 Quantum XY chain

A one-dimensional quantum spin model with transverse XY exchange interactions; $J>0$ is the ferromagnetic sign in the displayed convention. Unlike the classical XY model, its site variables are noncommuting spin operators in a fixed irreducible spin representation. With periodic boundary conditions and $L\ge 2$, $H=-\frac{J}{2}{\sum }_n(S_n^{+}{S}_{n+1}^{-}+S_n^{-}{S}_{n+1}^{+})$. It preserves total $S^z$ and the associated global U(1) symmetry.