Landau-Ginzburg theory

Landau-Ginzburg theory promotes an order parameter to a spatial field and includes an energy cost for its variation. For a scalar field,The local terms select preferred phases, while the gradient term makes abrupt spatial changes costly. Minimizing this free-energy functional can describe a smooth domain wall between two ordered regions.

Expanding around a uniform stable minimum gives a quadratic fluctuation kernel and an Ornstein--Zernike correlation function. The framework therefore connects an equilibrium phase diagram with measurable spatial correlations and interfaces. Minimization alone is a mean-field treatment; integrating fluctuations can change the predictions and is central to the renormalization group. The derivative expansion assumes fields varying slowly compared with microscopic lengths.

A diffuse-interface free energy balances the bulk preference for coexisting compositions against a positive penalty for sharp composition gradients. For dimensionless composition $\varphi$, $f_b$ is a bulk energy density and $K$ has units of energy per length. Subtract the bulk coexistence common tangent to define $\Delta \omega (\varphi )$, with equal zero minima at the two compositions. A planar equilibrium profile obeys $K{\varphi }^{\prime \prime }=\Delta {\omega }^{\prime }(\varphi )$ and $K({\varphi }^{\prime }{)}^2/2=\Delta \omega (\varphi )$. Its surface tension is $\int K({\varphi }^{\prime }{)}^{2}dz$. As a thermodynamic critical point is approached, the composition contrast and bulk curvature vanish, the correlation length grows, and the intrinsic interface broadens. A scalar quartic domain wall gives the mean-field hyperbolic-tangent profile. Thermally excited capillary waves can broaden a measured profile further.

For a real statistical source convention $Z[h]=\int \mathcal{D}\varphi e^{-H_0[\varphi ]+\int h\varphi }$, define $F=-\log Z$ and the displayed Legendre transform. On a differentiable stable branch, $\delta \Gamma /\delta m=h$. In the Landau approximation, the branch expression is ${\Gamma }_{\mathrm{L}}[m]=H_0[m]$. The exact Legendre effective free energy is convex; a bare nonconvex Landau free energy is a local approximation and requires branch or coexistence interpretation.

The Landau approximation evaluates a statistical field's partition function at a stable stationary configuration, omitting fluctuation-loop corrections to the free energy. Functional differentiation of that stationary value still gives a connected linear response: the inverse of the saddle's quadratic fluctuation operator. Freezing fields before differentiating would lose this response. A Gaussian field theory, in contrast, permits exact integration of its quadratic fluctuations.

With positive $g$, the quadratic coefficients select the component with the smaller negative coefficient. Equal negative coefficients give a circle of minima in the mean-field approximation; the negative diagonal is a spin-flop transition, ending at a bicritical point at the origin.

A Landau free energy can be expanded in local powers of the order parameter and its spatial derivatives. Internal and spatial symmetries constrain the allowed terms; truncation assumes slowly varying fields.

For a statistical scalar theory with gradient coefficient $K/2>0$, this rescaling gives a canonical $(\nabla \psi {)}^2/2$ term. A mass coefficient $r$ becomes $r/K$, a quartic coefficient $g$ becomes $g/K^2$, and a source $h$ becomes $h/\sqrt{K}$. Thus the simple propagator denominator $p^2+R$ and loop-coupling formulas assume these normalized coefficients. The associated measure Jacobian is field independent and belongs to the free-energy normalization.

A gradient energy penalizes spatial variation of an order parameter. Its leading isotropic form is usually proportional to the squared spatial gradient.

A gradient expansion orders a local effective free energy by the number of spatial derivatives of slowly varying fields. Symmetry determines the allowed derivative contractions.

Interfacial tension is the excess equilibrium free energy per unit area associated with an interface between phases, relative to the corresponding bulk phases. For a nearly flat interface of area $A$, its leading contribution is $F_{\mathrm{int}}=\sigma A$ with tension $\sigma$.

Increasing area therefore costs free energy, explaining why a droplet tends toward a compact shape when other forces are negligible. A curved interface also produces capillary pressure; for a spherical interface with isotropic tension, its magnitude is $2\sigma /R$. In nucleation, the area cost competes with the volume-dependent gain from forming the stable phase. In a Landau-Ginzburg theory, the tension is computed from the equilibrium order-parameter profile across the interface, including both gradient and local free-energy contributions.

