Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 3 / a

At a fixed point, the order parameter can be chosen as a dilation eigenoperator. Therefore, for $x^{\prime }=x/\zeta$,Interactions shift ${\Delta }_{\varphi }$ from $(d-2)/2$ to $(d-2+\eta )/2$.

Solved by gpt-5.6-sol high.

If $t^{\prime }={\zeta }^{{\Delta }_t}t$, lengths shrink by $\zeta$, so $\xi (t^{\prime })={\zeta }^{-1}\xi (t)$. Substituting $\xi (t)\sim t^{-\nu }$ gives ${\zeta }^{-\nu {\Delta }_t}={\zeta }^{-1}$ and hence

Solved by gpt-5.6-sol high.

The source term transforms asTherefore

Solved by gpt-5.6-sol high.

Choose $\zeta =t^{-\nu }$. The singular free-energy density scales as $t^{d\nu }$, giving the hyperscaling relation $\alpha =2-d\nu$. The order parameter givesThe susceptibility has exponentAt $t=0$, choosing $\zeta =B^{-1/{\Delta }_B}$ gives

Solved by gpt-5.6-sol high.

For $d>4$, the quartic coupling has $y_g=4-d<0$, but setting it to zero removes the stabilizing term. It is a dangerously irrelevant coupling. At $t=0$, minimizing $g{\varphi }^4-B\varphi$ gives $\varphi \sim (B/g{)}^{1/3}$ and $\delta =3$. This is consistent with Gaussian scaling becauseFor $d<4$, $g$ flows to a nonzero Wilson-Fisher fixed point, so no dangerously irrelevant zero-coupling limit spoils the ordinary scaling relations.

Solved by gpt-5.6-sol high.