Homogeneous and thermodynamic limits of the chiral Potts form-factor determinants: first-principles proof of the order parameter of the general homogeneous $Z_p$ chiral Potts model

The $Z_p$ chiral Potts model — the most general known solution of the star-triangle (Yang–Baxter) relation, reducing to the Ising model at $p=2$ — has order parameters $M_r$ whose closed form $M_r=(1-k'^2)^{r(p-r)/(2p^2)}$ was conjectured in 1983–1989, derived by Baxter (2005) for the general model under unproven analyticity assumptions, and proven from the finite lattice only on the superintegrable subvariety. Grosjean–Maillet–Niccoli (2013, published 2015) expressed finite-size, inhomogeneous chiral Potts matrix elements — including the order-parameter element $M_{p-1}$ — as determinants via Sklyanin's separation of variables, and announced a future analysis of the homogeneous and thermodynamic limits aiming to prove the order-parameter formulae for general homogeneous $Z_p$ representations from the finite lattice. That announced program remains open. The open problem is the limit analysis of these determinant representations and the resulting unconditional, general-representation proof of the order-parameter formulae, together with the auxiliary goal of single-determinant formulae for the $v_n$ generators.

Background

The $Z_p$ chiral Potts model is a two-dimensional lattice model in which each site carries a spin taking $p$ values (a clock variable $Z$ with $Z^p=1$), with Boltzmann weights invariant under the cyclic group $Z_p$. Unlike the Ising model, its weights depend on two points ('rapidities') lying on a high-genus algebraic curve, and it is the most general known solution of the star-triangle relation — the form of the Yang–Baxter equation for edge-interacting spin models. Its transfer matrix also generates a one-dimensional quantum chain of $p$-state clock variables. In the ordered (ferromagnetic) phase the $Z_p$ symmetry is spontaneously broken, and the order parameters (the spontaneous-magnetization components) are $M_r=\lim_{L\to\infty}\langle Z^r\rangle,\qquad r=1,\dots,p-1,$ the thermodynamic-limit ground-state expectations of the $r$-th power of the local clock variable.

Howes–Kadanoff–den Nijs (1983) evaluated $M_1$ and $M_2$ for $p=3$ by series expansion in the modulus and conjectured the closed form; Albertini–McCoy–Perk–Tang (1989) extended the series analysis to general $p$ and conjectured, for all $r$, $M_r=(1-k'^2)^{r(p-r)/(2p^2)},$ where $k'$ is the modulus of the model's rapidity curve ($0<k'<1$ in the ordered phase, $k'\to 1$ at criticality). In complementary-modulus notation $k^2+k'^2=1$ this is $M_r=k^{r(p-r)/p^2}$, the form used by Baxter; Grosjean–Maillet–Niccoli write it as $M_r=(1-k^2)^{r(p-r)/(2p^2)}$, their $k$ being Baxter's $k'$. For $p=2$ the formula reduces to the Onsager–Yang spontaneous magnetization of the Ising model.

Baxter (2005) derived the conjecture for the general chiral Potts model (generic rapidities), generalizing the broken-rapidity-line functional-relation method introduced by Jimbo–Miwa–Nakayashiki for the eight-vertex model. His derivation is, by his own statement, 'subject to a plausible analyticity assumption' — two numbered assumptions in §7 of the long paper — concerning analytic properties of correlation functions with broken rapidity lines; these are supported by small-modulus evidence but unproven. Unconditional proofs starting from finite lattices exist only on the superintegrable subvariety, the special self-dual class of representations where the chiral Potts quantum chain satisfies quartic commutation relations implying an Onsager-algebra symmetry (the Lie-algebraic structure underlying Onsager's 1944 Ising solution). There, Baxter's generalized Onsager algebra led to a determinantal formula for the spontaneous magnetization, and the Au-Yang–Perk eigenvector constructions, together with the Iorgov–Pakuliak–Shadura–Tykhyy–von Gehlen proof of the conjectured factorized form of the spin-operator matrix elements, gave a second finite-lattice route. Au-Yang–Perk (2011) used $Z$-invariance (insensitivity of the model to deformations of the rapidity lattice) to reduce pair correlations of the general model to superintegrable eigenvectors, obtaining the order parameter 'as a sum almost identical to the one given by Baxter' — still importing Baxter's final evaluation, and resting on eigenvector constructions whose algebraic underpinnings are proven only for some states. Thus the formula's value is settled knowledge in the community, while a derivation that is both unconditional and valid for general representations, starting from the finite lattice, is missing.

