Landau theory

Landau theory expands an effective free energy analytically in powers of an order parameter, constrained by symmetry.

The mean-field approximation replaces interactions with a self-consistent average field, neglecting correlated fluctuations around that average.

This is a use of mean-field theory, whose general framework also applies beyond a Landau free-energy model.

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.

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.

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.

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 phase is a local but nonglobal minimum of the thermodynamic potential. A finite nucleation barrier can delay conversion to the globally stable phase until fluctuations create a sufficiently large nucleus or the metastable minimum reaches a spinodal point.

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.

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.

Superheating is the metastable persistence of a low-temperature phase above its equilibrium transition temperature.

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 the metastable persistence of a high-temperature phase below its equilibrium transition temperature.

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.

Landau-Ginzburg theory supplements the local Landau potential with gradient terms and treats the order parameter as a spatial field.

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 free energy per unit area of an equilibrium interface relative to its coexisting bulk phases.

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 free energy quadratic in its field modes. If $H_0={\sum }_{\mathbf{q}}^{+}J(q)|{\varphi }_{\mathbf{q}}{|}^2$ for a real field, then $⟨|{\varphi }_{\mathbf{q}}{|}^2⟩=J(q{)}^{-1}$ and its partition function is a product of elementary Gaussian integrals.

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 $F_0=-\log Z_0$ and the expectation uses the trial Gibbs distribution $Z_0^{-1}{e}^{-H_0}$.

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

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 has one-dimensional translational order, represented at the simplest level by a periodically modulated field $\varphi (\mathbf{r})=A\cos ({\mathbf{q}}_0\cdot \mathbf{r})$.

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 requires order-parameter fluctuations inside a correlation volume to be small compared with the squared mean-field order parameter. Its failure marks the fluctuation-dominated critical region.

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 quadratic kernel $K(t,k)\sim t+k^2$, the singular heat-capacity correction is proportional to

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

For an ordinary quartic Landau transition, the heat-capacity and order-parameter exponents are $\alpha =0$ and $\beta =1/2$. At a tricritical sextic point they are $\alpha =1/2$ and $\beta =1/4$.

A tricritical point is where a line of continuous transitions meets a line of first-order transitions; in a ${\mathbb{Z}}_2$ Landau expansion both quadratic and quartic coefficients vanish and a positive sextic term stabilizes the free energy.

For the sextic Landau free energy $V=r{M}^2/2+v{M}^6/6-hM$, $v>0$, with $r$ linear in reduced temperature, the equation of state is $h=rM+v{M}^5$. Its ordered zero-field minimum satisfies $M^4=-r/v$, the inverse magnetic susceptibility is $4|r|$ below the transition and $r$ above, and the critical isotherm satisfies $h=v{M}^5$. The minimized singular free-energy density is $-|r{|}^{3/2}/(3\sqrt{v})$ below and zero above. These facts give the displayed mean-field critical exponents, obeying both the Rushbrooke scaling relation and Widom scaling relation. Both the quadratic and quartic coefficients must be tuned; a generic path with a positive quartic coefficient instead has ordinary critical behavior. Fluctuations change the result below the tricritical upper critical dimension three, and marginal corrections can add logarithms at three.

For an even sextic Landau free energy perturbed by a conjugate field, a tricritical wing is a phase coexistence surface at nonzero field between ordered minima of different magnitudes. Two symmetry-related wings emerge from the three-phase line at zero field and terminate at ordinary critical edges. They meet the zero-field ordered coexistence sheet at the tricritical point.

For $V=r{m}^2/2+u{m}^4/4+v{m}^6/6-hm$ with $v>0$, the displayed line has three global minima, $m=0$ and $m=\pm \sqrt{-3u/(4v)}$. The factorization $V=(v/6)m^2(m^2+3u/(4v){)}^2$ proves their coexistence. Two nonzero-field tricritical wings and the zero-field ordered coexistence sheet meet along this line. Its order-parameter discontinuity tends to zero as it terminates at the tricritical point.

Take two nonnegative coexisting minima $a\le b$, put $s=a+b$ and $p=ab$, and fix $v>0$. For the sextic Landau free energy with quadratic, quartic, sextic and linear terms, equal stationary minimum values occur atAt these coefficients,Thus both are global minima. The case $a=0$ gives the zero-field three-phase line, including the negative minimum; the limit $a=b$ gives the tricritical wing critical edge. Reflecting $M,h$ gives the negative-field wing.

For $f=r{M}^2/2+u{M}^4/4+v{M}^6/6-hM$ with $v>0$, the ordinary critical edge of a tricritical wing satisfies $f^{(1)}=f^{(2)}=f^{(3)}=0$. For $u<0$, $M_c^2=-3u/(10v)$, $r_c=9u^2/(20v)$ and $h_c=6u^2{M}_c/(25v)$. Its fourth derivative is $-12u>0$. The two signs of $M_c$ give the two wings.

For the stable sextic Landau free energy $f=r{m}^2/2+q{m}^4/4+s{m}^6/6-hm$, $s>0$, rescaling $m=(|r|/s{)}^{1/4}\psi$ gives the minimized order-parameter contribution $f_{\mathrm{eq}}=|r{|}^{3/2}{s}^{-1/2}{\Phi }_{\sigma }(X,Y)$, where $\sigma =\mathrm{sgn}r$, $X=q/(|r|s{)}^{1/2}$ and $Y=h{s}^{1/4}/|r{|}^{5/4}$. The two functions are ${\Phi }_{\sigma }={\min }_{\psi }(\sigma {\psi }^2/2+X{\psi }^4/4+{\psi }^6/6-Y\psi )$. In particular ${\Phi }_{-}(0,0)=-1/3$, but ${\Phi }_{+}(0,0)=0$. These separate branches and removal of smooth background terms are necessary when using the homogeneous form to infer mean-field critical exponents.

On the ordered branch of tricritical crossover scaling, at zero field and $X\to +\infty$, the minimizing square is $a={\psi }^2=(\sqrt{X^2+4}-X)/2\sim 1/X$. Its stationarity relation $Xa+a^2=1$ gives ${\Phi }_{-}(X,0)=-a/4-a^3/12\sim -1/(4X)$. Thus the minimized free-energy density becomes $-r^2/(4q)$ for fixed $q>0$ as $r\uparrow 0$, recovering the ordinary quartic continuous phase transition and its heat-capacity critical exponent $\alpha =0$. Replacing the scaling function by a nonzero constant in this limit would give an incorrect exponent.

In three dimensions, after tuning the quadratic and quartic relevant operators, the positive sextic coupling has a renormalization-group beta function $d\lambda /ds=-C{\lambda }^2+\cdots$ with $C>0$ for increasing length scale $s$. The Wick contractions connecting two sextic vertices with three propagators give the primitive logarithmic correction. The interaction is marginally irrelevant.