Landau–Ginzburg Theory Preview

Landau–Ginzburg theory promotes a uniform order parameter to a slowly varying field and assigns a free-energy cost both to its local value and to its spatial variation. For a real scalar field with an exact sign-reversal symmetry at zero source, a standard static functional is

$$\mathcal F[\phi;h] = \int_{\Omega} d^d x\, \bigg[ \frac c2 (\nabla\phi)^2 + \frac r2 \phi^2 + \frac u4 \phi^4 -h(\mathbf x)\phi(\mathbf x) \bigg] + \mathcal F_{\partial\Omega}.$$

Helpful background. Landau theory explains the uniform polynomial; connected correlations introduces the spatial decay measured below.

Here $\phi(\mathbf x)$ is a coarse-grained order-parameter field, $c>0$ is a stiffness, $r$ tunes the uniform instability, $u>0$ stabilizes the amplitude, $h$ is a conjugate source, and $\mathcal F_{\partial\Omega}$ records any surface physics. The functional is static: it ranks spatial configurations in equilibrium but does not specify how they move in time.

At temperature $T$, a statistical field theory uses

$$\begin{aligned} \mathcal Z_{\mathrm{LG}}[h] &= \int^{\Lambda}\!\mathcal D\phi\, \times \exp\!\left[-\beta\mathcal F[\phi;h]\right], \\ \beta &= \frac{1}{k_{\mathrm B}T}. \end{aligned}$$

with an ultraviolet cutoff $\Lambda$ fixed by the coarse-graining scale. Minimizing $\mathcal F$ is the saddle-point approximation to this integral. Integrating over fluctuations is the step that turns a spatial Landau functional into a genuine statistical field theory. Statistical Field Theory Preview develops the regulated measure, exact coarse-field construction, source functional, and distinctions among constrained, Wilsonian, and Legendre effective actions.

Static scalar fields and their free energy

Landau, Landau–Ginzburg, and Ginzburg–Landau

Naming varies across subfields.

• Landau theory often means a uniform polynomial in one or more order parameters.
• Landau–Ginzburg theory often means the spatial statistical-field extension used for critical phenomena.
• Ginzburg–Landau theory often refers specifically to a complex superconducting order parameter coupled to electromagnetism.

The mathematical structures overlap, but they are not interchangeable without stating the degrees of freedom, symmetries, controls, and fields. This page uses Landau–Ginzburg for the generic static spatial theory and reserves the superconducting application for a short extension below.

Free-energy convention

The symbol $\mathcal F[\phi]$ denotes an energy. Consequently $\beta\mathcal F$ is dimensionless. Some texts instead call the dimensionless object

$$\mathcal S_{\mathrm{LG}}[\phi] \equiv \beta\mathcal F[\phi]$$

the Landau–Ginzburg Hamiltonian or action and absorb $k_{\mathrm B}T$ into every coefficient. Both conventions are valid. Mixing them changes fluctuation amplitudes by factors of $k_{\mathrm B}T$.

Fourier convention

For a large translation-invariant region, use

$$\begin{aligned} \phi(\mathbf x) &= \int_{\lvert\mathbf k\rvert<\Lambda} \frac{d^d k}{(2\pi)^d}\, e^{i\mathbf k\cdot\mathbf x} \phi_{\mathbf k}, \\ \phi_{\mathbf k} &= \int d^d x\, e^{-i\mathbf k\cdot\mathbf x} \phi(\mathbf x). \end{aligned}$$

Reality implies $\phi_{-\mathbf k}=\phi_{\mathbf k}^{*}$. The cutoff $\Lambda$ is not decorative: a continuum functional obtained by coarse-graining is not intended to describe arbitrarily short wavelengths.

From One Number to a Field

Coarse-grained local order

A microscopic spin, density, displacement, or pair operator can vary strongly from site to site. A Landau–Ginzburg field is instead a local average over a cell of linear size $\ell$:

$$a_{\mathrm{micro}} \ll \ell \ll L_{\mathrm{variation}},$$

where $a_{\mathrm{micro}}$ is a microscopic range and $L_{\mathrm{variation}}$ is the scale on which the collective profile changes appreciably. Schematically,

$$\phi_{\ell}(\mathbf x) = \int d^d y\, W_{\ell}(\mathbf x-\mathbf y) \mathcal O(\mathbf y),$$

with a normalized smoothing kernel $W_{\ell}$. The precise kernel affects coefficient values but should not change long-distance universal predictions when the effective description is valid.

The field may represent an expectation value, a constrained coarse-grained variable, or an auxiliary variable that has acquired an order-parameter interpretation after microscopic degrees of freedom are integrated out. These constructions can lead to similar functionals, but their measures and matching conditions need not be identical.

Why spatial variation matters

Uniform Landau theory cannot answer questions involving distance. It does not determine:

• the width or energy of an interface;
• the response to a source at nonzero wavevector;
• the shape of a critical scattering peak;
• a correlation length;
• spatial textures, vortices, or domains;
• how fluctuations at different wavelengths compete.

Promoting $m$ to $\phi(\mathbf x)$ supplies the missing coordinates. The gradient expansion then organizes the cost of spatial structure.

The field is not the microscopic operator

The coarse-grained field generally has fewer degrees of freedom than the microscopic system. Its coefficients already contain short-distance physics. Treating $\phi(\mathbf x)$ as if it remained valid down to zero distance double-counts or invents ultraviolet modes.

The field normalization and source must match the microscopic order parameter. Hold them fixed while studying the spatial consequences.

The Local and Gradient Expansion

Local potential

For a scalar inversion symmetry at zero source, locality and analyticity suggest

$$V(\phi) = V_0 + \frac r2\phi^2 + \frac u4\phi^4 + \frac v6\phi^6 + \cdots.$$

This is the same invariant expansion as uniform Landau theory, now evaluated point by point. It is meaningful only over a field-amplitude range in which the omitted terms remain controlled. A negative quartic coefficient still requires a stabilizing higher-order term.

