Universal Chern number statistics in random matrix fields

TL;DR

This paper studies the distribution and correlations of Chern numbers in a random matrix model for disordered systems, revealing universal behavior and challenging previous assumptions about their independence.

Contribution

It provides the first comprehensive analysis of Chern number correlations and their universality across different regimes in a random matrix ensemble.

Findings

Chern number statistics agree with previous curvature predictions at small correlation lengths

Gap Chern numbers exhibit weak but persistent correlations

Weighted sums of Chern numbers deviate from uncorrelated predictions

Abstract

We investigate the probability distribution of Chern numbers (quantum Hall effect integers) for a parametric version of the GUE random matrix ensemble, which is a model for a chaotic or disordered system. The numerically-calculated single-band Chern number statistics agree well with predictions based on an earlier study [O. Gat and M. Wilkinson, SciPost Phys., 10, 149, (2021)] of the statistics of the quantum adiabatic curvature, when the parametric correlation length is small. However, contrary to an earlier conjecture, we find that the gap Chern numbers are correlated, and that correlation is weak but slowly-decaying. Also, the statistics of weighted sums of Chern numbers for many bands differs markedly from predictions based upon the hypothesis that gap Chern numbers are uncorrelated. All our results are consistent with the universality hypothesis described in the earlier paper, including in the previously unstudied regime of large correlation length, where the Chern statistics is highly non-Gaussian.