Topological quadrupolar semimetals

In this Rapid Communication we predict several types of topological semimetals that exhibit a bulk quadrupole moment. These semimetals are modeled with a three-dimensional extension of the two-dimensional 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 midgap 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.