N_{p}\sim\int_{\Sigma^{(p)}}F^{(p)}\sim F\mbox{Vol}\Sigma^{(p)}$ (99)
Thus $\int\sqrt{g}|F^{(p)}|^{2}\sim N_{p}^{2}L^{k-2p}$ in fundamental units,
and $T_{string}\sim 1$.
## Appendix B Computations of curvature
Following Wald appendix D, we write $\nabla_{A}$ for the covariant derivative
adapted to the metric $g$, and ${\tilde{\nabla}}_{A}$ for that adapted to
${\tilde{g}}$, so
$C^{c}_{~{}ab}\equiv\nabla_{a}\omega_{b}-{\tilde{\nabla}}_{a}\omega_{b}.$
(100)
satisfies
$C^{c}_{~{}ab}=\frac{1}{2}{\tilde{g}}^{cd}\left(\nabla_{a}{\tilde{g}}_{bd}+\nabla_{b}{\tilde{g}}_{ad}-\nabla_{d}{\tilde{g}}_{ab}\right).$
(101)
The curvature of ${\tilde{g}}$ is then (Wald 7.5.8)
${\tilde{R}}_{abc}^{~{}~{}~{}d}=R_{abc}^{~{}~{}~{}d}-2\nabla_{[a}C^{d}_{~{}b]c}+2C^{e}_{~{}c[a}C^{d}_{~{}b]e}.$
(102)
Note that Eq. (22) has a $\mathbb{Z}_{2}$ symmetry under
$e^{\mu}\rightarrow-e^{\nu}$, which constrains the connection and curvature
coefficients. Using $\nabla_{a}g_{bc}=R_{ij\mu}^{~{}~{}~{}\nu}=0$, we find
$\displaystyle C^{i}_{~{}jk}$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}\partial_{k}B+\delta^{i}_{k}\partial_{j}B-g_{jk}\nabla^{i}B$
(103) $\displaystyle C^{\mu}_{~{}\nu i}$ $\displaystyle=$
$\displaystyle\delta^{\mu}_{\nu}\partial_{i}A$ (104) $\displaystyle
C^{i}_{~{}\mu\nu}$ $\displaystyle=$
$\displaystyle-e^{2A-2B}\eta_{\mu\nu}\nabla^{i}A$ (105)
$\displaystyle{\tilde{R}}_{ijk}^{~{}~{}~{}l}$ $\displaystyle=$ $\displaystyle
R_{ijk}^{~{}~{}~{}l}+2\delta^{l}_{[i}\nabla_{j]}\nabla_{k}B-2g_{k[i}\nabla_{j]}\nabla^{l}B$
$\displaystyle+2(\nabla_{[i}B)\delta^{l}_{j]}\nabla_{k}B-2(\nabla_{[i}B)g_{j]k}\nabla^{l}B-2g_{k[i}\delta^{l}_{j]}|\nabla
B|^{2}$ $\displaystyle{\tilde{R}}_{ij\mu}^{~{}~{}~{}\nu}$ $\displaystyle=$
$\displaystyle-2\nabla_{[i}C^{\nu}_{~{}j]\mu}+2C^{\lambda}_{~{}\mu[i}C^{\nu}_{~{}j]\lambda}$
$\displaystyle=$ $\displaystyle
0\qquad(\sim\nabla_{[i}\nabla_{j]}A+\nabla_{[i}A\nabla_{j]}A)$
$\displaystyle{\tilde{R}}_{i\mu j}^{~{}~{}~{}\nu}$ $\displaystyle=$
$\displaystyle-\nabla_{i}C^{\nu}_{~{}\mu j}+C^{k}_{~{}ji}C^{\nu}_{~{}\mu
k}-C^{\lambda}_{~{}j\mu}C^{\nu}_{~{}i\lambda}$ $\displaystyle=$
$\displaystyle\delta^{\nu}_{\mu}\left(-\nabla_{i}\nabla_{j}A+C^{k}_{ij}\nabla_{k}A-(\nabla_{i}A)(\nabla_{j}A)\right)$
$\displaystyle=$
$\displaystyle\delta^{\nu}_{\mu}\left(-\nabla_{i}\nabla_{j}A-\nabla_{i}(A-B)\nabla_{j}(A-B)+\nabla_{i}B\nabla_{j}B-g_{ij}(\nabla
A\cdot\nabla B)\right)$
$\displaystyle{\tilde{R}}_{\mu\nu\lambda}^{~{}~{}~{}~{}\rho}$ $\displaystyle=$
$\displaystyle
R_{\mu\nu\lambda}^{~{}~{}~{}~{}\rho}+2C^{i}_{~{}\lambda[\mu}C^{\rho}_{~{}\nu]i}$
$\displaystyle=$ $\displaystyle
R_{\mu\nu\lambda}^{~{}~{}~{}~{}\rho}-2e^{2A-2B}\eta_{\lambda[\mu}\delta^{~{}\rho}_{\nu]}(\nabla
A)^{2}$
The other mixed components can be obtained using the antisymmetry
$R_{[ab][cd]}$. Here ${\tilde{R}}_{ijk}^{~{}~{}~{}l}$ is the same as for a
$k$-dimensional conformal transformation (Wald D.7).
One check is the case $A=B$. The only nontrivial one is
${\tilde{R}}_{i\mu
j}^{~{}~{}~{}\nu}=\delta^{\nu}_{\mu}\left(-\nabla_{i}\nabla_{j}B+\nabla_{i}B\nabla_{j}B-g_{ij}|\nabla
B|^{2}\right)$ (110)
from the standard formula, which agrees with Eq. (B) at $A=B$.
Contracting indices, we find
$\displaystyle{\tilde{R}}_{ik}=R_{ik}$
$\displaystyle-(k-2)\nabla_{i}\nabla_{k}B-g_{ik}\nabla^{2}B+(k-2)\nabla_{i}B\nabla_{k}B-(k-2)g_{ik}(\nabla
B)^{2}$
$\displaystyle+d\left(-\nabla_{i}\nabla_{k}A-\nabla_{i}(A-B)\nabla_{k}(A-B)+\nabla_{i}B\nabla_{k}B-g_{ik}(\nabla
A\cdot\nabla B)\right)$ $\displaystyle{\tilde{R}}_{\mu\lambda}=R_{\mu\lambda}$
$\displaystyle+e^{2A-2B}\eta_{\mu\lambda}\left(-d(\nabla
A)^{2}-\nabla^{2}A-(k-2)(\nabla A\cdot\nabla B)\right)$ (112)
Again, this agrees with the standard formula for $A=B$.
Finally, the scalar curvature is a sum of two terms obtained by tracing these
two contributions. Let us define the Laplacian with respect to the conformally
transformed 6d metric,
$\displaystyle{\tilde{\Delta}}$
$\displaystyle\equiv-\frac{1}{\sqrt{{\tilde{g}}}}\partial_{i}\sqrt{{\tilde{g}}}{\tilde{g}}^{ij}\partial_{j}$
(113) $\displaystyle=$ $\displaystyle-\nabla^{2}-(k-2)\nabla B\cdot\nabla,$
(114)
then
$\displaystyle{\tilde{R}}^{(k)}$ $\displaystyle=$ $\displaystyle
e^{-2B}\left(R^{(k)}-(2k-2)\nabla^{2}B-(k-2)(k-1)(\nabla
B)^{2}+d({\tilde{\Delta}}A-(\nabla A)^{2})\right)$ (115)
$\displaystyle{\tilde{R}}^{(d)}$ $\displaystyle=$ $\displaystyle
e^{-2A}R^{(d)}+de^{-2B}\left({\tilde{\Delta}}A-d(\nabla A)^{2}\right)$ (116)
Note that the term ${\tilde{\Delta}}A-d(\nabla A)^{2}$ is precisely the scalar
Laplacian associated to the metric Eq. (22), restricted to functions
independent of space-time. Thus, it is a total derivative, and its
contribution to $\int\sqrt{{\tilde{g}}}{\tilde{R}}^{(d)}$ vanishes (of course
the other term still depends on $A$).
Finally, the conformally invariant Laplacian is $\Delta_{c}=\Delta+\alpha R$
with
$s=-\frac{k-2}{2};\qquad\alpha=\frac{k-2}{4(k-1)};$ (117)
such that
${\tilde{\Delta}}_{c}e^{sB}\phi=e^{(s-2)B}\Delta_{c}\phi.$ (118)
## References
* [1] B. S. Acharya and M. R. Douglas, “A finite landscape?,” [arXiv:hep-th/0606212].
* [2] K. Akutagawa, M. Ishida and C.. LeBrun, “Perelman’s invariant, Ricci flow, and the Yamabe invariants of smooth manifolds,” Archiv der Mathematik 88 1, 71–76 (2007) [arXiv:math.DG/0610130].
* [3] M. Alishahiha, E. Silverstein and D. Tong, “DBI in the sky,” Phys. Rev. D 70, 123505 (2004) [arXiv:hep-th/0404084].
* [4] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “The hierarchy problem and new dimensions at a millimeter,” Phys. Lett. B 429, 263 (1998) [arXiv:hep-ph/9803315].
* [5] T. Banks, “Landskepticism or why effective potentials don’t count string models,” [arXiv:hep-th/0412129].
* [6] K. Becker and M. Becker, “M-Theory on Eight-Manifolds,” Nucl. Phys. B 477, 155 (1996) [arXiv:hep-th/9605053].
* [7] R. Bousso and J. Polchinski, “Quantization of four-form fluxes and dynamical neutralization of the cosmological constant,” JHEP 0006, 006 (2000) [arXiv:hep-th/0004134].
* [8] U. H. Danielsson, S. S. Haque, G. Shiu and T. Van Riet, “Towards Classical de Sitter Solutions in String Theory,” JHEP 0909, 114 (2009) [arXiv:0907.2041 [hep-th]].
* [9] S. P. de Alwis, “On potentials from fluxes,” Phys. Rev. D 68, 126001 (2003) [arXiv:hep-th/0307084].
* [10] F. Denef, M. R. Douglas and B. Florea, “Building a better racetrack,” JHEP 0406, 034 (2004) [arXiv:hep-th/0404257].
* [11] B. de Wit, D. J. Smit and N. D. Hari Dass, “Residual Supersymmetry of Compactified D=10 Supergravity,” Nucl. Phys. B 283, 165 (1987).
* [12] O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, “Type IIA moduli stabilization,” JHEP 0507, 066 (2005) [arXiv:hep-th/0505160].
* [13] M. Dine and N. Seiberg, “Is The Superstring Weakly Coupled?,” Phys. Lett. B 162, 299 (1985).
* [14] M. R. Douglas, “Statistical analysis of the supersymmetry breaking scale,” [arXiv:hep-th/0405279].
* [15] M. R. Douglas and S. Kachru, “Flux compactification,” Rev. Mod. Phys. 79, 733 (2007) [arXiv:hep-th/0610102].
* [16] M. R. Douglas and G. Torroba, “Kinetic terms in warped compactifications,” arXiv:0805.3700 [hep-th].
* [17] M. J. Duff, B. E. W. Nilsson and C. N. Pope, “Kaluza-Klein Supergravity,” Phys. Rept. 130, 1 (1986).
* [18] A. R. Frey, G. Torroba, B. Underwood and M. R. Douglas, “The Universal Kaehler Modulus in Warped Compactifications,” JHEP 0901, 036 (2009) [arXiv:0810.5768 [hep-th]].
* [19] G.W. Gibbons, “ Aspects Of Supergravity Theories,” Three lectures given at GIFT Seminar on Theoretical Physics, San Feliu de Guixols, Spain, Jun 4-11, 1984. Published in GIFT Seminar 1984:0123 (QCD161:G2:1984).
* [20] G. W. Gibbons, “Thoughts on tachyon cosmology,” Class. Quant. Grav. 20, S321 (2003) [arXiv:hep-th/0301117].
* [21] S. B. Giddings, S. Kachru and J. Polchinski, “Hierarchies from fluxes in string compactifications,” Phys. Rev. D 66, 106006 (2002) [arXiv:hep-th/0105097].
* [22] S. B. Giddings, “The fate of four dimensions,” Phys. Rev. D 68, 026006 (2003) [arXiv:hep-th/0303031].
* [23] S. B. Giddings and A. Maharana, “Dynamics of warped compactifications and the shape of the warped landscape,” Phys. Rev. D 73, 126003 (2006) [arXiv:hep-th/0507158].
* [24] M. P. Hertzberg, S. Kachru, W. Taylor and M. Tegmark, “Inflationary Constraints on Type IIA String Theory,” JHEP 0712, 095 (2007) [arXiv:0711.2512 [hep-th]].
* [25] S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, “De Sitter vacua in string theory,” Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240].
* [26] J. M. Lee and T. H. Parker, “The Yamabe problem,” Bull. Amer. Math. Soc. (N.S.) 17, 1 (1987), 37-91.
* [27] J. M. Maldacena and C. Nunez, “Supergravity description of field theories on curved manifolds and a no go theorem,” Int. J. Mod. Phys. A 16, 822 (2001) [arXiv:hep-th/0007018].
* [28] G. Perelman, “The entropy formula for the Ricci flow and its geometric applications,” [arXiv:math/0211159].
* [29] L. Randall and R. Sundrum, “A large mass hierarchy from a small extra dimension,” Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221].
* [30] A. Saltman and E. Silverstein, “A new handle on de Sitter compactifications,” JHEP 0601, 139 (2006) [arXiv:hep-th/0411271].
* [31] R. Schoen and S. T. Yau, “Positivity Of The Total Mass Of A General Space-Time,” Phys. Rev. Lett. 43, 1457 (1979).
* [32] E. Silverstein, “Simple de Sitter Solutions,” Phys. Rev. D 77, 106006 (2008) [arXiv:0712.1196 [hep-th]].
* [33] L. Susskind, “Supersymmetry breaking in the anthropic landscape,” [arXiv:hep-th/0405189].
* [34] T. R. Taylor and C. Vafa, “RR flux on Calabi-Yau and partial supersymmetry breaking,” Phys. Lett. B 474, 130 (2000) [arXiv:hep-th/9912152].
* [35] A. A. Tseytlin, “On sigma model RG flow, ’central charge’ action and Perelman’s entropy,” Phys. Rev. D 75, 064024 (2007) [arXiv:hep-th/0612296].
* [36] H. L. Verlinde, “Holography and compactification,” Nucl. Phys. B 580, 264 (2000) [arXiv:hep-th/9906182].
* [37] R. M. Wald, General Relativity, University of Chicago Press, 1984.
* [38] E. Witten, “A Simple Proof Of The Positive Energy Theorem,” Commun. Math. Phys. 80, 381 (1981).
|
arxiv-papers
| 2009-11-17T19:19:21 |
2024-09-04T02:49:06.530793
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael R. Douglas",
"submitter": "Michael R. Douglas",
"url": "https://arxiv.org/abs/0911.3378"
}
|
0911.3489
|
# On the Casimir entropy between ‘perfect crystals’
C. HENKEL∗ and F. INTRAVAIA444Present address: Theoretical Division, MS B213,
Los Alamos National Laboratory, Los Alamos NM 87545, U. S. A. Institut für
Physik und Astronomie, Universität Potsdam,
Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany
∗E-mail: henkel@uni-potsdam.de
###### Abstract
We give a re-interpretation of an ‘entropy defect’ in the electromagnetic
Casimir effect. The electron gas in a perfect crystal is an
electromagnetically disordered system whose entropy contains a finite Casimir-
like contribution. The Nernst theorem (third law of thermodynamics) is not
applicable.
###### keywords:
Temperature; entropy; dissipation; overdamped mode.
ξ π
## 1 Introduction
It is well known that fluctuation interactions at nonzero temperature are
entropic in character, a prominent example being the critical Casimir effect
in liquid mixtures close to a continuous phase transition (see Ref.Gambassi09
for an overview). The electromagnetic Casimir interaction is also associated
with an entropy that determines its limiting behaviour at high temperatures
and/or large distances[2, 3]. The Casimir entropy for two material plates has
recently attracted much interest also for low temperatures, as for certain
situations a violation of the third law of thermodynamics (the Nernst heat
theorem) has been claimed[4, 5]. This has been used to argue in favor of a
description where the DC conductivity of the metallic plates is ignored.
Although the result of this theoretical prescription provides a better fit to
recent experiments[6], the situation is, however, not satisfactory from the
physical point of view. In addition, a similar analysis for an experiment with
laser-irradiated semiconductors[7] leaves open the meaning of the threshold
value above which the DC conductivity should be included in the theory.
Much has been said about spatially dispersive mirrors where the third law is
verified, due to the anomalous skin effect[8], and where a continuous cross-
over from a dielectric to a perfectly conducting response has been found[9].
We focus in these proceedings on a strictly local framework, mainly for
simplicity, but also to show that this case is thermodynamically consistent as
well. We shall see that, indeed, spatial dispersion plays only a small role in
the range of wave vectors that are relevant for current Casimir experiments.
We take up the interpretation of Ref.Hoye03a where the nonzero Casimir entropy
found as $T\to 0$ was associated to two oscillators coupled via a third one.
Following this idea, we consider two half-spaces filled by an ideal electron
gas, separated by a distance $L$, and provide a direct calculation of the
entropy per area $S(L)=\lim\limits_{T\to 0}S(L,T)$ in one of the two field
polarizations. This calculation highlights the following point:
The Casimir entropy $S(L)<0$ results from the coupling between two systems
that are not in equilibrium as $T\to 0$. They are filled with a _frozen_
magnetization and, in the local limit, have separately divergent (bulk and
surface) entropies that characterize the disorder of the electromagnetic
configuration. The Casimir entropy is the correction to additivity when the
two bodies are close enough for their frozen currents to be mutually coupled
by the quasi-static magnetic fields that ‘leak’ through their surfaces. The
Nernst theorem is clearly not applicable for this disordered system. The
situation is quite similar to the ‘ideal conductor’ (in distinction to a
superconductor, see Ref.Landau10) that does not reach thermodynamical
equilibrium as it is cooled, because its random bulk currents freeze. (See
Ref.Evans65 where it is argued how special this ideal conductor case is.)
## 2 Casimir entropy from frozen medium currents
### 2.1 Motivation
We have analyzed in a recent paper[13] the overdamped field modes to which the
unusually large thermal corrections to the Casimir force between metals can be
attributed. Substantiating previous observations[14, 15], we have interpreted
the characteristic frequency, $\xi_{L}=D/L^{2}=(\mu_{0}\sigma L^{2})^{-1}$, in
terms of a diffusion equation (diffusion coefficient $D$) satisfied by the
magnetic field and electric currents in a medium with DC conductivity
$\sigma$.
There is no contradiction between thermodynamics and fluctuation
electrodynamics in this case. Fields and currents induced in the metal are
clearly damped and lose energy into the phonon bath, say. In equilibrium,
however, this is compensated by field and current fluctuations that are
created by the bath. This concept can be traced back to the Einstein–Langevin
theory of Brownian motion [16] and is also the very essence of the
fluctuation-dissipation theorem[17, 18]. Moreover, for the quantum field
theory, the quantum (or zero-point) fluctuations of the bath variables are an
essential tool to establish at all times the commutation relations for the
field operators[19, 20, 21, 22, 23]. In the field theory considered in
Ref.Intravaia09a, we dealt with overdamped modes: if the wave equation were
homogeneous, its eigenfrequencies would be purely imaginary, similar to free
Brownian particles. The quantum theory of Brownian motion[24] provides a
consistent scheme for the quantum thermodynamics of this damped system. In
this setting, nonzero entropies and even negative heat capacities find a quite
natural explanation (see, e.g., Refs.Ingold09a,Ingold09b).
In the particular case of an ideal electron gas (or ‘perfect crystal’), the
diffusion constant $D=D(T)=\mathcal{O}(T^{2})$. As the temperature drops to
zero, the diffusion-dominated modes of the electromagnetic field do not reach
a unique ground state, but remain in the classical regime $\hbar\xi_{L}\ll T$.
This motivates the present calculation where the electromagnetic Casimir
entropy between perfect crystals is re-derived within a classical model.
The basic ingredient are transverse modes that extend throughout the bulk of
the gas: static current waves interlocked with a magnetic field. The magnetic
fields associated to the bulk currents leave one medium, by continuity, and
cross the vacuum gap to the other medium in the form of (transverse)
evanescent waves. This coupling between the two media changes slightly the
wave vector of each current mode. Summing over all modes, we get a non-zero
change in entropy that depends, quite naturally, on the separation $L$. It
represents the distance-dependent change per unit area of the (much larger)
entropy of the two frozen bulk systems.
### 2.2 Lagrangian and conservation laws
We start with the Lagrangian density
$\mathcal{L}=\frac{nm}{2}\dot{\bfxi}^{2}+en\dot{\bfxi}\cdot{\bf
A}-\frac{1}{2\mu_{0}}(\nabla\times{\bf A})^{2}$ (1)
where the field $\bfxi$ describes the displacement of a charged fluid element,
${\bf A}$ is the vector potential, $n$ a constant background charge density
and $e$ a coupling constant with units of charge. The current density is ${\bf
j}=en\dot{\bfxi}$ [see Eq.(2) below], so that $\dot{\bfxi}$ represents a
velocity field. The first term in Eq.(1) is thus the kinetic energy (density),
the second one a bilinear coupling, and the third one the magnetic energy.
Note that we neglect electric fields here. This is consistent if we make the
assumption that $\nabla\cdot\bfxi=\nabla\cdot{\bf A}=0$. The first equality
ensures that the medium displacement does not produce any charge density, the
second one is the Coulomb gauge. The variation of the Lagrangian (1) with
respect to ${\bf A}$ gives the Faraday equation
$en\dot{\bfxi}-\frac{1}{\mu_{0}}\nabla\times(\nabla\times{\bf A})=0$ (2)
In addition, the displacement field $\bfxi$ is a cyclic variable, hence we get
a conserved momentum field
$\frac{\partial{\bfpi}}{\partial t}=0,\quad{\bfpi}=nm\dot{\bfxi}+en{\bf A}$
(3)
There are two ways to implement this conservation law physically.[27]
(i) In a _London superconductor_ , the current density is tied at all times to
the vector potential, with the momentum ${\bfpi}$ being zero:
$\mbox{superconductor:\ }\quad{\bf j}=en\dot{\bfxi}=-\frac{ne^{2}}{m}{\bf A}$
(4)
The Maxwell–Faraday equation (2) becomes
$\left(\lambda^{-2}-\nabla^{2}\right){\bf A}=0$ (5)
where the Meißner–London penetration depth $\lambda$ is given by the familiar
expression $\lambda^{-2}=\mu_{0}ne^{2}/m=\Omega^{2}/c^{2}$ ($\Omega$ is the
plasma frequency). Eq.(5) does only allow for solutions that start at the
surface and exponentially decay into the bulk on a length scale $\lambda$ (or
shorter). Except for a surface layer of thickness $\sim\lambda$, the interior
of the medium remains free of magnetic field: the Meißner–Ochsenfeld effect.
From the London equation (4), we can also conclude that the Meißner effect is
maintained in time-dependent fields (at least with sufficiently slow
variations; a detailed analysis clearly goes beyond the simple model
considered here). For a given frequency component $\omega$, the ‘dielectric
function’ of the London superconductor can be read off from the polarization
field associated to $\bfxi$:
${\bf P}=en\bfxi=\frac{{\bf j}}{-{\rm
i}\omega}=-\frac{\varepsilon_{0}\Omega^{2}}{\omega^{2}}{\bf E}$ (6)
leading to the so-called plasma model
$\varepsilon(\omega)=1-\Omega^{2}/\omega^{2}$. There is no violation of
causality here, if we read Eq.(4) as a retarded response function between the
current density and the time integral of the electric field (i.e., the vector
potential).
The option (ii) that complies with the conservation law (3) corresponds to an
_ideal conductor_ :
$\mbox{ideal conductor:\ }\quad\frac{\partial{\bf j}}{\partial
t}=0\quad\mbox{and}\quad\frac{\partial{\bf A}}{\partial t}=0$ (7)
which means that currents, once created, are not damped and that the magnetic
field is static. The value of the conserved momentum ${\bfpi}$ is not
restricted otherwise. The Faraday equation (2) then yields the field ${\bf A}$
in terms of its source ${\bf j}$. Note that the magnetic field is in this case
tied to the current density, similar to the scalar potential and the charge
density in Coulomb-gauge electrodynamics [28]. Let us switch to reciprocal
space with wavevector ${\bf q}$: the vector potential ${\bf A}_{\bf q}$
created by the current is
${\bf A}_{\bf q}=\mu_{0}\frac{{\bf j}_{\bf q}}{{\bf q}^{2}}$ (8)
so that the Lagrangian (1) becomes
$L=\frac{\mu_{0}V}{2}\sum_{\bf q}\left(\lambda^{2}+{\bf q}^{-2}\right)|{\bf
j}_{\bf q}|^{2}$ (9)
where $V$ is the quantization volume. The conjugate momentum becomes
$\bfpi_{\bf q}=\frac{m}{e}\left(1+\frac{1}{\lambda^{2}{\bf q}^{2}}\right){\bf
j}_{\bf q}$ (10)
### 2.3 Normal modes and entropy
The normal modes of the effective Lagrangian (9) can clearly be chosen as
plane waves, labelled by wave vector ${\bf q}$ and polarization index $\mu$.
The associated Hamiltonian, expressed in terms of the canonical momentum
field, is then
$H=\frac{V}{2nm}\sum_{{\bf q},\mu}\left(1+\frac{1}{\lambda^{2}{\bf
q}^{2}}\right)^{-1}|{\pi}_{{\bf q}\mu}|^{2}$ (11)
The (classical) thermodynamics of this system is determined by summing the
free energies of the normal modes over the quantum numbers ${\bf q}$, $\mu$.
For one mode, we find by calculating the classical partition function
($\beta=1/T$)
$F_{{\bf q}\mu}=-T\log\int\\!{\rm d}{\pi}_{{\bf
q}\mu}^{\phantom{*}}\exp\left(-\frac{\beta}{2}\epsilon_{q}|\pi_{{\bf
q}\mu}|^{2}\right)=\frac{T}{2}\log\frac{\beta\epsilon_{q}}{2\pi}$ (12)
where $\epsilon_{q}=(V/nm)(1+\lambda^{-2}{\bf q}^{-2})^{-1}$ determines the
mode’s energy. The entropy of this polarization mode is
$S_{{\bf q}\mu}=-\frac{\partial F_{{\bf q}\mu}}{\partial
T}=-\frac{1}{2}\log\frac{\beta\epsilon_{q}}{2\pi}+\frac{1}{2}$ (13)
where the equipartition term $+1/2$ comes from the $\beta$ in the logarithm.
When we sum this over all modes, the entropy becomes to leading order
extensive in the volume $V$ of the medium. The part that depends on the
surface is calculated in the usual way. Consider two media (total volume $V$)
with parallel surfaces of area $\mathcal{A}$ facing each other at a distance
$L$ and write the entropy of the total system in the form
$S=Vs+2{\cal A}S_{\rm surf}+{\cal A}S(L)$ (14)
where $s$ is the (intensive) bulk entropy density, $S_{\rm surf}$ is the
entropy per area of one (isolated) surface and $S(L)$ the Casimir entropy per
area. We can read the latter as the deviation from additivity in the system of
two media: it thus describes how the disorder (or information content) of the
two plates is changed by the coupling across the vacuum gap of thickness $L$.
The physical mechanism for this coupling is the penetration of magnetic fields
through the medium surface, as allowed for by the electromagnetic boundary
conditions. In the vacuum between the media, the fields satisfy the Laplace
equation $\nabla^{2}{\bf A}=0$: for a given wave vector ${\bf k}$ parallel to
the surface, they ‘propagate’ perpendicular to the surface (along the
$z$-axis, say) as evanescent waves $\sim\exp(\pm kz)$ where $k=|{\bf k}|$. For
a single surface, only solutions that decay into the vacuum are permitted. In
the gap $0\leq z\leq L$ between two surfaces, even and odd solutions $\cosh
k(z-L/2)$ and $\sinh k(z-L/2)$ can be constructed. This is illustrated
schematically in Fig.1.
file=standing-waves,width=2.5in
Figure 1: Illustration of standing waves at the surface of an ideally
conducting medium. We plot the component of the vector potential tangential to
the surface. Thin line: isolated surface, thick line: mode between two
surfaces with even parity.
Both the surface entropy and the Casimir entropy can be calculated from the
phase shifts of standing wave modes (see Refs.Barton79,Bordag01 for details).
For the surface entropy,
$S_{\rm surf}=\sum_{\mu}\int\frac{{\rm
d}^{2}k}{(2\pi)^{2}}\int\limits_{0}^{\infty}\\!{\rm d}k_{z}S_{{\bf
q}\mu}\left(-\frac{1}{\pi}\frac{\partial\theta_{\mu}}{\partial k_{z}}\right)$
(15)
where ${\bf q}=({\bf k},k_{z})$, and the mode functions in the medium ($z\leq
0$) are proportional to ${\rm e}^{{\rm i}{\bf k}\cdot{\bf
r}_{\|}}\sin(k_{z}z+\theta_{\mu})$ with ${\bf r}_{\|}=(x,y)$ the coordinates
parallel to the surface. From this form, we can also read off a ‘reflection
coefficient’ for (time-independent) waves from within the medium,
$r_{\mu}=-{\rm e}^{-2{\rm i}\theta_{\mu}}$. From a physical point of view, we
can interpret the phase derivative in Eq.(15) as a density of modes in ${\bf
q}$-space, more precisely, its change due to the surface.
For an isolated interface, the usual matching of the component of the vector
potential tangential to the surface and its derivative at the interface yields
in TE-polarization (current perpendicular to the plane of incidence spanned by
${\bf k}$ and the surface normal)
$r_{\rm TE}=\frac{k_{z}-{\rm i}k}{k_{z}+{\rm i}k},\qquad\tan\theta_{\rm
TE}=-\frac{k_{z}}{k}$ (16)
Here, $k=|{\bf k}|$ gives the decay constant of the evanescent wave on the
vacuum side. In the TM-polarization (current in the plane of incidence), the
current has to satisfy the boundary condition $\lim\limits_{z\to 0}j_{z}(z)=0$
to avoid the build-up of a surface charge sheet. (That case would require
electrical field energy in the Lagrangian (1) and is best described within a
spatially dispersive model.[29]) This boundary condition immediately leads to
$r_{\rm TM}=1$, and there is no phase shift. The surface entropy in TM-
polarization hence vanishes, while it is logarithmically divergent at large
${\bf q}$ in the TE-polarization: $S_{\rm
surf}\approx-(8\pi\lambda^{2})^{-1}\log(q_{c}\lambda)$ with a short-range
cutoff $q_{c}$. One needs a nonlocal description of the material response
(spatial dispersion) to get a finite result, see, e.g., Ref.Horing85 for the
surface self-energy.
For the Casimir entropy, a local calculation is sufficient, as we shall see
now: the reflection phases for even and odd modes in the vacuum gap are found
as (we henceforward suppress the TE-polarization label)
$\tan\theta_{\rm
even}(L)=-\frac{k_{z}}{k}\coth\frac{kL}{2},\qquad\tan\theta_{\rm
odd}(L)=-\frac{k_{z}}{k}\tanh\frac{kL}{2}$ (17)
The entropy per area for the two-surface system, $2S_{\rm surf}+S(L)$, is then
given by Eq.(15) with $\theta$ replaced by $\theta_{\rm even}(L)+\theta_{\rm
odd}(L)$. Subtract twice the single-interface phase shift and calculate the
quantity
$\exp 2{\rm i}[\theta_{\rm even}(L)+\theta_{\rm
odd}(L)-2\theta]=\frac{1-r^{2}\,{\rm e}^{-2kL}}{1-(r^{*})^{2}\,{\rm
e}^{-2kL}}$ (18)
as can be checked with straightforward algebra. The Casimir entropy from TE-
polarized bulk currents becomes [combining Eqs.(13, 15, 18)]
$S(L)=\int\limits_{0}^{\infty}\\!\frac{k{\rm
d}k}{2\pi}\int\limits_{0}^{\infty}\\!\frac{{\rm
d}k_{z}}{2\pi}\left(\log\frac{\beta V\lambda^{2}{\bf q}^{2}}{2\pi
nm(1+\lambda^{2}{\bf q}^{2})}-1\right)\frac{\partial}{\partial k_{z}}{\rm
Im}\,\log\left(1-r^{2}\,{\rm e}^{-2kL}\right)$ (19)
The terms independent of $k_{z}$ in the entropy per mode are irrelevant: after
a partial integration, the integrated terms vanish because for $k_{z}\to
0,\infty$, the reflection coefficient $r$ becomes real. Manifestly, short-
wavelength modes with $2kL\gg 1$ are suppressed, and a local theory is
sufficient unless $L$ becomes comparable to the length scales typical for
spatial dispersion (mean free path, Debye-Hückel screening length, Fermi
wavelength).
### 2.4 Calculation of the entropy
Integrating Eq.(19) by parts, we have to evaluate the integral:
$\displaystyle I_{k}$ $\displaystyle=$
$\displaystyle-\int\limits_{-\infty}^{\infty}\\!\frac{{\rm d}k_{z}}{2\pi{\rm
i}\,\lambda^{2}}\frac{k_{z}\,\log\left(1-r^{2}\,{\rm
e}^{-2kL}\right)}{(k_{z}^{2}+k^{2})(k_{z}^{2}+k^{2}+\lambda^{-2})}$ (20)
where we recall that $r$ is given by Eq.(16) above. We have extended the
integration domain to $-\infty<k_{z}<+\infty$, using the property
$r(-k_{z})=[r(k_{z})]^{*}$. Observe that $r(k_{z})$, as a function of complex
$k_{z}$, satisfies $|r(k_{z})|\leq 1$ in the upper half-plane, that the
integrand vanishes at infinity, and evaluate the integral by closing the
contour. There are simple poles at $k_{z}={\rm i}k$ and $k_{z}={\rm
i}(k^{2}+\lambda^{-2})^{1/2}$. At the first pole, the reflection coefficient
(16) vanishes, and we get from the second one:
$I_{k}=\frac{1}{2}\log\left[1-r_{\rm pl}^{2}(k,0){\rm e}^{-2kL}\right],\quad
r_{\rm
pl}(k,0)=\frac{(k^{2}+\lambda^{-2})^{1/2}-k}{(k^{2}+\lambda^{-2})^{1/2}+k}$
(21)
As it happens, the reflection coefficient $r_{\rm pl}(k,\omega)$ for
electromagnetic waves from a plasma half-space [dielectric function after
Eq.(6)] appears here, evaluated at zero frequency and in the TE-polarization.
If Eq.(21) is integrated over $k$, we get the ‘entropy defect’ calculated in
the Lifshitz theory of the Casimir effect using the local dielectric function
of a ‘perfect crystal’ [see, e.g., Eq.(20) of Ref.Bezerra04]:
$S(L)=\int\limits_{0}^{\infty}\\!\frac{k{\rm
d}k}{2\pi}I_{k}=\int\limits_{0}^{\infty}\\!\frac{k{\rm
d}k}{4\pi}\log\left[1-r_{\rm pl}^{2}(k,0)\,{\rm e}^{-2kL}\right]$ (22)
A switch to the integration variable to $y=2kL$ shows that
$S(L)=-\frac{\zeta(3)}{16\pi}\frac{f(L/\lambda)}{L^{2}}$ (23)
where the scaling function $f(L/\lambda)$, plotted in Fig.2, is dimensionless
and normalized to unity in the limit $L\gg\lambda$. Indeed, in this regime,
one may expand $r_{\rm pl}(k,0)$ in powers of $k$ to get the asymptotic series
[for higher terms, see Eq.(20) of Ref.Bezerra04]:
$L\gg\lambda:\quad f(L/\lambda)\approx
1-4\frac{\lambda}{L}+12\frac{\lambda^{2}}{L^{2}}+\mathcal{O}(\lambda/L)^{3}$
(24)
file=scale-function,width=2.5in
Figure 2: Scaling function $f(L/\lambda)$ for the Casimir entropy of for the
ideal electron gas, defined by Eqs.(22, 23). Dashed: Eq.(24).
$\lambda=c/\Omega$ is the London-Meißner penetration depth (plasma
wavelength). The same $f(L/\lambda)$ governs the ‘entropy defect’ for the
electromagnetic Casimir effect between perfect crystals (i.e., Lifshitz theory
with the Drude dielectric function and scattering rate
$\gamma(T)=\mathcal{O}(T^{2})$), see Eq.(20) in Ref.Bezerra04.
## 3 Conclusions
We have analyzed the Casimir entropy for the ideal electron gas, in particular
the contribution of electric currents frozen inside the bulk. This system
shows (electromagnetic) disorder, and the third law of thermodynamic does not
apply in its orthodox formulation. We have recovered the ‘entropy defect’
(negative Casimir entropy at zero temperature) reported in several places in
the literature. Its thermodynamically consistent interpretation is the measure
of the change in the disorder of the frozen currents due to their interaction
through quasi-static magnetic fields.
#### Acknowledgments.
We acknowledge financial support by the European Science Foundation within the
activity ‘New Trends and Applications of the Casimir Effect’ (www.casimir-
network.com). F.I. acknowledges financial support by the Alexander von
Humboldt Foundation.
## References
* [1] A. Gambassi, C. Hertlein, L. Helden, S. Dietrich and C. Bechinger, Europhys. News 40, 18 (Jan 2009).
* [2] R. Balian and B. Duplantier, Ann. Phys. (N. Y.) 112, 165 (1978).
* [3] J. Feinberg, A. Mann and M. Revzen, Ann. Phys. (N.Y.) 288, 103 (2001).
* [4] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A 63, 062108 (2001).
* [5] V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko and C. Romero, Phys. Rev. A 69, 022119 (2004).
* [6] R. Decca, D. López, E. Fischbach, G. Klimchitskaya, D. Krause and V. Mostepanenko, Eur. Phys. J. C 51, 963 (2007).
* [7] F. Chen, G. L. Klimchitskaya, V. M. Mostepanenko and U. Mohideen, Phys. Rev. B 76, 035338 (2007).
* [8] V. B. Svetovoy, Phys. Rev. Lett. 101, 163603 (2008).
* [9] L. P. Pitaevskii, Phys. Rev. Lett. 101, 163202 (2008).
* [10] J. S. Hoye, I. Brevik, J. B. Aarseth and K. A. Milton, Phys. Rev. E 67, 056116 (2003).
* [11] L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of continuous media, 2nd edn. (Pergamon, Oxford, 1984).
* [12] W. A. B. Evans and G. Rickayzen, Ann. Phys. (N.Y.) 33, 275 (1965).
* [13] F. Intravaia and C. Henkel, Phys. Rev. Lett. 103, 130405 (2009).
* [14] J. R. Torgerson and S. K. Lamoreaux, Phys. Rev. E 70, 047102 (2004).
* [15] V. B. Svetovoy, Phys. Rev. A 76, 062102 (2007).
* [16] A. Einstein, Ann. Physik (Leipzig), Vierte Folge 17, 549 (1905).
* [17] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
* [18] X. L. Li, G. W. Ford and R. F. O’Connell, Phys. Rev. E 48, 1547 (1993).
* [19] B. Huttner and S. M. Barnett, Europhys. Lett. 18, 487 (1992).
* [20] T. Gruner and D.-G. Welsch, Phys. Rev. A 51, 3246 (1995).
* [21] A. Tip, Phys. Rev. A 56, 5022 (1997).
* [22] L. G. Suttorp, J. Phys. A: Math. Gen. 40, 3697 (2007).
* [23] S. Scheel and S. Y. Buhmann, acta phys. slov. 58, 675 (2008).
* [24] U. Weiss, Quantum Dissipative Systems, Series in Modern Condensed Matter Physics, Vol. 10, third edn. (World Scientific, Singapore, 2007).
* [25] G.-L. Ingold, P. Hänggi and P. Talkner, Phys. Rev. E 79, 061105 (2009).
* [26] G.-L. Ingold, A. Lambrecht and S. Reynaud, Phys. Rev. E 80, 041113 (2009).
* [27] F. London and H. London, Proc. Roy. Soc. (London) A 149, 71 (1935).
* [28] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and Atoms — Introduction to Quantum Electrodynamics (Wiley, New York 1989).
* [29] G. Barton, Rep. Prog. Phys. 42, 963 (1979).
* [30] M. Bordag, U. Mohideen and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001).
* [31] N. J. Morgenstern Horing, E. Kamen and G. Gumbs, Phys. Rev. B 31, 8269 (1985).
|
arxiv-papers
| 2009-11-18T09:42:45 |
2024-09-04T02:49:06.543317
|
{
"license": "Public Domain",
"authors": "Carsten Henkel and Francesco Intravaia",
"submitter": "Carsten Henkel",
"url": "https://arxiv.org/abs/0911.3489"
}
|
0911.3490
|
# Mode contributions to the Casimir effect
F. INTRAVAIA∗444Present address: Theoretical Division, MS B213 Los Alamos
National Laboratory, Los Alamos NM 87545, U. S. A. and C. HENKEL Institut für
Physik und Astronomie, Universität Potsdam,
Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany
∗E-mail: francesco.intravaia@qipc.org
www.quantum.physik.uni-potsdam.de
###### Abstract
Applying a sum-over-modes approach to the Casimir interaction between two
plates with finite conductivity, we isolate and study the contributions of
surface plasmons and Foucault (eddy current) modes. We show in particular that
for the TE-polarization eddy currents provide a repulsive force that cancels,
at high temperatures, the Casimir free energy calculated with the plasma model
###### keywords:
Mode contributions, surface plasmons, eddy currents.
ζ π
## 1 Introduction
Intense theoretical effort is currently devoted to the understanding of the
Casimir effect for real experimental setups. This involves the impact of
temperature, finite conductivity, engineered materials, and may identify
routes to _design_ the final Casimir pressure. Almost all analyses rely on the
Lifshitz formula [1, 2] where the physical properties of the material are
encoded in the scattering amplitudes (i.e., reflection coefficients in planar
geometries). Their evaluation at imaginary frequencies obscures, however, how
the material objects modify the modes of the electromagnetic field. A ‘sum
over modes’ approach is nevertheless possible, even if the eigenfrequencies
$\omega_{m}$ are complex (due to material absorption, for example). For two
objects at distance $L$ the Casimir energy at zero temperature can be written
as [3]
$E=\frac{\hbar}{2}\sideset{}{{}^{\prime}}{\sum}_{p,\mathbf{k}}{\rm
Re}\,\Big{[}\sum_{m}\big{(}\omega_{m}-\frac{2\mathrm{i}\omega_{m}}{\pi}\ln\frac{\omega_{m}}{\Lambda}\big{)}\Big{]}^{L}_{\infty},\qquad\mathrm{Im}\left.\Big{[}\sideset{}{{}^{\prime}}{\sum}_{p,\mathbf{k},m}\omega_{m}\Big{]}_{\infty}^{L}\right.=0$
(1)
where the prime indicates that purely imaginary eigenfrequencies are weighted
with $1/2$. Eq.(1) generalizes Casimir’s formula for the vacuum energy between
two perfect reflectors [4] and is valid for generic (causal) mirrors with
arbitrary thickness. Note that one does not simply take real parts of the
complex eigenfrequencies, as suggested some time ago[5] (see also
Ref.Sernelius06). The logarithmic correction in Eq.(1) is consistent with the
‘system+bath’ paradigm that describes the thermodynamics of quantum
dissipative systems[7]. In this context, the frequency scale $\Lambda$ is
interpreted as the cutoff frequency of the bath spectral density. The Casimir
energy does not depend on this constant because of the sum rule in (1).
The sum-over-modes approach provides an ‘anatomic view’ of the Casimir effect
where contributions from different modes are clearly identified. This is
useful to understand unusual behaviours and may suggest new ways to taylor the
Casimir force[8, 9, 10]. In the following, we illustrate Eq.(1) with the help
of a few examples.
## 2 Dissipative Plasmons at short distance
One of the most interesting contributions to the Casimir force originates from
surface modes bound to the vacuum/medium interface[11]. These modes have a
dispersion relation that splits in two branches, $\omega=\Omega_{\pm}(k)$, as
two surfaces are approached. Substituting these frequencies in Eq.(1), we get
a plasmonic contribution to the Casimir energy ($A$: surface area)
$E_{\rm pl}=\frac{\hbar A}{2}\int\\!\frac{k{\rm d}k}{2\pi}{\rm
Re}\,\Big{[}\sum_{i=\pm}\big{(}\Omega_{i}(k)-\frac{2\mathrm{i}\Omega_{i}(k)}{\pi}\ln\frac{\Omega_{i}(k)}{\Lambda}\big{)}\Big{]}^{L}_{\infty}$
(2)
Consider the case of two metals at a distance smaller than the plasma
wavelength $\lambda_{\rm pl}=2\pi c/\omega_{\rm pl}$. We are then in the
quasi-electrostatic regime, and the surface plasmon modes are given by[12]
(red and blue points in Fig.1)
$\Omega_{\pm}=\sqrt{\omega^{2}_{\pm}-\frac{\gamma^{2}}{4}}-\mathrm{i}\frac{\gamma}{2},\qquad\omega^{2}_{\pm}=\frac{\omega^{2}_{\rm
pl}}{2}\left(1\pm e^{-kL}\right)$ (3)
where $\gamma$ is the damping rate in a Drude description of the metal.
Figure 1: (Left) Complex eigenfrequencies in the parallel plate geometry, for
a fixed wavevector $k$ (not to scale). Red and blue points: dissipative
surface plasmons. Red line: bulk continuum of eddy currents. Black crosses:
propagating modes in the cavity between the plates. (Right) A counter-
clockwise path around the eddy current continuum is equivalent to a clockwise
path around the whole complex plane, encircling all other modes.
One can easily check that the sum rule in Eq.(1) is automatically satisfied.
To leading order in $\gamma\ll\omega_{\rm pl}$ (good conductors) Eq.(2) yields
$E_{\rm pl}\approx-\frac{\pi^{2}\hbar
cA}{720L^{3}}\frac{3}{2}\left(\alpha\frac{L}{\lambda_{\rm
pl}}-\frac{15\zeta(3)}{\pi^{4}}\frac{\gamma
L}{c}\right),\qquad\alpha=1.193\ldots$ (4)
where $\zeta(3)\approx 1.202$ is a Zeta function. This corresponds exactly to
the total Casimir force calculated in Ref.Henkel04, including the dissipative
correction. In fact, in this short distance limit, the Casimir energy is
completely dominated by the plasmonic contribution[14, 15, 13]. Eq.(2) is
valid also beyond the good conductor limit, however, and could be used, e.g.,
to analyze semiconductors where surface plasmons appear in a different
frequency range and can have much stronger damping.
## 3 Eddy currents
As a second example, consider the contribution from eddy current modes. They
are connected with low-frequency currents that satisfy a diffusion equation in
the conducting metal[16] and are completely absent within the lossless
description of the so-called plasma model[2]. We have analyzed these modes
recently[10] and constructed from the ‘system+bath’ paradigm their quantum
thermodynamics. They behave like free Brownian particles, since the
eigenfrequencies of bulk eddy currents are purely imaginary
$\omega_{m}=-\mathrm{i}\xi_{m}$ ($\xi_{m}>0$). From Eq.(1), we get the Casimir
energy
$E_{\rm
eddy}=-\sum_{p,\mathbf{k}}\,\Big{[}\sum_{m}\frac{\hbar\xi_{m}}{2\pi}\ln\frac{\xi_{m}}{\Lambda}\Big{]}^{L}_{\infty}$
(5)
For these modes alone, the sum rule [Eq.(1)] is not satisfied, and the eddy
current contribution to the Casimir energy depends on the cutoff $\Lambda$.
This is also well-known from quantum Brownian motion where bath modes up to
$\Lambda$ are entangled to the particle.
Mathematically, eddy currents form a mode continuum that can be identified in
the complex frequency plane from the branch cut of the root
$k_{m}=\sqrt{\epsilon(\omega)\omega^{2}/c^{2}-k^{2}}$ which describes the
propagation of the electromagnetic field inside the medium. For a Drude metal,
the cut is located between
$\omega_{m}=-\mathrm{i}\xi_{0}(\mathbf{k})\approx-\mathrm{i}Dk^{2}$ (for
$k\ll\omega_{\rm pl}/c$) and $\omega_{m}=-\mathrm{i}\gamma$ (see Fig. 1),
where $D=\gamma(\lambda_{\rm pl}/2\pi)^{2}$ is the electromagnetic diffusion
constant. We get the $L$-dependent change in the mode density along the branch
cut by applying the logarithmic argument theorem to the Green function of the
electromagnetic field. Using the contour sketched in Fig.1(left), it is
possible to show that Eq.(5) can be written as
$E_{\rm eddy}=\int_{0}^{\infty}\\!\frac{{\rm d}\xi}{\pi}\sum_{p,\mathbf{k}}\
\partial_{\xi}\Big{(}\frac{\hbar\xi}{2\pi}\ln\frac{\xi}{\Lambda}\Big{)}\mathrm{Im}\left.\ln\left[1-r_{p}^{2}(-\mathrm{i}\xi-0^{+})e^{-2\kappa
L}\right]\right.,$ (6)
with $\kappa=\sqrt{\xi^{2}+k^{2}}$ and $r_{p}$ the reflection coefficient of
the mirrors in polarization $p={\rm TE,TM}$. This gives rise to a repulsive
Casimir force (Fig.1 of Ref. Intravaia09), provided $\Lambda$ is sufficiently
large, e.g., $\Lambda\geq\gamma$.
The structure of Eq.(6) allows for an immediate translation to the high-
temperature (classical) limit. Replace the zero-point energy with the
classical free energy per mode, $k_{B}T\ln(\hbar\xi/k_{B}T)$, and get
$\mathcal{F}_{\rm
eddy}\approx-\int_{0}^{\infty}\frac{d\xi}{\pi}\sum_{p,\mathbf{k}}\
\frac{k_{B}T}{\xi}\mathrm{Im}\left.\ln\left[1-r_{p}^{2}(-\mathrm{i}\xi-0^{+})e^{-2\kappa
L}\right]\right.,$ (7)
(A more rigorous proof follows from the representation for the free energy
given in Ref. Intravaia09.) Eq. (7) is thus the result of the logarithmic
argument theorem applied to the high-temperature limit of the free energy. Now
the contour around the eddy current continuum can also be interpreted as a
contour encircling the whole complex plane, i.e., the surface plasmon and
propagating modes [Fig. 1(right)]. This is particularly interesting in the TE-
polarization because there are no surface plasmons, and the residue at
$\omega=0$ vanishes [$r_{\rm TE}^{2}(\omega\to 0)=0$]. This means that eddy
currents and propagating modes give, up to a sign, the same Casimir energy at
high temperature (or large distance). Since propagating modes are only
slightly affected by conduction on the metal (i.e., they behave similarly in
the Drude and plasma models), we find the simple relation
$\mathcal{F}^{\rm TE}_{\rm eddy}\approx-\mathcal{F}^{\rm TE}_{\rm C}({\rm
pl.m.}),\qquad\gamma/\omega_{p}\ll 1$ (8)
where $\mathcal{F}^{\rm TE}_{\rm C}({\rm pl.m.})$ is the Casimir free energy
at high temperature calculated within the plasma model[2]. In the Drude model,
the two contributions are present and cancel each other when they are both in
the high-temperature regime (which happens at different distances, see Fig.4
of Ref.Intravaia09).
A different scenario occurs in the TM-polarization. The residue at $\omega=0$
does not vanish and corresponds exactly to the high-temperature limit of the
plasma model.[2] Indeed, we have checked that eddy currents give only a very
small contribution.
## 4 Conclusions
Using a mode-summation approach, we have isolated and analyzed the
contribution of two classes of modes to the Casimir effect, allowing for
complex eigenfrequencies of the electromagnetic field. A previous result for
the short-distance limit between good conductors[13] has been generalized to
any conductivity and distance by considering coupled surface plasmonic modes
(for the lossless case, see Refs.Intravaia05, Intravaia07). We also considered
eddy currents which are overdamped or diffusive modes in the bulk of a Drude
metal, and showed that they contribute a repulsive Casimir interaction, in
agreement with Ref.Intravaia09. At high temperature and for a good conductor,
we found in a simple way that their free energy in the TE-polarization differs
only slightly from the Casimir free energy within a dissipationless
description (the plasma model), but is of the opposite sign. In this way, eddy
currents nearly cancel out the attractive Casimir interaction from propagating
modes. This explains the strong difference between the Drude and plasma models
for the temperature correction of the electromagnetic Casimir effect[2].
We thank H. Haakh for a critical reading and acknowledge financial support by
the European Science Foundation within the activity ‘New Trends and
Applications of the Casimir Effect’ (www.casimir-network.com). F.I.
acknowledges financial support by the Alexander von Humboldt Foundation.
## References
* [1] E. Lifshitz, Sov. Phys. JETP (USA) 2, p. 73 (1956).
* [2] G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Rev. Mod. Phys. (2010 in press), arXiv:0902.4022v1.
* [3] F. Intravaia and C. Henkel, J. Phys. A 41, 164018 (2008).
* [4] H. Casimir, Proc. kon. Ned. Ak. Wet. 51, p. 793 (1948).
* [5] D. Langbein, Phys. Rev. B 2, p. 3371 (1970).
* [6] B. E. Sernelius, Phys. Rev. B 74, 233103 (2006).
* [7] U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore 2008).
* [8] F. Intravaia and A. Lambrecht, Phys. Rev. Lett. 94, 110404 (2005).
* [9] F. Intravaia, C. Henkel and A. Lambrecht, Phys. Rev. A 76, 033820 (2007).
* [10] F. Intravaia and C. Henkel, Phys. Rev. Lett. 103, 130405 (2009).
* [11] G. Barton, Rep. Prog. Phys. 42, p. 963 (1979).
* [12] E. N. Economou, Phys. Rev. 182, p. 539 (1969).
* [13] C. Henkel, K. Joulain, J.-P. Mulet and J.-J. Greffet, Phys. Rev. A 69, 023808 (2004).
* [14] N. V. Kampen, B. Nijboer and K. Schram, Phys. Lett. A 26, p. 307 (1968).
* [15] E. Gerlach, Phys. Rev. B 4, p. 393 (1971).
* [16] J. Jackson, Classical Electrodynamics (Wiley & Sons, New York, 1975).
|
arxiv-papers
| 2009-11-18T09:49:00 |
2024-09-04T02:49:06.548736
|
{
"license": "Public Domain",
"authors": "Francesco Intravaia and Carsten Henkel",
"submitter": "Carsten Henkel",
"url": "https://arxiv.org/abs/0911.3490"
}
|
0911.3524
|
# Radicals of symmetric cellular algebras
###### Abstract.
Using a slightly weaker definition of cellular algebra, due to Goodman ([4]
Definition 2.9), we prove that for a symmetric cellular algebra, the dual
basis of a cellular basis is again cellular. Then a nilpotent ideal is
constructed for a symmetric cellular algebra. The ideal connects the radicals
of cell modules with the radical of the algebra. It also reveals some
information on the dimensions of simple modules. As a by-product, we obtain
some equivalent conditions for a finite dimensional symmetric cellular algebra
to be semisimple.
This work is partially supported by the Research Fund of Doctor Program of
Higher Education, Ministry of Education of China.
Yanbo Li
Department of Information and Computing Sciences,
Northeastern University at Qinhuangdao;
Qinhuangdao, 066004, P.R. China
School of Mathematics Sciences, Beijing Normal University;
Beijing, 100875, P.R. China
E-mail: liyanbo707@163.com
2000 AMS Classification: 16G30, 16N20
Key words: radicals; symmetric cellular algebras; Gram matrix.
## 1\. Introduction
Cellular algebras were introduced by Graham and Lehrer [6] in 1996, motivated
by previous work of Kazhdan and Lusztig [9]. They were defined by a so-called
cellular basis with some nice properties. The theory of cellular algebras
provides a systematic framework for studying the representation theory of non-
semisimple algebras which are deformations of semisimple ones. One can
parameterize simple modules for a finite dimensional cellular algebra by
methods in linear algebra. Many classes of algebras from mathematics and
physics are found to be cellular, including Hecke algebras of finite type,
Ariki-Koike algebras, $q$-Schur algebras, Brauer algebras, Temperley-Lieb
algebras, cyclotomic Temperley-Lieb algebras, Jones algebras, partition
algebras, Birman-Wenzl algebras and so on, we refer the reader to [3, 6, 17,
19, 20] for details.
An equivalent basis-free definition of cellular algebras was given by Koenig
and Xi [10], which is useful in dealing with structural problems. Using this
definition, in [11], Koenig and Xi made explicit an inductive construction of
cellular algebras called inflation, which produces all cellular algebras. In
[12], Brauer algebras were shown to be iterated inflations of group algebras
of symmetric groups and then more information about these algebras was found.
There are some generalizations of cellular algebras, we refer the reader to
[2, 7, 8, 18] for details. Recently, Koenig and Xi [13] introduced affine
cellular algebras which contain cellular algebras as special cases. Affine
Hecke algebras of type A and infinite dimensional diagram algebras like the
affine Temperley-Lieb algebras are affine cellular.
It is an open problem to find explicit formulas for the dimensions of simple
modules of a cellular algebra. By the theory of cellular algebras, this is
equivalent to determine the dimensions of the radicals of bilinear forms
associated with cell modules. In [14], for a quasi-hereditary cellular
algebra, Lehrer and Zhang found that the radicals of bilinear forms are
related to the radical of the algebra. This leads us to studying the radical
of a cellular algebra. However, we have no idea for dealing with general
cellular algebras now. We will do some work on the radicals of symmetric
cellular algebras in this paper. Note that Hecke algebras of finite types,
Ariki-Koike algebras over any ring containing inverses of the parameters,
Khovanov’s diagram algebras are all symmetric cellular algebras. The trivial
extension of a cellular algebra is also a symmetric cellular algebra. For
details, see [1], [15], [21].
Throughout this paper, we will adopt a slightly weaker definition of cellular
algebra due to Goodman ([4] Definition 2.9). It is helpful to note that the
results of [6] remained valid with his weaker axiom. In case $2$ is
invertible, these two definitions are equivalent.
We begin with recalling definitions and some well-known results of symmetric
algebras and cellular algebras in Section 2. Then in Section 3, we prove that
for a symmetric cellular algebra, the dual basis of a cellular basis is again
cellular. In Section 4, a nilpotent ideal of a symmetric cellular algebra is
constructed. This ideal connects the radicals of cell modules with the radical
of the algebra and also reveals some information on the dimensions of simple
modules. As a by-product, in Section 5, we obtain some equivalent conditions
for a finite dimensional symmetric cellular algebra to be semisimple.
## 2\. Preliminaries
In this section, we start with the definitions of symmetric algebras and
cellular algebras (a slightly weaker version due to Goodman) and then recall
some well-known results about them.
Let $R$ be a commutative ring with identity and $A$ an associative
$R$-algebra. As an $R$-module, $A$ is finitely generated and free. Suppose
that there exists an $R$-bilinear map $f:A\times A\rightarrow R$. We say that
$f$ is non-degenerate if the determinant of the matrix
$(f(a_{i},a_{j}))_{a_{i},a_{j}\in B}$ is a unit in $R$ for some $R$-basis $B$
of $A$. We say $f$ is associative if $f(ab,c)=f(a,bc)$ for all $a,b,c\in A$,
and symmetric if $f(a,b)=f(b,a)$ for all $a,b\in A$.
###### Definition 2.1.
An $R$-algebra $A$ is called symmetric if there is a non-degenerate
associative symmetric bilinear form $f$ on $A$. Define an $R$-linear map
$\tau:A\rightarrow R$ by $\tau(a)=f(a,1)$. We call $\tau$ a symmetrizing
trace.
Let $A$ be a symmetric algebra with a basis $B=\\{a_{i}\mid i=1,\ldots,n\\}$
and $\tau$ a symmetrizing trace. Denote by $D=\\{D_{i}\mid i=i,\ldots,n\\}$
the basis determined by the requirement that $\tau(D_{j}a_{i})=\delta_{ij}$
for all $i,j=1,\ldots,n$. We will call $D$ the dual basis of $B$. For
arbitrary $1\leq i,j\leq n$, write $a_{i}a_{j}=\sum\limits_{k}r_{ijk}a_{k}$,
where $r_{ijk}\in R$. Fixing a symmetrizing trace $\tau$ for $A$, then we have
the following lemma.
###### Lemma 2.2.
Let $A$ be a symmetric $R$-algebra with a basis $B$ and the dual basis $D$.
Then the following hold:
$a_{i}D_{j}=\sum_{k}r_{kij}D_{k};\,\,\,\,\,D_{i}a_{j}=\sum_{k}r_{jki}D_{k}.$
###### Proof.
We only prove the first equation. The other one is proved similarly.
Suppose that $a_{i}D_{j}=\sum\limits_{k}r_{k}D_{k}$, where $r_{k}\in R$ for
$k=1,\cdots,n$. Left multiply by $a_{k_{0}}$ on both sides of the equation and
then apply $\tau$, we get $\tau(a_{k_{0}}a_{i}D_{j})=r_{k_{0}}$. Clearly,
$\tau(a_{k_{0}}a_{i}D_{j})=r_{k_{0},i,j}$. This implies that
$r_{k_{0}}=r_{k_{0},i,j}$. ∎
Given a symmetric algebra, it is natural to consider the relation between two
dual bases determined by two different symmetrizing traces. For this we have
the following lemma.
###### Lemma 2.3.
Suppose that $A$ is a symmetric $R$-algebra with a basis $B=\\{a_{i}\mid
i=1,\cdots,n\\}$. Let $\tau,\tau^{\prime}$ be two symmetrizing traces. Denote
by $\\{D_{i}\mid i=1,\cdots,n\\}$ the dual basis of $B$ determined by $\tau$
and $\\{D_{i}^{\prime}\mid i=1,\cdots,n\\}$ the dual basis determined by
$\tau^{\prime}$. Then for $1\leq i\leq n$, we have
$D_{i}^{\prime}=\sum_{j=1}^{n}\tau(a_{j}D_{i}^{\prime})D_{j}.$
###### Proof.
It is proved by a similar method as in Lemma 2.2. ∎
Graham and Lehrer introduced the so-called cellular algebras in [6] , then
Goodman weakened the definition in [4]. We will adopt Goodman’s definition
throughout this paper.
###### Definition 2.4.
([4]) Let $R$ be a commutative ring with identity. An associative unital
$R$-algebra is called a cellular algebra with cell datum $(\Lambda,M,C,i)$ if
the following conditions are satisfied:
(C1) The finite set $\Lambda$ is a poset. Associated with each
$\lambda\in\Lambda$, there is a finite set $M(\lambda)$. The algebra $A$ has
an $R$-basis $\\{C_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$.
(C2) The map $i$ is an $R$-linear anti-automorphism of $A$ with $i^{2}=id$ and
$i(C_{S,T}^{\lambda})\equiv
C_{T,S}^{\lambda}\,\,\,\,(\rm{mod}\,\,\,A(<\lambda))$
for all $\lambda\in\Lambda$ and $S,T\in M(\lambda)$, where $A(<\lambda)$ is
the $R$-submodule of $A$ generated by
$\\{C_{S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}}^{\mu}\mid
S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}\in M(\mu),\mu<\lambda\\}$.
(C3) If $\lambda\in\Lambda$ and $S,T\in M(\lambda)$, then for any element
$a\in A$, we have
$aC_{S,T}^{\lambda}\equiv\sum_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{\prime},S)C_{S^{{}^{\prime}},T}^{\lambda}\,\,\,\,(\rm{mod}\,\,\,A(<\lambda)),$
where $r_{a}(S^{{}^{\prime}},S)\in R$ is independent of $T$.
Apply $i$ to the equation in (C3), we obtain
(C3′) $C_{T,S}^{\lambda}i(a)\equiv\sum\limits_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{{}^{\prime}},S)C_{T,S^{{}^{\prime}}}^{\lambda}\,\,\,\,(\rm
mod\,\,\,A(<\lambda)).$
###### Remark 2.5.
Graham and Lehrer’s original definition in [6] requires that
$i(C_{S,T}^{\lambda})=C_{T,S}^{\lambda}$ for all $\lambda\in\Lambda$ and
$S,T\in M(\lambda)$. But Goodman pointed out that the results of [6] remained
valid with his weaker axiom. In case $2\in R$ is invertible, these two
definitions are equivalent.
It is easy to check the following lemma by Definition 2.4.
###### Lemma 2.6.
([6]) Let $\lambda\in\Lambda$ and $a\in A$. Then for arbitrary elements
$S,T,U,V\in M(\lambda)$, we have
$C_{S,T}^{\lambda}aC_{U,V}^{\lambda}\equiv\Phi_{a}(T,U)C_{S,V}^{\lambda}\,\,\,\,(\rm
mod\,\,\,A(<\lambda)),$
where $\Phi_{a}(T,U)\in R$ depends only on $a$, $T$ and $U$.
We often omit the index $a$ when $a=1$, that is, writing $\Phi_{1}(T,U)$ as
$\Phi(T,U)$.
Let us recall the definition of cell modules now.
###### Definition 2.7.
([6]) Let $A$ be a cellular algebra with cell datum $(\Lambda,M,C,i)$. For
each $\lambda\in\Lambda$, define the left $A$-module $W(\lambda)$ as follows:
$W(\lambda)$ is a free $R$-module with basis $\\{C_{S}\mid S\in M(\lambda)\\}$
and $A$-action defined by
$aC_{S}=\sum_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{{}^{\prime}},S)C_{S^{{}^{\prime}}}\,\,\,\,(a\in A,S\in
M(\lambda)),$
where $r_{a}(S^{{}^{\prime}},S)$ is the element of $R$ defined in Definition
2.4 (C3).
Note that $W(\lambda)$ may be thought of as a right $A$-module via
$C_{S}a=\sum_{S^{{}^{\prime}}\in
M(\lambda)}r_{i(a)}(S^{{}^{\prime}},S)C_{S^{{}^{\prime}}}\,\,\,\,(a\in A,S\in
M(\lambda)).$
We will denote this right $A$-module by $i(W(\lambda))$.
###### Lemma 2.8.
([6]) There is a natural isomorphism of $R$-modules
$C^{\lambda}:W(\lambda)\otimes_{R}i(W(\lambda))\rightarrow R{\rm-
span}\\{C_{S,T}^{\lambda}\mid S,T\in M(\lambda)\\},$
defined by $(C_{S},C_{T})\rightarrow C_{S,T}^{\lambda}$.
For a cell module $W(\lambda)$, define a bilinear form
$\Phi_{\lambda}:\,\,W(\lambda)\times W(\lambda)\longrightarrow R$ by
$\Phi_{\lambda}(C_{S},C_{T})=\Phi(S,T)$. It plays an important role for
studying the structure of $W(\lambda)$. It is easy to check that
$\Phi(T,U)=\Phi(U,T)$ for arbitrary $T,U\in M(\lambda)$.
Define
$\operatorname{rad}\lambda:=\\{x\in
W(\lambda)\mid\Phi_{\lambda}(x,y)=0\,\,\,\text{for all}\,\,\,y\in
W(\lambda)\\}.$
If $\Phi_{\lambda}\neq 0$, then $\operatorname{rad}\lambda$ is the radical of
the $A$-module $W(\lambda)$. Moreover, if $\lambda$ is a maximal element in
$\Lambda$, then $\operatorname{rad}\lambda=0$.
The following results were proved by Graham and Lehrer in [6].
###### Theorem 2.9.
[6] Let $K$ be a field and $A$ a finite dimensional cellular algebra. For any
$\lambda\in\Lambda$, denote the $A$-module
$W(\lambda)/\operatorname{rad}\lambda$ by $L_{\lambda}$. Let
$\Lambda_{0}=\\{\lambda\in\Lambda\mid\Phi_{\lambda}\neq 0\\}$. Then
$\\{L_{\lambda}\mid\lambda\in\Lambda_{0}\\}$ is a complete set of
(representative of equivalence classes of ) absolutely simple $A$-modules.
###### Theorem 2.10.
([6]) Let $K$ be a field and $A$ a cellular $K$-algebra. Then the following
are equivalent.
(1) The algebra $A$ is semisimple.
(2) The nonzero cell representations $W(\lambda)$ are irreducible and pairwise
inequivalent.
(3) The form $\Phi_{\lambda}$ is non-degenerate (i.e.
$\operatorname{rad}\lambda=0$) for each $\lambda\in\Lambda$.
For any $\lambda\in\Lambda$, fix an order on $M(\lambda)$ and let
$M(\lambda)=\\{S_{1},S_{2},\cdots,S_{n_{\lambda}}\\}$, where $n_{\lambda}$ is
the number of elements in $M(\lambda)$, the matrix
$G(\lambda)=(\Phi(S_{i},S_{j}))_{1\leq i,j\leq n_{\lambda}}$ is called Gram
matrix. It is easy to know that all the determinants of $G(\lambda)$ defined
with different order on $M(\lambda)$ are the same. By the definition of
$G(\lambda)$ and $\operatorname{rad}\lambda$, for a finite dimensional
cellular algebra $A$, it is clear that if $\Phi_{\lambda}\neq 0$, then
$\dim_{K}L_{\lambda}=\operatorname{rank}G(\lambda)$.
## 3\. Symmetric cellular algebras
In this section, we prove that for a symmetric cellular algebra, the dual
basis of a cellular basis is again cellular.
Let $A$ be a symmetric cellular algebra with a cell datum $(\Lambda,M,C,i)$.
Denote the dual basis by $D=\\{D_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$ throughout, which satisfies
$\tau(C_{S,T}^{\lambda}D_{U,V}^{\mu})=\delta_{\lambda\mu}\delta_{SV}\delta_{TU}.$
For any $\lambda,\mu\in\Lambda$, $S,T\in M(\lambda)$, $U,V\in M(\mu)$, write
$C_{S,T}^{\lambda}C_{U,V}^{\mu}=\sum\limits_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(S,T,\lambda),(U,V,\mu),(X,Y,\epsilon)}C_{X,Y}^{\epsilon}.$
A lemma which we now prove plays an important role throughout this paper.
###### Lemma 3.1.
Let $A$ be a symmetric cellular algebra with a cell datum $(\Lambda,M,C,i)$
and $\tau$ a given symmetrizing trace. For arbitrary $\lambda,\mu\in\Lambda$
and $S,T,P,Q\in M(\lambda)$, $U,V\in M(\mu)$, the following hold:
(1) $D_{U,V}^{\mu}C_{S,T}^{\lambda}=\sum\limits_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(S,T,\lambda),(Y,X,\epsilon),(V,U,\mu)}D_{X,Y}^{\epsilon}.$
(2) $C_{S,T}^{\lambda}D_{U,V}^{\mu}=\sum\limits_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(Y,X,\epsilon),(S,T,\lambda),(V,U,\mu)}D_{X,Y}^{\epsilon}.$
(3) $C_{S,T}^{\lambda}D_{T,Q}^{\lambda}=C_{S,P}^{\lambda}D_{P,Q}^{\lambda}.$
(4) $D_{T,S}^{\lambda}C_{S,Q}^{\lambda}=D_{T,P}^{\lambda}C_{P,Q}^{\lambda}.$
(5) $C_{S,T}^{\lambda}D_{P,Q}^{\lambda}=0\,\,if\,\,T\neq P.$
(6) $D_{P,Q}^{\lambda}C_{S,T}^{\lambda}=0\,\,if\,\,Q\neq S.$
(7) $C_{S,T}^{\lambda}D_{U,V}^{\mu}=0\,\,\,\,if\,\,\,\mu\nleq\lambda.$
(8) $D_{U,V}^{\mu}C_{S,T}^{\lambda}=0\,\,\,\,if\,\,\,\mu\nleq\lambda.$
###### Proof.
(1), (2) are corollaries of Lemma 2.2. The equations (5), (6), (7), (8) are
corollaries of (1) and (2). We now prove (3).
By (2), we have
$C_{S,T}^{\lambda}D_{T,Q}^{\lambda}=\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(Y,X,\epsilon),(S,T,\lambda),(Q,T,\lambda)}D_{X,Y}^{\epsilon}$
$C_{S,P}^{\lambda}D_{P,S}^{\lambda}=\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(Y,X,\epsilon),(S,P,\lambda),(Q,P,\lambda)}D_{X,Y}^{\epsilon}.$
On the other hand, by (C3) of Definition 2.4 we also have
$r_{(Y,X,\epsilon),(S,T,\lambda),(Q,T,\lambda)}=r_{(Y,X,\epsilon),(S,P,\lambda),(Q,P,\lambda)}$
for all $\epsilon\in\Lambda$ and $X,Y\in M(\epsilon)$. This completes the
proof of (3).
(4) is proved similarly. ∎
###### Lemma 3.2.
Let $A$ be a symmetric cellular algebra with a cell datum $(\Lambda,M,C,i)$.
Then the dual basis $D=\\{D_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$ is again a cellular basis of $A$ with respect
to the opposite order on $\Lambda$.
###### Proof.
Clearly, we only need to consider (C2) and (C3) of Definition 2.4. Now we
proceed in two steps.
Step 1. (C2) holds.
Let $i(D_{S,T}^{\lambda})=\sum\limits_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{X,Y,\epsilon}D_{X,Y}^{\epsilon}$ with $r_{X,Y,\epsilon}\in R$.
If there exists $\eta\ngeq\lambda$ such that $r_{P,Q,\eta}\neq 0$ for some
$P,Q\in M(\eta)$. Then
$\tau(i(D_{S,T}^{\lambda})C_{Q,P}^{\eta})=r_{P,Q,\eta}\neq 0$. This implies
that $i(D_{S,T}^{\lambda})C_{Q,P}^{\eta}\neq 0$. Thus
$C_{P,Q}^{\eta}D_{S,T}^{\lambda}\neq 0$. But we know $\eta\ngeq\lambda$, then
by Lemma 3.1 (7), $C_{P,Q}^{\eta}D_{S,T}^{\lambda}=0$, a contradiction. This
implies that
$i(D_{S,T}^{\lambda})\equiv\sum\limits_{X,Y\in
M(\lambda)}r_{X,Y,\lambda}D_{X,Y}^{\lambda}\,\,\,(\mod A_{D}(>\lambda)).$
Now assume $r_{U,V,\lambda}\neq 0$. Then
$i(D_{S,T}^{\lambda})C_{V,U}^{\lambda}\neq 0$, hence
$C_{U,V}^{\lambda}D_{S,T}^{\lambda}\neq 0$. By Lemma 3.1 (5), $V=S$. We can
get $U=T$ similarly.
Step 2. (C3) holds.
For arbitrary $C_{S,T}^{\lambda}$, by Lemma 3.1 (2), we have
$C_{S,T}^{\lambda}D_{U,V}^{\mu}=\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(Y,X,\epsilon),(S,T,\lambda),(V,U,\mu)}D_{X,Y}^{\epsilon}.$
By (C3) of Definition 2.4, if $\epsilon<\mu$, then
$r_{(Y,X,\epsilon),(S,T,\lambda),(V,U,\mu)}=0$. Therefore,
$C_{S,T}^{\lambda}D_{U,V}^{\mu}\equiv\sum_{X,Y\in
M(\mu)}r_{(Y,X,\mu),(S,T,\lambda),(V,U,\mu)}D_{X,Y}^{\mu}\,\,\,\,\,(\mod
A_{D}(>\mu)),$
where $A_{D}(>\mu)$ is the $R$-submodule of $A$ generated by
$\\{D_{S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}}^{\eta}\mid
S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}\in M(\lambda),\eta>\mu\\}.$
By (C3′) of Definition 2.4, if $Y\neq V$, then
$r_{(Y,X,\mu),(S,T,\lambda),(V,U,\mu)}=0$. So
$C_{S,T}^{\lambda}D_{U,V}^{\mu}\equiv\sum_{X\in
M(\mu)}r_{(V,X,\mu),(S,T,\lambda),(V,U,\mu)}D_{X,V}^{\mu}\,\,\,(\mod
A_{D}(>\mu)).$
Clearly, for arbitrary $X\in M(\mu)$, we have
$r_{(V,X,\mu),(S,T,\lambda),(V,U,\mu)}=r_{C_{T,S}^{\lambda}}(U,X)$
and which is independent of $V$. Since $C_{S,T}^{\lambda}$ is arbitrary, then
$aD_{U,V}^{\mu}\equiv\sum_{U^{\prime}\in
M(\mu)}r_{i(a)}(U,U^{\prime})D_{U^{\prime},V}^{\mu}\,\,\,\,\,(\mod
A_{D}(>\mu))$
for any $a\in A$. By Definition 2.4, $r_{i(a)}(U,U^{\prime})$ is independent
of $V$. ∎
###### Remark 3.3.
Using the original definition of cellular algebras, Graham proved in [5] the
dual basis of a cellular basis is again cellular in the case when
$\tau(a)=\tau(i(a))$, for all $a\in A$.
Since the dual basis is again cellular, for arbitrary elements $S,T,U,V\in
M(\lambda)$, it is clear that
$D_{S,T}^{\lambda}D_{U,V}^{\lambda}\equiv\Psi(T,U)D_{S,V}^{\lambda}\,\,\,\,(\rm
mod\,\,\,A(>\lambda)),$
where $\Psi(T,U)\in R$ depends only on $T$ and $U$. Then we also have Gram
matrices $G^{\prime}(\lambda)$ defined by the dual basis. Now it is natural to
consider the problem what is the relation between $G(\lambda)$ and
$G^{\prime}(\lambda)$. To study this, we need the following lemma.
