 O(3) quartic Landau minima

For $f=f_0+a(T-T_c)|\mathbf{m}{|}^2+b|\mathbf{m}{|}^4+c|\nabla \mathbf{m}{|}^2$ with positive coefficients and uniform free boundaries, the minimum is $\mathbf{m}=0$ at $T\ge T_c$. Below $T_c$ it is any constant vector with $|\mathbf{m}{|}^2=a(T_c-T)/(2b)$. A chosen direction leaves an $O(2)$ subgroup of $O(3)$ invariant; for proper rotations the pattern is $SO(3)\to SO(2)$. The mean-field transition is continuous, with square-root onset. Long-wavelength fluctuations can invalidate the ordered state in low dimensions.