Open problems for the free-fermion eight-vertex models on the square lattice: singularities of staggered free-fermion models, invariant variables, and the general even–odd finite-torus mapping

The free-fermion eight-vertex models on the square lattice — the even and odd 8-vertex subcases of the 16-vertex model satisfying Fan–Wu-type free-fermion conditions — are exactly solvable by Pfaffian methods, and Assis (2017) consolidated their solution theory while posing five explicit open questions. One of them, the search for Brascamp–Kunz-type boundary conditions making the finite-lattice partition functions single double products, has been resolved by Li and Wang (2025), who found such boundary conditions for both even and odd free-fermion models. The remaining four make up three independent problems, because two of them state the same singularity question — the extension of Hurst's 1963 theorem to staggered and other-lattice free-fermion models — at two levels, once as a conjecture to be proven and once as a request for an explicit analysis of the extra singularities. The surviving open core is therefore threefold, and its three parts are each independently required: (A) prove or disprove that for staggered free-fermion eight-vertex models, and for free-fermion vertex models on other lattices, the only thermodynamically relevant singularities of the free energy occur under the temperature-independent free-fermion conditions — extending Hurst's 1963 theorem for the checkerboard and triangular lattices — or, equivalently, determine whether singularities satisfying the general but not the temperature-independent free-fermion condition can generate physical critical points; (B) construct a canonical set of 16 variables invariant under the full 32-element finite-torus partition-function symmetry group of the 16-vertex model; (C) find a local mapping, valid on a finite toroidal lattice, between the general even and the general odd free-fermion 8-vertex models, which remains to be found and is known only in special cases, the existing weak-graph and $SL(2)\times SL(2)$ invariant-equality even–odd mappings holding only under constraints among each set of weights. These components share one objective: completing the exact-solution and equivalence theory of the free-fermion eight-vertex class.

Background

A square-lattice vertex model places arrows (or spin variables) on edges and assigns a Boltzmann weight to each allowed configuration of the four edges meeting at a vertex. With all $2^4=16$ configurations allowed one has the 16-vertex model; the even 8-vertex subcase allows only configurations with an even number of arrows pointing into each vertex (this includes the standard six-vertex model), and the odd 8-vertex subcase allows only configurations with an odd number of arrows in. Fan and Wu (1970) showed that when the weights satisfy a set of quadratic 'free-fermion' relations — for the even subcase, in the standard weight notation, $w_1w_2+w_3w_4=w_5w_6+w_7w_8$ — the partition function of the 8-vertex model on a finite torus can be written as a sum of products of Pfaffians (a Pfaffian being the square root of a determinant of an antisymmetric matrix, the standard combinatorial device for dimer-like counting problems), which makes the thermodynamic free energy an explicit integral. A commonly imposed strengthening is the 'temperature-independent' free-fermion condition, which additionally requires $w_1w_2w_3w_4=w_5w_6w_7w_8$, equivalently a linear relation among the bare interaction energies that involves no temperature dependence. Hurst (1963), developing the Pfaffian method for plane Ising lattices, proved for free-fermion models on the checkerboard and triangular lattices that the only singularities of the free energy that are physically relevant (reachable with positive real weights) arise at points where the temperature-independent specialization holds, while singularities obeying only the general condition occur merely in analytic continuations of the parameter space; this closed the singularity analysis for those uniform cases. For non-uniform ('staggered') models, whose weights alternate between sublattices or rows, Wu and Kunz (2004) solved the odd free-fermion 8-vertex model by a one-to-one configuration mapping to a row-by-row staggered even free-fermion model. Brascamp and Kunz (1974) exhibited special boundary conditions for the Ising model — periodic in one direction with fixed arrow configurations on the top and bottom edges — under which the finite-lattice partition function collapses to a single product over momenta, enabling exact finite-size studies of partition-function zeros. Assis (2017) gave a unified treatment of the even and odd free-fermion 8-vertex subcases of the 16-vertex model: exact free energies, correspondences between the subcases in the thermodynamic limit, weak-graph (a re-summation technique re-expressing vertex-model partition functions over alternative expansion variables) and Pfaffian constructions, a mapping criterion based on the invariance of the torus partition function under an $SL(2)\times SL(2)$ transformation of the weights (two independent copies of the group of $2\times2$ real matrices with unit determinant, a symmetry found by Gaaff, Hijmans, and Schram), whereby two models whose polynomial invariant quantities under this transformation can be made equal are mapped to one another — the 'invariant-equality' condition used to search for mappings, which the source stresses is sufficient but not necessary for two finite-torus partition functions to agree — and a determination that the finite-torus partition function of the 16-vertex model is invariant under a 32-element group isomorphic to $C_2\times C_2\times D_8$ (two independent two-fold operations plus the dihedral group of the square) acting on the 16 vertex weights. That paper posed five open questions, of which one — the search for Brascamp–Kunz-analogue boundary conditions giving single double-product finite-lattice partition functions — was answered by Li and Wang (2025). The remaining questions, formulated below, govern when the free-fermion parameterization is trustworthy in staggered and other-lattice settings, how to coordinatize the weight space by symmetry invariants, and whether the general even and odd free-fermion models can be mapped to one another on a finite torus.

