Ising Supercritical Regime

• ISR is a regime characterized by smooth crossover ridges replacing the traditional phase boundary above the CEP, defined by peaks in specific heat and susceptibility.
• Universal scaling laws with critical exponents (β ≈ 0.326, γ ≈ 1.237) govern ISR, revealing an emergent Z2 symmetry across magnetic, quantum, and fluid systems.
• ISR leads to nontrivial phenomena such as divergent magnetic Grüneisen ratios and singular magnetocaloric effects, enabling advanced cooling protocols and deeper phase transition insights.

The Ising Supercritical Regime (ISR) generalizes the critical crossover structure above the critical endpoint (CEP) in systems exhibiting Ising universality, unifying supercriticality in magnetic, quantum, and fluid systems. In ISR, incipient phase boundaries above the CEP are replaced by symmetry-protected thermodynamic ridges—crossovers defined by maxima in susceptibilities and response functions—with universal scaling and emergent $Z_2$ (Ising) symmetry. These features yield highly nontrivial phenomena such as divergent magnetic Grüneisen ratios, singular magnetocaloric effects, and robust corrections induced by field-mixing asymmetry. The ISR has been confirmed in a wide range of contexts, from frustrated antiferromagnets (e.g., Nd$_3$BWO$_9$) to classical fluids, quantum spin chains, and dynamical nonequilibrium settings.

1. Emergence of ISR and Definition of Supercritical Crossovers

In systems of the Ising universality class, the traditional phase boundary between ordered and disordered phases terminates at a finite-temperature CEP (e.g., $(\mu_0 H_c, T_c)$ in Nd$_3$BWO$_9$ (Liu et al., 12 Jan 2026)). Beyond this point, the first-order transition evolves into a pair of smooth crossover ridges—the ISR—which in field–temperature (or pressure–temperature) phase space demarcate regimes of distinct fluctuational character. In the Ising context, these lines are loci of maxima of specific heat and susceptibility:

$$\text{ISR boundaries:}\quad C(T, H) = \max, \quad \chi(T, H) = \max,$$

which, in reduced variables $t\equiv(T-T_c)/T_c$, $h\equiv(H-H_c)/H_c$, obey the universal scaling

$h \propto t^{\beta+\gamma}$

with critical exponents $(\beta \simeq 0.326,\, \gamma \simeq 1.237)$ of the 3D Ising class, yielding $\beta+\gamma\simeq1.563$ (Li et al., 2023, Liu et al., 12 Jan 2026, Cui et al., 21 Sep 2025). This structure is mirrored in classical fluids as the two Widom-type crossover lines $L^+$ and $L^-$ in the supercritical region, separating liquid-like, indistinguishable, and gas-like regimes with explicit $Z_2$ symmetry about the critical isochore (Li et al., 2023, Cui et al., 21 Sep 2025, Li et al., 9 Dec 2025).

2. Universal Scaling Structure and Symmetry in ISR

The singular part of the free energy in the ISR region above the CEP takes the scaling form

$$F_{\mathrm{sing}}(t, h) = t^{2-\alpha}\,\phi_F(h\,t^{-(\beta+\gamma)}),$$

where $\alpha$ is the specific heat exponent. From this, all thermodynamic quantities inherit their scaling:

• Specific heat: $C = t^{-\alpha}\,\phi_C(h\,t^{-(\beta+\gamma)})$
• Susceptibility: $$\chi = t^{-\gamma}\,\phi_\chi(h\,t^{-(\beta+\gamma)})$$

Crossover ridges are located by maximizing $C$ or $\chi$ with respect to $T$ or $H$, yielding the ridge loci $h \propto t^{\beta+\gamma}$ (Liu et al., 12 Jan 2026).

Importantly, the Ising $Z_2$ symmetry enforces the presence of two symmetric crossovers—a consequence of the oddness of the equation of state and the self-reciprocal property between subcritical coexistence branches and supercritical ridges. This dual-ridge structure has been rigorously established in both mean-field (van der Waals) and Ising models, as well as verified empirically in supercritical fluids (Ar, CO$_2$) (Li et al., 2023, Cui et al., 21 Sep 2025).