Nonlocal quadratic kernel

A more honest starting point for weak inhomogeneity is a quadratic form

$$\mathcal F_2 = \frac12 \int d^d x\,d^d y\, \phi(\mathbf x) K(\mathbf x-\mathbf y) \phi(\mathbf y).$$

Translation invariance diagonalizes it in momentum space:

$$\begin{aligned} \mathcal F_2 &= \frac12 \int_{\mathbf k} K(\mathbf k) \lvert\phi_{\mathbf k}\rvert^2, \\ \int_{\mathbf k} &\equiv \int_{\lvert\mathbf k\rvert<\Lambda} \frac{d^d k}{(2\pi)^d}. \end{aligned}$$

For a local bare or coarse-grained functional with sufficiently short-range interactions and a soft mode at $\mathbf k=0$, a derivative expansion gives

$$K(\mathbf k) = r + c k^2 + O(k^4).$$

Transforming back to position space gives

$$\mathcal F_2 = \int d^d x\, \left[ \frac r2\phi^2 + \frac c2(\nabla\phi)^2 + \cdots \right].$$

Thus the gradient term is the leading momentum dependence of the local quadratic kernel, and of its Gaussian inverse susceptibility. The exact interacting critical inverse susceptibility can instead scale as $k^{2-\eta}$; microscopic locality does not make that full critical quantity analytic. It is not added merely because smooth curves look plausible.

Why the leading term is quadratic in gradients

For a real scalar in a homogeneous inversion-symmetric medium, a bulk term linear in $\nabla\phi$ is either forbidden or a total derivative. Rotational symmetry then reduces the leading even term to $(\nabla\phi)^2$.

Lower crystal symmetry permits an anisotropic stiffness tensor:

$$\mathcal F_{\mathrm{grad}} = \frac12 \int d^d x\, K_{ij} (\partial_i\phi) (\partial_j\phi),$$

where the symmetric matrix $K_{ij}$ must be positive definite for a stable uniform soft mode. Its eigenvectors define principal correlation axes.

When the gradient expansion fails

The simple $r+c k^2$ kernel can fail for several distinct reasons:

• long-range interactions produce nonanalytic powers such as $\lvert k\rvert^{\sigma}$;
• gapless fermions or gauge fields generate nonlocal momentum and frequency dependence;
• the soft mode lies near a nonzero ordering wavevector $\mathbf Q$;
• broken inversion and multicomponent order allow Lifshitz invariants linear in derivatives;
• frustration can make the $k^2$ stiffness negative;
• disorder destroys translation invariance.

If

$$\begin{aligned} K(k) &= r + c k^2 + \kappa k^4, \\ c &< 0, \qquad \kappa>0. \end{aligned}$$

the quadratic kernel is minimized at

$$q_0 = \sqrt{-\frac{c}{2\kappa}},$$

not at zero momentum. The appropriate order parameter is then built around modes near $\lvert\mathbf k\rvert=q_0$. Expanding around $\mathbf k=0$ while discarding $k^4$ would make the theory unstable and miss modulated order.

The scalar functional and stability

With the constant background suppressed, the minimal stable model is

$$\mathcal F[\phi;h] = \int_{\Omega} d^d x\, \bigg[ \frac c2(\nabla\phi)^2 + \frac r2\phi^2 + \frac u4\phi^4 - h\phi \bigg].$$

with $c>0$ and $u>0$. The source may vary in space. At $h=0$, the functional is invariant under $\phi\mapsto-\phi$.

Engineering dimensions

Let $[E]$ denote energy, $[L]$ length, and $[\phi]$ the chosen order-parameter unit. Requiring each integral term to have units of energy gives

$$\begin{array}{c|c} \text{coefficient} & \text{physical dimension} \\ \hline c & [E][L]^{2-d}[\phi]^{-2} \\ r & [E][L]^{-d}[\phi]^{-2} \\ u & [E][L]^{-d}[\phi]^{-4} \\ h & [E][L]^{-d}[\phi]^{-1} \end{array}$$

These are physical units before any dimensionless field rescaling. RG scaling dimensions are a different concept because they describe how variables transform under a change of resolution.

Stability is a spectrum condition

Around a homogeneous configuration $\bar\phi$, set $\phi=\bar\phi+\eta$. The quadratic fluctuation operator is

$$\mathcal L_{\bar\phi} = -c\nabla^2 + V''(\bar\phi).$$

Local stability requires every allowed eigenvalue of $\mathcal L_{\bar\phi}$ to be nonnegative, with zero modes treated separately. Testing only $V''(\bar\phi)$ is sufficient in an infinite isotropic system with $c>0$ and a minimum at $k=0$, but not for negative stiffness, unusual boundaries, or coupled fields.

The quadratic coefficient is generally renormalized

The symbol $r$ in a coarse-grained functional need not equal a bare microscopic parameter. It depends on the coarse-graining scale, ensemble, and matching prescription. In a mean-field approximation one often writes

$$r = a_0 t, \qquad t = \frac{T-T_c^{(0)}}{T_c^{(0)}},$$

where $T_c^{(0)}$ is a mean-field critical temperature. Fluctuations can shift the physical critical point, so a calculation must say whether $r=0$ refers to a bare, mean-field, renormalized, or experimentally matched condition.

Functional-Integral Meaning

Configuration probability

For a classical equilibrium order-parameter field with a specified cutoff,

$$\mathcal P[\phi] = \frac{1}{\mathcal Z[0]} e^{-\beta\mathcal F[\phi;0]}.$$

This assigns probability to entire field configurations, not independently to each point. The gradient term couples neighboring values, and the quartic term couples Fourier modes through momentum-conserving convolutions.

The notation $\mathcal D\phi$ abbreviates a regulated multiple integral. On a coarse lattice with $N$ cells it is schematically

$$\int\mathcal D\phi \longrightarrow \prod_{j=1}^{N} \int d\phi_j$$

times a normalization fixed by the chosen measure. The continuum symbol alone does not define an absolute free energy.

Source derivatives

Define

$$W[h] \equiv \ln\mathcal Z[h],$$

with the source appearing as $-\int h\phi$ in $\mathcal F$. Then

$$\frac{\delta W}{\delta h(\mathbf x)} = \beta \langle\phi(\mathbf x)\rangle_h,$$

and

$$\frac{\delta^2 W} {\delta h(\mathbf x)\delta h(\mathbf y)} = \beta^2 \langle \phi(\mathbf x)\phi(\mathbf y) \rangle_{c,h}.$$

Equivalently, the static response kernel is

$$\chi(\mathbf x,\mathbf y) \equiv \frac{\delta\langle\phi(\mathbf x)\rangle_h} {\delta h(\mathbf y)} = \beta \langle \phi(\mathbf x)\phi(\mathbf y) \rangle_{c,h}.$$

This is the equilibrium fluctuation–response relation in the present normalization. Operator-valued quantum response and real-time causality require the Kubo framework rather than this static identity.

