Nodal points in three-dimensional carbon networks without inversion symmetry

Three-dimensional (3D) carbon networks provide a great platform to search and investigate various topological phases. A lot of nodal-line and nodal-surface topological phases have been found in the 3D carbon networks. However, nodal points are few in them because most carbon networks have high symmetries including inversion symmetry. Here, we propose that nodal points can be generated in low-symmetry carbon structures without inversion symmetry. Two sp(2)-hybridized 3D carbon networks, consisting of distorted six-membered rings and a double helix, respectively, are used as examples to show nodal points. There are several pairs of nodal points locating on the high-symmetry paths of the first Brillouin zone. These nodal points are Dirac points. However, they can also be treated as Weyl points because of the negligibly small spin-orbit coupling in the carbon structures. These nodal points only shift in the momentum space but do not disappear when perturbations such as strains are applied on the carbon structures. We have also studied the topological surface states and found Fermi arcs linking the nodal points. In addition, the synthesis of two carbon networks is discussed.