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An order parameter distinguishes thermodynamic phases. For a ferromagnet it is the magnetization; for a fluid near its liquid-gas critical point it can be the density relative to its critical value. In a system with a scalar order parameter, introduce a slowly varying real scalar field $\varphi (x)$ whose spatial average represents that order parameter. The Landau-Ginzburg theory organizes the effective free energy as a local expansion consistent with the symmetry:Here $K>0$ is the stiffness in the gradient energy, and $H$ is the field conjugate to an order parameter. The even powers assume a symmetry $\varphi \mapsto -\varphi$ at $H=0$. Without that symmetry further terms, including a cubic term, may be allowed. The coefficients are smooth functions of temperature and other controls; near a mean-field transition one usually writes $r=a(T-T_0)$ with $a>0$. A positive highest retained even coefficient makes the local potential bounded below.

The partition function is obtained by weighting configurations with $\exp [-\mathcal{F}/(k_{B}T)]$. The Landau approximation replaces this functional average by minimization, taking a uniform $\varphi =M$ in a homogeneous phase. In a symmetric finite system the zero-field average can vanish even below the transition because it averages both signs of $M$. The spontaneous order parameter selects a pure thermodynamic phase by taking the thermodynamic limit before removing the conjugate field:Thus the order parameter measures the ordering of a selected thermodynamic phase, rather than the average over all symmetry-related phases.

A first-order phase transition has a discontinuity in a first derivative of the equilibrium free energy, such as entropy or the order parameter; a thermal first-order transition normally has latent heat. At a continuous phase transition the order parameter goes continuously to zero and there is no latent heat, although response functions and higher derivatives of the free energy can be singular.

For the symmetric scalar Landau free energy at zero conjugate field, minimizeIf $u>0$, the sixth-order term is unnecessary near $r=0$. For $r>0$ the minimum is $M=0$, whereas for $r<0$ there are two symmetry-related minima with $M^2=-r/u$ to leading order. The order parameter vanishes continuously as $r\uparrow 0$. The equilibrium free-energy density changes from zero to $-r^2/(4u)$, so its first temperature derivative is continuous and its second derivative has a jump. This is the ordinary mean-field continuous phase transition.

If $u<0$, keep $v>0$ for stability. A nonzero stationary order parameter, with $s=M^2$, obeys $r+us+v{s}^2=0$. For phase coexistence with $M=0$ its free energy must also vanish. Substituting $r=-us-v{s}^2$ into $V$ givesThe nonzero solution of $V=0$ is thereforeThe competing minima exchange stability at positive $r_t$, with a finite jump of the order parameter. This is a first-order phase transition. The metastability region is also visible: nonzero stationary points first appear at the spinodal $r=u^2/(4v)$, and the disordered local minimum loses stability at $r=0$.

For example, holding $u,v$ fixed and taking $r=a(T-T_0)$, the ordered phase has entropy lower than the disordered phase by $a{M}_t^2/2$ at coexistence. Heating through the transition absorbs latent heat per volume $T_{t}a{M}_t^2/2$. More generally the temperature dependence of all coefficients contributes to this entropy difference. With no inversion symmetry, a cubic term can also produce a first-order phase transition, as comparison of the competing minima of the Landau free energy shows.

The relevant boundary here is the upper critical dimension: above it the critical interactions become negligible at long distances and the ordinary mean-field critical exponents apply; at it there can be logarithmic corrections. This differs from the lower critical dimension, below which a particular kind of ordering may fail to exist.

Normalize the coefficient of $(\nabla \varphi {)}^2$ in the Landau-Ginzburg theory. Under a change of length scale $x=b{x}^{\prime }$, its invariance gives the engineering dimension $x_{\varphi }=(d-2)/2$. The quartic interaction then scales asIt is a relevant operator below four dimensions and an irrelevant operator above four. Thus the ordinary scalar transition has upper critical dimension $d_c=4$. At a tricritical point, where the quartic interaction is tuned away, the sextic coupling scales as $v^{\prime }=b^{6-2d}v$, giving the tricritical upper critical dimension $d_c=3$. This is the upper critical dimension of an even scalar interaction applied to powers four and six.

