01282nas a2200121   4500008004100000245007500041210006900116260001000185520086400195100002901059700002101088856005101109       2015                        en  d00aStability of closed gaps for the alternating Kronig-Penney Hamiltonian0 aStability of closed gaps for the alternating KronigPenney Hamilt  bSISSA3 aWe consider the Kronig-Penney model for a quantum crystal with equispaced periodic delta-interactions of alternating strength. For this model all spectral gaps at the centre of the Brillouin zone are known to vanish, although so far this noticeable property has only been proved through a very delicate analysis of the discriminant of the corresponding
ODE and the associated monodromy matrix. We provide a new, alternative proof by showing that this model can be approximated, in the norm resolvent sense, by a model of regular periodic interactions with finite range for which all gaps at the centre of the Brillouin zone are still vanishing. In particular this shows that the vanishing gap property
is stable in the sense that it is present also for the "physical" approximants and is not only a feature of the idealised model of zero-range interactions.1 aMichelangeli, Alessandro1 aMonaco, Domenico  uhttp://urania.sissa.it/xmlui/handle/1963/3446001557nas 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