Quantum geometric exciton drift velocity

We show that the dipole moment of an exciton is uniquely determined by the quantum geometry of its eigenstates, and demonstrate its intimate connection with a quantity we call the dipole curvature. The dipole curvature arises naturally in semiclassical dynamics of an exciton in an electric field, adding a term additional to the anomalous velocity coming from the Berry's curvature. In a uniform electric field this contributes a drift velocity akin to that expected for excitons in crossed electric and magnetic fields, even in the absence of a real magnetic field. We compute the quantities relevant to semiclassical exciton dynamics for several interesting examples of bilayer systems with weak interlayer tunneling and Fermi energy in a gap, where the exciton may be sensibly described as a two-body problem. These quantities include the exciton dispersion, its dipole curvature, and Berry's curvature. For two gapped-graphene layers in a vanishing magnetic field, we find the dipole curvature vanishes if the layers are identical, but may be non-zero when the layers differ. We further analyze examples in the presence of magnetic fields, allowing us to examine cases involving graphene, in which a gap is opened by Landau level splitting. Heterostructures involving TMDs are also considered. In each case the dipole and/or Berry's curvatures play out differently. In some cases, the lowest energy exciton state is found to reside at finite momentum, with interesting possibilities for Bose condensation. We also find situations in which the dipole curvature increases monotonically with exciton momentum, suggesting that the quantum geometry can be exploited to produce photocurrents from initially bound excitons with electric fields. We speculate on further possible effects of the semiclassical dynamics in geometries where the constituent layers are subject to the same or different electric fields.