  We performed a theoretical investigation on the ground state properties of a
two dimensional ultra-cold Fermi superfluid with an anisotropic spin-orbit
coupling (SOC). In the absence of Zeeman field, the system evolves from weak
coupling BCS regime to strongly interacting BEC regime (BCS-BEC crossover) with
increasing either the two-particle interaction strength or SOC parameters. We
focused on the behaviors of pairing parameter and density of states (DOS) when
increasing the anisotropic parameter of the SOC. Surprisingly, we discovered
that the gap parameter decreases with increasing the anisotropic parameters,
but the DOS at the Fermi surface shows non-monotonic behavior as a function of
the anisotropic parameter. In the presence of the Zeeman field, we discussed a
particular type of topological phase transition by obtaining the analytical
result of the topological invariant and directly related this quantum phase
transition with a sudden change of the ground state wave-function. Effects of
higher partial wave pairing terms on this topological phase transition were
briefly discussed.
