Critical behaviour of the planar Ashkin–Teller model: rigorous critical exponents and the order of the $J < U$ transitions

The Ashkin–Teller (AT) model — two Ising copies coupled by a four-spin interaction — is the canonical planar system predicted to possess a line of critical points with continuously varying critical exponents. Aoun–Dober–Glazman rigorously established the complete ferromagnetic phase diagram (locations and sharpness of all transitions) and explicitly left the critical behaviour open. The exponents are expected to be Ising-like for the two transitions in the $J < U$ regime, and to vary continuously along the self-dual segment $J \ge U$ (from the decoupled-Ising point $U = 0$ to the four-state-Potts point $J = U = \frac14\ln 3$). The problem is open in its core: exponents and spin-field scaling limits are rigorous only perturbatively near $U = 0$ (Mastropietro 2004) and, following the 2026 Gaussian-free-field convergence of the six-vertex height function on part of the segment, at the four-state-Potts endpoint (Chen–Duminil-Copin–He–Jacopin–Krachun–Manolescu–Xia; Chen–Xia); the open interior of the segment and the continuity of the two $J < U$ transitions (Question 1 of the published source) remain unresolved, and the Ising-like decay predicted at those transitions, as well as the compactified-boson scaling limits along the segment, likewise remain unproven.

Background

The Ashkin–Teller (AT) model, introduced in 1943, is defined on the square lattice $\mathbb{Z}^2$: at each vertex sits a pair of Ising spins $(\tau_v, \tau'_v) \in \{\pm 1\}^2$, and for a finite subgraph the probability of a configuration $(\tau, \tau')$ is proportional to

$$\exp\Big( J \sum_{uv} (\tau_u \tau_v + \tau'_u \tau'_v) + U \sum_{uv} \tau_u \tau'_u \tau_v \tau'_v \Big),$$

where $J > 0$ is a ferromagnetic nearest-neighbour coupling acting within each of the two Ising copies and $U > 0$ is a ferromagnetic four-spin coupling between them, under which the product $\tau\tau'$ also serves as an order parameter. At $U = 0$ the model is two independent Ising models; on the line $J = U$ it coincides exactly with the four-state Potts model (the spin system whose local states are four colours, with energy favouring equal colours on an edge), and $J = U = \frac14\ln 3$ is that model's critical point. Throughout, a critical exponent refers to the power law governing an observable at or near a transition: the anomalous dimension $\eta$ via the critical two-point decay $\langle \tau_0 \tau_x\rangle \sim |x|^{-\eta}$, the correlation-length exponent $\nu$ via $\xi \sim |t|^{-\nu}$ where $t$ is the distance to criticality, and the specific-heat exponent $\alpha$ via $C \sim |t|^{-\alpha}$. The Ising values are $\eta = 1/4$, $\nu = 1$, $\alpha = 0$ (only a logarithmic divergence); the four-state-Potts values are $\eta = 1/4$, $\nu = 2/3$, $\alpha = 2/3$, with multiplicative logarithmic corrections.

The model has a Kramers–Wannier-type self-duality (a symmetry exchanging the high- and low-temperature regimes, generalizing the Ising duality), and the self-dual curve $\sinh(2J) = e^{-2U}$ runs from the decoupled-Ising point $(J, U) = (\frac12\ln(1+\sqrt2), 0)$ to the four-state-Potts point $J = U = \frac14\ln 3$, with $J > U$ along the entire open segment. At this curve the AT model is equivalent to the six-vertex model — an exactly solvable model of arrows on edges obeying an ice rule (two arrows in, two out at each vertex) — with weights $a = b = 1$, $c = \coth(2J)$, a correspondence behind Baxter's 1971 exact solution of the corresponding eight-vertex model (the six-vertex model enlarged so that all four arrows of a vertex may point simultaneously inwards or outwards). From that solution, Kadanoff–Wegner (1971) and Wu–Lin (1974) derived the physics picture that has stood since the 1970s: for $J \ge U$ the model has a single transition, located at the self-dual curve, and the critical exponents vary continuously along it — the AT line is the textbook instance of non-universality, a line of critical points whose exponents depend on the coupling rather than on symmetry alone, interpolating between Ising behaviour at the decoupled point and four-state-Potts behaviour (with logarithmic corrections) at $J = U$; for $0 < J < U$ the transition splits into two, at curves dual to each other, and both are predicted to be Ising-like, with critical correlations decaying as $|x|^{-1/4}$. The predicted scaling limit along the critical line is the compactified free boson — a massless Gaussian free field (the standard Gaussian random generalized function on the plane) whose values are identified modulo a period, i.e. a boson on a circle whose radius varies along the line; in conformal-field-theory language, a family of theories of central charge $c = 1$ (the central charge is the single number labelling the universality class of a two-dimensional critical point), equivalently the sine-Gordon quantum field theory (a standard integrable field theory of a periodic scalar field). Delfino–Grinza (2004) survey these predictions and compute universal amplitude ratios within this framework.

