Single-valued hyperelliptic representation of the automorphism-invariant order-parameter combination $L_{pq}(r)$ of the chiral Potts model for arbitrary rapidities

The $N$-state chiral Potts model is the prototypical exactly solvable two-dimensional lattice model without the rapidity-difference property: its Boltzmann weights satisfy the star-triangle (Yang-Baxter) relation, but correlation functions depend separately on two rapidities $p,q$ that are points on an algebraic curve of genus greater than two when $N>2$. Baxter (2005) derived the order parameters $M_r$ by solving, through broken-rapidity-line functional relations, only the superintegrable specialization $q=Vp$, where the generalized order-parameter ratios $G_{pq}(r)$ reduce to a single function $S(t_p)$ of one variable; in his 2006 paper he showed for $N=3$ that $x_p^{-1}S(t_p)^3$ is a single-valued meromorphic function of two hyperelliptic variables $(z_p,w_p)$, the first such representation of a thermodynamic quantity of the $N>2$ chiral Potts model. He left open whether $G_{pq}(r)$ for arbitrary rapidities admits a comparable representation. A concrete obstruction exists: the map $p\mapsto R^6p$ fixes $(z_p,w_p)$ while changing $G_{pq}(r)$, so $G_{pq}(r)$ is not single-valued in the hyperelliptic variables; the invariant combination $L_{pq}(r)=G_{pq}(r)G_{Rq,Rp}(r)$ is the natural candidate. No single-valued hyperelliptic representation of $G_{pq}(r)$, or of the invariant $L_{pq}(r)$, for arbitrary rapidities is known.

Background

The chiral Potts model is an $N$-state model of equilibrium statistical mechanics: spins $\sigma_i$ taking values $0,\ldots,N-1$ sit on the sites of the square lattice, and the Boltzmann weight of an edge depends on the spin difference $\sigma_i-\sigma_j \bmod N$, on two parameters $p,q$ called rapidities, and on a temperature-like modulus $k$ with $k^2+k'^2=1$ ($k'$ small at low temperature). The weights satisfy the star-triangle (Yang-Baxter) relation, which makes the model exactly solvable; the free energy was obtained in 1988. The model is, however, the standard example of a solvable model without the rapidity-difference property: thermodynamic quantities depend on $p$ and $q$ separately, not only on a difference $q-p$. The rapidities themselves are not free numbers but points on an algebraic curve of genus greater than two when $N>2$; for $N=2$ the curve is elliptic and everything reduces to Jacobi elliptic functions of a single uniformizing variable, whereas for $N>2$ no single-valued one-variable parametrization of the curve exists. One can instead parametrize using hyperelliptic functions, the higher-genus analogue of elliptic functions, which for $N=3$ can be written as infinite products resembling Jacobi elliptic functions of two related arguments.

The order parameters are $M_r=\langle\omega^{r\sigma_1}\rangle$ with $\omega=e^{2\pi i/N}$ and $\sigma_1$ a spin deep inside the lattice. Albertini, McCoy, Perk and Tang conjectured in 1989 that $M_r=k^{r(N-r)/N^2}$ (for $N=2$ this is the Onsager-Yang result $M_1=k^{1/4}$); Baxter derived it in 2005. The derivation used the broken rapidity line method of Jimbo, Miwa and Nakayashiki: one breaks the medial rapidity line passing through the central spin and assigns rapidity $p$ to the left half and $q$ to the right half. The resulting generalized correlations $\tilde F_{pq}(r)=\langle\omega^{r\sigma_1}\rangle$ then satisfy $\tilde F_{pp}(r)=M_r$, and their ratios $G_{pq}(r)=\tilde F_{pq}(r)/\tilde F_{pq}(r-1)$ obey functional relations under four automorphisms $R,S,V,M$ of the rapidity curve, where a rapidity is the variable set $p=\{x_p,y_p,\mu_p,t_p\}$ with $t_p=x_py_p$ and $x_p^N+y_p^N=k(1+x_p^Ny_p^N)$, and for instance $R:\{x_p,y_p,\mu_p,t_p\}\mapsto\{y_p,\omega x_p,1/\mu_p,\omega t_p\}$ and $V:\{x_p,y_p,\mu_p,t_p\}\mapsto\{x_p,\omega y_p,\mu_p,\omega t_p\}$. Two sample relations are $G_{Rp,Rq}(r)=1/G_{pq}(N-r+1)$ and $G_{Mp,q}(r)=G_{p,M^{-1}q}(r)=G_{pq}(r+1)$. These relations do not determine $G_{pq}(r)$: the missing ingredient is analyticity information on the many-sheeted Riemann surface of the curve, and for $N>2$ there is no uniformizing substitution. Baxter bypassed the difficulty in 2005 by solving only the superintegrable specialization $q=Vp$ (that is, $x_q=x_p$, $y_q=\omega y_p$, $\mu_q=\mu_p$), where $G_{pq}(r)=k^{(N+1-2r)/N^2}S(t_p)$ with $S(t_p)$ a single-valued, non-zero function of the single variable $t_p$, analytic in a cut plane and given by a definite integral.

