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Sublattice-enriched tunability of bound states in second-order topological insulators and superconductors

Di Zhu, Majid Kheirkhah, and Zhongbo Yan
Phys. Rev. B 107, 085407 – Published 9 February 2023
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  • INTRODUCTION
  • KANE-MELE MODEL WITH AN IN-PLANE…
  • HELICAL EDGE STATES AND CORNER BOUND…
  • BOUND STATES AT SUBLATTICE DOMAIN WALLS
  • MAJORANA ZERO MODES AT SUBLATTICE DOMAIN…
  • DISCUSSION AND CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Bound states at sharp corners have been widely viewed as the hallmark of two-dimensional second-order topological insulators and superconductors. In this paper, we show that the existence of sublattice degrees of freedom can enrich the tunability of bound states on the boundary and hence lift the constraint on their locations. We take the Kane-Mele model with honeycomb-lattice structure to illustrate the underlying physics. With the introduction of an in-plane exchange field to the model, we find that the boundary Dirac mass induced by the exchange field has a sensitive dependence on the boundary sublattice termination. We find that the sensitive sublattice dependence can lead bound states to emerge at a specific type of boundary defects named as sublattice domain walls if the exchange field is of ferromagnetic nature, even in the absence of any sharp corner on the boundary. Remarkably, this sensitive dependence of the boundary Dirac mass on the boundary sublattice termination allows the positions of bound states to be manipulated to any place on the boundary for an appropriately-designed sample. With a further introduction of conventional s-wave superconductivity to the model, we find that, no matter whether the exchange field is ferromagnetic, antiferromagnetic, or ferrimagnetic, highly controllable Majorana zero modes can be achieved at the sublattice domain walls. Our paper reshapes the understanding of boundary physics in second-order topological phases, and meanwhile opens potential avenues to realize highly controllable bound states for potential applications.

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    • Received 20 November 2022
    • Revised 1 February 2023
    • Accepted 3 February 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.085407

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Edge statesKane-Mele modelProximity effectSpin-singlet pairingTopological insulatorsTopological phases of matterTopological superconductors
    1. Physical Systems
    Magnetic systemsTwo-dimensional electron system
    1. Techniques
    Bogoliubov-de Gennes equationsTight-binding modelk dot p method
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Di Zhu1, Majid Kheirkhah2,3, and Zhongbo Yan1,*

    • 1Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
    • 2Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
    • 3Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

    • *yanzhb5@mail.sysu.edu.cn

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    Issue

    Vol. 107, Iss. 8 — 15 February 2023

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    Images

    • Figure 1
      Figure 1

      A schematic diagram of the sublattice domain walls and the dependence of boundary Dirac mass (m) on the sublattice termination. The upper boundary consists of two parts, with one part being a zigzag edge (top left-hand side in each panel) and the other being a beard edge (top right-hand side in each panel), leading to the formation of sublattice domain walls at their intersections. The blue and green parabolas represent massive Dirac energy spectra of the gapped edge states. Different color patterns are just used to intuitively illustrate whether the signs of Dirac masses on the two sides of the sublattice domain wall are the same (a) or opposite (b). On each site, the red arrow denotes the direction of the exchange field. (a) When the exchange field is antiferromagnetic, the boundary Dirac masses on the upper zigzag and beard edges have the same sign, accordingly, the sublattice domain wall is not a Dirac-mass domain wall and hence does not harbor any bound states. (b) When the exchange field is ferromagnetic, the boundary Dirac masses on the upper zigzag and beard edges have opposite signs, hence the sublattice domain wall is a Dirac-mass domain wall supporting bound state illustrated by the black star.

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    • Figure 2
      Figure 2

      Energy spectra for a ribbon with armchair edges. The ribbon has open boundary conditions in the x direction and periodic boundary conditions in the y direction. Chosen parameters are t=1, λso=0.1, Mx=My=0.2. (a) λν=0, γ=1, (b) λν=0.1, γ=1, (c) λν=0, γ=0.5, and (d) λν=0.1, γ=0.5.

