Inherited Berry curvature of phonons in Dirac materials with time-reversal symmetry

The Berry curvature of phonons is an active subject of research in condensed matter physics. Here, we present a model in which phonons acquire a Berry curvature through their coupling to electrons in crystals with time-reversal symmetry. We illustrate this effect for ${\mathrm{BaMnSb}}_{2}$, a quasi-two-dimensional Dirac insulator, whose low-energy massive Dirac fermions generate a phonon Berry curvature that is proportional to the electronic valley Chern number. We estimate the contribution of the phonon Berry curvature to the thermal Hall conductivity in ${\mathrm{BaMnSb}}_{2}$.