  The competition between the length scales associated with the periodicity of
a lattice potential and the cyclotron radius of a uniform magnetic field is
known to have dramatic effects on the single-particle properties of a quantum
particle, e.g., the fractal spectrum is known as the Hofstadter butterfly.
Having this intricate competition in mind, we consider a two-component Fermi
gas on a square optical lattice with opposite synthetic magnetic fields for the
components, and study its effects on the many-body BCS-pairing phenomenon. By a
careful addressing of the distinct superfluid transitions from the semi-metal,
quantum spin-Hall insulator or normal phases, we explore the low-temperature
phase diagrams of the model, displaying lobe structures that are reminiscent of
the well-known Mott-insulator transitions of the Bose-Hubbard model.
