Phys. Rev. B 106, L081118 (2022)

Linear level repulsions near exceptional points of non-Hermitian systems

C. Wang^1,* and X. R. Wang^2,3,†

1 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China

2 Physics Department, The Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowloon, Hong Kong

3 HKUST Shenzhen Research Institute, Shenzhen 518057, China

* Corresponding author: physcwang@tju.edu.cn

† Corresponding author: phxwan@ust.hk

Abstract

The nearest-neighbor level-spacing distributions are a fundamental quantity of disordered systems and are classified into different universality classes. They are the Wigner-Dyson and the Poisson functions for extended and localized states in Hermitian systems, respectively. The distributions follow the Ginibre functions for the non-Hermitian systems whose eigenvalues are complex and away from exceptional points (EPs). However, the level-spacing distributions of disordered non-Hermitian systems near EPs are still unknown, and a corresponding random matrix theory is absent. Here, we show another class of universal level-spacing distributions in the vicinity of EPs of non-Hermitian Hamiltonians. Two distribution functions, $P_{\text{SP}}\left(s\right)$ for the symmetry-preserved phase and $P_{\text{SB}}\left(s\right)$ for the symmetry-broken phase, are needed to describe the nearest-neighbor level-spacing distributions near EPs. Surprisingly, both $P_{\text{SP}}\left(s\right)$ and $P_{\text{SB}}\left(s\right)$ are proportional to $s$ for small $s$, or linear level repulsions, in contrast to cubic level repulsions of the Ginibre ensembles. For disordered non-Hermitian tight-binding Hamiltonians, $P_{\text{SP}}\left(s\right)$ and $P_{\text{SB}}\left(s\right)$ can be well described by a surmise ${\tilde{P}}_{\text{ep}}\left(s\right)={\tilde{c}}_{1}sexp[-{\tilde{c}}_2{s}^{\tilde{\alpha }}]$ in the thermodynamic limit (infinite systems) with a constant $\tilde{\alpha }$ that depends on the localization nature of states at EPs rather than the dimensionality of non-Hermitian systems and the order of EPs.