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A free-energy functional assigns a free energy to an entire spatial order parameter configuration rather than only to one uniform value. In Landau-Ginzburg theory, a local derivative expansion gives $\mathcal{A}[m]=\int d^{D}x\{\kappa (\nabla m{)}^2/2+V(m)\}$, with appropriate boundary terms and regular backgrounds. The Landau approximation minimizes this functional; a fluctuating statistical field theory instead integrates its Boltzmann weight over configurations. These procedures need not give identical critical exponents.

An order parameter distinguishes thermodynamic phases and transforms under the symmetry that may be broken. For a scalar ferromagnet it is the magnetization per site; a disordered zero-field phase has $m=0$, while ordered pure thermodynamic phases have $m=\pm m_0$. In a fluid one can instead use the density measured relative to its critical value; the field conjugate to it is then related to the chemical potential rather than literally a magnetic field. A nonzero value in the presence of an explicit conjugate field is not by itself evidence of a spontaneous transition.

The LG theory describes a slowly varying local order parameter by a symmetry-constrained free-energy functional, for exampleAt $h=0$, a ${\mathbb{Z}}_2$ symmetry excludes odd powers. The coefficients are assumed analytic functions of the controls near the transition, and the expansion is stabilized by a positive highest retained even coefficient. The Landau approximation obtains equilibrium by minimizing this functional, neglecting long-wavelength fluctuation corrections. Its nonconvex local potential describes distinct candidate phases and mean-field metastability; the exact thermodynamic potential is convexified when macroscopic mixtures are admitted.

In a finite symmetric system, the zero-field expectation of $m$ can vanish even below the transition because both ordered orientations are sampled. Spontaneous order is defined by taking the thermodynamic limit before removing a selecting field, for example $m_0={\lim }_{h\downarrow 0}{\lim }_{N\to \infty }⟨m⟩$. This distinction also matters for the connected correlations used below.

Critical exponents describe leading power-law singularities as the reduced temperature $t=(T-T_c)/T_c$ and the conjugate field tend to zero. Their amplitudes depend on microscopic details, while the exponents are characteristic of a universality class. Define $m_0\sim (-t{)}^{\beta }$ on a selected ordered branch, $\chi \sim |t{|}^{-\gamma }$ at zero field, $C_{V,s}\sim |t{|}^{-\alpha }$ for the singular heat capacity, and $m(t=0,h)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. Also define $\xi \sim |t{|}^{-\nu }$ and $G(r,t=0)\sim r^{-(D-2+\eta )}$. In these statements $\beta$ is the order-parameter critical exponent, not inverse temperature.

Assume $r=r_{t}t$ with $r_t>0$ and a nonzero positive quartic coefficient $u$. Minimizing the quartic potential gives $m_0=\sqrt{-r/u}$, hence $\beta =1/2$. Differentiating its equation of state givesso $\gamma =1$ with different amplitudes. At $r=0$, $h=u{m}^3$, giving $\delta =3$. The minimized potential is zero above the transition and $-r^2/(4u)$ below, so its second temperature derivative has a finite jump: $\alpha =0$.

The quadratic fluctuation kernel around the stable uniform minimum is $\kappa p^2+V^{\prime \prime }(m_0)$, so ${\xi }^2=\kappa /V^{\prime \prime }(m_0)$. Since $V^{\prime \prime }$ is $r$ above and $-2r$ below, $\nu =1/2$. Its critical momentum dependence is $p^{-2}$, giving $\eta =0$. Thus the ordinary mean-field critical exponents areThese are predictions of the Landau approximation, whose validity is constrained by the Ginzburg criterion; they need not be the exponents of the fluctuating theory below its upper critical dimension.

The parameters of a cutoff statistical field theory describe only the retained modes. Changing the ultraviolet cutoff changes which fluctuations have already been integrated out, so its effective mass, interaction coefficients and gradient normalization must change to preserve the same long-distance physics. They are not separately cutoff-independent observables. Because the source's weight is $e^{-H}$, its statistical Hamiltonian is dimensionless, with the physical inverse-temperature factor already absorbed.

