Spin-wave damping in the two-dimensional ferromagnetic XY model

[(LSqLS-qperpendicular to)-L-perpendicular to], which is related to the effect of damping of spin waves in a two-dimensional classical ferromagnetic XY model, is considered. The damping rate Gamma (q) is calculated using the leading diagrams due to the quartic-order deviations from the harmonic spin Hamiltonian. The resulting four-dimensional integrals are evaluated by extending the techniques developed by Gilat and others for spectral density types of integrals, Gamma (q) is included into the memory function formalism due to Reiter and Solander, and Menezes, to determine the dynamic structure function S(q,omega). For the infinite sized system, the memory function approach is found to give nondivergent spin-wave peaks, and a smooth nonzero background intensity (plateau or distributed intensity) for the whole range of frequencies below the spin-wave peak. The background amplitude relative to the spin-wave peak rises with temperature, and eventually becomes higher than the spin-wave peak, where it appears as a central peak. For finite-sized systems, there are multiple sequences of weak peaks on both sides of the spin-wave peaks whose number and positions depend on the system size and wave vector in integer units of 2 pi /L. These dynamical finite-size effects are explained in the memory function analysis as due to either spin-wave difference processes below the spin-wave peak or sum processes above the spin-wave peak. These features are also found in classical Monte Carlo-spin-dynamics simulations.