Spin Liquid States on the Triangular and Kagome Lattices: A Projective Symmetry Group Analysis of Schwinger Boson States

A symmetry based analysis (Projective Symmetry Group) is used to study spin liquid phases on the triangular and Kagom\'e lattices in the Schwinger boson framework. A maximum of eight distinct $Z_2$ spin liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest neighbor amplitudes which are studied in greater detail. On the triangular lattice, only two such states are present - the first (zero-flux state) is the well known state introduced by Sachdev, which on condensation of spinons leads to the 120 degree ordered state. The other solution which we call the $\pi$-flux state has not previously been discussed. Spinon condensation leads to an ordering wavevector at the Brillouin zone edge centers, in contrast to the 120 degree state. While the zero-flux state is more stable with just nearest-neighbor exchange, we find that the introduction of either next-neighbor antiferromagnetic exchange or four spin ring-exchange (of the sign obtained from a Hubbard model) tends to favor the $\pi$-flux state. On the Kagom\'e lattice four solutions are obtained - two have been previously discussed by Sachdev, which on spinon condensation give rise to the $q=0$ and $\sqrt{3}\times\sqrt{3}$ spin ordered states. In addition we find two new states with significantly larger values of the quantum parameter at which magnetic ordering occurs. For one of them this even exceeds unity, $\kappa_c\approx 2.0$ in a nearest neighbor model, indicating that if stabilized, could remain spin disordered for physical values of the spin. This state is also stabilized by ring exchange interactions with signs as derived from the Hubbard model.