The Ginzburg criterion makes the reason for failure of the Landau approximation quantitative. In the ordered quartic theory, $M^2=|r|/u$, $V^{\prime \prime }(M)=2|r|$ and ${\xi }^2=K/(2|r|)$. The fluctuation of the order parameter averaged over a correlation volume has the scaleCompared with the squared mean order parameter, the Ginzburg criterion therefore givesFor $d<4$ this ratio grows without bound as the correlation length diverges. The assumption that fluctuations are small relative to the ordered value fails arbitrarily near the transition, and the interacting renormalization-group fixed point determines the critical exponents. For $d>4$ the ratio tends to zero. At $d=4$ scale counting is marginal, and the running quartic coupling produces logarithms.

Similarly, at the tricritical point $M^2\asymp \sqrt{K/v}{\xi }^{-1}$, so the multicritical Ginzburg ratio scales as $(k_{B}T\sqrt{v}/K^{3/2}){\xi }^{3-d}$. The Landau-Ginzburg theory remains a useful effective description below its upper critical dimension; it is its fluctuation-free exponent calculation that fails, not the possibility of including fluctuations in the theory.

Spontaneous symmetry breaking occurs when the statistical Hamiltonian is invariant under a symmetry but a chosen equilibrium pure state is not. A small selecting field or a boundary condition chooses one of several states related by the symmetry. For a finite unconstrained system the symmetric ensemble generally has zero order parameter; spontaneous order is defined by taking the thermodynamic limit before removing the selecting field. A ferromagnet with a selected magnetization direction is the standard example.

If the broken symmetry is continuous, nearby symmetry-related states cost no energy for a uniform displacement along the manifold of states. Locality and a nonzero stiffness then make a slowly varying displacement cost gradient energy rather than a mass term. These gapless long-wavelength fluctuations are Goldstone modes. In the conventional relativistic internal-symmetry setting their count is the number of broken generators. Nonrelativistic dynamics can pair broken generators, as in a ferromagnet; gauge symmetries and broken space-time symmetries require additional qualifications. A discrete broken symmetry alone does not produce a Goldstone mode.

For an interface with a selected mean height, the continuous displacement is a rigid translation normal to the mean plane. The tension gives a stiffness for spatially varying translations. The following calculation concerns this static Goldstone sector. It determines the excitation energy and equilibrium fluctuations, but a frequency dispersion requires a separate inertial or dissipative dynamics. In low dimensions thermal roughness also limits the existence of a strictly localized flat interface in an infinite system.

The scaling hypothesis for critical phenomena describes the singular part of the equilibrium free-energy density, rather than its analytic background. For an isotropic system in spatial dimension $D$, with thermal scaling field $t$ and conjugate field $h$, writeThis follows from coarse graining near a renormalization-group fixed point: a correlation volume is enlarged by $b^D$, while the relevant thermal and field perturbations grow with their eigenvalues. Choosing $b=|t{|}^{-1/y_t}$ givesThe two functions describe the two sides of the transition. Nonuniversal metric factors multiplying $t,h,f_s$ must be allowed when comparing microscopic models. Exponents and suitably normalized scaling functions or amplitude ratios characterize a universality class.

Differentiating the free-energy density gives the order parameter $M=-{\partial }_h{f}_s$, magnetic susceptibility $\chi ={\partial }_{h}M$ and singular heat capacity proportional to $-{\partial }_t^2{f}_s$. ThusThe subscript distinguishes the order-parameter exponent from inverse temperature. At $t=0$, choose $b=|h{|}^{-1/y_h}$ to obtain $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, with $\delta =y_h/(D-y_h)$. These derivatives imply the Rushbrooke scaling relation $\alpha +2{\beta }_{\mathrm{mag}}+\gamma =2$ and the Widom scaling relation $\gamma ={\beta }_{\mathrm{mag}}(\delta -1)$. Equivalently, the equation of state has $h=|t{|}^{\Delta }{\mathcal{H}}_{\pm }(M/|t{|}^{{\beta }_{\mathrm{mag}}})$, with $\Delta ={\beta }_{\mathrm{mag}}\delta ={\beta }_{\mathrm{mag}}+\gamma$.

The correlation length scales as $\xi \sim |t{|}^{-\nu }$, $\nu =1/y_t$. At criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$, giving $y_h=(D+2-\eta )/2$. Integrating $G$ over a correlation volume gives $\chi \sim {\xi }^{2-\eta }$ and hence the Fisher scaling relation $\gamma =(2-\eta )\nu$. The hyperscaling relation $2-\alpha =D\nu$ expresses one singular free-energy contribution per correlation volume.

