Free-fermion solvability of the two-dimensional Ising model in a real magnetic field

The two-dimensional nearest-neighbour Ising model is exactly solved at zero field and at the special imaginary field $H=i(\pi/2)k_BT$ , but not in a real nonzero magnetic field — an open problem since Onsager's 1944 solution. Li, Wang and Yang (Entropy 27, 799 (2025)) mapped the honeycomb, triangular and Kagomé lattice models, at zero field and at $H=i(\pi/2)k_BT$ , onto even or odd free-fermion eight-vertex models via decorated-lattice, star-triangle and weak-graph transformations, and posed as an open question whether the mapping methodology extends to a real nonzero field. Because that work also demonstrates that the method's success depends on how the global field factor is repartitioned into local vertex contributions, the question turns on which repartitionings — if any — can yield symmetric or antisymmetric sixteen-vertex models amenable to free-fermion solution at a real field.

Background

The Ising model — spins $s_i=\pm1$ on a lattice with nearest-neighbour couplings $J_{ij}$ and an external field $H$ coupling to $\sum_i s_i$ — is the canonical exactly solvable lattice model of statistical mechanics. Onsager (1944) obtained the exact zero-field free energy of the square-lattice model, and zero-field solutions on the honeycomb, triangular, Kagomé and other lattices followed. In a real nonzero magnetic field, however, no exact solution is known for any two-dimensional lattice; this has been a famous open problem since Onsager's work. The only field value besides zero at which two-dimensional Ising models are exactly solvable is the special imaginary field $H=i(\pi/2)k_BT$ , first treated by Lee and Yang (1952) for the square lattice in their study of the zeros of the partition function in the complex-field plane (Lee-Yang zeros).

The solvable cases are understood through vertex models. An eight-vertex model on the square lattice assigns a Boltzmann weight to each allowed arrow configuration around a lattice vertex; the sixteen-vertex model allows all sixteen configurations, and its even (odd) subcase keeps the eight configurations with an even (odd) number of arrows pointing into the vertex. Fan and Wu (1970) showed that eight-vertex models whose weights satisfy a set of quadratic relations — the free-fermion condition, e.g. $\omega_1\omega_2+\omega_3\omega_4=\omega_5\omega_6+\omega_7\omega_8$ for the even subcase — are exactly solvable, essentially because the condition makes the model equivalent to noninteracting (free) fermions. A sixteen-vertex model is called symmetric (antisymmetric) if each weight is invariant under (changes sign under) simultaneous reversal of all arrows, $\omega_1=\omega_2,\ldots,\omega_{15}=\omega_{16}$ (respectively $\omega_1=-\omega_2,\ldots,\omega_{15}=-\omega_{16}$ ). The weak-graph expansion — a partition-function-preserving linear rearrangement of the sixteen weights, $\tilde\omega=(1/4)M\omega$ with a fixed matrix $M$ — transforms a symmetric model into an even eight-vertex model and an antisymmetric one into an odd eight-vertex model. The decorated-lattice technique (inserting auxiliary spins on edges) and the star-triangle transformation (eliminating a spin of coordination number three, replacing a triangular interaction unit by an equivalent three-leg unit) are classical partition-function-preserving devices for converting non-square Ising lattices into square-lattice vertex models.

