Berry electrodynamics: Anomalous drift and pumping from a time-dependent Berry connection

The Berry connection and curvature encode the geometry of a particle's wave functions, and they can modify the motion of particles in a nontrivial manner. They are also dual to the vector potential and magnetic field. A little-explored avenue is what happens when the Berry connection changes with time. This must have an effect on the anomalous velocity of particles, just as the time dependence of the vector potential produces an electric field in standard electrodynamics. In this work, the authors outline how these `Berry-electrodynamics' effects can be measured, how they relate to shift currents, and how both could be put together to produce a pumping cycle in honeycomb lattices.