Quantum XY chain ($H=-J{\sum }_n(S_n^x{S}_{n+1}^x+S_n^y{S}_{n+1}^y)$)

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.

With the Schwinger boson representation and $a_n=L^{-1/2}{\sum }_q{e}^{-iqn}{A}_q$, the Hamiltonian operator of a periodic quantum XY chain takes the displayed form. The four inverse discrete Fourier transforms and site sum give $1/L$ and lattice momentum conservation. The forward and reverse nearest-neighbour exchange phases are $e^{i(q-t)}$ and $e^{-i(q-t)}$, giving the cosine. All momenta are defined modulo $2\pi$; neighbouring sites must be distinct for the stated normal ordering.