Li, Wang and Yang (Entropy 27, 799 (2025)) used this machinery to map the anisotropic nearest-neighbour Ising models on the honeycomb, triangular and Kagomé lattices, at zero field and at $H=i(\pi/2)k_BT$ , onto even or odd free-fermion eight-vertex models, rederiving five known solutions and reporting the previously unreported exact solution of the Kagomé lattice model at $H=i(\pi/2)k_BT$ (with residual entropy $\frac{1}{6}\ln 3$ — the zero-temperature entropy left by frustrated interactions — for the isotropic antiferromagnetic case). A decisive structural ingredient at the imaginary field is the repartitioning of the global field Boltzmann factor $e^{\beta H\sum_i s_i}$ (here $\beta=1/k_BT$ is the inverse temperature; with $\beta H=i\pi/2$ this factor becomes $i^{\sum_i s_i}$ , a product of powers of $i$ ) into local contributions attached to vertex units — the local spin clusters that the decoration maps to single vertices of the auxiliary square lattice. Formally, an admissible repartitioning assigns to each vertex unit $u$ and each spin $s_i$ in $u$ a real coefficient $c_i^{(u)}$ , subject to $\sum_{u:\,i\in u} c_i^{(u)}=1$ for every spin $s_i$ , and attaches to unit $u$ the local field factor $e^{\beta H\sum_{i\in u} c_i^{(u)} s_i}$ , so that the product of these local factors over all units equals the global factor $e^{\beta H\sum_i s_i}$ . The paper shows that for the Kagomé lattice — where a vertex unit consists of four boundary spins $s_1,\ldots,s_4$ and one internal spin $s_5$ that is summed over in the vertex weight — the natural repartitioning $e^{\beta H(s_1+s_2+s_5)}$ yields an antisymmetric sixteen-vertex model, while the equally natural alternative $e^{\beta H[\frac{1}{2}(s_1+s_2+s_3+s_4)+s_5]}$ yields weights that are neither symmetric nor antisymmetric, for which the weak-graph route to a free-fermion model fails. The success of the whole pipeline therefore depends sensitively on a local choice for which no general rule is known.

The real-field case resists this machinery: for real $H$ the field factor does not reduce to powers of $i$ , and under the natural repartitionings the induced vertex weights lose the arrow-reversal structure underlying the symmetric/antisymmetric classification. Consistent with this, Martins (arXiv:2405.07341) showed that the square-lattice Ising model in a magnetic field is equivalent, at finite size, to a mixed eight-vertex model combining even and odd vertex configurations, and that its Yang-Baxter algebra (the algebraic structure underlying integrable vertex models) loses the difference property — dependence of the R-matrix, the elementary two-leg operator of that algebra, on only the difference of its two spectral parameters — required by standard integrability techniques. The exact treatment of the two-dimensional Ising model in a real nonzero field, and the systematic understanding of when field factors can be repartitioned into free-fermion-solvable vertex weights, remain open.

Problem Statement

Li, Wang and Yang's mapping pipeline (decorated lattice, star-triangle transformation, repartitioning of the field factor, weak-graph expansion) solves the two-dimensional nearest-neighbour Ising models on the honeycomb, triangular and Kagomé lattices exactly at zero field and at the imaginary field $H=i(\pi/2)k_BT$ , by producing symmetric or antisymmetric sixteen-vertex models that transform into even or odd eight-vertex models satisfying the free-fermion condition. The open question posed explicitly in that work is whether this mapping methodology extends to a physical (real) and nonzero magnetic field — a question of central physical interest, since no exact solution of the two-dimensional Ising model in a real nonzero field is known for any lattice (see Background).

In the specific terms of the mapping methodology: does there exist a repartitioning of the field Boltzmann factor $e^{\beta H\sum_i s_i}$ , with $\beta H\in\mathbb{R}\setminus\{0\}$ , into local vertex-unit contributions — on any of the square, honeycomb, triangular, or Kagomé lattices, with real anisotropic nearest-neighbour couplings — under which the induced sixteen-vertex model is symmetric or antisymmetric and its weak-graph transform satisfies the free-fermion condition, yielding a closed-form expression for the free energy, and hence the magnetization and other thermodynamics, in a nonzero real field? A positive answer would deliver an exact real-field solution of the two-dimensional Ising model on the corresponding lattices through the free-fermion mapping route. A rigorous negative answer resolves the posed question in the negative only for an explicitly stated class of mappings — the four named lattices, real anisotropic nearest-neighbour couplings, and all admissible repartitionings of the field factor (as defined in the Background) — and must exclude a symmetric or antisymmetric sixteen-vertex model with a free-fermion-satisfying weak-graph transform at every member of that class; the source paper's honeycomb, triangular and Kagomé constructions, with their natural and alternative repartitionings, are instances of this class. Such a result would establish that this route cannot reach real fields, while the general question of exact solvability of the two-dimensional Ising model in a real field would remain open in its own right.

