Ashkin–Teller for the Working Mathematician

1One Ising model, in five lines

Spins σ_i = ±1 on the square lattice, H = −K Σ_⟨ij⟩ σ_iσ_j, with β absorbed into K. Kramers–Wannier duality maps the model at coupling K onto itself at a dual coupling; on the square lattice the self-dualLeft unchanged by the duality map. For the Ising model the unique transition must sit there; for Ashkin–Teller the exact critical segment lies on the self-dual set (§4). point is sinh 2K = 1,^[2] so K_c = ½ ln(1 + √2) = 0.44069.^[1] At the end point X₀ of §4 the two copies decouple, so the formulas there give one Ising model’s exponents: ν = 1, α/ν = 0, γ/ν = 7/4, η = 1/4. In scaling dimensionsx: at criticality a local operator’s two-point function decays as r^−2x. A coupling h to that operator scales as b^2−x under coarse-graining by b (in two dimensions)., the spin has x_σ = η/2 = 1/8 and the energy density x_ε = 2 − 1/ν = 1. Already here a power-law fit misleads: at X₀, from the specific heat, Wiseman and Domany obtain α/ν = 0.23(1) with a log–log fit (log C against log ℓ) and 0 with a semi-log fit (C against log ℓ).^[1]

2Two copies: the double Ising model

Put a second Ising spin τ_i on every site, and give each copy its own coupling K = K_c with no term between them. The partition function factorises, 𝒵 = 𝒵_Ising(K)² (𝒵, so as not to clash with the coordinate Z of §4). The symmetry is Z₂ × Z₂ (flip every σ; flip every τ; independently), and there are three Ising-like order parameters: σ, τ and the product στ, which survives flipping both. Independence gives the product’s correlations for free:

Coarse-graining by b multiplies a small coupling to an operator of dimension x by b^y, with y = 2 − x. A coupling with y > 0 is relevantGrows under coarse-graining: switching it on drives the system away from the critical point it started at.; y < 0 is irrelevant; y = 0 is marginalNeither grows nor shrinks under coarse-graining, to linear order. Whether it stays so at higher order decides whether it generates a line of critical points..^[1][3] These are the local operators of the decoupled pair:

The bilinear row (στ) and the four-spin row (ε_σε_τ) are the inter-copy couplings this page compares. Both keep the σ ↔ τ swap. The four-spin term keeps the full Z₂ × Z₂ (flipping every σ leaves σ_iσ_jτ_iτ_j unchanged); the bilinear σ_iτ_i survives only the joint flip of both layers.

3Turn on the coupling

The Ashkin–Teller Hamiltonian adds the four-spin term, with β absorbed:^[1]

The new term is the product of the two copies’ bond energies, (σ_iσ_j)(τ_iτ_j). At the decoupled point it couples to ε_σε_τ: x = 2, y = 0, marginal. Marginal to linear order is not enough for a line, since a marginal coupling can still grow or die slowly at the next order. This one does neither: it is exactly marginal (Ginsparg’s four steps below), and Ginsparg identifies ε₁ε₂ (his notation for ε_σε_τ) as the operator that moves the theory along a line of critical theories.^[3] On the lattice, moving along that line changes K and L together: K falls from 0.44069 to 0.27465 as L rises from 0 (§4). With K held at K_c, any L ≠ 0 leaves the line (§4: e^−2L = sinh 2K = 1 forces L = 0). So the lattice term L is not purely the field-theory coupling to ε_σε_τ: switched on alone it also moves the system off criticality, and K must fall to compensate. That decomposition is not given here. At L = K the four states (σ, τ) = (±1, ±1) are all alike and the model is the four-state Potts modelq states per site; neighbours gain energy only when they are in the same state. q = 2 is Ising., where σ, τ and στ are interchangeable.^[1]

The contrast is the on-site bilinear coupling g Σ_i σ_iτ_i. It multiplies στ, with x = 1/4 and y = 7/4: strongly relevant. The decoupled point is unstable in the g direction and there is no line. Where that flow ends is not established on this page.

