Creation and manipulation of higher-order topological states by altermagnets

Letter

Creation and manipulation of higher-order topological states by altermagnets

Yu-Xuan Li, Yichen Liu, and Cheng-Cheng Liu

Phys. Rev. B 109, L201109 – Published 9 May 2024

Abstract

We propose to implement tunable higher-order topological states in a heterojunction consisting of a two-dimensional (2D) topological insulator and the recently discovered altermagnets, whose unique spin-polarization in both real and reciprocal space and null magnetization are in contrast to conventional ferromagnets and antiferromagnets. Based on symmetry analysis and effective edge theory, we show that the special spin splitting in altermagnets with different symmetries, such as $d$ wave, can introduce Dirac mass terms with opposite signs on the adjacent boundaries of the topological insulator, resulting in the higher-order topological state with mass-domain-bound corner states. Moreover, by adjusting the direction of the Néel vector, we can manipulate such topological corner states by moving their positions. By first-principles calculations, taking a 2D topological insulator bismuthene with a square lattice on an altermagnet ${MnF}_{2}$ as an example, we demonstrate the feasibility of creating and manipulating the higher-order topological states through altermagnets. Finally, we discuss the experimental implementation and detection of the tunable topological corner states, as well as the potential non-Abelian braiding of the Dirac corner fermions.

Received 25 July 2023
Revised 18 April 2024
Accepted 22 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.L201109

Physics Subject Headings (PhySH)

First-principles calculations Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yu-Xuan Li, Yichen Liu, and Cheng-Cheng Liu ^*

Center for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing 100081, China

^*ccliu@bit.edu.cn

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Issue

Vol. 109, Iss. 20 — 15 May 2024

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Images

Figure 1
Left panel: A 2D topological insulator with the helical edge states protected by time-reversal symmetry. Middle panel: A proximitized altermagnet induces altermagnetism in the 2D topological insulator and breaks time-reversal symmetry. We take a $d$ -wave altermagnet as an example. Right panel: When the Néel vector lies in the plane, the helical edge states open a gap with two in-gap states localized at the corners appearing, i.e., the topological corner states (in orange). The manipulation of these corner states (in red) can be achieved by adjusting the Néel vector around the $[1 \bar{1}]$ direction, as indicated by the red arrow.
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Figure 2
(a) The edge spectrum for a cylinder geometry. The blue dotted lines represent the gapless helical edge states of the topological insulator. The red solid lines denote the gapped helical edge states after the altermagnet is turned on. (b) Inset: Two in-gap states emerge with the Néel vector along the $[1 \bar{1}]$ direction. The real spatial distribution of their wave function is plotted. (c) Same as (b) but with the Néel vector along the [11] direction. (d) The tangents $L (α)$ , on which we will develop the generic edge theory, mark different boundaries with the clockwise rotation angle $α$ . (e), (f) The change of the boundary Dirac mass with the rotation angle $α$ when the Néel vector is along the $[1 \bar{1}]$ direction and the [11] direction, respectively. (g) The topological invariant $ν$ is plotted as a function of the $m_{0}$ . (h) Schematic diagram: The boundary of the mirror-symmetry operation connection has opposite Dirac mass, and the TCSs originate from different mirror subspaces. (i) The mirror-graded winding number $ν_{M_{x y}}$ is plotted as a function of the $m_{0}$ . Common parameters: $m_{0} = 1.0, t_{x} = t_{y} = A_{x} = A_{y} = 2.0, J_{0} = 0.5$ .
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Figure 3
(a) Side view of the optimized crystal structure of the ${MnF}_{2} / Bi / {MnF}_{2}$ configuration. (b) The 2D Brillouin zone with high-symmetric points and lines. (c) The spin-polarized energy bands of the ${MnF}_{2} / Bi / {MnF}_{2}$ sandwich structure, where the blue-solid/red-dashed lines represent the spin-up/spin-down. (d) The $d$ -wave spin-polarized band splitting induced in bismuthene from our DFT calculation. (e) Helical edge states are calculated via DFT in the absence of altermangetism. (f) Gapped edge states obtained by the DFT calculation with the Néel vector aligned along the $[1 \bar{1}]$ direction. (g) and (h) Néel vector along the $[1 \bar{1}]$ and [11] directions, respectively. Inset: Two in-gap topological corner states emerge. The small arrows depict the direction of the Néel vector. The real spatial distribution of their wave function is plotted.
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