orderparameter

An order parameter is a quantity that characterizes the degree of order across a phase transition. In many systems it is zero in the disordered phase and becomes nonzero when order develops, signaling spontaneous symmetry breaking.

Mathematically, the order parameter is often the expectation value of an operator that transforms nontrivially under the symmetry of the high-temperature phase. If the symmetry is unbroken, the expectation value ⟨O⟩ vanishes; if it is spontaneously broken, ⟨O⟩ is nonzero. In practice, the order parameter can be a scalar, vector, or complex quantity, and it may vary in space. In theory, the order parameter is treated as a field φ(x) and used in Landau-Ginzburg or Landau theories to build a free-energy functional F[φ]. The behavior of φ near the transition controls critical phenomena and universality classes.

Examples include magnetization M in a ferromagnet (scalar); the superconducting or superfluid order parameter, a complex pair amplitude ψ or Δ, reflecting phase coherence; the density difference between coexisting phases in a liquid–vapor transition; and a crystalline order parameter describing the emergence of periodic density in a solid.

In experiments, order parameters are inferred from measurements such as magnetization, scattering patterns that reveal long-range order, or phase coherence in superconductors. Some systems nevertheless possess no local order parameter, or feature topological order that is not captured by a conventional order parameter.