Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 81 / 1 / d / Solution

The linear spin-wave approximation expands about the rotated all-up state, with ${\hat{n}}_n=a_n^{†}{a}_n$:In the longitudinal product, $S_n^z{S}_{n+1}^z=S^2-S({\hat{n}}_n+{\hat{n}}_{n+1})+O(S^0)$. Each site belongs to two bonds, so the resulting quadratic Hamiltonian isThe omitted terms are of order $S^0$ at fixed small occupation number. The pair terms describe the quantum fluctuations that were missing from the classical Néel state.