Supersolid phase in a U (2) symmetric S = 1 magnet on the triangular lattice

A spin supersolid is characterized by the simultaneous breaking of lattice translation and continuous spin rotation symmetries. In this work, we study a spin-1 model with U (2) ≅ SU (2) × U (1)/Z₂ symmetry on the triangular lattice, and determine the phase diagram with a variational $ℂP^2$ approach. We identify a novel supersolid phase which contains a 3-sublattice solid order and a superfluid order with spontaneous SU (2)-symmetry breaking. Unlike usual supersolid phases having only one Goldstone mode, the SU (2)-supersolid phase has two Goldstone modes. Another important feature of this supersolid is that the spin excitation spectrum has symmetry protected double degeneracy in the whole Brillouin zone. As by-products, several other ordered phases are obtained, including the ferromagnetic and the antiferromagnetic states breaking the SU (2) symmetry, as well as genuine phases that completely break the U (2) symmetry. Furthermore, the instabilities of SU (3)-flavor linear spin-wave theory are consistent with the phase boundaries between different ordered phases.