Feature-energy duality of topological boundary states in multilayer quantum spin Hall insulator

TL;DR

This paper explores the relationship between topological boundary states and feature spectrum topology in multilayer quantum spin Hall insulators, revealing a feature-energy duality and identifying bismuth bromide as a promising material.

Contribution

It introduces the concept of feature-energy duality in topological boundary states and demonstrates its connection to the spin Chern number in multilayer quantum spin Hall insulators.

Findings

Feature-energy duality links edge states in E-k map and feature spectrum.

Bismuth bromide (Bi4Br4) identified as a candidate quantum spin Hall insulator.

The aggregate number of gapless edge states equals the spin Chern number.

Abstract

Gapless topological boundary states characterize nontrivial topological phases arising from the bulk-boundary correspondence in symmetry-protected topological materials, such as the emergence of helical edge states in a two-dimensional $\mathbb{Z}_2$ topological insulator. However, the incorporation of symmetry-breaking perturbation terms in the Hamiltonian leads to the gapping of these edge bands, resulting in missing these crucial topological boundary states. In this work, we systematically investigate the robustness of bulk-boundary correspondence in the quantum spin Hall insulator via recently introduced feature spectrum topology. Our findings present a comprehensive understanding of feature-energy duality, illustrating that the aggregate number of gapless edge states in the energy-momentum ($\it{E-k}$) map and the non-trivial edge states in the $\hat{S}_z$ feature spectrum equals the spin Chern number of multilayer quantum spin Hall insulator. We identify a van der Waals material bismuth bromide $\rm(Bi_4Br_4)$ as a promising candidate through first-principles calculations. Our work not only unravels the intricacies of bulk-boundary correspondence but also charts a course for exploring quantum spin Hall insulators with high spin-Chern number.