We report ultralow-temperature ${}^{23}\mathrm{Na}$ NMR measurements on the Ising triangular-lattice antiferromagnet ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$, which precisely resolve the phase diagram under a magnetic field applied along the crystalline $c$ axis. With increasing field, the NMR spectra resolve three ordered phases with distinct spin configurations: the Y, up-up-down (UUD), and V phases. In addition, separated ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ phases are also identified inside the V phase. The spin-lattice relaxation rate $1/T_1$ demonstrates gapless excitations in both the Y and V phases down to 0.04 K, strongly supporting the spin supersolid nature. However, the phase transitions from the UUD phase to the two supersolid phases exhibit dramatically different behaviors upon cooling. Prior to entering the Y phase, $1/T_1$ identifies a gapless regime within the UUD phase, suggesting a Berezinskii-Kosterlitz-Thouless-like phase above a second-order phase transition. In contrast, the coexistence of the UUD and V phases observed in our experiments provides evidence of a first-order phase transition between them.

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• Frustrated magnetism
• Quantum phase transitions
• Antiferromagnets
• Triangular lattice
• Nuclear magnetic resonance