Observations of acoustic Wannier configurations revealing topological corner anomaly.

Wannier configuration,which is the real-space distribution of the spatial locations of symmetric Wannier functions(SWFs),pro-vides a new viewpoint for the wave motions in crystalline struc-tures.In Wannier configurations,the deviation of the spatial location of SWF from the unit-cell center often indicates the exis-tence of nontrivial boundary signatures.Moreover,the excess charges extracted from Wannier configuration constitute genuine topological indicators in real space[1,2].In two-dimensional higher-order topological insulators(HOTIs)[3-5],the scheme of Wannier configurations can manifest fractional charges to reveal the corner modes even when these modes emerge in the bulk bands.Hence,Wannier configurations are of particular importance in classical metamaterials due to the lack of the chiral or particle-hole symmetry which is essential to pin the corner modes into the mid-gap[6].In optical topological systems,Wannier configura-tions have been utilized to detect the topological crystalline discli-nation[7],experimentally verify the emergence of half quantum number[8],and characterize hierarchical bulk topology in a sys-tem without complete bandgap[9].