173 research outputs found

### Random Matrix Theory and Entanglement in Quantum Spin Chains

We compute the entropy of entanglement in the ground states of a general
class of quantum spin-chain Hamiltonians - those that are related to quadratic
forms of Fermi operators - between the first N spins and the rest of the system
in the limit of infinite total chain length. We show that the entropy can be
expressed in terms of averages over the classical compact groups and establish
an explicit correspondence between the symmetries of a given Hamiltonian and
those characterizing the Haar measure of the associated group. These averages
are either Toeplitz determinants or determinants of combinations of Toeplitz
and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture
are used to compute the leading order asymptotics of the entropy as N -->
infinity . This is shown to grow logarithmically with N. The constant of
proportionality is determined explicitly, as is the next (constant) term in the
asymptotic expansion. The logarithmic growth of the entropy was previously
predicted on the basis of numerical computations and conformal-field-theoretic
calculations. In these calculations the constant of proportionality was
determined in terms of the central charge of the Virasoro algebra. Our results
therefore lead to an explicit formula for this charge. We also show that the
entropy is related to solutions of ordinary differential equations of
Painlev\'e type. In some cases these solutions can be evaluated to all orders
using recurrence relations.Comment: 39 pages, 1 table, no figures. Revised version: minor correction

### Localized eigenfunctions in Seba billiards

We describe some new families of quasimodes for the Laplacian perturbed by the
addition of a potential formally described by a Dirac delta function. As an application,
we find, under some additional hypotheses on the spectrum, subsequences
of eigenfunctions of Šeba billiards that localize around a pair of unperturbed
eigenfunctions

### The Mean Value of $L(\tfrac{1}{2},\chi)$ in the Hyperelliptic Ensemble

We obtain an asymptotic formula for the first moment of quadratic Dirichlet
$L$--functions over function fields at the central point $s=\tfrac{1}{2}$.
Specifically, we compute the expected value of $L(\tfrac{1}{2},\chi)$ for an
ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as
$g\rightarrow\infty$. Our approach relies on the use of the analogue of the
approximate functional equation for such $L$--functions. The results presented
here are the function field analogues of those obtained previously by Jutila in
the number-field setting and are consistent with recent general conjectures for
the moments of $L$--functions motivated by Random Matrix Theory.Comment: 22 pages, To appear in Journal of Number Theory Volume 132, Issue 12,
December 2012, Pages 2793-281

### Chaotic Diffusion on Periodic Orbits: The Perturbed Arnol'd Cat Map

Chaotic diffusion on periodic orbits (POs) is studied for the perturbed
Arnol'd cat map on a cylinder, in a range of perturbation parameters
corresponding to an extended structural-stability regime of the system on the
torus. The diffusion coefficient is calculated using the following PO formulas:
(a) The curvature expansion of the Ruelle zeta function. (b) The average of the
PO winding-number squared, $w^{2}$, weighted by a stability factor. (c) The
uniform (nonweighted) average of $w^{2}$. The results from formulas (a) and (b)
agree very well with those obtained by standard methods, for all the
perturbation parameters considered. Formula (c) gives reasonably accurate
results for sufficiently small parameters corresponding also to cases of a
considerably nonuniform hyperbolicity. This is due to {\em uniformity sum
rules} satisfied by the PO Lyapunov eigenvalues at {\em fixed} $w$. These sum
rules follow from general arguments and are supported by much numerical
evidence.Comment: 6 Tables, 2 Figures (postscript); To appear in Physical Review

