Winding Number Statistics for Chiral Random Matrices: Universal Correlations and Statistical Moments in the Unitary Case

TL;DR

This paper analyzes the statistical properties of the winding number in chiral unitary random matrices, revealing Gaussian distribution behavior and providing explicit formulas for moments and correlations.

Contribution

It provides a comprehensive analysis of winding number statistics in chiral unitary matrices, including Gaussian limits and explicit correlation functions.

Findings

Winding number distribution becomes Gaussian at large matrix size

Explicit formulas for statistical moments of the winding number

Derived k-point correlation functions for the winding number density

Abstract

The winding number is the topological invariant that classifies chiral symmetric Hamiltonians with one-dimensional parametric dependence. In this work we complete our study of the winding number statistics in a random matrix model belonging to the chiral unitary class AIII. We show that in the limit of large matrix dimensions the winding number distribution becomes Gaussian. Our results include expressions for the statistical moments of the winding number and for the k-point correlation function of the winding number density.