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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.

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.

The hyperscaling relation expresses the idea that the singular dimensionless free energy in one correlation volume is of order one. A macroscopic volume $V$ contains of order $V/{\xi }^d$ such correlated regions, soSubstitution of the homogeneous correlation length givesThis has exactly the required homogeneous field variable. Comparing its thermal power with that of the singular free-energy density provesThe argument concerns powers, not equality of the two scaling functions or their amplitudes. It assumes the correlation length supplies the only singular length scale and that no dangerously irrelevant coupling changes the free energy per correlated region. For ordinary short-range quartic theory, hyperscaling holds below the upper critical dimension four, with the usual qualifications for marginal logarithms at four. Above four, mean-field critical exponents $\alpha =0$, $\nu =1/2$ do not satisfy $d\nu =2-\alpha$; the quartic coupling is dangerously irrelevant. The stated homogeneous form of $\xi$ alone therefore does not prove hyperscaling without the correlation-volume assumption.

The Ginzburg criterion tests whether fluctuations on the scale of one correlation volume are small compared with the squared equilibrium order parameter. In the scalar quartic Landau-Ginzburg theory of part (a), the ordered mean-field value is $M^2=|r|/u$, and its longitudinal correlation length is proportional to $(c/|r|{)}^{1/2}$.

Average the fluctuation field over a region of size $\xi$. Its smoothing factor selects momenta of order ${\xi }^{-1}$ or smaller. The Gaussian correlation function then gives, up to a positive dimension-dependent constant,The last step follows directly by setting $q=\sqrt{|r|/c}k$. Dividing by $M^2$ proves the correlation-volume derivation of the scalar Ginzburg ratio:For $d<4$, the ratio diverges as the transition is approached, so there is a fluctuation-dominated critical region of a phase transition in which mean-field critical exponents cannot be trusted. With $r=r_{0}t$, an estimate of its width isin fixed microscopic units. Its numerical coefficient depends on the averaging convention and microscopic normalization.

For $d>4$, the ratio tends to zero and the long-distance mean-field approximation becomes self-consistent. Four is therefore the ordinary quartic upper critical dimension. At $d=4$ this test is marginal: its thermal power is zero, and the running quartic interaction produces logarithmic corrections. The marginal Ginzburg criterion alone cannot decide those logarithms. Short-distance contributions to an unaveraged variance renormalize the coefficients; they should not be mistaken for the long-wavelength fluctuations used in this criterion. Nor does the criterion by itself prove that a proposed ordered phase exists below its lower critical dimension.

The relevant boundary here is the upper critical dimension: above it the critical interactions become negligible at long distances and the ordinary mean-field critical exponents apply; at it there can be logarithmic corrections. This differs from the lower critical dimension, below which a particular kind of ordering may fail to exist.

Normalize the coefficient of $(\nabla \varphi {)}^2$ in the Landau-Ginzburg theory. Under a change of length scale $x=b{x}^{\prime }$, its invariance gives the engineering dimension $x_{\varphi }=(d-2)/2$. The quartic interaction then scales asIt is a relevant operator below four dimensions and an irrelevant operator above four. Thus the ordinary scalar transition has upper critical dimension $d_c=4$. At a tricritical point, where the quartic interaction is tuned away, the sextic coupling scales as $v^{\prime }=b^{6-2d}v$, giving the tricritical upper critical dimension $d_c=3$. This is the upper critical dimension of an even scalar interaction applied to powers four and six.

The Ginzburg criterion makes the reason for failure of the Landau approximation quantitative. In the ordered quartic theory, $M^2=|r|/u$, $V^{\prime \prime }(M)=2|r|$ and ${\xi }^2=K/(2|r|)$. The fluctuation of the order parameter averaged over a correlation volume has the scaleCompared with the squared mean order parameter, the Ginzburg criterion therefore givesFor $d<4$ this ratio grows without bound as the correlation length diverges. The assumption that fluctuations are small relative to the ordered value fails arbitrarily near the transition, and the interacting renormalization-group fixed point determines the critical exponents. For $d>4$ the ratio tends to zero. At $d=4$ scale counting is marginal, and the running quartic coupling produces logarithms.

Similarly, at the tricritical point $M^2\asymp \sqrt{K/v}{\xi }^{-1}$, so the multicritical Ginzburg ratio scales as $(k_{B}T\sqrt{v}/K^{3/2}){\xi }^{3-d}$. The Landau-Ginzburg theory remains a useful effective description below its upper critical dimension; it is its fluctuation-free exponent calculation that fails, not the possibility of including fluctuations in the theory.

