Statistical mechanics, structure of matter Lattice spin systems Theorem

Onsager's solution of the two-dimensional Ising model

Builds on Ising model Thermodynamic limit Kramers–Wannier duality

Statement

Consider the Ising model on the square lattice $\Z^2$ with nearest-neighbour coupling $J > 0$ and no external field, at inverse temperature $\beta > 0$ , and let $K = \beta J$ . For $n \ge 1$ let $Z_n$ be its partition function on the $n \times n$ box. Then the free energy per site exists in the thermodynamic limit and equals $\lim_{n \to \infty} \frac{1}{n^2} \ln Z_n = \ln 2 + \frac{1}{8 \pi^2} \int_0^{2\pi} \!\! \int_0^{2\pi} \ln\!\Big[ \cosh^2(2K) - \sinh(2K) \big( \cos\theta_1 + \cos\theta_2 \big) \Big] \, d\theta_1 \, d\theta_2.$ This function of $K$ is real analytic except at the single point $K_c$ with $\sinh(2K_c) = 1$ , that is $K_c = \frac12 \ln(1 + \sqrt 2)$ .

Remarks

Lars Onsager obtained the formula in 1944 by computing the largest eigenvalue of the transfer matrix of the model, using the algebra generated by its two factors. Later proofs write the partition function as a Pfaffian or determinant, through free fermions, dimer coverings or the Kac–Ward formula. Check at $K = 0$ : the integrand is $\ln 1 = 0$ , so the limit is $\ln 2$ , the entropy of independent spins. For large $K$ the right-hand side is about $2K$ , which matches the ground state energy $-2J$ per site, since each site has two bonds. The argument of the logarithm equals $(\sinh 2K - 1)^2$ at $\theta_1 = \theta_2 = 0$ , which explains the singularity at $K_c$ : the specific heat diverges logarithmically there. The spontaneous magnetization for $K > K_c$ is $\left(1 - \sinh(2K)^{-4}\right)^{1/8}$ ; Onsager announced this formula, and C. N. Yang published a proof in 1952.

Going deeper

For a student

The critical point agrees with the self-dual point of Kramers–Wannier duality, found three years earlier. Near $K_c$ the solution gives the critical exponent $0$ for the specific heat (a logarithmic divergence) and $1/8$ for the magnetization, which differ from mean-field values. These exact exponents became the main test case for scaling theory and the renormalization group. The method of transfer matrices used here also underlies the solutions of many later exactly solvable lattice models.

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Ising model example
Planar graph concept
Thermodynamic limit concept
Kramers–Wannier duality statement
Onsager's solution of the two-dimensional Ising model statement

Tag	`JJ4ZH5YD80`
URI	`m2://JJ4ZH5YD80`
Kind	statement, theorem
Writing	W0 stub
MSC	82B20, 82B23, 82B26
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About	Ising model
Uses	Ising model, Thermodynamic limit, Kramers–Wannier duality
Changed	3 November 2026