Saddle point is not the full integral

Let $\bar\phi$ make $\mathcal F$ stationary and write $\phi=\bar\phi+\eta$. Then

$$\mathcal F[\bar\phi+\eta] = \mathcal F[\bar\phi] + \frac12 \int d^d x\,d^d y\, \eta(\mathbf x) \times \mathcal L(\mathbf x,\mathbf y) \eta(\mathbf y) + O(\eta^3).$$

Keeping only $\bar\phi$ is mean field. Keeping the quadratic term adds Gaussian fluctuations. Formally, for a stable saddle without zero modes,

$$\mathcal Z \simeq{} e^{-\beta\mathcal F[\bar\phi]} \left[ \det\!\left(\beta\mathcal L\right) \right]^{-1/2} \times \text{measure factor}.$$

The determinant needs a regulator, and only properly normalized determinants or ratios have an invariant meaning. Cubic and higher fluctuation terms generate interactions and loop corrections.

Coarse-grained free energy versus exact effective action

A phenomenological Landau–Ginzburg functional is often deliberately nonconvex so that it can represent competing local phases and interfaces. The exact Legendre effective action at zero infrared cutoff has stronger convexity properties in equilibrium. A scale-dependent coarse-grained functional can remain nonconvex because long-wavelength phase mixing has not yet been integrated out.

Calling every one of these objects “the free energy” hides an important distinction. State whether the object is microscopic, constrained, coarse-grained at a finite scale, or fully Legendre transformed.

Stationary Profiles

Euler–Lagrange equation

Varying the scalar functional gives

$$\delta\mathcal F ={} \int_{\Omega} d^d x\, \big[ -c\nabla^2\phi + r\phi + u\phi^3 - h \big] \delta\phi + c \int_{\partial\Omega} dS\, (\mathbf n\cdot\nabla\phi) \delta\phi.$$

The bulk stationary equation is therefore

$$-c\nabla^2\phi + r\phi + u\phi^3 = h(\mathbf x).$$

The Laplacian couples the local order to its surroundings. Uniform solutions recover the Landau equation of state $r\phi+u\phi^3=h$.

Boundary conditions

The boundary term must be handled rather than silently discarded.

• Fixed $\phi$ gives a Dirichlet condition, so $\delta\phi=0$ at the boundary.
• A free surface with no surface functional gives the natural Neumann condition $\mathbf n\cdot\nabla\phi=0$.
• A surface energy can produce a Robin condition.

For example, with

$$\mathcal F_{\partial\Omega} = \int_{\partial\Omega} dS\, \left[ \frac{c_s}{2}\phi^2 - h_1\phi \right],$$

stationarity requires

$$c\,\mathbf n\cdot\nabla\phi + c_s\phi - h_1 = 0.$$

Surface enhancement, boundary fields, and finite geometry can therefore change profiles without changing the bulk polynomial.

Linearized response equation

In the disordered phase, linearizing around $\phi=0$ gives

$$\left( -c\nabla^2 + r \right) \phi(\mathbf x) = h(\mathbf x).$$

The Green function of this screened Poisson operator is simultaneously the static response kernel and, up to $k_{\mathrm B}T$, the Gaussian connected correlator.

Gaussian Fluctuations Above the Transition

Assume $r>0$, $u$ is negligible for a first fluctuation estimate, the system is translation invariant, and the source vanishes. The quadratic functional is

$$\mathcal F_{\mathrm G}[\phi] = \frac12 \int_{\mathbf k} \left(r+c k^2\right) \lvert\phi_{\mathbf k}\rvert^2.$$

Independent Fourier modes

Every independent real Fourier degree of freedom is Gaussian. Equipartition or direct integration gives

$$\left\langle \phi_{\mathbf k} \phi_{\mathbf k'} \right\rangle = (2\pi)^d \delta^{(d)}(\mathbf k+\mathbf k') \frac{k_{\mathrm B}T}{r+c k^2}.$$

Define the momentum-space connected correlator by removing the delta function:

$$G(\mathbf k) = \frac{k_{\mathrm B}T}{r+c k^2}.$$

This is a prediction of the Gaussian approximation, not an exact identity for an interacting critical system.

Static susceptibility

A weak source produces

$$\langle\phi_{\mathbf k}\rangle_h = \frac{h_{\mathbf k}}{r+c k^2},$$

exactly within the Gaussian model. Thus

$$\chi(\mathbf k) = \frac{1}{r+c k^2}, \qquad G(\mathbf k) = k_{\mathrm B}T\,\chi(\mathbf k).$$

Long-wavelength modes are the softest because their restoring coefficient is smallest. As $r\to0^+$, both the uniform susceptibility and the range of spatial correlations grow.

Correlation length

Factor the kernel as

$$r+c k^2 = r\left(1+k^2\xi^2\right),$$

where

$$\xi = \sqrt{\frac c r}.$$

Then

$$G(\mathbf k) = \frac{k_{\mathrm B}T}{r} \frac{1}{1+k^2\xi^2}.$$

This Lorentzian small-$k$ form is often called Ornstein–Zernike form in critical-scattering contexts. A measured structure factor may include normalization factors, form factors, tensor projections, regular backgrounds, and a peak centered at a nonzero ordering wavevector.

The upper profile solves the scalar quartic theory with $r<0$ and $u,c>0$, with wall width $\ell_w=\sqrt{2c/|r|}$. The lower Gaussian examples have $r>0$ and use $c=k_{\mathrm B}T=1$: decreasing $r$ from $1$ to $1/4$ quadruples the peak height and halves its half-width. Thus the same kernel controls both the susceptibility and the correlation length.

