Designing Topological Defect Lines Protected by Gauge-Dependent Symmetry Indicators

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Designing Topological Defect Lines Protected by Gauge-Dependent Symmetry Indicators

Erda Wen, Dia’aaldin J. Bisharat, Robert J. Davis, Xiaozhen Yang, and Daniel F. Sievenpiper

Phys. Rev. Applied 17, 064008 – Published 3 June 2022

Abstract

Symmetry indicators are a modern tool for characterizing topological phases that require minimal computational expense but provide an elegant means of designing practical devices. This paper demonstrates how a rotational symmetry indicator can be used to construct and characterize a topologically robust waveguide, verified experimentally on a printed circuit board platform. The design takes advantage of the real-space gauge dependency of the symmetry indicators and adopts a trivial $C_{6}$ lattice with simple shifts. The model illustrates the critical role real-space information plays in determining the topological properties of photonic crystals, enabling a wider range of possible realizations.

Received 17 February 2022
Revised 5 April 2022
Accepted 3 May 2022

DOI:https://doi.org/10.1103/PhysRevApplied.17.064008

Physics Subject Headings (PhySH)

Crystal symmetry Edge states Line defects Metamaterials Photonic crystal waveguide Topological effects in photonic systems

Topological materials

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Erda Wen ^*, Dia’aaldin J. Bisharat ^†, Robert J. Davis ^‡, Xiaozhen Yang ^§, and Daniel F. Sievenpiper^¶

Applied Electromagnetics Group, Electrical and Computer Engineering Department, University of California, San Diego, California 92093, USA

^*ewen@ucsd.edu
^†dbisharat@gc.cuny.edu; Advanced Science Research Center, The City University of New York, 85 St Nicholas Terrace, New York, 10031, New York, USA.
^‡rjdavis@ucsd.edu
^§xiy003@ucsd.edu
^¶dsievenpiper@ucsd.edu

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Vol. 17, Iss. 6 — June 2022

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Images

Figure 1
The lattice designs. (a) The design process of the metallic geometry resembling a TB model with hexagonal lattice consisting six sites in each unit. Blue and red lines represent intraunit and interunit coupling of different intensities. (b) Different unit-cell boundary selections lead to different topologies: (d) topological trivial unit with strong intraunit coupling and weak interunit coupling; (c),(e) nontrivial unit with strong interunit coupling an hybrid intraunit coupling. (f) Band diagram of the tight-binding (TB) model with $γ = 1$ and $λ = 0.5$ . (g) Dispersion relationship of the metallic geometry on the substrate, with gap size $g_{2} = 0.04 a$ . The dotted line represents the light cone. The bandgap being inspected is highlighted in (f) and (g).
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Figure 2
Two types of defect line arrangements. (a),(b) The “tight arrangement” and the “loose arrangement” and the numerical solutions of the band diagram for tight and loose supercells, respectively. The structures on both sides of the line defect are identical with different colors indicating different topologies. Modes in the shadowed area are uninterested bulk or corner modes. Purple arrows in the field phase plots indicate the direction of the energy flux.
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Figure 3
The experiments verifying the predicted edge state using the PCB defect line with sharp turns. Panels (a)–(g) show an illustration and results for the “loose arrangement” and (h)–(k) for the “tight arrangement.” (b),(i) Measured near-field distribution. The purple arrows represent the single excitation on one end of the waveguide. (c),(j) Transmission over frequency. Highlighted areas are the regions where edge modes locate on the band diagram. (d),(k) LDOS plot. Dashed lines are the simulated band structure. (e) Photo of the prototype. (f) Relationship between the directivity and the phase delay between the two sources that excites the loose defect line as in (a). (g) Measured near-field distribution with the paired excitation as in (a) with a $2 π / 3$ phase delay, from the dielectric side of the loose defect line.
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Figure 4
Waveguide arrangement integrated with a termination. (a) The geometry. (b) Near-field profile, measured from the dielectric side, with a single excitation. WG; Waveguide.
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Figure 5
Band structure of the PCB lattice with different inner gap size, compared with the band structure of the tight-binding model with different intracell coupling intensity.
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Figure 6
Exciting unidirectional mode for the tight arrangement: relationship between the directivity and the phase delay between the three sources that excites the defect line. It can be found that $| \pm ⟩ = | 1 ⟩ + e^{\mp i 2 π / 3} | 2 ⟩ + e^{\mp i 4 π / 3} | 3 ⟩$ .
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