These laws organize measurements: plotting $M/|t{|}^{{\beta }_{\mathrm{mag}}}$ against $h/|t{|}^{\Delta }$ collapses near-critical curves onto the appropriate scaling function. A finite system replaces the divergent length by its size $L$; the corresponding prediction is $f_s=L^{-D}\mathcal{F}(t{L}^{y_t},h{L}^{y_h})$ under the same scaling assumptions. Finite size rounds a sharp thermodynamic singularity.

The hypothesis has a domain of validity. At a renormalization-group fixed point one generally needs $f_s=b^{-D}{f}_s(b^{y_t}t,b^{y_h}h,b^{y_u}u,\dots )$, including other scaling fields. Ordinary irrelevant couplings can be set to zero in a nonsingular limiting scaling function. A dangerously irrelevant coupling cannot: above the upper critical dimension of scalar quartic theory, the quartic term is irrelevant at the Gaussian point but still stabilizes the ordered phase. Consequently naive hyperscaling fails there even though other exponent relations can remain valid. Marginal variables can produce logarithms, anisotropic systems require the appropriate volume exponent, and the Berezinskii–Kosterlitz–Thouless transition has an essential correlation-length singularity instead of a finite power-law $\nu$. Scaling theory relates observables through a small number of fixed-point dimensions; it does not make analytic backgrounds, marginal logarithms or dangerously irrelevant variables disappear.

The Berezinskii–Kosterlitz–Thouless transition is a defect-unbinding transition of a two-dimensional system with a compact continuous phase, such as the short-range XY model. At long wavelengths, smooth spin waves have energywhere ${\rho }_s$ is the phase stiffness. Gaussian averaging givesThus smooth fluctuations already destroy true nonzero magnetization in an infinite two-dimensional system, in accordance with the Mermin-Wagner theorem. The low-temperature phase instead has quasi-long-range order, with algebraically decaying correlations. The stiffness in its asymptotic exponent is the renormalized long-distance stiffness, not merely the microscopic coupling.

The phase is periodic modulo $2\pi$, so it also permits topological defects with integer winding $n$. A phase vortex has $|\nabla \theta |=|n|/r$, and integration givesA singly charged phase vortex can be placed in roughly $(L/a{)}^2$ locations, giving entropy $2k_B\log (L/a)$. This energy-entropy balance suggests that isolated vortices become favorable when the effective stiffness falls to $2k_{B}T/\pi$. Below the transition, vortex-antivortex binding predominates; above it their unbinding screens the logarithmic interaction and produces a finite correlation length. A single-vortex estimate identifies the mechanism but does not include screening by smaller pairs.

The dilute-vortex vortex-pair renormalization flow incorporates this screening. With $K={\rho }_s/(k_{B}T)$ and dimensionless core fugacity $y$, one conventional leading-order normalization givesThe first equation says polarizable phase vortex pairs reduce stiffness. The second compares the phase vortex's position scaling dimension two with its logarithmic energy cost $\pi K$. Numerical coefficients in $y$ depend on its definition, but the zero-fugacity threshold $K=2/\pi$ is invariant. For $K>2/\pi$, sufficiently small fugacity flows to zero, leaving a line of Gaussian spin-wave renormalization-group fixed points with varying algebraic exponent. On the other side of the separatrix, fugacity grows and the dilute approximation eventually ceases to apply.

Near the endpoint set $X=2-\pi K$ and $Y=4\pi y$. To leading order $X^{\prime }=Y^2$ and $Y^{\prime }=XY$, so $X^2-Y^2$ is invariant. The critical separatrix approaches $(0,0)$. Just on the disordered side this invariant is $-{\epsilon }^2$, proportional to the reduced temperature deviation for a generic path through the transition. Hence $X^{\prime }=X^2+{\epsilon }^2$, whose integration gives a time in logarithmic scale of order $1/\epsilon$. Stopping when the fugacity becomes order one yieldswith nonuniversal $B>0$. This derives the essential singularity and explains why assigning a finite ordinary correlation-length exponent is inappropriate.

At the transition the limiting stiffness from below has the universal stiffness jumpAbove $T_c$, the thermodynamic long-distance stiffness vanishes and correlations decay exponentially. The singular free-energy contribution is of order ${\xi }^{-2}$ and is exponentially small in $1/\sqrt{\tau }$, so this infinite-order transition is unlike a Landau transition with a discontinuous order parameter or an ordinary power-law heat-capacity singularity. The renormalization mechanism is developed in Kosterlitz's original vortex analysis. The transition separates bound vortices and algebraic order from unbound vortices and short-range order, while neither finite-temperature phase has conventional spontaneous XY magnetization.

