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arXiv · 1408.1030

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

Domenico Fiorenza·Domenico Monaco·Gianluca Panati
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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.

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Domenico Fiorenza, Domenico Monaco, Gianluca Panati. 2016-01-12. $\mathbb{Z}_2$ invariants of topological insulators as geometric obstructions. https://doi.org/10.1007/s00220-015-2552-0

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