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

TL;DR

This paper investigates the universal behavior of energy level correlations near a gap closing in the spectrum of a perturbed random matrix, revealing a new universality class at the singularity.

Contribution

It introduces a new universality class for energy correlations at the gap-closing point in a perturbed random matrix spectrum.

Findings

Universal correlations at the gap closure point.

New class of spectral universality identified.

Behavior independent of the deterministic part of the Hamiltonian.

Abstract

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.