For example split the scalar field into slow modes ${\varphi }_{<}$ with $|p|<\Lambda /b$ and shell modes ${\varphi }_{>}$ with $\Lambda /b<|p|<\Lambda$. Define the momentum-shell renormalization group step byThis integrates out short-wavelength fluctuations exactly if all generated terms are retained. The new Hamiltonian can be expanded in local symmetry-allowed operators when the retained external momenta are well below the shell scale. A cumulant expansion of a coarse-grained free energy gives perturbative coefficients. Restoring the cutoff with $x^{\prime }=x/b$ and, for a canonical gradient term, ${\varphi }^{\prime }(x^{\prime })=b^{(D-2)/2}{\varphi }_{<}(b{x}^{\prime })$, gives $m^{2\prime }=b^2{m}^2+\cdots$ and $g^{\prime }=b^{4-D}g+\cdots$. Interaction corrections also change the field normalization. Repeating the step produces the effective theory at successively longer distances.

To obtain the LG theory, assume a short-range scalar theory, slowly varying retained fields, analytic local couplings, positive gradient stiffness and stability, and a regime in which fluctuation corrections at the remaining scales are small. Keeping the leading gradient, quadratic and quartic operators gives a local Landau-Ginzburg theory functional. Its equilibrium in the Landau approximation is a uniform minimum, with a quadratic coefficient proportional to $T-T_c$ after the critical mass has been tuned. The RG explains why this is asymptotically consistent for the ordinary transition above four dimensions: the quartic interaction is irrelevant near the Gaussian fixed point, while it must still be retained to stabilize the ordered phase. Below four dimensions it cannot be dropped in the asymptotic critical region; an interacting Wilson-Fisher fixed point rather than the elementary saddle generally controls the transition. At four dimensions the interaction is marginal and produces logarithmic corrections. Tuning a quartic coefficient through zero requires a sextic stabilizing term and leads to tricriticality.

More concretely, in the Gaussian scaling regime let $u=g/6$ so the local quartic term is $u{\varphi }^4/4$. At a total blocking scale $\ell$, the leading couplings are $r^{\prime }={\ell }^{2}r$, $u^{\prime }={\ell }^{4-D}u$, $h^{\prime }={\ell }^{(D+2)/2}h$. Apply the saddle approximation to the blocked potential and multiply its minimum by ${\ell }^{-D}$ to convert back to original volume units. Rescaling the saddle field by $(|r^{\prime }|/u^{\prime }{)}^{1/2}$ givesAll blocking-scale factors cancel. This explicitly recovers mean-field scalar free-energy scaling with $r\propto t$. Although $u^{\prime }\to 0$ above four dimensions, the saddle free energy is proportional to $1/u^{\prime }$; setting it to zero before minimization would remove the ordered phase. This is the dangerously irrelevant coupling mechanism rather than homogeneous two-variable hyperscaling.

For the perturbative calculation make the source's kinetic convention explicit. Write $K={\alpha }^{-1}>0$. The canonical normalization of a scalar gradient term uses $\psi =\sqrt{K}\varphi$, so the canonical quadratic and quartic coefficients are $r/K$ and $g/K^2$. In what follows $m^2,g$ denote those canonical coefficients; then the reference propagator has denominator $p^2+M^2$. Without this normalization the propagator denominator is $K{p}^2+r$, and the unmodified printed integral would not apply. At one loop the quartic tadpole is momentum independent, so it produces no gradient renormalization at this order.

In the convention fixed by the displayed equation, the truncated two-point function is the one-particle-irreducible two-point vertex, not the connected two-point cumulant itself. If $W[J]=\log Z[J]$ and $\Gamma [\phi ]=\int J\phi -W[J]$ is the Legendre transform, then its second derivative is the inverse of $W^{(2)}=G$. For a translation-invariant background,Decompose the canonically normalized statistical Hamiltonian into a Gaussian part of mass $M^2$, a mass counterterm $\delta m^2=m^2(\Lambda ,T)-M^2$, and the quartic interaction $g{\varphi }^4/4!$. With the Euclidean sign convention in which a positive mass correction increases the inverse propagator, the self-energy expansion isHere $\Sigma$ contains loop corrections from proper two-point diagrams, excluding the separately displayed mass counterterm. The corresponding connected propagator begins $G=G_0-G_0(\delta m^2+\Sigma )G_0+\cdots$. This fixes the sign, which would be reversed if “self-energy” instead denoted the insertion added with a plus sign inside a Dyson series.

