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Séminaire Physique mathématique ICJ

# Optimal decay of Wannier functions in Chern and Quantum Hall insulators

## by Gianluca Panati (Università di Roma "La Sapienza")

vendredi 17 février 2017 de au (Europe/Paris)
at Institut Camille Jordan ( Fokko du Cloux )
Université Lyon 1, Bât. Braconnier, 21 av. Claude Bernard, 69100 Villeurbanne
 Description We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e.g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as |x| tends to infinity, of the composite Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is +infinity. In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity. The talk is based on a joint work with Domenico Monaco, Adriano Pisante, and Stefan Teufel. Organisé par Fabien Vignes-Tourneret Contact Email: vignes@math.univ-lyon1.fr