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Gapped periodic quantum systems exhibit an interesting LocalizationDichotomy, which emerges when one looks at the localization of the optimallylocalized Wannier functions associated to the Bloch bands below the gap. Asrecently proved, either these Wannier functions are exponentially localized, asit happens whenever the Hamiltonian operator is time-reversal symmetric, orthey are delocalized in the sense that the expectation value of$|\mathbf{x}|^2$ diverges. Intermediate regimes are forbidden.  Following the lesson of our Maestro, to whom this contribution is gratefullydedicated, we find useful to explain this subtle mathematical phenomenon in thesimplest possible model, namely the discrete model proposed by Haldane (Phys.Rev. Lett. 61, 2017 (1988)). We include a pedagogical introduction to the modeland we explain its Localization Dichotomy by explicit analytical arguments. Wethen introduce the reader to the more general, model-independent version of thedichotomy proved in (Commun. Math. Phys. 359, 61-100 (2018)), and finally weannounce further generalizations to non-periodic models.

The Haldane model and its localization dichotomy