Real-space correlator

Let

$$\kappa \equiv \xi^{-1} = \sqrt{\frac r c}.$$

In an infinite isotropic continuum,

$$G(\mathbf x) = \frac{k_{\mathrm B}T}{c} \int_{\mathbf k} \frac{e^{i\mathbf k\cdot\mathbf x}} {k^2+\kappa^2}.$$

Extending the local continuum kernel to all momenta gives the transform

$$G(x) ={} \frac{k_{\mathrm B}T}{c} \frac{1}{(2\pi)^{d/2}} \times \left(\frac{\kappa}{x}\right)^{d/2-1} K_{d/2-1}(\kappa x).$$

where $K_{\nu}$ is a modified Bessel function and $x=\lvert\mathbf x\rvert$. For $x\gg\xi$,

$$G(x) \propto x^{-(d-1)/2} e^{-x/\xi}.$$

In three dimensions this simplifies to

$$G(x) = \frac{k_{\mathrm B}T}{4\pi c x} e^{-x/\xi}.$$

The algebraic prefactor matters when fitting data. These are the infinite-cutoff continuum formulas, useful in the scaling regime. A literal sharp momentum cutoff also creates oscillatory long-distance tails from its artificial edge; it is not correct to dismiss every cutoff effect as a contact term. A smooth local regularization or the original lattice kernel should be used when testing exponential decay.

Gaussian critical exponents

If $r=a_0 t$ with $t=(T-T_c)/T_c$, then

$$\xi \propto t^{-1/2},$$

so the Gaussian value is

$$\nu_{\mathrm G} = \frac12.$$

At $r=0$ and for dimensions where the Fourier transform is infrared meaningful,

$$G(k) \propto k^{-2}.$$

Comparing with the critical definition

$$G(k) \propto k^{-2+\eta}$$

gives

$$\eta_{\mathrm G} = 0.$$

These are Gaussian or mean-field values. In dimensions below the upper critical dimension, the quartic interaction generally changes them at the interacting critical fixed point.

Fluctuations in the Ordered Phase

For $r<0$, $u>0$, and $h=0$, the uniform minima are

$$\phi_0 = \pm\sqrt{\frac{\lvert r\rvert}{u}}.$$

Choose one pure phase and write

$$\phi(\mathbf x) = \phi_0 + \sigma(\mathbf x).$$

The local curvature is

$$V''(\phi_0) = r + 3u\phi_0^2 = 2\lvert r\rvert.$$

The scalar amplitude fluctuation therefore has Gaussian kernel

$$2\lvert r\rvert + c k^2$$

and longitudinal correlation length

$$\xi_- = \sqrt{\frac{c}{2\lvert r\rvert}}.$$

Above the transition at the same $\lvert r\rvert$,

$$\xi_+ = \sqrt{\frac{c}{\lvert r\rvert}},$$

so the Gaussian amplitude ratio is

$$\frac{\xi_+}{\xi_-} = \sqrt2.$$

Amplitude ratios can be universal within a universality class, but this mean-field value is not exact below the upper critical dimension.

Phase selection matters

In a finite symmetric system at zero source, averaging over both minima gives $\langle\phi\rangle=0$. Expanding about one minimum presumes a selected phase, implemented by suitable boundary conditions, or by taking the thermodynamic limit at a nonzero selecting source and only then removing that source. Conditioning a finite sample on one order-parameter branch is a useful related approximation, whose relation to a pure bulk phase must be checked.

Expanding around a symmetry-restored mixture is not equivalent to expanding around either pure phase. Spontaneous symmetry breaking explains that limit order.

Multicomponent order and soft directions

For an $O(N)$ vector field,

$$\mathcal F[\boldsymbol\phi] = \int d^d x\, \bigg[ \frac c2 (\nabla\boldsymbol\phi)^2 + \frac r2 \boldsymbol\phi^2 + \frac u4 (\boldsymbol\phi^2)^2 \bigg].$$

Choose the ordered direction along component $1$ and write

$$\boldsymbol\phi = (\phi_0+\sigma,\boldsymbol\pi).$$

At quadratic order,

$$\mathcal F_2 = \frac12 \int d^d x\, \bigg[ c(\nabla\sigma)^2 + 2\lvert r\rvert\sigma^2 + c(\nabla\boldsymbol\pi)^2 \bigg].$$

The longitudinal fluctuation has a nonzero static restoring curvature, while the $N-1$ transverse directions have none at quadratic order. This static statement does not by itself supply real-time excitation frequencies. Beyond Gaussian order, coupling to transverse fluctuations can also make longitudinal correlations singular. Their consequences and the qualifications imposed by dimension, temperature, gauge coupling, and explicit symmetry breaking belong to Goldstone Modes in Many-Body Systems.

Interfaces and Healing

Gradient energy makes abrupt jumps costly. A domain wall illustrates how local ordering and stiffness balance.

Scalar wall equation

For $r<0$ and $h=0$, rewrite the potential relative to either minimum as

$$\begin{aligned} V(\phi)-V(\phi_0) &= \frac u4 \left(\phi^2-\phi_0^2\right)^2, \\ \phi_0^2 &= \frac{\lvert r\rvert}{u}. \end{aligned}$$

Consider a flat wall that depends only on $x$ and obeys

$$\phi(-\infty) = -\phi_0, \qquad \phi(+\infty) = \phi_0.$$

The profile equation is

$$-c\frac{d^2\phi}{dx^2} + r\phi + u\phi^3 = 0.$$

First integral

Multiplying by $d\phi/dx$ and integrating once gives

$$\frac c2 \left(\frac{d\phi}{dx}\right)^2 = V(\phi)-V(\phi_0).$$

The integration constant is fixed by the uniform limits. The equality has a mechanical analogy, but it is simply a first integral of the static Euler–Lagrange equation.

Kink profile

The exact wall is

$$\phi(x) = \phi_0 \tanh\!\left(\frac{x}{\ell_{\mathrm w}}\right),$$

with

$$\ell_{\mathrm w} = \sqrt{\frac{2c}{\lvert r\rvert}} = 2\xi_-.$$

The phrase “wall width” depends on the convention used to convert a smooth profile into one number. The intrinsic scale in the hyperbolic tangent is $\ell_{\mathrm w}$; other width definitions differ by numerical factors.

Surface tension

The excess free energy per unit wall area is

$$\sigma_{\mathrm w} = \int_{-\infty}^{\infty} dx\, \bigg[ \frac c2 \left(\frac{d\phi}{dx}\right)^2 + V(\phi)-V(\phi_0) \bigg].$$

Using the first integral and the kink solution yields

$$\sigma_{\mathrm w} = \frac{2\sqrt2}{3} \frac{\sqrt c\,\lvert r\rvert^{3/2}}{u}.$$

Within this mean-field theory, the wall broadens and its tension vanishes as $r\to0^-$. Close enough to a fluctuation-dominated critical point, both powers and amplitudes require corrections.

