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

An order parameter distinguishes phases through a macroscopic observable, such as the mean spin $m$ of a ferromagnet. Its vanishing can reflect an unbroken symmetry, while choosing one of several nonzero equilibrium values gives spontaneous symmetry breaking. Landau-Ginzburg theory treats it as a slowly varying field and expands the free energy in terms allowed by symmetry:Here $K>0$ penalizes spatial variation, $h$ is the conjugate field, and the highest retained even coefficient must stabilize large $|m|$. For a spin-flip-symmetric system, odd powers are forbidden at $h=0$. Equilibrium in the Landau approximation minimizes this functional; near an ordinary continuous transition $r=r_{t}t$ with $t=(T-T_c)/T_c$ and $r_t>0$.