Adjudicating conflicting exact spontaneous-magnetization formulae for the isotropic kagomé Ising lattice and deriving the closed-form anisotropic magnetization

For the Ising model on the two-dimensional kagomé lattice with isotropic coupling $J$, three incompatible 'exact' spontaneous magnetizations now coexist: the classical formula of Naya (1954), as analyzed by Matveev–Shrock (1995); the corrected Onsager-form formula of Kassan-Ogly (2023), who claims all previously published kagomé magnetizations are fallacious; and Eqs. (5.122)–(5.123) of the 2026 fermionic eight-vertex solution (arXiv:2608.02156, v1), which the authors state 'slightly differ' from the earlier results by factors that 'do not have zeros'. The three formulae disagree at intermediate temperatures, and the exact relation between the 2026 expression and Naya's formula is worked out in the Background below. The open problem is to (i) adjudicate among the three formulae by rigorous derivation and/or independent finite-lattice computation, and (ii) derive the closed-form spontaneous magnetization for anisotropic couplings $J_1,J_2,J_3$ (completion standard in the Problem Statement).

Background

The kagomé lattice is the 2D Archimedean lattice of corner-sharing triangles (coordination number 4, three inequivalent bond directions). The Ising model on it with couplings $J_1,J_2,J_3$ (isotropic when $J_1=J_2=J_3=J$) is one of the few exactly solvable planar Ising systems: via the star-triangle (decoration) transformation — a local reweighting that replaces each triangle of three spins by a star of three spins coupled to a new central spin, exactly preserving the partition function — or, in the 2026 approach of Khachatryan–Zhang–Sedrakyan (arXiv:2608.02156), via a unitary transformation of the Boltzmann weights to a non-symmetric eight-vertex model (a vertex model whose local weights are free parameters related by the free-fermion condition, an algebraic constraint on the vertex weights equivalent to those of non-interacting fermions, allowing Grassmann/fermionic solution) with an explicit R-matrix; the spontaneous magnetization $M(T)$ is then extracted from a Toeplitz determinant (a determinant whose entries depend only on the difference of their indices; Montroll–Potts–Ward first obtained the square-lattice magnetization in this form). The classical result is Naya's (1954) formula, best written in the complex-temperature variable $u=e^{-4K}$ ($K=J/T$): $M=\left[(1-6u-3u^2)^{1/8}(1+2u+5u^2)^{3/8}(1+3u)^{1/4}\right]/\left[(1-u)^{1/4}(1+u)\right]$, with critical point $u_c=2/\sqrt3-1\approx0.1547$ (equivalently $e^{4J_c}=3+2\sqrt3$, $\sinh(2J_c)=(4/3)^{1/4}$, $J_c=\ln(3+2\sqrt3)/4\approx0.46657$), Onsager-type exponent $\beta=1/8$ (the square-lattice value), and correct saturation $M\to 1$ as $T\to 0$ ($u\to 0$); Matveev–Shrock (1995) re-derived this formula and analyzed its full complex-temperature zero structure (zeros $u_c=2/\sqrt3-1$ with exponent $\beta=1/8$, $u_\ell=-1/3$ with $\beta=1/4$, and the conjugate pair $u_k,u_k^*$ with $\beta=3/8$). The 2026 paper instead obtains, in terms of $c_J=\sinh^2(2J)/(\sinh 2J-2\cosh 2J)^2$ (its Eq. 5.103) with critical condition $3c_J=1$, the magnetization $M^2=\left[(1+c_J)^3(1-3c_J)/((1-c_J)^3(1+3c_J))\right]^{1/4}$ (Eq. 5.122), i.e. $M=\left[(3+6e^{4J}-e^{8J})(5+2e^{4J}+e^{8J})^3/(128(3+e^{8J})(1+e^{4J})^3)\right]^{1/8}$ (Eq. 5.123). Kassan-Ogly (2023) contests the entire literature: using two necessary conditions (the magnetization must vanish exactly at the transition temperature obtained from the exact free energy, and must saturate to 1 as $T\to 0$) he shows numerically that the Syozi–Naya (1960) anisotropic formula yields a purely imaginary magnetization (already at isotropic coupling $J_1=J_2=J_3=1$), that the Syozi (1972) correction violates the transition-temperature condition on the test case $J_1=0.2,J_2=0.3,J_3=-0.8$ (where $T_c\approx0.392$), and that the Geǐlikman (1974) formula violates the saturation condition on the test case $J_1=J_2=0.3,J_3=2.9$; the surviving isotropic candidates (Syozi–Nakano 1955, Syozi 1972, Matveev–Shrock 1995) satisfy both endpoint conditions but disagree at intermediate temperatures. Kassan-Ogly proposes a corrected Onsager-form anisotropic magnetization $M=(1-K_k)^{1/8}$ built from $s_i=\sinh(2J_i/T)$, $u_i=e^{2J_i/T}$ and the algebraic combinations $K_i=(-u_i^2+u_j^2+u_k^2-u_1^2u_2^2u_3^2)/(4u_1u_2u_3)$, supported by the $J_3=0$ limit in which the kagomé lattice reduces to a doubly decorated square lattice. A decorated lattice is one obtained by inserting spins on the edges of a parent lattice, allowing exact reduction of its partition function to the parent's. The anisotropic expressions needed below are located as follows: Kassan-Ogly reproduces the Syozi–Naya (1960) anisotropic magnetization as his Eqs. (6)–(9) and states that the Syozi (1972) amendment replaces $\Delta_i^2$ by $\Delta_i$ in his Eq. (7); the Geǐlikman (1974) expression is not displayed in the cited literature, which reports its behaviour only through the corresponding numerical curve. Comparing the 2026 squared magnetization directly with Naya's gives, with $y=e^{4J}$, $M^8_{(5.123)}/M^8_{\mathrm{Naya}}=-R(y)^8$, hence a ratio of squared formulae of modulus $R(y)^2$, where $R(y)=2^{-7/8}(y-1)^{1/4}(y+1)^{5/8}/[(y^2+3)^{1/8}(y+3)^{1/4}]$ (the critical-temperature factors $y^2-6y-3$ and $y^2+2y+5$ cancel only up to the sign $(3+6y-y^2)=-(y^2-6y-3)$); $R(y)$ grows from 1.10 at $J=0.5$ to 2.43 at $J=1$ and diverges as $2^{-7/8}y^{3/8}$, while the factor $(3+6y-y^2)$ of Eq. (5.123) is negative throughout the ordered phase $y>3+2\sqrt3$. This relation is the discrepancy any adjudication has to account for; it is to be recomputed from the cited closed forms rather than assumed, and arXiv:2608.02156 states neither a branch convention for $b_J=\sqrt{c_J}$ nor for $\sqrt{g_J(q)}$ — while $c_J$ itself, its Eq. (5.103), is an unambiguous nonnegative rational function of $e^{4J}=y$ — and does not discuss the $T\to0$ saturation.

