Abstract
We consider a SU(3) model with antiferromagnetic three-site ring exchanges, in addition to two-site exchanges, on the triangular lattice. We first present numerical site-factorized state studies on the magnetic ordered states, which show two different three-sublattice-ordered states---the antiferroquadrupolar phase and the standard ${120}^{\ensuremath{\circ}}$ antiferromagnetic phase---along the axis of the strength of the three-site ring exchanges. We further study the model using slave-fermion mean field approaches in which we rewrite the exchange operators in terms of three flavors of fermions. At the mean-field level, we find that the main competing trial states are the trimer state (triangular plaquette state), gapped uniform $\ensuremath{\pi}/3$-flux chiral spin liquid, and gapped uniform $2\ensuremath{\pi}/3$-flux chiral spin liquid. The filled band of the $\ensuremath{\pi}/3$-flux chiral spin liquid has Chern number $+1$, and that of the $2\ensuremath{\pi}/3$-flux chiral spin liquid has Chern number $+2$. We also give the effective Chern-Simons theory for each chiral spin liquid at the mean-field level.
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