  We study the interplay between the Hofstadter butterfly, strong interactions
and Zeeman field within the mean-field Bogoliubov-de Gennes theory in real
space, and explore the ground states of the attractive single-band
Hofstadter-Hubbard Hamiltonian on a square lattice, including the exotic
possibility of imbalanced vector potentials. We find that the cooperation
between the vector potential and superfluid order breaks the spatial symmetry
of the system, and flourish stripe-ordered Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO)-like superfluid and supersolid phases that can be distinguished and
characterized according to their coexisting pair-density (PDW), charge-density
(CDW) and spin-density (SDW) wave orders. We also discuss confined systems and
comment on the likelihood of observing such stripe-ordered phases by loading
neutral atomic Fermi gases on laser-induced optical lattices under
laser-generated artificial gauge fields.
