Thermal Conductivity and Theory of Inelastic Scattering of Phonons by Collective Fluctuations

We study the coupling of phonons to any general operator field and its consequences on the thermal conductivity of phonons. Using a scattering approach, we find that the lowest-order diagonal scattering rate, which determines the longitudinal conductivity, is controlled by two-point correlation functions of the collective fluctuations the phonons couple to, while the off-diagonal scattering rates involve a minimum of four-point correlation functions. We take up the challenge of computing analytically these two- and four-point correlation functions in an ordered antiferromagnet and fermionic spinon spin liquid, hence providing expressions for the longitudinal and Hall conductivities in such systems. By explicit numerical computation of the phonon conductivity tensor from a given realistic interacting model of spins, we relate thermal conduction properties to features of the spin dynamics and the magnetoelastic coupling.