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A first-order phase transition exchanges two distinct global minima of the Landau free energy. The equilibrium free energy remains continuous, but a first derivative such as the magnetization or entropy jumps. A temperature-driven transition generically has latent heat; crossing $h=0$ between symmetry-related ordered phases instead has a magnetization jump without requiring latent heat. A continuous phase transition has no such order-parameter jump and usually has a divergent correlation length and nonanalytic higher responses.

For $u>0$, $h=0$ and $r$ crossing zero, minimizing $V=r{M}^2/2+u{M}^4/4$ gives $M=0$ for $r>0$ and $M=\pm \sqrt{-r/u}$ for $r<0$. The minima merge continuously at zero. The curvature vanishes there, so the long-distance response becomes singular.

For $u<0<v$, put $z=M^2$. A nonzero stationary minimum satisfies $r+uz+v{z}^2=0$. Equality with the $M=0$ free energy givesThe finite separation demonstrates the first-order phase transition. Metastability occurs where a nonglobal local minimum persists; its disappearance at a spinodal point is not the equilibrium coexistence criterion.

Use a selected translationally invariant pure thermodynamic phase and define the connected lattice correlation functionAway from criticality its long-distance tail has an exponential factor $e^{-r/\xi }$, possibly multiplied by a power. Equivalently ${\xi }^{-1}=-{\lim }_{r\to \infty }{r}^{-1}\log |G(r)|$ when this limit exists; different directions can have different lengths on an anisotropic lattice. At a thermodynamic critical point the correlation length diverges and the tail becomes algebraic. Differentiate $M=N^{-1}⟨{\sum }_n{\sigma }_n⟩$ with respect to the physical field appearing in the energy Statistical Hamiltonian. With ${\beta }_{\mathrm{th}}=1/(k_{B}T)$, the correlation-function susceptibility sum rule isIn physical continuum coordinates the sum becomes $a^{-D}\int d^{D}rG(r)$. Differentiating with respect to the dimensionless source ${\beta }_{\mathrm{th}}h$ instead removes the explicit inverse-temperature factor. Below the transition, take a one-sided pure-phase limit rather than a macroscopic symmetric mixture.

A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ assigns a probability or a delta constraint to a coarse configuration for each microscopic one, with ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$. Define the blocked Boltzmann weight bySumming over ${\sigma }^{\prime }$ proves exact equality of the partition functions. Repeated real-space renormalization group steps compose the kernels and integrate out shorter-scale variables. To retain equality one must keep all generated interactions and the field-independent constant; a finite truncation is an approximation. Long-distance observables are mapped through the corresponding coarse observables, not identified blindly with microscopic spins. In physical units,A subsequent coordinate rescaling restores the reference lattice spacing while expressing couplings in cutoff units.

Let $F(u,C)=-({\beta }_{\mathrm{th}}N{)}^{-1}\log Z=C+f(u)$ be free energy per microscopic site. Invariance of $Z$ givesThe constant transforms as $C_1=b^D{C}_0+c(u_0)$. Therefore, with $g(u)=b^{-D}c(u)$, the identity-operator contribution to renormalization-group free energy givesThe second equality follows by substitution and induction. The inhomogeneous term records eliminated-mode entropy, determinants, and normalization factors, all multiplying the identity operator; it is not an interaction with the retained spins. Strictly, this $f$ can contain regular background terms. After an analytic subtraction of an inhomogeneous renormalization recursion, the genuinely singular part satisfies the homogeneous scaling equation, except for possible logarithmic resonances.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. Linearize in scaling coordinates $s_i$: $s_i^{\prime }=b^{{\lambda }_i}{s}_i$. A relevant operator has ${\lambda }_i>0$, an irrelevant operator has ${\lambda }_i<0$, and a marginal operator has ${\lambda }_i=0$ and needs nonlinear analysis. These ${\lambda }_i$ are logarithmic scaling eigenvalues; the discrete Jacobian eigenvalues are $b^{{\lambda }_i}$, not ${\lambda }_i$. The critical surface is the stable manifold obtained by tuning all relevant coordinates to zero. A repulsive renormalization-group trajectory leaves the fixed point along a relevant direction as the coarse-graining length grows. Stable and unstable manifolds become curved away from linear order.

