Scaling in the one-dimensional Anderson localization problem in the region of fluctuation states

TL;DR

This paper investigates the distribution of conductivity in a one-dimensional Anderson model, revealing that while single parameter scaling fails in fluctuation states, a modified scaling approach based on two length scales can describe the distribution.

Contribution

The study introduces a new length scale related to the density of states and demonstrates a modified scaling approach for fluctuation states in Anderson localization.

Findings

Single parameter scaling is invalid in the fluctuation state region.

A new length scale $l_s$ is introduced related to the density of states.

Variance of the Lyapunov exponent can grow with system length $L$.

Abstract

We numerically study the distribution function of the conductivity (transmission) in the one-dimensional tight-binding Anderson model in the region of fluctuation states. We show that while single parameter scaling in this region is not valid, the distribution can still be described within a scaling approach based upon the ratio of two fundamental quantities, the localization length, $l_{loc}$, and a new length, $l_s$, related to the integral density of states. In an intermediate interval of the system's length $L$, $l_{loc}\ll L\ll l_s$, the variance of the Lyapunov exponent does not follow the predictions of the central limit theorem, and may even grow with $L$.