Non-perturbative arrow correlations of the critical six-vertex model: exact staggered exponents and the Thirring scaling limit beyond small anisotropy

For the isotropic zero-field six-vertex model at small anisotropy, Falco (2013) derived the large-distance asymptotics of arrow-arrow correlations — a uniform dipolar term plus a staggered term with anomalous exponent $\kappa_- = 1-2\lambda/\pi+O(\lambda^2)$, where $\Delta = 1-e^\lambda$ — and stated, without proof, that the scaling limits of arbitrary $n$-point arrow correlations are, up to staggering prefactors, linear combinations of massless Thirring model correlations. The derivation (six-vertex to interacting dimers to lattice fermions to renormalization group) is perturbative and confined to small $|\Delta|$, while the Bethe-ansatz exact solution provides no closed formulas for local arrow correlations. The open problem is the non-perturbative completion: establish, for the whole critical interval $|\Delta|\le 1$, the two-point asymptotics with the exact $\Delta$-dependent anomalous exponent and amplitudes, and prove the $n$-point Thirring representation. A 2026 breakthrough proved Gaussian-free-field convergence of the height field but only on $\Delta\in[-1,-1/2]$ and only for the Gaussian (uniform) sector; the anomalous staggered sector and the Thirring identification remain open.

Background

The six-vertex model places an orientation (an arrow) on every edge of the square lattice subject to the ice rule: at each vertex exactly two arrows point in and two point out, giving six admissible local arrangements with weights $p_1,\dots,p_6$. In the zero-field isotropic case one sets $p_1=p_2=p_3=p_4=a=b$, $p_5=a$, $p_6=2ae^\lambda$, and all Gibbs measures are parametrized by the anisotropy $\Delta=(a^2+b^2-p_5p_6)/(2ab)=1-e^\lambda$. The model is 'exactly solvable' in the sense that the free energy is known exactly from the Bethe Ansatz, but exact formulas for correlations of local bulk observables — such as the arrow-orientation variables $\sigma_0(x),\sigma_1(x)\in\{\pm 1\}$ of the horizontal or vertical edge centered at $x$ — are in general not available. In the critical regime $|\Delta|\le 1$ the transfer-matrix spectrum is gapless, and arrow correlations are expected to decay as power laws at large distance.

At the free-fermion point $\Delta=0$ the arrow-arrow correlation is exactly known (Sutherland, 1968). For $\Delta\ne 0$ the expected asymptotics were inferred non-perturbatively from the Luttinger-liquid and Coulomb-gas pictures (Youngblood–Axe–McCoy, 1980; den Nijs, 1983): a line of critical points with exponents varying continuously with $\Delta$ (a 'floating' anomalous exponent for the staggered part). A Luttinger liquid is the generic gapless one-dimensional fermionic (or, via mappings, two-dimensional height-model) universality class whose correlations decay with continuously varying exponents; here it emerges from the six-vertex model's interacting-fermion representation.

Falco (2013) gave a derivation of these asymptotics that does not use the exact solution: the six-vertex model is first mapped to an interacting dimer model (Baxter), then to lattice Grassmann fermions with a staggered quartic interaction, which are analyzed by Wilsonian renormalization group (RG) and compared with the exactly solvable massless Thirring model — two-dimensional Euclidean fermions with a current-current quartic interaction, solvable by bosonization. For small $|\Delta|$ (equivalently small $|\lambda|$) he derived, for the two-point arrow correlation, a two-term large-distance asymptotic: a uniform (dipolar) term of order $|x|^{-2}$ with amplitude $c_0$, plus a staggered term

$c_-\,\frac{(-1)^{x'_0\pm x'_1}}{(x_0'^2+x_1'^2)^{\kappa_-}}$

(in diagonal lattice coordinates $x=x'_0v_0+x'_1v_1$, the sign pattern depending on the pair of arrows) with anomalous exponent

$\kappa_- = 1-\frac{2\lambda}{\pi}+O(\lambda^2),$

and amplitudes $c_0,c_- = 2/\pi^2+O(\Delta)$; at $\Delta=0$ this reproduces Sutherland's exact result, and as an application the height variable (the integer scalar potential whose discrete gradient is the arrow field) has Gaussian-boson logarithmic covariance. His conclusion states, without proof, that 'using ideas in' earlier Thirring-model works 'one could show' that the scaling limit of arbitrary $n$-point arrow correlations is, apart from staggering prefactors, a linear combination of Thirring-model correlations.

