  Motivated by experimental advances in the synthesis of gauge potentials for
ultracold atoms, we consider the superfluid phase of interacting bosons on a
square lattice in the presence of a magnetic field. We show that superfluid
order implies spatial symmetry breaking, and predict clear signatures of
many-body effects in time-of-flight measurements. By developing a Bogoliubov
expansion based on the exact Hofstadter spectrum, we find the dispersion of the
quasiparticle modes within the superfluid phase, and describe the consequences
for Bragg spectroscopy measurements. The theory also provides an estimate of
the critical interaction strength at the transition to the Mott insulator
phase.
