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\({\mathbb{Z}_{2}}\) Invariants of Topological Insulators as Geometric Obstructions

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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 \({-\mathbb{1}}\). 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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Authors and Affiliations

  1. Dipartimento di Matematica, “La Sapienza” Università di Roma, Piazzale Aldo Moro 2, 00185, Rome, Italy

    Domenico Fiorenza & Gianluca Panati

  2. SISSA, International School for Advanced Studies, Via Bonomea 265, 34136, Trieste, Italy

    Domenico Monaco

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  1. Domenico Fiorenza
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  2. Domenico Monaco
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  3. Gianluca Panati
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Correspondence to Gianluca Panati.

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Communicated by R. Seiringer

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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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