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

TL;DR

This paper extends the study of $Z_2$ topological invariants to three-dimensional time reversal invariant systems, revealing a new invariant linked to momentum space monopoles and proposing a 3D quantum spin Hall effect.

Contribution

It introduces a novel fourth $Z_2$ invariant in three dimensions and explores its relation to momentum space monopoles and the 3D quantum spin Hall effect.

Findings

Discovery of a new $Z_2$ invariant in 3D systems.

Identification of a 'trapped monopole' in momentum space.

Potential realization of a three-dimensional quantum spin Hall effect.

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.