Onsager solution of the 2D Ising model (zero field)

Exact infinite-volume free energy for the 2D nearest-neighbor Ising model on the square lattice and identification of the critical point.

Onsager solution of the 2D Ising model (zero field)

Context

The Ising model on the 2D square lattice is the canonical exactly solvable interacting lattice system exhibiting a genuine phase transition at positive temperature.

Consider the nearest-neighbor ferromagnet (see ferromagnetic Ising ) with Hamiltonian

H(\sigma) = -J\sum_{\langle i,j\rangle}\sigma_i\sigma_j , \qquad \sigma_i\in\{-1,+1\},\quad J>0,

in finite volume with partition function (see canonical partition function )

Z_\Lambda(\beta)=\sum_{\sigma_\Lambda}\exp\bigl(-\beta H_\Lambda(\sigma)\bigr).

The (Helmholtz) free energy density is the thermodynamic limit (see statistical free energy )

f(\beta)= -\frac{1}{\beta}\lim_{|\Lambda|\to\infty}\frac{1}{|\Lambda|}\log Z_\Lambda(\beta),

when the limit exists.

Theorem (Onsager, zero external field)

Let $K=\beta J$ and define the modulus

k(K)=\frac{2\sinh(2K)}{\cosh^2(2K)}.

For the 2D square-lattice Ising model at zero field, the infinite-volume free energy density exists and satisfies the exact formula

-\beta f(\beta) ={} \ln\!\bigl(2\cosh(2K)\bigr) +\frac{1}{2\pi}\int_0^\pi \ln\!\left( \frac{1+\sqrt{1-k(K)^2\sin^2\theta}}{2} \right)\,d\theta.

Moreover, the nonanalyticity occurs precisely at

\sinh\!\bigl(2K_c\bigr)=1 \qquad\Longleftrightarrow\qquad K_c=\frac12\ln(1+\sqrt2),

equivalently at $T_c=J/(k_B K_c)$ (using temperature ).

Key consequences

Second-order transition: the free energy is nonanalytic at $T_c$ , yielding a continuous transition with a logarithmic divergence of the specific heat:
$C_V(T)\sim -A\log|T-T_c| \quad \text{as } T\to T_c,$
where $C_V$ is the heat capacity at constant volume .
Spontaneous magnetization (Yang): below $T_c$ the spontaneous magnetization is strictly positive and has the closed form
$m(\beta)=\bigl(1-\sinh^{-4}(2K)\bigr)^{1/8} \quad \text{for } K>K_c,$
while $m(\beta)=0$ for $K\le K_c$ .
Exact critical data: the solution provides a benchmark for universality class ideas, critical exponents , and scaling relations .

Prerequisites and connections (cross-links)

Lattice setup and Gibbs measures: lattice Hamiltonian , finite-volume Gibbs measure , infinite-volume Gibbs measure .
Thermodynamic potentials: Helmholtz free energy , pressure as log-partition density .
Order parameters and transitions: order parameter , phase transition indicators .