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We investigate the superfluid--Mott-insulator quantum phase transition ofspin-1 bosons in an optical lattice created by pairs of counterpropagatinglinearly polarized laser beams, driving an $F_g=1$ to $F_e=1$ internal atomictransition. The whole parameter space of the resulting two-componentBose-Hubbard model is studied. We find that the phase transition is not alwayssecond order as in the case of spinless bosons, but can be first order incertain regions of the parameter space. The calculations are done in themean-field approximation by means of exact numerical diagonalization as well aswithin the framework of perturbaton theory

First- and second-order superfluid–Mott-insulator phase transitions of spin-1 bosons with coupled ground states in optical lattices