Correlated random band matrices: localization-delocalization transitions

TL;DR

This paper investigates how correlations in random band matrices influence eigenvector behavior, revealing a localization-delocalization transition akin to quantum Hall phenomena when certain parameters scale with matrix size.

Contribution

It introduces a correlated random band matrix model and demonstrates a localization-delocalization transition driven by the correlation parameter's scaling.

Findings

Localization-delocalization transition observed

Rich phenomenology due to correlations

Transition resembles quantum Hall effect

Abstract

We study the statistics of eigenvectors in correlated random band matrix models. These models are characterized by two parameters, the band width B(N) of a Hermitian N times N matrix and the correlation parameter C(N) describing correlations of matrix elements along diagonal lines. The correlated band matrices show a much richer phenomenology than models without correlation as soon as the correlation parameter scales sufficiently fast with matrix size. In particular, for B(N) and C(N) increasing like the square root of N the model shows a localization-delocalization transition of the quantum Hall type.