Model Hamiltonian for hexagonal ABC topological semimetals

We introduce a lattice regularized k.p model Hamiltonian for a broad class of hexagonal ABC materials with space group P6₃mc. First-principles electronic band structure calculations have shown that specific members of this class realize Dirac, Weyl, or nodal line semimetals. By deriving a model Hamiltonian for the relevant low-energy bands, formulated in terms of two effective j=3/2 quartets related by the screw rotation, we show that the nature of the electronic structure of these materials can be understood from a conceptually unified picture. In particular, we elucidate the origin of the topological semi-metallic phases, and relate the parameters of our model to the properties of the ABC materials (e.g. crystal field splitting, spin-orbit coupling, inversion asymmetry). Since the hexagonal ABC structures lack inversion symmetry, they are expected to show a nonlinear optical response. We employ our model Hamiltonian to address second harmonic generation and photovoltaic effects in ABC materials, and interpret the response in terms of band structure topology.