$\mathbb{Z}_2$ invariants of topological insulators as geometric obstructions

TL;DR

This paper studies the topological invariants of time-reversal symmetric topological insulators, showing how $bZ_2$ invariants serve as geometric obstructions to constructing symmetric Bloch frames in 2D and 3D.

Contribution

It establishes a geometric interpretation of $bZ_2$ invariants as obstructions and provides explicit algorithms for constructing symmetric Bloch frames in both two and three dimensions.

Findings

$bZ_2$ invariant in 2D matches Fu-Kane index

Four $bZ_2$ invariants in 3D relate to Fu-Kane-Mele indices

Constructive algorithms for Bloch frames when no topological obstruction exists

Abstract

We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to -1. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a $\mathbb{Z}_2$-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four $\mathbb{Z}_2$ invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones.