Analytic and numerical toolkit for the Anderson model in one dimension

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.

Figures (1)

• Figure 1: Analytic predictions obtained by solving (\ref{['betaeq']}) and evaluating (\ref{['rhobeta']}), represented as black curves, superposed on top of the histograms generated by sampling 100 random matrices at $L=10000$ for each model specification: (a) for (\ref{['uniform']}) with $W=1.5$; (b) the 'cat' distribution for (\ref{['uniform']}) with $W=2.5$; (c) for (\ref{['Gaussian']}) with $W=0.5$; (d) for (\ref{['Gaussian']}) with $W=1.5$; (e) for (\ref{['asymm']}) with $W=0.1$: the extravagant shape of this curve, accurately captured by the analytic framework presented here, is understood from approaching a discrete distribution of on-site potentials at small values of $W$---the DoS is known to develop singularities for discrete distributions discr; (f) for (\ref{['gaussall']}).