Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$

TL;DR

This paper develops a theoretical framework using non-linear sigma models to analyze spectral correlations in non-Hermitian random matrices with time-reversal symmetry, relevant for understanding chaotic open quantum systems.

Contribution

It introduces a fermionic replica non-linear sigma model approach to derive integral formulas for spectral moments in symmetry classes AI$^{\

Findings

Reproduces density of states for non-Hermitian matrices with TRS$^{\

Derives integral expressions for spectral moments in classes AI$^{\

Qualitatively matches known spectral correlations in these symmetry classes.

Abstract

Symmetry of non-Hermitian matrices underpins many physical phenomena. In particular, chaotic open quantum systems exhibit universal bulk spectral correlations classified on the basis of time-reversal symmetry$^{\dagger}$ (TRS$^{\dagger}$), coinciding with those of non-Hermitian random matrices in the same symmetry class. Here, we analytically study the spectral correlations of non-Hermitian random matrices in the presence of TRS$^{\dagger}$ with signs $+1$ and $-1$, corresponding to symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$, respectively. Using the fermionic replica non-linear sigma model approach, we derive $n$-fold integral expressions for the $n$th moment of the one-point and two-point characteristic polynomials. Performing the replica limit $n\to 0$, we qualitatively reproduce the density of states and level-level correlations of non-Hermitian random matrices with TRS$^{\dagger}$.