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 ${\widehat{S}}_z$ feature spectrum equals the spin-Chern number of a multilayer quantum spin Hall insulator. We identify a van der Waals material bismuth bromide $\left({\mathrm{Bi}}_4{\mathrm{Br}}_4\right)$ 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.

Figure 1

Monolayer Kane-Mele model with and without time-reversal symmetry. (a) Black line: band structure with $t=4$, $\lambda =-2$, and ${\lambda }_{so}=0$. Brown line: band structure with $t=4$, $\lambda =-2$, and ${\lambda }_{so}=2$. (b) $\langle P{S}_{z}P\rangle$ feature spectrum of ${\widehat{S}}_z$-conserved (black line) and non-${\widehat{S}}_z$-conserved model (brown line). (c) The spin-resolved Wilson loop indicates $C_s=1$ and ${\mathbb{Z}}_2=1$. The red and blue lines indicate $C_s^{+}$ and $C_s^{-}$, respectively. (d) The edge states with time-reversal symmetry in the E-k map. (e) Spin-polarized edge states with time-reversal symmetry in the E-k map. The red and blue lines denote spin-up and spin-down, respectively. The green and orange arrows in the inset indicate the edge states from the LHS and RHS boundary, respectively. (f) ${\widehat{S}}_z$ feature edge states in the Kane-Mele model with time-reversal symmetry. (g) Same as (d), except with time-reversal symmetry breaking. (h) Same as (e), except with time-reversal symmetry breaking. The green and orange arrows indicate the direction of ${\widehat{S}}_z$ propagation on the edge states from LHS and RHS boundary, respectively. The green and orange arrows in the inset indicate the edge states from the LHS and RHS boundary, respectively. (i) Same as (f) except with time-reversal symmetry breaking. The green and orange lines denote the nontrivial edge states in the edge feature spectrum from LHS and RHS boundary, respectively. The green and orange arrows in the inset indicate the edge states from the LHS and RHS boundary, respectively.

Figure 2

The bilayer Kane-Mele model with interlayer coupling. (a) The band structure with $t_l=t_o=1$. (b) $\langle P{S}_{z}P\rangle$ feature spectrum. The feature spectrum is doubly degenerate in the layer degree of freedom. (c) The spin-resolved Wilson's loop indicates $C_s=2$ and ${\mathbb{Z}}_2=0$. Red and blue lines indicate $C_s^{+}$ and $C_s^{-}$, respectively. (d) The edge states with interlayer coupling in the E-k map. (e) Spin-polarized edge states with interlayer coupling in the E-k map. The red and blue lines denote spin-up and spin-down, respectively. The green and orange arrows indicate the direction of ${\widehat{S}}_z$ propagation on the edge states from the LHS and RHS boundary, respectively. (f) ${\widehat{S}}_z$ feature edge states in the bilayer Kane-Mele model with interlayer coupling. The green and orange lines denote the nontrivial edge states in the edge feature spectrum from the LHS and RHS boundary, respectively. The green and orange arrows in the inset indicate the edge states from the LHS and RHS boundary, respectively.

Figure 3

Monolayer and bilayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$. (a) Crystal structure of $\beta -{\mathrm{Bi}}_4{\mathrm{Br}}_4$. The inset is a 2D BZ. The red square indicates the monolayer structure; the black square is for the bilayer structure. (b) Band structure of monolayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ with contributions of the two type of Bi atoms. (c) $\langle P{S}_{z}P\rangle$ feature spectrum of monolayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$. The inset is calculated using the spin-resolved Wilson's loop, indicating $C_s=1$ and ${\mathbb{Z}}_2=1$. The red and blue lines indicate $C_s^{+}$ and $C_s^{-}$, respectively. (d) Spin-polarized edge states of monolayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ with time-reversal symmetry in the E-k map. The green arrow in the inset indicates the edge states from the LHS boundary. (e) Same as (d) except for time-reversal symmetry breaking. The green arrows indicate the direction of ${\widehat{S}}_z$ propagation on the edge states from the LHS boundary. The green arrow in the inset indicates the edge states from the LHS boundary. (f) The $\langle P{S}_{z}P\rangle$ edge feature spectrum of monolayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ with time-reversal symmetry breaking. The green and orange lines denote the nontrivial edge states in the edge feature spectrum from the LHS and RHS boundary, respectively. The green and orange arrows in the inset indicate the edge states from the LHS and RHS boundary, respectively. (g) Spin-polarized edge states of bilayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ with interlayer coupling in the E-k map. The green arrows indicate the direction of ${\widehat{S}}_z$ propagation on the edge states from the LHS boundary. The green arrow in the inset denotes the edge states from the LHS boundary. (h) The $\langle P{S}_{z}P\rangle$ edge feature spectrum of bilayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ with interlayer coupling. The green and orange lines denote the nontrivial edge states in the edge feature spectrum from the LHS and RHS boundary, respectively.

Figure 4

Trilayer Bi4Br4 and layer-dependent spin-Chern number. (a) Spin-polarized edge states of trilayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ with time-reversal symmetry in the E-k map. The green arrows indicate the direction of ${\widehat{S}}_z$ propagation on the edge states from the LHS boundary. The green arrow in the inset denotes the edge states from the LHS boundary. (b) $\langle P{S}_{z}P\rangle$ feature edge states in trilayer ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ with time-reversal symmetry. The green and orange lines denote the nontrivial edge states in the edge feature spectrum from the LHS and RHS boundary, respectively. The green and orange arrows in the inset indicate the edge states from the LHS and RHS boundary, respectively. (c) The spin-resolved Wilson's loop indicates $C_s=3$ and ${\mathbb{Z}}_1=0$. The red and blue line indicates $C_s^{+}$ and $C_s^{-}$, respectively. (d) Same as (a) except for time-reversal symmetry breaking. The green arrows indicate the direction of ${\widehat{S}}_z$ propagation on the edge states from LHS boundary. The green arrow in the inset indicates the edge states from the LHS boundary. (e) Same as (b) except for time-reversal symmetry breaking. The green and orange lines denote the nontrivial edge states in the edge feature spectrum from the LHS and RHS boundary, respectively. The green and orange arrows in the inset indicate the edge states from the LHS and RHS boundary, respectively. (f) The $C_s$ and ${\mathbb{Z}}_2$ as a function of the number of ${\mathrm{Bi}}_4{\mathrm{Br}}_4$ layers. [(g)–(h)] Schematic depiction of the variations in the ${\widehat{S}}_z$ component in the gapped/gapless edge band structure.

Figure 5

Feature-energy duality. (a) The examples shown illustrate that the total number of gapless edge states in the E-k map and nontrivial edge states in the ${\widehat{S}}_z$-feature spectrum equals the spin-Chern number $C_s$ of the multilayer quantum spin Hall insulator.