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    • Figure 3
      Figure 3

      Corner bound states in a rectangular sample with both x and y directions taking open boundary conditions. Chosen parameters are t=1, λso=0.1, Mx=My=0.2, γ=0.3, λν=0, and Nx=Ny=28. The geometries of the samples are depicted in (a2) and (b2), and a few corresponding eigenenergies near zero energy are shown in (a1) and (b1). The red dots in (a1) and (b1) correspond to the eigenenergies of the corner bound states. The shade of the red color on the lattice sites in (a2) and (b2) reflects the weight of the probability density of the corner bound states.

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    • Figure 4
      Figure 4

      Energy spectra for a ribbon with periodic boundary conditions in the x direction and beard edges in the y direction. Chosen parameters are t=1, λso=0.1, Mx=My=0.2, λν=0, and Ny=100. (a) A schematic diagram of a sample with beard edges in the y direction. Blue and red sites correspond to A-type and B-type sublattices, respectively. The red-solid lines in (b)–(d) denote energy spectra of the edge states on the beard edges. (b) γ=1, the helical edge states on the upper and lower beard edges are gapped out. (c) γ=0, the helical edge states on the upper beard edge are gapped out, while the helical edge states on the lower beard edge remain almost gapless. (d) γ=−1, the helical edge states on the upper and lower beard edges are gapped out.

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    • Figure 5
      Figure 5

      Energy spectra for a ribbon with periodic boundary conditions in the x direction and zigzag edges in the y direction. Chosen parameters are t=1, λso=0.1, Mx=My=0.2, λν=0, and Ny=100. (a) A schematic diagram of a sample with zigzag edges in the y direction. The red-solid lines in (b)–(d) denote energy spectra of the edge states on the zigzag edges. (b) γ=1, the helical edge states on the upper and lower zigzag edges are gapped out. (c) γ=0, the helical edge states on the upper zigzag edge remain almost gapless, while the helical edge states on the lower zigzag edge are gapped out. (d) γ=−1, the helical edge states on the upper and lower zigzag edges are gapped out.

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    • Figure 6
      Figure 6

      Tunable bound states at the sublattice domain walls. Chosen parameters are t=1, λso=0.1, Mx=My=0.2, γ=1, and λν=0. (a1) and (b1) show the energy spectra (only a few eigenenergies near zero energy are shown) corresponding to systems with the geometries shown in (a2) and (b2), respectively. Periodic boundary conditions are imposed in the x direction, namely, the left and right armchair edges in (a2) and (b2) are connected when diagonalizing the Hamiltonian. The red dots in (a1) and (b1) correspond to the eigenenergies of the bound states at the sublattice domain walls. The shade of the red color on the lattice sites in (a2) and (b2) reflects the weight of the probability density of the bound states.

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    • Figure 7
      Figure 7

      The evolution of boundary energy gap on the upper y-normal edge with respect to μ for a cylindrical geometry with periodic boundary conditions in the x direction. Chosen parameters are t=1, λso=0.1, Mx=My=0.2, γ=0.5, λν=0, and Δ=0.1. For (a1)–(a3), the open boundaries are beard edges, and the critical value of μ at which the boundary energy gap of the upper edge is equal to 0.265 according to the chosen parameters. For (b1)–(b3), the open boundaries are zigzag edges, and the critical value of μ is equal to 0.077. (a1) μ=0, (a2) μ=0.265, (a3) μ=0.35, (b1) μ=0, (b2) μ=0.07, and (b3) μ=0.15.

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    • Figure 8
      Figure 8

      Tunable MZMs at sublattice domain walls for a ferromagnetic exchange field. Chosen parameters are t=1, λso=0.1, Mx=My=0.2, γ=1, Δ=0.15, λν=0.2, and μ=0.2. The geometry considered is depicted in (a2) and (b2), and a few corresponding eigenenergies near zero energy are shown in (a1) and (b1). Periodic boundary conditions are imposed in the x direction. The red dots in the energy spectra denote MZMs (their energies are not exactly zero due to splitting induced by finite-size effects) at the sublattice domain walls. The right panels show their probability density profiles with the shade of the red color on the lattice sites reflecting the weight.

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    • Figure 9
      Figure 9

      Tunable MZMs at sublattice domain walls for an antiferromagnetic exchange field. All parameters are the same as in Fig. 8 except γ=−1.

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