### Entanglement entropy in quantum spin chains with finite range interaction

We study the entropy of entanglement of the ground state in a wide family of
one-dimensional quantum spin chains whose interaction is of finite range and
translation invariant. Such systems can be thought of as generalizations of the
XY model. The chain is divided in two parts: one containing the first
consecutive L spins; the second the remaining ones. In this setting the entropy
of entanglement is the von Neumann entropy of either part. At the core of our
computation is the explicit evaluation of the leading order term as L tends to
infinity of the determinant of a block-Toeplitz matrix whose symbol belongs to
a general class of 2 x 2 matrix functions. The asymptotics of such determinant
is computed in terms of multi-dimensional theta-functions associated to a
hyperelliptic curve of genus g >= 1, which enter into the solution of a
Riemann-Hilbert problem. Phase transitions for thes systems are characterized
by the branch points of the hyperelliptic curve approaching the unit circle. In
these circumstances the entropy diverges logarithmically. We also recover, as
particular cases, the formulae for the entropy discovered by Jin and Korepin
(2004) for the XX model and Its, Jin and Korepin (2005,2006) for the XY model.Comment: 75 pages, 10 figures. Revised version with minor correction

### Eigenfunction Statistics on Quantum Graphs

We investigate the spatial statistics of the energy eigenfunctions on large
quantum graphs. It has previously been conjectured that these should be
described by a Gaussian Random Wave Model, by analogy with quantum chaotic
systems, for which such a model was proposed by Berry in 1977. The
autocorrelation functions we calculate for an individual quantum graph exhibit
a universal component, which completely determines a Gaussian Random Wave
Model, and a system-dependent deviation. This deviation depends on the graph
only through its underlying classical dynamics. Classical criteria for quantum
universality to be met asymptotically in the large graph limit (i.e. for the
non-universal deviation to vanish) are then extracted. We use an exact field
theoretic expression in terms of a variant of a supersymmetric sigma model. A
saddle-point analysis of this expression leads to the estimates. In particular,
intensity correlations are used to discuss the possible equidistribution of the
energy eigenfunctions in the large graph limit. When equidistribution is
asymptotically realized, our theory predicts a rate of convergence that is a
significant refinement of previous estimates. The universal and
system-dependent components of intensity correlation functions are recovered by
means of an exact trace formula which we analyse in the diagonal approximation,
drawing in this way a parallel between the field theory and semiclassics. Our
results provide the first instance where an asymptotic Gaussian Random Wave
Model has been established microscopically for eigenfunctions in a system with
no disorder.Comment: 59 pages, 3 figure

### Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case

We consider the sample covariance matrices of large data matrices which have
i.i.d. complex matrix entries and which are non-square in the sense that the
difference between the number of rows and the number of columns tends to
infinity. We show that the second-order correlation function of the
characteristic polynomial of the sample covariance matrix is asymptotically
given by the sine kernel in the bulk of the spectrum and by the Airy kernel at
the edge of the spectrum. Similar results are given for real sample covariance
matrices

### Linear Statistics of Point Processes via Orthogonal Polynomials

For arbitrary $\beta > 0$, we use the orthogonal polynomials techniques
developed by R. Killip and I. Nenciu to study certain linear statistics
associated with the circular and Jacobi $\beta$ ensembles. We identify the
distribution of these statistics then prove a joint central limit theorem. In
the circular case, similar statements have been proved using different methods
by a number of authors. In the Jacobi case these results are new.Comment: Added references, corrected typos. To appear, J. Stat. Phy

### On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble

We consider the asymptotics of the correlation functions of the
characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$.
We show that for the correlation function of any even order the asymptotic
coincides with this for the GUE up to a factor, depending only on the forth
moment of the common probability law $Q$ of entries $\Im W_{jk}$, $\Re W_{jk}$,
i.e. that the higher moments of $Q$ do not contribute to the above limit.Comment: 20

### Zeta Function Zeros, Powers of Primes, and Quantum Chaos

We present a numerical study of Riemann's formula for the oscillating part of
the density of the primes and their powers. The formula is comprised of an
infinite series of oscillatory terms, one for each zero of the zeta function on
the critical line and was derived by Riemann in his paper on primes assuming
the Riemann hypothesis. We show that high resolution spectral lines can be
generated by the truncated series at all powers of primes and demonstrate
explicitly that the relative line intensities are correct. We then derive a
Gaussian sum rule for Riemann's formula. This is used to analyze the numerical
convergence of the truncated series. The connections to quantum chaos and
semiclassical physics are discussed

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