Hidden SU(2) symmetries, the symmetry hierarchy, and the emergent eightfold 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. In this work, 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) superset of SU(2) x 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 eightfold 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 eightfold way. An emergent S = 1/2 quantum spin liquid is proposed to coexist with gapful spin nematic order in one of the ground states. In analogy to quantum chromodynamics, we find the gap relation for hydrodynamic modes in quantum spin-orbital liquid states, which is nothing but the Gell-Mann-Okubo formula.