Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 3 / f / Solution

Identifying $g$ with temperature, write $g=g_c^{\ast }+\delta g$ near the nonzero critical fixed point. The derivative of the beta function there isThus the temperature-like perturbation has renormalization-group eigenvalue $y_t=ϵ$. Since the correlation-length critical exponent satisfies $\nu =1/y_t$,The Gaussian fixed point $g=0$ is the stable ordered-phase fixed point for $d>2$, rather than the finite-temperature transition. In exactly two dimensions the flow instead gives an essential, exponential correlation-length divergence, corresponding formally to $\nu =\infty$ rather than a finite power-law exponent.