The scaling hypothesis for critical phenomena describes the singular part of the equilibrium free-energy density, rather than its analytic background. For an isotropic system in spatial dimension $D$, with thermal scaling field $t$ and conjugate field $h$, writeThis follows from coarse graining near a renormalization-group fixed point: a correlation volume is enlarged by $b^D$, while the relevant thermal and field perturbations grow with their eigenvalues. Choosing $b=|t{|}^{-1/y_t}$ givesThe two functions describe the two sides of the transition. Nonuniversal metric factors multiplying $t,h,f_s$ must be allowed when comparing microscopic models. Exponents and suitably normalized scaling functions or amplitude ratios characterize a universality class.

Differentiating the free-energy density gives the order parameter $M=-{\partial }_h{f}_s$, magnetic susceptibility $\chi ={\partial }_{h}M$ and singular heat capacity proportional to $-{\partial }_t^2{f}_s$. ThusThe subscript distinguishes the order-parameter exponent from inverse temperature. At $t=0$, choose $b=|h{|}^{-1/y_h}$ to obtain $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, with $\delta =y_h/(D-y_h)$. These derivatives imply the Rushbrooke scaling relation $\alpha +2{\beta }_{\mathrm{mag}}+\gamma =2$ and the Widom scaling relation $\gamma ={\beta }_{\mathrm{mag}}(\delta -1)$. Equivalently, the equation of state has $h=|t{|}^{\Delta }{\mathcal{H}}_{\pm }(M/|t{|}^{{\beta }_{\mathrm{mag}}})$, with $\Delta ={\beta }_{\mathrm{mag}}\delta ={\beta }_{\mathrm{mag}}+\gamma$.

The correlation length scales as $\xi \sim |t{|}^{-\nu }$, $\nu =1/y_t$. At criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$, giving $y_h=(D+2-\eta )/2$. Integrating $G$ over a correlation volume gives $\chi \sim {\xi }^{2-\eta }$ and hence the Fisher scaling relation $\gamma =(2-\eta )\nu$. The hyperscaling relation $2-\alpha =D\nu$ expresses one singular free-energy contribution per correlation volume.

These laws organize measurements: plotting $M/|t{|}^{{\beta }_{\mathrm{mag}}}$ against $h/|t{|}^{\Delta }$ collapses near-critical curves onto the appropriate scaling function. A finite system replaces the divergent length by its size $L$; the corresponding prediction is $f_s=L^{-D}\mathcal{F}(t{L}^{y_t},h{L}^{y_h})$ under the same scaling assumptions. Finite size rounds a sharp thermodynamic singularity.

The hypothesis has a domain of validity. At a renormalization-group fixed point one generally needs $f_s=b^{-D}{f}_s(b^{y_t}t,b^{y_h}h,b^{y_u}u,\dots )$, including other scaling fields. Ordinary irrelevant couplings can be set to zero in a nonsingular limiting scaling function. A dangerously irrelevant coupling cannot: above the upper critical dimension of scalar quartic theory, the quartic term is irrelevant at the Gaussian point but still stabilizes the ordered phase. Consequently naive hyperscaling fails there even though other exponent relations can remain valid. Marginal variables can produce logarithms, anisotropic systems require the appropriate volume exponent, and the Berezinskii–Kosterlitz–Thouless transition has an essential correlation-length singularity instead of a finite power-law $\nu$. Scaling theory relates observables through a small number of fixed-point dimensions; it does not make analytic backgrounds, marginal logarithms or dangerously irrelevant variables disappear.

The Landau-Ginzburg theory for a scalar order parameter supplements a local symmetry-allowed potential with a gradient penalty. For a ${\mathbb{Z}}_2$ symmetry at zero field, a stable quartic model iswith $r=a(T-T_c)$ near a mean-field transition. Equilibrium fluctuations have weight $e^{-F/(k_{B}T)}$. In the Landau approximation, minimize the functional. A uniform stationary state obeys $r\varphi +u{\varphi }^3=h$. At zero field the minimum is $\varphi =0$ for $r>0$ and $\varphi =\pm \sqrt{-r/u}$ for $r<0$. The latter pair is spontaneous symmetry breaking of the discrete symmetry.

