  We investigate the localization properties of gapped periodic quantum
systems, modeled by a periodic or covariant family of projectors, as e.g. the
orthogonal projectors on the occupied orbitals at fixed crystal momentum for a
gas of non-interacting electrons. We prove a general localization dichotomy for
dimension $d\leq 3$: either the system is topologically trivial i.e. all the
Chern numbers vanish, or any arbitrary choice of composite Wannier functions
yields an infinite expectation value for the squared position operator.
Equivalently, in the topologically non-trivial phase, the localization
functional introduced by Marzari and Vanderbilt diverges, as already noticed in
the case of the Haldane model. Our result is formulated by using only the
relevant symmetries of the system, and it is thus largely model-independent.
Possible applications include both tight-binding and continuous models of
crystalline solids, cold gases in optical lattices as well as flat band
superconductivity.
