= Analytic vector Landau expansion
{title2=$f(\mathbf m)=b|\mathbf m|^2+d|\mathbf m|^4+\cdots$}
For an isotropic <vector> <order parameter>, an analytic scalar <Landau free energy> near the origin contains powers of $\mathbf m\cdot\mathbf m$. Odd powers of its magnitude are rotationally invariant but nonanalytic as functions of the Cartesian components at zero; excluding them requires analyticity, not merely writing the magnitude as a nonnegative scalar. With $d>0$, changing $b$ from positive to negative continuously creates minima at $|\mathbf m|=\sqrt{-b/(2d)}$, with an arbitrary direction.<Statistical Field Theory by David Tong>[§4.2, rotationally invariant vector expansion]{ref}
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