###### Lemma 3.4.
Let $A$ be a symmetric cellular algebra with cell datum $(\Lambda,M,C,i)$. For
every $\lambda\in\Lambda$ and $S,T,U,V,P\in M(\lambda)$, we have
$C_{S,T}^{\lambda}D_{T,U}^{\lambda}C_{U,V}^{\lambda}D_{V,P}^{\lambda}=\sum_{Y\in
M(\lambda)}\Phi(Y,V)\Psi(Y,V)C_{S,T}^{\lambda}D_{T,P}^{\lambda}.$
###### Proof.
By Lemma 3.1 (1), we have
$\displaystyle
C_{S,T}^{\lambda}D_{T,U}^{\lambda}C_{U,V}^{\lambda}D_{V,P}^{\lambda}=C_{S,T}^{\lambda}(D_{T,U}^{\lambda}C_{U,V}^{\lambda})D_{V,P}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(U,V,\lambda),(Y,X,\epsilon),(U,T,\lambda)}C_{S,T}^{\lambda}D_{X,Y}^{\epsilon}D_{V,P}^{\lambda}.$
If $\varepsilon>\lambda$, then by Lemma 3.1 (7),
$C_{S,T}^{\lambda}D_{X,Y}^{\epsilon}=0$; if $\varepsilon<\lambda$, by
Definition 2.4 (C3), $r_{(U,V,\lambda),(Y,X,\epsilon),(U,T,\lambda)}=0$. This
implies that
$\displaystyle\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(U,V,\lambda),(Y,X,\epsilon),(U,T,\lambda)}C_{S,T}^{\lambda}D_{X,Y}^{\epsilon}D_{V,P}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{X,Y\in
M(\lambda)}r_{(U,V,\lambda),(Y,X,\lambda),(U,T,\lambda)}C_{S,T}^{\lambda}D_{X,Y}^{\lambda}D_{V,P}^{\lambda}.$
By Definition 2.4 (C3), if $X\neq T$, then
$r_{(U,V,\lambda),(Y,X,\lambda),(U,T,\lambda)}=0$. Hence,
$\displaystyle\sum_{X,Y\in
M(\lambda)}r_{(U,V,\lambda),(Y,X,\lambda),(U,T,\lambda)}C_{S,T}^{\lambda}D_{X,Y}^{\lambda}D_{V,P}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{Y\in
M(\lambda)}r_{(U,V,\lambda),(Y,T,\lambda),(U,T,\lambda)}C_{S,T}^{\lambda}D_{T,Y}^{\lambda}D_{V,P}^{\lambda}.$
Note that
$D_{T,Y}^{\lambda}D_{V,P}^{\lambda}\equiv\Psi(Y,V)D_{T,P}^{\lambda}\,\,\,\,\,\,(\mod
A_{D}(>\lambda)).$
Moreover, by Lemma 3.1 (7), if $\epsilon>\lambda$, then
$C_{S,T}^{\lambda}D_{X,Y}^{\epsilon}=0$. Thus
$\sum_{Y\in
M(\lambda)}r_{(U,V,\lambda),(Y,T,\lambda),(U,T,\lambda)}C_{S,T}^{\lambda}D_{T,Y}^{\lambda}D_{V,P}^{\lambda}=\sum\limits_{Y\in
M(\lambda)}\Phi(Y,V)\Psi(Y,V)C_{S,T}^{\lambda}D_{T,P}^{\lambda}.$
This completes the proof. ∎
By Lemma 3.1, $C_{U,V}^{\lambda}D_{V,P}^{\lambda}$ is independent of $V$, so
is $\sum\limits_{Y\in M(\lambda)}\Phi(Y,V)\Psi(Y,V)$. Then for any
$\lambda\in\Lambda$, we can define a constant $k_{\lambda,\tau}$ as follows.
###### Definition 3.5.
Keep the notation above. For $\lambda\in\Lambda$, take an arbitrary $V\in
M(\lambda)$. Define
$k_{\lambda,\tau}=\sum\limits_{X\in M(\lambda)}\Phi(X,V)\Psi(X,V).$
Note that $\\{k_{\lambda,\tau}\mid\lambda\in\Lambda\\}$ is not independent of
the choice of symmetrizing trace. Fixing a symmetrizing trace $\tau$, we often
write $k_{\lambda,\tau}$ as $k_{\lambda}$. The following lemma reveals the
relation among $G(\lambda)$, $G^{\prime}(\lambda)$ and $k_{\lambda}$.
###### Lemma 3.6.
Let $A$ be a symmetric cellular algebra with cell datum $(\Lambda,M,C,i)$. For
any $\lambda\in\Lambda$, fix an order on the set $M(\lambda)$. Then
$G(\lambda)G^{\prime}(\lambda)=k_{\lambda}E$, where $E$ is the identity
matrix.
###### Proof.
For an arbitrary $\lambda\in\Lambda$, according to the definition of
$G(\lambda)$, $G^{\prime}(\lambda)$ and $k_{\lambda}$, we only need to show
that $\sum\limits_{Y\in M(\lambda)}\Phi(Y,U)\Psi(Y,V)=0$ for arbitrary $U,V\in
M(\lambda)$ with $U\neq V$.
In fact, on one hand, for arbitrary $S\in M(\lambda)$, by Lemma 3.1 (5),
$U\neq V$ implies that $C_{S,U}^{\lambda}D_{V,S}^{\lambda}=0$. Then
$C_{S,U}^{\lambda}D_{U,S}^{\lambda}C_{S,U}^{\lambda}D_{V,S}^{\lambda}=0$.
On the other hand, by a similar method as in the proof of Lemma 3.4,
$\displaystyle
C_{S,U}^{\lambda}D_{U,S}^{\lambda}C_{S,U}^{\lambda}D_{V,S}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(S,U,\lambda),(Y,X,\epsilon),(S,U,\lambda)}C_{S,U}^{\lambda}D_{X,Y}^{\epsilon}D_{V,S}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{Y\in
M(\lambda)}r_{(S,U,\lambda),(Y,U,\lambda),(S,U,\lambda)}C_{S,U}^{\lambda}D_{U,Y}^{\lambda}D_{V,S}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{Y\in
M(\lambda)}\Phi(Y,U)\Psi(Y,V)C_{S,U}^{\lambda}D_{U,S}^{\lambda}.$
Then $\sum\limits_{Y\in
M(\lambda)}\Phi(Y,U)\Psi(Y,V)C_{S,U}^{\lambda}D_{U,S}^{\lambda}=0$. This
implies that
$\tau(\sum\limits_{Y\in
M(\lambda)}\Phi(Y,U)\Psi(Y,V)C_{S,U}^{\lambda}D_{U,S}^{\lambda})=0.$
Since $\tau(C_{S,U}^{\lambda}D_{U,S}^{\lambda})=1$, then $\sum\limits_{Y\in
M(\lambda)}\Phi(Y,U)\Psi(Y,V)=0$. ∎
###### Corollary 3.7.
Let $A$ be a symmetric cellular algebra over an integral domain $R$. Then
$k_{\lambda}=0$ for any $\lambda\in\Lambda$ with
$\operatorname{rad}\lambda\neq 0$.
###### Proof.
Since $|G(\lambda)|=0$ is equivalent to $\operatorname{rad}\lambda\neq 0$,
then by Lemma 3.6, $\operatorname{rad}\lambda\neq 0$ implies that
$k_{\lambda}=0$. ∎
Using the dual basis, for each $\lambda\in\Lambda$, we can also define the
cell module $W_{D}(\lambda)$. Then the following lemma is clear.
###### Lemma 3.8.
There is a natural isomorphism of $R$-modules
$D^{\lambda}:W_{D}(\lambda)\otimes_{R}i(W_{D}(\lambda))\rightarrow R{\rm-
span}\\{D_{S,T}^{\lambda}\mid S,T\in M(\lambda)\\},$
defined by $(D_{S},D_{T})\rightarrow D_{S,T}^{\lambda}$.
## 4\. Radicals of Symmetric Cellular Algebras
To study radicals of symmetric cellular algebras, we need the following lemma.
###### Lemma 4.1.
Let $A$ be a symmetric cellular algebra. Then for any $\lambda\in\Lambda$, the
elements of the form $\sum\limits_{S,U\in
M(\lambda)}r_{SU}C_{S,V}^{\lambda}D_{V,U}^{\lambda}$ with $r_{SU}\in R$ make
an ideal of $A$.
###### Proof.
Denote the set of the elements of the form $\sum\limits_{S,U\in
M(\lambda)}r_{SU}C_{S,V}^{\lambda}D_{V,U}^{\lambda}$ by $I^{\lambda}$. Then
for any $\eta\in\Lambda$, $P,Q\in M(\eta)$, and $S,U\in M(\lambda)$, we claim
that the element $C_{P,Q}^{\eta}C_{S,V}^{\lambda}D_{V,U}^{\lambda}\in
I^{\lambda}$. In fact, by (C3) of Definition 2.4 and Lemma 3.1 (7),
$\displaystyle C_{P,Q}^{\eta}C_{S,V}^{\lambda}D_{V,U}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(P,Q,\eta),(S,V,\lambda),(X,Y,\epsilon)}C_{X,Y}^{\epsilon}D_{V,U}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{X\in
M(\lambda)}r_{(P,Q,\eta),(S,V,\lambda),(X,V\lambda)}C_{X,V}^{\lambda}D_{V,U}^{\lambda}$
The element $C_{S,V}^{\lambda}D_{V,U}^{\lambda}C_{P,Q}^{\eta}\in I^{\lambda}$
is proved similarly. ∎
We will denote $\sum\limits_{\lambda\in\Lambda,k_{\lambda}=0}I^{\lambda}$ by
$I^{\Lambda}$.
Similarly, for each $\lambda\in\Lambda$, the elements of the form
$\sum\limits_{S,U\in M(\lambda)}r_{U,S}D_{U,V}^{\lambda}C_{V,S}^{\lambda}$
with $r_{U,S}\in R$ also make an ideal $I_{D}^{\lambda}$ of $A$. Denote
$\sum\limits_{\lambda\in\Lambda,k_{\lambda}=0}I_{D}^{\lambda}$ by
$I_{D}^{\Lambda}$.
Define
$I=I^{\Lambda}+I_{D}^{\Lambda}$
and define
$\Lambda_{1}=\\{\lambda\in\Lambda\mid\operatorname{rad}\lambda=0\\},$
$\Lambda_{2}=\Lambda_{0}-\Lambda_{1},$
$\Lambda_{3}=\Lambda-\Lambda_{0},$ $\Lambda_{4}=\\{\lambda\in\Lambda_{1}\mid
k_{\lambda}=0\\}$.
Now we are in a position to give the main results of this paper.
###### Theorem 4.2.
Suppose that $R$ is an integral domain and that $A$ is a symmetric cellular
algebra with a cellular basis $C=\\{C_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$. Let $\tau$ be a symmetrizing trace on $A$
and let $\\{D_{T,S}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$ be
the dual basis of $C$ with respect to $\tau$. Then
(1) $I\subseteq\operatorname{rad}A$, $I^{3}=0$.
(2) $I$ is independent of the choice of $\tau$.
Moreover, if $R$ is a field, then
(3)
$\dim_{R}I\geq\sum\limits_{\lambda\in\Lambda_{2}}(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)\dim_{R}L_{\lambda}+\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2},$
where $n_{\lambda}$ is the number of the elements in $M(\lambda)$.
(4)
$\sum\limits_{\lambda\in\Lambda_{2}}(\dim_{K}L_{\lambda})^{2}-\sum\limits_{\lambda\in\Lambda_{3}}n_{\lambda}^{2}\leq\sum\limits_{\lambda\in\Lambda_{2}}(\dim_{K}\operatorname{rad}\lambda)^{2}-\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2}.$
###### Proof.
(1) $I\subseteq\operatorname{rad}A$ , $I^{3}=0$.
Firstly, we prove $(I^{\Lambda})^{2}=0$. Obviously, by the definition of
$I^{\Lambda}$, every element of $(I^{\Lambda})^{2}$ can be written as a linear
combination of elements of the form
$C_{S_{1},T}^{\lambda}D_{T,S_{2}}^{\lambda}C_{U_{1},V}^{\mu}D_{V,U_{2}}^{\mu}$(we
omit the coefficient here) with $k_{\lambda}=0$ and $k_{\mu}=0$.
If $\mu<\lambda$, then
$C_{S_{1},T}^{\lambda}D_{T,S_{2}}^{\lambda}C_{U_{1},V}^{\mu}D_{V,U_{2}}^{\mu}=0$
by Lemma 3.1 (8).
If $\mu>\lambda$, then by Lemma 3.1 (1) and (7),
$C_{S_{1},T}^{\lambda}D_{T,S_{2}}^{\lambda}C_{U_{1},V}^{\mu}D_{V,U_{2}}^{\mu}=\sum_{Y\in
M(\lambda)}r_{(U_{1},V,\mu),(Y,T,\lambda),(S_{2},T,\lambda)}C_{S_{1},T}^{\lambda}D_{T,Y}^{\lambda}D_{V,U_{2}}^{\mu}.$
However, by Lemma 3.2, every $D_{P,Q}^{\eta}$ with nonzero coefficient in the
expansion of $D_{T,Y}^{\lambda}D_{V,U_{2}}^{\mu}$ satisfies $\eta\geq\mu$.
Since $\mu>\lambda$, then $\eta>\lambda$. Now, by Lemma 3.1 (7), we have
$C_{S_{1},T}^{\lambda}D_{P,Q}^{\eta}=0$, that is,
$C_{S_{1},T}^{\lambda}D_{T,S_{2}}^{\lambda}C_{U_{1},V}^{\mu}D_{V,U_{2}}^{\mu}=0$
if $\mu>\lambda$.
If $\lambda=\mu$, by Lemma 3.1 (3) and (4), we only need to consider the
elements of the form
$C_{S_{1},T_{1}}^{\lambda}D_{T_{1},S_{2}}^{\lambda}C_{S_{2},T_{2}}^{\lambda}D_{T_{2},S_{3}}^{\lambda}.$
By Lemma 3.4 and Lemma 3.7,
$\displaystyle
C_{S_{1},T_{1}}^{\lambda}D_{T_{1},S_{2}}^{\lambda}C_{S_{2},T_{2}}^{\lambda}D_{T_{2},S_{3}}^{\lambda}=k_{\lambda}C_{S_{1},T_{1}}^{\lambda}D_{T_{1},S_{3}}^{\lambda}=0.$
Then we get that all the elements of the form
$C_{S_{1},T}^{\lambda}D_{T,S_{2}}^{\lambda}C_{U_{1},V}^{\mu}D_{V,U_{2}}^{\mu}$
are zero, that is, $(I^{\Lambda})^{2}=0$.
Similarly, we get $(I_{D}^{\Lambda})^{2}=0$.
To prove $I^{3}=0$, we now only need to consider the elements in
$I^{\Lambda}I_{D}^{\Lambda}I^{\Lambda}$ and
$I_{D}^{\Lambda}I^{\Lambda}I_{D}^{\Lambda}$. For $\lambda,\mu,\eta\in\Lambda$
with $k_{\lambda}=k_{\mu}=k_{\eta}=0$ and $S,T,M\in M(\lambda)$, $U,V,N\in
M(\mu)$, $P,Q,W\in M(\eta)$, suppose that
$C_{S,T}^{\lambda}D_{T,M}^{\lambda}D_{U,V}^{\mu}C_{V,N}^{\mu}C_{P,Q}^{\eta}D_{Q,W}^{\eta}\neq
0$. If $\lambda>\mu$, then any $D_{X,Y}^{\epsilon}$ with nonzero coefficient
in the expansion of $D_{T,M}^{\lambda}D_{U,V}^{\mu}$ satisfies
$\epsilon\geq\lambda$, so $\epsilon>\mu$, this implies that
$D_{X,Y}^{\epsilon}C_{V,N}^{\mu}=0$ by Lemma 3.1, a contradiction. If
$\lambda<\mu$, then any $D_{X,Y}^{\epsilon}$ with nonzero coefficient in the
expansion of $D_{T,M}^{\lambda}D_{U,V}^{\mu}$ satisfies $\epsilon\geq\mu$, so
$\epsilon>\lambda$, this implies that $C_{S,T}^{\lambda}D_{X,Y}^{\epsilon}=0$
by Lemma 3.1, a contradiction. Thus $\lambda=\mu$. Similarly, we get
$\eta=\mu$. By a direct computation, we can also get
$C_{S,T}^{\lambda}D_{T,M}^{\lambda}D_{U,V}^{\mu}C_{V,N}^{\mu}C_{P,Q}^{\eta}D_{Q,W}^{\eta}=0$.
This implies that $I^{\Lambda}I_{D}^{\Lambda}I^{\Lambda}=0$. Similarly
$I_{D}^{\Lambda}I^{\Lambda}I_{D}^{\Lambda}=0$ is proved. Then $I^{3}=0$
follows.
Now it is clear that $I\subseteq\operatorname{rad}A$ for $I$ is a nilpotent
ideal of $A$.
(2) $I$ is independent of the choice of $\tau$.
Let $\tau$ and $\tau^{\prime}$ be two symmetrizing traces and $D$, $d$ the
dual bases determined by $\tau$ and $\tau^{\prime}$ respectively. By Lemma
2.3, for arbitrary $d_{U,V}^{\lambda}\in d$,
$d_{U,V}^{\lambda}=\sum_{\varepsilon\in\Lambda,X,Y\in
M(\varepsilon)}\tau(C_{X,Y}^{\varepsilon}d_{U,V}^{\lambda})D_{Y,X}^{\varepsilon}.$
Then for arbitrary $S\in M(\lambda)$,
$C_{S,U}^{\lambda}d_{U,V}^{\lambda}=\sum_{\varepsilon\in\Lambda,X,Y\in
M(\varepsilon)}\tau(C_{X,Y}^{\varepsilon}d_{U,V}^{\lambda})C_{S,U}^{\lambda}D_{Y,X}^{\varepsilon}.$
By Lemma 3.1 (7), (8), if $\varepsilon<\lambda$, then
$C_{X,Y}^{\varepsilon}d_{U,V}^{\lambda}=0$; if $\varepsilon>\lambda$, then
$C_{S,U}^{\lambda}D_{Y,X}^{\varepsilon}=0.$ This implies that
$C_{S,U}^{\lambda}d_{U,V}^{\lambda}=\sum_{X,Y\in
M(\lambda)}\tau(C_{X,Y}^{\lambda}d_{U,V}^{\lambda})C_{S,U}^{\lambda}D_{Y,X}^{\lambda}.$
By Lemma 3.1 (5), if $Y\neq U$, then $C_{S,U}^{\lambda}D_{Y,X}^{\lambda}=0$.
Hence
$C_{S,U}^{\lambda}d_{U,V}^{\lambda}=\sum_{X\in
M(\lambda)}\tau(C_{X,U}^{\lambda}d_{U,V}^{\lambda})C_{S,U}^{\lambda}D_{U,X}^{\lambda}.$
Noting that
$\tau(C_{X,U}^{\lambda}d_{U,V}^{\lambda})=\tau(d_{U,V}^{\lambda}C_{X,U}^{\lambda})$,
it follows from Lemma 3.1 that $d_{U,V}^{\lambda}C_{X,U}^{\lambda}=0$ if
$X\neq V$. Thus
$C_{S,U}^{\lambda}d_{U,V}^{\lambda}=\tau(C_{V,U}^{\lambda}d_{U,V}^{\lambda})C_{S,U}^{\lambda}D_{U,V}^{\lambda}.$
Similarly, we obtain
$C_{S,U}^{\lambda}D_{U,V}^{\lambda}=\tau^{\prime}(C_{V,U}^{\lambda}D_{U,V}^{\lambda})C_{S,U}^{\lambda}d_{U,V}^{\lambda},$
$d_{V,U}^{\lambda}C_{U,S}^{\lambda}=\tau(C_{V,U}^{\lambda}d_{U,V}^{\lambda})D_{V,U}^{\lambda}C_{U,S}^{\lambda},$
$D_{V,U}^{\lambda}C_{U,S}^{\lambda}=\tau^{\prime}(C_{V,U}^{\lambda}D_{U,V}^{\lambda})d_{V,U}^{\lambda}C_{U,S}^{\lambda}.$
The above four formulas imply that $I$ is independent of the choice of
symmetrizing trace.
(3)
$\dim_{R}I\geq\sum\limits_{\lambda\in\Lambda_{2}}(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)\dim_{R}L_{\lambda}+\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2}.$
For any $\lambda\in\Lambda_{2}$ and $S,T\in M(\lambda)$, it follows from Lemma
3.1 that
$C_{S,T}^{\lambda}D_{T,T}^{\lambda}\equiv\sum\limits_{X\in
M(\lambda)}\Phi(X,S)D_{X,T}^{\lambda}\,\,\,\,\,(\mod A_{D}(>\lambda)),$
$D_{T,T}^{\lambda}C_{T,S}^{\lambda}\equiv\sum\limits_{Y\in
M(\lambda)}\Phi(Y,S)D_{T,Y}^{\lambda}\,\,\,\,\,(\mod A_{D}(>\lambda)).$
Let $V$ be the $R$-space generated by
$\\{\sum\limits_{X\in M(\lambda)}\Phi(X,S)D_{X,T}^{\lambda}\mid S,T\in
M(\lambda)\\}\cup\\{\sum\limits_{Y\in
M(\lambda)}\Phi(Y,S)D_{T,Y}^{\lambda}\mid S,T\in M(\lambda)\\}.$
Then it is easy to know from the definition of $I^{\lambda}$ and
$I_{D}^{\lambda}$ that
$\dim_{R}(I^{\lambda}+I_{D}^{\lambda})\geq\dim V.$
Note that by Lemma 3.8, $D^{\lambda}:(D_{S},D_{T})\rightarrow
D_{S,T}^{\lambda}$ is an isomorphism of $R$-modules. So we only need to
consider the dimension of $V^{\prime}$ generated by
$\\{\sum\limits_{X\in M(\lambda)}\Phi(X,S)D_{X}\otimes D_{T}\mid S,T\in
M(\lambda)\\}\cup\\{D_{T}\otimes\sum\limits_{Y\in
M(\lambda)}\Phi(Y,S)D_{Y}\mid S,T\in M(\lambda)\\}.$
Since $\Phi_{\lambda}\neq 0$,
$\operatorname{rank}G_{\lambda}=\dim_{R}L_{\lambda}$, we have $\dim
V^{\prime}=2n_{\lambda}\dim_{R}L_{\lambda}-(\dim_{R}L_{\lambda})^{2},$ that
is, $\dim
V^{\prime}=\dim_{R}L_{\lambda}\times(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)$.
Thus
$\dim_{R}(I^{\lambda}+I_{D}^{\lambda})\geq\dim_{R}L_{\lambda}\times(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda).$
Clearly, the above inequality holds true for any $\lambda\in\Lambda_{4}$, then
we have
$\dim_{R}(I^{\lambda}+I_{D}^{\lambda})\geq n_{\lambda}^{2}$
for any $\lambda\in\Lambda_{4}$.
It is clear from Lemma 3.2 that
$\dim_{R}I\geq\sum\limits_{\lambda\in\Lambda_{2}}\dim_{R}(I^{\lambda}+I_{D}^{\lambda})+\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2}$
and then item (3) follows.
(4)
$\sum\limits_{\lambda\in\Lambda_{2}}(\dim_{K}L_{\lambda})^{2}-\sum\limits_{\lambda\in\Lambda_{3}}n_{\lambda}^{2}\leq\sum\limits_{\lambda\in\Lambda_{2}}(\dim_{K}\operatorname{rad}\lambda)^{2}.$
By (1) and (3),
$\dim_{R}\operatorname{rad}A\geq\sum\limits_{\lambda\in\Lambda_{2}}(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)\dim_{R}L_{\lambda}+\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2}.$
By the formula
$\dim_{R}\operatorname{rad}A=\dim_{R}A-\sum_{\lambda\in\Lambda_{0}}(\dim_{R}L_{\lambda})^{2},$
we have
$\dim_{R}A-\sum_{\lambda\in\Lambda_{0}}(\dim_{R}L_{\lambda})^{2}\geq\sum\limits_{\lambda\in\Lambda_{2}}(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)\dim_{R}L_{\lambda}+\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2}.$
That is,
$\sum_{\lambda\in\Lambda_{3}}n_{\lambda}^{2}+\sum_{\lambda\in\Lambda_{0}}n_{\lambda}^{2}-\sum_{\lambda\in\Lambda_{0}}(\dim_{R}L_{\lambda})^{2}\geq\sum\limits_{\lambda\in\Lambda_{2}}(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)\dim_{R}L_{\lambda}+\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2},$
or
$\sum_{\lambda\in\Lambda_{3}}n_{\lambda}^{2}+\sum_{\lambda\in\Lambda_{2}}n_{\lambda}^{2}-\sum_{\lambda\in\Lambda_{2}}(\dim_{R}L_{\lambda})^{2}\geq\sum\limits_{\lambda\in\Lambda_{2}}(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)\dim_{R}L_{\lambda}+\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2},$
or
$\sum\limits_{\lambda\in\Lambda_{2}}(\dim_{K}L_{\lambda})^{2}-\sum\limits_{\lambda\in\Lambda_{3}}n_{\lambda}^{2}\leq\sum\limits_{\lambda\in\Lambda_{2}}n_{\lambda}^{2}-\sum\limits_{\lambda\in\Lambda_{2}}(n_{\lambda}+\dim_{R}\operatorname{rad}\lambda)\dim_{R}L_{\lambda}-\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2}.$
According to
$\dim_{R}L_{\lambda}=n_{\lambda}-\dim_{R}\operatorname{rad}\lambda$, the right
side of the above inequality is
$\sum\limits_{\lambda\in\Lambda_{2}}(\dim_{K}\operatorname{rad}\lambda)^{2}-\sum\limits_{\lambda\in\Lambda_{4}}n_{\lambda}^{2}$
and this completes the proof. ∎
###### Corollary 4.3.
Let $R$ be an integral domain and $A$ a symmetric cellular algebra. Let
$\lambda$ be the minimal element in $\Lambda$. If
$\operatorname{rad}\lambda\neq 0$, then $R-{\rm span}\\{C_{S,T}^{\lambda}\mid
S,T\in M(\lambda)\\}\subset\operatorname{rad}A$.
###### Proof.
If $a=\sum\limits_{X,Y\in M(\lambda)}r_{X,Y}C_{X,Y}^{\lambda}$ is not in
$\operatorname{rad}A$, then there exists some $D_{U,V}^{\mu}$ such that
$aD_{U,V}^{\mu}\notin\operatorname{rad}A$. If $\mu\neq\lambda$, then
$aD_{U,V}^{\mu}=0$ by Lemma 3.1, it is in $\operatorname{rad}A$. If
$\mu=\lambda$, then $aD_{U,V}^{\mu}\in\operatorname{rad}A$ by Theorem 4.2. It
is a contradiction. ∎
###### Corollary 4.4.
Let $A$ be a finite dimensional symmetric cellular algebra and
$r\in\operatorname{rad}A$. Assume that $\lambda\in\Lambda$ satisfies:
(1) There exists $S,T\in M(\lambda)$ such that $C_{S,T}^{\lambda}$ appears in
the expansion of $r$ with nonzero coefficient.
(2) For any $\mu>\lambda$ and $U,V\in M(\mu)$, the coefficient of
$C_{U,V}^{\mu}$ in the expansion of $r$ is zero.
Then $k_{\lambda}=0$.
###### Proof.
Since $r=\sum\limits_{\varepsilon\in\Lambda,X,Y\in
M(\varepsilon)}r_{X,Y,\varepsilon}C_{X,Y}^{\varepsilon}\in\operatorname{rad}A$,
we have $rD_{T,S}^{\lambda}\in\operatorname{rad}A$. The conditions (1) and (2)
imply that
$rD_{T,S}^{\lambda}=\sum\limits_{X\in
M(\lambda)}r_{X,T,\lambda}C_{X,T}^{\lambda}D_{T,S}^{\lambda}.$
It is easy to check that
$(rD_{T,S}^{\lambda})^{n}=(k_{\lambda}r_{S,T,\lambda})^{n-1}rD_{T,S}^{\lambda}$.
Applying $\tau$ on both sides of this equation, we get
$\tau((rD_{T,S}^{\lambda})^{n})=(k_{\lambda}r_{S,T,\lambda})^{n-1}r_{S,T,\lambda}$.
If $k_{\lambda}\neq 0$, then $\tau((rD_{T,S}^{\lambda})^{n})\neq 0$. Hence
$rD_{T,S}^{\lambda}$ is not nilpotent and then
$rD_{T,S}^{\lambda}\notin\operatorname{rad}A$, a contradiction. This implies
that $k_{\lambda}=0$. ∎
Example The group algebra $\mathbb{Z}_{3}S_{3}$.
The algebra has a basis
$\\{1,s_{1},s_{2},s_{1}s_{2},s_{2}s_{1},s_{1}s_{2}s_{1}\\}.$
A cellular basis is
$C_{1,1}^{(3)}=1+s_{1}+s_{2}+s_{1}s_{2}+s_{2}s_{1}+s_{1}s_{2}s_{1}$,
$C_{1,1}^{(2,1)}=1+s_{1},\,\,\,\,\,\,\,\,\,C_{1,2}^{(2,1)}=s_{2}+s_{1}s_{2}$,
$C_{2,1}^{(2,1)}=s_{2}+s_{2}s_{1},C_{2,2}^{(2,1)}=1+s_{1}s_{2}s_{1}$,
$C_{1,1}^{(1^{3})}=1$.
The corresponding dual basis is
$D_{1,1}^{(3)}=-s_{2}+s_{1}s_{2}+s_{2}s_{1}$,
$D_{1,1}^{(2,1)}=s_{1}+s_{2}-s_{1}s_{2}-s_{2}s_{1},D_{2,1}^{(2,1)}=s_{2}-s_{1}s_{2}$,
$D_{1,2}^{(2,1)}=s_{2}-s_{2}s_{1},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,D_{2,2}^{(2,1)}=s_{2}-s_{1}s_{2}-s_{2}s_{1}+s_{1}s_{2}s_{1}$,
$D_{1,1}^{(1^{3})}=1-s_{1}-s_{2}+s_{1}s_{2}+s_{2}s_{1}-s_{1}s_{2}s_{1}$.
It is easy to know that $\Lambda_{3}=(3)$ and $\Lambda_{1}=(1^{3})$. Then
$\dim_{K}\operatorname{rad}A=4$. Now we compute $I$.
$C_{1,1}^{(3)}D_{1,1}^{(3)}=1+s_{1}+s_{2}+s_{1}s_{2}+s_{2}s_{1}+s_{1}s_{2}s_{1}$,
$C_{1,2}^{(2,1)}D_{2,1}^{(2,1)}=1+s_{1}-s_{2}-s_{1}s_{2}s_{1}$,
$C_{1,2}^{(2,1)}D_{2,2}^{(2,1)}=s_{2}+s_{1}s_{2}-s_{2}s_{1}-s_{1}s_{2}s_{1}$,
$C_{2,1}^{(2,1)}D_{1,2}^{(2,1)}=1-s_{1}-s_{1}s_{2}+s_{1}s_{2}s_{1}$,
$C_{2,1}^{(2,1)}D_{1,1}^{(2,1)}=s_{2}+s_{2}s_{1}-s_{1}-s_{1}s_{2}$.
Then $\dim_{K}I=4$. This implies that $I=\operatorname{rad}A$.
## 5\. Semisimplicity of symmetric cellular algebras
As a by-product of the results on radicals, we will give some equivalent
conditions for a finite dimensional symmetric cellular algebra to be
semisimple.
###### Corollary 5.1.
Let $A$ be a finite dimensional symmetric cellular algebra. Then the following
are equivalent.
(1) The algebra $A$ is semisimple.
(2) $k_{\lambda}\neq 0$ for all $\lambda\in\Lambda$.
(3) $\\{C_{S,T}^{\lambda}D_{T,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$ is a basis of $A$.
(4) For any $\lambda\in\Lambda$, there exist $S,T\in M(\lambda)$, such that
$(C_{S,T}^{\lambda}D_{T,S}^{\lambda})^{2}\neq 0$.
(5) For any $\lambda\in\Lambda$ and arbitrary $S,T\in M(\lambda)$,
$(C_{S,T}^{\lambda}D_{T,S}^{\lambda})^{2}\neq 0$.
###### Proof.
(2)$\Longrightarrow$(1) If $k_{\lambda}\neq 0$ for all $\lambda\in\Lambda$,
then $\operatorname{rad}\lambda=0$ for all $\lambda\in\Lambda$ by Corollary
3.7. This implies that $A$ is semisimple by Theorem 2.10.
(1)$\Longrightarrow$(2) Assume that there exists some $\lambda\in\Lambda$ such
that $k_{\lambda}=0$. Then it is easy to check that $I^{\lambda}$ is a
nilpotent ideal of $A$. Obviously, $I^{\lambda}\neq 0$ because at least
$C_{U,V}^{\lambda}D_{V,U}^{\lambda}\neq 0$. This implies that
$I^{\lambda}\subseteq\operatorname{rad}A$. But $A$ is semisimple, a
contradiction. This implies that $k_{\lambda}\neq 0$ for all
$\lambda\in\Lambda$.
(2)$\Longrightarrow$(3) Let $\sum\limits_{\lambda\in\Lambda,S,T\in
M(\lambda)}k_{S,T,\lambda}C_{S,T}^{\lambda}D_{T,T}^{\lambda}=0$. Take a
maximal element $\lambda_{0}\in\Lambda$. For arbitrary $X,Y\in
M(\lambda_{0})$,
$\displaystyle
C_{X,X}^{\lambda_{0}}D_{X,Y}^{\lambda_{0}}(\sum_{\lambda\in\Lambda,S,T\in
M(\lambda)}k_{S,T,\lambda}C_{S,T}^{\lambda}D_{T,T}^{\lambda})=k_{\lambda_{0}}\sum_{T\in
M(\lambda_{0})}k_{Y,T,\lambda_{0}}C_{X,T}^{\lambda_{0}}D_{T,T}^{\lambda_{0}}=0.$
This implies that $\tau(k_{\lambda_{0}}\sum\limits_{T\in
M(\lambda_{0})}k_{Y,T,\lambda_{0}}C_{X,T}^{\lambda_{0}}D_{T,T}^{\lambda_{0}})=0$,
i.e., $k_{\lambda_{0}}k_{Y,X,\lambda_{0}}=0$. Since $k_{\lambda_{0}}\neq 0$,
then we get $k_{Y,X,\lambda_{0}}=0$.
Repeating the process as above, we get that all the $k_{S,T,\lambda}$ are
zeros.
(3)$\Longrightarrow$(2) Since
$\\{C_{S,T}^{\lambda}D_{T,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$ is a basis of $A$, we have
$1=\sum_{\lambda\in\Lambda,S,T\in
M(\lambda)}k_{S,T,\lambda}C_{S,T}^{\lambda}D_{T,T}^{\lambda}.$
For arbitrary $\mu\in\Lambda$ and $U,V\in M(\mu)$, we have
$\displaystyle C_{U,V}^{\mu}D_{V,V}^{\mu}$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\Lambda,S,T\in
M(\lambda)}k_{S,T,\lambda}C_{S,T}^{\lambda}D_{T,T}^{\lambda}C_{U,V}^{\mu}D_{V,V}^{\mu}$
$\displaystyle=$ $\displaystyle k_{\mu}\sum_{X\in
M(\mu)}k_{X,U,\mu}C_{X,V}^{\mu}D_{V,V}^{\mu}.$
This implies that $k_{\mu}\neq 0$ since $C_{U,V}^{\mu}D_{V,V}^{\mu}\neq 0$.
The fact that $\mu$ is arbitrary implies that $k_{\lambda}\neq 0$ for all
$\lambda\in\Lambda$.
(2)$\Longleftrightarrow$(4) and (2)$\Longleftrightarrow$(5) are clear by Lemma
3.4. ∎
###### Corollary 5.2.
Let $R$ be an integral domain and $A$ a symmetric cellular algebra with a cell
datum $(\Lambda,M,C,i)$. Let $K$ be the field of fractions of $R$ and
$A_{K}=A\bigotimes_{R}K$. If $A_{K}$ is semisimple, then
$\\{\mathcal{E}_{S,T}^{\lambda}=C_{S,S}^{\lambda}D_{S,T}^{\lambda}C_{T,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$
is a cellular basis of $A_{K}$. Moreover, if $\lambda\neq\mu$, then
$\mathcal{E}_{S,T}^{\lambda}\mathcal{E}_{U,V}^{\mu}=0$.
###### Proof.
Firstly, we prove that
$\\{\mathcal{E}_{S,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in M(\lambda)\\}$ is
a basis of $A_{K}$. We only need to show the elements in this set are
$K$-linear independent. By Lemma 3.1, we have
$\displaystyle\mathcal{E}_{S,T}^{\lambda}$ $\displaystyle=$
$\displaystyle\sum\limits_{X\in
M(\lambda)}r_{(T,T,\lambda),(X,S,\lambda),(T,S,\lambda)}C_{S,S}^{\lambda}D_{S,X}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum\limits_{X\in
M(\lambda)}\Phi(X,T)C_{S,X}^{\lambda}D_{X,X}^{\lambda}$
for all $\lambda\in\Lambda,S,T\in M(\lambda)$. Since $A_{K}$ is semisimple,
all $G(\lambda)$ are non-degenerate. Moreover,
$\\{C_{S,T}^{\lambda}D_{T,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$ is a basis of $A_{K}$ by Corollary 5.1, then
$\\{\mathcal{E}_{S,T}^{\lambda}=C_{S,S}^{\lambda}D_{S,T}^{\lambda}C_{T,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$
is a basis of $A_{K}$.
Secondly, $i(\mathcal{E}_{S,T}^{\lambda})\equiv\mathcal{E}_{T,S}^{\lambda}$
for arbitrary $\lambda\in\Lambda$, and $S,T\in M(\lambda)$. This is clear by
Lemma 3.1 and 3.2.
Thirdly, for arbitrary $a\in A$, since
$\\{C_{S,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in M(\lambda)\\}$ is a cellular
basis of $A$, we have
$\displaystyle a\mathcal{E}_{S,T}^{\lambda}$ $\displaystyle=$ $\displaystyle
aC_{S,S}^{\lambda}D_{S,T}^{\lambda}C_{T,T}^{\lambda}$ $\displaystyle=$
$\displaystyle\sum_{X\in
M(\lambda)}r_{a}(X,S)C_{X,S}^{\lambda}D_{S,T}^{\lambda}C_{T,T}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{X\in
M(\lambda)}r_{a}(X,S)C_{X,X}^{\lambda}D_{X,T}^{\lambda}C_{T,T}^{\lambda}$
$\displaystyle=$ $\displaystyle\sum_{X\in
M(\lambda)}r_{a}(X,S)\mathcal{E}_{X,T}^{\lambda}.$
Clearly, $r_{a}(X,S)$ is independent of $T$. Then
$\\{\mathcal{E}_{S,T}^{\lambda}=C_{S,S}^{\lambda}D_{S,T}^{\lambda}C_{T,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$
is a cellular basis of $A_{K}$.
Finally, for any $\lambda,\mu\in\Lambda$, $S,T\in M(\lambda)$, $U,V\in
M(\mu)$,
$\displaystyle\mathcal{E}_{S,T}^{\lambda}\mathcal{E}_{U,V}^{\mu}$
$\displaystyle=$ $\displaystyle
C_{S,S}^{\lambda}D_{S,T}^{\lambda}C_{T,T}^{\lambda}C_{U,U}^{\mu}D_{U,V}^{\mu}C_{V,V}^{\mu}$
$\displaystyle=$ $\displaystyle\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(T,T,\lambda),(U,U,\mu),(X,Y,\epsilon)}C_{S,S}^{\lambda}D_{S,T}^{\lambda}C_{X,Y}^{\epsilon}D_{U,V}^{\mu}C_{V,V}^{\mu}.$
By Lemma 3.1,
$C_{S,S}^{\lambda}D_{S,T}^{\lambda}C_{X,Y}^{\epsilon}D_{U,V}^{\mu}C_{V,V}^{\mu}\neq
0$ implies $\epsilon\geq\lambda,\epsilon\geq\mu$. On the other hand, by
Definition 2.4, $r_{(T,T,\lambda),(U,U,\mu),(X,Y,\epsilon)}\neq 0$ implies
$\epsilon\leq\lambda$ and $\epsilon\leq\mu$. Therefore, if $\lambda\neq\mu$,
then $\mathcal{E}_{S,T}^{\lambda}\mathcal{E}_{U,V}^{\mu}=0$. ∎
Acknowledgement
The author acknowledges his supervisor Prof. C.C. Xi. He also thanks Dr. Wei
Hu and Zhankui Xiao for many helpful conversations.
## References
* [1] J. Brundan and C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra I: cellularity, arxiv: math0806.1532v1.
* [2] J. Du and H.B. Rui, Based algebras and standard bases for quasi-hereditary algebras, Trans. Amer. Math.Soc., 350, (1998), 3207-3235.
* [3] M. Geck, Hecke algebras of finite type are cellular, Invent. math., 169, (2007), 501-517.
* [4] F. Goodman, Cellularity of cyclotomic Birman-Wenzl-Murakami algebras, J. Algebra, 321, (2009), 3299-3320.
* [5] J.J. Graham, Modular representations of Hecke algebras and related algebras, PhD Thesis, Sydney University, 1995.
* [6] J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math., 123, (1996), 1-34.
* [7] R.M. Green, Completions of cellular algebras, Comm. Algebra, 27, (1999), 5349-5366.
* [8] R.M. Green, Tabular algebras and their asymptotic versions, J. Algebra, 252, (2002), 27-64.
* [9] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, (1979), 165-184.
* [10] S. Koenig and C.C. Xi, On the structure of cellular algebras, In: I. Reiten, S. Smalo and O. solberg(Eds.): Algebras and Modules II. Canadian Mathematics Society Proceedings, Vol. 24, (1998), 365-386.
* [11] S. Koenig and C.C. Xi, Cellular algebras: Inflations and Morita equivalences, J. London Math. Soc. (2), 60, (1999), 700-722.
* [12] S. Koenig and C.C. Xi, A characteristic-free approach to Brauer algebras, Trans. Amer. Math. Soc., 353, (2001), 1489-1505.
* [13] S. Koenig and C.C. Xi, Affine cellular algebras, preprint.
* [14] G.I. Lehrer and R.B. Zhang, A Temperley-Lieb analogue for the BMW algebra, arXiv:math/08060687v1.
* [15] G. Malle and A. Mathas, Symmetric cyclotomic Hecke algebras, J. Algebra, 205, (1998), 275-293.
* [16] E. Murphy, The representations of Hecke algebras of type $A_{n}$, J. Algebra, 173, (1995), 97-121.
* [17] H.B. Rui and C.C. Xi, The representation theory of cyclotomic Temperley-Lieb algebras, Comment. Math. Helv., 79, no.2, (2004), 427-450.
* [18] B.W. Westbury, Invariant tensors and cellular categories, J. Algebra, 321, (2009), 3563-3567.
* [19] C.C. Xi, Partition algebras are cellular, Compositio math., 119, (1999), 99-109.
* [20] C.C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154, (2000), 280-298.
* [21] C.C. Xi and D.J. Xiang, Cellular algebras and Cartan matrices, Linear Algebra Appl., 365, (2003), 369-388.
|
arxiv-papers
| 2009-11-18T12:42:58 |
2024-09-04T02:49:06.553715
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanbo Li",
"submitter": "Yanbo Li",
"url": "https://arxiv.org/abs/0911.3524"
}
|
0911.3529
|
# Jucys-Murphy elements and centers of cellular algebras ††thanks: keywords:
Jucys-Murphy elements, cellular algebras, center.
Yanbo Li
Department of Information and Computing Sciences,
Northeastern University at Qinhuangdao;
Qinhuangdao, 066004, P.R. China
School of Mathematics Sciences, Beijing Normal University;
Beijing, 100875, P.R. China
E-mail: liyanbo707@163.com
###### Abstract
Let $R$ be an integral domain and $A$ a cellular algebra over $R$ with a
cellular basis $\\{C_{S,T}^{\lambda}\mid\lambda\in\Lambda,S,T\in
M(\lambda)\\}$. Suppose that $A$ is equipped with a family of Jucys-Murphy
elements which satisfy the separation condition in the sense of A. Mathas
[14]. Let $K$ be the field of fractions of $R$ and $A_{K}=A\bigotimes_{R}K$.
We give a necessary and sufficient condition under which the center of $A_{K}$
consists of the symmetric polynomials in Jucys-Murphy elements.
## 1 Introduction
Jucys-Murphy elements were constructed for the group algebras of symmetric
groups first. The combinatorics of these elements allow one to compute simple
representations explicitly and often easily in the semisimple case. Then
Dipper, James and Murphy [3], [4], [5], [6], [7] did a lot of work on
representations of Hecke algebras and produced analogues of the Jucys-Murphy
elements for Hecke algebras of types A and B. The constructions for other
algebras can be found in [11], [16] and so on. In [4], Dipper and James
conjectured that the center of a Hecke algebra of type A consists of symmetric
polynomials in Jucys-Murphy elements. The conjecture was proved by Francis and
Graham [8] in 2006. In [2], Brundan proved that the center of each degenerate
cyclotomic Hecke algebra consists of symmetric polynomials in the Jucys-Murphy
elements. An analogous conjecture for Ariki-Koike Hecke algebra is open in
non-semisimple case.
Cellular algebras were introduced by Graham and Lehrer [10] in 1996, motivated
by previous work of Kazhdan and Lusztig [13]. The theory of cellular algebras
provides a systematic framework for studying the representation theory of non-
semisimple algebras which are deformations of semisimple ones. Many classes of
algebras from mathematics and physics are found to be cellular, including
Hecke algebras of finite type, Ariki-Koike Hecke algebras, $q$-Schur algebras,
Brauer algebras, partition algebras, Birman-Wenzl algebras and so on, see [9],
[10], [17], [18] for details.
The fact that most of the algebras which have Jucys-Murphy elements are
cellular leads one to defining Jucys-Murphy elements for general cellular
algebras. In [14], Mathas did some work in this direction. By the definition
of Mathas, we investigate the relations between the centers and the Jucys-
Murphy elements of cellular algebras.
Let $R$ be an integral domain and $A$ a cellular $R$-algebra with a cellular
basis $\\{C_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$. Let
$K$ be the field of fractions of $R$ and $A_{K}=A\bigotimes_{R}K$. Suppose
that $A$ is equipped with a family of Jucys-Murphy elements
$L_{1},\ldots,L_{m}$ which satisfy the separation condition [14]. For any
$\lambda\in\Lambda$, $\\{c_{\lambda}(i)\mid 1\leq i\leq m\\}$ is a family of
elements in $R$. Then the main result of this paper is the following theorem.
Theorem. _Suppose that every symmetric polynomial in $L_{1},\ldots,L_{m}$
belongs to the center of $A_{K}$. Then the following are equivalent._
(1) _The center of $A_{K}$ consists of symmetric polynomials in Jucys-Murphy
elements._
(2) _$\\{c_{\lambda}(i)\mid 1\leq i\leq m\\}$ can not be obtained from
$\\{c_{\mu}(i)\mid 1\leq i\leq m\\}$ by permutations for arbitrary
$\lambda,\mu\in\Lambda$ with $\lambda\neq\mu$._
The condition in the above theorem is also a necessary condition for the
center of $A$ consisting of the symmetric polynomials in Jucys-Murphy
elements. Moreover, by this theorem, we can prove that the centers of Ariki-
Koike Hecke algebras consist of the symmetric polynomials in Jucys-Murphy
elements in semisimple case. The proof is different from Ariki's in [1] and A.
Ram's in [15].
## 2 Cellular algebras and Jucys-Murphy elements
In this section, we first recall the definition of cellular algebras and then
give a quick review of the results under the so-called separation condition in
A. Mathas' paper [14].
###### Definition 2.1.
([10] 1.1) Let $R$ be a commutative ring with identity. An associative unital
$R$-algebra is called a cellular algebra with cell datum $(\Lambda,M,C,i)$ if
the following conditions are satisfied:
(C1) The finite set $\Lambda$ is a poset. Associated with each
$\lambda\in\Lambda$, there is a finite set $M(\lambda)$. The algebra $A$ has
an $R$-basis $\\{C_{S,T}^{\lambda}\mid S,T\in
M(\lambda),\lambda\in\Lambda\\}$.
(C2) The map $i$ is an $R$-linear anti-automorphism of $A$ with $i^{2}=id$
which sends $C_{S,T}^{\lambda}$ to $C_{T,S}^{\lambda}$.
(C3) If $\lambda\in\Lambda$ and $S,T\in M(\lambda)$, then for any element
$a\in A$, we have
$aC_{S,T}^{\lambda}\equiv\sum_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{{}^{\prime}},S)C_{S^{{}^{\prime}},T}^{\lambda}\,\,\,\,(\rm{mod}\,\,\,A(<\lambda)),$
where $r_{a}(S^{{}^{\prime}},S)\in R$ is independent of $T$ and where
$A(<\lambda)$ is the $R$-submodule of $A$ generated by
$\\{C_{S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}}^{\mu}\mid
S^{{}^{\prime\prime}},T^{{}^{\prime\prime}}\in M(\mu),\mu<\lambda\\}$.
Apply $i$ to the equation in (C3), we obtain
$(C3^{{}^{\prime}})\,\,C_{T,S}^{\lambda}i(a)\equiv\sum\limits_{S^{{}^{\prime}}\in
M(\lambda)}r_{a}(S^{{}^{\prime}},S)C_{T,S^{{}^{\prime}}}^{\lambda}\,\,\,\,(\rm
mod\,\,\,A(<\lambda)).$
Let $R$ be an integral domain. Given a cellular algebra $A$, we will also
assume that $M(\lambda)$ is a poset with an order $\leq_{\lambda}$. Let
$M(\Lambda)=\bigsqcup_{\lambda\in\Lambda}M(\lambda)$, we consider $M(\Lambda)$
as a poset with an order $\leq$ as follows.
$S\leq T\Leftrightarrow\begin{cases}S\leq_{\lambda}T,&\text{if $S,T\in
M(\lambda)$;}\\\ \lambda\leq\mu,\,&\text{if $S\in M(\lambda)$, $T\in
M(\mu)$.}\end{cases}$
Let $K$ be the field of fractions of $R$ and $A_{K}=A\bigotimes_{R}K$. We will
consider $A$ as a subalgebra of $A_{K}$.
###### Definition 2.2.
([14] 2.4) Let $R$ be an integral domain and $A$ a cellular algebra. A family
of elements $L_{1},\ldots,L_{m}$ are called Jucys-Murphy elements of $A$ if
(1) $L_{i}L_{j}=L_{j}L_{i}$, for $1\leq i,j\leq m$;
(2) $i(L_{j})=L_{j}$, for $j=1,\cdots,m$;
(3) For all $\lambda\in\Lambda$, $S,T\in M(\lambda)$ and $L_{i}$,
$i=1,\cdots,m$,
$C_{S,T}^{\lambda}L_{i}\equiv
c_{T}(i)C_{S,T}^{\lambda}+\sum_{V<T}r_{L_{i}}(V,T)C_{S,V}^{\lambda}\,\,\,\,(\mod\,\,A({<\lambda})),$
where $c_{T}(i)\in R,\,\,r_{L_{i}}(T,V)\in R$. We call $c_{T}(i)$ the content
of $T$ at $i$. Denote $\\{c_{T}(i)\mid T\in M(\Lambda)\\}$ by $\mathscr{C}(i)$
for $i=1,2,\cdots,m$.
Example. Let $K$ be a field. Let $A$ be the group algebra of symmetric group
$S_{n}$. Set $L_{i}=\sum\limits_{j=1}^{i-1}(i,j)$ for $i=2,\cdots,n$. Then
$L_{i},\,\,\,i=2,\cdots,n$, is a family of Jucys-Murphy elements of $A$.
###### Definition 2.3.
([14] 2.8) Let $A$ be a cellular algebra with Jucys-Murphy elements
$\\{L_{1},\ldots,L_{m}\\}$. We say that the Jucys-Murphy elements satisfy the
separation condition if for any $S,T\in M(\Lambda),S\leq T,S\neq T$, there
exists some $i$ with $1\leq i\leq m$ such that $c_{S}(i)\neq c_{T}(i)$.
Remark. The separation condition forces $A_{K}$ to be semisimple (c.f. [14]).
From now on, we always assume that $A$ is a cellular algebra equipped with a
family of Jucys-Murphy elements which satisfy the separation condition. We now
recall some results of [14].
###### Definition 2.4.
([14] 3.1) Let $A$ be a cellular algebra with Jucys-Murphy elements
$\\{L_{1},\ldots,L_{m}\\}$. For $\lambda\in\Lambda$, $S,T\in M(\lambda)$,
define
$F_{T}=\prod_{i}\prod_{c\in\mathscr{C}(i),c\neq
c_{T}(i)}(L_{i}-c)/(c_{T}(i)-c)$
and $f_{S,T}^{\lambda}=F_{S}C_{S,T}^{\lambda}F_{T}$.
Note that the coefficient of $C_{S,T}^{\lambda}$ in the expansion of
$f_{S,T}^{\lambda}$ is $1$ for any $\lambda\in\Lambda$ and $S,T\in
M(\lambda)$, see [14] 3.3 (a). Then Mathas proved the following theorems.
###### Theorem 2.5.
([14] 3.7) Let $A$ be a cellular algebra with Jucys-Murphy elements
$\\{L_{1},\ldots,L_{m}\\}$. Let $\lambda,\mu\in\Lambda$, $S,T\in M(\lambda)$
and $U,V\in M(\mu)$. Then
(1)
$f_{S,T}^{\lambda}f_{U,V}^{\mu}=\begin{cases}\gamma_{T}f_{S,V}^{\lambda},&\text{$\lambda=\mu,\,\,\,T=U$,}\\\
0,\,&\text{otherwise,}\end{cases}$
where $\gamma_{T}\in K$ and $\gamma_{T}\neq 0$ for all $T\in M(\Lambda)$.
(2) $\\{f_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$ is a
cellular basis of $A_{K}$. $\Box$
###### Theorem 2.6.
([14] 3.16) Let $A$ be a cellular algebra with Jucys-Murphy elements
$\\{L_{1},\ldots,L_{m}\\}$. Then
(1) Let $\lambda\in\Lambda$ and $T\in M(\lambda)$. Then $F_{T}$ is a primitive
idempotent in $A_{K}$. Moreover, $\\{F_{T}\mid T\in M(\lambda)\\}$ is a
complete set of pairwise orthogonal primitive idempotents in $A_{K}$.
(2) $F_{\lambda}=\sum\limits_{T\in M(\lambda)}F_{T}$ is a central idempotent
in $A_{K}$ for any $\lambda\in\Lambda$. Moreover,
$\\{F_{\lambda}\mid\lambda\in\Lambda\\}$ is a complete set of central
idempotents which are primitive in $Z(A_{K})$.
(3) In particular,
$1=\sum\limits_{\lambda\in\Lambda}F_{\lambda}=\sum\limits_{T\in
M(\Lambda)}F_{T}$ and $L_{i}=\sum\limits_{T\in M(\Lambda)}c_{T}(i)F_{T}$.
$\Box$
## 3 Jucys-Murphy elements and centers of cellular algebras
In [14], A. Mathas gave a relation between the center and Jucys-Murphy
elements of a cellular algebra.
###### Proposition 3.1.
([14] 4.13) Let $A$ be a cellular algebra with Jucys-Murphy elements
$\\{L_{1},\ldots,L_{m}\\}$. For any $\lambda\in\Lambda$ and $S,T\in
M(\lambda)$, if $\\{c_{S}(i)\mid 1\leq i\leq m\\}$ can be obtained by
permutations from $\\{c_{T}(i)\mid 1\leq i\leq m\\}$, then every symmetric
polynomial in $L_{1},\ldots,L_{m}$ belongs to the center of $A_{K}$. $\Box$
In fact, the inverse proposition also holds.
###### Proposition 3.2.
Let $A$ be a cellular algebra with a family of Jucys-Murphy elements
$\\{L_{1},\ldots,L_{m}\\}$. Suppose that every symmetric polynomial in
$L_{1},\ldots,L_{m}$ belongs to the center of $A_{K}$. Let $\lambda\in\Lambda$
and $S,T\in M(\lambda)$. Then $\\{c_{S}(i)\mid 1\leq i\leq m\\}$ can be
obtained by permutations from $\\{c_{T}(i)\mid 1\leq i\leq m\\}$.
Proof: Suppose that there exists some $\lambda\in\Lambda$ and $S,T\in
M(\lambda)$ such that $\\{c_{S}(i)\mid 1\leq i\leq m\\}$ can not be obtained
by permutations from $\\{c_{T}(i)\mid 1\leq i\leq m\\}$. Then there exists a
symmetric polynomial $p$ such that
$p(c_{S}(1),\ldots,c_{S}(m))\neq p(c_{T}(1),\ldots,c_{T}(m)).$
Note that $L_{i}=\sum\limits_{X\in M(\Lambda)}c_{X}(i)F_{X}$, then
$p(L_{1},\cdots,L_{m})=\sum_{U\in
M(\Lambda)}p(c_{U}(1),\ldots,c_{U}(m))F_{U}.$
Multiply by $F_{T}$ on both sides, we get
$p(L_{1},\cdots,L_{m})F_{T}=p(c_{T}(1),\ldots,c_{T}(m))F_{T}$ from Theorem 2.6
(1), the equation
$p(L_{1},\cdots,L_{m})F_{S}=p(c_{S}(1),\ldots,c_{S}(m))F_{S}$ is obtained
similarly.
On the other hand, since $p(L_{1},\cdots,L_{m})\in Z(A_{K})$, then by Theorem
2.6 (3),
$p(L_{1},\cdots,L_{m})=\sum\limits_{\lambda\in\Lambda}r_{\lambda}F_{\lambda}$,
where $r_{\lambda}\in K$. Multiply by $F_{T}$ on both sides, we get
$p(L_{1},\cdots,L_{m})F_{T}=r_{\lambda}F_{T}$. The equation
$p(L_{1},\cdots,L_{m})F_{S}=r_{\lambda}F_{S}$ can be obtained similarly. Then
$p(c_{T}(1),\ldots,c_{T}(m))=r_{\lambda}=p(c_{S}(1),\ldots,c_{S}(m))$. It is a
contradiction. $\Box$
By the above proposition, if every symmetric polynomial in
$L_{1},\ldots,L_{m}$ belongs to the center of $A_{K}$, then for any
$\lambda\in\Lambda$ and $S,T\in M(\lambda)$, we have $\\{c_{S}(i)\mid 1\leq
i\leq m\\}$ and $\\{c_{T}(i)\mid 1\leq i\leq m\\}$ are the same if we do not
consider the order. So we can denote any of them by $\\{c_{\lambda}(i)\mid
1\leq i\leq m\\}$.
Now we are in a position to give the main result of this paper.
###### Theorem 3.3.
Let $R$ be an integral domain and $A$ a cellular $R$-algebra with a cellular
basis $\\{C_{S,T}^{\lambda}\mid S,T\in M(\lambda),\lambda\in\Lambda\\}$. Let
$K$ be the field of fractions of $R$ and $A_{K}=A\bigotimes_{R}K$. Suppose
that $A$ is equipped with a family of Jucys-Murphy elements
$L_{1},\ldots,L_{m}$ which satisfy the separation condition and all symmetric
polynomials in $L_{1},\ldots,L_{m}$ belong to the center of $A_{K}$. Then the
following are equivalent.
(1) The center of $A_{K}$ consists of all symmetric polynomials in the Jucys-
Murphy elements.
(2) For any $\lambda,\mu\in\Lambda$ with $\lambda\neq\mu$,
$\\{c_{\lambda}(i)\mid 1\leq i\leq m\\}$ can not be obtained from
$\\{c_{\mu}(i)\mid 1\leq i\leq m\\}$ by permutations.
To prove this theorem, we need the following two lemmas.
###### Lemma 3.4.
Let $X_{1},X_{2},\cdots,X_{m}$ be indeterminates over a field $K$ and let
$\\{x_{1},\ldots,x_{m}\\}$ and $\\{y_{1},\ldots,y_{m}\\}$ be two families of
elements in $K$. Suppose that there exists some $k\in K$, such that
$p(x_{1},\ldots,x_{m})=kp(y_{1},\ldots,y_{m})$ for any symmetric polynomial
$p(X_{1},X_{2},\cdots,X_{m})\in K[X_{1},X_{2},\cdots,X_{m}]$. Then
$\\{x_{1},\ldots,x_{m}\\}$ can be obtained by permutations from
$\\{y_{1},\ldots,y_{m}\\}$.
Proof: Clearly, if $p$ is a symmetric polynomial, then $p^{2}$ is also a
symmetric polynomial. Then
$(p(x_{1},\ldots,x_{m}))^{2}=(kp(y_{1},\ldots,y_{m}))^{2}=k(p(y_{1},\ldots,y_{m}))^{2}.$
Hence $(k^{2}-k)(p(y_{1},\ldots,y_{m}))^{2}=0$. Then $k^{2}-k=0$ since $p$ is
arbitrary. So we have $k=0$ or $k=1$. If $k=0$, then $p(x_{1},\ldots,x_{m})=0$
for any $p$. This is impossible. Then $k=1$. That is
$p(x_{1},\ldots,x_{m})=p(y_{1},\ldots,y_{m})$ for arbitrary $p$. $\Box$
Let $\\{k_{11},\ldots,k_{1m}\\}$, $\ldots$, $\\{k_{n1},\ldots,k_{nm}\\}$ be
$n$ families of elements in $K$ and $p$ a symmetric polynomial. We will denote
$p(k_{i1},\ldots,k_{im})$ by $p(i)$.
###### Lemma 3.5.
Suppose that $\\{k_{11},\ldots,k_{1m}\\}$, $\ldots$,
$\\{k_{n1},\ldots,k_{nm}\\}$ are $n$ families of elements in a field $K$ and
$X_{1},X_{2},\cdots,X_{m}$ indeterminates. Let
$p_{1}^{{}^{\prime}}(X_{1},X_{2},\cdots,X_{m})$, $\ldots$,
$p_{n}^{{}^{\prime}}(X_{1},X_{2},\cdots,X_{m})$ $\in
K[X_{1},X_{2},\cdots,X_{m}]$ be $n$ symmetric polynomials such that
$\begin{vmatrix}p_{1}^{{}^{\prime}}(1)&\ldots&p_{1}^{{}^{\prime}}(n)\\\
\ldots&\ldots&\ldots\\\
p_{n}^{{}^{\prime}}(1)&\ldots&p_{n}^{{}^{\prime}}(n)\end{vmatrix}\neq 0.$
Then there exist $n$ symmetric polynomials $p_{1},\ldots,p_{n}$ such that
$\begin{vmatrix}p_{1}(1)&p_{1}(2)&\ldots&p_{1}(n)\\\
0&p_{2}(2)&\ldots&p_{2}(n)\\\ \ldots&\ldots&\ldots&\ldots\\\
0&0&\ldots&p_{n}(n)\end{vmatrix}\neq 0.$
Proof: Without loss of generality, we assume that $p_{1}^{{}^{\prime}}(1)\neq
0$ and set $p_{1}=p_{1}^{{}^{\prime}}$. Then let
$p_{2}=p_{2}^{{}^{\prime}}-\dfrac{p_{2}^{{}^{\prime}}(1)}{p_{1}(1)}p_{1}$.
Clearly, $p_{2}$ is a symmetric polynomial and $p_{2}(1)=0$. Moreover,
$\begin{vmatrix}p_{1}(1)&p_{1}(2)&\ldots&p_{1}(n)\\\
0&p_{2}(2)&\ldots&p_{2}(n)\\\
p_{3}^{{}^{\prime}}(1)&p_{3}^{{}^{\prime}}(2)&\ldots&p_{3}^{{}^{\prime}}(n)\\\
\ldots&\ldots&\ldots&\ldots\\\
p_{n}^{{}^{\prime}}(1)&p_{n}^{{}^{\prime}}(2)&\ldots&p_{n}^{{}^{\prime}}(n)\end{vmatrix}=\begin{vmatrix}p_{1}^{{}^{\prime}}(1)&\ldots&p_{1}^{{}^{\prime}}(n)\\\
\ldots&\ldots&\ldots\\\
p_{n}^{{}^{\prime}}(1)&\ldots&p_{n}^{{}^{\prime}}(n)\end{vmatrix}.$
Repeat the above process similarly, we can find $p_{1},\ldots,p_{n}$ such that
$\begin{vmatrix}p_{1}(1)&p_{1}(2)&\ldots&p_{1}(n)\\\
0&p_{2}(2)&\ldots&p_{2}(n)\\\ \ldots&\ldots&\ldots&\ldots\\\
0&0&\ldots&p_{n}(n)\end{vmatrix}=\begin{vmatrix}p_{1}^{{}^{\prime}}(1)&\ldots&p_{1}^{{}^{\prime}}(n)\\\
\ldots&\ldots&\ldots\\\
p_{n}^{{}^{\prime}}(1)&\ldots&p_{n}^{{}^{\prime}}(n)\end{vmatrix}\neq 0.$
$\Box$
Proof of Theorem. Since
$p(L_{1},L_{2},\cdots,L_{m})=\sum\limits_{\lambda\in\Lambda}p(c_{\lambda}(1),\cdots,c_{\lambda}(m))F_{\lambda}$(see
the proof of Proposition 4.12 in [14]), then $(1)\Rightarrow(2)$ is obvious.
Now we prove $(2)\Rightarrow(1)$ by induction on the number of the elements in
the poset $\Lambda$. Denote the number by $\sharp(\Lambda)$ and denote the
elements in $\Lambda$ by natural numbers.
It is easy to know that we only need to find symmetric polynomials
$p^{{}^{\prime}}_{1},p^{{}^{\prime}}_{2},\cdots,p^{{}^{\prime}}_{n}$ such that
$\begin{vmatrix}p^{{}^{\prime}}_{1}(1)&p^{{}^{\prime}}_{1}(2)&\ldots&p^{{}^{\prime}}_{1}(n)\\\
p^{{}^{\prime}}_{2}(1)&p^{{}^{\prime}}_{2}(2)&\ldots&p^{{}^{\prime}}_{2}(n)\\\
\ldots&\ldots&\ldots&\ldots\\\
p^{{}^{\prime}}_{n}(1)&p^{{}^{\prime}}_{n}(2)&\ldots&p^{{}^{\prime}}_{n}(n)\end{vmatrix}\neq
0$
where $n=\sharp(\Lambda)$.
For $\sharp(\Lambda)=1$, it is clear.
We now assume that $(2)\Rightarrow(1)$ holds for $\sharp(\Lambda)=n$. Then by
Lemma 3.5, there exist symmetric polynomials $p_{1},\ldots,p_{n}$ such that
$\begin{vmatrix}p_{1}(1)&p_{1}(2)&\ldots&p_{1}(n)\\\
0&p_{2}(2)&\ldots&p_{2}(n)\\\ \ldots&\ldots&\ldots&\ldots\\\
0&0&\ldots&p_{n}(n)\end{vmatrix}\neq 0.$
We now assume that for any symmetric polynomial $p$,
$d:=\begin{vmatrix}p_{1}(1)&p_{1}(2)&\ldots&p_{1}(n)&p_{1}(n+1)\\\
0&p_{2}(2)&\ldots&p_{2}(n)&p_{2}(n+1)\\\ \ldots&\ldots&\ldots&\ldots&\ldots\\\
0&0&\ldots&p_{n}(n)&p_{n}(n+1)\\\
p(1)&p(2)&\ldots&p(n)&p(n+1)\end{vmatrix}=0.$
Then $p(n+1)=k_{1}p(1)+k_{2}p(2)+\ldots+k_{n}p(n)$, where $k_{i}\in K$ is
independent of $p$ for $i=1,\ldots,n$. Then we have
$p_{n}p(n+1)=k_{1}p_{n}p(1)+k_{2}p_{n}p(2)+\ldots+k_{n}p_{n}p(n)$ since
$p_{n}p$ is also a symmetric polynomial. Assume that $p_{n}(n+1)\neq 0$, then
$p_{n}(n+1)p(n+1)=k_{n}p_{n}(n)p(n)$, or $p(n+1)=kp(n)$, where $k\in K$ is
independent of the choice of $p$. This implies that $p(n+1)=p(n)$ by Lemma
3.4. It is a contradiction. Then $p_{n}(n+1)=0$. That is
$k_{n}p_{n}(n)p(n)=0$. Since $p_{n}(n)\neq 0$ and $p$ is arbitrary, then
$k_{n}=0$. Repeat this process similarly, we have $k_{i}=0$ for $i=1,\cdots,n$
and then $p(n+1)=0$. It is impossible for $p$ is arbitrary. Then there exists
a symmetric polynomial $p$ such that $d\neq 0$. This completes the proof.
$\Box$
###### Corollary 3.6.
Let $R$ be an integral domain and $A$ a cellular algebra. Suppose that $A$ is
equipped with a family of Jucys-Murphy elements which separate $A$. If the
center of $A$ consists of symmetric polynomials in Jucys-Murphy elements, then
$\\{c_{\lambda}(i)\mid 1\leq i\leq m\\}$ can not be obtained from
$\\{c_{\mu}(i)\mid 1\leq i\leq m\\}$ by permutations for arbitrary
$\lambda,\mu\in\Lambda$ with $\lambda\neq\mu$. $\Box$
## 4 An application on Ariki-Koike Hecke algebras
In this section, we prove that the center of a semisimple Ariki-Koike Hecke
algebra ($q\neq 1$) consists of the symmetric polynomials in Jucys-Murphy
elements. It is a new proof different from Ariki's in [1] and A. Ram's [15].
Firstly, we recall some notions of combinatorics. Recall that a partition of
$n$ is a non-increasing sequence of non-negative integers
$\lambda=(\lambda_{1},\cdots,\lambda_{r})$ such that
$\sum_{i=1}^{r}\lambda_{i}=n$. The diagram of a partition $\lambda$ is the
subset $[\lambda]=\\{(i,j)\mid 1\leq j\leq\lambda_{i},i\geq 1\\}$. The
elements of $\lambda$ are called nodes. Define the residue of the node
$(i,j)\in[\lambda]$ to be $j-i$. For any partition
$\lambda=(\lambda_{1},\lambda_{2},\cdots)$, the conjugate of $\lambda$ is
defined to be a partition
$\lambda^{\prime}=(\lambda^{\prime}_{1},\lambda^{\prime}_{2},\cdots)$, where
$\lambda^{\prime}_{j}$ is equal to the number of nodes in column $j$ of
$[\lambda]$ for $j=1,2,\cdots$. For partitions, we have the following simple
lemma.
###### Lemma 4.1.
Let $\lambda$ and $\mu$ be two partitions of $n$. Then $\lambda=\mu$ if and
only if all residues of nodes in $[\lambda]$ and $[\mu]$ are the same. $\Box$
Given two partitions $\lambda$ and $\mu$ of $n$, write
$\lambda\trianglerighteq\mu$ if
$\sum_{i=1}^{j}\lambda_{i}\geq\sum_{i=1}^{j}\mu_{i},\,\,\,\,\text{for all
$i\geq 1$}.$
This is the so-called dominance order. It is a partial order.
A $\lambda$-tableau is a bijection
$\mathfrak{t}:[\lambda]\rightarrow\\{1,2,\cdots,n\\}$. We say $\mathfrak{t}$ a
standard $\lambda$-tableau if the entries in $\mathfrak{t}$ increase from left
to right in each row and from top to bottom in each column. Denote by
$\mathfrak{t}^{\lambda}$ (resp., $\mathfrak{t}_{\lambda}$) the standard
$\lambda$-tableau, in which the numbers $1,2,\cdots,n$ appear in order along
successive rows (resp., columns), The row stabilizer of
$\mathfrak{t}^{\lambda}$, denoted by $S_{\lambda}$, is the standard Young
subgroup of $S_{n}$ corresponding to $\lambda$. Let
$\operatorname{Std}(\lambda)$ be the set of all standard $\lambda$-tableaux.
For a fixed positive integer $m$, a $m$-multipartitions of $n$ is an $m$-tuple
of partitions which sum to $n$. Let
$\lambda=((\lambda_{11},\lambda_{12},\cdots,\lambda_{1i_{1}}),(\lambda_{21},\lambda_{22},\cdots,\lambda_{2i_{2}}),\cdots,(\lambda_{m1},\lambda_{m2},\cdots,\lambda_{mi_{m}}))$
be a $m$-multipartitions of $n$, we denote
$\lambda_{j1}+\lambda_{j2}+\cdots+\lambda_{ji_{j}}$ by $n_{j\lambda}$ for
$1\leq j\leq m$. A standard $\lambda$-tableau is an $m$-tuple of standard
tableaux. We can define $\mathfrak{t}^{\lambda}$ similarly.
Let $R$ be an integral domain, $q,u_{1},u_{2},\cdots,u_{m}\in R$ and $q$
invertible. Fix two positive integers $n$ and $m$. Then Ariki-Koike algebra
$\mathscr{H}_{n,m}$ is the associative $R$-algebra with generators
$T_{0},T_{1},\cdots,T_{n-1}$ and relations
$\displaystyle(T_{0}-u_{1})(T_{0}-u_{2})\cdots(T_{0}-u_{m})=0,$ $\displaystyle
T_{0}T_{1}T_{0}T_{1}=T_{1}T_{0}T_{1}T_{0},$
$\displaystyle(T_{i}-q)(T_{i}+1)=0,\quad\text{for $1\leq i\leq n-1$,}$
$\displaystyle T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1},\quad\text{for $1\leq
i\leq n-2$,}$ $\displaystyle T_{i}T_{j}=T_{j}T_{i},\quad\text{for $0\leq
i<j-1\leq n-2$.}$
Denote by $\Lambda$ the set of $m$-multipartitions of $n$. For
$\lambda\in\Lambda$, let $M(\lambda)$ be the set of standard
$\lambda$-tableau. Then $\mathscr{H}_{n,m}$ has a cellular basis of the form
$\\{m_{\mathfrak{s}\mathfrak{t}}^{\lambda}\mid\lambda\in\Lambda,\mathfrak{s},\mathfrak{t}\in
M(\lambda)\\}$. See [6] for details.
Let $L_{i}=q^{1-i}T_{i-1}\cdots T_{1}T_{0}T_{1}\cdots T_{i-1}$. Then
$L_{1},L_{2},\cdots,L_{n}$ is a family of Jucys-Murphy elements of
$\mathscr{H}_{n,m}$. If $i$ is in row $r$ column $c$ of the $j$-th tableau of
$\mathfrak{t}$, then $m_{\mathfrak{s}\mathfrak{t}}^{\lambda}L_{i}\equiv
u_{j}q^{c-r}m_{\mathfrak{s}\mathfrak{t}}^{\lambda}$. If
$[1]_{q}\cdots[n]_{q}\prod_{1\leq i<j\leq
m}\prod_{|d|<n}(q^{d}u_{i}-u_{j})\neq 0$ and $q\neq 1$, then the Jucys-Murphy
elements separate $M(\Lambda)$. These were proved in [12].
Denote $\mathscr{H}_{n,m}\otimes_{R}K$ by $\mathscr{H}_{n,m,K}$. The following
result has been proved in [1] and [15]. We give a new proof here.
###### Theorem 4.2.
([1],[15]) The center of $\mathscr{H}_{n,m,K}$ is equal to the set of
symmetric polynomials in the Jucys-Murphy elements if
$[1]_{q}\cdots[n]_{q}\prod_{1\leq i<j\leq
m}\prod_{|d|<n}(q^{d}u_{i}-u_{j})\neq 0$ and $q\neq 1$.
Proof: The algebra $\mathscr{H}_{n,m,K}$ satisfies the conditions of the
Proposition 3.1 has been pointed out in [14]. By Theorem C, we only need to
show that for any $\lambda,\mu\in\Lambda$ with $\lambda\neq\mu$,
$\\{c_{\lambda}(i)\mid 1\leq i\leq n\\}$ can not be obtained from
$\\{c_{\mu}(i)\mid 1\leq i\leq n\\}$ by permutations. Note that we can obtain
these two sets by $\mathfrak{t}^{\lambda}$ and $\mathfrak{t}^{\mu}$
respectively.
Case 1. There exists $1\leq j\leq m$ such that $n_{j\lambda}\neq n_{j\mu}$.
Then by the separation condition, the number of the elements of the form
$u_{j}q^{x}$ is $n_{j\lambda}$ in $\\{c_{\lambda}(i)\mid 1\leq i\leq M\\}$ and
is $n_{j\mu}$ in $\\{c_{\mu}(i)\mid 1\leq i\leq M\\}$, where $x\in\mathbb{Z}$.
This implies that $\\{c_{\lambda}(i)\mid 1\leq i\leq M\\}$ can not be obtained
from $\\{c_{\mu}(i)\mid 1\leq i\leq n\\}$ by permutations.
