Continuous transitions on the $J<U$ critical curves and sharpness on the $J>0>U$ self-dual curve of the two-dimensional Ashkin–Teller model

The Ashkin–Teller (AT) model — two Ising spin copies $\tau,\tau'$ on $\mathbb{Z}^2$ coupled by a four-spin interaction $U$ — is the canonical planar lattice model with continuously varying critical exponents. Aoun, Dober and Glazman (Comm. Math. Phys. 405, 2024) proved the phase diagram conjectured by physicists in the 1970s: for $J\ge U>0$ the transitions of $\tau$ and of the polarization $\tau\tau'$ both lie on the self-dual curve $\sinh 2J=e^{-2U}$, while for $0<J<U$ they split into two dual critical curves $\gamma_\tau$ and $\gamma_{\tau\tau'}$; all located transitions are sharp. Their published version states two open questions on the nature of these transitions. First, for $0<J<U$, prove that the two transitions are continuous: $\langle\tau_0\tau_x\rangle^{+,+}\to0$ on $\gamma_\tau$ and $\langle\tau_0\tau'_0\tau_x\tau'_x\rangle^{+,+}\to0$ on $\gamma_{\tau\tau'}$ as $|x|\to\infty$, with physics predicting Ising-like decay $|x|^{-1/4}$. Second, for mixed-sign couplings $J>0>U$, prove that the transition is sharp and located at the self-dual curve for correlations of both $\tau$ and $\tau\tau'$, in a regime where monotonicity — the missing input for the decision-tree sharpness argument — fails.

Background

The Ashkin–Teller (AT) model, introduced in 1943, is a pair of interacting Ising models. At each site $u$ of the square lattice $\mathbb{Z}^2$ sit two spins $\tau_u,\tau'_u\in\{\pm1\}$, and the Hamiltonian is $H=-\sum_{uv}\bigl(J\,\tau_u\tau_v+J\,\tau'_u\tau'_v+U\,\tau_u\tau'_u\,\tau_v\tau'_v\bigr),$ where $J\ge0$ is the Ising coupling within each copy and $U$ couples the two copies through a four-spin term; the isotropic case (equal coupling in both copies) is considered throughout. Two special loci orient the phase diagram: $U=0$ gives two independent Ising models, and the line $U=J$ corresponds to the 4-state Potts model (the standard $q$-state generalization of the Ising model with $q=4$ spin values per site).

The critical behavior has been studied by physicists since the early 1970s. Kadanoff and Wegner (1971), Wu and Lin (1974) and others predicted that for $J<U$ the model undergoes two distinct transitions — one for the spins $\tau$ (and $\tau'$) and one for the polarization $\tau\tau'$ — while for $J\ge U$ both observables order at the same point. Along the critical line the critical exponents are predicted to vary continuously, staying Ising-like throughout the $J<U$ regime ('weak universality'); field-theoretic analyses predict in particular algebraic decay like $|x|^{-1/4}$ for the polarization correlations on the $J<U$ critical curves, exactly as in the Ising model (Delfino and Grinza (2004) survey this physics literature).

The rigorous phase diagram is recent. Aoun, Dober and Glazman (2024) define, for couplings $(J,U)$, critical inverse temperatures $\beta_c^\tau(J,U)$ and $\beta_c^{\tau\tau'}(J,U)$ as the thresholds above which the infinite-volume two-point function of $\tau$ (respectively of $\tau\tau'$), taken with $+$ boundary conditions on both copies (denoted $\langle\cdot\rangle^{+,+}$), fails to vanish at large distances. They prove the 1970s physics picture: for $0<J<U$ one has $\beta_c^\tau>\beta_{sd}>\beta_c^{\tau\tau'}$, where $\beta_{sd}$ is the unique inverse temperature with $(\beta J,\beta U)$ on the self-dual curve $\sinh 2J=e^{-2U}$; the critical curves $\gamma_\tau=\{(J,U):0<J<U,\ \beta_c^\tau(J,U)=1\}$ and $\gamma_{\tau\tau'}$ (defined analogously) are mapped to each other by a Kramers–Wannier-type duality involution of the $(J,U)$ plane (an analogue, for this model, of the self-map relating the Ising model's high- and low-temperature regimes); and for $J\ge U>0$ both transition points coincide with the self-dual curve. All located transitions are proved sharp via the OSSS inequality — an influence inequality for decision-tree algorithms that Duminil-Copin, Raoufi and Tassion (2019) extended to dependent measures, where it yields sharpness in the standard sense: exponential decay of correlations throughout the subcritical regime and an order parameter bounded away from zero throughout the supercritical regime. A transition is called continuous (second-order) if long-range order vanishes at the critical point itself — here, that the relevant two-point function tends to $0$ as $|x|\to\infty$ — as opposed to a first-order transition, at which residual long-range order persists at criticality.

