Hidden SU(2) Symmetries, Symmetry Hierarchy and Emergent Eight-Fold-Way in Spin-1 Quantum Magnets*

The allowed largest symmetry in quantum spin-1 systems is SU(3) rather than spin rotational SO(3). In this work, we study some SU(2) symmetries as subgroups of the SU(3), which was not bewared in literature to the best of our knowledge. We construct two-body Hamiltonians where these SU(2) symmetries are respected, and explore the ground phase diagram according to the symmetry hierarchy SU(3)/SU(2)×U(1). It is natural to treat the eight generators of the SU(3) symmetry on an equal footing, called eight-fold-way. And possible experimental consequences are discussed as well.