Statistics of quantum transmission in one dimension with broad disorder

TL;DR

This paper analyzes the statistical behavior of quantum transmission in one-dimensional disordered systems with broad or narrow disorder distributions, revealing different localization phases and non-self-averaging properties of conductance.

Contribution

It introduces a comprehensive statistical framework for quantum transmission in disordered systems with broad disorder distributions, identifying multiple localization phases and non-self-averaging conductance behavior.

Findings

Identification of four distinct phases based on disorder distribution parameters.

Demonstration of non-self-averaging behavior of conductance in three phases.

Classification of localization regimes: underlocalization, superlocalization, and fluctuating localization.

Abstract

We study the statistics of quantum transmission through a one-dimensional disordered system modelled by a sequence of independent scattering units. Each unit is characterized by its length and by its action, which is proportional to the logarithm of the transmission probability through this unit. Unit actions and lengths are independent random variables, with a common distribution that is either narrow or broad. This investigation is motivated by results on disordered systems with non-stationary random potentials whose fluctuations grow with distance. In the statistical ensemble at fixed total sample length four phases can be distinguished, according to the values of the indices characterizing the distribution of the unit actions and lengths. The sample action, which is proportional to the logarithm of the conductance across the sample, is found to obey a fluctuating scaling law, and therefore to be non-self-averaging, in three of the four phases. According to the values of the two above mentioned indices, the sample action may typically grow less rapidly than linearly with the sample length (underlocalization), more rapidly than linearly (superlocalization), or linearly but with non-trivial sample-to-sample fluctuations (fluctuating localization).