The periodic NN-state superintegrable chiral Potts spin chain is the canonical NN-state generalization of the Ising chain within Onsager's algebra: its spectrum is exactly known, and the infinite-distance limit of its two-point functions is fixed by Baxter's order-parameter formula. Yet no exact representation of the two-point functions ⟨Z0rZR†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle at any finite separation RR is known for any N≥3N\geq 3. Fabricius and McCoy computed these correlators exactly for the three-state chain at lengths L=3,4,5L=3,4,5 and conjectured a functional form for the nearest-neighbour correlator valid for arbitrary LL; a 2026 symmetry result of Zhu resolved only the reality of midpoint correlations, leaving the constructive problem untouched. The problem is to derive exact closed-form, determinant-type, or multiple-integral representations of these finite-distance correlators for arbitrary NN and chain length — in particular to prove or disprove the Fabricius–McCoy nearest-neighbour conjecture — for which no analogue of the Ising free-fermion/determinant machinery is currently available.

Background

The periodic NN-state superintegrable chiral Potts spin chain lives on the Hilbert space (CN)⊗L(\mathbb{C}^N)^{\otimes L} of LL sites. Its local operators are the Weyl (clock) operators ZZ and XX, defined by Z∣a⟩=ωa∣a⟩Z|a\rangle=\omega^a|a\rangle and X∣a⟩=∣a+1⟩X|a\rangle=|a+1\rangle with ω=e2πi/N\omega=e^{2\pi i/N} and labels modulo NN, so that ZX=ωXZZX=\omega XZ; ZjZ_j and XjX_j denote the same operators acting at site jj. The Hamiltonian is H=A0+λA1,A0=−∑j=0L−1∑r=1N−1eiπ(2r−N)/2Nsin⁡(πr/N) ZjrZj+1†r,A1=−∑j=0L−1∑r=1N−1eiπ(2r−N)/2Nsin⁡(πr/N) Xjr,H=A_0+\lambda A_1,\qquad A_0=-\sum_{j=0}^{L-1}\sum_{r=1}^{N-1}\frac{e^{i\pi(2r-N)/2N}}{\sin(\pi r/N)}\,Z_j^r Z_{j+1}^{\dagger r},\qquad A_1=-\sum_{j=0}^{L-1}\sum_{r=1}^{N-1}\frac{e^{i\pi(2r-N)/2N}}{\sin(\pi r/N)}\,X_j^r, with real coupling λ\lambda and periodic boundary conditions. The ZN\mathbb{Z}_N-symmetric family of chains with infinitely many conserved charges was found by von Gehlen and Rittenberg (1985), and its superintegrable representative — the member of the family carrying the maximal set of conserved charges — was singled out by Baxter (1988). The generators A0,A1A_0,A_1 obey the Dolan–Grady relations — a pair of cubic commutation relations guaranteeing that they generate a representation of the Onsager algebra, the algebra discovered by Onsager (1944) in his solution of the two-dimensional Ising model, generated by elements An,GmA_n,G_m with [Aj,Ak]=4Gj−k[A_j,A_k]=4G_{j-k}, [Gm,Al]=2Al+m−2Al−m[G_m,A_l]=2A_{l+m}-2A_{l-m}, [Gj,Gk]=0[G_j,G_k]=0. This forces the Ising-like square-root spectrum found by Albertini, McCoy, Perk and Tang (1989), a structure explained directly from the algebra by Davies (1990). For N=2N=2 the chain is the transverse-field Ising chain.

