= Field-polarized planar ferromagnetic spin wave
{title2=$\epsilon_k=S\sqrt{(|h|+2J)[|h|+2J(1-\cos k)]}$}
For the <XY model> with equal ferromagnetic exchange $J$ in its plane and <Magnetic field>[field] term $-hS\sum_j S_j^x$, rotate the local quantization axis along the <Classical spin>. The <Holstein–Primakoff transformation> gives the number coefficient $\alpha_k=S(|h|+2J-J\cos k)$ and anomalous coefficient $\Delta_k=JS\cos k$ with one transverse-axis convention. <Bosonic Bogoliubov diagonalization> gives $\epsilon_k=\sqrt{\alpha_k^2-\Delta_k^2}$ and <Magnetic field>[field] <Energy gap>[gap] $S\sqrt{|h|(|h|+2J)}$. A nonzero <Magnetic field>[field] stabilizes every oscillator mode.
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