Why exactly marginal, in four steps (Ginsparg’s argument, quoted, not derived; φ is a free boson, not the coordinate X of §4):^[3]

1. Two critical Ising models are two free Majorana fermions, and these bosonise onto the orbifoldA free boson φ on a circle of radius r, with φ identified with −φ (the reflection φ → −φ). Folding the circle this way leaves a segment with two fixed points, φ = 0 and φ = πr, each carrying a twist field of dimension (1/16, 1/16), that is total x = 1/8, which does not depend on r. V = ∂φ ∂̄φ is invariant under φ → −φ, so it survives the identification and changes the orbifold radius just as it changes the circle radius. at radius r = 1, with 𝒵_orb(1) = 𝒵_Ising².
2. The energy product ε₁ε₂ becomes the four-fermion term ψ₁ψ̄₁ψ₂ψ̄₂.
3. In boson language that term is V = ∂φ ∂̄φ, of dimension (1,1), that is total x = 2. Perturbing by V “just changes the overall normalization of the action, which by a rescaling of [φ] can be absorbed into a change in the radius r”.
4. The radius labels the critical theories, so a change of radius moves the theory along a line of them rather than carrying it off to some other fixed point.

4The phase diagram

In Wiseman and Domany’s coordinates the phase diagram is the unit square X, Z ∈ [0, 1]:^[1]

The parabola Z = X² is L = 0 (two decoupled Ising models); the diagonal Z = X is L = K (four-state Potts). The critical line lies exactly on the self-dual line Z = 1 − 2X, equivalently e^−2L = sinh 2K. Its segment from the point X₀ at (X, Z) = (√2 − 1, 3 − 2√2), the decoupled Ising point (K = 0.44069, L = 0), to X₄ = (1/3, 1/3), the Potts point (K = L = ¼ ln 3 = 0.27465), is a line of critical points with continuously varying exponents. Below X₀ (L < 0) the same line continues as the ferro–para boundary. Three phases: P, paramagnetic; F, ferromagnetic, with ⟨σ⟩ = ±⟨τ⟩ and ⟨στ⟩ ≠ 0; and a ⟨στ⟩ phase in which only the product orders (⟨σ⟩ = ⟨τ⟩ = 0), which lies within L > K. Two further boundaries leave X₄; the source describes them as flowing to Ising fixed points (Ising-type transitions). They have no exact equation and are not drawn.^[1]

What is proved. Aoun, Dober and Glazman prove, for K ≥ 0 and L ≥ 0, that when 0 < L ≤ K (the segment, its Ising end X₀ excluded from the theorem as written) the σ transition and the στ transition both sit exactly on the self-dual line and are sharp: correlations decay exponentially on one side and the order parameter is positive on the other. For L > K they prove that the two transitions split, occurring at two distinct temperatures on opposite sides of the self-dual line, and that duality maps the two transition curves onto each other; between them σ is disordered and στ ordered. So in this ferromagnetic, symmetric, square-lattice case the region labels of the diagram below are theorems. They do not locate the two curves, and they compute no exponent, so the exponents below remain exact-solution and Coulomb-gas results, not theorems of that paper.^[8] Claims below are tagged by kind: proved [8]; exact solution / Coulomb gas (Baxter; Wiseman and Domany [1]); numerical (the table [1]; §8c [6]).

Along the segment the exponents come from the Coulomb gasA standard exact method for 2D critical models (Nienhuis). Its single coupling, here g_R, labels the position on the line and fixes the exponents., quoted (not derived here) by Wiseman and Domany; exact solution / Coulomb gas, not proved in [8]:^[1]

At X₀, ½ coth 2K = 1/√2 and g_R = 2; at X₄ it is 1 and g_R = 4. By algebra, ν = 1/y_t runs from 1 to 2/3, α/ν from 0 to 1, and the thermal dimension x_ε = 2 − y_t from 1 to 1/2, while η = 2 − γ/ν = 1/4 never moves. Drag the handle (or focus it and use the arrow keys). It lives on the critical segment only: that is the range over which the source gives these exponents, and this page does not follow the line below X₀.

