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 $C^{\pm }(k_z)$ with momentum $k_z$ as a parameter. When $\mathrm{\mid }g\mathrm{\mid }$ 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 $\mathrm{\mid }g\mathrm{\mid }<g_c$, $C^{\pm }(k_z)$ undergo a transition from $\pm 1$ to 0 with increasing $\mathrm{\mid }{k}_z\mathrm{\mid }$ to a critical value $k_z^C$. Correspondingly, surface states exist for $\mathrm{\mid }{k}_z\mathrm{\mid }<k_z^C$, and vanish for $\mathrm{\mid }{k}_z\mathrm{\mid }\ge k_z^C$. The transition at $\mathrm{\mid }{k}_z\mathrm{\mid }=k_z^C$ is acompanied by closing of spin spectrum gap rather than energy gap.