  In this work, we investigate conditions which ensure the existence of an
exponentially localized Wannier basis for a given periodic hamiltonian. We
extend previous results in [Pan07] to include periodic zero flux magnetic
fields which is the setting also investigated in [Kuc09]. The new notion of
magnetic symmetry plays a crucial role; to a large class of symmetries for a
non-magnetic system, one can associate "magnetic" symmetries of the related
magnetic system. Observing that the existence of an exponentially localized
Wannier basis is equivalent to the triviality of the so-called Bloch bundle, a
rank m hermitian vector bundle over the Brillouin zone, we prove that magnetic
time-reversal symmetry is sufficient to ensure the triviality of the Bloch
bundle in spatial dimension d=1,2,3. For d=4, an exponentially localized
Wannier basis exists provided that the trace per unit volume of a suitable
function of the Fermi projection vanishes. For d>4 and d \leq 2m (stable rank
regime) only the exponential localization of a subset of Wannier functions is
shown; this improves part of the analysis of [Kuc09]. Finally, for d>4 and d>2m
(unstable rank regime) we show that the mere analysis of Chern classes does not
suffice in order to prove trivility and thus exponential localization.
