Topological phases and the quantum spin Hall effect in three dimensions

We show the existence of topological phases of Bloch insulators with time-reversal symmetry in three dimensions. These phases are characterized by topological $Z_2$ invariants whose stability is studied using band-touching arguments. Unlike insulators which break time-reveral symmetry, some of these topological phases are intrinsically three dimensional. The number of invariants (four) needed to specify the phase of these insulators also differs from the time-reversal-breaking case. The relation between these phases and the quantum spin Hall effect in three dimensions is investigated.

Figure 1

A schematic figure showing the diabolical points in the spectrum of the associated Hamiltonian.

Figure 2

A representation of the Brillouin zone for a three-dimensional insulator. The two-dimensional planes whose $Z_2$ invariants specify the topological phase are shaded for (a) the case when there are no diabolical points, and (b) when there are diabolical points in the spectrum of the associated Hamiltonian.