The generalized order parameter function $G_{pq}(r)$ of the solvable chiral Potts model for arbitrary rapidities $p,q$

The solvable (Yang–Baxter integrable) $N$ -state chiral Potts model is the archetypal exactly solved lattice model without the rapidity-difference property. Its order parameters $\mathcal{M}_r=k^{r(N-r)/N^2}$ were derived in 2005 through the broken-rapidity-line technique, which replaces the magnetization by a two-rapidity ratio function $G_{pq}(r)$ satisfying closed functional relations. That function is known in closed form only for $q=p$ and $q=Vp$ ( $V$ a standard automorphism of the rapidity curve); for arbitrary independent rapidities no closed form, analytic continuation, or branch structure on the many-sheeted rapidity surface is known, the conjectured product-of- $\mathcal{S}$ -factor form for $q=V^ip$ ( $i\ge2$ ) is unproved, the companion function $L_{pq}(r)$ lacks any structural description, and the analyticity and Wiener–Hopf uniqueness arguments underlying the known cases are not fully rigorous. Baxter's problem of determining $G_{pq}(r)$ in this full generality remains open.

Background

The chiral Potts model assigns to each site of a square lattice a spin $\sigma\in\{0,\dots,N-1\}$ and to each horizontal (vertical) edge a positive Boltzmann weight $W_{pq}(\sigma_i-\sigma_j)$ ( $\bar W_{pq}(\sigma_i-\sigma_j)$ ). Its solvable version satisfies the star-triangle (Yang–Baxter) relation, an algebraic identity among weights that makes the row-to-row transfer matrices commute and puts the model in the class of exactly solved models; it is called chiral because $W(-n)\ne W(n)$ . For $N=2$ it degenerates to the Ising model. The weights are parametrized by two rapidities $p,q$ , each a set $p=\{x_p,y_p,\mu_p,t_p\}$ of complex variables constrained by $kx_p^N=1-k'/\mu_p^N$ , $ky_p^N=1-k'\mu_p^N$ , $t_p=x_py_p$ , with fixed real $k,k'\in(0,1)$ , $k^2+k'^2=1$ . For $N>2$ this constraint defines an algebraic curve of genus greater than one: there is no uniformizing parametrization in single-valued functions of one variable (unlike $N=2$ , where everything is a Jacobi elliptic function of a single elliptic argument), and $x_p,y_p,\mu_p$ are multi-valued functions of $t_p$ with $N$ branch cuts $B_0,\dots,B_{N-1}$ about the points $1,\omega,\dots,\omega^{N-1}$ , where $\omega=e^{2\pi i/N}$ . The model also lacks the rapidity-difference property of the six- and eight-vertex models: the weights depend on $p$ and $q$ separately, which is the source of its exceptional difficulty.

For such models the classical hierarchy of exact results is free energy, then order parameters. The free energy was obtained in 1988–1990. The order parameter $\mathcal{M}_r=\langle\omega^{r\sigma}\rangle$ was conjectured by Albertini, McCoy, Perk and Tang in 1989 as $\mathcal{M}_r=k^{r(N-r)/N^2}$ for $0<k<1$ (vanishing for $k>1$ ), and derived by Baxter in 2005 using the broken rapidity line technique that Jimbo, Miwa and Nakayashiki had introduced for the eight-vertex model: one cuts a rapidity line adjacent to the central spin and gives the two halves different rapidities $p$ and $q$ . The central-spin probability $F_{pq}(a)$ then genuinely depends on both $p$ and $q$ ; from its discrete Fourier transform $\tilde F_{pq}(r)=\sum_{a=0}^{N-1}\omega^{ra}F_{pq}(a)$ one defines the generalized order parameter (ratio) function $G_{pq}(r)=\tilde F_{pq}(r)/\tilde F_{pq}(r-1)$ . When $q=p$ the line is unbroken, can be pulled to infinity, and $\mathcal{M}_r=G_{pp}(1)\cdots G_{pp}(r)$ , independent of $p$ .

Rotating and crossing the two half-lines yields closed functional relations for $G_{pq}(r)$ , involving the automorphisms of the rapidity curve $R,S,M,V$ , e.g. $V:\{x_p,y_p,\mu_p,t_p\}\mapsto\{x_p,\omega y_p,\mu_p,\omega t_p\}$ and $R:\{x_p,y_p,\mu_p,t_p\}\mapsto\{y_p,\omega x_p,1/\mu_p,\omega t_p\}$ , together with the normalization $\prod_{r=1}^N G_{pq}(r)=1$ . For $N=2$ these relations plus a strip-analyticity assumption determine $G_{pq}(r)$ and reproduce the Ising spontaneous magnetization. For $N>2$ the relations connect the physical sheet $\mathcal D$ (the branch with $\mu_p^N$ outside the unit circle and $x_p$ near $1$ ) to only a small subset of the sheets of the many-sheeted rapidity surface obtained by crossing the cuts $B_0,\dots,B_{N-1}$ — in Baxter's words there is a basic lack of information.

