Spin Supersolid Phase and Double Magnon-Roton Excitations in a Cobalt-based Triangular Lattice

Supersolid is an exotic quantum state of matter that hosts spontaneously the features of both solid and superfluidity, which breaks the lattice translational symmetry and U (1) gauge symmetry. Here we conduct inelastic neutron scattering (INS) measurements and tensor-network calculations on the triangular-lattice cobaltate Na₂BaCo (PO₄) ₂, which is proposed in Xiang et al. , Nature 625, 270-275 (2024) as a quantum magnetic analog of supersolid. We uncover characteristic dynamical signatures, which include distinct magnetic Bragg peaks indicating out-of-plane spin solidity and gapless Goldstone modes corresponding to the in-plane spin superfluidity, offering comprehensive spectroscopic evidence for spin supersolid in Na₂BaCo (PO₄) ₂. We also compute spin dynamics of the easy-axis triangular-lattice model, and reveal magnon-roton excitations containing U (1) Goldstone and roton modes associated with the in-plane spin superfluidity, as well as pseudo-Goldstone and roton modes related to the out-of-plane spin solidity, rendering double magnon-roton dispersions in the spin supersolid. Akin to the role of phonon-roton dispersion in shaping the helium thermodynamics, the intriguing magnetic excitations also strongly influence the low-temperature thermodynamics of spin supersolid down to sub-Kelvin regime, explaining the recently observed giant magnetocaloric effect in Na₂BaCo (PO₄) ₂.