Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 72 / 2 / a / Solution

With the dimensionless magnetic field conjugate to the magnetization, $m=-\partial f_s/\partial h$. Set $y=h/|t{|}^{\Delta }$ and differentiate the homogeneous free-energy density:On the ordered branch, the one-sided value $g_{m,-}(0^{+})$ is nonzero, so the order-parameter critical exponent isTo reach the critical isotherm, take $|t|\to 0$ at fixed nonzero $h$. Suppose $g_m(y)\sim A\mathrm{sgn}(y)|y{|}^p$ at large $|y|$. Cancellation of the residual thermal factor requires $2-\alpha -\Delta -p\Delta =0$. Hence $m(0,h)\propto \mathrm{sgn}(h)|h{|}^{(2-\alpha -\Delta )/\Delta }$ andThe singular scaling function determines the nonanalytic critical-isotherm contribution; any analytic response background is subleading for the usual $\delta >1$.