= Helical minimum of a complex Landau functional
{title2=$k^2=-c/2,\quad |\phi|^2=c^2/4-a$}
For $f=a|\phi|^2+|\phi|^4/2+c|\partial_x\phi|^2+|\partial_x^2\phi|^2$, a plane-wave <order parameter> has quadratic coefficient $a+ck^2+k^4$. If $c<0$, this is minimized at $k^2=-c/2$, and a nonzero amplitude appears when $a<c^2/4$. If $c\ge0$, the minimizing wave number is zero and the ordered mean-field phase requires $a<0$. The arbitrary <global phase> is a degeneracy, not a missing minimization condition. These are functional-minimization results; finite-temperature fluctuations can invalidate long-range order in low dimensions.
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