Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 3 / a / Solution

For an analytic, local Landau theory with independent global rotations of the magnetization vector, a uniform scalar potential can depend only on $\rho =\mathbf{m}\cdot \mathbf{m}$. Keeping terms through fourth order in the field, dropping an additive constant, and retaining the leading isotropic positive gradient term gives the displayed quadratic-plus-quartic functional. The stiffness has been absorbed into the normalization of length and field. Higher powers and higher derivatives are neglected in this long-wavelength approximation.

The quartic truncation is stable at large $|\mathbf{m}|$ when $u>0$. The coefficient $t$ changes sign at the mean-field critical point; close to it one commonly writes $t\propto T-T_c$. If $u<0$, higher stabilizing powers are essential, so the displayed truncation alone cannot determine a stable first-order phase diagram. For $u=0$ and $t<0$ it is likewise unbounded.

For $u>0$, spatially uniform stationary points obey $(t+4u\rho )\mathbf{m}=0$. The global minima areThe corresponding minimum free-energy density is zero and $-t^2/(16u)$, respectively. The disordered phase gives way continuously to a ferromagnet with an arbitrary direction of magnetization. At this mean-field transition the amplitude vanishes as $(-t{)}^{1/2}$; long-wavelength critical fluctuations can modify mean-field exponents in three dimensions.^[ref]