Spin Nematic and Spin Supersolid States on the Spin-1 Triangular Heisenberg Antiferromagnets

Motivated by recent experiments on the triangular lattice compound Na₂BaNi(PO₄)₂, we numerically study its effective spin-1 antiferromagnetic Heisenberg model on a triangular lattice in the presence of out-of-plane magnetic fields. Using density matrix renormalization group methods, we obtain its ground state properties such as the spin correlations and the quadrupole order parameters, which suggest a spin nematic that originates from the pair condensate. Furthermore, we study the superfluid stiffness that is calculated through a pi-phase twist at the cylindrical boundary. We also investigate the dynamical spin and quadrupolar structure factor of the ground states under various magnetic fields that can characterize the nature of their excitations. The finite superfluid stiffness as well as the gapless Goldstone mode provides a strong numerical evidence of the spin supersolid state that is relevant to this material.