Case 2. $n_{j\lambda}=n_{j\mu}$ for all $1\leq j\leq m$. Then there must exist
$1\leq s\leq m$, such that the partition of $n_{s}$ in $\lambda$ is not equal
to that in $\mu$ since $\lambda\neq\mu$. Denote the partitions by
$\lambda_{s}$ and $\mu_{s}$. Then the residues of $\lambda_{s}$ and $\mu_{s}$
are not the same. Now by Lemma 4.1 and the separation condition, the set of
all the elements of the form $u_{s}q^{x}$ in $\\{c_{\lambda}(i)\mid 1\leq
i\leq M\\}$ is different from that in $\\{c_{\mu}(i)\mid 1\leq i\leq n\\}$.
Then for any $\lambda,\mu\in\Lambda$ with $\lambda\neq\mu$,
$\\{c_{\lambda}(i)\mid 1\leq i\leq m\\}$ can not be obtained from
$\\{c_{\mu}(i)\mid 1\leq i\leq m\\}$ by permutations. $\Box$
Remark. If $q\neq 1$, then $[1]_{q}\cdots[n]_{q}\prod_{1\leq i<j\leq
m}\prod_{|d|<n}(q^{d}u_{i}-u_{j})\neq 0$ if and only if $\mathscr{H}_{n,m,K}$
is semisimple. See [1] for details.
Acknowledgments The author acknowledges his supervisor Prof. C.C. Xi and the
support from the Research Fund of Doctor Program of Higher Education, Ministry
of Education of China. He also acknowledges Dr. Wei Hu and Zhankui Xiao for
many helpful conversations.
## References
* [1] S. Ariki, On the semi-simplicity of the Hecke algebra of $(\mathbb{Z}/r\mathbb{Z})\wr S_{n}$, J. Algebra, 169, (1994), 216-225.
* [2] J. Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category, arXiv: math0607717.
* [3] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 52, (1986), 20-52.
* [4] R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 54, (1987), 57-82.
* [5] R. Dipper and G. James, Representations of Hecke algebras of type ${\rm B_{n}}$, J. Algebra, 146, (1992), 454-481.
* [6] R. Dipper, G. James and A. Mathas, Cyclotomic q-Schur algebras, M. Z., 229, (1998), 385-416.
* [7] R. Dipper, G. James and G. Murphy, Hecke algebras of type $\rm B_{n}$ at roots of unity, Proc. London Math. Soc. (3), 70, (1995), 505-528.
* [8] A. Francis and J.J. Graham, Centers of Hecke algebras: The Dipper-James conjecture, J. Algebra, 306, (2006), 244-267.
* [9] M. Geck, Hecke algebras of finite type are cellular, Invent. math., 169, (2007), 501-517.
* [10] J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math., 123, (1996), 1-34.
* [11] T. Halverson and Arun Ram, Partition Algebras, European Journal of Combinatorics 26, Issue 6, (2005), 869-921.
* [12] G. James and A. Mathas, The Jantzen sum formula for cyclotomic q-Schur algebras, Trans. Amer. Math. Soc., 352, (2000), 5381-5404.
* [13] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, (1979), 165-184.
* [14] A. Mathas, Seminormal forms and Gram determinants for cellular algebras, arXiv:math.RT/0604108v4.
* [15] A. Ram and J. Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theorem, in A tribute to C.S. Seshadri: Perspectives in Geometry and Representation theory, V. Lakshimibai et al eds., Hindustan Book Agency , New Delhi (2003), 428–466.
* [16] A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3), 75, (1997), 99-133.
* [17] C.C. Xi, Partition algebras are cellular, Compositio math., 119, (1999), 99-109.
* [18] C.C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math., 154, (2000), 280-298.
|
arxiv-papers
| 2009-11-18T13:03:16 |
2024-09-04T02:49:06.560223
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanbo Li",
"submitter": "Yanbo Li",
"url": "https://arxiv.org/abs/0911.3529"
}
|
0911.3659
|
# Derivation of a Relation for the Steepening of TeV Selected Blazar
$\gamma$-ray Spectra with Energy and Redshift
Floyd William Stecker NASA/Goddard Space Flight Center Greenbelt, MD 20771
Floyd.W.Stecker@nasa.gov Sean T. Scully Department of Physics, James Madison
University Harrisonburg, VA 22807 scullyst@jmu.edu
###### Abstract
We derive a relation for the steepening of blazar $\gamma$-ray spectra between
the multi-GeV Fermi energy range and the TeV energy range observed by
atmospheric Čerenkov telescopes. The change in spectral index is produced by
two effects: (1) an intrinsic steepening, independent of redshift, owing to
the properties of emission and absorption in the source, and (2) a redshift-
dependent steepening produced by intergalactic pair production interactions of
blazar $\gamma$-rays with low energy photons of the “intergalactic background
light” (IBL). Given this relation, with good enough data on the mean
$\gamma$-ray SED of TeV Selected BL Lacs, the redshift evolution of the IBL
can, in principle, be determined independently of stellar evolution models. We
apply our relation to the results of new Fermi observations of TeV selected
blazars.
Gamma-rays: general – blazars
## 1 Introduction
Stecker & Scully (2006) (SS06) derived a simple analytic expression for the
change in spectral index of a TeV $\gamma$-ray source in the redshift range
between 0.05 and 0.4. They showed that the change in the spectral index caused
by intergalactic absorption is given by an approximately linear relation in
redshift, i.e., $\Delta\Gamma_{a}\simeq C+Dz$.
The purpose of this letter is to generalze this relation by including the
effect of intrinsic steepening in the source spectra between the mutli-GeV
energy range observed by Fermi and the TeV range observed by atmospheric
Čherenkov telescopes. Our general result is roughly independent of the
specific model of the intergalactic background light (IBL) used, because it
only depends on the shape of the average galaxy spectral energy distribution
(SED) on the near IR side of the starlight peak that determines the absorption
in the TeV energy range.
We compare our relation for the specific baseline and fast evolution models of
Stecker, Malkan & Scully (2006) (SMS06) with the results of recent Fermi
observations of 13 TeV selected BL Lac AGN. We also show how it can be used to
independently determine the redshift evolution of the IBL.
## 2 Steepening by Absorption
In order to determine the effect of intergalactic absorption, we use the
results of SS06 demonstrating that $\tau(E_{\gamma},z)$ can be fitted to an
approximately logarithmic function in $E_{\gamma}$ in the energy range
$0.2~{}\rm TeV<E_{\gamma}<2~{}\rm TeV$ and one which is linear on $z$ over the
range $0.05<z<0.4$. It is important to note that our linear fit to the $z$
dependence is both qualitatively and quantitatively different from the linear
dependence on redshift which would be obtained for small redshifts $z<<1$ and
which simply comes from the fact that for small $z$ where luminosity evolution
is unimportant and where $\tau\propto d$, with the distance $d\simeq
cz/H_{0}\propto z$. Our quantitative fit for the higher redshift range
$0.05<z<0.4$ comes from the more complex calculations based on the models of
SMS. For this reason, the linear fits in SS06 are not simply proportional to
redshift.
SS06 found that $\tau(E_{\gamma},z)$ is well approximated by
$\tau(E_{\gamma},z)=(A+Bz)+(C+Dz)\ln[E_{\gamma}/(\rm 1~{}TeV)],$ (1)
where $A,B,C~{}$and $D$ are constants. This expression holds over the energy
and redshift ranges given above. The energy range of validity is the energy
range to which the atmospheric Čerenkov telescopes are sensitive.
Following SS06, we assume an intrinsic source spectrum that can be
approximated by a power law of the form
$\Phi_{s,TeV}(E_{\gamma})~{}\simeq~{}kE_{\gamma}^{-\Gamma_{s,TeV}}$ over a
limited energy range. Then the spectrum that will be observed at the Earth
following intergalactic absorption will be of the power-law form
$\Phi_{o,TeV}(E_{\gamma})~{}=~{}ke^{-(A+Bz)}E_{\gamma}^{-(\Gamma_{s,TeV}+C+Dz)}.$
(2)
This can be compared with the empirically observed TeV spectra that are
usually presented in the literature to be good fits to power-laws. The
observed spectral index, $\Gamma_{o,TeV}$, will then be given by
$\Gamma_{o,TeV}=\Gamma_{s,TeV}+\Delta\Gamma_{a,TeV}(z)$ (3)
where the intrinsic spectral index of the source is steepened by intergalactic
absorption by an amount $\Delta\Gamma_{a}(z)~{}=~{}C+Dz$.
On the other hand, in the multi-GeV range over which the Fermi Large Area
Telescope (LAT) is sensitive, we expect essentially no steepening from
absorption over the redshift range of validity, i.e.,
$\Delta\Gamma_{a,GeV}(z)\simeq 0$.
The parameters $C$, and $D$ obtained by fitting the optical depths derived for
the fast evolution (FE) and baseline (B) models of SMS06 are given in Table 1.
## 3 Intrinsic Steepening
The importance of synchrotron self Compton emission in producing high energy
$\gamma$-rays in astrophysical sources was pointed out by Rees (1967).
Particular applications to the Crab and other sources were discussed by Gould
(1965) and Rieke & Weekes (1969). Today, given present observational studies
of TeV-selected BL Lac AGN, it is generally accepted that the synchrotron self
Compton mechanism is the primary emission mechanism for producing
$\gamma$-rays in the TeV energy range in TeV-selected BL Lac AGN.
The $\gamma$-ray spectrum from Compton interactions is a smoothy varying one
(Blumenthal & Gould 1970). Good empirical fits the the SEDs of the low energy
peaks in blazars were obtained by using parabolic functions on log-log plots
(Landau, et al. 1986, Sambruna, Maraschi & Urry 1996). In fact, these
considerations led to the prediction of candidate BL Lac objects to be found
by atmospheric Čerenkov telescopes in the TeV energy range (Stecker, de Jager
& Salamon 1996; Costamante & Ghisellini 2002).
We assume such an approximation here, i.e.,
$E^{2}{{dN_{\gamma}}\over{dE}}=f(x),$ (4)
where $x=\log E$ with the simplification that $E$ is in dimensionless fiducial
units (i.e.,$~{}E/E_{f}$), and
$f(x)={-{(x-x_{0})^{2}}\over{2A}}+B.$ (5)
Here, $A\equiv\log<W>$ is a constant parameter based on the ensemble averaged
width of the parabola in $\log E$ space based on a set of SEDs of TeV selected
BL Lacs and $x_{0}\equiv\log E_{0}$ where $E_{0}$ is the energy at which the
Compton peak of the average BL Lac SED is a maximum. We note that for some BL
Lacs electron scattering in the Klein-Nishina regime will distort our
parabolic assumption on the high energy end of their SEDs, possibly reducing
$\log<W>$ by a small amount (Tavecchio, Maraschi & Ghisellini 1998). However,
this does not significantly affect our formalism.
We find that the change in the slope of the intrinsic source spectrum
occurring between some average observed energy in the GeV range $<E_{\rm
GeV}>$ and some average observed energy in the TeV range $<E_{\rm TeV}>$ is
approximately given by a constant,
${{\log<E_{TeV}>-~{}\log<E_{GeV}>}\over{A}}\equiv K$ (6)
Thus, the ensemble averaged intrinsic BL Lac source spectrum in going from the
GeV range to the TeV range is independent of redshift within the errors in the
observational data. We will therefore approximate the average intrinsic
steepening over an observed set of TeV selected blazars by
$\Gamma_{s,TeV}-\Gamma_{s,GeV}=\Delta\Gamma_{s}=K.$ (7)
## 4 The Relation for the Total Steepening between GeV and TeV Energies
The total steepening expected betwen the GeV and TeV energy ranges is then
given by the relation
$\Delta\Gamma=\Delta\Gamma_{s}+\Delta\Gamma_{a}=(C+K)+Dz$ (8)
The parameters $C$, and $D$ obtained by fitting the optical depths derived for
the fast evolution (FE) and baseline (B) models of SMS06 are given in Table 1.
The parameter $K$ is then derived by performing a $\chi^{2}$ fit to the
observed steepening data obtained by the Fermi collaboration (Abdo, et al.
2009) in order to find $(C+K)$. The parameter $K$ is also given in Table 1 for
the FE and B models. The data on the TeV selected BL Lacs from Abdo, et al.
(2009) are shown in figures 1 and 2, along with the best-fit linear relations
for the models indicated. It can be seen that, given the present limited data
set and large error bars, one cannot uniquely determine the parameters $C$,
$D$ and $K$. However, in principle, with a good enough data set, one could
determine $K$ uniquely from equation (6). Then, since $K\gg C$, one could use
equation (8) to determine the parameter $D$. This will then give a
determination of the redshift evolution of the IBL independently of models of
the star formation rate.
Table 1: Steepening Parameters Evolution Model | C | D | K
---|---|---|---
Fast Evolution | -0.0972 | 10.6 | 0.427
Baseline | -0.0675 | 7.99 | 0.716
Figures 1 and 2 show the fits of the parameters given in Table 1 to the linear
dependence in and redshift given by equation (8).
Figure 1: The fits obtained for the linear functions $C+Dz$ (dashed line) and
$C+Dz+K$ (solid line) shown for the SMS06 baseline model as descibed in the
text. These are fit to the data on 13 BL Lacs given by Abdo et al. (2009).
Figure 2: The fits obtained for the linear functions $C+Dz$ (dashed line) and
$C+Dz+K$ (solid line) shown for the SMS06 fast evolution model as descibed in
the text. These are fit to the data on 13 BL Lacs given by Abdo et al. (2009).
## 5 Conclusions
We have derived a simple analytic approximation for determining the steepening
in the spectra of TeV selected BL Lac AGN and compared our results with recent
observational data on 13 TeV selected BL Lac AGN. Our relation is in excellent
agreement with the observational data and provides a framework for
understanding and interpreting both these and observational data.
SS06 have shown that the effect of intergalactic absorption on the spectra of
AGN in the energy range 0.2 TeV $<E_{\gamma}<$ 2 TeV and the redshift range
$0.05<z<0.4$ is a simple power-law to power-law steepening. Absorption in this
energy range is primarily produced by interactions with near infrared photons
from on the low energy sides of the starlight peaks in galaxy SEDs. The shape
of the resulting peak in the intergalactic SED produces an approximately
logarithmic energy dependence for the function $\tau(E_{\gamma})$. This energy
dependence should be roughly the same for all models of the IBL. Therefore,
our general result of a linear $\Delta\Gamma=(C+K)+Dz$ relationship is roughly
independent of the specific IBL model used because it only depends on the
shape of the average galaxy SED on the near IR side of the IBL starlight peak.
The parameters $C,D$ and $K$ will be different for the different models since
the absolute value of $\tau$ varies from one IBL model to another.
Given our derived relation between $\Delta\Gamma$ and redshift, and with good
enough data on the mean $\gamma$-ray SED of TeV Selected BL Lacs used to
determine $K$, the redshift evolution of the IBL can, in principle, be
determined independently of stellar evolution models.
## Acknowledgment
We thank Stephen Fegan and David Sanchez for sending the BL Lac data table
used in our Figures 1 and 2. We thank Markos Georganopoulos for a helpful
discussion.
## References
* (1) Abdo, A. A. et al. 2009, e-print arXiv:0910.4881
* (2) Blumenthal, G. R. & Gould, R. J. 1970, Rev. Mod. Phys. 42, 237
* (3) Costamante, L. & Ghisellini, G. 2002, Astron. & Astrophys. 384, 56
* (4) Gould, R. J. 1965, Phys. Rev. Letters 15, 577
* (5) Landau, R. et al. 1986, ApJ 308, 78
* (6) Rees, M. J. 1967, Mon. Not. Roy. Astr. Soc. 137, 429
* (7) Rieke, G. H. & Weekes, T. C. 1969, ApJ 155, 429
* (8) Sambruna, R. M., Maraschi, L. and Urry, C. M. 1996, ApJ 463, 444
* (9) Stecker, F. W., De Jager, O. C. & Salamon, M. A. 1996, ApJ 473, L75
* (10) Stecker, F. W., Malkan, M. A. & Scully, S. T. 2006, ApJ 648, 774
* (11) Stecker, F. W.& Scully, S. T. 2006, ApJ 652, L9
* (12) Tavecchio, F., Maraschi, L. & Ghisellini, G. 1998, ApJ 509, 608
|
arxiv-papers
| 2009-11-18T21:05:06 |
2024-09-04T02:49:06.566874
|
{
"license": "Public Domain",
"authors": "Floyd W. Stecker (NASA/GSFC) and Sean T. Scully (JMU)",
"submitter": "Floyd Stecker",
"url": "https://arxiv.org/abs/0911.3659"
}
|
0911.3845
|
# A short note on $\infty$-groupoids and the period map for projective
manifolds
Domenico Fiorenza fiorenza@mat.uniroma1.it
www.mat.uniroma1.it/people/fiorenza/ and Elena Martinengo
elenamartinengo@gmail.com www.mat.uniroma1.it/dottorato/
###### Key words and phrases:
Differential graded Lie algebras, functors of Artin rings, $\infty$-groupoids,
projective manifolds, period maps
###### 1991 Mathematics Subject Classification:
18G55; 14D07
##
A common criticism of $\infty$-categories in algebraic geometry is that they
are an extremely technical subject, so abstract to be useless in everyday
mathematics. The aim of this note is to show in a classical example that quite
the converse is true: even a naïve intuition of what an $\infty$-groupoid
should be clarifies several aspects of the infinitesimal behaviour of the
periods map of a projective manifold. In particular, the notion of Cartan
homotopy turns out to be completely natural from this perspective, and so
classical results such as Griffiths’ expression for the differential of the
periods map, the Kodaira principle on obstructions to deformations of
projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson
quasi-abelianity theorem are easily recovered.
Since most of the statements and constructions we recall in the paper are well
known in the $(\infty,1)$-categorical folklore, despite our efforts in giving
credit, it is not unlikely we may have misattributed a few of the results; we
sincerely apologize for this. We thank Ezra Getzler, Donatella Iacono, Marco
Manetti, Jonathan Pridham, Carlos Simpson, Jim Stasheff, Bruno Vallette,
Gabriele Vezzosi and the $n$Lab for several inspiring conversations on the
subject of this paper.
Through the whole paper, ${\mathbb{K}}$ is a fixed characteristic zero field,
all algebras are defined over ${\mathbb{K}}$ and local algebras have
${\mathbb{K}}$ as residue field. In order to keep our account readable, we
will gloss over many details, particularly where the use of higher category
theory is required.
## 1\. From dglas to $\infty$-groupoids and back again
With any nilpotent dgla ${\mathfrak{g}}$ is naturally associated the
simplicial set
$\operatorname{MC}({\mathfrak{g}}\otimes\Omega_{\bullet}),$
where $\operatorname{MC}$ stands for the Maurer-Cartan functor mapping a dgla
to the set of its Maurer-Cartan elements, and $\Omega_{\bullet}$ is the
simplicial differential graded commutative associative algebra of polynomial
differential forms on algebraic $n$-simplexes, for $n\geq 0$. The importance
of this construction, which can be dated back to Sullivan’s [Su77], relies on
the fact that, as shown by Hinich and Getzler [Ge09, Hi97], the simplicial set
$\operatorname{MC}({\mathfrak{g}}\otimes\Omega_{\bullet})$ is a Kan complex,
or -to use a more evocative name- an $\infty$-groupoid. A convenient way to
think of $\infty$-groupoids is as homotopy types of topological spaces;
namely, it is well known111At least in higher categories folklore that any
$\infty$-groupoid can be realized as the $\infty$-Poincaré groupoid, i.e., as
the simplicial set of singular simplices, of a topological space, unique up to
weak equivalence. Therefore, the reader who prefers to can substitute homotopy
types of topological spaces for equivalence classes of $\infty$-groupoids. To
stress this point of view, we’ll denote the $k$-truncation of an
$\infty$-groupoid $\mathbf{X}$ by the symbol $\pi_{\leq k}\mathbf{X}$. More
explicitely, $\pi_{\leq k}\mathbf{X}$ is the $k$-groupoid whose $j$-morphisms
are the $j$-morphisms of $\mathbf{X}$ for $j<k$, and are homotopy classes of
$j$-morphisms of $\mathbf{X}$ for $j=k$. In particular, if $\mathbf{X}$ is the
$\infty$-Poincaré groupoid of a topological space $X$, then $\pi_{\leq
0}\mathbf{X}$ is the set $\pi_{0}(X)$ of path-connected components of $X$, and
$\pi_{\leq 1}\mathbf{X}$ is the usual Poincaré groupoid of $X$.
The next step is to consider an $(\infty,1)$-category, i.e., an
$\infty$-category whose hom-spaces are $\infty$-groupoids. This can be thought
as a formalization of the naïve idea of having objects, morphisms, homotopies
between morphisms, homotopies between homotopies, et cetera. In this sense,
endowing a category with a model structure should be thought as a first step
towards defining an $(\infty,1)$-category structure on it.
Turning back to dglas, an easy way to produce nilpotent dglas is the
following: pick an arbitrary dgla ${\mathfrak{g}}$; then, for any
(differential graded) local Artin algebra $A$, take the tensor product
${\mathfrak{g}}\otimes{\mathfrak{m}}_{A}$, where ${\mathfrak{m}}_{A}$ is the
maximal ideal of $A$. Since both constructions
$\displaystyle{\bf DGLA}\times{\bf Art}$ $\displaystyle\to{\bf
nilpotent\,\,DGLA}$ $\displaystyle(\mathfrak{g},A)$
$\displaystyle\mapsto\mathfrak{g}\otimes\mathfrak{m}_{A}$
and
$\displaystyle{\bf nilpotent\,\,DGLA}$ $\displaystyle\to{\bf\infty\text{\bf-
Grpd}}$ $\displaystyle\mathfrak{g}$
$\displaystyle\mapsto\operatorname{MC}({\mathfrak{g}}\otimes\Omega_{\bullet})$
are functorial, their composition defines a functor
$\operatorname{Def}:{\bf DGLA}\to{\bf\infty\text{\bf-Grpd}}^{\bf Art}.$
The functor of Artin rings $\operatorname{Def}({\mathfrak{g}})\colon{\bf
Art}\to{\bf\infty\text{\bf-Grpd}}$ is called the formal $\infty$-groupoid
associated with the dgla ${\mathfrak{g}}$. Note that $\pi_{\leq
0}(\operatorname{Def}({\mathfrak{g}}))$ is the usual set valued deformation
functor associated with ${\mathfrak{g}}$, i.e., the functor
$A\mapsto\operatorname{MC}({\mathfrak{g}}\otimes{\mathfrak{m}}_{A})\bigl{/}{\rm
gauge},$
where the gauge equivalence of Maurer-Cartan elements is induced by the gauge
action
$e^{\alpha}*x=x+\sum_{n=0}^{\infty}\frac{({\rm ad}_{\alpha})^{n}}{(n+1)!}\
([\alpha,x]-d\alpha)$
of $\exp(\mathfrak{g}^{0}\otimes\mathfrak{m}_{A})$ on the subset
$\operatorname{MC}({\mathfrak{g}}\otimes{\mathfrak{m}}_{A})$ of
$\mathfrak{g}^{1}\otimes\mathfrak{m}_{A}$. However, due to the presence of
nontrivial irrelevant stabilizers, the groupoid $\pi_{\leq
1}(\operatorname{Def}({\mathfrak{g}}))$ is not equivalent to the action
groupoid
$\operatorname{MC}({\mathfrak{g}}\otimes{\mathfrak{m}}_{A})\bigl{/}\bigl{/}\exp(\mathfrak{g}^{0}\otimes\mathfrak{m}_{A})$,
unless ${\mathfrak{g}}$ is concentrated in nonnegative degrees. We will come
back to this later. Also note that the zero in
$\mathfrak{g}^{1}\otimes\mathfrak{m}_{A}$ gives a natural distinguished
element in $\pi_{\leq 0}(\operatorname{Def}({\mathfrak{g}}))$: the isomorphism
class of the trivial deformation. Since this marking is natural, we will use
the same symbol $\pi_{0}(\operatorname{Def}({\mathfrak{g}}))$ to denote both
the set $\pi_{\leq 0}(\operatorname{Def}({\mathfrak{g}}))$ and the pointed set
$\pi_{0}(\operatorname{Def}({\mathfrak{g}});0)$.
A very good reason for working with $\infty$-groupoids valued deformation
functors rather than with their apparently handier set-valued or groupoid-
valued versions is the following folk statement, which allows one to move
homotopy constructions back and forth between dglas and (homotopy types of)
‘nice’ topological spaces.
###### Theorem.
The functor $\operatorname{Def}:{\bf DGLA}\to{\bf\infty\text{\bf-Grpd}}^{\bf
Art}$ induces an equivalence of $(\infty,1)$-categories.
Here the $(\infty,1)$-category structures involved are the most natural ones,
and they are both induced by standard model category structures. Namely, on
the category of dglas one takes surjective morphisms as fibrations and quasi-
isomorphisms as weak equivalences, just as in the case of differential
complexes, whereas the model category structure on the right hand side is
induced by the standard model category structure on Kan complexes as a
subcategory of simplicial sets. A sketchy proof of the above equivalence can
be found in [Lu09a]; see also [Pr10].
## 2\. Homotopy vs. gauge equivalent morphisms of dglas (with a detour into
$L_{\infty}$-morphisms)
Let ${\mathfrak{g}}$ and ${\mathfrak{h}}$ be two (nilpotent) dglas. Then, from
the $(\infty,1)$-category structure on dglas, we have a natural notion of
homotopy equivalence on the set of dgla morphisms
$\operatorname{Hom}(\mathfrak{g},\mathfrak{h})$. Actually, in this form this
is a too naïve statement. Indeed, in order to have a good notion of homotopy
classes of morphisms one first has to perform a fibrant-cofibrant replacement
of $\mathfrak{g}$ and $\mathfrak{h}$. In more colloquial terms, what one does
is moving from the too narrow realm of strict dgla morphisms to the more
flexible world of morphisms which preserve the dgla structure only up to
homotopy; the formalization of this idea leads to the notion of
$L_{\infty}$-morphism, see, e.g., [LS93, Ko03]. Now, a notion of homotopy (and
of higher homotopies) is well defined on the set of $L_{\infty}$-morphisms
between the dglas $\mathfrak{g}$ and $\mathfrak{h}$; this defines the
$\infty$-groupoid $\operatorname{Hom}_{\infty}(\mathfrak{g},\mathfrak{h})$.
The definition of $L_{\infty}$-morphism is best given in the language of
differential graded cocommutative coalgebras. Namely, for a graded vector
space $V$, let
$C(V)=\bigoplus_{n\geq
1}\left(\otimes^{n}V\right)^{\Sigma_{n}}\subseteq\bigoplus_{n\geq
1}\left(\otimes^{n}V\right)$
be the cofree graded cocommutative coalgebra without counit cogenerated by
$V$, endowed with the standard coproduct
$\Delta(v_{1}\otimes\cdots\otimes
v_{n})=\sum_{q_{1}+q_{2}=n}\sum_{\sigma\in{\rm
Sh}(q_{1},q_{2})}\pm(v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(q_{1})})\bigotimes(v_{\sigma(q_{1}+1)}\otimes\cdots\otimes
v_{\sigma(n)}),$
where $\sigma$ ranges in the set of $(q_{1},q_{2})$-unshuffles and $\pm$
stands for the Koszul sign. If $V$ is endowed with a dgla structure, then the
differential of $V$ can be seen as a linear morphism $Q_{1}^{1}:V[1]\to V[2]$
and the Lie bracket of $V$ as a linear morphism $Q_{2}^{1}:(V[1]\otimes
V[1])^{\Sigma_{2}}\to V[2]$, via the canonical identification $(V\wedge
V)[2]\cong(V[1]\otimes V[1])^{\Sigma_{2}}$. Since $C(V[1])$ is cofreely
generated by $V[1]$, the morphisms $Q_{1}^{1}$ and $Q_{2}^{1}$ uniquely extend
to a degree 1 coderivation $Q$ of $C(V[1])$, and the compatibility of the
differental and the bracket of $V$ translates into the condition $QQ=0$, i.e.,
$Q$ is a codifferential.
With the dglas $\mathfrak{g}$ and $\mathfrak{h}$ are therefore associated the
differential graded cocommutative coalgebras
$(C(\mathfrak{g}[1]),Q_{\mathfrak{g}})$ and
$(C(\mathfrak{h}[1]),Q_{\mathfrak{h}})$, respectively. An
$L_{\infty}$-morphism between $\mathfrak{g}$ and $\mathfrak{h}$ is then
defined as a coalgebra morphism $F:C(\mathfrak{g}[1])\to C(\mathfrak{h}[1])$
compatible with the codifferentials, i.e., such that
$FQ_{\mathfrak{g}}=Q_{\mathfrak{h}}F$. Since $C(\mathfrak{g}[1])$ is cofreely
cogenerated by $\mathfrak{g}[1]$, a coalgebra morphism is completely
determined by its Taylor coefficients, i.e. by the components
$F^{1}_{n}:(\otimes^{n}\mathfrak{g}[1])^{\Sigma_{n}}\to h[1]$. Similarly, the
codifferentials $Q_{\mathfrak{g}}$ and $Q_{\mathfrak{h}}$ are completely
determined by their Taylor coefficients which, as we have already remarked,
are nothing but the differentials and the brackets of $\mathfrak{g}$ and
$\mathfrak{h}$, respectively. Therefore, the equation
$FQ_{\mathfrak{g}}=Q_{\mathfrak{h}}F$ is equivalent to the following set of
equations involving only the morphisms $F_{n}^{1}$ and the dgla structures of
$\mathfrak{g}$ and $\mathfrak{h}$:
$\displaystyle
d_{\mathfrak{h}}F_{n}^{1}(\gamma_{1}\wedge\cdots\wedge\gamma_{n})+\frac{1}{2}\\!\\!\\!\sum_{\begin{subarray}{c}q_{1}+q_{2}=n\\\
\sigma\in{\rm
Sh}(q_{1},q_{2})\end{subarray}}\\!\\!\\!\\!\\!\\!\pm[F_{q_{1}}^{1}(\gamma_{\sigma(1)}\wedge\cdots\wedge\gamma_{\sigma(q_{1})}),F_{q_{2}}^{1}(\gamma_{\sigma(q_{1}+1)}\wedge\cdots\wedge\gamma_{\sigma(q_{1}+q_{2})})]_{\mathfrak{h}}$
$\displaystyle=\sum_{i}\pm F_{n}^{1}(\gamma_{1}\wedge\cdots\wedge
d_{\mathfrak{g}}\gamma_{i}\wedge\cdots\wedge\gamma_{n})$
$\displaystyle\qquad\qquad+\sum_{i<j}\pm
F_{n-1}^{1}([\gamma_{i},\gamma_{j}]_{\mathfrak{g}}\wedge\gamma_{1}\wedge\cdots\wedge\widehat{\gamma_{i}}\wedge\cdots\wedge\widehat{\gamma_{j}}\wedge\cdots\wedge\gamma_{q+1}).$
Note in particular that a dgla morphism $\varphi:\mathfrak{g}\to\mathfrak{h}$
is, in a natural way, an $L_{\infty}$-morphism between $\mathfrak{g}$ and
$\mathfrak{h}$, of a very special kind: all but the first one of its Taylor
coefficients vanish. One sometimes refers to this by saying that $\varphi$ is
a _strict_ $L_{\infty}$-morphisms between $\mathfrak{g}$ and $\mathfrak{h}$.
The equation defining $L_{\infty}$-morphisms above manifestly looks like the
Maurer-Cartan equation in a suitable dgla. This is not unexpected: by the
equivalence between dglas and (formal) $\infty$-groupoids stated at the end of
the previous section, there must be a dgla
$\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}})$ such that
$\operatorname{MC}(\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}})\otimes\Omega_{\bullet})$
is equivalent to $\operatorname{Hom}_{\infty}({\mathfrak{g}},{\mathfrak{h}})$.
What we see here is that the dgla
$\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}})$ arises in a
very natural way and admits a simple explicit description: it is the
Chevalley-Eilenberg-type dgla given by the total dgla of the bigraded dgla
$\underline{\operatorname{Hom}}^{p,q}({\mathfrak{g}},{\mathfrak{h}})=\operatorname{Hom}_{{\mathbb{Z}}-{\bf
Vect}}(\wedge^{q}{\mathfrak{g}},{\mathfrak{h}}[p])=\operatorname{Hom}^{p}(\wedge^{q}{\mathfrak{g}},{\mathfrak{h}}),$
endowed with the Lie bracket
$[\,,\,]_{\underline{\operatorname{Hom}}}\colon\underline{\operatorname{Hom}}^{p_{1},q_{1}}({\mathfrak{g}},{\mathfrak{h}})\otimes\underline{\operatorname{Hom}}^{p_{2},q_{2}}({\mathfrak{g}},{\mathfrak{h}})\to\underline{\operatorname{Hom}}^{p_{1}+p_{2},q_{1}+q_{2}}({\mathfrak{g}},{\mathfrak{h}})$
defined by
$\displaystyle[f,g]_{\underline{\operatorname{Hom}}}$
$\displaystyle(\gamma_{1}\wedge\cdots\wedge\gamma_{q_{1}+q_{2}})=$
$\displaystyle\hskip-10.00002pt=\sum_{\sigma\in{\rm
Sh}(q_{1},q_{2})}\pm[f(\gamma_{\sigma(1)}\wedge\cdots\wedge\gamma_{\sigma(q_{1})}),g(\gamma_{\sigma(q_{1}+1)}\wedge\cdots\wedge\gamma_{\sigma(q_{1}+q_{2})})]_{\mathfrak{h}},$
with $\sigma$ ranging in the set of $(q_{1},q_{2})$-unshuffles, and with the
differentials
$d_{1,0}\colon\underline{\operatorname{Hom}}^{p,q}({\mathfrak{g}},{\mathfrak{h}})\to\underline{\operatorname{Hom}}^{p+1,q}({\mathfrak{g}},{\mathfrak{h}})$
and
$d_{0,1}\colon\underline{\operatorname{Hom}}^{p,q}({\mathfrak{g}},{\mathfrak{h}})\to\underline{\operatorname{Hom}}^{p,q+1}({\mathfrak{g}},{\mathfrak{h}})$
given by
$(d_{1,0}{f})(\gamma_{1}\wedge\cdots\wedge\gamma_{q})=d_{\mathfrak{h}}(f(\gamma_{1}\wedge\cdots\wedge\gamma_{q}))+\sum_{i}\pm
f(\gamma_{1}\wedge\cdots\wedge
d_{\mathfrak{g}}\gamma_{i}\wedge\cdots\wedge\gamma_{q+1})$
and
$(d_{0,1}f)(\gamma_{1}\wedge\cdots\wedge\gamma_{q+1})=\sum_{i<j}\pm
f([\gamma_{i},\gamma_{j}]_{\mathfrak{g}}\wedge\gamma_{1}\wedge\cdots\wedge\widehat{\gamma_{i}}\wedge\cdots\wedge\widehat{\gamma_{j}}\wedge\cdots\wedge\gamma_{q+1}).$
These operations are best seen pictorially:
$\left[\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.2263pt\hbox{{\hbox{\kern-2.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\kern-9.81473pt\raise
1.42262pt\hbox{\hbox{\kern
0.0pt\raise-2.5pt\hbox{$\textstyle{f}$}}}}}}}}}\,,\,\,\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 8.53578pt\hbox{{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
4.26437pt\raise 1.42262pt\hbox{\hbox{\kern
0.0pt\raise-1.18056pt\hbox{$\textstyle{g}$}}}}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}\quad\right]_{\underline{\operatorname{Hom}}}\phantom{i}=\phantom{m}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
14.2263pt\hbox{{\hbox{\kern-2.5pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}{\hbox{\kern
14.57156pt\raise 7.11314pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-9.81473pt\raise-4.26788pt\hbox{\hbox{\kern
0.0pt\raise-2.5pt\hbox{$\textstyle{f}$}}}}}{\hbox{\kern
28.79787pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
28.79787pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
28.79787pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
36.13133pt\raise-4.26788pt\hbox{\hbox{\kern
0.0pt\raise-1.18056pt\hbox{$\textstyle{g}$}}}}}{\hbox{\kern
28.79787pt\raise-5.69052pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}{\hbox{\kern
14.57156pt\raise 7.11314pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
14.57156pt\raise 7.11314pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
17.07156pt\raise 22.76208pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\kern 21.33557pt\raise
9.9584pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle{[\,,\,]_{\mathfrak{h}}}}$}}}}}}}}}\qquad;$
$d_{1,0}\left(\,\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.2263pt\hbox{{\hbox{\kern-2.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\kern-9.81473pt\raise
1.42262pt\hbox{\hbox{\kern
0.0pt\raise-2.5pt\hbox{$\textstyle{f}$}}}}}}}}}\,\right)\phantom{i}=\phantom{i}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 19.91682pt\hbox{{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 19.91682pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\kern-9.81473pt\raise
1.42262pt\hbox{\hbox{\kern
0.0pt\raise-2.5pt\hbox{$\textstyle{f}$}}}}}{\hbox{\kern-2.5pt\raise
8.53578pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern
5.03392pt\raise 9.9584pt\hbox{\hbox{\kern
0.0pt\raise-1.73611pt\hbox{$\textstyle{\scriptstyle{d_{\mathfrak{h}}}}$}}}}}}}}}\phantom{i}+\phantom{i}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 20.74684pt\hbox{{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\kern-9.81473pt\raise
1.42262pt\hbox{\hbox{\kern
0.0pt\raise-2.5pt\hbox{$\textstyle{f}$}}}}}{\hbox{\kern-13.88104pt\raise-10.52745pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-20.74684pt\raise-6.8286pt\hbox{\hbox{\kern
0.0pt\raise-1.61111pt\hbox{$\textstyle{\scriptstyle{d_{\mathfrak{g}}}}$}}}}}}}}}\qquad;\qquad
d_{0,1}\left(\,\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.2263pt\hbox{{\hbox{\kern-2.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\kern-9.81473pt\raise
1.42262pt\hbox{\hbox{\kern
0.0pt\raise-2.5pt\hbox{$\textstyle{f}$}}}}}}}}}\,\right)\phantom{i}=\phantom{i}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 26.92273pt\hbox{{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-2.5pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\circ}$}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise 17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\kern-9.81473pt\raise
1.42262pt\hbox{\hbox{\kern
0.0pt\raise-2.5pt\hbox{$\textstyle{f}$}}}}}{\hbox{\kern-13.88104pt\raise-10.52745pt\hbox{\hbox{\kern
0.0pt\raise-2.22223pt\hbox{$\textstyle{\bullet}$}}}}}{\hbox{\kern-26.92273pt\raise-7.11314pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\textstyle{\scriptstyle{[\,,\,]_{\mathfrak{g}}}}$}}}}}}}}}\quad.$
It should be remarked that the above construction is an instance of a more
general phenomenon: if $\mathcal{O}$ is an operad, $A$ is an
$\mathcal{O}$-algebra, and $B$ is a (differential graded) cocommutative
coalgebra, then the space of linear mappings from $B$ to $A$ has a natural
$\mathcal{O}$-algebra structure, see [Do07].
At the zeroth level, the equivalence
$\operatorname{Hom}_{\infty}({\mathfrak{g}},{\mathfrak{h}})\simeq\operatorname{MC}(\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}})\otimes\Omega_{\bullet})$
implies the following:
###### Proposition.
Let $f,g:{\mathfrak{g}}\to{\mathfrak{h}}$ be two $L_{\infty}$-morphisms of
dglas. Then $f$ and $g$ are gauge equivalent in
$\operatorname{MC}(\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}}))$
if and only if $f$ and $g$ represent the same morphism in the homotopy
category of dglas.
Indeed, one immediately sees that
$\operatorname{MC}(\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}}))$
is the set of $L_{\infty}$-morphisms between ${\mathfrak{g}}$ and
${\mathfrak{h}}$ and, as we have already remarked, the set $\pi_{\leq
0}(\operatorname{MC}(\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}})\otimes\Omega_{\bullet}))$
is somorphic to the quotient
$\operatorname{MC}(\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}}))/{\rm
gauge}$. On the other hand, $\pi_{\leq
0}(\operatorname{Hom}_{\infty}({\mathfrak{g}},{\mathfrak{h}}))$ is the set of
homotopy classes of $L_{\infty}$-algebra morphisms between ${\mathfrak{g}}$
and ${\mathfrak{h}}$, i.e., the set of morphisms between ${\mathfrak{g}}$ and
${\mathfrak{h}}$ in the homotopy category of dglas.
We thank Jonathan Pridham for having shown us a proof of the equivalence
between $\operatorname{Hom}_{\infty}({\mathfrak{g}},{\mathfrak{h}})$ and
$\operatorname{MC}(\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}})\otimes\Omega_{\bullet})$,
and Bruno Vallette for having addressed our attention to [Do07]. The same
result holds, more in general, for the homotopy category of
${\mathcal{O}}$-algebras, where ${\mathcal{O}}$ is an operad, see [MV09,
Pr09].
## 3\. Cartan homotopies appear
Let now ${\mathfrak{g}}$ and ${\mathfrak{h}}$ be dglas and
$\boldsymbol{i}\colon{\mathfrak{g}}\to{\mathfrak{h}}[-1]$ be a morphism of
graded vector spaces. Then $\boldsymbol{i}$, and so also $-\boldsymbol{i}$, is
an element of
$\underline{\operatorname{Hom}}^{-1,1}({\mathfrak{g}},{\mathfrak{h}})$, and so
a degree zero element in the dgla
$\underline{\operatorname{Hom}}({\mathfrak{g}},{\mathfrak{h}})$. The gauge
transformation $e^{-\boldsymbol{i}}$ will map the $0$ dgla morphism to an
$L_{\infty}$-morphism $e^{-\boldsymbol{i}}*0$ between ${\mathfrak{g}}$ and
${\mathfrak{h}}$. This $L_{\infty}$-morphism will in general fail to be a dgla
morphism (i.e., it will not be a strict $L_{\infty}$-morphism) since its
nonlinear components will be nontrivial. This is conveniently seen as follows:
let $\boldsymbol{l}=d_{1,0}\boldsymbol{i}$; that is,
$\boldsymbol{l}_{a}=d_{\mathfrak{h}}\boldsymbol{i}_{a}+\boldsymbol{i}_{d_{\mathfrak{g}}a}$
for any $a\in{\mathfrak{g}}$. Then the $(0,1)$-component of
$\displaystyle e^{-\boldsymbol{i}}*0$
$\displaystyle=\sum_{n=0}^{+\infty}\frac{{({\rm
ad}_{-\boldsymbol{i}})}^{n}}{(n+1)!}\
(d_{\underline{\operatorname{Hom}}}\boldsymbol{i})=\sum_{n=0}^{+\infty}\frac{{({\rm
ad}_{-\boldsymbol{i}})}^{n}}{(n+1)!}\
(\boldsymbol{l}+\boldsymbol{i}_{[\,,\,]_{\mathfrak{g}}})$
is just $\boldsymbol{l}$; the $(-1,2)$-component is
$\boldsymbol{i}_{[\,,\,]_{\mathfrak{g}}}-\frac{1}{2}[\boldsymbol{i},\boldsymbol{l}]_{\underline{\operatorname{Hom}}}$
and, for $n\geq 3$ the $(1-n,n)$-component has two contributions, one of the
form
$[\boldsymbol{i},[\boldsymbol{i},\cdots,[\boldsymbol{i},\boldsymbol{l}]_{\underline{\operatorname{Hom}}}\cdots]_{\underline{\operatorname{Hom}}}]_{\underline{\operatorname{Hom}}}$
and the other of the form
$[\boldsymbol{i},[\boldsymbol{i},\cdots,[\boldsymbol{i},\boldsymbol{i}_{[\,,\,]_{\mathfrak{g}}}]_{\underline{\operatorname{Hom}}}\cdots]_{\underline{\operatorname{Hom}}}]_{\underline{\operatorname{Hom}}}$.
From this we see that all the nonlinear components of $e^{-\boldsymbol{i}}*0$
vanish as soon as one imposes the two simple conditions
$\boldsymbol{i}_{[a,b]_{\mathfrak{g}}}=\frac{1}{2}\bigl{(}[\boldsymbol{i}_{a},\boldsymbol{l}_{b}]_{\mathfrak{h}}\pm[\boldsymbol{i}_{b},\boldsymbol{l}_{a}]_{\mathfrak{h}}\bigr{)}\qquad\text{and}\qquad[\boldsymbol{i}_{a},[\boldsymbol{i}_{b},\boldsymbol{l}_{c}]_{\mathfrak{h}}]_{\mathfrak{h}}=0,\qquad\mbox{for
all}\ a,b,c\in{\mathfrak{g}}.$
A linear map $\boldsymbol{i}\colon{\mathfrak{g}}\to{\mathfrak{h}}[-1]$
satisfying the two conditions above will be called a Cartan homotopy. Up to
our knowledge, this terminology has been introduced in [FM06, FM09], where the
stronger conditions
$\boldsymbol{i}_{[a,b]_{\mathfrak{g}}}=[\boldsymbol{i}_{a},\boldsymbol{l}_{b}]_{\mathfrak{h}}$
and $[\boldsymbol{i}_{a},\boldsymbol{i}_{b}]_{\mathfrak{h}}=0$ were imposed.
The name Cartan homotopy has an evident geometric origin: if
${\mathcal{T}}_{X}$ is the tangent sheaf of a smooth manifold $X$ and
$\Omega_{X}$ is the sheaf of complexes of differential forms, then the
contraction of differential forms with vector fields is a Cartan homotopy
$\boldsymbol{i}\colon{\mathcal{T}}_{X}\to{\mathcal{E}}nd(\Omega_{X})[-1].$
In this case, $\boldsymbol{l}_{a}$ is the Lie derivative along the vector
field $a$, and the conditions
$\boldsymbol{i}_{[a,b]}=[\boldsymbol{i}_{a},\boldsymbol{l}_{b}]$ and
$[\boldsymbol{i}_{a},\boldsymbol{i}_{b}]=0$, together with the defining
equation $\boldsymbol{l}_{a}=[d_{\Omega_{X}},\boldsymbol{i}_{a}]$ and with the
equations $\boldsymbol{l}_{[a,b]}=[\boldsymbol{l}_{a},\boldsymbol{l}_{b}]$ and
$[d_{\Omega_{X}},\boldsymbol{l}_{a}]=0$ expressing the fact that
$\boldsymbol{l}\colon{\mathcal{T}}_{X}\to{\mathcal{E}}nd(\Omega_{X})$ is a
dgla morphism, are nothing but the well-known Cartan identities involving
contractions and Lie derivatives.
The above discussion can be summarized as follows.
###### Proposition.
Let ${\mathfrak{g}}$ and ${\mathfrak{h}}$ be two dglas. If
$\boldsymbol{i}\colon{\mathfrak{g}}\to{\mathfrak{h}}[-1]$ is a Cartan
homotopy, then
$\boldsymbol{l}=d_{1,0}\boldsymbol{i}\colon{\mathfrak{g}}\to{\mathfrak{h}}$ is
a dgla morphism gauge equivalent to the zero morphism via the gauge action of
$e^{\boldsymbol{i}}$.
## 4\. Homotopy fibers (and the associated exact sequence)
Let now $\boldsymbol{i}\colon{\mathfrak{g}}\to{\mathfrak{h}}[-1]$ be a Cartan
homotopy and $\boldsymbol{l}\colon{\mathfrak{g}}\to{\mathfrak{h}}$ be the
associated dgla morphism. Then, the equation
$e^{\boldsymbol{i}}*\boldsymbol{l}=0$ implies that, for any subdgla
${\mathfrak{n}}$ of ${\mathfrak{h}}$ containing the image of $\boldsymbol{l}$,
the morphism $\boldsymbol{l}\colon{\mathfrak{g}}\to{\mathfrak{n}}$ equalizes
the diagram
$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
incl.}}$$\scriptstyle{0}$$\textstyle{\mathfrak{h}}$ up to a homotopy provided
by the gauge action of $e^{\boldsymbol{i}}$. Hence we have a morphism to the
homotopy limit:
${\mathfrak{g}}\xrightarrow{(\boldsymbol{l},e^{\boldsymbol{i}})}\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$
Taking Def’s we obtain a natural transformation of $\infty$-groupoid valued
functors:
$\operatorname{Def}({\mathfrak{g}})\xrightarrow{(\boldsymbol{l},e^{\boldsymbol{i}})}\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
17.23615pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-17.23615pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Def}({\mathfrak{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
17.27086pt\raise 8.125pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.73611pt\hbox{$\scriptstyle{\operatorname{Def}_{\rm
incl.}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.23615pt\raise
2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
19.93753pt\raise-8.07498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{\operatorname{Def}_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.23615pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.23615pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Def}({\mathfrak{h}})}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$
The map
$\operatorname{Def}_{0}\colon\operatorname{Def}({\mathfrak{n}})\to\operatorname{Def}({\mathfrak{h}})$
is the constant map to the distinguished point $0$ in
$\operatorname{Def}({\mathfrak{h}})$; therefore, the homotopy limit above is
the homotopy fiber of $\operatorname{Def}_{\rm
incl.}\colon\operatorname{Def}({\mathfrak{n}})\to\operatorname{Def}({\mathfrak{h}})$
over the point $0$, and we obtain a natural transformation
$\operatorname{Def}({\mathfrak{g}})\xrightarrow{(\boldsymbol{l},e^{\boldsymbol{i}})}{\rm
hoDef}^{-1}_{\rm incl.}(0),$
which at the zeroth level gives a natural transformation of Set-valued
deformation functors
${\mathcal{P}}\colon\pi_{\leq 0}\operatorname{Def}({\mathfrak{g}})\to\pi_{\leq
0}{\rm hoDef}^{-1}_{\rm incl.}(0).$
The differential of ${\mathcal{P}}$ is easily computed: it is the linear map
$H^{1}({\mathfrak{g}})\xrightarrow{H^{1}((\boldsymbol{l},e^{\boldsymbol{i}}))}H^{1}(\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)).$
Since the model category structure on dglas is the same as on differential
complexes, we can compute the $H^{1}$ on the right hand side by taking the
holimit in complexes. Then the natural quasi-isomorphism
$\operatorname{holim}(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces)\simeq({\mathfrak{h}}/{\mathfrak{n}})[-1]$
tells us that the differential of ${\mathcal{P}}$ is just the map
$H^{1}({\boldsymbol{i}})\colon H^{1}({\mathfrak{g}})\to
H^{0}({\mathfrak{h}}/{\mathfrak{n}})$
induced by the morphism of complexes
${\boldsymbol{i}}\colon{\mathfrak{g}}\to({\mathfrak{h}}/{\mathfrak{n}})[-1]$.
Also, the map
$H^{2}({\boldsymbol{i}})\colon H^{2}({\mathfrak{g}})\to
H^{1}({\mathfrak{h}}/{\mathfrak{n}})$
maps the obstruction space of $\pi_{\leq 0}\operatorname{Def}({\mathfrak{g}})$
(as a subspace of $H^{2}({\mathfrak{g}})$) to the obstruction space of
$\pi_{\leq 0}{\rm hoDef}^{-1}_{\rm incl.}(0)$ (as a subspace of
$H^{1}({\mathfrak{h}}/{\mathfrak{n}})$). In particular, if $\pi_{\leq 0}{\rm
hoDef}^{-1}_{\rm incl.}(0)$ is smooth, and therefore unobstructed, the
obstructions of the deformation functor $\pi_{\leq
0}\operatorname{Def}({\mathfrak{g}})$ are contained in the kernel of the map
$H^{2}({\boldsymbol{i}})\colon H^{2}({\mathfrak{g}})\to
H^{1}({\mathfrak{h}}/{\mathfrak{n}})$.
To investigate the geometry of $\pi_{\leq 0}{\rm hoDef}^{-1}_{\rm incl.}(0)$
note that, by looking at it as a pointed set, it nicely fits into the homotopy
exact sequence
$\pi_{1}(\operatorname{Def}({\mathfrak{n}});0)\xrightarrow{\operatorname{Def}_{{\rm
incl.}*}}\pi_{1}(\operatorname{Def}({\mathfrak{h}});0)\to\pi_{0}({\rm
hoDef}^{-1}_{\rm
incl.}(0);0)\to\pi_{0}(\operatorname{Def}({\mathfrak{n}});0),$
so we get a canonical isomorphism
$\pi_{\leq 0}{\rm hoDef}^{-1}_{\rm
incl.}(0)\simeq\frac{\pi_{1}(\operatorname{Def}({\mathfrak{h}});0)}{{\operatorname{Def}_{{\rm
incl.}*}}\pi_{1}(\operatorname{Def}({\mathfrak{n}});0)}.$
The group $\pi_{1}(\operatorname{Def}({\mathfrak{h}});0)$ is the group of
automorphisms of $0$ in the groupoid $\pi_{\leq
1}(\operatorname{Def}({\mathfrak{h}}))$. We have already remarked that this
groupoid is not equivalent to the Deligne groupoid of ${\mathfrak{h}}$, i.e.,
the action groupoid for the gauge action of
$\exp({\mathfrak{h}}^{0}\otimes{\mathfrak{m}}_{A})$ on
$\operatorname{MC}({\mathfrak{h}}\otimes{\mathfrak{m}}_{A})$, since the
irrelevant stabilizer
${\rm Stab}(x)=\\{dh+[x,h]\mid
h\in\mathfrak{h}^{-1}\otimes{\mathfrak{m}}_{A}\\}\subseteq\\{a\in\mathfrak{h}^{0}\otimes\mathfrak{m}_{A}\mid
e^{a}*x=x\\}$
of a Maurer-Cartan element $x$ may be nontrivial. However, the group
$\pi_{1}(\operatorname{Def}({\mathfrak{h}});0)$ only sees the connected
component of $0$, and on this connected component the irrelevant stabilizers
are trivial as soon as the differential of the dgla ${\mathfrak{h}}$ vanishes
on $\mathfrak{h}^{-1}$. This immediately follows from noticing that irrelevant
stabilizers of gauge equivalent Maurer-Cartan elements are conjugate subgroups
of $\exp(\mathfrak{h}^{0}\otimes\mathfrak{m}_{A})$, see, e.g., [Ma07]. In
particular, if ${\mathfrak{h}}$ is a graded Lie algebra (which we can consider
as a dgla with trivial differential), then
$\pi_{1}(\operatorname{Def}({\mathfrak{h}});0)\simeq\exp({\mathfrak{h}}^{0})$,
where ${\mathfrak{h}}^{0}$ denotes the degree zero component of
${\mathfrak{h}}$. Similarly, since ${\mathfrak{n}}$ is a subdgla of
${\mathfrak{h}}$, one has
$\pi_{1}(\operatorname{Def}({\mathfrak{n}});0)\simeq\exp({\mathfrak{n}}^{0})$,
and the group homomorphism $\operatorname{Def}_{{\rm incl.}*}$ is just the
inclusion. Therefore, when ${\mathfrak{h}}$ has trivial differential, the map
induced at the zeroth level by $\operatorname{Def}({\mathfrak{g}})\to{\rm
hoDef}^{-1}_{\rm incl.}(0)$ is just the natural map
$e^{\boldsymbol{i}}\colon\pi_{\leq
0}\operatorname{Def}({\mathfrak{g}})\to\exp({\mathfrak{h}}^{0})/\exp({\mathfrak{n}}^{0})$
which sends a Maurer-Cartan element
$\xi\in{\mathfrak{g}}^{1}\otimes{\mathfrak{m}}_{A}$ to
$e^{\boldsymbol{i}_{\xi}}\mod\exp({\mathfrak{n}}^{0})$. A particularly
interesting case is when the pair $({\mathfrak{h}},{\mathfrak{n}})$ is
formal,222We are not sure whether this terminology is a standard one i.e., if
the inclusion of ${\mathfrak{n}}$ in ${\mathfrak{h}}$ induces an inclusion in
cohomology and the two inclusions $H^{*}({\mathfrak{n}})\hookrightarrow
H^{*}({\mathfrak{h}})$ and ${\mathfrak{n}}\hookrightarrow{\mathfrak{h}}$ are
homotopy equivalent. Indeed, in this case the pair
$(\operatorname{Def}({\mathfrak{h}}),\operatorname{Def}({\mathfrak{n}}))$ will
be equivalent to the pair
$(\operatorname{Def}(H^{*}({\mathfrak{h}})),\operatorname{Def}(H^{*}({\mathfrak{h}})))$
and there will be an induced isomorphism between
$\pi_{1}(\operatorname{Def}({\mathfrak{h}});0)/{\operatorname{Def}_{{\rm
incl.}*}}\pi_{1}(\operatorname{Def}({\mathfrak{n}});0)$ and the smooth
homogeneous space $\exp(H^{0}({\mathfrak{h}}))/\exp(H^{0}({\mathfrak{n}}))$.
We can summarize the results described in this section as follows:
###### Proposition.
Let $\boldsymbol{i}\colon{\mathfrak{g}}\to{\mathfrak{h}}[-1]$ be a Cartan
homotopy, let $\boldsymbol{l}\colon{\mathfrak{g}}\to{\mathfrak{h}}$ be the
associated dgla morphism, and let ${\mathfrak{n}}$ be a subdgla of
${\mathfrak{h}}$ containing the image of $\boldsymbol{l}$. Then, if the pair
$({\mathfrak{h}},{\mathfrak{n}})$ is formal, we have a natural
transformation333This natural transformation is not canonical: it depends on
the choice of a quasi isomorphism
$({\mathfrak{h}},{\mathfrak{n}})\simeq(H^{*}({\mathfrak{h}}),H^{*}({\mathfrak{n}}))$.
Also note that the tangent space at $0$ on the right hand side is
$H^{0}({\mathfrak{h}})/H^{0}({\mathfrak{n}})$; this is only apparently in
contrast with the general result mentioned above that the tangent space at $0$
to $\pi_{\leq 0}{\rm hoDef}^{-1}_{\rm incl.}(0)$ is
$H^{0}({\mathfrak{h}}/{\mathfrak{n}})$. Indeed, when
$({\mathfrak{h}},{\mathfrak{n}})$ is a formal pair, the two vector spaces
$H^{0}({\mathfrak{h}})/H^{0}({\mathfrak{n}})$ and
$H^{0}({\mathfrak{h}}/{\mathfrak{n}})$ are (non canonically) isomorphic. of
Set-valued deformation functors
${\mathcal{P}}\colon\pi_{\leq
0}(\operatorname{Def}({\mathfrak{g}}))\to\exp(H^{0}({\mathfrak{h}}))/\exp(H^{0}({\mathfrak{n}}))$
induced by the dgla map
${\mathfrak{g}}\xrightarrow{(\boldsymbol{l},e^{\boldsymbol{i}})}\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$
In particular, since $\exp(H^{0}({\mathfrak{h}}))/\exp(H^{0}({\mathfrak{n}}))$
is smooth, the obstructions of the Set-valued deformation functor $\pi_{\leq
0}(\operatorname{Def}({\mathfrak{g}});0)$ are contained in the kernel of the
map $H^{2}({\boldsymbol{i}})\colon H^{2}({\mathfrak{g}})\to
H^{1}({\mathfrak{h}}/{\mathfrak{n}})$.
This result can be nicely refined, by showing how the main result from [IM010]
naturally fits into the discussion above. We have:
###### Proposition.
Let $({\mathfrak{h}},{\mathfrak{n}})$ be a formal pair of dglas. Then, the
dgla $\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)$
is quasi-abelian. In particular there is a (non-canonical) quasi-isomorphism
of dglas between $\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)$
and the abelian dgla obtained by endowing the complex
$({\mathfrak{h}}/{\mathfrak{n}})[-1]$ with the trivial bracket.
To see this, notice that, since by hypothesis the inclusion
${\mathfrak{n}}\hookrightarrow{\mathfrak{h}}$ induces an inclusion
$H^{*}({\mathfrak{n}})\hookrightarrow H^{*}({\mathfrak{h}})$, the projection
${\mathfrak{h}}[-1]\to{\mathfrak{h}}/{\mathfrak{n}}[-1]$ admits a section
$\boldsymbol{i}$ which is a morphism of complexes. Denote by ${\mathfrak{g}}$
the dgla obtained from the complex ${\mathfrak{h}}/{\mathfrak{n}}[-1]$ by
endowing it with the trivial bracket. Then, the map of graded vector spaces
$\boldsymbol{i}\colon{\mathfrak{g}}\to{\mathfrak{h}}[-1]$ is a Cartan homotopy
whose associated dgla morphism is the zero map
$0\colon{\mathfrak{g}}\to{\mathfrak{h}}$. Therefore we have a dgla map
$({\mathfrak{h}}/{\mathfrak{n}})[-1]\xrightarrow{(0,e^{\boldsymbol{i}})}\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$
Since $\boldsymbol{i}$ is a section to
${\mathfrak{h}}[-1]\to{\mathfrak{h}}/{\mathfrak{n}}[-1]$, the map in
cohomology
$H^{*}({\mathfrak{h}}/{\mathfrak{n}})[-1]\xrightarrow{H^{*}(0,e^{\boldsymbol{i}})}H^{*}(\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right))$
is identified with the identity of $H^{*}({\mathfrak{h}}/{\mathfrak{n}})[-1]$
by the the natural quasi-isomorphism of complexes
$\operatorname{holim}(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-5.77779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathfrak{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.36111pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.02779pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{h}}$}}}}}}}\ignorespaces}}}}\ignorespaces)\xrightarrow{\sim}({\mathfrak{h}}/{\mathfrak{n}})[-1]$.
## 5\. From local to global, and classical periods
Assume now $\mathbb{K}$ is algebrically closed. Let $X$ be a projective
manifold, and let ${\mathcal{T}}_{X}$ and $\Omega_{X}^{*}$ be the tangent
sheaf and the sheaf of differential forms on $X$, respectively. The sheaf of
complexes $(\Omega_{X}^{*},d)$ is naturally filtered by setting
$F^{p}\Omega_{X}^{*}=\oplus_{i\geq p}\Omega^{i}_{X}$. Finally, let
${\mathcal{E}}nd^{*}(\Omega_{X}^{*})$ be the endomorphism sheaf of
$\Omega_{X}^{*}$ and ${\mathcal{E}}nd^{\geq 0}(\Omega_{X}^{*})$ be the
subsheaf consisting of nonnegative degree elements. Note that
${\mathcal{E}}nd^{\geq 0}(\Omega_{X}^{*})$ is a subdgla of
${\mathcal{E}}nd^{*}(\Omega_{X}^{*})$, and can be seen as the subdgla of
endomorphisms preserving the filtration on $\Omega_{X}^{*}$.
Recall that the prototypical example of Cartan homotopy was the contraction of
differential forms with vector fields
${\boldsymbol{i}}:\mathcal{T}_{X}\to\mathcal{E}nd^{*}(\Omega_{X}^{*})[-1]$;
the corresponding dgla morphism is $a\mapsto\boldsymbol{l}_{a}$, where
$\boldsymbol{l}_{a}$ the Lie derivative along $a$. Explicitly,
$\boldsymbol{l}_{a}=d\circ{\boldsymbol{i}}_{a}+{\boldsymbol{i}}_{a}\circ d$,
and so $\boldsymbol{l}_{a}$ preserves the filtration. Therefore, we have a
natural transformation444Of what? The correct answer would be of
$\infty$-sheaves, see [Lu09b], but to keep this note as far as possible at an
informal level we will content ourselves with noticing that, for any open
subset $U$ of $X$, there is a natural transformation of $\infty$-groupoids
induced by the dgla map
${\mathcal{T}}_{X}(U)\xrightarrow{(\boldsymbol{l},e^{\boldsymbol{i}})}\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
33.47154pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-33.47154pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{E}}nd^{\geq
0}(\Omega_{X}^{*})(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.04202pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 57.47154pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
35.7087pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 57.47154pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
57.47154pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{E}}nd^{*}(\Omega_{X}^{*})}$}}}}}}}\ignorespaces}}}}\ignorespaces(U)\right)$.
$\operatorname{Def}({\mathcal{T}}_{X})\xrightarrow{(\boldsymbol{l},e^{\boldsymbol{i}})}\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
37.08199pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-37.08199pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Def}({\mathcal{E}}nd^{\geq
0}(\Omega_{X}^{*}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.57642pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 61.08199pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
43.24309pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 61.08199pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
61.08199pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Def}({\mathcal{E}}nd^{*}(\Omega_{X}^{*}))}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$
The homotopy fiber on the right should be thought as a homotopy flag manifold.
Let us briefly explain this. At least naïvely, the functor
$\operatorname{Def}({\mathcal{E}}nd^{*}(\Omega_{X}^{*}))$ describes the
infinitesimal deformations of the differential complex $\Omega_{X}^{*}$,
whereas the functor $\operatorname{Def}({\mathcal{E}}nd^{\geq
0}(\Omega_{X}^{*}))$ describes the deformations of the filtered complex
$(\Omega_{X}^{*},F^{\bullet}\Omega_{X}^{*})$, i.e., of the pair consisting of
the complex $\Omega_{X}^{*}$ _and_ the filtration $F^{\bullet}\Omega_{X}^{*}$.
Therefore, the holimit describes a deformation of the pair (complex,
filtration) together with a trivialization of the deformation of the complex.
Summing up, the contraction of differential forms with vector fields induces a
map of deformation functors
$\operatorname{Def}({\mathcal{T}}_{X})\rightarrow{\rm
hoFlag}(\Omega_{X}^{*};F^{\bullet}\Omega_{X}^{*}),$
which we will call the _local periods map_ of $X$.
To recover from this the classical periods map, we just need to take global
sections. Clearly, since we are working in homotopy categories, these will be
derived global sections. The morphism of sheaves
${\boldsymbol{i}}:\mathcal{T}_{X}\to\mathcal{E}nd(\Omega_{X}^{*})[-1]$ induces
a Cartan homotopy ${\boldsymbol{i}}:{\bf R}\Gamma\mathcal{T}_{X}\to{\bf
R}\Gamma\mathcal{E}nd(\Omega_{X}^{*})[-1]$; composing this with the dgla
morphism ${\bf R}\Gamma\mathcal{E}nd(\Omega_{X}^{*})\to\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*})$ induced by the action of (derived) global sections of
the endomorphism sheaf of $\Omega_{X}^{*}$ on (derived) global sections of
$\Omega_{X}^{*}$, we get a Cartan homotopy
${\boldsymbol{i}}:{\bf R}\Gamma\mathcal{T}_{X}\to\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*})[-1].$
The image of the corresponding dgla morphism $\boldsymbol{l}$ (the derived
globalization of Lie derivative) preserves the filtration $F^{\bullet}{\bf
R}\Gamma\Omega_{X}^{*}$ induced by $F^{\bullet}\Omega_{X}^{*}$, so we have a
natural map of $\infty$-groupoids
$\operatorname{Def}({\bf R}\Gamma\mathcal{T}_{X})\to{\rm hoFlag}({\bf
R}\Gamma\Omega_{X}^{*};F^{\bullet}{\bf R}\Gamma\Omega_{X}^{*})$
and, at the zeroth level, a map of Set-valued deformation functors
${\mathcal{P}}\colon\pi_{\leq 0}\operatorname{Def}({\bf
R}\Gamma\mathcal{T}_{X})\to\pi_{\leq 0}{\rm hoFlag}({\bf
R}\Gamma\Omega_{X}^{*};F^{\bullet}{\bf R}\Gamma\Omega_{X}^{*})$
The functor on the left hand side is the Set-valued functor of (classical)
infinitesimal deformations of $X$; let us denote it by
$\operatorname{Def}_{X}$. If we denote by $\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*};F^{\bullet}{\bf R}\Gamma\Omega_{X}^{*})$ the subdgla of
$\operatorname{End}({\bf R}\Gamma\Omega_{X}^{*})$ consisting of endomorhisms
preserving the filtration, then the pair $(\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*}),\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*};F^{\bullet}{\bf R}\Gamma\Omega_{X}^{*}))$ is
formal.555This is essentially a consequence of the $E_{1}$-degeneration of the
Hodge-de Rham spectral sequence, see, e.g., [DI87, Fa88]. Moreover,
$H^{0}(\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*}))=\operatorname{End}^{0}(H^{*}(X;{\mathbb{K}}))$ and
$H^{0}(\operatorname{End}({\bf R}\Gamma\Omega_{X}^{*};F^{\bullet}{\bf
R}\Gamma\Omega_{X}^{*}))=\operatorname{End}^{0}(H^{*}(X;{\mathbb{K}});F^{\bullet}H^{*}(X;{\mathbb{K}}))$,where
$F^{\bullet}H^{*}(X;{\mathbb{K}})$ is the Hodge filtration on
$H^{*}(X;{\mathbb{K}})$. By results described in the previous section, this
means that
$\pi_{\leq 0}{\rm hoFlag}({\bf R}\Gamma\Omega_{X}^{*};F^{\bullet}{\bf
R}\Gamma\Omega_{X}^{*})\simeq\frac{\exp(\operatorname{End}^{0}(H^{*}(X;{\mathbb{K}})))}{\exp(\operatorname{End}^{0}(H^{*}(X;{\mathbb{K}});F^{\bullet}H^{*}(X;{\mathbb{K}})))}$
and we recover the classical periods map of $X$
${\mathcal{P}}\colon\operatorname{Def}_{X}\to{\rm
Flag}(H^{*}(X;{\mathbb{K}});F^{\bullet}H^{*}(X;{\mathbb{K}})).$
Also, the differential of ${\mathcal{P}}$ is the map induced in cohomology by
the contraction of differential forms with vector fields,
$H^{1}({\boldsymbol{i}})\colon
H^{1}(X,{\mathcal{T}_{X}})\to\int_{p}\operatorname{Hom}^{0}\left(F^{p}H^{*}(X;{\mathbb{K}});\frac{H^{*}(X;{\mathbb{K}})}{F^{p}H^{*}(X;{\mathbb{K}})}\right),$
a result originally proved by Griffiths [Gr68]. In the above formula,
$\int_{p}$ denotes the end of the diagram
$\operatorname{Hom}^{0}\left(F^{p}H^{*};\frac{H^{*}}{F^{p}H^{*}}\right)\rightarrow\operatorname{Hom}^{0}\left(F^{p}H^{*};\frac{H^{*}}{F^{p+1}H^{*}}\right)\leftarrow\operatorname{Hom}^{0}\left(F^{p+1}H^{*};\frac{H^{*}}{F^{p+1}H^{*}}\right)$
Also, we have the following version of the so-called Kodaira principle
(ambient cohomology annihilates obstructions): obstructions to classical
infinitesimal deformations of $X$ are contained in the kernel of
$H^{2}({\boldsymbol{i}})\colon
H^{2}(X,{\mathcal{T}_{X}})\to\int_{p}\operatorname{Hom}^{1}\left(F^{p}H^{*}(X;{\mathbb{K}});\frac{H^{*}(X;{\mathbb{K}})}{F^{p}H^{*}(X;{\mathbb{K}})}\right).$
In particular, if the canonical bundle of $X$ is trivial, then the contraction
pairing
$H^{2}(X,{\mathcal{T}_{X}})\otimes H^{n-2}(X,\Omega^{1}_{X})\to
H^{n}(X;{\mathcal{O}}_{X})\simeq{\mathbb{K}}$
is nondegenerate, and so classical deformations of $X$ are unobstructed
(Bogomolov-Tian-Todorov theorem, see [Bo78, Ti87, To89]). Following [IM010],
one immediately obtains the following refinement, due in its original
formulation to Goldman and Millson [GM90]: if the canonical bundle of $X$ is
trivial, then ${\bf R}\Gamma\mathcal{T}_{X}$ is a quasi-abelian dgla. To see
this, just notice that the dgla map
${\bf
R}\Gamma\mathcal{T}_{X}\xrightarrow{({\boldsymbol{l}},e^{\boldsymbol{i}})}\operatorname{holim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
49.29144pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-49.29144pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*};F^{\bullet}{\bf
R}\Gamma\Omega_{X}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
32.4116pt\raise 7.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\phantom{mmm}{\rm incl.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 73.29144pt\raise 2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.07828pt\raise-7.25555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{\phantom{mmm}0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 73.29144pt\raise-2.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.29144pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{End}({\bf
R}\Gamma\Omega_{X}^{*})}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)$
is injective in cohomology and the target is a quasi-abelian dgla. Indeed, if
$f\colon{\mathfrak{g}}\to{\mathfrak{h}}$ is a dgla morphism, with $H^{*}(f)$
injective and ${\mathfrak{h}}$ quasi-abelian, then the diagram of dglas
$\textstyle{{\mathfrak{g}}\,}$$\textstyle{{\mathfrak{h}}}$$\textstyle{{\mathfrak{k}}}$$\textstyle{{\mathfrak{h}}}$$\textstyle{{\mathfrak{k}}}$$\textstyle{V}$$\textstyle{\scriptstyle{f}}$
where $V$ is a graded vector space considered as a dgla with trivial
differential and bracket, can be completed to a homotopy commutative diagram
$\textstyle{{\mathfrak{g}}\,}$$\textstyle{{\mathfrak{h}}}$$\textstyle{{\mathfrak{k}}}$$\textstyle{{\mathfrak{h}}}$$\textstyle{{\mathfrak{k}}}$$\textstyle{V}$$\textstyle{{\mathfrak{l}}}$$\textstyle{{\mathfrak{g}}\,}$$\textstyle{{\mathfrak{l}}}$$\textstyle{{\mathfrak{k}}}$$\textstyle{V}$$\textstyle{W}$$\textstyle{\scriptstyle{f}}$
with $W$ a graded vector space, and the composition ${\mathfrak{l}}\to W$ a
quasi-isomorphism.
## References
* [Bo78] F. Bogomolov. _Hamiltonian Kählerian manifolds._ Dokl. Akad. Nauk SSSR 243 (1978) 1101-1104. Soviet Math. Dokl. 19 (1979) 1462-1465.
* [DI87] P. Deligne, L. Illusie. _Relévements modulo $p^{2}$ et décomposition du complexe de de Rham._ Invent. Math. 89 (1987) 247-270.
* [Do07] V. A. Dolgushev. _Erratum to: ”A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold”_ , arXiv:math/0703113.
* [Fa88] G. Faltings. _$p$ -adic Hodge theory._ J. Amer. Math. Soc. 1 (1988) 255-299.
* [FM06] D. Fiorenza, M. Manetti. _$L_{\infty}$ -algebras, Cartan homotopies and period maps._ math/0605297
* [FM09] D. Fiorenza, M. Manetti. _A period map for generalized deformations._ Journal of Noncommutative Geometry, Vol. 3, No. 4 (2009), 579-597; arXiv:0808.0140.
* [Ge09] E. Getzler. _Lie theory for nilpotent $L_{\infty}$-algebras._ Ann. of Math., 170, (1), (2009), 271-301; arXiv:math/0404003v4.
* [GM90] W. M. Goldman, J. J. Millson. _The homotopy invariance of the Kuranishi space._ Illinois J. Math. 34 (1990) 337-367.
* [Gr68] Ph. Griffiths. _Periods of integrals on algebraic manifolds. II. Local study of the period mapping._ Amer. J. Math. 90 (1968) 805-865.
* [Hi97] V. Hinich. _Descent of Deligne groupoids._ Internat. Math. Res. Notices, (1997), 5, 223-239.
* [IM010] D. Iacono, M. Manetti._An algebraic proof of Bogomolov-Tian-Todorov theorem_ Deformation Spaces. Vol. 39 (2010), p. 113-133; arXiv:0902.0732
* [Ko03] M. Kontsevich. _Deformation quantization of Poisson manifolds._ Lett. Math. Phys. 66 (2003), no. 3, 157–216; arXiv:q-alg/9709040
* [LS93] T. Lada and J. Stasheff. _Introduction to SH Lie algebras for physicists._ Internat. J. Theoret. Phys. 32 (1993), no. 7, 1087-1103; arXiv:hep-th/9209099
* [Lu09a] J. Lurie. _Moduli Problems for Ring Spectra._ http://www.math.harvard.edu/~lurie/papers/moduli.pdf
* [Lu09b] J. Lurie. _Higher Topos Theory._ Princeton University Press (2009); arXiv:math/0608040.
* [Lu09c] J. Lurie. _A survey of elliptic cohomology._ Algebraic topology, 219–277, Abel Symp., 4, Springer, Berlin, 2009; http://www.math.harvard.edu/~lurie/papers/survey.pdf
* [Ma07] M. Manetti. _Lie description of higher obstructions to deforming submanifolds._ Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2007) 631-659; math.AG/0507287
* [MV09] S. Merkulov, B. Vallette. _Deformation theory of representation of prop(erad)s I and II._ J. Reine Angew. Math. (Crelle), 634 (2009), 51 106 and J. Reine Angew. Math. (Crelle), to appear, arXiv:0707.0889
* [Pr10] J. P. Pridham. _Unifying derived deformation theories._ Adv. Math. 224 (2010), no.3, 772-826; arXiv:0705.0344
* [Pr09] J. P. Pridham. _The homotopy theory of strong homotopy algebras and bialgebras._ arXiv:0908.0116
* [Su77] D. Sullivan. _Infinitesimal computations in topology._ Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269 331.
* [Ti87] G. Tian. _Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric._ Mathematical Aspects of String Theory (San Diego, 1986), Adv. Ser. Math. Phys. 1, World Sci. Publishing, Singapore, (1987), 629-646.
