  We use microscopic Bogoliubov-de Gennes formalism to explore the ground-state
phase diagram of the single-band attractive Hofstadter-Hubbard model on a
square lattice. We show that the interplay between the Hofstadter butterfly and
superfluidity breaks spatial symmetry, and gives rise to stripe-ordered
superfluid and supersolid phases in large parameter spaces. We also discuss the
effects of a trapping potential and comment on the viability of observing
stripe-ordered phases with cold Fermi gases.
