Study Of Topological Phases And Topological Quantum Phase Transitions In Topological Insulators

Topological insulators (TIs) are novel states of matter, which have a bulk gap like an ordinary insulator but have gapless edge states or surface states on their edge or surface due to the topologically nontrivial band structures. The topological invariants are introduced to characterize such quantum states of matter and distinguish various topologically ordered states. Moreover, the topological quantum phase transition is ac-companied with a change of topological numbers, which can be used to describe the topological quantum phase transitions between topologically distinct phases. Topolog-ical insulators have attracted extensive attentions due to their unique physical proper-ties, and show great application prospect in spintronics and quantum computation. In recent years, remarkable progress has been made both theoretically and experimentally in topological insulators. However, there is still much room for further improvement in exploring more novel topologically quantum phenomenon. Therefore, it is highly significant for us to study the topological phases of the topological insulators and the related topological quantum phase transitions. The dissertation consists of four chap-ters:In chapter one, we introduce the theory and experimental study of the two-dimensional (2D) topological systems, which include the integer quantum Hall effect (IQHE), anomalous quantum Hall effect (QAHE) in the Haldane model and quantum spin Hall effect (QSHE) in the Kane-Mele model and HgTe/CdTe quantum well. Then we discuss the topological invariants characterizing these topological systems. Finally, we focus on the relationships between edge stats and topological invariants.In chapter two, we investigate the topological properties of a three-dimensional (3D) TI thin film in the presence of an exchange field, and explore the general relation between spin Chern numbers and edge state properties. The exchange field can break the time-reversal symmetry (TRS). Using the spin Chern number to characterize the topological quantum phase, we obtain a phase diagram consisting of a QSH phase (C±=±1), and a trivial insulator phase (C±=0). Very particularly, unlike the traditional phase transition, there is a topological quantum phase transition between the two phases, accompanied with spectral gap closing rather than the bulk gap. In the limit of g=0, the system is TR invariant, and a Z2index may also be defined, therefore we call it Z2phase. In order to show behavior of edge states, we construct a tight-binding model on a square lattice for the3DTI thin film. We consider a junction consisting of two strips of the3DTI thin films in different phases, then analyze the properties of the edge states at the boundary between the different phases. The QSH phases with TRS (Z2phase) and without TRS are shown to be topologically equivalent. Moreover, the appearance and disappearance of edge states is dependent on the band topology rather than any symmetry. It is provided with some reference value to further research of the properties of the edge states or surface states at the interface between two different samples of topological materials.In chapter three, we carefully study the topological quantum phase transitions in the Kane-Mele model with an exchange field and staggered magnetic fluxes (SMFs). Meanwhile, we introduce the Chern number and the spin Chern number to classify the topological properties of the system. Topological phases with high Chern number or spin Chern number are obtained in the presence of the staggered magnetic fluxes. Very interestingly, there are more rich features when SMFs are introduced into the QSH systems. By tuning the exchange field, the system may become a hybrid topological in-sulator, where the QSH and QAH phases coexist. We further analyze the properties of the edge-state spectrum for different parameter sets, and demonstrate that the topolog-ical characterization based on the Chern number and spin Chern number fully agrees with the characteristic spectrum of the edge states in different regimes. The last chapter presents a summary of this dissertation, and then gives some outlook for the investigation.