Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 1 / b / i / Solution

Landau-Ginzburg theory assumes a local gradient expansion for a slowly varying order parameter. At zero field the microscopic spin-flip symmetry sends $m\mapsto -m$, so only even powers occur. Rotational symmetry makes the leading derivative term $(\nabla m{)}^2$, while locality and analyticity near the transition organize higher powers and derivatives. Stability requires a positive gradient coefficient and, at an ordinary continuous transition, ${\alpha }_4(T_c)>0$; the coefficient ${\alpha }_2(T)$ changes sign at $T_c$.