For the square-lattice Ising model, the diagonal spin-spin correlation functions DnD_n are determined, for every separation nn, by a single nonlinear second-order ordinary differential equation - the Okamoto sigma form of the sixth Painlevé equation - proved by Jimbo and Miwa in 1980. Whether an analogous nonlinear ordinary differential equation exists for the row correlation functions Rn=⟨σ0,0σn,0⟩R_n=\langle\sigma_{0,0}\sigma_{n,0}\rangle of the isotropic model is an open problem posed by Shrock (J. Phys. A 55, 425001 (2022)), which contributed general-nn series coefficients of RnR_n as input data for the search: no such equation is known, and neither an existence proof nor a non-existence proof is available. The known partial results leave the question untouched: the Painlevé connection for row correlations holds only in the scaling limit (Painlevé III); the exact nonlinear relations valid for all correlations (Perk's quadratic identities, McCoy-Wu partial difference equations) are discrete rather than differential; form-factor expansions give series representations, not equations; and the nonlinear Painlevé VI equations found for general anisotropic correlations C(M,N)C(M,N) hold only on the special loci ν=−k\nu=-k and ν=−1/k\nu=-1/k where those correlations degenerate to diagonal ones, while the isotropic row case corresponds to ν=1\nu=1. The problem is to determine whether such an equation exists, to construct it and prove that it determines RnR_n for all nn in both temperature phases if it does, or to prove non-existence for a precisely specified class of candidate equations.

Background

The two-dimensional Ising model on the square lattice, with Hamiltonian H=−J∑i,j(σi,jσi+1,j+σi,jσi,j+1)H=-J\sum_{i,j}(\sigma_{i,j}\sigma_{i+1,j}+\sigma_{i,j}\sigma_{i,j+1}) and spins σ=±1\sigma=\pm1, is the canonical exactly solved statistical system. Among its exact results are the two-point spin-spin correlation functions, of which the row correlation Rn=⟨σ0,0σn,0⟩R_n=\langle\sigma_{0,0}\sigma_{n,0}\rangle (spins separated by nn lattice spacings along a row) and the diagonal correlation Dn=⟨σ0,0σn,n⟩D_n=\langle\sigma_{0,0}\sigma_{n,n}\rangle are the standard representatives. Both are expressible as Toeplitz determinants - determinants of matrices whose entries depend only on the difference of their row and column indices - with symbols determined by the free-fermion dispersion of the model. The correlations are naturally organized by the elliptic modulus: with K=J/(kBT)K=J/(k_BT) and v=tanh⁡Kv=\tanh K, one has k>=sinh⁡2(2K)k_{>}=\sinh^2(2K) for T≥TcT\ge T_c and k<=sinh⁡−2(2K)k_{<}=\sinh^{-2}(2K) for T≤TcT\le T_c, both lying in [0,1][0,1] and equal to 1 at the critical point Kc=12ln⁡(2+1)K_c=\frac{1}{2}\ln(\sqrt{2}+1). Exact calculations show that DnD_n is a homogeneous polynomial of degree nn in the complete elliptic integrals K(k)K(k) and E(k)E(k) (the standard special functions ∫0π/2(1−k2sin⁡2θ)∓1/2 dθ\int_0^{\pi/2}(1-k^2\sin^2\theta)^{\mp1/2}\,d\theta of the first and second kind), while RnR_n has the substantially more complicated form of a finite linear combination of products E(k)aKˉ(k)bE(k)^a\bar K(k)^b, with Kˉ(k)=(k−1)K(k)\bar K(k)=(k-1)K(k) and coefficients that are rational functions of kk, times a prefactor Bnk−pnB_n k^{-p_n} with pn=⌊n2/4⌋p_n=\lfloor n^2/4\rfloor (Ghosh and Shrock, 1985; Shrock, 2022).

