Theoretical Study Of The Phase Diagram And Dynamical Excitation Spectrum Of Low-dimensional Spin-orbit Coupled Superfluid Fermi Gas In Optical Lattice

In recent years,the exploration of topological superfluid has become a key focus in the research of ultracold atom physics.Topological superfluid is characterized by the presence of Majorana zero modes(Majorana fermions).One can tune the Majorana zero mode so that the system has a double degenerate ground state and non-locality.Non-local topological qubits are expected to solve the problem of decoherence of quantum states due to environmental disturbances in quantum computing.Therefore,it is believed that Majorana zero-mode can be applied to the study of fault-tolerant topological quantum computing.Despite intensive theoretical and experimental studies,implementing topological superfluid remains a significant challenge,and there is currently no good scheme to distinguish between ordinary superfluid states and topological superfluid states.These urgent scientific problems are the focus of this research topic,and we aim to study these problems from a theoretical perspective.In this thesis,we use the Green’s function method based on mean-field theory and random phase approximation theory to investigate the phase transition and dynamical excitations of low-dimensional spin-orbit coupled Fermi gases in optical lattices at the theoretical level.We focus on two different physical systems: one-dimensional Raman-type spin-orbit coupled Fermi gas in optical lattice system and two-dimensional Rashba spin-orbit coupled Fermi gas in optical lattice system.Theoretically,there are topological superfluids in both systems that have been of interest.In this thesis,we briefly introduce basic concepts such as superfluid phenomena,spin-orbit coupling,and optical lattices,and introduce the progress of related theoretical and experimental research in cold atom physics.Using Green’s function method,we derived the density and spin dynamical structure factors of one-dimensional Raman spin-orbit coupled Fermi gas in optical lattice.Then,we used the random phase approximation theory to derive the density dynamical structure factors of two-dimensional Rashba spin-orbit coupled Fermi gas in optical lattice.In the study of one-dimensional spin-orbit coupled Fermi gas in optical lattice,we calculated the density and spin dynamical structure factors and quasiparticle energy spectrum when the system is in the case of half-filling and away from half-filling,respectively.By changing the coupling strength h of the spin-orbit coupling,we found that when the system is in the case of half-filling,density and spin dynamical structure factors are very dependent on the change of the coupling strength h.With the increase of h,the structure of density and spin dynamical structure factors become more complex,and the differences between them become larger.When h reaches a critical value,the system undergoes a metal-insulator phase transition.We also explained the process of metal-insulator phase transition through the relationship between Fermi energy and quasiparticle energy spectrum.We studied the influence of doping on the dynamical excitations of the system.We found that when the doping concentration exceeded a certain value,the insulation phase in the system disappeared.In the research work of the two-dimensional Rashba spin-orbit coupled Fermi gas in optical lattice,we studied the h-λ phase diagrams of the system in the case of half-filling and away from half-filling.We obtained from the phase diagrams that as the gradual increase of Zeeman filed strength h,the system undergoes a topological phase transition from BCS superfluid to topological superfluid,and a phase transition from superfluid state to normal state.In the normal state,when the system is in the case of half-filling,there is a metal-insulator phase transition.The optimal theoretical parameters of topological superfluid can also be obtained from the phase diagrams,which provides an idea for realizing topological superfluid in this system experimentally.We calculated the band structures,density of states and quasiparticle energy spectrum,which can reflect the phase transition of the system.Finally,we further analyzed the dynamical excitations of the system by calculating the sound speed and dynamical structure factors of the system when it deviates from half-filling.We found that different phases can be distinguished by dynamical structure factors,which provides a theoretical reference for identifying topological superfluid states in experiments.