Conductance distribution near the Anderson transition

TL;DR

This paper introduces a two-parameter family of conductance distributions near the Anderson transition, unifying metallic, localized, and critical regimes through differential equations and analyzing their asymptotic behaviors.

Contribution

It develops a novel theoretical framework using differential equations to describe conductance distributions across different phases near the Anderson transition.

Findings

Distribution exhibits log-normal and exponential asymptotics.

Universal properties depend on two parameters related to system size and correlation length.

Critical distribution in three dimensions is well described by the theory.

Abstract

Using a modification of the Shapiro approach, we introduce the two-parameter family of conductance distributions W(g), defined by simple differential equations, which are in the one-to-one correspondence with conductance distributions for quasi-one-dimensional systems of size L^{d-1}\times L_z, characterizing by parameters L/\xi and L_z/L (\xi is the correlation length, d is the dimension of space). This family contains the Gaussian and log-normal distributions, typical for the metallic and localized phases. For a certain choice of parameters, we reproduce the results for the cumulants of conductance in the space dimension d=2+\epsilon obtained in the framework of the \sigma-model approach. The universal property of distributions is existence of two asymptotic regimes, log-normal for small g and exponential for large g. In the metallic phase they refer to remote tails, in the critical region they determine practically all distribution, in the localized phase the former asymptotics forces out the latter. A singularity at g=1, discovered in numerical experiments, is admissible in the framework of their calculational scheme, but related with a deficient definition of conductance. Apart of this singularity, the critical distribution for d=3 is well described by the present theory. One-parameter scaling for the whole distribution takes place under condition, that two independent parameters characterizing this distribution are functions of the ratio L/\xi.