Anomalies and non-log-normal tails in one-dimensional localization with power-law disorder

TL;DR

This paper investigates wave function statistics in a one-dimensional Anderson localization model with power-law disorder, revealing anomalies, power-law tails, and non-universal scaling behaviors that challenge traditional assumptions.

Contribution

It demonstrates the existence of anomalies and power-law tails in wave function and conductance distributions in the Lloyd model with Cauchy disorder, highlighting deviations from log-normal behavior.

Findings

Presence of anomalies at rational wavelength ratios

Distribution functions have power-law tails

Lack of universal single-parameter scaling

Abstract

Within a general framework, we discuss the wave function statistics in the Lloyd model of Anderson localization on a one-dimensional lattice with a Cauchy distribution for the random on-site potential. We demonstrate that already in leading order in the disorder strength, there exists a hierarchy of anomalies in the probability distributions of the wave function, the conductance, and the local density of states, for every energy which corresponds to a rational ratio of wave length to the lattice constant. We also show that these distribution functions do have power-law rather then log-normal tails and do not display universal single-parameter scaling. These peculiarities persist in any model with power-law tails of the disorder distribution function.