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Using the density matrix renormalization group method, we studied the groundstate of the one-dimensional $S=1$ Bose-Hubbard model with local three-bodyinteractions, which can be a superfluid or a Mott insulator state. We drew thephase diagram of this model for both ferromagnetic and antiferromagneticinteraction. Regardless of the sign of the spin-dependent coupling, we obtainedthat the Mott lobes area decreases as the spin-dependent strength increases,which means that the even-odd asymmetry of the two-body antiferromagnetic chainis absent for local three-body interactions. For antiferromagnetic coupling, wefound that the density drives first-order superfluid-Mott insulator transitionsfor even and odd lobes. Ferromagnetic Mott insulator and superfluid states wereobtained with a ferromagnetic coupling, and a tendency to a "long-range" orderwas observed.

Mott lobes of the $S=1$ Bose-Hubbard model with three-body interactions