A simple random matrix model for the vibrational spectrum of jammed packings

TL;DR

This paper introduces a simple random matrix model to analyze vibrational spectra in jammed packings, revealing regimes that match observed behaviors in structural glasses, including a universal $ ext{d}( extomega) extasciitilde extomega^4$ law.

Contribution

It presents a minimal random matrix model capturing key vibrational features of jammed systems and structural glasses, including the universal low-frequency behavior.

Findings

Identifies three spectral regimes: low-frequency, intermediate, and plateau.

Derives an analytical prediction for the low-frequency tail using extremal statistics.

Shows the intermediate regime exhibits a universal $ extomega^4$ law in certain cases.

Abstract

To better understand the surprising low-frequency vibrational modes in structural glasses, we study the spectra of a large ensemble of sparse random matrices where disorder is controlled by the distribution of bond weights and network coordination. We find $D(\omega)$ has three regimes: a very-low-frequency regime that can be predicted analytically using extremal statistics, an intermediate regime with quasi-localized modes, and a plateau with $D(\omega) \sim \omega^0$. In the special case of uniform bond weights, the intermediate regime displays $D(\omega) \sim \omega^4$, independent of network coordination and system size, just as recently discovered in simulations of structural glasses.