Past exam of the theoretical physics I course of the University of Cambridge / 2019 / 4 / c / Solution

For $b<0$ at $T=T_c$, the quartic term lowers the free energy near $m=0$, while the positive sextic term stabilizes it at larger $|m|$. The nonzero stationary points haveThey are global minima. The stationary point $m=0$ has zero quadratic curvature but is a flat local maximum because its leading nonzero term is negative. Thus the system is already ordered at $T_c$, with a finite magnetization.

More explicitly, a continuous onset from $m=0$ could occur only where $f^{\prime \prime }(0)=2r$ vanishes, namely at $r=0$. But the global minima there are the finite values above. For sufficiently large positive $r$, only $m=0$ is stable. Therefore the equilibrium minimum must change at some positive $r$ by crossing between a zero and a finite nonzero order parameter. This is a first-order phase transition. Its actual temperature lies above $T_c$ and is calculated next; $T_c$ itself is the disordered spinodal point, not the coexistence temperature.