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.

Figure 1

(a) A two-dimensional cylinder threaded by magnetic flux $\Phi$. When the cylinder has a circumference of a single lattice constant, $\Phi$ plays the role of the edge crystal momentum $k_x$ in band theory. (b) The time-reversal invariant fluxes $\Phi =0$ and $h∕2e$ correspond to edge time-reversal invariant momenta ${\Lambda }_1$ and ${\Lambda }_2$. ${\Lambda }_a$ are projections of pairs of the four bulk time-reversal momenta ${\Gamma }_{i=\left(a\mu \right)}$, which reside in the two-dimensional Brillouin zone indicated by the shaded region. (c) In three dimensions, the generalized cylinder can be visualized as a Corbino donut, with two fluxes, which correspond to the two components of the surface crystal momentum. (d) The four time-reversal invariant fluxes ${\Phi }_1$, ${\Phi }_2=0,h∕2e$ correspond to the four two-dimensional surface momenta ${\Lambda }_a$. These are projections of pairs of the eight ${\Gamma }_{i=\left(a\mu \right)}$ that reside in the bulk 3D Brillouin zone.

Figure 2

Schematic representations of the surface energy levels of a crystal in either two or three dimensions as a function of surface crystal momentum on a path connecting ${\Lambda }_a$ and ${\Lambda }_b$. The shaded region shows the bulk continuum states, and the lines show discrete surface (or edge) bands localized near one of the surfaces. The Kramers degenerate surface states at ${\Lambda }_a$ and ${\Lambda }_b$ can be connected to each other in two possible ways, shown in (a) and (b), which reflect the change in time-reversal polarization ${\pi }_a{\pi }_b$ of the cylinder between those points. Case (a) occurs in topological insulators and guarantees that the surface bands cross any Fermi energy inside the bulk gap.

Figure 3

Diagrams depicting four different phases indexed by ${\nu }_0$; $\left({\nu }_1{\nu }_2{\nu }_3\right)$. The top panel depicts the signs of ${\delta }_i$ at the points ${\Gamma }_i$ on the vertices of a cube. The bottom panel characterizes the band structure of a 001 surface for each phase. The solid and open circles depict the time-reversal polarization ${\pi }_a$ at the surface momenta ${\Lambda }_a$, which are projections of pairs of ${\Gamma }_i$ that differ only in their $z$ component. The thick lines indicate possible Fermi arcs that enclose specific ${\Lambda }_a$.

Figure 4

(a) Honeycomb lattice of graphene, with a unit cell indicated by the dashed lines. (b) Brillouin zone with the values of ${\delta }_i$ associated with the time-reversal invariant momenta labeled. ${\tau }_{1∕2}$ describes the loop enclosing half the zone used in Eq. (4.13).

Figure 5

Schematic representation of band energy evolution of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ as a function of $x$. Adapted from Ref. 43.

Figure 6

(a) Band structure of $\alpha \text{−}\mathrm{Sn}$ near the $\Gamma$ point, which describes zero gap semiconductor due to the inverted ${\Gamma }_8^{+}$ and ${\Gamma }_7^{-}$ bands. (b) In the presence of uniaxial strain, the degeneracy at $\Gamma$ is lifted, opening a gap in the spectrum. The parity eigenvalues remain unchanged.

Figure 7

Schematic uniaxial strain-composition phase diagram for ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$. Away from the inversion transition at $x\sim 0.4$, the material is a conventional insulator (I). Near the transition, it is a strong topological insulator (STI).