Extended scaling relations for the eight-vertex and Ashkin-Teller models beyond the small-coupling regime

The critical exponents of the two-dimensional eight-vertex (8V) and Ashkin-Teller (AT) models vary continuously along a line of fixed points, yet are conjectured to be rigidly constrained by exact algebraic extended scaling relations. Benfatto, Falco and Mastropietro (2009) proved rigorously that $x_-=1/x_+$, $\alpha=1/(2-x_+)$ and $x_T=(2-x_+)/(2-x_-)$, but only for small quartic coupling, for local energy-like observables, and for quartic interactions satisfying specific symmetry conditions. This problem asks for the exact non-perturbative domain of validity of these relations: their status at finite or large coupling, and a rigorous derivation of the conjectured polarization-correlation index relations for the lattice models themselves. Whether the same relations hold at finite or large coupling, and whether the conjectured polarization-correlation index relations admit a rigorous derivation for the lattice models, remain open.

Background

The models and their peculiar universality. The Ashkin-Teller (AT) model and the eight-vertex (8V) model are classical two-dimensional lattice models that can both be written as two nearest-neighbor Ising models coupled by a four-spin (quartic) interaction, $H=H_J(\sigma)+H_{J'}(\sigma')-J_4V(\sigma,\sigma')$, where $\sigma,\sigma'=\pm1$ are two Ising spin configurations, $H_J$ is the ferromagnetic nearest-neighbor Ising Hamiltonian, and the quartic interaction is either $V^{AT}=\sum_{x,j}\sigma_x\sigma_{x+e_j}\sigma'_x\sigma'_{x+e_j}$ (one bond of each Ising copy per term, with possibly $J\ne J'$ - the AT model) or the plaquette product $V^{8V}$ with $J=J'$ (the 8V model); $e_0,e_1$ are the lattice unit vectors. Baxter's exact solution of the 8V model revealed that, unlike the Ising model, its critical exponents vary continuously with the coupling $J_4$: the critical surface is not an isolated renormalization-group fixed point but a line of fixed points, a one-parameter family of fixed points along which the exponents change continuously. The AT model is expected to lie in the same universality class but is not exactly solvable.

The conjectured extended scaling relations. In the 1970s, Kadanoff, Luther and Peschel, den Nijs, Kadanoff and Brown, and Pruisken and Brown argued - via formal bosonization, i.e. the representation of the critical spin correlations in terms of an effective free boson, by analogy with the Luttinger liquid (the exactly solved model of one-dimensional interacting fermions whose correlation exponents vary continuously with the interaction) - that although the exponents of the 8V/AT class are non-universal functions of $J_4$, they are all determined by any single one of them through exact algebraic relations. Writing $x_\pm$ for the exponents of the energy and crossover correlations (the operators $O_x^\pm=\sum_j(\sigma_x\sigma_{x+e_j}\pm\sigma'_x\sigma'_{x+e_j})$, whose critical truncated correlations decay as $|x-y|^{-2x_\pm}$), $\alpha$ for the exponent of the inverse correlation length ($\xi^{-1}\sim|T-T_c|^\alpha$), $x_T$ for the transition index of the anisotropic AT model ($|T_{1,c}-T_{2,c}|\sim|J-J'|^{x_T}$), and $X_P$ for the polarization exponent (polarization $p_x=\sigma_x\sigma'_x$, with $\langle p_xp_y\rangle\sim|x-y|^{-2X_P}$ at criticality), the conjectured relations include $x_-x_+=1$, $\alpha=1/(2-x_+)$ and $x_T=(2-x_+)/(2-x_-)$, together with the polarization relation $X_P=x_+/4=1/(4x_-)$ and further conjectured relations for the spin (magnetization) correlations.

The rigorous result and its method. Mastropietro (2003, 2004) initiated a rigorous constructive renormalization-group (RG) program - a rigorous multiscale analysis of the Grassmann-integral (anticommuting-variable) representation of the correlations - proving that the 8V and isotropic AT models have the above power-law criticality with exponents given by convergent series in the coupling; Giuliani and Mastropietro (2004, 2005) treated the anisotropic AT model. Benfatto, Falco and Mastropietro (2009) then proved the extended scaling relations: for $|J_4|$ small enough, $x_-=1/x_+$, $\alpha=1/(2-x_+)$ and $x_T=(2-x_+)/(2-x_-)$, and the 8V and AT indices coincide for a suitable choice of $J_4^{AT}$ as a function of $J_4^{8V}$. The proof has three ingredients: (i) constructive RG expresses each exponent as a convergent series in the asymptotic coupling $\lambda_{-\infty}$; (ii) an equivalence theorem shows that the lattice exponents equal those of a continuum quantum-field-theoretic reference model whose bare coupling $\lambda_\infty$ is fixed through an analytic, locally invertible map $h$, with $\lambda_{-\infty}=h(\lambda_\infty)=\lambda_\infty+O(\lambda_\infty^2)$; (iii) the reference model enjoys extra chiral symmetries producing Ward identities (exact symmetry relations among correlation functions) and closed equations, which combined with the non-perturbative Adler-Bardeen theorem (anomaly non-renormalization: the coefficient of the chiral anomaly receives no corrections beyond one loop) proved by Mastropietro (2007) fixes the anomaly coefficient exactly ($\nu=\lambda_\infty/(4\pi)$) and forces the algebraic relations rather than determining them order by order.

