Dynamical Spin Susceptibility in the $t$–$J$ Model in the Superconducting Phase

Dynamical spin susceptibility is calculated for the $t$–$J$ model in the superconducting phase using the memory function method in terms of the Hubbard operators. The self-consistent system of equations for the memory function is obtained within the mode-coupling approximation. Both itinerant hole excitations and localized spin fluctuations contribute to the memory function. Moreover, the itinerant contribution itself consists of two parts, i.e., the contribution of Bogoliubov quasiparticles and that of Cooper pairs. The spin dynamics is diffusive in the hydrodynamic limit, but the itinerant part does not contribute to the spin diffusion. In the high frequency region, spin-wave-like excitations continue to exist. We discuss our analytic results in the light of neutron scattering experiments performed on the cuprate superconductors.