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. 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.