Single-valued hyperelliptic representation of the invariant order-parameter ratio combination $L_{pq}(r) = G_{pq}(r)\,G_{Rq,Rp}(r)$ of the $N=3$ chiral Potts model for arbitrary rapidities

The $N$ -state chiral Potts model is exactly solvable — its Boltzmann weights satisfy the star-triangle (Yang–Baxter) relation — yet it lacks the rapidity-difference property that underlies the solution of the Ising and eight-vertex models, and its correlation functions have never been uniformized. The broken-rapidity-line method produces a two-rapidity function $G_{pq}(r)$ whose specialization gives the order parameter $M_r = k^{r(N-r)/N^2}$ , conjectured in 1989 and proved by Baxter (2005) — a proof possible because $M_r$ is rapidity-independent, so it suffices to know $G_{pq}(r)$ in the superintegrable case $q = Vp$ , the only case where it has been obtained. He then (2006) expressed $(x_p^{-1/3}S(t_p))^3$ — the ingredient of $G_{pq}(r)$ in that case — as a product of functions of the hyperelliptic variables $z_p, w_p$ : the first thermodynamic quantity of the $N>2$ chiral Potts model ever so expressed. The open question he posed: does such a representation exist for arbitrary independent rapidities $p, q$ ? His own results prove an obstruction — the automorphism $R^6$ fixes $z_p, w_p$ but not $G_{pq}(r)$ , so $G_{pq}(r)$ is not single-valued in these variables — while the combination $L_{pq}(r) = G_{pq}(r)\,G_{Rq,Rp}(r)$ is invariant and 'may be so expressible'; the automorphism $M$ also fixes $z_p, w_p$ but shifts $r$ , so residual rapidity monodromy must enter any answer. The problem: construct such a hyperelliptic representation of $L_{pq}(r)$ for arbitrary independent $p, q$ — a single-valued meromorphic function of $z_p, w_p, z_q, w_q$ (and the nome $x$ ) times explicit residual monodromy factors of the fixed algebraic kind carried by the proven prefactor $x_p^{-1/3}$ — or prove that $L_{pq}(r)$ admits none.

