Topological phase transitions with and without energy gap closing

TL;DR

This paper investigates topological phase transitions in 3D topological insulators with an exchange field, highlighting transitions that occur without energy gap closing but with spin spectrum gap closing, and identifies conditions for Weyl semimetal phases.

Contribution

It reveals that topological phase transitions can occur via spin spectrum gap closing without energy gap closing, and characterizes the transition to Weyl semimetals through spin Chern number analysis.

Findings

Transition from topological insulator to Weyl semimetal at critical exchange field

Surface states vanish when spin spectrum gap closes

Transition involves spin spectrum gap closing, not energy gap closing

Abstract

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.