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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 ultraviolet cutoff specifies which fluctuations remain explicit. Changing it integrates out some modes; to preserve long-distance predictions, their effect must be absorbed into the mass, interactions, and other couplings. Thus bare coefficients depend on $\Lambda$. The long-distance inverse connected correlation function has the form $\tilde{\Gamma }(p)\simeq Z^{-1}(p^2+{\xi }^{-2})$. After fixing the gradient normalization, the infrared mass is an inverse correlation length. With the question's unrescaled gradient coefficient, ${\xi }^{-2}=\alpha m^2$ at Gaussian level, so proportionality rather than literal equality is appropriate until that normalization is chosen.

As a concrete momentum-shell renormalization group step, split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ into modes below $\Lambda /b$ and in the shell $\Lambda /b<|p|<\Lambda$, and defineThen rescale coordinates and the field to restore the cutoff and gradient coefficient. A local derivative expansion of $H_{\mathrm{eff}}$ gives new quadratic, quartic, and higher even couplings. If the system has short-range interactions, the field varies slowly, the effective potential is analytic, and remaining fluctuations are small, retaining the leading gradient and lowest stabilizing powers and minimizing the resulting functional gives Landau-Ginzburg theory. Suppressing the omitted fluctuations is the substantive Landau approximation; it is justified sufficiently far from the narrow fluctuation region or above the appropriate upper critical dimension.

The truncated two-point function in this question is the one-particle-irreducible two-point vertex, $\tilde{\Gamma }(p)=\tilde{G}(p{)}^{-1}$. It is the Hessian of the Legendre effective action, not merely the connected two-point function. In perturbation theory,Here ${\tilde{G}}_0$ is the Gaussian propagator about the chosen reference mass, $\delta m^2$ is the mass counterterm or bare-to-reference mass shift, and $\Sigma$ is the interaction self-energy, computed from proper diagrams. Fix the gradient normalization so ${\tilde{G}}_0(p)=(p^2+M^2{)}^{-1}$, and choose the infrared reference mass $M^2=m^2(0,T)$. Then $\delta m^2=m^2(\Lambda ,T)-M^2$ at this order. The mass renormalization condition is $\tilde{\Gamma }(0)=M^2$.

At one loop the only two-point diagram is the quartic tadpole diagram. There are $4\cdot 3$ choices for attaching its two external lines, leaving the other pair contracted; dividing by $4!$ gives its factor $1/2$. It is independent of external momentum, soThe renormalization condition therefore gives the self-consistent one-loop equationUsing $M$ inside the loop is a self-consistent resummation; replacing it by the bare mass gives the same formal first-order expansion away from infrared problems.

Subtract the equation at $T_c$, where $M=0$, and absorb regular temperature dependence of $g$ into the thermal coefficient. Let $\tau =m^2(\Lambda ,T)-m^2(\Lambda ,T_c)\propto T-T_c$ and $I(M)={\int }_{|q|<\Lambda }{d}^{D}q/(2\pi {)}^D(q^2+M^2{)}^{-1}$. ThenFor $D>4$, the integral divided by $M^2$ tends to a finite cutoff-dependent constant, since its small-momentum radial integrand is $q^{D-5}$. Thus $\tau$ is proportional to $M^2$, consistent with the Landau prediction. For $2<D<4$, set $q=Mz$: the correction instead scales as $M^{D-2}$ and dominates $M^2$. At $D=4$ it is proportional to $M^2\log (\Lambda /M)$. Hence a pure linear Landau mass law without logarithmic modification is consistent only aboveFor $D\le 2$, the massless integral itself is infrared divergent, so the subtraction already needs additional care. The sub-four-dimensional self-consistent approximation locates the breakdown of mean-field behavior; it is not an exact calculation of interacting critical exponents.

At a tricritical point the quartic coupling is tuned away and the leading stabilizing interaction is $g_6{\varphi }^6$. At the Gaussian fixed point, the field has engineering dimension $(D-2)/2$, so the sextic coefficient rescales as $g_6^{\prime }=b^{D-3(D-2)}{g}_6=b^{6-2D}{g}_6$. It changes from irrelevant to relevant atThis is the sextic case of the upper critical dimension of an even scalar interaction; marginality at three dimensions allows logarithmic corrections.

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.

Use the usual homogeneous, local, analytic Landau-Ginzburg theory; rotational symmetry and Z2 symmetry alone do not specify locality, analyticity, or the signs of the coefficients. The local derivative expansion contains every rotational scalar with an even total number of fields:The omitted terms include all allowed higher field powers and contracted spatial derivatives. Terms differing by a total derivative are equivalent for bulk behavior with suitable boundary conditions. The displayed gradient energy assumes spatial homogeneity. If only rotations about an origin are imposed, rotationally invariant position-dependent coefficients are also possible.

In the mean-field approximation, take a spatially uniform order parameter and minimize its Landau free energy. For $\gamma >0$, $u>0$ and $r=a(T-T_c)$ with $a>0$, keeping the quadratic and quartic terms givesThe single minimum splits into two minima as the temperature passes below the critical temperature. Choosing one minimum gives spontaneous symmetry breaking of the Z2 symmetry and a continuous phase transition. The order parameter is the expectation in a selected ordered state; averaging equally over both states gives zero. A Z2 symmetry does not guarantee a continuous phase transition: a negative quartic coefficient stabilized by a positive sextic coefficient can instead give a first-order phase transition.

Relaxing the Z2 symmetry permits odd terms. A nonzero linear source generally rounds this ordinary continuous phase transition. A cubic term can instead give a first-order phase transition: for $V(\varphi )=r{\varphi }^2/2+w{\varphi }^3/3+u{\varphi }^4/4$, $u>0$ and $w\ne 0$, equal minima occur atThe order parameter jumps from zero to ${\varphi }_{\ast }$. A continuous phase transition without an exact Z2 symmetry is still possible with additional tuning; odd terms do not logically exclude it.

For the multicritical even Landau potential, set $r={\mu }^2=a(T-T_c)$ and assume an integer $n\ge 2$, ${\lambda }_n>0$ and $\gamma >0$. For $n>2$, all lower even interactions must be absent or tuned to zero to obtain these mean-field critical exponents. The stationary equation isIts nonzero solutions are minima because $V^{\prime \prime }({\varphi }_0)=-(2n-2)r>0$. Thus the order-parameter critical exponent isSubstituting into the minimum free energy density givesThe singular heat capacity per unit volume is $c_s=-T{\partial }_T^2{f}_{\min }$. Smooth variation of the other coefficients does not change the leading power. ConsequentlyFor $n=2$, this means a finite heat capacity jump, with ${\alpha }_{\mathrm{MF}}=0$; for $n=3$, the tricritical point has ${\beta }_{\mathrm{MF}}=1/4$ and ${\alpha }_{\mathrm{MF}}=1/2$. The case $n=1$ would contain only a quadratic potential and would not stabilize an ordered phase.

The Gaussian fluctuation correction near a critical point is more singular than the mean-field approximation contribution whenThis is the upper critical dimension of an even scalar interaction. Equivalently, its coupling has engineering dimension $2n-(n-1)d$, positive below $d_c$. Dominant fluctuations signal breakdown of the mean-field approximation; the Gaussian expression is not then a calculation of the exact interacting heat-capacity critical exponent. At $d=d_c$, equality of the powers allows logarithmic corrections.