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 rĂ´le of additional 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 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.

%B Acta Applicandae Mathematicae, vol. 137, Issue 1, 2015, pages: 185-203 %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34468 %1 34642 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Domenico Monaco (dmonaco@sissa.it) on 2015-05-15T10:12:11Z No. of bitstreams: 1 CalaGonone.pdf: 500680 bytes, checksum: 00aa2839c4d737f45da3d828e5cc0a53 (MD5) %R 10.1007/s10440-014-9995-8