Landau theory

Landau theory describes phases using a symmetry-constrained expansion of an effective free energy in an order parameter. For a uniform scalar order parameter with sign-reversal symmetry, a simple example isAt zero conjugate field $h$, positive $a$ favors $m=0$, while negative $a$ produces minima $m=\pm \sqrt{-a/b}$. The change in the minimum gives a simple model of continuous ordering.

The coefficients summarize microscopic effects rather than following from symmetry alone. A negative quartic coefficient requires stabilizing higher powers and can permit a first-order transition. Minimizing the expansion gives mean-field critical exponents; sufficiently strong fluctuations invalidate that approximation near criticality. Landau-Ginzburg theory introduces spatial gradients so the same framework can describe interfaces and correlations.

An Arrott plot presents squared magnetisation against field divided by magnetisation. For the uniform SI Landau equation ${\mu }_{0}H=aM+b{M}^3$, it is a straight lineThe critical mean-field isotherm has $a=0$ and passes through the origin. Horizontal intercept $a/{\mu }_0$ changes sign across the transition; a positive vertical intercept estimates spontaneous magnetisation squared. The original criterion concerns internal magnetic field and the molecular-field regime. Demagnetizing fields and non-mean-field critical exponents require additional care.^[ref]

If physical ordinate is $Y=M^2$ and abscissa $X=H/M$, slope $S=dY/dX$ and horizontal intercept $X_0$ give $b={\mu }_0/S$ and $a={\mu }_0{X}_0$. Any displayed squared-magnetisation scale must be converted before using S. For example, an ordinate unit $(10^4\mathrm{A}{\mathrm{m}}^{-1}{)}^2$ equals $10^8{\mathrm{A}}^2{\mathrm{m}}^{-2}$, not $10^4$. With a free-energy density, the resulting units are $[a]=\mathrm{H}{\mathrm{m}}^{-1}$ and $[b]=\mathrm{J}\mathrm{m}{\mathrm{A}}^{-4}$. Curvature or differing isotherm slopes limits the accuracy of a constant-coefficient fit.

A long-wavelength order-parameter description extending a local Landau free energy by gradient terms. For a one-component complex field, a common isotropic form is $\mathcal{F}=\int [\alpha |\Psi {|}^2+\beta |\Psi {|}^4/2+K|\nabla \Psi {|}^2]d^{3}r$. A charged field instead uses a covariant gradient. Static minimization determines equilibrium profiles; a dynamical law requires additional input. Ginzburg-Landau theory of superconductivity and the Ginzburg-Landau functional of a dilute Bose gas realize this structure with different microscopic normalizations.

Take $\Psi =⟨\hat{\psi }⟩$ normalized so that $|\Psi {|}^2$ is condensate density. The coherent-field expectation of a dilute contact-interacting Bose Hamiltonian gives the grand-potential functional $\int [{\hslash }^2|\nabla \Psi {|}^2/(2m_B)+(V_{\mathrm{ext}}-\mu )|\Psi {|}^2+g_B|\Psi {|}^4/2]d^{3}r$. Its leading microscopic coefficients are therefore those displayed, with positive s-wave scattering length $a_s$ for repulsion. Varying with respect to ${\Psi }^{\ast }$ gives the stationary Gross–Pitaevskii equation. Thermal and quantum fluctuations can renormalize the coefficients; these mean-field identities are not exact universal critical coefficients.

For $f=f_0+a(T-T_c)q^2/2+b{q}^4/4$ with $a,b>0$, the minimum is $q=0$ above $T_c$ and $q=\pm \sqrt{a(T_c-T)/b}$ below it. The stationary solutions must be checked for stability; below $T_c$ the central solution is a maximum.

A scalar order parameter with symmetry $q\mapsto -q$ has only even powers in its zero-field free energy. A conjugate field contributes $-hq$. Without this symmetry, odd terms can be allowed and change the transition structure.