The left section has $h=0$, one irrelevant coordinate $w$ and a relevant thermal coordinate $t$. Its critical surface is $t=0$ in that section. With two relevant variables, the full critical surface has codimension two: the second section shows repulsion in the $(t,h)$ plane at $w=0$.

Assume an isolated fixed point with two relevant scaling fields, analytic changes from physical controls to $(t,h)$, short-range isotropic scaling, no dangerously irrelevant variable in the thermodynamic scaling function, and no marginal or resonant logarithm. Let ${\lambda }_t,{\lambda }_h>0$ denote their scaling eigenvalues. The homogeneous singular free-energy density obeysChoose a stopping scale $L=b^p\simeq |t{|}^{-1/{\lambda }_t}$ so the renormalized thermal field is order one. Irrelevant fields vanish at this scale. With $s=D/{\lambda }_t$ and $\Delta ={\lambda }_h/{\lambda }_t$,Continuous scale interpolation absorbs the immaterial integer stopping-scale convention. Nonuniversal thermal, field and free-energy metric factors can be included explicitly; the normalized functions depend only on the universality class. The $+$ and $-$ labels refer to $t>0$ and $t<0$, which flow to different sides of the critical surface.

For clarity about the additive terms, let $a(t,h)$ solve $a(t,h)=b^{-D}a(b^{{\lambda }_t}t,b^{{\lambda }_h}h)+g(t,h)$. If $g=\sum g_{mn}{t}^m{h}^n$, its analytic coefficients areprovided the denominators do not vanish. Subtracting $a$ makes the recursion homogeneous. A vanishing denominator is a resonance and generically produces logarithms, which the simple power-law hypothesis excludes. In a convention where the regular field-dependent background has been subtracted, write ${\Phi }_{\pm }=f_{\pm }+C_{\pm }$ and $I_{\pm }(t)=C_0+a(t,0)$. This yields the requested formThe $C_{\pm }$ contain branch-dependent matching contributions; only $f_{\pm }+C_{\pm }$ is physically determined. The $I_{\pm }$ are nonsingular functions analytic through the thermodynamic critical point and represent the same regular Taylor background on the two sides, though either branch may be used to describe them. For the full free energy at nonzero field, the regular background is generally $I(t,h)$, not just $I(t)$. Adding an analytic identity-operator contribution proportional to $h^2$ provides a direct counterexample to an exact unrestricted $I(t)$ formula; it changes no singular exponents. The printed expression is justified for the leading singular scaling form with that analytic field background removed. Confluent corrections from irrelevant fields are also omitted in this leading form.

Differentiate the singular scaling form, with one-sided derivatives in the ordered phase. It givesThe length transforms as $\xi (t,h)=L\xi (L^{{\lambda }_t}t,L^{{\lambda }_h}h)$, so $\nu =1/{\lambda }_t$. Thus the Rushbrooke scaling relation and hyperscaling relation follow by direct substitution:The same reasoning also gives $\gamma =\beta (\delta -1)$. Hyperscaling relation relies on the assumptions above and can fail above an upper critical dimension because of a stabilizing dangerously irrelevant coupling.

For the universal specific-heat amplitude ratio, write $F_s=A_f|a_{t}t{|}^s{\Phi }_{\pm }(0)$ at $h=0$. Two temperature derivatives multiply both branches by the same dimensional and thermal metric factors. Away from logarithmic resonances and exceptional zero amplitudes, $A_{+}/A_{-}={\Phi }_{+}(0)/{\Phi }_{-}(0)$ with the same heat-capacity sign convention on both sides. HenceThe functions and their matched branch constants are tied to the fixed-point theory; the constants are not independent arbitrary experimental parameters. If $\alpha <0$, this means the amplitudes of the subleading singular contribution after subtracting the analytic specific-heat background. A finite background ratio itself need not be universal.

