Winding Number Statistics of a Parametric Chiral Unitary Random Matrix Ensemble

TL;DR

This paper analyzes the statistical properties of the winding number, a topological index in chiral symmetric systems, using a parametric random matrix model to derive probability distributions and correlation functions.

Contribution

It introduces a random matrix model for chiral unitary systems with parametric dependence and analytically derives winding number distributions and correlations.

Findings

Derived the probability distribution of winding numbers.

Calculated parametric correlation functions of winding number density.

Proposed a universality conjecture for the two-point function.

Abstract

The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological quantity. To this end, we set up a random matrix model for a chiral unitary system with a parametric dependence. We analytically calculate the discrete probability distribution of the winding numbers, as well as the parametric correlations functions of the winding number density. Moreover, we address aspects of universality for the two-point function of the winding number density by identifying a proper unfolding procedure. We conjecture the unfolded two-point function to be universal.