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±(kz) with momentum kz as a parameter. When |g| exceeds a critical value gC, a transition of the 3D TI into a Weyl semimetal occurs, where two Weyl points appear as critical points separating kz regions with different first Chern numbers. For |g| < gC, C±(kz) undergo a transition from ±1 to 0 with increasing |kz| to a critical value kCz. Correspondingly, surface states exist for |kz| < kCz, and vanish for |kz| kCz. The transition at |kz| = kCz is accompanied by closing of the bulk spin spectrum gap rather than the energy gap.