Ginzburg criterion
= Ginzburg criterion
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The Ginzburg criterion estimates whether critical fluctuations are small enough for a <mean-field approximation>. Compare the mean-square fluctuation of the <order parameter> averaged over a <correlation volume> with the squared mean-field order parameter. The approximation requires the former to be much smaller.
For an ordinary scalar quartic theory below its transition, $m_0^2\sim|a|/b$ and $\xi\sim(\kappa/|a|)^{1/2}$. Long-wavelength Gaussian fluctuations then give a relative scale proportional to $k_BT\,b\,\kappa^{-d/2}|a|^{(d-4)/2}$. Below four dimensions this grows as $a\to0$, exposing a region where mean field fails. Numerical factors and microscopic cutoffs depend on the model. The criterion diagnoses a failure of the approximation, rather than a failure of the underlying transition to exist.