A deep chapter of this subject connects these correlations to the Painlevé equations - the six classical nonlinear second-order ordinary differential equations whose general solutions define new transcendental functions and whose defining property (the Painlevé property: no movable critical points, i.e. no singularities whose type depends on the initial data) makes them the nonlinear analogues of the classical special functions. In the scaling limit T→TcT\to T_c with n(T−Tc)n(T-T_c) held fixed, Wu, McCoy, Tracy and Barouch (1976) and McCoy, Tracy and Wu (1977) showed that the two-point functions are expressed through solutions of the third Painlevé equation. On the exact lattice, Jimbo and Miwa (1980) obtained the definitive result for the diagonal case: with the variable t=k>−2t=k_{>}^{-2} for T≥TcT\ge T_c and t=k<−2t=k_{<}^{-2} for T≤TcT\le T_c, the auxiliary functions

σn,+=t(t−1)dln⁡Dn,+dt−t4,σn,−=t(t−1)dln⁡Dn,−dt−14\sigma_{n,+}=t(t-1)\frac{d\ln D_{n,+}}{dt}-\frac{t}{4},\qquad \sigma_{n,-}=t(t-1)\frac{d\ln D_{n,-}}{dt}-\frac{1}{4}

satisfy the Okamoto sigma form of the sixth Painlevé equation (Painlevé VI), a Hamiltonian symmetric form for the logarithmic derivative of the equation's tau function:

[t(t−1)σn′′]2−n2[(t−1)σn′−σn]2+4σn′[(t−1)σn′−σn−14](tσn′−σn)=0.\big[t(t-1)\sigma_n''\big]^2-n^2\big[(t-1)\sigma_n'-\sigma_n\big]^2+4\sigma_n'\Big[(t-1)\sigma_n'-\sigma_n-\frac{1}{4}\Big]\big(t\sigma_n'-\sigma_n\big)=0.

This is a single nonlinear second-order equation in which the separation nn enters only as a parameter, and whose solutions, with boundary conditions selecting the physical branch, determine DnD_n for all nn at once; it is the engine behind exact results such as the general-nn coefficients of the high-temperature series of DnD_n.

For the row correlations no analogous statement is known. What exists is: (i) exact nonlinear relations of discrete type - the quadratic identities of Perk (1980) and the nonlinear partial difference equations of McCoy and Wu (1980), which determine all correlations C(M,N)C(M,N) recursively from initial data but are difference equations, not ordinary differential equations in the temperature variable; (ii) form-factor expansions (Lyberg and McCoy, 2007; Mangazeev and Guttmann, 2010), convergent infinite-series representations of RnR_n and DnD_n, which connect the diagonal case to Painlevé VI tau functions - and for which Mangazeev and Guttmann note that even for the nearest-neighbour row correlation C(0,1;λ)C(0,1;\lambda) - the lambda extension, i.e. the one-parameter deformation of the form-factor expansion in which each term carries an extra power of the parameter λ\lambda, reducing to the physical correlation at λ=1\lambda=1 - a connection to Painlevé VI theory "is not known"; and (iii) a family of nonlinear Painlevé VI sigma-form equations, depending explicitly on the integers MM and NN, that determine the anisotropic correlations C(M,N)C(M,N) with M≤NM\le N (Boukraa, Maillard and McCoy, 2020) - but only on the special locus ν=−k\nu=-k in the (k,ν)(k,\nu) parametrization of the anisotropic model, where EhE_h and EvE_v are the horizontal and vertical bond couplings, ν=sinh⁡(2Eh)/sinh⁡(2Ev)\nu=\sinh(2E_h)/\sinh(2E_v), and the correlations degenerate, up to a power of 1−t1-t, to the diagonal correlation DND_N (the companion locus ν=−1/k\nu=-1/k is obtained from ν=−k\nu=-k by the transposition symmetry C(M,N;ν=−k)=C(N,M;ν=−1/k)C(M,N;\nu=-k)=C(N,M;\nu=-1/k)). The isotropic row correlations RnR_n correspond to ν=1\nu=1 and are not covered by any of these results. Shrock (2022) contributed general-nn expressions for the coefficients of the high- and low-temperature series of RnR_n (for instance Rn=vn+n(n+1)vn+2+⋯R_n=v^n+n(n+1)v^{n+2}+\cdots in v=tanh⁡Kv=\tanh K, with higher coefficients explicit rational functions of nn), obtained by fitting polynomials in nn to exactly computed RnR_n for 1≤n≤61\le n\le6, explicitly as input data for the search for such an ordinary differential equation. The question is whether that search can ever succeed.