For $F=\int [a{\varphi }^2/2+b{\varphi }^4/4+\kappa |\nabla \varphi {|}^2/2]$ with $a<0<b,\kappa$, the planar coexistence interface is

The excess free energy per area of the phi-four diffuse interface is

A Gaussian field theory has a quadratic free energy and therefore a Gaussian measure on its field modes. For independent real coordinates $x_{\alpha }$ with positive quadratic coefficients ${\lambda }_{\alpha }$, writeHere $T>0$ is temperature and $k_B$ is the Boltzmann constant. Each elementary Gaussian integral givesFor example, differentiating $\log Z_0$ with respect to ${\lambda }_{\alpha }$ gives $-⟨x_{\alpha }^2⟩/(2k_{B}T)$ and proves the variance formula.

For a real scalar field, paired complex Fourier modes obey ${\varphi }_{-q}={\varphi }_q^{\ast }$. If each independent complex pair contributes $J(q)|{\varphi }_q{|}^2$ to the physical free energy, then $⟨|{\varphi }_q{|}^2⟩=k_{B}T/J(q)$. The often-used inverse-kernel formula $J(q{)}^{-1}$ treats $J$ as the dimensionless kernel in $F_0/(k_{B}T)$. Unpaired real modes require the real-coordinate convention above. ^[ref]

The inverse quadratic kernel is the two-point correlation function; Wick theorem then determines higher correlation functions. This makes a Gaussian field theory a solvable reference for perturbation theory and renormalization group calculations. A zero or negative quadratic eigenvalue requires separate treatment because the corresponding unrestricted Gaussian integral is not finite.

The uniform mode of a pure Gaussian field theory with negative squared mass has an unbounded quadratic energy. Its partition function diverges even with a finite-volume ultraviolet regulator, so it supplies no stable ordered branch. At positive squared mass the zero-source average is zero. Thus the scaling dimension of the field can yield a formal order-parameter critical exponent without establishing a negative-reduced-temperature magnetization law. A stabilizing quartic interaction is relevant below four dimensions and ordinarily changes the critical theory to the Wilson-Fisher fixed point; above four its coefficient is a dangerously irrelevant coupling for the ordered phase.

For the positive quadratic kernel $\tilde{\Delta }(p)={\kappa }^{-1}{p}^2+m^2$, $\kappa ,m>0$, the connected correlation function is its inverse. Fourier inversion gives $G(r)=\kappa (\sqrt{\kappa }m/r{)}^{D/2-1}{K}_{D/2-1}(\sqrt{\kappa }mr)/(2\pi {)}^{D/2}$. The Modified Bessel function of the second kind has an exponentially decaying large-argument tail, so the correlation length is $1/(\sqrt{\kappa }m)$ at fixed positive gradient coefficient. In three dimensions this becomes $\kappa e^{-\sqrt{\kappa }mr}/(4\pi r)$. The statement concerns the continuum long-distance theory; regulator-dependent contact terms do not define the physical correlation length.

For a real field, ${\varphi }_{-\mathbf{q}}={\varphi }_{\mathbf{q}}^{\ast }$. A sum ${\sum }_{\mathbf{q}}^{+}$ takes one representative from each nonzero pair $\{\mathbf{q},-\mathbf{q}\}$ so that every independent complex Fourier amplitude is counted once.

A Gaussian variational approximation chooses a quadratic trial kernel and minimizes a variational upper bound on the interacting free energy. The optimized kernel often equals the bare kernel plus a self-consistent, wavevector-independent mass shift.

For $H=\int [a{\varphi }^2/2+\kappa |\nabla \varphi {|}^2/2+\gamma ({\nabla }^2\varphi {)}^2/2+B{\varphi }^2|\nabla \varphi {|}^2]$ and an even positive trial kernel $J(q)$, define $S_0={\sum }_q^{+}1/J(q)$ and $S_2={\sum }_q^{+}{q}^2/J(q)$ using a positive-wavevector sum for a real field. Isserlis theorem gives $⟨H_4{⟩}_0=4B{S}_0{S}_2/V$. The Feynman-Bogoliubov inequality therefore has stationary kernelsThus both the mass and gradient coefficient shift. In the thermodynamic limit, their self-consistency equations areA coarse-graining ultraviolet cutoff $\Lambda$ is necessary for the first integral in three dimensions: its large-$k$ radial integrand tends to a constant. Physical solutions require $J(q)>0$.

