 Helical minimum of a complex Landau functional

For $f=a|\varphi {|}^2+|\varphi {|}^4/2+c|{\partial }_x\varphi {|}^2+|{\partial }_x^2\varphi {|}^2$, a plane-wave order parameter has quadratic coefficient $a+c{k}^2+k^4$. If $c<0$, this is minimized at $k^2=-c/2$, and a nonzero amplitude appears when $a<c^2/4$. If $c\ge 0$, 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.