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We study a spin system with both two- and four-spin exchange interactions onthe triangular lattice as a possible model for the nuclear magnetism of solid$^3$He layers adsorbed on grafoil. The ground state is analyzed by the use ofthe mean-field approximation. It is shown that the four-sublattice state isfavored by introduction of the four-spin exchange interaction. A possible phasetransition at a finite temperature into a phase with the scalar chirality ispredicted. Application of a magnetic field is shown to cause a variety of phasetransitions.Comment: 5 pages, Revte

Ground state of a spin system with two- and four-spin exchange interactions on the triangular lattice