The Novel Phases Of Ultracold Fermi Gases In Low-dimensional Optical Lattices

This thesis focuses on the numerical quests for the novel phases and phase transitions in ultracold Fermi gases in low-dimensional optical lattices. We dis-cuss the novel properties of various inhomogeneous Hubbard models in optical lattices, such as the phase separation, the pure Mott-insulator phase, the pure FFLO state as well as the properties of some novel phases in finite temperature, etc.Firstly, we discuss the phase separation in system with spin-dependent ex-ternal potentials. In Fermi gases of two different components, the system can be described by the Hubbard model with spin-dependent external potentials. The properties of phase separation in this system are discussed, including the influence of related parameters on this process and the property of the energy. Furthermore, the phase diagram is obtained, and the existence of the minimum critical point in the phase diagram is also discussed. Those studies are essential for the sympathetic cooling process as well as providing a scheme for demixing quantum gases of different components.Secondly, the novel phases in the off-diagonal confinement (ODC) system are discussed. For the ordinary confining potential used in experiment for trapping atoms, various quantum phases coexist in the system, while for the theoretic prediction and test, the homogeneous system is preferable. This problem, to some extent, can be solved by introducing the ODC system, which can be obtained by using spatially changed hopping parameter in the Hamiltonian. With the ODC configuration, we can mimic properties of the homogeneous system. For the repulsion interaction case, a kind of pure Mott phase can be achieved. The various quantum phases in the system are discussed as well as the phase diagram of the system. Furthermore, we study the correlation properties of the system, including the density-density correlation function, the spin correlation function, the charge correlation function, and their Fourier transition, respectively. For the attractive case, we discuss the phases in the system and obtain the phase diagram, in which a region of pure FFLO state can be identified. Moreover, we discuss the signature of FFLO state, including the pair correlation function and the magnetic structure factor, which helps the search of FFLO state in experiments.Finally, we develop the finite-temperature density-functional theory based on the thermodynamic Bethe-ansatz (TBA-FTDFT) and obtain the related prop-erties of the system in finite temperature. The numerical solution of thermody-namic Bethe-ansatz equations and its parametrization as well as the FTDFT are discussed. The parameterized formula for the exchange-correlation function of the Hubbard model in finite temperature with high accuracy are provided for the first time. Furthermore, with the multi-dimensional bisection method we solve the long existed problem in density-functional theory (DFT), i.e., the fail-ure of the convergence in the coupled Kohn-Sham equation because of the abrupt change of the exchange-correlation function at around the half-filled region. Our investigation covers the application of DFT in the simulation of inhomogeneous Hubbard model in all the general cases.The problems in this thesis are mainly discussed using numerical methods, including the density-matrix renormalization group (DMRG), exact diagonaliza-tion (ED), DFT, and the local spin-density approximation (LSDA) etc. The reliability of the results is guaranteed by the double check of different methods.