 Reduced-zone counting of bipartite spin waves

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.