Novel family of topological semimetals with butterflylike nodal lines

Xiaoting Zhou, Chuang-Han Hsu, Hugo Aramberri, Mikel Iraola, Cheng-Yi Huang, Juan L. Mañes, Maia G. Vergniory, Hsin Lin, and Nicholas Kioussis

Phys. Rev. B 104, 125135 – Published 22 September 2021

Abstract

In recent years, the exotic properties of topological semimetals (TSMs) have attracted great attention and significant efforts have been made in seeking new topological phases and material realization. In this work, we propose a family of TSMs which harbors an unprecedented nodal line (NL) landscape consisting of a pair of concentric intersecting coplanar ellipses (CICEs) at half-filling. Meanwhile, the CICE at half-filling guarantees the presence of a second pair of CICEs beyond half-filling. Both CICEs are linked at fourfold degenerate points at zone boundaries. In addition, we identify the generic criteria for the existence of the CICE in a time-reversal-invariant spinless fermion system or a spinful system with negligible spin-orbital coupling. Consequently, 9 out of 230 space groups (SGs) are feasible for hosting CICEs whose location centers in the first Brillouin zone (BZ) are identified. We provide a simple model with SG $P b a m$ (No. 55) which exhibits CICEs, and the exotic intertwined drumhead surface states, induced by double band inversions. Finally, we propose a series of material candidates that host butterflylike CICE NLs, such as $Zr X_{2}$ ( $X = P$ , As), ${Tl}_{2} {GeTe}_{5}, {CYB}_{2}$ , and ${Al}_{2} Y_{3}$ .

Received 29 April 2020
Revised 10 March 2021
Accepted 13 September 2021

DOI:https://doi.org/10.1103/PhysRevB.104.125135

Physics Subject Headings (PhySH)

Electronic structure Topological materials Topological phases of matter

Node-line semimetals Semimetals

Band structure methods Density functional theory Group theory Tight-binding model Wannier function methods k dot p method

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiaoting Zhou ^1,*, Chuang-Han Hsu², Hugo Aramberri ¹, Mikel Iraola ^3,4, Cheng-Yi Huang^5,1, Juan L. Mañes ⁴, Maia G. Vergniory^3,6, Hsin Lin ⁵, and Nicholas Kioussis^1,†

¹Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
²Department of Electrical and Computer Engineering, Faculty of Engineering, National University of Singapore, Singapore 117583
³Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain
⁴Department of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
⁵Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
⁶IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain

^*physxtzhou@gmail.com
^†nick.kioussis@csun.edu

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Vol. 104, Iss. 12 — 15 September 2021

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Images

Figure 1
Schematic band structures showing the mechanism of the formation of the CICE NLs, resulting from the DBI. (a) The half-filling CICE can be decomposed into two ellipses in red and blue, respectively, which are locked by symmetries. Each ellipse is the band crossing due to a single band inversion and protected by mirror (or glide) symmetry $M_{z}$ (or $G_{z}$ ). (b) The second pair of CICE NLs (labeled in magenta) emerge due to the crossings between the bands denoted in red and blue corresponding to crossings between two occupied (one-quarter filling) or two unoccupied (three-quarter filling) bands. These cross the Fermi surface (FS) leading to the intersection of four NLs at four points on the $k_{x}$ and $k_{y}$ axes. Band structures along the paths (c) $p_{1}$ and (d) the $k_{x}$ axis. The bands labeled by solid and dashed lines carry different eigenvalues of $M_{z}$ (or $G_{z}$ ), and opposite parity at the center TRIM point of the CICE. Bands are doubly degenerate along the $k_{x}$ axis (d) and similarly, along the $k_{y}$ axis. I, II, and III denote the regions divided by the NLs.
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Figure 2
(a) Orthorhombic crystal structure of the lattice model in Eq. (1) in SG $P b a m$ , consisting of a bipartite lattice with two sublattices $A$ (in blue) and $B$ (in gray). (b) Bulk BZ, the projected (001) surface BZ, and high-symmetry points. (c) Band structures of the model without spin-orbit coupling. (d) Schematic dispersion of the drumhead surface states (DSSs) of the (001) surface stemming from the NLs, with two saddlelike hyperbolic paraboloids intertwined with each other. (e), (f) Dispersion along high symmetric directions of two types of DSSs on the (001) surface. (e) Type-I DSSs stem from the CICE FS, while (f) type II from the NLs labeled in magenta in Fig. 1.
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Figure 3
Crystal structure of bulk (a) ${ZrAs}_{2}$ and (d) ${Tl}_{2} {GeTe}_{5}$ . Band structure of (b) ${ZrAs}_{2}$ and (e) ${Tl}_{2} {GeTe}_{5}$ close to the Fermi energy without spin-orbit coupling (SOC) along the symmetry lines in the BZ shown in the insets where high-symmetry points are marked. The relevant crossings on $U$ ( $M$ )- $Γ$ for the half-filling and the second pair of CICEs are indicated by $[red, blue]$ and magenta dots, respectively. Energy-momentum spread for the half-filing CICE nodal lines in the BZ for (c) ${ZrAs}_{2}$ centered at the $U$ point on the (010) plane (gray shaded) and (f) ${Tl}_{2} {GeTe}_{5}$ at the $M$ point on the (001) plane (gray shaded). The black dots shown on the projected CICEs in the $k$ space are the FDPs and one of them is indicated in the band along $X − U$ ( $M$ ) of (b) and (e). As suggested by the crossings in (b) and (e), the energy spreads for the second pair of CICEs are very similar.
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Figure 4
(a) CICE nodal lines around the $S$ point on the $k_{z} = 0$ plane. (b) The Zak phase $γ$ distributions for the model. The yellow (blue) color represents $π$ (0). Outside the projection of the NLs, no surface modes can be observed and the Zak phase $γ = 0$ , while within the area labeled in yellow, there is one DSS. However, within the overlapping area of the projections of two NLs, the Zak phase is 0 (mod $2 π$ ), although there are two DSSs within that area, which cannot be characterized by the $Z_{2}$ Zak phase.
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Figure 5
Crystal structure of bulk (a) ${CYB}_{2}$ with tetragonal SG $P 4_{2} / m b c$ (No. 135) and (c) ${Al}_{2} Y_{3}$ with tetragonal SG $P 4 / m b m$ (No. 136). Band structure of (b) ${CYB}_{2}$ and (d) ${Al}_{2} Y_{3}$ close to the Fermi energy without SOC along the symmetry lines in the Brillouin zone shown in the insets where high-symmetry points are marked. The relevant crossings on $M − Γ$ for the half-filling and the second pair of CICEs are indicated by [red, $blue]$ and magenta dots, respectively. The black dots along $X − M$ are the FDPs.
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