Properties of stable ensembles of Euclidean random matrices

TL;DR

This paper investigates the vibrational spectrum of Euclidean random matrices modeling disordered harmonic oscillators, revealing sound wave damping, localized modes, and the applicability of Wigner's law, supported by numerical and theoretical analysis.

Contribution

It provides a detailed numerical and theoretical analysis of the vibrational properties of Euclidean random matrix ensembles in disordered systems.

Findings

Identification of a low-frequency sound wave regime with Rayleigh damping

Existence of localized vibrational modes at high frequencies

Validation of Wigner's semicircle law for the density of states in certain regimes

Abstract

We study the spectrum of a system of coupled disordered harmonic oscillators in the thermodynamic limit. This Euclidean random matrix ensemble has been suggested as model for the low-temperature vibrational properties of glass. Exact numerical diagonalization is performed in three and two spatial dimensions, which is accompanied by a detailed finite size analysis. It reveals a low-frequency regime of sound waves that are damped by Rayleigh scattering. At large frequencies localized modes exist. In between, the central peak in the vibrational density of states is well described by Wigner's semicircle law for not too large disorder, as is expected for simple random matrix systems. We compare our results with predictions from two recent self-consistent field theories.