The uniform free-energy density at its minimum is zero above the transition and $-r^2/(4u)$ below, up to regular backgrounds. Differentiating the equation of state gives ${\chi }_{+}=1/r$ and ${\chi }_{-}=1/(2|r|)$. At $r=0$, $\varphi =(h/u{)}^{1/3}$. Gaussian fluctuations around the minimum have inverse covariance $(K{q}^2+r)/(k_{B}T)$ above and $(K{q}^2+2|r|)/(k_{B}T)$ below. Their correlation lengths are ${\xi }_{+}=(K/r{)}^{1/2}$ and ${\xi }_{-}=(K/(2|r|){)}^{1/2}$. These results give the usual mean-field critical exponentsThe heat capacity has a finite jump rather than a power-law divergence. Negative quartic coefficient requires higher stabilizing terms and can produce a first-order transition; a vanishing quartic coefficient requires a tricritical analysis. Thus the assumed positivity of $u$ matters.

The Ginzburg criterion tests whether this saddle description is self-consistent. Below the transition compare fluctuations of the order parameter averaged over a correlation volume with its squared mean-field value ${\varphi }_0^2=|r|/u$. Modes with $q\lesssim {\xi }^{-1}$ dominate this coarse-grained fluctuation estimate:where the positive dimensionless $A_D$ depends on the averaging window and the factor of two used in $\xi$. Dividing by ${\varphi }_0^2$ givesIt is the fluctuation of a correlation-volume average, not the full cutoff-dominated microscopic variance, that should be compared with the mean-field order parameter.

For $D<4$, $\mathcal{G}$ diverges as $r\to 0$, so the asymptotic critical region is fluctuation dominated. Away from it, mean-field theory can still work whenFor $D>4$, the long-wavelength ratio tends to zero near criticality, consistently with Gaussian control and mean-field thermodynamic exponents. At $D=4$, quartic coupling is marginal; scale-dependent logarithms require a renormalization-group analysis rather than interpreting a constant bare ratio as a proof of no corrections. This identifies four as the upper critical dimension of the short-range scalar quartic model.

The Landau-Ginzburg theory is valuable because symmetries restrict a manageable list of local operators, the gradient term describes correlations and interfaces, and the Ginzburg criterion identifies when neglected fluctuations become decisive. Below the upper critical dimension an interacting renormalization-group fixed point, rather than the bare saddle, determines universal exponents. For continuous order parameters the manifold of ordered minima also has Goldstone modes, and lower-critical-dimension fluctuations require separate attention. Mean-field minimization supplies an approximation; the Ginzburg ratio decides whether its neglect of fluctuations is justified.

Changing the ultraviolet cutoff changes which fluctuations remain explicit. Integrating out a shell of Fourier modes shifts the coefficients of all symmetry-allowed operators and a field-independent constant. Hence the cutoff-dependent coefficients are chosen so that the remaining partition function and long-distance observables reproduce the same microscopic system. This is the Wilsonian coarse-grained statistical Hamiltonian; retaining only the printed operators is a local derivative expansion, not an exact closure.

In a massive symmetric phase, the quadratic long-wavelength inverse two-point function has the form ${\alpha }^{-1}{p}^2+m^2$. Its Landau scalar correlation length satisfies ${\xi }^{-2}=\alpha m^2$ in this approximation. For canonical normalization of a scalar gradient term, set $\varphi =\sqrt{\alpha }\psi$; then the mass and quartic coefficients become $r=\alpha m^2$ and $g_c={\alpha }^{2}g$. With this normalization the infrared mass is the inverse correlation length, $m_R={\xi }^{-1}$, within the massive quadratic or one-loop approximation. More generally the decay rate is a pole mass, whereas the zero-momentum inverse magnetic susceptibility is a curvature mass; an anomalous dimension or nontrivial momentum dependence can distinguish them. In the ordered phase one must first expand about a selected stable equilibrium magnetization, rather than identify a negative quadratic coefficient at the unstable origin with a squared inverse correlation length. Thus the intended mass identification includes a field normalization and a stable-phase qualification.

For momentum-shell renormalization group, split $\varphi ={\varphi }_{<}+{\varphi }_{>}$, retaining $|p|<\Lambda /b$ and eliminating $\Lambda /b<|p|<\Lambda$. DefineTaking a logarithm gives the cumulant expansion of a coarse-grained free energy. For a centered fast Gaussian measure with $G_{>}(0)=⟨{\varphi }_{>}^2⟩$, the first quartic cumulant isThe Wick contractions give a mass shift $\Delta r=g{G}_{>}(0)/2$ and an identity-operator contribution, while leaving the quartic term at this order. The second cumulant produces higher interactions and a quartic correction. After the length and field rescalings $x=b{x}^{\prime }$ and ${\varphi }_{<}(x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$, the gradient coefficient remains one, andThe one-loop shell mass renormalization in scalar quartic theory explicitly shows why the critical bare mass is shifted by fluctuations.