Wiseman and Domany test five equally spaced points X_i = X₀ + (i/4)(X₄ − X₀) with their cluster algorithm on ℓ × ℓ lattices up to ℓ = 128 (ℓ, the linear size, is never written L on this page). They measure C = ℓ²(⟨E²⟩ − ⟨E⟩²), with E the energy per site, and χ = ℓ²⟨M²⟩ with M = ℓ⁻²Σσ (built on σ alone; ⟨M⟩ = 0 on a finite lattice by symmetry, so this is a variance), and read exponents by finite-size scalingAt a critical point, fluctuation quantities such as C and χ grow as powers of the linear size: C ~ ℓ^α/ν, χ ~ ℓ^γ/ν. Exponents are read from that growth., C ~ ℓ^α/ν and χ ~ ℓ^γ/ν.^[1] The errors are statistical only, and the results are numerical.

The fitted γ/ν stays within 0.013 of 7/4 at every point. The fitted α/ν misses the exact value by more than its statistical error at every point: by 0.23 at X₀ (log–log), 0.17 at X₁, 0.12 at X₂, 0.017 at X₃ (error 0.013) and 0.25 at X₄. At X₀ the semi-log entry fits C against log ℓ, the form that corresponds to α/ν = 0, so it is not an independent estimate of α/ν. The source explains only the Potts-end miss (§8b).

5Live Monte Carlo: two layers on the line

The simulation uses Wiseman and Domany’s cluster method.^[1] Freeze τ. Then σ alone is an Ising model with bond couplings J_jk = K + L τ_jτ_k, ferromagnetic whenever |L| ≤ K (here 0 ≤ L ≤ K). Grow one WolffWolff and Swendsen–Wang are cluster algorithms for Ising-type models: instead of single spins, they flip whole clusters at once, grown by adding aligned neighbours across each bond with a probability set by its coupling. cluster on σ and flip it; then swap the roles of σ and τ. The demo uses single Wolff clusters; Wiseman and Domany note that their method reduces to the standard Wolff and Swendsen–Wang methods at L = 0 and at L = K. The controls are the position t on the critical segment and a temperature factor f that sends (K, L) to (K/f, L/f): f = 1 is on the line, f < 1 colder and ordered, f > 1 hotter and disordered. Dividing both couplings by f keeps L/K fixed, so L ≤ K holds everywhere the sliders reach. One sweep is enough cluster flips to turn over about ℓ² spins, Wiseman and Domany’s unit of Monte Carlo time.

6The RG picture: a line of fixed points

In the single-fixed-point picture, criticality is one fixed point, and tuning its relevant directions to zero puts you on it. Here the critical theories form a one-parameter family. Near it, use two schematic local scaling coordinates: u, transverse to criticality (relevant), and v, which labels the critical theory (marginal). One u suffices for this model: its Z₂ × Z₂ symmetry (§2) excludes couplings to σ, τ and στ, and its σ ↔ τ swap excludes ε_σ − ε_τ, which leaves the symmetric energy, temperature. The RG time is ln b, with b the coarse-graining factor of §2; ℓ stays the lattice size.

Schematic: local scaling coordinates near the line, not flow equations derived for K and L.

Tuning temperature removes u. Moving along the line changes v, and nothing in the flow pushes v back, so every point of the line is a fixed point. Because y_t depends on v, so does ν(v) = 1/y_t(v), which slides from 1 to 2/3 (§4). Some scaling dimensions are pinned along the line and others move:

Two pictures should be kept apart. The critical segment drawn in §4 is a curve of lattice parameters in the (K, L) plane; the line of fixed points lives in the space of theories. They are related, not identical: each critical lattice point flows under coarse-graining to one fixed point on the line. The lattice L is not the coordinate v. As §3 noted, switching L on alone also moves the system off criticality, so in these coordinates the lattice L has a component along u as well as along v. What self-duality supplies is the location: the lattice sits on the critical curve for every L/K in [0, 1], and moving along that curve moves the fixed point it flows to along the line.