For an exact dimensionless Hamiltonian $H$ and trial Hamiltonian $H_0$, the free energies obeywhere $F=-\ln Z$, $F_0=-\ln Z_0$, and the expectation value uses the normalized trial equilibrium state. A physical Hamiltonian is multiplied by $\beta =1/(k_{B}T)$ in these formulas; multiplying the bound by $k_{B}T$ restores physical free energy units.

For classical configurations, the identity $e^{-F}=e^{-F_0}⟨e^{-(H-H_0)}{⟩}_0$ and the Jensen inequality for the convex exponential function give $\ln ⟨e^{-(H-H_0)}{⟩}_0\ge -⟨H-H_0{⟩}_0$. Taking minus the logarithm proves the bound.

The same bound holds for finite-dimensional, noncommuting quantum Hamiltonians, although the preceding exponential-factorization identity cannot generally be used. Define the density matrices $\rho =e^{-H}/Z$ and ${\rho }_0=e^{-H_0}/Z_0$. With natural logarithms, nonnegativity of quantum relative entropy givesRearrangement proves the quantum result without requiring $[H,H_0]=0$. Equality holds when ${\rho }_0=\rho$; in this finite-dimensional setting that means $H-H_0$ is a scalar multiple of the identity operator.^[ref]

A tractable trial family therefore supplies upper bounds that can be minimized over its parameters. This variational minimum need not reproduce exact correlations or phase transition locations.

A Brazovskii model has a quadratic Fourier kernel minimized on a sphere of nonzero wavevector magnitude. The large phase space of soft modes can prevent the Gaussian mass from vanishing and turn a mean-field continuous transition into a fluctuation-induced first-order transition.

A gradient-quartic Brazovskii model stabilizes a negative quadratic gradient coefficient with $\gamma ({\nabla }^2\varphi {)}^2/2$ and uses the quartic interaction $B{\varphi }^2|\nabla \varphi {|}^2$. For $\gamma ,B>0$ and a zero-average compositional order parameter, its nonzero-wavevector instability is a candidate for smectic phase ordering. The Gaussian variational kernel for a gradient quartic interaction shifts both the quadratic coefficients.

For $\varphi =A\cos (q_{0}z)$, the gradient-quartic Brazovskii model has volume-averaged free-energy densityThe averages are $⟨{\cos }^2⟩=⟨{\sin }^2⟩=1/2$ and $⟨{\cos }^2{\sin }^2⟩=1/8$. For $B>0$, the stable amplitude is zero for $a\ge a_c$ and satisfies $A^2=(a_c-a)/(B{q}_0^2)$ below it. The mean-field approximation therefore predicts a continuous phase transition.

For $G(q)=a+\kappa q^2+\gamma q^4$ with $\kappa <0<\gamma$, the minimum occurs on the sphere $q=q_0=\sqrt{-\kappa /(2\gamma )}$. The minimum value is $a-{\kappa }^2/(4\gamma )$.

In three dimensions, let $J(q)=r+\gamma (q^2-q_{\ast }^2{)}^2$, with $q_{\ast }>0$ and small $r>0$. Near the soft shell, $J(q)\simeq r+4\gamma q_{\ast }^2(q-q_{\ast }{)}^2$, soThe radial integral uses ${\int }_{-\infty }^{\infty }ds/(r+A{s}^2)=\pi /\sqrt{Ar}$. The $r^{-1/2}$ divergence obstructs the zero-mass limit of a self-consistent Gaussian variational approximation at a fixed nonzero ordering wavevector. This is the mechanism behind the Brazovskii fluctuation-induced first-order transition; Gross, Ignatiev and Chakraborty's study of fluctuation-driven ordering analyzes the corresponding standard Brazovskii model.

In a self-consistent Gaussian treatment, fluctuations near the soft-mode sphere generate a positive mass shift that diverges as the renormalized mass approaches zero. The disordered phase therefore remains locally stable until its free energy crosses that of a finite-amplitude modulated phase, producing a first-order transition.

