Analytic completion of the elliptic-integral representation of anisotropic Ising correlations: global validity of the third-kind duality identity, connection constants, and constructive operator factorization

For the two-dimensional anisotropic square-lattice Ising model, McCoy and Maillard (2016) proved that every two-point correlation $C(M,N)$ is a homogeneous polynomial of degree $N$ in the complete elliptic integrals of the first, second and third kind, exhibited an involution representing Kramers–Wannier duality (the low-temperature/high-temperature self-duality of the model, acting as $k\to 1/k$) on these integrals through a two-variable identity on the third-kind integral $\tilde\Pi$, and showed that the linear differential operator annihilating $C(M,N)$ factorizes into operators homomorphic to symmetric powers of the second-order operator $L_K$ annihilating $\tilde K$. Each of these results is local or existential: the identity's precise domain of validity in the two complex variables $(z,k)$ is undetermined, the connection constants expressing the duality-pulled-back integrals (naturally $1/k$ series) in the paper's local bases at $k=0$ are not computed, no general constructive formula for the factors is given, and the determinantal route to exact expressions is impractical for general separations. This record asks for the completion of this analytic and constructive theory: (i) the branch and singularity classification of the third-kind duality identity; (ii) closed-form connection constants from $k=\infty$ to $k=0$; and (iii) a general algorithmic construction of the operator factors for arbitrary $M,N$. None of these components is known to be resolved in the published literature.

Background

The two-dimensional Ising model on the square lattice with unequal horizontal and vertical couplings $E_h,E_v$ at inverse temperature $\beta$ has two-point spin correlation functions $C(M,N)=\langle\sigma_{0,0}\sigma_{M,N}\rangle$. Writing $s_h=\sinh 2\beta E_h$, $s_v=\sinh 2\beta E_v$, the natural variables are the modulus $k=s_h s_v$ (so the critical point is $k=1$; $k_<\,=1/k$ below $T_c$) and the anisotropy ratio $\nu=s_h/s_v$, which is invariant under Kramers–Wannier self-duality (the low-temperature/high-temperature duality of the model, acting as $(s_h,s_v)\to(1/s_v,1/s_h)$, i.e. $k\to 1/k$ with $\nu$ fixed).

The three kinds of (normalized) complete elliptic integrals are

$$\tilde K(k)=\frac{2}{\pi}\int_0^{\pi/2}\frac{d\phi}{(1-k^2\sin^2\phi)^{1/2}},\qquad \tilde E(k)=\frac{2}{\pi}\int_0^{\pi/2}(1-k^2\sin^2\phi)^{1/2}d\phi,$$ $$\tilde\Pi(n,k)=\frac{2}{\pi}\int_0^{\pi/2}\frac{d\phi}{(1-n\sin^2\phi)(1-k^2\sin^2\phi)^{1/2}},$$

so all three equal $1$ at $k=n=0$; here $n$ is called the characteristic.

McCoy and Maillard (2016) established, using the quadratic difference equations for the correlations (McCoy–Wu 1980; Perk 1980) rather than the Toeplitz-determinant representation (the exact representation of $C(M,N)$ as a determinant built from the Fourier coefficients of a generating function, which is impractical to evaluate for general non-diagonal separations), that $C(M,N)$ with $M<N$ is a homogeneous polynomial of degree $N$ in $\tilde K(k)$, $\tilde E(k)$ and $\tilde\Pi(-\nu k,k)$ (for $M>N$ the characteristic becomes $-k/\nu$; below $T_c$ one uses $k_<$). This elliptic-integral structure makes the Kramers–Wannier duality $k\to 1/k$ act directly on the correlations. For the first and second kind such dual actions were known from Ghosh (2007); McCoy and Maillard extended the action to the third kind through a two-variable identity

$$A(z,k)\,\tilde\Pi(z,k)+B(z,k)\,\tilde\Pi\!\left(R(z,k),k\right)=\tilde K(k),$$ $$R(z,k)=4k^2\,\frac{z(z-1)(z-k^2)}{(z^2-k^2)^2},\qquad A=\frac{4(z-1)(z-k^2)}{\rho},\qquad B=\frac{(z^2+k^2-2z)(z^2+k^2-2k^2z)}{(z^2-k^2)\,\rho},$$ $$\rho(z,k)=z^2-2z-2k^2z+3k^2.$$

The rational map $R$ is of infinite order: in terms of the Jacobi elliptic sine it is the doubling map $z=\mathrm{sn}^2(u)\mapsto \mathrm{sn}^2(2u)$; its iterates have degrees growing generically like $4^N$. The identity was verified by a local series expansion (valid when $z$ and $k$ are small with $|z/k|\ll 1$; in the complementary regime where $z$ and $k$ are small but $|z/k|\gg 1$, a different identity takes place) and proved by two local arguments (a homomorphism of third-order differential operators, and the $u=a$ limit of the Jacobi zeta addition formula), but the authors state that it is difficult to find the precise domain of validity of the identity in the two complex variables $z$ and $k$, the loci $z=\pm k$ and the poles of the pullback playing a visible role.