Take a positive gradient coefficient ${\kappa }^{-1}$, a positive squared mass, a finite-volume regulator and an ultraviolet cutoff. Use the dimensionless statistical action convention $Z=\int \mathcal{D}\varphi e^{-H}$: inverse temperature is absorbed into the coefficients. If $H$ denotes dimensional energy instead, the exponent below gains a factor ${\beta }_{\mathrm{th}}$ and the covariance gains ${\beta }_{\mathrm{th}}^{-1}$. These conventions cannot be mixed.

Choose the Fourier transform conventionIntegration of the quadratic gradient and source terms givesComplete the square with ${\tilde{\varphi }}_{\mathrm{cl}}(p)=\tilde{\Delta }(p{)}^{-1}\tilde{J}(p)$. Translation of the regulated real Gaussian variables leaves their measure unchanged, so the Gaussian integral givesHere $Z_G=Z_0[0]$ is the source-free Gaussian partition function. For $n$ independent regulated real coordinates and a positive quadratic matrix $A$, it is $(2\pi {)}^{n/2}(\det A{)}^{-1/2}$ under Lebesgue normalization. The continuum determinant needs the stated regulator and a measure convention. Reality pairs the nonzero Fourier modes, so they must not be counted as two independent real fields.

A connected correlation function subtracts the product of the means. The logarithm $W[J]=\log Z_0[J]$ generates its cumulants: differentiating $W$ twice with respect to the source, using the measure $d^{D}p/(2\pi {)}^D$ in the functional derivative convention, yieldsThe source changes the mean but not the covariance; the stated zero-source limit therefore follows immediately. Momentum conservation uses $p+q$, appropriate for a real field without complex conjugation on either displayed factor.

For the continuum massive Gaussian field correlation tail, set $\mu =\sqrt{\kappa }m>0$ and $r=|x|$. Fourier inversion givesThe large-argument Modified Bessel function of the second kind has $K_{\rho }(z)\sim \sqrt{\pi /(2z)}{e}^{-z}$. Hence $G_0(r)$ is proportional to $r^{-(D-1)/2}{e}^{-\mu r}$ at large distance andAt zero mass the exponential length diverges. For $D>2$ the critical continuum tail is algebraic, $G_0(r)\propto r^{2-D}$; in lower dimensions infrared regulation needs additional care.

For a uniform source $h$, perform a momentum-shell renormalization group step: separate slow modes $|p|<\Lambda /b$ and fast modes $\Lambda /b<|p|<\Lambda$. In a Gaussian field theory they decouple; the uniform source couples only to the retained zero mode. Integrating fast modes contributes only a field-independent determinant to the free energy. Restore the cutoff by $x^{\prime }=x/b$, $p^{\prime }=bp$ andThe factors from the measure, gradient and two fields cancel in the gradient term. The mass term gains $b^2$, and the linear source gains $b^{(D+2)/2}$. ThereforeHere $m$ is the positive mass; for signed thermal coordinates it is the squared-mass relation that applies. The Gaussian fixed point has thermal scaling eigenvalue ${\lambda }_t=2$, source eigenvalue ${\lambda }_h=(D+2)/2$ and zero anomalous dimension.

Under the formal homogeneous Gaussian scaling hypothesis, substitution of those eigenvalues givesThe last two values are independently visible in $\xi \propto t^{-1/2}$ and $\chi ={\tilde{G}}_0(0)=1/m^2\propto t^{-1}$, agreeing with Landau-Ginzburg theory. The specific-heat power follows directly from the determinant: after two thermal derivatives its singular contribution is proportional toThe change of variables $p=\sqrt{\kappa t}q$ proves this power because the rescaled integral converges at large $q$ exactly when $D<4$. The two-dimensional free energy itself has a $t\log t$ resonance, while its second derivative still has the indicated $t^{-1}$ power.

