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

TL;DR

The paper numerically studies wave-function and dipole-matrix-element statistics in the $d=2$ and $d=3$ Anderson model under weak disorder, focusing on regimes with $g \gg 1$ and the energy scale $E_D = g\Delta$. It analyzes two nontrivial cases: the GOE→GUE transition induced by an Aharonov-Bohm flux and incipient localization due to increasing disorder, comparing results to non-linear sigma-model and semiclassical predictions. The findings show good agreement for the GOE→GUE transition and reveal $g^{-1}$ corrections to wave-function statistics in $d=3$, with some discrepancies in the disorder-dependence of the correction amplitude. These results illuminate the limits of random-matrix theory in the localization crossover and guide future tail analyses and tests of eigenfunction statistics near localization.

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.

Figures (2)

• Figure 1: Velocities and matrix elements in the transition regime between GOE and GUE for the $27\times 27$ Anderson model with $W=1.7$ (symbols), compared to the corresponding semiclassical expressions (lines). The numerical values of $\epsilon$ are $0.158$ ($\bigtriangledown$), $0.224$ ($\times$), $0.316$ ($\lhd$), $0.447$ ($\circ$), $0.631$ ($\Box$), $0.891$ ($\Diamond$), $1.26$ ($\bigtriangleup$), $1.78$ ($+$), $2.51$ ($\ast$).
• Figure 2: Wave-function statistics in the $13\times 13 \times 13$ Anderson model. (a) GOE to GUE transition of the distribution function in the metallic regime. The Porter-Thomas distributions are shown as dashed lines. The predictions of fal94 are show as solid lines. (b) Deviations $\Delta f(t)$ from the Porter--Thomas distribution for the orthogonal case, and several values of disorder strength, $W=1,2,3,4$ and $5$. The results are averaged over the energy interval $[-1.7,-1.4]$. The dashed vertical lines denote the zeros of the first order correction term in Eq. (\ref{['eq:co']}). The solid lines are fits according to Eq. (\ref{['eq:co']}).