Axion insulators protected by ${C}_{2}\mathbb{T}$ symmetry, their $K$-theory invariants, and material realizations

Axion insulators protected by $C_{2} T$ symmetry, their $K$ -theory invariants, and material realizations

Rafael González-Hernández, Carlos Pinilla, and Bernardo Uribe

Phys. Rev. B 106, 195144 – Published 23 November 2022

Abstract

Axion insulators are generally understood as magnetic topological insulators whose Chern-Simons axion coupling term is quantized and equal to $π$ . Inversion and time reversal, or the composition of either one with a rotation or a translation, are symmetries which protect this invariant. In this work, we focus our attention on the composition of a twofold rotation with time reversal, and we show that insulators with this symmetry possess a $Z_{2}$ invariant arising from Atiyah's real $K$ theory. We call this invariant the $K$ -theory Kane-Mele invariant due to the similarities it has with the Kane-Mele invariant for systems with time reversal symmetry. Whenever all Chern numbers vanish, we demonstrate that this invariant is equivalent to the Chern-Simons axion coupling, and in the presence of the inversion symmetry, we show how this invariant could be obtained from the eigenvalues of the inversion operator on its fixed points in momentum space. For the general case of nontrivial Chern numbers, the Chern-Simons axion coupling term incorporates information of the $K$ -theory Kane-Mele invariant as well as information regarding bands with a nontrival Chern number. An explicit formula in terms of $K$ -theory generators is presented for the Chern-Simons axion coupling term, the relation with the $K$ -theory Kane-Mele invariant is explained, and a formula in terms of eigenvalues of the inversion operator is obtained. Using an effective Hamiltonian model and first-principles calculations, we also show that the occurrence of bulk-band inversion and nontrivial $K$ -theory Kane-Mele invariant index can be observed in axion insulators of the pnictides family. In particular, we demonstrate that NpBi can be classified as an axion insulator due to the detection of additional topological indicators such as the quantum spin Hall effect, gapped surface states, surface quantized anomalous Hall effect, and chiral hinge modes.

Received 23 June 2022
Revised 19 September 2022
Accepted 10 November 2022

DOI:https://doi.org/10.1103/PhysRevB.106.195144

Physics Subject Headings (PhySH)

Topological insulators

Topological materials

Electronic structure Tight-binding model Topology

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Rafael González-Hernández ^1,*, Carlos Pinilla^1,†, and Bernardo Uribe ^2,‡

¹Departamento de Física y Geociencias, Universidad del Norte, Km. 5 Vía Antigua Puerto Colombia, Barranquilla 080020, Colombia
²Departamento de Matemáticas y Estadística, Universidad del Norte, Km. 5 Vía Antigua Puerto Colombia, Barranquilla 080020, Colombia

^*rhernandezj@uninorte.edu.co
^†ccpinilla@uninorte.edu.co
^‡bjongbloed@uninorte.edu.co

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Vol. 106, Iss. 19 — 15 November 2022

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Figure 1
Energy bands of the local Hamiltonian of Eq. (66) preserving both inversion and $C_{2 z} T$ but breaking time reversal. (a) Energy bands along the $k_{z}$ axis with the projection on the first orbital of the upper valence band. (b) Upper valence band on the $k_{z} = 0$ with its projection on the first orbital. Note the change of orbital type along the energy bands.
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Figure 2
Generators of the $K$ -theory groups in the presence of inversion (a) and $C_{2 d} T$ and inversion (b). The $\pm 1$ eigenvalues of the inversion operator are labeled on the eight TRIMs. In (a), the first one represents the bundle $E$ of Eq. (44), the next three represent one band systems with odd Chern numbers across the planes $k_{x} = 0, k_{y} = 0$ , and $k_{z} = 0$ respectively, and the last three represent one band systems where the connection winds around the $k_{x}, k_{y}$ , and $k_{z}$ axes respectively with winding number 1. In (b), the first one labeled with $a$ represents the bundle whose $K$ -theory Kane-Mele invariant is nontrivial, the second labeled with $b$ represents a one band system with odd Chern number across the plane $k_{z} = 0$ , the third one labeled with $c$ represents a one band system with odd Chern numbers across $k_{y} = 0$ and $k_{x} = 0$ and the fourth labeled with $d$ represents a one band system with winding number 1 along the $k_{z}$ axis. These generators are labeled according to Eq. (100). In (c), all two different one band systems of (b) are superposed. The Chern-Simons axion coupling term is calculated as the parity of the number of pairs of negative $I$ eigenvalues on all TRIMs.
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Figure 3
(a) Crystal and magnetic structure of the NpBi noncollinear antiferromagnetic solid. (b) The first Brillouin zone of the cubic lattice. (c) Dynamic stability from phonon structure calculations. d) Calculated bulk band structure with Np- $f$ orbital projected states on the energy bands of NpBi. A double band inversion is noted around $- 0.66$ eV. Eigenvalues of the inversion symmetry operator change from $- 1$ and $+ 1$ in $Γ$ to $+ 1$ and $+ 1$ in $R$ point. (e) Spin Hall conductivity as a function of the Fermi level. It is noted a QSHC in the band gap of the material characterizes the topological nature of NpBi.
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Figure 4
(a) Calculated band structure of the (100), $(1 \bar{1} 0)$ , and (111) NpBi surface. Red and blue colors represent the projections onto the bottom and top surfaces, respectively. Bulk projected states are indicated in gray color. (b) Surface anomalous Hall conductivity (per slab thickness) as a function of the number of Np-Bi bilayers. A half-quantized anomalous Hall conductivity at the (111) surface is noted, which implies an axion insulator state for NpBi. (c) Gapless chiral hinge states calculated for the NpBi nanowire grown in the $(1 \bar{1} 0)$ direction.
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Figure 5
Gapless hinge states for the equivalent $〈 1 \bar{1} 0 〉$ directions of the NpBi material. Chiral hinge states generate an octahedral macroscopic crystal with eight faces which correspond to the equivalent ${111}$ surfaces. Blue and red surfaces indicate positive and negative hq-sAHC. Black arrows show the circulation of the chiral hinge state, where clockwise rotation corresponds to positive hq-sAHC.
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Physical Review B

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Axion insulators protected by $C_{2} T$ symmetry, their $K$ -theory invariants, and material realizations

Rafael González-Hernández, Carlos Pinilla, and Bernardo Uribe

Phys. Rev. B 106, 195144 – Published 23 November 2022

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Physical Review B

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Axion insulators protected by C2T symmetry, their K-theory invariants, and material realizations

Rafael González-Hernández, Carlos Pinilla, and Bernardo Uribe

Phys. Rev. B 106, 195144 – Published 23 November 2022

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Axion insulators protected by $C_{2} T$ symmetry, their $K$ -theory invariants, and material realizations