Gapless hinge states from adiabatic pumping of axion coupling

We demonstrate that gapless chiral hinge states naturally emerge in insulating crystals undergoing a slow cyclic evolution that changes the Chern-Simons axion coupling θ by 2π. This happens when the surface (not just the bulk) returns to its initial state at the end of the cycle, in which case it must pass through a metallic state to dispose of the excess quantum of surface anomalous Hall conductivity pumped from the bulk. If two adjacent surfaces become metallic at different points along the cycle, there is an interval where they are in topologically distinct insulating states, with gapless chiral modes propagating along the connecting hinge. We illustrate these ideas for a 3D tight-binding model consisting of coupled Haldane-model layers. The surface topology is determined in a slab geometry using two different markers, surface anomalous Hall conductivity and surface polarization, and we find that both correctly predict the appearance of gapless hinge states in a rod geometry.