What the wall does not calculate

The static profile does not determine domain-wall mobility, damping, nucleation rate, pinning, roughening, or coarsening. Those require a dynamical law and, often, noise. Nor does the planar wall by itself provide the critical droplet for a first-order transition, whose curvature and bulk free-energy bias are essential.

The Ginzburg Criterion

Landau minimization and Gaussian fluctuations are not automatically reliable arbitrarily close to a continuous transition. The Ginzburg criterion estimates where fluctuations become comparable with the mean-field order.

Fluctuation in a correlation volume

Use the disordered-side Gaussian propagator and retain modes with $k\lesssim\xi^{-1}$. Their contribution scales as

$$\begin{aligned} \langle(\delta\phi)^2\rangle_{\xi} &\sim k_{\mathrm B}T \int_{k\lesssim\xi^{-1}} \frac{d^d k}{(2\pi)^d} \frac{1}{r+c k^2} \\ &\sim \frac{k_{\mathrm B}T}{c} \xi^{2-d}. \end{aligned}$$

Only the scaling is claimed; the numerical coefficient depends on the cutoff convention and on how the correlation volume is defined.

Below the transition, mean field gives

$$\phi_0^2 = \frac{\lvert r\rvert}{u} \sim \frac{c}{u\xi^2}.$$

The ratio is therefore

$$\frac{\langle(\delta\phi)^2\rangle_{\xi}} {\phi_0^2} \sim \frac{k_{\mathrm B}T\,u}{c^2} \xi^{4-d}.$$

Define the dimensionless fluctuation strength at scale $\xi$ by

$$g_{\xi} \equiv \frac{k_{\mathrm B}T\,u}{c^2} \xi^{4-d}.$$

Mean-field reasoning is parametrically controlled only when $g_{\xi}\ll1$.

Upper critical dimension

As $\xi\to\infty$:

• for $d<4$, $g_{\xi}$ grows and the quartic interaction becomes important;
• for $d=4$, the coupling is marginal and logarithmic corrections can appear;
• for $d>4$, $g_{\xi}$ decreases and Gaussian critical behavior is stable for the short-range scalar problem, subject to dangerous-irrelevance qualifications.

This identifies

$$d_c = 4$$

as the upper critical dimension of the ordinary short-range $\phi^4$ theory. The Renormalization Group Preview derives the same result by power counting and explains what changes at an interacting fixed point.

Width of the Ginzburg region

If

$$r = a_0 t, \qquad \xi \simeq \left(\frac{c}{a_0\lvert t\rvert}\right)^{1/2},$$

then for $d<4$,

$$g_{\xi} \sim \mathcal G \lvert t\rvert^{-(4-d)/2},$$

where

$$\mathcal G = \frac{k_{\mathrm B}T_c u}{c^2} \left(\frac{c}{a_0}\right)^{(4-d)/2}.$$

The crossover estimate $g_{\xi}\sim1$ gives

$$\lvert t_{\mathrm G}\rvert \sim \mathcal G^{2/(4-d)}.$$

This is a nonuniversal scale estimate, not a sharp phase boundary. Numerical constants, the relevant side of the transition, anisotropy, component number, interaction range, and additional soft modes all modify it.

What the criterion says and does not say

The Ginzburg criterion says when a particular mean-field expansion loses parametric control. It does not calculate the true exponents, prove that the transition remains continuous, or identify the eventual fixed point. A large Ginzburg region means non-Gaussian fluctuations matter over an experimentally visible interval; a small region can make mean-field behavior look accurate until very close to criticality.

For continuous symmetries in low dimension, transverse fluctuations introduce additional infrared constraints. For gauge-coupled or fermionic systems, integrating out other soft fields can qualitatively modify the effective functional. The scalar estimate must not be transplanted unchanged.

Multicomponent and Complex Fields

General quadratic form

For components $\phi_a$, the leading functional has the schematic form

$$\mathcal F[\boldsymbol\phi] = \int d^d x\, \bigg[ \frac12 K_{ij}^{ab} (\partial_i\phi_a) (\partial_j\phi_b) + V(\boldsymbol\phi) \bigg].$$

Around a stationary uniform state, the Gaussian inverse susceptibility matrix is

$$\left[\chi^{-1}(\mathbf k)\right]_{ab} = M_{ab} + K_{ij}^{ab}k_i k_j + O(k^4),$$

where

$$M_{ab} = \left. \frac{\partial^2 V} {\partial\phi_a\partial\phi_b} \right|_{\bar{\boldsymbol\phi}}.$$

Diagonalizing only $M$ may be insufficient if the stiffness matrices do not commute with it. Different eigenmodes can have different correlation lengths, polarizations, and soft wavevectors.

Complex order parameter

For a global $U(1)$ order parameter $\psi$, a neutral isotropic functional is

$$\mathcal F[\psi] = \int d^d x\, \left[ c\lvert\nabla\psi\rvert^2 + r\lvert\psi\rvert^2 + \frac u2\lvert\psi\rvert^4 \right].$$

The amplitude and phase are both spatial fields. Where $\psi\ne0$, write

$$\psi = \rho e^{i\theta}.$$

Then

$$\lvert\nabla\psi\rvert^2 = (\nabla\rho)^2 + \rho^2(\nabla\theta)^2.$$

The phase stiffness is therefore proportional to the squared amplitude in this minimal model. Singular phase configurations, compactness, and vortices require more than a smooth small-fluctuation expansion.

Gauge-covariant superconducting extension

For a charged condensate, local gauge invariance replaces the ordinary gradient by a covariant derivative and includes magnetic-field energy. Schematically,

$$\mathcal F[\psi,\mathbf A] ={} \int d^3 x\, \bigg\{ c\left\lvert \left(\nabla-iq\mathbf A\right)\psi \right\rvert^2 + r\lvert\psi\rvert^2 + \frac u2\lvert\psi\rvert^4 + \frac{\mathbf B^2}{2\mu_0} \bigg\}.$$

In this SI form, $q=Q/\hbar$, where $Q$ is the condensate charge; the kinetic coefficient $c$ includes the corresponding $\hbar^2$ factor when derived from a mass. Specify whether $\mathbf A$ is varied at a saddle, integrated over as a thermal field, or prescribed externally. An equilibrium magnetic energy does not supply its real-time dynamics. Ginzburg–Landau theory for superconductors develops coherence and penetration lengths, critical fields, and vortices.

