 Solution

Choose the reference staggered direction so its classical moment is positive. The Bogoliubov vacuum has $⟨a_k^{†}{a}_k⟩={\sinh }^2{\theta }_k$, henceNear each soft point the integrand is proportional to $1/|k|$, so the one-dimensional thermodynamic integral diverges logarithmically. 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 ${\int }_{0}dk{k}^{d-2}$ 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 $d\le 2$. This thermal conclusion is distinct from the one-dimensional zero-point argument.