  The Hofstadter model describes non-interacting fermions on a lattice in the
presence of an external magnetic field. Motivated by the plethora of
solid-state phases emerging from electron interactions, we consider an
interacting version of the Hofstadter model including a Hubbard repulsion U. We
investigate this model in the large-U limit corresponding to a t-J Hamiltonian
with an external (orbital) magnetic field. By using renormalized mean field
theory supplemented by exact diagonalization calculations of small clusters, we
find evidence for competing symmetry-breaking phases, exhibiting (possibly
co-existing) charge, bond and superconducting orders. Topological properties of
the states are also investigated and some of our results are compared to
related experiments involving ultra-cold atoms loaded on optical lattices in
the presence of a synthetic gauge field.
