From Néel long-range order to spin liquids in the multiple-spin exchange model

The phase diagram of the multiple-spin exchange model on the triangular lattice is studied using exact diagonalizations. The two-spin ${(J}_2)$ and four-spin ${(J}_4)$ exchanges have been taken into account for 12-, 16-, 19-, 21-, 24-, and 27-site samples in the parameter region $J_4=0\mathrm{}$–0.25 (for a fixed $J_2=1).\mathrm{}$ It is found that the three-sublattice Néel-ordered state built up by the pure two-spin exchange can be destroyed by the four-spin exchange, forming a spin-liquid state. The different data suggest that the phase diagram in this range of parameters exhibits two phases. The pure $J_2$ phase is a three-sublattice Néel-ordered phase; a small $J_4$ drives it into a spin-liquid state with a spin gap filled of a large number of singlets. This spin-liquid phase is not of the same generic kind as the phase studied by Misguich et al. [Phys. Rev. B 60, 1064 (1999)]. It is observed on the finite-size samples that the spin-liquid phase, as the Néel-ordered phase, exhibits a magnetization plateau at $m=1/3,$ and for $J_4>0.15\mathrm{}$ a second plateau at $m=1/2.\mathrm{}$ These two plateaus are associated, respectively, with the semiclassical orderings $\mathrm{uud}$ and $\mathrm{uuud}.\mathrm{}$