Thermal conductivity and theory of inelastic scattering of phonons by collective fluctuations

We study the intrinsic scattering of phonons by a general quantum degree of freedom, i.e. a fluctuating "field" $Q$, which may have completely general correlations, restricted only by unitarity and translational invariance. From the induced scattering rates, we obtain the consequences on the thermal conductivity tensor of the phonons. We find that the lowest-order diagonal scattering rate, which determines the longitudinal conductivity, is controlled by two-point correlation functions of the $Q$ field, while the off-diagonal scattering rates involve a minimum of three to four point correlation functions. We obtain general and explicit forms for these correlations which isolate the contributions to the Hall conductivity, and provide a general discussion of the implications of symmetry and equilibrium. We evaluate these two- and four-point correlation functions and hence the thermal transport for the illustrative example of an ordered two dimensional antiferromagnet. In this case the $Q$ field is a composite of magnon operators arising from spin-lattice coupling. A numerical evaluation of the required integrals demonstrates that the results satisfy all the necessary symmetry restrictions but otherwise lead to non-vanishing scattering and Hall effects, and in particular that this mechanism leads to comparable thermal Hall conductivity for thermal currents within and normal to the plane of the antiferromagnetism.