Ground States of the Spin-1 Bose-Hubbard Model
  We prove basic theorems about the ground states of the S=1 Bose-Hubbard model. The results are quite universal and depend only on the coefficient U_2 of the spin-dependent interaction. We show that the ground state exhibits saturated ferromagnetism if U_2<0, is spin-singlet if U_2>0, and exhibits "SU(3)-ferromagnetism" if U_2=0, and completely determine the degeneracy in each region. 
Hosho Katsura and Hal Tasaki
cond-mat.quant-gas cond-mat.stat-mech