Adiabatic charge transport in extended SSH models

We explore the topological properties of extended SSH models, considering four sub-lattices in a unit cell and second-nearest-neighbor intercell hopping for SSH4 and SSH long-range (SSHLR) models, respectively. The additional tuning parameters cause the SSH4 (SSHLR) model to host chiral symmetry protected two (two and four) zero-energy modes producing a richer phase diagram that we characterize by momentum space, periodic-bulk and open-bulk real space winding numbers. We introduce time to study charge transport in the periodically driven SSH4 and SSHLR models under the adiabatic limit. We remarkably find that the whole parameter space turned topological for a certain choice of the remaining parameters leading to always finite quantized value of pumped charge at the end of a complete cycle. Considering time as another variable, we characterize these new phases of the driven models by momentum space Chern number, periodic-bulk and open-bulk real space Bott index. We also investigate the time evolution of pumped charge for these models and connect it with the intriguing windings of the mid-gap energy levels with time. Interestingly, the maximum value of Chern number or Bott index for the driven models is more than that of the winding number associated with the static model indicating the fact that there exist more zero-energy modes during the full course of a driving cycle compared to the underlying static models. We further extend our study to the quantum metric where the fluctuations in the above quantity can identify the presence of a topological phase boundary.