Spin-wave damping in the two-dimensional ferromagnetic $\mathrm{XY}$ model

$〈{\mathrm{LS}}_{\mathbf{q}}^{\perp }{\mathrm{LS}}_{-\mathbf{q}}^{\perp }〉,$ which is related to the effect of damping of spin waves in a two-dimensional classical ferromagnetic $\mathrm{XY}$ model, is considered. The damping rate ${\Gamma }_{\mathbf{q}}$ 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. ${\Gamma }_{\mathbf{q}}$ is included into the memory function formalism due to Reiter and Solander, and Menezes, to determine the dynamic structure function $S(\mathbf{q},\omega ).\mathrm{}$ 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 $2\pi /L.\mathrm{}$ 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.