Emergent spinless Weyl semimetals between the topological crystalline insulator and normal insulator phases with glide symmetry

We construct a theory describing phase transitions between the spinless topological crystalline insulator phase with glide symmetry and a normal insulator phase. We show that a spinless Weyl semimetal phase should intervene between these two phases. Here, because all the bands are free from degeneracy in general, a gap closing between a single conduction band and a single valence band at phase transition generally gives rise to a pair creation of Weyl nodes; hence the Weyl semimetal phase naturally appears. We show the relationship between the change of the ${\mathbb{Z}}₂$ topological number when the system goes through the Weyl semimetal phase, and the trajectory of the Weyl nodes.