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For a stable homogeneous scalar quartic saddle with gradient coefficient one, the Ornstein--Zernike correlation function has denominator $q^2+r_0+u_0{m}_0^2/2$. At zero source and $u_0>0$, the correlation length is $r_0^{-1/2}$ in the disordered phase and $(-2r_0{)}^{-1/2}$ on either selected ordered branch. Linear thermal tuning of $r_0$ gives correlation-length critical exponent $\nu =1/2$ with different amplitudes on the two sides.

Landau-Ginzburg theory describes a slowly varying order parameter by a local free-energy functional consistent with the microscopic symmetries. For a representative scalar system with spin-inversion symmetry, take the dimensionless functionalThe gradient term penalizes spatial variation, while the even local potential implements the zero-field symmetry $\varphi \mapsto -\varphi$. The positive quartic coefficient stabilizes a continuous transition. A negative quartic coefficient requires higher powers and can instead produce a first-order phase transition. A vector order parameter gives the corresponding rotationally invariant model with powers of $|\mathbf{\varphi }{|}^2$.

In the mean-field approximation, replace the field by a constant $M$ and minimize the local potential. Its equation of state and stable zero-field solutions areBelow the transition these two minima exhibit spontaneous symmetry breaking. Differentiating the equation of state gives $\chi =(r+3u{M}^2{)}^{-1}$: it is $1/r$ above the transition and $1/(2|r|)$ within either ordered phase. At $r=0$, $M=(h/u{)}^{1/3}$. The minimized zero-field free-energy density is zero above and $-r^2/(4u)$ below, apart from an analytic background; its second temperature derivative has a finite jump. Consequently

To include fluctuations, write $\varphi =M+\psi$. The quadratic part isA Gaussian functional integral therefore gives $⟨\psi (q)\psi (-q)⟩=(c{q}^2+r+3u{M}^2{)}^{-1}$, with the momentum-volume delta function understood. The Ornstein--Zernike correlation function has correlation length $\xi =\sqrt{c/(r+3u{M}^2)}$. Thus ${\nu }_{\mathrm{MF}}=1/2$, and at criticality the Gaussian covariance is proportional to $q^{-2}$, giving ${\eta }_{\mathrm{MF}}=0$. Continuous-symmetry ordered phases additionally have transverse Goldstone modes, for which the restoring mass vanishes.

The Gaussian integration contributes $\frac{1}{2}\int d^{d}q(2\pi {)}^{-d}\log (c{q}^2+r)$ to the disordered-phase free-energy density. Differentiating twice with respect to $r$ gives a critical contribution proportional to $\int d^{d}q(c{q}^2+r{)}^{-2}$; for $d<4$ its singular part grows as $r^{(d-4)/2}$ and at four dimensions it is logarithmic. Interacting fluctuations thus become important close to the critical point below four dimensions. The Ginzburg criterion tests this failure of the mean-field approximation; the renormalization group then explains modified exponents and universality. Mean-field theory finds the competing minima, while fluctuations determine whether its predicted critical behavior survives at long distance.

Use the connected correlation function $G(x)=⟨\varphi (x)\varphi (0)⟩-⟨\varphi {⟩}^2$, so that the constant ordered contribution is removed. Away from a thermodynamic critical point its long-distance decay defines the correlation length: $-{\lim }_{R\to \infty }{R}^{-1}\log |G(R)|={\xi }^{-1}$ when this limit exists. In a massive rotation-invariant phase the exponential decay is typically multiplied by an algebraic factor.

Expand the Landau-Ginzburg theory free energy about a stable uniform minimum $M$, writing $\varphi =M+\delta \varphi$. To quadratic order,integration of the Gaussian field theory gives the Fourier-space connected correlation function $\tilde{G}(q)=k_{B}T/(c+\kappa q^2)$. Thus the Ornstein--Zernike correlation function hasFor an ordinary quartic transition, $M=0$ above the transition and $M^2=-r/u$ below, so $c=r$ and $c=-2r$, respectively. The correlation length diverges as $|T-T_c{|}^{-1/2}$ on both sides, with different amplitudes. At criticality the mass vanishes and the Gaussian correlation function is proportional to $R^{-(D-2)}$ for $D>2$. Interactions can change this power to $R^{-(D-2+\eta )}$.

Let $t=(T-T_c)/T_c$, and suppose $r=r_{t}t$ with $r_t>0$ while $u>0$ and $\kappa >0$ remain finite at the transition. Critical exponents describe the leading singular powers: $M\sim (-t{)}^{\beta }$ below the transition, $\chi \sim |t{|}^{-\gamma }$, $\xi \sim |t{|}^{-\nu }$, $C_{V,s}\sim |t{|}^{-\alpha }$, $M(t=0)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, and $G(R,t=0)\sim R^{-(D-2+\eta )}$.

