The generalized order-parameter function $G_{pq}(r)$ of the solvable chiral Potts model at arbitrary rapidities

The solvable (star-triangle) $N$-state chiral Potts model is the prototypical exactly solvable lattice model without the rapidity-difference property. Baxter's 2005 derivation of its order parameters, which verified the 1989 Albertini–McCoy–Perk–Tang conjecture $M_r = k^{r(N-r)/N^2}$, proceeded through a generalized order-parameter ratio function $G_{pq}(r)$ defined with a broken rapidity line, and solved only the special rapidity relations $q=p$ and $q=Vp$ (the latter by Wiener–Hopf factorization, introducing the non-rational factor $S_p$). Baxter's 2006 review 'The challenge of the chiral Potts model' explicitly leaves open: the closed form and complete analytic continuation of $G_{pq}(r)$, and of the associated function $L_{pq}(r)$, for arbitrary independent rapidities on the many-sheeted rapidity surface; a proof of the conjectured product formula for $G_{p,V^i p}(r)$ over $i=0,\dots,N-1$ and $r=1,\dots,N-i$, together with the unresolved combinations outside that range; and a rigorous foundation for the analyticity assumptions and the Wiener–Hopf argument. No resolution is known in the literature to date: later work treats only the $q=Vp$ special case (in hyperelliptic variables for $N=3$), the superintegrable model, or adjacent questions, and Baxter stated in 2010 that the generalized order parameter at arbitrary rapidities remained unknown to him. The problem is open.

Background

The chiral Potts model is a two-dimensional classical lattice model of statistical mechanics in which each site carries a spin taking $N$ values $0,1,\dots,N-1$ and neighbouring spins interact through Boltzmann weight functions $W_{pq}(n)$ and $\bar W_{pq}(n)$ that depend on the spin difference $n$. The model is called solvable when the weights satisfy the star-triangle (Yang–Baxter) relation, which guarantees commuting transfer matrices. It is special among exactly solvable lattice models because for $N>2$ it lacks the rapidity-difference property: in most solvable models (Ising, six- and eight-vertex) the weights can be made to depend only on the difference of two spectral parameters, but here each rapidity $p$ is the whole set $\{x_p, y_p, \mu_p, t_p\}$ of variables constrained by $k\,x_p^N = 1 - k'/\mu_p^N, \qquad k\,y_p^N = 1 - k'\,\mu_p^N, \qquad t_p = x_p y_p,$ with $k^2 + k'^2 = 1$; a rapidity is thus a point on an algebraic curve of genus $N^3 - 2N^2 + 1$, which exceeds one as soon as $N > 2$. The parameter $k$ decreases from 1 to 0 as the temperature rises from zero to the critical point, and $\omega = e^{2\pi i/N}$.

The free energy of the infinite solvable model was obtained in 1988–1990, but the order parameters (spontaneous magnetizations) $M_r = \langle \omega^{ra} \rangle$, $r = 0,\dots,N$, for a spin $a$ deep inside the lattice, resisted calculation much longer. The corner transfer matrix method — which computes one-point functions by splitting the lattice into four quadrant transfer matrices that commute and factorize — fails for this model precisely because of the missing difference property. In 1989 Albertini, McCoy, Perk and Tang conjectured, from series expansions, the simple law $M_r = k^{r(N-r)/N^2}$. Baxter verified this conjecture in 2005 by adapting the broken rapidity line technique that Jimbo, Miwa and Nakayashiki had introduced for eight-vertex correlation functions: one breaks the rapidity line passing next to the spin $a$, assigning rapidity $p$ to the left half and $q$ to the right half of the break. The one-spin average $\tilde F_{pq}(r) = \langle \omega^{ra} \rangle$ then depends on both rapidities, and $M_r = \tilde F_{pp}(r)$.

The central object is the ratio function $G_{pq}(r) = \tilde F_{pq}(r)/\tilde F_{pq}(r-1).$ Rotating and crossing the two half-lines yields functional relations, for example $G_{Rp,Rq}(r) = 1/G_{pq}(N-r+1)$, $G_{Mp,q}(r) = G_{p,M^{-1}q}(r) = G_{pq}(r+1)$, and $\prod_{r=1}^{N} G_{pq}(r) = 1$, where $R$, $S$, $M$, $V$ are automorphisms of the rapidity curve (for instance $V$ maps $\{x_p, y_p, \mu_p, t_p\}$ to $\{x_p, \omega y_p, \mu_p, \omega t_p\}$). For $N=2$ the model reduces to the Ising model, the curve has genus one, all quantities become elliptic functions of arguments $u_p, u_q$ depending only on the difference $u_q - u_p$, and the functional relations plus a strip-analyticity property are solved by Fourier transforms, reproducing the Onsager–Yang magnetization. For $N>2$ there is no uniformizing substitution (no single-valued one-variable parametrization of the curve): regarded as a function of $t_p$, $G_{pq}(r)$ lives on a many-sheeted Riemann surface built from a central domain $D$ by repeated crossings of $N$ branch cuts $B_0,\dots,B_{N-1}$ running outward from the points $1,\omega,\dots,\omega^{N-1}$ of the $t$-plane, and the functional relations connect only a small subset of the sheets — in Baxter's words, there is a basic lack of information.

