Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 46 / 2 / b / Solution

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. A critical fixed point is scale invariant and has divergent correlation length. The critical surface consists of couplings flowing toward it after relevant variables have been tuned; noncritical flows ultimately leave its neighborhood toward finite-correlation-length phases.

Linearize the renormalization-group transformation near the fixed point and choose scaling coordinates. If the thermal and magnetic coordinates transform as $t^{\prime }=b^{y_t}t$ and $h^{\prime }=b^{y_h}h$, the singular free-energy density obeysChoose $b=|t{|}^{-1/y_t}$ to stop at an order-one thermal coordinate. This gives the scaling hypothesis for critical phenomenaConsequently $\nu =1/y_t$ and $\Delta =y_h/y_t$, with $2-\alpha =D/y_t$ when hyperscaling applies. The fixed point determines universal powers, while microscopic couplings set metric factors and amplitudes. Dangerously irrelevant variables and marginal logarithms can qualify the simple homogeneous form.