Topological Superfluid in P-band Optical Lattice*

By studying p-band fermionic system with nearest neighbor attractive interaction we find translation symmetry protected $Z_{2}$ topological superfluid (TSF) that is characterized by a special fermion parity pattern at high symmetry points in momentum space $k=$ $(0,0)$, ($0$, $\pi $), ($\pi $, $0$), ($\pi $, $\pi $). Such $Z_{2}$ TSF supports the robust Majorana edge modes and a new type of low energy excitation - (supersymmetric) $Z_{2}$ link-excitation.