Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 41 / 1 / iv / Solution

Critical exponents describe singular behavior near a continuous phase transition. Define $m(t,0)\sim (-t{)}^{\beta }$ below $T_c$, susceptibility $\chi (t,0)=\partial m/\partial h\sim |t{|}^{-\gamma }$, singular heat capacity $C_s\sim |t{|}^{-\alpha }$, and critical isotherm $m(0,h)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. The ordinary quartic Landau free energy gives the equation of stateAt zero field its ordered minimum is proportional to $(-t{)}^{1/2}$. Differentiating the equation of state gives ${\chi }^{-1}=r_{t}t+3u{m}^2$, proportional to $|t|$ on either side. At $t=0$, $h=u{m}^3$, and the minimized free energy below the transition is proportional to $-t^2$. ThusHere $\alpha =0$ denotes a finite heat-capacity jump, rather than a power divergence. Including gradients gives the Landau scalar correlation length $\xi \propto |t{|}^{-1/2}$ and correlation-length exponent $\nu =1/2$.