Phonon Thermal Hall Conductivity from Scattering with Collective Fluctuations

Because electrons and ions form a coupled system, it is a priori clear that the dynamics of the lattice should reflect symmetry breaking within the electronic degrees of freedom. Recently, this has been clearly evidenced for the case of time-reversal and mirror symmetry breaking by observations of a large phononic thermal Hall effect in many strongly correlated electronic materials. However, the mechanism by which time-reversal breaking and chirality is communicated to the lattice is far from evident. In this paper, we discuss how this occurs via many-body scattering of phonons by collective modes: a consequence of nonGaussian correlations of the latter modes. We derive fundamental new results for such skew (i.e., chiral) scattering and the consequent thermal Hall conductivity. We emphasize that these results apply to any collective variables in any phase of matter: electronic, magnetic, or neither; highly fluctuating and correlated, or not. As a proof of principle, we compute general formulas for the above quantities for ordered antiferromagnets. From the latter, we obtain the scaling behavior of the phonon thermal Hall effect in clean antiferromagnets. The calculations show several different regimes and give quantitative estimates of similar order to that seen in recent experiments.