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Overcomplete Bases for S = 1 Spin Liquids

For a $S=1$ system with even number of spins, the product states of two-body singlets, called the singlet pair states (SPSs), are overcomplete bases for the Hilbert space of many-body singlets. If the system contains odd number of spins, a singlet state can be decomposed as a superposition of all of the following configurations, in each configuration three of the spins form a three-body singlet and the remaining form two-body singlet pairs. This indicates that a $S=1$ spin liquid is especially a resonating-valence bond state. Although this conclusion is no longer valid for $SO(3)$ symmetric $S>1$ systems, it can be generalized to an integer spin-$S$ system if it has an enlarged $SO(2S+1)$ symmetry. Similar results can also be obtained for systems with $SU(n)$ symmetry.