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Based on density matrix renormalization group method, we investigate thespin-orbit coupled Fermi gas with attractive interactions in one-dimensionaloptical lattice and present a complete phase diagram for a quarter-fillingsystem with intermediate strong interactions. We unveil the exotic pairingsinduced by spin-orbit couplings and the phase transitions between differentpairing states, and further demonstrate the existence of the long-soughttopological superfluid state at moderate magnetic field and spin-orbit couplingstrength. For the particle conserved system, this topological superfluid statecan be fixed by its gapless single particle excitation, Luttinger parameter andnon-trivial boundary effect. Our study removes out the widespread doubt on theexistence of topological superfluid in one dimension and paves the way forsimulating topological superfluid states in cold atom settings.

Phase diagram and topological superfluid state of spin-orbit coupled Fermi gas with attractive interactions in a one-dimensional optical lattice