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 \({-\mathbb{1}}\). 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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Fiorenza, D., Monaco, D. & Panati, G. \({\mathbb{Z}_{2}}\) Invariants of Topological Insulators as Geometric Obstructions. Commun. Math. Phys. 343, 1115–1157 (2016). https://doi.org/10.1007/s00220-015-2552-0
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DOI: https://doi.org/10.1007/s00220-015-2552-0