Topological Quantization In Topological Insulators

Topological insulators are novel states of quantum matter, which have a bulk gap like an conventional insulators but have a metallic edge or surface states that are pro-tected topologically. Generally speaking, topological insulators can be divided into two class. One is the quantum Hall system which breaks time reversal symmetry, and another is newly discovered type which does not break reversal symmetry. The time-reversal invariant topological insulators have been theoretically predicted and ex-perimentally observed in a variety of systems, including HgTe/CdTe quantum wells, BixSb1-x alloys, and Bi2Se3crystals. In this doctoral dissertation, after a systematic introduction to the basic theory of topological insulators, we focus mainly on what I have finished, which include the following two topics:(1) We systematically study the topological magneto-optical rotation in topologi-cal insulators:The topological magnetoelectric effect (TME) is the fundamental quantization ef-fect for topological insulators in units of the fine structure constant a. The experimental observation of the TME, for instance by way of magneto-optical measurements, is a key goal in the field which is being actively pursued. Before this work, a topological quantization condition of the TME was given under orthogonal incidence of the the optical beam, in which the frequency of the light or the thickness of the topological insulator film must be tuned to some commensurate values. This fine tuning is difficult to realize experimentally.In order to solve this difficulty, we work out manifestly SL(2, Z) electric-magnetic duality covariant expression for the Kerr and Faraday angles at oblique incidence at a single surface between a trivial insulator and a semi-infinite topological insulator, as well as at a topological insulator thick film with two surfaces. In the limit of orthog-onal incidence, our results is in perfect agreement with the results in literature before. When light incident at a topological insulator thick film with a finite incidence an-gle, we give a generalized topological quantization condition by combining two sets of Faraday and Kerr angles in both sides which are all measured at reflectivity minima. The new topological quantization condition we obtained is independent of material de-tails, and can be easier to realize experimentally compared with the earlier proposal since the incidence angle can be continuously tuned. Moreover, the generalized topo- logical quantization condition can also be used to measure the total Hall conductance of the TI thick film as well as the fine structure constant a.(2) We systematically study the topological properties of the quasi-hole excita-tions in the fractional quantum spin Hall state:In the past few years, much of the work on time reversal invariant topological in-sulators are focused on non-interacting or weakly interacting systems, it is natural to consider the fate of this physics in the presence of strong interactions. Strongly inter-acting insulators can be divided into two classes:In the first class, insulators can be adiabatically deformed into(non-interacting) band insulators without closing the bulk gap. In the second class, they cannot, and those that cannot. In this thesis we fo-cus on the second case:strongly interacting, time reversal invariant electron systems whose ground state cannot be adiabatically connected to a band insulator. These sys-tems are typically fractionalized in the sense that they have quasi-particle excitations with fractional charge and fractional statistics. Levin and Stern has pointed that the sz conserving model is a fractional topological insulator(i.e. fractional quantum spin Hall insulators) if and only if σsH/e*is odd, where σsH is the spin-Hall conductance (in units of e/2π) and e*is the elementary charge (in units of e), but they don’t study the charge and anyonic statistics in this systems.We study the anyonic properties in fractional quantum spin Hall (FQSH) state based on sz conserving model. We consider there exist two types of fundamental quasi-hole excitation in the FQSH state and we calculate the fractional charge and anyonic statistics of them by two different methods. In the first method, we construct the full effective field theory with quasi-particle excitations of FQSH state and obtain a matrix version of the Hopf term, then we obtain the fractional charges and statistical angles of the quasi-hole excitations in the FQSH states. In the second method, we first get the charges of quasi-hole excitations in the FQSH states by generalized two components plasma analogy approach, and then we figure out the statistical angles of the quasi-hole excitations in the FQSH states by Berry phase technique. By the two different ways, we obtain the identical charge and statistical angle for each type of quasi-hole, as well as the identical mutual statistics between two different types of quasi-hole excitation, and we further point out the internal relations between the two methods.