caa a22 4500 606166432 CHVBK 20210128100658.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s10440-014-9995-8 doi (NATIONALLICENCE)springer-10.1007/s10440-014-9995-8 Symmetry and Localization in Periodic Crystals: Triviality of Bloch Bundles with a Fermionic Time-Reversal Symmetry [Elektronische Daten] [Domenico Monaco, Gianluca Panati] 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 rôle of additional $\mathbb{Z}_{2}$ -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 $\mathbb{Z}_{2}$ -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. Springer Science+Business Media Dordrecht, 2014 Periodic Schrödinger operators nationallicence Composite Wannier functions nationallicence Bloch bundle nationallicence Bloch frames nationallicence Time-reversal symmetry nationallicence Space-reflection symmetry nationallicence Invariants of topological insulators nationallicence Monaco Domenico SISSA, Via Bonomea 265, 34136, Trieste, Italy aut Panati Gianluca Dipartimento di Matematica, "La Sapienza” Università di Roma, Piazzale Aldo Moro 2, 00185, Rome, Italy aut Acta Applicandae Mathematicae Springer Netherlands 137/1(2015-06-01), 185-203 0167-8019 137:1<185 2015 137 10440 https://doi.org/10.1007/s10440-014-9995-8 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10440-014-9995-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Monaco Domenico SISSA, Via Bonomea 265, 34136, Trieste, Italy aut NATIONALLICENCE 700 1- Panati Gianluca Dipartimento di Matematica, "La Sapienza” Università di Roma, Piazzale Aldo Moro 2, 00185, Rome, Italy aut NATIONALLICENCE 773 0- Acta Applicandae Mathematicae Springer Netherlands 137/1(2015-06-01), 185-203 0167-8019 137:1<185 2015 137 10440