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.

FIG. 3. (Color online) The locus of wave vectors for different values of J2/J1 giving the minimal energy of the Jastrow variational wave functions. The results are shown for the XY model with L = 18 (i.e., 648 sites). FIG. 6. (Color on-line) Finite-size scaling of the energy of the spin-1/2 XY model for different values of J2/J1 in the Néel and collinear phases. Lines depict the results of fits according to Eq. (13). Red circles (blue squares) indicate the Néel (collinear) phase.J 2 /J 1 ∈ [0.0, 0.30) J 2 /J 1 ∈ (0.35, 0.65) J 2 /J 1 ∈ (0.70, 1.40) J 2 /J 1 = 1.45 J 2 /J 1 ∈ (1.50, 2.00) J 2 /J 1 ∈ (2.10, 3.30) J 2 /J 1 ∈ (3.40, 5.00] FIG. 7. (Color online) The same as Fig. 3, but for the Heisenberg model.