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We investigate a highly frustrated spin-1 model on the triangular lattice,with nearest- and next-nearest-neighbor antiferromagnetic $S\cdot S$interactions and nearest-neighbor $(S\cdot S)^2$ interactions. Using thedensity matrix renormalization group (DMRG) technique, we find threemagnetically ordered phases, namely 120$^\circ$ spiral order, stripe order, andtetrahedral order, as well as two spin nematic phases: ferroquadrupolar andantiferroquadrupolar. While our data could be consistent with a spin liquidphase between the 120$^\circ$ spiral and antiferroquadrupolar orders, the morelikely scenario is a direct continuous transition between these two orders.

Phase diagram of a bilinear-biquadratic spin-1 model on the triangular lattice from density matrix renormalization group simulations