@article {2015,
title = {Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry},
journal = {Acta Applicandae Mathematicae, vol. 137, Issue 1, 2015, pages: 185-203},
year = {2015},
note = {The article is composed of 23 pages and recorded in PDF format},
publisher = {Springer},
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 r{\^o}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.},
doi = {10.1007/s10440-014-9995-8},
url = {http://urania.sissa.it/xmlui/handle/1963/34468},
author = {Domenico Monaco and Gianluca Panati}
}