Abstract:
We study the ground state phase diagram and the quantum phase transitions in spin-1 Bose gases with Raman induced spin-orbit coupling. In addition to the Bose-Einstein condensates with uniform density, three types of stripe condensation phases that simultaneously break the U(1) and the translation symmetry are identified in presence of spin-dependent interactions. The transitions between these phases are characterized by the spin magnetization and different crystalline orders, and the occurrences of the various tricritical points are predicted. The excitation spectra in the plane-wave phase and the zero-momentum phase show rich roton-maxon structures, and their instabilities indicate the tendency to develop crystalline orders. We propose the atomic gas of $^{23}$Na could be a candidate for observing the stripe condensate with high contrast fringes.

Abstract:
We study two-component Bose gases with Raman induced spin-orbit coupling via a perturbation approach at finite temperature. For weak coupling, free energy is expanded in terms of Raman coupling strength up to the second order, where the coefficient (referred to as Raman susceptibility) is determined according to linear response theory. The equation of state for the stripe phase and the plane-wave phase are obtained in Popov approximation, and the first order transition between these two phases is investigated. As temperature increases, we find the phase boundary bends toward the stripe phase side in most temperature regions, which implies the ferromagnetic order is more robust than the crystalline order in presence of thermal fluctuations. Our results qualitatively agree with the recent experimental observation in rubidium atomic gases. A method to measure the Raman susceptibility through the two-photon Bragg scattering experiment is also discussed.

Abstract:
Ground-state phase diagram of two-component Bose gases with Rashba spin-orbit coupling is determined via a variational approach. A phase in which the fully polarized condensate occupies zero momentum is identified. This zero-momentum phase competes with the spin density wave phase when interspecies interaction is stronger than intraspecies interaction, and the former one is always the ground state for weak spin-orbit coupling. When the energies of these two phases are close, there is a phase separation between them. At finite temperature, such a zero-momentum condensation can be induced by a ferromagnetic phase transition in normal state. The spontaneous spin polarization removes the degeneracy of quasiparticles' energy minima, and consequently the modified density of state accommodates a Bose condensation to appear below a critical temperature.

Abstract:
In a half-filled Hubbard model on a square lattice, the next-nearest-neighbor hopping causes spin frustration, and the collinear antiferromagnetic (CAF) state appears as the ground state with suitable parameters. We find that there is a metal-insulator transition in the CAF state at a critical on-site repulsion. When the repulsion is small, the CAF state is metallic, and a van Hove singularity can be close to the Fermi surface, resulting in either a kink or a discontinuity in the magnetic moment. When the on-site repulsion is large, the CAF state is a Mott insulator. A first-order transition from the CAF phase to the antiferromagnetic phase and a second-order phase transition from the CAF phase to the paramagnetic phase are obtained in the phase diagram at zero temperature.

Abstract:
Bose-Einstein condensation (BEC) of Feshbach molecules in a homogeneous Bose gas is studied at finite temperatures in a single-channel mean-field approach where the Hartree-Fock energy and pairing gap are determined self-consistently. In the molecular-BEC state, the atomic excitation is gapped and the molecular excitation is gapless. The binding energy of Feshbach molecules is shifted from the vacuum value due to many-body effect. When the scattering length $a_s$ of atoms is negative, the system is subject to mechanical collapse due to negative compressibility. The system is stable in most regions with positive scattering lengths. However at low temperatures near the resonance, the molecular-BEC state vanishes, and the coherent mixture of atomic and molecular BEC is subject to mechanical collapse.

Abstract:
We study the effect of the induced interaction on the superfluidtransition temperature of a spin-polarized Fermi gas. In the BCS limit, the polarization is very small in the superfluid state, and the effect of the induced interaction is almost the same as in the spin-balanced case. The temperature Tt and the polarization Pt of the tricritical point are both reduced from mean-field results by a factor about 2.22. This reduction is also significant beyond the BCS limit. In the unitary limit, we find (Pt,Tt/TF)=(0.42,0.16), in comparison with mean-field and experimental results.

Abstract:
In this letter we study both ground state properties and the superfluid transition temperature of a spin-1/2 Fermi gas across a Feshbach resonance with a synthetic spin-orbit coupling, using mean-field theory and exact solution of two-body problem. We show that a strong spin-orbit coupling can significantly enhance the pairing gap for 1/(k_F a_s)<=0 due to increased density-of-state. Strong spin-orbit coupling also significantly enhances the superfluid transition temperature when 1/(k_F a_s)<=0, while suppresses it slightly when 1/(k_F a_s)>0. The universal interaction energy and pair size at resonance are also discussed.

Abstract:
We study the effect of the induced interaction on the superfluid transition temperature of a Fermi gas with a BEC-BCS crossover. The Gorkov-Melik-Barkhudarov theory about the induced interaction is extended from the BCS side to the entire crossover, and the pairing fluctuation is treated in the approach by Nozi\`{e}res and Schmitt-Rink. At unitarity, the induced interaction reduces the transition temperature by about twenty percent. In the BCS limit, the transition temperature is reduced by a factor about 2.22, as found by Gorkov and Melik-Barkhudarov. Our result shows that the effect of the induced interaction is important both on the BCS side and in the unitary region.

Abstract:
The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is derived by the path-integral method. In addition to the standard Ginzburg-Landau equation, a second equation describing the total atom density is obtained. These two coupled equations are necessary to describe both homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near the transition temperature $T_c$ on both sides of the crossover. In the weakly-interacting BEC region, it is also accurate at zero temperature where the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP) equation. The applicability of GP equation at finite temperature is discussed. On the BEC side, the fluctuation of the order parameter is studied and the renormalization to the molecule coupling constant is obtained.

Abstract:
In this paper we investigate the properties of Bose gases with Raman-induced spin-orbit(SO) coupling. It is found that the SO coupling can greatly modify the single particle density-of-state, and thus lead to non-monotonic behavior of the condensate depletion, the Lee-Huang-Yang correction of ground-state energy and the transition temperature of a non-interacting Bose-Einstein condensate. The presence of the SO coupling also breaks the Galilean invariance, and this gives two different critical velocities, corresponding to the movement of the condensate and the impurity respectively. Finally, we show that with SO coupling, the interactions modify the BEC transition temperature even at Hartree-Fock level, in contrast to the ordinary Bose gas without SO coupling. All results presented here can be directly verified in the current cold atom experiments using Raman laser-induced gauge field.