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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 spinorbit 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.

In recent years it has become clear that electronic Berry curvature (BC) is a key concept to understand and predict physical properties of crystalline materials. A wealth of interesting Hall-type responses in charge, spin and heat transport are caused by the BC associated to electronic bands inside a solid: anomalous Hall effects in magnetic materials 1, 2 , and various nonlinear Hall 3-5 and Nernst effects 6, 7 in non-magnetic systems that lack inversion symmetry. However, for the largest class of known materials -non-magnetic ones with inversion symmetry-electronic BC is strictly zero 8,9 . Here we show that precisely for these bulk BCfree materials, a finite BC can emerge at their surfaces and interfaces. This immediately activates certain surfaces in producing Hall-type transport responses. We demonstrate this by first principles calculations of the BC at bismuth, mercury-telluride (HgTe) and rhodium surfaces of various symmetries, revealing the presence of a surface Berry curvature dipole and associated quantum nonlinear Hall effects at a number of these. This opens up a plethora of materials to explore and harness the physical effects emerging from the electronic Berry curvature associated exclusively to their boundaries.

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