The MFA ground states for the extended Bose-Hubbard model with a three-body constraint

We address the intensively studied extended bosonic Hubbard model (EBHM) with truncation of the on-site Hilbert space to the three lowest occupation states $n=0,1,2$ in frames of the $S=1$ pseudospin formalism. Similar model was recently proposed to describe the charge degree of freedom in a model high-T ${}_c$ cuprate with the on-site Hilbert space reduced to the three effective valence centers, nominally Cu^1+;2+;3+. With small corrections the model becomes equivalent to a strongly anisotropic $S=1$ quantum magnet in an external magnetic field. We have applied a generalized mean-field approach and quantum Monte-Carlo technique for the model 2D $S=1$ system with a two-particle transport to find the ground state phase with its evolution under deviation from half-filling.