Gutzwiller study of spin-1 bosons in an optical lattice under a magnetic   field
  We study spin-1 bosons in an optical lattice under a magnetic field by the Gutzwiller approximation. Our results thus obtained join the discontinuous phase boundary curves obtained by perturbative studies through a first-order transition. On the phase boundary curve, we also find a peculiar cusp structure originating from the degeneracy between different spin Mott states under a magnetic field. The magnetic field dependence of both fluctuation in the total number of bosons and the spin magnetization clarifies that the superfluid phase is divided into two regions reflecting the coexisting first- and second-order superfluid transitions. 
Takashi Kimura, Shunji Tsuchiya, Makoto Yamashita, and Susumu Kurihara