The nearest-neighbor Ising model on the infinite square lattice is exactly solved only at zero magnetic field: Onsager (1944) obtained the exact free energy per site and Yang (1952) the spontaneous magnetization, while at any real nonzero field no exact expression for the free energy — or for any other thermodynamic quantity — is known. The problem asks for an exact expression, whether a closed form or a provably convergent representation, for the free energy per site f(T,H)f(T,H) of the square-lattice Ising ferromagnet at real field H≠0H \neq 0, together with the thermodynamics derived from it; any answer must reduce to Onsager's result as H→0H \to 0 and be consistent with the Lee–Yang circle theorem.

Background

The Ising model is the canonical lattice model of ferromagnetism and the archetypal exactly solvable statistical system. Spins si=±1s_i = \pm 1 occupy the sites of a square lattice with Hamiltonian H=−J∑⟨i,j⟩sisj−H∑isi\mathcal{H} = -J\sum_{\langle i,j\rangle} s_i s_j - H\sum_i s_i, where J>0J > 0 is the nearest-neighbor ferromagnetic coupling and HH a uniform external magnetic field; the free energy per site f(T,H)f(T,H) (with inverse temperature β=1/(kBT)\beta = 1/(k_B T)) controls all equilibrium thermodynamics through derivatives such as the magnetization m=−∂f/∂Hm = -\partial f/\partial H and the susceptibility χ=∂m/∂H\chi = \partial m/\partial H. Ising solved the one-dimensional chain in 1925 and found no phase transition. Onsager's 1944 solution of the two-dimensional square lattice at zero field gave the exact free energy and a logarithmically divergent specific heat at a critical temperature TcT_c; Kaufman's spinor reformulation (1949), the Schultz–Mattis–Lieb free-fermion solution (1964), Yang's exact spontaneous magnetization (1952), and the Wu–McCoy–Tracy–Barouch correlation functions (1976) completed an extensive zero-field solution that became the proving ground of scaling theory. All of these methods exploit a structural feature of the H=0H=0 transfer matrix (the finite matrix whose largest eigenvalue yields the thermodynamic free energy on a strip): after a suitable transformation it reduces to free fermions — noninteracting fermionic modes that can be diagonalized by a Fourier transform. A nonzero field destroys this structure: the H≠0H \neq 0 transfer matrix is non-centrosymmetric, meaning it lacks the reflection symmetry between opposite matrix elements on which the zero-field diagonalization relies, and it admits no known fermionic or Bethe-ansatz reduction (the Bethe ansatz being the exact trial-wavefunction method that solves one-dimensional integrable systems such as the Heisenberg chain). Yang and Lee (1952) proved that for ferromagnetic interactions the zeros of the finite-lattice partition function in the complex field plane lie on the imaginary axis (the Lee–Yang circle theorem), so the free energy is analytic for all real H≠0H \neq 0: there is no phase transition off the H=0H=0 axis, and any exact solution must respect this analytic structure. Later exact work only touches the in-field problem at its edges. Zamolodchikov (1989) solved the scaling limit — the continuum theory reached by zooming in on distances large compared with the lattice spacing near criticality — at T=TcT = T_c and small field, as an integrable relativistic field theory with eight massive particles transforming under the symmetry of E8E_8, the largest of the exceptional simple Lie algebras; this is a continuum result, not the lattice free energy. Fonseca and Zamolodchikov (2001) studied the analytic properties of that continuum free energy numerically. Khrapov (2020) found exact disorder solutions — special parameter loci on which correlations vanish identically and the free energy simplifies — but only for generalized multi-spin Ising models. Wei (2019) emphasized that no exact partition function in field is accepted and proposed an empirical fit. Orrick, Nickel, Guttmann and Perk (2001) showed that even the zero-field susceptibility — a strictly easier quantity — has no known closed form, its scaling functions being fixed only numerically from several-hundred-term series. Gaudreault (2025) develops a quaternion-based transfer-operator approximation to the in-field thermodynamics that is valid only where the angle φa=arctan⁡[sinh⁡(2βH)exp⁡(2βJ)]\varphi_a = \arctan[\sinh(2\beta H)\exp(2\beta J)] is small — high temperature, small field, far from criticality — and states plainly that the two-dimensional Ising model in an external field (H≠0H \neq 0) is unsolved and that an exact solution requires an approach able to deal with non-centrosymmetric matrices. A 2025 preprint by Zhang (reference 16) claims an exact in-field solution, but it predicts a critical point shifted to higher temperature and a first-order magnetization jump above TcT_c, contradicting the Lee–Yang circle theorem and the ferromagnetic bound m(T,H)>0m(T,H) > 0, so it is not accepted as a resolution; the same author's three-dimensional conjectures (reference 17) were publicly contested (reference 18). Centenary reviews (2024; 2026) likewise credit Onsager's solution only at zero field. The unsolved in-field thermodynamics is thus a famous open problem of statistical mechanics.

