Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 34D / i

Landau theory represents the appropriate thermodynamic potential as an analytic function of a small order parameter, retaining every low-order term allowed by the system's symmetry. At fixed control parameters, the equilibrium phase is the global minimum of the Landau free energy; local minima describe metastable phases. A continuous phase transition occurs when a minimum changes continuously and loses stability, whereas a first-order phase transition occurs when distinct minima exchange global stability.

Forthe stationary points satisfyIf $\epsilon >0$, $\varphi =0$ is the unique minimum. If $\epsilon <0$, $\varphi =0$ is a maximum and the two degenerate minima areThus the reflection symmetry $\varphi \mapsto -\varphi$ is spontaneously broken below $\epsilon =0$, and