Mean field gives
$$
C_{\rm MF}\sim|T-T_c|^{-\alpha}
=|T-T_c|^{-(n-2)/(n-1)}.
$$
The stated Gaussian fluctuation term is
$$
C_{\rm fl}\sim|T-T_c|^{d/2-2}.
$$
As $T\to T_c$, mean field is more singular precisely when
$$
-\frac{n-2}{n-1}<\frac d2-2,
$$
or
$$
d>\frac{2n}{n-1}.
$$
Thus the <upper critical dimension> for the $m^{2n}$ interaction is
$$
\boxed{d_c=\frac{2n}{n-1}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 1 / a / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-303/1/a/iv/solution