Separation of topological insulator and Dirac semimetal phases in ABX honeycomb structures

Topological insulators (TI) and Dirac semimetals (DSM) are quantum materials that have band inversion at time reversal invariant k-points. The difference is that TI has non-vanishing bulk band gap and surface Dirac cones protected by time reversal symmetry, whereas DSM has bulk Dirac cones protected by crystalline symmetry. Numerous previous theoretical studies of ABX compounds in the honeycomb lattice (space group P6₃/mmc) predicted many to be TIs, but the present examination of the dispersion relation in other areas of the Brillouin Zone reveals that (i) most are instead DSM (ii) We also predict a few true TI’s with good size gaps, and (iii) extending this study to hundreds of (real as well as hypothetical) ABX honeycomb structures, we separate them to be normal insulators vs metals, vs TI vs DSM. We discuss the chemical and physical rules that determine whether a band-inverted ABX honeycomb is a topological insulator or Dirac semimetal.