The basic order-parameter observables are the two-point functions ⟨Z0rZR†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle with 1≤r≤N−11\leq r\leq N-1, i.e. expectation values of products of powers of the clock operator at two sites, taken in an eigenstate of HH (the ground state in the original study); for N=2N=2 these are the familiar Ising spin-spin correlations. The contrast with the Ising case is stark. For N=2N=2, Kaufman's free-fermion (spinor) solution expresses the two-point functions as finite determinants, from which Montroll, Potts and Ward (1963) obtained the spontaneous magnetization via Szegő's theorem — the classical asymptotic formula for Toeplitz determinants, the determinants of matrices that are constant along each diagonal — confirming Yang's 1952 result. For N≥3N\geq 3 the chain is genuinely interacting: the Jordan–Wigner transformation (the substitution that rewrites Ising spins as free fermions) has no analogue, so no free-fermion structure survives. The infinite-distance limit is nevertheless exactly known: Baxter (2005) proved the order-parameter formula Mr=(1−λ2)r(N−r)/2N2M_r=(1-\lambda^2)^{r(N-r)/2N^2} — valid in the ordered regime ∣λ∣<1|\lambda|<1 — with Mr2=lim⁡R→∞⟨Z0rZR†r⟩M_r^2=\lim_{R\to\infty}\langle Z_0^r Z_R^{\dagger r}\rangle, by functional-equation methods that do not extend to finite RR; Baxter (2010) also established a determinantal form of the spontaneous magnetization; and Au-Yang and Perk (2011) showed how ZZ-invariance (Baxter's condition, following from the star-triangle relation, that local rearrangements of local spectral parameters leave local quantities invariant) reduces the integrable chiral Potts pair correlation at infinite distance to superintegrable eigenvector data. Thus everything is known about the R→∞R\to\infty endpoint, while no exact representation of ⟨Z0rZR†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle at any finite separation is known for any N≥3N\geq 3.

The only systematic exact data at finite LL are due to Fabricius and McCoy (2010), who computed the ground-state correlators of the three-state chain at lengths L=3,4,5L=3,4,5 by computer algebra. Writing xj=(λ2+1+ajλ)1/2x_j=(\lambda^2+1+a_j\lambda)^{1/2}, where the constants aja_j are determined by the roots of the polynomial fixing the ground-state energy (the Albertini–McCoy–Perk–Tang finite-size formula), they found that ⟨Z0Z1†⟩\langle Z_0 Z_1^{\dagger}\rangle is a rational function of λ\lambda and the xjx_j of a fixed structural form, and conjectured that this form persists for every chain length: ⟨Z0Z1†⟩=P0+∑jPj(λ)xj+(1−i3)∑j<kPj,k(λ)xjxk,\langle Z_0 Z_1^{\dagger}\rangle=P_0+\sum_j\frac{P_j(\lambda)}{x_j}+(1-i\sqrt{3})\sum_{j<k}\frac{P_{j,k}(\lambda)}{x_jx_k}, with P0P_0 constant, Pj(λ)P_j(\lambda) linear in λ\lambda, and Pj,k(λ)P_{j,k}(\lambda) at most quadratic in λ\lambda, all with real coefficients. They further expected that in the thermodynamic limit ⟨Z0rZR†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle becomes an (N−1)R(N-1)R-fold integral, and they observed and conjectured that for even LL the half-chain correlator ⟨Z0ZL/2†⟩\langle Z_0 Z_{L/2}^{\dagger}\rangle is real, with limiting value (1−λ2)r(N−r)/N2(1-\lambda^2)^{r(N-r)/N^2}. A useful consistency constraint on any candidate formula is the Feynman–Hellmann sum rule (obtained by differentiating the eigenvalue equation with respect to the coupling): the ground-state energy fixes a specific weighted sum of nearest-neighbour correlators via E0−λ ∂λE0E_0-\lambda\,\partial_\lambda E_0.

Two later developments delimit what is known. First, matrix-element building blocks exist: Iorgov, Pakuliak, Shadura, Tykhyy and von Gehlen (2010) computed spin-operator matrix elements between eigenstates in the sectors distinguished by the commuting Onsager-algebra charges, in factorized form with overall scalar factors undetermined outside the ground-state sectors; and Grosjean, Maillet and Niccoli (2015) used Sklyanin's separation of variables (a method that solves integrable transfer-matrix eigenvalue problems by factorizing them into single-variable pieces) to express scalar products and form factors (matrix elements of local operators between energy eigenstates) in the Bazhanov–Stroganov and chiral Potts models as single determinant formulae. However, the spectral sums over complete sets of intermediate states that would rebuild a two-point function have never been carried out. Second, Zhu (2026) proved an exact finite-volume symmetry identity: in every simultaneous eigenvector of HH and the one-site translation operator, the correlator ρr(R)=⟨Z0rZR†r⟩\rho_r(R)=\langle Z_0^r Z_R^{\dagger r}\rangle satisfies ρr(R)∗=ρr(−R)\rho_r(R)^{*}=\rho_r(-R), so midpoint correlators are real for even LL — resolving the Fabricius–McCoy midpoint-reality conjecture. The proof uses only conjugation and translation covariance and yields no explicit formula. Adjacent recent work — the coupled Temperley–Lieb algebra presentation (Temperley–Lieb algebras are diagrammatic algebras of non-crossing connectivity patterns acting on chain segments) of Adderton, Batchelor and Wedrich (2020), or the matrix-product-state eigenstates (variational wavefunctions built by contracting a chain of finite-dimensional tensors) near Onsager-algebra fixed points constructed by Camp and Jones (2026) — likewise provides structural or approximate descriptions, not exact correlators. The constructive problem, an exact theory of finite-distance correlations analogous to the Ising determinant theory, therefore remains open.

