One-determinant form factors for the Weyl generators $v_n$ and finite-lattice derivation of the order-parameter formulae in the Bazhanov–Stroganov and chiral Potts models

The Bazhanov–Stroganov (BS) construction realizes the integrable chiral Potts model through local L-operators acting on $p$-dimensional cyclic spaces tied together by clock (Weyl) operators $u_n$, $v_n$ with $v u=\omega u v$, $\omega=e^{2\pi i/p}$. Grosjean, Maillet and Niccoli (Ann. Henri Poincaré 16 (2015) 1103) built the separation-of-variables (SOV) solution of the BS/chiral Potts transfer matrices in finite inhomogeneous cyclic representations and derived determinant formulae for the form factors of $u_n^{-1}$, but for the generators $v_n$ they obtained only finite sums of determinants, and they left the homogeneous and thermodynamic limits — designed to prove the closed-form chiral Potts order parameter $M_r=(1-k^2)^{r(p-r)/(2p^2)}$ from the finite lattice in general representations — to future work. This problem asks for exactly that completion: single-determinant representations for the $v_n$ form factors, and a proof of the Albertini–McCoy–Perk–Tang order-parameter formula for the general homogeneous $Z_p$ chiral Potts model from finite-lattice SOV matrix elements, free of the analyticity assumptions entering Baxter's broken-rapidity derivation and of superintegrable-only structures, reproducing the superintegrable results and the $p=2$ Ising magnetization as checks. No published work addresses either core, so both remain open.

Background

Two-dimensional integrable lattice models are statistical-mechanical systems whose transfer matrix — the matrix whose trace generates the partition function — commutes with itself for all values of its spectral parameters, which is the structural property making exact solution possible. The Ising model ($p=2$) is the archetype; the chiral Potts model is its $p$-state generalization, integrable for Boltzmann weights built from variables living on a high-genus algebraic curve (the spectral curve), which is why the determinant technologies that solve six-vertex-type models do not transfer to it. In the cyclic gauge the chiral Potts transfer matrix can be written à la Bazhanov and Stroganov as a trace of a product of local L-operators acting on $p$-dimensional quantum spaces, chained together by clock (Weyl) operators $u$, $v$ satisfying $v u=\omega u v$ with $\omega=e^{2\pi i/p}$; the local operators of the associated quantum chain are built from the site generators $u_n$, $v_n$ of these local Weyl algebras. Correlation functions of integrable models are computed through form factors, i.e. matrix elements of local operators between eigenstates of the transfer matrix. Grosjean, Maillet and Niccoli constructed the separation-of-variables (SOV) representation for the BS/chiral Potts transfer matrices — Sklyanin's method, in which the transfer-matrix eigenproblem is reduced to scalar equations in single separated variables so that eigenstates and matrix elements become explicit — proved completeness of the SOV spectrum on finite chains in inhomogeneous cyclic representations (cyclic meaning that the local clock powers satisfy $u_n^p=v_n^p=1$), and obtained one-determinant formulae for the form factors of the generators $u_n^{-1}$ and of an elementary-operator basis; for the generators $v_n$ they obtained only finite sums of determinants, and they posed the search for one-determinant $v_n$ formulae as an open problem under investigation. The second target is the order parameter of the homogeneous chiral Potts quantum chain: the spontaneous magnetization $M_r=\langle \text{g.s.}|u_1^r|\text{g.s.}\rangle/\langle \text{g.s.}|\text{g.s.}\rangle$, $r=1,\dots,p-1$, i.e. the normalized ground-state expectation value of the $r$-th power of the clock operator at site 1, with the closed form $M_r=(1-k^2)^{r(p-r)/(2p^2)}$, where the modulus $k$ (with $k^2+k'^2=1$) parametrizes the chiral Potts spectral curve. This formula was conjectured in 1989 by Albertini, McCoy, Perk and Tang from finite-lattice series expansions. Baxter derived it in 2005 for the general integrable model from Jimbo–Miwa–Nakayashiki functional equations for broken rapidity lines (rapidity being the chiral Potts name for the spectral parameter, a lattice line being 'broken' when the two sides of a cut through the lattice carry different rapidities on it), explicitly subject to analyticity assumptions that the functional relations alone cannot replace; in the superintegrable chiral Potts chain — the special subcase introduced by von Gehlen and Rittenberg, solvable by a generalized Onsager algebra (the operator algebra underlying the Ising solution) — the formula was proven on the finite lattice through Baxter's determinant $D_{PQ}$ and factorized spin matrix elements (Iorgov et al.); Au-Yang and Perk re-derived the general-model result in 2011 from $Z$-invariance (invariance of the partition function under local star-triangle rearrangements of the planar lattice) combined with superintegrable eigenvectors and their inner products, the key generating-function identity being proved in their companion preprint. What is missing to this day is the proof from finite-lattice matrix elements in general cyclic representations — precisely the route that the SOV determinant formulae were designed to open, encouraged by the fact that the matrix element of $M_{p-1}$ already admits a simple determinant form in that formalism.

