Abstract:
We extend the original work of Ruderman, Kittel, Kasuya, and Yosida (RKKY) on the interaction between two magnetic moments embedded in an electron gas to the case where the electron gas is spin polarized. The broken symmetry of a host material %(here we have a broken time reversal symmetry) introduces the Dzyaloshinsky-Moriya (DM) vector and tensor interaction terms, in addition to the standard RKKY term, so that the net interaction energy has the form: $ {\cal H} = J \vec S_1 \cdot \vec S_2 + \vec D \cdot \vec S_1 \times \vec S_2 + \vec S_1 \cdot \stackrel{\leftrightarrow}{\Gamma} \cdot \vec S_2 $. We find that for the spin-polarized electron gas, a non-zero tensor interaction $\stackrel{\leftrightarrow}{\Gamma}$ is present in addition to the scalar RKKY interaction $J$, while $\vec D$ is zero due to the presence of inversion symmetry. Explicit expressions for these are derived for the electron gas both in 2D and 3D. The RKKY interaction exhibits a beating pattern, caused by the presence of the two Fermi momenta $k_{F\uparrow}$ and $k_{F\downarrow}$, while the $R^{-3}$ distance dependence of the original RKKY result for the 3D electron gas is retained. This model serves as a simple example of the magnetic interaction in systems with broken symmetry, which goes beyond the RKKY interaction.

Abstract:
We investigate the ground state of the two-dimensional polarized Fermi gas with spin-orbit coupling and construct the phase diagram at zero temperature. We find there exist phase separation when the binding energy is low. As the binding energy increasing, the topological nontrivial superfluid phase coexist with topologically trivial superfluid phase which is topological phase separation. The spin-orbit coupling interaction enhance the triplet pairing and destabilize the phase separation against superfluid phase.

Abstract:
The collective excitations of a zero-temperature, spin-polarized, harmonically trapped, two-dimensional dipolar Fermi gas are examined within the Thomas-Fermi von Weizs\"acker hydrodynamic theory. We focus on repulsive interactions, and investigate the dependence of the excitation frequencies on the strength of the dipolar interaction and particle number. We find that the mode spectrum can be classified according to bulk modes, whose frequencies are shifted upward as the interaction strength is increased, and an infinite ladder of surface modes, whose frequencies are {\em independent} of the interactions in the large particle limit. We argue quite generally that it is the {\em local} character of the two-dimensional energy density which is responsible for the insensitivity of surface excitations to the dipolar interaction strength, and not the precise form of the equation of state. This property will not be found for the collective excitations of harmonically trapped, dipolar Fermi gases in one and three dimensions, where the energy density is manifestly nonlocal.

Abstract:
We consider a mixture of a spin-polarized Fermi gas and a dipolar Bose-Einstein condensate in which s-wave scattering between fermions and the quasiparticles of the dipolar condensate can result in an effective attractive Fermi-Fermi interaction anisotropic in nature and tunable by the dipolar interaction. We show that such an interaction can significantly increase the prospect of realizing a superfluid with a gap parameter characterized with a coherent superposition of all odd partial waves. We formulate, in the spirit of the Hartree-Fock-Bogoliubov mean-field approach, a theory which allows us to estimate the critical temperature when the anisotropic Fock potential is taken into consideration and to determine the system parameters that optimize the critical temperature at which such a superfluid emerges before the system begins to phase separate.

Abstract:
We study the stability region of the topological superfluid phase in a trapped two-dimensional polarized Fermi gas with spin-orbit coupling and across a BCS-BEC crossover. Due to the competition between polarization, pairing interaction and spin-orbit coupling, the Fermi gas typically phase separates in the trap. Employing a mean field approach that guarantees the ground state solution, we systematically study the structure of the phase separation and investigate in detail the optimal parameter region for the preparation of the topologically non-trivial superfluid phase. We then calculate the momentum space density distribution of the topological superfluid state and demonstrate that the existence of the phase leaves a unique signature in the trap integrated momentum space density distribution which can survive the time-of-flight imaging process.

