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 ${\mathrm{BaMnSb}}_2$.

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• Berry curvature
• Dirac fermions
• Electron-phonon coupling
• Phonons
• Topological insulators
• Perturbation theory