Abstract
We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for three-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study (of independent interest) of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three \(\mathbb {Z}_2\)-valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a “multi-step logarithm” that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.
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Communicated by Vieri Mastropietro.
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Cornean, H.D., Monaco, D. On the Construction of Wannier Functions in Topological Insulators: the 3D Case. Ann. Henri Poincaré 18, 3863–3902 (2017). https://doi.org/10.1007/s00023-017-0621-y
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DOI: https://doi.org/10.1007/s00023-017-0621-y