Ginzburg Criterion and Upper Critical Dimension

Landau theory and its Ornstein–Zernike Gaussian extension describe a continuous phase transition by minimizing a coarse-grained free-energy functional of an order parameter, neglecting the thermal fluctuations of that order parameter about its most probable value. The result is a set of universal but dimension-independent mean-field critical exponents ( $\beta=\tfrac12$ , $\gamma=1$ , $\nu=\tfrac12$ , $\alpha=0$ , $\delta=3$ ). Experiment and exact solutions — the 2D Ising model above all — show these are wrong in low dimensions, so a good theory must contain the seeds of its own breakdown. The Ginzburg criterion is precisely that self-consistency check. One computes, within Gaussian theory, the size of order-parameter fluctuations averaged over one correlation volume and demands it be small compared with the mean-field order parameter squared. Because the correlation length $\xi(t)\propto|t|^{-1/2}$ diverges at the critical point, the relative fluctuation grows without bound for $d<4$ and the inequality is eventually violated no matter how weak the coupling. The temperature at which equality holds defines the Ginzburg temperature $t_G$ , the boundary of the genuine critical region. The dimension at which the relative fluctuation neither grows nor shrinks as $t\to0$ is the upper critical dimension $d_c$ . For a short-ranged $\phi^4$ theory $d_c=4$ . For $d>d_c$ mean-field theory is asymptotically exact (up to non-universal amplitudes), the quartic interaction is renormalization-group irrelevant, and the transition is governed by the Gaussian fixed point. For $d<d_c$ the Wilson–Fisher fixed point takes over and exponents acquire $\varepsilon=4-d$ corrections. Quantitatively the criterion is more than a scaling argument: it pins the prefactor $t_G\sim(k_BT_c/\Delta C\,\xi_0^d)^2$ , explaining why mean-field BCS superconductivity works superbly (coherence length $\xi_0\sim10^3\,$ Å gives $t_G\sim10^{-12}$ ) while critical opalescence in fluids and the lambda transition in helium are firmly in the fluctuation-dominated regime. The same logic generalizes to tricritical points ( $d_c=3$ ) and long-range interactions, where $d_c$ shifts.

Key takeaways

Mean-field (Landau) theory ignores spatial fluctuations of the order parameter; the Ginzburg criterion quantifies when those fluctuations become large compared with the mean-field order parameter itself.
The criterion compares the fluctuation of the order parameter integrated over a correlation volume $\xi^d$ to the square of the equilibrium order parameter; mean-field theory is self-consistent only when fluctuations are relatively small.
Because $\xi$ diverges as $t\to0$ , fluctuations always dominate sufficiently close to $T_c$ in low dimensions, opening a Ginzburg window $|t|<t_G$ where mean-field exponents fail.
The borderline is the upper critical dimension $d_c=4$ for short-range interactions and a scalar (or $n$ -component) order parameter: for $d>4$ mean-field exponents are exact, for $d<4$ they are corrected.
The relative fluctuation scales as $|t|^{(d-4)/2}$ , and the width $t_G\sim(a_0/\xi_0)^{2d/(4-d)}$ is set by the bare correlation length $\xi_0$ : huge in superconductors (window invisible), atomic in fluids and magnets (window wide).
Key idea: Mean-field theory is exact when the order-parameter fluctuation integrated over a correlation volume is small compared with the square of the order parameter; this ratio scales as $|t|^{(d-4)/2}$ , self-consistent for $d>4$ and diverging near $T_c$ for $d<4$ .

The core insight

Mean-field theory is exact when the order-parameter fluctuation integrated over a correlation volume is small compared with the square of the order parameter; this ratio scales as $|t|^{(d-4)/2}$ , so it is self-consistent for $d>4$ and breaks down near $T_c$ for $d<4$ .