For an isotropic vector order parameter, an analytic scalar Landau free energy near the origin contains powers of $\mathbf{m}\cdot \mathbf{m}$. Odd powers of its magnitude are rotationally invariant but nonanalytic as functions of the Cartesian components at zero; excluding them requires analyticity, not merely writing the magnitude as a nonnegative scalar. With $d>0$, changing $b$ from positive to negative continuously creates minima at $|\mathbf{m}|=\sqrt{-b/(2d)}$, with an arbitrary direction.^[ref]

For zero field, $F=F_0+b(T-T_c)M^2+d{M}^4$ with $b,d>0$ has ordered minima $M^2=b(T_c-T)/(2d)$ below $T_c$. Minimizing gives $F_{\min }=F_0-b^2(T-T_c{)}^2/(4d)$. Since $C=-T{\partial }_T^2{F}_{\min }$, the ordered-phase excess heat capacity is $T{b}^2/(2d)$ and its jump relative to the disordered side is $T_c{b}^2/(2d)$. The value is a total heat capacity or a heat-capacity density according to the normalization of $F$.

For density $F=\alpha (T-T_c)M^2/2+b{M}^4/4-{\mu }_{0}HM$, the zero-field ordered minimum is $-{\alpha }^2(T-T_c{)}^2/(4b)$. Its constant-field heat-capacity density jump is $\Delta c_H=T_c{\alpha }^2/(2b)$. In an Arrott plot, let physical slope be $S={\mu }_0/b$ and horizontal intercept be $X_0(T)=a(T)/{\mu }_0$. ThenThis makes numerical sensitivity explicit: uncertainty in the intercept slope is squared. A volumetric free energy determines a volumetric heat-capacity jump; total or molar values need further sample information.

A mean-field approximation replaces the fluctuating influence of many neighbors by a self-consistent average. For an Ising model with coordination number $z$, exchange $J$ and field $h$, a uniform treatment gives $m=\tanh [\beta (zJm+h)]$ when spins take values $\pm 1$.

The assumed average magnetization appears on both sides: solving the self-consistency equation chooses a state compatible with its own effective field. This makes a many-body problem tractable and can estimate ordering temperatures and qualitative phases. It neglects correlations between fluctuations, so it can overestimate ordering and predict incorrect behavior in low dimensions. The Ginzburg criterion assesses when critical fluctuations are too large to justify the simple mean-field result.

A self-consistency equation requires the average field assumed in a mean-field approximation to equal the expectation value calculated in that field.

A Landau free energy is a symmetry-constrained analytic expansion in an order parameter. Its minima determine the mean-field equilibrium phases.

For $u>0$ and $t>0$, minimizing a uniform field gives zero magnetization when $|r|\le t$. For $r<-t$ only component 1 orders, with $m_1^2=-(t+r)/(4u)$; for $r>t$ only component 2 orders, with $m_2^2=(r-t)/(4u)$. Both transition lines are continuous. The positive-$t$ disordered wedge separates the two easy-axis phases.

For $f=a|\varphi {|}^2+|\varphi {|}^4/2+c|{\partial }_x\varphi {|}^2+|{\partial }_x^2\varphi {|}^2$, a plane-wave order parameter has quadratic coefficient $a+c{k}^2+k^4$. If $c<0$, this is minimized at $k^2=-c/2$, and a nonzero amplitude appears when $a<c^2/4$. If $c\ge 0$, the minimizing wave number is zero and the ordered mean-field phase requires $a<0$. The arbitrary global phase is a degeneracy, not a missing minimization condition. These are functional-minimization results; finite-temperature fluctuations can invalidate long-range order in low dimensions.

A stationary point on the domain $\varphi \ge 0$ must be tested using allowed one-sided variations. For $F=A{\varphi }^2/2-B{\varphi }^3+\lambda {\varphi }^4/4$ with $B>0$, the boundary is a local minimum when $A>0$ and a local maximum when $A<0$. At $A=0$, the leading negative cubic term makes it a one-sided local maximum even though $F^{\prime \prime }(0)=0$; on an unrestricted real line it is instead a stationary inflection point.

