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.