Problem Statement

For the isotropic square-lattice Ising model with dimensionless coupling K=J/(kBT)K=J/(k_BT), consider the row spin-spin correlation functions Rn=⟨σ0,0σn,0⟩R_n=\langle\sigma_{0,0}\sigma_{n,0}\rangle, n≥1n\ge1, as exact functions of temperature through the elliptic-modulus variable t=k>−2t=k_{>}^{-2} for T≥TcT\ge T_c and t=k<−2t=k_{<}^{-2} for T≤TcT\le T_c, where k>=sinh⁡2(2K)k_{>}=\sinh^2(2K) and k<=sinh⁡−2(2K)k_{<}=\sinh^{-2}(2K) are the dual elliptic moduli (both in [0,1][0,1], equal to 1 at the critical point TcT_c, defined by sinh⁡(2Kc)=1\sinh(2K_c)=1).

Determine whether there exists a nonlinear ordinary differential equation, analogous to the Jimbo-Miwa Okamoto sigma-form Painlevé VI equation for the diagonal correlations DnD_n - that is, an equation of fixed (in the envisioned analogue, second) order in tt, nonlinear, with the separation nn entering only as a parameter and not altering the order or structure of the equation - which, together with boundary conditions singling out the physical solution, determines the row correlations RnR_n (equivalently, has the RnR_n among its solutions) for all n≥1n\ge1 in both temperature phases. The equation is required to be given explicitly, with coefficients that are functions of tt and the parameter nn alone - as in the sigma form - and fixed independently of RnR_n and of the elliptic integrals K(k),E(k)K(k),E(k) and the Toeplitz determinant from which RnR_n is constructed; an equation whose coefficients encode RnR_n itself does not answer the question.

An affirmative answer must supply: (a) the explicit equation and its boundary conditions; (b) a proof that the equation determines RnR_n for all nn in both phases T≥TcT\ge T_c and T≤TcT\le T_c - for example by derivation from the Toeplitz-determinant representation, from Perk's quadratic identities, from the form-factor expansion, or by an equivalent certified verification; and (c) a demonstration of consistency with the exact data available, including the general-nn series coefficients of RnR_n and the exactly computed values R1,…,R6R_1,\dots,R_6. For clarity: for each fixed nn, RnR_n is a finite combination of products of complete elliptic integrals and therefore satisfies some linear ODE whose order grows with nn; such per-nn linear equations do not answer the question, which asks for one uniform nonlinear equation with nn as a parameter, exactly as the sigma-form Painlevé VI equation serves all DnD_n at once.

A negative answer must prove, for a precisely specified class of candidate equations, that no equation of that class can determine the RnR_n - for example by showing that the required uniform nonlinear differential structure in tt, with nn as a parameter, fails for the row correlations. The specified class is admissible only if it contains every equation of that kind - nonlinear, of fixed order in tt, with nn entering only as a parameter and with coefficients that are functions of tt and nn alone; a non-existence result established only for a proper subclass of that kind does not resolve the question.

The question is stated for the isotropic row correlations RnR_n; resolving the broader question for the general anisotropic two-point function C(M,N)C(M,N) at general anisotropy ν\nu (of which RnR_n is the row case, ν=1\nu=1) also resolves it and is the form in which Boukraa, Maillard and McCoy posed the problem. The special loci ν=−k\nu=-k and ν=−1/k\nu=-1/k, where Painlevé VI equations are already known, do not by themselves resolve the problem.

References

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