Checkerboard Z-invariant eight-vertex weights: completing the classification and going beyond the free-fermion manifold

Melotti (2021) constructed what seems to be the first set of checkerboard Z-invariant weights for the eight-vertex model, a two-modulus elliptic family $(k,l)$ lying entirely inside the free-fermion manifold, and explicitly left open whether this parametrization extends outside free fermion, noting that no complete parametrization of checkerboard Yang-Baxter solutions is known. The problem asks for a complete classification, up to face-wise gauge, of real standard checkerboard Z-invariant eight-vertex weights on lozenge graphs, in particular whether families outside the free-fermion manifold exist, with verified star-triangle identities and a proof of completeness; the existence question is answered either by an explicit family outside free fermion or by a rigidity theorem showing that the free-fermion condition is forced, and the complete parametrization is required in either case.

Background

The eight-vertex (8V) model lives on a quadrangulation $Q$, a planar graph whose faces are quadrilaterals: a configuration is a subset of the dual edges such that an even number of them meet at every face, giving eight allowed local patterns with real Boltzmann weights $A(f),B(f),C(f),D(f)$ attached to each face $f$. The six-vertex (6V) model is the subcase $D=0$. The weights are called free-fermion when $A(f)^2+B(f)^2=C(f)^2+D(f)^2$ at every face, the condition introduced by Fan and Wu (1970); at this point the model becomes exactly solvable by Pfaffian and determinant methods, like the Ising model. Eight-vertex configurations are in bijection with perfect matchings (dimers) on a decorated graph obtained by replacing each face of $Q$ with a small gadget; this decorated graph is in general non-bipartite, whereas free-fermion 6V models correspond to bipartite dimers, for which a far-reaching analytic theory exists (Kasteleyn matrices, i.e. signed adjacency matrices whose Pfaffians — square roots of determinants of antisymmetric matrices — count matchings; spectral curves and Gibbs measures; locality of correlations on isoradial graphs) through the work of Kenyon, Okounkov and Sheffield and of Boutillier, de Tilière and Raschel. A model is called checkerboard (or staggered) when its weights depend on the black/white bipartite coloring of the faces of $Q$, so that two different weight systems alternate across the lattice. A model is called Z-invariant when its weights satisfy the star-triangle (Yang-Baxter) relations: under a local rhombus move transforming three adjacent faces of $Q$ into three new ones, there is a coupling of the configurations before and after the move agreeing outside the moved region; this is equivalent, by conditioning on all boundary conditions, to a system of 16 algebraic relations on the weights of the six faces involved (equations (45) of Melotti 2021). Z-invariance is the signature of integrability (commuting transfer matrices) and, on isoradial embeddings, it produces locality of correlations. On a lozenge graph, that is, a quadrangulation whose faces are rhombi of a common side length, each face carries a half-angle $\theta$ at its black vertices, and Z-invariant weights are sought as explicit functions of $\theta$. Baxter parametrized the Z-invariant 8V weights in the non-checkerboard setting in terms of elliptic functions, covering the general 8V model, and Perk and Au-Yang noted that the checkerboard Yang-Baxter equations can admit additional solutions; yet before Melotti (2021) no checkerboard Z-invariant 8V weights were known; Melotti introduced what seems to be the first such set, together with a switching symmetry relating free-fermion 8V models to pairs of free-fermion 6V (bipartite dimer) models. His family is parametrized by two moduli $k,l$ with $k^2,l^2\in(-\infty,1)$: writing $\theta_k=2K(k)\theta/\pi$ with $K$ the complete elliptic integral of the first kind, the weights at a face of half-angle $\theta$ are $A=\mathrm{sn}(\theta_k|k)+\mathrm{sn}(\theta_l|l),\quad B=\mathrm{cn}(\theta_k|k)+\mathrm{cn}(\theta_l|l),$ $C=1+\mathrm{sn}(\theta_k|k)\,\mathrm{sn}(\theta_l|l)+\mathrm{cn}(\theta_k|k)\,\mathrm{cn}(\theta_l|l),\quad D=\mathrm{cn}(\theta_k|k)\,\mathrm{sn}(\theta_l|l)-\mathrm{sn}(\theta_k|k)\,\mathrm{cn}(\theta_l|l),$ in Jacobi elliptic functions, where $\mathrm{sn}$ and $\mathrm{cn}$ are the doubly periodic analogues of $\sin$ and $\cos$ with modulus $k$ or $l$. These satisfy the checkerboard Yang-Baxter equations on lozenge graphs; when $(1-k^2)(1-l^2)=1$ the dependence on the bipartite coloring disappears and one recovers Baxter's free-fermion solution, and when $k=l$ one obtains a Z-invariant 6V model corresponding to bipartite dimers. All of these weights lie inside the free-fermion manifold. The open question is whether this is the whole story: whether the checkerboard star-triangle system forces free fermion, and more generally what the complete set of checkerboard Z-invariant 8V solutions is.

