For the zero-field square-lattice Ising model, the Kramers–Wannier duality k→1/kk\to 1/k of the elliptic modulus, acting together with the classical transformations of the complete elliptic integrals KK, EE (and Π\Pi in the anisotropic case), is known to map the exact closed-form expressions of every even-spin correlation function from the high-temperature phase to the low-temperature phase — observed for the known closed forms by Ghosh (2007) and proved for all anisotropic two-point correlations C(M,N)C(M,N), hence all even multipoint correlations, by McCoy and Maillard (2016). The duality is nevertheless not constructive one level down: the high- and low-temperature series expansions of the correlations are local solutions of the associated linear Fuchsian and nonlinear sigma-form Painlevé VI equations about different singular points, they are not related term by term, and no explicit duality connection between them is known — for the diagonal correlations the two phases are classical, while at the locus ν=−k\nu=-k with M+NM+N even the exhibited low-to-high-temperature transformation of the sigma-form equations is an involutive change of variables rather than the duality itself, the locus ν=−k\nu=-k not being duality-invariant. The problem is to construct this connection explicitly — at its source level, the connection matrices of the pulled-back complete elliptic integrals Π(−ν/k,1/k)\Pi(-\nu/k,1/k), E(1/k)E(1/k), K(1/k)K(1/k), i.e. the analytic continuation from k=∞k=\infty to k=0k=0, the third-kind case Π\Pi being the uncomputed core — and thence, through the polynomial structure of C(M,N)C(M,N) in KK, EE and Π\Pi, the explicit two-phase series relation for the general anisotropic two-point correlations; together with, at the nonlinear level, determining whether and in what precise sense the duality relates the low- and high-temperature sigma-form Painlevé VI equations, known at the locus ν=−k\nu=-k (equivalently ν=−1/k\nu=-1/k), and their physical boundary conditions.

Background

The zero-field Ising model on the square lattice, with interaction energy −∑(Evσi,jσi+1,j+Ehσi,jσi,j+1)-\sum(E_v\sigma_{i,j}\sigma_{i+1,j}+E_h\sigma_{i,j}\sigma_{i,j+1}) over nearest-neighbour pairs of spins σi,j=±1\sigma_{i,j}=\pm1, is the canonical exactly solved statistical system: Onsager (1944) computed its free energy and Yang (1952) its spontaneous magnetization. Its two-point spin correlation functions C(M,N)=⟨σ0,0σM,N⟩C(M,N)=\langle\sigma_{0,0}\sigma_{M,N}\rangle are given by Toeplitz or bordered-Toeplitz determinant representations — a Toeplitz determinant being the determinant of a matrix whose entries depend only on the difference between its row and column indices, and the bordered variants adjoining extra rows and columns that carry the spin insertions (Montroll–Potts–Ward 1963 for the diagonal and near-diagonal cases; Wu 1966 for the rows), whose entries are Fourier coefficients of algebraic functions of the couplings; they can be computed recursively from Perk's quadratic identities (1980), and correlations of any even number of spins reduce to sums of products of two-point functions (Groeneveld–Boel–Kasteleyn 1978), so the two-point theory carries the general even case.

