Fourier vertex of a Schwinger boson chain ($H=-\frac{J}{L}{\sum }_{p,q,r,t}{\delta }_{p+q,r+t}^{(L)}\cos (q-t)A_p^{†}{B}_q^{†}{B}_r{A}_t$)

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.