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

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