Two-eigenfunction correlation in a multifractal metal and insulator

TL;DR

This paper investigates eigenfunction correlations in disordered systems near the Anderson transition, revealing persistent multifractal features and discovering new phenomena like eigenfunction mutual avoiding and enhanced correlations in insulators.

Contribution

It identifies random matrix ensembles that capture multifractal eigenfunction correlations in both metallic and insulating phases near the Anderson transition.

Findings

Eigenfunction correlations show multifractality away from criticality.

Discovered eigenfunction mutual avoiding at large energy separations.

Found logarithmic enhancement of correlations at small energy separations.

Abstract

We consider the correlation of two single-particle probability densities $|\Psi_{E}({\bf r})|^{2}$ at coinciding points ${\bf r}$ as a function of the energy separation $\omega=|E-E'|$ for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the models in the parameter range where they are close but not exactly at the Anderson localization transition. We show that even far away from the critical point the eigenfunction correlation show the remnant of multifractality which is characteristic of the critical states. By a combination of the numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we were able to identify the Gaussian random matrix ensembles that describe the multifractal features in the metal and insulator phases. In particular those random matrix ensembles describe new phenomena of eigenfunction correlation we discovered from simulations on the Anderson model. These are the eigenfunction mutual avoiding at large energy separations and the logarithmic enhancement of eigenfunction correlations at small energy separations in the two-dimensional (2D) and the three-dimensional (3D) Anderson insulator. For both phenomena a simple and general physical picture is suggested.