Problem Statement

For the Ising model on the kagomé lattice: (1) Isotropic adjudication — determine the correct spontaneous magnetization $M(J)$ of the isotropic ferromagnetic kagomé Ising model, by rigorous derivation and/or exact finite-lattice computation, deciding among (a) Naya's formula as analyzed by Matveev–Shrock, (b) Kassan-Ogly's corrected Onsager-form formula (his Eqs. 11–17), and (c) Eqs. (5.122)–(5.123) of arXiv:2608.02156. Any adjudication must account for the exact relation between the 2026 expression and Naya's formula worked out in the Background, and must settle whether Eqs. (5.122)–(5.123) can be made consistent with the required saturation $M\to1$ as $T\to0$ under some choice of branch convention for $b_J=\sqrt{c_J}$ and $\sqrt{g_J(q)}$ — a convention the paper does not state, while $c_J$ itself (its Eq. (5.103)) is an unambiguous nonnegative rational function of $e^{4J}$. (2) Anisotropic closed form — derive, by an independent method rather than by restating a formula already published (for instance Kassan-Ogly's Eqs. (11)–(17)), a closed-form spontaneous magnetization $M(J_1,J_2,J_3)$ for anisotropic kagomé couplings satisfying both of Kassan-Ogly's necessary conditions, recovering the correct isotropic limit and the $J_3=0$ doubly-decorated-square-lattice limit; clarify why the previously published anisotropic formulas (Syozi–Naya 1960, Syozi 1972) fail them, and state what the cited numerical critique establishes for the Geǐlikman (1974) formula, whose expression the cited literature does not display.

References

1. S. Naya, 'On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices', Progress of Theoretical Physics 11, 53–62 (1954). DOI: 10.1143/PTP.11.53, https://doi.org/10.1143/PTP.11.53
2. I. Syozi and H. Nakano, 'Statistical Models of Ferrimagnetism', Progress of Theoretical Physics 13, 69–78 (1955). DOI: 10.1143/PTP.13.69, https://doi.org/10.1143/PTP.13.69 (reproduces Naya's isotropic kagomé magnetization without its leading multiplier, per Kassan-Ogly's account)
3. I. Syozi and S. Naya, 'Spontaneous Magnetization, A Generalized Square Lattice', Progress of Theoretical Physics 23, 374–376 (1960). DOI: 10.1143/PTP.23.374, https://doi.org/10.1143/PTP.23.374 (with the companion paper below, the source of the Syozi–Naya anisotropic kagomé magnetization formula criticized by Kassan-Ogly)
4. I. Syozi and S. Naya, 'Symmetrical Properties of Two-Dimensional Ising Lattices', Progress of Theoretical Physics 24, 829–839 (1960). DOI: 10.1143/PTP.24.829, https://doi.org/10.1143/PTP.24.829
5. V. Matveev and R. Shrock, 'Complex-temperature properties of the Ising model on 2D heteropolygonal lattices', Journal of Physics A: Mathematical and General 28, 5235–5256 (1995). DOI: 10.1088/0305-4470/28/18/014, arXiv:hep-lat/9503005, https://arxiv.org/abs/hep-lat/9503005
6. F. A. Kassan-Ogly, 'Spontaneous magnetization of Kagome lattice in Ising model', Journal of Magnetism and Magnetic Materials 572, 170568 (2023). DOI: 10.1016/j.jmmm.2023.170568, arXiv:2206.09915, https://arxiv.org/abs/2206.09915
7. Sh. A. Khachatryan, Z. Zhang, and A. G. Sedrakyan, 'Inhomogeneous Ising Model on 2D kagomé Lattice: Fermionic field approach' (2026). arXiv:2608.02156, https://arxiv.org/abs/2608.02156 (v1, submitted 2026-08-03)
8. Sh. A. Khachatryan and A. G. Sedrakyan, 'Characteristics of two-dimensional lattice models from a fermionic realization: Ising and XYZ models', Physical Review B 80, 125128 (2009). DOI: 10.1103/PhysRevB.80.125128, arXiv:0712.0273, https://arxiv.org/abs/0712.0273 (the earlier fermionic-realization method paper; cited as [28] of arXiv:2608.02156)
9. I. Syozi, 'Transformation of Ising Models', in C. Domb and M. S. Green (eds.), Phase Transitions and Critical Phenomena, Vol. 1 (Exact Results), Academic Press, London, 1972, pp. 270–329.
10. M. B. Geǐlikman, 'Spontaneous Magnetization and Correlation Functions in the Kagome Lattice', Soviet Physics JETP 39 (3), 570–573 (1974).