Energy correlations for a random matrix model of disordered bosons

TL;DR

This paper models the statistical properties of characteristic frequencies in disordered bosonic systems using a random matrix approach, revealing universal bulk correlations and unique low-frequency behavior.

Contribution

It introduces a random matrix model for disordered bosonic systems and analyzes their spectral correlations, highlighting universal and novel scaling behaviors.

Findings

Bulk correlations follow sine-kernel or GUE universality.

Low-frequency spectrum exhibits unusual scaling behavior.

Spectral correlations are described by a determinantal process with bi-orthogonal polynomial kernel.

Abstract

Linearizing the Heisenberg equations of motion around the ground state of an interacting quantum many-body system, one gets a time-evolution generator in the positive cone of a real symplectic Lie algebra. The presence of disorder in the physical system determines a probability measure with support on this cone. The present paper analyzes a discrete family of such measures of exponential type, and does so in an attempt to capture, by a simple random matrix model, some generic statistical features of the characteristic frequencies of disordered bosonic quasi-particle systems. The level correlation functions of the said measures are shown to be those of a determinantal process, and the kernel of the process is expressed as a sum of bi-orthogonal polynomials. While the correlations in the bulk scaling limit are in accord with sine-kernel or GUE universality, at the low-frequency end of the spectrum an unusual type of scaling behavior is found.