Kagome-Layered Spiral Ising Compound

• Kagome-layered spiral Ising compounds are frustrated antiferromagnets characterized by rare-earth elements, strong Ising anisotropy, and intricate spiral magnetic structures.
• They demonstrate complex field–temperature phase diagrams with metamagnetic transitions, a critical endpoint, and an Ising supercritical regime analogous to liquid–gas transitions.
• Experimental studies on Nd₃BWO₉ reveal divergent magnetocaloric effects and universal 3D Ising scaling, highlighting their potential for efficient sub-Kelvin cooling applications.

A kagome-layered spiral Ising compound is a class of frustrated antiferromagnets characterized by geometrically intricate magnetic structures and the realization of Ising supercriticality. Nd$_3$BWO$_9$ serves as a prototypical example, crystallizing in a rare-earth kagome lattice motif and exhibiting pronounced field-induced critical phenomena, including a metamagnetic critical endpoint (CEP), an Ising supercritical regime (ISR), and a divergent magnetocaloric response. The compound’s field–temperature phase diagram closely parallels the liquid–gas critical point paradigm, making it a uniquely valuable platform for studying universal scaling in highly frustrated, Ising-anisotropic magnets (Liu et al., 12 Jan 2026).

1. Crystal and Magnetic Structure

Nd$_3$BWO$_9$ crystallizes in a trigonal space group (P3$_1$21), comprising alternating planes of nonmagnetic BO$_3$ and WO$_6$ polyhedra intercalated with magnetic Nd$^{3+}$ kagome layers. Within each magnetic plane, Nd$^{3+}$ ions—subject to strong Ising-type single-ion anisotropy induced by crystal electric field (CEF) splitting—occupy the vertices of a corner-sharing triangular network, creating a two-dimensional kagome net.

Successive kagome layers are coupled along the $c$-axis via two distinct Ising exchange pathways: an interlayer antiferromagnetic “rung” ($J_b > 0$, $J_b \simeq +0.24$ meV) and an interlayer ferromagnetic “leg” ($J_r < 0$, $J_r \simeq –0.084$ meV). This arrangement produces columnar “spiral tubes” in which each triangular plaquette in one layer is connected to two spins in the next layer. The Ising-like character of Nd$^{3+}$ arises from the well-separated Kramers doublet ($J = 9/2$), acting as an effective spin-$\frac{1}{2}$ system with a principal $g$–tensor axis tilted by approximately 54° relative to the external field direction. This configuration yields a highly frustrated, locally anisotropic magnetic network (Liu et al., 12 Jan 2026).

2. Spiral Antiferromagnetic Order and Frustration

Below the Néel temperature $T_N \simeq 0.29$ K, Nd$_3$BWO$_9$ orders into a collinear up-up-down (UUD) spiral antiferromagnetic phase. Within each kagome triangle, two spins align parallel and one antiparallel to their local $z$-axis. This motif is shifted by 120° between successive layers, producing a triple-braid spiral structure. The competition between $J_b$, $J_r$, and weaker intertube coupling channels generates nearly degenerate spin manifolds, which, in turn, lead to a pronounced susceptibility to perturbations by external fields and result in a sequence of first-order metamagnetic transitions (Liu et al., 12 Jan 2026).

3. Field–Temperature Phase Diagram and Ising Supercritical Regime

Application of a magnetic field $H$ along the $c$-axis induces two low-temperature first-order transitions:

• A spin-flip transition at $\mu_0 H_\mathrm{SF} \simeq 0.65$ T, which transforms the UUD spiral phase into a macroscopically degenerate plateau manifold.
• A metamagnetic transition at $\mu_0 H_c \simeq 1.04$ T ($T \lesssim 0.3$ K), where the system jumps from a 1/3-magnetization “liquid-like” plateau to a partially polarized “gas-like” phase.

The first-order line $H_c(T)$ terminates at a finite-temperature CEP: $\mu_0 H_c \simeq 1.04(4)$ T, $T_c \simeq 0.30(2)$ K. For $T > T_c$ and $H > H_c$, the system enters the ISR, in which the contrast between plateau and polarized phases is lost. This regime features supercritical crossover lines (ridges of specific heat maxima $T^*_L(H)$ and $T^*_R(H)$) emanating from the CEP, directly reminiscent of the liquid–gas crossover structure (Liu et al., 12 Jan 2026).

