The <Ginzburg criterion> compares fluctuations of the order parameter averaged over one correlation volume with the squared mean-field order parameter. The stated correlation function gives
$$
\langle(\delta m)^2\rangle_\xi
\sim\frac1{\xi^d}\int_0^\xi r^{d-1}r^{-(d-2)},dr
\sim\xi^{2-d}.
$$
Meanwhile $m_0^2\sim|t|\sim\xi^{-2}$, so their ratio scales as $\xi^{4-d}$. It vanishes at criticality for $d>4$, grows for $d<4$, and is marginal at $d=4$. Thus the ordinary scalar Landau-Ginzburg model has <upper critical dimension> $d_c=4$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 1 / b / iii / Solution

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The Ginzburg criterion requires order-parameter fluctuations inside a correlation volume to be small compared with the squared mean-field order parameter. Its failure marks the fluctuation-dominated critical region.


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