 Solution

At a critical point, a local field operator $\mathcal{O}$ has scaling dimension ${\Delta }_{\mathcal{O}}$ if $\mathcal{O}(x)\mapsto b^{-{\Delta }_{\mathcal{O}}}\mathcal{O}(x/b)$ under coarse-graining by $b$. The coupling $u$ in $\int d^{d}xu\mathcal{O}$ has renormalization-group eigenvalueA perturbation is a relevant operator when $y_u>0$, an irrelevant operator when $y_u<0$, and a marginal operator when $y_u=0$; nonlinear terms in its renormalization-group beta function decide the fate of a marginal coupling. Irrelevant microscopic interactions decay under coarse-graining, so many distinct systems approach the same fixed point and share one universality class.

Near criticality the singular free energy density is of order one per correlation volume:The heat capacity contains two derivatives with respect to temperature, soConsequently the hyperscaling relation is