  The Fulde-Ferrell (FF) superfluid phase, in which fermions form
finite-momentum Cooper pairings, is well studied in spin-singlet superfluids in
past decades. Different from previous works that engineer the FF state in
spinful cold atoms, we show that the FF state can emerge in spinless Fermi
gases confined in optical lattice associated with nearest-neighbor
interactions. The mechanism of the spinless FF state relies on the split Fermi
surfaces by tuning the chemistry potential, which naturally gives rise to
finite-momentum Cooper pairings. The phase transition is accompanied by changed
Chern numbers, in which, different from the conventional picture, the band gap
does not close. By beyond-mean-field calculations, we find the finite-momentum
pairing is more robust, yielding the system promising for maintaining the FF
state at finite temperature. Finally we present the possible realization and
detection scheme of the spinless FF state.
