  In this work we predict several new types of topological semimetals that
exhibit a bulk quadrupole moment. These semimetals are modeled with a 3D
extension of the 2D quadrupole topological insulator. One type of semimetal has
bulk nodes and gapped, topological surfaces. A second type, which we may call a
higher order topological semimetal, has a gapped bulk, but harbors a Dirac
semimetal with an even number of nodes on one or more surfaces. The final type
has a gapped bulk, but harbors half of a Dirac semimetal on multiple surfaces.
Each of these semimetals gives rise to mid-gap hinge states and hinge charge,
as well as surface polarization, which are all consequences of a bulk
quadrupole moment. We show how the bulk quadrupole moments of these systems can
be calculated from the momentum-locations of bulk or surface nodes in the
energy spectrum. Finally, we illustrate that in some cases it is useful to
examine nodes in the Wannier bands, instead of the energy bands, to extract the
bulk quadrupole moment.

Mesoscale and Nanoscale Physics