Vibrational excitations in systems with correlated disorder

TL;DR

This paper studies vibrational excitations in correlated disordered systems using analytical and numerical methods, revealing how disorder and correlation length influence the density of states and spectral features.

Contribution

It introduces an analytical effective-medium theory for correlated disorder and validates it with numerical simulations, highlighting the impact of correlation length on vibrational properties.

Findings

Disorder enhances the density of states over Debye's law (boson peak).

Correlation length amplifies the boson peak and affects spectral scaling.

Universal scaling relation between correlation length, wavenumber, and linewidth is observed.

Abstract

We investigate a $d$-dimensional model ($d$ = 2,3) for sound waves in a disordered environment, in which the local fluctuations of the elastic modulus are spatially correlated with a certain correlation length. The model is solved analytically by means of a field-theoretical effective-medium theory (self-consistent Born approximation) and numerically on a square lattice. As in the uncorrelated case the theory predicts an enhancement of the density of states over Debye's $\omega^{d-1}$ law (``boson peak'') as a result of disorder. This anomay becomes reinforced for increasing correlation length $\xi$. The theory predicts that $\xi$ times the width of the Brillouin line should be a universal function of $\xi$ times the wavenumber. Such a scaling is found in the 2d simulation data, so that they can be represented in a universal plot. In the low-wavenumber regime, where the lattice structure is irrelevant there is excellent agreement between the simulation at small disorder. At larger disorder the continuum theory deviates from the lattice simulation data. It is argued that this is due to an instability of the model with stronger disorder.