Gianluca Panati

### The Localization Dichotomy for gapped periodic quantum systems

Abstract: The talk concerns the localization properties of independent electrons in a periodic  background, possibly including a periodic magnetic field, as e.g. in ordinary insulators, 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 (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to lie 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 Chern numbers, or the decay of \emph{any} composite Wannier function is such that the expectation value of the squared position operator is infinite. Equivalently, in the topologically non-trivial phase the localization functional introduced by Marzari and Vanderbilt diverges, as numerically observed in the case of the Haldane model. The result is formulated by using only the relevant symmetries of the system, and extends to the case of interacting electrons within the Hartree-Fock approximation.