Topologically quantized current in quasiperiodic Thouless pumps

Thouless pumps are topologically nontrivial states of matter with quantized charge transport, which are analogous to the quantum Hall state and can be realized by ultracold atomic gases in optical lattices. However, contrarily to the exact and extremely precise quantization of the Hall conductance, the pumped charge is quantized only when the pumping time is a multiple of a characteristic time scale, i.e., the pumping cycle duration. Here we consider the quasiperiodic regime realized via the superposition of two distinct optical lattices with incommensurate wavelengths. We show that in this case the Bloch bands and the Berry curvature become flat, the pumped charge becomes linear in time, while the current becomes steady, topologically quantized, and proportional to the Chern number, independently from the pumping time[1]. The current quantization is exact up to exponentially small corrections. This has to be contrasted with the case of the non-quasiperiodic regime, where the current is not constant, and the pumped charge is quantized only at integer multiples of the pumping cycle.
[1] Marra, Nitta, arXiv:2001.11022