Chiral Potts order parameters from finite-lattice determinant representations: homogeneous and thermodynamic limits in general $Z_p$ representations, and one-determinant form factors for the generators $v_n$

The chiral Potts model — the $Z_p$-symmetric, star-triangle-solvable generalization of the Ising model whose rapidities live on high-genus algebraic curves — is the prominent exactly solved lattice model for which correlation functions remain essentially inaccessible in the general (non-superintegrable) case. Grosjean, Maillet and Niccoli (Ann. Henri Poincare 16 (2015) 1103) used Sklyanin's separation of variables to derive, for general cyclic representations on finite inhomogeneous chains, determinant representations for the form factors of all local operators of the associated Bazhanov-Stroganov ($\tau_2$) model: one-determinant formulae for the generators $u_n^{-1}$ and for an elementary-operator basis, but only finite sums of determinants for the generators $v_n$. They posed two linked open problems: (i) take the homogeneous and thermodynamic limits of these finite-lattice matrix elements so as to prove the order-parameter formulae $M_r=(1-k^2)^{r(p-r)/(2p^2)}$, $r=1,\dots,p-1$, for the general homogeneous $Z_p$ chiral Potts model directly from the finite lattice — these formulae having previously been established only through superintegrable-case techniques plus a $Z$-invariance reduction, or under unproven analyticity assumptions; and (ii) find one-determinant formulae for the $v_n$ form factors. Both problems remain open.

Background

The chiral Potts model is the $Z_p$-symmetric ($p$-state) generalization of the Ising model found in 1987-88 as a new solution of the star-triangle (Yang-Baxter) equations (the functional equations that make a two-dimensional lattice model exactly solvable): its Boltzmann weights are parameterized by points on a high-genus algebraic curve ("rapidities") rather than by difference variables, so the corner-transfer-matrix techniques (which compute correlations from the partition functions of quadrant sublattices) that solved correlation functions of the six-vertex and Ising models do not apply. The commuting family of transfer matrices defines an extensive set of conserved quantities, and the free energy is known exactly; correlation functions, however, are not. The model contains a distinguished "superintegrable" subcase in which the rapidities obey additional constraints, the local Hamiltonians generate an Onsager algebra (the Lie algebra underlying the exact solution of the two-dimensional Ising model), and the associated $Z_p$ quantum spin chain can be diagonalized in closed form.

The Bazhanov-Stroganov model (also called the $\tau_2$ model) interpolates between the six-vertex model and the chiral Potts model: its Lax operator (the local site matrix from which the transfer matrix is built as an ordered product over lattice sites) is built from local Weyl algebras with generators $u_n$, $v_n$ (site index $n=1,\dots,N$) satisfying $u_n v_m = q^{\delta_{n,m}} v_m u_n$ with $q=e^{2\pi i/p}$. Cyclic representations of these Weyl algebras — $p$-dimensional representations in which $u_n$ and $v_n$ act with period $p$ — at general (non-superintegrable) parameter values define the general chiral Potts representations, and the $\tau_2$ transfer matrix specializes to the chiral Potts transfer matrix on a two-dimensional subvariety of parameters.

The order parameters of the homogeneous $Z_p$ chiral Potts quantum chain are the normalized ground-state expectations $M_r=\langle g.s.|u_1^r|g.s.\rangle/\langle g.s.|g.s.\rangle$, $r\in\{1,\dots,p-1\}$, interpretable as spontaneous magnetizations of the spin chain. The closed form $M_r=(1-k^2)^{r(p-r)/(2p^2)}$, with $k$ the modulus entering the superintegrable constraint, was conjectured from series expansions by Howes, Kadanoff and den Nijs (1983, $p=3$) and by Albertini, McCoy, Perk and Tang (1989, general $p$), and derived by Baxter (2005) using the broken-rapidity-line method of Jimbo, Miwa and Nakayashiki together with two unproven analyticity assumptions of the kind standard in Yang-Baxter solvable models. Au-Yang and Perk (2011) later recovered the general-model magnetization by $Z$-invariance (a consequence of the star-triangle equation allowing correlation functions to be transplanted between integrable lattices), reducing general-model pair correlations to inner products of superintegrable ground-state eigenvectors and finishing with the superintegrable determinantal evaluations of Baxter (2009-10) and Iorgov, Pakuliak, Shadura, Tykhyy and von Gehlen (2010). Thus every existing route passes through the superintegrable case or through unproven analyticity assumptions; no evaluation of $M_r$ inside general representations is known.

