= One-dimensional planar spin-wave depletion
{title2=$\delta S=(2L)^{-1}\sum_k(\alpha_k/\epsilon_k-1)$}
The <Bogoliubov vacuum> has original-boson <Bosonic occupation number>[occupation] $v_k^2=(\alpha_k/\epsilon_k-1)/2$, reducing the ordered <spin> by its average over <momenta>. For the zero-field planar ferromagnetic chain, $v_k^2\sim1/(2\sqrt2|k|)$. Even after excluding the exact uniform <zero mode>, the sum grows as $(2\pi\sqrt2)^{-1}\log L$. It invalidates a thermodynamic fixed-orientation expansion with finite dilute <One-dimensional planar spin-wave depletion>[depletion]; it does not represent a negative physical <magnetization>.
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