On the localization dichotomy for gapped quantum systems

Since their introduction by G. Wannier in 1937, Wannier functions have been extensively used in solid state physics to analyze and understand the physical properties of perfect crystalline quantum systems. In 2016, D. Monaco, G. Panati, A. Pisante and S. Teufel proved a localization dichotomy result for periodic Schrödinger operators, namely that the localization properties of Wannier functions are deeply connected to the topological properties of the quantum system. The original results presented in this thesis concern the possibility of extending such localization dichotomy to generic gapped quantum systems. First of all, by reviewing and analyzing the different few existing results about generalized Wannier functions, we give a precise definition of generalized Wannier functions for generic gapped quantum systems. Moreover, we prove the existence of Parseval frames of exponentially localized generalized Wannier functions for a large class of magnetic systems, as a byproduct we show the existence of a generalized Wannier basis for magnetic Hamiltonians. Furthermore, we analyze the Chern number in position space, namely the Chern character, by proving a gap labelling theorem for Bloch-Landau Hamiltonians using gauge covariant magnetic perturbation theory and investigating the validity of the gap labelling theorem in a non-covariant setting. We also explicitly show how to connect the Chern character to the Středa formula. Finally, we show that an ultra generalized type of Wannier basis is not capable to encode the physical properties of the systems and we prove that the existence of a well-localized localized generalized Wannier basis implies the vanishing of the Chern character.