We analyze the ground-state phase diagram of two-component Fermi gases loaded into a two-dimensional checkerboard superlattice, i.e., a double-well optical lattice, potential within the BCS mean-field theory. We show that, by coupling the two s-wave sublattice superfluid order parameters, a checkerboard potential gives rise to an effectively two-band model with three (two intraband and an interband) nonlocal order parameters. We study the evolution of these order parameters as a function of particle filling, interaction strength, and checkerboard potential and find that the system always prefers the 0-phase solutions, i.e., where the phase difference between sublattice order parameters is 0, but never the pi-phase one. In addition, we find that the ground state of the system undergoes a superfluid-normal quantum phase transition at half fillings beyond a critical checkerboard potential C, the threshold of which is precisely determined by the magnitude of the order parameter at C = 0, and that the normal state rapidly turns into a checkerboard insulator as C increases.
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