Statistical mechanics, structure of matter Phase transitions and critical phenomena Fact

Ginzburg criterion

Builds on Landau theory Mean-field theory Correlation function

Statement

Let $d \ge 1$ , and consider a real order parameter field $m \colon \R^d \to \R$ . Its Landau–Ginzburg free energy is $\mathcal{F}[m] = \int_{\R^d} \Big( a t \, m(x)^2 + b \, m(x)^4 + c \, \abs{\nabla m(x)}^2 \Big) \, dx.$ Here $a, b, c > 0$ are constants, $T_c > 0$ is the mean-field critical temperature and $t = (T - T_c)/T_c$ . Let $k_B$ be the Boltzmann constant and define the Ginzburg number $\mathrm{Gi}$ by $\mathrm{Gi}^{(4-d)/2} = \frac{k_B T_c \, b}{a^{2 - d/2} \, c^{d/2}} \quad (d \ne 4).$ Let $m_0^2 = a \abs{t} / (2b)$ be the squared mean-field order parameter for $t < 0$ . In the Gaussian approximation, consider the mean square fluctuation of $m$ averaged over a correlation volume. Its ratio to $m_0^2$ is of order $(\mathrm{Gi} / \abs{t})^{(4-d)/2}$ , up to a factor that depends only on $d$ . Hence mean-field theory is self-consistent near $T_c$ exactly when this ratio is small:

if $d < 4$ , it holds only for $\abs{t} \gg \mathrm{Gi}$ and fails as $t \to 0$ ;
if $d > 4$ , the ratio tends to $0$ as $t \to 0$ , and mean-field theory holds up to the critical point.

Remarks

The idea: the mean-field correlation length is $\xi \sim (c / (a \abs{t}))^{1/2}$ and the susceptibility is $\chi \sim 1/(a \abs{t})$ . The integral of the correlation function over a ball of radius $\xi$ is of order $k_B T_c \chi$ , so the fluctuation per unit volume there is about $k_B T_c \chi / \xi^d$ . Comparing it with $m_0^2 = a \abs{t} / (2b)$ gives the condition $k_B T_c b \ll a^{2 - d/2} c^{d/2} \abs{t}^{2 - d/2}$ . The borderline dimension $d = 4$ is the upper critical dimension of the Ising universality class, where mean-field exponents hold with logarithmic corrections.

Example: in conventional superconductors the coherence length is long, so $c$ is large and $\mathrm{Gi}$ is tiny. So the mean-field Ginzburg–Landau theory works there except extremely close to $T_c$ .

Going deeper

For a student

The criterion is a self-consistency test of Landau theory with a gradient term, done inside that theory. It explains why mean-field exponents are wrong for the three-dimensional Ising model, but right above four dimensions. The renormalization group later confirmed the picture: below four dimensions the Gaussian fixed point is unstable to the $m^4$ term, and above four dimensions this term is irrelevant. In a temperature window $\abs{t} \gg \mathrm{Gi}$ one sees mean-field behaviour, and in the window $\abs{t} \ll \mathrm{Gi}$ one sees the true critical exponents.

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Tag	`S3HXP33ZP8`
URI	`m2://S3HXP33ZP8`
Kind	statement, fact
Writing	W0 stub
MSC	82B27, 82B26
Status	This copy of the site has no build data. Run `m2 status S3HXP33ZP8`.
About	Mean-field theory, Landau theory
Uses	Landau theory, Mean-field theory, Correlation function
Same as	Wikidata Q5563311 Wikipedia Ginzburg criterion
Changed	3 November 2026