Stably protected gapless edge states without Wannier obstruction

Abstract

Bulk-boundary correspondence serves as an important feature of topological insulators, including Chern insulators and ${\mathbb{Z}}_{2}$ topological insulators. These topological insulators have spectral flow in the Wilson-loop spectrum, and they encounter Wannier obstruction. Recent studies show that the bulk topological features may not imply the existence of protected gapless boundary states. Here we address the opposite question: Does the existence of stably protected gapless edge states necessarily imply Wannier obstruction or Wilson-loop winding? We provide an example in which stably protected gapless edge states arise without the aforementioned bulk topological features. This trivialized topological insulator belongs to a new class of systems with non-$\ensuremath{\delta}$-like Wannier functions. Interestingly, the gapless edge states are not protected by crystalline symmetry; instead, the protection originates from mirror antisymmetry, a combination of chiral and mirror symmetries. Although the protected gapless edge states cannot be captured by bulk topological features, they can be characterized by the spectral flow in the entanglement spectrum.