Assume short-range interactions, an analytic even local potential, a positive gradient coefficient, and a weak-coupling trajectory in the basin of the Gaussian fixed point. Above four dimensions the quartic interaction has negative engineering dimension and its dimensionless strength decreases under coarse graining; higher local powers and higher derivative terms also decrease. After absorbing analytic short-distance shifts into the coefficients, long-wavelength fluctuations are small compared with the ordered order parameter. The remaining coarse Landau free energy can therefore be evaluated at its stable saddle:The quartic stabilizer must still be retained for the ordered minimum even though it is dangerously irrelevant. Stationarity, $r_{R}M+g_R{M}^3/6=h$, gives the ordinary Landau-Ginzburg theory and its mean-field critical exponents. This derives the saddle approximation from decreasing fluctuation strength under the stated RG assumptions; it does not assume that every exact coarse-grained functional becomes a quartic polynomial. Below four dimensions the quartic coupling grows, long-wavelength loop corrections cease to be small, and the argument fails. At four dimensions the interaction is marginal and generates logarithmic corrections. The tuned sextic theory has the analogous boundary at three dimensions.

Use a consistent Fourier transform pair for the connected correlation function:The printed forward-transform expression integrates over $p$ while leaving $x$ free; that integration variable is an error. The above convention resolves it and produces the loop measure used below. At zero magnetization, $G_c(x)=⟨\varphi (0)\varphi (x)⟩$; otherwise the disconnected product of expectations must be subtracted.

In the inverse-propagator identity used here, the truncated two-point function means the full one-particle-irreducible two-point vertex, including its free quadratic part. This convention differs from using “truncated” for a connected cumulant. To derive the inverse relation, introduce $W[J]=\log Z[J]$, $\phi =\delta W/\delta J$ and the effective action $\Gamma [\phi ]=\int J\phi -W[J]$. Then $\delta \Gamma /\delta \phi =J$ and the chain rule gives ${\Gamma }^{(2)}{W}^{(2)}=1$. Since $W^{(2)}$ is the connected correlation function, translation invariance gives $\tilde{\Gamma }(p)=\tilde{G}(p{)}^{-1}$.

Now use canonically normalized coefficients, denoting the cutoff mass by $m_b^2=m^2(\Lambda ,T)$ and the renormalized infrared mass by $m_R^2=m^2(0,T)$. Choose the free propagator ${\tilde{G}}_0(p)=(p^2+m_R^2{)}^{-1}$ and put $\delta m^2=m_b^2-m_R^2$ in the interaction as a quadratic counterterm. With the Euclidean self-energy convention in which an insertion changes the connected propagator by $-G_0\Sigma G_0$, the perturbative series beginsThe self-energy sums one-particle-irreducible loop insertions; the explicit mass counterterm is kept separate. Repeated insertions generate the geometric Dyson resummation, which explains why reducible chains occur in $G$ but not as independent terms in its inverse. Beyond one loop a momentum-dependent self-energy and wave-function renormalization must also be included.

At one loop the only quartic two-point graph is the tadpole diagram. At a $g{\varphi }^4/4!$ vertex, attaching the two labeled external fields uses $4\cdot 3=12$ choices and contracts the remaining two fields together. The factor is $12/4!=1/2$. Equivalently this is the same quadratic term in the shell cumulant expansion above. ThusIt is positive for $g>0$ and independent of external momentum, so there is no one-loop gradient renormalization. The condition $\tilde{\Gamma }(0)=m_R^2$ implies $\delta m^2+\Sigma (0)=0$, henceThe cutoff is necessary: the unregulated integral is ultraviolet divergent for $D\ge 2$. Using the renormalized mass internally is a self-consistent one-loop organization, not an exact resummation of all critical fluctuations. With nonunit $\alpha$, this formula instead applies to $r=\alpha m^2$, $g_c={\alpha }^{2}g$ after canonical normalization of a scalar gradient term.

