Possible spin liquid states with parton Fermi surfaces in the SU(3) ring-exchange model on the triangular lattice

We consider a SU(3) ring-exchange model on a triangular lattice. Unlike the SU(2) case, under perturbation expansion of the SU(3) Hubbard model, the three-site ring exchange is present as well as the usual four-site ring exchange. Interestingly, the three-site ring exchange differs from the usual two-site and four-site exchanges by a minus sign and is ferromagnetic. We first present numerical site-factorized state studies on this model which shows a three-sublatticed order phase and a ferromagnetic phase. We further study the model using slave-fermion mean field in which we rewrite the exchange operators in terms of three flavors of fermions. We find the main competing trial states are the trimer state (triangular plaquette state) and the gapless U(1) spin liquid states with parton Fermi surfaces which include both the uniform zero-flux spin liquid state, and the uniform $\pi$-flux spin liquid state. Furthermore, we find there are possible pairing instabilities of the zero-flux (Fermi surface) spin liquid state toward a $f$-wave gapless (nodal) spin liquid state and the $\pi$-flux (Fermi surface) spin liquid state toward an interesting exotic $s$-wave “gapless” spin liquid state with two flavors of fermions paired up while one flavor of fermions remains gapless.