What is known: for $q=p$ , $G_{pp}(r)=D_r/D_{r-1}$ with $D_r$ pure powers of $k$ , giving $\mathcal{M}_r=k^{r(N-r)/N^2}$ . For $q=Vp$ (i.e. $x_q=x_p$ , $y_q=\omega y_p$ , $\mu_q=\mu_p$ ), a Wiener–Hopf factorization — the decomposition of a function analytic in a cut plane into factors analytic inside and outside the cut, the standard tool for scalar Riemann–Hilbert problems — gives $G_{p,Vp}(r)=k^{(N+1-2r)/N^2}\,\mathcal{S}_p^{\epsilon(r)}$ for $r=1,\dots,N$ , with $\epsilon(r)=1-N\delta_{r,0}$ (the label $r$ being counted modulo $N$ , so $\epsilon(N)=\epsilon(0)=1-N$ ) and $\log\mathcal{S}_p=-\frac{2}{N^2}\log k+\frac{1}{2N\pi}\int_0^{2\pi}\frac{k'e^{i\theta}}{1-k'e^{i\theta}}\log[\Delta(\theta)-t_p]\,d\theta$ , $\Delta(\theta)=[(1-2k'\cos\theta+k'^2)/k^2]^{1/N}$ . For $q=V^2p$ only low-temperature series (four orders, $N=3$ ) exist, consistent with $G_{p,V^2p}(1)=k^{2/9}\mathcal{S}_p\mathcal{S}_{Vp}$ . Baxter conjectured for general $N$ that $G_{p,V^ip}(r)=k^{(N+1-2r)/N^2}\,\mathcal{S}_p\mathcal{S}_{Vp}\cdots\mathcal{S}_{V^{i-1}p}$ for $i=0,\dots,N-1$ and $r=1,\dots,N-i$ , the remaining $(i,r)$ combinations being unexplained except $i=1,r=N$ , which follows from the normalization. He also introduced $L_{pq}(r)=G_{pq}(r)G_{Rq,Rp}(r)$ in 1998 hoping it would be rational in the rapidity variables for all $N$ (its square is for $N=2$ ); the $q=Vp$ solution shows it must involve the non-rational factor $\mathcal{S}_p$ , contingent on the $N=3$ $i=2$ conjecture. For $N=3$ , $\mathcal{S}_p$ can be written in terms of hyperelliptic (generalized Jacobi elliptic) variables $z_p,w_p$ , but $G_{pq}(r)$ cannot be single-valued in these variables (the automorphism $R^6p$ fixes $z,w$ while changing $G$ ), whereas $L_{pq}(r)$ is invariant and may be. Finally, the derivations of the known cases rest on analyticity assumptions about the thermodynamic-limit functions and on the uniqueness of the Wiener–Hopf factorization on the multi-sheeted surface; Baxter himself notes such derivations are believable but hard to make fully mathematically rigorous.

Problem Statement

Determine the generalized order parameter 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 $p,q$ — the case Baxter states 'continues to defy calculation'. Here $\tilde F_{pq}(r)=\sum_{a=0}^{N-1}\omega^{ra}F_{pq}(a)$ , with $\omega=e^{2\pi i/N}$ , is the discrete Fourier transform of the central-spin probability $F_{pq}(a)$ in the thermodynamic limit of the lattice with a broken rapidity line (the two half-lines adjacent to the central spin carrying the rapidities $p$ and $q$ ), and $V$ , $R$ and $\mathcal{S}_p$ are the automorphisms of the rapidity curve and the Wiener–Hopf factor defined in the background above. The common scientific objective is a complete analytic description of the two-rapidity order-parameter data of the model, with these jointly supporting components: (a) a closed-form expression for $G_{pq}(r)$ valid for general $p,q$ , together with its complete analytic continuation and branch/monodromy structure on the many-sheeted rapidity surface obtained from the physical domain $\mathcal{D}$ by crossing the $N$ branch cuts of the $t$ -plane; (b) as the structured ladder toward (a), 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 general $N$ and all $i=0,\dots,N-1$ , $r=1,\dots,N-i$ (the cases $i=0,1$ being already derived), together with an explanation of the remaining $(i,r)$ combinations other than the known exception $i=1$ , $r=N$ fixed by $\prod_{r=1}^N G_{pq}(r)=1$ ; and (c) a structural characterization of $L_{pq}(r)=G_{pq}(r)G_{Rq,Rp}(r)$ for arbitrary $p,q$ — for instance whether it is single-valued in, and expressible through, hyperelliptic variables — given that, if the $N=3$ , $i=2$ series result $G_{p,V^2p}(1)=k^{2/9}\mathcal{S}_p\mathcal{S}_{Vp}$ is correct, $L_{pq}(r)$ cannot be rational in the rapidity variables for $N>2$ . Any answer must satisfy the functional relations of Baxter (1998/2005), including $\prod_{r=1}^N G_{pq}(r)=1$ , and must reduce to the known special cases: $q=p$ giving $\mathcal{M}_r=k^{r(N-r)/N^2}$ , and $q=Vp$ giving $G_{p,Vp}(r)=k^{(N+1-2r)/N^2}\mathcal{S}_p^{\epsilon(r)}$ with $\epsilon(r)=1-N\delta_{r,0}$ (the label $r$ being counted modulo $N$ , so $\epsilon(N)=1-N$ ), as well as to the $N=2$ Ising limit. Component (a) is the mandatory deliverable: because a closed form valid for general $p,q$ specialises to the $q=V^ip$ subcase, it also settles (b). Components (b) and (c) are independently refutable supporting results — proving or refuting (b) alone, or characterising $L_{pq}(r)$ alone, resolves only that component and not the task. Concretely, (a) counts as a closed form only for an explicit formula for $G_{pq}(r)$ , valid for general $N$ and arbitrary independent $p,q$ , built from the model data — $k$ , $\omega$ , the rapidity-curve data $x_p,y_p,\mu_p,t_p$ , and the functions of a single rapidity already known — rather than an implicit characterisation such as 'the unique solution of the functional relations carrying the prescribed monodromy'; and 'complete branch/monodromy structure' must locate the singularities and the sheets of analyticity of $G_{pq}(r)$ and give how $G_{pq}(r)$ transforms under analytic continuation around the branch cuts $B_0,\dots,B_{N-1}$ , relative to an explicitly named generating set of loops on the rapidity surface.

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