Density of states in random lattices with translational invariance

TL;DR

This paper introduces a random matrix model for vibrational excitations in disordered lattices with translational invariance, revealing conditions where phonons cannot propagate and the system exhibits jamming-like behavior.

Contribution

It presents a novel random matrix approach to model vibrational properties in disordered systems with translational invariance, connecting to jamming phenomena.

Findings

Phonons are suppressed at small frequencies under certain disorder conditions.

The density of states becomes constant at low frequencies, indicating a breakdown of elasticity.

Young's modulus approaches zero in the thermodynamic limit.

Abstract

We propose a random matrix approach to describe vibrational excitations in disordered systems. The dynamical matrix M is taken in the form M=AA^T where A is some real (not generally symmetric) random matrix. It guaranties that M is a positive definite matrix which is necessary for mechanical stability of the system. We built matrix A on a simple cubic lattice with translational invariance and interaction between nearest neighbors. We found that for certain type of disorder phonons cannot propagate through the lattice and the density of states g(w) is a constant at small w. The reason is a breakdown of affine assumptions and inapplicability of the elasticity theory. Young modulus goes to zero in the thermodynamic limit. It strongly reminds of the properties of a granular matter at the jamming transition point. Most of the vibrations are delocalized and similar to diffusons introduced by Allen, Feldman et al., Phil. Mag. B v.79, 1715 (1999).