Closed form and analytic structure of the generalized order-parameter function $G_{pq}(r)$ of the chiral Potts model for arbitrary rapidities

The solvable $N$-state chiral Potts model is the paradigm exactly solvable two-dimensional lattice model whose Boltzmann weights lack the rapidity-difference property. Its free energy (1988) and its order parameters $M_r = k^{r(N-r)/N^2}$ (conjectured 1989, derived 2005) are known, but the generalized order-parameter function $G_{pq}(r)$ — the ratio of successive expectations $\langle\omega^{ra}\rangle$ around a broken rapidity line whose two halves carry arbitrary rapidities $p$ and $q$ — is known only on special loci ($q=p$, $q=Vp$, and series-evidenced cases $q=V^2p$). Baxter's review 'The challenge of the chiral Potts model' (2006) closes by leaving open: a closed form and complete branch structure for $G_{pq}(r)$ on the many-sheeted rapidity surface; and the unproved conjecture that $G_{p,V^ip}(r)$ is a simple product of $\mathcal S$-factors for $i=0,\ldots,N-1$, $r=1,\ldots,N-i$. The same review records the negative structural result that the companion function $L_{pq}(r)$ is not rational for $N>2$, and the companion 2006 hyperelliptic paper leaves open whether $L_{pq}(r)$ — unlike $G_{pq}(r)$ — can be simply expressed in terms of hyperelliptic variables. The subsequent literature (2006–2026) shows that the order-parameter formula has been independently established by an algebraic Onsager-algebra route, and that frameworks for the correlations exist ($Z$-invariance, separation-of-variables form factors), but no work determines $G_{pq}(r)$ or $L_{pq}(r)$ for arbitrary $p,q$ or proves the product-form conjecture. The problem remains open.

Background

Two-dimensional lattice models that can be solved exactly usually owe their solvability to the star-triangle (Yang–Baxter) relation, an algebraic identity satisfied by their Boltzmann weights which allows the auxiliary 'rapidity' lines threading the lattice to be moved around freely. The chiral Potts model is the $N$-state spin model of this type: each spin $\sigma$ takes the values $0,\ldots,N-1$, nearest-neighbour weights $W_{pq}(\sigma_i-\sigma_j)$ and $\bar W_{pq}(\sigma_i-\sigma_j)$ depend on the spin difference modulo $N$ and on the rapidities $p,q$ of the two lines crossing at that edge. A rapidity is a set $p=\{x_p,y_p,\mu_p,t_p\}$ of complex numbers 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 $k^2+k'^2=1$; $k'$ plays the role of temperature, the system being ferromagnetically ordered for $0<k'<1$. Unlike the Ising, six-vertex and eight-vertex models, for $N>2$ the weights cannot be made to depend only on the difference of the two rapidities: a rapidity is a point on an algebraic curve of genus greater than one for $N>2$, and although the curve can be parametrized by hyperelliptic functions, these involve $N-1$ related arguments rather than a single uniformizing variable, so no difference-type spectral parameter exists. The free energy was nevertheless obtained in 1988–1990.

The order parameters (spontaneous magnetizations) $M_r=\langle\omega^{ra}\rangle$, where $\omega=e^{2\pi i/N}$ and $a$ is a spin deep inside the lattice, were conjectured in 1989 by Albertini, McCoy, Perk and Tang from series expansions to be $M_r=k^{r(N-r)/N^2}$. Baxter's 2005 derivation used the 'broken rapidity line' technique that Jimbo, Miwa and Nakayashiki had introduced for the eight-vertex model: one cuts the rapidity line passing next to the central spin $a$ and gives the two halves different rapidities $p$ and $q$. The star-triangle relation can then no longer move this line away, so the expectation

$\tilde F_{pq}(r)=\langle\omega^{ra}\rangle$

becomes a non-trivial function of both $p$ and $q$ — the generalized order parameter — with $\tilde F_{pp}(r)=M_r$. One works with the ratio $G_{pq}(r)=\tilde F_{pq}(r)/\tilde F_{pq}(r-1)$, so that $M_r=G_{pp}(1)\cdots G_{pp}(r)$. In 1998 Baxter showed that $G_{pq}(r)$ satisfies closed functional relations associated with automorphisms of the rapidity curve, for instance $V:\{x_p,y_p,\mu_p,t_p\}\mapsto\{x_p,\omega y_p,\mu_p,\omega t_p\}$ (and similarly defined $R,S,M$), together with the periodicity $\prod_{r=1}^{N}G_{pq}(r)=1$ and reflection symmetries. For $N=2$ (the Ising model) these relations plus a strip-analyticity condition determine $G_{pq}(r)$ and give the Onsager–Yang magnetization. For $N>2$ they do not: $G_{pq}(r)$ lives on a many-sheeted Riemann surface built from a central sheet $\mathcal D$ by crossing the $N$ branch cuts $B_0,\ldots,B_{N-1}$ in the $t_p$- and $t_q$-planes, and the analyticity information needed to pin down the function is exactly what is missing.