Where this lives today

The Ginzburg criterion is the first question asked of any newly discovered continuous transition: is it mean-field or fluctuation-dominated? In conventional superconductors the window $t_G\sim10^{-12}$ makes BCS mean-field theory exact for all practical purposes, whereas high- $T_c$ cuprates, with short coherence lengths, show a visible fluctuation regime that is routinely analysed with 3D-XY critical exponents. The lambda transition of $^4$ He, measured to extraordinary precision (including in microgravity to suppress gravitational rounding), sits entirely below its Ginzburg temperature and provides one of the most accurate confirmations of non-classical exponents in all of physics. The same power counting that gives $d_c=4$ tells experimenters that tricritical points in $d=3$ sit exactly at their marginal dimension and therefore show mean-field exponents with logarithmic corrections — a prediction confirmed in $^3$ He– $^4$ He mixtures and metamagnets.

The physical picture

Picture the order parameter $\phi(\mathbf x)$ as a slowly varying field. Mean-field theory replaces it everywhere by its average $\bar\phi$ . This is sensible only if the typical wobble of $\phi$ within a region the size of the correlation length $\xi$ is much smaller than $\bar\phi$ itself. As $T\to T_c$ , $\bar\phi\to0$ while the wobbles, fed by an ever-larger $\xi$ , grow — eventually the wobble wins and the mean-field picture collapses. The dimension decides the winner because fluctuation "energy" is shared among modes: the number of fluctuating modes inside a correlation volume scales as $\xi^d$ , while the order parameter squared scales as $|t|$ (mean field). Counting powers gives the relative fluctuation $\sim|t|^{(d-4)/2}$ — harmless for $d>4$ , divergent for $d<4$ . Four is the magic dimension where the two competing effects exactly balance.

Figure

The Ginzburg budget — mean-field order φ̄² shrinks while the fluctuation per correlation volume grows

Think of the criterion as a budget: the heat-capacity jump $\Delta C$ measures how much free energy the ordering provides per unit volume, while the fluctuation cost grows as the correlation volume shrinks the available phase space. When the ordering budget can no longer pay for the fluctuations, criticality goes non-classical. The punchline is that whether it can pay is decided by one integer — the dimension — through the sign of $(d-4)$ .

Figure

Ginzburg number vs dimension — fluctuations win below d=4, mean field survives above, marginal at d=4

The core equations

The starting point is the Landau–Ginzburg free-energy functional: a gradient stiffness, a symmetric double-well, and the field coupling. Fluctuations are the deviations of $\phi$ from the value that minimizes it.

\mathcal F[\phi]=\int\mathrm d^d x\left[\tfrac12 c(\nabla\phi)^2+\tfrac12 a\,t\,\phi^2+\tfrac14 b\,\phi^4-h\phi\right] \qquad (1)

where: $\phi(\mathbf x)$ is the coarse-grained order-parameter field; $t=(T-T_c)/T_c$ ; $a,b,c>0$ are phenomenological constants ( $c$ the stiffness); $h$ the conjugate field; $d$ the dimension. Physically: the first term penalizes spatial variation, the double-well drives ordering. Minimizing the uniform part below $T_c$ gives the mean-field order parameter and the exponent $\beta=\tfrac12$ .

\bar\phi^2=\frac{a|t|}{b}\ (t<0),\qquad \bar\phi\propto|t|^{\beta},\ \beta=\tfrac12 \qquad (2)

where: $\bar\phi$ is the equilibrium uniform order parameter below $T_c$ . Physically: the ordered value grows as the square root of the distance below $T_c$ . Gaussian fluctuations about the minimum have the Ornstein–Zernike propagator, whose $q\to0$ value is the susceptibility and whose pole sets the correlation length.

\tilde G(\mathbf q)=\frac{k_BT}{c\,(q^2+\xi^{-2})},\qquad \xi=\xi_0|t|^{-\nu},\ \xi_0=\sqrt{c/a},\ \nu=\tfrac12 \qquad (3)

where: $\tilde G(\mathbf q)$ is the Fourier-space correlation function; $\xi$ the correlation length; $\xi_0$ the bare (microscopic) amplitude; $\nu=\tfrac12$ the mean-field correlation-length exponent. The fluctuation of the order parameter averaged over one correlation volume follows from the $q\to0$ propagator, and scales as $\xi^{2-d}$ .

