Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect

The $Z_2$ invariant for filled bands in the ground states of systems with time reversal invariance characterizes the number of stable pairs of edge states. Here we study the $Z_2$ invariant using band touching methods discussed in a recent previous work \cite{roy2006zcq} and extend the study to three dimensions. Band collisions preserve the $Z_2$ invariant both in two and three dimensions, but there are crucial differences in the two cases. In three dimensions,we find a novel fourth $Z_2$ invariant which is characterized by a “trapped monopole” in momentum space. If the monopole charge in half the Brillouin zone is odd, then atleast one of the monopoles cannot recombine with another monopole and vanish unlike the case when the monopole charge is even. We also point out the possibility of a three dimensional quantum spin Hall effect and discuss the connection of various topological invariants to such an effect.