Topological charge pumping by sliding moiré pattern

We theoretically study the adiabatic topological charge pumping driven by interlayer sliding in the moiré superlattices. We show that, when we slide a single layer of the twisted bilayer system relatively to the other, a moiré pattern flow and a quantized transport of electrons occurs.[1] When the Fermi energy is in a spectral gap, the number of pumped charges in the interlayer sliding process is quantized to a sliding Chern number, which is related to the interlayer sliding degree of freedom and obeys a Diophantine equation analogous to the quantum Hall effect [2]. We apply the argument to the twisted bilayer graphene, and find that energy gaps above and below the nearly-flat bands has non-zero sliding Chern numbers. When the Fermi energy is in either of those gaps, the slide-driven topological pumping occurs perpendicularly to the sliding direction.
[1] M. Fujimto, H. Koschke and M. Koshino, arXiv:1910.04654 (2019)
[2] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)