\langle\delta\phi^2\rangle_{\xi}\equiv\frac{1}{\xi^d}\int_{|\mathbf x|<\xi}\mathrm d^d x\,G(\mathbf x)\sim\frac{k_BT_c}{c}\,\xi^{\,2-d} \qquad (4)

where: $\langle\delta\phi^2\rangle_{\xi}$ is the mean-square fluctuation over one correlation volume. Physically: the integral picks up the $q=0$ propagator $k_BT\xi^2/c$ , then divides by the volume $\xi^d$ . The criterion itself demands this fluctuation be small compared with the mean-field order squared.

\frac{\langle\delta\phi^2\rangle_{\xi}}{\bar\phi^2}\ll1 \qquad (5)

where: the left side is the relative fluctuation. Physically: mean field is trustworthy only where its own neglected fluctuations are subdominant. Inserting the $|t|$ -scalings turns the criterion into a power law whose exponent changes sign at $d=4$ .

\frac{\langle\delta\phi^2\rangle_{\xi}}{\bar\phi^2}\propto|t|^{(d-4)/2},\qquad t_G\sim\left(\frac{a_0}{\xi_0}\right)^{2d/(4-d)} \qquad (6)

where: $(d-4)/2$ is the controlling power; $t_G$ is the Ginzburg temperature; $a_0$ a microscopic length; large $\xi_0$ gives a tiny $t_G$ . Physically: for $d>4$ the ratio vanishes at $T_c$ (mean field exact), for $d<4$ it diverges. In three dimensions the width can be recast through the measurable specific-heat jump.

t_G\sim\left(\frac{k_B}{\Delta C\,\xi_0^{\,d}}\right)^{2}\quad(d=3) \qquad (7)

where: $\Delta C$ is the mean-field specific-heat jump per unit volume at $T_c$ . Physically: a large condensation energy times a large correlation volume shrinks the fluctuation window.

Derivation APower counting for the upper critical dimension d = 4

The goal is to determine when the relative order-parameter fluctuation stays finite as the critical point is approached, which fixes the upper critical dimension at four for a short-range scalar theory. We compute the fluctuation amplitude averaged over a correlation volume using the Ornstein–Zernike propagator, take the ratio to the mean-field order parameter squared, and substitute the mean-field temperature scalings of the correlation length and the order parameter. Tracking the net power of the reduced temperature reveals a single controlling exponent whose sign flips exactly at $d=4$ , which is why four is not an arbitrary integer but the balance point of two competing dimensional effects.

Step 1. Fluctuation amplitude: $\langle\delta\phi^2\rangle_{\xi}\sim\frac{k_BT_c}{c}\xi^{2-d}\propto|t|^{-\nu(2-d)}=|t|^{(d-2)/2}$ , using $\xi\propto|t|^{-1/2}$ . Why: the correlation-volume-averaged fluctuation is $k_BT\xi^2/c$ divided by $\xi^d$ , and $\nu=\tfrac12$ converts $\xi$ -powers to $|t|$ -powers.

Step 2. Order parameter squared: $\bar\phi^2\propto|t|^{2\beta}=|t|^{1}$ with mean-field $\beta=\tfrac12$ . Why: below $T_c$ , $\bar\phi\sim|t|^{1/2}$ so its square is one power of $|t|$ .

Step 3. Ratio: $\dfrac{\langle\delta\phi^2\rangle_{\xi}}{\bar\phi^2}\propto\dfrac{|t|^{(d-2)/2}}{|t|}=|t|^{(d-4)/2}$ . Why: dividing the two temperature powers leaves the single controlling exponent $(d-4)/2$ .