A smectic phase is a layered liquid crystal with periodic density or composition variation along one direction and fluid-like positional organization within a layer. A simple order-parameter profile is $\varphi (\mathbf{r})=A\cos ({\mathbf{q}}_0\cdot \mathbf{r})$, with layer spacing $2\pi /|{\mathbf{q}}_0|$.

Layer displacement and orientational fluctuations are additional degrees of freedom, so a fixed cosine is only the simplest mean profile. A smectic A phase has average molecular alignment along the layer normal; a smectic C phase has a tilted director. Scattering can reveal the preferred layer spacing and orientation through peaks near ${\mathbf{q}}_0$. The layer structure does not imply a full three-dimensional crystal lattice of fixed molecular positions.

A smectic C phase has layered positional order with the nematic director tilted away from the layer normal. The tilt supplies additional orientational information beyond the density modulation of a smectic phase.

A smectic A phase has layered positional order with the average molecular orientation along the layer normal. Molecular positions within each layer remain liquid-like. Its orientational axis is described by a nematic director.

A polar helical smectic has a vector order parameter of constant magnitude whose direction rotates at a nonzero wavevector. For an isotropic Brazovskii free energy, all constant rotations of the helical plane are degenerate until a longitudinal-gradient term such as $|\nabla \cdot \mathbf{p}{|}^2$ selects its orientation.

The Ginzburg criterion estimates whether critical fluctuations are small enough for a mean-field approximation. Compare the mean-square fluctuation of the order parameter averaged over a correlation volume with the squared mean-field order parameter. The approximation requires the former to be much smaller.

For an ordinary scalar quartic theory below its transition, $m_0^2\sim |a|/b$ and $\xi \sim (\kappa /|a|{)}^{1/2}$. Long-wavelength Gaussian fluctuations then give a relative scale proportional to $k_{B}Tb{\kappa }^{-d/2}|a{|}^{(d-4)/2}$. Below four dimensions this grows as $a\to 0$, exposing a region where mean field fails. Numerical factors and microscopic cutoffs depend on the model. The criterion diagnoses a failure of the approximation, rather than a failure of the underlying transition to exist.

For dimensionless scalar Landau-Ginzburg theory with quadratic coefficient $r=a_{t}t$, gradient coefficient $c>0$ and quartic term $u{\varphi }^4/4$, the correlation-volume derivation of the scalar Ginzburg ratio gives $\mathcal{G}\sim u{c}^{-d/2}{a}_t^{(d-4)/2}|t{|}^{-(4-d)/2}$. For $d<4$, solving $\mathcal{G}\sim 1$ gives the displayed nonuniversal window. Mean-field self-consistency requires $|t|\gg \mathrm{Gi}$ while still staying in the domain of the local expansion. In three dimensions $\mathrm{Gi}\sim u^2/(c^3{a}_t)$. Numerical prefactors depend on the averaging convention and microscopic normalization; no universal material value follows from dimensional power counting. At four dimensions use the marginal Ginzburg criterion instead of this singular exponent formula.

For the dimensionless scalar Landau-Ginzburg theory with stiffness $c$, quadratic coefficient $r<0$ and quartic term $u{\varphi }^4/4$, the mean-field approximation gives ${\varphi }_0^2=|r|/u$ and correlation length $\xi \asymp (c/|r|{)}^{1/2}$. Restricting the Gaussian Fourier transform covariance $(c{q}^2+2|r|{)}^{-1}$ to $|q|\lesssim {\xi }^{-1}$ estimates the variance of the field averaged over a correlation volume as $c^{-d/2}|r{|}^{(d-2)/2}$, up to a dimension-dependent constant. Dividing by ${\varphi }_0^2$ proves the displayed Ginzburg criterion ratio. It diverges below four dimensions, vanishes above four and is marginal at four. Short-distance fluctuations renormalize microscopic coefficients and should not be confused with this long-distance critical test.

The critical region is the range of control parameters near a thermodynamic critical point where correlated order parameter fluctuations invalidate a simple mean-field approximation. For short-range scalar quartic theory below four dimensions, the dimensionless Ginzburg criterion ratio grows as $g{\xi }^{4-D}$ in fixed microscopic units. The range where this ratio is of order one or larger is fluctuation dominated. Its width depends on nonuniversal microscopic coefficients even though the limiting critical exponents can be universal.

