Competing superfluid orders in spin-orbit-coupled fermionic cold-atom optical lattices

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, a superconducting state with nonzero total momentum Cooper pairs in a large magnetic field, was first predicted about 50 years ago and has since become an important concept in many branches of physics. Despite an intensive search in various materials, unambiguous experimental evidence for the FFLO phase is still lacking in experiments. In this paper we show that both Fulde-Ferrell (FF) (a uniform order parameter with a plane-wave phase) and Larkin-Ovchinnikov (LO) phases (a spatially varying order parameter amplitude) can be observed using fermionic cold atoms in spin-orbit-coupled optical lattices. The increasing spin-orbit coupling enhances the FF phase over the LO phase. The coexistence of superfluid and magnetic orders is also found in the normal BCS phase. The pairing mechanism for different phases is understood by visualizing superfluid pairing densities in different spin-orbit bands. The possibility of observing similar physics using spin-orbit-coupled superconducting ultrathin films is also discussed.