Step 4. Read off: $(d-4)/2>0\iff d>4$ , so the ratio vanishes for $d>4$ (mean field exact), diverges for $d<4$ , and is marginal (logarithmic) at $d=4$ . Hence $d_c=4$ . Why: the sign of $(d-4)$ decides whether fluctuations grow or die at the critical point. *Verified: verify_aq-sm-ginzburg-criterion.py reproduces the exponent $(d-4)/2$ symbolically from ν=½, β=½ and confirms it changes sign at d=4.*

Go deeper · Derivation B · 1 minGinzburg criterion in general exponents and its hyperscaling form

Derivation BGinzburg criterion in general exponents and its hyperscaling form

The goal is to write the criterion for arbitrary critical exponents rather than the mean-field values, showing that the condition for mean-field self-consistency is exactly the hyperscaling statement $\alpha<0$ . We replace the integrated fluctuation by the susceptibility using the fluctuation–response relation, insert the general power laws for the susceptibility, correlation volume, and order parameter, and demand that the ratio vanish as the critical point is approached. Rewriting the resulting inequality with the hyperscaling relation collapses it to a condition on the single specific-heat exponent, exposing why mean field with $\alpha=0$ sits exactly at the marginal case.

Step 1. General ratio: $\dfrac{\langle\delta\phi^2\rangle_{\xi}}{\bar\phi^2}\sim\dfrac{\chi}{\xi^d\bar\phi^2}$ , since the integrated correlation equals the susceptibility $\chi\propto|t|^{-\gamma}$ . Why: the fluctuation–response relation makes $\int G=\chi$ ; dividing by the correlation volume and order squared gives the relative fluctuation.

Step 2. Insert exponents: $\sim\dfrac{|t|^{-\gamma}}{|t|^{-d\nu}|t|^{2\beta}}=|t|^{\,d\nu-2\beta-\gamma}$ . Why: $\xi^d\propto|t|^{-d\nu}$ and $\bar\phi^2\propto|t|^{2\beta}$ .

Step 3. Self-consistency requires the ratio to vanish: $d\nu-2\beta-\gamma>0$ . Mean-field check: $\tfrac d2-1-1=\tfrac d2-2>0\iff d>4$ . Why: a positive exponent means the ratio $\to0$ as $t\to0$ .

Step 4. Rewrite the right side of $d\nu>2\beta+\gamma$ with Rushbrooke ( $2\beta+\gamma=2-\alpha$ ): the condition becomes $d\nu>2-\alpha$ . On the mean-field/hyperscaling line $d\nu=2-\alpha$ this is an equality — the marginal boundary — and mean-field $\alpha=0$ sits exactly there, giving $d_c=4$ . For a true critical set with $\alpha>0$ (as in 3D Ising) the inequality fails and fluctuations dominate; $\alpha<0$ (as in the 3D-XY $\lambda$ -transition) means mean-field-like mildness. Why: the specific-heat exponent packages whether fluctuations are strong enough to invalidate mean field. *Verified: verify_aq-sm-ginzburg-criterion.py confirms the mean-field boundary $d\nu=2\beta+\gamma\Rightarrow d/2=2\Rightarrow d_c=4$ , that mean-field $\alpha=0$ is marginal, and that the 3D Ising hyperscaling set has $\alpha>0$ (fluctuation-dominated).*

Go deeper · Derivation C · 1 minGinzburg number for a BCS superconductor

Derivation CGinzburg number for a BCS superconductor

The goal is to estimate the Ginzburg window for a conventional superconductor and show why BCS mean-field theory is effectively exact, in sharp contrast to helium and simple fluids. We use the three-dimensional form of the Ginzburg temperature expressed through the coherence length and the condensation energy, insert realistic numbers for a clean elemental superconductor, and obtain a fluctuation window many orders of magnitude narrower than any achievable temperature resolution. The same formula with the atomic-scale coherence length of helium or a critical fluid gives a window of order unity, so the single expression accounts for both extremes of observed critical behaviour.

