Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 50 / 1 / a / Solution

An order parameter distinguishes phases by an equilibrium expectation value. For an Ising model, the local field $M(x)$ is the coarse-grained magnetization, and the uniform value is $M=N^{-1}⟨{\sum }_r{\sigma }_r⟩$. The spin inversion symmetry maps $M$ to $-M$, so at zero field the Landau free energy contains even powers. Its disordered minimum is $M=0$; ordered pure thermodynamic phases have $M=\pm M_0$. Their sign is selected by taking the thermodynamic limit before sending $h$ to zero. A symmetric mixture can have zero average without being a single disordered phase. The term $-hM$ identifies $h$ as the field conjugate to an order parameter.