Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence

TL;DR

This paper extends the analysis of winding number statistics in chiral random matrices from a specific model to multiple symmetry classes, providing analytical ensemble averages for ratios of determinants with parametric dependence without superspace mapping.

Contribution

It introduces a method to compute winding number statistics for chiral ensembles AIII and CII, generalizing previous models and avoiding superspace mapping.

Findings

Analytical ensemble averages for ratios of determinants obtained.

Method applicable to multiple chiral symmetry classes.

Supersymmetric structures identified without superspace mapping.

Abstract

Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding numbers for a specific parametric model of the chiral Gaussian Unitary Ensemble to other chiral random matrix ensembles. Especially, we address the two chiral symmetry classes, unitary (AIII) and symplectic (CII), and we analytically compute ensemble averages for ratios of determinants with parametric dependence. To this end, we employ a technique that exhibits reminiscent supersymmetric structures while we never carry out any map to superspace.