Anomalous Behavior of the Non-Hermitian Topological System with an Asymmetric Coupling Impurity

A notable feature of systems with non-Hermitian skin effects is the sensitivity to boundary conditions. In this work, we introduce one type of boundary condition provided by a coupling impurity. We consider a system where a two-level system as an impurity couples to a nonreciprocal Su-Schrieffer-Heeger chain under periodic boundary conditions at two points with asymmetric couplings. We first study the spectrum of the system and find that asymmetric couplings lead to topological phase transitions. Meanwhile, a striking feature is that the coupling impurity can act as an effective boundary, and asymmetric couplings can also induce a flexibly adjusted zero mode. It is localized at one of the two effective boundaries or both of them by tuning coupling strengths. Moreover, we uncover three types of localization behaviors of eigenstates for this non-Hermitian impurity system with on-site disorder. These results corroborate the potential for control of a class of non-Hermitian systems with coupling impurities.