Introduction

Abstract

Since the discovery of quantum Hall effect (von Klitzing et al., Phys. Rev. Lett. 45:494, 1980 [1]), researches on new physical phenomena originating from topological order (Thouless et al., Phys. Rev. Lett. 49:405, 1982 [3]; Wen, Adv. Phys. 44:405, 1995 [4]) have been intensively done in condensed matter physics. In these phenomena, interestingly, the boundary states are characterized by information of the bulk, and they are insensitive to details of the geometry of the system. The relation between such a boundary state and the topological characterization based on the bulk states is called bulk-boundary correspondence. When a topological order exists under certain symmetry, the bulk-boundary correspondence leads to existence of robust boundary states because to annihilate the boundary states, one needs to change the topological phase of the bulk. In this chapter we refer to the integer quantum Hall effect and topological insulators as an introduction to topological orders, and we will review them in Chap. 2 in detail.