Cluster mean field theory for 2-dimensional spin-1 Bose Hubbard model

Abstract

The Cluster Mean Field Theory(CMFT) for 2-dimensional Spin-1 Bose-Hubbard model is applied to study the superfluid (SF) to Mott insulator (MI) phase transitions and various magnetic phases, that arise in the presence of spin-dependent anti-ferromagnetic and ferromagnetic interactions. This study clearly shows the nematic behavior of the SF and odd density MI phases, and a continuous phase transition between them for the anti-ferromagnetic interaction. The even density MI shows nematic or singlet phase depending upon the strength of the interaction, and a continuous crossover between them. The SF to singlet MI transition is discontinuous. The dependence of cluster size on the critical on-site interaction ($U_0^C$) for the SF to MI transitions is also studied. In case of anti-ferromagnetic interaction, it is found that $U_0^C$ for the SF to odd density MI transition decreases with cluster size, however, no such changes are observed for SF to singlet MI transition. In case of ferromagnetic interactions, SF to MI transitions are continuous, and Mott lobes get enlarged with cluster size. The second order R\'{e}nyi Entanglement Entropy (EE) in different phases are calculated. The results show that R\'{e}nyi EE is large in the nematic MI phase.