  The modern theory of charge polarization in solids is based on a
generalization of Berry's 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.
