Attractive Hofstadter-Hubbard model with imbalanced chemical and vector potentials

We study the interplay between the Hofstadter butterfly, strong interactions, and the 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 stripe-ordered Fulde-Ferrell-Larkin-Ovchinnikov-like superfluid and supersolid phases occur that can be distinguished and characterized according to their coexisting pair-density-, charge-density-, and spin-density-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.