The quartic equation of state $h=rM+u{M}^3$ immediately gives $M=\sqrt{-r/u}$ at zero field below the transition, so $\beta =1/2$. Differentiating with respect to the conjugate field gives ${\chi }^{-1}=r+3u{M}^2$, which is $r$ above and $2|r|$ below; therefore $\gamma =1$. At $r=0$, $M=(h/u{)}^{1/3}$ with its real sign, giving $\delta =3$. The quadratic correlation function has $\xi \propto |r{|}^{-1/2}$ and critical Ornstein--Zernike correlation function proportional to $q^{-2}$, giving $\nu =1/2$ and $\eta =0$.

Finally the minimized potential below the transition is $V_s=-r^2/(4u)$, while it is zero above. Its second temperature derivative is a finite step, so the mean-field heat-capacity critical exponent is $\alpha =0$. In particular,These are derived mean-field critical exponents. In the fluctuation-dominated critical region of a phase transition, one instead derives exponents from the scaling eigenvalues of the renormalization-group fixed point, as in the next question. A zero heat-capacity power exponent can also accompany logarithms in other theories, so specifying the power alone does not completely describe that singularity.

Use the connected correlation function $G(x)=⟨M(x)M(0)⟩-⟨M{⟩}^2$, since the disconnected constant in an ordered phase does not measure fluctuation decay. Away from criticality its large-distance exponential defines the correlation length:when that limit exists. Algebraic prefactors do not affect this definition.

Expand the Landau-Ginzburg theory about a stable homogeneous minimum $M_0$. With $A=V^{\prime \prime }(M_0)>0$, the quadratic fluctuation energy is $\frac{1}{2}\int d^{D}x[\kappa |\nabla \delta M{|}^2+A(\delta M{)}^2]$. The Gaussian functional integral givesThis Ornstein--Zernike correlation function shows why the curvature of the free-energy minimum controls correlations. At a continuous phase transition that curvature vanishes, allowing long-range fluctuations. In the ordinary quartic theory with $u>0$, $A=r$ above the transition and $A=-2r$ on either ordered branch below, so $\xi \propto |r{|}^{-1/2}$ on both sides. At an equilibrium first-order transition the competing minima generally retain nonzero curvature, and the correlation length remains finite. The limit of local stability at a spinodal point is a distinct mean-field phenomenon.

Work in a translation-invariant pure thermodynamic phase, and write ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ to distinguish inverse temperature from the order-parameter exponent. The connected correlation function isAway from criticality in the massive scalar order channel, it decays exponentially, possibly multiplied by an algebraic factor. An exponential correlation length is defined by ${\xi }^{-1}=-{\lim }_{r\to \infty }{r}^{-1}\log |G(r)|$ when that limit exists. At a continuous transition $\xi$ diverges and the critical decay is algebraic. In an ordered symmetric mixture, a nondecaying contribution can remain; selecting a pure branch prevents confusing it with connected critical fluctuations.

Let $M={\sum }_{\mathbf{n}}{\sigma }_{\mathbf{n}}$ and $m=⟨M⟩/N$. Differentiating the partition function with the energy term $-hM$ gives the correlation-function susceptibility sum ruleIf the source is instead the dimensionless ${\beta }_{\mathrm{th}}h$, the explicit inverse-temperature factor is absent. In continuum physical coordinates the lattice sum is approximately $a^{-D}\int d^{D}rG(\mathbf{r})$.

Choose a normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )\ge 0$ such that ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$, replacing the sum by an integral for continuous variables. A deterministic example is a product of delta functions imposing that each coarse variable equals the average of spins in a block of $b^D$ sites, with its field normalization included consistently. Define the effective Boltzmann weight byKernel normalization makes the blocked partition function exactly equal to the original. Its coarse-grained variables retain the long-distance observables through their defining relation to the original fields. An exact RG step generally generates all symmetry-allowed operators; keeping only a small coupling set is an approximation, not an exact closure assumption. The geometrical coarse lattice hasso its physical volume is unchanged. A subsequent coordinate rescaling may restore the numerical lattice spacing to its initial value, which is the equivalent rescaled-coordinate convention.

Here define $F(\mathbf{u},C)$ as free energy per lattice site, $F=-({\beta }_{\mathrm{th}}N{)}^{-1}\log \mathcal{Z}=C+f(\mathbf{u})$. Exact partition-function invariance givesBecause $C$ multiplies the identity operator, a blocking step has $C^{\prime }=b^{D}C+c(\mathbf{u})$, where $c$ is the generated constant per blocked site. With $g(\mathbf{u})=b^{-D}c(\mathbf{u})$ this impliesThe identity-operator contribution to renormalization-group free energy includes the eliminated modes' entropy, connected vacuum terms and field-measure normalization. It cannot be discarded when calculating an absolute free energy, even though it cancels out of normalized correlation functions.