Under the duality pullback $k\to 1/k$ ($\nu$ fixed) each elliptic integral is naturally a series in $1/k$ (a low-temperature expansion). McCoy and Maillard construct, for each of $\tilde\Pi(-\nu k,k)$, $\tilde E$, $\tilde K$, the linear differential operator annihilating the pulled-back function and a basis of its solutions at $k=0$ (for the third kind: an algebraic solution $S_0$, an analytic solution $k\cdot\tilde\Pi$, and a formal logarithmic series). Expressing the pulled-back function as a definite linear combination of that basis requires the connection constants of the analytic continuation from $k=\infty$ to $k=0$ (the connection matrix between bases of solutions at the two singular points), which the paper indicates but does not compute for any of the three integrals.

Finally, viewing $C(M,N)$ with $\nu$ fixed as a function of $k$ alone, it satisfies a linear ordinary differential equation of order $\frac{1}{2}(N-M+1)(N+M+2)$ — for row correlations $C(0,N)$, order $\frac{1}{2}(N+1)(N+2)$ — which factorizes canonically (stated in the paper for $M<N$; the case $M>N$ follows by permuting $s_h\leftrightarrow s_v$, exchanging the characteristic $-\nu k$ with $-k/\nu$) as a product $M_{N+1}\cdots M_{M+1}$ of operators of decreasing orders $N+1,\dots,M+1$, where $M_n$ is homomorphic to (equivalent, via explicit rational-function maps, to) the $(n-1)$-th symmetric power of $L_K$, the second-order Fuchsian operator (an ordinary differential operator all of whose singular points are regular) annihilating $\tilde K(k)$ (equivalently, the Gauss hypergeometric equation with parameters $(1/2,1/2;1)$ in $t=k^2$). The $n$-th symmetric power of a second-order operator with solution space $\langle f,g\rangle$ is the order-$n+1$ operator annihilating all products $f^i g^{n-i}$, so this identifies the solution space of each factor with homogeneous polynomials of degree $n-1$ in $\tilde K,\tilde E$. The paper proves this homomorphism structure and gives explicit operators only for low orders; no general closed-form or algorithmic construction of the $M_n$ for arbitrary $(M,N)$ is provided. In the isotropic limit $\nu=1$ the factorization degenerates to a direct sum.

Problem Statement

Complete the analytic and constructive theory of the elliptic-integral representation of the anisotropic Ising correlations $C(M,N)$ established by McCoy and Maillard (2016). The common objective is to convert that paper's local and existential statements - a locally verified two-variable identity, an indicated but uncomputed connection problem, and an existence theorem for operator factorizations - into a complete global analytic theory of the same objects. Concretely, resolve the following three interdependent issues, which together constitute the completion of that theory.

(1) Global validity of the third-kind duality identity. Determine the precise domain of validity, in the two complex variables $(z,k)$, of the identity

$$A(z,k)\,\tilde\Pi(z,k)+B(z,k)\,\tilde\Pi\!\left(R(z,k),k\right)=\tilde K(k)$$

with $A,B,R$ as defined in the Background (the map $R$ being the infinite-order doubling map on $z=\mathrm{sn}^2(u)$). This requires a classification of the branch-cut and singularity loci of the two multivalued third-kind integrals - in particular the role of $z=\pm k$ and of the poles of the rational pullback $R$ - specifying for which branches in which regions of $\mathbb{C}^2$ the identity holds, going beyond the local series verification at $|z/k|\ll 1$ given in the paper (a different identity taking place in the complementary regime where $z$ and $k$ are small but $|z/k|\gg 1$).

(2) Connection constants of the duality pullback. Compute in closed form the connection constants (equivalently, the connection matrix from $k=\infty$ to $k=0$) expressing each pulled-back integral - the third-kind $\tilde\Pi(-\nu/k,\,1/k)$ with its algebraic, analytic and formal-logarithmic basis solutions of the order-three operator, and likewise $\tilde E(1/k)$ and $\tilde K(1/k)$ in their respective bases - as explicit linear combinations of these local basis solutions at $k=0$ (in general including the formal-logarithmic one; only combinations of the algebraic and analytic basis solutions are themselves analytic at $k=0$), with the constants given as explicit functions of $\nu$ where relevant.