4. Universal Scaling and Critical Behavior

In the ISR, the thermodynamics are governed by the 3D Ising universality class. Define dimensionless reduced variables: $$t \equiv (T - T_c)/T_c,\quad h \equiv (H - H_c)/H_c.$$

The singular part of the free energy above the CEP is: $$F_\mathrm{sing}(t, h) = t^{2 - \alpha}\, \Phi_F(h\, t^{-(\beta + \gamma)}), \quad (t > 0)$$ with $\alpha \approx 0.110$, $\beta \approx 0.326$, $\gamma \approx 1.237$.

From this, key response functions acquire scaling forms: $$\begin{align*} C(T, H) &= -T \frac{\partial^2 F}{\partial T^2} = t^{-\alpha} \Phi_C(h\, t^{-(\beta + \gamma)}), \ \chi(T, H) &= - \frac{\partial^2 F}{\partial H^2} = t^{-\gamma} \Phi_\chi(h\, t^{-(\beta + \gamma)}). \end{align*}$$ Maxima of $C$ (“supercritical crossovers”) trace the locus $h \propto t^{\beta + \gamma}$. Magnetization data for $\chi(H)$ at $T > T_c$ collapse onto a universal function, consistent with the 3D Ising model as obtained in Monte Carlo simulations. This affirms the universal critical scaling in the kagome-layered spiral Ising framework (Liu et al., 12 Jan 2026).

5. Divergent Grüneisen Ratio and Magnetocaloric Response

The magnetic Grüneisen ratio,

$$\Gamma_H \equiv \frac{1}{T}\left(\frac{\partial T}{\partial H}\right)_S = - \frac{\partial^2 F / \partial H \partial T}{T\, \partial^2 F / \partial T^2},$$

exhibits a universal scaling near the CEP: $$\Gamma_H = t^{1-\beta-\gamma} \Phi_\Gamma\left(h t^{-(\beta+\gamma)}\right).$$ For $H \approx H_c$, the peak values diverge as: $$\Gamma_H^\mathrm{peak} \propto t^{-(\beta + \gamma - 1)} \qquad (\beta + \gamma \simeq 1.563),$$ demonstrating a universally divergent magnetocaloric effect as the CEP is approached from above.

6. Experimental Magnetocalorics and Sub-Kelvin Cooling

Adiabatic demagnetization experiments reveal pronounced isentropic dips in temperature within the ISR when ramping down the field from $(\mu_0 H_i = 4 \text{ T}, T_i = 4 \text{ K})$ or $(4 \text{ T}, 2 \text{ K})$. Cooling to the CEP enables $T_\mathrm{min} \simeq 586$ mK for $T_i = 4$ K, while for $T_i = 2$ K further demagnetization through the spin-flip field $H_\mathrm{SF}$ leads to an observed $T_\mathrm{min} \simeq 195$ mK. The UUD spiral tube manifold at $H_\mathrm{SF}$ contains a zero-point entropy $S_0 \simeq 0.481\, R$ per formula unit, due to extensive domain-wall degeneracy.

The volumetric magnetic entropy change,

$$-\Delta S_m \simeq 83\, \text{mJ}\cdot\text{K}^{-1}\cdot\text{cm}^{-3}$$

for a field change $\Delta H \approx 1$ T ($2$ T $\to H_c$), substantially exceeds that of comparable materials such as Na$_2$BaCo(PO$_4$)$_2$ ($$\sim 37\, \text{mJ}\cdot\text{K}^{-1}\cdot\text{cm}^{-3}$$) or standard paramagnetic salts. This is attributed to the high Nd$^{3+}$ spin density $N \simeq 16.9\, \text{nm}^{-3}$, facilitating efficient sub-Kelvin cooling (Liu et al., 12 Jan 2026).

7. Broader Implications and Related Materials

Nd$_3$BWO$_9$ exemplifies a broader family of rare-earth kagome magnets, RE$_3$BWO$_9$ (RE = Pr–Sm, Gd–Ho), distinguished by strong local Ising anisotropy, spiral geometries, and competing exchange interactions. These compounds manifest metamagnetic CEPs and extended ISRs, closely paralleling the thermodynamic behavior of classical liquid–gas transitions. The magnetocaloric divergence near the CEP and high entropy density position RE$_3$BWO$_9$ and analogous Ising-anisotropic materials (notably spin ices such as Dy$_2$Ti$_2$O$_7$, Pr$_2$Zr$_2$O$_7$, and LiHoF$_4$) as promising candidates for efficient cryogenic refrigeration, providing an alternative to $^3$He-based technologies and enabling “supercritical” cooling strategies in frustrated antiferromagnets (Liu et al., 12 Jan 2026).