The scaling of conductance in the Anderson model of localization in one dimension is a two-parameter scaling

TL;DR

This paper derives an exact two-parameter scaling law for conductance in the one-dimensional Anderson model, showing deviations from the traditional one-parameter scaling by analyzing cumulants of the logarithm of conductance.

Contribution

It provides an exact calculation of the cumulants of ln conductance in the localized regime, revealing a two-parameter scaling behavior contrary to standard models.

Findings

Variance and mean of ln conductance vary linearly with length

The conductance distribution is log-normal

Standard one-parameter scaling is contradicted

Abstract

The cumulants of the logarithm of the conductance (lng) in the localized regime in the one-dimensional Anderson model are calculated exactly in the second Born approximation for weak disorder. Only the first two cumulants turn out to ne non-zero since the third and fourth cumulants vanish identically and the higher cumulants are of higher order in the disorder. The variance and the mean of lng vary linearly with length $L$ while their ratio is proportional to the inverse localization length. The resulting exact log-normal distribution of conductance thus corresponds to a special form of two-parameter scaling. This contradicts the standard one-parameter scaling in the random phase approximation.