Superfluid-Mott insulator transition of spin-1 bosons in optical lattice under magnetic field

We study the first- and second-order superfluid-Mott insulator transitions of spin-1 bosons in an optical lattice under a magnetic field by the Gutzwiller approximation. We find that, in contrast to previous perturbative studies, the phase boundary curve continuously changes as a function of magnetic field. We also find a sharp cusp structure on the phase boundary curve under some circumstances. When the phase boundary curve deviates from that obtained with the perturbative studies, the phase transition is a first-order one. We further clarify the specific features of the phase transition in the present system. The order parameters exactly at the boundary point, where the transition changes from first to second order, remain finite under a finite magnetic field and continuously vanish as the magnetic field approaches zero.