The printed magnetization exponent is a formal Gaussian scaling index, not a proved ordered-phase law of the purely quadratic model. The ordered-phase obstruction in a Gaussian scalar model is explicit: the uniform field has energy density $m^2{M}^2/2-hM$. At $h=0,m^2<0$ it tends to $-\infty$ as $|M|\to \infty$, and the partition function diverges. For $m^2>0$, the mean is $M=h/m^2$ and tends to zero as $h\to 0$, so there is no spontaneously ordered branch. Thus no literal $M\sim |t{|}^{\beta }$ for $t<0$ can be established from this Statistical Hamiltonian alone. The value $(D-2)/4$ follows from the scaling dimension of the field if one assumes such a branch; for $D\le 2$ its nonpositive value further warns against that interpretation. Adding a stabilizing quartic interaction defines an ordered phase, but below four dimensions that interaction is relevant and generally leads to the Wilson-Fisher fixed point, not these Gaussian thermodynamic powers.

For the Gaussian specific-heat infrared threshold, at $D=4$ the same determinant integral grows like $\log ({\Lambda }^2/t)$, so $\alpha =0$ as a power exponent, with a logarithmic divergence. For $D>4$, it has a finite cutoff-dependent limit; the leading heat capacity is a regular constant and has no divergent power, again described by $\alpha =0$ in the requested convention. After subtraction of this background the Gaussian singular term can still have the negative power index $2-D/2$; it is not correct to erase that distinction. In a stable quartic Landau-Ginzburg theory above four dimensions, the quartic coefficient is a dangerously irrelevant coupling: it is needed for the ordered phase and contributes $f_s\propto -t^2/u$, giving the mean-field specific-heat jump and $\alpha =0$ while invalidating naive hyperscaling relation. At four dimensions, marginal interactions can add further logarithmic corrections. The agreement with Landau theory is in the leading heat-capacity power; it does not assert the absence of logarithms or supply a missing stable Gaussian ordered phase.

Along an isolated two-phase coexistence line, two distinct stable minima have equal free energy. If this line ends at an ordinary interior endpoint without encountering another phase, the distinction between the two phases must disappear: their minima and the intervening barrier merge. For separated nondegenerate minima, their stationary values are smooth functions of the controls. The difference $\Delta V=V(m_1)-V(m_2)$ has ${\partial }_h\Delta V=-m_1+m_2\ne 0$. The implicit function theorem therefore continues the equation $\Delta V=0$ as a local coexistence curve. It cannot end while those distinct stable minima persist.

At the merger the potential satisfiesThe third-derivative condition is essential: a stationary zero-curvature point with nonzero third derivative is an inflection, not a stable equilibrium. Shifting to $\psi =m-m_c$ gives an ordinary quartic Landau free energy with two independent control directions, temperature-like and field-like. The curvature and the order-parameter discontinuity vanish, the magnetic susceptibility diverges, and the correlation length diverges. The endpoint is therefore associated with a continuous transition. This is the critical endpoint of a quartic Landau free energy mechanism.

In the symmetric quartic example, the coexistence line is $h=0,r<0$. Its jump $2\sqrt{-r/u}$ vanishes at $(r,h)=(0,0)$, where $V=u{m}^4/4$. This supplies a concrete realization. The word “must” requires the stated ordinary-endpoint assumptions: a first-order line can instead meet a third phase, a multicritical point, a boundary of parameter space, or continue without a finite endpoint. A spinodal point alone is not an equilibrium critical endpoint.

A first-order phase transition has a discontinuity in a first derivative of the equilibrium free energy, such as the entropy or order parameter. It can have latent heat when the entropy jumps. A continuous phase transition has a continuously vanishing order parameter and no latent heat, with singular higher derivatives and a diverging correlation length. The LG theory compares global minima, not merely the points where a local minimum loses stability.

For the uniform quartic Landau free energythe equation of state is $rm+u{m}^3=h$. At zero field the stable minimum is $m=0$ for $r>0$, and $m=\pm \sqrt{-r/u}$ for $r<0$. Thus tuning $r$ through zero gives a continuous transition. For a fixed $r<0$, varying $h$ through zero instead switches between the two ordered minima and makes $m$ jump, giving a first-order phase transition in the conjugate field.

