Observation of Non-Abelian Charged Nodes Linking Nonadjacent Gaps

Dongyang Wang, Biao Yang, Mudi Wang, Ruo-Yang Zhang, Xiao Li, Z. Q. Zhang, Shuang Zhang, and C. T. Chan

Phys. Rev. Lett. 129, 263604 – Published 23 December 2022

Abstract

Nodal links are special configurations of band degeneracies in the momentum space, where nodal line branches encircle each other. In $P T$ symmetric systems, nodal lines can be topologically characterized using the eigenvector frame rotations along an encircling loop and the linking structure can be described with non-Abelian frame charges involving adjacent bands. While the commutation rules between the frame charges are well established, the underlying relationship between distant band gap closing nodes remains to be explored. In this Letter, we present a photonic multiple nodal links system, where the nodal lines of nonadjacent bands are investigated with symmetry constraints on frame charges. Through an orthogonal nodal chain, the nodal line from the lower two bands predicts the existence of nodal lines formed between the higher bands. We designed and fabricated a metamaterial, with which the multiple nodal links and the topological connection between nonadjacent nodal lines are experimentally demonstrated.

Received 9 July 2022
Accepted 22 November 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.263604

Physics Subject Headings (PhySH)

Metamaterials Symmetry protected topological states Topological effects in photonic systems

Atomic, Molecular & Optical

Authors & Affiliations

Dongyang Wang ¹, Biao Yang^1,2, Mudi Wang¹, Ruo-Yang Zhang ¹, Xiao Li ¹, Z. Q. Zhang¹, Shuang Zhang^3,4,*, and C. T. Chan^1,†

¹Department of Physics and Center for Metamaterials Research, Hong Kong University of Science and Technology, Hong Kong, China
²College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073, China
³Department of Physics, The University of Hong Kong, Hong Kong, China
⁴Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China

^*Corresponding author. shuzhang@hku.hk
^†Corresponding author. phchan@ust.hk

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Issue

Vol. 129, Iss. 26 — 23 December 2022

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Images

Figure 1
Multiple nodal links in the momentum space. (a) Resonators for building multiple nodal links. The bianisotropic couplings in two rings cancel out. (b) Band dispersions along the $k_{y}$ and $k_{z}$ directions. Degeneracy points are colored consistent with nodal lines. The adopted parameters are $l = 1$ , $A = 0.5$ , $ω_{0} = 2$ , $ω_{p x, y, z} = 0$ . (c) Dispersions of $ϵ_{y y}$ and $u_{z z}$ , the frequencies at the zero values correspond to the flat bands in (b). (d) Nodal structure in momentum space, and multiple nodal links are formed. The two $π_{1}$ loops define the “double $- 1$ ” charge at the $Γ$ point. (e)–(f) $2 π$ eigenpolarization state rotations for the 1st and 2nd bands on the $k_{z} = 0$ and $k_{y} = 0$ planes, verifying the “double $- 1$ ” charge.
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Figure 2
Nonadjacent nodal line connection. (a) Orthogonally chained nodal rings between the 2nd and 3rd bands in the momentum space. The initial arrows (representing the non-Abelian frame charges $q = \pm g_{2}$ ) are adopted in later analyzation. (b) The arrow configuration after considering the braiding with adjacent blue nodal line, with the viewpoint indicated with an “eye” symbol. Inconsistency is found for the two TRS-related $π_{1}$ loops. (c) Possible configuration of the predicted purple nodal lines, which counterbalances the braiding induced by the blue nodal line.
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Figure 3
Photonic metamaterial realization of multiple nodal links. (a) Band structures for the designed metamaterial (shown as inset, with background $ϵ_{b} = 2$ ). Resonator sizes are $L_{x} = 2 mm$ , $L_{y} = 2.5 mm$ , and gap $g = 1 mm$ . (b) First BZ of the metamaterial. (c) Nodal structure in momentum space, with colors specifying the interacting bands. (d) Calculated band projection on the $k_{x} − k_{z}$ plane along the $k_{θ}$ directions, with $θ$ defined in (c). (e) Experimentally measured band projections along the $k_{θ}$ directions, bulk bands are shown on top as red lines. (f) Calculated and measured EFCs in the $k_{y} - k_{z}$ plane. The marked points form into the blue nodal line in (c).
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Figure 4
Experimental verification of the nonadjacent nodal line connection. (a) Photo of the fabricated metamaterial in the $x - y$ plane. (b) Nodal structure in the $k_{z} = 0$ plane, the nodal ring from the 2nd and 3rd bands is shown in red, the predicted nodal lines from the 3rd and 4th bands are shown in purple. (c) Calculated band projections at fixed values of $k_{x} = [0.7, 0.8, 0.9]$ $π / p_{x}$ , where red nodal ring and purple nodal lines are manifested as colored points. Bulk bands at $k_{z} = 0$ plane are shown as red lines. (d) Experimentally measured band projections, with bulk bands shown in black.
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