Pair-correlation functions of the two-dimensional integrable N-state chiral Potts model

For the two-dimensional integrable N-state chiral Potts model - the archetypal Yang-Baxter-solvable lattice model beyond the free-fermion class, with Boltzmann weights parametrized on a higher-genus spectral curve - Perk and Au-Yang noted in 2006 that only the order parameter (one-point function) had been proved, while the pair-correlation (two-point) functions at finite spin separation remained unsolved. Later work produced a Z-invariance strategy reducing general-model correlations to superintegrable-chain matrix elements (Au-Yang and Perk, 2010), exact finite-length data and a nearest-neighbour conjecture (Fabricius and McCoy, 2010), factorized matrix elements with only partially known normalizations (Iorgov et al., 2010), finite-lattice determinant form-factor representations (Grosjean, Maillet and Niccoli, 2015), and a proven finite-volume symmetry property (Zhu, 2026); none evaluates the two-point function at arbitrary separation in the thermodynamic limit, so the problem remains open. The problem asks for explicit, systematically evaluable two-point functions for arbitrary N, whose infinite-separation limit is the proved order parameter and whose N=2 specialization reproduces the known Ising correlations, thereby unlocking wavevector-dependent susceptibility computations for chiral Potts-type systems.

Background

The two-dimensional Ising model set the standard for what 'exactly solved' means in equilibrium statistical mechanics: after Onsager's 1944 computation of the free energy, its spin-spin correlation functions were also obtained explicitly (Wu, McCoy, Tracy and Barouch, 1976), and these two-point functions are the input from which the wavevector-dependent susceptibility $\chi(\mathbf{q})$ - the Fourier transform of the pair correlation - can be computed to arbitrary precision. Extending such correlation-function results to integrable models that are not of free-fermion type (that is, whose correlations do not reduce to determinants or Pfaffians of Clifford-algebra operators) has been a recognized open frontier ever since. The integrable chiral Potts model (Au-Yang, McCoy, Perk, Tang and Yan, 1987; Baxter, Perk and Au-Yang, 1988) is the most prominent such model. Its spins take the $N$ values $\omega^\sigma$, $\sigma=0,\ldots,N-1$, $\omega=e^{2\pi i/N}$, on a square lattice, with interaction weights that break the spin-inversion symmetry of the ordinary Potts model (whence 'chiral'). The weights satisfy the star-triangle (Yang-Baxter) equation, the algebraic condition making the transfer matrices commute and the model solvable; it was the first solved model whose spectral parameters ('rapidities') live on an algebraic curve of genus larger than one. Because the weights are single-valued only on this higher-genus curve and do not depend on rapidity differences, the corner transfer matrix - Baxter's device that represents one quadrant of the lattice and yields one-point functions as its matrix elements - fails here, and the model became the recognized hard case of the discipline. The order parameter (the one-point function $\langle s^n\rangle$, nonzero only in the ordered phase, also called the spontaneous magnetization) was nonetheless conquered: Albertini, McCoy, Perk and Tang conjectured in 1989 that $M_r=(1-k^2)^{r(N-r)/(2N^2)}$ for $r=1,\ldots,N-1$, with $k$ the modulus of the rapidity curve, and Baxter proved this in 2005 by functional-relation methods; algebraic proofs via the Onsager algebra - the infinite-dimensional Lie algebra generated by the two parts of the superintegrable transfer matrix, first appearing in Onsager's 1944 paper - followed (Au-Yang and Perk; Iorgov et al.). Here 'superintegrable' denotes the special submanifold of the full integrable model (Baxter, 1988), equivalent to the $Z_N$-symmetric quantum chain of von Gehlen and Rittenberg with an Onsager-algebra spectrum, which reduces to the Ising model at $N=2$. Pair correlations are another matter. A two-point function $\langle s_A^{(n)} s_B^{(n)}\rangle$ measures the joint statistics of two spins at sites $A$ and $B$ separated by a finite distance; its large-distance limit factorizes into order parameters, so the solved one-point problem is only the asymptotic tip. By $Z$-invariance - the consequence of commuting transfer matrices that correlations are unchanged under deformations of the rapidity lattice - a bulk pair correlation of the general integrable model depends only on the $2m$ rapidity lines passing between the two spins, through universal functions $g_{2m}$; hence general-model correlations can be reconstructed from ground-state expectations $\langle Z_0^r Z_R^{\dagger r}\rangle$ of the Weyl ('clock') operators $Z$ at separation $R$ in the superintegrable chain. Perk and Au-Yang (2006) stated the resulting open problem: the pair-correlation functions of the two-dimensional integrable $N$-state chiral Potts model 'have not yet been solved', and detailed pair correlations for the eight-vertex model, the full (incommensurate-phase) chiral Potts model, and three-dimensional Yang-Baxter integrable models were likewise unavailable - the missing input, they noted, for extending their exact wavevector-dependent-susceptibility program for $Z$-invariant Ising models (including quasiperiodic pentagrid/Penrose-tiling systems) to these harder integrable models. Three partially successful routes to the chiral Potts correlations exist. The Onsager-algebra route constructs the eigenvectors of the superintegrable chain in 'Onsager sectors' and factorized matrix elements of the spin operators between them (matrix elements of local operators between eigenstates are called form factors). The functional-relation route proved the order parameter but does not directly give finite-separation behavior. The separation-of-variables (SOV) route of the quantum inverse scattering method reconstructs all local operators on finite lattices and expresses their form factors as finite sums of determinants. Each route stops short of the summed, thermodynamic-limit two-point function at finite separation. The 'incommensurate' phases of the full chiral Potts model are phases in which the ordering wavevector varies continuously instead of being fixed by the lattice, and a 'Lifshitz point' would be a multicritical point where such a phase meets the commensurate ordered and disordered phases (Huse and Fisher, 1982).

Problem Statement

Obtain the pair-correlation (two-point) functions of the two-dimensional integrable $N$-state chiral Potts model at arbitrary finite separation between the two spins, in the thermodynamic limit, for arbitrary $N$ (the case $N=2$ reduces to the exactly solved Ising model and serves as the consistency limit). In the superintegrable formulation this is the ground-state two-point function $\langle Z_0^r Z_R^{\dagger r}\rangle$ of the superintegrable chiral Potts chain at arbitrary separation $R$ and $r=1,\ldots,N-1$; by $Z$-invariance it determines the universal functions $g_{2m}$ of the $2m$ rapidity lines passing between the two spins for the general integrable model, including the full chiral Potts model with its incommensurate phases. 'Obtain' is meant in the sense established for the Ising model: explicit closed-form, multiple-integral, or systematically convergent series representations from which numerical values at any prescribed separation and rapidity configuration can be computed and large-distance asymptotics extracted. Any correct solution necessarily satisfies the standard consistency limits: at infinite separation in the ordered phase the correlation factorizes into the product of the proved order parameters $M_r=(1-k^2)^{r(N-r)/(2N^2)}$, and at $N=2$ it reproduces the known Ising two-point functions. The resulting two-point functions are precisely the input required for the wavevector-dependent susceptibility computations in chiral Potts-type systems.

References

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