Sparse random matrices and vibrational spectra of amorphous solids

TL;DR

This paper uses a random matrix approach to analyze vibrational spectra of amorphous solids, revealing delocalized vibrational modes that resemble diffusons and align with molecular dynamics simulations.

Contribution

It introduces a sparse random matrix model to describe vibrational properties of amorphous solids, connecting spectral statistics with physical vibrational modes.

Findings

Vibrational density follows the Wigner quarter circle law.

Eigenfrequency repulsion indicates delocalized vibrational modes.

Model results agree with molecular dynamics simulations.

Abstract

A random matrix approach is used to analyze the vibrational properties of amorphous solids. We investigated a dynamical matrix M=AA^T with non-negative eigenvalues. The matrix A is an arbitrary real NxN sparse random matrix with n independent non-zero elements in each row. The average values <A_{ij}>=0 and dispersion <A_{ij}^2>=V^2 for all non-zero elements. The density of vibrational states g(w) of the matrix M for N,n >> 1 is given by the Wigner quarter circle law with radius independent of N. We argue that for n^2 << N this model can be used to describe the interaction of atoms in amorphous solids. The level statistics of matrix M is well described by the Wigner surmise and corresponds to repulsion of eigenfrequencies. The participation ratio for the major part of vibrational modes in three dimensional system is about 0.2 - 0.3 and independent of N. Together with term repulsion it indicates clearly to the delocalization of vibrational excitations. We show that these vibrations spread in space by means of diffusion. In this respect they are similar to diffusons introduced by Allen, Feldman, et al., Phil. Mag. B 79, 1715 (1999) in amorphous silicon. Our results are in a qualitative and sometimes in a quantitative agreement with molecular dynamic simulations of real and model glasses.