Ginzburg criterion

Mean field theory gives sensible results as long as one is able to neglect fluctuations in the system under consideration. The Ginzburg criterion tells quantitatively when mean field theory is valid. It also gives the idea of an upper critical dimension, a dimensionality of the system above which mean field theory gives proper results, and the critical exponents predicted by mean field theory match exactly with those obtained by numerical methods.

Example: Ising model

If $\phi$ is the order parameter of the system, then mean field theory requires that the fluctuations in the order parameter are much smaller than the actual value of the order parameter near the critical point.

Quantitatively, this means that [1]

\displaystyle {\mathcal {\langle }}(\delta \phi )^{2}\rangle \quad {\ll }\quad \langle \phi \rangle ^{2}

Using this in the Landau theory, which is identical to the mean field theory for the Ising model, the value of the upper critical dimension comes out to be 4. If the dimension of the space is greater than 4, the mean-field results are good and self-consistent. But for dimensions less than 4, the predictions are less accurate. For instance, in one dimension, the mean field approximation predicts a phase transition at finite temperatures for the Ising model, whereas the exact analytic solution in one dimension has none (except for $T=0$ and $T\rightarrow \infty$ ).

Example: Classical Heisenberg model

In the classical Heisenberg model of magnetism, the order parameter has a higher symmetry, and it has violent directional fluctuations which are more important than the size fluctuations. They overtake to the Ginzburg temperature interval over which fluctuations modify the mean-field description thus replacing the criterion by another, more relevant one.

Footnotes

K., Pathria, R. (2011). Statistical mechanics. Beale, Paul D. (3rd ed.). Boston: Academic Press. p. 460. ISBN 9780123821881. OCLC 706803528.

References

V. L. Ginzburg (1961). "Some remarks on phase transitions of the 2nd kind and the microscopic theory of ferroelectric materials". Soviet Physics - Solid State. 2: 1824.
D. J. Amit (1974). "The Ginzburg criterion-rationalized". J. Phys. C: Solid State Phys. 7 (18): 3369–3377. Bibcode:1974JPhC....7.3369A. doi:10.1088/0022-3719/7/18/020.
J. Als-Nielsen and R. J. Birgeneau (1977). "Mean field theory, the Ginzburg criterion, and marginal dimensionality of phase transitions". American Journal of Physics. AAPT. 45 (6): 554–560. Bibcode:1977AmJPh..45..554A. doi:10.1119/1.11019. Archived from the original on 2013-02-23. Retrieved 2019-12-28.
H. Kleinert (2000). "Criterion for Dominance of Directional over Size Fluctuations in Destroying Order". Phys. Rev. Lett. 84 (2): 286–289. arXiv:cond-mat/9908239. Bibcode:2000PhRvL..84..286K. doi:10.1103/physrevlett.84.286. PMID 11015892. S2CID 24140115. Archived from the original on 2013-02-23.

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[1] K., Pathria, R. (2011). Statistical mechanics. Beale, Paul D. (3rd ed.). Boston: Academic Press. p. 460. ISBN 9780123821881. OCLC 706803528.