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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 $mathbb{Z}_2$-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.
Much of the recent enthusiasm directed towards topological insulators as a new state of matter is motivated by their hallmark feature of protected chiral edge states. In fermionic systems, Kramers degeneracy gives rise to these entities in the presen
For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to $0$) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all
We provide numerical evidence that the Onsager symmetry remains valid for systems subject to a spatially dependent magnetic field, in spite of the broken time-reversal symmetry. In addition, for the simplest case in which the field strength varies on
This paper is devoted to the construction of differential geometric invariants for the classification of Quaternionic vector bundles. Provided that the base space is a smooth manifold of dimension two or three endowed with an involution that leaves f
We propose a two-dimensional time-reversal invariant system of essentially non-interacting electrons on a square lattice that exhibits configurations with fractional charges e/2. These are vortex-like topological defects in the dimerization order par