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 1D RM model, we observe two independent tuning knobs that drive the topological transition, viz., non-Hermitian parameter $\gamma$ and system size $N$. To elucidate the system-size dependency, we made use of the finite-size generalized Brillouin zone (GBZ) scheme. This scheme captures the state pumping of topological edge modes in the static 1D RM model and provides further insight into engineering novel gapless exceptional edge modes with the help of adiabatic drive. Finally, we apply three types of adiabatic protocols to study TCP in the 1+1D RM model. We further explain the number of pumped charges (in each period) using a non-Bloch topological invariant. This exactly explains the presence of different pumping phases in the non-Hermitian RM model as we tune the non-Hermitian parameter $\gamma$. We observe that in a non-Hermitian system, even a trivial adiabatic protocol can lead to pumping that has no Hermitian counterpart.