A temperature-like first-order transition can occur at zero field when $u<0$ and a positive sextic term $v{m}^6/6$ stabilizes the potential. Write $q=m^2$. Stationarity of a nonzero phase gives $r+uq+v{q}^2=0$, while equality with $V(0)=0$ gives $rq/2+u{q}^2/4+v{q}^3/6=0$. Solving these two conditions yieldsThe order parameter jumps from zero to $\pm \sqrt{q_{\mathrm{coex}}}$. At this pointso the competing stationary points are genuinely global minima. This phase coexistence condition differs from the spinodal points $r=0$ and $r=u^2/(4v)$, which mark loss or creation of local stability, not equilibrium coexistence.

If the symmetry permits a cubic term $w{m}^3/3$, a positive quartic coefficient does not preclude a first-order transition. For $V=r{m}^2/2+w{m}^3/3+u{m}^4/4$ with $w\ne 0,u>0$, stationarity and coexistence give $m=-2w/(3u)$ and $r=2w^2/(9u)$. This is the first-order transition in a cubic-quartic Landau potential; the symmetry restriction on the expansion is therefore part of the prediction.

The scaling hypothesis for critical phenomena says that the singular part of the equilibrium free energy is a generalized homogeneous function of its thermal and field-like controls. Smooth backgrounds must first be removed; the hypothesis does not claim that the entire free energy, including arbitrary regular terms, has a pure scaling form.

For the ordinary scalar quartic LG theory, set $r=r_{t}t$ with $r_t>0$ and $u>0$, and rescale the uniform order parameter asThe order-parameter-dependent free-energy density becomesDefine $f_{\pm }(H)$ as the minimum of the braces, with the sign of the quadratic term respectively positive or negative. ConsequentlyFor an extensive $A_s$, $a$ additionally contains the system volume. The printed double-inequality subscript labels the two temperature branches: the upper-temperature function is $f_{+}$ and the lower-temperature function is $f_{-}$. It does not classify positive and negative magnetic fields. In particular $f_{+}(0)=0$ and $f_{-}(0)=-1/4$; below the transition the field dependence has a cusp at zero, so derivatives are taken on a selected branch.

For the following derivatives, take $A_s$ to be a density, so $m$ is the magnetization density; for total free energy the derivative gives total magnetization instead. Differentiate this mean-field scalar free-energy scaling form. The magnetization is $m=-\partial A_s/\partial h=-ab|t{|}^{1/2}{f}_{\pm }^{\prime }(H)$, giving $m_0\propto (-t{)}^{1/2}$ and $\beta =1/2$. The magnetic susceptibility is $\chi =-{\partial }_h^2{A}_s=-a{b}^2|t{|}^{-1}{f}_{\pm }^{\prime \prime }(H)$, giving $\gamma =1$. At zero field the nonzero curvature amplitudes are $-f_{+}^{\prime \prime }(0)=1$ and $-f_{-}^{\prime \prime }(0^{+})=1/2$.

To allow nonclassical critical exponents, replace the fixed powers byThe heat-capacity critical exponent is defined by $C_{V,s}\sim |t{|}^{-\alpha }$; temperature differentiation gives the thermal exponent $2-\alpha$ in $A_s$. The order-parameter critical exponent has $m_0\sim (-t{)}^{\beta }$, and the magnetic-susceptibility critical exponent has $\chi \sim |t{|}^{-\gamma }$. Differentiating the scaling form givesEliminating $\Delta$ proves the Rushbrooke scaling relationThe critical-isotherm exponent is defined by $m(0,h)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. At fixed small $h$, the $t\to 0$ limit requires $f_{\pm }(H)\sim c|H{|}^{(2-\alpha )/\Delta }$, so that the temperature factors cancel. Therefore $1/\delta =(2-\alpha -\Delta )/\Delta =\beta /\Delta$, yielding $\Delta =\beta \delta$. But the two differentiated identities also give $\Delta =\beta +\gamma$, and hence the Widom scaling relationThese are relations among the leading power indices. At marginal dimensions, multiplicative logarithms can accompany them, and an additive analytic background must not be mistaken for the singular scaling contribution.

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 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.