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(Replaced content with "==Problem 1: Ginzburg criterion in arbitrary spatial dimension and upper critical dimension == The general solution for the correlation function in arbitrary spatial dimensi...") |
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==Problem 1: Ginzburg criterion in arbitrary spatial dimension and upper critical dimension == | ==Problem 1: Ginzburg criterion in arbitrary spatial dimension and upper critical dimension == | ||
The general solution for the correlation function in arbitrary spatial dimension within the mean-field theory can be written as: | The general solution for the correlation function in arbitrary spatial dimension <math> d </math> within the mean-field theory can be written as: | ||
<math> G(r) \sim \frac{e^- | <math> G(r) \sim \frac{e^{-r/\xi}}{r^{d-2}} </math> | ||
assuming that distance <math> r \gg a </math> is much larger than the lattice spacing <math> a </math>. | assuming that distance <math> r \gg a </math> is much larger than the lattice spacing <math> a </math>. | ||
(a) Generalize the Ginzburg criterion | |||
<math> \frac{G(r)}{\int m^2 d\mathbf{r}} \ll 1 </math> | |||
for the validity of the mean-field theory to arbitrary spatial dimension <math> d </math> to show that this is satisfied if | |||
<math> d>2+2\beta/\nu </math>. | |||
(b) Using your result in (a), find the ''upper critical dimension'' for the Ising model above which its critical behavior near temperature <math> T_c </math> is governed by the mean-field theory. | |||
==Problem 2: == | ==Problem 2: == | ||
Revision as of 16:22, 29 April 2011
Problem 1: Ginzburg criterion in arbitrary spatial dimension and upper critical dimension
The general solution for the correlation function in arbitrary spatial dimension within the mean-field theory can be written as:
assuming that distance is much larger than the lattice spacing .
(a) Generalize the Ginzburg criterion
for the validity of the mean-field theory to arbitrary spatial dimension to show that this is satisfied if
.
(b) Using your result in (a), find the upper critical dimension for the Ising model above which its critical behavior near temperature is governed by the mean-field theory.
