Self-consistent theory for sound propagation in a simple model of a disordered, harmonic solid

TL;DR

This paper develops a self-consistent theoretical model to understand sound propagation in disordered harmonic solids, capturing the unjamming transition through the vanishing of sound speed.

Contribution

It introduces a novel self-consistent approach combining projection operators and approximations for Euclidean random matrices in disordered solids.

Findings

Speed of sound remains non-negative in the model.

Unjamming transition corresponds to the speed of sound approaching zero.

The theory provides insights into vibrational properties near the transition.

Abstract

We present a self-consistent theory for sound propagation in a simple model of a disordered solid. The solid is modeled as a collection of randomly distributed particles connected by harmonic springs with strengths that depend on the interparticle distances, i.e the Euclidean random matrix model of Mezard et al. [Nucl. Phys. 559B, 689 (1999)]. The derivation of the theory combines two exact projection operator steps and a factorization approximation. Within our approach the square of the speed of sound is non-negative. The unjamming transition manifests itself through vanishing of the speed of sound.