Existence of a nonlinear Painlevé-type ordinary differential equation for the row correlation functions of the two-dimensional Ising model

The two-point correlation functions of the zero-field square-lattice Ising model are among the most studied exact objects in statistical mechanics. For the diagonal correlations $D_n=\langle\sigma_{0,0}\sigma_{n,n}\rangle$, Jimbo and Miwa (1980) proved a nonlinear second-order ordinary differential equation of Painlevé VI type in the sigma form, with the spin separation $n$ entering only as a parameter, whose solutions determine $D_n$ at all temperatures. For the row correlations $R_n=\langle\sigma_{0,0}\sigma_{n,0}\rangle$ no analogous equation is known: $R_n$ are Toeplitz determinants with a more complicated elliptic-integral structure, and Shrock (2022) states explicitly that it remains open whether any nonlinear (second-order) ODE of the Jimbo–Miwa type exists whose solutions yield the general $R_n$ — no construction, no existence proof, and no non-existence theorem is known. The question is whether such an equation exists for the isotropic model at general temperature, in the precise sense specified in the Problem Statement below. Known partial results delimit but do not resolve the question: coupled Garnier-system (isomonodromic) descriptions exist through the anisotropic triangular lattice (Witte 2016) but are multi-variable systems, not a single second-order equation, and explicit sigma-form Painlevé VI equations are known only on the special anisotropic locus $\nu=-k$, which does not contain the isotropic model.

Background

The Ising model is the paradigmatic exactly solved statistical system: spins $\sigma=\pm1$ occupy the sites of a square lattice with nearest-neighbor coupling $J$, and in zero magnetic field its free energy (Onsager 1944), spontaneous magnetization (Yang 1952), and two-point correlation functions are known exactly. The spin-spin correlation between sites separated by the lattice vector $(m,n)$ is the thermal average $C(m,n)=\langle\sigma_{0,0}\sigma_{m,n}\rangle$. Two families have been studied in depth: the diagonal correlations $D_n=C(n,n)$ and the row correlations $R_n=C(n,0)$. Montroll, Potts and Ward (1963) expressed both as Toeplitz determinants — determinants whose entry in position $(j,k)$ depends only on the difference $j-k$ — with entries built from the model's dual variables.

Standard temperature variables are $v=\tanh(\beta J)$ (with $\beta=1/k_BT$) for $T\ge T_c$ and $z=e^{-2\beta J}$ for $T\le T_c$, or equivalently the elliptic moduli $k_> = \sinh^2(2\beta J)$ and its dual $k_< = 1/k_>$, which both equal 1 exactly at the critical temperature $T_c$. Here $K(k)$ and $E(k)$ denote the complete elliptic integrals of the first and second kind, $K(k)=\int_0^{\pi/2}(1-k^2\sin^2\theta)^{-1/2}d\theta$ and $E(k)=\int_0^{\pi/2}(1-k^2\sin^2\theta)^{1/2}d\theta$, special functions of the modulus $k$ that pervade all exact Ising results.

The deepest structural result concerns the diagonal family. Painlevé VI (PVI) is the sixth and most general of the classical Painlevé equations — nonlinear second-order ordinary differential equations whose general solutions have no movable branch-point singularities; its Okamoto sigma form is an equation for a logarithmic-derivative combination $\sigma$ of the unknown. Jimbo and Miwa (1980) proved that each $D_n$ is determined by such an equation: with $t=k_>^{-2}$ for $T\ge T_c$ (respectively $t=k_<^{-2}$ for $T\le T_c$) and $\sigma_{n}=t(t-1)\,d\ln D_{n}/dt-t/4$ for $T\ge T_c$ (respectively minus $1/4$ for $T\le T_c$), $\sigma_n$ satisfies (as transcribed in Shrock 2022, Appendix A)

$\big[t(t-1)\,\sigma_n''\big]^2-n^2\big[(t-1)\sigma_n'-\sigma_n\big]^2+4\sigma_n'\Big[(t-1)\sigma_n'-\sigma_n-\tfrac{1}{4}\Big]\big(t\sigma_n'-\sigma_n\big)=0,\qquad \sigma_n'\equiv d\sigma_n/dt.$

The separation $n$ enters this single second-order equation only through the parameter $n^2$, and $D_n$ is recovered from $\sigma_n$ by one quadrature. This made the Ising diagonal correlations tau-functions of PVI — a tau-function is a potential-like object whose logarithmic derivatives build the solutions of the associated equation, exactly as $D_n$ builds $\sigma_n$ above — and opened the large interface between Ising correlations, Painlevé equations, and isomonodromic deformation theory (the study of deformations of a linear differential equation that preserve its monodromy, i.e. the analytic-continuation data of its solutions; the Garnier systems are the multi-variable generalizations of PVI arising in this framework).

