Non-Hermitian Disordered Systems

TL;DR

This paper reviews the physics and mathematics of non-Hermitian disordered systems, emphasizing symmetry classifications, spectral statistics, chaos, and phase transitions, with applications across physics and complex systems.

Contribution

It provides a comprehensive overview of non-Hermitian disordered systems, highlighting the 38-fold symmetry classification and its implications for spectral properties and phase transitions.

Findings

Classification of non-Hermitian systems into 38 symmetry classes.

Analysis of spectral statistics and universality classes.

Discussion of Anderson transitions and chaos in non-Hermitian systems.

Abstract

Non-Hermitian disordered systems have emerged as a central arena in modern physics, with ramifications spanning condensed matter, quantum, statistical, and high energy contexts. The same principles also underlie phenomena beyond physics, such as network science, complex systems, and biophysics, where dissipation, nonreciprocity, and stochasticity are ubiquitous. Here, we review the physics and mathematics of non-Hermitian disordered systems, with particular emphasis on non-Hermitian random matrix theory. We begin by presenting the 38-fold symmetry classification of non-Hermitian systems, contrasting it with the 10-fold way for Hermitian systems. After introducing the classic Ginibre ensembles of non-Hermitian random matrices, we survey various diagnostics for complex-spectral statistics and distinct universality classes realized by symmetry. As a key application to physics, we discuss how non-Hermitian random matrix theory characterizes chaos and integrability in open quantum systems. We then turn to the criticality due to the interplay of disorder and non-Hermiticity, including Anderson transitions in the Hatano-Nelson model and its higher-dimensional extensions. We also discuss the effective field theory description of non-Hermitian disordered systems in terms of nonlinear sigma models.