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

TL;DR

The work extends the $Z_2$ topological invariant for time-reversal invariant ground states from two to three dimensions using band-touching arguments. It identifies four independent $Z_2$ invariants: three on the faces of the Brillouin-zone torus $T^3$ and a novel fourth tied to trapped monopole charges in momentum space, with stability enforced by TR symmetry. It connects these invariants to a proposed three-dimensional quantum spin Hall effect via spin Chern numbers defined under twisted boundary conditions, clarifying when a quantized spin response can arise. A key result is that a true 3D spin Hall quantization requires the fourth $Z_2$ invariant to vanish, while edge-state robustness remains governed by the bulk TR-protected indices.

Abstract

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.