Linear level repulsions near exceptional points of non-Hermitian systems

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.

Figure 1

The nearest-neighbor level-spacing distributions of small random matrices in the symmetry-preserved [labeled as $P_{\text{SP}}\left(s\right)$] and symmetry-broken [labeled as $P_{\text{SB}}\left(s\right)$] phases for the $K$-symmetric classes of ${\mathrm{\Theta }}_k^2=I$ [(a), (d)] and ${\mathrm{\Theta }}_k^2=-I$ [(b), (e)] and the $Q$-symmetric class [(c), (f)]. Open and solid circles are numerical data obtained by diagonalizing Eqs. (1), (6), and (8) for $\sigma =1$ and ${10}^{10}$ random ensembles, and solid lines are Eqs. (5), (7), and (9). For comparisons, $P\left(s\right)$ of the Ginibre unitary distributions of $2\times 2$ matrices are also plotted (blue lines) [21].

Figure 2

(a) Eigenvalues of the real-space Hamiltonian $H_{\text{2D}}$ of Eq. (10) with disorders in the complex-energy plane for $L=20$. (b) $P\left(s\right)$ of $H_{\text{2D}}$ of $L=200$ in two energy windows $[{\epsilon }_0-\mathrm{\Delta }\epsilon ,{\epsilon }_0+\mathrm{\Delta }\epsilon ]$ with ${\epsilon }_0=3.5$ (triangles) and 3.99 (circles) and $\mathrm{\Delta }\epsilon \sim {10}^{-2}$. The red line in (b) is the Wigner surmise for the Gaussian unitary ensemble. The black lines in (b) and (d) are ${\tilde{P}}_{\text{ep}}\left(s\right)={\tilde{c}}_{1}sexp[-{\tilde{c}}_2{s}^{\tilde{\alpha }}]$ with $\tilde{\alpha }=3$. (c), (d) $P_{\text{SP}}\left(s\right)$ and $P_{\text{SB}}\left(s\right)$ of $H_{\text{2D}}$ for (c) $L=20$ and (d) $L=200$. Other model parameters are $v_0=4, \alpha =0.2, \kappa =0.1, \sigma =0.1$. Each point in (b)–(d) is averaged over more than ${10}^4$ ensembles.

Figure 3

(a) $\tilde{\alpha }$ as a function of $L$ for $H_{\text{2D}}$ in the symmetry-preserved (the blue circles) and symmetry-broken (the red squares) phases. (b) $\tilde{\alpha }$ as a function of $ln\left[L\right]$ for $H_{\text{1D}}$. Here, ${\nu }_0=4, \alpha =0.2, \kappa =0.1, \sigma =0.1$. The black dashed lines in (a) and (b) locate $\tilde{\alpha }=3$ and 2, respectively.

Figure 4

(a) Eigenvalues of $H_{\text{3D}}$ of Hamiltonian Eq. (13) with disorders in the complex-energy plane for $\alpha =0.2, \kappa =0.1, \sigma =0.1, L=8$. (b) $P_{\text{SP}}\left(s\right)$ (the orange circles) and $P_{\text{SB}}\left(s\right)$ (the purple squares) of levels near the EP in (a). The solid lines are fitted by Eq. (11) with $\tilde{\alpha }=2.96\pm 0.08$ and $2.6\pm 0.2$ for $P_{\text{SP}}\left(s\right)$ and $P_{\text{SB}}\left(s\right)$, respectively. Insert: Zoom-in of the peaks of $P_{\text{SP}}\left(s\right)$ and $P_{\text{SB}}\left(s\right)$. (c) Same as (b) but for $L=12$. The black solid line is Eq. (11) of $\tilde{\alpha }=3$. (d) $\tilde{\alpha }$ as a function of $L$ for the symmetry-preserved and symmetry-broken phases. The black dashed line is $\tilde{\alpha }=3$. Each data is averaged over more than ${10}^4$ ensembles.