Graphene, a two-dimensional (2D) carbon sheet, acquires many of its amazing properties from the Dirac point nature of its electronic structures with negligible spin-orbit coupling. Extending to 3D space, graphene networks with negative curvature, called Mackay-Terrones crystals (MTCs), have been proposed and experimentally explored, yet their topological properties have yet to be discovered. Based on the first-principle calculations, we report an all-carbon MTC with topologically nontrivial electronic states by exhibiting node lines in bulk. When the node lines are projected onto surfaces to form circles, “drumhead”-like flat surface bands nestled inside of the circles are formed. The bulk node line can evolve into a 3D Dirac point in the absence of inversion

symmetry, the existence of which has been shown to be plausible in recent experiments.

http://dx.doi.org/10.1103/PhysRevB.92.045108