Anomalous pumping in the non-Hermitian Rice-Mele model

We study topological charge pumping (TCP) in the Rice-Mele (RM) model with irreciprocal hopping. The non-Hermiticity gives rise to interesting pumping physics, owing to the presence of skin effect and exceptional points. In the static one-dimensional (1D) RM model, we find two independent tuning knobs that can drive the topological transition, namely, non-Hermitian parameter $\gamma$ and system size $N$. To elucidate the system-size dependency, we use a finite-size generalized Brillouin zone (GBZ) scheme to show that the edge modes can be distinguished from the non-hermiticity induced skin modes. Moreover, this scheme can capture the state pumping of topological edge modes as a function of $\gamma$ in the static 1D RM model and it further provides insight into engineering novel gapless exceptional edge modes with the help of adiabatic drive. Furthermore, we show that the standard topological pumping due to the adiabatic and periodic variation of the model parameters survives even with finite $\gamma$. However, we observe that it depends upon the driving protocols and strength of the non-Hermiticity ($\gamma$). With increasing $\gamma$, the adiabatic pumping for non-trivial protocols is destroyed first and then re-emerges as an anomalous pumping which does not have any Hermitian counterpart. Additionally, we observe that even a trivial adiabatic protocol can give rise to pumping as opposed to the Hermitian system. This is attributed to the inherent point gap physics of non-Hermitian system which we explain by reformulating a non-Bloch topological invariant for the 1+1D RM model. This invariant explains the number of pumped charges (in each period) for all the driving protocols.