Statistics of transmission in one-dimensional disordered systems: universal characteristics of states in the fluctuation tails

TL;DR

This paper investigates the distribution of conductance in one-dimensional disordered systems, revealing how scaling properties depend on system length and introducing a new scaling approach for fluctuation states.

Contribution

It introduces a novel scaling framework for conductance distribution in fluctuation states, accounting for the relation between system length and the density-of-states-derived length.

Findings

Distribution depends on system length and density-of-states length

New scaling approach valid for long systems where L >> l_s

Variance of Lyapunov exponent deviates from central limit theorem predictions in intermediate regimes

Abstract

We numerically study the distribution function of the conductance (transmission) in the one-dimensional tight-binding Anderson and periodic-on-average superlattice models in the region of fluctuation states where single parameter scaling is not valid. We show that the scaling properties of the distribution function depend upon the relation between the system's length $L$ and the length $l_s$ determined by the integral density of states. For long enough systems, $L \gg l_s$, the distribution can still be described within a new scaling approach based upon the ratio of the localization length $l_{loc}$ and $l_s$. 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 this scaling becomes invalid.