Rigorous results long remained perturbative or partial. Pfister (1982) proved early results on the transition structure. Mastropietro (2004) considered exactly two coupled Ising models (the spin representation of the eight-vertex/AT model) and, by renormalization-group methods, wrote a convergent perturbative expansion for the specific heat and the energy–energy correlation up to the critical temperature, proving a form of non-universality: the critical behaviour is governed by critical indices that are non-trivial functions of the coupling, with the Ising logarithmic singularity of the specific heat changed into a power law with a non-universal index — the only rigorous verification of the varying-exponent prediction, confined to a perturbative neighbourhood of the decoupled point ($U$ sufficiently small). Giuliani–Mastropietro (2005) proved the Kadanoff–Wegner–Wu–Lin–Baxter universality conjecture for the anisotropic AT model ($J \ne J'$, weak coupling) — Ising exponents for the two transitions, together with anomalous non-universal features (an anomalous exponent for the distance between the critical points and a renormalized logarithmic amplitude in the specific heat) in the approach to the isotropic case.

The transition locations in the isotropic model were established only in 2024 by Aoun–Dober–Glazman (ADG), confirming the 1970s picture. The transitions are taken in the inverse temperature $\beta$ scaling both couplings, $(J, U) \mapsto (\beta J, \beta U)$; expectations $\langle \cdot \rangle^{+,+}$ refer to the infinite-volume Gibbs state obtained as the thermodynamic limit of finite-volume expectations with plus boundary conditions on both Ising copies (all boundary spins of $\tau$ and $\tau'$ fixed to $+1$). For $0 < J < U$ there are two transition curves, $\gamma_\tau = \{(J, U) : 0 < J < U,\ \beta_c^\tau(J, U) = 1\}$ and its analogue $\gamma_{\tau\tau'}$ (where $\beta_c^\tau(J, U)$ and $\beta_c^{\tau\tau'}(J, U)$ are the critical inverse temperatures for the onset of $\tau$- and $\tau\tau'$-ordering), dual to each other under the Kramers–Wannier-type parameter involution — cooling from high temperature, the product $\tau\tau'$ orders first (on $\gamma_{\tau\tau'}$) and the spins $\tau, \tau'$ order on $\gamma_\tau$ — while for $J \ge U$ both transitions occur at the self-dual curve; all transitions are sharp, via the OSSS inequality (a decision-tree influence inequality from theoretical computer science). Moreover, through the equivalence to the six-vertex model, whose height function — the integer-valued function on faces encoding the arrow configuration — was proven by Duminil-Copin–Karrila–Manolescu–Oulamara (2024) to delocalize (have unbounded fluctuations) for $a = b = 1$, $1 \le c \le 2$ (an alternative proof without Bethe Ansatz is due to Glazman–Lammers 2025), ADG deduce that the $J \ge U$ transition is continuous. The order of the $J < U$ transitions, and all critical exponents and scaling limits, lie outside their methods; the published version poses the order of the $J < U$ transitions as its Question 1 and records the continuously varying exponents for $J \ge U$ as an expectation established so far only for $U$ sufficiently small (Mastropietro 2004).

