Low-temperature phonon dynamics of the one-dimensional XY model

We study the influence of the anharmonicity of the phonons, caused by the coupling of phonons and spins, on the dynamic structure factor of the quantum one-dimensional XY model. Our calculations are performed at low temperature using the memory function formalism. The spin variables are treated by means of the Villain's representation. In the classical limit we have obtained an analytical expression for the time-dependent memory function.