Level statistics of real eigenvalues in non-Hermitian systems

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. We show that these universal distributions of non-Hermitian random matrices describe the real-eigenvalue spacings of ergodic (metallic) physical models in the same symmetry classes, such as free fermionic systems and bosonic many-body systems with disorder and dissipation, whereas the spacings in the Anderson-localized and many-body localized phases show the Poisson statistics. We also find that the ergodic (metallic) and localized phases in these symmetry classes are characterized by the unique scaling relations between the number of real eigenvalues and the dimensions of their physical Hamiltonians. These results serve as effective tools for detecting quantum chaos, many-body localization, and real-complex transitions in non-Hermitian systems with symmetries.