Probing the Topology in Band Insulators

Topological Insulator is a newly found state of matter. Unlike phases described by the traditional Landau theory of symmetry breaking, the topological phases do not break symmetry, and it is not obvious in which measurable quantity will the topological index manifest itself. In this thesis, our main goal is to understand how topological classification produces measurable consequences in periodic insulators. We first warm up by investigating the charge conjugation invariant insulator in one spatial dimension. We show there are two topological distinct classes and derive an integral formula for the topological index that distinguishes between them. We then show that the topological index appear as a Berry's phase when one adiabatically turns on a electric field. We then study the effective theory induced by this Berry's phase and show that there are measurable consequences. We then generalize the discussion to three spatial dimensions. It is hard to capture the topological terms in the effective theory by conventional perturbation methods. We then introduce a new formalism to calculate properties produced by those topological terms such as the polarization and the magnetization, in a unified way. The formalism is based on a perturbative expansion of the Green's functions in powers of a uniform field strength, instead of the potential. In particular, this formalism allows us to capture the effective action describing the three dimensional topological insulator defined under time reversal symmetry, which previously can only be calculated via pumping. Finally, we discuss measurable consequences from the effective theory, in various different boundary settings. Among the properties we have calculated, we find we can identify part of them as of bulk nature, and some other part of them more as an effect associated with boundaries. For the part that are associated with boundaries, the Maxwell relation in the bulk can be violated. For example, the isotropic orbital magneto-polarizability and the orbital electric-susceptibility are different with periodic boundary conditions. However, they become identical whenever there is a boundary. Thesis Supervisor: Patrick A. Lee Title: Professor