A numerical study of wave-function and matrix-element statistics in the Anderson model of localization

TL;DR

This paper investigates the statistical properties of wave functions and matrix elements in the Anderson localization model, analyzing fluctuations under external flux and incipient localization, with comparisons to analytical theories.

Contribution

It provides a detailed numerical analysis of eigenfunction and matrix-element statistics in the Anderson model, highlighting non-trivial correlations and comparing results with theoretical predictions.

Findings

Eigenfunction statistics are non-trivial and do not factorize.

Fluctuations depend on external Aharonov-Bohm flux.

Results agree with non-linear sigma model and semiclassical theory.

Abstract

We have calculated wave functions and matrix elements of the dipole operator in the two- and three-dimensional Anderson model of localization and have studied their statistical properties in the limit of weak disorder. In particular, we have considered two cases. First, we have studied the fluctuations as an external Aharonov-Bohm flux is varied. Second, we have considered the influence of incipient localization. In both cases, the statistical properties of the eigenfunctions are non-trivial, in that the joint probability distribution function of eigenvalues and eigenvectors does no longer factorize. We report on detailed comparisons with analytical results, obtained within the non-linear sigma model and/or the semiclassical approach.