The row correlations have resisted every analogous treatment. Their exact structure is genuinely more complicated: Ghosh and Shrock (1984, 1985) showed that while $D_n$ is a homogeneous polynomial of degree $n$ in $K(k)$ and $E(k)$, each $R_n$ is instead a multi-'level' sum — for even $n$,

$R_{n,\pm}=B_n\,k^{-p_n}\sum_{\ell=0}^{n/2}\pi^{-2\ell}\sum_{s=0}^{2\ell}\mathcal R^{(n,\pm)}_{2\ell-s,s}(k)\,E(k)^{\,2\ell-s}\,\bar K(k)^{\,s},$

with $\bar K(k)=(k-1)K(k)$, $p_n=\lfloor n^2/4\rfloor$, a numeric prefactor $B_n$, and coefficient polynomials $\mathcal R^{(n,\pm)}_{a,b}(k)$; odd $n$ takes an analogous form with an additional factor $(1+k^{-1})^{1/2}$. An immediate consequence is that each fixed $R_n$ is D-finite — it satisfies some linear ODE with rational coefficients, of order growing with $n$ — so 'an ODE' in the trivial sense always exists. The meaningful open question is therefore a uniform nonlinear second-order equation with $n$ as a parameter, exactly parallel to Jimbo–Miwa. What is known for $R_n$ is of a different character: Perk (1980) discovered quadratic identities, and McCoy and Wu (1980) nonlinear partial difference equations, that determine all $C(m,n)$ recursively in the separations — discrete relations, not differential equations. Lyberg and McCoy (2007) gave form-factor (multi-integral) expansions of the row and diagonal correlations. Witte (2007, 2016) connected the next-to-diagonal correlations, and — through the diagonal correlations of the anisotropic triangular lattice, which include the square-lattice row/column correlations as specializations — the row correlations themselves, to Garnier-system isomonodromy; but the associated differential equations were not written down, and they are coupled multi-variable systems rather than a single second-order equation. Boukraa, Maillard and McCoy (2020) did derive explicit sigma-form PVI equations for $C(M,N)$ including the row case, but only on the special anisotropic locus $\nu=-k$ relating the anisotropy ratio $\nu$ to the modulus $k$ — a locus that does not contain the isotropic model ($\nu=1$). The anisotropic correlations themselves are built from elliptic integrals including those of the third kind (McCoy and Maillard 2016). In this state of knowledge, Shrock (2022) posed the existence question for $R_n$ sharply and contributed general-$n$ coefficients of the high- and low-temperature series of $R_n$ as new data for the quest.

Problem Statement

For the zero-field isotropic square-lattice Ising model in the thermodynamic limit, settle whether there exists a nonlinear second-order ordinary differential equation, in a single temperature variable (for example $t=k_>^{-2}$ for $T\ge T_c$ and $t=k_<^{-2}$ for $T\le T_c$, where $k_>=\sinh^2(2\beta J)$ above $T_c$ and $k_<=1/k_>$ below it, or an equivalent single-variable reparametrization), with the spin separation $n$ entering only as a parameter of the equation — in precise analogy to the Jimbo–Miwa sigma-form Painlevé VI equation for the diagonal correlations $D_n$ — whose solutions determine the row correlation functions $R_n=\langle\sigma_{0,0}\sigma_{n,0}\rangle$ for general $n$ and both temperature phases. Throughout this statement, 'Painlevé-type' in the title refers to this precise analogy with the Jimbo–Miwa sigma-form equation: the operative criteria for both branches below are the uniform-in-$n$ nonlinear second-order properties specified here, exactly as in Shrock's (2022) formulation of the open question, and no further singularity property is imposed on admissible equations beyond the stated analogy. The question has two directions. (i) Constructive branch: exhibit the equation explicitly — specifying the auxiliary function of $R_n$ that plays the role of the sigma function, the equation itself and its $n$-dependence, and the boundary or normalization conditions that single out the Ising solution — together with a proof that the solutions so identified equal $R_n$ for all $n\ge 1$ in both phases. (ii) Negative branch: prove that no member of the admissible class determines the $R_n$. The admissible class is fixed as the class of all nonlinear second-order equations — explicit nonlinear ordinary differential equations, or implicit algebraic relations in the dependent variable and its first two derivatives that are not linear in them — in a single independent variable (the temperature variable, or an equivalent single-variable reparametrization), with coefficients rational in that variable and in $n$, so that linear second-order equations lie outside the class; this class contains the Jimbo–Miwa equation as the analogous member for the diagonal case. The class is non-vacuous: it is specified by general structural properties, independently of the non-existence proof, and contains infinitely many distinct equations (the Jimbo–Miwa equation being one of them); non-existence must not be established merely by individually excluding a finite enumerated set of equations whose solutions are already known to differ from the $R_n$ — in particular, the singleton class consisting of the Jimbo–Miwa equation itself, whose solutions yield $D_n\neq R_n$, is vacuous and inadmissible.

References

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