Fate of pseudo mobility-edge and multiple states in non-Hermitian Wannier-Stark lattice

Abstract: The interaction between non-reciprocity and disorder-free localization has emerged as a fascinating open question. Here, we explore the effects of pseudo mobility edges (MEs) along with different types of eigenstates in a one-dimensional (1D) lattice subjected to a non-reciprocal finite-height Wannier-Stark ladder. Utilizing the transfer matrix method, we analytically investigate the pseudo mobility edges under non-reciprocity, which accurately describe the boundary between ergodic and non-ergodic states. The ergodic states, under nonreciprocity, form topological point gaps in the complex plane, with the corresponding eigenstates localized at the boundaries. The localization of mixed states induced by the skin effect and Wannier-Stark ladder is further amplified under non-reciprocity. Through similarity transformations, the fate of multiple eigenstates under non-reciprocal transitions can be captured. Finally, we use wave packet dynamics as a means to detect these emerging states. Our findings broaden the understanding of disorder-free localization in non-Hermitian systems.