Critical behaviour of the ferromagnetic square-lattice Ashkin–Teller model: rigorous verification of the predicted critical exponents and transition order along the transition curves

The Ashkin–Teller (AT) model — two Ising copies coupled by a four-spin interaction with pair coupling $J$ and inter-copy coupling $U$ — is one of the canonical lattice models of two-dimensional criticality. Its ferromagnetic phase diagram on the square lattice is now rigorous: for $J\ge U$ a single transition lies on the self-dual curve $\sinh(2\beta J)=e^{-2\beta U}$, while for $J<U$ two dual transition curves separate the ferro phase from a partially ordered intermediate phase (products ordered, spins disordered) and from the paramagnet; along the whole $J\ge U$ segment the transition is, moreover, already known to be continuous, as a consequence of delocalization results for the height function of the associated six-vertex model. The critical behaviour is nevertheless verified rigorously only in perturbative regimes and at the endpoints of the critical line: constructive renormalization-group methods prove continuously varying exponents for sufficiently small $U$ along the self-dual curve (Mastropietro) and Ising exponents for the product-spin transition for sufficiently small $J$ in the $J<U$ regime (Giuliani–Mastropietro); very recently, the six-vertex height function was proven to converge to the Gaussian free field on the part of the self-dual curve adjacent to the four-state Potts endpoint, and at that endpoint the scaling limit of the AT spin field was identified. The open problem is to extend the rigorous account of the exponents and the order of the transitions to the whole phase diagram: prove the physics prediction that along the self-dual critical curve the exponents vary continuously between the Ising universality class at the decoupling point $U=0$ and four-state Potts behaviour at $J=U$, and prove that both transitions in the $J<U$ regime are Ising-universal and continuous.

Background

The Ashkin–Teller (AT) model, introduced in 1943, places two Ising spin configurations $\tau,\tau'\in\{\pm1\}^{V}$ on the same finite subgraph $\Omega=(V,E)$ of the square lattice and couples them through a four-spin term: $H=\sum_{uv\in E}\big(J\,\tau_u\tau_v+J\,\tau_u'\tau_v'+U\,\tau_u\tau_u'\tau_v\tau_v'\big).$ Here $J$ is the Ising (pair) coupling, common to the two copies, and $U$ is the strength of the interaction between the copies; the model is ferromagnetic when $J,U>0$, and one studies the family of Gibbs measures at inverse temperature $\beta$ applied to the couplings $(\beta J,\beta U)$. At $U=0$ it is two independent Ising models; on the line $J=U$ it is exactly the four-state Potts model (comparing edge weights shows the Potts inverse temperature is $4J$, whose square-lattice critical value $\ln 3$ gives the AT critical point $\beta J=\beta U=(\ln 3)/4$); for general $(J,U)$ it interpolates between them. A universality class is the set of models sharing the same critical exponents — the power-law indices governing the divergence of the correlation length and of susceptibilities at a continuous phase transition — and the critical behaviour of a transition comprises these exponents, its order (continuous versus discontinuous, i.e. whether the spontaneous magnetization vanishes or jumps at the critical point), and its scaling limit, meaning the random fields or curves obtained from lattice observables under diffusive rescaling at criticality; for two-dimensional models the scaling limit is predicted to be conformally invariant, described by a conformal field theory whose central charge $c$ counts its effective degrees of freedom.

The square-lattice AT model is self-dual: a Kramers–Wannier-type duality maps it to itself and leaves invariant the curve $\sinh(2\beta J)=e^{-2\beta U}$, which passes through the decoupled-Ising point ($\sinh(2\beta J)=1$, $U=0$) and the four-state Potts point. A long line of physics work — Kadanoff and Wegner, together with Baxter's exact solution of the zero-field eight-vertex model (an ice-type model of arrows on edges with exactly solvable local weights, to which the AT model is equivalent on the self-dual curve, a correspondence made explicit by Fan) — established the expected picture: along the self-dual critical curve the scaling limit is a free compactified boson (the Gaussian free field taking values on a circle), the canonical family of $c=1$ conformal field theories parametrized by the compactification radius, which is set by the couplings; consequently the critical exponents vary continuously with $(J,U)$, interpolating between Ising values at $U=0$ and four-state Potts values (with logarithmic corrections from a marginally irrelevant operator) at $J=U$. A notable feature, sometimes called weak universality, is that the anomalous dimension of the individual spins keeps the Ising value $\eta_\sigma=1/4$ along the whole curve while the thermal and product-field exponents drift. Wu and Lin moreover predicted that when the four-spin coupling dominates ($J<U$) the model has two distinct transitions: as the temperature is raised the individual spins $\tau,\tau'$ disorder first, and the products $\tau\tau'$ (which at $J=0$ behave as an independent Ising model with coupling $U$) disorder later, so that an intermediate phase exists in which the products are ferromagnetically ordered while the spin correlations decay exponentially.

