Topological phase transitions with and without energy gap closing

Topological phase transitions in a three-dimensional (3D) topological insulator (TI) with an exchange field of strength g are studied by calculating spin Chern numbers C ^± ( k _z ) with momentum k _z as a parameter. When | g | exceeds a critical value g _C , a transition of the 3D TI into a Weyl semimetal occurs, where two Weyl points appear as critical points separating k _z regions with different first Chern numbers. For | g |