arXiv subjects

E. Brezin

Publications and source records attributed to E. Brezin.

28 records · Page 2Linked to original sources

arXiv1999

Level statistics inside the vortex of a superconductor and symplectic random matrix theory in an external source

In the core of the vortex of a superconductor, energy levels appear inside the gap. We discuss here through a random matrix approach how these levels are broadened by impurities. It is first shown that the level statistics is governed by an ensemble consisting of a symplectic random potential added to a non-random matrix. A generalization of previous work on the unitary ensemble in the presence of an external source (which relied on the Itzykson-Zuber integral) is discussed for this symplectic case through the formalism introduced by Harish-Chandra and Duistermaat-Heckman. This leads to explicit formulae for the density of states and for the correlation functions, which describe the cross-over from the clean to the dirty limits.

cond-mat.mes-hall↗

arXiv1998

Smoothed Correlators For Symmetric Double-Well Matrix Models: Some Puzzles and Resolutions

Some puzzles which arise in matrix models with multiple cuts are presented. They are present in the smoothed eigenvalue correlators of these models. First a method is described to calculate smoothed eigenvalue correlators in random matrix models with eigenvalues distributed in a single-cut, previous known results are reproduced. The method is extended to symmetric two-cut random matrix models. The correlators are written in a form suitable for application to mesoscopic systems. Connections are made with the smooth correlators derived using the Orthogonal Polynomial (OP) method. A few interesting observations are made regarding even and odd density-density correlators and cross-over correlators in $Z_2$ symmetric random matrix models.

cond-mat↗

arXiv1998

Universal singularity at the closure of a gap in a random matrix theory

We consider a Hamiltonian $ H = H_0+ V $, in which $ H_0$ is a given non-random Hermitian matrix,and $V$ is an $N \times N$ Hermitian random matrix with a Gaussian probability distribution.We had shown before that Dyson's universality of the short-range correlations between energy levels holds at generic points of the spectrum independently of $H_{0}$. We consider here the case in which the spectrum of $H_{0}$ is such that there is a gap in the average density of eigenvalues of $H$ which is thus split into two pieces. When the spectrum of $H_{0}$ is tuned so that the gap closes, a new class of universality appears for the energy correlations in the vicinity of this singular point.

cond-mat.stat-mech↗

arXiv1998

Level Spacing of Random Matrices in an External Source

In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials, or on the Coulomb gas representation, are not available. Nevertheless the n-point correlation functions are still given in terms of the determinant of a kernel, known through an explicit integral representation. This kernel is no longer symmetric though and is not readily accessible to standard methods. In particular finding the level spacing probability is always a delicate problem in Fredholm theory, and we have to reconsider the problem within our model. We find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state. The problem is solved through coupled non-linear differential equations, which turn out to form a Hamiltonian system. As a result we find that the level spacing probability $p(s)$ behaves like $\exp[ - C s^{8\over{3}}]$ for large spacing $s$; this is consistent with the asymptotic behavior $\exp[ - C s^{2 β+ 2}]$, whenever the density of state behaves near the edge as $ρ(λ)\sim λ^β$.

cond-mat.stat-mech↗

arXiv1997

Non-Hermitean De-Localization: Multiple Scattering and Bounds

We study localization and delocalization in a class of non-hermitean Hamiltonians inspired by the problem of vortex pinning in superconductors. We show how to take into account multiple scattering. We also obtain some bounds on the complex energy spectrum.

cond-mat↗

arXiv1997

An Extension of Level-spacing Universality

Dyson's short-distance universality of the correlation functions implies the universality of P(s), the level-spacing distribution. We first briefly review how this property is understood for unitary invariant ensembles and consider next a Hamiltonian H = H_0+ V , in which H_0 is a given, non-random, N by N matrix, and V is an Hermitian random matrix with a Gaussian probability distribution. n-point correlation function may still be expressed as a determinant of an n by n matrix, whose elements are given by a kernel $K(λ,μ)$ as in the H_0=0 case. From this representation we can show that Dyson's short-distance universality still holds. We then conclude that P(s) is independent of H_0.

cond-mat.mes-hall↗

arXiv1996

Fancy and facts in the (d - 2) expansion of non-linear sigma models

We review the existing results on the scaling dimensions of operators with more than two derivatives in the non-linear sigma models. We argue that the speculations on the relevance of these operators, and correspondingly on the breakdown of the $(d-2)$ expansion for the classical Heisenberg model, or for the one-parameter scaling theory of localization, are based on a dubious mathematical analysis.

cond-mat.stat-mech↗

arXiv1994

Universal correlations for deterministic plus random Hamiltonians

We consider the (smoothed) average correlation between the density of energy levels of a disordered system, in which the Hamiltonian is equal to the sum of a deterministic H0 and of a random potential $φ$. Remarkably, this correlation function may be explicitly determined in the limit of large matrices, for any unperturbed H0 and for a class of probability distribution P$(φ)$ of the random potential. We find a compact representation of the correlation function. From this representation one obtains readily the short distance behavior, which has been conjectured in various contexts to be universal. Indeed we find that it is totally independent of both H0 and P($φ$).

hep-th↗

arXiv1993

Correlation Functions in Disordered Systems

{Recently, we found that the correlation between the eigenvalues of random hermitean matrices exhibits universal behavior. Here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous work to the case in which the random matrix evolves in time or varies as some external parameters vary. We compute the current-current correlation function, discuss various generalizations, and compare our work with the work of other authors. We study the distribution of eigenvalues of Hamiltonians consisting of a sum of a deterministic term and a random term. The correlation between the eigenvalues when the deterministic term is varied is calculated.}

cond-mat↗

arXiv1992

A naive matrix-model approach to two-dimensional quantum gravity coupled to matter of arbitrary central charge

In the usual matrix-model approach to random discretized two-dimensional manifolds, one introduces n Ising spins on each cell, i.e. a discrete version of 2D quantum gravity coupled to matter with a central charge n/2. The matrix-model consists then of an integral over $2^{n}$ matrices, which we are unable to solve for $n>1$. However for a fixed genus we can expand in the cosmological constant g for arbitrary values of n, and a simple minded analysis of the series yields for n=0,1 and 2 the expected results for the exponent $γ_{string}$ with an amazing precision given the small number of terms that we considered. We then proceed to larger values of n. Simple tests of universality are successfully applied; for instance we obtain the same exponents for n=3 or for one Ising model coupled to a one dimensional target space. The calculations are easily extended to states Potts models, through an integration over $q^{n}$ matrices. We see no sign of the tachyonic instability of the theory, but we have only considered genus zero at this stage.

hep-th↗