The localization dichotomy for gapped periodic systems and its relevance for macroscopic transport

We review recent results concerning the localization of gapped periodic systems of independent fermions, as e.g. elec- trons in Chern and Quantum Hall insulators. We show that there is a “localization dichotomy” which shows some analogies with phase transitions in Statistical Mechanics: either there exists a system of exponentially localized composite Wannier functions for the Fermi projector, or any possible system of composite Wannier functions yields a diverging expectation value for the squared position operator. This fact is largely model independent, covering both tight-binding and continuous models. The results are discussed with emphasis on the main ideas and the broader context, avoiding most of the technical details.