Topological phases and pumps in the Su-Schrieffer-Heeger model periodically modulated in time

By the Floquet theory, we transform the Su-Schrieffer-Heeger model with the periodically modulated nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions into an effective 2D model, which holds the total Chern number of  ±1 modulated by the parameter θ. Under a staggered electric potential, the topological phase diagrams of the effective 2D model are reshaped and similar to the well-known Haldane model. While under a staggered Zeeman field, the topological phase diagram has the same shape as the former case, but with different Chern numbers, such as the spin and valley Chern numbers. With the combination of the staggered Zeeman field and the electric field, the effective 2D model holds even richer topological phases. At last, we find some types of topological pump, which can generate the time-averaged current without any bias voltage. The current depends on their different Chern numbers. In other words, we can modulate the parameters to obtain various Chern numbers to control the topological pump.