Order parameter

An order parameter is a map from equilibrium states (typically infinite-volume Gibbs measures ) to a finite-dimensional quantity (often $\mathbb{R}$ or $\mathbb{R}^k$) that separates phases.

In lattice spin systems, a common choice is the expectation of a (quasi-)local observable $O$:

where $O$ is chosen so that $\mathcal{M}$ differs across distinct pure phases (extremal Gibbs measures ). For the Ising model at $h=0$, the canonical example is magnetization with $O=\sigma_0$, leading to spontaneous magnetization .

Key properties

• Phase discrimination: If $\mu_1,\mu_2$ are distinct equilibrium phases, a good order parameter satisfies $\mathcal{M}(\mu_1)\neq \mathcal{M}(\mu_2)$.
• Symmetry behavior: In models with a symmetry group $G$, an order parameter is often chosen to transform nontrivially under $G$ so that $\mathcal{M}(\mu)=0$ in a $G$-invariant phase but $\mathcal{M}(\mu)\neq 0$ in phases exhibiting spontaneous symmetry breaking .
• Dependence on state selection: When multiple Gibbs measures exist, $\mathcal{M}(\mu)$ can depend on boundary conditions and on how limits are taken (see boundary conditions and finite-volume Gibbs measures ).
• Fluctuations and response: The response of an order parameter to a conjugate field is captured by linear response, often tied to the susceptibility and to two-point correlation functions .

Physical interpretation

An order parameter measures “how ordered” the system is with respect to a proposed pattern (alignment, staggered alignment, rotational ordering, clustering, etc.). In disordered phases, microscopic fluctuations average out so the order parameter is typically zero (or symmetry-forced to vanish). In ordered phases, correlations persist over long distances, producing a stable macroscopic signal in the order parameter.