Emergence of topological semimetals in gap closing in semiconductors without inversion symmetry

A band gap for electronic states in crystals governs various properties of solids, such as transport, optical and magnetic properties. Its estimation and control have been an important issue in solid state physics. The band gap can be controlled externally by various parameters, such as pressure, atomic compositions and external field. Sometimes, the gap even collapses by tuning some parameter. In the field of topological insulators, such closing of the gap at a time-reversal invariant momentum indicates a band inversion, i.e. it leads to a topological phase transition from a normal insulator to a topological insulator. Here we show that the gap losing in inversion-asymmetric crystals is universal, in the sense that the gap closing always leads either to a Weyl semimetal or a nodal-line semimetal, from an exhaustive study on possible space groups. We here consider three-dimensional spinful systems with time-reversal symmetry. The space group of the system and the wavevector at the gap closing uniquely determine which possibility occurs and where the gap-closing points or lines lie in the wavevector space after closing of the gap. In particular, we show that an insulator-to-insulator transition never happens, which is in sharp contrast with inversion-symmetric systems.