Single parameter scaling in one-dimensional localization revisited

TL;DR

This paper provides an exact calculation of the Lyapunov exponent variance in a 1D Anderson model with Cauchy-distributed disorder, introduces a new scaling parameter, and revises the criterion for single parameter scaling.

Contribution

It introduces a new significant scaling parameter and an exact analytical criterion for single parameter scaling in 1D localization, differing from traditional phase randomization conditions.

Findings

Exact variance calculation for Lyapunov exponent

Identification of a new scaling parameter

Revised criterion for single parameter scaling

Abstract

The variance of the Lyapunov exponent is calculated exactly in the one-dimensional Anderson model with random site energies distributed according to the Cauchy distribution. We find a new significant scaling parameter in the system, and derive an exact analytical criterion for single parameter scaling which differs from the commonly used condition of phase randomization. The results obtained are applied to the Kronig-Penney model with the potential in the form of periodically positioned $\delta$-functions with random strength.