The one-loop proper diagram is the tadpole diagram. Attaching two external fields to the quartic vertex gives $4\cdot 3$ contractions, divided by $4!$, so its symmetry factor is $1/2$. With the loop momentum restricted by the cutoff,It is independent of external momentum. Impose the zero-momentum mass condition $M^2=\tilde{\Gamma }(0)=m^2(0,T)$. This gives $\delta m^2+(g/2)I_D(M^2;\Lambda )=0$, henceThis is the one-loop relation using a renormalized mass in the reference propagator, or the corresponding self-consistent tadpole approximation if solved without expanding in $g$. It is not an exact all-orders gap equation. Away from the critical infrared problem, replacing the loop mass by the bare one changes a strict perturbative result only at higher order.

Take $g>0$ smooth and nonzero for an ordinary stable quartic transition. To test the assumed linear thermal mass, work from the disordered side and put $R=m^2(0,T)>0$. For $D>2$, $I_D(0;\Lambda )$ is infrared finite. The critical bare mass is shifted, not generically zero: $m^2(\Lambda ,T_c)=-(g/2)I_D(0;\Lambda )$. Absorb the smooth temperature dependence of couplings into an analytic thermal tuning $\tau =A_t(T-T_c)+\cdots$, with $A_t>0$. Critical subtraction gives the one-loop critical-mass subtractionwhereThe infrared asymptotics of the critical-mass subtraction now distinguish the dimensions. For $D>4$, $J_D(0)=K_D{\Lambda }^{D-4}/(D-4)$ is finite, soFor $D=4$, $J_4(R)=(K_4/2)\log [({\Lambda }^2+R)/R]$ diverges logarithmically. For $2<D<4$, substitution $p=\sqrt{R}q$ givesso the correction to $\tau$ scales as $R^{(D-2)/2}$ and dominates the analytic linear term. Pure mean-field linear mass scaling is therefore consistent only aboveAt the boundary dimension logarithms modify the simple power law. Below it this calculation diagnoses the failure of the Gaussian expansion; the exponent obtained by treating the self-consistent one-loop equation as exact is not automatically the exponent of the interacting scalar theory. For $D\le 2$, the massless subtraction itself has an infrared divergence, so this perturbative argument cannot establish absence of a transition. In particular it does not rule out the two-dimensional Ising critical point.

The same upper dimension follows by engineering dimension counting: a canonical scalar field has dimension $(D-2)/2$, so $[g]=D-4(D-2)/2=4-D$. For a tricritical point tune the renormalized quadratic and quartic terms to zero and retain a positive sextic interaction $v{\varphi }^6$. Its engineering dimension isIt is marginal at $D_c=3$, irrelevant above three, and relevant below three. More generally the upper critical dimension of an even scalar interaction ${\varphi }^{2n}$ is $2n/(n-1)$.

A Ginzburg criterion check gives the same result: at tricritical mean-field scaling $m_0^2\sim |t{|}^{1/2}$ and $\xi \sim |t{|}^{-1/2}$, whereas fluctuations in a correlation volume scale as ${\xi }^{2-D}\sim |t{|}^{(D-2)/2}$. Their ratio to $m_0^2$ is $|t{|}^{(D-3)/2}$, which tends to zero only for $D>3$. ThusThe tricritical tuning concerns renormalized couplings: shell contractions of a sextic term can regenerate quadratic and quartic terms even when their bare coefficients vanish. At three dimensions the marginal sextic coupling produces logarithmic corrections rather than a strictly fluctuation-free mean-field limit.

The statistical Hamiltonian in this question is already in thermal units, as indicated by $e^{-H}$ without an additional ${\beta }_T$. Define $W[h]=\log Z[h]$ and $F[h]=-W[h]$. The mean field and its linear response areThese are functional derivatives of the connected generating functional; the subtraction defines the connected correlation function. In the Landau approximation, neglect loop corrections and evaluate the field integral at a stable saddle $m_h$. It satisfies the Euler-Lagrange equationThis follows by varying the gradient term and integrating by parts, with periodic, decaying, or otherwise appropriate boundary conditions.

The two requested free energy functionals, following the source and Legendre conventions of the question, areIn the scalar-field source Legendre transform on the chosen stable branch, $h$ is chosen to produce $m$, and $\delta {\Gamma }_{\mathrm{L}}/\delta m=h$. Thus the imposed-source Helmholtz free energy and the fixed-order-parameter Gibbs free energy have the appropriate opposite source derivatives. These names are used in the question's magnetic-ensemble convention; the defining sign relation is what fixes the calculation. At leading Landau approximation there is no fluctuation-determinant term in $F_{\mathrm{L}}$.

