= Solution
At mean-field level, positive gradient stiffness favours a spatially uniform <order parameter>. Writing $r=|\phi|$, minimization of $f=ar^2+r^4/2$ gives
$$
\boxed{\phi=0\quad(a\ge0),\qquad
\phi=\sqrt{-a}\,e^{i\delta}\quad(a<0),}
$$
with arbitrary constant $\delta$. The minimized density is zero above the transition and $-a^2/2$ below it. The <order parameter> vanishes continuously as $(-a)^{1/2}$; the first <derivative> of the minimized free <energy> is continuous while the second changes. This is the continuous mean-field transition.
The broken symmetry is \b[<global U(1) symmetry>], not merely the Ising sign symmetry. Selecting one ordered phase breaks this continuous symmetry and leaves a soft phase direction. These are minima of the supplied <Landau free energy>, not an exact finite-temperature long-range-order claim in one dimension: phase fluctuations invalidate that inference for a short-range one-dimensional system.<Statistical Field Theory by David Tong>[§4.2.3, continuous symmetry, phase fluctuations and the Mermin-Wagner theorem]{ref}
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