Topological insulators with inversion symmetry

Topological insulators are materials with a bulk excitation gap generated by the spin-orbit interaction that are different from conventional insulators. This distinction is characterized by $Z_2$ topological invariants, which characterize the ground state. In two dimensions, there is a single $Z_2$ invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four $Z_2$ invariants that distinguish the ordinary insulator from “weak” and “strong” topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the two-dimensional quantum spin-Hall phase and the three-dimensional strong topological insulator, these states are robust and are insensitive to weak disorder and interactions. In this paper, we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the $Z_2$ invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wave functions at the time-reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials that are strong topological insulators, including the semiconducting alloy ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ as well as $\alpha \text{−}\mathrm{Sn}$ and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments.