= Easy-axis XXZ antiferromagnetic spin-wave spectrum
{title2=$\epsilon_k=2JS\sqrt{\Delta^2-\cos^2k},\quad\Delta\geq1$}
After a <bipartite spin rotation>, a leading <Holstein–Primakoff transformation> expansion gives $H_2=-J\Delta NS^2+JS\sum_k[2\Delta n_k+\cos k(a_ka_{-k}+a_k^\dagger a_{-k}^\dagger)]$. The <bosonic Bogoliubov diagonalization> gives the displayed spectrum and <vacuum energy> $-J\Delta NS^2+JS\sum_k[\sqrt{\Delta^2-\cos^2k}-\Delta]$. The anisotropy gap is $2JS\sqrt{\Delta^2-1}$. At $\Delta=1$ the finite-angle transformation is singular at its <zero modes>, and <quantum depletion of Néel order> is infrared divergent in one dimension.
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