Observation of returning Thouless pumping

Introduced by David Thouless in 1983, Thouless pumping is a driving mechanism for topological systems where the pumped charge is quantized by the Chern number. The recent theoretical demonstration of returning Thouless pumping describes a system where the quantized charge pumped during the first half of the cycle returns to zero in the second half. This mechanism leads to crystalline symmetry-protected delicate topological insulators that, unlike conventional topological bands, are not atomically obstructed and can be described by Wannier functions. More precisely, delicate topologies feature multicellular Wannier functions, extending beyond a single unit cell. Here, by using adding synthetic dimension, we realize a two-dimensional delicate topological insulator consisting of a set of one-dimensional acoustic crystals with finely tuned geometric parameters. Measuring acoustic bands and wavefunctions, we directly observe returning Thouless pumping and symmetric multicellular Wannier functions and establish a bulk-boundary correspondence between Chern numbers of the sub-Brillouin zone and gapless boundary modes. Our experimental demonstration of returning Thouless pumping expands the current understanding of topological phases of matter, enriching it with crystalline symmetries.