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

Gapless topological boundary states characterize nontrivial topological phases that arise from the bulk-boundary correspondence in symmetry-protected topological materials. However, symmetry-breaking perturbations gap these edge bands, resulting in the loss of these crucial boundary states. In this connection, we systematically examine the robustness of the bulk-boundary correspondence in the case of a quantum spin Hall insulator via the feature-spectrum topology approach. Our findings provide a comprehensive understanding of feature-energy duality and show that the aggregate number of gapless edge states in the energy-momentum (E-k) map and the nontrivial edge states in the { \^{}S}ₙ feature spectrum equals the spin-Chern number of a multilayer quantum spin Hall insulator. We identify a van der Waals material bismuth bromide (Bi₄Br₄) as a promising candidate through our first-principles calculations. Our paper not only unravels the intricacies of the bulk-boundary correspondence, but it also provides a pathway for exploring quantum spin Hall insulators with high spin-Chern numbers.