Unconventional pumping in non-Hermitian Rice-Mele model

We study the Rice-Mele (RM) model in the presence of an asymmetric hopping-induced non-Hermitian parameter, $\gamma$. In particular, we examine the effect of non-Hermiticity on the topological boundary modes and topological pumping. For weak and moderate values of $\gamma$, the inherent topological edge modes remain localized at the boundaries, whereas the bulk modes are pumped to the boundary, leading to the typical non-Hermitian skin modes. Using generalized Brillouin zone (GBZ) scheme, we show that the non-Hermiticity-induced skin modes can be distinguished from the topological boundary modes. Upon increasing $\gamma$, the topological boundary mode localized at one edge is pumped to the other edge, leading to an unconventional state pumping. This is in contrast to the standard topological pumping where adiabatic evolution of the parameters of the RM model leads to the pumping. We further show that the usual topological pumping due to the adiabatic and periodic variation of the model parameters survives even with finite $\gamma$. However, it depends upon the driving protocols and strength of the non-Hermiticity. With increasing $\gamma$, the adiabatic pumping is destroyed first and then re-emerges as an unconventional pumping which does not have any Hermitian counterpart.