The Effect Of Disorder On Quantum Gas In One Dimension

All particles in the nature can be divided into fermions and bosons depending on the total spin. Bosons can occupy the same energy level. The weakly inter-acting Bose gas will condensate into Bose Einstein Condensation when T＝0. The same Fermions which can not occupy the same energy level dominated by the Pauli exclusion principle. In nowadays, low dimensional system can be built with the experimental technique in optical lattices, where new physics emerges. In low dimensions, we can not only get the answer by theory, but also check it in experiments. The cold atoms in optical lattices give a new way in simulating the Hamiltonians of condensed state system by easily controlling the interaction, the strength of confined potential, dimensionality and even disorder, which gives as hope to answer to some long unsolved problems.Based on the Bethe-ansatz technique and the local density approximation, we use a spin-density functional theory to solve some model on quantum gases in a one-dimension system. The paper is organized as follows:In chapterⅠ, we introduce the background for the cold atom in one-dimensional systems. In chapterⅡ, we study the confined two-component polarized Fermi gases. This chapter is divided into three parts. The first part is about a nu-merical study on a one-dimensional attractive Gaudin model of confined two-component polarized Fermi gases in the presence of the speckle disorder. For a uniform system without trapping, the ground state is either a partially polar-ized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, or a fully paired Bardeen-Cooper-Schrieffer (BCS) state, or a fully polarized normal state. For a clean system in the presence of a harmonic trap, a two-shell structure exists with a polarized superfluid core surrounded by wings composed of either an unpolarized superfluid (FFLO-BCS) or a fully polarized normal (FFLO-N) gas. Between these two different phases, there exists a critical polarization where a pure FFLO state happens. In the presence of speckle disorder with fixed polarization, the sys-tem can be changed from FFLO-BCS into FFLO-N upon increasing the disorder amplitude. In the second part we discuss the Wigner crystallization of confined one-dimensional Fermi gases in a continuous space. We numerically investigate the 2kF�'4kF transformation and the formation of antiferromagnetic quasi-order with increasing repulsive coupling strength for the unpolarized and polarized systems. The third part is about the phase separation in one-dimensional Fermi gases with spin-dependent harmonic trap. The density distributions are calcu-lated. The phase separation between spin-up and spin-down atoms is induced by the interplay of the spin-dependent harmonic confinement and the strong re-pulsive interaction between intercomponent fermions. We find the existence of a critical repulsive interaction strength above which the phase separation evolves.In chapter III, we study the thermodynamic Hubbard model. After solving the infinite coupled equations from the thermodynamic Bethe-ansatz, we get the relation of density and chemical potential. Then we use both the Euler-Lagrange and thermodynamic density-functional-theory to get the density distribution. Both agree very well. In this model, we study the Mott and band insulating phases for the nonzero temperature.In chapter IV, we discuss the thermodynamic boson systems. We also use Euler-Lagrange and thermodynamic density-functional theory to obtain the den-sity distribution. We find that when the interaction is weak, there are some offset between those two results; but when the interaction is strong, both have a good agreements. We also study the density distribution in the strong interaction limit. A good agreement is found between the two methods.