  The mathematical field of topology has become a framework to describe the
low-energy electronic structure of crystalline solids. A typical feature of a
bulk insulating three-dimensional topological crystal are conducting
two-dimensional surface states. This constitutes the topological bulk-boundary
correspondence. Here, we establish that the electronic structure of bismuth, an
element consistently described as bulk topologically trivial, is in fact
topological and follows a generalized bulk-boundary correspondence of
higher-order: not the surfaces of the crystal, but its hinges host
topologically protected conducting modes. These hinge modes are protected
against localization by time-reversal symmetry locally, and globally by the
three-fold rotational symmetry and inversion symmetry of the bismuth crystal.
We support our claim theoretically and experimentally. Our theoretical analysis
is based on symmetry arguments, topological indices, first-principle
calculations, and the recently introduced framework of topological quantum
chemistry. We provide supporting evidence from two complementary experimental
techniques. With scanning-tunneling spectroscopy, we probe the unique
signatures of the rotational symmetry of the one-dimensional states located at
step edges of the crystal surface. With Josephson interferometry, we
demonstrate their universal topological contribution to the electronic
transport. Our work establishes bismuth as a higher-order topological
insulator.