Differentiate the saddle equation with respect to $h(y)$. The response obeysThereforeEquivalently, the inverse Hessian relation for a connected two-point function states that $G$ is the inverse kernel of ${\delta }^2{\Gamma }_{\mathrm{L}}/\delta m(x)\delta m(y)$. The factor $u_0/2$ follows from twice differentiating the quartic term $u_0{m}^4/4!$; it is not $u_0/6$. This tree-level connected response is obtained by varying the saddle. It does not require replacing the exact connected correlator by a product of the saddle values, which would incorrectly give zero.

For the requested single-momentum formula, assume a homogeneous source and a translationally invariant equilibrium phase, so $m_h(x)=m_0$ and $G(x,y)=G(x-y)$. SetThe Fourier transform with the printed positive sign sends $-{\nabla }^2$ to $q^2$, while the Dirac delta function transforms to one. Hence the Ornstein--Zernike correlation function hasThe inverse convention is $G(x)=\int d^{D}q(2\pi {)}^{-D}{e}^{-iq\cdot x}\tilde{G}(q)$. For general inhomogeneous $h(x)$, $K$ has nonconstant coefficients and $G$ depends separately on its two positions; the displayed momentum-diagonal formula then does not follow. The preceding differential equation still holds in that case.

At zero source and for $u_0>0$, the homogeneous saddle is $m_0=0$ for $r_0>0$, and $m_0^2=-6r_0/u_0$ on either selected stable ordered branch for $r_0<0$. Thus the Landau scalar correlation length isIn the ordered phase the negative bare quadratic coefficient is compensated by the positive curvature at the nonzero saddle. Retaining $m_0=0$ below the transition would instead give an unstable kernel and is not a physical correlation length. With the usual analytic thermal tuning $r_0=a_0(T-T_c)$, $a_0>0$, both branches diverge as $|T-T_c{|}^{-1/2}$, so the correlation-length critical exponent isThe high-temperature amplitude is $\sqrt{2}$ times the low-temperature amplitude for the same $a_0$. This statement is within Landau theory; fluctuations can change critical behavior outside the mean-field regime.

When $u_0=0$, the field theory is a Gaussian field theory, so integrating out short-wavelength modes can be done exactly rather than merely at a saddle. Introduce an ultraviolet momentum cutoff $\Lambda$ and a scale factor $b>1$, then split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ into $|q|<\Lambda /b$ and $\Lambda /b<|q|<\Lambda$. Orthogonality of Fourier modes eliminates cross terms in the quadratic Hamiltonian. A constant source couples only to the zero-momentum mode and therefore does not couple to ${\varphi }_{>}$.

Before rescaling, the Gaussian functional integral factorizes:where $H_{<}$ has the original $r_0$ and $h$, restricted to low momenta. Up to a field-independent normalization, the eliminated modes contributeThe determinant affects the additive free energy, but generates neither a mass correction nor a source correction, since no interaction mixes low and high modes. This remains true beyond the Landau approximation: it is an exact integration of a Gaussian field theory, not neglect of the high-mode fluctuations.

To restore the original cutoff, use $x=b{x}^{\prime }$, $q^{\prime }=bq$, and choose the field rescalingSubstitution into the low-mode Hamiltonian gives the Gaussian momentum-shell scalingThe gradient coefficient is unchanged: the volume factor is $b^D$, the two derivatives contribute $b^{-2}$, and the two fields contribute $b^{2-D}$. The mass term gains $b^2$, while the source term gains $b^{(D+2)/2}$. ThusFor a nonconstant source the corresponding formula is $h^{\prime }(x^{\prime })=b^{(D+2)/2}h(b{x}^{\prime })$, after restricting its coupling to retained modes. The constant-source case avoids that extra source filtering.

Writing $b=e^{\ell }$, the exact Gaussian renormalization-group flow hasBoth perturbations are relevant at the massless zero-source Gaussian fixed point, with $y_t=2$ and $y_h=(D+2)/2$. There is no anomalous field rescaling here, so $\eta =0$ and the correlation-length critical exponent is again $\nu =1/y_t=1/2$. The coefficient changes arise from rescaling, not from an interaction-induced shift in $r_0$ during shell integration.

A stable real Gaussian functional integral requires $r_0>0$, or an infrared prescription that treats the zero mode at the massless point. With $u_0=0$ and $r_0<0$, the field energy is unbounded below; the Gaussian field theory alone cannot describe a stable ordered phase. The scaling laws are therefore interpreted about the Gaussian fixed point from the stable side, with finite-volume zero-mode regularization when needed.

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.