  The recent realization of two-dimensional (2D) synthetic spin-orbit (SO)
coupling opens a broad avenue to study novel topological states for ultracold
atoms. Here, we propose a new scheme to realize exotic chiral Fulde-Ferrell
superfluid for ultracold fermions, with a generic theory being shown that the
topology of superfluid pairing phases can be determined from the normal states.
The main findings are two fold. First, a semimetal is driven by a new type of
2D SO coupling whose realization is even simpler than the recent experiment,
and can be tuned into massive Dirac fermion phases with or without inversion
symmetry. Without inversion symmetry the superfluid phase with nonzero pairing
momentum is favored under an attractive interaction. Furthermore, we show a
fundamental theorem that the topology of a 2D chiral superfluid can be uniquely
determined from the unpaired normal states, with which the topological chiral
Fulde-Ferrell superfluid with a broad topological region is predicted for the
present system. This generic theorem is also useful for condensed matter
physics and material science in search for new topological superconductors.