The nature of the transition is understood on the ferromagnetic self-dual segment $J\ge U>0$. There the AT model couples to the six-vertex model — a vertex model on $\mathbb{Z}^2$ whose six allowed arrow configurations satisfy the ice rule of two arrows in and two out at each vertex — equivalently to its height function, an integer-valued function on the faces whose local increments encode the arrow pattern. Delocalization of this height function (unbounded fluctuations) in the regime $1\le c\le2$ of the vertex-weight parameter, proved by Duminil-Copin, Karrila, Manolescu and Oulamara (2020) with an alternative proof, avoiding the Bethe ansatz (an exact-solution technique for two-dimensional vertex models), by Glazman and Lammers (2023), implies that at the self-dual curve both $\langle\tau_0\tau_x\rangle^{+,+}$ and $\langle\tau_0\tau'_0\tau_x\tau'_x\rangle^{+,+}$ vanish as $|x|\to\infty$: the two transitions are continuous on this segment. Inside the $J<U$ regime, at the self-dual curve itself, Glazman and Peled (2023) proved that the polarization $\tau\tau'$ is ordered while $\tau$ has exponentially decaying correlations, consistent with the self-dual curve lying strictly between the two critical curves there.

What remains open is the nature of the transitions at the two critical curves for $0<J<U$ and in the mixed-sign regime $J>0>U$. The latter is antiferromagnetic territory: for $U<0$ the FKG lattice condition (positive association of the Gibbs measure) fails, so the monotonicity input required by the decision-tree sharpness argument is missing; part of this regime is analyzed by Dober (2025).

Problem Statement

Let $\mathrm{AT}_{J,U}$ denote the isotropic Ashkin–Teller model on $\mathbb{Z}^2$ with couplings $(J,U)$, and write $\langle\cdot\rangle^{+,+}$ for the infinite-volume expectation taken with $+$ boundary conditions imposed on both Ising copies. Let $\beta_c^\tau(J,U)$ and $\beta_c^{\tau\tau'}(J,U)$ be the critical inverse temperatures for the correlations of $\tau$ and of the polarization $\tau\tau'$, set $\gamma_\tau=\{(J,U):0<J<U,\ \beta_c^\tau(J,U)=1\}$ and $\gamma_{\tau\tau'}=\{(J,U):0<J<U,\ \beta_c^{\tau\tau'}(J,U)=1\}$, and let $\mathrm{SD}$ be the self-dual curve $\sinh 2J=e^{-2U}$. Resolve the two open questions posed in the 'Open questions' section of Aoun–Dober–Glazman (2024), which share one objective — determining the nature (continuous versus first-order, and sharpness) of the AT transitions in the two regimes left open by the rigorous phase diagram:

(1) Continuity of the two transitions for $0<J<U$: show that $\langle\tau_0\tau_x\rangle^{+,+}\to0$ as $|x|\to\infty$ for every $(J,U)\in\gamma_\tau$, and $\langle\tau_0\tau'_0\tau_x\tau'_x\rangle^{+,+}\to0$ as $|x|\to\infty$ for every $(J,U)\in\gamma_{\tau\tau'}$. (The physics prediction — algebraic decay like $|x|^{-1/4}$, as in the Ising model — is the expected refinement and is not required for the claim.)

(2) Sharpness at the self-dual curve for $J>0>U$: show that the Ashkin–Teller model undergoes a sharp transition at SD in terms of correlations of $\tau$ and of $\tau\tau'$ — that is, the transition points for both observables lie on the self-dual curve, with exponential decay of the corresponding correlations throughout the disordered side and a non-vanishing order parameter throughout the ordered side. As the source notes, such sharpness would also imply continuity of this transition, since the six-vertex height-delocalization results cover the $U<0$ part of the self-dual curve.

Both parts are required: together they determine the nature of the transitions in all regimes left open by the Aoun–Dober–Glazman phase diagram.

References

1. Yacine Aoun, Moritz Dober, Alexander Glazman, 'Phase Diagram of the Ashkin–Teller Model', Communications in Mathematical Physics 405, article 37 (2024). DOI: 10.1007/s00220-023-04925-0; arXiv:2301.10609. https://doi.org/10.1007/s00220-023-04925-0
2. Moritz Dober, 'On antiferromagnetic regimes in the Ashkin–Teller model', Electronic Journal of Probability 30 (2025). DOI: 10.1214/25-EJP1331; arXiv:2406.06266. https://doi.org/10.1214/25-EJP1331
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