  We examine a system of doubly degenerate $p$-orbital polarized fermions on a
two-dimensional square lattice with a strong on-site interaction. We consider
the system density at the half filling limit and tackle the strong attractive
interaction using a perturbation theory. We treat the four-site square
plaquette interaction term generated from the directional tunneling dependence
of $p$-orbitals using the fourth order in perturbation theory. We map the
strong coupling particle Hamiltonian into an effective spin-Hamiltonian and
then use a variational mean field approach and a linear spin-wave theory to
study the phase diagram. Further, we discuss the experimental signatures of
these phases within the context of current cold-atom experimental techniques.
