Phonons in electron crystals with Berry curvature

When electrons in clean two-dimensional materials experience strong Coulomb interactions, they freeze into a crystal. However, when these electrons originate from bands with nonzero Berry curvature—the quantum-geometric property responsible for anomalous Hall effects—the resulting “topological crystal” displays dramatically distinct properties. In this work, we investigate how these crystals vibrate. We derive the general, quantitative theory of those vibrations, identifying a previously overlooked “kineo-elastic coupling” that links lattice deformations to its momentum. Applying the framework to a topological crystal in rhombohedral multilayer graphene, we find that the kineo-elastic coupling results in a striking anisotropy in the phonon velocity. Our work points to the wealth of phenomena that can arise when electrons crystallize in the presence of quantum geometry.