  We investigate the thermal conductivity of the classical Heisenberg spin
chain coupled to a variety of phonon systems using a Green-Kubo approach as
well as coupling to heat reservoirs. The decay of energy current correlations
is power law or exponential, depending on the type of (an)harmonicity of the
phonon system and the presence of a substrate. In particular, we find that
coupling the spin system to displacements with an acoustic type harmonic
inter-site potential leads to a diverging thermal conductivity. Adding a
substrate, harmonic or periodic such as sine-Gordon, drives the system to
diffusive behavior. With anharmonic potential without a substrate the generic
transport is diffusive even for the integrable Toda potential, while for the
FPU chain is ballistic.
