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Abundant quadrupolar or nematic phases driven by the Heisenberg interactions in a spin-1 dimer system forming a bilayer

We explore several classes of quadrupolar ordering in a system of antiferromagnetically coupled quantum spin-1 dimers, which are stacked in the triangular lattice geometry forming a bilayer. Low-energy properties of this model is described by an $\mathcal{S}=1$ hard-core bosonic degrees of freedom defined on each dimer-bond, where the singlet and triplet states of the dimerized spins are interpreted as the vacuum and the occupancy of boson, respectively. The number of bosons per dimer and the magnetic and density fluctuations of bosons are controlled by the inter-dimer Heisenberg interactions. In a solid phase where each dimer hosts one boson and the inter-dimer interaction is weak, a conventional spin nematic phase is realized by the pair-fluctuation of bosons. Larger inter-dimer interaction favors Bose Einstein condensates (BEC) carrying quadrupolar moments. Among them, we find one exotic phase where the quadrupoles develop a spatially modulated structure on the top of a uniform BEC, interpreted in the original dimerized spin-1 model as coexistent $p$-type nematic and 120$^\circ$-magnetic correlations. This may explain an intriguing nonmagnetic phase found in Ba$_{3}$ZnRu$_{2}$O$_{9}$.