What remains open. Every ingredient is proved only in a neighborhood of $J_4=0$, for quartic interactions satisfying specific symmetry conditions, and for local energy-like observables. The authors state explicitly that the indices of the polarization correlations are much more difficult to study with their methods and that new ideas seem to be required for such cases. Whether the same exact algebra holds at finite or large coupling - where the RG flow is genuinely non-perturbative and the convergent expansions are unavailable - is unknown. Baxter's exact solution of the 8V model determines the free energy, and hence the thermal exponent $\alpha$, in closed form, but it does not determine the correlation exponents $x_\pm$ at arbitrary coupling - Benfatto, Falco and Mastropietro [1] note that Kadanoff's extended scaling law was conjectured 'mainly on the basis of numerical evidence' - and the AT model has no exact solution at all.

Problem Statement

Determine, with mathematical rigor, the domain of validity of the extended scaling relations of the eight-vertex/Ashkin-Teller line of fixed points beyond the small-coupling regime in which they were proved by Benfatto, Falco and Mastropietro (2009).

Let $H=H_J(\sigma)+H_{J'}(\sigma')-J_4V(\sigma,\sigma')$ be the coupled-Ising Hamiltonian defining the 8V or AT model, with the conventions of the background: $\sigma,\sigma'=\pm1$ are two Ising spin configurations, the quartic interaction is the bond product $V^{AT}$ (possibly $J\ne J'$) or the plaquette product $V^{8V}$ ($J=J'$), and $e_0,e_1$ are the lattice unit vectors. The critical exponents at issue are: $x_\pm$, defined by the critical truncated correlations of the energy and crossover operators $O_x^\pm=\sum_{j=0,1}(\sigma_x\sigma_{x+e_j}\pm\sigma'_x\sigma'_{x+e_j})$, which decay as $|x-y|^{-2x_\pm}$; $\alpha$, defined by the inverse correlation length $\xi^{-1}\sim|T-T_c|^\alpha$ as $T\to T_c$; $x_T$, the transition index of the anisotropic AT model, defined by $|T_{1,c}-T_{2,c}|\sim|J-J'|^{x_T}$ as $J-J'\to0$; and $X_P$, the polarization exponent, defined by the critical decay $\langle p_xp_y\rangle\sim|x-y|^{-2X_P}$ of the polarization $p_x=\sigma_x\sigma'_x$. The relations proved so far, for $|J_4|$ small and under symmetry hypotheses on $V$, are

$x_- = \frac{1}{x_+},\qquad \alpha = \frac{1}{2-x_+},\qquad x_T = \frac{2-x_+}{2-x_-},$

together with the matching of 8V and AT indices for a suitable choice of the quartic couplings; the extended scaling framework additionally conjectures, but does not prove for the lattice models, the polarization-correlation relation

$X_P = \frac{1}{4}x_+ = \frac{1}{4x_-}.$

The problem comprises two linked facets of one common objective - the exact, non-perturbative algebra of this line of fixed points:

1. Finite and large coupling. Prove or disprove that the above three relations hold uniformly for finite or large $|J_4|$, outside the small-coupling neighborhood in which the exponents are controlled by convergent expansions in the effective coupling $\lambda_{-\infty}$; if a relation fails, identify the mechanism (for example, a modification of the anomaly structure of the continuum reference theory) and demonstrate the failure rigorously in an explicit model of the class.

2. Non-local observables. Provide a rigorous derivation, for the 8V/AT lattice models themselves, of the conjectured polarization-correlation index relation $X_P = \frac{1}{4}x_+ = \frac{1}{4x_-}$, and of the analogous conjectured index relations for the other non-local observables - the spin (magnetization) correlations - for which a rigorous derivation of any one of the relations counts as resolving that part. The current fermionic constructive methods, which reach only local energy-like observables, do not reach any of them.

References

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