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Topological semimetals and insulators in three-dimensional honeycomb materials

Semimetals, in which conduction and valence bands touch but do not form Fermi surfaces, have attracted considerable interest for their anomalous properties starting with the discovery of Dirac matter in graphene and other two-dimensional honeycomb materials. Here we introduce a family of three-dimensional honeycomb systems whose electronic band structures exhibit a variety of topological semimetals with Dirac nodal lines. We show that these nodal lines appear in varying numbers and mutual geometries, depending on the underlying lattice structure. They are stabilized, in most cases, by a combination of time-reversal and inversion symmetries and are accompanied by topologically protected ``drumhead'' surface states. In the bulk, these nodal line systems exhibit Landau level quantization and flat bands upon applying a magnetic field. In the presence of spin-orbit coupling, these topological semimetals are found to generically form (strong) topological insulators. This comprehensive classification of the electronic band structures of three-dimensional honeycomb systems might serve as guidance for future material synthesis.