Observation of returning Thouless pumping

Introduced by David Thouless in 1983, Thouless pumping exemplifies topological properties in topological systems, where the transported charge is quantized by the Chern number. Recently, returning Thouless pumping was theoretically proposed, in which quantized charge is pumped during the first half of the cycle but returns to zero in the second half. This mechanism leads to crystalline symmetry-protected delicate topological insulators. Unlike conventional topological bands, a delicate topological band is Wannierizable but not atomically obstructed, which features multicellular Wannier functions extending beyond a single unit cell. Here, by replacing the second dimension with a synthetic dimension, we realize a two-dimensional delicate topological insulator via a set of one-dimensional acoustic crystals with fine-tuned geometric parameters. Through acoustic bands and wavefunction measurements, we directly observe returning Thouless pumping and symmetric multicellular Wannier functions, followed by establishing the bulk-boundary correspondence between sub-Brillouin zone Chern numbers and gapless boundary modes. As enriched by crystalline symmetries, our experimental demonstration of returning Thouless pumping expands the current understanding of topological phases of matter.