Insulator-superfluid transition of spin-1 bosons in an optical lattice in magnetic field

We study the insulator-superfluid transition of spin-1 bosons in an optical lattice in a uniform magnetic field. Based on a mean-field approximation we obtained a zero-temperature phase diagram. We found that depending on the particle number the transition for bosons with antiferromagnetic interaction may occur into different polar superfluid phases with spins aligned along or opposite to the field direction. This is qualitatively different from the field-free transition for which the mean-field theory predicts a unique polar superfluid state for any particle number