Non-necessity of band inversion process in two-dimensional topological insulators for bulk gapless states and topological phase transitions

In commonly employed models for two-dimensional (2D) topological insulators, bulk gapless states are well known to form at the band inversion points where the degeneracy of the states is protected by symmetries. It is thus sometimes quite tempting to consider this feature, the occurrence of gapless states, a result of the band inversion process under protection of the symmetries. Similarly, the band inversion process might even be perceived as necessary to induce 2D topological phase transitions. To clarify these misleading perspectives, we propose a simple model with a flexible Chern number to demonstrate that the bulk gapless states emerge at the phase boundary of topological phase transitions, despite the absence of a band inversion process. Furthermore, the bulk gapless states do not need to occur at the special $k$ points protected by symmetries. Given the significance of these fundamental conceptual issues and their widespread influence, our clarification should generate strong general interests and significant impacts. Furthermore, the simplicity and flexibility of our general model with an arbitrary Chern number should prove useful in a wide range of future studies of topological states of matter.

Figure 1

Illustration of occurrence of the bulk gapless states during the band inversion process in typical models. The dispersion of $d_3\left(k\right)$ (dashed lines) makes clear that as $M$ moves from negative (a) through zero (c) to positive (e), the bands become inverted in the yellow region near $k=0$. While the gap opens up upon introduction of the off-diagonal coupling $d_1$ and $d_2$ in (b) and (d), bulk gapless states are preserved at $M=0$ (c) due to a priori absence of coupling at $k=0$ in the models (presumably protected by some unspecified symmetry).

Figure 2

Illustration of pseudospins (upper panels) and band dispersion in the 2D $k$ space (middle panels) and along the ($k_x$,0) path (lower panels), across the $(n,n^{\prime })=(0,1)$ topological phase transition from $C=0$ (two left panels) to $C=1$ (two right panels) across the phase boundary at ${\theta }_c$ (center panels). No band inversion process takes place, since with a fixed $M=10$ the bands remain inverted inside the $d_3=0$ contour (dashed lines in the upper panels and the yellow region in the lower panels). Blue (red) arrows represent the $xy$ components of the pseudospins having positive (negative) $z$ components. At the phase boundary $\theta ={\theta }_c$, ${\vec{k}}_D$ denotes the location of the nonanalytic $\pi$ jumps of pseudospins, where the bulk gapless states appear.

Figure 3

Same as Fig. 2 but with $(n,n^{\prime })=(1,3)$.