Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 51 / 3 / f / Solution

Let $t,h$ be the thermal and magnetic scaling fields, with eigenvalues $b^{y_t}$ and $b^{y_h}$. After discarding irrelevant variables whose vanishing is nonsingular, the homogeneous singular free-energy density satisfiesChoose $b=|t{|}^{-1/y_t}$. ThenTwo thermal derivatives give the heat-capacity critical exponent; one and two field derivatives give the order-parameter critical exponent and magnetic-susceptibility critical exponent. The correlation-length critical exponent follows directly from the length rescaling. ThusAt $t=0$, choose $b=|h{|}^{-1/y_h}$. Differentiating $f_s(0,h)\propto |h{|}^{D/y_h}$ gives $M\propto \mathrm{sgn}(h)|h{|}^{(D-y_h)/y_h}$, so $\delta =y_h/(D-y_h)$. The order parameter has scaling dimension $x_M=D-y_h$; therefore its critical connected correlation function has power $r^{-2x_M}$. Comparing with $r^{-(D-2+\eta )}$ yieldsThese formulas express the requested critical exponents in terms of the relevant scaling exponents. Their hyperscaling assumptions must be respected: above an upper critical dimension, a dangerously irrelevant coupling can invalidate the free-energy homogeneity used here. Applying the formulas to the Gaussian thinning transformation in the root solution gives the requested Gaussian indices and their physical qualifications.