Problem Statement

Complete the separation-of-variables (SOV) form-factor solution theory of the Bazhanov–Stroganov and chiral Potts models down to physical observables, as posed at the end of Grosjean–Maillet–Niccoli (Ann. Henri Poincaré 16 (2015) 1103–1153, arXiv:1309.4701). The problem has two linked deliverables serving this single objective. (1) Determinantal form factors for the generators $v_n$: derive single-determinant representations for the form factors of the Weyl generators $v_n$ of the local Weyl algebras between SOV eigenstates of the finite inhomogeneous Bazhanov–Stroganov/chiral Potts transfer matrices, in the same generality in which one-determinant formulae were obtained there for $u_n^{-1}$ (at present the $v_n$ form factors are known only as finite sums of determinants). (2) Order parameter from the finite lattice: starting from these finite-lattice SOV determinant matrix elements, perform the homogeneous limit and then the thermodynamic limit in general cyclic representations, and prove the closed-form order-parameter formula

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

for the general homogeneous $Z_p$ chiral Potts model — without restricting to the superintegrable subcase and without inputting unverified analyticity assumptions — with the derivation reproducing the known superintegrable results and the $p=2$ Ising spontaneous magnetization as limiting checks. A third item in the source's outlook — extending the SOV determinant program to additional integrable models and representations and systematizing its interplay with Onsager-algebra and algebraic-Bethe-ansatz frameworks — is a program direction without a completion boundary and is context for this problem, not a deliverable of it.

References

1. N. Grosjean, J.-M. Maillet, G. Niccoli, 'On the form factors of local operators in the Bazhanov–Stroganov and chiral Potts models', Annales Henri Poincaré 16 (2015) 1103–1153. DOI: 10.1007/s00023-014-0358-9. arXiv:1309.4701. https://arxiv.org/abs/1309.4701
2. G. Albertini, B. M. McCoy, J. H. H. Perk, S. Tang, 'Excitation spectrum and order parameter for the integrable N state chiral Potts model', Nuclear Physics B 314 (1989) 741–763. DOI: 10.1016/0550-3213(89)90415-X. https://doi.org/10.1016/0550-3213(89)90415-X
3. R. J. Baxter, 'Derivation of the order parameter of the chiral Potts model', Physical Review Letters 94 (2005) 130602. DOI: 10.1103/PhysRevLett.94.130602. arXiv:cond-mat/0501227. https://arxiv.org/abs/cond-mat/0501227
4. R. J. Baxter, 'The order parameter of the chiral Potts model', Journal of Statistical Physics 120 (2005) 1–36. DOI: 10.1007/s10955-005-5534-3. arXiv:cond-mat/0501226. https://arxiv.org/abs/cond-mat/0501226
5. H. Au-Yang, J. H. H. Perk, 'Quantum loop subalgebra and eigenvectors of the superintegrable chiral Potts transfer matrices', Journal of Physics A 44 (2011) 025205. DOI: 10.1088/1751-8113/44/2/025205. arXiv:0907.0362. https://arxiv.org/abs/0907.0362
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7. H. Au-Yang, J. H. H. Perk, 'Superintegrable chiral Potts model: proof of the conjecture for the coefficients of the generating function G(t,u)', arXiv:1108.4713 (2011). https://arxiv.org/abs/1108.4713
8. R. J. Baxter, 'Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ', Journal of Physics A 43 (2010) 145002. DOI: 10.1088/1751-8113/43/14/145002. arXiv:0912.4549. https://arxiv.org/abs/0912.4549
9. N. Iorgov, S. Pakuliak, V. Shadura, Yu. Tykhyy, G. von Gehlen, 'Spin operator matrix elements in the superintegrable chiral Potts quantum chain', Journal of Statistical Physics 139 (2010) 743–768. DOI: 10.1007/s10955-010-9972-1. arXiv:0912.5027. https://arxiv.org/abs/0912.5027
10. G. von Gehlen, V. Rittenberg, 'Zn-symmetric quantum chains with an infinite set of conserved charges and Zn zero modes', Nuclear Physics B 257 (1985) 351–370. DOI: 10.1016/0550-3213(85)90350-5. https://doi.org/10.1016/0550-3213(85)90350-5
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