Analytic and numerical toolkit for the Anderson model in one dimension

TL;DR

This paper develops an analytic and numerical toolkit for the one-dimensional Anderson model, enabling precise calculation of the density-of-states and localization length for arbitrary potential distributions.

Contribution

It introduces a supersymmetry-based integral equation method to compute the density-of-states, bridging the gap between analytic and numerical approaches.

Findings

Analytic integral equation matches numerical sampling results

Provides explicit control over the density-of-states for arbitrary distributions

Enables analytic determination of localization length via Thouless formula

Abstract

The Anderson model in one dimension is a quantum particle on a discrete chain of sites with nearest-neighbor hopping and random on-site potentials. It is a progenitor of many further models of disordered systems, and it has spurred numerous developments in various branches of physics. The literature does not readily provide, however, practical analytic tools for computing the density-of-states of this model when the distribution of the on-site potentials is arbitrary. Here, supersymmetry-based techniques are employed to give an explicit linear integral equation whose solutions control the density-of-states. The output of this analytic procedure is in perfect agreement with numerical sampling. By Thouless formula, these results immediately provide analytic control over the localization length.