Searching for the first topological superfluid (TSF) remains a primary goal of modern science. Here we study the system of attractively interacting fermions hopping in a square lattice with any linear combinations of Rashba or Dresselhaus spin-orbit coupling (SOC) in a normal Zeeman field. By imposing self-consistence equations at half filling, we find there are 3 phases: Band insulator ( BI ), Superfluid (SF) and Topological superfluid (TSF) with a Chern number $ C=2 $. The $ C=2 $ TSF happens in small Zeeman fields and weak interactions which is in the experimentally most easily accessible regime. The transition from the BI to the SF is a first order one due to the multi-minima structure of the ground state energy landscape. There is a new class of topological phase transition from the SF to the $ C=2 $ TSF at the low critical field $ h_{c1} $, then another one from the $ C=2 $ TSF to the BI at the upper critical field $ h_{c2} $. We derive effective actions to describe the two new classes of topological phase transitions, then use them to study the Majorana edge modes and the zero modes inside the vortex core of the $ C=2 $ TSF near both $ h_{c1} $ and $ h_{c2} $, especially explore their spatial and spin structures. We find the edge modes decay into the bulk with oscillating behaviors and determine both the decay and oscillating lengths. We compute the bulk spectra and map out the Berry Curvature distribution in momentum space near both $ h_{c1} $ and $ h_{c2} $. We also elaborate some intriguing bulk-Berry curvature-edge-vortex correspondences. Experimental implications in both 2d non-centrosymmetric materials under a periodic substrate and cold atoms in an optical lattice are given.
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