For $\varphi \ge 0$ and positive $c,B,\lambda ,T_0$, define $\gamma =2B^2/(\lambda c)$. Completing the square givesFor $0<\gamma <1$, phase coexistence occurs at $T^{\ast }=T_0/\sqrt{1-\gamma }$ with a discontinuous minimum at ${\varphi }^{\ast }=2B{T}^{\ast }/\lambda$. For $\gamma \ge 1$, a nonzero global minimum has lower free energy at every finite temperature. Following the global equilibrium branch then does not force a restoration transition.

When $\gamma \ge 1$ in the stated model, the broken, nonzero minimum remains globally preferred at every finite temperature. A metastable zero state can still exist for $T>T_0$, but its lifetime and a nucleation temperature are kinetic questions. Its disappearance at the spinodal point is not an equilibrium coexistence temperature.

At an equilibrium minimum, $d{F}_{\min }/dT={\partial }_{T}F$ because ${\partial }_{\varphi }F=0$. For $0<\gamma <1$, the entropy decrease at coexistence is $S_0-S_{+}=c(1-\gamma )T^{\ast }({\varphi }^{\ast }{)}^2$. Multiplying by $T^{\ast }$ gives the positive latent heat released on cooling. The units are total heat or heat density according to the normalization of $F$.

For $f=f_0+a(T-T_c)|\mathbf{m}{|}^2+b|\mathbf{m}{|}^4+c|\nabla \mathbf{m}{|}^2$ with positive coefficients and uniform free boundaries, the minimum is $\mathbf{m}=0$ at $T\ge T_c$. Below $T_c$ it is any constant vector with $|\mathbf{m}{|}^2=a(T_c-T)/(2b)$. A chosen direction leaves an $O(2)$ subgroup of $O(3)$ invariant; for proper rotations the pattern is $SO(3)\to SO(2)$. The mean-field transition is continuous, with square-root onset. Long-wavelength fluctuations can invalidate the ordered state in low dimensions.

Fix the spatial mean of a scalar order parameter to $M$ and sum the Boltzmann weights of all configurations with that mean. The resulting constrained partition function $Z_M$ defines this free-energy density, with an additive normalization fixed consistently. For an applied conjugate field $h$, $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the chosen measure normalization. The thermodynamic limit gives $f(h)={\inf }_M[F_c(M)-hM]$. A homogeneous Landau free energy is an approximation to this constrained potential; phase-separated configurations can convexify it in the full thermodynamic limit.

In a finite box, retain the constant mode $M$ while integrating all nonzero field modes using the Wilsonian coarse-grained statistical Hamiltonian. Below the smallest nonzero momentum, the remaining effective energy is $V{U}_{\Lambda }(M)$ and its Boltzmann weight is precisely the constrained order-parameter free energy weight. Thus the displayed equality holds with consistent field-independent normalization. It is not an identity between a bare statistical Hamiltonian and an unconstrained equilibrium free energy. An analytic polynomial Landau free energy further assumes a homogeneous branch and regular effective coefficients; long-distance fluctuations near and below an upper critical dimension can invalidate that analytic approximation.

For a real unrestricted scalar order parameter and $A=A_2{m}^2/2+A_3{m}^3/3+A_4{m}^4/4$ with $A_3\ne 0$ and $A_4>0$, the disordered global minimum is replaced discontinuously by a nonzero minimum at the displayed phase coexistence condition. At coexistence $A=A_4{m}^2[m+2A_3/(3A_4){]}^2/4$. Ordered stationary points first appear at the spinodal point $A_2=A_3^2/(4A_4)$, while the disordered point loses local stability at $A_2=0$; neither local limit is the equilibrium transition condition.

