Linear level repulsions near exceptional points of non-Hermitian systems

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, ${P}_{\text{SP}}(s)$ for the symmetry-preserved phase and ${P}_{\text{SB}}(s)$ for the symmetry-broken phase, are needed to describe the nearest-neighbor level-spacing distributions near EPs. Surprisingly, both ${P}_{\text{SP}}(s)$ and ${P}_{\text{SB}}(s)$ 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, ${P}_{\text{SP}}(s)$ and ${P}_{\text{SB}}(s)$ can be well described by a surmise ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{\text{ep}}(s)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{1}sexp[\ensuremath{-}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{2}{s}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}}]$ in the thermodynamic limit (infinite systems) with a constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}$ that depends on the localization nature of states at EPs rather than the dimensionality of non-Hermitian systems and the order of EPs.