= Solution
Choose the reference staggered direction so its classical moment is positive. The Bogoliubov vacuum has $\langle a_k^\dagger a_k\rangle=\sinh^2\theta_k$, hence
$$
\frac{M_s}{N}=S-\delta S,\qquad\boxed{\delta S=\frac1{2N}\sum_k\left[\frac1{\sqrt{1-\gamma_k^2}}-1\right].}
$$
Near each soft point the integrand is proportional to $1/|k|$, so the one-dimensional thermodynamic integral diverges logarithmically. \b[The ordered-reference expansion fails; it does not predict a physical negative magnetization.] Long-range Néel order is destroyed in this isotropic one-dimensional setting. The exact finite symmetric ground state also has zero staggered expectation without selecting a direction.
For linearly dispersing modes in $d$ dimensions, zero-temperature depletion behaves as $\int_0dk\,k^{d-2}$ and is infrared finite for $d>1$, allowing order in suitable two- and three-dimensional antiferromagnets. Finite depletion is not a guarantee for every frustrated quantum magnet. At nonzero temperature, the extra Bose occupation changes the singularity to $\int dk\,k^{d-3}$: short-range isotropic systems cannot sustain continuous-symmetry long-range order for $d\le2$. This thermal conclusion is distinct from the one-dimensional zero-point argument.
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