To test a linear thermal mass, approach the ordinary transition from the symmetric side and set $R=m_R^2>0$. For $D>2$ the massless tadpole is infrared finite. Subtract its critical value, absorb regular temperature dependence of the coupling into a nonzero thermal coefficient $A$, and write $m_b^2(T)-m_b^2(T_C)=A(T-T_C)+\cdots$. The one-loop critical-mass subtraction giveswhereFor $D>4$, $J_D(0)=c_D{\Lambda }^{D-4}/(D-4)$ is finite, so the bracket tends to a finite constant and $R\propto T-T_C$ is self-consistent. At $D=4$,and the growing logarithm obstructs an asymptotically constant linear coefficient at nonzero coupling. For $2<D<4$, put $k=\sqrt{R}q$ to obtainThe divergent bracket then invalidates the assumed linear thermal mass. The radial critical-mass subtraction in dimensions three to five gives explicit checks of all three behaviors. For $D\le 2$, the massless tadpole subtraction itself is infrared divergent; that failure of the Gaussian expansion does not prove absence of an interacting transition. Therefore the ordinary upper critical dimension is $D_C=4$, with the unmodified Landau-Ginzburg theory asymptotically justified by this test only for $D>4$. Exponents inferred by solving the self-consistent one-loop equation below four dimensions would not be exact interacting exponents.

For a tricritical point the renormalized quartic coefficient must be tuned to zero, and a positive sextic interaction stabilizes the order parameter. The gradient term gives the scalar field engineering dimension $(D-2)/2$. A coefficient of ${\varphi }^6$ therefore has dimension $D-6(D-2)/2=6-2D$, which becomes marginal at $D=3$. This gives $D_C^{\mathrm{tri}}=3$. A direct Ginzburg criterion reaches the same result: long-wavelength variance in a correlation volume iswhile the tricritical saddle has $M^2\sim |r{|}^{1/2}$. Their ratio scales as $|r{|}^{(D-3)/2}$, vanishing only above three dimensions. At three dimensions it is marginal; below three it grows. In contrast the ordinary saddle $M^2\sim |r|/g$ gives a ratio $g|r{|}^{(D-4)/2}$. Both calculations assume the thermal mass is measured relative to its shifted critical value. A sextic interaction generates quartic terms under coarse graining, so tricriticality requires tuning the renormalized quartic scaling field as well as the mass.

Use ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ for inverse temperature to distinguish it from the order-parameter exponent. The decaying two-point correlation function needed for the response identity is connected:Away from the critical point in a short-range Ising model pure phase, the correlation length controls the exponential long-distance falloff, $G(r)\sim A(r)e^{-|r|/\xi }$ with a possible algebraic prefactor. The full two-spin expectation below the transition instead tends to $M^2$. The spin-mixture contribution to zero-field susceptibility explains why a symmetric mixture is unsuitable for defining a finite connected correlation length in the ordered phase.

Let $F=-{\beta }_{\mathrm{th}}^{-1}\log Z$ be the total Helmholtz free energy. Take magnetization and susceptibility per site, soDifferentiating the Boltzmann weights at fixed temperature givesTranslation invariance then yields the correlation-function susceptibility sum ruleTotal magnetization and total susceptibility multiply these expressions by $N$; per physical volume they have an additional factor $a^{-D}$ relative to the per-site convention. These normalizations do not change the critical powers.

For the subsequent scaling requests, let $F_s$ denote singular free energy per site; the fixed extensive normalization is suppressed. The labels $+$ and $-$ denote $t>0$ and $t<0$, with distinct scaling functions and amplitudes on the two sides. Below the transition a one-sided derivative at $h=0$ selects a spontaneous-magnetization branch. Writing $x=h/|t{|}^{\Delta }$, differentiation of the scaling hypothesis for critical phenomena givesThe field-independent $I_{\pm }$ makes no contribution to these field derivatives, although it can change heat-capacity amplitudes. Assuming nonzero leading amplitudes and the indicated pure-phase limits, comparison with the exponent definitions givesThis is the Rushbrooke scaling relation. A temperature-independent critical isotherm requires the large-$|x|$ magnetization scaling function to grow as $\mathrm{sgn}(x)|x{|}^{\beta /\Delta }$: otherwise $M_s=|t{|}^{\beta }{\mathcal{M}}_{\pm }(x)$ would retain a temperature power as $t\to 0$ at fixed small $h$. Therefore $1/\delta =\beta /\Delta$. Since the preceding two equalities also give $\Delta =\beta +\gamma$, the Widom scaling relation follows:For the hyperscaling relation, additionally assume that one correlation volume supplies an order-one singular free-energy contribution and that there is a single isotropic correlation length, with no dangerously irrelevant coupling spoiling this argument. Then $F_s\sim {\xi }^{-D}\sim |t{|}^{D\nu }$. Two temperature derivatives give the singular heat capacity, soHyperscaling is an additional physical assumption, not a consequence of the two-variable equation alone, and generally fails above the upper critical dimension.

