Temporal and field evolution of spin excitations in the disorder-free triangular antiferromagnet ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$

The realization of quantum spin liquids on a triangular lattice magnet remains challenging due to quenched disorders present in real materials. Here, we report local-probe signatures of spin-liquid-like excitations in the structurally perfect triangular antiferromagnet ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$. Our zero-field and longitudinal-field muon spin relaxation measurements reveal both spatially and temporally correlated spins with no hint of static magnetism down to 80 mK, consistent with a quantum spin liquid. Comprehensive ${}^{23}\mathrm{Na}$ nuclear magnetic resonance experiments enable us to identify a field-induced crossover of magnetic excitations from a spinonlike to a gapped magnon at the critical field ${\mu }_0{H}_C=1.65$ T. Our findings uncover the temporal, thermal, and magnetic-field characteristics of the magnetic excitations in the triangular-lattice quantum spin liquid candidate ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$.

Figure 1

Crystal structure of ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$. The triangular layers consist of the coupled Co spins through the super-superexchange interaction Co-O-O-Co in the $ab$ plane, which are separated by ${\mathrm{BaO}}_{12}$ and ${\mathrm{NaO}}_{10}$ polyhedra. The asymmetric Na polyhedron comprises the combined structure of a triangular cupola and hexagonal pyramid. The solid blue lines connecting the Co atoms in the $ab$ plane represent the intratriangular exchange coupling $J$. The ${\mathrm{PO}}_4$ tetrahedra, ${\mathrm{CoO}}_6$ octahedra, and ${\mathrm{BaO}}_{12}$ polyhedra are omitted for clarity.

Figure 2

(a) Representative ZF-$\mu \mathrm{SR}$ spectra of ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$ at selected temperatures. The colored solid curves represent the fits to the data. (b) Temperature dependence of the muon spin relaxation rate ${\lambda }_{Z}F$ on a log-log scale. The solid blue line denotes the power-law behavior ${\lambda }_{Z}F\sim T^{-0.22\left(1\right)}$. (c) The stretched exponent ${\beta }_{Z}F$ as a function of temperature.

Figure 3

(a) LF-$\mu \mathrm{SR}$ spectra of ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$ collected at $T=80$ mK in various applied fields. The colored solid curves represent the fits to the data. (b) LF dependence of the muon spin relaxation rate ${\lambda }_{L}F\left(H\right)$. The black dotted and solid blue curves denote the fits using the Redfield model ($x=0$) and the general expression of ${\lambda }_{L}F\left(H\right)$ with $x=0.54$, respectively, as detailed in the main text. (c) The stretched exponent ${\beta }_{L}F$ as a function of LF.

Figure 4

${}^{23}\mathrm{Na}$ NMR spectra at various temperatures measured at a fixed magnetic field of 4 T (a) perpendicular and (b) parallel to the $c$ axis of the single crystal. Two symmetric satellite peaks arise from the nonzero local electric field gradient. C and ${\mathrm{S}}_i$ ($i=1,2$) indicate the center peak and satellite peaks, respectively. (c) Angular dependence of the frequency difference $f_C-f_{S_i}$ between the main peak and the satellite peaks. Note that $\theta =0^{\circ }\left({90}^{\circ }\right)$ corresponds to the field direction of ${\mu }_{0}H//c\left({\mu }_{0}H//ab\right)$. The solid curve represents the fit using the electric field gradient model for the nuclear spin $I=3/2$. (d) Temperature dependence of the FWHM for three NMR signals.

Figure 5

(a) Temperature dependence of the Knight shift in magnetic fields of ${\mu }_{0}H=1.5,2,2.6,3$, and 4 T with the dc magnetic susceptibility ${\chi }_{a}b\left(T\right)$ (blue solid curve). The orange dashed line is a guide to the eye for the power-law $T^{-0.45\left(1\right)}$. (b) Plot of the nuclear magnetization recovery curves vs time at selected temperatures at ${\mu }_{0}H=4$ T. The solid curves represent the fits to the data, as described in the experimental section. (c) Temperature dependence of the stretching exponent $\beta$ in magnetic fields of ${\mu }_{0}H=2,3$, and 4 T obtained from the fitting of the nuclear magnetization recovery curves. (d) Log-log plot of the spin-lattice relaxation rate $1/T_1$ as a function of temperature and field. The blue arrows indicate the onset temperature of the thermal activation behavior $T^{*}$ at the respective magnetic fields. (e) Temperature and field dependences of $1/T_{1}T$ on a double-logarithmic scale. The dotted lines represent the power-law behavior $1/T_{1}T\sim T^{-n}$, with $n=0.99\left(1\right)–1.10\left(1\right)$ for ${\mu }_{0}H\le 2$ T. (f) Semilog plot of $1/T_1$ vs inverse temperature for different magnetic fields. The colored dashed lines denote the fits to the Arrhenius equation $1/T_1\propto exp(-\mathrm{\Delta }/k_{B}T)$.

Figure 6

$H-T$ phase diagram of ${\mathrm{Na}}_2\mathrm{BaCo}{\left({\mathrm{PO}}_4\right)}_2$ based on the contour plot of $1/T_{1}T$ vs $T$ in various magnetic fields. $T^{*}$ represents the onset temperature for the thermal activation behavior of $1/T_1$. $\mathrm{\Delta }/k_B$ denotes the spin gap. The dashed curve indicates the crossover temperature from the paramagnetic to the field-induced polarized state. The dotted line is the linear fit to the field dependence of the spin excitation gap.