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We consider a mixture of a two-component Fermi gas and a single-componentdipolar Bose gas in a square optical lattice and reduce it into an effectiveFermi system where the Fermi-Fermi interaction includes the attractiveinteraction induced by the phonons of a uniform dipolar Bose-Einsteincondensate. Focusing on this effective Fermi system in the parameter regimethat preserves the symmetry of $D_4$, the point group of a square, we explore,within the Hartree-Fock-Bogoliubov mean-field theory, the phase competitionamong density wave orderings and superfluid pairings. We construct the matrixrepresentation of the linearized gap equation in the irreduciblerepresentations of $D_4$. We show that in the weak coupling regime, each matrixelement, which is a four-dimensional (4D) integral in momentum space, can beput in a separable form involving a 1D integral, which is only a function oftemperature and the chemical potential, and a pairing-specific "effective"interaction, which is an analytical function of the parameters thatcharacterize the Fermi-Fermi interactions in our system. We analyze thecritical temperatures of various competing orders as functions of differentsystem parameters in both the absence and presence of the dipolar interaction.We find that close to half filling, the d_{x^{2}-y^{2}}-wave pairing with acritical temperature in the order of a fraction of Fermi energy (at halffilling) may dominate all other phases, and at a higher filling factor, thep-wave pairing with a critical temperature in the order of a hundredth of Fermienergy may emerge as a winner. We find that tuning a dipolar interaction candramatically enhance the pairings with $d_{xy}$- and g-wave symmetries but notenough for them to dominate other competing phases.

the european physical journal. d, atomic, molecular and optical physics/the european physical journal. d, atomic, molecular, optical and plasma physics

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