Finally the correlation scaling function $f_G(x)$ falls exponentially, up to algebraic factors, for $x\to \infty$ in a massive short-range pure phase. At $x\to 0$ it approaches a finite nonzero value in the scaling regime. Approximating the lattice sum at distances much larger than $a$, and denoting the area of the unit sphere by $S_{D-1}$, givesIf $\eta <2$ and the integral has a finite nonzero limiting value, microscopic distances contribute only a regular background and the leading divergence is ${\xi }^{2-\eta }$. Thus the Fisher scaling relation isThese statements concern leading singular powers; logarithmic corrections and cancellations of amplitudes require separate treatment.

The partition function sums the Boltzmann weights of all statistical configurations. For discrete states $s$ of energy $E_s$, $Z={\sum }_s{e}^{-{\beta }_{\mathrm{th}}{E}_s}$; for a regulated scalar field it is a functional integral, $Z=\int \mathcal{D}\varphi e^{-{\beta }_{\mathrm{th}}H[\varphi ]}$. It normalizes probabilities and generates thermodynamic averages. Its Helmholtz free energy isA quantum version uses $Z=\mathrm{Tr}{e}^{-{\beta }_{\mathrm{th}}\hat{H}}$. In the following identification, $F(M)$ is a constrained free-energy density, so its units match those of a Hamiltonian density rather than a total extensive energy.

Fix a microscopic cutoff ${\Lambda }_0$. Split the field into retained modes ${\varphi }_{<}$ with momenta below $\Lambda$ and eliminated modes ${\varphi }_{>}$ between $\Lambda$ and ${\Lambda }_0$. Define the Wilsonian coarse-grained statistical Hamiltonian exactly byIts coefficients and additive constant depend on $\Lambda$ so that integrating the retained modes gives precisely the original $Z$. With sources and observables carried along, the same construction preserves long-wavelength responses. Thus changing this intermediate ultraviolet cutoff is a change of description, not a change of the physical system. It is advantageous because one can integrate out fluctuations irrelevant to the observable length scale, organize a local gradient expansion, and use renormalization-group flow toward simpler universal fixed-point descriptions. Exact invariance retains all generated operators; truncating them is an approximation.

The quadratic gradient coefficient generally becomes $K_{\Lambda }$, not a constant. Field renormalization sets a chosen kinetic normalization, for example ${\varphi }_R=\sqrt{K_{\Lambda }}\varphi$ when this makes the gradient coefficient one in the energy convention. Sources and observables must be transformed inversely so that their physical meaning is preserved. Under rescaled coordinates $x^{\prime }=x/b$, the critical field scaling isIts anomalous part is the anomalous dimension; it gives $G(r)\sim r^{-2x_{\varphi }}$ and the conjugate-field exponent $y_h=D-x_{\varphi }$. Different conventions for the wave-function factor require corresponding changes in its logarithmic derivative. A fixed physical magnetization must not be confused with a cutoff-dependent dimensionless renormalized field.

To justify the zero-cutoff identification of constrained free energy, work first in a finite periodic volume $V$. Define the physical spatial mean $M=V^{-1}\int \varphi (x)d^{D}x$ and constrain that mean in the microscopic integral:Retain the constant mode $M$ while eliminating every nonzero mode. When $\Lambda$ is below the smallest nonzero box momentum, the remaining energy is $H_{\Lambda }[M]=V{U}_{\Lambda }(M)$. Consequently $e^{-{\beta }_{\mathrm{th}}V{U}_{\Lambda }(M)}=Z_M$ with a consistently fixed, possibly field-independent measure normalization. This provesFor a uniform field, $U_{\Lambda }(M)$ is the Hamiltonian density $\mathcal{H}(\Lambda ,M)$ in the question. The statement assumes that the physical zero mode is retained, all other modes are integrated exactly, field-rescaling factors are undone, and density and additive-normalization conventions agree. Taking the cutoff limit in the finite box before its thermodynamic limit makes this derivation unambiguous. With an applied uniform source, $Z(h)=\int dM{e}^{-{\beta }_{\mathrm{th}}V[F_c(M)-hM]}$ up to the same measure convention; its large-volume saddle minimizes $F_c(M)-hM$.