Nor do two lattice couplings mean two tuning parameters. Hold the ratio L/K fixed in [0, 1] and raise both couplings together, which is lowering the temperature: sinh 2K rises strictly from 0 without bound while e^−2L starts at 1 and never rises, so the ray meets the self-dual curve sinh 2K = e^−2L exactly once. Temperature alone crosses the line, once, and the ratio chooses which fixed point. That is the sweep the temperature factor f of §5 performs: it divides K and L by f at fixed L/K, and crosses criticality at f = 1.

7Weak universality: what slides and what is pinned

Suzuki’s weak universality, as Khan et al. state it: along a line generated by a marginal parameter, β (the exponent here, not the inverse temperature absorbed into K), γ and ν vary, but β/ν and γ/ν stay fixed, and with them (in their statement) δ and η.^[4] Ashkin–Teller obeys it: ν slides from 1 to 2/3, so γ = (7/4)ν slides with it, while γ/ν = 7/4 and η = 1/4 are pinned along the whole segment.^[1] The pinned η is a statement about σ and τ. The product sector is not pinned: x_στ falls from 1/4 at X₀ to 1/8 at X₄ while x_σ = 1/8 holds throughout (table in §6).

Khan et al. report a magnet that does not: in (Sm_1−yNd_y)_0.52Sr_0.48MnO₃ (y is the Nd fraction here, not an RG eigenvalue), β, γ and δ all vary with composition, δ included.^[4]

8For the indicator-builder

Four readings of the facts above. Each is a statement about this model and these measurements; the page cites no early-warning literature and makes no claim about it.

9Where else the line lives

The eight-vertex model. Ashkin–Teller maps to Baxter’s eight-vertex model, and its critical segment is, in Wiseman and Domany’s word, isomorphic to the eight-vertex critical line (Wegner; Fan and Wu; Kadanoff).^[1][7]

The c = 1 picture. Among the critical theories of central charge c = 1 are two lines, the circle line (the Gaussian model) and the orbifold line (§3). Critical Ashkin–Teller runs along the orbifold line from r = 1 (Ising × Ising) to r = √2 (four-state Potts). The two lines meet at the Kosterlitz–Thouless point of the XY model, circle r = √2 = orbifold r = 1/√2, outside the Ashkin–Teller range r ∈ [1, √2]; this page does not say where that branch meets the lattice model. Radii are in Ginsparg’s normalisation; other sources scale them differently.^[3]

A frustrated Ising model in disguise. Jin, Sen and Sandvik’s J₁–J₂ Ising model on the square lattice is Ashkin–Teller critical for g = J₂/|J₁| ≥ g* = 0.67(1), with the four-state Potts point at g*. Its critical curve g ∈ [g*, ∞) corresponds one-to-one with the Ashkin–Teller segment, L/K running from 1 towards 0. At the Potts point the Binder crossing takes the universal value U* = 0.792(4). The same model shows the pseudo-first-order signals of §8c.^[6]

10What to remember

• Between two critical Ising copies, the bilinear στ coupling (y = 7/4, relevant) gives no line; the four-spin ε_σε_τ coupling (y = 0, exactly marginal) gives a line of RG fixed points: it only changes a free boson’s radius.
• The line is exact, and its location and sharpness are proved: Z = 1 − 2X, from X₀ (Ising × Ising, α/ν = 0) to X₄ (four-state Potts, α/ν = 1).
• Along it ν, γ and α/ν slide; γ/ν = 7/4 and η = 1/4 are pinned. That is weak universality; Khan et al. report a magnet that departs from it.
• At criticality, each layer’s susceptibility grows as ℓ^7/4 all along the line, so its exponent cannot locate you there (the per-site ⟨M²⟩ falls as ℓ^−1/4); the energy’s exponent α/ν can.
• Fits at ℓ ≤ 128 miss the exact α/ν, worst at the Potts end, where a multiplicative logarithm sits.
• Double peaks and negative Binder minima appear at continuous transitions up to ℓ = 256.