Step 1. Length scale: the BCS coherence length in a clean elemental superconductor is $\xi_0\sim10^{3}\,$ Å $=10^{-7}\,$ m — enormous on atomic scales. Why: Cooper pairs are spatially extended, so the correlation length is huge.

Step 2. Assemble the ratio $\dfrac{k_BT_c}{\Delta C\,\xi_0^{3}}\ll1$ : with $T_c$ of a few kelvin and a typical condensation energy density, the vast correlation volume $\xi_0^3$ makes the fluctuation budget negligible. Why: the denominator carries $\xi_0^3\sim10^{-21}\,$ m $^3$ , dwarfing the thermal numerator.

Step 3. Square it: $t_G\sim(k_BT_c/\Delta C\,\xi_0^3)^2\sim10^{-12}$ – $10^{-16}$ , far narrower than any temperature resolution, so BCS mean-field exponents are what is observed. Why: the criterion enters squared in $d=3$ , amplifying the smallness.

Step 4. Contrast: high- $T_c$ cuprates have $\xi_0\sim10\,$ Å and high $T_c$ , giving $t_G\sim10^{-2}$ , so critical fluctuations are visible; the $\lambda$ -transition of $^4$ He has $\xi_0\sim$ few Å and $t_G\sim O(1)$ , entirely fluctuation-dominated. Why: shrinking $\xi_0$ by four orders of magnitude widens $t_G$ by many. *Verified: verify_aq-sm-ginzburg-criterion.py evaluates the order-of-magnitude BCS estimate t_G~10^{-12}–10^{-16} from ξ₀~10³ Å and contrasts it with the ξ₀~few-Å helium/fluid case giving t_G~O(1).*

Limiting cases

Behaviour of the relative fluctuation $Q(t)\equiv\langle\delta\phi^2\rangle_{\xi}/\bar\phi^2$ as $t\to0$ in dimension $d$ :

Case	Result	Meaning
$d>4$ (above $d_c$ )	$Q(t)\sim\|t\|^{(d-4)/2}\to0$	Fluctuations negligible; mean-field exponents asymptotically exact; Gaussian fixed point stable, quartic coupling irrelevant.
$d=4$ (marginal)	$Q(t)\sim\|\ln t\|$	Logarithmic corrections to mean field; the borderline upper critical dimension.
$d<4$ (below $d_c$ )	$Q(t)\sim\|t\|^{(d-4)/2}\to\infty$	Fluctuations dominate near $T_c$ ; mean field fails inside $\|t\|<t_G$ ; Wilson–Fisher fixed point governs true exponents.
Tricritical $\phi^6$	$d_c=3$	With the relevant coupling being $\phi^6$ , power counting gives $d_c=3$ ; tricritical points in $d=3$ show only logarithmic corrections.
Long-range $J(r)\sim r^{-(d+\sigma)}$	$d_c=2\sigma$	Long-range interactions soften fluctuations and shift $d_c$ ; mean field valid for $d>2\sigma$ with $0<\sigma\le2$ .

Worked examples

Example 1 — the width of the critical region in $d=3$ from the exponent. Estimate how the Ginzburg window depends on the ratio of microscopic to correlation length.

Step 1. In $d=3$ the relative fluctuation is $Q(t)\sim|t|^{(3-4)/2}=|t|^{-1/2}$ times a prefactor $\sim(a_0/\xi_0)^{3}$ (from $t_G\sim(a_0/\xi_0)^{2d/(4-d)}$ with $d=3$ : exponent $2\cdot3/1=6$ , so $t_G\sim(a_0/\xi_0)^{6}$ ).

Step 2. Take a superconductor with $a_0\sim3\,$ Å and $\xi_0\sim10^{3}\,$ Å, so $a_0/\xi_0=3\times10^{-3}$ . Then $t_G\sim(3\times10^{-3})^{6}=7\times10^{-16}$ . The Ginzburg window is $t_G\approx10^{-15}$ — utterly invisible.

