Near the critical point of the anisotropic square-lattice Ising model, Wu, McCoy, Tracy and Barouch (WMTB, 1976) showed that the two-point spin correlation in arbitrary lattice position has the expansion ⟨σ0,0σM,N⟩=R−1/4F±(t~)+R−5/4F1±(t~)+o(R−5/4)\langle\sigma_{0,0}\sigma_{M,N}\rangle=R^{-1/4}F_{\pm}(\tilde{t})+R^{-5/4}F_{1\pm}(\tilde{t})+o(R^{-5/4}), where RR is the anisotropically rescaled separation and the scaling functions F±F_{\pm} are expressed through Painlevé transcendents. WMTB raised, and conjecturally answered, the question of how the leading finite-separation corrections to these scaling functions can be characterized as solutions of differential equations. Forrester, Perk, Trinh and Witte (2019) resolved this question only for the diagonal correlation C(N,N)C(N,N): the corrections of order N−1N^{-1} and N−2N^{-2} satisfy inhomogeneous linear second-order ODEs with coefficients given by the Painlevé transcendents of the scaling limit, and the N−1N^{-1} term is trivial. The open problem is to extend this characterization to the two-point correlation in general position — arbitrary lattice direction and anisotropy, above and below TcT_c — including the status of WMTB's claims about the correction functions and the subleading susceptibility amplitudes.

Background

The ferromagnetic Ising model on the square lattice, with nearest-neighbour couplings E1,E2E_1,E_2 along the two lattice directions and β=1/kBT\beta=1/k_BT, is the paradigmatic exactly solved model of a phase transition: its free energy was computed by Onsager (1944) and its spontaneous magnetization by Yang (1952). For the spin-spin correlation function ⟨σ0,0σM,N⟩\langle\sigma_{0,0}\sigma_{M,N}\rangle, exact finite-separation formulas exist but become analytically subtle near the critical point, where the correlation length diverges and the correlation admits a scaling limit: the separation NN tends to infinity while the scaling variable s=N(1−t)/2s=N(1-t)/2 is held fixed, where tt is a temperature variable with t=1t=1 at criticality.

For the diagonal correlation C(N,N)=⟨σ0,0σN,N⟩C(N,N)=\langle\sigma_{0,0}\sigma_{N,N}\rangle the starting point is a Toeplitz determinant — a determinant whose entry in position (i,j)(i,j) depends only on the difference i−ji-j — as derived by Wu (1966) from the Kaufman–Onsager representation. Jimbo and Miwa (1980) showed that C(N,N)C(N,N) is a τ\tau-function (a function generating solutions) of the sixth Painlevé equation in so-called σ\sigma-form, an equation for the logarithmic derivative of the correlation; the Painlevé equations are six classical second-order nonlinear ODEs whose solutions are the 'non-elementary' special functions. In the scaling limit this σ\sigma-form degenerates to a σ\sigma-form of the degenerate fifth Painlevé equation, equivalent to a Painlevé III equation, whose transcendents specify the two scaling functions (one for T>TcT>T_c, one for T<TcT<T_c). Independent access is provided by the form factor expansion of Lyberg and McCoy (2007) — a multi-particle integral representation of the correlation — and by high-order series in the coupling (Orrick–Nickel–Guttmann–Perk).

Wu, McCoy, Tracy and Barouch (WMTB, 1976) treated the correlation in general position on the anisotropic lattice. Writing zj=tanh⁡βEjz_j=\tanh\beta E_j, R2=(sinh⁡2βE1/sinh⁡2βE2)1/2M2+(sinh⁡2βE2/sinh⁡2βE1)1/2N2R^2=(\sinh 2\beta E_1/\sinh 2\beta E_2)^{1/2}M^2+(\sinh 2\beta E_2/\sinh 2\beta E_1)^{1/2}N^2 and t~=∣z1z2+z1+z2−1∣ [z1z2(1−z12)(1−z22)]−1/4R\tilde t=|z_1z_2+z_1+z_2-1|\,[z_1z_2(1-z_1^2)(1-z_2^2)]^{-1/4}R for their scaling variable, they found

