Ising Supercriticality and Universal Magnetocalorics in Spiral Antiferromagnet Nd$_3$BWO$_9$

Abstract

The celebrated analogy between the pressure-temperature phase diagram of a liquid-gas system and the field-temperature phase diagram of an Ising ferromagnet has long been a cornerstone for understanding universality in critical phenomena. Here we extend this analogy to a highly frustrated antiferromagnet, the kagome-layered spiral Ising compound Nd$_3$BWO$_9$. In its field-temperature phase diagram, we identify a critical endpoint (CEP) and an associated Ising supercritical regime (ISR). The CEP of the metamagnetic transition is located at ${μ_0H_c} \simeq 1.04$ T and $T_c \simeq 0.3$ K. Above this point, the ISR emerges with supercritical crossover lines that adhere to a universal scaling law, as evidenced by the specific heat and magnetic susceptibility measurements. Remarkably, we observe a universally divergent Grueneisen ratio near the emergent CEP, $Γ_H \propto 1/t^{β+γ-1}$, with $β + γ \simeq 1.563$ the critical exponents of the 3D Ising universality class and $t \equiv (T - T_c)/T_c$ the reduced temperature. Our adiabatic demagnetization measurements on Nd$_3$BWO$_9$ reveal a lowest temperature of 195 mK achieved from 2 K and 4 T. Our work opens new avenues for studying supercritical physics and efficient cooling in layered-kagome, rare-earth RE$_3$BWO$_9$ family and, more broadly, in Ising-anisotropic magnets like spin ices.

Figures (11)

• Figure 1: Schematic phase diagrams of (a) water and (b) Nd$_3$BWO$_9$, where the solid line represents a first-order transition and red star denotes the CEP. For Nd$_3$BWO$_9$, the metamagnetic transition line separates 1/3-plateau ("liquid") and partially polarized ("gas") phases, and the dashed lines enclose the Ising supercritical regime (ISR). The universal supercritical crossover scaling is $\delta \hat{P}^{\pm}\propto t^{\beta+\gamma}$ for water li2024supfluid, and $h\propto t^{\beta+\gamma}$ for Nd$_3$BWO$_9$. $\delta\hat{P}^{\pm}$ is the reduced pressure, $h \equiv (H-H_{\rm c})/H_{\rm c}$ is the reduced field, and $t \equiv (T-T_{\rm c})/T_{\rm c}$ is the reduced temperature. $\beta$ and $\gamma$ are critical exponents of 3D Ising universality class. (c) Coupled spin tubes of Nd$^{3+}$ and distorted NdO$_8$ octahedron. (d-f) Schematic plots of magnetization curves above, at, and below $T_{\rm c}$.
• Figure 2: (a) Low-temperature specific heat ($C_p/T$) of Nd$_3$BWO$_9$ under $c$-axis magnetic fields from 0 T to 2 T. (b) Color map of $C_p$, where the maxima for $H < H_{\rm c}$ (red squares) and $H > H_{\rm c}$ (blue circles) denote the supercritical crossovers (dashed black lines). The uncertainty (error bar) is estimated by the temperature step size in the specific heat measurements. The star marks the CEP at $\mu_0H_{\rm c} = 1.04(4)$ T and $T_{\rm c} = 0.30(2)$ K. The uncertainties reflect the variation across different measurements, including the specific heat, magnetization, and magnetocalorics. The solid black line below the CEP represents the first-order metamagnetic transition line. The red diamonds mark the peak positions of $\chi \equiv dM/dH$ below $T_{\rm c}$, which are used to identify the first-order line (see Supplementary Fig. \ref{['figS1']}). Above the CEP, there exists an ISR which separates the "liquid" and "gas" states. The gray dot indicates the spin-flip field $\mu_0H_{\rm SF} \simeq 0.65$ T, and the solid gray line is the AF phase boundary. (c) Supercritical crossover scaling law $h\propto t^{\beta+\gamma}$, with $\beta$ and $\gamma$ the critical exponents of 3D Ising universality class. The reduced temperature $t = |T^*_{\rm L, R}-T_{\rm c}|/T_{\rm c}$, with $T^{*}_{\rm L}$ and $T^{*}_{\rm R}$ the peak positions in the left and right branches of $C_p$ in panel (b), respectively. (d) Magnetic susceptibility $\chi$ as a function of magnetic field at various temperatures above $T_{\rm c}$. (e) Data collapse of the measured $\chi$ results in the supercritical regime, whose profile is in excellent agreement with the universal scaling function $\phi_{\chi}(x)$ theoretically obtained from calculating the 3D Ising model (see Appendix). Data in the gray region are omitted from the collapse due to limited resolution in this highly field-sensitive regime, where the $\chi$ divergence is suppressed by the finite field step.
• Figure 3: (a) The isentropes obtained via adiabatic demagnetization measurements, with legends specifying the initial conditions ($H_{\rm i},T_{\rm i}$). The supercritical crossover line (blue dotted line) $h \propto t^{\beta+\gamma}$ are determined from the peak/dip positions of the magnetic Grüneisen ratio $\Gamma_H\equiv \frac{1}{T}\left(\frac{\partial T}{\partial H}\right)_S$ shown in (b), which are derived from the isentropic lines. Blue and red triangles mark the peaks and dips of $\Gamma_H$, respectively. (c) Isentropic lines of Nd$_3$BWO$_9$ below $T_{\rm c} \simeq 0.3$ K. Hollow triangles denote the peak/dip positions of corresponding $\Gamma_H$. (d) Temperature dependence of the peak/dip values of $\Gamma_H$. The black dashed line indicates the supercritical magnetocaloric scaling $\Gamma_H \propto 1/t^{\beta+\gamma-1}$ for $t > 0$ (i.e., $T > T_{\rm c}$). Very close to $T_{\rm c}$, we find deviation of $\Gamma_H$ from this scaling, which owes to the measurement accuracy. On the subcritical side ($t < 0$), these extrema decrease rapidly.
• Figure 4: (a) ADR process under fields along the $c$-axis. The green and red solid lines are for different initial temperatures $T_{\rm i} = 4$ K and 2 K, respectively, both at the initial field $\mu_0H_{\rm i} = 4$ T. The black dashed lines represent the cooling curves of ideal paramagnetic (PM) salt. The purple shaded areas highlight the enhanced cooling effect in Nd$_3$BWO$_9$ compared to the PM salt, characterized by $\Delta T$, which equals 414 mK and 249 mK for curves from $T_{\rm i}$ = 4 K and 2 K, respectively. (b) Volumetric magnetic entropy change ($-\Delta S_{\rm m}$) for Nd$_3$BWO$_9$ under different field changes (final field at $\mu_0H_{\rm f} = \mu_0H_{\rm c}\simeq 1$ T). The dots mark the maximum of each curve. The black dotted line shows the spin supersolid compound Na_2BaCo(PO_4)_2 (NBCP) under a field change from 4 T to its critical field 1.7 T Xiang2024Nature; and the gray dashed line shows a prototypical PM coolant Ce$_2$Mg$_3$(NO$_3$)$_{12}\cdot$24H$_2$O (CMN) under a field change of 5 T.
• Figure 5: X-ray diffraction pattern of Nd$_3$BWO$_9$ single crystal. Left inset: an optical image of typical single crystals. Right inset: the rocking curve (black line) of the (200) Bragg peak, together with the Gaussian fitting (red line).