* [To89] A. N. Todorov. _The Weil-Petersson geometry of the moduli space of $SU(3)$ (Calabi-Yau) Manifolds I._ Commun. Math. Phys., 126, (l989), 325-346.
|
arxiv-papers
| 2009-11-19T19:02:05 |
2024-09-04T02:49:06.574297
|
{
"license": "Public Domain",
"authors": "Domenico Fiorenza and Elena Martinengo",
"submitter": "Domenico Fiorenza",
"url": "https://arxiv.org/abs/0911.3845"
}
|
0911.3849
|
# Static Charged Black Hole Solutions in Hořava-Lifshitz Gravity
Jin-Zhang Tang111Electronic address:JinzhangTang@pku.edu.cn Department of
Physics, and State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, China
(August 27, 2024
)
###### Abstract
In the present work, we search static charged black hole solutions to Hořava-
Lifshitz gravity with or without projectability condition. We consider the
most general form of action which electromagnetic field couples with Hořava-
Lifshitz gravity. With the projectability condition, we find (A)dS-Reissner-
Nordstrom black hole solution in Painlevé-Gullstrand type coordinates in the
IR region and a de-Sitter space-time solution in the UV region. Without the
projectability condition, in the IR region, we find an especial static charged
black hole solution.
###### pacs:
98.80.Cq
## I introduction
The Hořava-Lifshitz gravity which was introduced in Horava:2008ih ;
Horava:2009uw is intended to be a power-counting renormalizable gravity
theory. The basic idea behind Hořava’s theory is that time and space may have
different dynamical scaling in UV limit. This was inspired by the development
in quantum critical phenomena in condensed matter physics, with the typical
model being Lifshitz scalar field theoryLifshitz ; Chen:2009ka . In this
Hořava-Lifshitz theory, time and space will take different scaling behavior as
$\mathbf{x}\rightarrow b\mathbf{x},\;\;\;\;t\rightarrow b^{z}t,$ (1)
where $z$ is the dynamical critical exponent characterizing the anisotropy
between space and time. Due to the anisotropy, instead of diffeomorphism, we
have the so-called foliation-preserving diffeomorphism. The transformation is
now just
$\displaystyle t$ $\displaystyle\rightarrow$ $\displaystyle\tilde{t}(t),$
$\displaystyle x^{i}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{x^{i}}(x^{j},t).$ (2)
As a result, there is one more dynamical degree of freedom in Hořava-Lifshitz-
like gravity than in the usual general relativity. Such a degree of freedom
could play important role in UV physics, especially in early
cosmologyCai:2009dx ; Chen:2009jr . At IR, due to the emergence of new gauge
symmetry, this degree of freedom is not dynamical any more such that the
kinetic part of the theory recovers the one of the general relativity.
Since time direction plays a privileged role in the whole construction, it is
more convenient to work with ADM metric
$ds^{2}=-N^{2}dt^{2}+h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (3)
in which $N$ and $N_{i}$ are called “lapse” and “shift” variables
respectively.
Taking Hořava-Lifshitz gravity as a new gravitational theory, it is an
important issue to study its black hole solutions. In papers Lu2009 ;
Nastase2009 ; Kehagias:2009is ; AhmadGhodsi2009 ; Colgain:2009fe ; park2009 ;
RongCai-2009 ; Hatefi2009 , it was assumed that the metric of the black
solutions had the following Schwarzschild coordinates form
$ds^{2}=-N(r)^{2}dt_{S}^{2}+\frac{dr^{2}}{g(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (4)
From this metric ansatz, it was found that there were new black hole
solutions, even at IR. For example, in Kehagias:2009is , based on a modified
Hořava-Lifshitz -type action, an asymptotically flat solution with
$g=N^{2}=1+\omega r^{2}-\sqrt{r(\omega^{2}r^{3}+4\omega M)}$ (5)
was found. And paper RongCai-2009 find a new static charged black hole
solution in the IR region, paper Hatefi2009 had studied the extremal rotating
no-charged black hole solutions. However, in the above ansatz (4) the “lapse
function” $N(r)$ obviously breaks the “projectability condition” which means
that $N$ only is the function of $t$. This is introduced in Horava:2008ih ;
Horava:2009uw . For the metric of the form (4), we can work in the Painlevé-
Gullstrand coordinates by making a transformation
$dt_{S}=dt_{PG}-\frac{\sqrt{1-N^{2}}}{N^{2}}dr.$ (6)
Then the ansatz (4) becomes
$ds^{2}=-dt_{PG}^{2}+(dr+\sqrt{1-N^{2}}dt_{PG})^{2}+(\frac{1}{g}-\frac{1}{N^{2}})dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (7)
Comparing with the ADM metric, we find that $N(t_{PG})=1$ which is in accord
with the projectability condition and if $g=N^{2}$ we reach (3). For instance,
a Reissner-Nordstrom black hole after the transform (6) has the form
$ds^{2}=-dt_{PG}^{2}+(dr\pm\sqrt{\frac{2GM}{r}-\frac{Q^{2}}{r^{2}}}dt_{PG})^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (8)
So paper static_black_hole_of_HL search non-charge black hole solutions to
modified Hořava-Lifshitz gravity which was introduced by Visser_2009 with the
metric ansatz as
$ds^{2}=-N^{2}dt^{2}+\frac{1}{f(r)}(dr+N^{r}dt)(dr+N^{r}dt)+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (9)
where $N$ only is a function of $t$. Paper static_black_hole_of_HL found
maximally symmetric space solution with curvature $\Lambda_{W}$ and in the IR
region, and (A)dS-Schwarzschild black hole solution in Painlevé-Gullstrand
type coordinates was found. In the UV region, it found a de-Sitter space-time
solution.
Then papers Kiritsis_blackhole ; Capasso_blackhole search black hole
solutions with the same metric ansatz as (9), but $N$ only is the function of
$r$. They seem get some new black hole solutions. Actually the solutions of
paper Capasso_blackhole in the IR region is very similar as paper park2009 .
Furthermore, paper Setare2009 had studied the plane symmetric solutions in
Hořava-Lifshitz theory.
In the present work, we search static charged black hole solutions to Hořava-
Lifshitz gravity coupling with electromagnetic field. We work on the same
metric ansatz as (9). Whether the metric ansatz respects the projectability
condition is determined on whether $N$ only is the function of $t$. With the
projectability condition, we find (A)dS-Reissner-Nordstrom black hole solution
in Painlevé-Gullstrand type coordinates in the IR region and a de-Sitter
space-time solution in the UV region. Without the projectability condition, in
the IR region, we find a static charged black hole solution which is similar
as the results of RongCai-2009 . In the UV region, we find the same solution
as Lu2009 .
## II The Hořava-Lifshitz gravity
In this section, we give a brief review of Hořava-Lifshitz gravity. Using the
ADM formalism, the action of this Hořava-Lifshitz gravitational theory is
given byHorava:2008ih ; Horava:2009uw
$\displaystyle S_{HL}$ $\displaystyle=$ $\displaystyle\int
dtd^{3}\mathbf{x}(\mathcal{L}_{K}+\mathcal{L}_{V}),$
$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})\right\\},$ $\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda^{2}_{W})}{8(1-3\lambda)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}\right.$
(10)
$\displaystyle\left.-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\},$
where $\mathcal{L}_{K}$ is the kinetic term and $\mathcal{L}_{V}$ is the
potential term. If we consider the term which represents a “soft” violation of
the “detailed balance” condition in Horava:2009uw and add the term in the
action, The kinetic term $\mathcal{L}_{K}$ is the same, and the potential term
$\mathcal{L}_{V}$ becomes
$\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda^{2}_{W})}{8(1-3\lambda)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}\right.$
(11)
$\displaystyle\left.-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)+\frac{\Omega\kappa^{2}\mu^{2}}{8(3\lambda-1)}R\right\\}.$
The last term has been introduced in Horava:2009uw ; Nastase2009 ;
Kehagias:2009is . In the action, $\lambda,\kappa,\mu,\omega$,$\Lambda_{W}$ and
$\Omega$ are the coupling parameters, and $C_{ij}$ is the Cotton tensor
defined by
$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{j}_{l}-\frac{1}{4}R\delta^{j}_{l}\right).$
(12)
The study of the perturbations around the Minkowski vacuum shows that there is
ghost excitation when $\frac{1}{3}<\lambda<1$. This indicates that the theory
is only well-defined in the region $\lambda\leq\frac{1}{3}$ and $\lambda\geq
1$. Since the theory should be RG flow to IR with $\lambda=1$, we expect that
at UV, $\lambda>1$ to have a well-defined RG flow. At IR, $\lambda=1$, the
kinetic term recovers the one of standard general relativity. Comparing (II)
to the action of the general relativity in the ADM formalism, the speed of
light, the Newton’s constant and the cosmological constant emerge as
$\displaystyle
c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda_{W}}{1-3\lambda}},\hskip
12.91663ptG=\frac{\kappa^{2}}{32\pi c},\hskip
12.91663pt\Lambda=\frac{3}{2}\Lambda_{W}.$ (13)
It follows from (13) that for $\lambda>1/3$ ,the cosmological constant
$\Lambda_{W}$ has to be negative. It was noticed in Lu2009 that if we make an
analytic continuation of the parameters
$\mu\to i\mu,\hskip 17.22217pt\omega^{2}\to-i\omega^{2},$ (14)
the four-dimensional action (II) remains real as
$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})\right\\},$ $\displaystyle\mathcal{L}_{V}$ $\displaystyle=$
$\displaystyle\sqrt{h}N\left\\{\frac{\kappa^{2}\mu^{2}(\Lambda_{W}R-3\Lambda^{2}_{W})}{8(3\lambda-1)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(3\lambda-1)}R^{2}\right.$
(15)
$\displaystyle\left.+\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\}.$
In this case, the emergent speed of light becomes
$c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda_{W}}{3\lambda-1}}.$ (16)
The requirement that this speed be real implies that $\Lambda_{W}$ must be
positive for $\lambda>\frac{1}{3}$.
## III Electromagnetic Field in Hořava-Lifshitz Static Curved Spacetime
The electrodynamics in curved spacetime has been discussed many times in the
old days. Ellis(1973) first wrote down the Maxwell’s equations in $3+1$
congruence language. Later the Maxwell’s equations in $3+1$ form were fully
discussed by Thorne, Price, and Macdonald(1986). The Maxwell’s equations in a
curved spacetime could be written as
$\displaystyle\nabla_{\nu}F^{\mu\nu}=0,$ (18)
$\displaystyle\partial_{\mu}F_{\nu\rho}+\partial_{\nu}F_{\rho\mu}+\partial_{\rho}F_{\mu\nu}=0,$
where $F_{\mu\nu}$ is the antisymmetric electromagnetic tensor and
$F^{\mu\nu}=F_{\alpha\beta}g^{\alpha\mu}g^{\beta\nu}$. The electric field
$E_{i}$ and magnetic field $H_{i}$ are related to $F_{\mu\nu}$ as
$E_{i}=F_{ti},\;H_{i}=-\frac{\epsilon_{ijk}}{2}\sqrt{-g}F^{jk},$ (19)
where $E_{i}$ and $H_{i}$ are spatial vectors. The electromagnetic action in
the curved spacetime could be written in $3+1$ form
$S_{em}=\int
dtd^{3}\mathbf{x}\sqrt{h}Ng_{em}\left[F_{tk}F^{tk}+F_{kt}F^{kt}+F_{ij}F^{ij}\right],$
(20)
where $g_{em}$ is a constant. But the action at the Lifshitz point may be very
different. This has been discussed by HořavaHorava_Gugue_field , Chen and
Huang Chen_Huang_Lishitz_Point . In the paper Chen_Huang_Lishitz_Point , Chen
and Huang showed the action of Yang-Mills gauge field at the Lifshitz point in
the flat spacetime as
$S_{YM}=\frac{1}{2}\int
dtd^{d}\mathbf{x}\left[\frac{1}{g^{2}_{E}}Tr(E_{i}E_{i})-\sum_{J\geqslant
2}\mathcal{O}_{J}\star F^{J}\right],$ (21)
where
$\mathcal{O}_{J}=\frac{1}{g^{J}_{E}}\sum^{n_{J}}_{n=0}(-1)^{n}\frac{\lambda_{J,n}}{M^{2n+\frac{d+1}{2}J-d-1}}D^{2n}.$
(22)
Here $F$ and $D$ are the abbreviated denotation for $F_{ij}$ and $D_{k}$
respectively, and $\lambda_{J,n}$ are the the coupling with zero energy
dimension. Similarly $D^{2n}\star F^{J}$ also contains all possible
independent combinations of $D_{k}$ and $F_{ij}$. The action of Yang-Mills
theory at Lifshitz point in curved spacetime should be similar with action
show above. To the static charged black hole, $F_{tr}$ component of
$F_{\mu\nu}$ is not zero and it must be functions of $r$. If there are
independent magnetic charges in the world and the static black hole absorbs
some magnetic charges, the magnetic field component $H_{r}$ should not be
zero. From (19), $F^{\theta\phi}$ should not be zero. So the action of
electromagnetic field to static charged black hole will be reduced to
$S_{em}=\int dtd^{3}\mathbf{x}\mathcal{L}_{em}=\int
dtd^{3}\mathbf{x}\sqrt{h}N2g_{em}\left[F_{tr}F^{tr}+F_{\theta\phi}F^{\theta\phi}\right].$
(23)
From the equations (18) and (18) we get four independent equations
$\partial_{r}\left(\sqrt{h}NF^{tr}\right)=0.$ (24)
$\partial_{\theta}\left(\sqrt{h}NF^{\theta\phi}\right)=0,\;\;\;\partial_{\phi}\left(\sqrt{h}NF^{\theta\phi}\right)=0.$
(25)
$\partial_{r}F_{\theta\phi}+\partial_{\phi}F_{r\theta}+\partial_{\theta}F_{\phi
r}=0.$ (26)
## IV Static Charged Black Hole Solutions
As the discussion in the first paragraph, we now seek the static charged black
hole solutions with the metric ansatz
$ds^{2}=-N(t,r)^{2}dt^{2}+\frac{1}{f(r)}(dr+N^{r}dt)(dr+N^{r}dt)+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (27)
With this metric ansatz, from (24), $F^{tr}$ should satisfy the equation
$\partial_{r}\left(\frac{N}{\sqrt{f}}r^{2}F^{tr}\right)=0.$ (28)
The solution of this equation is
$F^{tr}=\frac{Q_{e}\sqrt{f}}{Nr^{2}},$ (29)
where $Q_{e}$ is an integration constant. From (25) and (26), $F^{\theta\phi}$
and $F_{\theta\phi}$ should satisfy
$\partial_{\theta}\left(\frac{N}{\sqrt{f}}r^{2}\sin{\theta}F^{\theta\phi}\right)=0,\;\partial_{\phi}\left(\frac{N}{\sqrt{f}}r^{2}\sin{\theta}F^{\theta\phi}\right)=0,\;\partial_{r}F_{\theta\phi}=0.$
(30)
The solution of the three equations are
$F^{\theta\phi}=\frac{Q_{m}}{r^{4}\sin{\theta}},\;F_{\theta\phi}=Q_{m}\sin{\theta},$
(31)
where $Q_{m}$ is an integration constant. So from the action (23), the
Lagrangian of electromagnetic field is
$\mathcal{L}_{em}=-2g_{em}\frac{N}{\sqrt{f}}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(32)
The whole action of electromagnetic field couples with Hořava-Lifshitz gravity
is $S=S_{HL}+S_{em}$. Substituting the metric ansatz (27) into the Lagrangians
(II) and (23), up to an overall scaling constant, we get
$\displaystyle\mathcal{L}_{K}=$
$\displaystyle\frac{1}{N\sqrt{f}}\left\\{(1-\lambda)r^{2}f^{2}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)^{2}+2(1-2\lambda)f^{2}N_{r}^{2}\right.$
(33) $\displaystyle\hskip 8.61108pt\left.-4\lambda
rf^{2}N_{r}\left(N^{{}^{\prime}}_{r}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)\right\\},$
$\displaystyle\mathcal{L}_{V}=$
$\displaystyle\frac{N}{\sqrt{f}}\left\\{2-3\Lambda_{W}r^{2}-2f-2rf^{{}^{\prime}}+\frac{\lambda-1}{2\Lambda_{W}}f^{{}^{\prime}2}\right.$
(34) $\displaystyle\hskip
8.61108pt\left.-\frac{2\lambda(f-1)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{W}r^{2}}\right\\},$
$\displaystyle\mathcal{L}_{em}=$
$\displaystyle-2\tilde{g}_{em}\frac{N}{\sqrt{f}}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(35)
where $N_{r}=N^{r}/f$ and ′ means derivative to $r$. Here we have set $c=1$
and $\tilde{g}_{em}=\kappa^{2}g_{em}/2$. The full Lagrangian is
$\mathcal{L}=\mathcal{L}_{K}+\mathcal{L}_{V}+\mathcal{L}_{em}$. By varying the
action with respect to the functions $N_{r}$ , $f$ and $N(t)$, we obtain three
equations of motions,
$\displaystyle 0=2(1-\lambda)r^{2}f^{2}\frac{1}{N}$
$\displaystyle\left\\{N_{r}^{{}^{\prime\prime}}+\frac{f^{{}^{\prime\prime}}}{2f}N_{r}+\frac{3}{2}\frac{f^{{}^{\prime}}}{f}N_{r}^{{}^{\prime}}+2\frac{N_{r}^{{}^{\prime}}}{r}+\frac{1-2\lambda}{1-\lambda}\frac{f^{{}^{\prime}}}{f}\frac{N_{r}}{r}-2\frac{N_{r}}{r^{2}}\right.$
(36)
$\displaystyle\left.-\frac{N^{{}^{\prime}}}{N}\left(N_{r}^{{}^{\prime}}+N_{r}\frac{f^{{}^{\prime}}}{2f}-\frac{2\lambda}{1-\lambda}\frac{N_{r}}{r}\right)\right\\},$
$\displaystyle 0$
$\displaystyle=-\left(\frac{f^{{}^{\prime}}}{2f}\frac{1}{N}+\frac{N^{{}^{\prime}}}{N^{2}}\right)\left\\{(1-\lambda)r^{2}fN_{r}\left(N_{r}^{{}^{\prime}}+N_{r}\frac{f^{{}^{\prime}}}{2f}\right)-2\lambda
rfN_{r}^{2}\right\\}$ (37)
$\displaystyle+\left(N^{{}^{\prime}}-\frac{f^{{}^{\prime}}}{2f}N\right)\left\\{-2r+\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime}}-\frac{2\lambda(f-1)}{\Lambda_{W}r}\right\\}+N\left\\{\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime\prime}}+\frac{2(1-\lambda)(f-1)}{\Lambda_{W}r^{2}}\right\\}$
$\displaystyle+\frac{1}{N}\left\\{(1-\lambda)r^{2}fN_{r}N_{r}^{{}^{\prime\prime}}+\frac{1}{2}(1-\lambda)r^{2}f^{{}^{\prime\prime}}N_{r}^{2}-(1-\lambda)r^{2}fN_{r}^{{}^{\prime}2}+(1-\lambda)r^{2}f^{{}^{\prime}}N_{r}N_{r}^{{}^{\prime}}\right.$
$\displaystyle+\left.2(1+\lambda)rfN_{r}N_{r}^{{}^{\prime}}+(1-\lambda)rf^{{}^{\prime}}N_{r}^{2}+(6\lambda-4)fN_{r}^{2}\right\\}+\frac{1}{2\sqrt{f}}\left\\{\mathcal{L}_{K}+\mathcal{L}_{V}\right\\}-\tilde{g}_{em}\frac{N}{f}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}},$
$0=-\mathcal{L}_{K}+\mathcal{L}_{V}+\mathcal{L}_{em}.$ (38)
### IV.1 Solutions with Projectability Condition
For the projectability condition, lapse function $N$ only is a function of
$t$, $N^{{}^{\prime}}$ is zero and we can set $N(t)=1$ with the time
rescaling. Obviously for all $\lambda$, $N_{r}=0$ is a solution of (36). In
this case, we assume ansatz $f(r)=1+yr^{2}$, just when $Q=0$, from (37),(38),
we get quadratic equations of $y$,
$\displaystyle y^{2}-2\Lambda_{W}y-3\Lambda_{W}^{2}=0,$ (39) $\displaystyle
y^{2}+2\Lambda_{W}y+\Lambda_{W}^{2}=0.$ (40)
Their solution is $y=-\Lambda_{W}$. This solutions corresponds to a maximally
symmetric space with curvature $\Lambda_{W}$. This result has been shown in
paper static_black_hole_of_HL .
In the IR region where $\lambda=1$, the equation (36) is reduced to
$\frac{f^{{}^{\prime}}}{f}\frac{N_{r}}{r}=0.$ (41)
Its $N_{r}=0$ solution has been discussed above. It has another solution as
$f=constant$. when $f$ is a constant, we could get the same equation from
(37),(38) as
$(N_{r}^{2})^{\prime}+\frac{N_{r}^{2}}{r}+\frac{1}{2f^{2}}\left\\{-3\Lambda_{W}r+\frac{2(1-f)}{r}+\frac{(1-f)^{2}}{\Lambda_{W}r^{3}}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{3}}\right\\}=0.$
(42)
The solutions of this equation are
$N_{r}=\pm\frac{1}{f}\sqrt{\frac{M}{r}+\frac{\Lambda_{W}}{2}r^{2}+(f-1)+\frac{(1-f)^{2}}{2\Lambda_{W}r^{2}}-\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}}.$
(43)
Especially when $f=1$, the solution reduces to
$N_{r}=\pm\sqrt{\frac{M}{r}+\frac{\Lambda_{W}}{2}r^{2}-\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}}.$
(44)
It is a (A)dS-Reissner-Nordstrom black hole written in Painlevé-Gullstrand
type coordinates. Especially when $Q_{e}=Q_{m}=0$, it is (A)dS-Schwarzschild
black hole which has been shown in static_black_hole_of_HL . The electric
field of the black hole (44) is $E_{r}=F_{tr}=(-Q_{e})/r^{2}$. Its charge is
$Q_{BH}=\frac{1}{4\pi}\int_{S}\vec{E}\cdot
d\vec{\sigma}=\frac{1}{4\pi}E_{r}\cdot 4\pi r^{2}=-Q_{e},$ (45)
where $S$ is a closed surface everywhere with the same $r$ and $d\vec{\sigma}$
is surface integral element. We should choose $\tilde{g}_{em}=1$ when the
solution (44) is a charged black hole with charge $\pm Q_{e}$.
In the UV region where $\lambda\neq 1$, when $f$ is a constant, from (36),(37)
and (38), we get three equations,
$\displaystyle 0=$ $\displaystyle
N_{r}^{{}^{\prime\prime}}+2\frac{N_{r}^{{}^{\prime}}}{r}-2\frac{N_{r}}{r^{2}},$
(46) $\displaystyle 0=$
$\displaystyle(1-\lambda)r^{2}N_{r}^{{}^{\prime}2}-4\lambda
rN_{r}N_{r}^{{}^{\prime}}+2(1-2\lambda)N_{r}^{2}-\frac{4(1-\lambda)(f-1)}{\Lambda_{W}r^{2}f}$
(47)
$\displaystyle-\frac{1}{f^{2}}\left\\{2(1-f)-3\Lambda_{W}r^{2}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{2}r^{2}}\right\\}+2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{f^{2}r^{2}},$
$\displaystyle 0=$ $\displaystyle(1-\lambda)r^{2}N_{r}^{{}^{\prime}2}-4\lambda
rN_{r}N_{r}^{{}^{\prime}}+2(1-2\lambda)N_{r}^{2}$ (48)
$\displaystyle-\frac{1}{f^{2}}\left\\{2(1-f)-3\Lambda_{W}r^{2}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{2}r^{2}}\right\\}+2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{f^{2}r^{2}}.$
Just when $f=1$ and $Q_{e}^{2}-Q_{m}^{2}=0$, they have a solution as
$N_{r}=\pm\;\sqrt{\frac{\Lambda_{W}}{(3\lambda-1)}}r.$ (49)
This solution actually describes the same de-Sitter space-time. One easy way
to see this point is to change inversely into the Schwarzschild coordinates.
This result is the same as paper static_black_hole_of_HL .
### IV.2 Solutions without Projectability Condition
Without projectability condition, $N$ is function of $r$, $N^{\prime}$ isn’t
zero. In the IR region where $\lambda=1$, the equation (36) is reduced to
$\left(2\frac{N^{{}^{\prime}}}{N}-\frac{f^{{}^{\prime}}}{f}\right)\frac{N_{r}}{r}=0.$
(50)
Its solutions are $N_{r}=0$ or $N^{2}=f$. In the case $N_{r}=0$, with the same
discussion above, we find a solution
$f=1-\Lambda_{W}r^{2},$ (51)
and the function $N(r)$ is unconstrained. This result is the same as paper
Lu2009 .
When $\lambda=1$, in the case $N^{2}=f$, from (37),(38), we get one same
equation,
$\displaystyle
0=2\left[rfN_{r}^{2}\right]^{{}^{\prime}}+\left\\{2(1-f)-3\Lambda_{W}r^{2}-2rf^{{}^{\prime}}+\frac{2(1-f)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(f-1)^{2}}{\Lambda_{W}r^{2}}\right\\}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(52)
This equation has a solution as
$N_{r}^{2}=\frac{\beta}{rf},\;f=1-\Lambda_{W}r^{2}-\sqrt{cr+2\tilde{g}_{em}\Lambda_{W}\left(Q_{e}^{2}-Q_{m}^{2}\right)},$
(53)
Where $\beta,c$ are integration constants. If we reconsider the action (11)
which contains the “soft” violation term, we could get a equation similar as
(52) as
$\displaystyle
0=2\left[rfN_{r}^{2}\right]^{{}^{\prime}}+\left\\{-2\frac{\Omega-\Lambda_{W}}{\Lambda_{W}}(1-f-rf^{{}^{\prime}})-3\Lambda_{W}r^{2}+\frac{2(1-f)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(f-1)^{2}}{\Lambda_{W}r^{2}}\right\\}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}}.$
(54)
Its solution is
$N_{r}^{2}=\frac{\beta}{rf},\;f=1+\left(\Omega-\Lambda_{W}\right)r^{2}-\sqrt{\Omega(\Omega-2\Lambda_{W})r^{4}+cr+2\tilde{g}_{em}\Lambda_{W}\left(Q_{e}^{2}-Q_{m}^{2}\right)},$
(55)
Where $\beta,c$ are integration constants. When $Q_{m}=0$, this solution is
similar as paper RongCai-2009 and When $Q_{e}^{2}-Q_{m}^{2}=0$ it is the
similar as park2009 . The solutions (53),(55) are two especial static charged
black hole solutions.
In the UV region where $\lambda\neq 1$, when $N_{r}=0$, from (37).(38), we get
two equations,
$\displaystyle 0=$ $\displaystyle
2-3\Lambda_{W}r^{2}-2f-2rf^{{}^{\prime}}+\frac{\lambda-1}{2\Lambda_{W}}f^{{}^{\prime}2}-\frac{2\lambda(f-1)}{\Lambda_{W}r}f^{{}^{\prime}}+\frac{(2\lambda-1)(f-1)^{2}}{\Lambda_{W}r^{2}}-2\tilde{g}_{em}\frac{Q_{e}^{2}-Q_{m}^{2}}{r^{2}},$
(56) $\displaystyle 0=$
$\displaystyle\left(N^{{}^{\prime}}-\frac{f^{{}^{\prime}}}{2f}N\right)\left\\{-2r+\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime}}-\frac{2\lambda(f-1)}{\Lambda_{W}r}\right\\}+N\left\\{\frac{\lambda-1}{\Lambda_{W}}f^{{}^{\prime\prime}}+\frac{2(1-\lambda)(f-1)}{\Lambda_{W}r^{2}}\right\\}$
(57)
They have two new solutions as
$Q_{e}^{2}-Q_{m}^{2}=0,\;f=1-\Lambda_{W}r^{2}-\alpha
r^{\frac{2\lambda\pm\sqrt{6\lambda-2}}{\lambda-1}},\;N=ar^{-\frac{1+3\lambda\pm
2\sqrt{6\lambda-2}}{\lambda-1}}\sqrt{f},$ (58)
where $\alpha,a$ are constants. The solutions has been got by paper Lu2009 .
## Acknowledgments
The work was partially supported by NSFC Grant No.10535060, 10775002, 10975005
and RFDP. I would like to thank Bin Chen for drawing my attention to Hořava-
Lifshitz gravity and giving me many useful suggestions.
## References
* (1) P. Horava, “Membranes at Quantum Criticality,” JHEP 0903, 020 (2009) [arXiv:0812.4287 [hep-th]].
* (2) P. Horava, “Quantum Gravity at a Lifshitz Point,” Phys. Rev. D 79, 084008 (2009) [arXiv:0901.3775 [hep-th]].
* (3) E.M. Lifshitz, “On the Theory of Second-Order Phase Transitions I & II”, Zh. Eksp. Teor. Fiz 11 (1941)255 & 269\.
* (4) B. Chen and Q. G. Huang, “Field Theory at a Lifshitz Point,” arXiv:0904.4565 [hep-th].
* (5) R. G. Cai, B. Hu and H. B. Zhang, “Dynamical Scalar Degree of Freedom in Horava-Lifshitz Gravity,” Phys. Rev. D 80, 041501 (2009) [arXiv:0905.0255 [hep-th]].
* (6) B. Chen, S. Pi and J. Z. Tang, “Scale Invariant Power Spectrum in Hořava-Lifshitz Cosmology without Matter,” JCAP 08 (2009)007, arXiv:0905.2300 [hep-th].
* (7) M. Li and Y. Pang, “A Trouble with Hořava-Lifshitz Gravity,” arXiv:0905.2751 [hep-th].
* (8) H. Lu, J. Mei and C. N. Pope, “Solutions to Horava Gravity,” arXiv:0904.1595 [hep-th].
* (9) Horatiu Nastase, “On IR solutions in Horava gravity theories,” arXiv:0904.3604 [hep-th]
* (10) A. Kehagias and K. Sfetsos, “The black hole and FRW geometries of non-relativistic gravity,” Phys. Lett. B 678, 123 (2009) [arXiv:0905.0477 [hep-th]].
* (11) Ahmad Ghodsi , “Toroidal solutions in Horava Gravity,” arXiv:0905.0836 [hep-th].
* (12) Mu-in Park, “The Black Hole and Cosmological Solutions in IR modified Horava Gravity,” arXiv:0905.4480 [hep-th]
* (13) E. O. Colgain and H. Yavartanoo, “Dyonic solution of Horava-Lifshitz Gravity,” JHEP 0908, 021 (2009) [arXiv:0904.4357 [hep-th]].
* (14) Rong-Gen Cai, Li-Ming Cao, Nobuyoshi Ohta, “Topological Black Holes in Horava-Lifshitz Gravity,” Phys. Rev. D 80, 024003 (2009) [arXiv:0904.3670 [hep-th]]
* (15) Ahmad Ghodsi and Ehsan Hatefi, “Extremal rotating solutions in Horava Gravity,” arXiv:0906.1237 [hep-th].
* (16) T. Harko, Z. Kovacs and F. S. N. Lobo, “Testing Hořava-Lifshitz gravity using thin accretion disk properties,” Phys. Rev. D 80, 044021 (2009) [arXiv:0907.1449 [gr-qc]].
T. Harko, Z. Kovacs and F. S. N. Lobo, “Solar system tests of Hořava-Lifshitz
gravity,” arXiv:0908.2874 [gr-qc].
L. Iorio and M. L. Ruggiero, “Horava-Lifshitz gravity and Solar System orbital
motions,” arXiv:0909.2562 [gr-qc].
* (17) Thomas P. Sotiriou, Matt Visser, Silke Weinfurtner, “Phenomenologically viable Lorentz-violating quantum gravity”, Phys.Rev.Lett.102:251601,2009, arXiv:0904.4464 [hep-th];“Quantum gravity without Lorentz invariance,” arXiv:0905.2798 [hep-th].
* (18) Painlevé P. La mécanique classique el la theorie de la relativité(Classical mechanics of the theory of relativity).C. R. Acad. Sci. (Paris), 173 (1921), 677 C680.
* (19) Gullstrand A. Allegemeine l$\ddot{o}$sung des statischen eink$\ddot{o}$rper-problems in der einsteinshen gravitations theorie (General solution for static onebody problems in Einstein s theory of gravity)._Arkiv. Mat. Astron. Fys._ ,16(8) (1922), 1-15.
* (20) Lema$\hat{i}$tre G. L’nivers en expansion (The universe in expansion)._Ann. Soc. Sci. (Bruxelles)_ ,A53 (1933), 51-85.
* (21) Hawking S W and Israel S W (editors)._Three hundred years of gravitation_. Cambridge University Press, England (1987). See especially the discussion on page 234.
* (22) A.A.Kocharyan, “Is nonrelativistic gravity possible?” Phys. Rev. D 80, 024026 (2009) [arXiv:0905.4204 [hep-th]]
* (23) P. Hořava, “Quantum Criticality and Yang-Mills Gauge Theory,” arXiv:0811.2217 [hep-th].
* (24) Bin Chen, Qing-Guo Huang, “Quantum Criticality and Yang-Mills Gauge Theory,” arXiv: 0904.4565 [hep-th].
* (25) Bin Chen, Jin-Zhang Tang, “Static Charged Black Hole to modified Hořava-Lifshitz Gravity with Projectability Condition,” arXiv:0909.4127 [hep-th]
* (26) Elias Kiritsis, Georgios Kofinas, “On Horava-Lifshitz Black Holes,” arXiv:0910.5487[hep-th]
* (27) Dario Capasso, Alexios P. Polychronakos, “General static spherically symmetric solutions in Horava gravity,” arXiv:0911.1535[hep-th]
* (28) M. R. Setare, D. Momeni “Plane symmetric solutions in Horava-Lifshitz theory,” arXiv:0911.1877 [hep-th].
|
arxiv-papers
| 2009-11-19T17:53:49 |
2024-09-04T02:49:06.581511
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jin-Zhang Tang",
"submitter": "Jinzhang Tang",
"url": "https://arxiv.org/abs/0911.3849"
}
|
0911.3895
|
# Strong approximations in a charged-polymer model
Yueyun Hu
Université Paris 13 Davar Khoshnevisan
University of Utah
(November 19, 2009)
###### Abstract
We study the large-time behavior of the charged-polymer Hamiltonian $H_{n}$ of
Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian
case], using strong approximations to Brownian motion. Our results imply,
among other things, that in one dimension the process $\\{H_{[nt]}\\}_{0\leq
t\leq 1}$ behaves like a Brownian motion, time-changed by the intersection
local-time process of an independent Brownian motion. Chung-type LILs are also
discussed.
Keywords: Charged polymers, strong approximation, local time.
AMS 2000 subject classification: Primary: 60K35; Secondary: 60K37.
Running Title: Polymer measures, strong approximations.
## 1 Introduction
Consider a sequence $\\{q_{i}\\}_{i=1}^{\infty}$ of independent, identicallly-
distributed mean-zero random variables, and let
$S:=\\{S_{i}\\}_{i=0}^{\infty}$ denote an independent simple random walk on
${\mathbf{Z}}^{d}$ starting from $0$. For $n\geq 1$, define
$H_{n}:=\mathop{\sum\sum}_{1\leq i<j\leq
n}q_{i}q_{j}\mathbf{1}_{\\{S_{i}=S_{j}\\}};$ (1.1)
this is the Hamiltonian of a socalled “charged polymer model.” See Kantor and
Kadar [12] in the case that the $q_{i}$’s are Bernoulli, and Derrida,
Griffiths, and Higgs [8] for the case of Gaussian random variables. Roughly
speaking, $q_{1},q_{2},\ldots$ are random charges that are placed on a polymer
path modeled by the trajectories of $S$; and one can construct a Gibbs-type
polymer measure from the Hamiltonian $H_{n}$.
We follow Chen [4] (LIL and moderate deviations), Chen and Khoshnevisan [5]
(comparison between $H_{n}$ and the random walk in random scenery model), and
Asselah [1] (large deviations in high dimensional case), and continue the
analysis of the Hamiltonian $H_{n}$. We assume here and in the sequel that
${\mathrm{E}}(q_{1}^{2})=1\ \text{and}\
{\mathrm{E}}\left(|q_{1}|^{p(d)}\right)<\infty\ \text{ where }\
p(d):=\begin{cases}6&\text{if $d=1$},\\\ 4&\text{if $d\geq 2$}.\end{cases}$
(1.2)
###### Theorem 1.1.
On a possibly-enlarged probability space, we can define a version of
$\\{H_{n}\\}_{n=1}^{\infty}$ and a one-dimensional Brownian motion
$\\{\gamma(t)\\}_{t\geq 0}$ such that the following holds almost surely:
$H_{n}=\begin{cases}\displaystyle\frac{1}{\sqrt{2}}\,\gamma\left(\int_{-\infty}^{\infty}(\ell_{n}^{x})^{2}\,{\rm
d}x\right)+o(n^{\frac{3}{4}-\epsilon})&\text{if $d=1$ and $0<\epsilon<{1\over
24}$},\\\\[8.53581pt] \displaystyle\frac{1}{\sqrt{2\pi}}\,\gamma(n\log
n)+O(n^{\frac{1}{2}}\log\log n)&\text{if $d=2$},\\\\[8.53581pt]
\sqrt{\kappa}\,\gamma(n)+o(n^{\frac{1}{2}-\epsilon})&\text{if $d\geq 3$ and
$0<\epsilon<{1\over 8}$},\end{cases}$ (1.3)
where $\\{\ell_{t}^{x}\\}_{t\geq 0,x\in{\mathbf{R}}}$ denotes the local times
of a linear Brownian motion $B$ independent of $\gamma$, and
$\kappa:=\sum_{k=1}^{\infty}{\mathrm{P}}\\{S_{k}=0\\}$.
It was shown in [5] that when $d=1$ the distribution of $H_{n}$ converges,
after normalization, to the “random walk in random scenery.” The preceding
shows that the stochastic process $\\{H_{[nt]}\\}_{0\leq t\leq 1}$ does not
converge weakly to the random walk in random scenery; rather, we have the
following consequence of Brownian scaling for all $T>0$: As $n\to\infty$,
$\left\\{\frac{H_{[nt]}}{n^{3/4}}\right\\}_{0\leq t\leq T}\mathop{\hbox
spread10.0pt{\rightarrowfill}}\limits^{D([0,T])}\,\left\\{\frac{1}{\sqrt{2}}\,\gamma\left(\int_{-\infty}^{\infty}(\ell^{x}_{t})^{2}\,{\rm
d}x\right)\right\\}_{0\leq t\leq T}.$ (1.4)
With a little bit more effort, we can also obtain strong limit theorems. Let
us state the following counterpart to the LILs of Chen [4], as it appears to
have novel content.
###### Theorem 1.2.
Almost surely: (i) If $d=1$, then
$\liminf_{n\to\infty}\left({\log\log n\over n}\right)^{3/4}\max_{0\leq k\leq
n}|H_{k}|={(a^{*})^{3/4}\frac{\pi}{4}},$
where $a^{*}=2.189\pm 0.0001$ is a numerical constant [11, (0.6)];
(ii) If $d=2$, then
$\liminf_{n\to\infty}\sqrt{\frac{\log\log n}{n\log n}}\max_{0\leq k\leq
n}|H_{k}|={\frac{\sqrt{\pi}}{4}};$
(iii) If $d\geq 3$, then
$\liminf_{n\to\infty}\sqrt{\frac{\log\log n}{n}}\max_{0\leq k\leq
n}|H_{k}|=\pi\sqrt{\frac{\kappa}{8}},$
where $\kappa$ was defined in Theorem 1.1.
Theorems 1.1 and 1.2 are proved respectively in Sections 2 and 3.
## 2 Proof of Theorem 1.1
Let $W$ be a one-dimensional Brownian motion starting from $0$. By the
Skorohod embedding theorem, there exists a sequence of stopping times
$\\{T_{n}\\}_{n=1}^{\infty}$ such that $\\{T_{n}-T_{n-1}\\}_{n=1}^{\infty}$
(with $T_{0}=0$) are i.i.d., and:
$\begin{split}&{\mathrm{E}}(T_{1})={\mathrm{E}}(q_{1}^{2})=1,\quad\text{Var}(T_{1})\leq\text{const}\cdot{\mathrm{E}}(q_{1}^{4})<\infty,\quad\text{and}\\\
&\quad\left\\{W(T_{n})-W(T_{n-1})\right\\}_{n=1}^{\infty}\,{\buildrel{\rm
law}\over{=}}\,\\{q_{n}\\}_{n=1}^{\infty}.\end{split}$ (2.1)
Throughout this paper, we take the following special construction of the
charges $\\{q_{i}\\}_{i=1}^{\infty}$:
$q_{n}:=W(T_{n})-W(T_{n-1})\qquad\text{for $n\geq 1$}.$ (2.2)
Next, we describe how we choose a special construction of the random walk $S$,
depending on $d$.
If $d=1$, then on a possibly-enlarged probability space let $B$ be another
one-dimensional Brownian motion, independent of $W$. By using a theorem of
Révész [14], we may construct a one-dimensional simple symmetric random walk
$\\{S_{i}\\}_{i=1}^{\infty}$ from $B$ such that almost surely,
$\sup_{x\in{\mathbf{Z}}}\left|L_{n}^{x}-\ell_{n}^{x}\right|=n^{\frac{1}{4}+o(1)}\quad\text{as
$n\to\infty$,\ where}\ L_{n}^{x}:=\sum_{i=1}^{n}\mathbf{1}_{\\{S_{i}=x\\}},$
(2.3)
and $\ell_{n}^{x}$ denotes the local times of $B$ at $x$ up to time $n$.
If $d\geq 2$, then we just choose an independent simple symmetric random walk
$\\{S_{n}\\}_{n=1}^{\infty}$, after enlarging the probability space, if we
need to.
Now we define the Hamiltonians $\\{H_{n}\\}_{n=1}^{\infty}$ via the preceding
constructions of $\\{q_{i}\\}_{i=1}^{\infty}$ and
$\\{S_{n}\\}_{n=1}^{\infty}$. That is,
$\begin{split}H_{n}&=\mathop{\sum\sum}_{1\leq i<j\leq
n}(W(T_{i})-W(T_{i-1}))(W(T_{j})-W(T_{j-1}))\mathbf{1}_{(S_{i}=S_{j})}\\\
&=\int_{0}^{T_{n}}G_{n}\,{\rm d}W,\end{split}$ (2.4)
where, for all integers $n\geq 1$ and reals $s\geq 0$,
$G_{n}(s):=\mathop{\sum\sum}_{1\leq i<j\leq
n}\mathbf{1}_{(S_{i}=S_{j})}(W(T_{i})-W(T_{i-1}))\mathbf{1}_{(T_{j-1}\leq
s<T_{j})}.$ (2.5)
By the Dambis, Dubins–Schwarz representation theorem [15, Theorem 1.6, p.
170], after possibly enlarging the underlying probability space, we can find a
one-dimensional Brownian motion $\gamma$ such that $\int_{0}^{t}G_{n}\,{\rm
d}W$ is equal to $\gamma(\int_{0}^{t}|G_{n}(s)|^{2}\,{\rm d}s)$ for $t\geq 0$.
We stress the fact that if $d=1$, then $\gamma$ is independent of $B$. This is
so, because the bracket between the two continuous martingales vanishes:
$\langle\int_{0}^{\bullet}G_{n}\,{\rm d}W\,,\,B\rangle_{t}=0$ for $t\geq 0$.
Consequently, the following holds for all $n\geq 1$: Almost surely,
$H_{n}=\gamma(\Xi_{n}),\quad\text{where}\quad\Xi_{n}:=\int_{0}^{T_{n}}|G_{n}(s)|^{2}\,{\rm
d}s.$ (2.6)
###### Proposition 2.1.
The following holds almost surely:
$\begin{split}\Xi_{n}=\begin{cases}\displaystyle\dfrac{1}{2}\int_{-\infty}^{\infty}(\ell_{n}^{x})^{2}\,{\rm
d}x+O(n^{\frac{3}{2}-\epsilon})&\text{if $d=1$ and
$0<\epsilon<\frac{1}{12}$},\\\\[5.69054pt] \dfrac{1}{2\pi}\,n\log
n+O(n\log\log n)&\text{if $d=2$},\\\\[5.69054pt] \kappa
n+O(n^{1-\epsilon})&\text{if $d\geq 3$ and
$0<\epsilon<\frac{1}{4}$}.\end{cases}\end{split}$ (2.7)
We prove this proposition later. First, we show that in case $d=1$, the
preceding proposition estimates $\Xi_{n}$ correctly to leading term.
###### Lemma 2.2.
If $d=1$, then a.s., $\int_{-\infty}^{\infty}(\ell_{n}^{x})^{2}\,{\rm
d}x=n^{\frac{3}{2}+o(1)}$ as $n\to\infty$.
###### Proof.
This is well known; we include a proof for the sake of completeness.
Because $\int_{-\infty}^{\infty}\ell_{n}^{x}\,{\rm d}x=n$, we have
$\int_{-\infty}^{\infty}(\ell_{n}^{x})^{2}\,{\rm d}x\leq
n\sup_{-\infty<x<\infty}\ell_{n}^{x}$, and this is $n^{\frac{3}{2}+o(1)}$
[13]. For the converse bound we apply the Cauchy–Schwarz inequality to find
that $n^{2}=(\int_{-\infty}^{\infty}\ell_{n}^{x}\,{\rm
d}x)^{2}\leq\int_{-\infty}^{\infty}(\ell_{n}^{x})^{2}\,{\rm
d}x\cdot\text{Osc}_{[0,n]}B,$ where
$\text{Osc}_{[0,n]}B:=\sup_{[0,n]}B-\inf_{[0,n]}B=n^{\frac{1}{2}+o(1)}$ by
Khintchine’s LIL. This completes the proof. $\Box$
Let us complete the proof of Theorem 1.1, first assuming Proposition 2.1. That
proposition will then be proved subsequently.
###### Proof of Theorem 1.1.
We shall consider only the case $d=1$; the cases $d=2$ and $d\geq 3$ are
proved similarly. We apply the Csörgő–Révész modulus of continuity of Brownian
motion [7, Theorem 1.2.1] to $H_{n}=\gamma(\Xi_{n})$—see (2.6)—with the
changes of variables, $t=n^{\frac{3}{2}+o(1)}$ and
$a(t)=n^{\frac{3}{2}-\epsilon}$; then apply Lemma 2.2 to see that
$|\gamma(\Xi_{n})-\gamma(\frac{1}{2}\int_{-\infty}^{\infty}(\ell_{n}^{x})^{2}\,{\rm
d}x)|=O(n^{\frac{3}{4}-\epsilon})$ as $n\to\infty$ a.s. $\Box$
###### Lemma 2.3.
The following holds almost surely:
$\begin{split}\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}=\begin{cases}\displaystyle\dfrac{1}{2}\int_{-\infty}^{\infty}(\ell_{n}^{x})^{2}\,{\rm
d}x+n^{\frac{5}{4}+o(1)}&\text{if $d=1$},\\\\[5.69054pt]
\dfrac{1}{2\pi}\,n\log n+O(n\log\log n)&\text{if $d=2$},\\\\[5.69054pt] \kappa
n+n^{\frac{1}{2}+o(1)}&\text{if $d\geq 3$}.\end{cases}\end{split}$ (2.8)
###### Proof.
In the case that $d=2$, this result follows from Bass, Chen and Rosen [2]; and
in the case $d\geq 3$, from Chen [4, Theorem 5.2]. Therefore, we need to only
check the case $d=1$.
We begin by writing $\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}=\frac{1}{2}\sum_{x\in{\mathbf{Z}}}(L_{n}^{x})^{2}-\frac{n}{2}$.
According to Bass and Griffin [3, Lemma 5.3],
$\sup_{x\in{\mathbf{Z}}}\,\sup_{y\in[x,x+1]}|\ell_{n}^{x}-\ell_{n}^{y}|=n^{\frac{1}{4}+o(1)}$
as $n\to\infty$ a.s. This and (2.3) together imply that
$|L_{n}^{x}-\ell_{n}^{y}|=n^{\frac{1}{4}+o(1)}$ uniformly over all
$y\in[x\,,x+1]$ and $x\in{\mathbf{Z}}$ [a.s.], whence
$\begin{split}\left|\sum_{x\in{\mathbf{Z}}}(L_{n}^{x})^{2}-\int_{-\infty}^{\infty}(\ell^{y}_{n})^{2}\,{\rm
d}y\right|&\leq
n^{\frac{1}{4}+o(1)}\cdot\sum_{x\in{\mathbf{Z}}}\int_{x}^{x+1}(L_{n}^{x}+\ell^{y}_{n})\,{\rm
d}y.\end{split}$ (2.9)
Since the latter sum is equal to $2n$, the lemma follows. $\Box$
###### Lemma 2.4.
The following holds a.s.: As $n\to\infty$,
$\begin{split}\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}\left(q_{i}^{2}-1\right)=\begin{cases}n^{\frac{7}{6}+o(1)}&\text{if
$d=1$},\\\\[5.69054pt] n^{\frac{3}{4}+o(1)}&\text{if $d\geq
2$}.\end{cases}\end{split}$ (2.10)
###### Proof.
We can let $M_{n}$ denote the double sum in the lemma, and check directly that
$M_{n}=\sum_{1\leq i\leq n-1}(L^{S_{i}}_{n}-L^{S_{i}}_{i})(q_{i}^{2}-1)$. Let
$\mathcal{S}$ denote the $\sigma$-algebra generated by the entire process $S$.
Then, conditionally on $\mathcal{S}$, each $M_{n}$ is a sum of independent
random variables. By Burkholder’s inequality [10, Theorem 2.10, p. 34], for
all even integers $p\geq 2$,
${\mathrm{E}}\left(|M_{n}|^{p}\right)\leq\text{const}\cdot{\mathrm{E}}\left(\left|\sum_{1\leq
i\leq
n-1}\left(L_{n}^{S_{i}}-L_{i}^{S_{i}}\right)^{2}(q_{i}^{2}-1)^{2}\right|^{p/2}\right).$
(2.11)
According to the generalized Hölder inequality,
${\mathrm{E}}\left(\prod_{k=1}^{p/2}(q_{i_{k}}^{2}-1)^{2}\right)\leq\prod_{k=1}^{p/2}\left\\{{\mathrm{E}}\left(\left|q_{i_{k}}^{2}-1\right|^{p}\right)\right\\}^{2/p}={\mathrm{E}}\left(|q_{1}^{2}-1|^{p}\right).$
(2.12)
Another application of the generalized Hölder inequality, together with an
appeal to the Markov property, yields
$\begin{split}{\mathrm{E}}\left(\prod_{k=1}^{p/2}\left(L_{n}^{S_{i_{k}}}-L_{i_{k}}^{S_{i_{k}}}\right)^{2}\right)&\leq\prod_{k=1}^{p/2}\left\\{{\mathrm{E}}\left(\left|L_{n}^{S_{i_{k}}}-L_{i_{k}}^{S_{i_{k}}}\right|^{p}\right)\right\\}^{2/p}\\\
&=\prod_{k=1}^{p/2}\left\\{{\mathrm{E}}\left(\left|L_{n-i_{k}}^{0}\right|^{p}\right)\right\\}^{2/p}.\end{split}$
(2.13)
Therefore, we can apply the local-limit theorem to find that
$\begin{split}{\mathrm{E}}\left(|M_{n}|^{p}\right)&\leq\text{const}\cdot\begin{cases}n^{p}&\text{if
$d=1$},\\\ n^{\frac{p}{2}+o(1)}&\text{if $d\geq 2$}.\end{cases}\end{split}$
(2.14)
The lemma follows from this and the Borel–Cantelli lemma. $\Box$
###### Lemma 2.5.
The following holds almost surely: As $n\to\infty$,
$\begin{split}\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}q_{i}^{2}\left(T_{k}-T_{k-1}-1\right)=\begin{cases}n^{1+o(1)}&\text{if
$d=1$},\\\\[5.69054pt] n^{\frac{1}{2}+o(1)}&\text{if $d\geq
2$}.\end{cases}\end{split}$ (2.15)
###### Proof.
Let $N_{n}$ denote the double sum in the lemma, and note that
$\begin{split}N_{n}=&\sum_{2\leq k\leq
n}\beta_{k-1}\left(T_{k}-T_{k-1}-1\right),\ \text{where}\\\
\beta_{k-1}:=&\sum_{1\leq i\leq
k-1}q_{i}^{2}\mathbf{1}_{\\{S_{i}=S_{k}\\}}.\end{split}$ (2.16)
Recall that $\mathcal{S}$ denotes the $\sigma$-algebra generated by the entire
process $S$ and observe that, conditionally on $\mathcal{S}$,
$\\{N_{n}\\}_{n=1}^{\infty}$ is a mean-zero martingale with
$\begin{split}{\mathrm{E}}(N_{n}^{2}\,|\,\mathcal{S})&=\text{Var}(T_{1})\cdot\sum_{2\leq
k\leq n}{\mathrm{E}}\left(\beta_{k-1}^{2}\,\big{|}\,\mathcal{S}\right)\\\
&=\text{Var}(T_{1})\cdot\sum_{2\leq k\leq
n}\left(\text{Var}(q_{1}^{2})+\left|{\mathrm{E}}(q_{1}^{2})\right|^{2}\cdot
L_{k-1}^{S_{k}}\right)L_{k-1}^{S_{k}}.\end{split}$ (2.17)
This and Doob’s inequality together show that
${\mathrm{E}}\left(\max_{1\leq k\leq n}N_{k}^{2}\right)\leq\text{const}\cdot
n\left(\max_{a\in{\mathbf{Z}}}{\mathrm{E}}(L^{a}_{n})+\max_{a\in{\mathbf{Z}}}{\mathrm{E}}(|L^{a}_{n}|^{2})\right).$
(2.18)
By the local-limit theorem, the preceding is at most a constant multiple of
$n(\sum_{1\leq i\leq n}i^{-d/2})^{2}$. The Borel–Cantelli lemma finishes the
proof. $\Box$
###### Proof of Proposition 2.1.
Recall the definition of each $q_{i}$. With that in mind, we can decompose
$\Xi_{n}$ as follows:
$\Xi_{n}=\sum_{1\leq k\leq n}\int_{T_{k-1}}^{T_{k}}{\rm
d}s\left(\mathop{\sum}_{1\leq
i<k}\mathbf{1}_{\\{S_{i}=S_{k}\\}}q_{i}\right)^{2}=\Xi_{n}^{(1)}+\Xi_{n}^{(2)},$
(2.19)
where,
$\Xi^{(1)}_{n}:=\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}q_{i}^{2}(T_{k}-T_{k-1}),$ (2.20)
and
$\Xi^{(2)}_{n}:=2\mathop{\sum\sum\sum}_{1\leq i<j<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{j}=S_{k}\\}}q_{i}q_{j}(T_{k}-T_{k-1}).$ (2.21)
Since
$\begin{split}\Xi^{(1)}_{n}&=\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}+\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}\left(q_{i}^{2}-1\right)\\\ &\hskip
108.405pt+\mathop{\sum\sum}_{1\leq i<k\leq
n}\mathbf{1}_{\\{S_{i}=S_{k}\\}}q_{i}^{2}\left(T_{k}-T_{k-1}-1\right),\end{split}$
(2.22)
Lemmas 2.2, 2.3, 2.4, and 2.5 together imply that $\Xi^{(1)}_{n}$ has the
large-$n$ asymptotics that is claimed for $\Xi_{n}$. In light of Lemma 2.2, it
suffices to show that almost surely the following holds as $n\to\infty$:
$\Xi^{(2)}_{n}=\begin{cases}n^{\frac{17}{12}+o(1)}&\text{if $d=1$ },\\\
n^{\frac{3}{4}+o(1)}&\text{if $d\geq 2$}.\end{cases}$ (2.23)
We can write $\Xi^{(2)}_{n}=2\sum_{k=3}^{n}\tau_{k-1}(S_{k})(T_{k}-T_{k-1})$,
where
$\tau_{k-1}(z):=\mathop{\sum\sum}_{1\leq i<j\leq
k-1}\mathbf{1}_{\\{S_{i}=S_{j}=z\\}}q_{i}q_{j}\qquad\text{for
$z\in{\mathbf{Z}}$ and $k>1$}.$ (2.24)
In particular, we can write
$\Xi^{(2)}_{n}:=2\left(a_{n}+b_{n}\right),\qquad\text{where}$ (2.25)
$a_{n}:=\sum_{k=3}^{n}\tau_{k-1}(S_{k})\left(T_{k}-T_{k-1}-1\right)\quad\text{and}\quad
b_{n}:=\sum_{k=3}^{n}\tau_{k-1}(S_{k}).$ (2.26)
Recall that $\mathcal{S}$ denotes the $\sigma$-algebra generated by the
process $S$. It follows that, conditional on $\mathcal{S}$, the process
$\\{a_{n}\\}_{n=1}^{\infty}$ is a mean-zero martingale, and
${\mathrm{E}}\left(a_{n}^{2}\,\big{|}\,\mathcal{S}\right)=\text{Var}(T_{1})\cdot\sum_{k=3}^{n}{\mathrm{E}}\left(|\tau_{k-1}(S_{k})|^{2}\,\big{|}\,\mathcal{S}\right).$
(2.27)
The latter conditional expectation is also computed by a martingale
computation. Namely, we write
$\tau_{k-1}(z)=\sum_{j=2}^{k-1}(\sum_{i=1}^{j-1}\mathbf{1}_{\\{S_{i}=S_{j}=z\\}}q_{i})q_{j}$
for all $z\in{\mathbf{Z}}$ in order to deduce that
${\mathrm{E}}\left(|\tau_{k-1}(z)|^{2}\,\big{|}\,\mathcal{S}\right)=\sum_{j=2}^{k-1}\mathbf{1}_{\\{S_{j}=z\\}}L^{z}_{j-1}\leq\left(L^{z}_{k-1}\right)^{2}.$
(2.28)
It follows from Doob’s maximal inequality that
${\mathrm{E}}\left(\max_{1\leq k\leq n}a_{k}^{2}\right)\leq
4\text{Var}(T_{1})\cdot{\mathrm{E}}\left(\sum_{k=3}^{n}(L^{S_{k}}_{k-1})^{2}\right).$
(2.29)
By time reversal, we can replace $L^{S_{k}}_{k-1}$ by $L^{0}_{k-1}$.
Therefore, the local-limit theorem implies that ${\mathrm{E}}(\max_{1\leq
k\leq n}a_{k}^{2})\leq\text{const}\cdot n(\sum_{i=1}^{n}i^{-d/2})^{2}$, and
hence almost surely as $n\to\infty$, (2.23) is satisfied with $\Xi^{(2)}_{n}$
replaced by $a_{n}$ [the Borel–Cantelli lemma]. It suffices to prove that
(2.23) holds if $\Xi_{n}$ is replaced by $b_{n}$.
We can write $b_{n}:=b_{n,n}$, where
$b_{n,k}=\sum_{j=2}^{n-1}\theta_{j-1,k}\,q_{j}\quad\text{for}\quad\theta_{j-1,k}:=\sum_{i=1}^{j-1}\mathbf{1}_{\\{S_{i}=S_{j}\\}}q_{i}\left(L^{S_{i}}_{k}-L^{S_{i}}_{j}\right).$
(2.30)
For each fixed integer $k\geq 1$, $\\{b_{n,k}\\}_{n\geq 3}$ is a mean-zero
martingale, conditional on $\mathcal{S}$. Therefore, Burkholder’s inequality
yields
${\mathrm{E}}\left(|b_{n,k}|^{p}\,\big{|}\,\mathcal{S}\right)\leq\text{const}\cdot{\mathrm{E}}\left(\left.\left[\sum_{j=2}^{n-1}\theta_{j-1,k}^{2}\,q_{j}^{2}\right]^{p/2}\,\right|\
\mathcal{S}\right),$ (2.31)
where the implied constant is nonrandom and depends only on $p$. Since
$|\sum_{j=1}^{n-1}x_{j}|^{p/2}\leq
n^{\frac{p}{2}-1}\sum_{j=1}^{n-1}|x_{j}|^{p/2}$ for all real
$x_{1},\ldots,x_{n-1}$, we can apply the preceding with $k:=n$ to obtain
${\mathrm{E}}\left(|b_{n}|^{p}\,\big{|}\,\mathcal{S}\right)\leq\text{const}\cdot{\mathrm{E}}(|q_{1}|^{p})\cdot
n^{\frac{p}{2}-1}\sum_{j=2}^{n-1}{\mathrm{E}}\left(|\theta_{j-1,n}|^{p}\,\big{|}\
\mathcal{S}\right).$ (2.32)
Yet another application of Burkholder’s inequality yields
$\begin{split}{\mathrm{E}}\left(|\theta_{j-1,n}|^{p}\,\big{|}\
\mathcal{S}\right)&\leq\text{const}\cdot{\mathrm{E}}\left(\left|\sum_{i=1}^{j-1}\mathbf{1}_{\\{S_{i}=S_{j}\\}}q_{i}^{2}\left(L^{S_{i}}_{n}-L^{S_{i}}_{j}\right)^{2}\right|^{p/2}\right)\\\
&\leq\text{const}\cdot{\mathrm{E}}(|q_{1}|^{p})\cdot\left(L_{j-1}^{S_{j}}\right)^{p/2}\left(L^{S_{j}}_{n}-L^{S_{j}}_{j}\right)^{p},\end{split}$
(2.33)
since ${\mathrm{E}}(q_{i_{1}}^{2}\cdots
q_{i_{p/2}}^{2})\leq{\mathrm{E}}(|q_{1}|^{p})$ for all $1\leq
i_{1},\ldots,i_{p/2}<j$. We take expectations and apply the Markov property
and time reversal to find that
${\mathrm{E}}\left(|\theta_{j-1,n}|^{p}\right)\leq\text{const}\cdot{\mathrm{E}}(|q_{1}|^{p})\cdot{\mathrm{E}}\left[\left(L^{0}_{j-1}\right)^{p/2}\right]{\mathrm{E}}\left[\left(L^{0}_{n-j}\right)^{p}\right].$
(2.34)
It follows readily that
${\mathrm{E}}\left(|b_{n}|^{p}\right)\leq\text{const}\cdot{\mathrm{E}}(|q_{1}|^{p})\cdot
n^{p/2}{\mathrm{E}}\left[\left(L^{0}_{n}\right)^{p/2}\right]{\mathrm{E}}\left[\left(L^{0}_{n}\right)^{p}\right]\\\
$ (2.35)
This, the local-limit theorem, and the Borel–Cantelli lemma together imply
that (2.23) holds with $b_{n}$ in place of $\Xi^{(2)}_{n}$. The proposition
follows. $\Box$
## 3 Proof of Theorem 1.2
In view of Theorem 1.1 and the LIL for the Brownian motion, it suffices to
consider only the case $d=1$, and to establish the following:
$\liminf_{n\to\infty}\left(\frac{\L_{2}n}{n}\right)^{3/4}\max_{0\leq k\leq
n}|\gamma(\alpha(k))|=(a^{*})^{3/4}\frac{\pi}{\sqrt{8}},$ (3.1)
where
$\L_{2}x:=\log\log(x\vee
1)\quad\text{and}\quad\alpha(t):=\int_{-\infty}^{\infty}(\ell^{x}_{t})^{2}\,{\rm
d}x.$ (3.2)
It is known that [6, Theorem 3], $\sup_{x\in{\mathbf{R}}}\sup_{1\leq k\leq
n}(\ell^{x}_{k}-\ell^{x}_{k-1})=o((\log n)^{-1/2})$ almost surely
$[{\mathrm{P}}]$. This implies readily that $\max_{0\leq k\leq
n}(\alpha(k+1)-\alpha(k-1))=O(n\sqrt{\log n})$ a.s. Therefore, it follows from
[7, Theorem 1.2.1] that (3.1) is equivalent to the following:
$\liminf_{t\to\infty}\left(\frac{\L_{2}t}{t}\right)^{3/4}\sup_{0\leq s\leq
t}|\gamma(\alpha(s))|=(a^{*})^{3/4}\frac{\pi}{\sqrt{8}}.$ (3.3)
Brownian scaling implies that $\alpha(t)$ and $t^{3/2}\alpha(1)$ have the same
distribution. On one hand, Proposition 1 of [11] tells us that the limit
$C:=\lim_{\lambda\to\infty}\exp\\{a^{*}\,\lambda^{2/3}\\}\,{\mathrm{E}}\exp(-\lambda\alpha(1))$
exists and is positive and finite. On the other hand, we can write
$\gamma^{*}(t):=\sup_{0\leq s\leq t}|\gamma(s)|$ and appeal to Lemma 1.6.1 of
[7] to find that for all $t,y>0$,
$\frac{2}{\pi}\exp\left(-\frac{\pi^{2}t}{8y^{2}}\right)\leq{\mathrm{P}}\left\\{\gamma^{*}(t)<y\right\\}\leq\frac{4}{\pi}\exp\left(-\frac{\pi^{2}t}{8y^{2}}\right).$
(3.4)
Therefore uniformly for all $t>0$ and $x\in(0\,,1]$,
$\begin{split}{\mathrm{P}}\left\\{\sup_{0\leq s\leq
t}|\gamma(\alpha(s))|<xt^{3/4}\right\\}&={\mathrm{P}}\left\\{\gamma^{*}(\alpha(1))<x\right\\}\leq{4\over\pi}{\mathrm{E}}\,{\rm
e}^{-\pi^{2}\alpha(1)/(8x^{2})}\\\ &\hskip
28.90755pt\leq\text{const}\cdot\exp\left(-\frac{a^{*}}{x^{4/3}}\left(\frac{\pi^{2}}{8}\right)^{2/3}\right).\end{split}$
(3.5)
This and an application of the Borel–Cantelli lemma together yield one half of
the (3.3); namely, (3.3) where “=” is replaced by “$\geq$.” In order to derive
the other half we choose $t_{n}:=n^{n}$ and $c>(a^{*})^{3/4}\pi/\sqrt{8}$, and
define
$A_{n}:=\left\\{\omega:\,\sup_{0\leq s\leq
t_{n}}|\gamma(\alpha(s))|<c\left(\frac{t_{n}}{\L_{2}t_{n}}\right)^{3/4}\right\\}.$
(3.6)
Every $A_{n}$ is measurable with respect to ${\cal
F}_{t_{n}}:=\sigma\\{\gamma(u):\,u\leq\alpha(t_{n})\\}\vee\sigma\\{B_{v}:\,v\leq
t_{n}\\}$. In light of the 0-1 law of Paul Lévy, and since
$c>(a^{*})^{3/4}\pi/\sqrt{8}$ is otherwise arbitrary, it suffices to prove
that
$\sum_{n=1}^{\infty}{\mathrm{P}}\left(\left.A_{n}\ \right|\,{\cal
F}_{t_{n-1}}\right)=\infty\qquad\rm a.s.$ (3.7)
The argument that led to (3.5) can be used to show that for all $v\geq 0$,
$\displaystyle{\mathrm{P}}\left\\{\gamma^{*}(v+\alpha(t))<xt^{3/4}\right\\}$
$\displaystyle\geq\frac{2}{\pi}\exp\left(-\frac{\pi^{2}v}{8x^{2}t^{3/2}}\right)\,{\mathrm{E}}\,{\rm
e}^{-\pi^{2}\alpha(1)/(8x^{2})}$ (3.8)
$\displaystyle\geq\text{const}\cdot\exp\left(-{\pi^{2}v\over
8x^{2}t^{3/2}}-\frac{a^{*}}{x^{4/3}}\left(\frac{\pi^{2}}{8}\right)^{2/3}\right).$
In order to prove (3.7), let us choose and fix a large integer $n$
temporarily. We might note that
$\sup_{0\leq s\leq
t_{n}}|\gamma(\alpha(s))|=\gamma^{*}(\alpha(t_{n}))\leq\gamma^{*}(\alpha(t_{n-1}))+\widetilde{\gamma}^{*}(\alpha(t_{n})-\alpha(t_{n-1})),$
(3.9)
where
$\widetilde{\gamma}(s)=\widetilde{\gamma}_{n}(s):=\gamma(s+\alpha(t_{n-1}))-\gamma(\alpha(t_{n-1}))$
and $\widetilde{\gamma}^{*}(s):=\sup_{0\leq v\leq s}|\widetilde{\gamma}(v)|$
for $s\geq 0$. Of course, $\widetilde{\gamma}$ is a Brownian motion
independent of ${\cal F}_{t_{n-1}}$. Moreover, we can write
$\ell_{t_{n}}^{x}=\ell_{t_{n-1}}^{x}+\widetilde{\ell}_{t_{n}-t_{n-1}}^{x-B_{t_{n-1}}}$,
where $\widetilde{\ell}$ denotes the local time process of the Brownian motion
$\widetilde{B}(s):=B(s+t_{n-1})-B(t_{n-1}),s\geq 0$. Clearly,
$(\widetilde{\gamma}\,,\widetilde{B})$ is a two-dimensional Brownian motion,
independent of ${\cal F}_{t_{n-1}}$. Observe that
$\displaystyle\alpha_{t_{n}}-\alpha_{t_{n-1}}$ $\displaystyle=\int{\rm
d}x\left[(\ell_{t_{n}}^{x})^{2}-(\ell_{t_{n-1}}^{x})^{2}\right]=\int{\rm d}x\
\widetilde{\ell}_{t_{n}-t_{n-1}}^{x-B_{t_{n-1}}}\left(\ell_{t_{n-1}}^{x}+\widetilde{\ell}_{t_{n}-t_{n-1}}^{x-B_{t_{n-1}}}\right)$
$\displaystyle\leq(t_{n}-t_{n-1})\ell^{*}_{t_{n-1}}+\widetilde{\alpha}_{t_{n}-t_{n-1}},$
(3.10)
where $\ell^{*}_{t_{n-1}}:=\sup_{x\in{\mathbf{R}}}\ell^{x}_{t_{n-1}}$.
Therefore, we obtain
$\gamma^{*}(\alpha_{t_{n}})\leq\gamma^{*}(\alpha_{t_{n-1}})+\widetilde{\gamma}^{*}\left((t_{n}-t_{n-1})\ell^{*}_{t_{n-1}}+\widetilde{\alpha}_{t_{n}-t_{n-1}}\right).$
(3.11)
Let $\varepsilon>0$ be such that $2\varepsilon<c-(a^{*})^{3/4}\pi/\sqrt{8}$,
and define
$D_{n}:=\left\\{\gamma^{*}(\alpha_{t_{n-1}})<\varepsilon\left({t_{n}\over\L_{2}t_{n}}\right)^{3/4},\,\ell^{*}_{t_{n-1}}\leq\sqrt{3t_{n-1}\L_{2}t_{n-1}}\right\\}.$
(3.12)
Clearly, $D_{n}$ is ${\cal F}_{t_{n-1}}$-mesurable. Let
$v_{n}:=t_{n}\sqrt{3t_{n-1}\L_{2}t_{n-1}}$. Since
$A_{n}\supset
D_{n}\cap\left\\{\widetilde{\gamma}^{*}(v_{n}+\widetilde{\alpha}_{t_{n}})\leq(c-\varepsilon)\left({t_{n}\over\L_{2}t_{n}}\right)^{3/4}\right\\},$
(3.13)
we can deduce from (3.8) that
$\displaystyle{\mathrm{P}}(A_{n}\,|\,{\cal F}_{t_{n-1}})$
$\displaystyle\geq\text{const}\cdot\mathbf{1}_{D_{n}}\exp\left(-\frac{\pi^{2}v_{n}}{8}\left(\frac{\L_{2}t_{n}}{t_{n}}\right)^{3/2}-a^{*}\left(\frac{\pi^{2}}{8(c-\varepsilon)^{2}}\right)^{2/3}\L_{2}t_{n}\right)$
$\displaystyle\geq\text{const}\cdot\mathbf{1}_{D_{n}}\,(n\ln
n)^{-a^{*}(\pi^{2}/8)^{2/3}(c-\varepsilon)^{-4/3}},$ (3.14)
where we have used the fact that $v_{n}/t_{n}^{3/2}\sim 1/n$. Because
$a^{*}(\pi^{2}/8)^{2/3}(c-\varepsilon)^{-4/3}<1$, (3.7) implies that almost
surely, $\mathbf{1}_{D_{n}}=1$ for all $n$ large. Indeed, the LIL tells us
that almost surely for all large $n$,
$\ell^{*}_{t_{n-1}}\leq\sqrt{3t_{n-1}\L_{2}t_{n-1}}$, and
$\gamma^{*}(\alpha_{t_{n-1}})\leq\sqrt{3\alpha_{t_{n-1}}\L_{2}\alpha_{t_{n-1}}}$.
Since $\alpha(t)=\int(\ell_{t}^{x})^{2}dx\leq t\,\ell^{*}_{t}$, we find that
$\alpha_{t_{n-1}}\leq t_{n-1}^{3/2}\sqrt{3\L_{2}t_{n-1}}$. Since
$t_{n-1}/t_{n}\sim 1/n$, it follows that almost surely,
$\ell^{*}_{t_{n-1}}\leq\varepsilon(t_{n}/\L_{2}t_{n})^{3/4}$ for all large $n$
and prove that $D_{n}$ realizes eventually for all large $n$. The proof of
Theorem 1.2 is complete. $\Box$
Acknowledgements. We thank the University of Paris for generously hosting the
second author (D.K.), during which time much of this research was carried out.
## References
* [1] Asselah, Amine, Annealed large deviation estimates for the energy of a polymer, Preprint (2009) arXiv:0812.0443v2.
* [2] Bass, R.F., Chen, X. and Rosen, J., Moderate deviations and laws of the iterated logarithm for the renormalized self-intersection local times of planar random walks. Electron. J. Probab. 11 (2006), no. 37, 993–1030 (electronic).
* [3] Bass, Richard F. and Griffin, Philip S., The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 417–436.
* [4] Chen, X., Limit laws for the energy of a charged polymer. Ann. Inst. Henri Poincaré: Probab. Stat. 44 (2008), no. 4, 638–672.
* [5] Chen, X. and Khoshnevisan, D., From charged polymers to random walk in random scenary. IMS Lecture Notes. Optimality: The 3rd E.L. Lehmann Symposium Vol. 57 (2009) 237–251.
* [6] Csáki, E., Csörgő, M., Földes, A., and Révész, P., How big are the increments of the local time of a Wiener process? Ann. Probab. 11 (1983), no. 3, 593–608.
* [7] Csörgő, M. and Révész, P., Strong Approximations in Probability and Statistics, Academic Press, Inc., New York-London, 1981.
* [8] Derrida, B., Griffiths, B. and Higgs, P.G., A Model of Directed Walks with Random Self-Interactions. Europhys. Lett., 18, no. 4 (1992) pp. 361–366.
* [9] Erdős, P.; and Taylor, S.J., Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11 (1960) 137–162.
* [10] Hall, P. and Heyde, C. C., Martingale Limit Theory and Its Applications, Academic Press, New York, 1980.
* [11] van der Hofstad, R., den Hollander, F. and König, W., Central limit theorem for Edwards model. Ann. Probab. 25 (1997), no. 2 573–597.
* [12] Kantor, Y. and Kardar, M., Polymers with Random Self-Interactions. Europhys. Lett. 14, No. 5 (1991) 421–426.
* [13] Kesten, Hary, An iterated logarithm law for local time. Duke Math. J. 32 (1965) 447–456.
* [14] Révész, P., Local time and invariance. In: Analytical Methods in Probability Theory (Oberwolfach, 1980), pp. 128–145, Lecture Notes in Math., 861, Springer, Berlin-New York, 1981.
* [15] Revuz, Daniel and Yor, Marc, Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1991.
Yueyun Hu. Département de Mathématiques, Université Paris XIII, 99 avenue J-B
Clément, F-93430 Villetaneuse, France, _Email:_ yueyun@math.univ-paris13.fr
Davar Khoshnevisan. Department of Mathematics, University of Utah, 155 South
1440 East, JWB 233, Salt Lake City, Utah 84112-0090, USA,
_Email:_ davar@math.utah.edu
|
arxiv-papers
| 2009-11-19T21:20:08 |
2024-09-04T02:49:06.587345
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yueyun Hu and Davar Khoshnevisan",
"submitter": "Davar Khoshnevisan",
"url": "https://arxiv.org/abs/0911.3895"
}
|
0911.4113
|
# On $K$-theory automorphisms related to bundles of finite order
A.V. Ershov ershov.andrei@gmail.com
###### Abstract.
In the present paper we describe the action of (not necessarily line) bundles
of finite order on the $K$-functor in terms of classifying spaces. This
description might provide with an approach for more general twistings in
$K$-theory than ones related to the action of the Picard group.