Separation of variables (SOV) is a basis construction for quantum integrable models, due to Sklyanin and coworkers, in which the transfer matrix acts diagonally — wave functions are values of functions of one quantum separate variable per site — so that scalar products and form factors (matrix elements of local operators between transfer-matrix eigenstates) can be written as determinants. Grosjean and Niccoli (2012) proved completeness of the SOV Bethe equations (the finite algebraic equations whose solutions characterize the transfer-matrix eigenstates) for the $\tau_2$/chiral Potts models in general cyclic representations, and Grosjean, Maillet and Niccoli (2013; published 2015; the source of this problem) built the form-factor theory on it: a reconstruction of all local operators in terms of the separate variables, one-determinant formulae for the form factors of $u_n^{-1}$, of the related local generator $\alpha_{0,n}$, and of a basis of "elementary operators", finite sums of determinants for everything else — in particular for the generators $v_n$ — and, already at finite size and in the inhomogeneous regime, a simple determinant formula for the matrix element defining $M_{p-1}$. These results hold in general cyclic representations but only away from the homogeneous point and at finite lattice size.

Taking a general representation to the homogeneous chiral Potts chain requires coalescing the inhomogeneities, at which point the separate-variable spectra degenerate and the determinant formulae become singular; the thermodynamic limit $N\to\infty$ additionally requires controlling solutions of the chiral Potts Bethe equations in general representations, whose structure is not known in closed form. Because correlation functions decompose into spectral series of form factors, these two limits of the determinant matrix elements are the gateway to exact correlation functions of the general model — the central open target of the program, and the reason the source paper poses the problems recorded here.

Problem Statement

Starting from the finite-size, inhomogeneous SOV determinant representations for matrix elements of local operators in general cyclic representations of the Bazhanov-Stroganov/chiral Potts models obtained by Grosjean, Maillet and Niccoli (arXiv:1309.4701), solve the two linked problems that the paper's conclusions pose.

(1) Core — homogeneous and thermodynamic limits: perform the homogeneous limit (the inhomogeneity parameters coalesced to the values defining the homogeneous $Z_p$ chiral Potts quantum chain) and the thermodynamic limit ($N\to\infty$) of these determinant matrix elements, and derive from the finite lattice the order-parameter formulae

$M_r=\bigl(1-k^2\bigr)^{\,r(p-r)/(2p^2)},\qquad r\in\{1,\dots,p-1\},$

for the general homogeneous $Z_p$ chiral Potts model — that is, prove them from matrix elements computed inside general (non-superintegrable) representations, without imposing the superintegrable constraint, without reducing the computation to superintegrable data via $Z$-invariance, and without postulating Baxter-type analyticity assumptions.

(2) Linked companion — determinant simplification: obtain one-determinant formulae for the form factors of the Weyl generators $v_n$ in general cyclic representations, replacing their published expressions as finite sums of determinants, in the same way one-determinant formulae were obtained for $u_n^{-1}$ and for the elementary-operator basis.

The common objective of the two components is to complete the determinant-representation program for the chiral Potts model in general representations; the motivation the source gives for seeking simple one-determinant forms is their use as efficient tools for the computation of correlation functions through form-factor spectral series. An answer that re-derives $M_r$ for the general model by mapping it back to superintegrable eigenvectors (the Au-Yang-Perk route) does not meet the stated objective, since removing the restriction to superintegrable data is precisely what is asked. The formulae themselves are not in doubt — they are established in the literature — what is asked is their derivation from the finite-lattice determinant representations in general representations.

