Thouless pumps and universal geometry-induced drift velocity in multi-sliding quasi-periodic lattices

Abstract

Quantized Thouless pumps in periodic systems, set by Chern numbers or Wannier-center winding, is by now fairly well established, whereas its quasi-periodic extensions still require further clarification. Here, we develop a general quantitative paradigm for bulk Thouless pumps in continuous models with spacetime quasi-periodicity, applicable to arbitrary spatial dimensions. Within this framework, the bulk pumping turns out to be governed by an emergent long wave-length effective potential. Based on this mechanism, we obtain our main result a universal relation between topological drifting and the geometry of quasi Brillouin zone. Reduced to periodic systems, our result gives an explicit and compact formula which enables us to directly calculate Chern numbers by microscopic data. These proposals are corroborated by simulations of one- and two-dimensional continuous moir\'e-type spacetime quasi-periodic lattices, which exhibit stable, localized, directional drift in excellent agreement with the theory.