Corner- and sublattice-sensitive Majorana zero modes on the kagome lattice

Majid Kheirkhah, Di Zhu, Joseph Maciejko, and Zhongbo Yan

Phys. Rev. B 106, 085420 – Published 24 August 2022

Abstract

In a first-order topological phase with sublattice degrees of freedom, a change in the boundary sublattice termination has no effect on the existence of gapless boundary states in dimensions higher than one. However, such a change may strongly affect the physical properties of those boundary states. Motivated by this observation, we perform a systematic study of the impact of sublattice terminations on the boundary physics on the two-dimensional kagome lattice. We find that the energies of the Dirac points of helical edge states in two-dimensional first-order topological kagome insulators sensitively depend on the terminating sublattices at the edge. Remarkably, this property admits the realization of a time-reversal invariant second-order topological superconducting phase with highly controllable Majorana Kramers pairs at the corners and sublattice domain walls by putting the topological kagome insulator in proximity to a $d$ -wave superconductor. Moreover, substituting the $d$ -wave superconductor with a conventional $s$ -wave superconductor, we find that highly controllable Majorana zero modes can also be realized at the corners and sublattice domain walls if an in-plane Zeeman field is additionally applied. Our study reveals promising platforms to implement highly controllable Majorana zero modes.

Received 14 June 2022
Revised 3 August 2022
Accepted 16 August 2022

DOI:https://doi.org/10.1103/PhysRevB.106.085420

Physics Subject Headings (PhySH)

Kane-Mele model Symmetry protected topological states Topological insulators Topological superconductors

Two-dimensional electron system

Bogoliubov-de Gennes equations Discrete symmetries in condensed matter Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Majid Kheirkhah ¹, Di Zhu², Joseph Maciejko ^1,3, and Zhongbo Yan ^2,*

¹Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1
²School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
³Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2E1

^*yanzhb5@mail.sysu.edu.cn

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Vol. 106, Iss. 8 — 15 August 2022

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Images

Figure 1
(a) Schematic figure of different sublattice terminations at the edges of the kagome lattice for $N_{x} = 4, N_{y} = 3$ , and $w = 3$ . With open boundary conditions (see Sec. 4), the left, bottom, and right edges of the lattice respectively terminate with sublattices $A$ (orange dots) and $C$ (blue dots), $A$ and $B$ (green dots), and $B$ . With periodic boundary conditions in the $x$ direction, sublattice terminations refer to the top and bottom edges only. We vary the sublattice termination of the top edge by tuning $w$ , i.e., it can be terminated with sublattice $C$ ( $A$ and $B$ ) for $w = N_{x}$ ( $w = 0$ ). A domain wall between the $C$ and $A B$ terminations occurs for $0 < w < N_{x}$ . The triangular unit vectors are denoted by $a_{1} = \hat{x}$ and $a_{2} = (\hat{x} + \sqrt{3} \hat{y}) / 2$ . (b),(c) Energy spectrum of the normal-state Hamiltonian for a cylindrical geometry with open (periodic) boundary conditions in the $y$ ( $x$ ) direction. Chosen parameters are $t = 1, λ = 0.2, h_{x} = 0, w = 0$ in (b) and $w = N_{x}$ in (c). In (c), the top edge dispersion is plotted in red and the bottom edge dispersion in blue. The edge states tangentially merge with the bulk states in the energy spectrum.
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Figure 2
Energy spectrum of the BdG Hamiltonian with $d$ -wave superconducting pairing for a cylindrical geometry with open (periodic) boundary conditions in the $y$ ( $x$ ) direction, and the bottom (top) edge terminates with sublattices $A$ and $B$ ( $C$ ). The top edge dispersion is plotted in red and the bottom edge dispersion in blue. The chosen parameters are $t = 1, λ = 0.2, Δ_{d} = 0.1$ , and $μ = 1.5$ in (a), $μ = 1.6$ in (b), $μ = 1.7$ in (c), and $μ = 1.8$ in (d). Tuning $μ$ closes and reopens the boundary energy gap, signaling a boundary topological phase transition at $μ = 1.7$ .
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Figure 3
Majorana Kramers pairs at the corners and sublattice domain walls of the kagome lattice, for top-edge sublattice terminations with (a) $w = 0$ , (b) $w = 20$ , and (c) $w = 60$ [see Fig. 1]. The two (four) red stars in the left panels represent one (two) Majorana Kramers pair; their corresponding probability densities and positions are plotted in the right panels. In (b) and (c), there is one Majorana Kramers pair at the $C - A B$ sublattice domain wall. Chosen parameters are $t = 1, λ = 0.2, μ = 1.442, Δ_{d} = 0.1, h_{x} = 0, N_{x} = 100$ , and $N_{y} = 50$ . The positions of the Majorana Kramers pairs can be precisely manipulated by locally controlling the sublattice terminations.
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Figure 4
Energy spectrum of the BdG Hamiltonian with on-site $s$ -wave superconducting pairing for a cylindrical geometry with open (periodic) boundary conditions in the $y$ ( $x$ ) direction, and the bottom (top) edge terminates with sublattices $A$ and $B$ ( $C$ ). The top edge dispersion is plotted in red and the bottom edge dispersion in blue. Chosen parameters are $t = 1, λ = 0.2$ , and $μ = 1.442, Δ_{0} = 0.1$ , and $h_{x} = 0$ in (a), $h_{x} = 0.05$ in (b), $h_{x} = 0.1$ in (c), and $h_{x} = 0.2$ in (d). On-site $s$ -wave pairing opens a gap in the edge dispersion and increasing the Zeeman field $h_{x}$ closes and reopens this gap, signaling a boundary topological phase transition at $h_{x} = 0.1$ .
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Figure 5
MZMs at the corners and sublattice domain walls of the kagome lattice, for top-edge sublattice terminations with (a) $w = 0$ , (b) $w = 20$ , and (c) $w = 60$ [see Fig. 1]. The red stars in the left panels represent MZMs; their corresponding probability densities and positions are plotted in the right panels. In (b) and (c), there is one MZM at the $C - A B$ sublattice domain wall. Chosen parameters are $t = 1, λ = 0.2, μ = 1.442, Δ_{0} = 0.1, h_{x} = 0.2, N_{x} = 100$ , and $N_{y} = 50$ . The positions of the MZMs can be precisely manipulated by locally controlling the sublattice terminations.
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