Ising model

A lattice spin model with binary spins (±1) and nearest-neighbor pair interactions, used as a paradigmatic model for phase transitions and symmetry breaking.

Ising model

The (nearest-neighbor) Ising model on $\mathbb{Z}^d$ has:

spin space $\{-1,+1\}$ ,
configurations $\sigma=(\sigma_x)_{x\in\mathbb{Z}^d}$ with $\sigma_x\in\{-1,+1\}$ ,
nearest-neighbor edges given by nearest neighbors .

For a finite region $\Lambda$ (see finite boxes ) and boundary condition $\eta$ , the standard finite-volume Hamiltonian is

H_{\Lambda}^{\eta}(\sigma)\;=\;-J\sum_{\substack{\{x,y\}\\ x\sim y\\ \{x,y\}\cap \Lambda\neq \emptyset}}\sigma_x\sigma_y\;-\;h\sum_{x\in\Lambda}\sigma_x,

where $J\in\mathbb{R}$ is the coupling and $h\in\mathbb{R}$ is the external field . The sum over edges crossing the boundary is interpreted using $\eta$ for spins outside $\Lambda$ (see boundary conditions ).

The associated finite-volume Gibbs measure at inverse temperature $\\beta$ is

\mu_{\Lambda,\beta,h}^{\eta}(\sigma)\;=\;\frac{1}{Z_{\Lambda}(\beta,h,\eta)}\exp\!\big(-\beta H_{\Lambda}^{\eta}(\sigma)\big),

with partition function $Z_{\Lambda}(\beta,h,\eta)$ .

Key properties

Spin-flip symmetry: At $h=0$ the Hamiltonian is invariant under $\sigma\mapsto -\sigma$ , leading to a $\mathbb{Z}_2$ symmetry that may be broken via SSB .
Interactions viewpoint: The model is specified by a finite-range, translation-invariant interaction potential (pair interaction on nearest-neighbor edges), fitting the framework of finite-range interactions and translation invariance .
Thermodynamics: The infinite-volume free energy density is encoded by the pressure as a thermodynamic limit of finite-volume pressures.
Correlations: Spatial dependence of order is studied via two-point correlations and the correlation length .
Phase transition behavior: For $d\ge 2$ and ferromagnetic coupling ( $J>0$ ), the model exhibits a low-temperature ordered regime with nonzero spontaneous magnetization (details depend on $d$ ). For $d=1$ with finite-range interactions, there is no finite-temperature phase transition.

Physical interpretation

Each spin represents a two-state degree of freedom (e.g. up/down magnetic moment). The pair term favors alignment if $J>0$ (ferromagnetism) or anti-alignment if $J<0$ (antiferromagnetism). Competition between energetic preference and thermal fluctuations produces sharp changes in macroscopic behavior, making the Ising model a central testbed for critical phenomena.