Magnetic phases of bosons with synthetic spin-orbit coupling in optical lattices

We investigate magnetic properties in the superfluid and Mott-insulating states of two-component bosons with spin-orbit (SO) coupling in 2D square optical lattices. The spin-independent hopping integral $t$ and SO coupled one $\lambda$are fitted from band structure calculations in the continuum, which exhibit oscillations as increasing SO coupling strength. The magnetic superexchange model is derived in the Mott-insulating state with one-particle per-site, characterized by the Dzyaloshinsky-Moriya (DM) interaction. In the limit of $|\lambda|\ll |t|$, we find a spin spiral Mott state whose pitch value is the same as that in the incommensurate superfluid state, while in the opposite limit $|t| \ll |\lambda|$, the ground state can be found by a dual transformation to the $|\lambda|\ll|t|$ limit.