The zero-field (symmetric) eight-vertex model on the square lattice — Baxter's elliptic generalization of the six-vertex ice-type models — has, besides its exactly known free energy, two order parameters in its ordered phases: the spontaneous magnetization and the spontaneous polarization. Baxter and Kelland conjectured in 1974 an infinite-product formula for the spontaneous polarization; the physics-level derivations subsequently found — via corner-transfer-matrix methods, difference equations, free-field/tail-operator constructions, and Sklyanin-algebra trace functionals — all rest on unproved structural assumptions, most notably the identification of the corner-transfer-matrix spectrum. Only the trigonometric limit — the staggered polarization of the six-vertex F-model (the antiferroelectric six-vertex model) — was proved, by Baxter in 1973. Through F. Y. Wu's 1975 relations the same formula fixes the sublattice-B spontaneous-magnetization exponent of the Hintermann–Merlini model (the three-spin Ising model on the Union Jack lattice), a prediction confirmed numerically by Ding et al. (2013), who emphasize that the polarization conjecture remains unproved. The problem posed here is to give a mathematically rigorous proof — or a rigorous refutation — of the Baxter–Kelland spontaneous-polarization formula for the zero-field eight-vertex model in its ordered phases, including the construction of the infinite-volume Gibbs measure and of the order parameter; Baxter's rigorously proved trigonometric (F-model) limit at q→0q\to0 is the consistency limit the formula must reproduce, not a separate mandatory deliverable.

Background

The eight-vertex model (Baxter, 1971) places arrows on the edges of the square lattice with the rule that an even number of arrows points into each site, allowing the eight vertex configurations with weights a,b,c,da,b,c,d; in the zero-field (symmetric) case the weights are invariant under global arrow reversal. Baxter solved its free energy and showed that its critical exponents vary continuously with a parameter uu (0<u≤π0<u\leq\pi). In the ordered regimes the model has two symmetry-breaking order parameters: a spin-type spontaneous magnetization and an electric-type spontaneous polarization (the staggered expectation of an arrow variable in the antiferroelectric regime, the uniform one in the ferroelectric regime). Barber and Baxter conjectured the spontaneous magnetization in 1973, and Stewart Kelland and Baxter the spontaneous polarization in 1974; the latter conjecture states that the polarization equals the infinite product P0=(−q;q)∞2(x2;x2)∞2(q;q)∞2(−x2;x2)∞2P_0=\frac{(-q;q)_\infty^2(x^2;x^2)_\infty^2}{(q;q)_\infty^2(-x^2;x^2)_\infty^2}, where (a;p)∞=∏n≥0(1−apn)(a;p)_\infty=\prod_{n\geq0}(1-ap^n) and qq (the elliptic nome) and xx are the standard parameters of the zero-field eight-vertex weights (0<q1/2<x<10<q^{1/2}<x<1 in the main antiferroelectric region; the ferroelectric regime is described by the same product under the weight identification aF=ca_F=c, bF=db_F=d, cF=ac_F=a, dF=bd_F=b and the corresponding parametrization given by Lashkevich (1996)). Baxter's corner-transfer-matrix (CTM) method — the partition function of a quadrant of the lattice, written as a transfer matrix built by rotating wedges about a central site, which for solvable models factorizes into commuting local factors — was introduced in 1976 and treats the magnetization at physics level; Baxter's retrospectives count the polarization among the results of this program, his 2006 account noting that the method worked readily for the magnetization (a single-site correlation) but had to be supplemented, for the polarization (a single-edge correlation), by the broken-rapidity-line technique of Jimbo, Miwa and Nakayashiki. However, the CTM route contains a load-bearing step that Baxter himself formulates as an expectation: the CTM eigenvalue spectrum, verified in the low-temperature limit, is asserted to persist throughout the ordered phase by a continuity argument that has never been made rigorous. Later rederivations did not close the gap: Jimbo, Miwa and Nakayashiki (1993) recovered the formula from difference equations of reduced qqKZ type (the elliptic analogue of the quantum Knizhnik–Zamolodchikov equations satisfied by correlation functions) but explicitly presented the derivation as heuristic, stating that the formula was conjectured by Baxter and Kelland and proved by Baxter only in the trigonometric case q=0q=0 (the six-vertex F-model, i.e. the antiferroelectric six-vertex model, 1973; Baxter's 2006 retrospective informally describes the 1993 derivation as a proof, but the Jimbo–Miwa–Nakayashiki paper itself calls its argument heuristic); Lashkevich (1996) and Lashkevich and Pugai (1998) reproduced the formula, together with the CTM spectrum, from a free-field (tail-operator) representation that they themselves describe as proposed; and the Boos–Jimbo–Miwa–Smirnov–Takeyama formulation of eight-vertex correlation functions as traces on the Sklyanin algebra (the elliptic quantum algebra generated by the eight-vertex R-matrix relations) remains a conjectural Ansatz. The formula is nevertheless universally trusted: it underlies Wu's exact results for the Hintermann–Merlini model — the Ising model on the Union Jack (centered-square) lattice with pure three-spin couplings, introduced by Hintermann and Merlini in 1972 and mapping to the symmetric eight-vertex model — where the two sublattice magnetizations obey MA∝∣T−Tc∣π/16uM_A\propto|T-T_c|^{\pi/16u} (via the eight-vertex magnetization) and MB∝∣T−Tc∣(π−u)/4uM_B\propto|T-T_c|^{(\pi-u)/4u} (via the conjectured polarization), and it passed the high-precision finite-size-scaling test of Ding et al. (2013).

