Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 6 / a / Solution

At mean-field level, positive gradient stiffness favours a spatially uniform order parameter. Writing $r=|\varphi |$, minimization of $f=a{r}^2+r^4/2$ giveswith arbitrary constant $\delta$. The minimized density is zero above the transition and $-a^2/2$ below it. The order parameter vanishes continuously as $(-a{)}^{1/2}$; the first derivative of the minimized free energy is continuous while the second changes. This is the continuous mean-field transition.

The broken symmetry is global U(1) symmetry, not merely the Ising sign symmetry. Selecting one ordered phase breaks this continuous symmetry and leaves a soft phase direction. These are minima of the supplied Landau free energy, not an exact finite-temperature long-range-order claim in one dimension: phase fluctuations invalidate that inference for a short-range one-dimensional system.^[ref]