At coexistence, the landau free energy can factor as $F(\varphi )=\lambda {\varphi }^2(\varphi -v{)}^2/4$. The minima at $0,v$ have equal free energy. Differentiating shows the intermediate maximum at $v/2$, with barrier height $\lambda v^4/64$. The barrier is a local free-energy barrier; predicting spatial nucleation also requires interfacial or gradient information.

For $a,c>0$, $b\le 0$ and the molecular bound $Q\in [-1/2,1]$, coexistence with $Q=0$ occurs at $b_t=-{\min }_{0<Q\le 1}(a/Q+3cQ)$. Negative $Q$ has positive free energy and cannot be the stable phase in this coefficient range. Thus $a\le 3c$ gives $b_t=-\sqrt{12ac}$ and jump $0\to \sqrt{a/(3c)}$, whereas $a>3c$ gives $b_t=-(a+3c)$ and jump $0\to 1$. The interior case is the usual small-order-parameter Landau theory result. A formal unconstrained quartic minimization can violate the molecular bounds when extrapolated too far.

For the stable scalar Landau free energy $f=b{\lambda }^4+c{\lambda }^3+a{\lambda }^2+d\lambda$, $b>0$, a stable zero-curvature equilibrium must have $f^{\prime }=f^{\prime \prime }=f^{\prime \prime \prime }=0$. Solving these equations givesAt this point $f=f({\lambda }_c)+b(\lambda -{\lambda }_c{)}^4$. Varying two control parameters allows a line of first-order phase transitions to terminate here. A spinodal point satisfies zero curvature but need not satisfy the third-derivative condition or remain a local minimum.

For a scalar Landau free energy $f=a{\varphi }^2/2+b{\varphi }^4/4+c{\varphi }^3/3+d\varphi$ with $b>0$, the shift $\varphi =\psi -c/(3b)$ removes the cubic term. The quadratic coefficient becomes $a^{\prime }=a-c^2/(3b)$ and the linear coefficient becomes $d^{\prime }=d-ac/(3b)+2c^3/(27b^2)$. In a closed binary fluid mixture, the linear term is constant at fixed total composition, leaving the symmetric quartic phase coexistence construction in the shifted variable.

For $n\ge 2$ and ${\lambda }_n>0$, this Landau free energy has mean-field critical exponents $\beta =1/[2(n-1)]$ and $\alpha =(n-2)/(n-1)$ when $r$ crosses zero linearly with temperature. For $n>2$, obtaining this behavior requires tuning the lower even interactions to zero.

For the Landau free energy $G(\varphi )={\varphi }^6/6+g{\varphi }^4/4+\epsilon {\varphi }^2/2$, positive $g$ gives a continuous phase transition at $\epsilon =0$. For negative $g$, phase coexistence between the disordered global minimum $\varphi =0$ and the ordered global minima occurs on $\epsilon =3g^2/16$, with ${\varphi }^2=-3g/4$. Indeed, on that locus the polynomial is $G(\varphi )={\varphi }^2({\varphi }^2+3g/4{)}^2/6\ge 0$, so these three zeros are global minima. For $g<0$, the disordered and ordered spinodal points are respectively $\epsilon =0$ and $\epsilon =g^2/4$; equality of stationary free energies determines phase coexistence, while loss of a local minimum determines a spinodal point.

The equilibrium magnetization is a value of the magnetization that minimizes the Landau free energy. A stable equilibrium has nonnegative second derivative with respect to the magnetization; degenerate nonzero minima describe symmetry-related magnetized phases.

The spinodal is the boundary where a metastable phase loses local stability. In a mean-field free-energy description the relevant curvature vanishes there, separating local minima from unstable configurations.

Spinodal decomposition is spontaneous growth of composition fluctuations in a homogeneous mixture whose local mixing free energy has negative composition curvature. For a mean $\varphi$, equal regions at $\varphi \pm \delta \varphi$ change the mean free energy by $f^{\prime \prime }(\varphi )(\delta \varphi {)}^2/2+O((\delta \varphi {)}^4)$, which is negative inside the spinodal. Interfacial costs suppress sufficiently short wavelengths, while conservation of composition constrains the dynamics. Outside the spinodal but inside the binodal, the uniform state is instead metastable and requires nucleation to begin separation.

