Thermal conductivity of a classical one-dimensional spin-phonon system

We investigate the thermal conduction properties of a one-dimensional lattice of atoms carrying classical spins and coupled vibrationally. The spin degrees of freedom interact via a classical Heisenberg interaction, while the vibrational degrees of freedom are coupled through nearest-neighbor linear as well as nonlinear forces. The thermal conductivity in spin-phonon systems has both a phononic as well as a magnetic contribution. We use extensive numerical simulations and evaluate the magnetic and phononic thermal current correlation functions as well as the combined thermal conductivity coefficient. We employ two distinct numerical approaches: The first is based on the linear response theory and proceeds through an evaluation of the energy current correlation function using the Green-Kubo formula. The second is through a simulation of the stochastic baths and a subsequent direct numerical evaluation of the magnetic and phononic heat currents. We find an anomalous thermal conductivity when the spins are coupled to a harmonic acoustic phonon chain. However, when the harmonic phonon chain contains, additionally, an optical mode, we find that the thermal conductivity is normal for a certain regime of on-site force parameters, while it becomes anomalous when the on-site frequency becomes larger than a certain value. Coupling thus to a harmonic system with an optical mode provides a case of tunable conductivity that switches from being diffusive to ballistic as a function of structural model parameters or of the temperature. When the spins are coupled to anharmonic chains, we find an anomalous conductivity when the phonon chain is acoustic, for instance, in the Fermi-Past-Ulam case, or a normal one when the nonlinearity is of optic type. For the cases analyzed, we provide quantitative information on the exponent characterizing the power law decay of the energy current correlation function and determine size and temperature dependencies of the conductivity coefficient. Finally, we also address the dependence of the thermal conductivity of spin-phonon chains on an externally applied magnetic field and find that in the harmonic case it generally increases with the field.