M

distinguishes macroscopic phases: for an Ising model it is the magnetization per site, while near a liquid–gas endpoint it can be a shifted density. An ordered pure thermodynamic phase has a nonzero value even when the conjugate field is removed. For a symmetry-related pair, take the thermodynamic limit before

h \to 0^{\pm}

; a finite-volume symmetric ensemble instead has zero average.

Coarse-grain to a slowly varying real field and use a local, symmetry-constrained free-energy density. A useful convention is

F [M] = \int d^{D} x [\frac{K}{2} (\nabla M)^{2} + \frac{r}{2} M^{2} + \frac{u}{4} M^{4} + \frac{v}{6} M^{6} - h M], K > 0.

For spin inversion symmetry, odd powers other than the externally imposed source are absent. Take

v > 0

when the quartic coefficient can be negative; for an ordinary transition

u > 0

already stabilizes the quartic truncation. The gradient energy penalizes spatial variations. In the Landau approximation, minimize this functional, usually first over constant fields. The corresponding statistical field theory also integrates fluctuations with weight