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We consider a SU(3) model with antiferromagnetic three-site ring exchanges,in addition to two-site exchanges, on the triangular lattice. We first presentnumerical site-factorized state studies on the magnetic ordered states, whichshows two different three-sublattice-ordered states, the antiferro-quadrupolarphase and the standard 120^o anti-ferromagnetic phase, along the axis of thestrength of the three-site ring exchanges. We further study the model usingslave-fermion mean field approaches in which we rewrite the exchange operatorsin terms of three flavors of fermions. At the mean-field level, we find themain competing trial states are the trimer state (triangular plaquette state),gapped uniform \pi/3-flux chiral spin liquid, and gapped uniform 2\pi/3-fluxchiral spin liquid. The filled band of the \pi/3-flux chiral spin liquid hasChern number +1, and that of the 2\pi/3-flux chiral spin liquid has Chernnumber +2. We also give the effective Chern-Simons theory for each chiral spinliquid at the mean-field level.

Possible uniform-flux chiral spin liquid states in the SU(3) ring-exchange model on the triangular lattice