Field-polarized planar ferromagnetic spin wave (${ϵ}_k=S\sqrt{(|h|+2J)[|h|+2J(1-\cos k)]}$)

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.

The Bogoliubov vacuum has original-boson occupation $v_k^2=({\alpha }_k/{ϵ}_k-1)/2$, reducing the ordered spin by its average over momenta. For the zero-field planar ferromagnetic chain, $v_k^2\sim 1/(2\sqrt{2}|k|)$. Even after excluding the exact uniform zero mode, the sum grows as $(2\pi \sqrt{2}{)}^{-1}\log L$. It invalidates a thermodynamic fixed-orientation expansion with finite dilute depletion; it does not represent a negative physical magnetization.