## Introduction
The complex $K$-theory is a generalized cohomology theory represented by the
$\Omega$-spectrum $\\{K_{n}\\}_{n\geq 0}$, where
$K_{n}=\mathbb{Z}\times\mathop{\rm BU}\nolimits$, if $n$ is even and
$K_{n}={\rm U}$ if $n$ is odd. $K_{0}=\mathbb{Z}\times\mathop{\rm
BU}\nolimits$ is an $E_{\infty}$-ring space, and the corresponding space of
units $K_{\otimes}$ (which is an infinite loop space) is
$\mathbb{Z}/2\mathbb{Z}\times\mathop{\rm BU}\nolimits_{\otimes}$, where
$\mathop{\rm BU}\nolimits_{\otimes}$ denotes the space $\mathop{\rm
BU}\nolimits$ with the $H$-space structure induced by the tensor product of
virtual bundles of virtual dimension $1$. Twistings of the $K$-theory over a
compact space $X$ are classified by homotopy classes of maps $X\rightarrow{\rm
B}(\mathbb{Z}/2\mathbb{Z}\times\mathop{\rm
BU}\nolimits_{\otimes})\simeq\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,1)\times\mathop{\rm
BBU}\nolimits_{\otimes}$ (where $\rm B$ denotes the functor of classifying
space). The theorem that $\mathop{\rm BU}\nolimits_{\otimes}$ is an infinite
loop space was proved by G. Segal [5]. Moreover, the spectrum $\mathop{\rm
BU}\nolimits_{\otimes}$ can be decomposed as follows: $\mathop{\rm
BU}\nolimits_{\otimes}=\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)\times\mathop{\rm BSU}\nolimits_{\otimes}$. This
implies that the twistings in the $K$-theory can be classified by homotopy
classes of maps $X\rightarrow\mathop{\rm
K}\nolimits(\mathbb{Z}/2\mathbb{Z},\,1)\times\mathop{\rm
K}\nolimits(\mathbb{Z},\,3)\times\mathop{\rm BBSU}\nolimits_{\otimes}$. In
other words, for a compact space $X$ the twistings correspond to elements in
$H^{1}(X,\,\mathbb{Z}/2\mathbb{Z})\times
H^{3}(X,\,\mathbb{Z})\times[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}],\;[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}]=bsu^{1}_{\otimes}(X),$ where
$\\{bsu^{n}_{\otimes}\\}_{n}$ is the generalized cohomology theory
corresponding to the infinite loop space $\mathop{\rm
BSU}\nolimits_{\otimes}$.
The twisted $K$-theory corresponding to the twistings coming from
$H^{1}(X,\,\mathbb{Z}/2\mathbb{Z})\times H^{3}(X,\,\mathbb{Z})$ has been
intensively studied during the last decade, but not the general case (as far
as the author knows). It seems that the reason is that there is no known
appropriate geometric model for “nonabelian” twistings from $[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}].$ In this paper we make an attempt to give such a
model for elements of finite order in $[X,\,\mathop{\rm
BBSU}\nolimits_{\otimes}].$ In particular, we are based on the model of the
$H$-space $\mathop{\rm BSU}\nolimits_{\otimes}$ given by the infinite matrix
grassmannian $\mathop{\rm Gr}\nolimits$ [7] (see also subsection 4.5 below).
A brief outline of this paper is as follows. In section 1 we recall the well-
known result that the action of the projective unitary group of the separable
Hilbert space $\mathop{\rm PU}\nolimits({\mathcal{H}})$ on the space of
Fredholm operators $\mathop{\rm Fred}\nolimits({\mathcal{H}})$ (which is the
representing space of $K$-theory) by conjugation corresponds to the action of
the Picard group $Pic(X)$ on $K(X)$ by group automorphisms (Theorem 1). The
key result of section 2 is Theorem 7 which is in some sense a counterpart of
Theorem 1. Roughly speaking, it asserts that in terms of representing space
$\mathop{\rm Fred}\nolimits({\mathcal{H}})$ the tensor multiplication of
$K$-functor by (not necessarily line) bundles of finite order $k$ can be
described by some maps $\gamma_{kl^{m},\,l^{n-m}}^{\prime}\colon\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n-m}}\times\mathop{\rm
Fred}\nolimits({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits({\mathcal{H}}),$ where $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n-m}}$ are the spaces parametrizing unital
$*$-homomorphisms of matrix algebras $M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{n}}(\mathbb{C})$. Then by arranging these maps
$\gamma_{kl^{m},\,l^{n-m}}^{\prime}$ we should construct an action of the
$H$-space $\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
Fr}\nolimits_{kl^{n},\,l^{n}}$ on $\mathop{\rm Fred}\nolimits({\mathcal{H}})$.
We also have shown that $\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
Fr}\nolimits_{kl^{n},\,l^{n}}$ is a well-pointed grouplike topological monoid
and therefore there exists the classifying space $\mathop{\rm
B}\nolimits\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
Fr}\nolimits_{kl^{n},\,l^{n}}$ (see subsection 3.2). More precisely, we
consider the direct limit of matrix algebras
$M_{kl^{\infty}}(\mathbb{C})=\lim\limits_{\longrightarrow\atop{m}}M_{kl^{m}}(\mathbb{C})$
and the monoid of its unital endomorphisms. We fix an increasing filtration
$A_{kl^{m}}\subset A_{kl^{m+1}}\subset\ldots,\quad
A_{kl^{m}}=M_{kl^{m}}(\mathbb{C})$ in $M_{kl^{\infty}}(\mathbb{C})$ such that
$A_{kl^{m+1}}=M_{l}(A_{kl^{m}})$. Then we consider endomorphisms of
$M_{kl^{\infty}}(\mathbb{C})$ that are induced by unital homomorphisms of the
form $h\colon A_{kl^{m}}\rightarrow A_{kl^{n}}$ (for some $m,\,n$), i.e. have
the form $M_{l^{\infty}}(h).$ Such endomorphisms form the above mentioned
topological monoid which is homotopy equivalent to the direct limit
$\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{m,\,n}}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}$, which is not contractible provided
$(k,\,l)=1$. Moreover, the automorphism subgroup in the monoid (corresponding
to $n=0$) is $\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
PU}\nolimits(kl^{m}).$ Furthermore, this monoid naturally acts on the space of
Fredholm operators and this action corresponds to the tensor multiplication of
the $K$-functor by bundles of order $k$. In subsection 3.3 we also sketch the
idea of the definition of the corresponding version of the twisted $K$-theory.
Roughly speaking, the “usual” (Abelian) twistings of order $k$ correspond to
the group of automorphisms while the nonabelian ones correspond to the monoid
of endomorphisms of $M_{kl^{\infty}}(\mathbb{C})$. Note that these
endomorphisms act on the localization of the space of Fredholm operators over
$l$ by homotopy auto-equivalences, i.e. they are invertible in the sense of
homotopy.
Although some technical difficulties remain we hope that this approach will be
useful in order to define a general version of the twisted $K$-theory.
## 1\. $K$-theory automorphisms related to line bundles
In this section we describe well-known results about the action of $Pic(X)$ on
the group $K(X)$. We also consider the special case of the subgroup of line
bundles of finite order.
Let $X$ be a compact space, $Pic(X)$ its Picard group consisting of
isomorphism classes of line bundles with respect to the tensor product. The
Picard group is represented by the $H$-space $\mathop{\rm
BU}\nolimits(1)\cong\mathbb{C}P^{\infty}\cong\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)$ whose multiplication is given by the tensor
product of line bundles or (in the appearance of the Eilenberg-MacLane space)
by addition of two-dimensional integer cohomology classes. In particular, the
first Chern class $c_{1}$ defines the isomorphism $c_{1}\colon
Pic(X)\stackrel{{\scriptstyle\cong}}{{\rightarrow}}H^{2}(X,\,\mathbb{Z}).$ The
group $Pic(X)$ is a subgroup of the multiplicative group of the ring $K(X)$
and therefore it acts on $K(X)$ by group automorphisms. This action is
functorial on $X$ and therefore it can be described in terms of classifying
spaces (see Theorem 1).
As a representing space for the $K$-theory we take $\mathop{\rm
Fred}\nolimits({\mathcal{H}}),$ the space of Fredholm operators in the
separable Hilbert space ${\mathcal{H}}$. It is known [2] that for a compact
space $X$ the action of $Pic(X)$ on $K(X)$ is induced by the conjugate action
$\gamma\colon\mathop{\rm PU}\nolimits({\mathcal{H}})\times\mathop{\rm
Fred}\nolimits({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits({\mathcal{H}}),\;\gamma(g,\,T)=gTg^{-1}$
of $\mathop{\rm PU}\nolimits({\mathcal{H}})$ on $\mathop{\rm
Fred}\nolimits({\mathcal{H}}).$ More precisely, there is the following theorem
(recall that $\mathop{\rm
PU}\nolimits({\mathcal{H}})\simeq\mathbb{C}P^{\infty}\simeq\mathop{\rm
K}\nolimits(\mathbb{Z},\,2)$).
###### Theorem 1.
If $f_{\xi}\colon X\rightarrow\mathop{\rm Fred}\nolimits({\mathcal{H}})$ and
$\varphi_{\zeta}\colon X\rightarrow\mathop{\rm PU}\nolimits({\mathcal{H}})$
represent $\xi\in K(X)$ and $\zeta\in Pic(X)$ respectively, then the composite
map
(1) $X\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\longrightarrow}}X\times
X\stackrel{{\scriptstyle\varphi_{\zeta}\times
f_{\xi}}}{{\longrightarrow}}\mathop{\rm
PU}\nolimits({\mathcal{H}})\times\mathop{\rm
Fred}\nolimits({\mathcal{H}})\stackrel{{\scriptstyle\gamma}}{{\rightarrow}}\mathop{\rm
Fred}\nolimits({\mathcal{H}})$
represents $\zeta\otimes\xi\in K(X)$.
Proof see [2].$\quad\square$
It is essential for the theorem that the group $\mathop{\rm
PU}\nolimits({\mathcal{H}})$, on the one hand having the homotopy type of
$\mathbb{C}P^{\infty}$ is the base of the universal $\mathop{\rm
U}\nolimits(1)$-bundle (which is related to the exact sequence of groups
$\mathop{\rm U}\nolimits(1)\rightarrow\mathop{\rm
U}\nolimits({\mathcal{H}})\rightarrow\mathop{\rm PU}\nolimits({\mathcal{H}})$,
because $\mathop{\rm U}\nolimits({\mathcal{H}})$ is contructible in the
considered norm topology), on the other hand being a group acts in the
appropriate way on the representing space of $K$-theory (the space of Fredholm
operators).
Then in order to define the corresponding version of the twisted $K$-theory
one considers the $\mathop{\rm Fred}\nolimits({\mathcal{H}})$-bundle
$\widetilde{\mathop{\rm Fred}\nolimits}({\mathcal{H}})\rightarrow\mathop{\rm
BPU}\nolimits({\mathcal{H}})$ associated (by means of the action $\gamma$)
with the universal $\mathop{\rm PU}\nolimits({\mathcal{H}})$-bundle over the
classifying space $\mathop{\rm BPU}\nolimits({\mathcal{H}})\simeq\mathop{\rm
K}\nolimits(\mathbb{Z},\,3)$, i.e. the bundle
(2) $\textstyle{\mathop{\rm
Fred}\nolimits({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
EPU}\nolimits({\mathcal{H}}){\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits({\mathcal{H}})}}\mathop{\rm
Fred}\nolimits({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
BPU}\nolimits({\mathcal{H}}).}$
Then for any map $f\colon X\rightarrow\mathop{\rm
BPU}\nolimits({\mathcal{H}})$ the corresponding twisted $K$-theory $K_{f}(X)$
is the set (in fact the group) of homotopy classes of sections
$[X,\,f^{*}\widetilde{\mathop{\rm Fred}\nolimits}({\mathcal{H}})]^{\prime}$ of
the pullback bundle (here $[\ldots,\,\ldots]^{\prime}$ denotes the set of
fiberwise homotopy classes of sections). The group $K_{f}(X)$ depends up to
isomorphism only on the homotopy class $[f]$ of the map $f$, i.e. in fact on
the corresponding third integer cohomology class.
In this paper we are interested in the case of bundles (more precisely, of
elements in $bsu_{\otimes}^{0}$) of finite order, therefore let us consider
separately the specialization of the mentioned result to the case of line
bundles of order $k$ in $Pic(X).$ For this we should consider subgroups
$\mathop{\rm PU}\nolimits(k)\subset\mathop{\rm PU}\nolimits({\mathcal{H}})$.
Let us describe the corresponding embedding.
Let ${\mathcal{B}}({\mathcal{H}})$ be the algebra of bounded operators on the
separable Hilbert space ${\mathcal{H}}$,
$M_{k}({\mathcal{B}}({\mathcal{H}})):=M_{k}(\mathbb{C}){\mathop{\otimes}\limits_{\mathbb{C}}}{\mathcal{B}}({\mathcal{H}})$
the matrix algebra over ${\mathcal{B}}({\mathcal{H}})$ (of course, it is
isomorphic to ${\mathcal{B}}({\mathcal{H}})$). Let $\mathop{\rm
U}\nolimits_{k}({\mathcal{H}})\subset M_{k}({\mathcal{B}}({\mathcal{H}}))$ be
the corresponding unitary group (which is isomorphic to $\mathop{\rm
U}\nolimits({\mathcal{H}})$). It acts on $M_{k}({\mathcal{B}}({\mathcal{H}}))$
by conjugations (which are $*$-algebra isomorphisms), moreover, the kernel of
the action is the center, i.e. the subgroup of scalar matrices
$\cong\mathop{\rm U}\nolimits(1).$ The corresponding quotient group we denote
by $\mathop{\rm PU}\nolimits_{k}({\mathcal{H}})$ (of course, it is isomorphic
to $\mathop{\rm PU}\nolimits({\mathcal{H}})$).
$M_{k}(\mathbb{C})\otimes\mathop{\rm
Id}\nolimits_{{\mathcal{B}}({\mathcal{H}})}$ is a $k$-subalgebra (i.e. a
unital $*$-subalgebra isomorphic to $M_{k}(\mathbb{C})$) in
$M_{k}({\mathcal{B}}({\mathcal{H}}))$. Then $\mathop{\rm
PU}\nolimits(k)\subset\mathop{\rm PU}\nolimits_{k}({\mathcal{H}})$ is the
subgroup of automorphisms of this $k$-subalgebra. Thereby we have defined the
injective group homomorphism
$i_{k}\colon\mathop{\rm PU}\nolimits(k)\hookrightarrow\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})$
induced by the group homomorphism $\mathop{\rm
U}\nolimits(k)\hookrightarrow\mathop{\rm
U}\nolimits_{k}({\mathcal{H}}),\>g\mapsto g\otimes\mathop{\rm
Id}\nolimits_{{\mathcal{B}}({\mathcal{H}})}$.
Let $[k]$ be the trivial $\mathbb{C}^{k}$-bundle over $X$.
###### Proposition 2.
For a line bundle $\zeta\rightarrow X$ satisfying the condition
(3) $\zeta\otimes[k]=\zeta^{\oplus k}\cong X\times\mathbb{C}^{k}$
the classifying map $\varphi_{\zeta}\colon X\rightarrow\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})\cong\mathop{\rm PU}\nolimits({\mathcal{H}})$
can be lifted to a map $\widetilde{\varphi}_{\zeta}\colon
X\rightarrow\mathop{\rm PU}\nolimits(k)$ such that
$i_{k}\circ\widetilde{\varphi}_{\zeta}\simeq\varphi_{\zeta}$.
Proof. Consider the exact sequence of groups
(4) $1\rightarrow\mathop{\rm U}\nolimits(1)\rightarrow\mathop{\rm
U}\nolimits(k)\stackrel{{\scriptstyle\chi_{k}}}{{\rightarrow}}\mathop{\rm
PU}\nolimits(k)\rightarrow 1$
and the fibration
(5) $\mathop{\rm
PU}\nolimits(k)\stackrel{{\scriptstyle\psi_{k}}}{{\rightarrow}}\mathop{\rm
BU}\nolimits(1)\stackrel{{\scriptstyle\omega_{k}}}{{\rightarrow}}\mathop{\rm
BU}\nolimits(k)$
obtained by its extension to the right. In particular,
$\psi_{k}\colon\mathop{\rm PU}\nolimits(k)\rightarrow\mathop{\rm
BU}\nolimits(1)\simeq\mathbb{C}P^{\infty}$ is the classifying map for the
$\mathop{\rm U}\nolimits(1)$-bundle $\chi_{k}$ (4). It is easy to see that the
diagram
$\textstyle{\mathop{\rm
PU}\nolimits(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{k}}$$\scriptstyle{\psi_{k}}$$\textstyle{\mathop{\rm
BU}\nolimits(1)}$$\textstyle{\mathop{\rm
PU}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$
commutes.
Let $\zeta\rightarrow X$ be a line bundle satisfying the condition (3),
$\varphi_{\zeta}\colon X\rightarrow\mathop{\rm BU}\nolimits(1)$ its
classifying map. Since $\omega_{k}$ (see (5)) is induced by taking the direct
sum of a line bundle with itself $k$ times (and the extension of the
structural group to $\mathop{\rm U}\nolimits(k)$), we see that
$\omega_{k}\circ\varphi_{\zeta}\simeq*$. Now it is easy to see from exactness
of (5) that $\varphi_{\zeta}\colon X\rightarrow\mathop{\rm BU}\nolimits(1)$
can be lifted to $\widetilde{\varphi}_{\zeta}\colon X\rightarrow\mathop{\rm
PU}\nolimits(k).\quad\square$
Note that the choice of a lift $\widetilde{\varphi}_{\zeta}$ corresponds to
the choice of a trivialization (3): two lifts differ up to a map
$X\rightarrow\mathop{\rm U}\nolimits(k)$. Thus, a lift is defined up to the
action of $[X,\,\mathop{\rm U}\nolimits(k)]$ on $[X,\,\mathop{\rm
PU}\nolimits(k)].$ The subgroup in $Pic(X)$ consisting of line bundles
satisfying the condition (3) is $\mathop{\rm
im}\nolimits\\{\psi_{k*}\colon[X,\,\mathop{\rm
PU}\nolimits(k)]\rightarrow[X,\,\mathbb{C}P^{\infty}]\\}$ or the quotient
$[X,\,\mathop{\rm PU}\nolimits(k)]/[X,\,\mathop{\rm U}\nolimits(k)].$
Let $\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})$ be the subspace of
Fredholm operators in $M_{k}({\mathcal{B}}({\mathcal{H}}))$. Clearly,
$\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})\cong\mathop{\rm
Fred}\nolimits({\mathcal{H}})$. Acting on $M_{k}(\mathbb{C})$ by
$*$-automorphisms, the group $\mathop{\rm PU}\nolimits(k)$ acts on
$M_{k}(\mathbb{C})\otimes\mathcal{B}(\mathcal{H})=M_{k}({\mathcal{B}}({\mathcal{H}}))$
through the first tensor factor. Let $\gamma^{\prime}_{k}\colon\mathop{\rm
PU}\nolimits(k)\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})$ be the restriction of this action on
$\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})$. Then the diagram
$\textstyle{\mathop{\rm PU}\nolimits({\mathcal{H}})\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\gamma}$$\textstyle{\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})}$$\textstyle{\mathop{\rm
PU}\nolimits(k)\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{k}\times\mathop{\rm
id}\nolimits}$$\scriptstyle{\gamma^{\prime}_{k}}$
commutes. Now one can consider the $\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})$-bundle
(6) $\begin{array}[]{c}\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.40698pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\\\\}}}\ignorespaces{\hbox{\kern-22.40698pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.40698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
46.40698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
46.40698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
EPU}\nolimits(k){\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits(k)}}\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
97.63693pt\raise-6.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
97.63693pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-40.5pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
77.29314pt\raise-40.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\rm
BPU}\nolimits(k)}$}}}}}}}\ignorespaces\ignorespaces}}}}}\end{array}$
associated by means of the action $\gamma^{\prime}_{k}$. This bundle is the
pullback of (2) by $\mathop{\rm B}\nolimits i_{k}$.
It is easy to see from the definition of the embedding $i_{k}$ that the action
$\gamma^{\prime}_{k}$ is trivial on elements of the form $k\xi.$ Indeed, a
classifying map for $k\xi$ can be decomposed into the composite
$X\stackrel{{\scriptstyle f_{\xi}}}{{\rightarrow}}\mathop{\rm
Fred}\nolimits({\mathcal{H}})\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\rightarrow}}\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}}).$
From the other hand, $(1+(\zeta-1))\cdot k\xi=k\xi+0=k\xi$ or
$\zeta\otimes([k]\otimes\xi)=(\zeta\otimes[k])\otimes\xi=[k]\otimes\xi.$
###### Remark 3.
Note that if we choose an isomorphism ${\mathcal{B}}({\mathcal{H}})\cong
M_{k^{\infty}}({\mathcal{B}}({\mathcal{H}}))$ and hence the isomorphism
$\mathop{\rm Fred}\nolimits({\mathcal{H}})\cong\mathop{\rm
Fred}\nolimits_{k^{\infty}}({\mathcal{H}}),$ we can define the limit action
$\gamma^{\prime}_{k^{\infty}}\colon\mathop{\rm
PU}\nolimits(k^{\infty})\times\mathop{\rm
Fred}\nolimits_{k^{\infty}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{k^{\infty}}({\mathcal{H}}),$ etc.
## 2\. The case of bundles of dimension $\geq 1$
As was pointed out in the previous section, the group $\mathop{\rm
PU}\nolimits({\mathcal{H}})$, from one hand acts on the representing space of
$K$-theory $\mathop{\rm Fred}\nolimits({\mathcal{H}})$, from the other hand it
is the base of the universal line bundle. This two facts lead to the result
that the action of $\mathop{\rm PU}\nolimits({\mathcal{H}})$ on $K(X)$
corresponds to the tensor product by elements of the Picard group $Pic(X)$
(i.e. classes of line bundles). This action can be restricted to subgroups
$\mathop{\rm PU}\nolimits(k)\subset\mathop{\rm PU}\nolimits({\mathcal{H}})$
which classify elements of finite order $k,\,k\in\mathbb{N}.$
In what follows the role of groups $\mathop{\rm PU}\nolimits(k)$ will play
some spaces $\mathop{\rm Fr}\nolimits_{k,\,l}$ (defined below). From one hand,
they “act” on $K$-theory (more precisely, their direct limit (which has the
natural structure of an $H$-space) acts), from the other hand, they are bases
of some nontrivial $l$-dimensional bundles of order $k$. We will show that
their “action” on $K(X)$ corresponds to the tensor product by those
$l$-dimensional bundles (see Theorem 7). The key result of this section is
Theorem 7 which can be regarded as a counterpart of Theorem 1.
Fix a pair of positive integers $k,\,l>1.$ Let ${\rm
Hom}_{alg}(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))$ be the space of all unital
$*$-homomorphisms $M_{k}(\mathbb{C})\rightarrow M_{kl}(\mathbb{C})$ [8]. It
follows from Noether-Skolem’s theorem [3] that it can be represented in the
form of a homogeneous space of the group $\mathop{\rm PU}\nolimits(kl)$ as
follows:
(7) ${\rm Hom}_{alg}(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))\cong\mathop{\rm
PU}\nolimits(kl)/(E_{k}\otimes\mathop{\rm PU}\nolimits(l))$
(here $\otimes$ denotes the Kronecker product of matrices). This space we
shall denote by $\mathop{\rm Fr}\nolimits_{k,\,l}$. We will be interested in
the case $(k,\,l)=1$ (we have to impose a condition of such a kind to make the
direct limit of the above spaces noncontractible and the construction below
nontrivial [7]).
###### Proposition 4.
A map $X\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$ is the same thing as an
embedding
(8) $\mu\colon X\times M_{k}(\mathbb{C})\hookrightarrow X\times
M_{kl}(\mathbb{C}),$
whose restriction to a fiber is a unital $*$-homomorphism of matrix algebras.
Proof follows directly from the natural bijection
$\mathop{\rm Mor}\nolimits(X\times
M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))\rightarrow\mathop{\rm
Mor}\nolimits(X,\,\mathop{\rm
Mor}\nolimits(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))).\quad\square$
Let $\mathop{\rm Gr}\nolimits_{k,\,l}$ be the “matrix grassmannian”, i.e. the
space whose points parameterize unital $*$-subalgebras in $M_{kl}(\mathbb{C})$
isomorphic to $M_{k}(\mathbb{C})$ (“$k$-subalgebras”) [7]. It follows from
Noether-Skolem’s theorem [3] that
(9) $\mathop{\rm Gr}\nolimits_{k,\,l}=\mathop{\rm
PU}\nolimits(kl)/(\mathop{\rm PU}\nolimits(k)\otimes\mathop{\rm
PU}\nolimits(l)).$
The matrix grassmannian $\mathop{\rm Gr}\nolimits_{k,\,l}$ is the base of the
tautological $M_{k}(\mathbb{C})$-bundle (its fiber over a point
$\alpha\in\mathop{\rm Gr}\nolimits_{k,\,l}$ is the $k$-subalgebra
$M_{k,\,\alpha}\subset M_{kl}(\mathbb{C})$ corresponding to this point) which
we denote by ${\mathcal{A}}_{k,\,l}\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$. More precisely, ${\mathcal{A}}_{k,\,l}$ is a subbundle
in $\mathop{\rm Gr}\nolimits_{k,\,l}\times M_{kl}(\mathbb{C})$ consisting of
all pairs $(\alpha,\,T),\>\alpha\in\mathop{\rm Gr}\nolimits_{k,\,l},\,T\in
M_{k,\,\alpha},$ where $M_{k,\,\alpha}$ is the $k$-subalgebra in
$M_{kl}(\mathbb{C})$ corresponding to the point $\alpha\in\mathop{\rm
Gr}\nolimits_{k,\,l}$.
Let ${\mathcal{B}}_{k,\,l}\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ be the
bundle of centralizers for the subbundle
${\mathcal{A}}_{k,\,l}\subset\mathop{\rm Gr}\nolimits_{k,\,l}\times
M_{kl}(\mathbb{C})$. It is easy to see that
${\mathcal{B}}_{k,\,l}\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ has fiber
$M_{l}(\mathbb{C})$.
It follows from representations (7) and (9) that $\mathop{\rm
Gr}\nolimits_{k,\,l}$ is the base of the principal $\mathop{\rm
PU}\nolimits(k)$-bundle $\pi_{k,\,l}\colon\mathop{\rm
Fr}\nolimits_{k,\,l}\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}.$ Clearly,
${\mathcal{A}}_{k,\,l}\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ is
associated with this principal bundle with respect to the action $\mathop{\rm
PU}\nolimits(k)\stackrel{{\scriptstyle\cong}}{{\rightarrow}}\mathop{\rm
Aut}\nolimits(M_{k}(\mathbb{C}))$ (recall that we consider $*$-automorphisms
only). Hence the pullback $\pi^{*}_{k,\,l}({\mathcal{A}}_{k,\,l})$ has the
canonical trivialization (while the bundle
$\pi^{*}_{k,\,l}({\mathcal{B}}_{k,\,l})\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l}$ is nontrivial, see below).
In general, $\mu$ (see (8)) is a nontrivial embedding, in particular, it can
be nonhomotopic to the choice of a constant $k$-subalgebra in $X\times
M_{kl}(\mathbb{C})$ (in this case the homotopy class of
$X\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$ is nontrivial). In particular,
the subbundle $B_{l}\rightarrow X$ (with fiber $M_{l}(\mathbb{C})$) in
$X\times M_{kl}(\mathbb{C})$ of centralizers for $\mu(X\times
M_{k}(\mathbb{C}))\subset X\times M_{kl}(\mathbb{C})$ can be nontrivial.
The fibration
(10) $\mathop{\rm PU}\nolimits(l)\stackrel{{\scriptstyle
E_{k}\otimes\ldots}}{{\longrightarrow}}\mathop{\rm
PU}\nolimits(kl)\stackrel{{\scriptstyle\chi_{k}^{\prime}}}{{\longrightarrow}}\mathop{\rm
Fr}\nolimits_{k,\,l}$
(cf. (7)) can be extended to the right
(11) $\mathop{\rm
Fr}\nolimits_{k,\,l}\stackrel{{\scriptstyle\psi_{k}^{\prime}}}{{\longrightarrow}}\mathop{\rm
BPU}\nolimits(l)\stackrel{{\scriptstyle\omega_{k}^{\prime}}}{{\longrightarrow}}\mathop{\rm
BPU}\nolimits(kl),$
where $\psi_{k}^{\prime}$ is the classifying map for the
$M_{l}(\mathbb{C})$-bundle
$\widetilde{\mathcal{B}}_{k,\,l}:=\pi^{*}_{k,\,l}({\mathcal{B}}_{k,\,l})\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l}$ (which is associated with the principal $\mathop{\rm
PU}\nolimits(l)$-bundle (10)).
Let $[M_{k}]$ be the trivial $M_{k}(\mathbb{C})$-bundle $X\times
M_{k}(\mathbb{C})$ over $X$.
###### Proposition 5.
(Cf. Proposition 2) For an $M_{l}(\mathbb{C})$-bundle $B_{l}\rightarrow X$
such that
(12) $[M_{k}]\otimes B_{l}\cong X\times M_{kl}(\mathbb{C})$
(cf. (3)) a classifying map $\varphi_{B_{l}}\colon X\rightarrow\mathop{\rm
BPU}\nolimits(l)$ can be lifted to $\widetilde{\varphi}_{B_{l}}\colon
X\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$ (i.e.
$\psi_{k}^{\prime}\circ\widetilde{\varphi}_{B_{l}}=\varphi_{B_{l}}$ or
$B_{l}=\widetilde{\varphi}_{B_{l}}^{*}(\widetilde{\mathcal{B}}_{k,\,l})$).
Proof follows from the analysis of fibration (11).$\quad\square$
Moreover, the choice of such a lift corresponds to the choice of
trivialization (12) and we return to the interpretation of the map
$X\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$ given in Proposition 4. We
stress that a map $X\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$ is not just
an $M_{l}(\mathbb{C})$-bundle, but an $M_{l}(\mathbb{C})$-bundle together with
a particular choice of trivialization (12).
It is not difficult to show [8] that the bundle $B_{l}\rightarrow X$ as in the
statement of Proposition 5 has the form $\mathop{\rm End}\nolimits(\eta_{l})$
for some (unique up to isomorphism) $\mathbb{C}^{l}$-bundle
$\eta_{l}\rightarrow X$ with the structural group $\mathop{\rm
SU}\nolimits(l)$ (here the condition $(k,\,l)=1$ is essential).
Let $\widetilde{\zeta}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$ be the line
bundle associated with the universal covering
$\rho_{k}\rightarrow\widetilde{\mathop{\rm
Fr}\nolimits}_{k,\,l}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l}$, where
$\rho_{k}$ is the group of $k$th roots of unity. Note that
$\widetilde{\mathop{\rm Fr}\nolimits}_{k,\,l}=\mathop{\rm
SU}\nolimits(kl)/(E_{k}\otimes\mathop{\rm SU}\nolimits(l))$. Put
$\zeta^{\prime}:=\widetilde{\varphi}_{B_{l}}^{*}(\widetilde{\zeta})\rightarrow
X$ and $\eta_{l}^{\prime}:=\eta_{l}\otimes\zeta^{\prime}.$
Recall that $\mathop{\rm Fred}\nolimits_{n}({\mathcal{H}})$ is the subspace of
Fredholm operators in $M_{n}({\mathcal{B}}({\mathcal{H}}))$. The evaluation
map
(13) $ev_{k,\,l}\colon\mathop{\rm Fr}\nolimits_{k,\,l}\times
M_{k}(\mathbb{C})\rightarrow M_{kl}(\mathbb{C}),\quad ev_{k,\,l}(h,\,T)=h(T)$
(recall that $\mathop{\rm Fr}\nolimits_{k,\,l}:=\mathop{\rm
Hom}\nolimits_{alg}(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))$) induces the map
(14) $\gamma_{k,\,l}^{\prime}\colon\mathop{\rm
Fr}\nolimits_{k,\,l}\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl}({\mathcal{H}}).$
###### Remark 6.
Note that map (13) can be decomposed into the composition
$\mathop{\rm Fr}\nolimits_{k,\,l}\times
M_{k}(\mathbb{C})\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l}{\mathop{\times}\limits_{\mathop{\rm
PU}\nolimits(k)}}M_{k}(\mathbb{C})={\mathcal{A}}_{k,\,l}\rightarrow
M_{kl}(\mathbb{C}),$
where the last map is the tautological embedding
$\mu\colon{\mathcal{A}}_{k,\,l}\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}\times M_{kl}(\mathbb{C})$ followed by the projection onto
the second factor.
Let $f_{\xi}\colon X\rightarrow\mathop{\rm Fred}\nolimits_{k}({\mathcal{H}})$
represent some element $\xi\in K(X).$
###### Theorem 7.
(Cf. Theorem 1). With respect to the above notation the composite map (cf.
(1))
$X\stackrel{{\scriptstyle\mathop{\rm
diag}\nolimits}}{{\longrightarrow}}X\times
X\stackrel{{\scriptstyle\widetilde{\varphi}_{B_{l}}\times
f_{\xi}}}{{\longrightarrow}}\mathop{\rm Fr}\nolimits_{k,\,l}\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\stackrel{{\scriptstyle\gamma_{k,\,l}^{\prime}}}{{\longrightarrow}}\mathop{\rm
Fred}\nolimits_{kl}({\mathcal{H}})$
represents the element $\eta_{l}^{\prime}\otimes\xi\in K(X).$
Proof (cf. [2], Proposition 2.1). By assumption the element $\xi\in K(X)$ is
represented by a family of Fredholm operators $F=\\{F_{x}\\}$ in a Hilbert
space ${\mathcal{H}}^{k}$. Then the element $\eta_{l}^{\prime}\otimes\xi\in
K(X)$ is represented by the family of Fredholm operators $\\{\mathop{\rm
Id}\nolimits_{(B_{l})_{x}}\otimes\,F_{x}\\}$ in the Hilbert bundle
$\eta_{l}^{\prime}\otimes({\mathcal{H}}^{k})$ (recall that $\mathop{\rm
End}\nolimits(\eta_{l})=B_{l}\,\Rightarrow\,\mathop{\rm
End}\nolimits(\eta_{l}^{\prime})=B_{l}$). A trivialization
$\eta_{l}^{\prime}\otimes({\mathcal{H}}^{k})\cong{\mathcal{H}}^{kl}$ is the
same thing as a map $\widetilde{\varphi}_{B_{l}}\colon X\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l}$, i.e. a lift of the classifying map
$\varphi_{B_{l}}\colon X\rightarrow\mathop{\rm BPU}\nolimits(l)$ for $B_{l}$
(see (11)).$\quad\square$
###### Remark 8.
In order to separate the “$\mathop{\rm SU}\nolimits$”-part of the “action”
$\gamma_{k,\,l}^{\prime}$ from its “line” part, one can use the space
$\widetilde{\mathop{\rm Fr}\nolimits}_{k,\,l}=\mathop{\rm
SU}\nolimits(kl)/(E_{k}\otimes\mathop{\rm SU}\nolimits(l))$ [8] in place of
$\mathop{\rm Fr}\nolimits_{k,\,l}$. Then one would have the representing map
for $\eta_{l}\otimes\xi\in K(X)$ instead of $\eta_{l}^{\prime}\otimes\xi$ in
the statement of Theorem 7.
###### Remark 9.
Note that $\mathop{\rm Fr}\nolimits_{k,\,1}=\mathop{\rm PU}\nolimits(k)$ and
the action $\gamma_{k,\,1}^{\prime}$ coincides with the action
$\gamma_{k}^{\prime}$ from the previous section.
Now using the composition of algebra homomorphisms we are going to define maps
$\phi_{k,\,l}\colon\mathop{\rm Fr}\nolimits_{k,\,l}\times\mathop{\rm
Fr}\nolimits_{k,\,l}\rightarrow\mathop{\rm Fr}\nolimits_{k,\,l^{2}},$ i.e.
$\phi_{k,\,l}\colon\mathop{\rm
Hom}\nolimits_{alg}(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))\times\mathop{\rm
Hom}\nolimits_{alg}(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))\rightarrow\mathop{\rm
Hom}\nolimits_{alg}(M_{k}(\mathbb{C}),\,M_{kl^{2}}(\mathbb{C})).$
First let us define a map
$\iota_{k,\,l}\colon\mathop{\rm
Hom}\nolimits_{alg}(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C}))\rightarrow\mathop{\rm
Hom}\nolimits_{alg}(M_{kl}(\mathbb{C}),\,M_{kl^{2}}(\mathbb{C})),\;\iota_{k,\,l}(h)=h\otimes\mathop{\rm
id}\nolimits_{M_{l}(\mathbb{C})}.$
Then $\phi_{k,\,l}$ is defined as the composition of homomorphisms:
$\phi_{k,\,l}(h_{2},\,h_{1})=\iota_{k,\,l}(h_{2})\circ h_{1},$ where
$h_{i}\in\mathop{\rm
Hom}\nolimits_{alg}(M_{k}(\mathbb{C}),\,M_{kl}(\mathbb{C})).$ Then we have
$ev_{k,\,l^{2}}(\phi_{k,\,l}(h_{2},\,h_{1}),\,T)=ev_{kl,\,l}(\iota_{k,\,l}(h_{2}),\,ev_{k,\,l}(h_{1},\,T)),$
i.e. $\phi_{k,\,l}(h_{2},\,h_{1})(T)=\iota_{k,\,l}(h_{2})(h_{1}(T)),$ where
$T\in M_{k}(\mathbb{C})$.
Now suppose there is an $M_{l}(\mathbb{C})$-bundle $C_{l}\rightarrow X$ with
the corresponding vector bundle $\theta_{l}^{\prime}$ such that
$C_{l}\cong\mathop{\rm End}\nolimits(\theta_{k}^{\prime})$ (cf. a few
paragraphs after Proposition 5). Suppose that
$\widetilde{\varphi}_{C_{l}}\colon X\rightarrow\mathop{\rm
Fr}\nolimits_{k,\,l}$ is its classifying map.
###### Proposition 10.
(cf. Theorem 7.) The composition
$X\stackrel{{\scriptstyle\mathop{\rm diag}\nolimits}}{{\rightarrow}}X\times
X\times
X\stackrel{{\scriptstyle\widetilde{\varphi}_{C_{l}}\times\widetilde{\varphi}_{B_{l}}\times
f_{\xi}}}{{\longrightarrow}}\mathop{\rm Fr}\nolimits_{k,\,l}\times\mathop{\rm
Fr}\nolimits_{k,\,l}\times\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{2}}({\mathcal{H}})$
where the last map is the composition
$\gamma^{\prime}_{k,\,l^{2}}\circ(\phi_{k,\,l}\times\mathop{\rm
id}\nolimits_{\mathop{\rm
Fred}\nolimits_{k}({\mathcal{H}})})=\gamma^{\prime}_{kl,\,l}\circ(\iota_{k,\,l}\times\gamma^{\prime}_{k,\,l})$
represents the element
$\eta_{l}^{\prime}\otimes\theta_{l}^{\prime}\otimes\xi\in K(X).$
Proof is evident.$\quad\square$
Clearly, the results of this section can be generalized to the case of spaces
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}},\;m,\,n\in\mathbb{N}.$ In the next
section we will construct a genuine action of their direct limit $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on the space of Fredholm operators.
## 3\. A construction of the classifying space
A simple calculation with homotopy groups shows that the direct limit
$\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
Fr}\nolimits_{kl^{n},\,l^{n}}$ for $(k,\,l)=1$ is not contractible because its
homotopy groups are $\mathbb{Z}/k\mathbb{Z}$ in odd dimensions (and $0$ in
even ones). In this section we show that it is a topological monoid and
construct its classifying space.
### 3.1. The category ${\mathcal{C}}_{k,\,l}$
First, we define some auxiliary category ${\mathcal{C}}_{k,\,l}$. Fix a pair
of positive integers $\\{k,\,l\\},\;k,\,l>1,(k,\,l)=1.$ By
${\mathcal{C}}_{k,\,l}$ denote the category with the countable number of
objects which are matrix algebras of the form $M_{kl^{m}}(\mathbb{C})$
($m=0,\,1,\,\ldots)$ and morphisms in ${\mathcal{C}}_{k,\,l}$ from
$M_{kl^{m}}(\mathbb{C})$ to $M_{kl^{n}}(\mathbb{C})$ are all unital
$*$-homomorphisms $M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{n}}(\mathbb{C})$
(this set is nonempty iff $m\leq n$). (Since $k>1$ we see that this category
does not contain the initial object, therefore there is no reason to expect
that its classifying space (i.e. the “geometric realization”) is
contractible). Thus, morphisms $\mathop{\rm
Mor}\nolimits(M_{kl^{m}}(\mathbb{C}),\>M_{kl^{n}}(\mathbb{C}))$ form the space
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n-m}}=\mathop{\rm
Hom}\nolimits_{alg}(M_{kl^{m}}(\mathbb{C}),\,M_{kl^{n}}(\mathbb{C}))$, hence
${\mathcal{C}}_{k,\,l}$ is a topological category. (Note that for $n=m$ the
space $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n-m}}=\mathop{\rm
Fr}\nolimits_{kl^{m},\,1}$ is the group $\mathop{\rm PU}\nolimits(kl^{m}).$)
In particular, there is the collection of continuous maps $\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}$ for all $m,\,n,\,r\geq 0$ given by the
composition of morphisms.
Recall ([6], Chapter IV) that there is an appropriate modification of the
construction of the geometric realization $\mathop{\rm
B}\nolimits{\mathcal{C}}$ for topological categories ${\mathcal{C}}$. More
precisely, the nerve in this case is a simplicial topological space and
$\mathop{\rm B}\nolimits{\mathcal{C}}$ is its appropriate geometric
realization. Now we are going to describe the classifying space of the
category $\mathop{\rm B}\nolimits{\mathcal{C}}_{k,\,l}$.
So let $\mathop{\rm B}\nolimits{\mathcal{C}}_{k,\,l}$ be the classifying space
of the topological category ${\mathcal{C}}_{k,\,l}$. Its $0$-cells (vertices)
are objects of ${\mathcal{C}}_{k,\,l}$, i.e. actually positive integers. Its
$1$-cells (edges) are morphisms in ${\mathcal{C}}_{k,\,l}$ (excluding identity
morphisms) attached to their source and target, i.e. $\coprod_{m,\,n\geq
0}\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}$ (recall that for
$n=0\;\mathop{\rm Fr}\nolimits_{kl^{m},\,1}=\mathop{\rm
PU}\nolimits(kl^{m})$). For each pair of composable morphisms $h_{0},\,h_{1}$
in ${\mathcal{C}}_{k,\,l}$ there is a $2$-simplex:
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}\circ
h_{0}}$$\scriptstyle{h_{0}}$$\textstyle{2}$
attached to the $1$-skeleton, etc. Thus, at the third step we obtain the space
$\coprod_{m,\,n,\,r\geq 0}(\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}})$ consisting of all pairs of composable
morphisms. The face maps $\partial_{0},\,\partial_{2}$ are defined by the
deletion of the corresponding morphism ($h_{0}$ and $h_{1}$ respectively in
the above simplex) and $\partial_{1}$ is defined by the composition map
$\coprod_{m,\,n,\,r\geq 0}(\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}})\rightarrow\coprod_{m,\,n+r\geq 0}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}=\coprod_{m,\,n\geq 0}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}.$ The nerve $N{\mathcal{C}}_{k,\,l}$ of the
category ${\mathcal{C}}_{k,\,l}$ is
(15) $(\mathbb{N},\,\coprod_{m,\,n\geq 0}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}},\,\coprod_{m,\,n,\,r\geq 0}(\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}),\ldots).$
Recall that there is a construction of the classifying space of a
(topological) group $G$ as the geometric realization of the simplicial
topological space $(\mathop{\rm pt}\nolimits,\,G,\,G\times G,\ldots)$. Then
the total space $\mathop{\rm EG}\nolimits$ of the universal principal
$G$-bundle is the geometric realization of the simplicial space $(G,\,G\times
G,\ldots)$. Consider the simplicial topological space
${\mathcal{E}}_{k,\,l}:=(\coprod_{m,\,n\geq 0}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}},\,\coprod_{m,\,n,\,r\geq 0}(\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}),\ldots),$
whose faces and degeneracies are defined by analogy with the construction of
$\mathop{\rm EG}\nolimits$. There is the map of simplicial spaces
$p\colon{\mathcal{E}}_{k,\,l}\rightarrow N{\mathcal{C}}_{k,\,l}.$
Now we are going to give a construction of the corresponding “universal
bundle”. The idea is to construct a “simplicial bundle” associated with the
“universal principal bundle” $p$.
More precisely, again consider the space ${\mathcal{E}}_{k,\,l}$:
$(\coprod_{m,\,n\geq 0}\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}},\,\coprod_{m,\,n,\,r\geq 0}(\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}),\,\coprod_{m,\,n,\,r,\,s\geq 0}(\mathop{\rm
Fr}\nolimits_{kl^{m+n+r},\,l^{s}}\times\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}),\ldots).$
Applying the natural (“evaluation”) maps $\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times M_{kl^{m}}(\mathbb{C})\rightarrow
M_{kl^{m+n}}(\mathbb{C})$, we obtain
(16) $(\coprod_{m\geq 0}M_{kl^{m}}(\mathbb{C}),\,\coprod_{m,\,n\geq
0}(\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})),\,\coprod_{m,\,n,\,r\geq 0}(\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times M_{kl^{m}}(\mathbb{C})),\ldots).$
The obtained object can be regarded as a “simplicial bundle” over (15). Now
performing the appropriate factorizations we should define its geometric
realization. Namely, the matrix algebras from the first disjoint union in (16)
are fibers of our bundle over $0$-cells of the space $\mathop{\rm
B}\nolimits{\mathcal{C}}_{k,\,l}$, from the second over $1$-cells, etc. The
attachment of $1$-cells to their source corresponds to the projection
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{m}}(\mathbb{C})$ onto the second
factor, and the one to their target corresponds to the natural map
$\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{m+n}}(\mathbb{C})$.
$M_{kl^{m}}(\mathbb{C})$-bundles over $2$-simplices correspond to the products
$\mathop{\rm Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times M_{kl^{m}}(\mathbb{C})$ and they are
identified over the boundary as follows:
$\overline{\partial}_{0}\colon\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times M_{kl^{m+n}}(\mathbb{C})$ is induced by
the natural map $\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{m+n}}(\mathbb{C})$,
$\overline{\partial}_{1}\colon\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}\times M_{kl^{m}}(\mathbb{C})$ by the
composition of morphisms $\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}$, $\overline{\partial}_{2}\colon\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\times M_{kl^{m}}(\mathbb{C})$ by the projection
onto the product of the second and the third factors. For bundles over
simplices of greater dimension the attaching maps are defined analogously.
The obtained topological space has the natural projection onto the space
$\mathop{\rm B}\nolimits{\mathcal{C}}_{k,\,l}$ with fibres
$M_{kl^{m}}(\mathbb{C})$.
Note that in the same way as ${\mathcal{C}}_{k,\,l}$ one can define categories
${\mathcal{C}}_{k^{m},\,l},\;m\geq 1.$ The objects of
${\mathcal{C}}_{k^{m},\,l}$ are matrix algebras
$M_{k^{m}l^{n}}(\mathbb{C}),\;n\geq 0$. Note that the tensor product of matrix
algebras gives rise to the bifunctor
$T_{m,\,n}\colon{\mathcal{C}}_{k^{m},\,l}\times{\mathcal{C}}_{k^{n},\,l}\rightarrow{\mathcal{C}}_{k^{m+n},\,l}$.
Thus, on objects we have:
$T_{m,\,n}(M_{k^{m}l^{r}}(\mathbb{C}),\,M_{k^{n}l^{s}}(\mathbb{C}))=M_{k^{m}l^{r}}(\mathbb{C})\otimes
M_{k^{n}l^{s}}(\mathbb{C})=M_{k^{m+n}l^{r+s}}(\mathbb{C})$, and on morphisms
$h_{1}\colon M_{k^{m}l^{r}}(\mathbb{C})\rightarrow
M_{k^{m}l^{r+t}}(\mathbb{C}),\;h_{2}\colon
M_{k^{n}l^{s}}(\mathbb{C})\rightarrow M_{k^{n}l^{s+u}}(\mathbb{C})\quad
T_{m,\,n}(h_{1},\,h_{2})=h_{1}\otimes h_{2}\colon
M_{k^{m+n}l^{r+s}}(\mathbb{C})\rightarrow M_{k^{m+n}l^{r+s+t+u}}(\mathbb{C}).$
Moreover, the bifunctor $T_{m,\,n}$ determines the continuous map of the
topological spaces $\mathop{\rm
Fr}\nolimits_{k^{m}l^{r},\,l^{t}}\times\mathop{\rm
Fr}\nolimits_{k^{n}l^{s},\,l^{u}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{m+n},\,l^{r+s+t+u}},\;(h_{1},\,h_{2})\mapsto h_{1}\otimes
h_{2}$.
For the topological bicategory
${\mathcal{C}}_{k^{m},\,l}\times{\mathcal{C}}_{k^{n},\,l}$ one can define the
bisimplicial topological space $X=\\{X_{p,\,q}\\},$ where $X_{p,\,q}$ consists
of all pairs of functors ${\bf p+1}\rightarrow{\mathcal{C}}_{k^{m},\,l},\;{\bf
q+1}\rightarrow{\mathcal{C}}_{k^{n},\,l}.$ There are two types of face and
degeneracy maps: “horizontal” and “vertical” which correspond to the category
${\mathcal{C}}_{k^{m},\,l}$ and ${\mathcal{C}}_{k^{n},\,l}$ respectively.
Clearly that its geometric realization $\mathop{\rm B}\nolimits X$ (to any
point of $X_{p,\,q}$ we attach $\Delta^{p}\times\Delta^{q}$) is $\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{m},\,l}\times\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{n},\,l}.$ The bifunctor $T_{m,\,n}$ determines
the continuous map $\widetilde{T}_{m,\,n}\colon\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{m},\,l}\times\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{n},\,l}\rightarrow\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{m+n},\,l}.$ It seems that the family of such
bifunctors with different $m,\,n$ defines a structure of $H$-space on the
direct limit $\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{m},\,l}$ (the maps $\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{m},\,l}\rightarrow\mathop{\rm
B}\nolimits{\mathcal{C}}_{k^{n},\,l}$ are given by functors
${\mathcal{C}}_{k^{m},\,l}\rightarrow{\mathcal{C}}_{k^{n},\,l}$ which on
objects are defined by the tensor product of matrix algebras
$M_{k^{m}l^{r}}(\mathbb{C}),\;r\geq 0$ by the fixed $M_{k^{n-m}}(\mathbb{C})$
and on morphisms by the assignment $h\mapsto h\otimes\mathop{\rm
id}\nolimits_{M_{k^{n-m}}(\mathbb{C})}$).
In conclusion of this subsection we describe yet another two properties of the
category ${\mathcal{C}}_{k,\,l}$.
Let ${\mathcal{C}}_{k,\,l}^{a}$ be the category with the same objects as
${\mathcal{C}}_{k,\,l}$ but with morphisms that are automorphisms in
${\mathcal{C}}_{k,\,l}$. Clearly, $\mathop{\rm
B}\nolimits{\mathcal{C}}_{k,\,l}^{a}\simeq\coprod_{n\geq 0}\mathop{\rm
BPU}\nolimits(kl^{n})$ and the embedding $\coprod_{n\geq 0}\mathop{\rm
BPU}\nolimits(kl^{n})\rightarrow\mathop{\rm B}\nolimits{\mathcal{C}}_{k,\,l}$
corresponds to the inclusion of the subcategory
${\mathcal{C}}_{k,\,l}^{a}\rightarrow{\mathcal{C}}_{k,\,l}$.
Let ${\bf N}$ be the category with countable set of objects
$\\{0,\,1,\,2,\,\ldots\\}$ and there is a morphism from $i$ to $j$ iff $i\leq
j$, and such morphism is unique. It is easy to see that its classifying space
$\mathop{\rm B}\nolimits{\bf N}$ is the infinite simplex $\Delta$.
There is the obvious functor $F\colon{\mathcal{C}}_{k,\,l}\rightarrow{\bf
N},\;F(M_{kl^{m}}(\mathbb{C}))=m$ and for a morphism $h\colon
M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{n}}(\mathbb{C})$ the morphism $F(h)$
is the unique morphism $m\rightarrow n$ in ${\bf N}$. Therefore there is the
corresponding map of classifying spaces $\mathop{\rm
B}\nolimits{\mathcal{C}}_{k,\,l}\rightarrow\mathop{\rm B}\nolimits{\bf N}.$
The subspace $\coprod_{n\geq 0}\mathop{\rm
BPU}\nolimits(kl^{n})\subset\mathop{\rm B}\nolimits{\mathcal{C}}_{k,\,l}$
corresponds to the vertices of the simplex $\mathop{\rm B}\nolimits{\bf N}$
(more precisely, to the corresponding discrete category). In general,
simplices degenerate under this map of classifying spaces, moreover, their
degenerations correspond to automorphisms of objects of
${\mathcal{C}}_{k,\,l}$ and nondegenerate simplices correspond to chains
$\\{h_{0},\,h_{1},\,\ldots h_{n}\\}$ of composable morphisms such that for all
$h_{r}$ the source and the target are different objects (in other words,
$h_{i}\in\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n}},\;n\neq 0$).
### 3.2. The definition of $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$
Now consider new topological category $\overline{\mathcal{C}}_{k,\,l}$. It has
a unique object
$M_{kl^{\infty}}(\mathbb{C})=\lim\limits_{\longrightarrow\atop{m}}M_{kl^{m}}(\mathbb{C})$
(i.e. $\overline{\mathcal{C}}_{k,\,l}$ is actually a monoid), where the direct
limit is taken over unital $*$-homomorphisms
$M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{m+1}}(\mathbb{C}),\;X\mapsto
X\otimes E_{l}.$ More precisely, we assume that the matrix algebra
$M_{kl^{\infty}}(\mathbb{C})$ is given together with the infinite family of
$*$-subalgebras $A_{k}\subset A_{kl}\subset A_{kl^{2}}\subset\ldots,$ where
$A_{kl^{m}}=M_{kl^{m}}(\mathbb{C})$ and $A_{kl^{m+1}}=M_{l}(A_{kl^{m}})$ for
every $m\geq 0$, which form its filtration.
By definition, for each morphism $h\colon
M_{kl^{\infty}}(\mathbb{C})\rightarrow
M_{kl^{\infty}}(\mathbb{C}),\;h\in\mathop{\rm
Mor}\nolimits(\overline{\mathcal{C}}_{k,\,l})$ there exists a pair
$m,\,n,\>n\geq m$ such that 1) $h|_{A_{kl^{m}}}$ is a unital $*$-homomorphism
$h|_{A_{kl^{m}}}\colon A_{kl^{m}}\rightarrow A_{kl^{n}}$ and 2)
$h=M_{l^{\infty}}(h|_{A_{kl^{m}}})$, i.e.
$h|_{A_{kl^{m+1}}}=M_{l}(h|_{A_{kl^{m}}})\colon
A_{kl^{m+1}}=M_{l}(A_{kl^{m}})\rightarrow A_{kl^{n+1}}=M_{l}(A_{kl^{n}}),$
etc. In other words, $h$ is induced by some $h^{\prime}\in\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n-m}}=\mathop{\rm
Hom}\nolimits_{alg}(A_{kl^{m}},\,A_{kl^{n}})$. For example, the composition of
morphisms induced by $h_{1}\colon A_{kl}\rightarrow A_{kl^{2}}$ and
$h_{2}\colon A_{k}\rightarrow A_{kl}$ is defined by the following diagram
$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{\ldots}$$\textstyle{A_{kl^{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{4}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l^{2}}(h)}$$\textstyle{A_{kl^{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l^{3}}(h_{2})}$$\textstyle{A_{kl^{3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l}(h_{1})}$$\textstyle{A_{kl^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l^{2}}(h_{2})}$$\textstyle{A_{kl^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{kl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{h_{1}}$$\textstyle{A_{kl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\qquad\qquad\qquad\qquad
M_{l}(h_{2})}$$\textstyle{A_{kl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{A_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{h_{2}}$$\textstyle{A_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$
as the class of the homomorphism $M_{l^{2}}(h_{2})\circ h_{1}.$ Clearly, the
composition of morphisms is well-defined and associative and the identity
morphism is $M_{l^{\infty}}(\mathop{\rm id}\nolimits_{A_{k}}),$ i.e. the
family $\\{\mathop{\rm id}\nolimits_{A_{k}},\,\mathop{\rm
id}\nolimits_{A_{kl}},\,\mathop{\rm id}\nolimits_{A_{kl^{2}}},\,\ldots\,\\}$.
Now we define the functor
$\Phi\colon{\mathcal{C}}_{k,\,l}\rightarrow\overline{\mathcal{C}}_{k,\,l}$
which sends every object $M_{kl^{m}}(\mathbb{C})$ in ${\mathcal{C}}_{k,\,l}$
to the unique object $M_{kl^{\infty}}(\mathbb{C})$ in
$\overline{\mathcal{C}}_{k,\,l}$. For a morphism $h\in\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n-m}}$ we put $\Phi(h)=M_{l^{\infty}}(h).$
Thus, we see that $\mathop{\rm Mor}\nolimits(\overline{\mathcal{C}}_{k,\,l})$
is the well-pointed grouplike (because $\pi_{0}(\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}})=0$) topological monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}.$ Recall [4] that for such a monoid
$M$ there exists the classifying space $\mathop{\rm B}\nolimits M$. Thus we
have the classifying space $\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ which is defined uniquely up to
$CW$-equivalence and there is the Whitehead equivalence $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}\rightarrow\Omega\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$; in
particular, $\pi_{i}(\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}})=\pi_{i+1}(\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}})$. Note that
$\Phi$ defines a continuous map $\widetilde{\Phi}\colon\mathop{\rm
B}\nolimits{\mathcal{C}}_{k,\,l}\rightarrow\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}.$ Moreover, the maps
$\widetilde{T}_{m,\,n}$ correspond to the maps $\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{m}l^{\infty},\,l^{\infty}}\times\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{n}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{m+n}l^{\infty},\,l^{\infty}}$ which
are given by maps $\mathop{\rm
Fr}\nolimits_{k^{m}l^{\infty},\,l^{\infty}}\times\mathop{\rm
Fr}\nolimits_{k^{n}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{m+n}l^{\infty},\,l^{\infty}}$ induced by the tensor product
of matrix algebras.
Note that there is the subgroup $\mathop{\rm
PU}\nolimits(kl^{\infty})=\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
PU}\nolimits(kl^{m})$ in the monoid $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ which corresponds to automorphisms of
$M_{kl^{\infty}}(\mathbb{C})$. The corresponding map $\mathop{\rm
BPU}\nolimits(kl^{\infty})\rightarrow\mathop{\rm B}\nolimits\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ has the homotopy fiber $\mathop{\rm
Gr}\nolimits_{kl^{\infty},\,l^{\infty}}$ (this follows from the fibration
$\mathop{\rm PU}\nolimits(kl^{m})\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}=\mathop{\rm
PU}\nolimits(kl^{m+n})/(E_{kl^{m}}\otimes\mathop{\rm
PU}\nolimits(l^{n}))\rightarrow\mathop{\rm Gr}\nolimits_{kl^{m},\,l^{n}}).$
###### Remark 11.
Retrospectively, we note that we have used the maps $\mathop{\rm
Fr}\nolimits_{kl^{m+n},\,l^{r}}\times\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n}}\rightarrow\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n+r}}$ (given by the composition of morphisms) in
order to define the monoid structure on $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ and thereby the classifying space
$\mathop{\rm B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$,
and the maps $\mathop{\rm Fr}\nolimits_{k^{m}l^{r},\,l^{s}}\times\mathop{\rm
Fr}\nolimits_{k^{n}l^{t},\,l^{u}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{m+n}l^{r+t},\,l^{l^{s+u}}}$ given by the tensor product in
order to define the additional structure on the spaces $\mathop{\rm
Fr}\nolimits_{k^{m}l^{\infty},\,l^{\infty}}$ (which gives rise to the
$H$-space structure on $\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
Fr}\nolimits_{k^{m}l^{\infty},\,l^{\infty}}$). From the category-theoretic
point of view the first corresponds to the composition of morphisms in the
category and the second to the monoidal structure on it.
### 3.3. The action of $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$
on the space of Fredholm operators
Recall (13) that there are evaluation maps
$ev_{kl^{m},\,l^{n-m}}\colon\mathop{\rm Fr}\nolimits_{kl^{m},\,l^{n-m}}\times
M_{kl^{m}}(\mathbb{C})\rightarrow M_{kl^{n}}(\mathbb{C})$
and the corresponding maps (recall that $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ is the subspace of Fredholm operators
in $M_{kl^{m}}({\mathcal{B}}({\mathcal{H}}))$)
$\gamma^{\prime}_{kl^{m},\,l^{n-m}}\colon\mathop{\rm
Fr}\nolimits_{kl^{m},\,l^{n-m}}\times\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{n}}({\mathcal{H}})$
(see (14)). Using the filtration in
$M_{kl^{\infty}}({\mathcal{B}}({\mathcal{H}}))$ (and hence in
$\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$) corresponding to the above filtration
$A_{k}\subset A_{kl}\subset A_{kl^{2}}\subset\ldots$ in the matrix algebra
$M_{kl^{\infty}}(\mathbb{C})$ one can define the action of the monoid
$\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$ on
$\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$. Note that since the direct limit is
taken over maps induced by the tensor product, we see that $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}}):=\lim\limits_{\longrightarrow\atop{m}}\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})$ is the localization in which $l$
becomes invertible (in particular, the index takes values in
$\mathbb{Z}[\frac{1}{l}]$, not in $\mathbb{Z}$). Thereby we have defined the
required action
(17) $\gamma^{\prime}_{kl^{\infty},\,l^{\infty}}\colon\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}\times\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$
of the monoid $\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$.
Note that the action (17) gives rise to the action on $K$-theory (which is
recall represented by the space of Fredholm operators) which corresponds to
the action of the $k$-torsion subgroup in $\mathop{\rm BU}\nolimits_{\otimes}$
by tensor products (cf. Proposition 14 and Theorem 7). In fact, this action is
defined on $K$-theory $K[\frac{1}{l}]$ localized over $l$ (in the sense that
$l$ becomes invertible). This is not surprising because in (14) we take the
tensor product of $K(X)$ by some $l$-dimensional bundle, $l>1$. It is not
difficult to show that in fact our construction does not depend on the choice
of $l,\,(k,\,l)=1$.
Note that the restriction of the action
$\gamma^{\prime}_{kl^{\infty},\,l^{\infty}}$ on $\mathop{\rm
Fr}\nolimits_{kl^{m},\,1}\cong\mathop{\rm PU}\nolimits(kl^{m})$ coincides with
the composition of the action $\gamma^{\prime}_{kl^{m}}$ (see Remark 3) and
the localization map $\mathop{\rm
Fred}\nolimits_{kl^{m}}({\mathcal{H}})\rightarrow\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$ on $l$.
Using this action (17) we can define the $\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})$-bundle
$\textstyle{\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
EFr}_{kl^{\infty},\,l^{\infty}}{\mathop{\times}\limits_{\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}}}\mathop{\rm
Fred}\nolimits_{kl^{\infty}}({\mathcal{H}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
BFr}_{kl^{\infty},\,l^{\infty}}}$
“associated” with the universal principal $\mathop{\rm
Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$-bundle (more precisely, with the
universal principal quasi-fibration, see [4]) over $\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{kl^{\infty},\,l^{\infty}}$. This allows
us to define a more general version of the twisted $K$-theory than the one
given by the action of $Pic(X)$ on $K(X)$. The above defined maps $\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{m}l^{\infty},\,l^{\infty}}\times\mathop{\rm
B}\nolimits\mathop{\rm
Fr}\nolimits_{k^{n}l^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
B}\nolimits\mathop{\rm Fr}\nolimits_{k^{m+n}l^{\infty},\,l^{\infty}}$ give
rise to the operation on twistings which is an analog of the one induced by
maps $\mathop{\rm BPU}\nolimits(k^{m})\times\mathop{\rm
BPU}\nolimits(k^{n})\rightarrow\mathop{\rm BPU}\nolimits(k^{m+n})$ in the
Abelian case (i.e. the Brauer group), etc.
## 4\. Appendix: $H$-space $\mathop{\rm
Fr}\nolimits_{k^{\infty},\,l^{\infty}}$
In this section we give a category-theoretic description of the structure of
$H$-space on $\mathop{\rm Fr}\nolimits_{k^{\infty},\,l^{\infty}}$ and
$\mathop{\rm Gr}\nolimits_{k^{\infty},\,l^{\infty}}$.
### 4.1. $k^{m}$-frames
###### Definition 12.
A $k^{m}$-frame $\alpha$ in the algebra $M_{k^{m}l^{n}}(\mathbb{C})$ is an
ordered collection of $k^{2m}$ linearly independent matrices
$\\{\alpha_{i,\,j}\\}_{1\leq i,\,j\leq k^{m}}$ such that
* (i)
$\alpha_{i,\,j}\alpha_{r,\,s}=\delta_{j,\,r}\alpha_{i,\,s}$ for all $1\leq
i,\,j,\,r,\,s\leq k^{m}$;
* (ii)
$\sum_{i=1}^{k^{m}}\alpha_{i,\,i}=E,$ where $E=E_{k^{m}l^{n}}$ is the unit
$k^{m}l^{n}\times k^{m}l^{n}$-matrix which is the unit of the algebra
$M_{k^{m}l^{n}}(\mathbb{C})$;
* (iii)
matrices $\\{\alpha_{i,\,j}\\}$ form an orthonormal basis with respect to the
hermitian inner product $(x,\,y):=\mathop{\rm tr}\nolimits(x\overline{y}^{t})$
on $M_{k^{m}l^{n}}(\mathbb{C})$.
For instance, the collection of “matrix units” $\\{e_{i,\,j}\\}_{1\leq
i,\,j\leq k^{m}}$ (where $e_{i,\,j}$ is the $k^{m}\times k^{m}$-matrix whose
only nonzero element is 1 on the intersection of ith row with jth column) is a
$k^{m}$-frame in $M_{k^{m}}(\mathbb{C})$, and the collection
$\\{e_{i,\,j}\otimes E_{l^{n}}\\}_{1\leq i,\,j\leq k^{m}}$ is a $k^{m}$-frame
in $M_{k^{m}l^{n}}(\mathbb{C})$. Clearly, every $k^{m}$-frame in
$M_{k^{m}l^{n}}(\mathbb{C})$ is a linear basis in some $k^{m}$-subalgebra.
###### Proposition 13.
The set of all $k^{m}$-frames in $M_{k^{m}l^{n}}(\mathbb{C})$ is the
homogeneous space $\mathop{\rm
PU}\nolimits(k^{m}l^{n})/(E_{k^{m}}\otimes\mathop{\rm PU}\nolimits(l^{n})).$
Proof follows from two facts: 1) the group $\mathop{\rm
PU}\nolimits(k^{m}l^{n})$ of $*$-automorphisms of the algebra
$M_{k^{m}l^{n}}(\mathbb{C})$ acts transitively on the set of $k^{m}$-frames,
and 2) the stabilizer of the $k^{m}$-frame $\\{e_{i,\,j}\otimes
E_{l^{n}}\\}_{1\leq i,\,j\leq k^{m}}$ is the subgroup
$E_{k^{m}}\otimes\mathop{\rm PU}\nolimits(l^{n})\subset\mathop{\rm
PU}\nolimits(k^{m}l^{n}).\quad\square$
In fact, the space of $k^{m}$-frames $\mathop{\rm
Fr}\nolimits_{k^{m},\,l^{n}}$ in $M_{k^{m}l^{n}}(\mathbb{C})$ is isomorphic to
the space of unital $*$-homomorphisms $\mathop{\rm
Hom}\nolimits_{alg}(M_{k^{m}}(\mathbb{C}),\,M_{k^{m}l^{n}}(\mathbb{C})).$ More
precisely, let $\\{e_{i,\,j}\\}_{1\leq i,\,j\leq k^{m}}$ be the frame in
$M_{k^{m}}(\mathbb{C})$ consisting of matrix units. Then the isomorphism
$\mathop{\rm Fr}\nolimits_{k^{m},\,l^{n}}\cong\mathop{\rm
Hom}\nolimits_{alg}(M_{k^{m}}(\mathbb{C}),\,M_{k^{m}l^{n}}(\mathbb{C}))$ is
given by the assignment
$\alpha\mapsto h_{\alpha}\colon M_{k^{m}}(\mathbb{C})\rightarrow
M_{k^{m}l^{n}}(\mathbb{C}),\;(h_{\alpha})_{*}(\\{e_{i,\,j}\\})=\alpha\quad\forall\alpha\in\mathop{\rm
Fr}\nolimits_{k^{m},\,l^{n}}.$
Let $\beta$ be a $k^{r}$-frame in $M_{k^{r}l^{s}}(\mathbb{C})$ and $m\leq r.$
Then one can associate with $\beta$ some new $k^{m}$-frame
$\alpha:=\pi_{1}^{m}(\beta)$ as follows:
$\alpha_{i,\,j}=\beta_{(i-1)k^{r-m}+1,\,(j-1)k^{r-m}+1}+\beta_{(i-1)k^{r-m}+2,\,(j-1)k^{r-m}+2}+\ldots+\beta_{ik^{r-m},\,jk^{r-m}},\quad
1\leq i,\,j\leq k^{m}.$
Also one can associate with $\beta$ some $k^{r-m}$-frame
$\gamma:=\pi_{2}^{r-m}(\beta)$ by the following rule:
$\gamma_{i,\,j}=\beta_{i,\,j}+\beta_{i+k^{r-m},\,j+k^{r-m}}+\ldots+\beta_{i+(k^{m}-1)k^{r-m},\,j+(k^{m}-1)k^{r-m}},\quad
1\leq i,\,j\leq k^{r-m}.$
The idea of the definition of $\pi_{1}^{m}(\beta)$ and $\pi_{2}^{r-m}(\beta)$
is the following. If one takes the $k^{r}$-frame $\epsilon$ in
$M_{k^{r}}(\mathbb{C})=M_{k^{m}}(\mathbb{C})\otimes M_{k^{r-m}}(\mathbb{C})$
consisting of the matrix units, then the $k^{m}$ and $k^{r-m}$-frames in
subalgebras $M_{k^{m}}(\mathbb{C})\otimes\mathbb{C}E_{k^{r-m}}\subset
M_{k^{r}}(\mathbb{C})$ and $\mathbb{C}E_{k^{m}}\otimes
M_{k^{r-m}}(\mathbb{C})\subset M_{k^{r}}(\mathbb{C})$ consisting of the matrix
units tensored by the corresponding unit matrices are $\pi_{1}^{m}(\epsilon)$
and $\pi_{2}^{r-m}(\epsilon)$ respectively. From the other hand it is easy to
see that the frame $\epsilon$ (under the appropriate ordering) is the tensor
product of the frames of matrix units in the tensor factors
$M_{k^{m}}(\mathbb{C})$ and $M_{k^{r-m}}(\mathbb{C})$. The matrices from
$\pi_{1}^{m}(\epsilon)$ commute with the matrices from
$\pi_{2}^{r-m}(\epsilon)$, moreover, all possible pairwise products of the
matrices from $\pi_{1}^{m}(\epsilon)$ by the matrices from
$\pi_{2}^{r-m}(\epsilon)$ (we have exactly $k^{2m}\cdot k^{2(r-m)}=k^{2r}$
such products) give all matrices from the frame $\epsilon.$ If we order the
collection of products in the appropriate way, we get the frame $\epsilon.$
The operation which to a pair consisting of commuting $k^{m}$ and
$k^{r-m}$-frames assigns (according to this rule) the $k^{r}$-frame we will
denote by dot $\cdot$. In particular,
$\beta=\pi_{1}^{m}(\beta)\cdot\pi_{2}^{r-m}(\beta)$ for any $k^{r}$-frame
$\beta.$
Thereby we have defined the continuous maps $\pi_{1}^{m}\colon\mathop{\rm
Fr}\nolimits_{k^{r},\,l^{s}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{m},\,k^{r-m}l^{s}}$ and $\pi_{2}^{r-m}\colon\mathop{\rm
Fr}\nolimits_{k^{r},\,l^{s}}\rightarrow\mathop{\rm
Fr}\nolimits_{k^{r-m},\,k^{m}l^{s}}.$ In terms of algebra homomorphisms they
correspond to the assignment to a homomorphism $h\colon
M_{k^{r}}(\mathbb{C})\rightarrow M_{k^{r}l^{s}}(\mathbb{C})$ its compositions
with homomorphisms $M_{k^{m}}(\mathbb{C})\rightarrow
M_{k^{r}}(\mathbb{C}),\;X\mapsto X\otimes E_{k^{r-m}}$ and
$M_{k^{r-m}}(\mathbb{C})\rightarrow M_{k^{r}}(\mathbb{C}),\;X\mapsto
E_{k^{m}}\otimes X$ respectively.
### 4.2. Functor $\mathop{\rm Fr}\nolimits$
In this subsection we define a functor $\mathop{\rm Fr}\nolimits$ from some
monoidal category $\mathcal{C}_{k,\,l}$ to the category of topological spaces
with a chosen basepoint.
Let us fix an ordered pair of positive integers $k,\,l,\>(k,\,l)=1,\>k,\,l>1.$
Define the category $\mathcal{C}_{k,\,l}$ whose objects are pairs of the form
$(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)$, consisting of a matrix algebra
$M_{k^{m}l^{n}}(\mathbb{C}),\,m,\,n\geq 0$ and a $k^{m}$-frame $\alpha$ in it.
A morphism
$f\colon(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\rightarrow(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta)$
is a unital $*$-homomorphism of matrix algebras $f\colon
M_{k^{m}l^{n}}(\mathbb{C})\rightarrow M_{k^{r}l^{s}}(\mathbb{C})$ such that
$f_{*}(\alpha)=\pi_{1}^{m}(\beta),$ where by $f_{*}$ we denote the map induced
on frames by $f$. Equivalently, we have the commutative diagram
$\textstyle{M_{k^{m}l^{n}}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M_{k^{r}l^{s}}(\mathbb{C})}$$\textstyle{M_{k^{m}}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{r,\,m}}$$\scriptstyle{h_{\alpha}}$$\textstyle{M_{k^{r}}(\mathbb{C}),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\beta}}$
where $i_{r,\,m}\colon M_{k^{m}}(\mathbb{C})\rightarrow M_{k^{r}}(\mathbb{C})$
is the homomorphism $X\mapsto X\otimes E_{k^{r-m}}$.
Note that $\mathcal{C}_{k,\,l}$ is a symmetric monoidal category with respect
to the bifunctor $\otimes$:
$((M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha),\,(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta))\mapsto(M_{k^{m}l^{n}}(\mathbb{C})\otimes
M_{k^{r}l^{s}}(\mathbb{C}),\,\alpha\otimes\beta)$
and the unit object $e:=(M_{1}(\mathbb{C})=\mathbb{C},\,\varepsilon),$ where
$\varepsilon=1$ is the unique $k^{0}=1$-frame.
Now let us define a functor $\mathop{\rm Fr}\nolimits$ from
$\mathcal{C}_{k,\,l}$ to the category of topological spaces with a chosen
basepoint. On objects $\mathop{\rm
Fr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)$ is the space of
$k^{m}$-frames in $M_{k^{m}l^{n}}(\mathbb{C})$, where $\alpha$ gives the
basepoint. For a morphism
$f\colon(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\rightarrow(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta)$
put
$\mathop{\rm Fr}\nolimits(f)\colon\mathop{\rm
Fr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\rightarrow\mathop{\rm
Fr}\nolimits(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta),\quad\mathop{\rm
Fr}\nolimits(f)(\alpha^{\prime})=f_{*}(\alpha^{\prime})\cdot\pi_{2}^{r-m}(\beta).$
Then $\mathop{\rm Fr}\nolimits(f)$ is a well-defined continuous map preserving
basepoints.
Consider a few particular cases.
1) Suppose $m=0,$ then $\mathop{\rm
Fr}\nolimits(M_{l^{n}}(\mathbb{C}),\,\varepsilon)=\\{\varepsilon\\}$ is the
space consisting of one point, and for a morphism
$f\colon(M_{l^{n}}(\mathbb{C}),\,\varepsilon)\rightarrow(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta)$
the induced map
$\mathop{\rm Fr}\nolimits(f)\colon\mathop{\rm
Fr}\nolimits(M_{l^{n}}(\mathbb{C}),\,\varepsilon)\rightarrow\mathop{\rm
Fr}\nolimits(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta),\quad\varepsilon\mapsto\varepsilon\cdot\beta=\beta$
is the inclusion of the basepoint (note that $\pi^{0}_{1}(\beta)=\varepsilon,$
cf. Definition 12, (ii)).
2) Suppose $n=0,$ then $\mathop{\rm
Fr}\nolimits(M_{k^{m}}(\mathbb{C}),\,\alpha)=\mathop{\rm PU}\nolimits(k^{m})$
and $\alpha$ corresponds to the unit in the group $\mathop{\rm
PU}\nolimits(k^{m})$. For a morphism
$f\colon(M_{k^{m}}(\mathbb{C}),\,\alpha)\rightarrow(M_{k^{r}}(\mathbb{C}),\,\beta)$
the diagram
$\textstyle{\mathop{\rm
Fr}\nolimits(M_{k^{m}}(\mathbb{C}),\,\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\rm
Fr}\nolimits(f)}$$\scriptstyle{=}$$\textstyle{\mathop{\rm
Fr}\nolimits(M_{k^{r}}(\mathbb{C}),\,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathop{\rm
PU}\nolimits(k^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ldots\otimes
E_{k^{r-m}}}$$\textstyle{\mathop{\rm PU}\nolimits(k^{r})}$
is commutative (the lower row corresponds to the homomorphism $X\mapsto
X\otimes E_{k^{r-m}}$).