Baxter's 2005 breakthrough was the observation that the order parameters only require the restricted locus $q=Vp$ (the 'superintegrable' case), where $g(p;r)=G_{p,Vp}(r)$ depends effectively on the single variable $t_p$ and is obtained by a Wiener–Hopf factorization — the decomposition of a function analytic in a cut plane into factors analytic on complementary sides of the cut — giving $g(p;r)=k^{(N+1-2r)/N^2}\mathcal S_p^{\varepsilon(r)}$ with $\varepsilon(r)=1$ for $r=1,\ldots,N-1$, $\varepsilon(N)=1-N$, and $\mathcal S_p$ an explicit logarithmic integral over $t_p$. For $q=V^2p$ and $N=3$ the low-temperature series support $g_2(p;1)=k^{2/9}\mathcal S_p\mathcal S_{Vp}$, and generally Baxter conjectured

$G_{p,V^ip}(r)=k^{(N+1-2r)/N^2}\,\mathcal S_p\,\mathcal S_{Vp}\cdots\mathcal S_{V^{i-1}p},\qquad i=0,\ldots,N-1,\; r=1,\ldots,N-i,$

of which only $i=0$ (constants) and $i=1$ are proved. A companion function $L_{pq}(r)=G_{pq}(r)G_{Rq,Rp}(r)$, originally hoped to be rational in the rapidity variables (as its square is for $N=2$), was shown by the same work to involve the non-rational factor $\mathcal S_p$ for $N>2$, so no simple algebraic ansatz can describe it. The 2005 derivation also rests on analyticity assumptions and a uniqueness assumption for the Wiener–Hopf factorization that Baxter himself characterized as believable but hard to make fully mathematically rigorous.

Problem Statement

Determine the generalized order-parameter function $G_{pq}(r)$ of the ferromagnetic solvable $N$-state chiral Potts model in the thermodynamic limit for arbitrary rapidities $p,q$ — that is, for general $N\ge 3$ and $r=1,\ldots,N$, without restricting $q$ to the loci $q=V^ip$ — by supplying, as one coherent determination:

1. a closed-form expression for $G_{pq}(r)$ as a function of the rapidity variables of both $p$ and $q$ (here and below, 'closed form' denotes an explicit analytic expression that determines $G_{pq}(r)$, of the same explicitness as the known $q=Vp$ result $k^{(N+1-2r)/N^2}\mathcal S_p^{\varepsilon(r)}$ with $\mathcal S_p$ an explicit logarithmic integral: explicit integral, product and special-function representations of that kind, including theta and hyperelliptic ones, all qualify, whereas an implicit characterization that does not determine the function does not);

2. the complete analytic continuation and branch structure of $G_{pq}(r)$ on the many-sheeted rapidity surface — which of the cuts $B_0,\ldots,B_{N-1}$ in the $t_p$- and $t_q$-planes are present and the monodromy around them — which is precisely the analyticity input needed to make the 1998 functional relations determine the function;

All four items are required. Items 1 and 2 are jointly required and together constitute the primary determination of $G_{pq}(r)$; items 3 and 4 are the accompanying questions, each to be answered in its own right:

3. a proof or refutation of the conjectured product form $G_{p,V^ip}(r)=k^{(N+1-2r)/N^2}\mathcal S_p\mathcal S_{Vp}\cdots\mathcal S_{V^{i-1}p}$ for $i=0,\ldots,N-1$ and $r=1,\ldots,N-i$ (the cases $i=0,1$ are known and the case $i=1$, $r=N$ is deducible from the functional relations; $i\ge 2$ is open); deriving it as the special locus $q=V^ip$ of the determination in items 1–2 satisfies this item, as does an independent proof or refutation;

4. a corresponding structural characterization of the companion function $L_{pq}(r)=G_{pq}(r)G_{Rq,Rp}(r)$ for arbitrary $p,q$, given that it is not rational in the rapidity variables for $N>2$: in particular, whether — unlike $G_{pq}(r)$, which cannot be a single-valued function of the hyperelliptic variables $(z_p,w_p,z_q,w_q)$ — $L_{pq}(r)$ can be simply expressed as a single-valued function of those variables.

A complete answer must be consistent with the functional relations, must reproduce the solved cases $q=p$ (the constants $k^{(N+1-2r)/N^2}$) and $q=Vp$ (the factor $\mathcal S_p$), must yield $M_r=k^{r(N-r)/N^2}$ in the unbroken limit $q\to p$, and must agree with the available $N=3$ low-temperature series.

References

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