Baxter's 2006 paper takes a further step for $N=3$. Using an alternative hyperelliptic parametrization in which a rapidity is described by two variables $(z_p,w_p)$ linked to $x_p,y_p$ by infinite-product relations $y_p/x_p=\omega h(z_p)=h(-1/w_p)=\omega^2 h(-w_p/z_p)$ (with $h$ a generalized elliptic function of nome $x$, the small parameter with $|x|<1$ in whose powers the infinite products are expanded), and under which the automorphisms act by $z_{Rp}=-xw_p$, $w_{Rp}=w_p/z_p$, and analogous rules for $S,V,M$, Baxter proved that $x_p^{-1}S(t_p)^3=k^{-1/3}G(-1/w_p)/G(z_p)$, where $G(z)=F(\omega z)/F(\omega^2 z)$ and $F(z)=\prod_{j\ge1}(1-x^jz)^j/(1-x^jz^{-1})^j$. The exponent $j$ on the factors makes $F$ a generalized elliptic function rather than an ordinary theta quotient; the proof is a Liouville-theorem argument on the lattice of Riemann sheets, which for $(z_p,w_p)$ forms a honeycomb. This was the first time a thermodynamic property of the $N>2$ chiral Potts model was expressed in hyperelliptic form.

The question left open concerns arbitrary rapidities. For general $p,q$ the function $G_{pq}(r)$ depends on both rapidities, and the candidate variables are the four hyperelliptic coordinates $(z_p,w_p,z_q,w_q)$. There is a concrete obstruction: since $t_{Rp}=\omega t_p$, the map $p\mapsto R^6p$ (for $N=3$) returns $t_p$ to itself and fixes $z_p$ and $w_p$; but $G_{R^6p,q}(r)\neq G_{pq}(r)$ and $G_{p,R^6q}(r)\neq G_{pq}(r)$ in general, so $G_{pq}(r)$ cannot be a single-valued function of $(z_p,w_p,z_q,w_q)$. The combination $L_{pq}(r)=G_{pq}(r)G_{Rq,Rp}(r)=G_{pq}(r)/G_{qp}(N-r+1)$, introduced in Baxter's 1998 functional-relations paper, is invariant under $p\mapsto R^6p$ and $q\mapsto R^6q$ and is the natural candidate for such a representation. For the wider program a companion obstruction is known: Baxter's companion paper of the same year applies the same parametrization to the free energy of the $\tau_2(t_q)$ model (an auxiliary exactly solved model, introduced by Bazhanov and Stroganov, whose transfer matrix is built from the chiral Potts rapidity variables), and in it he notes that extending hyperelliptic product representations beyond $N=3$ appears to require generalizations of hyperelliptic theta functions whose zeros have orders growing linearly with position on the sheet lattice, but that no such generalization is known.

Problem Statement

For the $N=3$ chiral Potts model, determine whether the generalized order-parameter ratios $G_{pq}(r)$ for arbitrary rapidities $p,q$ on the rapidity curve - or, since $G_{pq}(r)$ is provably not a single-valued function of the four hyperelliptic variables $(z_p,w_p,z_q,w_q)$ because the map $p\mapsto R^6p$ fixes $z_p,w_p$ while changing $G_{pq}(r)$, the automorphism-invariant combination $L_{pq}(r)=G_{pq}(r)\,G_{Rq,Rp}(r)=G_{pq}(r)/G_{qp}(N-r+1)$ - can be expressed as single-valued meromorphic functions of the hyperelliptic variables of the alternative parametrization, in the way achieved for the superintegrable specialization $q=Vp$ by the formula $x_p^{-1}S(t_p)^3=k^{-1/3}G(-1/w_p)/G(z_p)$, where $G(z)=F(\omega z)/F(\omega^2 z)$ is the single-variable function built from the product $F$ defined above (a different object from the ratios $G_{pq}(r)$). An affirmative answer supplies explicit formulae - for example finite products and quotients of generalized elliptic functions of single hyperelliptic arguments - valid for arbitrary rapidities $p,q$ and reducing correctly to the known cases $q=p$ (where $G_{pp}(1)\cdots G_{pp}(r)=M_r=k^{r(N-r)/N^2}$) and $q=Vp$. A negative answer also resolves the task: a proof that no such single-valued meromorphic representation of $L_{pq}(r)$ in $(z_p,w_p,z_q,w_q)$ exists is an equally valid resolution.

References

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