Quantum Phase Transition Of Fermi Gas Driven By Spin-orbit Coupling In A One-dimensional Optical Lattice

In recent years,the Spin-orbit coupling of ultracold atomic gas becomes a research hotspot.With the change of external condition,system can present different quantum phase.In this paper,firstly we make a brief introduction about the cold atoms in optical lattice,the effection of Spin-orbit coupling and Fulde-Ferrell-Larkin-Ovchinnikov(FFLO)state.Subsequently the method of Density Matrix Renormalization Group we use is introduced and in one-dimensional optical lattice bose gas we carry on some operation and obtain relevant results in good agreement with the derived results.Afterward we discuss one-dimensional attraction Fermi gas in the optical lattice when there exists the effection of Spin-orbit coupling and in-plane Zeeman field,then discover that the whole system in a certain section could generate a novel state with the coexistence of inter-band FFLO pairing and intra-band BCS pairing,named FFLO-BCS phase,which in the momentum space presents a three-peak structure of the pairing correlation.In addition the in-plane Zeeman field can enhance FFLO pairing and oppositely Spin-orbit coupling can strengthen BCS pairing.The existence of external potential makes the system shift from FFLO-BCS state to BCS state,which reinforces the system BCS pairing.Then we did a simple research of MPS.MPS is based on DMRG,but it can be applied to the quantum system that over one-dimensional.We utilize matrix product configuration to calculate ultra-cold Fermi gas system in 1-D optical lattice and the result is consistent with the one gained with the method of density matrix renormalization group.The two methods can not only validate each other so that improve the accuracy of the work,but also prepare for the future study of the two-dimensional system.Finally,we make a summary about the whole article,and envision the next step of work.