Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of |x| 2 diverges. Intermediate regimes are forbidden.Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane [15]. We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in [25], and finally we announce further generalizations to non-periodic models.Date: September 6, 2019. Extended version of the paper published in Rend. Mat. Appl. 39, 307-327 (2018). In comparison with the published version, we added some details and the whole Chapter 5.2010 Mathematics Subject Classification. 81Q70, 81V70, 47A56, 47A10.

Motivated by the analysis of Hofstadter-like Hamiltonians for a 2-dimensional crystal in presence of a uniform transverse magnetic field, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states.When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model m occupied energy bands by a real-analytic and Z 2periodic family {P (k)} k∈R 2 of orthogonal projections of rank m. More generally, in dimension d ≤ 3, a moving orthonormal basis of Ran P (k) consisting of real-analytic and Z d -periodic Bloch vectors can be constructed if and only if the first Chern number(s) of P vanish(es). Here we are mainly interested in the topologically obstructed case.First, by dropping the generating condition, we show how to construct a collection of m − 1 orthonormal, real-analytic, and Z d -periodic Bloch vectors. Second, by dropping the orthonormality condition, we can construct a Parseval frame of m + 1 real-analytic and Z d -periodic Bloch vectors which generate Ran P (k). Both constructions are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case.A moving Parseval frame of analytic, Z d -periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of Hofstadter-like Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting.

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We investigate the relation between the localization of generalized Wannier bases and the topological properties of two-dimensional gapped quantum systems of independent electrons in a disordered background, including magnetic fields, as in the case of Chern insulators and quantum Hall systems. We prove that the existence of a well-localized generalized Wannier basis for the Fermi projection implies the vanishing of the Chern character, which is proportional to the Hall conductivity in the linear response regime. Moreover, we state a localization dichotomy conjecture for general non-periodic gapped quantum systems.