Step 3. Take helium with $a_0\sim\xi_0\sim3\,$ Å, so $a_0/\xi_0\sim1$ and $t_G\sim1$ . The whole accessible range is fluctuation-dominated, which is why $^4$ He shows the 3D-XY exponent $\alpha\approx-0.013$ , not the mean-field $\alpha=0$ .

Example 2 — the marginal case and logarithmic corrections at $d=4$ . Show that exactly at the upper critical dimension the relative fluctuation grows only logarithmically.

Step 1. At $d=4$ the power $(d-4)/2=0$ , so the naive scaling gives a constant. The next-order momentum integral $\int^{1/\xi_0}_{1/\xi}\frac{d^4q}{q^2+\xi^{-2}}$ picks up a logarithm.

Step 2. Carrying the integral, $Q(t)\sim\ln(\xi/\xi_0)=\ln(|t|^{-1/2})=\tfrac12|\ln|t||$ . So $Q(t)\sim|\ln t|$ , diverging but only logarithmically. This is why $d=4$ shows mean-field exponents dressed with logarithmic factors (for example $C\sim|\ln t|^{1/3}$ ), not power-law corrections.

Origins · 1 minginzburg criterion and how this was discovered

ginzburg criterion

self-consistency test for mean-field theory · The Ginzburg criterion, formulated by Vitaly Ginzburg around 1960, is the self-consistency condition that decides whether the mean-field (Landau) description of a continuous transition can be trusted near the critical point. It compares the order-parameter fluctuation averaged over one correlation volume with the square of the equilibrium order parameter and demands the former be small. Because the correlation length diverges at T_c, the condition fails for spatial dimension below four, defining both the upper critical dimension d_c = 4 for short-range φ⁴ systems and the width t_G of the true critical region. The criterion explains at a stroke why conventional superconductors obey mean-field BCS theory while fluids, magnets, and superfluid helium show non-classical exponents, and it set the stage for Wilson and Fisher's ε-expansion below d = 4.

History

1937

— Landau formulates the general mean-field theory of continuous transitions

1944

— Onsager's exact 2D Ising solution shows mean-field exponents are wrong in low dimensions

1950

— Ginzburg and Landau write the phenomenological theory of superconductivity

1960

— Ginzburg states the fluctuation self-consistency criterion and the width of the critical region

1966

— Kadanoff's block-spin picture reframes the correlation-length divergence

1971

— Wilson and Fisher launch the ε = 4−d expansion below the upper critical dimension

2003

— Ginzburg shares the Nobel Prize for the theory of superconductors and superfluids

Going further

References. Kardar, Statistical Physics of Fields, Ch. 3 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 5 · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 13 · Chaikin & Lubensky, Principles of Condensed Matter Physics, Ch. 4–5 · Tinkham, Introduction to Superconductivity, §4 · Amit & Martín-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena, Ch. 1.

Provenance. The criterion follows Ginzburg (1960); the upper-critical-dimension power counting matches Kardar SPF Ch. 3 and Goldenfeld Ch. 5. The general-exponent form $d\nu>2\beta+\gamma\Leftrightarrow\alpha<0$ uses hyperscaling ( $d\nu=2-\alpha$ ) and Rushbrooke ( $2\beta+\gamma=2-\alpha$ ) from Pathria & Beale Ch. 13. The BCS estimate ( $\xi_0\sim10^3\,$ Å, $t_G\sim10^{-12}$ ) and the cuprate/helium contrast are standard (Tinkham §4). Long-range and tricritical shifts ( $d_c=2\sigma$ , $d_c=3$ ) follow standard power counting.

Builds on: Landau Theory of Phase Transitions · Correlation Function · Continuous Phase Transitions · Critical Exponents. Leads to: Universality and Universality Classes · Scaling Hypothesis and Scaling Relations · Critical Exponents.

