2D Ising model: finite-temperature phase transition

Square-lattice Ising model with Onsager’s critical temperature, spontaneous magnetization below Tc, and diverging correlation length at criticality.

2D Ising model: finite-temperature phase transition

Prerequisites and notation

Model definition: Ising model , ferromagnetic Ising interaction
Gibbs formalism: finite-volume Gibbs measure , infinite-volume Gibbs measure , DLR equations
Phase transition notions: phase transition (Gibbs measure non-uniqueness) , order parameter
Correlations: correlation length , structure factor

We consider spins $\sigma_x\in\{-1,+1\}$ on $\mathbb{Z}^2$ (or on an $L\times L$ torus for finite volume).

Example: 2D square-lattice nearest-neighbor Ising model

Hamiltonian

For a finite region $\Lambda\subset\mathbb{Z}^2$ ,

H_\Lambda(\sigma) = -J\sum_{\langle x,y\rangle\subset\Lambda} \sigma_x\sigma_y - h\sum_{x\in\Lambda}\sigma_x, \qquad J>0,

where $\langle x,y\rangle$ are nearest-neighbor edges.

Critical temperature

At zero field $h=0$ , the 2D nearest-neighbor model has a critical inverse temperature

\beta_c J = \frac{1}{2}\log(1+\sqrt{2}), \qquad T_c = \frac{1}{k_B\beta_c}.

For $T>T_c$ the equilibrium state is unique; for $T<T_c$ there are (at least) two distinct infinite-volume Gibbs measures corresponding to the two magnetized phases.

Spontaneous magnetization (order parameter)

The magnetization per site (in the thermodynamic limit) is the natural order parameter :

m(\beta) = \lim_{h\downarrow 0}\lim_{|\Lambda|\to\infty}\frac{1}{|\Lambda|}\sum_{x\in\Lambda}\langle \sigma_x\rangle_{\Lambda,\beta,h}.

At $h=0$ , the exact spontaneous magnetization for $\beta>\beta_c$ is

m(\beta) = \left(1-\sinh^{-4}(2\beta J)\right)^{1/8}, \qquad \beta>\beta_c,

and $m(\beta)=0$ for $\beta\le \beta_c$ . This sharp onset is a hallmark of a phase transition and contrasts with the 1D chain .

Correlation length and criticality

For $T\neq T_c$ , correlations decay exponentially and define a finite correlation length $\xi(T)$ ; at criticality $T=T_c$ the correlation length diverges, and correlations decay by a power law. Equivalently, the small- $k$ behavior of the structure factor becomes singular at $T_c$ .

Thermodynamic singularities

The free energy density is nonanalytic at $T_c$ . A standard thermodynamic signature is the specific heat: in 2D it diverges logarithmically at criticality (in contrast to the smooth behavior in 1D).

Gibbs-measure viewpoint (DLR and coexistence)

Below $T_c$ , distinct infinite-volume Gibbs measures $\mu^+$ and $\mu^-$ solve the DLR equations and satisfy

\mu^+(\sigma_0)=+m(\beta),\qquad \mu^-(\sigma_0)=-m(\beta).

This non-uniqueness is the equilibrium meaning of a phase transition .