Thermal drag revisited: Boltzmann versus Kubo

The effect of mutual drag between phonons and spin excitations on the thermal conductivity of a quantum spin system is discussed. We derive general expression for the drag component of the thermal current using Boltzmann equation as well as Kubo linear-response formalism to leading order in the spin-phonon coupling. We demonstrate that aside from higher-order corrections which appear in the Kubo formalism both approaches yield identical result for the drag thermal conductivity. We discuss the range of applicability of our result and provide a generalization of our consideration to the cases of fermionic excitations and to anomalous forms of boson-phonon coupling. Several asymptotic regimes of our findings relevant to realistic situations are highlighted.