What is the difference between the correlation length and the Ginzburg length?

The correlation length $\xi(t)=\xi_0|t|^{-\nu}$ is the actual size of correlated regions and diverges at $T_c$ . The Ginzburg criterion uses $\xi$ but adds a second scale: the bare amplitude $\xi_0$ , a microscopic length that does not diverge. The Ginzburg temperature $t_G$ is where the correlation length has grown enough (relative to $\xi_0$ ) that fluctuations averaged over $\xi^d$ overwhelm the order. So $\xi_0$ sets how close to $T_c$ you must get before fluctuations matter — a large $\xi_0$ pushes $t_G$ to unobservably small values.

Why does the specific-heat exponent α control the criterion?

Through Rushbrooke and hyperscaling. The general Ginzburg condition $d\nu>2\beta+\gamma$ becomes, using Rushbrooke ( $2\beta+\gamma=2-\alpha$ ), the statement $d\nu>2-\alpha$ ; on the hyperscaling line $d\nu=2-\alpha$ this is the marginal equality, so mean-field validity is equivalent to $\alpha<0$ . Physically, $\alpha<0$ means the specific heat does not diverge (only a cusp), which happens when fluctuations are mild enough for mean field to survive; $\alpha>0$ signals strong fluctuations. Mean-field $\alpha=0$ sits exactly at the marginal case, consistent with the logarithmic corrections that appear at $d_c=4$ .

Does the Ginzburg criterion give the correct exponents below d = 4?

No — and it does not claim to. The criterion only tells you whether mean-field exponents are valid; it flags their breakdown but says nothing about the replacement values. Computing the true non-classical exponents below $d_c$ requires the renormalization group and the Wilson–Fisher fixed point, whose $\varepsilon=4-d$ expansion gives $\nu=\tfrac12+\tfrac{\varepsilon}{12}+\dots$ and $\eta=\tfrac{\varepsilon^2}{54}+\dots$ . The Ginzburg criterion is the signpost that says "here mean field fails"; the RG is what computes what happens instead.

How does the upper critical dimension change for long-range interactions?

If the interaction decays as $J(r)\sim r^{-(d+\sigma)}$ with $0<\sigma<2$ , the propagator's stiffness term changes from $q^2$ to $q^{\sigma}$ , softening the fluctuations. Redoing the power counting gives $d_c=2\sigma$ instead of 4. For $\sigma\ge2$ the interaction is effectively short-range and the standard $d_c=4$ returns. This is why long-range Ising and spin systems can show mean-field behaviour in dimensions where short-range versions are strongly fluctuating — the upper critical dimension is a property of the dispersion of the relevant modes, not just the space dimension.

Is the Ginzburg criterion an exact statement or an estimate?

It is an order-of-magnitude self-consistency estimate, not an exact theorem. It correctly identifies the upper critical dimension $d_c$ (which the renormalization group confirms rigorously) and the scaling of the width $t_G$ with $\xi_0$ , but the prefactor in $t_G$ is only approximate — it depends on how one defines "averaged over a correlation volume" and on the precise cutoff. For practical purposes the exponent $(d-4)/2$ and the dependence $t_G\sim(a_0/\xi_0)^{2d/(4-d)}$ are reliable; the numerical coefficient should be trusted only within an order of magnitude.

Why is the lambda transition of helium such a good test of non-classical exponents?

Because its Ginzburg temperature is of order unity, so essentially the entire measurable range near $T_\lambda$ is fluctuation-dominated, and because the transition can be measured with extraordinary temperature resolution (down to nanokelvin, including in microgravity to remove the gravitational rounding of $T_\lambda$ across the sample). This combination lets experimenters follow the specific heat over many decades of reduced temperature and extract the 3D-XY exponent $\alpha\approx-0.013$ with tiny error bars — a decisive falsification of the mean-field $\alpha=0$ and one of the most precise confirmations of renormalization-group predictions.