= Reduced-zone counting of bipartite spin waves
{title2=$E_{\rm vac}=E_{\rm cl}+\sum_{k\in\mathrm{MBZ}}(E_k-A)$}
Two sublattices give two <bosonic> modes for each of N/2 wavevectors in a magnetic <Brillouin zone>. For a symmetric quadratic pairing block with number coefficient A and anomalous coefficient B, a bosonic <Bogoliubov transformation> gives two equal positive energies $E_k=\sqrt{A^2-B_k^2}$ and the displayed vacuum shift. There are N modes overall. The <quantum depletion of Néel order> per site is $(2/N)\sum_{\mathrm{MBZ}}v_k^2$, where $v_k^2=(A/E_k-1)/2$. Equivalently one can use one branch over a full zone with the corresponding factors of one half. Mixing these conventions doubles the vacuum energy or depletion.
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