Rigorous status of the conjectured odd-spin correlation forms underlying the triangular- and cubic-lattice Ising spontaneous magnetization relations

T. Kaya (Chin. J. Phys. 77, 2676 (2022); arXiv:2103.13723) derived closed-form spontaneous magnetization relations for the two-dimensional triangular and the three-dimensional simple-cubic Ising lattices by combining an exact single-site identity with a conjectured two-term closed form for local odd-spin correlation functions, $a\,m+(1-a)\,m^{1+\beta^{-1}}$, where $m$ is the spontaneous magnetization and $\beta$ the critical exponent; the distinct correlator coefficients are grouped into a single parameter fixed self-consistently at an input critical coupling, and $\beta$ itself is an input. The paper acknowledges that these steps are conjectural: exact closed forms for the required three- and five-spin correlations are unknown, the grouping error is unquantified, and whether the cubic-lattice formula is exact or only an accurate approximation — tested only against numerical data for reduced temperature $0.0005<t<0.26$ — is unsettled. The open problem is to settle that status: prove or disprove the ansatz (for instance against exactly known planar odd-spin correlations, or by rigorous or high-precision numerics), quantify the grouping error, and establish independently whether the cubic formula is exact, or determine its accuracy. No published work resolves, refutes, or independently validates the conjectures, and no exact three-dimensional Ising solution is known.

Background

The Ising model places spins $\sigma_i=\pm1$ on a lattice with nearest-neighbor coupling $J$, dimensionless coupling $K=J/(k_BT)$, and optional external field $H$; above a critical coupling $K_c$ (that is, below the critical temperature $T_c$) the zero-field system develops a spontaneous magnetization $m=\langle\sigma\rangle$, the order parameter of the second-order phase transition. In two dimensions the zero-field model is exactly solved — Onsager's partition function, Yang's square-lattice magnetization, and Potts' triangular-lattice magnetization — while no three-dimensional lattice Ising model has an exact solution: its critical quantities are known only from high-temperature series expansions, renormalization-group methods, Monte Carlo simulation and conformal-bootstrap calculations (the last a numerical method that determines critical exponents from consistency conditions on correlation functions).

Correlations of an odd number of spins vanish above $T_c$ in zero field but are nonzero in the ordered phase. For planar lattices, local odd-spin correlations have been obtained exactly in closed form: Baxter's triplet order parameter (the local three-spin correlation) for the triangular lattice, Enting's honeycomb triplet order parameters, the odd-number correlations of Barry, Múnera and Tanaka on honeycomb, triangular and general planar lattices, and the local three-spin correlations of Baxter and Choy for free-fermion models (a solvable class whose partition function reduces to a determinant, the technique underlying Onsager's solution) and planar models. These exact studies established the physical property on which the program rests: local odd-spin correlations vanish at criticality with the same critical exponent $\beta$ as the magnetization itself ($m\sim t^{\beta}$ in the reduced temperature $t$). Nothing analogous is known in three dimensions.

Kaya's starting point is an exact single-site decimation identity (known in the effective-field literature as the Callen identity), $\langle\sigma_0\rangle=\langle\tanh[K(\sigma_1+\dots+\sigma_z)+H]\rangle$, where $\sigma_1,\dots,\sigma_z$ are the nearest neighbors of the central spin $\sigma_0$. For $z=6$ — both the triangular and the simple-cubic lattice — the hyperbolic tangent can be rewritten exactly, as an identity on the finite set of neighbor-spin configurations, as a linear combination of the sums of odd spin monomials of degrees one, three and five, with closed-form $K$-dependent coefficients $A$, $B$, $C$; thermal averaging then expresses $m$ in terms of itself plus local three-spin and five-spin correlation functions of specific neighbor clusters, such as $\langle\sigma_1\sigma_2\sigma_3\rangle$ and $\langle\sigma_1\sigma_2\sigma_3\sigma_4\sigma_5\rangle$. The conjectural step replaces each such correlator by the two-term closed form

$$\langle\sigma_{j_1}\cdots\sigma_{j_k}\rangle=a_i\,m+(1-a_i)\,m^{1+\beta^{-1}},\qquad k=3,5,$$

