Field theory of non-Hermitian disordered systems

TL;DR

This paper develops a field-theoretical framework for non-Hermitian disordered systems, classifying universality classes and analyzing spectral properties and phase transitions, including unique one-dimensional Anderson transitions.

Contribution

It introduces a fermionic replica nonlinear sigma model approach for non-Hermitian systems, classifies all symmetry classes, and connects Anderson transitions with topological and symmetry considerations.

Findings

Classified all 38 symmetry classes of non-Hermitian systems.

Linked non-Hermitian Anderson transitions to Hermitized systems with chiral symmetry.

Analyzed the origin of the nonreciprocal Anderson transition in 1D models like Hatano-Nelson.

Abstract

The interplay between non-Hermiticity and disorder gives rise to unique universality classes of Anderson transitions. Here, we develop a field-theoretical description of non-Hermitian disordered systems based on fermionic replica nonlinear sigma models. We classify the target manifolds of the nonlinear sigma models across all the 38-fold symmetry classes of non-Hermitian systems and corroborate the correspondence of the universality classes of Anderson transitions between non-Hermitian systems and Hermitized systems with additional chiral symmetry. We apply the nonlinear sigma model framework to study the spectral properties of non-Hermitian random matrices with particle-hole symmetry. Furthermore, we demonstrate that the Anderson transition unique to nonreciprocal disordered systems in one dimension, including the Hatano-Nelson model, originates from the competition between the kinetic and topological terms in a one-dimensional nonlinear sigma model. We also discuss the critical phenomena of non-Hermitian disordered systems with symmetry and topology in higher dimensions.