⟨σ0,0σM,N⟩=R−1/4F±(t~)+R−5/4F1±(t~)+o(R−5/4),\langle\sigma_{0,0}\sigma_{M,N}\rangle=R^{-1/4}F_{\pm}(\tilde t)+R^{-5/4}F_{1\pm}(\tilde t)+o(R^{-5/4}),

where the leading scaling functions F±F_{\pm} depend only on t~\tilde t (and anisotropy through its definition), not on the direction of the separation — the exact statement of universality of the scaling function across lattice directions. WMTB furthermore claimed (their Eq. (2.24)) that the first correction functions satisfy F1+(t~)/F+(t~)=−F1−(t~)/F−(t~)=−t~R1F_{1+}(\tilde t)/F_{+}(\tilde t)=-F_{1-}(\tilde t)/F_{-}(\tilde t)=-\tilde t R_1 with R1R_1 a direction- and anisotropy-dependent but t~\tilde t-independent algebraic factor, a claim with direct consequences for the ratio of subleading to leading amplitudes of the magnetic susceptibility (C1+/C0+=−C1−/C0−=−βCR0C_{1+}/C_{0+}=-C_{1-}/C_{0-}=-\beta_C R_0, where βC\beta_C is the critical value of β\beta and R0R_0 is a direction- and anisotropy-dependent algebraic factor analogous to R1R_1), since the susceptibility sums the two-point function over all separations.

Forrester, Perk, Trinh and Witte (2019) revisited this question for the diagonal specialization. Their Proposition 1 shows that the leading large-NN corrections to the diagonal scaling functions, at orders N−1N^{-1} and N−2N^{-2}, are characterized as solutions of inhomogeneous linear second-order ODEs whose coefficients are the Painlevé transcendents specifying the scaling limit itself, or equivalently as a coupled system involving the degenerate Painlevé V σ\sigma-form plus a linear ODE. They find that the order N−1N^{-1} correction is trivial — it can be eliminated by an appropriate choice of scaling variables (essentially a correlation length accurate beyond first order) — so the leading non-trivial correction is of order N−2N^{-2}, and they confirm the precise sense in which WMTB's claims hold on the diagonal, cross-validating the ODE solutions against form-factor asymptotics, small-ss series to order s15s^{15}, and direct numerical evaluation of the Toeplitz determinants with NN-extrapolation. Crucially, their derivation perturbs the Jimbo–Miwa isomonodromic (τ\tau-function/Painlevé VI) structure, which is available at finite NN only for special separations: the row and diagonal correlations are Toeplitz determinants, the next-to-diagonal ones embed in an isomonodromic system of bi-orthogonal polynomials on the unit circle (Witte 2007) and in bordered Toeplitz determinants (Basor–Ehrhardt–Gharakhloo–Its–Li 2020), while the correlation in fully general position is a Pfaffian-type object (a determinant of an antisymmetric kernel) for which no comparable finite-NN isomonodromic description is known.

Problem Statement

For the anisotropic square-lattice Ising model near criticality, extend the differential-equation characterization of Forrester–Perk–Trinh–Witte (2019) from the diagonal correlation to the two-point correlation ⟨σ0,0σM,N⟩\langle\sigma_{0,0}\sigma_{M,N}\rangle in general lattice position: in the scaling regime where the rescaled separation R→∞R\to\infty at fixed scaling variable (with the direction ratio M/NM/N held at an arbitrary fixed value and arbitrary anisotropy E1/E2E_1/E_2), determine the order structure of the finite-size corrections to the WMTB expansion ⟨σ0,0σM,N⟩=R−1/4F±(t~)+R−5/4F1±(t~)+o(R−5/4)\langle\sigma_{0,0}\sigma_{M,N}\rangle=R^{-1/4}F_{\pm}(\tilde t)+R^{-5/4}F_{1\pm}(\tilde t)+o(R^{-5/4}) — identifying which corrections are trivial, i.e. removable by an appropriate choice of scaling variables such as a direction-dependent correlation length accurate beyond first order, and which are non-trivial — and characterize the leading non-trivial correction functions as solutions of explicit differential equations (expected to be linear, with coefficients built from the scaling functions F±F_{\pm} and their underlying Painlevé-type transcendents where available), with the boundary or initial conditions they satisfy, for both T>TcT>T_c and T<TcT<T_c. The answer must also determine the status in general position of WMTB's claimed relation F1+/F+=−F1−/F−=−t~R1F_{1+}/F_{+}=-F_{1-}/F_{-}=-\tilde t R_1 and its consequence for the susceptibility amplitude ratio C1±/C0±C_{1\pm}/C_{0\pm}, and must reduce to the known diagonal results (FPTW Proposition 1) under the diagonal specialization.

References

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