Analytic vector Landau expansion ($f(\mathbf{m})=b|\mathbf{m}{|}^2+d|\mathbf{m}{|}^4+\cdots$)

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.^[ref]