References

1. N. Grosjean, J.-M. Maillet, G. Niccoli, "On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models", Ann. Henri Poincare 16 (2015) 1103-1153. DOI 10.1007/s00023-014-0358-9, arXiv:1309.4701, https://arxiv.org/abs/1309.4701
2. N. Grosjean, G. Niccoli, "The $\tau_2$-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method", J. Stat. Mech. (2012) P11005. DOI 10.1088/1742-5468/2012/11/P11005, arXiv:1205.4614, https://arxiv.org/abs/1205.4614
3. H. Au-Yang, J. H. H. Perk, "Spontaneous magnetization of the integrable chiral Potts model", J. Phys. A: Math. Theor. 44 (2011) 445005. DOI 10.1088/1751-8113/44/44/445005, arXiv:1003.4805, https://arxiv.org/abs/1003.4805
4. R. J. Baxter, "The order parameter of the chiral Potts model", J. Stat. Phys. 120 (2005) 1-36. DOI 10.1007/s10955-005-5534-3, arXiv:cond-mat/0501226, https://arxiv.org/abs/cond-mat/0501226
5. R. J. Baxter, "Derivation of the order parameter of the chiral Potts model", Phys. Rev. Lett. 94 (2005) 130602. DOI 10.1103/PhysRevLett.94.130602, arXiv:cond-mat/0501227, https://arxiv.org/abs/cond-mat/0501227
6. S. Howes, L. P. Kadanoff, M. den Nijs, "Quantum model for commensurate-incommensurate transitions", Nucl. Phys. B 215 (1983) 169-208. DOI 10.1016/0550-3213(83)90212-2, https://doi.org/10.1016/0550-3213(83)90212-2
7. G. Albertini, B. M. McCoy, J. H. H. Perk, S. Tang, "Excitation spectrum and order parameter for the integrable N-state chiral Potts model", Nucl. Phys. B 314 (1989) 741-763. DOI 10.1016/0550-3213(89)90415-X, https://doi.org/10.1016/0550-3213(89)90415-X
8. H. Au-Yang, J. H. H. Perk, "Quantum loop subalgebra and eigenvectors of the superintegrable chiral Potts transfer matrices", J. Phys. A: Math. Theor. 44 (2011) 025205. DOI 10.1088/1751-8113/44/2/025205, arXiv:0907.0362, https://arxiv.org/abs/0907.0362
9. R. J. Baxter, "Some remarks on a generalization of the superintegrable chiral Potts model", J. Stat. Phys. 137 (2009) 798-813. DOI 10.1007/s10955-009-9778-1, arXiv:0906.3551, https://arxiv.org/abs/0906.3551
10. R. J. Baxter, "Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant $D_{PQ}$", J. Phys. A: Math. Theor. 43 (2010) 145002. DOI 10.1088/1751-8113/43/14/145002, arXiv:0912.4549, https://arxiv.org/abs/0912.4549
11. N. Iorgov, S. Pakuliak, V. Shadura, Yu. Tykhyy, G. von Gehlen, "Spin operator matrix elements in the superintegrable chiral Potts quantum chain", J. Stat. Phys. 139 (2010) 743-768. DOI 10.1007/s10955-010-9972-1, arXiv:0912.5027, https://arxiv.org/abs/0912.5027
12. K. Fabricius, B. M. McCoy, "Correlation functions for the three-state superintegrable chiral Potts spin chain of finite lengths", J. Phys. A: Math. Theor. 43 (2010) 085209. DOI 10.1088/1751-8113/43/8/085209, https://doi.org/10.1088/1751-8113/43/8/085209
13. M. Jimbo, T. Miwa, A. Nakayashiki, "Difference equations for the correlation functions of the eight-vertex model", J. Phys. A: Math. Gen. 26 (1993) 2199-2210. DOI 10.1088/0305-4470/26/9/015, arXiv:hep-th/9211066, https://arxiv.org/abs/hep-th/9211066
14. H. Zhu, "A symmetry formula for correlation functions in the superintegrable chiral Potts spin chain", arXiv:2603.27930 (2026), https://arxiv.org/abs/2603.27930
15. G. Niccoli, V. Terras, "On the gauge invariant boundary condition in open XXZ chains: SoV bases, elementary blocks and overlaps", arXiv:2608.22365 (2026), https://arxiv.org/abs/2608.22365