Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2015 / Paper 1 / 1 / a

Use $\hslash =1$ and $S_n^{\pm }=S_n^x\pm i{S}_n^y$. Operators on different sites commute, and expanding the two products gives $S_n^{+}{S}_{n+1}^{-}+S_n^{-}{S}_{n+1}^{+}=2(S_n^x{S}_{n+1}^x+S_n^y{S}_{n+1}^y)$. Therefore the quantum XX chain hasEach exchange moves a reversed spin between neighbours, conserving the total number of reversed spins. The factor of one half follows from defining $S^{\pm }$ without a normalizing square root.