Winding number statistics for chiral random matrices: Averaging ratios of determinants with parametric dependence

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. In particular, 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.

Correlation functions between singular values and eigenvalues

“…Both decompositions enjoy a multitude of applications, usually either only the eigenvalues or only the singular values. However, in some situations such as in Time Series Analysis of time-lagged matrices [15,43,45,47,53,55], in Quantum Chaos [16,27,48,49], in Quantum Chromodynamics [39,40] as well as topological statistics of Hamiltonians [14,28,29] both spectral quantities are useful. Born out of these motivations, we would like to address the question about the relation between the statistics of the eigenvalues and those of the singular values of a random matrix.…”