Band crossings in honeycomb-layered transition metal compounds

Two-dimensional electron dispersions with peculiar band crossings provide a platform for realizing topological phases of matter. Here we theoretically show that the $e_g$-orbital manifold of honeycomb-layered transition metal compounds accommodates a plethora of peculiar band crossings, such as multiple Dirac point nodes, quadratic band crossings, and line nodes. From the tight-binding analysis, we find that the band topology is systematically changed by the orbital dependent transfer integrals on the honeycomb network of edge-sharing octahedra, which can be modulated by distortions of the octahedra as well as chemical substitutions. The band crossings are gapped out by the spin-orbit coupling, which brings about a variety of topological phases distinguished by the spin Chern numbers. The results provide a comprehensive understanding of the previous studies on various honeycomb compounds. We also propose another candidate materials by ab initio calculations.