On the mathematical side, the location of these transitions was settled only recently. Aoun, Dober and Glazman proved the complete ferromagnetic phase diagram conjectured in the 1970s: for $J\ge U$ both the spin and the product transitions occur at the self-dual curve, while for $J<U$ there are two transition curves that are dual to each other, with the intermediate partially ordered phase in between (the spin side of the picture at self-duality is due to Glazman and Peled, via the antiferroelectric phase of the six-vertex model); all transitions are sharp, meaning correlations decay exponentially on the disordered side, by the OSSS inequality (an influence inequality for randomized decision-tree algorithms). The proof goes through the random-cluster representation of the AT model (ATRC), a Fortuin–Kasteleyn-type graphical representation in which two coupled percolation configurations encode the spin and product correlations. Sharpness, however, says nothing about critical exponents or scaling limits. Two rigorous critical-behaviour results exist, both perturbative: Mastropietro proved by convergent renormalization-group methods that for sufficiently small $U$ at criticality the specific-heat and energy–energy critical indices are non-trivial functions of the coupling — the exponents really do vary, but only near the decoupled point; and Giuliani and Mastropietro proved Ising universality for the anisotropic AT model in a weak-interaction regime, which (by exchanging the roles of the layers, $(\tau,\tau')\mapsto(\tau\tau',\tau)$) establishes Ising exponents for the product transition in the $J<U$ regime when $J$ is small enough.

About the order of the transitions the following is known. At self-duality the AT model couples to the six-vertex model with weights $a=b=1$, $c=\coth(2\beta J)$, and the height function of that model — an integer-valued function on the faces of the lattice whose level lines follow the paths turning with the six-vertex arrows — delocalizes for $1\le c\le2$, a range covering the entire $J\ge U$ segment; as [16] records, this implies that the AT transition is continuous throughout this regime, in the sense that both the spin and the product critical correlations vanish at infinity (at the four-state Potts endpoint this is also the classical $q\le4$ Potts continuity). The order of the two transitions in the $J<U$ regime is open: proving their continuity is posed as an explicit question in [16]. As for scaling limits, rigorous conformal invariance in two dimensions is known essentially only for the Ising model, Bernoulli percolation and closely related models. Very recently, the height function of the six-vertex model was proven to converge, after rescaling, to the Gaussian free field for $-1\le\Delta\le-1/2$ in the parameter $\Delta=(a^2+b^2-c^2)/(2ab)$ of the isotropic six-vertex model; since the AT self-dual segment has $\Delta\in[-1,0]$ ($\Delta=-1$ at the Potts endpoint $J=U$, $\Delta=0$ at the decoupled-Ising endpoint), this covers the sub-segment of the AT critical line adjacent to the Potts endpoint. At the Potts endpoint itself, the scaling limit of the AT spin field has been identified as a conformally covariant Coulomb-gas-type field. For the AT model in the interior of the varying-exponent line, along the $J<U$ curves, and for the product field, no scaling limits are known.

Problem Statement

Determine, rigorously, the critical exponents and the order of the transitions of the ferromagnetic isotropic Ashkin–Teller model on the square lattice (couplings $J,U>0$, inverse temperature $\beta$) along the transition curves established by Aoun, Dober and Glazman. The problem has one scientific objective — a complete rigorous account of the universality classes and the order of these transitions — with three interlocking components:

(1) The regime $J\ge U$. Prove that at the critical point $\beta=\beta_{\mathrm{sd}}(J,U)$ on the self-dual curve $\sinh(2\beta J)=e^{-2\beta U}$ the critical exponents vary continuously with the position $(J,U)$ on the curve, as predicted by the exact eight-vertex solution and the $c=1$ conformal field theory: determine the correlation-length and thermal (specific-heat/energy-density) exponents and the anomalous dimensions of the spin field $\tau$ and of the product field $\tau\tau'$ as functions of $(J,U)$ interpolating between their Ising values at the decoupling point $U=0$ and their four-state Potts values (with logarithmic corrections) at $\beta J=\beta U=(\ln 3)/4$ (i.e. on the line $J=U$), the individual-spin anomalous dimension remaining at the Ising value $\eta_\sigma=1/4$ along the curve. This must extend Mastropietro's verification, valid only for $U$ sufficiently small, to the entire curve.

(2) The regime $0<J<U$. Prove that both sharp transitions — the disordering of the spins $\tau,\tau'$ at $\beta_c^{\tau}$ and the disordering of the products $\tau\tau'$ at $\beta_c^{\tau\tau'}$, whose transition curves are dual to each other — belong to the Ising universality class (Ising critical exponents) throughout the regime $0<J<U$, extending the perturbative result of Giuliani and Mastropietro, which covers only the product transition for $J$ sufficiently small.

(3) Order of the transitions. Prove that the two transitions in the $J<U$ regime are continuous — that the spontaneous magnetizations of $\tau$ and of $\tau\tau'$ vanish, rather than jump, at the respective transition curves $\gamma_\tau$ and $\gamma_{\tau\tau'}$, with critical correlations decaying as $|x|^{-1/4}$ as predicted (Question 1 of [16]). (On the $J\ge U$ segment the transition is already known to be continuous, by delocalization of the associated six-vertex height function, and the four-state Potts endpoint has its known marginal logarithmic behaviour.)

The components are linked by the common objective of determining the critical exponents and the order of the transitions left open by the rigorous phase diagram; an answer that verifies only a perturbative neighbourhood of a special point (small $U$ along the self-dual curve, or small $J$ in the $J<U$ regime) does not resolve the problem. The scaling limits of the spin and product fields (conformal invariance, convergence to the Gaussian free field) are not part of the required objective; they are discussed in the Background as context only.

References

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