Two-Dimensional Topological Insulators

In the preceding chapter, we used simple intuitive arguments to

introduce the two-dimensional topological insulators (2DTIs) as the

2D systems with an insulating interior and conducting helical edge

states (see Section 1.3). Now we intend to construct a theoretical

model that would elucidate the topological origin of the helical edge

states. We have already mentioned the original theoretical works

on the 2DTIs in the context of graphene [1, 2] and semiconductor

quantum wells [3, 4]. These models are closely related to the notion

of Chern insulators introduced byHaldane [42]. Chern insulators are

nontrivial band insulators in which broken time-reversal symmetry

(TRS) induces a quantum Hall effect without any external magnetic

field. This topological state is characterized by a gapless edge mode

and the nonzero Thouless-Kohmoto-Nightingale-den Nijs (TKNN)

invariant. These ingredients are instrumental for understanding the

TR-invariant TIs as well. The chapter opens with a simple model

of the Chern insulator and the derivation of the TKNN topological

invariant. This will lead us to a TR-invariant theory of 2DTIs based

on a low-energy continuum model for the 2D Dirac matter. We

shall see that the helical edge states emerge from a local boundary

condition, that is topologically equivalent to a band-gap domain wall

describing the inversion of the band structure at the edge.