What is known concerns special relations between $p$ and $q$. Baxter observed that the restriction $q = Vp$ already suffices to obtain the $M_r$: for that case $G_{p,Vp}(r) = g(p;r)$ can be found by a Wiener–Hopf factorization (the factorization of a function analytic in a cut plane into factors analytic inside and outside a contour), giving $g(p;r) = k^{(N+1-2r)/N^2} S_p^{\epsilon(r)}$ for $r = 1,\dots,N$, with $\epsilon(r) = 1$ for $1 \le r \le N-1$ and $\epsilon(N) = 1-N$, where $S_p = S(t_p)$ is the single-valued function (not to be confused with the automorphism $S$ of the rapidity curve) $\log S(t_p) = -\frac{2}{N^2}\log k + \frac{1}{2N\pi}\int_0^{2\pi} \frac{k' e^{i\theta}}{1 - k' e^{i\theta}} \log\big[\Delta(\theta) - t_p\big]\, d\theta, \qquad \Delta(\theta) = \big[(1 - 2k'\cos\theta + k'^2)/k^2\big]^{1/N}.$ The function $S_p$ is meromorphic in $t_p$ but is not a rational or algebraic function of the rapidity variables. For $N=3$, Baxter moreover expressed $S(t_p)$ through hyperelliptic variables $z_p, w_p$ (a parametrization of the curve by a genus-two cover, with infinite products generalizing elliptic functions) — the first thermodynamic quantity of the $N>2$ model so expressed — but again only for the $q = Vp$ slice. The next case $q = V^2 p$ has been probed only by low-temperature series for $N=3$, whose first four terms support $G_{p,V^2 p}(1) = k^{2/9} S_p S_{Vp}$. Baxter also introduced in 1998 the combination $L_{pq}(r) = G_{pq}(r)\,G_{Rq,Rp}(r)$, hoping that its square would be rational in the rapidity variables as happens for $N=2$; the $q = Vp$ results show this cannot be true for $N>2$, since they involve $S_p$. Finally, the whole 2005 derivation of the order parameters — like the standard free-energy calculation — rests on assumed analyticity properties of functions of an auxiliary $\tau_2$ model (a simplified lattice model whose weights are linear in $t_q$ and whose transfer matrix is the product of two chiral Potts transfer matrices with related rapidities) and on a Wiener–Hopf factorization, supported by series checks; Baxter describes such derivations as believable but hard to make fully mathematically rigorous.

Problem Statement

Determine the generalized order-parameter ratio function $G_{pq}(r) = \tilde F_{pq}(r)/\tilde F_{pq}(r-1)$ of the solvable $N$-state chiral Potts model for arbitrary, independent rapidities $p, q$ on the rapidity curve, for $N > 2$ (for $N = 2$ the problem is solved by elliptic uniformization). As posed by Baxter in The challenge of the chiral Potts model and his related papers, the task comprises the following linked components. A complete answer must both (i) determine $G_{pq}(r)$ for general $p, q$ (components 1–3, where component 1 subsumes components 2 and 3) and (ii) place the analyticity and Wiener–Hopf argument on a rigorous footing (component 4); a result on a proper subset of the components is a partial result. The components are:

1. Obtain a closed-form or structural expression for $G_{pq}(r)$ — and for the associated function $L_{pq}(r) = G_{pq}(r)\,G_{Rq,Rp}(r)$ — valid for general $p, q$, together with the complete analytic continuation of these functions as single-valued meromorphic functions on the full many-sheeted rapidity surface obtained from the domain $D$ by repeated crossings of the branch cuts $B_0,\dots,B_{N-1}$; in particular, establish which of the cuts and sheets are genuinely present, supplying the analyticity information that the functional relations alone do not contain.
2. Prove (or refute) the conjectured product formula $G_{p,V^i p}(r) = k^{(N+1-2r)/N^2}\, S(t_p)\,S(\omega t_p)\cdots S(\omega^{i-1} t_p)$ for $i = 0,\dots,N-1$ and $r = 1,\dots,N-i$, and determine the form of $G_{p,V^i p}(r)$ for the remaining combinations of $i$ and $r$, which remain a puzzle (the case $i=1$, $r=N$ being an exception already deducible from the functional relations).
3. Give a satisfactory structural description of $L_{pq}(r)$ for arbitrary $p, q$ — for instance, as suggested by Baxter's $N=3$ hyperelliptic analysis, as a single-valued function of hyperelliptic variables of $p$ and $q$.
4. Place the analyticity assumptions and the Wiener–Hopf factorization argument underlying the 2005 derivation of the order parameters on a fully rigorous mathematical footing in the multi-sheeted rapidity setting.

