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arXiv:2509.17038 · Journal of High Energy Physics

Emergent Ising symmetry and supercritical fluids

Hong-Ming Cui, Zhong-Ying Fan

Published Feb 13, 2026• 1 citation
arXivHTMLPDF

Abstract

A bstract The symmetry of Ising model questions any single crossover scenario for supercritical fluids. In this work, we firstly study a pair of thermodynamic crossovers L ± analytically for the Van der Waals class fluids. We uncover an emergent Z 2 symmetry in addition to the universal scalings in the scaling regime for this class fluids. By using the self-reciprocal property between coexistenct phases, we further establish that under suitable conditions, the Ising symmetry generally emerges in the scaling regime for a general universality class. As a consequence, the thermodynamic crossovers L ± generally exhibit an emergent Z 2 symmetry in the scaling regime. This partly resolves the symmetry puzzle raised by the Ising model. The results also imply that the physical importance of the Ising model in critical phenomenon is far beyond the scope of magentic transitions.

DOI
10.1007/jhep02(2026)145

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A bstract The symmetry of Ising model questions any single crossover scenario for supercritical fluids. In this work, we firstly study a pair of thermodynamic crossovers L ± analytically for the Van der Waals class fluids. We uncover an emergent Z 2 symmetry in addition to the universal scalings in the scaling regime for this class fluids. By using the self-reciprocal property between coexistenct phases, we further establish that under suitable conditions, the Ising symmetry generally emerges in the scaling regime for a general universality class. As a consequence, the thermodynamic crossovers L ± generally exhibit an emergent Z 2 symmetry in the scaling regime. This partly resolves the symmetry puzzle raised by the Ising model. The results also imply that the physical importance of the Ising model in critical phenomenon is far beyond the scope of magentic transitions.
Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.
We report the discovery of a multicritical point that extends the liquid-gas paradigm to systems with competing symmetry-breaking orders. Using large-scale Monte Carlo simulations of a frustrated bilayer Ising antiferromagnet with tunable couplings, we map out a rich finite-temperature phase diagram hosting three ordered phases separated by both continuous and first-order transitions. By tuning the couplings, a tricritical line and a critical end-point line converge into a single multicritical line. At all points along the multicritical line, symmetry-distinct phases exhibit identical leading critical behavior---consistent with the tricritical Ising universality class---while the subleading exponent exhibits a sharp shift from yg=0.8{y}_{g}=0.8yg​=0.8 to yg=1{y}_{g}=1yg​=1. This shift reflects an emergent Z2{Z}_{2}Z2​ symmetry akin to that of the liquid-gas critical point, but realized here at a genuine multicritical point involving simultaneous microscopic symmetry breaking. Our results establish a universality scenario in which emergent symmetry preserves the leading class but reorganizes subleading scaling, providing a general mechanism for symmetry-enforced multicriticality.
The near-critical behavior of (d = 3)-dimensional Ising-model ferromagnets or simple lattice gases with equivalent first, second, and third nearest-neighbor interactions is studied through Monte Carlo simulations using histogram reweighting techniques and comparisons with series expansions. By carefully analyzing numerical data from relatively small finite systems using scaling and extrapolation methods, it is demonstrated that one can reliably estimate critical exponents, critical temperatures, and universal amplitude ratios, thereby distinguishing convincingly between different "nearby" universality classes and revealing systematic crossover effects. This study is preparatory to extending similar techniques to study criticality in continuum fluid models lacking symmetries, with Coulomb interactions, etc.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
The critical behavior of simple single-component fluids near the liquid-vapor critical point (and of other systems in the same universality class) is examined via an asymmetric spin Hamiltonian in Landau-Ginzburg-Wilson form. The leading important odd terms to be added to an Ising-type system are a five-spin interaction $\ensuremath{\int}\ensuremath{\phi}^{5}(x)$ and a nonlocal cubic interaction $\ensuremath{-}\ensuremath{\int}\ensuremath{\phi}^{2}\ensuremath{\nabla}^{2}\ensuremath{\phi}$. These two operators can be combined into two different eigenoperators with distinct physical interpretations. One combination can be shown to be exactly equivalent to the mixing of field variables (e.g., chemical potential and temperature) in the temperaturelike variable of an Ising or Ising-type system. This mixing has been used phenomenologically, found in certain models, and justified on general geometric grounds. This is the first proof that all asymmetric models, assuming universality and barring accidents, will have such mixing; this implies the universal occurrence of a ${t}^{1\ensuremath{-}a}$ fluid diameter. The second combination is treated by renormalization-group methods and its contributions to the Helmholtz free energy, magnetic equation of state, and correlation length are calculated to $O({\ensuremath{\epsilon}}^{2}), \ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. There are several novel features of this interaction which persist to all orders in perturbation theory. In addition to the expected renormalized M5{M}^{5}M5 contribution to the free energy, there are also terms linear in hhh (the magnetic field) and MMM which could be described as nonanalytic fluctuation-induced shifts in the order parameter and field. These might appear experimentally as apparent shifts in the background terms for hhh and MMM.
We show that supersymmetry emerges in a large class of models in 1+1 dimensions with both Z_{2} and U(1) symmetry at the multicritical point where the Ising and Berezinskii-Kosterlitz-Thouless transitions coincide. To arrive at this result we perform a detailed renormalization group analysis of the multicritical theory including all perturbations allowed by symmetry. This analysis reveals an intricate flow with a marginally irrelevant direction that preserves part of the supersymmetry of the fixed point. The slow flow along this special line has significant consequences on the physics of the multicritical point. In particular, we show that the scaling of the U(1) gap away from the multicritical point is different from the usual Berezinskii-Kosterlitz-Thouless exponential gap scaling.