Background

The chiral Potts model assigns spins $\sigma \in \{0,\ldots,N-1\}$ to the sites of a lattice with $Z_N$ -symmetric interactions ( $\omega = e^{2\pi i/N}$ ). Its integrable version has Boltzmann weights satisfying the star-triangle (Yang–Baxter) relation, so transfer matrices $T(p)$ , $T(q)$ commute; each matrix is labeled by a rapidity $p$ , a point on a high-genus algebraic curve with coordinates $(x_p, y_p, \mu_p)$ (together with the dependent symmetric variable $t_p = x_p y_p$ ). Crucially, the weights are not functions of any difference of rapidities (no 'rapidity-difference property'), which is why the Bethe-ansatz and corner-transfer-matrix machinery that solves the Ising and eight-vertex models fails here. The order parameters are $M_r = \langle \omega^{r\sigma}\rangle$ ; Albertini–McCoy–Perk–Tang (1989) conjectured $M_r = k^{r(N-r)/N^2}$ , where the modulus $k$ and complementary modulus $k'$ are fixed by the nome $x$ through $-k'^2 = 27x\prod_{n\ge1}\big((1-x^{3n})/(1-x^n)\big)^{12}$ ; for $N=2$ this reduces to the Onsager–Yang magnetization $k^{1/4}$ . The broken rapidity line method (adapted from Jimbo–Miwa–Nakayashiki) cuts one rapidity line of the lattice so the two sides carry different rapidities $p, q$ , producing a two-rapidity correlation function $\tilde F_{pq}(r)$ ; its ratios $G_{pq}(r) = \tilde F_{pq}(r)/\tilde F_{pq}(r-1)$ satisfy $G_{pq}(1)\cdots G_{pq}(N) = 1$ and give $M_r = G_{pp}(1)\cdots G_{pp}(r)$ when the line is restored ( $q \to p$ ). In the superintegrable case $q = Vp$ — $V$ an automorphism of the rapidity curve with $x_q = x_p$ , $y_q = \omega y_p$ , $\mu_q = \mu_p$ — Baxter (2005) proved the conjecture (under an analyticity assumption of the same type as in the free-energy derivation), obtaining $G_{pq}(r) = C_r\,S(t_p)^{\epsilon(r)}$ with $C_r = k^{(N+1-2r)/N^2}$ , $\epsilon(0) = \epsilon(N) = 1-N$ and $\epsilon(r) = 1$ otherwise; here $S$ is a function of the single variable $t_p$ , living on the branched $t_p$ -plane, normalized by $S(0) = 1$ and defined by a contour integral in Baxter (2005). For $N=3$ the rapidity admits a hyperelliptic parametrization by variables $(z_p, w_p)$ , tied to $(x_p, y_p, \mu_p)$ through elliptic-like functions $h, \varphi$ ; Baxter (2006) showed one must use his 'alternative' (section-5) parametrization, in which the curve automorphisms $R, S, V, M$ act by $z_{Rp} = -xw_p$ , $w_{Rp} = w_p/z_p$ , $z_{Sp} = -1/(xw_p)$ , $w_{Sp} = -1/(xz_p)$ , $z_{Vp} = -1/w_p$ , $w_{Vp} = z_p/w_p$ , while the automorphism $M$ (which multiplies $\mu_p$ by $\omega$ , leaving $x_p, y_p, t_p$ fixed) also fixes $z_p, w_p$ . Analytic continuation of $z_p, w_p$ at fixed $t_p$ takes values $x^m z_{p_0}^{\pm1}$ , $x^n w_{p_0}^{\pm1}$ with $m + n \equiv 0 \pmod 3$ on even sheets and $\equiv 1 \pmod 3$ on odd sheets; the Riemann sheet graph is a honeycomb lattice. The main result of Baxter (2006) is $(x_p^{-1/3}S(t_p))^3 = k^{-1/3}G(-1/w_p)/G(z_p)$ with $G(z) = F(\omega z)/F(\omega^2 z)$ and $F(z) = \prod_{j\ge1}(1-x^j z)^j/(1-x^j z^{-1})^j$ — infinite products generalizing elliptic functions in that factors carry the power $j$ . The $R^6$ obstruction: the automorphism rules give $z_{R^6p} = z_p$ and $w_{R^6p} = w_p$ , yet $G_{R^6p,q}(r) \neq G_{pq}(r)$ and $G_{p,R^6q}(r) \neq G_{pq}(r)$ ; hence $G_{pq}(r)$ cannot be a single-valued function of $z_p, w_p, z_q, w_q$ . However the combination $L_{pq}(r) = G_{pq}(r)G_{Rq,Rp}(r)$ , introduced in Baxter (1998), satisfies closed functional relations ( $L_{R^{2N}p,q}(r) = (-1)^{N-1}L_{pq}(r)$ , $L_{qp}(r) = 1/L_{pq}(N-r+1)$ , $L_{Mp,q}(r) = L_{p,M^{-1}q}(r) = L_{pq}(r+1)$ , $L_{pq}(1)\cdots L_{pq}(N) = 1$ ) and is unchanged by $p \to R^6p$ and $q \to R^6q$ , so it may admit such a representation. One further necessary condition follows from these same ingredients: $M$ also fixes $z_p, w_p$ for each rapidity, while the 1998 relations give $L_{p,M^{-1}q}(r) = L_{pq}(r+1)$ , which differs from $L_{pq}(r)$ in general — so at fixed $r$ even $L_{pq}(r)$ is not a single-valued function of $z_p, w_p, z_q, w_q$ alone. The four hyperelliptic variables leave the $\mu$ -branch of each rapidity undetermined (the choice among $p, Mp, M^2p$ ), and an explicit answer must carry that residual monodromy by explicit factors — fixed algebraic functions of the rapidity coordinates, of the same kind as the prefactor $x_p^{-1/3}$ carried by the proven single-rapidity identity — with all essential rapidity dependence residing in the single-valued meromorphic part.