Problem Statement

Provide a mathematically rigorous proof — or a rigorous refutation — of the Baxter–Kelland formula for the spontaneous polarization of the zero-field (symmetric) eight-vertex model on the square lattice. For the model with weights in the standard theta-function parametrization of the zero-field eight-vertex model (nome qq and parameter xx, in the conventions of the cited Jimbo–Miwa–Nakayashiki (1993) paper, in whose notation the product formula below is written), the claim to be established is that in the ordered phases the spontaneous polarization equals P0=(−q;q)∞2(x2;x2)∞2(q;q)∞2(−x2;x2)∞2P_0=\frac{(-q;q)_\infty^2(x^2;x^2)_\infty^2}{(q;q)_\infty^2(-x^2;x^2)_\infty^2}, where (a;p)∞=∏n≥0(1−apn)(a;p)_\infty=\prod_{n\geq0}(1-ap^n); in the main antiferroelectric region 0<q1/2<x<10<q^{1/2}<x<1 this is the staggered polarization, defined as the spontaneous (ground-state-sector) infinite-volume limit P0=G+−−(x−1)/G+−+(x−1)P_0=G^-_{+-}(x^{-1})/G^+_{+-}(x^{-1}) of the edge-spin order parameter in the sense of the cited Jimbo–Miwa–Nakayashiki (1993) paper; and in the ferromagnetic region F1F_1 (a>b+c+da>b+c+d), under the weight identification aF=ca_F=c, bF=db_F=d, cF=ac_F=a, dF=bd_F=b and the parametrization (3.4) of Lashkevich (1996), with 0<q1/2<x<ζ<10<q^{1/2}<x<\zeta<1, the same product gives the uniform polarization ⟨ε⟩\langle\varepsilon\rangle — a branch which, by Lashkevich's exact staggered-spin substitution, follows from the antiferroelectric-region formula rather than constituting an independent additional target. The phase list used here (Lashkevich (1996), eq. (1.2)) contains two further ordered regions — the second antiferroelectric region (d>a+b+cd>a+b+c) and the second ferromagnetic region (b>a+c+db>a+c+d) — which, like the branch F1F_1 above, are related to the main antiferroelectric region by the weight relabelings the cited parametrizations rest on and are therefore not independent additional targets. The main antiferroelectric region 0<q1/2<x<10<q^{1/2}<x<1 is accordingly the sole mandatory target, and a rigorous refutation must address the claim in that region. A complete answer must (i) construct the infinite-volume Gibbs measure(s) (equivalently, the ground-state-sector infinite-volume limit) in the ordered regimes and define the polarization as a spontaneous symmetry-breaking order parameter of that state; and (ii) prove the product formula for that limit, with full control of the thermodynamic limit. Baxter's rigorously proved staggered polarization of the six-vertex F-model at q→0q\to0 is the trigonometric limit the formula must reproduce; reproducing it is a consistency expectation, not a separate mandatory deliverable. As an immediate corollary the proof puts Wu's sublattice-B magnetization exponent βb=(π−u)/4u\beta_b=(\pi-u)/4u of the Hintermann–Merlini model on rigorous footing. A related, explicitly secondary direction flagged by the source literature but not required for completion is the critical behaviour of the Hintermann–Merlini model in the regimes where π/2u\pi/2u is an integer, where the generic singularity ∣T−Tc∣π/u|T-T_c|^{\pi/u} degenerates and logarithmic or anomalous corrections are expected; these regimes were excluded from the numerical analyses and remain unanalyzed.

References

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