Spin-supersolidity induced quantum criticality and magnetocaloric effect in the triangular-lattice antiferromagnet Rb$_2$Co(SeO$_3$)$_2$

We performed high-field magnetization, magnetocaloric effect (MCE), and NMR measurements on the Ising triangular-lattice antiferromagnet Rb$_2$Co(SeO$_3$)$_2$. The observations of the 1/3-magnetization plateau, the split NMR lines, and the thermal activation behaviors of the spin-lattice relaxation rate $1/T_1$ between 2 T and 15.8 T provide unambiguous evidence of a gapped up-up-down (UUD) magnetic ordered phase. For fields between 15.8 T and 18.5 T, the anomaly in the magnetic susceptibility, the slow saturation of the NMR line spectral ratio with temperature, and the power-law temperature dependence of $1/T_1$ suggest the ground state to be a spin supersolid with gapless spin excitations. With further increasing the field, the Gr\"{u}neisen ratio, extracted from the MCE data, reveals a continuous quantum phase transition at $H_{\rm C}\approx$ 19.5 T and a universal quantum critical scaling with the exponents ${\nu}z~\approx~$1. Near $H_{\rm C}$, the large high-temperature MCE signal and the broad peaks in the NMR Knight shift and $1/T_1$, manifest the strong spin fluctuations driven by both magnetic frustration and quantum criticality. These results establish Rb$_2$Co(SeO$_3$)$_2$ as a candidate platform for cryogenic magnetocaloric cooling.