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

The largest allowed symmetry in a spin-1 quantum system is an SU(3) symmetry rather than the SO(3) spin rotation. We reveal some SU(2) symmetries as subgroups of SU(3) that, to the best of our knowledge, have not previously been recognized. Then, we construct SU(2) symmetric Hamiltonians and explore the ground-state phase diagram in accordance with the SU(3)⊃SU(2)×U(1) symmetry hierarchy. It is natural to treat the eight generators of the SU(3) symmetry on an equal footing; this approach is called the eight-fold way. We find that the spin spectral functions and spin quadrupole spectral functions share the same structure, provided that the elementary excitations are flavor waves at low energies, which serves as a clue to the eight-fold way. An emergent S=1/2 gapless quantum spin liquid is found to coexist with spin nematic order in one of the ground states.