  We consider theoretically the problem of an artificial gauge potential
applied to a cold atomic system of interacting neutral bosons in a
tight-binding optical lattice. Using the Bose-Hubbard model, we show that an
effective magnetic field leads to superfluid phases with simultaneous spatial
order, which we analyze using Bogoliubov theory. This gives a consistent
expansion in terms of quantum and thermal fluctuations, in which the lowest
order gives a Gross-Pitaevskii equation determining the condensate
configuration. We apply an analysis based on the magnetic symmetry group to
show how the spatial structure of this configuration depends on commensuration
between the magnetic field and the lattice. Higher orders describe the
quasiparticle excitations, whose spectrum combines the intricacy the Hofstadter
butterfly with the characteristic features of the superfluid phase. We use the
depletion of the condensate to determine the range of validity of our
approximations and also to find an estimate for the onset of the Mott insulator
phase. Our theory provides concrete experimental predictions, for both
time-of-flight imagery and Bragg spectroscopy.
