Hopf-link topological nodal-loop semimetals

Yao Zhou, Feng Xiong, Xiangang Wan, and Jin An

Phys. Rev. B 97, 155140 – Published 18 April 2018

Abstract

We construct a generic two-band model which can describe topological semimetals with multiple closed nodal loops. All the existing multi-nodal-loop semimetals, including the nodal-net, nodal-chain, and Hopf-link states, can be examined within the same framework. Based on a two-nodal-loop model, the corresponding drumhead surface states for these topologically different bulk states are studied and compared with each other. The connection of our model with Hopf insulators is also discussed. Furthermore, to identify experimentally these topologically different semimetal states, especially to distinguish the Hopf-link from unlinked ones, we also investigate their Landau levels. It is found that the Hopf-link state can be characterized by the existence of a quadruply degenerate zero-energy Landau band, regardless of the direction of the magnetic field.

Received 15 January 2018

DOI:https://doi.org/10.1103/PhysRevB.97.155140

Physics Subject Headings (PhySH)

Electronic structure

Node-line semimetals Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yao Zhou¹, Feng Xiong¹, Xiangang Wan^1,2, and Jin An^1,2,*

¹National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, China
²Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

^*anjin@nju.edu.cn

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Vol. 97, Iss. 15 — 15 April 2018

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Images

Figure 1
Two-nodal-loop semimetals with their structures adjusted by parameter $k_{y}^{0}$ . (a1) The nodal-net state with $k_{y}^{0} = 0$ . (b1) The Hopf-link state with $k_{y}^{0} = \frac{π}{3}$ . (c1) The nodal-chain state with $k_{y}^{0} = \frac{2 π}{3}$ . (d1) The unlinked state with $k_{y}^{0} = π$ . (a2)–(d2) The corresponding bulk energy bands along $k_{y}$ axis, where $\pm$ denotes the signs of the eigenvalues of rotation operator $C_{2}$ for the bands. (a3)–(d3) The corresponding drumhead surface states for boundary along direction $(1, 0, - 1)$ . The integers are the 1D winding numbers $N_{L}$ with path $L$ chosen along $(1, 0, - 1)$ , with their absolute values representing degeneracies of the boundary states. The colored dots in (a1)–(c1) are the nodes of the corresponding Floquet gapless states driven by a circularly polarized light propagating along $y$ direction. Here $m = 2.5$ .
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Figure 2
Comparison of the surface bands for the Hopf-link seimimetal with those of the unlinked one. (a), (b) The surface bands for the Hopf-link state with $k_{-} = 0, k_{+} = 0$ , respectively, where $k_{\pm} = (k_{x} \pm k_{z}) / 2$ . (c), (d) The surface bands for the unlinked state with $k_{y} = 0$ . Open-boundary conditions are applied along $(1, 0, 1)$ for (a) and (c), while along $(1, 0, - 1)$ for (b) and (d). The insets show schematically the linking structures for the two bulk loops. The numbers labeled at the zero-energy flat band represent the 1D winding numbers $N_{L}$ with path $L$ chosen along the surface normal.
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Figure 3
(a) The linking structure of two nodal loops with linking number 3. To calculate the linking number of this complex structure, a 2D projection of the two closed loops is exhibited in (b) with each crossing point labeled by $\pm 1$ . The linking number of the two loops is one half the absolute value of the sum over all these integers at crossing points.
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Figure 4
Three nodal loops with any two of them linked with linking number 1 for (a) and 2 for (b). (c) The drumhead surface states of case (b) for boundary along $(0, 0, 1)$ direction, where the inset is the blowup of the central area. The corresponding surface bands scanning along $k_{x}$ axis and $k_{y}$ axis are exhibited in (d) and (e), respectively. The integers labeled in (c), (d), and (e) denote the 1D winding number $N_{L}$ with path $L$ chosen along $(0, 0, 1)$ .
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Figure 5
The Landau spectra for the semimetals with two nodal loops. (a) The nodal-net state with $k_{y}^{0} = 0$ . (b)–(d) The Hopf-link state with $k_{y}^{0} = 0.3, 1, 1.7$ , respectively. (e) The nodal-chain state with $k_{y}^{0} = 2$ . (f) The unlinked state with $k_{y}^{0} = 2.3$ . Here $k_{y}$ is measured in unit of the radius $k_{0}$ of the loops. The insets schematically show the topological configurations for the two loops, where the colored dotted lines indicate the $k_{y}$ regimes possessing a quadruply degenerate zero-energy landau level. Here $l_{B} k_{0} = 14$ .
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