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. In returning Thouless pumping, the quantized charge is pumped during the first half of the cycle and returns to zero during the second. Here, authors demonstrate returning Thouless pumping experimentally with a symmetry-protected delicate topological insulator, made of a two-dimensional acoustic crystal.