Order parameter

An order parameter is a quantity chosen to measure how much order a system has: zero in the symmetric, disordered phase and non-zero in the ordered phase. Its appearance is the phase transition. For a ferromagnet it is the magnetisation m; for a liquid–vapour system it is the density difference ρ_liq − ρ_gas; for a binary alloy it is the excess of one species on a sublattice; for a superfluid or superconductor it is a complex-valued wavefunction whose phase matters as well as its magnitude.

The concept is due to Landau, and its power is that it lets a transition be described without describing the substance. Writing the free energy as an expansion in the order parameter, keeping only terms the symmetry permits — F = F₀ + a t m² + b m⁴ for a system symmetric under m → −m — reproduces the transition mechanically: above T_c the minimum sits at m = 0, and below it the coefficient of m² turns negative and the minimum splits into two symmetric wells. The system must fall into one, breaking the symmetry the rule itself possessed.

The symmetry of the order parameter — whether it is a single real number, a two-component vector, or something larger — is one of the two things that determine a system's universality class, the other being spatial dimension. This is why an Ising magnet and a fluid, whose order parameters are both single real scalars, share critical exponents despite having nothing else in common.

Introduced by Lev Landau in his 1937 theory of phase transitions, which reframed the subject around symmetry rather than substance. The framework was extended to superconductivity by Ginzburg and Landau in 1950 and underlies the modern treatment of spontaneous symmetry breaking throughout physics.