For a scalar quartic interaction $g{\varphi }^4/4!$, the tadpole relation $R=r_{\Lambda }+(g/2)I_D(R)$ can be subtracted at the critical point. When $I_D(0)$ exists, $A(T-T_C)=R[1+(g/2)J_D(R)]$, with $J_D(R)=\int d^{D}p(2\pi {)}^{-D}/[p^2(p^2+R)]$. This integral is finite as $R\to 0$ only above four dimensions and is logarithmic at four. A self-consistent one-loop relation is not an exact interacting critical-exponent formula.

For the scalar quartic one-loop critical-mass subtraction, the subtracted tadpole is $I_D(0)-I_D(R)=R{J}_D(R)$. Above four dimensions $J_D(0)=K_D{\Lambda }^{D-4}/(D-4)$ is finite. At four dimensions $J_4=(K_4/2)\log [({\Lambda }^2+R)/R]$. For $2<D<4$, $J_D\sim K_D{R}^{(D-4)/2}{\int }_0^{\infty }{q}^{D-3}(1+q^2{)}^{-1}dq$, which diverges as $R\to 0$. Hence a smooth linear thermal mass survives this test only above the ordinary upper critical dimension four. The massless subtraction is infrared divergent for $D\le 2$ and cannot prove absence of an interacting transition there.

For $R>0$, a spherical ultraviolet cutoff gives the following exact examples of the one-loop critical-mass subtraction integral:The areas of the unit spheres in dimensions three, four and five are $4\pi$, $2{\pi }^2$ and $8{\pi }^2/3$. Dividing by $(2\pi {)}^D$ reduces the integrals to constants times ${\int }_0^{\Lambda }dp/(p^2+R)$, ${\int }_0^{\Lambda }pdp/(p^2+R)$ and ${\int }_0^{\Lambda }{p}^{2}dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to 0$, the first has a power divergence, the second a logarithmic divergence, and the third a finite limit. These examples make the ordinary upper critical dimension four visible without treating a self-consistent one-loop equation as an exact interacting critical exponent prediction.

At an upper critical dimension, the leading Ginzburg criterion ratio has zero thermal power rather than a negative one. It does not vanish asymptotically, but its magnitude can depend on the coupling. The criterion alone does not establish a power-law divergence or new critical power exponents. Running marginal interactions generally produce logarithmic corrections; the ordinary quartic and tricritical sextic scalar theories are examples.

For a tuned multicritical even Landau potential with leading positive $M^{2n}$ term, $M^2\sim |t{|}^{1/(n-1)}$ and $\xi \sim |t{|}^{-1/2}$. Long-wavelength Gaussian fluctuations in a correlation volume have variance proportional to ${\xi }^{2-D}$. Their ratio to $M^2$ therefore scales as $|t{|}^{(D-2)/2-1/(n-1)}$, vanishing only for $D>2n/(n-1)$. At equality the criterion is marginal, requiring further fluctuation analysis.

For a Gaussian field theory with quadratic kernel proportional to $t+k^2$, let $t>0$ be the distance from the critical point in rescaled units. The relevant heat capacity correction contains the integralwith fixed ultraviolet cutoff $\Lambda$. Radial integration followed by $k=\sqrt{t}x$ giveswhere $C_d$ is the area of the unit sphere in $d$ dimensions.

For $0<d<4$, the rescaled integral has a finite nonzero limit, so $I_d\propto t^{d/2-2}$. At $d=4$, its large-$x$ integrand behaves as $1/x$, givingFor $d>4$, the leading term instead tends to $I_d(0,\Lambda )=C_d{\Lambda }^{d-4}/(d-4)$. It is dominated by the short-distance cutoff. A bare power law cannot describe the full leading integral in every dimension. ^[ref]

The divergent correction below four dimensions warns that the neglected interactions can become important. It is not an exact interacting critical exponent; the Ginzburg criterion tests the validity of the approximation.

A Gaussian field with free-energy kernel $A+\kappa q^2$ has structure factor proportional to $(A+\kappa q^2{)}^{-1}$ and correlation length $\sqrt{\kappa /A}$.