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A symmetry generated by $Q^a$ is spontaneously broken when the vacuum is not invariant:A local order parameter is obtained by finding a field ${\phi }_m$ for whichwith the sign depending on the commutator convention. This criterion is useful in infinite volume because it can remain finite even when the spatially integrated charge itself is not a well-defined normalizable operator.

Landau theory treats the order parameter as spatially uniform and expands the free-energy density in powers allowed by its symmetries. The Landau-Ginzburg theory promotes it to a field $\varphi (\mathbf{x})$ and adds gradient terms such as $|\nabla \varphi {|}^2$. It therefore describes spatial fluctuations, interfaces, defects, and correlation lengths, while reducing to Landau theory for uniform fields.

Put $t=T-T_c$, so $a_4\sim t$. At zero field the stationary equation factors asFor $t>0$, $m=0$ is the unique minimum. For $t<0$, it is unstable and the two minima areThe order parameter therefore tends continuously to zero, and its order-parameter critical exponent is $\boxed{\beta =1/2}$.

At either ordered minimum the singular free-energy density iswhereas it is zero for $t>0$. Two temperature derivatives give a singular heat capacity proportional to $|t|$ below the transition and zero above it, so the heat-capacity critical exponent is $\boxed{\alpha =-1}$.

The inverse magnetic susceptibility at a stable minimum is the curvature ${\chi }^{-1}={\partial }^{2}f/\partial m^2$. Below $T_c$,and hence ${\chi }_{-}\sim |t{|}^{-2}$ and $\boxed{{\gamma }_{-}=2}$. Above $T_c$, however, the curvature at $m=0$ vanishes for every $t>0$. Indeed, at small field $B=4a_4{m}^3+O(m^5)$, so $m\sim (B/(4a_4){)}^{1/3}$ and the linear susceptibility is already infinite away from the critical point. Consequently the usual magnetic-susceptibility critical exponent ${\gamma }_{+}$ is not defined for this exceptional free energy; assigning it a finite value would incorrectly assume a quadratic term.

At $T=T_c$, the equation of state is $B=6a_6{m}^5$, so $m\sim |B{|}^{1/5}$ and the critical-isotherm exponent is $\boxed{\delta =5}$. Thus the transition is continuous, although its missing quadratic term makes the high-temperature linear response singular throughout that phase.

Subtract the zero-temperature term by using $\coth (x/2)=1+2/(e^x-1)$. In three dimensions the thermal part iswhere the last integral uses the Bose-Einstein distribution. Gaussian phase fluctuations reduce the order parameter by $⟨e^{-i\theta }⟩=e^{-⟨{\theta }^2⟩/2}$. Taking symmetry restoration to occur when the thermal variance is of order one givesThe effective action in part c has $\chi =1/(2\lambda )$, and hence $T_{\ast }\sim \sqrt{6v^3/\lambda }$. This is an order-of-magnitude estimate because near restoration the phase-only effective field theory omits large amplitude fluctuations and sensitivity to its ultraviolet cutoff.

The thermodynamic order parameter is the conserved scalar composition difference $\varphi$ of the binary fluid mixture. The momentum density $\rho \mathbf{v}$ is an additional conserved hydrodynamic mode; it is not a symmetry-breaking order parameter, but it must be retained because momentum relaxes only through spatial transport.

Composition conservation gives a continuity equation,The current $\mathbf{J}=-M\nabla \mu$ is the leading isotropic, dissipative constitutive law: it drives material down gradients of the chemical potential. For the Landau-Ginzburg theorythe functional derivative isThese statements give the advective Cahn--Hilliard equation.

Constant mass density and incompressible flow require $\nabla \cdot \mathbf{v}=0$. Momentum conservation gives the Navier-Stokes equation: material acceleration equals the sum of the Newtonian viscous force $\eta {\nabla }^2\mathbf{v}$, the pressure force $-\nabla P$, and the Korteweg force $-\varphi \nabla \mu$. The pressure is the Lagrange multiplier enforcing incompressibility. Together these equations are Model H dynamics; isothermality removes the need for a separate energy equation.