Three structures organize the exact answers. First, the natural elliptic modulus k=sinh⁡(2βEv) sinh⁡(2βEh)k=\sinh(2\beta E_v)\,\sinh(2\beta E_h) (β=1/kBT\beta=1/k_BT; k=(sinh⁡2βJ)2k=(\sinh 2\beta J)^2 for the isotropic model with coupling JJ) obeys k<1k<1 above the critical temperature TcT_c and k>1k>1 below it, so the high- and low-temperature regimes are two charts of the same elliptic curve. Second, the determinant entries — and in fact every two-point correlation itself — reduce to complete elliptic integrals, defined by K(k)=∫0π/2(1−k2sin⁡2ϕ)−1/2dϕK(k)=\int_0^{\pi/2}(1-k^2\sin^2\phi)^{-1/2}d\phi (first kind), E(k)=∫0π/2(1−k2sin⁡2ϕ)1/2dϕE(k)=\int_0^{\pi/2}(1-k^2\sin^2\phi)^{1/2}d\phi (second kind) and Π(n,k)=∫0π/2[(1−nsin⁡2ϕ)(1−k2sin⁡2ϕ)1/2]−1dϕ\Pi(n,k)=\int_0^{\pi/2}[(1-n\sin^2\phi)(1-k^2\sin^2\phi)^{1/2}]^{-1}d\phi (third kind): diagonal correlations are homogeneous polynomials in KK and EE (Ghosh–Shrock 1984), isotropic row correlations have a multi-level structure in EE and (k−1)K(k-1)K (Ghosh–Shrock 1985), and the general anisotropic C(M,N)C(M,N) are homogeneous polynomials in KK, EE and Π\Pi (McCoy–Maillard 2016). Third, as functions of kk the correlations satisfy linear Fuchsian ordinary differential equations (ODEs with only regular singular points; orders growing with M,NM,N — Boukraa–Hassani–Maillard–McCoy–Weil–Zenine 2006 for the diagonal case) and, nonlinearly, equations of Painlevé VI type (the sixth of the classical second-order nonlinear Painlevé equations, whose general solutions define new transcendental functions) in Okamoto sigma form — a symmetric rewriting of that equation carrying four constant parameters, the Okamoto parameters — in which dln⁡C/dtd\ln C/dt (with tt an algebraic function of kk) is the dependent variable of a second-order nonlinear ODE: Jimbo–Miwa (1980) for the diagonal correlations, and four two-parameter families of such equations for general C(M,N)C(M,N) at the special anisotropic locus ν=−k\nu=-k, where ν=sinh⁡2βEh/sinh⁡2βEv\nu=\sinh 2\beta E_h/\sinh 2\beta E_v measures the anisotropy (Boukraa–Maillard–McCoy 2020). These differential equations — linear and nonlinear alike — also admit one-parameter families of deformed solutions, the lambda extensions, which specialize to the physical correlation at a distinguished value of the parameter (Boukraa–Maillard 2022, 2023).

The Kramers–Wannier duality (Kramers–Wannier 1941) is the involution of the coupling constants sinh⁡2βEh→1/sinh⁡2βEv\sinh 2\beta E_h\to 1/\sinh 2\beta E_v, sinh⁡2βEv→1/sinh⁡2βEh\sinh 2\beta E_v\to 1/\sinh 2\beta E_h, which maps the high-temperature expansion of the partition function onto the low-temperature one and, in the isotropic case, sends k→1/kk\to 1/k. Acting simultaneously on the elliptic integrals, K(k)→kK(k)K(k)\to kK(k) and E(k)→[E(k)+(k2−1)K(k)]/kE(k)\to [E(k)+(k^2-1)K(k)]/k (with Π→kΠ\Pi\to k\Pi anisotropically), it transforms the exact closed-form expression of any even-spin correlation in the high-temperature phase into the corresponding low-temperature expression — observed for the known closed forms by Ghosh (2007) and proved for all anisotropic C(M,N)C(M,N), and hence all even multipoint correlations, by McCoy–Maillard (2016), for whom the transformation is an involution representing the duality.

The gap this problem addresses sits one level down, at the level of series. The physically used expansions in the isotropic model are the high-temperature series in v=tanh⁡βJv=\tanh\beta J and the low-temperature series in u=e−4βJu=e^{-4\beta J} — single-coupling variables, the anisotropic model having two independent high-temperature expansion variables (in tanh⁡βEv\tanh\beta E_v and tanh⁡βEh\tanh\beta E_h); on the elliptic side these correspond, in the isotropic case, to expansions about k=0k=0 and about k=∞k=\infty. The duality described above acts on closed forms, and Ghosh (2007) noted explicitly that it does not act term by term on the series solutions of the Jimbo–Miwa Painlevé VI equation. McCoy–Maillard (2016, Appendix B) made the obstruction precise: the pulled-back integrals K(1/k)K(1/k), E(1/k)E(1/k), Π(−ν/k,1/k)\Pi(-\nu/k,1/k) are series in 1/k1/k, but as functions near k=0k=0 they are specific linear combinations of the local Frobenius solutions (the power-series solutions constructed by the Frobenius method about a regular singular point) of the associated linear differential operators (combinations involving logarithmic formal series); identifying them is exactly a connection-matrix problem — expressing the local solutions at one singular point of the equation as linear combinations of the local solutions at another, i.e. analytic continuation from k=∞k=\infty to k=0k=0 — a problem solved classically only for KK and EE. What the duality does to the series coefficients, and to the low- and high-temperature Painlevé VI equations themselves, is not known: for the diagonal correlations both phases are explicit and classically related, while at the locus ν=−k\nu=-k with M+NM+N even an explicit transformation of the sigma-form equations exists but is an involutive change of variables rather than the Kramers–Wannier duality, which does not leave the locus ν=−k\nu=-k invariant.

