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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.

At a tricritical point, both $r$ and $u$ vanish, while $v>0$ stabilizes the Landau free energy. It separates ordinary continuous transitions for $u>0$ from first-order coexistence for $u<0$. The zero-field order-parameter jump shrinks to zero as $u\uparrow 0$, and the tricritical wings terminate there in the full $(r,u,h)$ diagram. For $r<0$, $u=0$, minimizing gives $M^4=-r/v$ andTwo thermal derivatives give the heat-capacity critical exponent $\alpha =1/2$, compared with the finite mean-field heat-capacity jump, $\alpha =0$, at an ordinary quartic transition. The remaining tricritical mean-field critical exponents are $(\beta ,\gamma ,\delta )=(1/4,1,5)$, and the mean-field correlation-length critical exponent is $\nu =1/2$. To observe tricritical scaling one must tune the quartic scaling field as well as the thermal field: a generic path with fixed $u>0$ instead has ordinary critical behavior. Its upper critical dimension is three, as the multicritical Ginzburg ratio shows.

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 $D_c$.

For $D<D_c$, the fluctuation ratio diverges and the saddle calculation cannot predict the generic interacting critical exponents. At $D=D_c$ the test is marginal, not a proof of changed power exponents: the ratio has zero thermal power and marginal interactions can produce logarithmic corrections. Thus the PDF's inclusion of equality in a blanket claim about wrong power exponents needs this qualification. The Landau-Ginzburg theory as a fluctuating field theory remains useful; what loses validity is its fluctuation-free Landau approximation.

In the convention implied by the printed expansion, the truncated two-point quantity is the inverse connected propagator, or one-particle-irreducible two-point vertex:It is the second functional derivative of the quantum effective action (or statistical Legendre effective action) about a translationally invariant zero-field equilibrium. The relation follows from the inverse Hessian relation for a connected two-point function. The subscript $c$ in $G$ already removes disconnected one-point products; the word “truncated” here is not a request to subtract that product a second time. Nor is a general amputated connected correlation function interchangeable with a one-particle-irreducible correlation function.

Use the zero-momentum renormalized mass $R=m^2(0,T)$ in the free propagator ${\tilde{G}}_0(p)=(p^2+R{)}^{-1}$, and split the quadratic coupling into $R$ plus a mass counterterm $\delta m^2=m^2(\Lambda ,T)-R$. The self-energy $\Sigma (p)$ is the sum of loop 1PI insertions, excluding the separately displayed counterterm. Summing repeated insertions by Dyson resummation givesThe sign convention is that a positive tadpole shifts the inverse propagator upwards. At first order, the connected correlation function correction is $-G_0^2\Sigma$, consistent with this inverse-propagator convention. A different split between the reference mass and counterterm produces the same renormalized result.

For the positive quartic interaction $g{\varphi }^4/4!$, the one-loop tadpole diagram has no external-momentum dependence. Its symmetry factor is $1/2$: assigning the two external legs to four vertex fields gives $4\cdot 3$ contractions, and division by $4!$ gives $1/2$. With the dimensionless statistical-action convention,The physical zero-momentum condition $\tilde{\Gamma }(0)=R$ sets $\delta m^2+{\Sigma }_1(0)=0$, hence the one-loop mass relation isWriting the internal line with $R$ is a renormalized or self-consistent one-loop convention. Away from critical infrared singularities it differs from a bare-mass insertion only at higher perturbative order. This equation does not by itself provide exact critical exponents once loop corrections become large.

For $D>2$, subtract the critical-temperature condition $0=m^2(\Lambda ,T_C)+(g/2)I_D(0)$. Take $g$ and the regular coefficients at their critical values, absorbing smooth changes into a coefficient $A>0$. Sincethe one-loop critical-mass subtraction becomesLet $K_D$ be the area of the unit $(D-1)$-sphere divided by $(2\pi {)}^D$. Radial integration gives $J_D(R)=K_D{\int }_0^{\Lambda }{k}^{D-3}(k^2+R{)}^{-1}dk$.

For $D>4$, $J_D(0)=K_D{\Lambda }^{D-4}/(D-4)$ is infrared finite, so it merely renormalises the coefficient and $R\propto T-T_C$ is consistent. For $2<D<4$, setting $k=\sqrt{R}x$ yieldswith a finite positive dimensionless integral. The correction is singular relative to the term $R$, invalidating the finite-coefficient linear-mass assumption. At $D=4$,so the boundary is logarithmically marginal. For $D\le 2$, even the subtraction using $I_D(0)$ needs an infrared regulator; it cannot be used to restore a finite linear critical expansion. Thus the ordinary upper critical dimension isA fixed-coupling self-consistent one-loop formula is not the full marginal renormalization group analysis, but its logarithm already shows why an uncorrected linear power law is not generic at $D=4$.

At a tricritical point, both the quadratic and quartic scaling directions must be tuned; the leading stabilising interaction is sextic. With canonical scalar-field engineering dimension $(D-2)/2$, the sextic coupling has eigenvalue $D-6(D-2)/2=6-2D$. It becomes marginal at $D=3$, givingLower even couplings generated by coarse-graining must remain tuned. This is why using an untuned quartic tadpole to diagnose a tricritical point would give the wrong boundary. The tricritical sextic beta function supplies marginal logarithmic corrections at three dimensions.

For a general multicritical even Landau potential, assume the lower stabilising even terms have been tuned away and the first remaining one is $A_{2n}{M}^{2n}$, with $n\ge 2$ and $A_{2n}>0$. Minimising the potential on its ordered branch givesIts curvature at the minimum is $4(n-1)a_2|t|$. For a finite positive gradient stiffness $c$, the longitudinal correlation length therefore scales as $\xi \propto |t{|}^{-1/2}$. The Ginzburg criterion compares the order-parameter fluctuation averaged over a correlation volume with this squared mean-field value. Keeping momenta of order ${\xi }^{-1}$ or less,Consequently the multicritical Ginzburg ratio behaves asOnly for a positive exponent do these relative fluctuations vanish on approaching the critical point. Thus the general upper critical dimension isEquivalently the interaction eigenvalue $y_{2n}=2n-(n-1)D$ vanishes there. The cases $n=2$ and $n=3$ reproduce 4 and 3 respectively.

For $D<D_C$, the ratio diverges and the mean-field assumptions lose self-consistency arbitrarily close to the transition. At $D=D_C$ the marginal Ginzburg criterion is scale-independent at this leading estimate, rather than tending to zero; the criterion alone does not prove a divergence or force new power indices. Marginal interactions require a renormalization group calculation and generally give logarithmic corrections, as for the quartic and sextic cases above. The boundary case is marginal, not a strict power-law divergence. This qualifies the printed wording at equality while recovering the requested upper critical dimensions.