3)
$r=m,\;f\colon(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\rightarrow(M_{k^{m}l^{s}}(\mathbb{C}),\,\beta)$.
Then $\beta=f_{*}(\alpha),\,\pi_{2}^{0}(\beta)=\varepsilon$ (cf. Definition
12, (ii)) $\mathop{\rm
Fr}\nolimits(f)(\alpha^{\prime})=f_{*}(\alpha^{\prime}).$
### 4.3. Natural transformation $\mu\colon\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm
Fr}\nolimits(\ldots)\rightarrow\mathop{\rm
Fr}\nolimits((\ldots)\otimes(\ldots))$
Using the bifunctor $\otimes$ on the category $\mathcal{C}_{k,\,l}$ we define
a natural transformation of functors $\mu\colon\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm
Fr}\nolimits(\ldots)\rightarrow\mathop{\rm
Fr}\nolimits((\ldots)\otimes(\ldots))$ from the category
$\mathcal{C}_{k,\,l}\times\mathcal{C}_{k,\,l}$ to the category of topological
spaces with a chosen basepoint. More precisely,
$\mu\colon\mathop{\rm
Fr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\times\mathop{\rm
Fr}\nolimits(M_{k^{p}l^{q}}(\mathbb{C}),\,\varphi)\rightarrow\mathop{\rm
Fr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C})\otimes
M_{k^{p}l^{q}}(\mathbb{C})),\,\alpha\otimes\varphi),$
$\mu(\alpha^{\prime},\,\varphi^{\prime})=\alpha^{\prime}\otimes\varphi^{\prime}$
(recall that $M_{k^{m}l^{n}}(\mathbb{C})\otimes
M_{k^{p}l^{q}}(\mathbb{C})\cong M_{k^{m+p}l^{n+q}}(\mathbb{C})$), where
$\alpha^{\prime}\otimes\varphi^{\prime}$ is the $k^{m+p}$-frame which is the
tensor product of the $k^{m}$-frame $\alpha^{\prime}$ and the $k^{p}$-frame
$\beta^{\prime}$.
In fact, $\mu$ is a natural transformation, because for any two morphisms in
$\mathcal{C}_{k,\,l}$
$f\colon(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\rightarrow(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta),\quad
g\colon(M_{k^{p}l^{q}}(\mathbb{C}),\,\varphi)\rightarrow(M_{k^{t}l^{u}}(\mathbb{C}),\,\psi)$
the diagram
(18) $\textstyle{\mathop{\rm
Fr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\times\mathop{\rm
Fr}\nolimits(M_{k^{p}l^{q}}(\mathbb{C}),\,\varphi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\mathop{\rm
Fr}\nolimits(f)\times\mathop{\rm Fr}\nolimits(g)}$$\textstyle{\mathop{\rm
Fr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C})\otimes
M_{k^{p}l^{q}}(\mathbb{C}),\,\alpha\otimes\varphi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\rm
Fr}\nolimits(f\otimes g)}$$\textstyle{\mathop{\rm
Fr}\nolimits(M_{k^{r}l^{s}}(\mathbb{C}),\,\beta)\times\mathop{\rm
Fr}\nolimits(M_{k^{t}l^{u}}(\mathbb{C}),\,\psi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\mathop{\rm
Fr}\nolimits(M_{k^{r}l^{s}}(\mathbb{C})\otimes
M_{k^{t}l^{u}}(\mathbb{C}),\,\beta\otimes\psi)}$
is commutative. Indeed, $\mu\circ(\mathop{\rm Fr}\nolimits(f)\times\mathop{\rm
Fr}\nolimits(g))(\alpha^{\prime},\,\varphi^{\prime})=\mu(f_{*}(\alpha^{\prime})\cdot\gamma,\,g_{*}(\varphi^{\prime})\cdot\chi)=(f_{*}(\alpha^{\prime})\cdot\gamma)\otimes(g_{*}(\varphi^{\prime})\cdot\chi),$
where $\gamma,\,\chi$ are $\pi_{2}^{r-m}(\beta)$ and $\pi_{2}^{t-p}(\psi)$
respectively. On the other hand, $\mathop{\rm Fr}\nolimits(f\otimes
g)\circ\mu(\alpha^{\prime},\,\varphi^{\prime})=\mathop{\rm
Fr}\nolimits(f\otimes
g)(\alpha^{\prime}\otimes\varphi^{\prime})=(f_{*}(\alpha^{\prime})\otimes
g_{*}(\varphi^{\prime}))\cdot\Xi,$ where $\Xi$ is the unique
$k^{r+t-m-p}$-frame such that $(f_{*}(\alpha)\otimes
g_{*}(\varphi))\cdot\Xi=\beta\otimes\psi$. But
$(f_{*}(\alpha^{\prime})\cdot\gamma)\otimes(g_{*}(\varphi^{\prime})\cdot\chi)=(f_{*}(\alpha^{\prime})\otimes
g_{*}(\varphi^{\prime}))\cdot(\gamma\otimes\chi)$ by virtue of the
commutativity of frames, and moreover
$(f_{*}(\alpha)\cdot\gamma)\otimes(g_{*}(\varphi)\cdot\chi)=\beta\otimes\psi.$
Hence $\Xi=\gamma\otimes\chi$ and the diagram commutes, as claimed.
### 4.4. Properties of the natural transformation $\mu$
First, the natural transformation $\mu$ is associative in the sense that the
functor diagram
$\textstyle{\mathop{\rm Fr}\nolimits(\ldots)\times(\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm Fr}\nolimits(\ldots))\cong(\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm Fr}\nolimits(\ldots))\times\mathop{\rm
Fr}\nolimits(\ldots)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\rm
id}\nolimits\times\mu}$$\scriptstyle{\quad\qquad\qquad\qquad\mu\times\mathop{\rm
id}\nolimits}$$\textstyle{\mathop{\rm
Fr}\nolimits((\ldots)\otimes(\ldots))\times\mathop{\rm
Fr}\nolimits(\ldots)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm
Fr}\nolimits((\ldots)\otimes(\ldots))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\mathop{\rm
Fr}\nolimits((\ldots)\otimes(\ldots)\otimes(\ldots))}$
commutes (we have used the natural isomorphism $\mathop{\rm
Fr}\nolimits((\ldots)\otimes((\ldots)\otimes(\ldots)))\cong\mathop{\rm
Fr}\nolimits(((\ldots)\otimes(\ldots))\otimes(\ldots))$ in the lower right
corner).
Secondly, we need the diagram for identity. Recall that in the monoidal
category $\mathcal{C}_{k,\,l}$ $\>e=(\mathbb{C},\,\varepsilon)$ is the unit
object, and it is also the initial object. In particular, for any object
$A=(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)$ there is a unique morphism
$\iota_{A}\colon e\rightarrow A$, i.e.
$\iota_{A}\colon(\mathbb{C},\,\varepsilon)\rightarrow(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)$.
The identity diagram has the following form:
$\textstyle{\mathop{\rm Fr}\nolimits(e)\times\mathop{\rm
Fr}\nolimits(\ldots)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\rm
Fr}\nolimits(\iota)\times\mathop{\rm id}\nolimits}$$\textstyle{\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm
Fr}\nolimits(\ldots)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm
Fr}\nolimits(e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\rm
id}\nolimits\times\mathop{\rm Fr}\nolimits(\iota)}$$\textstyle{\mathop{\rm
Fr}\nolimits((\ldots)\otimes(\ldots))}$
(note that $\mathop{\rm Fr}\nolimits(\iota)\colon\mathop{\rm
Fr}\nolimits(e)\rightarrow\mathop{\rm Fr}\nolimits(\ldots)$ is the inclusion
of the basepoint). It is easy to see that (for any pair of objects $A,\,B$ of
$\mathcal{C}_{k,\,l}$) the slanted arrows are homeomorphisms on their images.
There is also the commutativity diagram
$\textstyle{\mathop{\rm Fr}\nolimits(\ldots)\times\mathop{\rm
Fr}\nolimits(\ldots)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\mu}$$\textstyle{\mathop{\rm
Fr}\nolimits(\ldots)\times\mathop{\rm
Fr}\nolimits(\ldots)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\mathop{\rm
Fr}\nolimits((\ldots)\otimes(\ldots))}$
(where $\tau$ is the map which switches the factors) which is commutative up
to isomorphism. This gives us a homotopy $\mu\circ\tau\simeq\mu$.
For any pair $A,\,B$ of objects of ${\mathcal{C}}_{k,\,l}$ the natural
transformation $\mu$ determines a continuous map
$\widetilde{\mu}_{A,\,B}\colon\mathop{\rm Fr}\nolimits(A)\times\mathop{\rm
Fr}\nolimits(B)\to\mathop{\rm Fr}\nolimits(A\otimes B)$ of topological spaces.
Put $\mathop{\rm
Fr}\nolimits_{k^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{\\{f\\}}}\mathop{\rm
Fr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)$. Then the above diagrams
show that $\mathop{\rm Fr}\nolimits_{k^{\infty},\,l^{\infty}}$ is a homotopy
associative and commutative $H$-space with multiplication given by
$\widetilde{\mu}:=\lim\limits_{\longrightarrow\atop{A,\,B}}\widetilde{\mu}_{A,\,B}$
and with the homotopy unit
$\widetilde{\eta}:=\lim\limits_{\longrightarrow\atop{A}}\mathop{\rm
Fr}\nolimits(\iota_{A})\colon*=\mathop{\rm
Fr}\nolimits(e)\rightarrow\mathop{\rm Fr}\nolimits_{k^{\infty},\,l^{\infty}}$.
### 4.5. $H$-space structure on the matrix grassmannian
Note that the analogous construction can be applied to matrix grassmannians
(in place of frame spaces). Namely, consider the category
$\mathcal{D}_{k,\,l}$ whose objects are pairs of the form
$(M_{k^{m}l^{n}}(\mathbb{C}),\,A)$, consisting of a matrix algebra
$M_{k^{m}l^{n}}(\mathbb{C}),\;m,\,n\geq 0$ and a $k^{m}$-subalgebra $A\subset
M_{k^{m}l^{n}}(\mathbb{C})$ in it (recall that a $k$-subalgebra is a unital
$*$-subalgebra isomorphic $M_{k}(\mathbb{C})$). A morphism
$f\colon(M_{k^{m}l^{n}}(\mathbb{C}),\,A)\rightarrow(M_{k^{r}l^{s}}(\mathbb{C}),\,B)$
in $\mathcal{D}_{k,\,l}$ is a unital $*$-homomorphism of matrix algebras
$f\colon M_{k^{m}l^{n}}(\mathbb{C})\rightarrow M_{k^{r}l^{s}}(\mathbb{C})$
such that $f(A)\subset B.$ We define the $k^{r-m}$-subalgebra $C\subset
M_{k^{r}l^{s}}(\mathbb{C})$ as the centralizer of the subalgebra $f(A)$ in
$B$.
Define the functor $\mathop{\rm Gr}\nolimits$ from the category
$\mathcal{D}_{k,\,l}$ to the category of topological spaces with a chosen
basepoint as follows. On objects the space $\mathop{\rm
Gr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,A)$ is the space of all
$k^{m}$-subalgebras in $M_{k^{m}l^{n}}(\mathbb{C})$ and $A$ corresponds to its
basepoint. For a morphism
$f\colon(M_{k^{m}l^{n}}(\mathbb{C}),\,A)\rightarrow(M_{k^{r}l^{s}}(\mathbb{C}),\,B)$
as above we put
$\mathop{\rm Gr}\nolimits(f)\colon\mathop{\rm
Gr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,A)\rightarrow\mathop{\rm
Gr}\nolimits(M_{k^{r}l^{s}}(\mathbb{C}),\,B),\quad\mathop{\rm
Gr}\nolimits(f)(A^{\prime})=f(A^{\prime})\cdot C,$
where $C$ is the centralizer of the subalgebra $f(A)$ in $B$ and
$f(A^{\prime})\cdot C$ denotes the subalgebra in $M_{k^{r}l^{s}}(\mathbb{C})$
generated by subalgebras $f(A^{\prime})$ and $C$ (clearly, $B=f(A)\cdot C$).
Then one can define the analog $\mu^{\prime}\colon\mathop{\rm
Gr}\nolimits(\ldots)\times\mathop{\rm
Gr}\nolimits(\ldots)\rightarrow\mathop{\rm
Gr}\nolimits((\ldots)\otimes(\ldots))$ of the natural transformation $\mu$,
etc. (For example, the commutativity of the analog of diagram (18) follows
from the coincidence $Z_{B\otimes\Psi}(f(A)\otimes g(\Phi))=Z_{B}(f(A))\otimes
Z_{\Psi}(g(\Phi))$ for any two morphisms
$f\colon(M_{k^{m}l^{n}}(\mathbb{C}),\,A)\rightarrow(M_{k^{r}l^{s}}(\mathbb{C}),\,B)$
and
$g\colon(M_{k^{p}l^{q}}(\mathbb{C}),\,\Phi)\rightarrow(M_{k^{t}l^{u}}(\mathbb{C}),\,\Psi)$
(which is an analog of the above formula $\Xi=\gamma\otimes\chi$ for frames),
where $Z_{B}(A)$ denotes the centralizer of a subalgebra $A$ in an algebra
$B$.) This allows us to equip the direct limit $\mathop{\rm
Gr}\nolimits_{k^{\infty},\,l^{\infty}}:=\lim\limits_{\longrightarrow\atop{\\{f\\}}}\mathop{\rm
Gr}\nolimits(M_{k^{m}l^{n}}(\mathbb{C}),\,A)$ with the structure of a homotopy
associative and commutative $H$-space with a homotopy unit.
Note that there is the functor
$\lambda\colon\mathcal{C}_{k,\,l}\rightarrow\mathcal{D}_{k,\,l},\quad(M_{k^{m}l^{n}}(\mathbb{C}),\,\alpha)\mapsto(M_{k^{m}l^{n}}(\mathbb{C}),\,M(\alpha)),$
where $M(\alpha)$ is the $k^{m}$-subalgebra spanned on the $k^{m}$-frame
$\alpha$. There is the obvious natural transformation of functors
$\theta\colon\mathop{\rm Fr}\nolimits\rightarrow\mathop{\rm
Gr}\nolimits\circ\lambda$ from the category $\mathcal{C}_{k,\,l}$ to the
category of topological spaces with a chosen basepoint which gives rise to the
$H$-space homomorphism $\mathop{\rm
Fr}\nolimits_{k^{\infty},\,l^{\infty}}\rightarrow\mathop{\rm
Gr}\nolimits_{k^{\infty},\,l^{\infty}}$. Recall that $\mathop{\rm
Gr}\nolimits_{k^{\infty},\,l^{\infty}}=\mathop{\rm
Gr}\nolimits\cong\mathop{\rm BSU}\nolimits_{\otimes}$, and the image of the
just constructed homomorphism is the $k$-torsion subgroup in it, as the next
proposition claims.
###### Proposition 14.
Let $X$ be a compact space. Then the image of the homomorphism
$[X,\,\mathop{\rm
Fr}\nolimits_{k^{\infty},\,l^{\infty}}]\rightarrow[X,\,\mathop{\rm
Gr}\nolimits_{k^{\infty},\,l^{\infty}}]$ is the $k$-torsion subgroup in the
group $bsu^{0}_{\otimes}(X).$
Proof. This proposition follows from Proposition 5. Another way is to pass to
the direct limit in fibration
$\mathop{\rm Fr}\nolimits_{k^{n},\,l^{n}}\to\mathop{\rm
Gr}\nolimits_{k^{n},\,l^{n}}\to\mathop{\rm BPU}\nolimits(k^{n}),$
and to notice that the limit map $\mathop{\rm
Gr}\nolimits_{k^{\infty},\,l^{\infty}}\to\mathop{\rm
BPU}\nolimits(k^{\infty}):=\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
BPU}\nolimits(k^{n})$ actually is a localization on $k.\quad\square$
## References
* [1] M. Atiyah, G. Segal Twisted K-theory // arXiv:math/0407054v2 [math.KT]
* [2] M. Atiyah, G. Segal Twisted K-theory and cohomology // arXiv:math/0510674v1 [math.KT]
* [3] R.S. Pierce: Associative Algebras. Springer Verlag, 1982.
* [4] Yu.B. Rudyak: On Thom Spectra, Orientability and Cobordism. Springer Monogr. in Math., Springer (1998).
* [5] G.B. Segal: Categories and cohomology theories. Topology 13 (1974).
* [6] C. Weibel The K-book: An introduction to algebraic K-theory.
* [7] A.V. Ershov Homotopy theory of bundles with fiber matrix algebra // arXiv:math/0301151v1 [math.AT]
* [8] A.V. Ershov Topological obstructions to embedding of a matrix algebra bundle into a trivial one // arXiv:0807.3544v13 [math.KT]
|
arxiv-papers
| 2009-11-20T20:54:33 |
2024-09-04T02:49:06.598209
|
{
"license": "Public Domain",
"authors": "A.V. Ershov",
"submitter": "Andrey V. Ershov",
"url": "https://arxiv.org/abs/0911.4113"
}
|
0911.4179
|
# Estimation of characteristic size of ferromagnetic clusters forming above
$T_{C}$ in Nd0.75Ba0.25MnO3 manganite
A.V. Lazuta, V.A. Ryzhov, and V.V. Runov Petersburg Nuclear Physics Institute
of RAS, Gatchina, St. Petersburg, 188300, Russia I.O. Troaynchuk Institute
of Physics of Solids and Semiconductors, National Academy of Sciences, ul. P.
Brovki 17, Minsk, 220072, Belarus
(recieved; accepted)
###### Abstract
We present the data on depolarization of polarized neutron beam and second
harmonic of magnetization ($M_{2}$) for Nd1-xBaxMnO3 (x = 0.23, the Curie
temperature $T_{C}\approx$ 124 K; x = 0.25, $T_{C}\approx$ 129 K) manganites.
The depolarization starts to develop below $T\mbox{*}\approx$ 147 K $>T_{C}$
for both samples, being larger in x = 0.25 compound. This evidences the
arising of a ferromagnetic (F) cluster phase below $T$* and a growth of its
relative volume fraction with increasing doping concentration that agrees with
the previously published results of $M_{2}$ study. A characteristic size of
the F clusters and their concentration are estimated combining the neutron
depolarization and $M_{2}$ data for x = 0.25 manganite.
The interest in the study of the doped perovskite manganites is due to their
unusual magnetic and electronic properties, some aspects of which are not well
understood. An important problem is the origination of an inhomogeneous
magnetic state above Curie temperature, $T_{C}$, in these compounds [1].
Nd1-xBaxMnO3 is a series of doped manganites which exhibits a transition from
paramagnetic (P) to ferromagnetic (F) state for 0.2 $\leq x\leq$ 0.35
($T_{C}\sim$ 120 K) [2]. For $x\geq x_{IM}\approx$ 0.3, these compounds show
metallic behavior below $T_{C}$, whereas for $x<$ 0.3 they remain the
insulators in the magnetic ordered phase. The arising of the inhomogeneous
magnetic state above $T_{C}$ in process of a development of the second order
transition was observed in the insulating Nd1-xBaxMnO3 ($x$ = 0.23, 0.25)
single crystals. It is characterized by the appearance of a magnetic phase
with the strong nonlinear properties in the weak magnetic fields below
$T\mbox{*}(\approx T_{C}$ \+ 20 K) [3-7]. The unconventional behavior was
attributed to the F metallic regions which originate in the P matrix. The F
clusters as well as the unconventional behavior above $T_{C}$ were found in
the other doped manganites [8-13]. This suggests that this phenomenon is an
intrinsic property of these compounds. At the same time, the properties of
this clustered state are not well elucidated, specifically a characteristic
size of the F clusters is under question.
The data on the magnetic, structural and transport properties for $x$ = 0.23
and 0.25 NdBa manganites were published earlier [3-7]. According to the second
harmonic of magnetization ($M_{2}$) measurements, the signal arising below
$T$* from the F metallic regions indicated an increasing in a volume fraction
of the F phase at $T\rightarrow T_{C}$, this fraction being larger in the $x$
= 0.25 compound.
In this paper the data on a depolarization of a transmitted polarized neutron
beam are presented for both manganites. Also the additional data on the field
dependence of the $M_{2}$ at $T$ =142.6 K for $x$ = 0.25 crystal are reported.
The combined data on the depolarization and on $M_{2}(H)$ for this compound
are used in the quantitative analysis. It shows that the single-domain F
regions, which above $T_{C}$ are in a close to superparamagnetic (SPM) regime,
can reveal the strong nonlinear behavior in the weak magnetic fields. The
analysis allows us to estimate the parameters of these F clusters (a
magnetization, a characteristic size, a concentration, a magnetic anisotropy
and a relaxation rate of the magnetization).
The same Nd1-xBaxMnO3 ($x$ = 0.25) single crystal as in Refs. (6,7) was
employed at the $M_{2}$ measurements. It was grown by the flux melt technique,
using the BaO-B2O3-BaF2 ternary system as a solvent [14]. Structural single
phase of the crystal was confirmed by the X-ray and neutron diffractions. The
cation composition of the crystal was determined by X-ray fluorescent
analysis. The average oxidation state of manganese determined by photometry
[15] was found to be close to the value expected for the stoichiometric oxygen
content. The powder samples used in the depolarization measurements were
prepared by the traditional manner [2]. Structural single phase of the samples
was controlled by X-ray diffraction, and the oxygen content was determined by
a thermogravimetric analysis.
The polarization measurements were performed using a small-angle polarized
neutron setup “Vector” with a wavelength $\lambda$ = 9.2 Å,
$\triangle\lambda/\lambda$ = 0.25 (WWR-M reactor Gatchina) [16].
The additional measurements of the second harmonic of the longitudinal
component of magnetization $M_{2}$ were performed in the parallel steady and
alternating magnetic fields $H+h$sin$\omega t(h\leq$ 37 Oe,
$\omega/2\pi\approx$ 15.7 MHz) in the temperature range $T*\geq T\geq T_{C}$.
The Re$M_{2}$ and Im$M_{2}$ parts of $M_{2}$ were simultaneously recorded as
the functions of $H$. This field was scanned symmetrically relative to the
point $H$=0 for detecting a field hysteresis of the signal. The amplitude of
$H$-scan was 300 Oe. A condition $M_{2}\propto h^{2}$ was satisfied in the
measurements. An installation and a method of separation of the $M_{2}$-phase
components have been described previously [17].
Fig. 1 shows the temperature dependence of the polarization for
Nd1-xBaxMnO${}_{3}\;x$ = 0.23, 0.25 polycrystalline manganites. It is seen
that in both samples the depolarization appears just below $T\mbox{*}\approx$
147 K $>T_{C}\,(T_{C}\approx$ 124 K for $x$ =0.23 and 129 K for $x$ =0.25),
indicating the origination of the F regions, and it increases at decreasing
temperature. The depolarization is larger for $x$ = 0.25 sample. These results
confirm the $M_{2}(T,H)$ data [4,6].
The $M_{2}$ response of the F clusters coexists with the critical contribution
of the paramagnetic matrix, which also increases with decreasing temperature.
Therefore, it is convenient for the analysis to choose the $M_{2}(H,T)$ curves
of the $x$ = 0.25 compound at $T$ = 142.6 K (well above $T_{C}$), which are
displayed in Fig. 2. A characteristic kink in the Re$M_{2}(H)$ dependence,
which is due to a competition of the paramagnetic critical contribution and
the signal of the F regions, is clearly seen at this temperature. The
Im$M_{2}(H)$ is mainly determined by the F clusters [6]. Additionally, the
critical paramagnetic contribution is proportional to $H$ (see below) and the
demagnetization corrections are small here. The subsystem of F clusters can be
considered as an ensemble of single domain F particles in nearly
superparamagnetic regime since the field hysteresis of $M_{2}(H)$ dependences
is small [4,6]. Nevertheless, the presence of this small $H$-hysteresis can be
seen in Im$M_{2}(H)$ component (Fig. 2b) as an incomplete inversion symmetry
of the curve relative to the point $H$ = 0.
Let us go to the analysis. The neutron polarization $P$ after passing of the
sample with the ferromagnetic regions of a small concentration is given by
[18]
$P=P_{0}exp[-4/3(\gamma_{n}B/\nu)^{2}R\widetilde{C}^{1/3}L],$ (1)
where $P_{0}$ is the initial polarization directed along the beam,
$\gamma_{n}$ is the gyromagnetic ratio of the neutron, $\nu\approx$ 4.3$\cdot
10^{4}$ cm/s is the neutron velocity, $B=4\pi g\mu_{B}<S>/V_{0}$ is the
induction in the F clusters, $<S>$ is the value of the spin at temperature
$T,V_{0}$ is the volume per magnetic atom (58.6 Å3 [7]), $R$ is the mean
radius of the F regions, $\tilde{C}$ is the relative concentration of the
superparamagnetic phase and $L\approx$ 0.2 cm is the thickness of the sample.
We have three unknown parameters $B,R,\widetilde{C}$ in (1) and need the
additional $M_{2}$ results to find them from experimental data.
Going over the $M_{2}$ analysis, let us consider a possible magnetic
anisotropy of the F regions. Our compound has the Pbnm space group with a
relationship between the lattice parameters $a>b\approx c/\sqrt{2}$ [7] that
suggests a dominating uniaxial anisotropy directed along the $a$-axis. We
assume an easy axis character of the anisotropy. This assumption is supported
by our preliminarily magnetic resonance measurements in the $x$ = 0.3 single
crystal (with a close to the $x$ = 0.25 manganite crystal structure) where the
resonance signal from the F regions is observed above $T_{C}$. Such a signal
is not detected at $x$ = 0.25 [7] because of a small concentration of the F
clusters. As a result, the Hamiltonian of the single-domain F region is given
by
$H=-N[\vec{\mu}\vec{H}+K(\vec{e}\vec{e}_{\mu})^{2}],$ (2)
where $N$ is the number of the spins in the region, $\mu=g\mu_{B}<S>,g\approx$
2 is the $g$ \- factor, $H$ is the steady magnetic field, $K$ is the effective
uniaxial anisotropy per a magnetic atom which can include a shape anisotropy,
$\vec{e}$ is the direction of the uniaxial anisotropy and
$\vec{e}_{\mu}=\vec{\mu}/\mu$. The $M_{2}(H)$ data, which obtained for the
magnetic fields directed along the different crystal axes, do not reveal an
orientation dependence. It suggests a twin structure with a nearly equal
population of the structural domains. We will show below that the domains with
easy axis give the main contribution to $M_{2}(H)$ for $\vec{H}$ directed
along the crystallographic axes. In this case, the longitudinal response of
the second order for single-domain F cluster can be written as [4]:
$\widetilde{M}_{2}(H,\omega)/h^{2}=\frac{\Gamma(H)}{-2i\omega+\Gamma(H)}\chi_{2}(H)-\frac{i\omega}{\Gamma(H)}\frac{\partial\Gamma(H)/\partial
H}{(-2i\omega+\Gamma(H))(-i\omega+\Gamma(H))}\chi_{1}(H).$ (3)
Here 2$\chi_{2}(H)=\partial^{2}M/\partial H^{2},\chi_{1}(H)=\partial
M/\partial H,M$ is the magnetization of the single-domain region and
$\Gamma(H)$ is its relaxation rate. The magnetization of the domain is given
by
$M(mH/T,KN/T)=mL(H/C_{1},\alpha),$ (4)
where $m=\mu N,\alpha=KN/T,C_{1}=T/m$ and
$L(H/C_{1},\alpha)=C_{1}\partial\ln Z/\partial H,\qquad
Z=\int\limits_{0}^{1}\coth(xH/C_{1})\exp(\alpha x^{2})dx.$ (5)
Here Z is the partition function. In the first approximation, we perform a fit
of the data by neglecting the second net dynamic term in Eq. (3) and ignoring
an $H$-dependence of $\Gamma$. The $M_{2}$ of the sample, which is the sum of
the response from all the F clusters
$\widetilde{M}_{2S}=\widetilde{C}\widetilde{M}_{2}$ and a contribution from
the paramagnetic matrix $M_{2C}$, can be written as
$M_{2}(H,\omega)/h^{2}=(C^{\prime}_{2}+iC^{\prime\prime}_{2})\chi_{2}(H)/m+(C^{\prime}_{3}+iC^{\prime\prime}_{3})H,$
(6)
with
$\displaystyle
2C^{\prime}_{2}=\frac{1}{3}\mu\widetilde{C}\frac{1}{(2\omega/\Gamma)^{2}+1},\qquad
C^{\prime\prime}_{2}=\frac{2\omega}{\Gamma}C^{\prime}_{2},$ (7) $\displaystyle
6C^{\prime}_{3}=(1-\widetilde{C})\frac{\partial^{2}M_{C}}{\partial
H^{2}}\bigg{|}_{H=0},\qquad
C^{\prime\prime}_{3}=\frac{2\omega}{\Gamma_{C}}C^{\prime}_{3},\qquad\mbox{at}\quad\Gamma_{C}\gg
2\omega.$ (8)
Here a factor (1/3) in Eq. (7) takes into account the twinning structure of
the crystal, $M_{C}$ is the magnetization of the paramagnetic matrix in a weak
field when $M_{2C}\propto H$ [4], $\Gamma_{C}$ is the relaxation rate of the
uniform magnetization of the paramagnetic matrix. The factor $C^{\prime}_{3}$
corresponds to Re$M_{2C}$ at $\omega$ = 0 because
$(2\omega/\Gamma_{C})^{2}\approx 7.8\cdot 10^{-4}$ at $T\approx$ 143 K [6].
One can perform a fit of the Re$M_{2}$ and Im$M_{2}$ component of the signal
using Eqs. (4)-(8) where
$C_{1},C^{\prime}_{2},C^{\prime\prime}_{3},C^{\prime}_{3},C^{\prime\prime}_{3}$
and $\alpha$ are the fitting parameters. It allows one to find the following
characteristics of the F regions:
$\widetilde{C}_{fit}=\widetilde{C}<S>,N_{fit}=N<S>,KN$ and $\Gamma$ which are
determined by $C^{\prime}_{2},C_{1},\alpha$ and
$C^{\prime\prime}_{2}/C^{\prime}_{2}$, respectively. The value of $<S>$ can be
found by exploiting the data on the neutron depolarization (see below). As a
result, we obtain all the needed parameters in this approach. A combined fit
of the Re$M_{2}$, Im$M_{2}$ and neutron depolarization data gives
$C_{1}\approx 42,\alpha\approx 3,N\approx 4.23\cdot
10^{4},\widetilde{C}\approx 6.75\cdot 10^{-5},<S>\approx$ 0.595 as well as
$\Gamma/\omega\approx$ 5\. We have ignored above an $H$-dependence of
$\Gamma$. Now we consider an effect of $\Gamma(H)$. At further evaluation a
variation of $\alpha$ makes the fit worse and we exploit below the found above
$\alpha$ = 3 value. Using the numerical data for $\Gamma(H)$ which is governed
by thermoactivation over the barrier related to the magnetic anisotropy [19],
one can obtain an approximate expression at $\alpha$ = 3
$2\Gamma(H)=\Gamma_{0}\frac{\exp[-3(1+\widetilde{H})^{2}]+\exp[-3(1-\widetilde{H})^{2}]}{1+0.25\widetilde{H}^{2}+1.5\widetilde{H}^{2}(1-\widetilde{H}^{2})},$
(9)
where $\widetilde{H}=H/2H_{A},H_{A}=K/\mu$ is the anisotropy field,
$\widetilde{H}\leq 1,\;\partial\Gamma(H)/\partial H>0$ and
$\Gamma(H)/\Gamma(0)\approx 8$. At a following fit we use expression (3) for
Re$\widetilde{M}_{2}$ and Im$\widetilde{M}_{2}$ with $\Gamma(H)$ (Eq. (9)) in
Eq. (6) instead of the first term. The fitting parameters are
$\widetilde{C}_{fit}=\widetilde{C}<S>,C_{1},\Gamma_{0}$ and $C^{\prime}_{3}$.
The $M_{2}$ data give $\widetilde{C}_{fit}=\widetilde{C}<S>$ and
$N_{fit}=N<S>$. As a result, one can find the $<S>$ from Eq.(1) assuming a
spherical form of the regions
$(R=((3/4\pi)NV_{0})^{1/3}\propto(N_{fit}/<S>)^{1/3}\propto
1/<S>^{1/3},\widetilde{C}^{1/3}=(\widetilde{C}_{fit}/<S>)^{1/3}\propto
1/<S>^{1/3},B^{2}R\widetilde{C}^{1/3}\propto<S>^{4/3})$.
Let us consider the experimental results. Fig. 2 presents the $M_{2}(H)$
dependences with the fitting curves ($h\approx$ 36.6 Oe). The parameters of
the F regions are found to be $C_{1}\approx$ 47 Oe, $N\approx 3.26\cdot
10^{4},<S>\approx 0.59,R\approx$ 77.4 Å, $\widetilde{C}\approx 8.3\cdot
10^{-5},H_{A}\approx 165$ Oe, $K\approx 1.31\cdot 10^{-2}$ K and
$\Gamma(0)/\omega=\Gamma_{0}e^{-3}/\omega\approx 6.33,\Gamma\approx$ 624 MHz.
We used $P_{0}\approx$ 0.938 and $P\approx$ 0.93 at 142.6 K (Fig. 1). Note
that Re$\widetilde{M}_{2}(H)$ (Eq. (3)) is determined mainly by the static
amplitude $\chi_{2}(H)$ because: (i) $(2\omega/\Gamma(0))^{2}\approx$ 0.1 and
$\Gamma(H)$ increases with increasing $H$; (ii) as the calculations show, the
contribution of the second dynamical term in Eq. (3) to Re$M_{2}$ contains a
small factor $[\omega/\Gamma(H)]^{2}<0.02$ in comparison with the first one.
At the same time, the imaginary parts of both terms have the comparable
magnitudes and the similar $H$-dependences. We have neglected above a
contribution of the twins with the anisotropy axes directed perpendicular to
the field which is characterized by $\chi_{2\perp}(H)$. A ratio
$r(H)=2\chi_{2\perp}(H)/\chi_{2}(H)$ increases with increasing $H$, remaining,
however, small in an essential region of changing $\chi_{2}(H):r(H)<0.1$ from
$H$ = 0 up to a maximum of $|\chi_{2}(H)|$ at $H^{\mbox{{\scriptsize
Re}}}_{m}\approx$ 40 Oe and $r(H)\approx 1$ only at 120 Oe where
$|\chi_{2\perp}(H)|$ reaches a maximum. However, at this field $|\chi_{2}(H)|$
becomes small $\chi_{2}(120\mbox{Oe})/\chi_{2}(H^{\mbox{\scriptsize
Re}}_{m})\approx$ 0.15 and Re$M_{2}(H)$ is determined here by the large linear
critical contribution. As a result, the including of $|\chi_{2\perp}(H)|$ does
not affect the parameters of the F regions. A frequency dependence of
$\chi_{2\perp}(\omega,H)$ was not analyzed. It is known a linear
susceptibility $\chi_{1\perp}(\omega)$ at $H$ = 0 whose $\omega$-dependence is
determined by a temperature spread of the resonance frequencies of the
magnetic moments in the field of anisotropy [20] so that
Im$\chi_{1\perp}(\omega)\sim(\omega/2K)\chi_{1\perp}$ for
$\omega<2K\;(\approx$ 613 MHz). One can expect that in the magnetic field
Im$\chi_{2\perp}(\omega,H)\sim(\omega/K)\chi_{2\perp}(H)$ at the small
$\omega$ and $H<2H_{A}\;(\approx$ 330 Oe). Since Im$\chi_{2}(\omega,H)\sim
2(\omega/\Gamma(H))\chi_{2}(H)$ and $\Gamma(H)<2K$ at $H<120$ Oe we find, as
for Re$\widetilde{M}_{2}(\omega,H)$, that the contribution of the “orthogonal”
twins can only affect the tail of Im$\widetilde{M}_{2}(\omega,H)$ at $H>120$
Oe where $r(H)>1$. These corrections are unessential because the parameters of
the F domains are mainly determined by a region of $H<120$ Oe and the linear
critical paramagnetic contribution starts to dominate above 120 Oe. Note, the
taking into account the $\chi_{2\perp}(\omega,H)$ contribution to the response
could probably improve the fit of the tails of the experimental curves but
this question is out the scope of this paper.
Let us discuss the results. The found attempt frequency $\Gamma_{0}\approx
1.25\cdot 10^{4}$ MHz is within the typical interval of the values
($10^{3}-10^{5}$ MHz). The magnetization of the F regions characterized by
$<S>$ is small in comparison with a saturation magnetization ($<S>/S\approx
0.3,\;S=1.875$ is the average spin value for the given composition which
corresponds to the saturation magnetization). It seems to be the natural
result for the temperature slightly below $T\mbox{*}\approx$ 146 K. In the
La0.9Sr0.1MnO3 manganite, the magnetization of the F clusters increased with
decreasing temperature and was also small just below $T$* [10]. In regard to
the magnetic anisotropy, it is known only an upper limit of the single ion
magnetocrystalline anisotropy in the bulk ($J_{A}S^{2}_{\alpha}$ with
$J_{A}<0.5$ K) from the ESR measurements above $T_{C}$ [7]. For the F regions,
$K$ can be written as $K=J_{A}<S>^{2}$, leading to $J_{A}\approx 0.04$ K in
agreement with the estimation. In addition, according to our preliminarily ESR
data on the $x$ = 0.3 NdBa single crystal, the anisotropy field of the F
regions is $H_{A}\sim 250$ Oe which is not far from $H_{A}\approx 165$ Oe in
our compound. The anisotropy fields of the F clusters were reported for
La0.9Sr0.1MnO3 [10] and La0.88Ba0.12MnO3 [21]. The dominating uniaxial
components, which correspond to the hard and easy axes anisotropies in the
first and second manganite, respectively, were found to be about 2.4 kOe. The
small value of $H_{A}\propto J_{A}<S>$ in our case ($H_{A}/(2.4$ kOe) $\approx
0.07$) can be explained by a smallness of $<S>$ and $J_{A}$. The presented
data on the anisotropy fields [10, 21] were obtained at temperatures of 50 K
below $T$* that suggested a larger value of $<S>$ (2-3 times) than that at
temperature of the several degrees below $T$*. The constant $J_{A}$ is related
to the distortions of the oxygen octahedron and decreases with doping
increasing. This constant can be essentially smaller (7-5 times) near the
border of the insulator to metal transition in our system than that in the two
above mentioned manganites where doping is slightly above a border of an
antiferromagnet to ferromagnet transition.
We obtain the rather large value for the size of the F regions $R/a\approx$
19.9, where $a=(V_{0})^{1/3}\approx$ 3.884 Å is the magnetic lattice constant.
It is essential, therefore, to compare it with a critical radius of a single F
domain $R_{C}/a\approx 2\cdot(3J/\omega_{D})^{1/2}\;(J$ is the exchange
interaction and $\omega_{D}=4\pi(g\mu_{B})^{2}/V_{0}$ is the dipolar energy)
for a weakly anisotropic ferromagnet when $2J_{A}/\omega_{D}\ll
1\;(\omega_{D}\approx$ 0.75 K and $2J_{A}/\omega_{D}\approx$ 0.035 in our
system) [22]. Using the mean - field expression for $T_{C}\;(\approx
T\mbox{*}\approx$ 150 K) of the F regions, we find $J\approx$ 14 K and
$R_{C}/a\approx$ 15\. This result suggests that the F regions are close to the
uniformly magnetized domains.
As it was discussed in Refs. (4,6), temperature evolution of the signal of the
F regions corresponded mainly to an increase in their volume at decreasing
temperature from $T$* down to $T_{C}$. The matter is that a field position of
the extreme ($H^{\mbox{\scriptsize Im}}_{m}$ 30 Oe, Fig. 2b) in
Im$M_{2}(T,H)$, which is due to the F regions, remains temperature independent
with decreasing temperature whereas the extreme value of
Im$M_{2}(T,H^{\mbox{\scriptsize Im}}_{m})$ increases sharply down to $T_{C}$.
This behavior reflects an increase in the concentration of the F clusters
without the noticeable changes in their parameters. The same peculiarity of
the Im$M2(T,H^{\mbox{\scriptsize Im}}_{m})$ dependence with
$H^{\mbox{\scriptsize Im}}_{m}\approx$ 22 Oe was also observed in the $x$ =
0.23 NdBa crystal [4]. The closeness of $H^{\mbox{\scriptsize Im}}_{m}$ and a
strong resemblance between the Re$M_{2}(T,H)$ functions in both systems
suggests that the F regions are characterized by the close values of the
parameters. The main difference between the compounds is the concentration of
the F clusters which is about 10 times larger at $x$ = 0.25 than that at $x$ =
0.23 [6].
Magnetization of the F regions is characterized by the small scale of the
field nonlinearity $C_{1}\approx$ 47 Oe (Eq. 4). As a result, the $M(T,H)$
measurements at $H$ = 1 kOe performed above $T_{C}$ in the $x$ = 0.25 NdBa
compound reveal only the critical temperature dependence of the paramagnetic
matrix [7] since the magnetization of the F clusters, being near to a
saturation for the superparamagnetic regime, gives a temperature independent
correction which is proportional to their rather small concentration. An
extreme sensitivity of the clustered state to magnetic field was found in
La1-xBaxMnO${}_{3}\;(x$ = 0.27 and 0.3) [13]. The anomalous contribution to
$\chi_{1}$ was suppressed by $H\sim$ 150 Oe in the paramagnetic region (for
$T$ = 275-350 K, $T_{C}\approx$ 245 K, $x=0.27$ and $T$ = 310-345 K,
$T_{C}\approx$ 310 K, $x=0.3$). This behavior can be explained by a small
characteristic $H$-scale for magnetization of the F clusters which has the
comparable value with one in our system.
In conclusion, the analysis of the data on the nonlinear longitudinal response
and the depolarization of a polarized neutron beam for the Nd0.75Ba0.25MnO3
manganite allows us to estimate the characteristic size of the F clusters
($R\approx$ 77.4 Å), which form in critical paramagnetic regime below
$T\mbox{*}\approx$ 147 K and lead to a formation of heterogeneous magnetic
state above $T_{C}$.
This work was supported by the Program No 27 of the PRAS and by the RFBR
(grant No 09-02-01509).
## References
* (1) E. Dagotto, Phase Separation and Colossal Magnetoresistance (Springer, Berlin, 2002); Colossal Magnetoresistive Oxides, edited by Y. Tokura (Gordon and Breach, London, 2003).
* (2) I.O. Troyanchuk, D.D. Khalyavin, and H. Szymczak, J. Phys.: Condens. Matter 11, 8707 (1999).
* (3) I.D. Luzyanin, V.P. Khavronin , V.A. Ryzhov, I.I. Larionov, A.V. Lazuta, Pis’ma Zh. Eksp.Teor. Fiz. 73 (2001) 369 [JETP Lett. 73 (2001) 327].
* (4) V.A. Ryzhov, A.V. Lazuta, I.D. Luzyanin, I.I. Larionov, V.P. Khavronin, Yu.P. Chernencov, I.O. Troyanchuk, D.D. Khalyavin, Zh. Exsp. Teor. Fiz. 121 (2002) 678 [JETP 94 (2002) 581].
* (5) V.A. Ryzhov, A.V. Lazuta, I.A. Kiselev, Yu.P. Chernenkov, O.P. Smirnov, S.A. Borisov, I.O. Troyanchuk, D.D. Khalyavin, Solid State Commun. 128, 141 (2003).
* (6) V.A. Ryzhov, A.V. Lazuta, V.P. Khavronin, I.I. Larionov, I.O. Troyanchuk, and D.D. Khalyavin, Solid State Commun. 130, 803 (2004).
* (7) V.A. Ryzhov, A.V. Lazuta, O.P. Smirnov, I.A. Kiselev, Yu.P. Chernenkov, S.A. Borisov, I.O. Troaynchuk and D.D. Khalyavin, Phys. Rev. B 72, 134427 (2005).
* (8) M.B. Salamon, P. Lin, S.H. Chun, Phys. Rev. Lett. 88 (2002) 197203.
* (9) M.B. Salamon, S.H. Chun, Phys. Rev. B 68 (2003) 014411.
* (10) J. Deisenhofer, D. Braak, H.A. Krug von Nidda, J. Hemberger, R.M. Eremina, V.A. Ivanshin, A.M. Balbashov, G. Jug, A. Loidl, T. Kimura, Y. Tokura, Phys. Rev. Lett. 95 (2005) 257202.
* (11) V.A. Ryzhov, A.V. Lazuta, I.A. Kiselev, V.P. Khavronin, P.L. Molkanov, I.O. Troaynchuk, S.V. Trukhanov, JMMM 300, e159 (2006).
* (12) Wanjun Jiang, Xue Zhi Zhou, Gwyn Williams, Y. Mukovskii, K. Glazyrin, Phys. Rev. Lett. 99 (2007) 177203.
* (13) Wanjun Jiang, X.Z. Zhou, Gwyn Williams, Y. Mukovskii, K. Glazyrin, Phys. Rev. B 76 (2007) 092404.
* (14) D.D. Khalyavin, M. Pekala, G.L. Bychkov, S.V. Shiryaev, S.N. Barilo, I.O. Troyanchuk, J. Mucha, R. Szymczak, M. Baran, and H. Szymczak, J. Phys.: Condens. Matter 15, 925 (2003).
* (15) A.G. Soldatov, S.V. Shiryaev, and S.N. Barilo, J. Analytic. Chem. 56, 1077 (2001).
* (16) S.V. Grigoriev, O.A. Gubin, G.P. Kopitsa et al., Preprint PNPI-2028 Gatchina, 1995, (in Russian).
* (17) V.A. Ryzhov, I.I. Larionov, V.N. Fomichev, Zh. Tekh. Fiz. 66, 183 (1996) [Sov. Phys. Tech. Phys. 41, 620 (1996)].
* (18) S.V. Maleyev, J. Phys. (France) 43, C7 (1982).
* (19) A. Aharony, Phys. Rev. 177, 793 (1969).
* (20) D.A. Garanin, V.V. Ishchenko, L.V. Panina, Teor. Mat. Fiz. 82, 242 (1990) [Theor. Math. Phys. 82, 169 (1990)].
* (21) P.M. Eremina, I.V. Yatsyk, Ya.M. Mukovskii, H.A. Krug von Nidda, A. Loidl, JETP Lett. 85, 51 (2007) [Pisma Zh. Eksp. Teor. Fiz. 85, 57 (2007)].
* (22) I.A. Privorotskii, Sov. Phys. Usp. 15, 555 (1973) [Usp. Fiz. Nauk 108, 43, (1972)].
Figure captions
Fig. 1. Temperature dependence of the depolarization of the polarized neutron
beam for Nd1-xBaxMnO3 ($x$ = 0.23, 0.25) powdered samples. Neutron wavelength
is $\lambda$ = 9.2 Å, $\triangle\lambda/\lambda$ = 0.25 and the thickness of
the sample is $L\approx$ 0.2 cm.
Fig. 2. Second harmonic of magnetization, $M_{2}(H)$, as a function of $H$ for
the Nd1-xBaxMnO3 single crystal. Panels show $H$-dependence of the phase
components ((a) - Re$M_{2}(H)$, (b) - Im$M_{2}(H)$) at $T$ = 142.6 K and their
fits. The fits are described in the text.
|
arxiv-papers
| 2009-11-23T10:18:08 |
2024-09-04T02:49:06.608021
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.V. Lazuta (1), V.A. Ryzhov (1), V.V. Runov (1), I.O. Troyanchuk (2)\n ((1) Petersburg Nuclear Physics Institute RAS, (2) Institute of Solid State\n and Semiconductor Physics of National Academy of Sciences)",
"submitter": "Vyacheslav Ryzhov A.",
"url": "https://arxiv.org/abs/0911.4179"
}
|
0911.4240
|
# Entanglement Dynamics and Spin Squeezing of The non-linear Tavis-Cummings
model mediated by a Nonlinear Binomial Field
M. S. Ateto111E-mail: omersog@yahoo.com, mohamed.ali11@sci.svu.edu.eg
Mathematics Department, Faculty of Science at Qena, South Valley University,
83523 Qena, Egypt
###### Abstract
We show that spin squeezing implies entanglement for quantum tripartite-state,
where the subsystem of the bipartite-state is identical. We study the relation
between spin squeezing parameters and entanglement through the quantum entropy
of a system starts initially in a pure state when the cavity is binomial. We
show that spin squeezing can be a convenient tool to give some insight into
the subsystems entanglement dynamics when the bipartite subsystem interacts
simultaneously with the cavity field subsystem, specially when the interaction
occurs off-resonantly without and with a nonlinear medium contained in the
cavity field subsystem. We illustrate that, in case of large off-resonance
interaction, spin squeezing clarifies the properties of entanglement almost
with full success. However, it is not a general rule when the cavity is
assumed to be filled with a non-linear medium. In this case, we illustrate
that the insight into entanglement dynamics becomes more clearly in case of a
weak nonlinear medium than in strong nonlinear medium. In parallel, the role
of the phase space distribution in quantifying entanglement is also studied.
The numerical results of Husimi $Q$-function show that the integer strength of
the nonlinear medium produces Schrödinger cat states which is necessary for
quantum entanglement.
###### pacs:
03.67.-a, 03.67.Bg, 05.30.-d
Accepted in Int. J. Quant. Inf.
## I Overview
The squeezing of the quantum fluctuations is one of the most fundamental
manifestations of the Heisenberg uncertainty relation, which is among the most
important principles of quantum mechanics. A long time ago, great effort has
been paid to squeezing of the radiation field due to its strong application in
the optical communication YS78 and weak signal detections CMC81 .
Accordingly, it was established the relationship between the squeezing of the
atoms and that of the radiation field WX03 and the possibility of squeezed
atoms to radiate a squeezed field POUMO01 . Moreover, much attention has been
devoted to atomic spin squeezing MB02 ; GB03 ; SM01 ; VPG00 ; WX01 ; Wx04 ;
YWS07 ; ZPP91 . Spin squeezed states WX01 ; VPG00 ; AALU02 ; WBI94 ; KIUE93 ;
AGPU90 ; LUYEFL02 ; KUMOPO97 ; WEMO02 ; HEYO01 ; MUZHYO02 ; POMO01 ; THMAWI02
; GAROBU02 ; STGEDOMA03 ; ZHSOLI02 ; WASOMO01 are quantum correlated states
with reduced fluctuations in one of the collective components. Spin squeezed
states offer an interesting possibility for reducing quantum noise in
precision measurements WBI94 ; WBI92 ; UOMK01 ; AALU02 ; DPJN03 ; JKP01 ;
MRKSIMW01 , with potentially possible applications in ultra-precise
spectroscopy, atomic interferometers and high precision atomic clocks WBI94 ;
WBI92 .
Interestingly, it was found that spin squeezing is closely related to a key
ingredient in quantum information theory and implies quantum entanglement
SDCZ01 ; SORENSEN02 ; BERSAN02 ; RAFUSAGU0137 . As there are various kinds of
entanglement, a question arises: what kind of entanglement does spin squeezing
imply? In Ref. UOMK01 , it has been found that, for a two-qubit symmetric
state, spin squeezing is equivalent to its bipartite entanglement, i.e., spin
squeezing implies bipartite entanglement and vice versa. Wang and Sanders
WASA03 , presented a generalization of the results of Ref. UOMK01 to the
multiqubit case, where the authors showed that spin squeezing implies pairwise
entanglement for arbitrary symmetric multiqubit states. More quantitatively,
for spin states with a certain parity, if a state is spin squeezed according
to the definition for Kitagawa and Ueda KIUE93 , a quantitative relation is
obtained between the spin squeezing parameter and the concurrence HW97 ; HW98
; XHCY02 ; ZL03 , quantifying the entanglement of two half-spin particles
UOMK01 ; WASA03 .
The close relation between the entanglement and spin squeezing enhances the
importance of spin squeezing which motivate us to explore the role of Kerr
medium MSAT07 ; MSAT109 ; MSAT209 ; MSAT309 and the nonlinear binomial state
on the spin squeezing and entanglement.
A binomial state is one of the important nonclassical states of light which
has attracted much attention in the field of quantum optics in the last few
decades, see for example Buzek90 ; STSATE85 ; Vidiella94 . This, in fact, is
due to the importance of these states which have been experimentally produced
and characterized. This state is regarded as one of the intermediate states of
the coherent state Buzek90 ; STSATE85 ; Vidiella94 . It can be simply produced
from the action of a single mode squeeze operator on the state $\mid
p,0\rangle$ STSATE85 , where $p$ is the Bargmann number. It also represents a
linear combination from the number states with coefficients chosen such that
the photon counting probability distribution is the binomial distribution with
mean $M|p|$, where $0<|p|<1$.
The scope of this communication is to employ the spin squeezing and quantum
entropy to elucidate entanglement for two-atom system prepared initially in
Bell-state interacting with single cavity field prepared initially in a
binomial state. We introduce our Hamiltonian model and give exact expressions
for the full wave function of the atomic Bell-state and field systems,
shedding light on the important question of the relation between spin
squeezing and quantum entropy behaviors. We also examine the evolution of the
quasiprobability distribution Husimi $Q$-function for the model under
consideration. With the help of the quasiprobability distribution, one gains
further insight into the mechanism of the interaction of the model subsystems.
In the language of quantum information theory a definition of spin squeezing
is presented for this system. The utility of the definition is illustrated by
examining spin squeezing from the point of view of quatum information theory
for the present system. This analysis is applicable to any quantum tripartite-
state subject to spin squeezing with appropriate initial conditions.
## II model solution and the reduced density operator
Our system is as follows: A light field is initially in the binomial state
(the state which interpolates between the nonlinear coherent and the number
states )
$|\psi_{F}(0)\rangle=|p,M\rangle=\sum_{n=0}^{\infty}~{}b_{n}|n\rangle,$ $None$
where $|n\rangle$ is the eigenstate of number operator
$\widehat{a}^{{\dagger}}\widehat{a}=\widehat{n}$,
$\widehat{n}|n\rangle=n|n\rangle$, while, the coefficients $b_{n}$ are given
by
$b_{n}=\sqrt{\binom{M}{n}~{}|p|^{n}(1-|p|)^{M-n}},$ $None$
where $M>0$ (i.e., $M$ in general is any real positive number), $0<|p|<1$,
interacts with two identical two-level atoms are initially in the following
Bell state
$|\psi_{A}^{\uparrow\uparrow\downarrow\downarrow}(0)\rangle=\gamma_{1}|\uparrow\uparrow\rangle+\gamma_{4}|\downarrow\downarrow\rangle,$
$None$
The binomial states have the properties
$|p,M\rangle=\begin{cases}\mid M\rangle&|p|\rightarrow 1\cr\mid
0\rangle&|p|\rightarrow 0\cr\mid\alpha\rangle&|p|\rightarrow
0,~{}~{}~{}M\rightarrow\infty,~{}~{}~{}M|p|=\alpha^{2}.\cr\end{cases}$ $None$
For this system, the atoms-field interaction is governed by the ($N=2$) Tavis-
Cummings model(TCM) TCM68 . The Tavis-Cummings model (TCM) TCM68 describes
the simplest fundamental interaction between a single mode of quantized
electromagnetic field and a collection of $N$ atoms under the usual two-level
and rotating wave approximations when all of the atoms couple identically to
the field. The two-atom ($N=2$) TCM is governed by the Hamiltonian
$\widehat{H}=\omega\widehat{n}+\frac{1}{2}\omega_{0}(\widehat{\sigma}_{z}^{(1)}+\widehat{\sigma}_{z}^{(2)})+f(\chi,\widehat{n})+\lambda\sum_{j=1}^{2}\biggl{(}\widehat{\sigma}_{-}^{(j)}~{}\widehat{a}^{{\dagger}}+\widehat{\sigma}_{+}^{(j)}~{}a\biggr{)},$
$None$
where
$f(\chi,\widehat{n})=\chi\widehat{n}(\widehat{n}-1)+2\sqrt{\chi}~{}\widehat{n}$
represents the nonlinear term with,
$f(\chi,\widehat{n})\mid n\rangle=\bigl{(}\chi
n(n-1)+2\sqrt{\chi}n\bigr{)}\mid n\rangle=f(\chi,n)\mid n\rangle.$ $None$
We denote by $\chi$ the dispersive part of the third order susceptibility of
the Kerr-like medium MSAT07 ; MSAT109 ; MSAT209 ; MSAT309 . The parameter
$\lambda$ represents the atoms-field coupling constant. The operators
$\widehat{\sigma}_{\pm}^{(i)}$ and $\widehat{\sigma}_{z}^{(i)}$,
($i\in\\{1,2\\}$) display a local $SU(2)$ algebra for the $i$-th atom in the
2D supspace spanned by the ground (excited) state
$|\downarrow\rangle$,($|\uparrow\rangle$) and obey the commutation relations
$[\widehat{\sigma}_{+}^{(i)},\widehat{\sigma}_{-}^{(i)}]=\widehat{\sigma}_{z}^{(i)}$,
($i\in\\{1,2\\}$), and $\widehat{a}$($\widehat{a}^{{\dagger}}$) is bosonic
annihilation (creation) operator for the single mode field of frequency
$\omega$. To make calculation to be more clear and simple, we put
$\kappa\sum_{j=1}^{2}~{}\widehat{\sigma}_{s}^{(j)}=\widehat{J}_{s};~{}~{}~{}~{}~{}s=z,-,+,$
$None$
the parameter $\kappa=\frac{1}{2},1,1$ if $s=z,-,+$, respectively.
The Hamiltonian (5), with the detuning parameter $\Delta=\omega_{0}-\omega$,
takes the form
$\widehat{H}=\widehat{H}_{0}+\widehat{H}_{INT}=\omega\bigl{(}\widehat{n}+\widehat{J}_{z}\bigr{)}+\Delta~{}\widehat{J}_{z}+f(\chi,\widehat{n})+\lambda~{}\bigl{(}\widehat{J}_{-}~{}\widehat{a}^{{\dagger}}+\widehat{J}_{+}~{}\widehat{a}\bigr{)},$
$None$
Consider, at $t=0$, the two atoms are in the Bell state, Eq. (3). The initial
state of the system is a decoupled pure state, and the sate vector can be
written as
$|\psi_{AF}(0)\rangle=\sum_{n=0}^{\infty}b_{n}\bigl{[}\gamma_{1}|\uparrow\uparrow,n\rangle+\gamma_{4}|\downarrow\downarrow,n\rangle\bigr{]},$
$None$
As the time goes, the evolution of the system in the interaction picture can
be obtained by solving the Schrödinger equation
$i\frac{d}{dt}|\psi_{AF}(t)\rangle=H|\psi_{AF}(t)\rangle,$ $None$
and using the condition Eq. (9) to obtain the time-dependent wave function in
the form
$|\psi_{AF}(t)\rangle=|U\rangle~{}|1\rangle+|R\rangle~{}|2\rangle+|S\rangle~{}|3\rangle+|T\rangle~{}|4\rangle,$
$None$
where
$|U\rangle=\sum_{n=0}^{\infty}~{}b_{n}\biggl{[}A_{n}(t)|n\rangle+H_{n-2}(t)|n-2\rangle\biggr{]},$
$None$
$|R\rangle=\sum_{n=0}^{\infty}~{}b_{n}\biggl{[}B_{n+1}(t)|n+1\rangle+G_{n-1}(t)|n-1\rangle\biggr{]}=|S\rangle,$
$None$
and
$|T\rangle=\sum_{n=0}^{\infty}~{}b_{n}\biggl{[}D_{n+2}(t)|n+2\rangle+E_{n}(t)|n\rangle\biggr{]},$
$None$
where we used the notations
$|1\rangle=|\uparrow\uparrow\rangle,~{}~{}~{}~{}|2\rangle=|\uparrow\downarrow\rangle~{}~{}~{}~{}|3\rangle=|\downarrow\uparrow\rangle~{}~{}~{}~{}~{}|4\rangle=|\downarrow\downarrow\rangle.$
$None$
the group of the complex amplitudes $A_{n}(t)$, $D_{n+2}(t)$, $B_{n+1}(t)$ and
$C_{n+1}(t)$ are given, respectively by
$A_{n}(t)=\frac{1}{2\Gamma_{1}\Gamma_{2}}\sum_{k=0}^{2}D^{k}_{n+2}(0)[\eta^{2}_{k}-2\Gamma_{2}^{2}+(\alpha_{2}+\alpha_{3})\eta_{k}+\alpha_{2}\alpha_{3}]e^{i\eta_{k}t},$
$None$
$B_{n+1}(t)=-\frac{1}{2\Gamma_{2}}\sum_{k=0}^{2}D^{k}_{n+2}(0)(\eta_{k}+\alpha_{3})e^{i\eta_{k}t}=C_{n+1}(t),$
$None$ $D_{n+2}(t)=\sum_{k=0}^{2}D^{k}_{n+2}(0)e^{i\eta_{k}t},$ $None$
with
$\alpha_{1}=\Delta+f(\chi,n),~{}~{}~{}\alpha_{2}=f(\chi,n+1),~{}~{}~{}\alpha_{3}=-\Delta+f(\chi,n+2)$
$None$ $\Gamma_{1}=\lambda\sqrt{n+1},~{}~{}~{}\Gamma_{2}=\lambda\sqrt{n+2},$
$None$
where
$\eta_{k}=-\frac{X_{1}}{3}+\frac{2}{3}\biggl{(}\sqrt{X_{1}^{2}-3X_{2}}\biggr{)}\cos(\theta^{k}),$
$None$
with
$\theta^{k}=\biggl{(}\frac{1}{3}\cos^{-1}\biggl{[}\frac{9X_{1}X_{2}-2X_{1}^{3}-27X_{3}}{2(X_{1}^{2}-3X_{2})^{\frac{3}{2}}}\biggr{]}+\frac{2k\pi}{3}\biggr{)},k=0,1,2,$
$None$
and
$X_{1}=\alpha_{1}+\alpha_{2}+\alpha_{3},~{}~{}~{}X_{2}=\alpha_{2}\alpha_{3}+\alpha_{1}(\alpha_{2}+\alpha_{3})-2(\Gamma_{1}^{2}+\Gamma_{2}^{2}),~{}~{}~{}X_{3}=\alpha_{1}\alpha_{2}\alpha_{3}-2(\alpha_{1}\Gamma_{2}^{2}+\alpha_{3}\Gamma_{1}^{2}),$
$None$
where the complex coefficients $D^{k}_{n+2}(0)$, $k=0,1,2$ are given by
$D^{k}_{n+2}(0)=\frac{2\gamma_{1}\Gamma_{1}\Gamma_{2}}{\eta_{kr}\eta_{ks}}~{}~{}~{}~{},k,r,s=0,1,2;~{}~{}~{}~{}k\neq
r\neq s$ $None$
If in Eqs. (16-24), we let $\gamma_{1}\rightarrow\gamma_{4}$ and
$\alpha_{1}=-\Delta+f(\chi,n),~{}~{}~{}\alpha_{2}=f(\chi,n-1),~{}~{}~{}\alpha_{3}=\Delta+f(\chi,n-2),$
$None$ $\Gamma_{1}=\lambda\sqrt{n},~{}~{}~{}\Gamma_{2}=\lambda\sqrt{n-1},$
$None$
and replacing the group of complex amplitudes $A_{n}(t)$, $D_{n+2}(t)$,
$B_{n+1}(t)$ and $C_{n+1}(t)$ by the other group of complex amplitudes
$E_{n}(t)$, $F_{n-1}(t)$, $G_{n-1}(t)$, $H_{n-2}(t)$, we obtain easily the
last group, respectively.
The reduced density operator of the subsystem is given by
$\rho(t)_{A(F)}=\mathbf{Tr}_{F(A)}\rho(t)_{AF}=\mathbf{Tr}_{F(A)}|\psi_{AF}(t)\rangle\langle\psi_{AF}(t)|.$
$None$
Then the reduced atomic density operator in matrix form is given by
$\mathbf{\rho_{A}}=\left(\begin{array}[]{cccc}\rho_{A}^{11}{}&~{}\rho_{A}^{12}{}&~{}\rho_{A}^{13}{}&~{}\rho_{A}^{14}\\\
\\\ \rho_{A}^{21}{}&~{}\rho_{A}^{22}{}&~{}\rho_{A}^{23}{}&~{}\rho_{A}^{24}\\\
\\\ \rho_{A}^{31}{}&~{}\rho_{A}^{32}{}&~{}\rho_{A}^{33}{}&~{}\rho_{A}^{34}\\\
\\\ \rho_{A}^{41}{}&~{}\rho_{A}^{42}{}&~{}\rho_{A}^{43}{}&~{}\rho_{A}^{44}\\\
\end{array}\right),$ $None$
where the elements $\rho_{A}^{sr}$, $s,r\in\\{1,2,3,4\\}$ are given by
$\rho_{A}^{11}=\langle
U|U\rangle=\sum_{n=0}^{\infty}\biggl{\\{}|b_{n}|^{2}\biggl{(}|A_{n}|^{2}+|H_{n-2}|^{2}\biggr{)}+b_{n}b_{n+2}^{\ast}A_{n}H_{n}^{\ast}+b_{n}b_{n-2}^{\ast}A_{n-2}^{\ast}H_{n-2}\biggr{\\}},$
$None$ $\rho_{A}^{22}=\langle
R|R\rangle=\sum_{n=0}^{\infty}\biggl{\\{}|b_{n}|^{2}\biggl{(}|B_{n+1}|^{2}+|G_{n-1}|^{2}\biggr{)}+b_{n}b_{n+2}^{\ast}B_{n+1}G_{n+1}^{\ast}+b_{n}b_{n-2}^{\ast}B_{n-1}^{\ast}G_{n-1}\biggr{\\}}=\langle
S|S\rangle=\rho_{A}^{33},$ $None$ $\rho_{A}^{44}=\langle
T|T\rangle=\sum_{n=0}^{\infty}\biggl{\\{}|b_{n}|^{2}\biggl{(}|D_{n+2}|^{2}+|E_{n}|^{2}\biggr{)}+b_{n}b_{n+2}^{\ast}D_{n+2}E_{n+2}^{\ast}+b_{n}b_{n-2}^{\ast}D_{n}^{\ast}E_{n}\biggr{\\}},$
$None$ $\rho_{A}^{12}=\langle
R|U\rangle=\sum_{n=0}^{\infty}\biggl{\\{}b_{n}b_{n-1}^{\ast}A_{n}B_{n}^{\ast}+b_{n}b_{n-3}^{\ast}B_{n-2}^{\ast}H_{n-2}$
$+b_{n}b_{n+1}^{\ast}A_{n}G_{n}^{\ast}+b_{n}b_{n-1}^{\ast}G_{n-2}^{\ast}H_{n-2}\biggr{\\}}=(\rho_{21})^{\ast}=\langle
S|U\rangle=\rho_{A}^{13},$ $None$ $\rho_{A}^{14}=\langle
T|U\rangle=\sum_{n=0}^{\infty}\biggl{\\{}b_{n}b_{n-2}^{\ast}A_{n}D_{n}^{\ast}+b_{n}b_{n-4}^{\ast}D_{n-2}^{\ast}H_{n-2}$
$+|b_{n}|^{2}A_{n}E_{n}^{\ast}+b_{n}b_{n-2}^{\ast}E_{n-2}^{\ast}H_{n-2}\biggr{\\}}=(\rho_{41})^{\ast},$
$None$ $\rho_{A}^{23}=\langle
S|R\rangle=\sum_{n=0}^{\infty}\biggl{\\{}|b_{n}|^{2}|B_{n+1}|^{2}+b_{n}b_{n-2}^{\ast}C_{n-1}^{\ast}G_{n-1}$
$+b_{n}b_{n+2}^{\ast}B_{n+1}F_{n+1}^{\ast}+|b_{n}|^{2}|F_{n-1}|^{2}\biggr{\\}}=\langle
R|S\rangle=\rho_{32},$ $None$ $\rho_{A}^{24}=\langle
T|R\rangle=\sum_{n=0}^{\infty}\biggl{\\{}b_{n}b_{n-1}^{\ast}B_{n+1}D_{n+1}^{\ast}+b_{n}b_{n-3}^{\ast}D_{n-1}^{\ast}G_{n-1}$
$+b_{n}b_{n+1}^{\ast}B_{n+1}E_{n+1}^{\ast}+b_{n}b_{n-1}^{\ast}E_{n-1}^{\ast}G_{n-1}\biggr{\\}}=(\rho_{42})^{\ast}=\rho_{A}^{34},$
$None$
## III Spin Squeezing
Spin squeezing phenomenon reflects the reduced quantum fluctuations in one of
the field quadratures at the expense of the other corresponding stretched
quadrature. In the literature, KIUE93 ; WBI94 ; PZM02 ; SDCZ01 ; ZPP91 ;
WIZO81 , there are several definitions of spin squeezing and which one is the
best is still an unsolved issue. Squeezing or reduction of quantum
fluctuations, for arbitrary operators $A$ and $B$ which obey the commutation
relation $[A,B]=C$, is the product of the uncertainties in determining their
expectation values as follows WIZO81 :
$\Delta A\Delta B\geq\frac{1}{2}|\langle C\rangle|,$ $None$
where $(\Delta A)^{2}=\langle A^{2}\rangle-\langle A\rangle^{2}$ and $(\Delta
B)^{2}=\langle B^{2}\rangle-\langle B\rangle^{2}$.
In this work spin squeezing parameters are based on angular momentum
commutation relations. From the commutation relation $[J_{x},J_{y}]=iJ_{z}$
the uncertainty relation between different componenets of the angular momentum
given by
$\Delta J_{x}\Delta J_{y}\geq\frac{1}{2}|\langle J_{z}\rangle|,$ $None$
where
$J_{x}=\frac{1}{2}(J_{+}+J_{-}),$ $None$ $J_{y}=\frac{1}{2i}(J_{+}-J_{-}),$
$None$
where the operators $J_{+}$, $J_{-}$ and $J_{z}$ are given by (6). Without
violating Heisenberg’s uncertainty relation, it is possible to redistribute
the uncertainty unevenly between $J_{x}$ and $J_{y}$, so that a measurement of
either $J_{x}$ or $J_{y}$ becomes more precise than the standard quantum limit
$\sqrt{|\langle J_{z}\rangle|/2}$. States with this property are called spin
squeezed states in analogy with the squeezed states of a harmonic oscillator.
Consequently, the two squeezing parameters can be written as
$F_{1}=(\Delta J_{x})^{2}-\frac{1}{2}|\langle
J_{z}\rangle|=\frac{1}{2}\biggl{(}1-\frac{1}{2}\biggl{\langle}(J_{+}+J_{-})\biggr{\rangle}^{2}-|\langle
J_{z}\rangle|\biggr{)},$ $None$ $F_{2}=(\Delta J_{y})^{2}-\frac{1}{2}|\langle
J_{z}\rangle|=\frac{1}{2}\biggl{(}1-\frac{1}{2i}\biggl{\langle}(J_{+}-J_{-})\biggr{\rangle}^{2}-|\langle
J_{z}\rangle|\biggr{)},$ $None$
If the parameter $F_{1}$ ($F_{2}$) satisfies the condition $F_{1}<0$
($F_{2}<0$), the fluctuation in the component $J_{x}$ ($J_{y}$) is said to be
squeezed.
Using the wave function (11), we can easily compute the following time-
dependent expectation values of the operators $J_{-}$, $J_{+}$ and $J_{z}$,
respectively in the forms
$\langle J_{-}\rangle=2(\rho_{A}^{12}+\rho_{A}^{24})=\langle
J_{+}\rangle^{\ast},$ $None$ $\langle
J_{z}\rangle=\rho_{A}^{11}-\rho_{A}^{44},$ $None$
## IV Quantum Entropy
Quantum entropy, as a natural generalization of Boltzmann classical entropy,
was proposed by von Neumann NEUMANN27 . It has been applied, in particular, as
a measure of quantum entanglement, quantum decoherence, quantum optical
correlations, purity of states, quantum noise or accessible information in
quantum measurement (the capacity of the quantum channel). Entropy is related
to the density matrix, which provides a complete statistical description of
the system. Since we have assumed that the two two-level atoms and the single-
mode binomial field are initially in a disentangled pure state, the total
entropy of the system is zero. In terms of Araki $\&$ Lieb inequality of the
entropy ARLE70
$|S_{A}(t)-S_{F}(t)|\leq S_{AF}(t)\leq|S_{A}(t)+S_{F}(t)|,$ $None$
we can find that the reduced entropies of the two subsystems are identical,
namely, $S_{A}(t)=S_{F}(t)$. Here, the field entropy can be obtained by
operating the atomic entropy. The quantum field entropy can be defined as
follows VPRK97
$S(\rho_{A})=-\sum_{s=1}^{4}~{}\Pi_{A}^{(s)}\ln\Pi_{A}^{(s)},$ $None$
where $\Pi_{A}^{(s)}$, ($s=1,2,3,4$) is the eigenvalue of the reduced density
matrix, Eq. (27), and can be represented by the roots of forth order algebraic
equation
$c_{0}+c_{1}\Pi_{A}+c_{2}\Pi_{A}^{2}+c_{3}\Pi_{A}^{3}+\Pi_{A}^{4}=0,$ $None$
with the coefficients, $c$’s, are given by
$c_{3}=-\rho_{11}-\rho_{22}-\rho_{33}-\rho_{44},$ $None$
$c_{2}=-|\rho_{41}|^{2}-2|\rho_{42}|^{2}-2|\rho_{12}|^{2}-|\rho_{23}|^{2}+2\rho_{22}(\rho_{11}+\rho_{44})+\rho_{22}^{2}+\rho_{11}\rho_{44},$
$None$
$c_{1}=2|\rho_{14}|^{2}\rho_{22}+2|\rho_{24}|^{2}(\rho_{11}+\rho_{22})+2|\rho_{12}|^{2}(\rho_{22}+\rho_{44})-\rho_{22}^{2}(\rho_{11}+\rho_{44})$
$-2\rho_{11}\rho_{22}\rho_{44}-\Re(\rho_{23})(|\rho_{42}|^{2}+|\rho_{12}|^{2})-2\Re(\rho_{41}\rho_{12}\rho_{24})+|\rho_{32}|^{2}(\rho_{11}+\rho_{44}),$
$None$
$c_{0}=|\rho_{41}|^{2}|\rho_{23}|^{2}-\rho_{22}\rho_{33}|\rho_{41}|^{2}-\rho_{11}\rho_{33}|\rho_{42}|^{2}-\rho_{44}\rho_{22}|\rho_{12}|^{2}-\rho_{11}\rho_{44}|\rho_{23}|^{2}-\rho_{33}\rho_{44}|\rho_{12}|^{2}$
$+\rho_{11}\rho_{22}\rho_{33}\rho_{44}-\rho_{11}\rho_{22}|\rho_{24}|^{2}+(\rho_{11}|\rho_{24}|^{2}+\rho_{44}|\rho_{12}|^{2})\Re(\rho_{23})+(\rho_{22}+\rho_{33}-\Re(\rho_{23}))\Re(\rho_{41}\rho_{12}\rho_{24}),$
$None$
where we used the notations
$\rho_{22}=\rho_{33},~{}~{}~{}~{}~{}\rho_{12}=\rho_{13},~{}~{}~{}~{}~{}\rho_{24}=\rho_{34}.$
$None$
The eigenvalues $\Pi_{A}^{(s)}$, ($s=1,2,3,4$) are given, respectively by
$\Pi_{A}^{(s)}=\frac{U_{s}+(-1)^{s}V_{s+1}}{2};~{}~{}~{}~{}s=1,2$ $None$
$\Pi_{A}^{(s)}=\frac{U_{s}+(-1)^{s+1}V_{s+2}}{2};~{}~{}~{}~{}s=3,4$ $None$
where
$U_{s}=-\frac{c_{3}}{2}+(-)^{s}f,V_{s}=\sqrt{z_{3}+(-1)^{s}z_{4}},$ $None$
with
$z_{3}=-2~{}c_{2}+\frac{3~{}c_{3}^{2}}{4}-f^{2},z_{4}=\frac{8c_{1}-4c_{2}c_{3}+c_{3}^{3}}{4~{}f}$
$None$
## V Husimi $Q$-function
For measuring the quantum state of the radiation field, balanced homodyning
has become a well established method, it directly measures phase sensitive
quadrature distributions. The homodyne measurement of an electromagnetic field
gives all possible linear combinations of the field quadratures. The average
of the random outcomes of the measurement is connected with the marginal
distribution of any quasi-probability used in quantum optics. It has been
shown from earlier studies EWIGNER32 ; ZWIGNer32 ; KAGL69 ; HICOSCWG84 that
the quasi-probability functions $W$-, (Husimi) $Q$-, and (Glauber-Sudershan)
$P$-function, are important for the statistical description of a microscopic
system and provide insight into the nonclassical features of the radiation
fields. Therefore, we devote the present section to concentrate on one of
these functions, that is the Husimi $Q$-function which has the nice property
of being always positive and further advantage of being readily measurable by
quantum tomographic techniques BOTAWA98 ; MANTOM97 . In fact, Husimi
$Q$-function is not only a convenient tool to calculate the expectation values
of anti-normally ordered products of operators, but also to give some insight
into the mechanism of interaction for the model under consideration. The
relation between the phase-space measurement; Husimi $Q$-function; and the
classical information-theoretic entropy associated with quantum fields was
introduced by Wehrl WEHRL79 . However, on expanding the von Neumann quantum
entropy in a power series of classical entropies, it was shown explicity
PEKRPELUSZ86 that the first term of this expansion is the Wehrl entropy.
