= O(3) quartic Landau minima
{c}
{title2=$|\mathbf m|^2=a(T_c-T)/(2b)$}
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.
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