The Ashkin–Teller (AT) model — two Ising spin copies τ,τ′\tau,\tau' on Z2\mathbb{Z}^2 coupled by a four-spin interaction UU — 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≥U>0J\ge U>0 the transitions of τ\tau and of the polarization ττ′\tau\tau' both lie on the self-dual curve sinh⁡2J=e−2U\sinh 2J=e^{-2U}, while for 0<J<U0<J<U they split into two dual critical curves γτ\gamma_\tau and γττ′\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<U0<J<U, prove that the two transitions are continuous: ⟨τ0τx⟩+,+→0\langle\tau_0\tau_x\rangle^{+,+}\to0 on γτ\gamma_\tau and ⟨τ0τ0′τxτx′⟩+,+→0\langle\tau_0\tau'_0\tau_x\tau'_x\rangle^{+,+}\to0 on γττ′\gamma_{\tau\tau'} as ∣x∣→∞|x|\to\infty, with physics predicting Ising-like decay ∣x∣−1/4|x|^{-1/4}. Second, for mixed-sign couplings J>0>UJ>0>U, prove that the transition is sharp and located at the self-dual curve for correlations of both τ\tau and ττ′\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 uu of the square lattice Z2\mathbb{Z}^2 sit two spins τu,τu′∈{±1}\tau_u,\tau'_u\in\{\pm1\}, and the Hamiltonian is H=−∑uv(J τuτv+J τu′τv′+U τuτu′ τvτv′),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≥0J\ge0 is the Ising coupling within each copy and UU 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=0U=0 gives two independent Ising models, and the line U=JU=J corresponds to the 4-state Potts model (the standard qq-state generalization of the Ising model with q=4q=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<UJ<U the model undergoes two distinct transitions — one for the spins τ\tau (and τ′\tau') and one for the polarization ττ′\tau\tau' — while for J≥UJ\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<UJ<U regime ('weak universality'); field-theoretic analyses predict in particular algebraic decay like ∣x∣−1/4|x|^{-1/4} for the polarization correlations on the J<UJ<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)(J,U), critical inverse temperatures βcτ(J,U)\beta_c^\tau(J,U) and βcττ′(J,U)\beta_c^{\tau\tau'}(J,U) as the thresholds above which the infinite-volume two-point function of τ\tau (respectively of ττ′\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<U0<J<U one has βcτ>βsd>βcττ′\beta_c^\tau>\beta_{sd}>\beta_c^{\tau\tau'}, where βsd\beta_{sd} is the unique inverse temperature with (βJ,βU)(\beta J,\beta U) on the self-dual curve sinh⁡2J=e−2U\sinh 2J=e^{-2U}; the critical curves γτ={(J,U):0<J<U, βcτ(J,U)=1}\gamma_\tau=\{(J,U):0<J<U,\ \beta_c^\tau(J,U)=1\} and γττ′\gamma_{\tau\tau'} (defined analogously) are mapped to each other by a Kramers–Wannier-type duality involution of the (J,U)(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≥U>0J\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 00 as ∣x∣→∞|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≥U>0J\ge U>0. There the AT model couples to the six-vertex model — a vertex model on Z2\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≤c≤21\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 ⟨τ0τx⟩+,+\langle\tau_0\tau_x\rangle^{+,+} and ⟨τ0τ0′τxτx′⟩+,+\langle\tau_0\tau'_0\tau_x\tau'_x\rangle^{+,+} vanish as ∣x∣→∞|x|\to\infty: the two transitions are continuous on this segment. Inside the J<UJ<U regime, at the self-dual curve itself, Glazman and Peled (2023) proved that the polarization ττ′\tau\tau' is ordered while τ\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<U0<J<U and in the mixed-sign regime J>0>UJ>0>U. The latter is antiferromagnetic territory: for U<0U<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 ATJ,U\mathrm{AT}_{J,U} denote the isotropic Ashkin–Teller model on Z2\mathbb{Z}^2 with couplings (J,U)(J,U), and write ⟨⋅⟩+,+\langle\cdot\rangle^{+,+} for the infinite-volume expectation taken with ++ boundary conditions imposed on both Ising copies. Let βcτ(J,U)\beta_c^\tau(J,U) and βcττ′(J,U)\beta_c^{\tau\tau'}(J,U) be the critical inverse temperatures for the correlations of τ\tau and of the polarization ττ′\tau\tau', set γτ={(J,U):0<J<U, βcτ(J,U)=1}\gamma_\tau=\{(J,U):0<J<U,\ \beta_c^\tau(J,U)=1\} and γττ′={(J,U):0<J<U, βcττ′(J,U)=1}\gamma_{\tau\tau'}=\{(J,U):0<J<U,\ \beta_c^{\tau\tau'}(J,U)=1\}, and let SD\mathrm{SD} be the self-dual curve sinh⁡2J=e−2U\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<U0<J<U: show that ⟨τ0τx⟩+,+→0\langle\tau_0\tau_x\rangle^{+,+}\to0 as ∣x∣→∞|x|\to\infty for every (J,U)∈γτ(J,U)\in\gamma_\tau, and ⟨τ0τ0′τxτx′⟩+,+→0\langle\tau_0\tau'_0\tau_x\tau'_x\rangle^{+,+}\to0 as ∣x∣→∞|x|\to\infty for every (J,U)∈γττ′(J,U)\in\gamma_{\tau\tau'}. (The physics prediction — algebraic decay like ∣x∣−1/4|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>UJ>0>U: show that the Ashkin–Teller model undergoes a sharp transition at SD in terms of correlations of τ\tau and of ττ′\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<0U<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
  3. Tomás Alcalde López, Avelio Sepúlveda, 'Excursion decomposition of the XOR-Ising model', arXiv:2602.06011 (2026). https://arxiv.org/abs/2602.06011
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