Competing Orders in a Dipolar Bose - Fermi Mixture on a Square Optical Lattice: Mean-Field Perspective*

We study superfluid pairings of two-component fermions interacting by exchanging virtual phonons of a dipolar condensate in an optical lattice that preserves the symmetry of D4. We construct, within the Hartree-Fock-Bogoliubov theory, the matrix representation of the linearized gap equation in the irreducible representations of D4. We find that each matrix element, which is a four-dimensional (4D) integral in momentum space, can be put in a separable form involving a 1D integral, which is only a function of temperature and the chemical potential, and a pairing-specific ``effective'' interaction, which is an analytical function of the parameters that characterize Fermi-Fermi interactions. We analyze the critical temperatures of various competing orders (superfluids with s-, d$_{x^2-y^2}$-, d$_{xy}$-, and g-wave symmetries and density waves) as functions of different system parameters in both the absence and presence of the dipolar interaction. We find that tuning a dipolar interaction can dramatically enhance various unconventional pairings.