A spinodal point is the limit of local metastability of a phase. At a mean-field spinodal, a local minimum merges with a neighboring maximum and the corresponding second derivative of the free energy vanishes.

For the Landau free energy $V(M)=r{M}^2/2+u{M}^4/4-hM$ with $r<0$ and $u>0$, stationary branches satisfy $h=rM+u{M}^3$. Their local stability requires $r+3u{M}^2>0$. At a spinodal point, $M_{\mathrm{sp}}=\pm \sqrt{|r|/(3u)}$ and $h_{\mathrm{sp}}=(2r/3)M_{\mathrm{sp}}$, giving the displayed field magnitude. Following local minima without nucleation produces a hysteresis loop switching at these limits. Equilibrium instead switches between the two global minima at $h=0$.

A metastable state is locally stable against small disturbances but has a higher thermodynamic potential than another accessible phase. In a free-energy landscape it occupies a local minimum separated from the stable state by a barrier.

A liquid undergoing supercooling illustrates the distinction: small fluctuations can relax without freezing, but a sufficiently large crystal nucleus can grow. The nucleation barrier depends on the competition between interfacial cost and bulk free-energy gain. Waiting time is therefore affected by sample volume, impurities, surfaces and observation duration. A spinodal point is the mean-field limit where the metastable minimum loses local stability; it is distinct from the coexistence condition where the stable phases have equal appropriate intensive potentials. Metastability explains hysteresis without making the equilibrium phase boundary path dependent.

A vapor whose partial pressure exceeds its vapor pressure at the local temperature; nucleation can permit condensation.

The fractional excess of water-vapor pressure above equilibrium saturation over a flat pure-water surface at the same temperature: $s=p_v/p_{\mathrm{sat}}(T)-1$. Positive $s$ can drive droplet growth, but activation of a particular aerosol particle requires exceeding its critical supersaturation. Cooling, condensation and mixing jointly determine parcel supersaturation.

The ambient water vapor partial pressure divided by the equilibrium saturation pressure over a stated reference surface at the same temperature. Relative to flat pure water, atmospheric supersaturation is $s=S-1$. A curved solute-containing cloud droplet generally has a different equilibrium saturation ratio from unity.

Nucleation forms a sufficiently large region of a new phase within an old metastable phase. In classical nucleation theory, positive surface energy competes with bulk free-energy gain, producing a critical nucleus and a barrier to conversion. The Arrhenius nucleation time depends exponentially on this barrier; boundaries and impurities can alter it. This explains how metastability and hysteresis can persist despite a different globally stable phase.

Formation of most nuclei within an interval short compared with subsequent particle growth. A common birth period can help narrow a particle size distribution if mixing and later growth are sufficiently uniform and secondary nucleation is suppressed.

A schematic time history of growth-species concentration showing buildup, a short nucleation interval above an effective nucleation threshold, then growth above the equilibrium solubility but below substantial new nucleation. The concentration history suggests how to separate particle birth from growth; it does not establish that every particle synthesis follows that mechanism.

Formation of a critical nucleus at a pre-existing surface or interface. Favorable wetting can replace part of the cost of a free nucleus and lower its barrier; later deposition of a free particle on a support is a different process.

Ice forms from ice-supersaturated vapour at a suitable foreign surface. The relevant temperature and surface properties differ from liquid-droplet immersion freezing.

A suitable particle colliding with a supercooled droplet can trigger heterogeneous freezing. This is distinguished from nucleation around an inclusion already inside it.

Ice nucleates on an inclusion already immersed in supercooled water. The interfacial nucleation barrier, inclusion properties, droplet volume and observation time affect freezing probability.