Thus, Husimi $Q$-function can be related to the von Neumann quantum entropy in
different approaches WEHRL79 ; PEKRPELUSZ86 ; FAGU06 ; CAALCARA09 ; HUFAN09 ;
MIMAWA00 ; MIWAIM01 ; BERETA84 .
The Husimi $Q$-function can be given in the form as HICOSCWG84 ; HUSIMI40 ;
FuSOLO001
$Q(\alpha)=\frac{\langle\alpha\mid\rho_{F}\mid\alpha\rangle}{\pi},$ $None$
where $\rho_{F}$ is the reduced density operator of the cavity field given by
(27). The state $\mid\alpha\rangle$ represents the well-known coherent state
with amplitude $\alpha=X+iY$. Inserting $\rho_{F}$ into Eq. (56), we can
easily obtain the Husimi $Q$-function of the cavity field
$Q(\alpha)=\frac{1}{\pi}(\mid\langle\alpha\mid
U\rangle\mid^{2}+\mid\langle\alpha\mid T\rangle\mid^{2})$ $None$
where
$\langle\alpha\mid
U\rangle=e^{-\mid\alpha\mid^{2}/2}\sum_{n=0}^{\infty}\biggl{[}b_{n}\frac{\alpha^{\ast
n}}{\sqrt{n!}}A_{n}(t)+b_{n+2}\frac{\alpha^{\ast
n}}{\sqrt{n!}}H_{n}(t)\biggr{]}$ $None$
and
$\langle\alpha\mid
T\rangle=e^{-\mid\alpha\mid^{2}/2}\sum_{n=0}^{\infty}b_{n}\biggl{[}\frac{\alpha^{\ast
n}}{\sqrt{n!}}E_{n}(t)+\frac{\alpha^{\ast
n+2}}{\sqrt{(n+2)!}}D_{n+2}(t)\biggr{]}$ $None$
## VI Discussion of the numerical results
Using different sets of parameters for the initial state of the system we
calculate numerically the quantum entropy $S_{A}$, spin squeezing parameters
$F_{1}$ and $F_{2}$ and atomic population $\langle\sigma_{z}\rangle$ as a
reference function. All results are plotted as functions of the Rabi angle
$\lambda t$. For each set of parameters four pictures are displayed. The
pictures (a and b) show, respectively, squeezing parameters $F_{1}$ and
$F_{2}$, while the pictures (c and d) show the quantum entropy $S_{A}$ and the
atomic population $\langle\sigma_{z}\rangle$. For all our plots we set the
Bell-state parameters $\gamma_{1}=\frac{1}{\sqrt{2}}$ and
$\gamma_{4}=i\gamma_{1}$. In figures 1 and 2 we have plotted the above
mentioned functions with $|p|=0.9$, $\chi/\lambda=0$ and various values of the
detuning parameter $\Delta/\lambda$. From these figures we can easily notice
that, just after the onset of interaction these functions fluctuate for short
period of time. This short period of revival is followed by a long period of
collapse. The period of revival starts longer for one period of time with high
amplitude to become wider with smaller amplitude as time goes. This is because
the width and heights of the revivals belonging to the different series of
eigenvalues are different. Furthermore, as we increase the detuning parameter
$\Delta/\lambda$ from its value $\Delta/\lambda=0$ (resonance case), the
overlap of revivals noticeably decreases with the increase in
$\Delta/\lambda$. Also, the periods of oscillations within the revival
decrease with the increase in the detuning parameter $\Delta/\lambda$. For the
population, $\langle\sigma_{z}\rangle$, the period of revival depends on the
average number $M|p|$ of photons whereas the time of collapse depends on the
dispersion, $M|p|(1-|p|)$, in the photon number distribution JoshPur87 .
Figure 1: Spin squeezing parameters $F_{1}$ (a), $F_{2}$ (b), atomic entropy
$S_{A}$ (c), and atomic inversion (d) with $|p|=0.9$, $M=50$, $\chi/\lambda=0$
and $\Delta/\lambda=0$
Moreover, from these figures we can see that spin squeezing parameters $F_{1}$
and $F_{2}$ experience collapse and revival where atomic population exhibits
collape and revival as time going. When we turn our attention to the role that
spin squeezing parametres play to discover entanglement properties, we can
realize that the behaviors of both squeezing parameter $F_{1}$ and atomic
entropy $S_{A}$ are equivalent, i.e., entanglemet implies spin squeezing and
vice versa. This can be understood as follows: quantum entropy $S_{A}$
oscillates when $F_{1}$ exhibits oscillations with same periods of time.
Figure 2: The same as Fig. (1) but for $\Delta/\lambda=10$ Figure 3: The same
as Fig. (1) but for $\chi/\lambda=0.5$
Furthermore, the function $S_{A}$ goes to its maximum when $F_{1}$ shows
oscillations around very small value (between 0.45 and 0.5) of its maximum and
when $F_{2}$ shows collapse equal to its maximum, while $S_{A}$ reaches its
minimum when squeezing occurs. This behavior occurs periodically for both
$S_{A}$ and $F_{1}$. This means that, on on-resonance atomic-system-field
interaction, we can, with full success, understand entanglement dynamics from
the dynamic of spin squeezing parameters $F_{1}$ and $F_{2}$ and vice versa.
Figure 4: The same as Fig. (1) but for $\chi/\lambda=5.0$
Let us now come to the case of off-resonance interaction between the atomic
system and the cavity field. In this case the same general behavior (with
periods shift to right when $\Delta/\lambda=10$) is noticed. Additionally, the
oscillations become more dense with reduced maximum of $S_{A}$ corresponding
to the increase of the oscillation interval of $F_{1}$ (between 0.4 and 0.5
and become longer as $\Delta/\lambda$ increase) and when some intervals of
collapse begin to appear.
The surprising and very interesting is the effect of the nonlinear medium
individually and in the presence of the detuning parameter $\Delta/\lambda$.
Figure 5: The same as Fig. (3) but for $\Delta/\lambda=5.0$
To examine the effect of these mentioned parameters, we recall figures (3-5).
These figures have been pictured by the setting of different values of the
parameter $\chi/\lambda$ individually and in the presence of the detuning
parameter $\Delta/\lambda$. It is easy to see the change in figures shape
where the standard behavior was changed completely. For a weak Kerr medium
such that $\chi/\lambda=0.5$, our reference function,
$\langle\sigma_{z}\rangle$, shows behavior similar to the modified Jayned-
Cumming model with Kerr medium ETCU63 ; JOSHPUR92 accompanied with reduced
amplitude of oscillations.
Figure 6: The same as Fig. (1) but for $|p|=0.98$, $M=100$
Furthermore, the population $\langle\sigma_{z}\rangle$ and spin squeezing
parameter $F_{2}$ oscillate periodically with fixed periods are equivalent but
$F_{1}$ does not. A quick look at the squeezing parameter $F_{1}$ one can
realize easily that it oscillates rapidly. The oscillations overlap for a
period of time (except for some instances) to become dense to show
periodically wave packets of Gaussian envelope with amplitude decreases as
time develops. The more the Gaussian-packet-envelope amplitude decreases the
more the entropy increases, i.e., stronger entanglement can be showed, see
figure 3. This behavior becomes more clear when we consider the detuning
parameter in our numerical computations, see figure 5.
Figure 7: The same as Fig. (6) but for $\chi/\lambda=0.5$
Furthermore, when the nonlinear medium becomes stronger,
$\langle\sigma_{z}\rangle$, $F_{1}$ and $F_{2}$ show chaotic behavior with no
indications of revivals or any other regular structure. This is accompanied
with change in the entropy maximum from slow to rapid increase as
$\chi/\lambda$ increases with time develops (see Fig. 4). This behavior is
dominant without or with high values of $\Delta/\lambda$.
With the increase of $|p|$ and $M$, such that $|p|=0.98$, and $M=100$, i.e.;
on increasing the average number $M|p|$ of photon, the oscillations of
squeezing parametres $F_{1}$, $F_{2}$, $\langle\sigma_{z}\rangle$ and quantum
entropy $S_{A}$ becomes more dense. This means that every two neighbor
overlapping revivals start to overlap again. This is because with larger mean
photon $M|p|$, these functions have bigger values with time evolution, which
causes dense oscillations of the cavity field parametres. In other words,
there are more revival series since more eigenvalues can be found in this
model. However, the same behavior, we saw before for $|p|=0.9$, is seen again
except for the envelope width becomes wider which resulted in fewer packets of
oscillation appear in the same period of time, see figure 6 and 7. It is worth
to note that each revival series of oscillations corresponds to a beat
frequency. Moreover, on weak Kerr medium the relation between the atomic
entropy and spin squeezing parametres seems more complicated. In this case all
of them oscillate with no indications of revivals or any other regular
structure where the Gaussian envelope completely disappears. The effect of
strong Kerr medium individually and with the coexistence of small detuning
exhibits behavior similar to that when $|p|=0.9$.
At the end we are going to focus our attention on the representation of the
field in phase space which provides some aspects of the field dynamics. Figure
8 shows mesh plots (left) and the corresponding contour plots (right) of the
Husimi $Q$-function in the complex $\alpha$-plane $X=Re(\alpha)$,
$Y=Im(\alpha)$ for the Rabi angle $\lambda t=\pi/4$ and different values of
Kerr parameter $\chi/\lambda$, while all other parameters are kept without
change as in Fig. 1. From figure 8a, it is clear that the state of the field
is a squeezed state, since the Husimi $Q$-function has different widths in the
$X$ and $Y$ directions. On the other hand, the squeezing is generated by the
nonlinearity inherent to the system by the binomial state. This can be
explained in terms of a superposition of different numbers of states of
different phases, creating a deviation from the classical phase. It is well
known that the squeezed states is a general class of the minimum-uncertainty
states SACHNEBU03 . Bearing in mind that the nonlinearity of the binomial
state yields squeezing of the quantum field and, as a result, entanglement
between the two sub-systems is also produced BUCHDADUMORU01 . It is of
particular interest to see how the Husimi $Q$-function behaves once the Kerr
medium is added. When $\chi/\lambda$ increases by a fraction value, we notice
clearly the single blob become almost perfectly circular with radius
$|\alpha|\approx 7.5$ rotates in the counterclockwise direction, see Figs. 8a,
b, d and 8f. This behavior of the quantum field distribution is similar to
that of the thermal state SACHNEBU03 which means that once the Kerr medium is
added, makes it clear that rethermalization of the binomial field is indeed
taking place. Now, it is perfectly sensible to ask, what is the situation if
the Kerr Medium parameter is increased by an integer value? In this case, when
the state evolves further, a multi-component structure develops as shown in
Figs 8c, e and 8g, respectively. In these figures, the Husimi $Q$-function
demonstrates that the quantum states obtained corresponding to Schrödinger cat
states. Moreover, we see that the cat states have different number of
components at different values of $\chi/\lambda$. Different mechanisms
demonstrated that, for the case of a radiation field propagating in a
nonlinear medium, Schrödinger cat states are generated FuSOLO001 ; MIMAWA00
with different number of components at different times in the evolution DYAO97
. It was shown that the splitting of the Husimi function, which is the
signature of the formation of Schrödinger cat states, is related strongly to
quantum entanglement VAOR95 ; ORPAKA95 ; JEOR94 ; MIMAWA00 ; MIWAIM01 .
To discuss the evolution of the Husimi $Q$-function in the case of resonance
and fixed value of the Kerr parameter, i.e., $\chi/\lambda=5.0$ (strong Kerr
medium), we have setted various values for the Rabi angle $\lambda t$ in our
computations, i.e., $\lambda t=0.0,\pi/6,\pi/4,\pi/3,\pi/2$ and $\pi$. The
results are displayed in figure 9. A collision of six blobs occurs gradually
when $\lambda t=\pi/6$, which implies the rethermalisation of the quantum
field. At the time evolution of $\lambda t=\pi,\pi/2,\pi/3$ and $\pi/4$ the
distribution of the Husimi $Q$-function splits into two, three and four blobs
corresponding to Schrödinger cat states corresponding to different number of
components at different times in the evolution. The center of blobs lies on a
circle with radius $|\alpha|$ centered at $X=Y=0.0$.
Figure 8: Husimi Q-function with $|p|=0.9$, $M=50$, $t=\pi/4$,
$\Delta/\lambda=0$ and (a)$\chi/\lambda=0.0$, (b)$\chi/\lambda=0.5$, (c)
$\chi/\lambda=1.0$, (d) $\chi/\lambda=1.5$, (e) $\chi/\lambda=2.0$, (f)
$\chi/\lambda=2.5$ and (g) $\chi/\lambda=5.0$
FIG. 8: continued
Figure 9: Husimi Q-function with $|p|=0.9$, $M=50$, $\chi=5.0$,
$\Delta/\lambda=0$ and (a) $t=0.0$, (b)$t=\pi/6$, (c) $t=\pi/4$, (d)
$t=\pi/3$, (e) $t=\pi/2$ and (f) $t=\pi$
FIG. 9: continued
## VII Conclusion
In Conclusion, we have shown that spin squeezing implies entanglement for
quantum tripartite-state, where the subsystem includes the bipartite-state is
identical. We have proved that spin squeezing parameters can be a convenient
tool to give some insight into the mechanism of entanglement for the model
under consideration. Moreover, a subsystem cavity field contains a nonlinear
medium enhances noticeably the dynamics of entanglement specially when
interacts with atomic subsystem off-resonantly. More clear insight into the
relation between entanglement and the phase space distribution i.e., Husimi
$Q$-function, is also illustrated. In this situation, the strong Kerr medium
stimulates the creation of Schrödinger cat states which is necessary for the
generation of entanglement.
## References
* (1) H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978); IT-25, 179 (1979); IT-26, 78 (1980);
* (2) C. M. Caves, Phys. Rev. D 23, 1693 (1981); J. Gea-Banacloche, and G. Leuchs, J. Mod. Opt. 34, 709 (1987).
* (3) X. G. Wang and B. C. Sanders, Phys. Rev. A 68, 033821 (2003).
* (4) U. V. Poulsen and K. Mölmer, Phys. Rev. Lett. 87, 123601 (2001).
* (5) I. Bouchoule and K. Molmer, Phys. Rev. A 65, 041803(R) (2002).
* (6) C. Genes, P. R. Berman and A. G. Rojo, Phys. Rev. A 68, 043809 (2003).
* (7) A. S. Sorensen and K. Molmer, Phys. Rev. Lett. 86, 4431 (2001).
* (8) L. Vernac, M. Pinard and E. Giacobino, Phys. Rev. A 62, 063812 (2000).
* (9) X. G. Wang, Opt. Commun. 200, 277 (2001).
* (10) X. G. Wang, Phys. Rev. A 331, 164 (2004).
* (11) D. Yan, X. G. Wang, L. J. Song and Z. G. Zong, Central Eur. J. Phys. 5(3), 367 (2007).
* (12) P. Zhou and J. S. Peng, Phys. Rev. A 72, 3331(1991).
* (13) A. Andre and M. D. Lukin, ibid. 65, 053819 (2002).
* (14) D. J. Wineland, J. J. Bollinger, W. M. Itano and D. J. Heinzen, Phys. Rev. A 50, 67 (1994).
* (15) M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).
* (16) G. S. Agarwal and R. R. Puri, Phys. Rev. A 41, 3782(1990).
* (17) M.D. Lukin, S.F. Yelin, and M. Fleischhauer, Phys. Rev. Lett. 84, 4232 (2000).
* (18) A. Kuzmich, K. Mölmer, and E.S. Polzik, Phys. Rev. Lett. 79, 4782 (1997)
* (19) J. Wesenberg and K. Mölmer, Phys. Rev. A 65, 062304 (2002)
* (20) K. Helmerson and L. You, Phys. Rev. Lett. 87, 170402 (2001)
* (21) Ö. E. Müstecaplio$\breve{g}$lu, M. Zhang, and L. You, Phys. Rev. A 66, 033611 (2002)
* (22) U. Poulsen and K. Mölmer, Phys. Rev. A 64, 013616 (2001)
* (23) L. K. Thomsen, S. Mancini, and H.M. Wiseman, Phys. Rev. A 65, 061801 (2002)
* (24) T. Gasenzer, D.C. Roberts, and K. Burnett, Phys. Rev. A 65, 021605 (2002).
* (25) J. K. Stockton, J.M. Geremia, A.C. Doherty, and H. Mabuchi, Phys. Rev. A 67, 022112 (2003).
* (26) L. Zhou, H.S. Song and C. Li, J. Opt. B: Quantum Semiclass. Opt. 4, 425 (2002).
* (27) X. Wang, A. Sörensen, and K. Mölmer, Phys. Rev. A 64, 053815 (2001)
* (28) D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A 64, 052106 (2001).
* (29) A. Dantan, M. Pinard, V. Josse, N. Nayak and P. R. Breman, ibib. 67, 045801 (2003).
* (30) B. Julsgaard, A. Kozhekin and E. S. Polzik, Nature (London) 413, 400 (2001).
* (31) V. Meyer, M. A. Rowe, D. Keilpinski, C. A. Sackett, W. M. Itano, C. M. Monroe and D. J. Wineland, Phys. Rev. Lett. 86, 5870 (2001).
* (32) D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore and D. J. Heinzen Phys. Rev. A 46, R6797 (1992).
* (33) A. Sorensen, M. L. Duan, J. I. Cirac and P. Zoller, Nature (London) 409, 1044 (2001).
* (34) A. Sörensen, Phys. Rev. A 65, 043610 (2002).
* (35) D. W. Berry and B.C. Sanders, New J. Phys. 4, 8 (2002).
* (36) M. G. Raymer, A.C. Funk, B.C. Sanders, and H. de Guise, quant-ph/0210137.
* (37) X. G. Wang and B. C. Sanders, Phys. Rev. A 68, 012101 (2003).
* (38) M. S. Ateto, Int. J. Quant. Inf. 5, 535(2007).
* (39) M. S. Ateto, Int. J. Theor. Phys. 48, 545 (2009).
* (40) M. S. Ateto, Int. J. Theor. Phys. 49, 276 (2010); DOI 10.1007/s10773-009-0201-0.
* (41) M. S. Ateto, Applied Mathematics $\&$ Information Sciences, 3, 41 (2009).
* (42) S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022(1997).
* (43) W. K. Wootters, Phys. Rev. Lett. 80, 2245(1998).
* (44) X. Q. Xi, S. R. Hao, W. X. Chen and R. H. Yue, Chin. Phys. Lett. 19, 1044(2002).
* (45) G. F. Zhang and J. Q. Liang, Chin. Phys. Lett. 20, 452 (2003).
* (46) V. Buzek, J. Mod. Optics, 37, 303 (1990)
* (47) D. Stoler, B. E. H. Saleh, and M. C. Teich, Optica Acta, 32, 345 (1985)
* (48) A. Vidiella-Barranco, J. A. Roversi, Phys. Rev. A 50, 5233 (1994).
* (49) M. Tavis and F. W. Cummings, Phys. Rev. 170(2), 379 (1968).
* (50) H. Pu, W. Zhang and P. Meystre, Phys. Rev. Lett. 89, 090401 (2002).
* (51) D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
* (52) J. von Neumann, Götingger Nachr., 273 (1927)
* (53) H. Araki and E. H. Lieb, Commun. Math. Phys. 18, 160 (1970).
* (54) V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knoght, Phys. Rev. Lett. 78, 2275 (1997).
* (55) E. Wigner, Phys. Rev. 40, 749 (1932).
* (56) Z. Wigner, Phys. Chem. B19, 203 (1932).
* (57) K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
* (58) M. Hillary, R. F. O’Conell, M. O. Scully and E. wigner, Phys. Rep. 106, 121 (1984).
* (59) E. L. Bolda, S. M. Tan and D. Walls, Phys. Rev. A 57, 4686 (1998).
* (60) S. Mancini and P. Tombesi, Europhys. Lett. 40, 352 (1997)
* (61) A. Wehrl, Rep. Math. Phys. 16, 353 (1979)
* (62) V. Pe$\breve{r}$inov$\acute{a}$, J. K$\breve{r}$epelka, J. Pe$\breve{r}$ina, J. Luk$\breve{s}$ and P. Szlachetkta, Opt. Acta 33, 15 (1986)
* (63) Hong-yi Fan and Qin Guo, quant-ph/0611206v1 (2006)
* (64) C. P.-Campos, J. R. G.-Alonso, O. Casta$\tilde{n}$os and R. L.-Pe$\tilde{n}$a, cond-mat.guant-gas/0910.3256v1 (2009)
* (65) Li-yun Hu and Hong-yi Fan, quant-ph/0911.0125v1, (2009)
* (66) A. Miranowicz,J. Bajer, M . R. B. Wahiddin and N. Imoto, J. Phys A: Math. Gen. 34, 3887 (2001).
* (67) A. Miranowicz, H. Matsueda and M. R. B. Wahiddin, J. Phys. A: Math. Gen. 33, 5159 (2000)
* (68) G. P. Beretta, J. Math. Phys. 25, 1507 (1984)
* (69) K. Husimi, Proc. Phys. Math. Soc. Japan 22, 264 (1940)
* (70) H. Fu and A. I. Solomom, J. Mod. Opt. 49, 259 (2002).
* (71) A. Joshi and R. R. Puri, J. Mod. Opt., 34(11), 1421 (1987).
* (72) E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
* (73) A. Joshi and R. R. Puri, Phys. Rev. A 45, 5056 (1992).
* (74) J. Rogel-Salazar, S. Choi, G. H. C. New and K. Burnett, cond-mat.guant-soft/0302066v1 (2003)
* (75) K. Burnett, S. Choi, M. Davis, J. A. Dunninghman, S. A. Morgan and M. Rusch, C. R. Acad. Sci. IV 2, 399 (2001).
* (76) D. Yao, Phys. Rev. A 55, 701 (1997).
* (77) J. A. Vaccaro and A. Orlowski, Phys. Rev. A 51, 4172 (1995)
* (78) A. Orlowski, H. Paul and G. Kastelewicz, Phys. Rev. A 52, 1621 (1995)
* (79) I. Jex and A. Orlowski, J. Mod. Opt. 41, 2301 (1994)
|
arxiv-papers
| 2009-11-22T08:56:44 |
2024-09-04T02:49:06.614769
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. S. Ateto",
"submitter": "Mohammed Ateto Dr.",
"url": "https://arxiv.org/abs/0911.4240"
}
|
0911.4256
|
# Improved variational approach for the Cornell potential
V.V. Kudryashov kudryash@dragon.bas-net.by V.I. Reshetnyak
v.reshetnyak@dragon.bas-net.by Institute of Physics, National Academy of
Sciences of Belarus
68 Nezavisimosti Ave., 220072, Minsk, Belarus
###### Abstract
The approximate radial wave functions for the Cornell potential describing
quark-antiquark interaction are constructed in the framework of a variational
method. The optimal values of the variational parameters are fixed by the
fulfillment of the requirements of the virial theorem and the minimality
condition for integral discrepancy. The results of calculation with the simple
trial function are in a good agreement with exact numerical results.
variational method, trial functions, integral discrepancy, Cornell potential
###### pacs:
03.65.Db; 03.65.Ge
Many investigations have been devoted to the quantum mechanical non-
relativistic description of quark-antiquark bound states with the Cornell
potential 1 by means of different approximation procedures (see, e.g., 2 ; 3
; 4 ; 5 and references therein) based on perturbation theory, the variational
method and their combinations. Recently a new improved variational approach 6
was proposed. This method was applied to the radial Schrödinger equation with
nonsingular power-law potentials $r^{s}$ with positive $s$ 7 . In the present
work, we apply our approach to the Cornell potential with the Coulomb
singularity.
Following 1 we study the radial Schrödinger equation in the dimensionless
form
$\hat{H}\psi(r)=E\psi(r),\quad\hat{H}=-\frac{d^{2}}{dr^{2}}+\frac{l(l+1)}{r^{2}}+V(r)$
(1)
for the Cornell (Coulomb plus linear) potential
$V(r)=-\frac{k}{r}+r.$ (2)
We use the simple functions 8
$<r|\psi_{l,n}(a,b)>=\psi_{l,n}(a,b,r)=N\sqrt{a}G_{l,n}(b,ar)$ (3)
as the trial functions, where
$G_{l,n}(b,x)=x^{l+1}\exp(-x^{b})L_{n}^{\frac{2l+1}{b}}(2x^{b}).$ (4)
Here $L_{n}^{\frac{2l+1}{b}}(2x^{b})$ is the Laguerre polynomial, $a$ and $b$
are variational parameters, $N$ is a normalization factor
$(<\psi_{l,n}(a,b)|\psi_{l,n}(a,b)>=1)$. Note that these functions reproduce
the exact results for the Coulomb potential if we choose $b=1$. In order to
fix the parameter values we consider two conditions to which the trial
functions should satisfy.
The first is the virial theorem. The virial condition
$<\psi_{l,n}(a,b)|-\frac{d^{2}}{dr^{2}}+\frac{l(l+1)}{r^{2}}-\frac{1}{2}\frac{k}{r}-\frac{1}{2}r|\psi_{l,n}(a,b)>=0$
(5)
leads to the following equation for $a$ :
$a^{3}=ku_{2}(b)a^{2}+u_{3}(b)$ (6)
where
$u_{2}(b)=\frac{\int_{0}^{\infty}G_{l,n}^{2}(b,x)x^{-1}dx}{2\int_{0}^{\infty}G_{l,n}(b,x)(-\frac{d^{2}}{dx^{2}}+\frac{l(l+1)}{x^{2}})G_{l,n}(b,x)dx},$
$u_{3}(b)=\frac{\int_{0}^{\infty}G_{l,n}^{2}(b,x)xdx}{2\int_{0}^{\infty}G_{l,n}(b,x)(-\frac{d^{2}}{dx^{2}}+\frac{l(l+1)}{x^{2}})G_{l,n}(b,x)dx}.$
The exact solution of this equation allows us to express the parameter $a$ via
the parameter $b$ :
$\displaystyle a_{0}(b)=\frac{1}{3}ku_{2}(b)$ $\displaystyle+$
$\displaystyle\left(\frac{k^{3}u^{3}_{2}(b)}{27}+\frac{u_{3}(b)}{2}+\sqrt{\frac{k^{3}u^{3}_{2}(b)u_{3}(b)}{27}+\frac{u^{2}_{3}(b)}{4}}\right)^{1/3}$
(7) $\displaystyle+$
$\displaystyle\left(\frac{k^{3}u^{3}_{2}(b)}{27}+\frac{u_{3}(b)}{2}-\sqrt{\frac{k^{3}u^{3}_{2}(b)u_{3}(b)}{27}+\frac{u^{2}_{3}(b)}{4}}\right)^{1/3}$
Thus, the considered problem is transformed into a one-parameter problem. Note
that the virial condition is equivalent to the usual condition
$\frac{\partial<\psi_{l,n}(a,b)|\hat{H}|\psi_{l,n}(a,b)>}{\partial a}{}=0.$
(8)
Denote the one-parameter trial functions with $a=a_{0}(b)$ as
$<r|\psi^{0}_{l,n}(b)>=\psi^{0}_{l,n}(b,r)=<r|\psi_{l,n}(a_{0}(b),b)>=\psi_{l,n}(a_{0}(b),b,r).$
Then the energy has the form
$E^{(1)}_{l,n}(b)=<\psi^{0}_{l,n}(b)|\hat{H}|\psi^{0}_{l,n}(b)>.$ (9)
In addition to the energy, expectation values of the squared Hamiltoninan
characterizing the goodness of the approximate eigenfunctions, can be
calculated:
$E_{l,n}^{(2)}(b)=\sqrt{<\psi^{0}_{l,n}(b)|\hat{H}^{2}|\psi^{0}_{l,n}(b)>}.$
(10)
As the second requirement imposed on the trial function, we select the
requirement that integral discrepancy
$d_{l,n}(b)=\frac{<\psi^{0}_{l,n}(b)|\hat{H}^{2}|\psi^{0}_{l,n}(b)>}{(<\psi^{0}_{l,n}(b)|\hat{H}|\psi^{0}_{l,n}(b)>)^{2}}-1,$
(11)
is minimum 6 . The quantity $d_{l,n}(b)$ characterizes goodness of the
approximation and is equal to zero for an exact solution of the Schrödinger
equation. The integral discrepancy is simply connected with the local
discrepancy 9
$<r|D_{l,n}(b)>=D_{l,n}(b,r)=\frac{\hat{H}\psi^{0}_{l,n}(b,r)}{{<\psi^{0}_{l,n}(b,r)|\hat{H}|\psi^{0}_{l,n}(b,r)>}}-\psi^{0}_{l,n}(b,r)$
(12)
by relation
$d_{l,n}(b)=<D_{l,n}(b)|D_{l,n}(b)>.$ (13)
The corresponding equation for selection of parameter $b$ is
$\frac{d}{db}d_{nl}(b)=0.$ (14)
Note that we must find the absolute minimum of $d_{l,n}(b)$ corresponding to
the minimal rotation of a trial vector under the action of the Hamilt0nian in
Hilbert space 9 .
The minimality condition for integral discrepancy is alternative to the
minimal sensitivity condition 10
$\frac{d}{db}E^{(1)}_{l,n}(b)=0.$ (15)
There can be several stationary points in the case of the function
$E^{(1)}_{l,n}(b)$, and there is no method to select one of them. Besides,
energy is not a preferred quantity in comparison with the other expectation
values. It is also well known, that calculation of the energy with increased
accuracy does not always lead to the improvement of other characteristics.
We compare our results $E^{(1)}_{l,n}$ with the results $E^{vfm}_{l,n}$ of
some variational method (VFM) 2 and with the results $E^{num}_{l,n}$ of
numerical integration of the Schrödinger equation 1 . We also compare our
values of the quantity
$<v^{2}>_{l,n}=\int_{0}^{\infty}\left(\frac{d\psi^{0}_{l,n}(b,r)}{dr}\right)^{2}dr$
(16)
with values from 1 .
Table 1: Linear potential $(k=0)$. $l$ | $n$ | $E^{num}_{l,n}$ | $E^{(1)}_{l,n}$ | $E^{(2)}_{l,n}$ | $d_{l,n}$ | $<v^{2}>^{num}_{l,n}$ | $<v^{2}>_{l,n}$
---|---|---|---|---|---|---|---
$0$ | $0$ | $2.3381$ | $2.3383$ | $2.3387$ | $2.9465\cdot 10^{-4}$ | $0.7794$ | $0.7794$
$0$ | $1$ | $4.0879$ | $4.0881$ | $4.0890$ | $4.4511\cdot 10^{-4}$ | $1.3626$ | $1.3627$
$1$ | $0$ | $3.3613$ | $3.3614$ | $3.3615$ | $5.5573\cdot 10^{-5}$ | $0.4921$ | $0.4923$
$1$ | $1$ | $4.8845$ | $4.8846$ | $4.8849$ | $1.5180\cdot 10^{-4}$ | $1.1151$ | $1.1057$
$2$ | $0$ | $4.2482$ | $4.2483$ | $4.2483$ | $1.8679\cdot 10^{-5}$ | $0.4089$ | $0.4090$
$2$ | $1$ | $5.6297$ | $5.6298$ | $5.6300$ | $6.9838\cdot 10^{-5}$ | $1.0097$ | $0.9997$
Table 2: Cornell potential. $k$ | $l$ | $n$ | $E^{num}_{l,n}$ | $E^{VFM}_{l,n}$ | $E^{(1)}_{l,n}$ | $E^{(2)}_{l,n}$ | $d_{l,n}$ | $<v^{2}>^{num}_{l,n}$ | $<v^{2}>_{l,n}$
---|---|---|---|---|---|---|---|---|---
$0.2$ | $0$ | $0$ | $2.1673$ | $2.1409$ | $2.1673$ | $2.1676$ | $2.4358\cdot 10^{-4}$ | $0.8389$ | $0.8389$
$0.2$ | $0$ | $1$ | $3.9702$ | $3.9643$ | $3.9704$ | $3.9708$ | $1.6310\cdot 10^{-4}$ | $1.4028$ | $1.4032$
$0.2$ | $1$ | $0$ | $3.2582$ | $3.1565$ | $3.2580$ | $3.2580$ | $2.7539\cdot 10^{-5}$ | $0.5028$ | $0.5031$
$0.2$ | $1$ | $1$ | $4.8019$ | $4.7414$ | $4.8019$ | $4.8021$ | $9.5487\cdot 10^{-5}$ | $1.1284$ | $1.1212$
$0.2$ | $2$ | $0$ | $4.1703$ | $4.0037$ | $4.1703$ | $4.1703$ | $1.3279\cdot 10^{-5}$ | $0.4136$ | $0.4137$
$0.2$ | $2$ | $1$ | $5.5634$ | $5.4491$ | $5.5634$ | $5.5636$ | $5.4213\cdot 10^{-5}$ | $1.0172$ | $1.0086$
$1$ | $0$ | $0$ | $1.3979$ | $1.3460$ | $1.4009$ | $1.4173$ | $2.3504\cdot 10^{-2}$ | $1.1716$ | $1.1801$
$1$ | $0$ | $1$ | $3.4751$ | $3.4765$ | $3.4742$ | $3.4841$ | $5.7230\cdot 10^{-3}$ | $1.5870$ | $1.5809$
$1$ | $1$ | $0$ | $2.8255$ | $2.7185$ | $2.8257$ | $2.8258$ | $5.5640\cdot 10^{-5}$ | $0.5524$ | $0.5520$
$1$ | $1$ | $1$ | $4.4619$ | $4.3940$ | $4.4619$ | $4.4620$ | $2.7702\cdot 10^{-5}$ | $1.1864$ | $1.1868$
$1$ | $2$ | $0$ | $3.8506$ | $3.6835$ | $3.8506$ | $3.8506$ | $1.5440\cdot 10^{-6}$ | $0.4340$ | $0.4341$
$1$ | $2$ | $1$ | $5.2930$ | $5.1730$ | $5.2930$ | $5.2930$ | $1.1421\cdot 10^{-5}$ | $1.0492$ | $1.0456$
Table 1 shows that the proposed approximation gives fairly accurate results in
the case of the purely linear potential. Table 2 demonstrates the efficiency
of our approach in the case of the combined Coulomb plus linear potential.
For example we compare our variant (14) and usual variant (15) in the case
$k=1,l=1,n=1$:
$E^{(1)}_{1,1}(14)=4.4619,~{}E^{(2)}_{1,1}(14)=4.4620,~{}<v^{2}>_{1,1}(14)=1.1868~{}(a=0.5439,b=1.6769),$
$E^{(1)}_{1,1}(15)=4.4583,~{}E^{(2)}_{1,1}(15)=5.0658,~{}<v^{2}>_{1,1}(15)=0.9905~{}(a=1.3352,b=0.9999)$
while the numerical solution 1 gives $E^{num}_{1,1}=4.4619$,
$<v^{2}>^{num}_{1,1}=1.1864$ .
We see that in spite of the simplicity of the used trial functions, the
optimal choice of the variatonal parameters leads to the satisfactory
approximation. At the same time we assume that the approximation can be
improved by means of some complication of the trial function reproducing the
behavior at the origin more accurately.
## References
* (1) E. Eichten et al. Charmonium: The model. Phys. Rev. D17, 3090-3117 (1978).
* (2) G. A. Arteca, F. M. Fernandez, E. A. Castro. Simple variational approaches to eigenvalues in quantum theory. J. Phys. A: Math. Gen. 20, 2221-2224 (1987).
* (3) A. V. Turbiner. On eigenfunctions in quarkonium potential models (Perturbation theory and variational method). Yad. Fiz. 46, 204-218 (1987) (Russian).
* (4) H. Ciftci, R. L. Hall, Q. D. Katadbeh. Coulomb plus power-law potentials in quantum mechanics. J. Phys. A: Math. Gen. 36, 7001-7008 (2003).
* (5) F. M. Ferrnandez. Variationally improved perturbation theory for central-field models. Eur. J. Phys. 24, 289-296 (2003).
* (6) V. V. Kudryashov, V. I. Reshetnyak. Modification of the perturbative-variational approach. Proc. of 9-th Annual Seminar ”Nonlinear Phenomena in Complex Systems”, eds. L. Babichev and V. Kuvshinov, Minsk, 2000, pp. 168-171.
* (7) V. V. Kudryashov, V. I. Reshetnyak. Improved variational approach for radial wave functions. Nonlineae Dynamics and Applications, eds. L. Babichev and V. Kuvshinov, Minsk, 2007, pp. 81-84.
* (8) H. Ciftci, E. Ateser and H. Koru. The power-law and logarithmic potentials. J.Phys.A: Math.Gen. 36, 3621-3628 (2003).
* (9) V. V. Kudryashov, V. I. Reshetnyak. Minimization of relative discrepancy and a variational method for excited states. Foundation and Advances in Nonlinear Science, eds. V. Kuvshinov and G. Krylov, Minsk, 2004, pp. 43-47.
* (10) I. D. Feranchuk et al. Operator method in the problem of quantum anharmonic ocsllator. Ann. of Phys. 238, 370-440 (1995).
|
arxiv-papers
| 2009-11-22T14:49:34 |
2024-09-04T02:49:06.621031
|
{
"license": "Public Domain",
"authors": "V.V. Kudryashov and V.I. Reshetnyak",
"submitter": "Vladimir Kudryashov",
"url": "https://arxiv.org/abs/0911.4256"
}
|
0911.4343
|
# Visiting Horava-Lifshitz gravity in extra dimensions
Qasem Exirifard exirifard@gmail.com
###### Abstract
The Horava-Lifshitz gravity, having broken the symmetry of space and time,
includes three objects: the spatial metric $g_{ij}$, the lapse variable $N$,
and the shift variable $N_{i}$. Each of these objects have their own scaling
dimensions. The action of the theory is required to be invariant under this
scaling, and spatial diffeomorphism, and the temporal foliation symmetry.
Noting that action can non-trivially depend on the lapse variable, we suggest
to consider the Horva’s approach to quantum gravity in higher dimensions such
that a set of extra spacial coordinates possess a scaling dimension different
from that of rest. In so so doing, we propose a new power counting
renormalizable theory for quantum gravity in its UV point. We show that the IR
point and UV point of the proposal possess the same number of degrees of
freedom in $3$, $8$ and $27$ extra-dimensional space-time geometry.
††preprint: IPM/P-2009/051
From the far east to the far west, once for a long time, all the known great
minds of the time unanimously agreed on the first theory of every thing: the
doctrine of four elements, a logical extension of the one element (water)
theory of Babylonia and Assyria. Today we have transcended/left this doctrine,
and each of us keeps smiling at that theory of every thing. But the moment we
halt smiling we find ourselves standing on nothing but the empty void of
between the two peaks of understandings. One peak is the theory of general
relativity, and the other one is quantum field theory. Even at their very
basic of their foundations, these two peaks of knowledge contradict the nature
of each other. At the top of the peaks, this contradiction is manifested by
our total lack of ability to quantise general relativity. The weakest form of
how to quantise gravity is to find a path from one peak to the other.
Unfortunately, so far, we have not yet found any connecting path between these
two that can be argued to be reasonably free of obstacles.
Petr Horava recently has suggested, to move away from one peak, and to break
the symmetry between time and space in the UV point in order to have a power-
counting renormalizable theory at that point. In this short note we would like
to provide a deconstruction of the Horava’s proposal which retain time and
most of the space symmetric, but makes the space asymmetric. The
deconstruction is implemented first by reviewing the terms that Petr Horava
has missed to include in his proposal, then using these missing terms to
create the deconstruction.
## I Revising the Horava’s proposal
Ref. Horava has considered only functionals of the metric when possible forms
for the potential term were being investigated. However, it addition to the
metric, the theory includes the lapse and shift variable. The potential could
be a non-trivial functional of these variables. We first show/review the non-
trivial dependence on the lapse variable.
Let us start with a $D+1$ dimensional metric of the space-time
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\,.$ (1)
Using the coordinate $(t,x^{i})$, the metric can be re-written in the ADM
form:
$ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)\,.$ (2)
Ref. Horava assigns the following scaling dimensions to the coordinates and
the variables appeared in the ADM form of the metric:
$[t]=-z,\,[x]=-1,\,[N]=0,\,[N_{i}]=z-1,\,[g_{ij}]=0\,.$ (3)
It then requires the physical quantities to be invariant under the scaling
symmetry. Furthermore ref.Horava demands the physical quantities to be
invariant under the following coordinate transformations (foliation symmetry):
$\displaystyle t$ $\displaystyle\to$ $\displaystyle f(t)\,,$ (4)
$\displaystyle x^{i}$ $\displaystyle\to$ $\displaystyle\xi^{i}(x^{j},t)\,.$
(5)
Under $x^{i}\to\xi^{i}$, $N$ transforms as a scalar. Under $t\to f(t)$, $N$
transforms as a scalar density. We can construct the following quantify out
from $N$ which transforms covariantly (as a tensor) under both of these
transformations:
$\displaystyle\nabla^{i}\ln N\,,$ (6)
The scaling dimension of the above tensor read
$[\nabla^{i}\ln N]=1\,,$ (7)
Ref. Horava first constructs the kinetic term for $g_{ij}$. It then adds the
potential term for the metric to the kinetic term. The potential term can be
any scalar respecting the symmetries of the theory constructed out of the
$g_{ij}$, $R_{ijkl}[g_{ij}]$, $N_{i}$ and $N$ and their spacial covariant
derivatives;
$S_{V}=S_{V}[g_{ij},R_{ijkl},N_{i},N,\nabla_{i}]\,,$ (8)
A subclass of which reads
$S_{V}=\int dtd^{D}x\sqrt{\det g}NV[g_{ij},R_{ijkl},N_{i},N,\nabla_{i}]$ (9)
where $V[g_{ij},R_{ijkl},N_{i},N,\nabla_{i}]$ is a scalar under spacial
transformation and preserves the foliation symmetry. Ref.Horava from outset
presumes that $V[g_{ij},R_{ijkl},N_{i},N,\nabla_{i}]$ does not depend on the
lapse and shift variables. However in contrary to the assumption of the
ref.Horava , the potential can non-trivially depend on the lapse variable and
its covariant derivatives. In fact any spacial scalar constructed out of
$g_{ij}$, $g^{ij}$, $R_{ijkl}[g_{ij}]$, $\nabla_{i}\ln N$ and their spatial
covariant derivatives will preserve the foliation symmetry. So the ‘general’
potential can be written as:
$\displaystyle S_{V}=\int dtd^{D}x\sqrt{g}\,N\,V\,,$ (10) $\displaystyle
V=V[g_{ij},R_{ijkl},\nabla_{i}\ln N,\nabla_{i}]\,,$ (11)
where $V$ is a scalar constructed out from its arguments. Because $S_{V}$
should be of vanishing scaling dimension, $V$ must satisfy $[V]=D+z$. For sake
of simplicity we consider only the choice of $D=3$ and $z=3$ but our
conclusion apparently remains valid for other cases of interest. For this
choice we have
$[\nabla_{i}]=1,[\nabla_{i}N]=1,[g_{ij}]=0\,.$ (12)
Let $f^{i_{1}\cdots i_{n}}=f^{i_{1}\cdots i_{n}}[\nabla^{i}\ln N,\nabla^{i}]$
be defined as the most general polynomial constructed out from $\nabla\ln N$
and $\nabla$ that carries the “$i_{1}\cdots i_{n}$” indices:
$\displaystyle f^{i_{1}}$ $\displaystyle=$ $\displaystyle
c_{1}\nabla^{i_{1}}\ln N\,,$ (13) $\displaystyle f^{i_{1}i_{2}}$
$\displaystyle=$ $\displaystyle c_{1}\nabla^{i_{1}}\ln N\nabla^{i_{2}}\ln
N+c_{2}\nabla^{i_{1}}\nabla^{i_{2}}\ln N\,,$ (14) $\displaystyle
f^{i_{1}i_{2}i_{3}}$ $\displaystyle=$ $\displaystyle c_{1}\nabla^{i_{1}}\ln
N\nabla^{i_{2}}\ln N\nabla^{i_{3}}\ln N+c_{2}\nabla^{i_{1}}\nabla^{i_{2}}\ln
N\,\nabla^{i_{3}}\ln N+c_{3}\nabla^{i_{1}}\nabla^{i_{2}}\nabla^{i_{2}}\ln
N\,,$ (15) $\displaystyle f^{i_{1}i_{2}i_{3}i_{4}}$ $\displaystyle=$
$\displaystyle c_{1}\nabla^{i_{1}}\ln N\nabla^{i_{2}}\ln N\nabla^{i_{3}}\ln
N\nabla^{i_{4}}\ln N+c_{2}\nabla^{i_{1}}\nabla^{i_{2}}\ln N\,\nabla^{i_{3}}\ln
N\nabla^{i_{4}}\ln N+$ (16)
$\displaystyle+c_{3}\nabla^{i_{1}}\nabla^{i_{2}}\nabla^{i_{2}}\ln
N\,\nabla_{i_{4}}\ln
N+c_{4}\nabla^{i_{1}}\nabla^{i_{2}}\nabla^{i_{2}}\nabla^{i_{4}}\ln
N+c_{5}\nabla^{i_{1}}\nabla^{i_{2}}\ln N\,\nabla^{i_{3}}\nabla^{i_{4}}\ln N,$
where $c_{i}$’s are real constant parameters, and the explicit form of
$f^{i_{1}\cdots i_{5}}$ and $f^{i_{1}\cdots i_{6}}$ have not been written but
they can be induced. Using the above definition it follows that $V$ reads
$V=\sum_{n=0}^{6}V^{6-n}_{i_{1}\cdots i_{n}}[g]\,f^{i_{1}\cdots
i_{n}}[\nabla^{i}\ln N,\nabla^{i}]\,,$ (17)
wherein $V^{6-n}_{i_{1}\cdots i_{n}}[g]$ is a tensor of rank $n$ containing
$6-n$ derivatives constructed out from the spatial metric, and its spatial
derivatives. For $D=3$ these terms can easily be constructed because the Ricci
tensor carries all the degrees of freedom of the Riemann tensor. We suffice to
present $V^{5}_{i_{1}}[g]$ and $V^{4}_{i_{1}i_{2}}[g]$
$\displaystyle V^{5}_{i_{1}}[g]$ $\displaystyle=$ $\displaystyle
d_{1}\nabla_{i_{1}}\Box
R+d_{2}R\nabla_{i_{1}}R\,+d_{3}R_{\alpha\beta}\nabla_{i_{1}}R^{\alpha\beta}+d_{4}R_{i_{1}\beta}\nabla^{\beta}R,$
(18) $\displaystyle V^{4}_{i_{1}i_{2}}[g]$ $\displaystyle=$ $\displaystyle
d_{1}R_{i_{1}i_{2}}R+d_{2}R_{i_{1}\alpha}R^{\alpha}_{~{}i_{2}}+d_{3}R^{2}\,g_{i_{1}i_{2}}+d_{4}\nabla_{i_{1}}\nabla_{i_{2}}R+d_{5}\nabla_{i_{2}}\nabla_{i_{1}}R+d_{6}\Box
R_{i_{1}i_{2}}+d_{7}\Box R\,g_{i_{1}i_{2}}$ (19)
where $R$ stands for the Ricci scalar and $d_{i}$’s are real constant
parameters. In higher dimensions one may use the classification of ref.
Fulling to write the algebraically independent terms in $V^{6-n}_{i_{1}\cdots
i_{n}}[g]$. Note that some of the above terms are related to each other by
partial integration in the action. We leave implementing the simplification by
partial integration to interested reader. These terms has been proposed in
Blas:2009yd and the physical significance to resolve the problems like that
of highlighted in Charmousis:2009tc ; Bogdanos:2009uj ; Bogdanos:2009uj is
investigated in Blas:2009qj , some claims of which are yet being debated
Li:2009bg ; Blas:2009ck . In the following we aim to use these missing terms
in order to propose a different Lifshitz point serving as a candidate for the
UV completion of Einstein general relativity.
To this aim we recall that, so far, different scaling dimensions have been
assigned to time and space in order to make the theory power counting
renormalizable. Let’s assume that there exists at least one quantum-
mechanically effective dimension beside the ordinary space-time dimensions. We
represents this dimension as $q$. Using coordinate $(q,x^{\mu})$, the
effective metric in the ADM-like coordinates reads
$ds^{2}\,=\,N^{2}dq^{2}+g_{\mu\nu}(dx^{\mu}+N^{\mu}dq)(dx^{\nu}+N^{\nu}dq)$
(20)
where $\mu$ runs from $0$ to $D$. From this time on, $D$ will represent the
dimension of the ordinary space-time. As before, we can assign the following
scaling coordinates to the variables appeared above
$[q]=-z,\,[x]=[t]=-1,\,[N]=0,\,[N_{\mu}]=z-1,\,[g_{\mu\nu}]=0\,.$ (21)
In the same analogy, we define the foliation symmetry over the $q$ coordinate:
$\displaystyle q$ $\displaystyle\to$ $\displaystyle f(q)\,,$ (22)
$\displaystyle x^{\mu}$ $\displaystyle\to$
$\displaystyle\xi^{\mu}(x^{\nu},q)\,.$ (23)
We then require the theory to be invariant under scaling and foliation of the
$q$ coordinate. This theory can be made power counting renormalizable. Having
defined
$K_{\mu\nu}\,=\,\frac{1}{2N}(\frac{\partial g_{\mu\nu}}{\partial
q}-\nabla_{\mu}N_{\nu}-\nabla_{\nu}N_{\mu})$ (24)
the part of the action of the theory that includes only two derivatives
follows
$S_{k}\,=\,\frac{2}{k^{2}}\int dqd^{D}x\sqrt{-\det
g}N(K_{\mu\nu}K^{\mu\nu}-\lambda K^{2})\,.$ (25)
In contrary to the original Horava-Lifshitz theory, this part does not include
the space-time derivatives of the metric. We set $z=D$ and make the coupling
constant for this action dimensionless. In our realisation, the term that is
known as potential term of the Horava theory generates the dynamics of the
metric. That term reads
$S_{V}\,=\,\int dqd^{D}x\sqrt{-\det g}NV[g_{\mu\nu},\nabla_{\mu}\ln
N,\nabla_{\mu}]$ (26)
while
$V=\sum_{n=0}^{2D}V^{2D-n}_{i_{1}\cdots i_{n}}[g]\,f^{i_{1}\cdots
i_{n}}[\nabla^{i}\ln N,\nabla^{i}]\,,$ (27)
wherein $V^{2D-n}_{i_{1}\cdots i_{n}}[g]$ is a tensor of rank $n$ containing
$2D-n$ derivatives constructed out from the space-time metric, and its
covariant derivatives. $f^{i_{1}\cdots i_{n}}$ again is the most general
polynomial constructed out from $\nabla\ln N$ and $\nabla$ that carries the
“$i_{1}\cdots i_{n}$” indices. Let $V$ be linear in term of the Riemann
tensor. In other words, consider just $n=2D-2$ term in (27). Let
$f^{i_{1}\cdots i_{2D-2}}$ be
$f^{i_{1}\cdots i_{2D-2}}\,=\,\prod_{p=1}^{2D-2}\nabla^{i_{p}}\ln N.$ (28)
These choices lead to the following ‘potential’ term:
$V=V^{2}_{i_{1}\cdots i_{2D-2}}[g]\prod_{p=1}^{2D-2}\nabla^{i_{p}}\ln N.$ (29)
the first variation of which leads to third order differential equations of
motion for $g$ and $N$. Usually it is the appearance of four order derivatives
and beyond that leads to fluctuations which are hard to control or to be
considered consistent with causality. Third order equations should not be
discarded based on the ghost-free criterion from outset. Besides there exist
some possibilities in $V=V^{2}_{i_{1}\cdots i_{2D-2}}[g]$ to further simplify
or demand additional properties on the equations of motion.
To summarize,
$S=\frac{2}{k^{2}}\int dqd^{D}x\sqrt{-\det g}N(K_{\mu\nu}K^{\mu\nu}-\lambda
K^{2}+V^{2}_{i_{1}\cdots i_{2D-2}}[g]\prod_{p=1}^{2D-2}\nabla^{i_{p}}\ln N)$
(30)
is a power counting renormalizable theory that possesses all symmetries of a
$D$ dimensional space-time (enhanced by foliation symmetry over the $q$
coordinate). So it can serves as a candidate for UV fixed point of gravity in
a $D$ dimensional space-time. In contrast to the original Horava’s proposal,
it does not break the symmetry between time and space. Instead it breaks the
symmetry in space, it divides the space into the $q$ coordinate and the
ordinary directions. Its low energy limit is gravity in $D+1$ dimensional
space-time where the symmetry group for the spacial directions is broken.
Since in the low energy limit the lowest derivative terms are the leading
terms, one expects to have the usual Einstein-Hilbert action in $D+1$
dimension corrected by $S_{k}$ (25)
$S_{\text{Low-Energy}}\,=\,\int
d^{D}xdq\sqrt{-\det\tilde{g}}\tilde{R}+S_{k}\,,$ (31)
where $\tilde{g}$ and $\tilde{R}$ are the metric and the Ricci scalar in/of
the higher dimensional space-time. Note that $S_{k}$ manifestly breaks the
symmetry between the ordinary space and the $q$ coordinate. The phenomenology
of original Horava’s proposal has investigated to almost fine details, for
example Saridakis:2009bv ; Cai:2009in ; Leon:2009rc ; Dutta:2009jn . There
exists a fair amount of literature/models about phenomenology of extra
dimensions. In each of these models, it sounds interesting how the inclusion
of $S_{k}$ term affects the consistency of the model with cosmological
observation, and possibly alters its prediction.
Also note that we can increase the number of extra dimensions from one to an
arbitrary number, call it $n$, where the metric reads
$ds^{2}\,=\,N_{ab}dq^{a}dq^{b}+g_{\mu\nu}(dx^{\mu}+N^{\mu}_{a}dq^{a})(dx^{\nu}+N^{\nu}_{a}dq^{a})$
(32)
where $\mu$ denotes the ordinary $D$ dimensional space-time, and $a$ runs from
$1$ to $n$, and the scaling dimension of the variables read
$[q_{a}]=-z,\,[x]=[t]=-1,\,[N_{ab}]=0,\,[N^{\mu}_{a}]=z-1,\,[g_{\mu\nu}]=0\,.$
(33)
Note that we shall use $N_{ab}$ to brings down/up the indices of $q$
dimensions. The generalised foliation symmetry on q-dimensions reads
$\displaystyle q_{n}$ $\displaystyle\to$ $\displaystyle f_{n}(q_{m})$ (34)
$\displaystyle x_{\mu}$ $\displaystyle\to$
$\displaystyle\xi_{\mu}(x_{\nu},q_{n})$
The ‘potential’ term follows
$S_{V}\,=\,\int d^{n}qd^{D}x\sqrt{-\det g}\sqrt{\det
N_{ab}}\,V[g_{\mu\nu},\nabla_{\mu}\ln N_{ab},\nabla_{\mu}]\,,$ (35)
where $V[g_{\mu\nu},\nabla_{\mu}N_{ab},\nabla_{\mu}]$ is an scalar with
scaling dimension of $[V]=D+nz$ constructed out from its argument. Since
$N_{ab}$ is a matrix, $V$ includes the trace operator. $V$ can be decomposed
as follows
$V[g_{\mu\nu},\nabla_{\mu}\ln
N_{ab},\nabla_{\mu}]=\sum_{p=0}^{D+nz}V^{D+nz-p}_{i_{1}\cdots
i_{p}}[g]\,f^{i_{1}\cdots i_{p}}[\nabla^{i}\ln N_{ab},\nabla^{i}]\,,$ (36)
wherein $V^{D+nz-p}_{i_{1}\cdots i_{p}}[g]$ is a tensor of rank $p$ containing
$D+nz-p$ derivatives constructed out from the space-time metric, and its
covariant derivatives. $f^{i_{1}\cdots i_{p}}$ again being an scalar, is the
most general polynomial constructed out from $\nabla\ln N_{ab}$ and $\nabla$
that carries the “$i_{1}\cdots i_{p}$” indices. Note that it is (35) which
governs the dynamics of the theory in the UV point. Again we choose this term
such that the equations of motion for $g_{\mu\nu}$ and $N_{ab}$ are third
order.
There exists another quantity that transforms nicely under (34)
$K_{\mu\nu a}\,=\,\frac{\partial g_{\mu\nu}}{\partial
q_{a}}-g_{\eta\nu}\nabla_{\mu}N^{\eta}_{a}-g_{\eta\mu}\nabla_{\nu}N^{\eta}_{a}\,.$
(37)
When we had only one extra q-dimension, the part of the action that included
$K$ was quadratic in term of $K$. This is not the case when more than one
q-dimension exists. Suppose that the kinetic part of the action was quadratic
in term of $K$, then making this part dimensionless would have required
$D+nz-2z=0$ the solution of which results to negative scaling dimension for
$V$ in case of $n>2$. Since we like $V$ to be polynomial of positive degree in
term of derivatives of $g_{\mu\nu}$ and $N_{ab}$, we are discard that action
is quadratic in term of $K$. We further choose the ‘kinetic’ part of the
action in the UV point to include $2n$ numbers of the $K$ tensor
$S_{k}\,=\,\frac{1}{k^{2}}\int d^{n}qd^{D}x\sqrt{-\det
g}\sqrt{N_{ab}}\,f_{1}^{\mu_{1}\cdots\mu_{2n}\nu_{1}\cdots\nu_{2n}}[g]\,f_{2}^{a_{1}\cdots
a_{2n}}[N]\prod_{i=1}^{2n}K_{\mu_{i}\nu_{i}a_{i}}\,,$ (38)
where $f_{1}^{\mu_{1}\cdots\mu_{2n}\nu_{1}\cdots\nu_{2n}}[g]$ is a tensor
polynomial in term of $g^{\mu\nu}$, and $f_{2}^{a_{1}\cdots a_{2n}}[N]$ is
polynomial in term of $N^{ab}$ where $N^{ab}$ is the components of the inverse
of $N_{ab}$. We currently do not need to present an explicit form for $f_{1}$
and $f_{2}$. Requiring $k$ to be dimensionless then results to
$D+nz-2nz=0\to D=nz$ (39)
The equations of motion in the IR limit by definition are second order and we
have $(D+n)(D+n+1)/2$ variables. So to single out a unique solution in the IR
limit we need $(D+n)(D+n+1)$ boundary conditions. As we apply our procedure to
(32), we get third order equations in the UV point for each variable. In the
UV point, we have $D(D+1)/2$ plus to $n(n+1)/2$ variables. So in total we need
$\frac{3}{2}(D(D+1)+n(n+1))$ (40)
‘boundary conditions’ to single out a unique solution in the UV point. Perhaps
the dynamical consistency of the theory requires the same number of boundary
conditions in order to single out a single solution in the UV and IR point. So
the consistency demands
$\frac{3}{2}(D(D+1)+n(n+1))=(D+n)(D+n+1)$ (41)
For $n=1$, the above is solved by $D=1$ and $D=2$. For $n=2$, it is solved by
$D=1$ and $D=6$. Note that for $n=3,4,5$, (41) does not have any solution for
$D$ ($D$ should be a natural number). For $n=6$, (41) is solved by $D=2$ and
$D=21$. From physical point of view, larger values for $n+D$ ( number of the
total dimensions ) sounds not plausible. So for
$(D,n)\in\\{(1,2),(2,1),(6,2),(2,6),(6,21),(21,6)\\}\,,$ (42)
there exists the same amount of physical information available at the UV and
IR point of the theory. Note that for each of the above choices, (39) gives
the scaling dimension of the q-dimensions. Noticing that all the problems of
original Horava gravity are due to not having the same number of degrees of
freedom around the IR and UV points, we count on the possibility to have the
same number of degrees freedom in IR and UV points of our realisation (beside
not needing to recover full diff. symmetry in the IR limit) as an indication
that our proposal will pass further consistency check between the UV and IR
point of the theory. Knowing how the nonlinear theory at the UV point of the
proposal explicitly runs toward the IR point, however, proceeds any attempt
that aims to further investigate the dynamical consistency between UV and IR
point. The UV point, however, is linear in term of the Riemann tensor. This
might simplify the effort to find how the UV point runs toward the IR point.
## References
* (1) P. Horava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79, 084008 (2009) [arXiv:0901.3775 [hep-th]].
* (2) S. A. Fulling, R.C. King, B.G. Wybourne, C.J. Cummins, Normal forms for tensor polynomials: I. The Riemann tensor, Class. Quantum Grav. 9 (1992) 1151–1197.
* (3) D. Blas, O. Pujolas and S. Sibiryakov, On the Extra Mode and Inconsistency of Horava Gravity, JHEP 0910 (2009) 029 [arXiv:0906.3046 [hep-th]].
* (4) D. Blas, O. Pujolas and S. Sibiryakov, A healthy extension of Horava gravity, arXiv:0909.3525 [hep-th].
* (5) M. Li and Y. Pang, A Trouble with Hořava-Lifshitz Gravity, JHEP 0908 (2009) 015 [arXiv:0905.2751 [hep-th]].
* (6) D. Blas, O. Pujolas and S. Sibiryakov, Comment on ‘Strong coupling in extended Horava-Lifshitz gravity’, arXiv:0912.0550 [hep-th].
* (7) C. Bogdanos and E. N. Saridakis, Perturbative instabilities in Horava gravity, arXiv:0907.1636 [hep-th].
* (8) C. Charmousis, G. Niz, A. Padilla and P. M. Saffin, Strong coupling in Horava gravity, JHEP 0908 (2009) 070 [arXiv:0905.2579 [hep-th]].
* (9) E. N. Saridakis, Horava-Lifshitz Dark Energy, arXiv:0905.3532 [hep-th].
* (10) Y. F. Cai and E. N. Saridakis, Non-singular cosmology in a model of non-relativistic gravity, JCAP 0910 (2009) 020 [arXiv:0906.1789 [hep-th]].
* (11) G. Leon and E. N. Saridakis, Phase-space analysis of Horava-Lifshitz cosmology, JCAP 0911, 006 (2009) [arXiv:0909.3571 [hep-th]].
* (12) S. Dutta and E. N. Saridakis, Observational constraints on Horava-Lifshitz cosmology, arXiv:0911.1435 [hep-th].
|
arxiv-papers
| 2009-11-23T09:28:03 |
2024-09-04T02:49:06.625896
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qasem Exirifard",
"submitter": "Qasem Exirifard",
"url": "https://arxiv.org/abs/0911.4343"
}
|
0911.4379
|
# Black Holes in Gravity with Conformal Anomaly and Logarithmic Term in Black
Hole Entropy
Rong-Gen Cai cairg@itp.ac.cn Key Laboratory of Frontiers in Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O.
Box 2735, Beijing 100190, China Department of Physics, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan Li-Ming Cao caolm@itp.ac.cn Department
of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan Nobuyoshi
Ohta ohtan@phys.kindai.ac.jp Department of Physics, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan
###### Abstract
We present a class of exact analytic and static, spherically symmetric black
hole solutions in the semi-classical Einstein equations with Weyl anomaly. The
solutions have two branches, one is asymptotically flat and the other
asymptotically de Sitter. We study thermodynamic properties of the black hole
solutions and find that there exists a logarithmic correction to the well-
known Bekenstein-Hawking area entropy. The logarithmic term might come from
non-local terms in the effective action of gravity theories. The appearance of
the logarithmic term in the gravity side is quite important in the sense that
with this term one is able to compare black hole entropy up to the subleading
order, in the gravity side and in the microscopic statistical interpretation
side.
††preprint: KU-TP 039
Conformal (Weyl) anomalies have a long history Duff1 (For a nice review see
Duff2 ). The conformal anomaly is not only a quite important concept in
quantum field theory in curved spaces, but also has a variety of applications
in cosmology, black hole physics, string theory and statistical mechanics. For
example, Christensen and Fulling showed that in two spacetime dimensions, the
Hawking radiation has a very close relation to the trace anomaly CF . Recently
this idea has been employed to study Hawking radiation of higher dimensional
black holes RW . In the aspect of application of conformal anomaly in
cosmology, a well-known result is that the anomaly can lead to a significant
inflation model Star ; Hawking ; Nojiri . The trace anomaly might also be
related to the well-known cosmological constant problem Duff2 .
In a Friedmann-Robertson-Walker (FRW) spacetime setting, the trace anomaly can
completely determine its corresponding energy-momentum tensor due to the fact
that the spacetime is homogeneous and isotropic. In a black hole spacetime
background, the trace anomaly can also completely determine the energy-
momentum tensor in the two-dimensional case; in the four-dimensional static,
spherically symmetric black hole case, however, it determines the energy-
momentum tensor up to an arbitrary function of position CF , which causes the
difficulty in studying the backreaction of the trace anomaly to the black hole
spacetime AMV .
On the other hand, in quantum field theory in curved spaces, it is required to
include the backreaction of the quantum fields to the spacetime geometry
itself BD
$R_{ab}-\frac{1}{2}Rg_{ab}=8\pi G\langle T_{ab}\rangle,$ (1)
where $\langle T_{ab}\rangle$ is the effective energy-momentum tensor by
quantum loops. However, the expression of the effective energy-momentum tensor
is usually quite complicated so that in general one is not able to find exact
analytic solutions to the semi-classical Einstein equations (1). Therefore,
one usually considers perturbative solutions to equations (1) in the
literature: one first assumes some background solutions to the vacuum Einstein
equations, then calculates the vacuum expectation value of the effective
energy-momentum tensor $\langle T_{ab}\rangle$ in the fixed background, and
finally discusses the backreaction effect of the quantum fields
perturbatively.
In this paper we report on some exact solutions of the semi-classical Einstein
equations (1) with conformal anomaly. To our knowledge, it is the first time
to obtain exactly nontrivial black hole solutions to the Einstein equations
with conformal anomaly. In addition, we find that the entropy of the black
hole solutions has a logarithmic term, in addition to the well-known horizon
area term. The appearance of the logarithmic term is quite interesting.
According to the statistical interpretation of black hole entropy in some
quantum theories of gravity such as string theory and loop quantum gravity, on
a very general ground, it can be argued that usually the leading term of
statistical degrees of freedom gives the Bekenstein-Hawking horizon area term,
while the subleading term is a logarithmic term. In the gravity side,
according to Wald’ entropy formula of black holes Wald , however, it is quite
difficult to produce such a logarithmic term in the black hole entropy in some
effective local theory of gravity even with higher derivative curvature terms.
It means that they do not get matched to the entropy in the subleading order
in the gravity side and in the microscopic statistical interpretation side.
Here the conformal anomaly corrected black hole solutions provide a possible
interpretation for the appearance of the logarithmic term in the field theory
side.
Let us start our discussions on our theory. In even dimensions, quantum
effects will destroy the conformal invariance of classical conformal field
theories. One loop quantum corrections lead to a trace anomaly of the energy-
momentum tensor of conformal field theories. In general, the trace anomaly has
the form Duff2 ; Deser
$g^{ab}\langle T_{ab}\rangle=\lambda I_{(2n)}-\alpha E_{(2n)}\,,$ (2)
where $\lambda$ and $\alpha$ are two positive constants depending on the
degrees of freedom of quantum fields and their values will not affect our
discussions. The term $I_{(2n)}$ is a polynomial of Weyl tensor, which is
called type B anomaly, while $E_{(2n)}$ is Euler characteristic of
$2n$-dimensional spacetime, named type A anomaly in Deser . In four
dimensions, $I_{(4)}$ is given by
$I_{(4)}=C_{abcd}C^{abcd}\,,$ (3)
and $E_{(4)}$ is just Gauss-Bonnet term
$E_{(4)}=R^{2}-4R_{ab}R^{ab}+R_{abcd}R^{abcd}\,.$ (4)
We now look for exact solutions to equations of motion (1) with corresponding
energy-momentum tensor to the trace anomaly (2). For simplicity, consider a
static, spherically symmetric spacetime of the form
$g_{ab}dx^{a}dx^{b}=-f(r)dt^{2}+\frac{1}{g(r)}dr^{2}+r^{2}d\Omega_{2}^{2}\,,$
(5)
where $f(r)$ and $g(r)$ are two functions of the radial coordinate $r$ only,
and $d\Omega_{2}^{2}$ is the line element of a two-dimensional sphere with
unit radius. As for the vacuum expectation value $\langle T_{ab}\rangle$ of
effective energy-momentum tensor, we only know the following two constraint
conditions: (i) its trace is given by (2), and (ii) it must be covariantly
conserved, namely, $\nabla_{a}\langle T^{ab}\rangle=0$. As was known in CF ,
in that case, one is not able to determine completely the energy-momentum
tensor. To go on, we have to impose another condition to determine the
corresponding energy-momentum tensor to the trace anomaly. Note that to have a
nontrivial black hole solution with metric (5), at a black hole horizon, say
$r_{+}$, the energy-momentum tensor must satisfy CJ : $\langle T^{t}_{\
t}(r)\rangle|_{r=r_{+}}=\langle T^{r}_{\ r}(r)\rangle|_{r=r_{+}}$. However,
this condition does not fix the energy-momentum tensor which is necessary to
find exact solutions. Henceforth we make a further assumption that this
relation holds not only at the black hole horizon, but also in the whole
spacetime, namely,
$\langle T^{t}_{\ t}(r)\rangle=\langle T^{r}_{\ r}(r)\rangle.$ (6)
Though at this point this is merely an assumption to find exact solutions,
there are certain examples that satisfy this relation, and we will discuss
this further towards the end of this paper.
With the symmetry of the metric (5), we can define
$\langle T^{t}_{\ t}\rangle=-\rho(r)\,,\qquad\langle T^{r}_{\
r}\rangle=p(r)\,,\qquad\langle T^{\theta}_{\ \theta}\rangle=\langle
T^{\phi}_{\ \phi}\rangle=p_{\bot}(r)\,.$ (7)
With the help of assumption (6), one can easily show that $f(r)=g(r)$. On the
other hand, the constraints (i) and (ii) turn out to be
$-\rho+p+2p_{\bot}=-\alpha E_{(4)}\,,$ (8)
and
$4f(p-p_{\bot})+(\rho+p)rf^{\prime}+2rp^{\prime}f=0\,,$ (9)
where we have set $\lambda=0$ and a prime denotes the derivative with respect
to $r$ and $E_{(4)}=\frac{2}{r^{2}}\left((1-f)^{2}\right)^{\prime\prime}$. Now
put $p_{\bot}$ from (9) and $\rho=-p$ from (6) into (8), one has
$rp^{\prime}+4p+\alpha E_{(4)}=0\,,$ (10)
which has the solution
$p=\frac{2\alpha}{r^{4}}(1-f)(1-f+2rf^{\prime})-\frac{q}{r^{4}}\,,$ (11)
where $q$ is an integration constant. We can then easily obtain the energy
density $\rho=-p$ and the transverse pressure $p_{\bot}=-\alpha E_{(4)}/2-2p$.
Substituting these results together with (7) into the semi-classical Einstein
equations (1), we obtain a solution with metric function
$f(r)=1-\frac{r^{2}}{4\tilde{\alpha}}\left(1\pm\sqrt{1-\frac{16\tilde{\alpha}GM}{r^{3}}+\frac{8\tilde{\alpha}Q}{r^{4}}}\right)\,,$
(12)
where $M$ is an integration constant, $\tilde{\alpha}=8\pi G\alpha$ and
$Q=8\pi Gq$. We have checked indeed the solution (5) with $g(r)=f(r)$
satisfies all components of the semi-classical Einstein equations (1). The
solution has two branches with $``\pm"$ in (12), respectively. That is, there
exist two vacuum solutions when $M=Q=0$: one is the Minkowski spacetime for
the branch with $``-"$ sign, and the other is a de Sitter space with the
effective radius $\ell_{\rm eff}=\sqrt{2\tilde{\alpha}}$ for the branch with
$``+"$ sign. The solution (5) with (12) looks very like the Gauss-Bonnet black
hole solution presented in BD2 , but here our solution is four dimensional. In
fact, our solution looks more like the black hole solution with a negative
constant curvature space in the Gauss-Bonnet gravity DM . But again, those
solutions presented in DM describe higher dimensional ($D\geq 6$) objects.
In the large $r$ limit, for the branch with $``-"$ sign, we have
$f(r)\approx 1-\frac{2GM}{r}+\frac{Q}{r^{2}}+{\cal O}(r^{-4})\,.$ (13)
We can clearly see that the solution behaves like the Reissner-Norström (RN)
solution if $Q>0$ and that the integration constant $M$ is nothing but the
mass of the solution. The meaning of the integration constant $Q$ will be
discussed later. On the other hand, for the branch with $``+"$ sign, the
behavior of large $r$ looks like
$f(r)\approx 1-\frac{r^{2}}{\ell_{\rm
eff}^{2}}+\frac{2GM}{r}-\frac{Q}{r^{2}}+{\cal O}(r^{-4})\,.$ (14)
This is the behavior of a Reissner-Nordström-de Sitter solution if $M<0$ and
$Q<0$.
It is expected that the trace anomaly will not change the vacuum of the
theories, therefore we believe that the branch with $``+"$ is unstable like
the black hole solutions in the Gauss-Bonnet gravity BD2 . In addition, in the
cosmology setup, the de Sitter solution is found unstable Star . Therefore we
will restrict ourselves to the branch with $``-"$ sign in what follows.
Now we turn to study the meaning of the integration constant $Q$. Taking the
limit $\tilde{\alpha}\to 0$, one has from the solution (12) that
$f(r)=1-\frac{2GM}{r}+\frac{Q}{r^{2}}.$ (15)
Clearly the vacuum solution to the Einstein equations (1) without the trace
anomaly term must be the Schwarzschild solution in the metric form (5). This
means one should have $Q=0$. Does it mean that one has to always take $Q=0$
when the trace anomaly term appears. The answer is negative. In fact, we can
see from the solution (11) that the term associated with the integration
constant $q$ is nothing but a “dark radiation” term with
$\rho_{d}=p_{\bot}=-p=q/r^{4}$. Such a term satisfies the two constraints (i)
and (ii), and is also consistent with the symmetry of the metric (5).
Therefore with the two constraints (i) and (ii), and the assumption (6), one
cannot exclude the existence of the “dark radiation”. In addition, we will see
that when a Maxwell field is present, the electric charge square $Q_{e}^{2}$
will appear in the same place as $Q$ in the solution (12) [see the solution
(22)] . Therefore the integration constant $Q$ corresponds to a quantity to be
explained as $U(1)$ conserved charge square of some conformal field theory.
Thus we keep this term $Q$ and have $Q>0$ in order to have a positive energy
density associated with this term.
Next let us have a look at the singularity of the solution. The square of
Riemann tensor is given by
$R_{abcd}R^{abcd}\sim\frac{2}{\tilde{\alpha}}\left(\frac{Q}{r^{4}}-\frac{2GM}{r^{3}}\right)+\cdots\,,$
(16)
in the small $r$ limit. There is therefore a singularity at the origin of $r$
as the RN solution in general relativity. Besides, there exists another
potential singularity where the square root vanishes in (12). It is quite
interesting to compare the small $r$ behavior of the square of Riemann tensor
between the solution (12) and the usual RN solution, which has the expansion
behavior
$R_{abcd}R^{abcd}\sim\frac{56Q^{2}}{r^{8}}-\frac{96GMQ}{r^{7}}+\frac{48G^{2}M^{2}}{r^{6}}+\cdots\,.$
(17)
We can clearly see that the backreaction of the Weyl anomaly drastically
softens the singularity at the origin.
Now we turn to thermodynamic properties of the black hole solution (12). The
black hole horizon satisfies $f(r)=0$, which has two roots
$r_{\pm}=GM\pm\sqrt{G^{2}M^{2}-(Q-2\tilde{\alpha})}$. This implies that the
black hole could have two horizons, a degenerated horizon and naked
singularity if $G^{2}M^{2}>Q-2\tilde{\alpha}$, $G^{2}M^{2}=Q-2\tilde{\alpha}$
and $G^{2}M^{2}<Q-2\tilde{\alpha}$, respectively. We assume the existence of
the black hole horizon, and then the ADM mass of the black hole can be
expressed as
$GM=\frac{r_{+}}{2}+\frac{Q}{2r_{+}}-\frac{\tilde{\alpha}}{r_{+}}\,.$ (18)
The Hawking temperature is easy to give by calculating surface gravity at the
horizon
$T=\frac{1}{4\pi}f^{\prime}(r_{+})=\frac{r_{+}}{4\pi(r_{+}^{2}-4\tilde{\alpha})}\left(1-\frac{Q}{r_{+}^{2}}+\frac{2\tilde{\alpha}}{r_{+}^{2}}\right).$
(19)
We can see from the solution (12) that in order to have a horizon, the horizon
radius must satisfy $r_{+}^{2}>4\tilde{\alpha}$. Therefore the behavior of the
Hawking temperature of the black hole is quite similar to the one for the
usual RN black hole: the temperature starts from zero for an extremal black
hole, monotonically increases and reaches its maximum at some horizon radius,
and then monotonically decreases forever as the horizon radius is increased.