where $\beta$ is the magnetic critical exponent: since $m\sim t^{\beta}$, the first term carries the leading critical singularity while the second, $m^{1+1/\beta}\sim t^{\beta+1}$, represents a correction-to-scaling contribution. The coefficients $a_i$ are intended as constants independent of the coupling (temperature), fixed once and then used at all $K$. The individual coefficients $a_i$ are then collapsed into grouped parameters, $z_1$ collecting the three-spin coefficients and $z_2$ the five-spin ones, with $z_2$ tied to $z_1$ ($z_2\simeq(6/20)z_1$) by the assumption that the $a_i$ are all close in value; the single remaining parameter $z_1$ is fixed self-consistently by requiring the coefficient of $m$ to vanish at an input critical coupling $K_c$, and $\beta$ is also an input (the exact $1/8$ in two dimensions, the numerical $0.32694$ in three). The output is a closed-form magnetization for each lattice: for the triangular lattice the result is claimed to be 'almost equivalent' to Potts' exact formula, and for the cubic lattice — for which no exact analytic magnetization exists — the result agrees with the high-precision numerical data and interpolation of Talapov and Blöte, valid for $0.0005<t<0.26$ with $t=1-K_c/K$. The paper itself acknowledges that every load-bearing step beyond the initial identity is conjectural or uncontrolled.

Problem Statement

The problem is to replace the conjectural steps in this derivation chain with rigorous results, settling in particular three linked questions.

(1) Settle the status of the conjectured closed form for local odd-spin correlation functions,

$$\langle\sigma_{j_1}\cdots\sigma_{j_k}\rangle=a_i\,m+(1-a_i)\,m^{1+\beta^{-1}},\qquad k=3,5,$$

for the specific neighbor clusters that occur in the triangular- and cubic-lattice expansions: prove it, disprove it (for example by comparison with the exactly known planar odd-spin correlation functions, or by rigorous or high-precision numerics on the cubic lattice). This includes checking the consistency of the ansatz's built-in critical structure — leading singularity $t^{\beta}$ with a single correction term $t^{\beta+1}$ — against the actual correction structure of the Ising magnetization.

(2) Quantify the error introduced when the distinct correlator coefficients $a_i$ are replaced by the grouped combinations $z_1$ (three-spin) and $z_2$ (five-spin), with $z_2$ tied to $z_1$ by the assumption that the $a_i$ are close in value; equivalently, determine the individual $a_i$, or establish a criterion for when the grouping fails.

(3) Determine whether the resulting cubic-lattice spontaneous magnetization formula — constructed from these conjectures together with a self-consistency condition at the input critical coupling $K_c=0.2216$ and an input exponent $\beta$ — is exact or only an accurate approximation, by an independent analytic proof or disproof, or by high-precision numerical validation extending beyond the tested window $0.0005<t<0.26$ (with reduced temperature $t=1-K_c/K$).

These three questions form one scientific objective: the ansatz, together with the grouping approximation, is the engine of the derivation and the cubic-lattice formula is its output, so settling (1) and (2) settles the status of the derivation and (3) settles the status of its principal novel output. The source additionally records two further open directions of the same program that are not required for the core objective but inherit the same unresolved control question: extension to nonzero external field $H\neq0$ and to other three-dimensional lattice types (the body-centered and face-centered cubic cases have since been treated under the same conjectures, without establishing accuracy conditions), and a self-consistent derivation of $\beta$ within the same algebraic framework rather than as an input, which would subsume the still-open problem of the exact three-dimensional Ising critical exponents.