The source demonstrates that the method's success depends sensitively on a local choice for which no general rule is known: on the Kagomé lattice the natural repartitioning $e^{\beta H(s_1+s_2+s_5)}$ yields an antisymmetric sixteen-vertex model, while the equally natural alternative $e^{\beta H[\frac{1}{2}(s_1+s_2+s_3+s_4)+s_5]}$ yields weights that are neither symmetric nor antisymmetric, for which the weak-graph route fails. The instances established in the source — the honeycomb mapping symmetric (even free-fermion), the triangular and Kagomé mappings antisymmetric (odd free-fermion) at $H=i(\pi/2)k_BT$ , and the alternative Kagomé repartitioning neither symmetric nor antisymmetric — mark the currently known behavior of repartitionings at the solvable field values.

The objective is to determine the field-dependence of free-fermion solvability of the two-dimensional Ising model in a real nonzero field: whether a repartitioning of the field Boltzmann factor into local vertex-unit contributions yields a symmetric or antisymmetric sixteen-vertex model whose weak-graph transform satisfies the free-fermion condition, either by exhibiting such a repartitioning on one of the named lattices or by proving that none exists within a precisely stated class of mappings that provably contains the source's methodology.

References

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, 799 (2025), DOI 10.3390/e27080799 (arXiv:2406.00232), https://arxiv.org/abs/2406.00232
T. D. Lee and C. N. Yang, 'Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model', Physical Review 87, 410–419 (1952), DOI 10.1103/PhysRev.87.410, https://doi.org/10.1103/PhysRev.87.410
Lars Onsager, 'Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition', Physical Review 65, 117–149 (1944), DOI 10.1103/PhysRev.65.117, https://doi.org/10.1103/PhysRev.65.117
Chungpeng Fan and F. Y. Wu, 'General Lattice Model of Phase Transitions', Physical Review B 2, 723–733 (1970), DOI 10.1103/PhysRevB.2.723, https://doi.org/10.1103/PhysRevB.2.723
M. J. Martins, 'On the equivalence between n-state spin and vertex models on the square lattice', arXiv:2405.07341 (2024, v3 2025), https://arxiv.org/abs/2405.07341
De-Zhang Li and Xin Wang, 'Free-fermion approach to the partition function zeros: Special boundary conditions and product form of solution', arXiv:2507.21943 (2025), https://arxiv.org/abs/2507.21943
De-Zhang Li and Xin Wang, 'Solution of the Ising model with Brascamp-Kunz boundary conditions by the transfer matrix method', arXiv:2604.16992 (2026), https://arxiv.org/abs/2604.16992
Kazuhiko Minami, 'The free energy of the square lattice Ising model with interactions alternating in horizontal and vertical directions', arXiv:2606.16225 (2026), https://arxiv.org/abs/2606.16225
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
Zhidong Zhang, 'Exact solution of the two-dimensional (2D) Ising model at an external magnetic field', arXiv:2512.16935 (2025), https://arxiv.org/abs/2512.16935
Jacques H. H. Perk, 'Erroneous solution of three-dimensional (3D) simple orthorhombic Ising lattices', arXiv:1209.0731 (2012), https://arxiv.org/abs/1209.0731
M V Vismaya and M V Sangaranarayanan, 'Partition function and magnetization of two-dimensional Ising models in non-zero magnetic field: A semi-empirical approach', arXiv:2512.18611 (2025), https://arxiv.org/abs/2512.18611