Correlations of quantum curvature and variance of Chern numbers

TL;DR

This paper investigates the correlation functions of quantum curvature in complex systems, revealing universal behaviors and their relation to Chern number variance, with implications for understanding quantized Hall conductance.

Contribution

It introduces a universal correlation function for quantum curvature using a random matrix model and relates it to the variance of Chern numbers, supported by Monte-Carlo simulations.

Findings

Correlation function diverges as inverse of distance at small separations.

Correlation function of mixed states is finite but singular at small separations.

Universal scaling form supported by Monte-Carlo simulations.

Abstract

We analyse the correlation function of the quantum curvature in complex quantum systems, using a random matrix model to provide an exemplar of a universal correlation function. We show that the correlation function diverges as the inverse of the distance at small separations. We also define and analyse a correlation function of mixed states, showing that it is finite but singular at small separations. A scaling hypothesis on a universal form for both types of correlations is supported by Monte-Carlo simulations. We relate the correlation function of the curvature to the variance of Chern integers which can describe quantised Hall conductance.