 Field-polarized planar ferromagnetic spin wave

For the XY model with equal ferromagnetic exchange $J$ in its plane and 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 ${ϵ}_k=\sqrt{{\alpha }_k^2-{\Delta }_k^2}$ and field gap $S\sqrt{|h|(|h|+2J)}$. A nonzero field stabilizes every oscillator mode.