References

1. T. Kaya, 'Relevant spontaneous magnetization relations for the triangular and the cubic lattice Ising model', Chinese Journal of Physics 77, 2676-2683 (2022). DOI: 10.1016/j.cjph.2022.03.043, https://doi.org/10.1016/j.cjph.2022.03.043; preprint arXiv:2103.13723, https://arxiv.org/abs/2103.13723.
2. T. Kaya, 'Exact three spin correlation function relations for the square and the honeycomb Ising lattices', Chinese Journal of Physics 66, 415-421 (2020). DOI: 10.1016/j.cjph.2020.04.021, https://doi.org/10.1016/j.cjph.2020.04.021; preprint arXiv:2004.14662, https://arxiv.org/abs/2004.14662.
3. T. Kaya, 'Analytic average magnetization expression for the body-centered cubic Ising lattice', The European Physical Journal Plus 137, 1130 (2022). DOI: 10.1140/epjp/s13360-022-03369-0, https://doi.org/10.1140/epjp/s13360-022-03369-0; preprint arXiv:2205.07132, https://arxiv.org/abs/2205.07132.
4. T. Kaya, 'Magnetic susceptibility of the square lattice Ising model', The European Physical Journal Plus 137, 886 (2022). DOI: 10.1140/epjp/s13360-022-03101-y, https://doi.org/10.1140/epjp/s13360-022-03101-y; preprint arXiv:2205.07127, https://arxiv.org/abs/2205.07127.
5. B. Tambaş, 'Relevant Analytic Spontaneous Magnetization Relation for the Face-Centered-Cubic Ising Lattice', Entropy 25, 197 (2023). DOI: 10.3390/e25020197, https://doi.org/10.3390/e25020197; preprint arXiv:2207.14185, https://arxiv.org/abs/2207.14185.
6. R. Baxter, 'Triplet order parameter of the triangular Ising model', Journal of Physics A: Mathematical and General 8, 1797-1805 (1975). DOI: 10.1088/0305-4470/8/11/015, https://doi.org/10.1088/0305-4470/8/11/015.
7. I. G. Enting, 'Triplet order parameters in triangular and honeycomb Ising models', Journal of Physics A: Mathematical and General 10, 1737-1743 (1977). DOI: 10.1088/0305-4470/10/10/008, https://doi.org/10.1088/0305-4470/10/10/008.
8. J. H. Barry, C. H. Múnera, T. Tanaka, 'Exact solutions for Ising model odd-number correlations on the honeycomb and triangular lattices', Physica A 113, 367-387 (1982). DOI: 10.1016/0378-4371(82)90145-5, https://doi.org/10.1016/0378-4371(82)90145-5.
9. J. H. Barry, T. Tanaka, M. Khatun, C. H. Múnera, 'Exact solutions for Ising-model odd-number correlations on planar lattices', Physical Review B 44, 2595-2608 (1991). DOI: 10.1103/PhysRevB.44.2595, https://doi.org/10.1103/PhysRevB.44.2595.
10. R. J. Baxter, T. C. Choy, 'Local three-spin correlations in the free-fermion and planar Ising models', Proceedings of the Royal Society of London A 423, 279-300 (1989). DOI: 10.1098/rspa.1989.0055, https://doi.org/10.1098/rspa.1989.0055.
11. L. 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.
12. C. N. Yang, 'The Spontaneous Magnetization of a Two-Dimensional Ising Model', Physical Review 85, 808-816 (1952). DOI: 10.1103/PhysRev.85.808, https://doi.org/10.1103/PhysRev.85.808.
13. R. B. Potts, 'Spontaneous Magnetization of a Triangular Ising Lattice', Physical Review 88, 352 (1952). DOI: 10.1103/PhysRev.88.352, https://doi.org/10.1103/PhysRev.88.352.
14. A. L. Talapov, H. W. J. Blöte, 'The magnetization of the 3D Ising model', Journal of Physics A: Mathematical and General 29, 5727-5733 (1996). DOI: 10.1088/0305-4470/29/17/042, https://doi.org/10.1088/0305-4470/29/17/042.
15. A. J. Guttmann, I. G. Enting, 'Solvability of Some Statistical Mechanical Systems', Physical Review Letters 76, 344-347 (1996). DOI: 10.1103/PhysRevLett.76.344, https://doi.org/10.1103/PhysRevLett.76.344.
16. M. Acharyya, Y. Holovatch, F. Iglói, '100 Glorious years of the Ising model', The European Physical Journal B 99, 16 (2026). DOI: 10.1140/epjb/s10051-026-01129-y, https://doi.org/10.1140/epjb/s10051-026-01129-y; preprint arXiv:2601.11671, https://arxiv.org/abs/2601.11671.
17. D. Olascoaga-Rodríguez, F. Sastre, V. Romero-Rochín, 'Mean-field behavior of the finite size Ising model near its critical point', Physica A 697, 131734 (2026). DOI: 10.1016/j.physa.2026.131734, https://doi.org/10.1016/j.physa.2026.131734; preprint arXiv:2602.05212, https://arxiv.org/abs/2602.05212.
18. E. Kantar, M. Ertaş, 'Triangular Ising nanowire system with core/shell structured', The European Physical Journal Plus 137, 1119 (2022). DOI: 10.1140/epjp/s13360-022-03315-0, https://doi.org/10.1140/epjp/s13360-022-03315-0.
19. H. B. Callen, 'Green Function Theory of Ferromagnetism', Physical Review 130, 890-898 (1963). DOI: 10.1103/PhysRev.130.890, https://doi.org/10.1103/PhysRev.130.890.