The free energy of the Ising model on the kagome lattice — the two-dimensional lattice of corner-sharing triangles — has been known exactly since Kanô and Naya (1953), yet its spontaneous magnetization remains without an accepted rigorous derivation. Competing closed-form expressions (Naya 1954; Syozi and Nakano 1955; Syozi and Naya 1960; Syozi 1972; Lin 1992; Matveev and Shrock 1995; Kassan-Ogly 2023) disagree at intermediate temperatures, and each has been checked only against two necessary conditions: the temperature at which the magnetization vanishes must coincide with the exactly known critical temperature, and M→1M\to1 as T→0T\to0. A 2026 fermionic-field derivation for the homogeneous case yields yet another expression that differs from Naya's and Matveev–Shrock's by non-vanishing factors of the coupling. This problem asks for a rigorous derivation of the anisotropic kagome spontaneous magnetization M(T;J1,J2,J3)M(T;J_1,J_2,J_3) from the exact Kanô–Naya free energy that resolves the standing disagreement among the competing expressions at intermediate temperatures. Beyond the two necessary conditions above, no general algorithm or verity criterion is known for obtaining, or certifying the verity of, a spontaneous-magnetization expression from an exactly solved planar-Ising free energy.

Background

Onsager's 1944 exact solution of the square-lattice Ising free energy stood for eight years before Yang's 1952 derivation of the spontaneous magnetization M=[1−sinh⁡−4(2J/T)]1/8M=[1-\sinh^{-4}(2J/T)]^{1/8}, and Montroll, Potts and Ward (1963) showed how MM follows from the limiting behavior of the two-point spin correlation function, evaluated through Pfaffians (determinant-like polynomials of antisymmetric matrices) and a Toeplitz determinant (a determinant whose entries depend only on the difference of their indices), whose large-size asymptotics are given by the Szegö theorem. The spontaneous magnetization MM — the per-site magnetization that survives in the thermodynamic limit below the critical temperature TcT_c — is not contained in the free energy itself, and no general mechanism is known that extracts it from the free energy alone. For the kagome lattice, the two-dimensional Archimedean lattice of corner-sharing triangles, the situation is markedly worse. Kanô and Naya (1953) obtained the exact free energy of the (in general anisotropic) kagome Ising net via Kramers–Wannier-type lattice transformations — star-triangle and decoration maps that rewrite the kagome partition function as one on the honeycomb lattice with renormalized effective couplings — but no analogue of the Montroll–Potts–Ward route from that free energy to MM exists. Naya (1954) gave an expression for the isotropic case, restated by Syozi and Nakano (1955); the anisotropic generalization of Syozi and Naya (1960, Prog. Theor. Phys. 24, 829) was amended by Syozi in 1972, and Kassan-Ogly (2023) reports that the amended version shifts the magnetization-vanishing temperature away from the true TcT_c in the anisotropic case; the companion Syozi–Naya expression (1960, Prog. Theor. Phys. 23, 374), repeated by Lin (1992, Chin. J. Phys. 30, 287), is reported to be purely imaginary at all temperatures; Gerlikman's (1974, Sov. Phys. JETP 39, 570) expression was reported to violate the T→0T\to0 saturation M→1M\to1; and Matveev and Shrock (1995) gave further expressions in a study of the complex-temperature (analytically continued coupling) phase diagram of kagome and related heteropolygonal (mixed-polygon) lattices. These historical verdicts are the source author's assessments, quoted from Kassan-Ogly (2023). Kassan-Ogly (2023) emphasized that 'there are currently no algorithms for deriving spontaneous magnetization, even if there is an exact solution for free energy, as there are no criteria for the verity of expressions for spontaneous magnetization, except for two natural conditions' — the two necessary conditions above — and proposed an anisotropic formula Mkag=(1−Kk)1/8M_{kag}=(1-K_k)^{1/8} (with KkK_k a function of ui=e2Ji/Tu_i=e^{2J_i/T}) that he validated only by its reduction to the decorated-square limit at J3=0J_3=0 and by numerical satisfaction of the two necessary conditions. Most recently, Khachatryan, Zhang and Sedrakyan (2026) re-derived, for the homogeneous kagome lattice only, a spontaneous magnetization via a mapping onto an eight-vertex model (a two-dimensional edge model with an even-parity rule at each vertex), a two-dimensional Jordan–Wigner (spin-to-fermion) Grassmann representation, and a Szegö-theorem evaluation of a Toeplitz determinant, obtaining cJ=sinh⁡2(2J)/(sinh⁡2J−2cosh⁡2J)2c_J=\sinh^2(2J)/(\sinh 2J-2\cosh 2J)^2 with M2=[(1+cJ)3(1−3cJ)/((1−cJ)3(1+3cJ))]1/4M^2=[(1+c_J)^3(1-3c_J)/((1-c_J)^3(1+3c_J))]^{1/4}, critical at 3cJ=13c_J=1; they state explicitly that their expressions 'slightly differ' from those of Naya and of Matveev–Shrock by 'cJc_J dependent factors, which are not critical: do not have zeros'. Thus as of 2026 several mutually inconsistent families of closed-form expressions coexist, none rigorously derived from the Kanô–Naya free energy in a form the community has accepted, and the inter-formula discrepancy at intermediate temperatures is unresolved.

