Interaction driven transverse thermal resistivity in a phonon gas

The amplitude of the Hall response of electrons can be understood without invoking interactions. Most theories of the phonon thermal Hall effect have likewise opted for a non-interacting picture. Here, we challenge this approach. Our study of WS$_2$, a transition metal dichalcogenide (TMD) insulator, finds that longitudinal, $\kappa_{xx}$, and transverse, $\kappa_{xy}$, thermal conductivities peak at almost the same temperature. Their ratio obeys an upper bound, as in other insulators. We then compare transverse thermal transport in a phonon gas and in a molecular gas. In the latter, the Senftleben-Beenakker effect is driven by the competition between molecular collisions and applied magnetic field in setting the distribution of molecular angular momenta. An off-diagonal transport response arises thanks to interactions between non-spherical particles, which do not need to be chiral. By analogy, we argue that in a phonon gas, magnetic field will influence phonon-phonon interactions, and generates a transverse thermal \emph{resistivity}, whose order of magnitude can be accounted for by invoking a Berry force on the drift velocity of the nuclei in the presence of a finite heat. This simple picture gives a reasonable account of the experimentally measured transverse thermal resistivity of seven different crystalline insulators.