  We study superconductivity in an ultracold Bose-Fermi mixture loaded into a
square optical lattice subjected to a staggered flux. While the bosons form a
superfluid at very low temperature and weak interaction, the interacting
fermions experience an additional long-ranged attractive interaction mediated
by phonons in the bosonic superfluid. This leads us to consider a generalized
Hubbard model with on-site and nearest-neighbor attractive interactions, which
give rise to two competing superconducting channels. We use the
Bardeen-Cooper-Schrieffer theory to determine the regimes where distinct
superconducting ground states are stabilized, and find that the non-local
pairing channel favors a superconducting ground state which breaks both the
gauge and the lattice symmetries, thus realizing unconventional
superconductivity. Furthermore, the particular structure of the single-particle
spectrum leads to unexpected consequences, for example, a dome-shaped
superconducting region in the temperature versus filing fraction phase diagram,
with a normal phase that comprises much richer physics than a Fermi-liquid.
Notably, the relevant temperature regime and coupling strength is readily
accessible in state of the art experiments with ultracold trapped atoms.