Problem Statement

Classify, up to face-wise gauge transformation (multiplying the four weights at a face by a common constant), all local real weight functions $A(\theta),B(\theta),C(\theta),D(\theta)$, with $C\neq 0$ (standard weights) and $A(\theta)^2+B(\theta)^2\neq 0$, attached to the black and white faces of lozenge graphs (isoradial rhombic quadrangulations, with $\theta$ the rhombus half-angle at black vertices) that satisfy the checkerboard Yang-Baxter equations, i.e. the 16 star-triangle identities (45) of Melotti (2021) for the weights of the three star faces and the three triangle faces under the move with angles $\theta_1+\theta_2+\theta_3=\pi/2$, in which the weights alternate between the two face colors. The objective has two linked components that jointly complete the classification program opened by Baxter in the checkerboard setting:

1. Decide whether the solution set contains weights outside the free-fermion manifold $A(\theta)^2+B(\theta)^2=C(\theta)^2+D(\theta)^2$: either exhibit an explicit family (for example in Jacobi elliptic functions of $\theta$ with finitely many parameters) satisfying the checkerboard star-triangle identities, violating the free-fermion condition, and not gauge-equivalent to a non-checkerboard Baxter-type solution; or prove that the checkerboard Yang-Baxter equations force the free-fermion condition, up to the degenerations (such as $(1-k^2)(1-l^2)=1$) under which the checkerboard structure collapses to known non-checkerboard solutions.

2. Provide a complete parametrization of all such solutions with a proof of completeness: an effective finite list of families such that every standard checkerboard Z-invariant eight-vertex solution on lozenge graphs is gauge-equivalent to a member of the list, identifying in particular whether Melotti's $(k,l)$ family and its degenerations exhaust the checkerboard Z-invariant eight-vertex models.

A solution may take the form of new parametrized families, a rigidity/impossibility theorem, or both, but must settle both components for the standard real-weight setting; no restriction on the number of parameters, the function class, or the lattice beyond the lozenge-graph (isoradial) setting of the source is permitted.

