Bulk-hinge correspondence and three-dimensional quantum anomalous Hall effect in second-order topological insulators

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Bulk-hinge correspondence and three-dimensional quantum anomalous Hall effect in second-order topological insulators

Bo Fu, Zi-Ang Hu, and Shun-Qing Shen

Phys. Rev. Research 3, 033177 – Published 20 August 2021

Abstract

The chiral hinge modes are the key feature of a second-order topological insulator in three dimensions. Here we propose a quadrupole index in combination of a slab Chern number in the bulk to characterize the flowing pattern of chiral hinge modes along the hinges at the intersection of the surfaces of a sample. We further utilize the topological field theory to present a picture of three-dimensional quantum anomalous Hall effect as a consequence of chiral hinge modes. The two bulk topological invariants can be measured in electric transport and magneto-optical experiments. In this way we establish the bulk-hinge correspondence in a three-dimensional second-order topological insulator.

Received 3 February 2021
Revised 29 April 2021
Accepted 7 July 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.033177

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Hall effect Quantum anomalous Hall effect Topological insulators Topological phases of matter

Chiral symmetry

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Bo Fu , Zi-Ang Hu , and Shun-Qing Shen ^*

Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

^*sshen@hku.hk

Article Text

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Vol. 3, Iss. 3 — August - October 2021

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Images

Figure 1
Illustration of selected patterns of chiral hinge modes and their projection in a second-order topological insulator in three dimensions. (a) 3D antiferromagnetic quantum anomalous Hall insulator: A double-loop pattern with the quadrupole indices $Δ_{x y} = - Δ_{z x} = 1$ and $Δ_{y z} = 0$ and the slab Chern number $n_{x} = n_{y} = n_{z} = 0$ . (b) 3D rotoinversion quantum anomalous Hall insulator: A single-loop pattern with $Δ_{x y} = 1$ and $Δ_{y z} = Δ_{z x} = 0$ and $n_{x} = n_{y} = 0$ and $n_{z} = - 1$ . (c) 3D inversion quantum anomalous Hall insulator: A single-loop pattern with $Δ_{x y} = Δ_{y z} = Δ_{z x} = 0$ and $n_{x} = n_{y} = - n_{z} = 1$ .
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Figure 2
(a) Schematic view of the hinge current. The planar surfaces of the topological insulator are characterized by integers $n_{1}$ and $n_{2}$ , describing the integer change of the $θ$ value nearby the surfaces. (b) Schematic view of $θ$ -term as a function of the angle $φ$ for topologically nontrivial and trivial cases. (c, d) Plots of the layer-resolved Hall response $σ_{α β} (r_{γ})$ and panels (e, f) are plots of the $θ$ -angle as function of the layer index for antiferromagnetic and rotoinversion quantum anomalous Hall insulator, respectively, from a layer-resolved Kubo formula in a slab geometry for 10 layers.
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Figure 3
(a) Schematic illustration of the measurement of Kerr and Faraday angle. Incident linearly polarized light becomes elliptically polarized after transmission (Faraday effect) and reflection (Kerr effect), with polarization angles as $θ_{F}$ and $θ_{K}$ , respectively. (b) The reflectivity $R$ as a function of the slab thickness $L_{z}$ along $z$ direction in the units of half of photon wavelength $λ_{b} / 2$ for suspended single loop case with $ε = 10$ and $μ = 1$ .
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Figure 4
The band structure of the rotoinversion QAHI and the antiferromagnetic QAHI in three directions. Panels (a–c) are the band structures of the 40 lowest bands along three directions of the rotoinversion QAHI. We use the parameters $c = 1$ and $b_{z} = 0.3, b_{x} = 0, b_{y} = 0$ . Panels (d–f) are band structures of the antiferromagnetic QAHI along three directions. We use the parameters $d = 0.2, c = 1$ . The parameters in $H_{0}$ are same for two cases: $v_{z} = v_{⊥} = 1, m_{z} = m_{⊥} = - 0.5, m_{0} = 0.5$ .
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Figure 5
The winding of quadrupole moments for the rotoinversion QAHI (a–c) and the antiferromagnetic QAHI (d–f) along three directions. The parameters are the same as described in the caption of Fig. 4.
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Figure 6
All possible patterns of chiral hinge modes as the boundary of colored face on a cubic.
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