Grosjean–Maillet–Niccoli (2013, published 2015) attacked the model with Sklyanin's quantum separation of variables (SOV), a framework in which the commuting transfer matrix acts as multiplication by the eigenvalues of an auxiliary separate-variable operator, so that scalar products of eigenstates become determinants. Working from the Bazhanov–Stroganov model (the six-vertex Yang–Baxter algebra with general cyclic representations, from which the chiral Potts transfer matrices are built) on finite inhomogeneous lattices (lattices whose sites carry independent rapidity parameters), they proved single-determinant formulae for scalar products of separate states, solved the quantum inverse problem — every local operator is reconstructed as a polynomial in the separate variables — and showed that the form factors (normalized matrix elements between transfer-matrix eigenstates) of any local operator are finite sums of determinants, with single-determinant formulae for a basis of elementary operators and for the generators $u_n^{-1}$ of the local Weyl algebra. The Weyl generators satisfy $u_n^p=v_n^p=1$ and $u_nv_n=\omega\,v_nu_n$ with $\omega$ a primitive $p$-th root of unity, and $u_1^{-1}$ reduces to the clock-spin variable whose ground-state expectation gives the order parameter: $M_{p-1}=\frac{\langle g.s.|u_1^{-1}|g.s.\rangle}{\langle g.s.|g.s.\rangle},$ for which their approach yields a simple determinant formula. They observe that these determinant formulae are exactly what 'will reproduce the chiral Potts order parameters in the thermodynamic limit', state that the known formulae were 'proven with techniques working only in the super-integrable case but they are expected to be true for the general homogeneous $Z_p$ chiral Potts models', and announce: 'In a future paper, we will analyze the homogeneous and thermodynamic limits focusing the attention on the derivation of the order parameter formulae for the general homogeneous $Z_p$ chiral Potts models.' They also pose the auxiliary problem of simplifying the form factors of the $v_n$ generators — currently finite sums of determinants — to single determinants.

Problem Statement

Analyze the homogeneous and thermodynamic limits of the finite-lattice SOV determinant representations of $Z_p$ chiral Potts matrix elements, and prove from the finite lattice the order-parameter formulae $M_r=(1-k'^2)^{r(p-r)/(2p^2)},\qquad r=1,\dots,p-1,$ for general homogeneous $Z_p$ chiral Potts representations — that is, without restricting to the superintegrable subvariety and without postulating the unproven analyticity properties on which Baxter's broken-rapidity-line derivation rests. Concretely, this program entails constructing the homogeneous limit of the separate-variable basis and of the determinant formulae, controlling the coalescence of the inhomogeneity parameters that defines the separate-variable spectrum; isolating the ground-state sector for general cyclic representations and taking the thermodynamic limit of the relevant matrix elements — in particular the order-parameter element $M_{p-1}=\langle g.s.|u_1^{-1}|g.s.\rangle/\langle g.s.|g.s.\rangle$ (where g.s. denotes the ground state), which admits a single-determinant formula — with controlled interchange of the homogeneous and infinite-volume limits; and evaluating the limit in closed form, recovering the conjectured expression, with the $p=2$ (Onsager–Yang Ising) and superintegrable specializations reproduced as consistency checks. As an auxiliary goal of the same determinant program, obtain single-determinant representations for the form factors of the Weyl generators $v_n$ (currently available only as finite sums of determinants), a simplification that would also lighten the subsequent limit analysis. The common objective is to make the SOV determinant representation of chiral Potts matrix elements yield the physics of the homogeneous model — its order parameters first, and by the same apparatus its correlation functions, which are beyond current techniques for the general model. Proving the order-parameter formulae for all $r=1,\dots,p-1$ in general homogeneous $Z_p$ representations resolves the task; the single-determinant form factors of the $v_n$ generators and the extension to correlation functions are optional extensions and are not required for completion.

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