Formation of a critical nucleus within the bulk parent phase without assistance from a pre-existing interface. Its interfacial cost competes with the bulk free-energy gain.

Ice embryos form without a foreign nucleating surface once thermal fluctuations can overcome the free-energy barrier on the relevant timescale. An apparent freezing threshold depends on droplet volume and measurement duration.

Dependence of a system's observed state on the path by which its control parameters were reached. In a first-order phase transition, a metastable local minimum may persist after another phase becomes globally stable, so increasing and decreasing the control parameter give different switching points. Nucleation and sweep rate determine actual switching; a mean-field spinodal point is a local-stability limit, not the equilibrium phase coexistence condition.

Magnetic hysteresis means that a material's magnetization depends on its field history as well as the present applied field. A closed quasistatic magnetic cycle dissipates energy per unit volume equal to its $H$–$B$ loop area. It is relevant to remanence in permanent magnets and core losses in electrical transformers.

Magnetic coercivity is the magnitude of a reverse applied magnetic field intensity required to bring a previously magnetized specimen's magnetization to zero on a specified hysteresis branch. It depends on material, preparation and measurement history. It differs from remanent magnetization, which is the remaining magnetization at zero applied field.

The magnitude of reverse magnetic field strength at which a previously magnetized specimen's magnetic flux density $B$ vanishes on a specified magnetic hysteresis branch. Intrinsic magnetic coercivity instead refers to zero magnetization. Because $B={\mu }_0(H+M)$, their zero conditions generally occur at different applied fields. The preparation, branch and specimen geometry must be specified when comparing measurements.

Superheating is persistence of a phase above the equilibrium temperature at which a higher-temperature phase becomes favorable. The familiar case is a liquid above its boiling point at the imposed pressure, although delayed melting can also be discussed as superheating.

The liquid can remain in a metastable state because producing a vapor bubble costs interfacial energy. A sufficiently large bubble overcomes the nucleation barrier and grows, converting the liquid and consuming latent heat. Smooth clean containers can delay nucleation, while cavities and boiling chips provide heterogeneous nucleation sites. The observed temperature of sudden bubble formation therefore need not equal the equilibrium boiling point. The amount of superheating is a kinetic and preparation-dependent observation, rather than a new equilibrium phase boundary.

A liquid maintained above its equilibrium boiling temperature at the imposed pressure. A barrier to nucleation can delay its conversion into vapor, making this a metastable state.

A solid can locally exceed its solidus because a thin solute-diffusion layer changes composition faster than thermal conduction changes temperature. Near a melting phase boundary, a depleted solid can join the equilibrium solidus at the interface while the adjacent, more concentrated solid remains above its own lower solidus. The assumed single sharp interface is then thermodynamically unstable to additional melting and may be replaced by a mushy layer.

Supercooling is persistence of a phase below the equilibrium temperature where a lower-temperature phase becomes favorable. A liquid remaining unfrozen below its equilibrium melting temperature is the standard example.

Creating a crystal nucleus gains bulk free energy but costs interfacial energy. Small embryos can disappear, while one exceeding the critical size can grow; this nucleation barrier permits a metastable liquid. A surface or suspended inclusion can lower the barrier through heterogeneous nucleation, whereas homogeneous nucleation occurs within the bulk. Measured freezing onset consequently depends on droplet size, purity and waiting time. Growth releases latent heat and can temporarily warm the remaining material toward the coexistence temperature, separating the onset of nucleation from the later heat-removal problem.

Two compositions coexist when one straight line is tangent to the free-energy density at both compositions. Equality of its slope gives equal chemical potentials, and equality of its intercept gives equal pressures.

The binodal is the coexistence boundary: its two branches give the compositions of phases joined by the common-tangent construction.

For a dilute-to-dense liquid coexistence problem, the dilute-phase coexistence concentration at specified conditions. It is a branch of the coexistence boundary, not generally the concentration at the critical endpoint.

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 $a-{\kappa }^2/(4\gamma )$.