Problem Statement

The three unresolved questions posed by Assis (2017) for the free-fermion 8-vertex subcases of the 16-vertex model on the square lattice, sharing the single objective of completing the exact-solution and equivalence theory of this class of models:

(A) Singularity completeness for staggered and other-lattice free-fermion models. Prove, or disprove, that for staggered free-fermion 8-vertex models and for free-fermion vertex models on lattices other than the square lattice, the only thermodynamically relevant singularities of the free energy occur under the temperature-independent free-fermion conditions, thereby extending Hurst's 1963 checkerboard and triangular theorem. In disproof form: exhibit a staggered (or other-lattice) free-fermion model whose weights satisfy the general free-fermion condition but not its temperature-independent specialization and which nevertheless possesses a physical critical point — a singularity reachable with positive real Boltzmann weights — rather than a singularity confined to an analytic continuation of the parameter space; a complete analysis of all such possible singularities is requested. (A) is a single general claim that quantifies over both model classes at once.

(B) Canonical invariant variables. Construct a convenient canonical set of 16 variables for the weights of the 16-vertex model that is invariant under the full finite-torus partition-function symmetry group (32 transformations forming $C_2\times C_2\times D_8$), so that model classification and the search for mappings become algebraically simpler; currently no such compact invariant variable set is known.

(C) Finite-torus mapping between the general even and odd free-fermion models. Find a local mapping, valid on a finite toroidal lattice, between the general even and the general odd free-fermion 8-vertex models. In the source such a mapping "remains to be found between the general even and odd free-fermion models on the finite lattice", and it is known only in special cases, such as Baxter's mapping from close-packed dimers (an odd free-fermion specialization) to the even free-fermion model; the checkerboard and union-jack lattice Ising models, by contrast, each already have a local finite-torus mapping to the even free-fermion model. The weak-graph and $SL(2)\times SL(2)$ invariant-equality mappings between the even and odd models given in the source are exact on the finite lattice only under constraints among each set of weights, and equality of the algebraic invariants is a sufficient but not necessary condition for two models to have the same finite-torus partition function, so a mapping of the requested generality need not respect the invariants.

The source question on Brascamp–Kunz-type boundary conditions giving single double-product finite-lattice partition functions is recorded as resolved by Li and Wang (2025) and is not part of the open core.

References

1. Michael Assis, 'The 16-vertex model and its even and odd 8-vertex subcases on the square lattice', J. Phys. A: Math. Theor. 50, 395001 (2017). DOI 10.1088/1751-8121/aa842c; arXiv:1702.02110. https://doi.org/10.1088/1751-8121/aa842c
2. De-Zhang Li and Xin Wang, 'Free-fermion approach to the partition function zeros: special boundary conditions and product form of solution' (2025). arXiv:2507.21943. https://arxiv.org/abs/2507.21943
3. C. A. Hurst, 'Solution of plane Ising lattices by the Pfaffian method', J. Chem. Phys. 38, 2558–2571 (1963). DOI 10.1063/1.1733541. https://doi.org/10.1063/1.1733541
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5. H. J. Brascamp and H. Kunz, 'Zeroes of the partition function for the Ising model in the complex temperature plane', J. Math. Phys. 15, 65–66 (1974). DOI 10.1063/1.1666505. https://doi.org/10.1063/1.1666505
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7. Paul Melotti, 'The free-fermion eight-vertex model: couplings, bipartite dimers and Z-invariance', Commun. Math. Phys. 381, 33–82 (2020). DOI 10.1007/s00220-020-03901-2; arXiv:1811.02026. https://doi.org/10.1007/s00220-020-03901-2
8. De-Zhang Li, Xin Wang, and Xiao-Bao Yang, 'Free-Fermion Models and Two-Dimensional Ising Models Under Zero Field and Imaginary Field i(π/2)kBT', Entropy 27(8), 799 (2025). DOI 10.3390/e27080799. https://doi.org/10.3390/e27080799
9. Pete Rigas, 'Operator formalism for discretely holomorphic parafermions of the two-color Ashkin-Teller, loop O(1), staggered eight-vertex, odd eight-vertex, and abelian sandpile models' (2023). arXiv:2310.08212. https://arxiv.org/abs/2310.08212