Finite-separation pair correlations of the integrable chiral Potts model: non-maximal eigenvector overlaps in the superintegrable spectral sum

The integrable $Z_N$ chiral Potts model is the archetypal exactly solved two-dimensional lattice model beyond the free-fermion and rapidity-difference classes. Au-Yang and Perk (2011) showed that $Z$-invariance (Baxter's invariance of the partition function and of local expectations under re-routing the rapidity lines of a solvable lattice) reduces the pair correlation function of the general integrable chiral Potts model at equal horizontal rapidities to a spectral sum over eigenvectors of the superintegrable chiral Potts transfer matrices; the infinite-separation limit of this sum reproduces Baxter's spontaneous magnetization through the inner products of the $N$ asymptotically degenerate maximum-eigenvalue eigenvectors alone. At finite spin separation, however, the sum runs over all eigenvectors weighted by their overlaps with the maximum eigenvectors, and these non-maximal overlaps are not known in closed form. The open problem is to compute exactly the finite-separation pair correlation $g^{(r)}_{2\ell}$ -- equivalently, the inner products $\langle Y^Q_{\max}|Y^P_j\rangle$ for every eigenvector not orthogonal to the maximum eigenvectors, or the equivalent vertical-row expectations of strings of $E^\pm$ creation-annihilation operators with intervening $\hat Z^r$ insertions -- for general $N$ and system size. Partial results exist (factorized matrix elements known up to normalization factors, finite-lattice determinant form factors via separation of variables, exact small-system computations, and symmetry constraints proved in 2026), but no exact finite-distance correlation function of the chiral Potts model is known, and recent literature explicitly describes the finite-distance regime as remaining far less explicit than the Ising case.

Background

The integrable chiral Potts model is an $N$-state two-dimensional classical lattice model in which spins $\sigma\in\mathbb{Z}_N$ sit on the sites of a square lattice and interact through pair Boltzmann weights $W_{pq}(\sigma-\sigma')$ and $\overline{W}_{pq}(\sigma-\sigma')$ on horizontal and vertical bonds; here $\omega=e^{2\pi i/N}$ is the primitive $N$-th root of unity. Its weights satisfy the star-triangle (Yang-Baxter) relation, so the row-to-row transfer matrices commute, but the rapidity variables $p,q$ live on an algebraic curve of genus $(N-1)^2$ and the weights lack the rapidity-difference property -- dependence on $p$ and $q$ only through the difference $p-q$, the structural feature underlying the solvability of the six- and eight-vertex models. It is consequently regarded as the most difficult of the two-dimensional solvable models, and its correlation functions have never been obtained in the way the Ising model's were. A special superintegrable case -- vertical rapidities alternating between $p$ and $p'$ with $x_{p'}=y_p$, $y_{p'}=x_p$, $\mu_{p'}=1/\mu_p$ -- has Ising-like properties: its transfer matrices generate an Onsager algebra (the associative algebra introduced in Onsager's 1944 solution of the Ising model, generated by elements obeying the Dolan-Grady relations), and its associated quantum spin chain is the $Z_N$-symmetric chain of von Gehlen and Rittenberg (1985) with Hamiltonian $H=A_0+\lambda A_1$.

The order parameter (spontaneous magnetization) $M_r=\langle\omega^{r\sigma}\rangle$ in the ordered phase was conjectured in 1989 by Albertini, McCoy, Perk and Tang and first derived by Baxter in 2005 through functional relations for a generalized magnetization with one broken rapidity line; Baxter noted that the general two-rapidity case remained beyond his method. Au-Yang and Perk (2011) gave an independent derivation inside the superintegrable framework. Writing the transfer matrices in the spin-shift (momentum) sectors $Q=0,\dots,N-1$ -- the Fourier basis $|Q;\{n_j\}\rangle$ that diagonalizes the spin-shift operator (the operator multiplying every spin by $\omega$, $\sigma\mapsto\omega\sigma$) -- the pair correlation at equal horizontal rapidities $q$ of two spins separated by $\ell$ transfer-matrix rows takes, in the thermodynamic limit, the form of a spectral sum

$$g^{(r)}_{2\ell}=\frac{1}{N}\sum_{Q=0}^{N-1}\sum_j\Big(\frac{\Delta^P_j}{\Delta^0_{\max}}\Big)^{2\ell}\langle Y^Q_{\max}|Y^P_j\rangle\langle Y^P_j|Y^Q_{\max}\rangle,\qquad P\equiv Q-r\ (\mathrm{mod}\ N),$$

