Ideal hourglass-type charge-three Weyl fermions in spinless systems

Xiaoliang Xiao, Yuanjun Jin, Da-Shuai Ma, Weixiang Kong, Jing Fan, Rui Wang, and Xiaozhi Wu

Phys. Rev. B 109, 075160 – Published 26 February 2024

Abstract

Hourglass-type Weyl fermions with the maximally topological charge of $| C | = 3$ , namely hourglass-type charge-three Weyl fermions (CTWFs), have seldom been reported due to the nonnegligible spin-orbit coupling effect and Pauli exclusion principles. Here, based on symmetry arguments and the low-energy effective $k \cdot p$ model, we confirm that two chiral space groups (SGs), i.e., SG $P 6_{3}$ (No. 173) and SG $P 6_{3} 22$ (No. 182), can host ideal hourglass-type CTWFs in spinless systems. Moreover, we take ${Li}_{2} {CO}_{4}$ as a concrete example to show the ideal hourglass-type CTWFs, which possess ultralong sextuple- and triple-helicoid Fermi arcs, making them easier to be detected by experiments. Our work not only offers an avenue to search for ideal hourglass-type CTWFs but also provides a promising platform for experiments to verify this kind of unconventional quasiparticles with the ultralong sextuple- and triple-helicoid surface arcs.

Received 15 August 2023
Revised 11 October 2023
Accepted 7 February 2024

DOI:https://doi.org/10.1103/PhysRevB.109.075160

Physics Subject Headings (PhySH)

Electronic structure Fermions First-principles calculations Symmetry protected topological states

3-dimensional systems Crystal structures Weyl semimetal

High-throughput calculations Symmetries in condensed matter Wannier function methods k dot p method

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiaoliang Xiao ¹, Yuanjun Jin ², Da-Shuai Ma^1,3, Weixiang Kong¹, Jing Fan ⁴, Rui Wang^1,3,5, and Xiaozhi Wu^1,5,*

¹Institute for Structure and Function & Department of Physics, Chongqing University, Chongqing 400044, People's Republic of China
²Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
³Center of Quantum Materials and Devices, Chongqing University, Chongqing 400044, People's Republic of China
⁴Center for Computational Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, People's Republic of China
⁵Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing 400044, People's Republic of China

^*xiaozhiwu@cqu.edu.cn

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Issue

Vol. 109, Iss. 7 — 15 February 2024

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Images

Figure 1
(a) and (b) The band dispersions of the neck crossing of the hourglass-type CTWF in the plane and out of the plane, which show the nonlinear and linear dispersions. (c) and (d) Two modes of the hourglass-type CTWF along the $Γ − A$ path. The $Γ$ and $A$ points possess two natural 2D IRs.
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Figure 2
(a) Top and (b) Side views of the crystal material ${Li}_{2} {CO}_{4}$ . (c) The bulk BZ and projected on the (001) and ( $10 \bar{1} 0$ ) surface BZs. The red lines of the (001) and ( $10 \bar{1} 0$ ) surface BZs represent the probable Fermi arcs. The marked circles in blue and red are the WPs with topological charges of $- 3$ and $+ 1$ , respectively.
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Figure 3
(a) The bulk band structure of crystal material ${Li}_{2} {CO}_{4}$ along high-symmetry paths. The band crossings near the Fermi level are marked by ${WF}_{1}, {WF}_{2}$ , and ${WF}_{3}$ , respectively. The inset is a 3D representation. (b) The enlarged drawings of energy dispersion along the $Γ − A$ path and relevant IRs with PG $C_{6}$ . (c) Another possible mode of energy dispersion along the $Γ − A$ path.
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Figure 4
(a)–(c) The evolutions of WCCs for ${WF}_{1}, {WF}_{2}$ , and ${WF}_{3}$ . Here, $φ \in$ [0, $π]$ is the polar angle, and $θ \in$ [0, 2 $π]$ represents the evolution of the WCCs. The WPs are characterized by topological charges of $- 3, + 1$ , and $+ 1$ , respectively. (d) and (e) The Berry curvature distributions of the (001) and $(10 \bar{1} 0)$ surface BZs. The ${WF}_{2}$ and ${WF}_{3}$ with $C = + 1$ as the “source” flow into the “sink” generated by ${WF}_{1}$ with $C = - 3$ .
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Figure 5
(a) The LDOS along $\bar{M} − \bar{Γ} − \bar{K} − \bar{M}$ on the (001) surface BZ, where TSSs denote topological surface states. The black dashed line represents the isoenergy contours at the energy ${cut}_{1}$ ( ${Ecut}_{1}$ = 0 meV). (b) The corresponding Fermi arcs at ${Ecut}_{1}$ = 0 meV and the positions of the WPs with the associated topological charges. (c) and (d) The surface LDOSs along two white clockwise loops ( ${loop}_{1}$ and ${loop}_{2}$ ) centered at $\bar{Γ}$ and $\bar{K}$ in (b). The black dashed lines are the reference line. It occurs in sextuple- and triple-helicoid TSSs, respectively. (e) The LDOS along $\bar{M} − \bar{Γ} − \bar{A}$ on the ( $10 \bar{1} 0$ ) surface BZ. The black dashed line represents the isoenergy contours at the energy ${cut}_{2}$ ( ${Ecut}_{2} = - 3$ meV). (f) The corresponding Fermi arcs at ${Ecut}_{2} = - 3$ meV and the positions of the WPs with the associated topological charges. (g) and (h) The surface LDOSs along the white anticlockwise loop ( ${loop}_{3}$ ) and clockwise loop ( ${loop}_{4}$ ) centered at ${WF}_{1}$ in (f). It only occurs in triple-helicoid TSSs.
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