The Landau approximation further assumes that this potential can be approximated by a regular local polynomial on the homogeneous branches, and that the remaining fluctuations are negligible for the observables sought. It is not an assertion that an exact zero-cutoff free energy is the bare polynomial. In a coexistence region the exact constrained density can be convexified by domains, while a nonconvex Landau branch potential describes local phases and their metastability.

For the breakdown criterion use a short-range scalar theory with positive quadratic gradient stiffness. At the Gaussian fixed point, invariance of $\int (\nabla \varphi {)}^2{d}^{D}x$ gives field dimension $(D-2)/2$. A coupling $g_{2n}$ multiplying ${\varphi }^{2n}$ therefore has engineering scaling exponentIt is irrelevant only above its upper critical dimension. For the ordinary critical point the leading stabilizing interaction is quartic, while at a tricritical point the renormalized quartic term is tuned away and the leading interaction is sextic. ThereforeOne can also see the failure through the Ginzburg criterion. Long-wavelength fluctuations averaged over a correlation volume obey $⟨(\delta M{)}^2{⟩}_{\xi }\sim {\xi }^{2-D}$, up to temperature and stiffness factors. The quartic saddle has $M^2\sim {\xi }^{-2}$, giving a fluctuation/mean-square ratio proportional to ${\xi }^{4-D}$. The tricritical sextic saddle has $M^2\sim {\xi }^{-1}$, giving a ratio proportional to ${\xi }^{3-D}$. Both ratios grow without bound below their respective critical dimension, invalidating the fluctuation-free Landau exponents sufficiently close to a transition.

At equality the ratios are marginal, not power-law divergent. The loop expansion resolves this boundary: the quartic vertex correction contains $\int d^{D}q/(q^2+{\xi }^{-2}{)}^2$, logarithmic at $D=4$; the sextic vertex correction contains a two-loop three-propagator integral whose overall degree is $2D-6$, logarithmic at $D=3$. Marginal running therefore generally adds logarithmic corrections at these dimensions. Below them, where the corresponding transition exists, interactions control the long-distance critical behavior. Above them, leading thermodynamic powers are mean-field, although stabilizing dangerously irrelevant couplings explain why naive hyperscaling need not hold.

The mean-field approximation replaces each site's fluctuating surroundings by a self-consistent average. On a cubic lattice each spin has coordination number of a lattice $z=2D$, even though summing only over positive basis directions counts each bond once. Write ${\sigma }_n=M+\delta {\sigma }_n$ and drop products of deviations. Each Ising bond becomes ${\sigma }_n{\sigma }_m\simeq M{\sigma }_n+M{\sigma }_m-M^2$, so the approximate statistical Hamiltonian isThe constant corrects counting each bond twice. The resulting independent-spin partition function is $Z_{\mathrm{MF}}=e^{-{\beta }_{T}NzJ{M}^2/2}[2\cosh ({\beta }_{T}zJM){]}^N$, where ${\beta }_T=1/(k_{B}T)$. The one-spin probabilities giveEquating this average with the assumed $M$ givesHere $J>0$ is assumed; a ferromagnetic order parameter is not the appropriate choice for an antiferromagnetic coupling. Put $A=T_c/T$. For $A\le 1$, $\tanh (AM)<AM\le M$ at any $M>0$, so only $M=0$ is possible. For $A>1$, the function $\tanh (AM)-M$ has positive slope at zero, is strictly concave on $M>0$ and is negative at $M=1$. It therefore has exactly one positive zero and a spin-reversed negative zero. Expanding at the bifurcation givesThe transition is continuous, with order-parameter critical exponent $\beta =1/2$. Stability can also be checked from the Bragg-Williams free energy of the Ising model per Ising spin:Its derivative is $-zJM+k_{B}T\mathrm{arctanh}M$, giving the same self-consistency equation. Near zero,The quartic coefficient is positive at $T_c$, the stable minima approach zero continuously, and the minimized free energy has no first-derivative jump. Thus mean-field approximation predicts a second-order transition at $T_c=2DJ/k_B$. This is the mean-field Ising critical bifurcation.