where $|Y^Q_j\rangle$ are the eigenvectors of $T^Q(x_q,y_q)\hat T^Q(x_q,y_q)$ with eigenvalues $(\Delta^Q_j)^2$ and $|Y^Q_{\max}\rangle$ is the eigenvector of the maximal eigenvalue $\Delta^Q_{\max}$. The $N$ maximal eigenvalues are asymptotically degenerate as the circumference $L\to\infty$, so only the maximal-eigenvector terms survive as $\ell\to\infty$: the order parameter reduces to $N$ inner products of maximal eigenvectors, which Au-Yang and Perk computed from their explicit construction of the ground-state-sector eigenvectors. That construction uses nilpotent creation-annihilation operators $E^\pm_{j,Q}$ with $(E^\pm_{j,Q})^2=0$, indexed by the roots of a polynomial $P_Q$ and generating a direct sum of spin-$1/2$ $sl_2$ (quantum loop) representations, which create the known superintegrable eigenvectors from the maximal state $|\Omega\rangle$. The ground-state overlaps were summed through a two-variable generating function $\mathcal{G}(t,u)$ whose coefficients rested on a conjecture; that conjecture was proved by Au-Yang and Perk themselves in 2011. For the Ising model ($N=2$) the analogue of the spectral sum is the Kaufman-Onsager form-factor expansion (a sum over form factors, the matrix elements of local operators between transfer-matrix eigenstates), which yields explicit finite-distance correlations as Toeplitz determinants; for the chiral Potts model nothing comparable exists, and Baxter's 2010 review emphasizes both the difficulty of the model and that even the generalized (broken-rapidity-line) order parameter is unknown away from special points. The superintegrable transfer matrix is equivalent to the Bazhanov-Stroganov $\tau_2$ model (a closely related exactly solved lattice model built from the same cyclic representations), whose full eigenvector spectrum is accessible through Tarasov's Bethe-ansatz prescription (the Bethe ansatz constructs eigenvectors of integrable models as superpositions labelled by parameter sets solving algebraic equations); the obstacle is that its spectrum is highly degenerate, so the Bethe vectors must be recombined into the correct linear combinations inside each degenerate eigenspace before any overlap with the maximal eigenvectors can be formed.

Problem Statement

The problem is to evaluate, exactly, the separation dependence of the pair correlation function of the integrable chiral Potts model -- the spectral sum of Au-Yang and Perk at finite separation $\ell$ -- for example, via the two ingredients that the paper of Au-Yang and Perk identifies as missing.

(i) Closed or otherwise exact usable forms, for general $N$ and circumference $L$, of the inner products between the maximum-eigenvalue eigenvectors of the superintegrable chiral Potts transfer matrices and every eigenvector not orthogonal to them,

$$\langle Y^Q_{\max}|Y^P_j\rangle,\qquad P\equiv Q-r\ (\mathrm{mod}\ N),$$

including the correct linear combinations inside each degenerate eigenspace, so that the finite-separation pair correlation (for each spin component $r$, taken modulo $N$)

$$g^{(r)}_{2\ell}=\frac{1}{N}\sum_{Q=0}^{N-1}\sum_j\Big(\frac{\Delta^P_j}{\Delta^0_{\max}}\Big)^{2\ell}\langle Y^Q_{\max}|Y^P_j\rangle\langle Y^P_j|Y^Q_{\max}\rangle$$

can be evaluated at finite $\ell$, either on the finite lattice or with a controlled thermodynamic limit $L\to\infty$ at fixed $\ell$.

(ii) Closed or otherwise exact usable forms of the vertical-row operator expectations with intervening local operators,

$$\langle\Omega|\prod_{k=1}^{n}\mathbf{E}^{-}_{j_k,P}\big(\hat Z^{r}_{1}\cdots\hat Z^{r}_{R}\big)\prod_{k=1}^{n}\mathbf{E}^{+}_{\ell_k,P}|\Omega\rangle,$$

for general $R$, $n$, $L$ and $N$, which enter the spin-chain pair correlations $\langle\hat Z^r_1\hat Z^{-r}_{R+1}\rangle$ and seed the combinatorial route proposed by Au-Yang and Perk.

Any exact route is admissible: a full construction of all eigenvectors via Tarasov's Bethe-ansatz prescription for the superintegrable $\tau_2$ model and the related higher-spin XXZ chain, combined with the resolution of the degenerate eigenspaces; new combinatorial or algebraic methods exploiting the $E^\pm$/Onsager-algebra structure; or finite-lattice determinant form factors together with an exact thermodynamic-limit summation. The common scientific objective uniting these ingredients is the finite-separation pair correlation itself: an accepted answer must deliver $g^{(r)}_{2\ell}$ at finite $\ell$ (or the equivalent chain two-point function) in exact, usable form for general $N$, and must reproduce the known spontaneous magnetization in the $\ell\to\infty$ limit as well as the known exact small-system values (for instance the $N=3$, $L\le 5$ results of Fabricius and McCoy) at finite size. The coefficients conjecture for the generating function $\mathcal{G}(t,u)$ was proved by Au-Yang and Perk (2011) and is excluded from the objective.

References

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