  High-order topological insulators (TIs) are a family of recently-predicted
topological phases of matter obeying an extended topological bulk-boundary
correspondence principle. For example, a two-dimensional (2D) second-order TI
does not exhibit gapless one-dimensional (1D) topological edge states, like a
standard 2D TI, but instead has topologically-protected zero-dimensional (0D)
corner states. So far, higher-order TIs have been demonstrated only in
classical mechanical and electromagnetic metamaterials exhibiting quantized
quadrupole polarization. Here, we experimentally realize a second-order TI in
an acoustic metamaterial. This is the first experimental realization of a new
type of higher-order TI, based on a breathing Kagome lattice, that has zero
quadrupole polarization but nontrivial bulk topology characterized by quantized
Wannier centers (WCs). Unlike previous higher-order TI realizations, the corner
states depend not only on the bulk topology but also on the corner shape; we
show experimentally that they exist at acute-angled corners of the Kagome
lattice, but not at obtuse-angled corners. This shape dependence allows corner
states to act as topologically-protected but reconfigurable local resonances.