Coupled orders

Several order parameters can share gradients and local couplings. For two scalars,

$$\mathcal F = \int d^d x\, \bigg[ \frac{c_\phi}{2}(\nabla\phi)^2 + \frac{c_\psi}{2}(\nabla\psi)^2 + V(\phi,\psi) \bigg].$$

A local term $w\phi^2\psi^2$ can favor competition or coexistence, while derivative couplings can mix spatial modes. The uniform phase structure follows from Landau theory; the present extension determines how the coupled orders vary and fluctuate in space.

Thermal, Quantum, and Dynamical Theories

Static thermal field theory

The integral

$$\mathcal Z = \int\mathcal D\phi\, e^{-\beta\mathcal F[\phi]}$$

is a $d$-dimensional classical statistical field theory. It can describe the long-distance static sector of a finite-temperature quantum many-body system when nonzero Matsubara modes and other massive fields have been integrated out appropriately.

The resulting coefficients contain quantum and thermal short-distance physics. Calling the final field theory “classical” refers to the statistical form of the remaining static modes, not to the microscopic constituents.

Quantum critical action

A zero-temperature quantum transition generally needs an imaginary-time-dependent field:

$$\mathcal Z = \int\mathcal D\phi\, e^{-S_{\mathrm E}[\phi]/\hbar}.$$

A schematic local action might contain

$$S_{\mathrm E} = \int d\tau\,d^d x\, \bigg[ \frac{Z_\tau}{2} (\partial_\tau\phi)^2 + \frac c2(\nabla\phi)^2 + V(\phi) \bigg].$$

but many-body systems can instead produce first-order time derivatives, Landau damping, Berry phases, nonlocal kernels, or gauge constraints. The dynamical exponent $z$ and the effective dimension $d+z$ cannot be inferred from the static functional alone.

Quantum phase transitions explains the scaling distinction between thermal and quantum criticality. Why Many-Body QM Leads to QFT develops the broader route from microscopic, auxiliary, and collective fields to effective actions.

Static functional does not imply kinetics

Two systems can share the same equilibrium $\mathcal F[\phi]$ but have different dynamics because the order parameter may be conserved in one system and nonconserved in another, or because it couples to momentum, energy, gauge fields, or other slow variables.

An equation such as

$$\partial_t\phi = -\Gamma \frac{\delta\mathcal F}{\delta\phi}$$

is an additional relaxational model, not a consequence of the equilibrium functional. Noise and fluctuation–dissipation consistency must also be specified. Dynamic universality is a separate classification problem.

From Microscopic Physics to the Functional

Symmetry constrains; matching determines

Symmetry tells which local and derivative operators may appear. It does not determine their coefficients. Microscopic matching may use:

• a controlled expansion near weak coupling;
• a Hubbard–Stratonovich transformation followed by integrating out matter fields;
• a cumulant or linked-cluster expansion;
• density-functional or effective-action methods;
• measured susceptibilities, stiffnesses, and nonlinear response;
• numerical coarse-graining.

The same symmetry class can therefore have very different $c$, $u$, cutoff, Ginzburg window, and nonuniversal amplitudes.

Auxiliary and collective fields are not synonyms

An auxiliary field can be introduced through an exact integral identity. It becomes a collective order-parameter field only after one identifies its transformation properties, source coupling, saddle, fluctuations, and relation to observables. Integrating out microscopic matter can also generate a nonlocal or complex effective action rather than the simple local real functional assumed here.

Derivative and amplitude expansions are independent

Small gradients do not guarantee small amplitude, and small amplitude does not guarantee slow variation. A controlled derivative expansion needs the relevant Fourier wavevectors to satisfy $k\ll\Lambda$, together with suppression of omitted local powers over the field amplitudes used. A ratio such as $|\nabla\phi|/(\Lambda|\phi|)$ is not a general test: a perfectly smooth sinusoid crosses zero, where that ratio diverges even when its wavevector is small. Near a defect core, boundary, or first-order interface, one expansion can fail before the other.

Coefficients depend on scale

Integrating out fluctuations between two cutoffs changes the effective coefficients and generates every operator allowed by symmetry. A polynomial that is accurate at one coarse-graining scale need not retain the same numerical coefficients at another.

This is why the Landau–Ginzburg functional is the starting point, not the endpoint, of the Wilsonian analysis. Renormalization-group flow describes how the coefficients change.

Representative Uses

Ising-like magnets and binary mixtures

A real scalar field can represent coarse-grained magnetization or a composition difference. The source is a magnetic field or chemical-potential difference, the stiffness penalizes spatial variation, and the Gaussian peak describes long-wavelength critical scattering above the transition.

The mapping identifies symmetry and soft modes, not numerical equality of coefficients. A fluid near its liquid–gas critical endpoint has no microscopic spin-inversion symmetry, but field mixing can produce an emergent scalar critical description.

Structural transitions

A displacement, strain component, or orientational tensor can serve as the order parameter. Crystal symmetry controls invariant polynomials and anisotropic gradients. Compatibility constraints and elastic long-range interactions can make the effective kernel nonlocal, so a naive local $c k^2$ approximation may be insufficient.

Neutral condensates

A complex field describes amplitude and phase variations of a neutral condensate. The stationary Gross–Pitaevskii equation has a related gradient-plus-local-nonlinearity structure, but its coefficients and interpretation arise from a condensate wavefunction and particle-number constraint. The Gross–Pitaevskii equation develops that microscopic mean-field setting and condensate dynamics.

Superconductors

The charged complex functional predicts coherence and magnetic screening scales, interfaces, vortices, and critical fields once electromagnetic coupling and material parameters are included. Near an ordinary thermal transition it can often be matched to microscopic BCS theory, but the phenomenological construction itself does not assume BCS pairing.

Modulated phases

Competing interactions may soften modes near $\mathbf Q\ne0$. Then the correct field may be a complex amplitude multiplying $e^{i\mathbf Q\cdot\mathbf x}$, a collection of symmetry-related wavevectors, or a tensor. Keeping the wavevector structure prevents the gradient expansion from misclassifying translation-breaking order.