Two 2026 developments close part of the segment. Duminil-Copin–Kozlowski–Lammers–Manolescu proved that for $a = b = 1$ and $c \in [\sqrt3, 2]$ — corresponding, via $c = \coth(2J)$ and $\sinh(2J) = e^{-2U}$, to the four-state-Potts half of the AT self-dual segment — the six-vertex height function converges in the scaling limit to a properly scaled full-plane Gaussian free field. Building on this convergence and on the Baxter–Kelland–Wu coupling between the six-vertex model and the random-cluster model (the Fortuin–Kasteleyn percolation representation of the Potts model), Chen–Duminil-Copin–He–Jacopin–Krachun–Manolescu–Xia extracted the critical exponents of the planar random-cluster model with cluster weight $q = 4$ and of the four-state Potts model (for instance, the second derivative of the free energy behaves as $|\beta - \beta_c|^{-2/3 + o(1)}$ and the spontaneous magnetization as $(\beta - \beta_c)^{1/12 + o(1)}$), and Chen–Xia proved that all even multi-point correlation functions of the fuzzy 4-Potts model — the spin system obtained by assigning independent balanced $\{\pm 1\}$ spins to the open clusters of a critical $q = 4$ random-cluster configuration, which is exactly the single-spin projection of the isotropic AT model at its Potts point — converge to explicit conformally covariant Coulomb-gas sums. These results resolve the four-state-Potts endpoint of the AT segment. Neither work reaches the open interior of the segment — where the predicted continuously varying exponents live — nor the $J < U$ regime, and the Gaussian-free-field limit itself is not yet available on the half of the segment closer to the decoupled-Ising point ($c \in (\sqrt2, \sqrt3)$).

Problem Statement

Determine, rigorously, the critical behaviour of the ferromagnetic square-lattice Ashkin–Teller model at the transitions whose locations were established by Aoun–Dober–Glazman (2024), in the regimes where it remains open. The common scientific objective is to replace the 1970s physics predictions for this model — which are exact in the underlying eight-vertex solution but unproven as lattice theorems — by rigorous statements; its two components correspond to the two regimes of the phase diagram.

(1) Continuously varying exponents on the self-dual segment. For $J \ge U > 0$ (equivalently, along $\sinh(2J) = e^{-2U}$, $0 < U \le \frac14\ln 3$), prove that the critical exponents of the AT observables — the anomalous dimensions governing the decay of $\langle \tau_0 \tau_x\rangle$ and $\langle \tau_0 \tau'_0 \tau_x \tau'_x\rangle$ at the critical curve, and the correlation-length and specific-heat exponents governing the approach to the curve from either side — vary continuously along the open segment $0 < U < \frac14\ln 3$, taking their Ising values at the decoupling point $U = 0$ and interpolating to their four-state-Potts values (with logarithmic corrections) at $J = U = \frac14\ln 3$. This component makes rigorous the expectation recorded in the published source — that the exponents should vary continuously when $J \ge U$ — which the source notes has been established only for $U$ sufficiently small; the construction of the full compactified-boson scaling limits, a stronger goal not posed in the source, is not part of this problem.

(2) Order of the two $J < U$ transitions (Question 1 of the published source). For $0 < J < U$, prove that the model undergoes two continuous phase transitions — $\langle \tau_0 \tau_x\rangle^{+,+} \to 0$ as $|x| \to \infty$ at the curve $\gamma_\tau$ and $\langle \tau_0 \tau'_0 \tau_x \tau'_x\rangle^{+,+} \to 0$ as $|x| \to \infty$ at the curve $\gamma_{\tau\tau'}$, the dual curves located by ADG. The published source additionally remarks the physics expectation (Delfino–Grinza 2004) that at each transition the ordering correlations decay as $|x|^{-1/4}$, as in the Ising model; that precise power law goes beyond the posed Question 1 and is not a requirement of this problem.

The order of the $J \ge U$ transition (its continuity) is already resolved via six-vertex height-function delocalization and is not part of the problem.

References

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