One-dimensional Anderson Localization: Devil's staircase of Statistical Anomalies

TL;DR

This paper investigates the statistical anomalies in wavefunctions within the 1D Anderson localization model, revealing rational filling factors cause specific anomalies and providing an exact solution for the generating function at the principal anomaly.

Contribution

It introduces a detailed analysis of statistical anomalies at rational filling factors and derives an integrable transfer-matrix equation for the generating function in the 1D Anderson model.

Findings

Anomalies occur at rational filling factors f=p/q.

The principal anomaly at f=1/2 is characterized and solved.

A transfer-matrix equation of Fokker-Planck type is derived and shown to be integrable.

Abstract

The statistics of wavefunctions in the one-dimensional (1d) Anderson model of localization is considered. It is shown that at any energy that corresponds to a rational filling factor f=p/q there is a statistical anomaly which is seen in expansion of the generating function (GF) to the order (q-2) in the disorder parameter. We study in detail the principle anomaly at $f=1/2$ that appears in the leading order. The transfer-matrix equation of the Fokker-Planck type with a two-dimensional internal space is derived for GF. It is shown that the zero-mode variant of this equation is integrable and a solution for the generating function is found in the thermodynamic limit.