Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

TL;DR

The paper investigates the eigenvalue statistics of a disordered, non-Hermitian SSH model with imaginary on-site potentials. By analyzing the spectral structure and symmetry properties, it shows that the spectrum can be entirely real in a parameter window without $\ ext{PT}$ symmetry, and that in this regime the spectrum maps to a Hermitian $\tilde{H}$ whose inherited symmetries yield GOE-level statistics for extended states. When imaginary eigenvalues are present, the DOS on the imaginary axis diverges at $E_I=\pm\gamma$ due to a Dyson singularity, while both real and imaginary DOS vanish linearly at the origin; the results connect non-Hermitian spectral features to the Dyson mechanism known from chiral Hermitian systems. Overall, the work clarifies how pseudo-Hermiticity and symmetry inheritance govern universal spectral properties in disordered non-Hermitian systems and provides analytically tractable links between non-Hermitian DOS and Dyson singularities, with potential relevance for experimental platforms such as waveguides and microwave resonators.

Abstract

We explore the eigenvalue statistics of a non-Hermitian version of the Su-Schrieffer-Heeger model, with imaginary on-site potentials and randomly distributed hopping terms. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of parity and time-reversal symmetry. As it turns out, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble. This demonstrates a general feature which we clarify that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we show that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems and derive analytically the asymptotes of the DOS which is different from that in Hermitian systems.

Figures (8)

• Figure 1: The non-Hermitian SSH model. One unit-cell, enclosed by a dashed square, contains two sublattices $A$ (red) and $B$ (blue). While $\gamma$ is independent of $x$, $t_1(x)$ and $t_2(x)$ have random position-dependent values (which are suppressed in the figure for brevity). There are $N$ unit-cells in the chain, and we have imposed periodic boundary conditions in the numerical calculations.
• Figure 2: Schematics that show the spectral change of the non-Hermitian SSH model in Eq. (\ref{['eq:Hamiltonian_AB']}) due to the increase of $\gamma$. Blue solid lines and green arrows represent eigenvalues and the direction to which eigenvalues shift with increasing $\gamma$, respectively. (a) The spectrum for the Hermitian case $\gamma=0$; series of eigenvalues on the real axis with a possible gap $\pm\Delta E_0$. (b) As we turn on $\gamma$, the gap around the origin is narrowed. (c) At the point $\Delta E_0=\gamma$, the gap closes. (d) The eigenvalues that reached the origin move onto the imaginary axis and away from the origin to up and down. (e) All eigenvalues are now on the imaginary axis. (f) A gap opens up on the imaginary axis.
• Figure 3: Eigenvalues of $H$ when $N=360$ and periodic boundary conditions are imposed, with (a) $\bar{t}_1=1.3,\ \bar{t}_2=1.0,\ w=0.35$, and (b) $\bar{t}_1=1.0,\ \bar{t}_2=1.0,\ w=0.7$. In the left column, the values of $\gamma$ are (a-1) $\gamma=0.0$, (a-2) $\gamma=0.1$, and (a-3) $\gamma=0.5$. In the right column, $\gamma$ is varied as (b-1) $\gamma=0.0$, (b-2) $\gamma=0.3$, and (b-3) $\gamma=2.4$.
• Figure 4: (a) The localization length $\xi$ for the Hamiltonian $H$ with the same parameters as in Fig. \ref{['fig:eigenvalue']} (a-2): $\gamma=0.1,\ \bar{t}_1=1.3,\ \bar{t}_2=1.0$, and $w=0.35$. The system size is $N=240$ and the number of ensembles is $50\,000$. The horizontal solid line represents $\xi_c=20=N/12$. We take the data in the range of $E$ between the two vertical broken lines, namely, $0.31\leq E \leq1.94$. (b) Examples of eigenstates $|\psi(x)|^2=|\psi_A(x)|^2+|\psi_B(x)|^2$. (b-1) An eigenstate with $E\simeq2.10$, which is regarded as a localized state. (b-2) An eigenstate with $E\simeq1.29$, which is regarded as an extended state.
• Figure 5: The level-spacing distribution $P(s)$ are plotted as green dots with $\gamma=0.1,\ \bar{t}_1=1.3,\ \bar{t}_2=1.0$, and $w=0.35$, corresponding to the parameters in Fig. \ref{['fig:eigenvalue']} (a-2). The system size is $N=240$ and the number of ensembles is $50000$. The red broken line indicates the level-spacing distribution of the GOE in Eq. (\ref{['eq:P(s)']}). In the inset, $P(s)$ near $s=0$ is depicted in a logarithmic scale, where the blue solid line indicates $P(s) \propto s$.