Localization and adiabatic pumping in a generalized Aubry-Andr\'e-Harper model

A generalization of the Aubry-Andr\'e-Harper (AAH) model is developed, containing a tunable phase shift between on-site and off-diagonal modulations. A localization transition can be induced by varying just this phase, keeping all other model parameters constant. The complete localization phase diagram is obtained. Unlike the original AAH model, the generalized model can exhibit a transition between topologically trivial bandstructures and topologically non-trivial bandstructures containing protected boundary states. These boundary states can be pumped across the system by adiabatic variations in the phase shift parameter. The model can also be used to demonstrate the phenomenon of adiabatic pumping breakdown due to localization.