= Quantum depletion of Néel order
{title2=$M_s=S-N^{-1}\sum_kv_k^2$}
For a <linear spin-wave approximation>, the ground state is empty of the diagonal <quasiparticles> but contains the original spin bosons. If $a_k=u_k\alpha_k-v_k\alpha_{-k}^\dagger$, their <occupation number> is $v_k^2$. The <staggered magnetization> is reduced from $S$ to $S-N^{-1}\sum_kv_k^2$. For the one-dimensional nearest-neighbor <Heisenberg antiferromagnet>, $v_k^2=(|\sin k|^{-1}-1)/2$ and the integral diverges logarithmically. This invalidates finite ordered magnetization within the approximation; it is not itself a determination of the exact spectral gap.
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