The modern theory of charge polarization in solids is based on a generalization of Berrys phase. Its possible quantization lies at the heart of our understanding of all systems with topological band structures that were discovered over the last decades. While based on the concept of the charge polarization, the same theory can be used as an elegant tool to characterize the Bloch bands of neutral bosonic systems such as photonic or phononic crystals. Recently, the theory of this quantized polarization was extended from the dipole- to higher multipole-moments. In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge-modes, which in turn stabilize zero-dimensional in-gap corner states. However, such a state of matter has not been observed experimentally. Here, we provide the first measurements of a phononic quadrupole insulator. We experimentally characterize the bulk, edge, and corner physics of a mechanical metamaterial and find the predicted gapped edge and in-gap corner states. We further corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases predicted by the quadrupole theory. From an application point of view, these topological corner states are an important stepping stone on the way to topologically protected wave-guides in higher dimensions and thereby open a new design path for metamaterials.
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