Landau-Ginzburg theory assumes a local gradient expansion for a slowly varying order parameter. At zero field the microscopic spin-flip symmetry sends $m\mapsto -m$, so only even powers occur. Rotational symmetry makes the leading derivative term $(\nabla m{)}^2$, while locality and analyticity near the transition organize higher powers and derivatives. Stability requires a positive gradient coefficient and, at an ordinary continuous transition, ${\alpha }_4(T_c)>0$; the coefficient ${\alpha }_2(T)$ changes sign at $T_c$.

A nonrelativistic Landau-Ginzburg theory for a complex order parameter with global $U(1)$ symmetry isIn the broken phase $r<0$, write $\psi =\sqrt{n}{e}^{i\theta }$. The first term isup to the sign convention for the phase. Thus the momentum conjugate to $\theta$ is $-n$: this is number-phase conjugacy. Canonical quantization gives the corresponding local commutator, while the spatial integral relates the global phase to total particle number.

Consequently a state with a sharply selected phase, as used to describe spontaneous symmetry breaking, cannot be an eigenstate of particle number. It is a coherent superposition of charge sectors with number fluctuations. Conversely, an exact finite-volume number eigenstate has no definite phase; the phase-selected broken-symmetry states emerge in the thermodynamic limit.

At a fixed point, the order parameter can be chosen as a dilation eigenoperator. Therefore, for $x^{\prime }=x/\zeta$,Interactions shift ${\Delta }_{\varphi }$ from $(d-2)/2$ to $(d-2+\eta )/2$.

A system can split into macroscopic regions with distinct local order parameters, densities or compositions. In a short-range system at phase coexistence, regions of two phases with means $M_1,M_2$ and volume fractions $q,1-q$ have spatial mean $M=q{M}_1+(1-q)M_2$. Their interface contribution is subextensive, so their thermodynamic free-energy density is the corresponding convex combination of the phase densities. This can lie below the homogeneous-branch Landau free energy at the same $M$. The equilibrium constrained order-parameter free energy is consequently convexified, with a straight coexistence segment described by the Maxwell construction. A nonconvex local potential remains useful for homogeneous branches, metastability and interfaces, but must not be identified with the full equilibrium constrained density throughout a coexistence interval.

A vortex is a topological defect around which the phase of a complex order parameter has nonzero integer winding. In two dimensions an isolated vortex in a phase-only model has energy proportional to the logarithm of the system size.

A quench is a change of control parameters faster than the system can equilibrate. The subsequent relaxation can trap domains and topological defects.

A quench abruptly changes temperature or another control parameter and then studies the subsequent nonequilibrium relaxation of the order parameter.

For a Landau-Ginzburg theory with positive gradient stiffness $\kappa$, $r<0$ and quartic coupling $u>0$, the degenerate minima are $\pm M_0$, $M_0^2=|r|/u$. A planar domain wall solves $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$. Multiplying by ${\varphi }^{\prime }$ and using the limiting minima gives $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$. Separating this equation yields the displayed profile, and integrating its excess energy gives the positive surface tension $2\sqrt{2\kappa }|r{|}^{3/2}/(3u)$. Its area cost is subextensive compared with bulk volume, permitting phase-separated common-tangent construction states when the average order parameter is constrained.

A thermal phase fluctuation is a finite-temperature fluctuation of the phase of a complex order parameter. For a linearly dispersing phase mode, its infrared variance is proportional to $T\int d^{d}k/k^2$, which prevents true finite-temperature long-range order when $d\le 2$.

At an ordinary thermodynamic critical point, two coexisting phases become indistinguishable: their order parameter difference vanishes, and the correlation length and appropriate response functions typically diverge. A continuous phase transition can be approached there by tuning the thermal and conjugate-field variables. A renormalization-group fixed point controls its singular long-distance behavior. A triple point with three separated minima is not an ordinary thermodynamic critical point, and a tricritical point requires an additional tuning and has a different scaling theory.

A thermodynamic phase is a form of macroscopic equilibrium characterized by its structure, symmetries and thermodynamic response. Away from phase transitions, its free-energy density varies smoothly with the control parameters. Different phases can share a phase coexistence boundary. An ordered magnetic phase is distinguished from a disordered phase by its spontaneous order parameter; positive and negative ordered states are different pure thermodynamic phases related by spin inversion symmetry.

For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

A vector order parameter has several components transforming together as a vector under spatial rotations. A polar order parameter is an example. Its functional derivative $\delta F/\delta \mathbf{p}$ has one component for each component of the order parameter.