Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 2 / c

Expand around the $+z$ polarized ground state. With $n_j=a_j^{†}{a}_j$ fixed as $S\to \infty$, the Holstein–Primakoff transformation gives $S_j^{+}=\sqrt{2S}{a}_j+O(S^{-1/2})$, $S_j^{-}=\sqrt{2S}{a}_j^{†}+O(S^{-1/2})$ and $S_j^z=S-n_j$. The longitudinal exchange contributesUseThe transverse exchange then gives $-JS(a_{j+1}^{†}{a}_j+a_j^{†}{a}_{j+1})$, and the Dzyaloshinskii-Moriya interaction gives $-iDS(a_{j+1}^{†}{a}_j-a_j^{†}{a}_{j+1})$. Discarding the constant $-LJ(1+u)S^2$ and all $O(S^0)$ interactions yieldsHere periodic boundary conditions make each $n_j$ occur twice in the longitudinal sum. The approximation is the linear spin-wave approximation, controlled by small local magnon occupation compared with $S$.