There is a useful convention behind the word “singular” in this inhomogeneous equation. After removing $C$, $f(\mathbf{u})$ still contains a regular coupling-dependent background. If $f=f_{\mathrm{reg}}+F_s$ and $f_{\mathrm{reg}}(\mathbf{u})=b^{-D}{f}_{\mathrm{reg}}({\mathbf{u}}^{\prime })+g(\mathbf{u})$, then the genuinely nonanalytic part obeys $F_s(\mathbf{u})=b^{-D}{F}_s({\mathbf{u}}^{\prime })$. Thus the printed inhomogeneous representative and homogeneous singular scaling below are consistent after background subtraction. At resonances or marginal points this subtraction may leave additive or multiplicative logarithms; a strictly homogeneous pure-power form is then qualified accordingly.

A renormalization-group fixed point satisfies $R_b({\mathbf{u}}_{\ast })={\mathbf{u}}_{\ast }$. Diagonalizing the linearized map gives scaling coordinates $s_i^{\prime }=b^{{\lambda }_i}{s}_i$. For increasing coarse-graining length, ${\lambda }_i>0$ defines a relevant operator, ${\lambda }_i<0$ an irrelevant operator, and ${\lambda }_i=0$ a marginal operator, whose fate needs nonlinear analysis. Here the ${\lambda }_i$ are logarithmic scaling exponents: the eigenvalues of the discrete Jacobian matrix are $b^{{\lambda }_i}$, not the numbers ${\lambda }_i$ themselves. The critical surface is the stable manifold flowing into the critical fixed point after every relevant scaling field is tuned to zero. A repulsive renormalization-group trajectory leaves the fixed point along relevant directions and flows toward a different long-distance phase.

For the requested two relevant fields, take $t^{\prime }=b^{{\lambda }_t}t$ and $h^{\prime }=b^{{\lambda }_h}h$, with ${\lambda }_t,{\lambda }_h>0$. Ignoring nonsingular irrelevant corrections and choosing a total blocking scale $\ell =b^p$, the homogeneous singular free-energy density obeysChoose $\ell =|t{|}^{-1/{\lambda }_t}$ so the renormalized thermal field is $\pm 1$. This provesMetric factors and regular analytic redefinitions of the scaling fields only change amplitudes. Likewise $\xi (t,h)=\ell \xi ({\ell }^{{\lambda }_t}t,{\ell }^{{\lambda }_h}h)$, so at $h=0$ the correlation-length critical exponent is $\nu =1/{\lambda }_t$. Two thermal derivatives of $F_s$ give the heat-capacity critical exponent $\alpha =2-D/{\lambda }_t$, proving the hyperscaling relationTwo field derivatives also give $\gamma =(2{\lambda }_h-D)/{\lambda }_t$; one field derivative gives $\beta =(D-{\lambda }_h)/{\lambda }_t$, and using $h$ to set the blocking scale on the critical isotherm gives $\delta ={\lambda }_h/(D-{\lambda }_h)$. These statements assume that no dangerously irrelevant coupling makes the scaling function singular as it is removed. In particular the naive hyperscaling form need not describe the quartic ordered phase above four dimensions, where its stabilizing quartic coupling is dangerously irrelevant. That restriction reconciles this derivation with the mean-field exponents and Question 3.

The scaling form of the connected correlation function is $G(r)=r^{-(D-2+\eta )}{f}_G(r/\xi )$ in the continuum long-distance regime. For $r\ll \xi$, $f_G$ tends to a finite nonzero constant. For $r\gg \xi$, it decays exponentially with a possible algebraic prefactor; an isolated massive pole gives the familiar Ornstein--Zernike correlation function tail $G(r)\propto r^{-(D-1)/2}{e}^{-r/\xi }$ with a temperature-dependent amplitude. Exponential decay is the essential feature; the same critical algebraic power need not remain the exact large-distance prefactor.

Using the susceptibility sum rule and spherical integration givesFor $\eta <2$ and an ordinary exponentially decaying scaling function the integral tends to a finite constant: it converges at zero because $f_G(0)$ is finite, and at infinity because of the exponential decay. Microscopic distances add a regular background. Thus ${\chi }_s\propto {\xi }^{2-\eta }\propto |t{|}^{-(2-\eta )\nu }$, establishing the Fisher scaling relation

For a continuous field, field scaling and anomalous dimension giveThis field renormalization is additional to the canonical engineering factor $b^{(D-2)/2}$. It makes $G(r)=b^{-2x_{\sigma }}{G}^{\prime }(r/b)$. Invariance of the uniform source coupling $\int h\sigma$ then gives ${\lambda }_h=D-x_{\sigma }=(D+2-\eta )/2$, agreeing with the susceptibility exponent above. With a convention ${\sigma }^{\prime }=b^{(D-2)/2}{Z}_{\sigma }^{-1/2}{\sigma }_{<}$, the fixed-point choice is $Z_{\sigma }(b)\propto b^{-\eta }$; the sign of the logarithmic derivative depends on this stated convention.

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 $b={\kappa }_1=0$, stationarity in $\varphi$ givesSubstituting this value, or completing the square in the Gaussian integral over $\varphi$, givesWhen $A>0$ and the quartic term is negligible, each Fourier mode is Gaussian with the Ornstein--Zernike correlation functionup to the chosen Fourier normalization. Its correlation length is ${\xi }_p=\sqrt{\kappa /A}$.