The Landau-Ginzburg theory for a scalar order parameter supplements a local symmetry-allowed potential with a gradient penalty. For a ${\mathbb{Z}}_2$ symmetry at zero field, a stable quartic model iswith $r=a(T-T_c)$ near a mean-field transition. Equilibrium fluctuations have weight $e^{-F/(k_{B}T)}$. In the Landau approximation, minimize the functional. A uniform stationary state obeys $r\varphi +u{\varphi }^3=h$. At zero field the minimum is $\varphi =0$ for $r>0$ and $\varphi =\pm \sqrt{-r/u}$ for $r<0$. The latter pair is spontaneous symmetry breaking of the discrete symmetry.

The uniform free-energy density at its minimum is zero above the transition and $-r^2/(4u)$ below, up to regular backgrounds. Differentiating the equation of state gives ${\chi }_{+}=1/r$ and ${\chi }_{-}=1/(2|r|)$. At $r=0$, $\varphi =(h/u{)}^{1/3}$. Gaussian fluctuations around the minimum have inverse covariance $(K{q}^2+r)/(k_{B}T)$ above and $(K{q}^2+2|r|)/(k_{B}T)$ below. Their correlation lengths are ${\xi }_{+}=(K/r{)}^{1/2}$ and ${\xi }_{-}=(K/(2|r|){)}^{1/2}$. These results give the usual mean-field critical exponentsThe heat capacity has a finite jump rather than a power-law divergence. Negative quartic coefficient requires higher stabilizing terms and can produce a first-order transition; a vanishing quartic coefficient requires a tricritical analysis. Thus the assumed positivity of $u$ matters.

The Ginzburg criterion tests whether this saddle description is self-consistent. Below the transition compare fluctuations of the order parameter averaged over a correlation volume with its squared mean-field value ${\varphi }_0^2=|r|/u$. Modes with $q\lesssim {\xi }^{-1}$ dominate this coarse-grained fluctuation estimate:where the positive dimensionless $A_D$ depends on the averaging window and the factor of two used in $\xi$. Dividing by ${\varphi }_0^2$ givesIt is the fluctuation of a correlation-volume average, not the full cutoff-dominated microscopic variance, that should be compared with the mean-field order parameter.

For $D<4$, $\mathcal{G}$ diverges as $r\to 0$, so the asymptotic critical region is fluctuation dominated. Away from it, mean-field theory can still work whenFor $D>4$, the long-wavelength ratio tends to zero near criticality, consistently with Gaussian control and mean-field thermodynamic exponents. At $D=4$, quartic coupling is marginal; scale-dependent logarithms require a renormalization-group analysis rather than interpreting a constant bare ratio as a proof of no corrections. This identifies four as the upper critical dimension of the short-range scalar quartic model.

The Landau-Ginzburg theory is valuable because symmetries restrict a manageable list of local operators, the gradient term describes correlations and interfaces, and the Ginzburg criterion identifies when neglected fluctuations become decisive. Below the upper critical dimension an interacting renormalization-group fixed point, rather than the bare saddle, determines universal exponents. For continuous order parameters the manifold of ordered minima also has Goldstone modes, and lower-critical-dimension fluctuations require separate attention. Mean-field minimization supplies an approximation; the Ginzburg ratio decides whether its neglect of fluctuations is justified.

An order parameter distinguishes equilibrium thermodynamic phases. For a scalar magnetic system, a slowly varying coarse-grained scalar field $\varphi (x)$ represents local magnetization, and its uniform expectation $M=⟨\varphi ⟩$ is the order parameter. Under spin inversion symmetry, $\varphi \mapsto -\varphi$ and $h\mapsto -h$. The disordered phase has $M=0$ at $h=0$, while a pure ordered phase has $M\ne 0$. In a finite symmetric system the average still vanishes at $h=0$; spontaneous symmetry breaking means taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$ to select a pure thermodynamic phase.

Landau-Ginzburg theory assigns a local free-energy functionalSpin inversion symmetry forbids odd powers at zero conjugate field; locality and slow variation motivate the gradient expansion. The gradient energy penalizes interfaces and determines long-wavelength response. Integrating $e^{-\mathcal{F}/(k_{B}T)}$ over fields includes fluctuations, whereas the Landau approximation replaces that integral by its stable stationary configuration. For a uniform phase this reduces to minimizing $V(M)$. If $f$ is the equilibrium free-energy density, then $M=-\partial f/\partial h$, so the order parameter is also the thermodynamic response to its conjugate field.