We study cold atoms in an optical lattice with synthetic spin-orbit coupling in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model using Peierls substitution and "localized Wannier states method" and derive the low-energy spin Hamiltonian for bosons and fermions. The spin Hamiltonian is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interactions and it has a rich classical phase diagram with collinear, spiral and vortex phases. We discuss the state of the art of experiments to realize and detect magnetic orderings in strongly correlated optical lattices.

Realistic methods to create vortices in spin-orbit-coupled Bose-Einstein condensates are discussed. It is shown that, contrary to common intuition, rotation of the trap containing a spin-orbit condensate does not lead to an equilibrium state with static vortex structures, but gives rise instead to non-equilibrium behavior described by an intrinsically time-dependent Hamiltonian. We propose here the following alternative methods to induce thermodynamically stable static vortex configurations: (1) to rotate both the lasers and the anisotropic trap; and (2) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. Effective Hamiltonians for spin-orbit condensates under such perturbations are derived for most currently known realistic laser schemes that induce synthetic spin-orbit couplings. The Gross-Pitaevskii equation is solved for several experimentally relevant regimes. The new interesting effects include spatial separation of left-and right-moving spin-orbit condensates, the appearance of unusual vortex arrangements, and parity effects in vortex nucleation where the topological excitations are predicted to appear in pairs. All these phenomena are shown to be highly non-universal and depend strongly on a specific laser scheme and system parameters.

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.The physics of spin-orbit coupling is responsible for a wide range of physical phenomena such as atomic spectra, the spin Hall effect and topological insulators [1][2][3]. More recently, the creation of artificial gauge fields [4,5] and spin-orbit coupling [6] in ultra-cold Bose and Fermi systems [7][8][9] has brought spin-orbit-coupled systems to the forefront of research in atomic, molecular, and optical physics. Furthermore, cold atoms bring to bear new tools for studying correlated systems. One such tool is the ability to dynamically tune the single-particle and many body energy scales [10]. By varying system parameters adiabatically or diabatically, experimentalists can probe interacting states both in and out of equilibrium. Such studies have not only provided access to thermodynamic quantities, but have also yielded insights into how correlations develop and spread across a system following a parameter quench, how long-range order is established, and the mechanisms underlying thermalization in isolated interacting systems [11][12][13][14][15]. Here we ask: what is the interplay between spin-orbit coupling and interactions in a gas which is driven out-of-equilibrium?We study spin dynamics occurring in a weakly interacting, uniform spin-orbit coupled Fermi gas, which is initially spin polarized. We assume that the momentum distribution in the initial state is the classical Maxwell distribution. Interplay between classical motion in the thermal gas, quantum spin degrees of freedom subject to spin-orbit coupling (SOC) and interactions gives rise to interesting dynamical regimes, which are the focus of this work. Our main results are summarized in Fig. 1: we find that there are three possible distinct steady states which can be labelled as: unpolarized state, oscillating magnetization state, and partially polarized state. Different steady states correspond to different values of the ratio of the interaction strength to the SOC strength (denoted as λ), and we find clear transitions between different regimes as we change this λ. FIG. 1:(color online) Schematic plot s...