Non-Hermiticity-induced size-dependent effects in Su-Schrieffer-Heeger chain topological states

We investigate a non-reciprocal Su-Schrieffer-Heeger (SSH) chain with intra-unit-cell and inter-unit-cell non-reciprocity and a staggered imaginary on-site potential. The system exhibits four distinct phases comprising the anti parity-time symmetric, broken parity-time symmetric, complex (C), and parity-time symmetric phases depending on the non-reciprocity of the couplings and the staggered imaginary on-site potential. Moreover, the topologically non-trivial edge states that emerge in certain regions of the parameter space exhibit intriguing size-dependent behaviors. Specifically, the eigenenergies of the topological modes switch between being complex, purely real, or purely imaginary as the system size varies, with topological zero modes occurring at some critical system size. Interestingly, varying the system size also modifies the competition between topological edge localization and the non-Hermitian skin effect, and may cause the edge localization to switch sides. We explain analytically the unexpectedly complex behavior of these non-Hermitian SSH chains.