  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
$|\mathbf{x}|^2$ 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 (Phys.
Rev. Lett. 61, 2017 (1988)). 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 (Commun. Math. Phys. 359, 61-100 (2018)), and finally we
announce further generalizations to non-periodic models.
