Double magnon-roton excitations in the triangular-lattice spin supersolid

Supersolid is an exotic quantum state of matter that spontaneously hosts the features of both solid and superfluid, which breaks the translation and U(1) gauge symmetries. Here we study the spin dynamics in the triangular-lattice compound ${\mathrm{Na}}_{2}\mathrm{BaCo}{({\mathrm{PO}}_{4})}_{2}$, which is revealed in [Xiang et al., Nature (London) 625, 270 (2024)] as a quantum magnetic analog of supersolid. We simulate the easy-axis Heisenberg model with tensor network approach and uncover unique dynamic traits. These features are manifested in two branches of excitations that can be associated with the spin solidity and superfluidity, respectively. One branch contains the U(1) Goldstone and roton modes, while the other comprises pseudo-Goldstone and roton modes. The gapless Goldstone modes of the in-plane superfluid order are confirmed by our inelastic neutron scattering measurements. Together with the evident out-of-plane solid order indicated by the magnetic Bragg peaks, our findings provide spectroscopic evidence for spin supersolidity in this easy-axis antiferromagnet. Akin to the role of phonon-roton modes---Landau elementary excitations---in shaping the helium superfluid thermodynamics, the intriguing double magnon-roton dispersion here determines the low-temperature thermodynamics of spin supersolids down to sub-Kelvin regime, explaining the recently observed giant magnetocaloric effect in ${\mathrm{Na}}_{2}\mathrm{BaCo}{({\mathrm{PO}}_{4})}_{2}$.