Time-reversal-breaking and $d$-wave superfluidity of ultracold dipolar fermions in optical lattices

We describe possible superfluid phases of ultracold dipolar fermions in optical lattices for two-dimensional systems. Considering the many-body screening of dipolar interactions at larger filling factors, we show that several superfluid phases with distinct pairing symmetries naturally emerge in the singlet channel: local $s$-wave $(sl)$, extended $s$-wave $(se)$, $d$-wave $(d)$ or time-reversal-symmetry breaking $(sl + se \pm id)$-wave. The temperature versus filling factor phase diagram indicates that $d$-wave is favored near half-filling, that $(sl + se)$-wave is favored near zero or full filling, and that time-reversal-breaking $(sl + se \pm id)$-wave is favored in between. When a harmonic trap is included a sequence of phases can exist in the cloud depending on the filling factor at the center of the trap. Most notably in the region where the $(sl + se \pm id)$-wave superfluid exists, spontaneous currents are generated, and may be detected using velocity sensitive Bragg spectroscopy.