01557nas a2200121   4500008004100000245012000041210006900161260001300230520109900243100002101342700002101363856005101384       2015                        en  d00aSymmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry0 aSymmetry and localization in periodic crystals triviality of Blo  bSpringer3 a<p>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 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.</p>1 aMonaco, Domenico1 aPanati, Gianluca  uhttp://urania.sissa.it/xmlui/handle/1963/34468