  We show how the weakly trapped Harper-Hofstadter model can be mapped onto a
Harper-Hofstadter model in momentum space. In this momentum-space model, the
band dispersion plays the role of the periodic potential, the Berry curvature
plays the role of an effective magnetic field, the real-space harmonic trap
provides the momentum-space kinetic energy responsible for the hopping, and the
trap position sets the boundary conditions around the magnetic Brillouin zone.
Spatially local interactions translate into nonlocal interactions in momentum
space: within a mean-field approximation, we show that increasing interparticle
interactions leads to a structural change of the ground state, from a single
rotationally symmetric ground state to degenerate ground states that
spontaneously break rotational symmetry.
