paper

Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry

arXiv:1601.02906 · doi:10.1007/s10440-014-9995-8

Abstract

We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The role of additional -symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same -symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.

Contribution to the proceedings of the conference "SPT2014 - Symmetry and Perturbation Theory", Cala Gonone, Italy (2014). Keywords: Periodic Schrödinger operators, composite Wannier functions, Bloch bundle, Bloch frames, time-reversal symmetry, space-reflection symmetry, invariants of topological insulators

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