Effective spin models for spinor lattice gases with gauge fields

We study the effective spin models for a Mott insulating Bose-Hubbard model in the presence of gauge fields. We first introduce a simple method based on adiabatic elimination to determine the low-energy effective Hamiltonian. Demonstration of a two-component one-dimensional model with synthetic magnetic field is shown. We further consider a generalized Bose-Hubbard model which covers the cases where spin-orbit coupling and synthetic flux are present. The resulting XYZ model contains Dzyaloshinskii-Moriya interactions and symmetric anisotropic couplings. Under certain conditions, this model reduces to various interesting magnetic Hamiltonians. Phase diagrams for two typical situations are obtained. We investigate magnetic phase transitions by focusing on order parameters and spin-spin correlations. Results are validated by numerical calculations using tensor network algorithms.