Level statistics of real eigenvalues in non-Hermitian systems

TL;DR

This paper uncovers universal statistical patterns in the real eigenvalues of non-Hermitian systems influenced by symmetries, with implications for understanding quantum chaos and localization.

Contribution

It identifies universal level-spacing distributions for real eigenvalues in non-Hermitian matrices with specific symmetries and connects these to physical models and phases.

Findings

Universal level-spacing distributions depend on symmetry class.

Ergodic phases follow non-Hermitian random matrix statistics.

Localized phases exhibit Poisson statistics.

Abstract

Symmetries associated with complex conjugation and Hermitian conjugation, such as time-reversal symmetry and pseudo-Hermiticity, have great impact on eigenvalue spectra of non-Hermitian random matrices. Here, we show that time-reversal symmetry and pseudo-Hermiticity lead to universal level statistics of non-Hermitian random matrices on and around the real axis. From the extensive numerical calculations of large random matrices, we obtain the five universal level-spacing and level-spacing-ratio distributions of real eigenvalues, each of which is unique to the symmetry class. Furthermore, we analyse spacings of real eigenvalues in physical models, such as bosonic many-body systems and free fermionic systems with disorder and dissipation. We clarify that the level spacings in ergodic (metallic) phases are described by the universal distributions of non-Hermitian random matrices in the same symmetry classes, while the level spacings in many-body localized and Anderson localized phases show the Poisson statistics. We also find that the number of real eigenvalues shows distinct scalings in the ergodic and localized phases in these symmetry classes. These results serve as effective tools for detecting quantum chaos, many-body localization, and real-complex transitions in non-Hermitian systems with symmetries.