Inducing Shifted Edge Modes by the Spin-Dependent Time-Periodical Driving and the Corresponding Topological Phase Transitions

Time periodic driving can serve as synthetic gauge fields and plays a key role in simulating dynamical topological materials. The periodically driven systems, where different spins (or sublattices) are engaged in the different dynamical driving processes are investigated. It is demonstrated that spin-dependent time-periodical periodic driving can result in shifted topological edge modes with non-zero (nor +/- omega/2) quasi-energies. Such shifted topological edge modes are not only related to the spin imbalance at each instantaneous time, but also the details of the dynamical driving. Here, it is also illustrated that the spin-dependent time-periodical driving can be conceived as the time-spin coupling, and similar to the static spatial spin-orbit coupling, tuning time-spin coupling parameters can lead to topological phase transitions. Experimental simulations on the spin-dependent time-periodic driving are proposed by shaking optical super-lattices and Raman assisted tunneling.