Collective modes of a Wigner crystal with Berry curvature

Inspired by recent experiments in multilayer graphene, we consider the collective modes of a two-dimensional Wigner crystal with nonzero Berry curvature. The traditional Wigner crystal has independent longitudinal (plasmon) and transverse (shear) modes, but in the presence of Berry curvature these modes are mixed and hybridize. We derive the dispersion relation for these modes using both a long-wavelength hydrodynamic approach and a more exact calculation based on diagonalizing the elastic tensor. We find that Berry curvature has the effect of softening the shear mode and stiffening the plasmon mode. We also consider the effect of a magnetic field B, and among our findings is a remarkable divergence of the mode frequency at a particular value of B that is antialigned with the Berry curvature.