Euclidean random matrix theory: low-frequency non-analyticities and Rayleigh scattering

TL;DR

This paper rigorously calculates the vibrational spectrum of disordered systems, revealing a non-analytic low-frequency behavior that confirms Rayleigh scattering and long-time tails in diffusion, correcting previous theoretical claims.

Contribution

It provides a detailed high-density expansion analysis of Euclidean random matrix theory, establishing the correct low-frequency self-energy behavior and its physical implications.

Findings

Low-frequency self-energy scales as rac{k^2 z^{d/2}}

Confirms presence of Rayleigh scattering in disordered systems

Identifies long-time tails in velocity autocorrelation functions

Abstract

By calculating all terms of the high-density expansion of the euclidean random matrix theory (up to second-order in the inverse density) for the vibrational spectrum of a topologically disordered system we show that the low-frequency behavior of the self energy is given by $\Sigma(k,z)\propto k^2z^{d/2}$ and not $\Sigma(k,z)\propto k^2z^{(d-2)/2}$, as claimed previously. This implies the presence of Rayleigh scattering and long-time tails of the velocity autocorrelation function of the analogous diffusion problem of the form $Z(t)\propto t^{(d+2)/2}$.