Statistical Properties of the Non-Hermitian SSH Model and Symmetry Inheritance owing to Real Spectra

We explore statistical properties of eigenvalues in Su-Schrieffer-Heeger model with imaginary on site potentials (non-Hermitian SSH model), whose hopping terms are randomly distributed spatially. We prove that, originating from a structure of the Hamiltonian, eigenvalues can be entirely real without Parity-Time symmetry [1] in a certain parameter region [2]. Also, we clarify that level statistics obey that of Gaussian orthogonal ensemble (GOE) when the Hamiltonian has entirely real spectra [2]. To this end, we show a general fact that a non-Hermitian Hamiltonian whose eigenvalues are real is mapped to a Hermitian Hamiltonian which shares the same symmetries with the original Hamiltonian [2]. When imaginary eigenvalues exist, we show that the density of states (DOS) becomes zero at the origin and diverges along the imaginary axis. We reveal that the divergence of DOS originates from Dyson singularity in chiral symmetric 1D Hermitian systems [2].

[1] C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243 (1998).
[2] K. Mochizuki, N. Hatano, J. Feinberg, and H. Obuse, Physical Review E 102, 012101 (2020).