Problem Statement

Derive the exact thermodynamics of the nearest-neighbor ferromagnetic Ising model on the infinite square lattice (coupling J>0J > 0) in a uniform real external magnetic field H≠0H \neq 0. Specifically, obtain an exact expression for the free energy per site

f(T,H)=−kBT lim⁡L→∞1L2ln⁡ZL(T,H),β=(kBT)−1,f(T,H) = -k_B T \, \lim_{L \to \infty} \frac{1}{L^2} \ln Z_L(T,H), \qquad \beta = (k_B T)^{-1},

where ZL(T,H)Z_L(T,H) is the partition function of the finite L×LL \times L square lattice with periodic boundary conditions. The target is a closed form or a provably convergent representation (for example an exact series or integral representation with proven convergence for all T>0T > 0 and real H≠0H \neq 0), together with the thermodynamics derived from it — the magnetization m=−∂f/∂Hm = -\partial f/\partial H, the susceptibility χ=∂m/∂H\chi = \partial m/\partial H, and the critical equation of state near the critical temperature TcT_c (given exactly by sinh⁡(2J/kBTc)=1\sinh(2J/k_B T_c) = 1, i.e. kBTc=2J/ln⁡(1+2)k_B T_c = 2J/\ln(1+\sqrt{2})) as H→0H \to 0. Any proposed solution must (i) reduce to Onsager's exact zero-field free energy in the limit H→0H \to 0, (ii) respect the Lee–Yang circle theorem, i.e. analyticity of ff in real H≠0H \neq 0 with nonanalyticities confined to the H=0H = 0 axis for T≤TcT \leq T_c, and (iii) satisfy the ferromagnetic bound m(T,H)>0m(T,H) > 0 for all H>0H > 0.

References

  1. Mathieu Gaudreault, 'Approximate thermodynamics of the two-dimensional Ising model in an external magnetic field', Journal of Physics A: Mathematical and Theoretical 58, 185001 (2025), DOI 10.1088/1751-8121/adcd32; preprint arXiv:2607.16735 (2026), https://arxiv.org/abs/2607.16735
  2. Ernst Ising, 'Beitrag zur Theorie des Ferromagnetismus', Zeitschrift für Physik 31, 253 (1925), DOI 10.1007/BF02980577, https://doi.org/10.1007/BF02980577
  3. Lars Onsager, 'Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition', Physical Review 65, 117 (1944), DOI 10.1103/PhysRev.65.117, https://doi.org/10.1103/PhysRev.65.117
  4. Bruria Kaufman, 'Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis', Physical Review 76, 1232 (1949), DOI 10.1103/PhysRev.76.1232, https://doi.org/10.1103/PhysRev.76.1232
  5. Chen Ning Yang, 'The Spontaneous Magnetization of a Two-Dimensional Ising Model', Physical Review 85, 808 (1952), DOI 10.1103/PhysRev.85.808, https://doi.org/10.1103/PhysRev.85.808
  6. C. N. Yang and T. D. Lee, 'Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation', Physical Review 87, 404 (1952), DOI 10.1103/PhysRev.87.404, https://doi.org/10.1103/PhysRev.87.404
  7. 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 (1952), DOI 10.1103/PhysRev.87.410, https://doi.org/10.1103/PhysRev.87.410
  8. T. D. Schultz, D. C. Mattis and E. H. Lieb, 'Two-Dimensional Ising Model as a Soluble Problem of Many Fermions', Reviews of Modern Physics 36, 856 (1964), DOI 10.1103/RevModPhys.36.856, https://doi.org/10.1103/RevModPhys.36.856
  9. T. T. Wu, B. M. McCoy, C. A. Tracy and E. Barouch, 'Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region', Physical Review B 13, 316 (1976), DOI 10.1103/PhysRevB.13.316, https://doi.org/10.1103/PhysRevB.13.316
  10. W. P. Orrick, B. Nickel, A. J. Guttmann and J. H. H. Perk, 'The Susceptibility of the Square Lattice Ising Model: New Developments', Journal of Statistical Physics 102, 795 (2001), DOI 10.1023/A:1004850919647, https://doi.org/10.1023/A:1004850919647
  11. W. P. Orrick, B. G. Nickel, A. J. Guttmann and J. H. H. Perk, 'Critical Behavior of the Two-Dimensional Ising Susceptibility', Physical Review Letters 86, 4120 (2001), DOI 10.1103/PhysRevLett.86.4120, https://doi.org/10.1103/PhysRevLett.86.4120
  12. A. B. Zamolodchikov, 'Integrable Field Theory from Conformal Field Theory', Advanced Studies in Pure Mathematics 19, 641 (1989), DOI 10.2969/aspm/01910641, https://doi.org/10.2969/aspm/01910641
  13. P. Fonseca and A. Zamolodchikov, 'Ising field theory in a magnetic field: analytic properties of the free energy', preprint arXiv:hep-th/0112167 (2001), https://arxiv.org/abs/hep-th/0112167
  14. Rong Qiang Wei, 'An empirical partition function for two-dimensional Ising Model in an external magnetic field', preprint arXiv:1905.04295 (2019), https://arxiv.org/abs/1905.04295
  15. Pavel Khrapov, 'Disorder solutions for generalized 2D Ising Model with multi-spin interaction', preprint arXiv:2004.02427 (2020), https://arxiv.org/abs/2004.02427
  16. Zhidong Zhang, 'Exact solution of the two-dimensional (2D) Ising model at an external magnetic field', preprint arXiv:2512.16935 (2025; v4 2026), https://arxiv.org/abs/2512.16935
  17. Z-D Zhang, 'Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices', Philosophical Magazine 87, 5309 (2007), DOI 10.1080/14786430701646325, https://doi.org/10.1080/14786430701646325
  18. Jacques H. H. Perk, 'Comment on "Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices"', Philosophical Magazine 89, 761 (2009), DOI 10.1080/14786430902776970, https://doi.org/10.1080/14786430902776970
  19. Zoe Budrikis, '100 years of the Ising model', Nature Reviews Physics 6, 530 (2024), DOI 10.1038/s42254-024-00760-x, https://doi.org/10.1038/s42254-024-00760-x
  20. Muktish Acharyya, Yurij Holovatch and Ferenc 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