Black hole entropy is an important quantity in black hole thermodynamics. Due
to the appearance of the Weyl anomaly term, the well-known area formula must
no longer hold. Usually Wald’s entropy formula is a powerful tool to calculate
black hole entropy Wald . Unfortunately Wald’s formula cannot be used either
here since we do not know the effective action of the anomaly term. Instead we
try to obtain the black hole entropy by employing the first law of black hole
thermodynamics, $dM=TdS+\dots$, where $\dots$ stands for some work terms, if
any. Integrating the first law yields
$S=\int T^{-1}\left(\frac{\partial M}{\partial
r_{+}}\right)_{Q}dr_{+}=\frac{\pi
r_{+}^{2}}{G}-\frac{4\pi\tilde{\alpha}}{G}\ln r_{+}^{2}+S_{0}\,.$ (20)
Here $S_{0}$ is an integration constant, which unfortunately we cannot fix
because of the existence of the logarithmic term. The entropy can also be
expressed as
$S=\frac{A}{4G}-\frac{4\pi\tilde{\alpha}}{G}\ln\frac{A}{A_{0}}\,,$ (21)
where $A=4\pi r_{+}^{2}$ is the horizon area and $A_{0}$ is a constant with
dimension of area. Clearly the first term is just the well-known Bekenstein-
Hawking area term of black hole entropy, while the appearance of the
logarithmic term in (21) is worth saying a few words.
First, we notice that our entropy formula (21) is completely the same as the
one associated with the apparent horizon of an FRW universe with the anomaly
term Lidsey , which comes out by investigating the relation between the
modified Friedmann equation and the first law of thermodynamics CCH . This
partially supports the conjecture that the entropy formula associated with the
apparent horizon of an FRW universe is the same as the one of black hole
horizon in the same gravity theory ACK . Second, as we mentioned above, a
logarithmic term is universally present as a subleading correction to the
Bekenstein-Hawking area entropy, in the microscopic statistical interpretation
of black hole entropy such as in string theory, loop quantum gravity, thermal
and/or quantum fluctuations in a fixed black hole background (see references
cited in CCH ). However, in the gravity side, it is quite difficult (if not
impossible) to produce such a term from a local effective gravity theory,
based on Wald’s entropy formula Wald , which says that black hole entropy is a
Noether charge, given by an integral of the variation of the Lagrangian of the
effective gravity theory with respect to Riemann tensor over the spatial cross
section of black hole horizon. Thus this is a serious challenge to match the
subleading term of black hole entropy in both sides. Here it seems the first
time to give the logarithmic term from the gravity side 111We notice that such
logarithmic term also appears in the entropy of black holes of Hořava-Lifshitz
gravity CCO . The Hořava-Lifshitz theory is a nonrelativistic gravity theory,
and therefore a non-local gravity theory.. Such a logarithmic term coming from
the non-local trace anomaly may shed some lights for the origin in the gravity
side. Indeed, we notice that in the paper Solo , Solodukhin argued on scaling
and dimensionality ground that such a logarithmic correction to a four-
dimensional Schwarzschild black hole entropy could come from rather
complicated non-local functionals in the low-energy effective action of string
theory and the coefficient of the logarithmic term is proportional to the
four-dimensional central charge which comes from the integrated conformal
anomaly for the zero-mass fields in the theory. Therefore the logarithmic term
in our case has seemingly the same origin as that discussed in Solo . However,
we here obtain such a term by directly solving the semi-classical Einstein
equations (1) without any approximation.
Some remarks are in order. (i) Note that we have set $\lambda=0$ in order to
find the exact analytic solution (12). Namely we have only considered the type
A anomaly Deser . At the moment it is not clear whether an analytic solution
can be derived once the type B anomaly (the Weyl tensor square term) is
included. In particular it involves higher derivative terms. But we believe
that with the same assumption, one is able to find some exact solutions at
least numerically in that case. (ii) In this paper we only discussed the
vacuum solution of the semi-classical Einstein equations without any classical
fields. If the cosmological constant $\Lambda$ and some gauge fields are
present, in the static, spherically symmetric case, we can easily find the
exact solution as
$f(r)=g(r)=1-\frac{r^{2}}{4\tilde{\alpha}}\left(1\pm\sqrt{1+\frac{4\tilde{\alpha}}{\ell^{2}}-\frac{16\tilde{\alpha}GM}{r^{3}}+\frac{8\tilde{\alpha}(Q+Q_{e}^{2})}{r^{4}}}\right)\,,$
(22)
where $Q_{e}$ is the electric charge of Maxwell field and $\ell^{2}=3\Lambda$.
(iii) In higher even ($2n(n>2)$) dimensions, the type A anomaly will be given
by Euler Characteristic of $2n$-dimensional spacetime. With the same approach,
one is able to get exactly analytic solutions. In that case, a logarithmic
term will also appear in the black hole entropy. (iv) In the spirit of AdS/CFT
correspondence, a conformal field theory occurs in a brane world. Hence
generalizing our solution to the brane world scenarios is of great interest.
These issues are currently under investigation. (v) The another remark is
concerned with the assumption (6). First, let us note that the condition that
$\langle T^{t}_{t}\rangle|_{r+r_{+}}=\langle T^{r}_{r}\rangle|_{r=r_{+}}$
holds at a black hole horizon is a requirement for a regular horizon CJ , and
it is satisfied by the stress-energy tensor of the trace anomaly in
Schwarzschild black hole and Reissner-Nordström black hole backgrounds (see,
for example, AMV ). Second, Eq. (6) is just an assumption in this paper.
However, this is indeed satisfied by some conformal field theories. One
typical example is the Maxwell field: for the electric field produced by an
electric charge $q$ at the origin $r=0$, its stress-energy tensor has the form
$T^{t}_{t}=T^{r}_{r}\sim-q^{2}/r^{4}$. Furthermore, the stress-energy tensor
for a non-linear extension of Maxwell field also satisfies the assumption in
the background (5). In addition, this assumption holds as well even for the
effective energy-momentum tensor of higher derivative terms such as Gauss-
Bonnet term and more general Lovelock terms BD2 . The nonlcal nature of the
anomaly term gives complicated stress-energy tensor in general, but we could
consider the special situation in which this relation is effectively
satisfied. It is true that in this case our solution (12) is not a general
static spherically symmetric solution, but represents a limited kind of black
hole solutions with backreaction of trace anomaly. The consistent entropy
formula (21) with the one in the cosmological setting signifies that our
solution (12) captures some features of the backreaction of trace anomaly.
Needless to say, it is of great interest to find exact analytic solutions
without the assumption (6). (vi) Finally we mention that when the temperature
(19) vanishes, the solution (5) with (12) stands for an extremal black hole,
whose near horizon geometry is $AdS_{2}\times S^{2}$. Indeed it is shown in
Sol2 that $AdS_{2}\times S^{2}$ is an exact solution of the semi-classical
Einstein equations (1) in presence of the Maxwell field. Thus our result is
completely consistent with that of Sol2 .
In summary we have presented a class of exact analytic black hole solutions in
the semi-classical Einstein equations with Weyl anomaly. The set of solutions
is parameterized by two integration constants, one is the mass of the
solutions and the other is a $U(1)$ conserved charge of some classical
conformal field theory. The solutions have two branches, one is asymptotically
flat and the other asymptotically de Sitter. We have argued that the branch
with asymptotic de Sitter behavior is unstable. We have studied thermodynamic
properties of the black hole solutions and found that there exists a
logarithmic correction to the well-known Bekenstein-Hawking area entropy. We
have discussed the implications of the logarithmic term. It might come from
non-local terms in the effective action of gravity theories. The appearance of
the logarithmic term in black hole entropy implies that a full quantum theory
of gravity must be a non-local theory.
## Acknowledgements
RGC thanks S.P. Kim, K. Maeda, S. Mukohyama, B. Wang and Z.Y. Zhu for helpful
discussions for some relevant topics, and Kinki University for warm
hospitality. RGC is supported partially by grants from NSFC, China (No.
10821504 and No. 10975168) and a grant from MSTC, China (No. 2010CB833004).
LMC is supported by JSPS fellowship No. P 09225. NO is supported in part by
the Grant-in-Aid for Scientific Research Fund of the JSPS No. 20540283, and
also by the Japan-U.K. Research Cooperative Program.
## References
* (1) D. M. Capper and M. J. Duff, Nuovo Cim. A 23, 173 (1974); M. J. Duff, Nucl. Phys. B 125, 334 (1977).
* (2) M. J. Duff, Class. Quant. Grav. 11, 1387 (1994) [arXiv:hep-th/9308075].
* (3) S. M. Christensen and S. A. Fulling, Phys. Rev. D 15, 2088 (1977).
* (4) S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [arXiv:gr-qc/0502074]; S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. Lett. 96, 151302 (2006) [arXiv:hep-th/0602146].
* (5) A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
* (6) S. W. Hawking, T. Hertog and H. S. Reall, Phys. Rev. D 63, 083504 (2001) [arXiv:hep-th/0010232].
* (7) S. Nojiri and S. D. Odintsov, Phys. Lett. B 484, 119 (2000) [arXiv:hep-th/0004097].
* (8) P. R. Anderson, E. Mottola and R. Vaulin, Phys. Rev. D 76, 124028 (2007) [arXiv:0707.3751 [gr-qc]].
* (9) N. D. Birrell and P. C. W. Davies, “Quantum Fields In Curved Space,” Cambridge University Press (1982) 340p
* (10) R. M. Wald, Phys. Rev. D 48, 3427 (1993) [arXiv:gr-qc/9307038]; V. Iyer and R. M. Wald, Phys. Rev. D 50, 846 (1994) [arXiv:gr-qc/9403028].
* (11) S. Deser and A. Schwimmer, Phys. Lett. B 309, 279 (1993) [arXiv:hep-th/9302047].
* (12) R. G. Cai, J. Y. Ji and K. S. Soh, Phys. Rev. D 58, 024002 (1998) [arXiv:gr-qc/9708064].
* (13) D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985); J. T. Wheeler, Nucl. Phys. B 268, 737 (1986); R. G. Cai, Phys. Rev. D 65, 084014 (2002) [arXiv:hep-th/0109133].
* (14) H. Maeda and N. Dadhich, Phys. Rev. D 74, 021501 (2006) [arXiv:hep-th/0605031]; H. Maeda and N. Dadhich, Phys. Rev. D 75, 044007 (2007) [arXiv:hep-th/0611188]; R. G. Cai, L. M. Cao and N. Ohta, arXiv:0911.0245 [hep-th].
* (15) J. E. Lidsey, arXiv:0911.3286 [hep-th]; J. E. Lidsey, Class. Quant. Grav. 26, 147001 (2009) [arXiv:0812.2791 [gr-qc]].
* (16) R. G. Cai, L. M. Cao and Y. P. Hu, JHEP 0808, 090 (2008) [arXiv:0807.1232 [hep-th]]; R. G. Cai, L. M. Cao and Y. P. Hu, Class. Quant. Grav. 26, 155018 (2009) [arXiv:0809.1554 [hep-th]].
* (17) R. G. Cai and L. M. Cao, Nucl. Phys. B 785, 135 (2007) [arXiv:hep-th/0612144]; R. G. Cai and L. M. Cao, Phys. Rev. D 75, 064008 (2007) [arXiv:gr-qc/0611071]. M. Akbar and R. G. Cai, Phys. Rev. D 75, 084003 (2007) [arXiv:hep-th/0609128]; R. G. Cai and S. P. Kim, JHEP 0502, 050 (2005) [arXiv:hep-th/0501055].
* (18) R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 80, 024003 (2009) [arXiv:0904.3670 [hep-th]]; R. G. Cai, Y. Liu and Y. W. Sun, JHEP 0906, 010 (2009) [arXiv:0904.4104 [hep-th]]; R. G. Cai, L. M. Cao and N. Ohta, Phys. Lett. B 679, 504 (2009) [arXiv:0905.0751 [hep-th]]; R. G. Cai and N. Ohta, arXiv:0910.2307 [hep-th]; Y. S. Myung, arXiv:0908.4132 [hep-th].
* (19) S. N. Solodukhin, Phys. Rev. D 57, 2410 (1998) [arXiv:hep-th/9701106].
* (20) S. N. Solodukhin, Phys. Lett. B 448, 209 (1999) [arXiv:hep-th/9808132].
|
arxiv-papers
| 2009-11-23T11:57:09 |
2024-09-04T02:49:06.631443
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Gen Cai, Li-Ming Cao, Nobuyoshi Ohta",
"submitter": "Rong-Gen Cai",
"url": "https://arxiv.org/abs/0911.4379"
}
|
0911.4459
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ Interval edge colorings of
some products of graphs Petros A. Petrosyan
# Interval edge colorings of some products of graphs
Petros A. Petrosyan email: pet_petros@{ipia.sci.am, ysu.am, yahoo.com}
Institute for Informatics and Automation Problems, National Academy of
Sciences, 0014, Armenia Department of Informatics and Applied Mathematics,
Yerevan State University, 0025, Armenia
###### Abstract
An edge coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an
interval $t$-coloring if for each $i\in\\{1,2,\ldots,t\\}$ there is at least
one edge of $G$ colored by $i$, and the colors of edges incident to any vertex
of $G$ are distinct and form an interval of integers. A graph $G$ is interval
colorable, if there is an integer $t\geq 1$ for which $G$ has an interval
$t$-coloring. Let $\mathfrak{N}$ be the set of all interval colorable graphs.
In 2004 Kubale and Giaro showed that if $G,H\in\mathfrak{N}$, then the
Cartesian product of these graphs belongs to $\mathfrak{N}$. Also, they
formulated a similar problem for the lexicographic product as an open problem.
In this paper we first show that if $G\in\mathfrak{N}$, then
$G[nK_{1}]\in\mathfrak{N}$ for any $n\in\mathbf{N}$. Furthermore, we show that
if $G,H\in\mathfrak{N}$ and $H$ is a regular graph, then strong and
lexicographic products of graphs $G,H$ belong to $\mathfrak{N}$. We also prove
that tensor and strong tensor products of graphs $G,H$ belong to
$\mathfrak{N}$ if $G\in\mathfrak{N}$ and $H$ is a regular graph.
Keywords: edge coloring, interval coloring, products of graphs
AMS Subject Classification: 05C15
## 1 Introduction
An edge coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an
interval $t$-coloring if for each $i\in\\{1,2,\ldots,t\\}$ there is at least
one edge of $G$ colored by $i$, and the colors of edges incident to any vertex
of $G$ are distinct and form an interval of integers. Interval edge colorings
naturally arise in scheduling problems and are related to the problem of
constructing timetables without “gaps”for teachers and classes. The notion of
interval edge colorings was introduced by Asratian and Kamalian [1] in 1987.
In [1] they proved that if a triangle-free graph $G=\left(V,E\right)$ has an
interval $t$-coloring, then $t\leq\left|V\right|-1$. In [19] interval edge
colorings of complete bipartite graphs and trees were investigated.
Furthermore, Kamalian [20] showed that if $G$ admits an interval $t$-coloring,
then $t\leq 2\left|V\right|-3$. Giaro, Kubale and Malafiejski [12] proved that
this upper bound can be improved to $2\left|V\right|-4$ if $\left|V\right|\geq
3$. For a planar graph $G$, Axenovich [5] showed that if $G$ has an interval
$t$-coloring, then $t\leq\frac{11}{6}\left|V\right|$. In general, it is an
$NP$-complete problem to decide whether a given bipartite graph $G$ admits an
interval edge coloring [35]. In papers [2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 19, 20, 21, 22, 23, 29, 30, 32, 34] the problem of existence and
construction of interval edge colorings was considered and some bounds for the
number of colors in such colorings of some classes of graphs were given.
Surveys on this topic can be found in some books [3, 18, 25].
The different products of graphs were introduced by Berge [6], Sabidussi [33]
and Vizing [36]. There are many papers [17, 24, 26, 27, 28, 31, 38] devoted to
edge colorings of various products of graphs. In this paper we investigate
interval edge colorings of various products of graphs.
## 2 Definitions and preliminary results
All graphs considered in this paper are finite, undirected and have no loops
or multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges
of $G$, respectively. The maximum degree of a vertex of $G$ is denoted by
$\Delta(G)$ and the chromatic index of $G$ by $\chi^{\prime}\left(G\right)$. A
partial edge coloring of $G$ is a coloring of some of the edges of $G$ such
that no two adjacent edges receive the same color. If $\alpha$ is a partial
edge coloring of $G$ and $v\in V(G)$ then $S\left(v,\alpha\right)$ denotes the
set of colors of colored edges incident to $v$.
A graph $G$ is interval colorable, if there is an integer $t\geq 1$, for which
$G$ has an interval $t$-coloring. Let $\mathfrak{N}$ be the set of all
interval colorable graphs [1, 20]. For a graph $G\in\mathfrak{N}$, the least
and the greatest values of $t$ for which $G$ has an interval $t$-coloring are
denoted by $w\left(G\right)$ and $W\left(G\right)$, respectively.
Let $G=(V(G),E(G))$ and $H=(V(H),E(H))$ be two graphs.
The Cartesian product $G\square H$ is defined as follows:
$V(G\square H)=V(G)\times V(H)$, $E(G\square
H)=\\{((u_{1},v_{1}),(u_{2},v_{2}))|$
$u_{1}=u_{2}~{}and~{}(v_{1},v_{2})\in
E(H)~{}or~{}v_{1}=v_{2}~{}and~{}(u_{1},u_{2})\in E(G)\\}$.
The tensor (direct) product $G\times H$ is defined as follows:
$V(G\times H)=V(G)\times V(H)$,
$E(G\times H)=\\{((u_{1},v_{1}),(u_{2},v_{2}))|~{}(u_{1},u_{2})\in
E(G)~{}and~{}(v_{1},v_{2})\in E(H)\\}$.
The strong tensor (semistrong) product $G\otimes H$ is defined as follows:
$V(G\otimes H)=V(G)\times V(H)$, $E(G\otimes
H)=\\{((u_{1},v_{1}),(u_{2},v_{2}))|$
$~{}(u_{1},u_{2})\in E(G)~{}and~{}(v_{1},v_{2})\in
E(H)~{}or~{}v_{1}=v_{2}~{}and~{}(u_{1},u_{2})\in E(G)\\}$.
The strong product $G\boxtimes H$ is defined as follows:
$V(G\boxtimes H)=V(G)\times V(H)$, $E(G\boxtimes
H)=\\{((u_{1},v_{1}),(u_{2},v_{2}))|~{}(u_{1},u_{2})\in E(G)$
$~{}and~{}(v_{1},v_{2})\in E(H)~{}or~{}u_{1}=u_{2}~{}and~{}(v_{1},v_{2})\in
E(H)~{}or~{}v_{1}=v_{2}~{}and~{}(u_{1},u_{2})\in E(G)\\}$.
The lexicographic product (composition) $G[H]$ is defined as follows:
$V(G[H])=V(G)\times V(H)$,
$E(G[H])=\\{((u_{1},v_{1}),(u_{2},v_{2}))|~{}(u_{1},u_{2})\in
E(G)~{}or~{}u_{1}=u_{2}~{}and~{}(v_{1},v_{2})\in E(H)\\}$.
The terms and concepts that we do not define can be found in [37].
Asratian and Kamalian proved the following:
###### Theorem 1
[1]. Let $G$ be a regular graph. Then
(1)
$G\in\mathfrak{N}$ if and only if $\chi^{\prime}(G)=\Delta(G)$.
(2)
If $G\in\mathfrak{N}$ and $\Delta(G)\leq t\leq W(G)$, then $G$ has an interval
$t$-coloring.
###### Corollary 2
If $G$ is an $r$-regular bipartite graph, then $G\in\mathfrak{N}$ and
$w(G)=r$.
Kubale and Giaro proved the following:
###### Theorem 3
[25]. If $G,H\in\mathfrak{N}$, then $G\square H\in\mathfrak{N}$. Moreover,
$w(G\square H)\leq w(G)+w(H)$ and $W(G\square H)\geq W(G)+W(H)$.
The $k$-dimensional grid $G(n_{1},n_{2},\ldots,n_{k})$, $n_{i}\in\mathbf{N}$
is the Cartesian product of paths $P_{n_{1}}\square
P_{n_{2}}\square\cdots\square P_{n_{k}}$. The cylinder $C(n_{1},n_{2})$ is the
Cartesian product $P_{n_{1}}\square C_{n_{2}}$ and the torus $T(n_{1},n_{2})$
is the Cartesian product $C_{n_{1}}\square C_{n_{2}}$, where $C_{n_{i}}$ is
the cycle of length $n_{i}$. For these graphs Kubale and Giaro proved the
following:
###### Theorem 4
[10]. If $G=G(n_{1},n_{2},\ldots,n_{k})$ or $G=C(m,2n)$, $m\in\mathbf{N}$,
$n\geq 2$, or $G=T(2m,2n)$, $m,n\geq 2$, then $G\in\mathfrak{N}$ and
$w(G)=\Delta(G)$.
For the greatest possible number of colors in interval edge colorings of grid
graphs Petrosyan and Karapetyan proved the following theorems:
###### Theorem 5
[29]. If $G=C(m,2n)$, $m\in\mathbf{N}$, $n\geq 2$, then $W(G)\geq 3m+n-2$.
###### Theorem 6
[29]. If $G=T(2m,2n)$, $m,n\geq 2$, then $W(G)\geq\max\\{3m+n,3n+m\\}$.
In [30] Petrosyan investigated interval edge colorings of complete graphs and
$n$-dimensional cubes $Q_{n}$. In particular, he proved the following
theorems:
###### Theorem 7
$~{}W\left(Q_{n}\right)\geq\frac{n\left(n+1\right)}{2}$ for any
$n\in\mathbf{N}$.
###### Theorem 8
Let $n=p2^{q}$, where $p$ is odd and $q$ is nonnegative. Then
$W\left(K_{2n}\right)\geq 4n-2-p-q$.
The Hamming graph $H(n_{1},n_{2},\ldots,n_{k})$, $n_{i}\in\mathbf{N}$ is the
Cartesian product of complete graphs $K_{n_{1}}\square
K_{n_{2}}\square\cdots\square K_{n_{k}}$. The graph $H_{n}^{k}$ is the
Cartesian product of the complete graph $K_{n}$ by itself $k$ times. It is
easy to see that from Theorems 1, 3 and 8, we have the following result:
###### Theorem 9
Let $n=p2^{q}$, where $p$ is odd and $q$ is nonnegative. Then
(1)
$H_{2n}^{k}\in\mathfrak{N}$,
(2)
$w(H_{2n}^{k})=(2n-1)k$,
(3)
$W(H_{2n}^{k})\geq(4n-2-p-q)k$.
It is known that there are graphs $G$ and $H$ for which $G\square
H\in\mathfrak{N}$ ($G[H]\in\mathfrak{N}$), but $G\in\mathfrak{N}$,
$H\notin\mathfrak{N}$ or $G,H\notin\mathfrak{N}$. For example, $K_{2}\square
C_{3}\in\mathfrak{N}$ and $K_{1,1,3}\square C_{3}\in\mathfrak{N}$
($K_{2}[C_{5}]\in\mathfrak{N}$ and $C_{5}[P]\in\mathfrak{N}$), but
$K_{1,1,3},C_{3}\notin\mathfrak{N}$ ($P,C_{5}\notin\mathfrak{N}$, where $P$ is
the Petersen graph). Moreover, general results can be obtained from the
following theorems:
###### Theorem 10
(Kotzig [24], Pisanski, Shawe-Taylor, Mohar [31]) If $G$ and $H$ are two
regular graphs for which at least one of the following conditions holds:
(1)
$G$ and $H$ contain a perfect matching,
(2)
$\chi^{\prime}(G)=\Delta(G)$,
(3)
$\chi^{\prime}(H)=\Delta(H)$,
then $\chi^{\prime}(G\square H)=\Delta(G\square H)$ and
$\chi^{\prime}(G[H])=\Delta(G[H])$.
###### Theorem 11
(Kotzig [24], Pisanski, Shawe-Taylor, Mohar [31]) Let $G$ be a cubic graph.
Then $\chi^{\prime}(G\square C_{n})=\Delta(G\square C_{n})=5$ and
$\chi^{\prime}(C_{n}[G])=\Delta(C_{n}[G])$ for any $n\geq 4$.
###### Corollary 12
If $G$ and $H$ are two regular graphs for which at least one of the following
conditions holds:
(1)
$G$ and $H$ contain a perfect matching,
(2)
$G\in\mathfrak{N}$,
(3)
$H\in\mathfrak{N}$,
then $G\square H,G[H]\in\mathfrak{N}$ and $w(G\square H)=\Delta(G\square H)$,
$w(G[H])=\Delta(G[H])$.
###### Corollary 13
Let $G$ be a cubic graph. Then $G\square C_{n},C_{n}[G]\in\mathfrak{N}$ and
$w(G\square C_{n})=\Delta(G\square C_{n})=5$, $w(C_{n}[G])=\Delta(C_{n}[G])$
for any $n\geq 4$.
###### Theorem 14
The torus $T(n_{1},n_{2})\in\mathfrak{N}$ if $n_{1}\cdot n_{2}$ is even,
$T(n_{1},n_{2})\notin\mathfrak{N}$ if $n_{1}\cdot n_{2}$ is odd and the
Hamming graph $H(n_{1},n_{2},\ldots,n_{k})\in\mathfrak{N}$ if $n_{1}\cdot
n_{2}\cdots n_{k}$ is even, $H(n_{1},n_{2},\ldots,n_{k})\notin\mathfrak{N}$ if
$n_{1}\cdot n_{2}\cdots n_{k}$ is odd.
* Proof.
Since $T(n_{1},n_{2})$ and $H(n_{1},n_{2},\ldots,n_{k})$ are regular graphs,
by Theorem 1 and Corollary 12, we have $T(n_{1},n_{2})\in\mathfrak{N}$ when
$n_{1}\cdot n_{2}$ is even and $H(n_{1},n_{2},\ldots,n_{k})\in\mathfrak{N}$
when $n_{1}\cdot n_{2}\cdots n_{k}$ is even.
Let us show that $T(n_{1},n_{2})\notin\mathfrak{N}$ when $n_{1}\cdot n_{2}$ is
odd and $H(n_{1},n_{2},\ldots,n_{k})\notin\mathfrak{N}$ when $n_{1}\cdot
n_{2}\cdots n_{k}$ is odd.
Since $T(n_{1},n_{2})$ and $H(n_{1},n_{2},\ldots,n_{k})$ are regular graphs,
we have
$|E(T(n_{1},n_{2}))|=2n_{1}\cdot n_{2}$ and
$|E(H(n_{1},n_{2},\ldots,n_{k}))|=\frac{n_{1}\cdot n_{2}\cdots
n_{k}\cdot\Delta(H(n_{1},n_{2},\ldots,n_{k}))}{2}$.
If $\chi^{\prime}(T(n_{1},n_{2}))=\Delta(T(n_{1},n_{2}))=4$, then
$|E(T(n_{1},n_{2}))|\leq 2(n_{1}\cdot n_{2}-1)$, since $n_{1}\cdot n_{2}$ is
odd.
This shows that $\chi^{\prime}(T(n_{1},n_{2}))=\Delta(T(n_{1},n_{2}))+1=5$
and, by Theorem 1, $T(n_{1},n_{2})\notin\mathfrak{N}$.
Similarly, if
$\chi^{\prime}(H(n_{1},n_{2},\ldots,n_{k}))=\Delta(H(n_{1},n_{2},\ldots,n_{k}))$,
then
$|E(H(n_{1},n_{2},\ldots,n_{k}))|\leq\frac{\left(n_{1}\cdot n_{2}\cdots
n_{k}-1\right)\cdot\Delta(H(n_{1},n_{2},\ldots,n_{k}))}{2}$, since $n_{1}\cdot
n_{2}\cdots n_{k}$ is odd.
This shows that
$\chi^{\prime}(H(n_{1},n_{2},\ldots,n_{k}))=\Delta(H(n_{1},n_{2},\ldots,n_{k}))+1$
and, by Theorem 1, $H(n_{1},n_{2},\ldots,n_{k})\notin\mathfrak{N}$. $\square$
## 3 Main results
First, we consider interval edge colorings of the tensor product of graphs. In
[25] Kubale and Giaro noted that there are graphs $G,H\in\mathfrak{N}$, such
that $G\times H\notin\mathfrak{N}$. Here, we prove that if one of the graphs
belongs to $\mathfrak{N}$ and the other is regular, then $G\times
H\in\mathfrak{N}$.
###### Theorem 15
If $G\in\mathfrak{N}$ and $H$ is an $r$-regular graph, then $G\times
H\in\mathfrak{N}$. Moreover, $w(G\times H)\leq w(G)\cdot r$ and $W(G\times
H)\geq W(G)\cdot r$.
* Proof.
Let $V(G)=\\{u_{1},u_{2},\ldots,u_{n}\\}$,
$V(H)=\\{v_{1},v_{2},\ldots,v_{m}\\}$ and
$V\left(G\times H\right)=\left\\{w_{j}^{(i)}|~{}1\leq i\leq n,1\leq j\leq
m\right\\}$,
$E\left(G\times
H\right)=\left\\{\left(w_{p}^{(i)},w_{q}^{(j)}\right)|\left(u_{i},u_{j}\right)\in
E(G)~{}and~{}\left(v_{p},v_{q}\right)\in E(H)\right\\}$.
Let us consider the graph $K_{2}\times H$. Clearly, $K_{2}\times H$ is an
$r$-regular bipartite graph, thus, by Corollary 2, $K_{2}\times
H\in\mathfrak{N}$ and $w\left(K_{2}\times H\right)=r$. Let $\alpha$ be an
interval $t$-coloring of the graph $G$, $\beta$ be an interval $r$-coloring of
the graph $K_{2}\times H$ and
$V\left(K_{2}\times
H\right)=\\{x_{1},x_{2},\ldots,x_{m},y_{1},y_{2},\ldots,y_{m}\\}$,
$E\left(K_{2}\times
H\right)=\left\\{\left(x_{i},y_{j}\right)|\left(v_{i},v_{j}\right)\in
E(H),1\leq i\leq m,1\leq j\leq m\right\\}$.
Define an edge coloring $\gamma$ of the graph $G\times H$ in the following
way:
for every $\left(w_{p}^{(i)},w_{q}^{(j)}\right)\in E(G\times H)$
$\gamma\left(\left(w_{p}^{(i)},w_{q}^{(j)}\right)\right)=\left(\alpha\left(\left(u_{i},u_{j}\right)\right)-1\right)\cdot
r+\beta\left(\left(x_{p},y_{q}\right)\right)$,
where $1\leq i\leq n,1\leq j\leq n,1\leq p\leq m,1\leq q\leq m$.
It is not difficult to see that $\gamma$ is an interval $t\cdot r$-coloring of
the graph $G\times H$. By the definition of $\gamma$, we have $w(G\times
H)\leq w(G)\cdot r$ and $W(G\times H)\geq W(G)\cdot r$. $\square$
The Figure 1 shows the interval $6$-coloring $\gamma$ of the graph
$P_{4}\times C_{5}$ described in the proof of Theorem 15.
Figure 1: The interval $6$-coloring $\gamma$ of the graph $P_{4}\times C_{5}$.
Note that from Theorems 1 and 15, we have the following result:
###### Corollary 16
(Pisanski, Shawe-Taylor, Mohar [31]) If $G$ is $1$-factorable and $H$ is a
regular graph, then $G\times H$ is also $1$-factorable.
We showed that if $G\in\mathfrak{N}$ and $H$ is regular, then $G\times
H\in\mathfrak{N}$. Now we prove a similar result for the strong tensor product
of graphs.
###### Theorem 17
If $G\in\mathfrak{N}$ and $H$ is an $r$-regular graph, then $G\otimes
H\in\mathfrak{N}$. Moreover, $w(G\otimes H)\leq w(G)\cdot(r+1)$ and
$W(G\otimes H)\geq W(G)\cdot(r+1)$.
* Proof.
Let $V(G)=\\{u_{1},u_{2},\ldots,u_{n}\\}$,
$V(H)=\\{v_{1},v_{2},\ldots,v_{m}\\}$ and
$V\left(G\otimes H\right)=\left\\{w_{j}^{(i)}|~{}1\leq i\leq n,1\leq j\leq
m\right\\}$,
$E\left(G\otimes H\right)=E\left(G\times
H\right)\cup\left\\{\left(w_{p}^{(i)},w_{p}^{(j)}\right)|~{}1\leq p\leq
m~{}and~{}\left(u_{i},u_{j}\right)\in E(G)\right\\}$.
Let us consider the graph $K_{2}\otimes H$. Clearly, $K_{2}\otimes H$ is an
$(r+1)$-regular bipartite graph, thus, by Corollary 2, $K_{2}\otimes
H\in\mathfrak{N}$ and $w\left(K_{2}\otimes H\right)=r+1$. Let $\alpha$ be an
interval $t$-coloring of the graph $G$, $\beta$ be an interval
$(r+1)$-coloring of the graph $K_{2}\otimes H$ and
$V\left(K_{2}\otimes
H\right)=\\{x_{1},x_{2},\ldots,x_{m},y_{1},y_{2},\ldots,y_{m}\\}$,
$E\left(K_{2}\otimes H\right)=\left\\{\left(x_{i},y_{i}\right)|~{}1\leq i\leq
m\right\\}\cup E\left(K_{2}\times H\right)$.
Define an edge coloring $\gamma$ of the graph $G\otimes H$ in the following
way:
for every $\left(w_{p}^{(i)},w_{q}^{(j)}\right)\in E(G\otimes H)$
$\gamma\left(\left(w_{p}^{(i)},w_{q}^{(j)}\right)\right)=\left(\alpha\left(\left(u_{i},u_{j}\right)\right)-1\right)\cdot(r+1)+\beta\left(\left(x_{p},y_{q}\right)\right)$,
where $1\leq i\leq n,1\leq j\leq n,1\leq p\leq m,1\leq q\leq m$.
It is not difficult to see that $\gamma$ is an interval $t\cdot(r+1)$-coloring
of the graph $G\otimes H$. By the definition of $\gamma$, we have $w(G\otimes
H)\leq w(G)\cdot(r+1)$ and $W(G\otimes H)\geq W(G)\cdot(r+1)$. $\square$
The Figure 2 shows the interval $9$-coloring $\gamma$ of the graph
$P_{4}\otimes C_{5}$ described in the proof of Theorem 17.
Figure 2: The interval $9$-coloring $\gamma$ of the graph $P_{4}\otimes
C_{5}$.
Note that from Theorems 1 and 17, we have the following result:
###### Corollary 18
(Pisanski, Shawe-Taylor, Mohar [31]) If $G$ is $1$-factorable and $H$ is a
regular graph, then $G\otimes H$ is also $1$-factorable.
Next, we consider interval edge colorings of the strong product of graphs. In
[25] Kubale and Giaro noted that there are graphs $G,H\in\mathfrak{N}$, such
that $G\boxtimes H\notin\mathfrak{N}$. Here, we prove that if two graphs
belong to $\mathfrak{N}$ and one of them is regular, then $G\boxtimes
H\in\mathfrak{N}$.
###### Theorem 19
If $G,H\in\mathfrak{N}$ and $H$ is an $r$-regular graph, then $G\boxtimes
H\in\mathfrak{N}$. Moreover, $w(G\boxtimes H)\leq w(G)\cdot(r+1)+r$ and
$W(G\boxtimes H)\geq W(G)\cdot(r+1)+r$.
* Proof.
Let $V(G)=\\{u_{1},u_{2},\ldots,u_{n}\\}$,
$V(H)=\\{v_{1},v_{2},\ldots,v_{m}\\}$ and
$V\left(G\boxtimes H\right)=\bigcup_{i=1}^{n}V^{i}(H)$, where
$V^{i}(H)=\left\\{w_{j}^{(i)}|~{}1\leq j\leq m\right\\}$,
$E\left(G\boxtimes H\right)=E\left(G\otimes
H\right)\cup\bigcup_{i=1}^{n}E^{i}(H)$, where
$E^{i}(H)=\left\\{\left(w_{p}^{(i)},w_{q}^{(i)}\right)|~{}\left(v_{p},v_{q}\right)\in
E(H)\right\\}$.
For $i=1,2,\ldots,n$, define a graph $H_{i}$ as follows:
$H_{i}=\left(V^{i}(H),E^{i}(H)\right)$.
First of all note that $\chi^{\prime}(H)=\Delta(H)=r$ since $H\in\mathfrak{N}$
and $H$ is an $r$-regular graph. This implies that there exists an interval
$r$-coloring of the graph $H$. Let us consider the graph $K_{2}\otimes H$.
Clearly, $K_{2}\otimes H$ is an $(r+1)$-regular bipartite graph, thus, by
Corollary 2, $K_{2}\otimes H\in\mathfrak{N}$ and $w\left(K_{2}\otimes
H\right)=r+1$. Let $\alpha$ be an interval $t$-coloring of the graph $G$,
$\beta$ be an interval $(r+1)$-coloring of the graph $K_{2}\otimes H$.
Define an edge coloring $\gamma$ of the graph $G\boxtimes H$ in the following
way:
(1)
for every $\left(w_{p}^{(i)},w_{q}^{(j)}\right)\in E(G\otimes H)$
$\gamma\left(\left(w_{p}^{(i)},w_{q}^{(j)}\right)\right)=\left(\alpha\left(\left(u_{i},u_{j}\right)\right)-1\right)\cdot(r+1)+\beta\left(\left(x_{p},y_{q}\right)\right)$,
where $1\leq i\leq n,1\leq j\leq n,1\leq p\leq m,1\leq q\leq m$.
(2)
for $i=1,2,\ldots,n$, the edges of the subgraph $H_{i}$ we color properly with
colors
$\max S\left(u_{i},\alpha\right)\cdot(r+1)+1,\max
S\left(u_{i},\alpha\right)\cdot(r+1)+2,\ldots,\max
S\left(u_{i},\alpha\right)\cdot(r+1)+r$
It is easy to see that $\gamma$ is an interval $(t\cdot(r+1)+r)$-coloring of
the graph $G\boxtimes H$. By the definition of $\gamma$, we have $w(G\boxtimes
H)\leq w(G)\cdot(r+1)+r$ and $W(G\boxtimes H)\geq W(G)\cdot(r+1)+r$. $\square$
The Figure 3 shows the interval $11$-coloring $\gamma$ of the graph
$P_{4}\boxtimes C_{4}$ described in the proof of Theorem 19.
Figure 3: The interval $11$-coloring $\gamma$ of the graph $P_{4}\boxtimes
C_{4}$.
Note that there are graphs $G$ and $H$ for which $G\boxtimes
H\in\mathfrak{N}$, but $G\in\mathfrak{N},H\notin\mathfrak{N}$. For example,
$K_{2}\boxtimes C_{3}\in\mathfrak{N}$, but $C_{3}\notin\mathfrak{N}$. For
regular graphs the following result was obtained by Zhou [38].
###### Theorem 20
If $G$ is $1$-factorable and $H$ is a regular graph, then $G\boxtimes H$ is
also $1$-factorable.
###### Corollary 21
Let $G$ and $H$ be two regular graphs and $G\in\mathfrak{N}$. Then $G\boxtimes
H\in\mathfrak{N}$.
Finally, we turn our attention to interval edge colorings of the lexicographic
product of graphs. In [25] Kubale and Giaro posed the following question:
###### Problem 1
Does $G[H]\in\mathfrak{N}$ if $G,H\in\mathfrak{N}$?
We start by focusing on the special case of this problem, when
$G\in\mathfrak{N}$ and $H=nK_{1}$ for any $n\in\mathbf{N}$.
###### Theorem 22
If $G\in\mathfrak{N}$, then $G[nK_{1}]\in\mathfrak{N}$ for any
$n\in\mathbf{N}$. Moreover, $w(G[nK_{1}])\leq w(G)\cdot n$ and
$W(G[nK_{1}])\geq(W(G)+1)\cdot n-1$.
* Proof.
Let $V(G)=\\{u_{1},u_{2},\ldots,u_{m}\\}$ and
$V\left(G[nK_{1}]\right)=\left\\{v_{j}^{(i)}|~{}1\leq i\leq m,1\leq j\leq
n\right\\}$,
$E\left(G[nK_{1}]\right)=\left\\{\left(v_{p}^{(i)},v_{q}^{(j)}\right)|~{}\left(u_{i},u_{j}\right)\in
E(G)~{}and~{}p,q=1,2,\ldots,n\right\\}$.
Let $\alpha$ be an interval $t$-coloring of the graph $G$.
Define an edge coloring $\beta$ of the graph $G[nK_{1}]$ in the following way:
for every $\left(v_{p}^{(i)},v_{q}^{(j)}\right)\in E\left(G[nK_{1}]\right)$
$\beta\left(\left(v_{p}^{(i)},v_{q}^{(j)}\right)\right)=\left\\{\begin{tabular}[]{ll}$\left(\alpha((u_{i},u_{j}))-1\right)\cdot
n+p+q-1\pmod{n}$, if $p+q\neq n+1$,\\\ $\alpha((u_{i},u_{j}))\cdot n$, if
$p+q=n+1$.\\\ \end{tabular}\right.$
where $1\leq i\leq m,1\leq j\leq m,1\leq p\leq n,1\leq q\leq n$.
It can be verified that $\beta$ is an interval $t\cdot n$-coloring of the
graph $G[nK_{1}]$. By the definition of $\beta$, we have $w(G[nK_{1}])\leq
w(G)\cdot n$.
Now we show that $W(G[nK_{1}])\geq(W(G)+1)\cdot n-1$.
Let $\phi$ be an interval $W(G)$-coloring of the graph $G$.
Define an edge coloring $\psi$ of the graph $G[nK_{1}]$ in the following way:
for every $\left(v_{p}^{(i)},v_{q}^{(j)}\right)\in E\left(G[nK_{1}]\right)$
$\psi\left(\left(v_{p}^{(i)},v_{q}^{(j)}\right)\right)=\left(\phi((u_{i},u_{j}))-1\right)\cdot
n+p+q-1$,
where $1\leq i\leq m,1\leq j\leq m,1\leq p\leq n,1\leq q\leq n$.
It is easy to see that $\psi$ is an interval $(W(G)\cdot n+n-1)$-coloring of
the graph $G[nK_{1}]$. $\square$
The Figure 4 shows the interval $6$-coloring $\beta$ of the graph
$(K_{1,3}+e)[2K_{1}]$ described in the proof of Theorem 22.
Figure 4: The interval $6$-coloring $\beta$ of the graph
$(K_{1,3}+e)[2K_{1}]$.
###### Corollary 23
(Kamalian, Petrosyan [22]) If $k$ is even, then $C_{k}[nK_{1}]\in\mathfrak{N}$
and
$W(C_{k}[nK_{1}])\geq 2n+\frac{n\cdot k}{2}-1$.
###### Corollary 24
(Kamalian, Petrosyan [23]) Let $k=p2^{q}$, where $p$ is odd and
$q\in\mathbf{N}$. Then $K_{k}[nK_{1}]\in\mathfrak{N}$ and
$W(K_{k}[nK_{1}])\geq(2k-p-q)\cdot n-1$.
Now we show that $G[H]\in\mathfrak{N}$ if $G,H\in\mathfrak{N}$ and $H$ is
regular.
###### Theorem 25
If $G,H\in\mathfrak{N}$ and $H$ is an $r$-regular graph, then
$G[H]\in\mathfrak{N}$. Moreover, if $|V(H)|=n$, then $w(G[H])\leq w(G)\cdot
n+r$ and $W(G[H])\geq W(G)\cdot n+r$.
* Proof.
Let $V(G)=\\{u_{1},u_{2},\ldots,u_{m}\\}$,
$V(H)=\\{v_{1},v_{2},\ldots,v_{n}\\}$ and
$V\left(G[H]\right)=\bigcup_{i=1}^{m}V^{i}(H)$, where
$V^{i}(H)=\\{w_{j}^{(i)}|~{}1\leq j\leq n\\}$,
$E\left(G[H]\right)=\left\\{\left(w_{p}^{(i)},w_{q}^{(j)}\right)|~{}\left(u_{i},u_{j}\right)\in
E(G)~{}and~{}p,q=1,2,\ldots,n\right\\}\cup\bigcup_{i=1}^{m}E^{i}(H)$, where
$E^{i}(H)=\left\\{\left(w_{p}^{(i)},w_{q}^{(i)}\right)|~{}\left(v_{p},v_{q}\right)\in
E(H)\right\\}$.
Let $\alpha$ be an interval $t$-coloring of the graph $G$ and
$H_{i}=\left(V^{i}(H),E^{i}(H)\right)$ for $i=1,2,\ldots,m$.
Note that $\chi^{\prime}(H)=\Delta(H)=r$ since $H\in\mathfrak{N}$ and $H$ is
an $r$-regular graph. This implies that there exists an interval $r$-coloring
of the graph $H$.
Define an edge coloring $\beta$ of the graph $G[H]$ in the following way:
(1)
for every $\left(w_{p}^{(i)},w_{q}^{(j)}\right)\in E(G[H])$
$\beta\left(\left(w_{p}^{(i)},w_{q}^{(j)}\right)\right)=\left\\{\begin{tabular}[]{ll}$r+\left(\alpha((u_{i},u_{j}))-1\right)\cdot
n+p+q-1\pmod{n}$,\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}if
$p+q\neq n+1$,\\\ $r+\alpha((u_{i},u_{j}))\cdot n$, if $p+q=n+1$,\\\
\end{tabular}\right.$
where $1\leq i\leq m,1\leq j\leq m,i\neq j,1\leq p\leq n,1\leq q\leq n$.
(2)
for $i=1,2,\ldots,m$, the edges of the subgraph $H_{i}$ we color properly with
colors
$\left(\min S\left(u_{i},\alpha\right)-1\right)\cdot n+1,\left(\min
S\left(u_{i},\alpha\right)-1\right)\cdot n+2,\ldots,\left(\min
S\left(u_{i},\alpha\right)-1\right)\cdot n+r$
It can be verified that $\beta$ is an interval $(t\cdot n+r)$-coloring of the
graph $G[H]$. By the definition of $\beta$, we have $w(G[H])\leq w(G)\cdot
n+r$ and $W(G[H])\geq W(G)\cdot n+r$. $\square$
The Figure 5 shows the interval $9$-coloring $\beta$ of the graph
$K_{4}[K_{2}]$ described in the proof of Theorem 25.
Figure 5: The interval $9$-coloring $\beta$ of the graph $K_{4}[K_{2}]$.
## 4 Problems
We conclude with the following problems on interval edge colorings of products
of graphs.
###### Problem 2
Are there graphs $G,H\notin\mathfrak{N}$, such that $G\times
H\in\mathfrak{N}$?
###### Problem 3
Are there graphs $G,H\notin\mathfrak{N}$, such that $G\otimes
H\in\mathfrak{N}$?
###### Problem 4
Are there graphs $G,H\notin\mathfrak{N}$, such that $G\boxtimes
H\in\mathfrak{N}$?
* Acknowledgement
We would like to thank the anonymous referees for useful suggestions.
## References
* [1] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987) 25-34 (in Russian).
* [2] A.S. Asratian, R.R. Kamalian, Investigation on interval edge-colorings of graphs, J. Combin. Theory Ser. B 62 (1994) 34-43.
* [3] A.S. Asratian, T.M.J. Denley, R. Haggkvist, Bipartite Graphs and their Applications, Cambridge University Press, Cambridge, 1998.
* [4] A.S. Asratian, C.J. Casselgren, J. Vandenbussche, D.B. West, Proper path-factors and interval edge-coloring of $\left(3,4\right)$-biregular bigraphs, J. Graph Theory 61 (2009) 88-97.
* [5] M.A. Axenovich, On interval colorings of planar graphs, Congr. Numer. 159 (2002) 77-94.
* [6] C. Berge, Theorie des Graphes et ses Applications, Dunod, Paris, 1958 (in French).
* [7] M. Bouchard, A. Hertz, G. Desaulniers, Lower bounds and a tabu search algorithm for the minimum deficiency problem, J. Comb. Optim. 17 (2009) 168-191.
* [8] Y. Feng, Q. Huang, Consecutive edge-coloring of the generalized $\theta$-graph, Discrete Appl. Math. 155 (2007) 2321-2327.
* [9] K. Giaro, The complexity of consecutive $\Delta$-coloring of bipartite graphs: $4$ is easy, $5$ is hard, Ars Combin. 47 (1997) 287-298.
* [10] K. Giaro, M. Kubale, Consecutive edge-colorings of complete and incomplete Cartesian products of graphs, Congr, Numer. 128 (1997) 143-149.
* [11] K. Giaro, M. Kubale, M. Malafiejski, On the deficiency of bipartite graphs, Discrete Appl. Math. 94 (1999) 193-203.
* [12] K. Giaro, M. Kubale, M. Malafiejski, Consecutive colorings of the edges of general graphs, Discrete Math. 236 (2001) 131-143.
* [13] K. Giaro, M. Kubale, Compact scheduling of zero-one time operations in multi-stage systems, Discrete Appl. Math. 145 (2004) 95-103.
* [14] H.M. Hansen, Scheduling with minimum waiting periods, Master’s Thesis, Odense University, Odense, Denmark, 1992 (in Danish).
* [15] D. Hanson, C.O.M. Loten, A lower bound for interval coloring of bi-regular bipartite graphs, Bull. ICA 18 (1996) 69-74.
* [16] D. Hanson, C.O.M. Loten, B. Toft, On interval colorings of bi-regular bipartite graphs, Ars Combin. 50 (1998) 23-32.
* [17] P.E. Himmelwright, J.E. Williamson, On 1-factorability and edge-colorability of cartesian products of graphs, Elem. Der Math. 29 (1974) 66-67.
* [18] T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995.
* [19] R.R. Kamalian, Interval colorings of complete bipartite graphs and trees, preprint, Comp. Cen. of Acad. Sci. of Armenian SSR, Erevan, 1989 (in Russian).
* [20] R.R. Kamalian, Interval edge colorings of graphs, Doctoral Thesis, Novosibirsk, 1990.
* [21] R.R. Kamalian, A.N. Mirumian, Interval edge colorings of bipartite graphs of some class, Dokl. NAN RA, 97 (1997) 3-5 (in Russian).
* [22] R.R. Kamalian, P.A. Petrosyan, Interval colorings of some regular graphs, Math. probl. of comp. sci. 25 (2006) 53-56.
* [23] R.R. Kamalian, P.A. Petrosyan, On interval colorings of complete $k$-partite graphs $K_{n}^{k}$, Math. probl. of comp. sci. 26 (2006) 28-32.
* [24] A. Kotzig, 1-Factorizations of cartesian products of regular graphs, J. Graph Theory 3 (1979) 23-34.
* [25] M. Kubale, Graph Colorings, American Mathematical Society, 2004.
* [26] E.S. Mahamoodian, On edge-colorability of cartesian products of graphs, Canad. Math. Bull. 24 (1981) 107-108.
* [27] B. Mohar, T. Pisanski, Edge-coloring of a family of regular graphs, Publ. Inst. Math. (Beograd) 33 (47) (1983) 157-162.
* [28] B. Mohar, On edge-colorability of products of graphs, Publ. Inst. Math. (Beograd) 36 (50) (1984) 13-16.
* [29] P.A. Petrosyan, G.H. Karapetyan, Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori, Proceedings of the CSIT Conference (2007) 86-88.
* [30] P.A. Petrosyan, Interval edge-colorings of complete graphs and $n$-dimensional cubes, Discrete Mathematics 310 (2010) 1580-1587.
* [31] T. Pisanski, J. Shawe-Taylor, B. Mohar, 1-Factorization of the composition of regular graphs, Publ. Inst. Math. (Beograd) 33 (47) (1983) 193-196.
* [32] A.V. Pyatkin, Interval coloring of $\left(3,4\right)$-biregular bipartite graphs having large cubic subgraphs, J. Graph Theory 47 (2004) 122-128.
* [33] G. Sabidussi, Graph multiplication, Math. Z. 72 (1960) 446-457.
* [34] A. Schwartz, The deficiency of a regular graph, Discrete Math. 306 (2006) 1947-1954.
* [35] S.V. Sevast’janov, Interval colorability of the edges of a bipartite graph, Metody Diskret. Analiza 50 (1990) 61-72 (in Russian).
* [36] V.G. Vizing, The Cartesian product of graphs, Vych. Sis. 9 (1963) 30-43 (in Russian).
* [37] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 1996.
* [38] M.K. Zhou, Decomposition of some product graphs into 1-factors and Hamiltonian cycles, Ars Combin. 28 (1989) 258-268.
|
arxiv-papers
| 2009-11-23T18:26:31 |
2024-09-04T02:49:06.638423
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Petros A. Petrosyan",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/0911.4459"
}
|
0911.4527
|
††thanks: Present address: Laser Centre Vrije Universiteit Amsterdam, The
Netherlands
# Frequency Comparison of Two High-Accuracy Al+ Optical Clocks
C. W. Chou chinwen@nist.gov D. B. Hume J. C. J. Koelemeij D. J. Wineland
T. Rosenband Time and Frequency Division, National Institute of Standards and
Technology, Boulder, Colorado 80305
###### Abstract
We have constructed an optical clock with a fractional frequency inaccuracy of
$8.6\times 10^{-18}$, based on quantum logic spectroscopy of an Al+ ion. A
simultaneously trapped Mg+ ion serves to sympathetically laser-cool the Al+
ion and detect its quantum state. The frequency of the 1S0$\leftrightarrow$3P0
clock transition is compared to that of a previously constructed Al+ optical
clock with a statistical measurement uncertainty of $7.0\times 10^{-18}$. The
two clocks exhibit a relative stability of $2.8\times 10^{-15}\tau^{-1/2}$,
and a fractional frequency difference of $-1.8\times 10^{-17}$, consistent
with the accuracy limit of the older clock.
Optical clocks based on petahertz ($10^{15}$ Hz) transitions in isolated atoms
have demonstrated significant improvements over the current cesium primary-
frequency-standards at 9.2 GHz. They also shed light on fundamental physics,
such as the possible variation of physical constants. While the merits of
laser-cooled ion optical frequency standards were known HGD1982monoion ;
DJW1987LaserCoolingLimits , further developments were required to permit their
use. Sub-hertz linewidth lasers BCY1999subhertz enabled single ions to be
probed with sufficient resolution for high-stability clock operation, and
control of external-field perturbations allowed such clocks to operate with an
inaccuracy below $10^{-16}$ HgCs2006 ; TR2008AlHg . For comparison, cesium
standards that realize the SI second have reached an inaccuracy of $3\times
10^{-16}$ 2008ParkerEFTF , and an optical lattice clock based on Sr atoms has
been reported Campbell2008 with an inaccuracy of $1.5\times 10^{-16}$. Here
we describe an Al+ ion clock with an inaccuracy of $8.6\times 10^{-18}$.
The 1S0$\leftrightarrow$3P0 transition in Al+ at 1.121 PHz is of interest due
to its low sensitivity to electromagnetic perturbations and its narrow natural
linewidth of 8 mHz Dehmelt1992 ; TR2007Al3P0observed . Al+ has the smallest
sensitivity to blackbody radiation TR2006BBRshift ; Mitroy2009BBR among
atomic species currently under consideration for clocks, thus relaxing the
requirement on ambient temperature control. However, the absence of an
accessible allowed optical transition prevents the internal state of Al+ ions
from being detected by conventional methods, and the ion cannot currently be
directly laser-cooled. Quantum logic spectroscopy (QLS) POS2005BeAl overcomes
these difficulties by trapping a “logic ion” that can be directly laser-cooled
together with the Al+ clock ion. The coupled motion of the two ions allows for
sympathetic cooling, as well as the transfer of the clock ion’s quantum state
to the logic ion, where the state can be measured.
The clock described here shares features with our previously-constructed Al-Be
clock TR2007Al3P0observed ; TR2008AlHg , but also includes many changes,
making a comparison of the two clocks a valuable test of systematic errors. In
the new clock, the 9Be+ logic ion has been replaced by 25Mg+, whose mass
closely matches that of 27Al+. Laser-cooling inefficiencies due to mass-
mismatch are thus suppressed. The Al-Mg ion trap is a linear Paul trap built
from tool-machined all-metal electrodes (Fig. 1). This construction differs
from the Al-Be trap that was built from laser-machined and gold-coated alumina
electrodes Rowe2002 , and it exhibits reduced RF-micromotion-inducing electric
fields.
Figure 1: Setup for comparing the frequencies of the two Al+ clocks. The 4th
harmonic of a fiber laser is locked to the Al+ 1S0$\leftrightarrow$3P0 clock
transition in the Al-Mg apparatus with a fixed offset frequency (applied to
AOM1). Another laser beam derived from the same laser probes the clock
transition in the Al-Be apparatus, where the laser frequency is locked to the
clock transition in a separate digital feedback loop that controls AOM2. The
record of frequencies applied to AOM2 represents the difference in clock
frequencies. BS: beam splitter; AOM: acousto-optic modulator; x2: frequency
doubler.
QLS with a Mg+ logic ion proceeds in the same way as with Be+, but the ground-
state-cooling process CM1995GScooling has been modified to require only two
lasers rather than three TwoLaserGSC . This cools the out-of-phase axial
motional mode to an average quantum number of $\bar{n}<0.05$. It also enables
quantum-non-demolition transfer (QNDT) of the Al+-clock-ion state to Mg+ with
approximately 80 % fidelity in a single QNDT repetition and over 99 % fidelity
after typically five QNDT repetitions DBH2007detection .
The trap utilized in the new Al+ clock has blade-shaped gold-coated beryllium-
copper electrodes (Fig. 1) whose edges are approximately 400 $\mu$m from the
ions. The 25Mg+ logic ion ($I=5/2$) is manipulated with light of 279.5 nm
wavelength from two independent lasers. A frequency-doubled dye laser
resonantly drives $|^{2}S_{1/2},F\in\\{2,3\\}\rangle$ $\rightarrow$
${}^{2}P_{3/2}$ cycling and repumping transitions, while a frequency-
quadrupled fiber laser is detuned by 40-60 GHz to drive Raman transitions
between the 25Mg+ hyperfine qubit states. Both ions are created via multi-
photon-ionization of neutral atoms from ovens. A 396 nm diode laser produces
27Al+ Hurst1979PI , while a frequency-doubled dye laser at 285 nm creates
25Mg+ Drewsen2000PI . Typically the clock transition is probed with 150 ms
duration $\pi$-pulses with a duty cycle of approximately 65 %. The remaining
35 % is occupied with state preparation and state detection functions as well
as interleaved experiments that allow real-time measurement and reduction of
micromotion.
Table 1: Systematic effects that shift the clock from its ideal unperturbed frequency. Shifts and uncertainties given are in fractional frequency units ($\Delta\nu/\nu$). See text for discussion. Effect | Shift | Uncertainty
---|---|---
| ($10^{-18}$) | ($10^{-18}$)
Excess micromotion | -9 | 6
Secular motion | -16.3 | 5
Blackbody radiation shift | -9 | 3
Cooling laser Stark shift | -3.6 | 1.5
Quad. Zeeman shift | -1079.9 | 0.7
Linear Doppler shift | 0 | 0.3
Clock laser Stark shift | 0 | 0.2
Background gas collisions | 0 | 0.5
AOM freq. error | 0 | 0.2
Total | -1117.8 | 8.6
Systematic shifts of the Al-Mg clock are listed in Table 1. Two types of
residual motion cause time-dilation shifts: micromotion near the trap drive
frequency of $\nu_{RF}=59$ MHz, and harmonic-oscillator (secular) motion at
the ion’s normal mode frequencies (Table 2). In both cases the clock frequency
shifts by $\frac{\Delta\nu}{\nu}=-\frac{\langle
v^{2}\rangle}{2c^{2}}\left(1+(\frac{f}{\mathrm{400MHz}})^{2}\right)$, where we
add to the relativistic time-dilation $\langle v^{2}\rangle/(2c^{2})$ a
frequency-dependent term that corresponds to the Stark shift from the motion-
inducing electric fields. Here $\langle v^{2}\rangle$ is the mean-squared
Al+-ion velocity and $f$ is the frequency of motion. For the highest motional
frequencies the Stark shift correction is 2 %.
Table 2: Motional modes of the Al-Mg ion pair. For each of the six normal modes, the oscillation frequency and zero-point motional amplitude for each ion is listed. The trap axis corresponds to $\hat{z}$, and $\hat{x},\hat{y}$ are two orthogonal radial directions whose orientation is determined by the trap geometry. Also shown are the calculated ($\bar{n}_{C}$) and measured ($\bar{n}_{M}$) Doppler-cooled average motional quantum numbers, and the time-dilation (TD) per motional quantum as well as the total TD per mode. f | [MHz] | 6.53 | 5.66 | 5.20 | 4.64 | 3.41 | 3.00
---|---|---|---|---|---|---|---
25Mg+ | [nm] | 4.9$\hat{y}$ | 2.9$\hat{y}$ | 4.6$\hat{z}$ | 5.5$\hat{x}$ | -4.2$\hat{x}$ | 5.6$\hat{z}$
27Al+ | [nm] | 2.6$\hat{y}$ | -5.0$\hat{y}$ | -4.1$\hat{z}$ | 3.5$\hat{x}$ | 6.2$\hat{x}$ | 5.8$\hat{z}$
$\bar{n}_{M}$ | | 2.9 | 4.5 | 3.4 | 6.3 | 10.0 | 7.0
$\bar{n}_{C}$ | | 3.3 | 3.8 | 3.4 | 5.9 | 8.0 | 5.9
TD/quantum | $[10^{-18}]$ | 0.226 | 0.731 | 0.197 | 0.290 | 0.771 | 0.133
Total TD | $[10^{-18}]$ | 0.77 | 3.66 | 0.77 | 1.97 | 8.10 | 1.00
Excess micromotion (EMM) refers to the rapid ion motion at $\nu_{RF}$
Berkeland1998MM . It is caused by electric fields that force the ion away from
the RF-minimum of the ion trap, or phase shifts between trap electrodes that
cause the RF-fields to be non-zero at the pseudo-potential minimum. We measure
the amplitude of this motion at $\nu_{RF}$ by observing the motional-sideband
strength of the Al+ 1S0 $\rightarrow$ 3P1 transition in three orthogonal
directions. For small amplitude of motion, the time dilation shift is
$\Delta\nu/\nu=-|\vec{\eta}\nu_{RF}/\nu_{L}|^{2}=-2.8\times
10^{-15}|\vec{\eta}|^{2}$, where $\nu_{L}=1.12$ PHz is the probe laser
frequency, and $\vec{\eta}=(\eta_{1},\eta_{2},\eta_{3})$ is the measured EMM
Lamb-Dicke parameter (the ratio of sideband and carrier Rabi rates) for the
three orthogonal directions. Typical values for $\eta_{1,2,3}$ are 0.01 to
0.04. We note that this EMM measurement method detects slow electric field
fluctuations such as those caused by the migration of photo-electrons, as well
as faster fluctuations caused by periodic line noise (50 to 60 Hz).
Fluctuations that are shorter than the laser probe period of 0.05 to 0.1 ms
will not be detected.
In a perfect linear Paul trap EMM along the trap axis does not occur, but
imperfections in the ion-trap geometry can lead to axial EMM . For the Al-Mg
trap there exists a sharp minimum of axial EMM at one spatial location. The
Al+ ion is held at this point, but random background-gas collisions cause the
Al-Mg ion pair to spontaneously re-order, which causes the Al+ ion to move by
3 $\mu$m every 200 s, on average. The Al+ ion is maintained at the position of
minimal micromotion by adjusting the electrode voltages every 10 s to force
the ion-pair into the desired order. When the ions are in the wrong order
(about 5 % of the time), the Al+ ion experiences excess axial micromotion and
a corresponding clock shift of $\Delta\nu/\nu=-2.7\times 10^{-17}$. This
additional shift is included in Table 1.
During each Al+ clock interrogation pulse, the Mg+ ion is simultaneously
Doppler cooled by a laser that is tuned 21 MHz below the
$|^{2}S_{1/2},F=3,m=-3\rangle$ $\rightarrow$ $|^{2}P_{3/2},F=4,m=-4\rangle$
cycling transition. The amplitude of secular motion (corresponding to the
motional temperature) is extracted from the ratio of amplitudes for the red-
and blue-sidebands of a Raman transition for all six normal modes
CM1995GScooling . Measured values are shown in Table 2, together with values
calculated from laser-cooling theory for a single ion Itano1982Cooling . The
single-Mg+-ion Doppler cooling limit is also valid when applied to each of the
six normal modes of the Al-Mg ion pair. The measured and calculated motional
quantum numbers agree within the measurement uncertainty, and we consider the
stated 30 % uncertainty for this shift to be a conservative limit.
The Mg+ Doppler cooling laser beam maintains the Al-Mg ion pair at a constant
motional temperature during the clock interrogation pulse, but because it also
overlaps the Al+ ion, it causes an AC Stark shift by coupling off-resonantly
to allowed transitions that connect to the ground (3s${}^{2})$1S0 and excited
(3s3p)3P0 clock states. Following the evaluation of the blackbody radiation
shift TR2006BBRshift , we estimate the differential clock polarizability at
$279.5$ nm as $\Delta\nu/\nu=(-3.5\pm 0.6)\times 10^{-17}S$, where
($S=I/I_{S}$) is the saturation parameter for 25Mg+ ($I_{S}\approx 2470$
W/m2). The intensity $I$ of the Mg+ Doppler cooling laser is estimated from
the rate at which this laser repumps the $|^{2}S_{1/2},F=2,m=-2\rangle$ dark
hyperfine ground state. The ion fluorescence photo-multiplier counts $F(t)$
collected in a duration $t$ due to repumping of the dark state may be written
as $F(t)=b(t+\tau(e^{-t/\tau}-1))$ where $\tau=(0.217/S)$ ms and $b$ is the
bright-state counting rate. We extract $S$ by fitting $F(t)$ to the observed
ion fluorescence. Typically, we measure $\tau=2.1\pm 0.8$ ms, and find
$\Delta\nu/\nu=(-3.6\pm 1.5)\times 10^{-18}$.
Another Stark shift is caused by thermal blackbody radiation. The temperature
of the Al-Mg ion trap is measured with two platinum sensors at opposite ends
of the trap structure which are expected to be at temperature extremes. Heat
is removed primarily through thermal conduction at one end of the trap where
we measure a temperature of 35 ∘C. At the higher thermal resistance trap-end
the temperature is 40 ∘C, and the thermal radiation field impinging upon the
ion is bounded by this maximum temperature and the laboratory room temperature
of 22 ∘C: $T_{ion}=(31\pm 9)$ ∘C.
During operation, the Al+ clock servo alternates between probing of the
$|^{1}S_{0},m_{F}=5/2\rangle$$\leftrightarrow$$|^{3}P_{0},m_{F}=5/2\rangle$
and
$|^{1}S_{0},m_{F}=-5/2\rangle$$\leftrightarrow$$|^{3}P_{0},m_{F}=-5/2\rangle$
transitions every 5 s, and the apparatus synthesizes an average of these two
frequencies to eliminate first-order Zeeman shifts Madej1998 ;
TR2007Al3P0observed . Each transition’s resonance is probed several times at
the high- and low-frequency half-maximum points to derive a frequency-
correction signal. The frequency-difference between the transitions is
proportional to the mean magnetic field $\langle B\rangle$, which allows an
accurate estimate of the quadratic Zeeman shift due to the quasi-static
quantization field of typically $\langle B\rangle=0.1$ mT. However, the
quadratic Zeeman shift is proportional to $\langle B^{2}\rangle=\langle
B\rangle^{2}+B_{AC}^{2}$, where $B_{AC}^{2}$ is the variance of the magnetic
field about its mean. The dominant sources of varying magnetic fields are
currents at $\nu_{RF}$ in conductors near the ion. We vary the trap RF drive
power $P$ and measure the frequency of the hyperfine clock transition in 25Mg+
$|F=3,m_{F}=0\rangle\rightarrow|F=2,m_{F}=0\rangle$ near 1.789 GHz, which has
a strong quadratic dependence on the magnetic field, to find
$B_{AC}^{2}=1.45\times 10^{-12}(P/\textrm{W})$ T2. For Al+ clock operation
$P=15$ W, and $B_{AC}^{2}=2.17\times 10^{-11}\textrm{T}^{2}$, which alters the
quadratic Zeeman shift by $\Delta\nu/\nu=(-1.4\pm 0.3)\times 10^{-18}.$
Other potential systematic shifts are listed in Table 1. When stabilizing the
clock laser to the ion we probe the clock transition alternately with counter-
propagating laser beams to observe and cancel potential first-order Doppler
shifts TR2008AlHg . For the clock comparison described below, we observe a
differential shift for the two probe directions of $(1.2\pm 0.7)\times
10^{-17}$ but this effect is suppressed by taking the average. The suppression
factor is limited because the atomic line-shape and hence the servo gains
differ slightly for the two probe directions. During clock operation the mean
fractional gain imbalance was 1.5 %, thereby reducing the possible first order
Doppler shift to $3\times 10^{-19}$.
We have looked for Stark shifts due to the clock pulse itself by raising the
intensity of the probe beam in one clock, and comparing the frequency to that
produced by the other Al+ clock. With an increase of 40 dB in the clock probe
intensity, we observe no statistically significant frequency difference with a
fractional uncertainty of $2\times 10^{-15}$, constraining any effect that
scales linearly with this intensity to $2\times 10^{-19}$. We observe a rate
of background gas collisions similar to the Al-Be clock and assign a $5\times
10^{-19}$ uncertainty to this potential shift TR2008AlHg . Thermally induced
frequency errors from beam switching AOMs (Fig. 1) have been evaluated
previously TR2008AlHg , and this uncertainty is reduced to $2\times 10^{-19}$
by operating the AOMs at less than 1 mW.
Figure 2: Clock stability. Fractional frequency uncertainty vs. averaging
period ($\tau$) for a comparison between the two Al+ clocks (10700 s
duration). Overlapping Allan deviation and N-sample standard deviation are
shown Riley2008Stability . For each comparison measurement the coefficient of
the $\tau^{-1/2}$ asymptote is estimated and used to derive the measurement’s
statistical uncertainty. The $2.8\times 10^{-15}\tau^{-1/2}$ asymptote is
reached for averaging periods that are longer than the servo time constant of
10 s.
We directly compared the Al-Mg clock with the Al-Be clock to perform
independent tests that could reveal unaccounted-for clock shifts. The Al-Be
clock was evaluated with an accuracy of $2.3\times 10^{-17}$ TR2008AlHg , and
this evaluation remained valid during the two-clock comparison. The two clocks
were compared in 56 separate measurements, each of duration $1000-11000$ s. As
shown in Fig. 1, the frequency of the 1S0$\leftrightarrow$3P0 probe laser was
actively steered to the Al+ ion in the Al-Mg apparatus with a servo time-
constant of about 10 s. A portion of this laser light simultaneously probed
the Al+ ion in the Al-Be clock, where it was servoed to the clock transition
in a separate digital feedback loop. The frequency produced by this feedback
loop represents the frequency difference between the two clocks, and was
recorded and analyzed for stability (Fig. 2). Average frequencies of the
individual measurements corrected for known shifts are shown in Fig. 3, where
the overall weighted mean is $(\nu_{AlMg}-\nu_{AlBe})/\nu=$ $(-1.8\pm
0.7)\times 10^{-17}$. This value is consistent with the 1-$\sigma$ error of
$2.5\times 10^{-17}$ that is calculated by adding in quadrature the
inaccuracies of the two clocks and the statistical uncertainty. The
reduced-$\chi^{2}$ for this data set is $1.02$, indicating that the error bars
derived from the estimated stability correctly capture the scatter of the
data. For this data set the total in-loop servo error for the Al-Mg clock was
$\Delta\nu/\nu=6\times 10^{-19}$, and for a well-designed servo loop this
error declines faster than the statistical error. It is therefore not included
as a systematic shift in Table 1.
Figure 3: Measurements of the fractional frequency difference between the two
Al+ clocks (blue points). Error bars represent the statistical uncertainty
(see Fig. 2). The horizontal line shows the weighted mean of $-1.8\times
10^{-17}$ with an overall statistical uncertainty of $\pm$$7.0\times 10^{-18}$
(shaded band).
In summary, we have built an Al+ ion clock with a fractional frequency
inaccuracy of $8.6\times 10^{-18}$. Its frequency is compared to that of a
previously constructed Al+ clock, and the measured fractional frequency
difference of $(-1.8\pm 0.7)\times 10^{-17}$ is consistent with the inaccuracy
of the previous clock. Significantly, the statistical uncertainty in the
frequency comparison of $7.0\times 10^{-18}$ is smaller than the inaccuracy of
either clock, and the average measurement stability was $2.8\times
10^{-15}\tau^{-1/2}$ (total measurement duration: 164 967 s). This result may
be compared to other direct same-species atomic clock comparisons, where two
Cs microwave clocks have reached an agreement of $(4\pm 3)\times 10^{-16}$
Vian2005Cs , and two Yb+ single ion clocks showed agreement of $(3.8\pm
6.1)\times 10^{-16}$ Schneider2005 . Future frequency ratio measurements of
the Al+ clock and the NIST Hg+ optical clock would enable improved constraints
on present-era changes in the fine-structure constant TR2008AlHg .
We thank J. C. Bergquist for use of his stable Fabry-Perot cavities, T. M.
Fortier for optical frequency measurements, W. M. Itano for atomic structure
calculations, and S. Bickman and D. Leibrandt for helpful comments on the
manuscript. This work was supported by ONR. J.C.J.K. acknowledges support from
the Netherlands Organisation for Scientific Research (NWO). Contribution of
NIST, not subject to U.S. copyright.
## References
* (1) H. G. Dehmelt, IEEE Trans. Inst. Meas. 31, 83 (1982)
* (2) D. J. Wineland, W. M. Itano, J. C. Bergquist, and R. G. Hulet, Phys. Rev. A 36, 2220 (1987)
* (3) B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, Phys. Rev. Lett. 82, 3799 (1999)
* (4) W. H. Oskay et al., Phys. Rev. Lett. 97, 020801 (2006).
* (5) T. Rosenband et al. Science 319, 1808 (2008)
* (6) T. E. Parker, in _Proc. 2008 EFTF Conf._ (2008)
* (7) G. K. Campbell et al. Metrologia 45, 539 (2008)
* (8) N. Yu, H. Dehmelt, and W. Nagourney, in _Proc. Nat. Acad. Sci_ (1992) p. 7289
* (9) T. Rosenband et al. Phys. Rev. Lett. 98, 220801 (2007)
* (10) T. Rosenband et al. arXiv:physics/0611125 (2006)
* (11) J. Mitroy, J. Y. Zhang, M. W. J. Bromley, and K. G. Rollin, European Physical Journal D 53, 15 (2009)
* (12) P. O. Schmidt et al. Science 309, 749 (2005)
* (13) M. A. Rowe et al. Quant. Inform. Comp. 2, 257 (2002)
* (14) C. Monroe et al. Phys. Rev. Lett. 75, 4011 (1995)
* (15) The entropy-reducing laser that repumps the states $|^{2}S_{1/2},F\in\\{2,3\\},m=-2\rangle$ via $|^{2}P_{1/2},F=3,m=-3\rangle$ into $|^{2}S_{1/2},F=3,m=-3\rangle$ is replaced by the repeated application of a pulse sequence that first drives the $|^{2}S_{1/2},F=2,m=-2\rangle\rightarrow|^{2}P_{3/2},F=3,m=-3\rangle$ transition and subsequently the transition $|^{2}S_{1/2},F=3,m=-2\rangle\rightarrow|^{2}S_{1/2},F=2,m=-2\rangle$.
* (16) D. B. Hume, T. Rosenband, and D. J. Wineland, Phys. Rev. Lett. 99, 120502 (2007)
* (17) G. S. Hurst, M. G. Payne, S. D. Kramer, and J. P. Young, Rev. Mod. Phys. 51, 767 (1979)
* (18) N. Kjaergaard et al. Appl. Phys. B 71, 207 (2000)
* (19) D. J. Berkeland et al. J. Appl. Phys. 83, 5025 (1998)
* (20) W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982)
* (21) J. E. Bernard, L. Marmet, and A. A. Madej, Opt. Comm. 150, 170 (1998)
* (22) W. J. Riley, _NIST Special Publication 1065_ (2008).
* (23) C. Vian et al. IEEE Trans. Inst. Meas. 54, 833 (2005)
* (24) T. Schneider, E. Peik, and C. Tamm, Phys. Rev. Lett. 94, 230801 (2005)
|
arxiv-papers
| 2009-11-24T01:57:07 |
2024-09-04T02:49:06.644721
|
{
"license": "Public Domain",
"authors": "C.-W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T.\n Rosenband",
"submitter": "Chin-wen Chou",
"url": "https://arxiv.org/abs/0911.4527"
}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.