(3) Constructive factorization. Provide a general closed-form formula or explicit terminating algorithm producing the coefficients of the factors $M_n$ in the canonical factorization $M_{N+1}\cdots M_{M+1}$ of the linear differential operator annihilating the general anisotropic correlation $C(M,N)$, together with the explicit homomorphisms to the $(n-1)$-th symmetric powers of $L_K$, valid for arbitrary $M,N$ - the factorization being stated in the paper for $M<N$, with the $M>N$ case obtained by permuting $s_h\leftrightarrow s_v$ (equivalently $\nu\to 1/\nu$ and $M\leftrightarrow N$, the characteristic becoming $-k/\nu$) - and going beyond the low-order explicit examples of the 2016 paper.

Components (1) and (2) are the analytic-continuation core of the duality action on the correlation representation; component (3) is the differential-operator counterpart of the same structure. An answer must address all three components.

References

1. B. M. McCoy and J-M. Maillard, 'The anisotropic Ising correlations as elliptic integrals: duality and differential equations', J. Phys. A: Math. Theor. 49 (2016) 434004. DOI 10.1088/1751-8113-49-43-434004; arXiv:1606.08796, https://arxiv.org/abs/1606.08796
2. S. Boukraa, J-M. Maillard and B. M. McCoy, 'The Ising correlation $C(M,N)$ for $\nu=-k$', J. Phys. A: Math. Theor. 53 (2020) 465202. DOI 10.1088/1751-8121/abbb61; arXiv:2008.06912, https://arxiv.org/abs/2008.06912
3. S. Boukraa, C. Cosgrove, J-M. Maillard and B. M. McCoy, 'Factorization of Ising correlations $C(M,N)$ for $\nu=-k$ and $M+N$ odd, $M\le N$, $T<T_c$ and their lambda extensions', J. Phys. A: Math. Theor. 55 (2022) 405204. DOI 10.1088/1751-8121/ac8ef6; arXiv:2204.10096, https://arxiv.org/abs/2204.10096
4. S. Boukraa and J-M. Maillard, 'The lambda extensions of the Ising correlation functions $C(M,N)$', J. Phys. A: Math. Theor. 56 (2023) 085201. DOI 10.1088/1751-8121/acb574; arXiv:2209.07434, https://arxiv.org/abs/2209.07434
5. Y. Abdelaziz and J-M. Maillard, 'Modular forms, Schwarzian conditions, and symmetries of differential equations in physics', J. Phys. A: Math. Theor. 50 (2017) 215203. DOI 10.1088/1751-8121/aa6cba; arXiv:1611.08493, https://arxiv.org/abs/1611.08493
6. R. Shrock, 'On general-$n$ coefficients in series expansions for row spin-spin correlation functions in the two-dimensional Ising model', J. Phys. A: Math. Theor. 55 (2022) 425001. DOI 10.1088/1751-8121/ac9654; arXiv:2207.09268, https://arxiv.org/abs/2207.09268
7. B. M. McCoy, 'Connection formulas for the lambda generalized Ising correlation functions', J. Phys. A: Math. Theor. 52 (2019) 375203. DOI 10.1088/1751-8121/ab2ff1; arXiv:1902.08550, https://arxiv.org/abs/1902.08550
8. J-M. Maillard, 'Modular correspondences and replicable functions (unabridged version)', arXiv:2505.16873 (2025), https://arxiv.org/abs/2505.16873
9. R. K. Ghosh, 'Duality and even number spin-correlation functions in the two dimensional square lattice Ising model', arXiv:cond-mat/0703602 (2007), https://arxiv.org/abs/cond-mat/0703602
10. J. H. H. Perk, 'Quadratic identities for Ising model correlations', Phys. Lett. A 79 (1980) 3-5. DOI 10.1016/0375-9601(80)90299-6, https://doi.org/10.1016/0375-9601(80)90299-6
11. B. M. McCoy and T. T. Wu, 'Nonlinear partial difference equations for the two-dimensional Ising model', Phys. Rev. Lett. 45 (1980) 675-678. DOI 10.1103/PhysRevLett.45.675, https://doi.org/10.1103/PhysRevLett.45.675
12. G. V. Dunne, 'Resurgence, Painleve equations and conformal blocks', J. Phys. A: Math. Theor. 52 (2019) 463001. DOI 10.1088/1751-8121/ab3142; arXiv:1901.02076, https://arxiv.org/abs/1901.02076
13. C. Boutillier, B. de Tiliere and K. Raschel, 'The Z-invariant Ising model via dimers', Probab. Theory Relat. Fields 174 (2019) 235-305. DOI 10.1007/s00440-018-0861-x; arXiv:1612.09082, https://arxiv.org/abs/1612.09082