Anomalous electrodynamics and quantum geometry in the Dirac-Harper model for a graphene bilayer

Graphene bilayers with layer antisymmetric strains are studied using the Dirac-Harper model for a pair of single-layer Dirac Hamiltonians coupled by a one-dimensional moire-periodic interlayer tunneling amplitude. This model hosts low-energy, nearly dispersionless bands near charge neutrality that support anomalous polarizations of its charge multipole distributions. These are analyzed introducing a generalized Berry curvature that encodes the field-induced dynamics of multipole fields allowed in a chiral medium with time-reversal symmetry. The formulation identifies a reciprocity relation between responses to layer-symmetric and layer-antisymmetric in-plane electric fields and reveals momentum-space quantum oscillations produced by a spatial pattern of band inversions on the moire scale.