Problem Statement

Extend the Kramers–Wannier duality of the square-lattice Ising model from exact closed forms to series solutions and their differential equations, for the general two-point correlations C(M,N)C(M,N).

(1) Linear/series level. Under the duality k→1/kk\to 1/k, which leaves the anisotropy ν=sinh⁡2βEh/sinh⁡2βEv\nu=\sinh 2\beta E_h/\sinh 2\beta E_v invariant, the pulled-back complete elliptic integrals Π(−ν/k,1/k)\Pi(-\nu/k,1/k), E(1/k)E(1/k), K(1/k)K(1/k) are naturally series in 1/k1/k (low temperature); re-expanded about k=0k=0 each is a linear combination of the local Frobenius solutions — including logarithmic formal series — of the associated linear differential operators, whose local bases are displayed in McCoy–Maillard (2016, Appendix B). Construct explicitly which linear combination, i.e. perform the analytic continuation from k=∞k=\infty to k=0k=0 (equivalently, compute the connection matrices in the sense of that appendix). In the KK and EE sectors this reduces to the classical elliptic-integral connection identities; the uncomputed core is the third-kind case Π(−ν/k,1/k)\Pi(-\nu/k,1/k). Thence, using the fact that every anisotropic C(M,N)C(M,N) is a homogeneous polynomial in KK, EE and Π(−νk,k)\Pi(-\nu k,k) (with Π(−k/ν,k)\Pi(-k/\nu,k) in the M>NM>N sector) with algebraic coefficients in (k,ν)(k,\nu), derive the explicit two-phase series relation for the general anisotropic two-point correlation C(M,N)C(M,N). In the isotropic model the same question reads as a coefficient-level transformation between the high-temperature series in v=tanh⁡βJv=\tanh\beta J and the low-temperature series in u=e−4βJu=e^{-4\beta J}. Even multipoint correlations reduce to sums of products of two-point functions; extending the two-phase relation to them would add a combinatorial layer and is a possible further step, not part of the required target.

(2) Nonlinear level. At the locus where the Okamoto sigma-form Painlevé VI equations of C(M,N)C(M,N) are known — ν=−k\nu=-k for M≤NM\leq N, equivalently ν=−1/k\nu=-1/k for M≥NM\geq N (Boukraa–Maillard–McCoy 2020) — determine whether, and in what precise sense, the Kramers–Wannier duality relates the low- and high-temperature sigma-form equations and selects their physical boundary conditions, including the physical interpretation of the formal Kramers–Wannier-type involution on the sigma variable that maps each of the four families to itself with M,NM,N permuted. Note that the duality leaves ν\nu invariant and sends k→1/kk\to 1/k, so the locus ν=−k\nu=-k is not duality-invariant and the high-temperature equations "have no reason to be deduced" from the low-temperature ones by a Kramers–Wannier-like duality; the low-to-high-temperature transformation exhibited at M+NM+N even is a separate involutive change of variables. For unrestricted anisotropy the sigma-form equations themselves are not yet known — a distinct open quest — so at general (k,ν)(k,\nu) the duality question arises only conditionally on that construction.

Both parts pursue one objective: making the phase duality constructive on the series solutions and their differential equations, as it already is on the closed forms.

References

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