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We study a quantum version of the three-state Potts model that includes asspecial cases the effective models of bosons and fermions on the square latticein the Mott insulating limit. It can be viewed as a model of quantumpermutations with amplitudes J_parallel and J_perp for identical and differentcolors, respectively. For J_parallel=J_perp>0, it is equivalent to the SU(3)Heisenberg model, which describes the Mott insulating phase of 3-colorfermions, while the parameter range J_perp<min(0,-J_parallel) can be realizedin the Mott insulating phase of 3-color bosonic atoms. Using linear flavor wavetheory, infinite projected entangled-pair states (iPEPS), and continuous-timequantum Monte-Carlo simulations, we construct the full T=0 phase diagram, andwe explore the T>0 properties for J_perp<0. For dominant antiferromagneticJ_parallel interactions, a three-sublattice long-range ordered stripe state isselected out of the ground state manifold of the antiferromagnetic Potts modelby quantum fluctuations. Upon increasing |J_perp|, this state is replaced by auniform superfluid for J_perp<0, and by an exotic three-sublattice superfluidfollowed by a two-sublattice superfluid for J_perp>0. The transition out of theuniform superfluid (that can be realized with bosons) is shown to be a peculiartype of Kosterlitz-Thouless transition with three types of elementary vortices.

physical review. b, condensed matter and materials physics

Competition between three-sublattice order and superfluidity in the  quantum 3-state Potts model of ultracold bosons and fermions on a square  optical lattice