Topological States And Its Transport Properties Of Condensed Matter System Under Time Periodic Driving

Great attention has been paid to the topological phases and topological transport phenomena in condensed matter system under time periodic driving over the past few years. This thesis mainly contains two topics, the first topic is that we propose to use off-resonant circularly polarized light to realize topological quantum phase transition and spin polarized transport phenomena in a two dimensional topological insulator (TI). The second topic is that we theoretically find a topological quantum spin pumping effect in a two-dimensional TI electrically modulated in adiabatic cycles. The thesis is organized as follows.In chapter one, we give a brief introduction to topological insulators and an ex-tended concept named symmetry protected topological phases. In chapter two, we first introduce the topological band theory which developed from the study of topological insulators, the Berry curvature of a two-band model Hamiltonian is calculated. We then give a symmetry argument about the intrinsic spin-orbit coupling in Graphene lattice and derive the second quantization form of the Kane-Mele Hamiltonian. The theo-retical study and experiment realization of the Bernevig-Hughs-Zhang model is also introduced briefly. At the end of this chapter, we derive the Keyldsh Green's function and linear response theory by using the path integral formulism, we then derive the form of the charge conductivity tensor. We also prove the relation between Hall conductivity and the topological invariant at zero temperature.In chapter three, we first give a framework about the Floquet theory, the form of the effective Hamiltonian under off-resonant approximation is derived. Applying this theoretical framework, we study the Bernevig-Hughs-Zhang model under off-resonant circularly polarized light, we find the circularly polarized light effectively creates a Zeeman exchange field and a renormalized Dirac mass, which are tunable by varying the intensity of the light and drive the quantum phase transition. This provides an alternative method to realize the quantum anomalous Hall (QAH) effect, other than magnetic doping. Both the transverse and longitudinal Hall conductivities are studied, and the former is consistent with the topological phase transition, when the Fermi level lies in the band gap. A highly controllable spin-polarized longitudinal electrical current can be generated, when the Fermi level is in the conduction band, which may be useful for designing topological spintronics.In chapter four, we report the theoretical discovery of a bulk quantum pumping effect in a two-dimensional TI electrically modulated in adiabatic cycles. In each cycle, an amount of spin proportional to the sample width can be pumped into a nonmagnetic electrode, which is attributed to nonzero spin Chern numbers. Moreover, by using a half-metallic electrode, universal quantized charge pumping conductivities can be measured. This discovery paves the way for direct utilization of the robust topological properties of the TIs.The final section contains a summary and a forecast.