Limitations and Breakdown

Fluctuation-dominated criticality

Below $d_c=4$ for short-range scalar $\phi^4$ theory, Gaussian exponents are generally not the asymptotic ones. The Ginzburg criterion locates a crossover estimate but does not replace RG or high-precision numerical and experimental analysis.

Low-dimensional continuous symmetry

Massless angular fluctuations can prevent conventional finite-temperature long-range order under the hypotheses of the Mermin–Wagner–Hohenberg results. A saddle with $\lvert\boldsymbol\phi\rvert\ne0$ does not by itself prove that the thermal expectation value remains ordered.

Topological defects

Smooth Gaussian expansion around one field value misses changes of winding sector, vortex cores, domain-wall proliferation, and other defect physics. Defects can control transitions even when a local field remains useful.

Additional soft degrees of freedom

Coupling an order parameter to gauge fields, conserved densities, phonons, or gapless fermions can generate nonanalytic kernels, alter exponents, or drive a transition first order. Integrating these modes out is legitimate only when their effects remain local and controlled at the scales of interest.

Strong first-order transitions

An amplitude expansion around zero may not cover distant competing minima. Interfaces can sample field values outside the polynomial’s reliable range. A phenomenological fit to equilibrium minima does not automatically predict the barrier or nucleation path.

Disorder and inhomogeneity

Random mass, random field, dilution, strain, and boundaries introduce spatially varying coefficients and new fluctuation regimes. Replacing disorder by its average can erase rare regions or change the symmetry of the effective problem.

Ultraviolet dependence

Coincident correlators and fluctuation determinants are cutoff dependent. A continuum integral extended to infinite momentum can diverge even though the original lattice model is finite. Regulate first, match coefficients consistently, and distinguish universal infrared behavior from ultraviolet-sensitive amplitudes.

Nonequilibrium and open systems

An equilibrium free-energy functional need not exist for a driven steady state. Even when a Lyapunov functional exists, it may not generate the probability distribution or response relations assumed above. Real-time contours, noise statistics, and conservation laws must be supplied explicitly.

Phases beyond local symmetry breaking

Topological order and intrinsically nonlocal organization cannot be classified completely by minimizing a local order-parameter field. Topological order explains this limitation through locally indistinguishable ground states.

Exercises

Exercise 1: Physical units

For

$$\mathcal F = \int d^d x\, \bigg[ \frac c2(\nabla\phi)^2 + \frac r2\phi^2 + \frac u4\phi^4 - h\phi \bigg].$$

derive the physical units of $c$, $r$, $u$, and $h$ in terms of energy, length, and the unit of $\phi$. Show directly that

$$\frac{k_{\mathrm B}T\,u}{c^2} \xi^{4-d}$$

is dimensionless.

Solution

The gradient term gives

$$[c] [\phi]^2 [L]^{d-2} = [E],$$

so

$$[c] = [E][L]^{2-d}[\phi]^{-2}.$$

The remaining terms give

$$\begin{aligned} [r] &= [E][L]^{-d}[\phi]^{-2}, \\ [u] &= [E][L]^{-d}[\phi]^{-4}, \\ [h] &= [E][L]^{-d}[\phi]^{-1}. \end{aligned}$$

Therefore

$$\left[ \frac{k_{\mathrm B}T\,u}{c^2} \right] = [L]^{d-4},$$

and multiplication by $\xi^{4-d}$ removes the remaining length unit. This check would fail if $\mathcal F$ and $\beta\mathcal F$ conventions were mixed.

Exercise 2: Sinusoidal source

In the disordered Gaussian theory, apply

$$h(\mathbf x) = h_0\cos(\mathbf q\cdot\mathbf x).$$

Find the stationary linear-response profile. Explain the limits $q\xi\ll1$ and $q\xi\gg1$.

Solution

The linear equation is

$$\left(-c\nabla^2+r\right)\phi = h_0\cos(\mathbf q\cdot\mathbf x).$$

Because the source is a Laplacian eigenfunction, take

$$\phi(\mathbf x) = \phi_q\cos(\mathbf q\cdot\mathbf x).$$

Substitution gives

$$\phi_q = \frac{h_0}{r+c q^2} = \frac{h_0/r}{1+q^2\xi^2}.$$

For $q\xi\ll1$, the source varies slowly across a correlation length and the response approaches the uniform value $h_0/r$. For $q\xi\gg1$, stiffness suppresses the response as $h_0/(c q^2)$. The crossover directly measures the pole length $\xi$ in the Gaussian approximation.

Exercise 3: Three-dimensional Gaussian correlator

Starting from

$$G(\mathbf k) = \frac{k_{\mathrm B}T}{r+c k^2},$$

show that for $x>0$ in three dimensions,

$$G(x) = \frac{k_{\mathrm B}T}{4\pi c x} e^{-x/\xi}.$$

Why is the expression not trustworthy at arbitrarily small $x$?

Solution

Choose the polar axis along $\mathbf x$. The angular integral gives

$$G(x) = \frac{k_{\mathrm B}T}{2\pi^2 c x} \int_0^{\infty} dk\, \frac{k\sin(kx)}{k^2+\xi^{-2}}.$$

For $x>0$,

$$\int_0^{\infty} dk\, \frac{k\sin(kx)}{k^2+\xi^{-2}} = \frac{\pi}{2}e^{-x/\xi}.$$

Combining factors yields

$$G(x) = \frac{k_{\mathrm B}T}{4\pi c x} e^{-x/\xi}.$$

The derivation extended the small-$k$ kernel to infinite momentum. At distances comparable with the coarse-graining cell, the cutoff and higher-gradient terms change the result. The $1/x$ divergence is therefore not a literal prediction at $x=0$.

Exercise 4: Domain wall

For $r<0$, verify the profile

$$\phi(x) = \sqrt{\frac{\lvert r\rvert}{u}} \tanh\!\left( x\sqrt{\frac{\lvert r\rvert}{2c}} \right).$$

Then derive its surface tension.

