Finite-time scaling at the Anderson transition for vibrations in solids

TL;DR

This paper investigates the Anderson localization transition in a 3D elastic medium modeled as a network of masses and springs, using finite-time scaling to estimate critical parameters and confirm universality class.

Contribution

It introduces a model with a dynamical matrix as a product of a sparse Gaussian matrix and its transpose, and applies finite-time scaling to analyze vibrational localization transition.

Findings

Critical exponent ν = 1.57 ± 0.02 consistent with 3D orthogonal universality class.

Finite-time scaling effectively estimates critical parameters of the localization transition.

Model reproduces Anderson transition behavior in vibrational systems.

Abstract

A model in which a three-dimensional elastic medium is represented by a network of identical masses connected by springs of random strengths and allowed to vibrate only along a selected axis of the reference frame, exhibits an Anderson localization transition. To study this transition, we assume that the dynamical matrix of the network is given by a product of a sparse random matrix with real, independent, Gaussian-distributed non-zero entries and its transpose. A finite-time scaling analysis of system's response to an initial excitation allows us to estimate the critical parameters of the localization transition. The critical exponent is found to be $\nu = 1.57 \pm 0.02$ in agreement with previous studies of Anderson transition belonging to the three-dimensional orthogonal universality class.