Abstract:
We consider a trapped atomic system in the presence of spatially varying laser fields. The laser-atom interaction generates a pseudospin degree of freedom (referred to simply as spin) and leads to an effective spin-orbit coupling for the fermions in the trap. Reflections of the fermions from the trap boundaries provide a physical mechanism for effective momentum relaxation and non-trivial spin dynamics due to the emergent spin-orbit coupling. We explicitly consider evolution of an initially spin-polarized Fermi gas in a two-dimensional harmonic trap and derive non-equilibrium behavior of the spin polarization. It shows periodic echoes with a frequency equal to the harmonic trapping frequency. Perturbations, such as an asymmetry of the trap, lead to the suppression of the spin echo amplitudes. We discuss a possible experimental setup to observe spin dynamics and provide numerical estimates of relevant parameters.

Abstract:
We show that spin-orbit coupling (SOC) gives rise to pairing instability in a highly polarized two-dimensional Fermi gas for arbitrary interaction strength. The pairing instability can lead to a Fulde-Ferrell-Larkin-Ovchinnikov-like molecular state, which undergoes a first-order transition into a pairing state with zero center-of-mass momentum as the parameters are tuned. These pairing states are metastable against a polaron state dressed by particle-hole fluctuations for small SOC. At large SOC, a polaron-molecule transition exists, which suggests a phase transition between the topological superfluid state and the normal state for a highly polarized Fermi gas in the thermodynamic limit. As polarization in a Fermi gas with SOC is induced by the effective Zeeman field, we also discuss the influences of the effective Zeeman field on the ground state of the system. Our findings may be tested directly in future experiments.

Abstract:
We consider a time-dependent non-linear Schr\"odinger equation in one dimension (1D) with a fifth-order interaction term and external harmonic confinement, as a model for both (i) a Bose gas with hard-core contact interactions in local-density approximation, and (ii) a spin-polarized Fermi gas in the collisional regime. We evaluate analytically in the Thomas-Fermi limit the density fluctuation profiles and the collective excitation frequencies, and compare the results for the low-lying modes with those obtained from the numerical solution of the Schr\"odinger equation. We find that the excitation frequencies are multiples of the harmonic-trap frequency even in the strong-coupling Thomas-Fermi regime. This result shows that the hydrodynamic and the collisionless collective spectra coincide in the harmonically confined 1D Fermi gas, as they do for sound waves in its homogeneous analogue. It also shows that in this case the local-density theory reproduces the exact collective spectrum of the hard-core Bose gas under harmonic confinement.

Abstract:
We systematically develop a density functional description for the equilibrium properties of a two-dimensional, harmonically trapped, spin-polarized dipolar Fermi gas based on the Thomas-Fermi von Weizs\"acker approximation. We pay particular attention to the construction of the two-dimensional kinetic energy functional, where corrections beyond the local density approximation must be motivated with care. We also present an intuitive derivation of the interaction energy functional associated with the dipolar interactions, and provide physical insight into why it can be represented as a local functional. Finally, a simple, and highly efficient self-consistent numerical procedure is developed to determine the equilibrium density of the system for a range of dipole interaction strengths.

Abstract:
We consider the time evolution of the magnetization in a Rashba spin-orbit-coupled Fermi gas, starting from a fully-polarized initial state. We model the dynamics using a Boltzmann equation, which we solve in the Hartree-Fock approximation. The resulting non-linear system of equations gives rise to three distinct dynamical regimes with qualitatively different asymptotic behaviors of the magnetization at long times. The distinct regimes and the transitions between them are controlled by the interaction strength: for weakly interacting fermions, the magnetization decays to zero. For intermediate interactions, it displays undamped oscillations about zero and for strong interactions, a partially magnetized state is dynamically stabilized. The dynamics we find is a spin analog of interaction induced self-trapping in double-well Bose Einstein condensates. The predicted phenomena can be realized in trapped Fermi gases with synthetic spin-orbit interactions.