Localization of disordered harmonic chain with long-range correlation

TL;DR

This paper studies how long-range correlations in disordered harmonic chains affect phonon localization, revealing that correlations induce positive Lyapunov exponents and alter phonon dynamics, with implications for wave propagation in disordered media.

Contribution

It extends previous work on electronic systems to phonons, demonstrating the impact of long-range correlations on phonon localization and dynamics in disordered harmonic chains.

Findings

Lyapunov exponent is positive for almost all frequencies except zero

Exponential decay of B-dependence for B > 2

Phonon wave packet spread transitions from disordered to periodic behavior as B increases

Abstract

In the previous paper [Yamada, Chaos, Solitons $\&$ Fractals, {\bf 109},99(2018)], we investigated localization properties of one-dimensional disordered electronic system with long-range correlation generated by modified Bernoulli (MB) map. In the present paper, we report localization properties of phonon in disordered harmonic chains generated by the MB map. Here we show that Lyapunov exponent becomes positive definite for almost all frequencies $\omega$ except $\omega=0$, and the $B-$dependence changes to exponential decrease for $B > 2 $, where $B$ is a correlation parameter of the MB map. The distribution of the Lyapunov exponent of the phonon amplitude has a slow convergence, different from that of uncorrelated disordered systems obeying a normal central-limit theorem. Moreover, we calculate the phonon dynamics in the MB chains. We show that the time-dependence of spread in the phonon amplitude and energy wave packet changes from that in the disordered chain to that in the periodic one, as the correlation parameter $B$ increases.