  The geometric structure of an energy band in a solid is fundamental for a
wide range of many-body phenomena in condensed matter and is uniquely
characterized by the distribution of Berry curvature over the Brillouin zone.
In analogy to an Aharonov-Bohm interferometer that measures the magnetic flux
penetrating a given area in real space, we realize an atomic interferometer to
measure Berry flux in momentum space. We demonstrate the interferometer for a
graphene-type hexagonal lattice, where it has allowed us to directly detect the
singular $\pi$ Berry flux localized at each Dirac point. We show that the
interferometer enables one to determine the distribution of Berry curvature
with high momentum resolution. Our work forms the basis for a general framework
to fully characterize topological band structures and can also facilitate
holonomic quantum computing through controlled exploitation of the geometry of
Hilbert space.