For short-range interactions, large dimension gives many neighbours and suppresses the relative fluctuations of their average. More quantitatively, Gaussian field theory fluctuations averaged over a correlation volume have variance of order ${\xi }^{2-D}$, while the squared ordinary mean-field order parameter is of order $|t|\sim {\xi }^{-2}$. Their ratio is proportional to ${\xi }^{4-D}$. The Ginzburg criterion therefore makes neglect of critical fluctuations asymptotically consistent above four dimensions, inconsistent below four, and marginal at four, where logarithms occur. Large dimension is a route to validity; sufficiently long-range or infinite-range interactions can also produce mean-field behaviour. In low-dimensional short-range systems even the existence and location of a transition can differ from this approximation.

A phase diagram locates stable pure thermodynamic phases and their phase coexistence boundaries in a space of control parameters. For a concrete three-dimensional example, take the temperature, crystal-field parameter, and magnetic field of the Blume–Capel model. Its spins take values $0,\pm 1$; changing the cost of the zero-spin state can turn a continuous ordering transition into a discontinuous one. A local scalar description near the resulting tricritical point is the Landau-Ginzburg theoryThe three local control coordinates are $(r,u,h)$, with $r$ varying linearly with the thermal scaling field. For $u>0$, $h=0$, the transition at $r=0$ is continuous. For $u<0$, zero-field coexistence is at $r=3u^2/(16v)$, and the magnetization jumps between zero and $\pm \sqrt{-3u/(4v)}$. The two loci meet at $r=u=h=0$, the tricritical point.

The full three-dimensional phase diagram has an ordered coexistence sheet at $h=0$, where the two opposite magnetizations exchange stability when $h$ changes sign. On its three-phase line it meets two nonzero-field tricritical wings. These wings are first-order phase transition surfaces between minima of the same sign but different magnitude, terminating at ordinary thermodynamic critical points. Their tricritical wing critical edges are found by $V^{\prime }=V^{\prime \prime }=V^{\prime \prime \prime }=0$:Indeed $V^{\prime \prime \prime }=6uM+20v{M}^3$ fixes $M_c^2$, and substituting into $V^{\prime \prime }=0$ and $V^{\prime }=0$ gives $r_c,h_c$; $V^{\prime \prime \prime \prime }=-12u>0$ ensures a stable critical quartic minimum. The two signs give the two wings. All transition surfaces and critical edges end together at the tricritical point. The following plot uses $v=1$; its wing surfaces come from the exact tricritical wing coexistence factorization, rather than schematic unconstrained surfaces.

The unheaded requests for critical exponents follow from the uniform equation of stateFor an ordinary thermodynamic critical point, $u>0$ is fixed and $r=At$, $A>0$. At $h=0$, $t<0$, the small ordered minimum obeys $M^2\sim -r/u$, so the order-parameter critical exponent is $\beta =1/2$. Above the transition $\chi =(\partial M/\partial h{)}_{h=0}=1/r$, giving the magnetic-susceptibility critical exponent $\gamma =1$. At $r=0$, $h\sim u{M}^3$, so the critical-isotherm exponent is $\delta =3$. At the tuned tricritical point, $u=0$, instead $M^4=-r/v$ below the transition and $h=v{M}^5$ on its critical isotherm, while the disordered susceptibility remains $1/r$. ThereforeBelow the transition the inverse susceptibility is $r+3u{M}^2+5v{M}^4$, giving $2|r|$ in the ordinary limit and $4|r|$ in the tricritical limit. Thus both sides give the same susceptibility power, with different amplitudes.

To derive the general Ginzburg criterion, tune away all lower even nonlinearities and let the leading stabilizer be $v_n{M}^{2n}/(2n)$, $n\ge 2$. The equation of state gives $M_0^2\asymp |r{|}^{1/(n-1)}$, and its curvature at the ordered saddle is $2(n-1)|r|$, so $\xi \asymp |r{|}^{-1/2}$. The fluctuations of the order parameter averaged over a correlation volume are estimated by its long-wavelength quadratic covariance:The integral has a fixed dimensionless upper limit after $q=p/\xi$; this is a coarse-volume fluctuation estimate, not the ultraviolet-dominated point variance. Consequently the multicritical Ginzburg ratio isThe Landau approximation is asymptotically controlled by this test only when the ratio tends to zero. This gives the upper critical dimensionThe same threshold follows by giving $M$ its Gaussian engineering dimension $(D-2)/2$: the coupling to $M^{2n}$ has scaling exponent $D-n(D-2)=2n-(n-1)D$, which becomes relevant below $D_c$.