Problem Statement

Construct an exact theory of the finite-distance two-point functions ⟨Z0rZR†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle (1≤r≤N−11\leq r\leq N-1, 1≤R≤L−11\leq R\leq L-1) of the periodic NN-state superintegrable chiral Potts chain with Hamiltonian H=A0+λA1H=A_0+\lambda A_1 (real λ\lambda), where an exact theory means closed-form, determinant-type, or multiple-integral representations valid for arbitrary chain length, together with the asymptotics and evaluation schemes derivable from them. The concrete, source-faithful targets, in increasing order of generality, are:

  1. The Fabricius–McCoy nearest-neighbour conjecture (N=3N=3). Prove or disprove that for every chain length LL the ground-state correlator ⟨Z0Z1†⟩\langle Z_0 Z_1^{\dagger}\rangle has the form ⟨Z0Z1†⟩=P0+∑jPj(λ)xj+(1−i3)∑j<kPj,k(λ)xjxk,xj=(λ2+1+ajλ)1/2,\langle Z_0 Z_1^{\dagger}\rangle=P_0+\sum_j\frac{P_j(\lambda)}{x_j}+(1-i\sqrt{3})\sum_{j<k}\frac{P_{j,k}(\lambda)}{x_jx_k},\qquad x_j=(\lambda^2+1+a_j\lambda)^{1/2}, with P0P_0 constant, Pj(λ)P_j(\lambda) linear in λ\lambda, Pj,k(λ)P_{j,k}(\lambda) at most quadratic in λ\lambda, all with real coefficients, and the constants aja_j those determined by the ground-state energy. This form is established by direct computation only for L=3,4,5L=3,4,5.

  2. General finite separation and general NN. Derive exact representations of ⟨Z0rZR†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle for all separations 1≤R≤L−11\leq R\leq L-1, arbitrary N≥3N\geq 3, and arbitrary finite LL, in the ground state and — since the symmetry identity holds in every translation eigensector — in general translation eigensectors. Any proposed representation must be consistent with: the Feynman–Hellmann sum rule relating a weighted sum of nearest-neighbour correlators to the ground-state energy; Zhu's symmetry identity ρr(R)∗=ρr(−R)\rho_r(R)^{*}=\rho_r(-R); the exactly known L=3,4,5L=3,4,5 values for N=3N=3; and the approach to Baxter's order-parameter formula Mr2=(1−λ2)r(N−r)/N2M_r^2=(1-\lambda^2)^{r(N-r)/N^2} in the large-distance (and conjectured half-chain) limit, for real λ\lambda in the ordered regime ∣λ∣<1|\lambda|<1 (the regime in which the order parameter is nonzero; the disordered regime is reached by the λ↦1/λ\lambda\mapsto 1/\lambda self-duality of the chain).

  3. Asymptotics and evaluation. From such representations, obtain controlled asymptotics in RR and LL, and exact or provably convergent finite-size evaluation schemes replacing brute-force diagonalization.

These targets are facets of one overarching objective: an exact, constructive description of the order-parameter correlations of the superintegrable chiral Potts chain — the interacting NN-state counterpart of the Kaufman–Montroll–Potts–Ward determinant theory of Ising correlations. The nearest-neighbour conjecture is the R=1R=1, N=3N=3 instance; the general-RR and general-NN representations subsume and extend it; the asymptotics and evaluation schemes are consequences of the same representations. Resolution requires all three targets jointly: the general-RR, general-NN representations, together with the asymptotics and evaluation schemes derived from them, and a proof or disproof of the nearest-neighbour conjecture — no single target on its own, including the conjecture, is sufficient. Answers must be exact (rigorous derivations or closed-form expressions); purely numerical or perturbative results do not settle the problem.

References

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