Phase diagram of p-orbital attractive fermions in a two-dimensional optical lattice

We study multi-orbital system of polarized fermions on a two-dimensional square lattice with attractive on-site interaction. We assume that the atoms are loaded to the lattice such that the $s$-orbital is completely filled and dynamic of the system is determined by the $p$-orbital atoms. By including the four-site square plaquette interaction term generated from the directional tunneling dependence at half filling, we derive an effective spin-Hamiltonian using forth order perturbation theory at the strongly interacting limit. We then use a variational mean field approach to map out the phase diagram.