Solution

Let

$$\phi_0 = \sqrt{\frac{\lvert r\rvert}{u}}, \qquad \ell_{\mathrm w} = \sqrt{\frac{2c}{\lvert r\rvert}}.$$

For $\phi=\phi_0\tanh(x/\ell_{\mathrm w})$, direct differentiation gives

$$-c\phi'' - \lvert r\rvert\phi + u\phi^3 = 0.$$

The first integral implies

$$\frac c2(\phi')^2 = V(\phi)-V(\phi_0),$$

so the two contributions to the wall energy are equal and

$$\sigma_{\mathrm w} = c \int_{-\infty}^{\infty} dx\, (\phi')^2.$$

Using

$$\phi' = \frac{\phi_0}{\ell_{\mathrm w}} \operatorname{sech}^2 \!\left(\frac{x}{\ell_{\mathrm w}}\right)$$

and

$$\int_{-\infty}^{\infty}dy\, \operatorname{sech}^4 y = \frac43$$

gives

$$\sigma_{\mathrm w} = \frac{2\sqrt2}{3} \frac{\sqrt c\,\lvert r\rvert^{3/2}}{u}.$$

Exercise 5: Ginzburg criterion and dimension

Use

$$\langle(\delta\phi)^2\rangle_{\xi} \sim \frac{k_{\mathrm B}T}{c}\xi^{2-d}$$

and

$$\phi_0^2 \sim \frac{c}{u\xi^2}$$

to classify the fluctuation correction for $d=3$, $d=4$, and $d=5$. Why does the estimate alone not prove mean-field theory exact in five dimensions for every observable?

Solution

The ratio scales as

$$g_{\xi} \sim \frac{k_{\mathrm B}T u}{c^2} \xi^{4-d}.$$

Thus:

$$\begin{array}{c|c|c} d & g_{\xi}\text{ as }\xi\to\infty & \text{interpretation} \\ \hline 3 & \propto\xi & \text{grows} \\ 4 & \propto\xi^0 & \text{marginal estimate} \\ 5 & \propto\xi^{-1} & \text{decreases} \end{array}$$

In three dimensions the Gaussian approximation eventually fails. In four dimensions the scaling estimate is inconclusive about logarithms, so nonlinear RG is required. In five dimensions Gaussian exponents are stable for the short-range scalar critical point, but the quartic coupling can be dangerously irrelevant for order-parameter and finite-size scaling. The Ginzburg estimate also assumes that no other soft fields or nonlocal interactions intervene.

Exercise 6: Anisotropic correlation lengths

Let

$$\mathcal F_2 = \frac12 \int d^d x\, \left[ r\phi^2 + \sum_{i=1}^{d} c_i(\partial_i\phi)^2 \right],$$

with $r>0$ and every $c_i>0$. Find the Gaussian susceptibility and the principal correlation lengths. Show how a coordinate rescaling makes the quadratic kernel isotropic.

Solution

Fourier transformation gives

$$\chi(\mathbf k) = \frac{1} {r+\sum_i c_i k_i^2}.$$

Along principal axis $i$, the pole length is

$$\xi_i = \sqrt{\frac{c_i}{r}}.$$

Choose a reference stiffness $c_*$ and define

$$x_i' = x_i\sqrt{\frac{c_*}{c_i}}.$$

Then derivatives transform so that the gradient kernel becomes proportional to

$$c_* \sum_i (\partial_i'\phi)^2,$$

while the Jacobian changes the overall coefficient normalization. This removes quadratic anisotropy in the bulk continuum theory, but lattice anisotropies can survive in higher derivatives, boundaries, observables, or as relevant perturbations at other fixed points.

Exercise 7: Finite-wavevector instability

Consider

$$K(k) = r + c k^2 + \kappa k^4, \qquad \kappa>0.$$

Find the soft wavevector for $c>0$, $c=0$, and $c<0$. For $c<0$, determine the value of $r$ at which the minimum of $K(k)$ first reaches zero.

Solution

Stationary wave numbers satisfy

$$\frac{dK}{dk} = 2k \left(c+2\kappa k^2\right) = 0.$$

For $c>0$, the minimum is at $k=0$. At $c=0$, the leading dispersion is quartic and $k=0$ remains the minimum. For $c<0$, the nonzero minimum is

$$q_0^2 = -\frac{c}{2\kappa}.$$

At this wave number,

$$K(q_0) = r - \frac{c^2}{4\kappa}.$$

The Gaussian instability therefore occurs at

$$r = \frac{c^2}{4\kappa}.$$

This estimate requires $q_0$ below the cutoff and ideally $q_0\ll\Lambda$ so higher derivatives are negligible. With no ultraviolet regulator, dropping the quartic-gradient term when $c<0$ makes the continuum theory unbounded. A finite cutoff can bound that truncated model but does not recover the correct modulation scale.

Exercise 8: Classify four field theories

Classify each expression as a static thermal functional, a Euclidean quantum action, or an added dynamical law. State one piece of information missing from each schematic formula.

$$\text{(a)}\qquad \mathcal Z = \int\mathcal D\phi\, e^{-\beta\mathcal F[\phi]}.$$ $$\text{(b)}\qquad S_{\mathrm E} = \int d\tau\,d^d x\, \big[ (\partial_\tau\phi)^2 + (\nabla\phi)^2 + V(\phi) \big].$$ $$\text{(c)}\qquad \partial_t\phi = -\Gamma \frac{\delta\mathcal F}{\delta\phi}.$$ $$\text{(d)}\qquad \mathcal F = \int d^3 x\, \big[ \lvert(\nabla-iq\mathbf A)\psi\rvert^2 + V(\lvert\psi\rvert) + \frac{\mathbf B^2}{2\mu_0} \big].$$

Solution

• (a) is a static thermal field integral. It still needs a cutoff, measure, coefficient convention, and boundary conditions.
• (b) is a schematic Euclidean quantum action. It still needs the coefficient normalization, temporal boundary conditions, regulator, and justification for the quadratic time derivative.
• (c) is an added deterministic relaxational law. It still needs the conservation class, noise, mobility structure, and fluctuation–dissipation relation if equilibrium is intended.
• (d) is a static gauge-coupled Ginzburg–Landau functional. It still needs charge and $\hbar$ conventions, electromagnetic boundary conditions, material coefficients, and a statement of whether gauge-field fluctuations are integrated over or treated classically.

The exercise emphasizes that similar-looking gradient terms do not determine the physical interpretation by themselves.

Further reading

Statistical field theory develops the regulated measure and effective action. Vortex matter discusses the fluctuations, disorder, and dynamics that follow after finding a static superconducting vortex profile.

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