One Dimensional Topological Insulators And Toplogical Superconductors Through Periodic Driving

A topological insulator is a mixture of ordinary insulator and low dimensional metal. Like an ordinary insulator, it has a bulk energy gap separating the highest oc-cupied electronic band from the lowest empty band. The surface in (or the edge in two and one dimensional) of a topological insulator, however, necessarily has gapless states that are protected by symmetry. In addition to their fundamental interest, these states are predicted to have special properties that could be useful for applications ranging form semiconductor spintronics to topological quantum computation. Despite recent advancements in the field, our ability to control topological transitions remains limited, and usually requires changing material or structural properties. Floquet topological insulators in the presence of time-periodic driving have attracted much attention in re-cent years. Using Floquet theory, topological states can be induced in materials which are initially in the trivial phase. This can be achieved by irradiation with light or mi-crowave. Quite often, the phase diagram in the Floquet system is remarkably richer than its stationary counterpart.In this thesis, we studied three one dimensional Floquet topological insulator mod-els both analytically and numerically.Firstly, we explored the topological phase transition in kicked 1D optical super-lattices. We consider a 1D polarized Fermi gas (spinless) loaded in a bichromatic opti-cal lattice with the commensurate potential turning on in a stroboscopic sense periodi-cally. Providing that the qusi-energy bands are defined on a 2-torus spanned by lattice momentum and a periodic phase, the band can be characterized by topological Chern numbers. However, Chern number can not uniquely determine the topological phase since phases with the same Chern number may have different edge states when subject to open boundary condition, including anomalous edge states in gap around π/T and counter-propagating edge states. Those edge states are closely related to the emergence of multiple Dirac cones around critical points. Contrary to its stationary counterpart, changing of time periodic or modulation amplitude can induce topological phase tran-sition. This make this model a prototype to study the topological phase transition.Secondly, We investigate the temporal evolution of the Floquet Majorana zero and π modes in a periodic driving spinless p-wave superconducting chain following a sud-den or slow change of its driving period. Starting from one of the topological phases that have Majorana edge modes, the system is suddenly driven to the other topological phase or the trivial phases by crossing the quantum critical points (QCPs) separating these phases. When quenching to the QCP separating topological and trivial phase or weak topological phase, the FMF propagating between two ends of the chain. But when quenching to topological phase with different number of FMFs, there is no such oscil-lating behavior. When quenching to the weak topological phase, the FMF is oscillating at one edge of the chain. The quenching dynamics is strongly path-dependent, its be-havior reflects the evolution of the edge state under topological phase transition. When the zero (or π) edge mode is largely survived as the edge state with zero quasi-energy in the new phase, the original Majorona edge state is stable, otherwise it de-coheres. When the zero (or π) edge mode is mainly survived as the edge state with non-zero (π) quasi-energy in the new phase, the MSP oscillating with a period proportional to its quasi-energy. The different behaviors of the MSP can be used as a probe for topological phase transition in period driving systems.Thirdly, we study the effects of electron-electron interaction in a Floquet topo-logical superconducting chain. There are two main effects of the interaction:the sup-pression of the Majorana edge states and the induction of quantum chaos. The chaotic and topological phenomena have opposite characteristics, the former being sensitive to disturbances and the latter exhibiting robustness. In the intermediate-frequency re-gion, the analytic procedures based on high-frequency and low-frequency expansions are invalid. We apply the exact diagonalization approach and the dynamics simula-tion of quenching process to investigate the topological properties of a finite chain, while the chaos degree is characterized by the level statistics. It is found that a weak interaction can lead to a multitude of tiny avoided crossings in quasi-energy spectra, indicating the emergence of chaos, and that the Floquet topological superconducting state can be robust against the weak chaos. With increasing the interaction strength, there is a crossover from weak to strong chaos. On the other hand, the survival proba-bility of Majorana edge modes vanishes and so the Floquet system is transformed into a topologically trivial phase.