Problem Statement

Let $p, q$ be arbitrary independent rapidities of the $N=3$ integrable chiral Potts model, and let $G_{pq}(r)$ ( $r = 1, \ldots, N$ with $N = 3$ ; $G_{pq}(1)\cdots G_{pq}(N) = 1$ ) be the generalized order-parameter ratios of the broken rapidity line method, $G_{pq}(r) = \tilde F_{pq}(r)/\tilde F_{pq}(r-1)$ , where $\tilde F_{pq}(r)$ is the spin correlation at the broken rapidity line. Known: (i) in the superintegrable case $q = Vp$ , $G_{pq}(r) = k^{(N+1-2r)/N^2}S(t_p)^{\epsilon(r)}$ with $\epsilon(0) = \epsilon(N) = 1-N$ , $\epsilon(r) = 1$ otherwise, and $(x_p^{-1/3}S(t_p))^3 = k^{-1/3}G(-1/w_p)/G(z_p)$ with $G(z) = F(\omega z)/F(\omega^2 z)$ , $F(z) = \prod_{j\ge1}(1-x^j z)^j/(1-x^j z^{-1})^j$ ; (ii) $G_{pq}(r)$ cannot be a single-valued meromorphic function of the hyperelliptic variables $(z_p, w_p, z_q, w_q)$ of the section-5 parametrization, because $R^6$ fixes all four variables while changing $G$ ; (iii) the automorphism $M$ , which likewise fixes $z_p, w_p$ (of each rapidity) while sending $\mu \to \omega\mu$ , shifts $r$ through the 1998 relation $L_{p,M^{-1}q}(r) = L_{pq}(r+1)$ , so at fixed $r$ even $L_{pq}(r)$ is not a single-valued function of these four variables alone — the variables leave the $\mu$ -branch of each rapidity undetermined, and residual factors of the kind specified below must carry it, just as the proven identity (i) carries the prefactor $x_p^{-1/3}$ . Open question: can $L_{pq}(r) = G_{pq}(r)\,G_{Rq,Rp}(r)$ — the combination proposed by Baxter (2006), unchanged by $p \to R^6p$ and $q \to R^6q$ — be simply expressed for arbitrary independent $p, q$ in terms of the section-5 hyperelliptic variables, that is, as a single-valued meromorphic function of $z_p, w_p, z_q, w_q$ (and the nome $x$ ) times explicit residual monodromy factors, where these factors are fixed algebraic functions of the rapidity coordinates $(x_p, y_p, \mu_p)$ , $(x_q, y_q, \mu_q)$ — of the same kind as the prefactor $x_p^{-1/3}$ in (i) — which carry only the monodromy under the honeycomb sheet structure ( $z_p \to x^m z_p^{\pm1}$ , $w_p \to x^n w_p^{\pm1}$ , $m+n \equiv 0,1 \pmod 3$ ) and under the automorphism fibers ( $R^6$ , $M$ ) that fix the hyperelliptic variables, while all essential rapidity dependence resides in the single-valued meromorphic function (so a factorization with constant single-valued part, absorbing $L_{pq}(r)$ itself into the residual factor, is not admissible)? Construct such a representation of $L_{pq}(r)$ , or prove that $L_{pq}(r)$ admits none (for instance by showing that its monodromy under the honeycomb sheet structure and the automorphism fibers is incompatible with any such factorization). Any such representation must reduce to (i) at $q = Vp$ , give $G_{pp}(r) = k^{(N+1-2r)/N^2}$ and $M_r = k^{r(N-r)/N^2}$ when the line is restored, and satisfy the 1998 functional relations for $L_{pq}$ (including $L_{R^{2N}p,q}(r) = (-1)^{N-1}L_{pq}(r)$ , $L_{qp}(r) = 1/L_{pq}(N-r+1)$ , $L_{Mp,q}(r) = L_{p,M^{-1}q}(r) = L_{pq}(r+1)$ ).

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