Any answer must be consistent with the functional relations satisfied by $G_{pq}(r)$, must reproduce the known special cases: $q = p$ (giving $M_r = k^{r(N-r)/N^2}$), $q = Vp$ (giving $G_{p,Vp}(r) = k^{(N+1-2r)/N^2} S_p^{\epsilon(r)}$), and the series-evidenced $q = V^2 p$ results for $N=3$ — and must reduce for $N=2$ to the elliptic (Ising) solution.

References

1. R. J. Baxter, The challenge of the chiral Potts model, J. Phys.: Conf. Ser. 42, 11-24 (2006). DOI: 10.1088/1742-6596/42/1/003; arXiv:cond-mat/0510683. https://doi.org/10.1088/1742-6596/42/1/003
2. R. J. Baxter, The order parameter of the chiral Potts model, J. Stat. Phys. 120, 1-36 (2005). DOI: 10.1007/s10955-005-5534-3; arXiv:cond-mat/0501226. https://doi.org/10.1007/s10955-005-5534-3
3. R. J. Baxter, Derivation of the order parameter of the chiral Potts model, Phys. Rev. Lett. 94, 130602 (2005). DOI: 10.1103/PhysRevLett.94.130602; arXiv:cond-mat/0501227. https://doi.org/10.1103/PhysRevLett.94.130602
4. G. Albertini, B. M. McCoy, J. H. H. Perk and S. Tang, Excitation spectrum and order parameter for the integrable N-state chiral Potts model, Nucl. Phys. B 314, 741-763 (1989). DOI: 10.1016/0550-3213(89)90415-X. https://doi.org/10.1016/0550-3213(89)90415-X
5. R. J. Baxter, Functional relations for the order parameters of the chiral Potts model, J. Stat. Phys. 91, 499-524 (1998). DOI: 10.1023/A:1023096408679. https://doi.org/10.1023/A:1023096408679
6. R. J. Baxter, Functional relations for the order parameters of the chiral Potts model: low-temperature expansions, Physica A 260, 117-130 (1998). DOI: 10.1016/S0378-4371(98)00307-0. https://doi.org/10.1016/S0378-4371(98)00307-0
7. R. J. Baxter, Hyperelliptic parametrisation of the generalised order parameter of the $N=3$ chiral Potts model, The ANZIAM Journal 47, 289-307 (2006). DOI: 10.1017/S1446181100009858; arXiv:cond-mat/0601325. https://doi.org/10.1017/S1446181100009858
8. H. Au-Yang and J. H. H. Perk, Spontaneous magnetization of the integrable chiral Potts model, J. Phys. A: Math. Theor. 44, 445005 (2011). DOI: 10.1088/1751-8113/44/44/445005; arXiv:1003.4805. https://doi.org/10.1088/1751-8113/44/44/445005
9. R. J. Baxter, Some comments on developments in exact solutions in statistical mechanics since 1944, J. Stat. Mech. (2010) P11037. DOI: 10.1088/1742-5468/2010/11/P11037; arXiv:1010.0710. https://doi.org/10.1088/1742-5468/2010/11/P11037
10. R. J. Baxter, The $\tau_2$ model and parafermions, J. Phys. A: Math. Theor. 47, 315001 (2014). DOI: 10.1088/1751-8113/47/31/315001; arXiv:1310.7074. https://doi.org/10.1088/1751-8113/47/31/315001
11. M. Jimbo, T. Miwa and A. Nakayashiki, Difference equations for the correlation functions of the eight-vertex model, J. Phys. A: Math. Gen. 26, 2199-2209 (1993). DOI: 10.1088/0305-4470/26/9/015. https://doi.org/10.1088/0305-4470/26/9/015
12. S.-s. Roan, Fermat surface and group theory in symmetry of rapidity family in chiral Potts model (2013). arXiv:1301.5260. https://arxiv.org/abs/1301.5260
13. N. Grosjean, J.-M. Maillet and G. Niccoli, On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models, Ann. Henri Poincare 16, 1103-1153 (2015). DOI: 10.1007/s00023-014-0358-9; arXiv:1309.4701. https://doi.org/10.1007/s00023-014-0358-9
14. H. Zhu, A symmetry formula for correlation functions in the superintegrable chiral Potts spin chain (2026). arXiv:2603.27930. https://arxiv.org/abs/2603.27930