Study of the localization-delocalization transition for phonons via transfer matrix method techniques

TL;DR

This study investigates the transition between localized and delocalized phonon states in a disordered lattice using transfer matrix techniques, providing precise estimates of critical parameters.

Contribution

It introduces a transfer-matrix approach to analyze phonon localization in a mass-spring model with disorder, accurately determining the critical transition frequency and exponent.

Findings

Critical transition frequency: ω_c^2 = 12.54 ± 0.03

Critical exponent: ν = 1.55 ± 0.002

Localization properties characterized for disordered phonons

Abstract

We use a transfer-matrix method to study the localization properties of vibrations in a `mass and spring' model with simple cubic lattice structure. Disorder is applied as a box-distribution to the force-constants $k$ of the springs. We obtain the reduced localization lengths $\Lambda_M$ from calculated Lyapunov exponents for different system widths to roughly locate the squared critical transition frequency $\omega_{\text{c}}^2$. The data is finite-size scaled to acquire the squared critical transition frequency of $\omega_{\text{c}}^2 = 12.54 \pm 0.03$ and a critical exponent of $\nu = 1.55 \pm 0.002$.