All of this is perturbative in $\lambda$. Later rigorous work sharpened but did not remove this restriction: the Giuliani–Mastropietro–Toninelli program for interacting (non-integrable) dimer models proved Gaussian height fluctuations, multipoint dimer correlations with anomalous exponents, and Haldane-type amplitude–exponent scaling relations, but only for sufficiently weak interaction, with the six-vertex model covered only 'close to the free-fermion point'. In 2026, Duminil-Copin–Kozlowski–Lammers–Manolescu proved the first genuinely non-perturbative scaling-limit result for the six-vertex model: on $\Delta\in[-1,-1/2]$ the height function converges to a full-plane Gaussian free field with the Coulomb-gas amplitude, using spectral analysis of the transfer matrix plus RSW-type (percolation-style crossing) estimates rather than constructive RG. Their paper states explicitly that constructive renormalization 'remain[s] confined to perturbative regimes' and that part of the critical interval ($c<1$, i.e. $\Delta>1/2$) is 'beyond the reach of current methods'. This concerns only the Gaussian height sector: the staggered part of the arrow correlations with its anomalous exponent — the content of Falco's formulas — is a vertex-operator (non-Gaussian) sector not reached by any existing non-perturbative method, and the projected $n$-point Thirring representation has not been performed anywhere in the literature.

Problem Statement

Extend Falco's analysis of zero-field six-vertex arrow correlations beyond the perturbative small-$|\Delta|$ regime and complete it non-perturbatively over the critical interval. Concretely, for the isotropic zero-field six-vertex model ($a=b$, $p_5=a$, $p_6=2ae^\lambda$, $\Delta=1-e^\lambda$), establish for every fixed $\Delta$ with $|\Delta|<1$ (with the Kosterlitz–Thouless endpoints $\Delta=\pm1$, where logarithmic corrections are expected, treated as limiting cases) the following two linked statements:

(A) Two-point asymptotics. The arrow-arrow correlations of two vertical arrows and of one vertical with one horizontal arrow admit the two-term large-distance decomposition

$\langle\sigma(x+y)\sigma(y)\rangle \;\sim\; c_0(\Delta)\cdot F(x) \;+\; c_-(\Delta)\,\frac{(-1)^{x'_0\pm x'_1}}{(x_0'^2+x_1'^2)^{\kappa_-}},$

where $F$ is the uniform dipolar term of order $|x|^{-2}$ of the $\Delta=0$ free-fermion correlation, the sign pattern $(-1)^{x'_0\pm x'_1}$ alternates on the diagonal sublattices, and the anomalous exponent $\kappa_-(\Delta)$ and amplitudes $c_0(\Delta),c_-(\Delta)$ take their exact $\Delta$-dependent values (of which Falco's $\kappa_-=1-2\lambda/\pi+O(\lambda^2)$ and $c_0,c_-=2/\pi^2+O(\Delta)$ are the first-order expansions), thereby settling whether the two-term decomposition and staggering structure persist for all $\Delta$ in the critical interval, with the associated higher-order coefficients and remainder control obtained to all orders.

(B) $n$-point Thirring representation. Prove that the scaling limit of arbitrary $n$-point arrow correlations is, up to staggering prefactors, a linear combination of massless Thirring model correlations, with the Thirring coupling identified exactly in terms of $\Delta$ — upgrading the source paper's unperformed 'one could show' statement to a theorem.

The exact Bethe-ansatz solution provides the free energy and finite-size spectral data but no closed formulas for local arrow correlations, so a solution cannot be read off from integrability and must extend existing machinery (constructive renormalization group beyond weak coupling, or non-perturbative spectral/RSW-type techniques pushed from the Gaussian height sector to the anomalous vertex-operator sector). The two deliverables are components of one construction — a non-perturbative Luttinger-liquid description of the critical six-vertex model — and an answer must establish both the exact exponents of the staggered sector and the Thirring identification over the stated parameter range; partial weak-coupling extensions do not answer the problem.

References

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