Temporary HW

Problem 1: Ginzburg criterion in arbitrary spatial dimension and upper critical dimension

The general solution for the correlation function in arbitrary spatial dimension $d$ within the mean-field theory can be written as:

$G(r)\sim \frac{e^{-r/\xi }}{r^{d-2}}$

assuming that distance $r\gg a$ is much larger than the lattice spacing $a$.

(a) Generalize the Ginzburg criterion

$\frac{G(r)}{\int m^{2}d\mathbf{𝐫}}\ll 1$

for the validity of the mean-field theory to arbitrary spatial dimension $d$ to show that this is satisfied if

$d>2+2\beta /\nu$.

(b) Using your result in (a), find the upper critical dimension for the Ising model above which its critical behavior near temperature $T_c$ is governed by the mean-field theory.