References

1. Paul Melotti, The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z-Invariance, Communications in Mathematical Physics 381(1):33-82, 2021. DOI: 10.1007/s00220-020-03901-2 (arXiv:1811.02026). https://doi.org/10.1007/s00220-020-03901-2
2. R. J. Baxter, Partition function of the eight-vertex lattice model, Annals of Physics 70(1):193-228, 1972. DOI: 10.1016/0003-4916(72)90335-1. https://doi.org/10.1016/0003-4916(72)90335-1
3. R. J. Baxter, Solvable eight-vertex model on an arbitrary planar lattice, Philosophical Transactions of the Royal Society of London A 289(1359):315-346, 1978. DOI: 10.1098/rsta.1978.0062. https://doi.org/10.1098/rsta.1978.0062
4. R. J. Baxter, Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics, Proceedings of the Royal Society of London A 404(1826):1-33, 1986. DOI: 10.1098/rspa.1986.0016. https://doi.org/10.1098/rspa.1986.0016
5. Chungpeng Fan and F. Y. Wu, General Lattice Model of Phase Transitions, Physical Review B 2(3):723-733, 1970. DOI: 10.1103/PhysRevB.2.723. https://doi.org/10.1103/PhysRevB.2.723
6. J. H. H. Perk and H. Au-Yang, Yang-Baxter Equations, in Encyclopedia of Mathematical Physics, Elsevier, 2006, pp. 465-473. DOI: 10.1016/B0-12-512666-2/00191-7. https://doi.org/10.1016/B0-12-512666-2/00191-7
7. Richard Kenyon, Andrei Okounkov and Scott Sheffield, Dimers and amoebae, Annals of Mathematics 163(3):1019-1056, 2006. DOI: 10.4007/annals.2006.163.1019. https://doi.org/10.4007/annals.2006.163.1019
8. Cédric Boutillier, Béatrice de Tilière and Kilian Raschel, The Z-invariant massive Laplacian on isoradial graphs, Inventiones mathematicae 208(1):109-189, 2017. DOI: 10.1007/s00222-016-0687-z. https://doi.org/10.1007/s00222-016-0687-z
9. Cédric Boutillier, Béatrice de Tilière and Kilian Raschel, The Z-invariant Ising model via dimers, Probability Theory and Related Fields 174(1-2):235-305, 2019. DOI: 10.1007/s00440-018-0861-x. https://doi.org/10.1007/s00440-018-0861-x
10. Richard Kenyon, Conformal invariance of loops in the double-dimer model, Communications in Mathematical Physics 326(2):477-497, 2014. DOI: 10.1007/s00220-013-1881-0 (arXiv:1105.4158). https://doi.org/10.1007/s00220-013-1881-0
11. Julien Dubédat, Double dimers, conformal loop ensembles and isomonodromic deformations, Journal of the European Mathematical Society 21(1):1-54, 2019. DOI: 10.4171/jems/830. https://doi.org/10.4171/jems/830
12. Mikhail Basok and Dmitry Chelkak, Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4), Journal of the European Mathematical Society 23(8):2787-2832, 2021. DOI: 10.4171/JEMS/1072 (arXiv:1809.00690). https://doi.org/10.4171/JEMS/1072
13. Mikhail Basok and Konstantin Izyurov, Nesting of double-dimer loops: local fluctuations and convergence to the nesting field of CLE(4), arXiv:2501.01574, 2025. https://arxiv.org/abs/2501.01574
14. David B. Wilson, XOR-Ising Loops and the Gaussian Free Field, arXiv:1102.3782, 2011. https://arxiv.org/abs/1102.3782
15. Cédric Boutillier and Béatrice de Tilière, Height representation of XOR-Ising loops via bipartite dimers, Electronic Journal of Probability 19, 2014, 33 pp. DOI: 10.1214/ejp.v19-2449. https://doi.org/10.1214/ejp.v19-2449
16. Marcin Lis, Lucas Rey and Kieran Ryan, Conformally invariant boundary arcs in double dimers, arXiv:2409.18015, 2024. https://arxiv.org/abs/2409.18015
17. Tomás Alcalde López and Avelio Sepúlveda, Excursion decomposition of the XOR-Ising model, arXiv:2602.06011, 2026. https://arxiv.org/abs/2602.06011
18. Vladimir Bošković, Newton polygons for the non-bipartite dimer model, arXiv:2607.07503, 2026. https://arxiv.org/abs/2607.07503
19. Vladimir Bošković, Flip dynamics on perfect matchings beyond bipartite and planar graphs, arXiv:2607.16101, 2026. https://arxiv.org/abs/2607.16101
20. Béatrice de Tilière and Lucas Rey, Classification of (non)-frustrated 2D Ising models in genus 1 on isoradial graphs, arXiv:2602.13526, 2026. https://arxiv.org/abs/2602.13526
21. De-Zhang Li, Xin Wang and Xiao-Bao Yang, Free-Fermion Models and Two-Dimensional Ising Models Under Zero Field and Imaginary Field i(pi/2)kBT, Entropy 27(8):799, 2025. DOI: 10.3390/e27080799 (arXiv:2406.00232). https://doi.org/10.3390/e27080799
22. Ze Tao and Fujun Liu, Integrable construction of a two-dimensional lattice model with anisotropic Hubbard couplings, arXiv:2512.06310, 2025. https://arxiv.org/abs/2512.06310
23. Shahane A. Khachatryan, Zhidong Zhang and Ara G. Sedrakyan, Inhomogeneous Ising Model on 2D kagomé Lattice: Fermionic field approach, arXiv:2608.02156, 2026. https://arxiv.org/abs/2608.02156