  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.