Problem Statement

Derive, rigorously and in closed form, the spontaneous magnetization M(T;J1,J2,J3)M(T;J_1,J_2,J_3) of the Ising model on the kagome lattice with (in general anisotropic) nearest-neighbor couplings J1,J2,J3J_1,J_2,J_3 of arbitrary values and signs, at absolute temperature TT (units with kB=1k_B=1), in the ordered-phase regime where a spontaneous magnetization exists, starting from the exact Kanô–Naya free energy (or an equivalent exact planar-Ising representation). The result must be real and non-negative throughout the ordered phase, reduce to the known decorated-square expression at J3=0J_3=0, and either reproduce or explicitly refute the expressions of Naya (1954), Matveev–Shrock (1995), Kassan-Ogly (2023), and Khachatryan–Zhang–Sedrakyan (2026), thereby resolving the standing disagreement among them at intermediate temperatures. Terminology: 'spontaneous magnetization' MM is the thermodynamic-limit expectation ∣⟨σ0⟩∣|\langle\sigma_0\rangle| per site in a weak symmetry-breaking field as the field tends to zero; 'verity criteria' are checkable conditions certifying correctness of a candidate expression; the 'decorated-square limit' J3=0J_3=0 is the regime in which the kagome lattice reduces to a square lattice singly decorated in both directions with zero nodal–nodal coupling, where the magnetization of the decorating (kagome) spins, Msq=[1−D12D22/(S12S22)]1/8M_{sq}=[1-D_1^2D_2^2/(S_1^2S_2^2)]^{1/8} at decoration degree d1=d2=1d_1=d_2=1 (Kassan-Ogly 2023, Eqs. (18)–(20)), is independently known.

References

  1. F. A. Kassan-Ogly, Spontaneous magnetization of Kagome lattice in Ising model, J. Magn. Magn. Mater. 572, 170568 (2023), doi:10.1016/j.jmmm.2023.170568, https://doi.org/10.1016/j.jmmm.2023.170568 (preprint arXiv:2206.09915, https://arxiv.org/abs/2206.09915)
  2. K. Kanô and S. Naya, Antiferromagnetism. The Kagome Ising Net, Prog. Theor. Phys. 10, 158–172 (1953), doi:10.1143/PTP.10.158, https://doi.org/10.1143/PTP.10.158
  3. S. Naya, On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices, Prog. Theor. Phys. 11, 53–62 (1954), doi:10.1143/PTP.11.53, https://doi.org/10.1143/PTP.11.53
  4. I. Syozi and H. Nakano, Statistical Models of Ferrimagnetism, Prog. Theor. Phys. 13, 69–78 (1955), doi:10.1143/PTP.13.69, https://doi.org/10.1143/PTP.13.69
  5. I. Syozi and S. Naya, Symmetrical Properties of Two-Dimensional Ising Lattices, Prog. Theor. Phys. 24, 829–839 (1960), doi:10.1143/PTP.24.829, https://doi.org/10.1143/PTP.24.829
  6. I. Syozi and S. Naya, Spontaneous Magnetization, A Generalized Square Lattice, Prog. Theor. Phys. 23, 374–376 (1960), doi:10.1143/PTP.23.374, https://doi.org/10.1143/PTP.23.374
  7. C. N. Yang, The Spontaneous Magnetization of a Two-Dimensional Ising Model, Phys. Rev. 85, 808–816 (1952), doi:10.1103/PhysRev.85.808, https://doi.org/10.1103/PhysRev.85.808
  8. E. W. Montroll, R. B. Potts and J. C. Ward, Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model, J. Math. Phys. 4, 308–322 (1963), doi:10.1063/1.1703955, https://doi.org/10.1063/1.1703955
  9. V. Matveev and R. Shrock, Complex-temperature properties of the Ising model on 2D heteropolygonal lattices, J. Phys. A: Math. Gen. 28, 5235–5256 (1995), doi:10.1088/0305-4470/28/18/014, https://doi.org/10.1088/0305-4470/28/18/014
  10. S. A. Khachatryan, Z. Zhang and A. G. Sedrakyan, Inhomogeneous Ising Model on 2D kagomé Lattice: Fermionic field approach, preprint arXiv:2608.02156 (2026), https://arxiv.org/abs/2608.02156
  11. Sh. Khachatryan and A. Sedrakyan, Critical couplings of two dimensional Ising model on various lattices, preprint arXiv:2608.16949 (2026), https://arxiv.org/abs/2608.16949
  12. V. A. Mutailamov and A. K. Murtazaev, Phase diagram of the decorated Ising model on the Kagome lattice with antiferromagnetic nearest-neighbor and ferromagnetic next-nearest-neighbor interactions, J. Magn. Magn. Mater. 655, 174367 (2026), doi:10.1016/j.jmmm.2026.174367, https://doi.org/10.1016/j.jmmm.2026.174367