Topological charge pumping by a sliding moire pattern

We study the adiabatic topological charge pumping driven by interlayer sliding in the moire superlattices. We show that, when we slide a single layer of the twisted bilayer system relatively to the other, a moire pattern flow and a quantized transport of electrons occur. The number of pumped charges is quantized to a sliding Chern number, which is related to the interlayer sliding degree of freedom. In the twisted bilayer graphene, the topological pumping current flows perpendicularly to the sliding direction when the Fermi energy is in the energy gap above or below the nearly-flat bands