Supercritical Observable	Scaling at ISR	Physical Meaning
$C$, $\chi$ Peak	$h\propto t^{\beta+\gamma}$	Crossover ridge location
Magnetocaloric Ratio $\Gamma_H$	$t^{1-(\beta+\gamma)}$	Divergence at CEP

3. Quantum and Dynamical Generalizations of ISR

In quantum Ising systems (e.g., TFIM), ISR emerges as a “quantum supercritical regime” (QSR), controlled by symmetry-breaking longitudinal field $h$ in the $h$–$T$ plane. Crossover boundaries are given by

$T_{c1,2}(h) \propto h^{z\nu/(\beta+\gamma)}$

where $z$ is the dynamical exponent and $\nu$ the correlation length exponent (Lv et al., 2024, Wang et al., 2024). The singular part of the free energy yields enhanced divergences in, e.g., the Grüneisen ratio:

$\Gamma_h(T) \propto T^{-(\beta+\gamma)/(z\nu)}$

This supercritical region is accompanied by robust data collapse in scaling variables and distinctive dynamical phenomena including nonanalytic cusps in the Loschmidt rate function at quenches to the QSC boundary. The boundary is determined by the intersection of Lee-Yang-Fisher zeros with the real-time axis (Wang et al., 2024, Ouyang et al., 2023).

4. ISR in Fluids: Crossover Loci and Emergent Symmetry

In classical fluids, ISR is manifested via two lines $L^+$ and $L^-$ in the ($T,P$) or ($T,\rho$) supercritical plane, defined by maxima of isothermal compressibility along isochoric paths parallel to $\rho_c$. These lines obey the same Ising scaling:

$$\delta P^\pm \propto (\Delta T)^{\Delta},\quad \Delta\rho^\pm \propto (\Delta T)^{\beta}$$

with $\Delta = \beta + \gamma$. The emergent $Z_2$ symmetry in the scaling domain implies these branches are related by a simple inversion and their physical characterization (e.g., liquid-like/indistinguishable/gas-like) precisely mirrors the Ising magnetic case (Cui et al., 21 Sep 2025, Li et al., 9 Dec 2025).

Experimental validation comes from correlation between dynamic (inelastic X-ray/neutron scattering) and thermodynamic markers of the crossovers (compressibility, $C_p$ peaks), confirming the predicted exponents and symmetry structure (Li et al., 2023).

5. Asymmetric Corrections and Antisymmetric Scaling in ISR

Deviations from ideal Ising symmetry, especially in real fluids, are encapsulated by linear mixing of field variables (chemical potential, pressure, temperature) into the scaling fields. This produces universal antisymmetric corrections to the scaling of ISR boundaries and their “diameters” (mean lines between $L^+$ and $L^-$):

$P_{\text{d}}^> - P_c = d_P\,|\Delta T|^{2\Delta-1}$

$$\rho_{\text{d}}^> - \rho_c = d_\rho\,|\Delta T|^{2\beta} + e_\rho\,|\Delta T|^{1-\alpha}$$

Higher-order cumulants (e.g., $\kappa_3$, $\kappa_4$ of the order parameter distribution) exhibit analogous scaling structure, with the “symmetry line” $\kappa_3=0$ tracking the leading antisymmetric corrections and $\kappa_4$ maxima coalescing near the ISR crossover lines (Li et al., 9 Dec 2025). Such corrections are tightly constrained by the field-mixing parameters and can be used to quantify the degree of asymmetry in real systems.

ISR Correction	Scaling Exponent	Parameter Dependence
Diameter in $P$	$2\Delta-1$	$a_3$, $b_2$, $\tan\varphi$
Diameter in $\rho$	$2\beta$, $1-\alpha$	$a_3$, $b_2$, $\tan\varphi$

6. ISR in Dynamical and Driven Ising Models

In nonequilibrium kinetic Ising models subjected to time-dependent fields, the supercritical regime is defined for $T<T_c$, period $P>P_c$ (slow driving). Under a nonzero bias field $H_b$, the system displays novel “sideband” peaks in the fluctuation spectrum and dynamic susceptibility, linked to resonant delay in magnetization switching governed by droplet nucleation dynamics. The scaling of the sideband positions and their dependence on system size, period, and effective interface tension have been quantitatively confirmed, both in the two-state Ising model and in the Blume–Capel generalization (where the presence of intermediate spin states reduces interface tension and modulates sideband structure) (Buendía et al., 11 Apr 2025).

7. Physical Consequences and Technological Implications

ISR has direct implications for magnetocaloric cooling and quantum thermodynamics. In Kagome antiferromagnets such as Nd$_3$BWO$_9$, the universality-protected divergence of the Grüneisen ratio within ISR yields record cooling rates, high entropy extraction, and sub-Kelvin refrigeration capabilities exceeding traditional paramagnetic salts and ferromagnets. The universality and robustness of ISR scaling point to broad utility in rare-earth–based Ising magnets and potentially quantum refrigerants where traditional $^3$He-based systems are impractical (Liu et al., 12 Jan 2026, Lv et al., 2024).

ISR analysis enables unified understanding and predictive power across classical and quantum phase transitions, guiding experimental searches for new phases, critical cooling protocols, and advanced quantum simulation platforms. Emergent $Z_2$ symmetry and scaling extend across diverse systems, resolving long-standing questions on asymmetry, singular corrections, and the boundaries of universality in critical phenomena.