Why do phonons in many insulators show a thermal Hall effect?

In insulators including $\mathrm{SrTiO_3}$, cuprate parent compounds, Kitaev candidates, garnets, quartz, MgO and intrinsic Si and Ge, fields of tens of kG produce a thermal Hall conductivity $\kappa_{xy}$ carried by phonons, with thermal Hall angle $\kappa_{xy}/\kappa_{xx}$ up to $\sim 3 \times 10^{-3}\text{ at }9 \times 10^{4}\,\mathrm{G}$, typically peaking near the maximum of $\kappa_{xx}$. Identify the coupling: intrinsic Berry curvature of phonon bands from spin-lattice or ionic Lorentz-type coupling, skew scattering off charged or magnetic defects, or resonant scattering by dynamical defects. An answer predicts magnitude, sign and temperature dependence across materials.

A theory predicting $\kappa_{xy}/\kappa_{xx}$ with sign in several unrelated insulators from measured parameters, confirmed by controlled defect doping.