Observation of Flat Frequency Bands at Open Edges and Antiphase Boundary Seams in Topological Mechanical Metamaterials

Kai Qian, Linghua Zhu, Keun Hyuk Ahn, and Camelia Prodan

Phys. Rev. Lett. 125, 225501 – Published 23 November 2020

Abstract

Motivated by the recent theoretical studies on a two-dimensional (2D) chiral Hamiltonian based on the Su-Schrieffer-Heeger chains [, , and , Phys. Rev. B 99, 041117(R) (2019)], we experimentally and computationally demonstrate that topological flat frequency bands can occur at open edges of 2D planar metamaterials and at antiphase boundary seams of ring-shaped or tubular metamaterials. Specifically, using mechanical systems made of magnetically coupled spinners, we reveal that the presence of the edge or seam bands that are flat in the entire projected reciprocal space follows the predictions based on topological winding numbers. The edge-to-edge distance sensitively controls the flatness of the edge bands and the localization of excitations, consistent with the theoretical analysis. The analog of the fractional charge state is observed. Possible realizations of flat bands in a large class of metamaterials, including photonic crystals and electronic metamaterials, are discussed.

Received 12 May 2020
Accepted 15 October 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.225501

Physics Subject Headings (PhySH)

Photonic crystals Topological effects in photonic systems Topological insulators

2-dimensional systems Metamaterials Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kai Qian¹, Linghua Zhu ², Keun Hyuk Ahn ^1,*, and Camelia Prodan^1,†

¹Department of Physics, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
²Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

^*kenahn@njit.edu
^†cprodan@njit.edu

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Issue

Vol. 125, Iss. 22 — 27 November 2020

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Images

Figure 1
(a) $6 \times 6$ spinner system, where rotatable spinners and magnetically coupled arms are highlighted and fixed spinners and arms without magnets are shaded. (b) Illustration of the spinner system pictured in (a), where the orange (gray) balls represent rotatable (fixed) spinners. The purple ellipse represents a unit cell. The blue (red) lines indicate the couplings within (between) the unit cells within the 1D SSH chains along the 0° direction. The green lines indicate the constant interchain coupling, which is smaller than the average intrachain coupling. Coordinates $(n_{1}, n_{2})$ describe the position of the spinners.
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Figure 2
Results for the topological systems. (a)–(e) Solid lines: Bulk mode spectra experimentally obtained by actuating and measuring at the $(N_{1} - 1, 1)$ spinner for the $N_{1} \times 6$ systems with $N_{1} = 4$ , 6, 8, 10, and 12, respectively. Dotted lines: Spectra obtained with the $(N_{1} / 2 + 1, 5)$ spinner to reveal the top (bottom) of the upper (lower) bulk bands better. The blue areas indicate the lower and upper bulk bands. (f)–(j) Edge mode spectra experimentally obtained with the $(N_{1}, 2)$ spinner for the same systems as in (a)–(e), respectively. Red areas indicate the edge bands. A red arrow in (a) [blue arrows in (f)] indicates edge (bulk) modes appearing in the bulk (edge) spectra due to the short edge-to-edge distance. (k) Experimental and theoretical bulk and edge bandwidths versus the system size $N_{1} \times N_{2}$ . (l) Logarithm of experimental edge bandwidth, $\ln (Δ f_{0})$ , versus $N_{1}$ and fitting to the theory, resulting in the localization length at $k_{2} = 0$ , $ξ_{0} \approx 3.2$ , close to the theory value of 3.1 [24, 28, 29].
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Figure 3
Spectra experimentally obtained by actuating and measuring at the (5,1), (6,2), and (6,1) spinners for the $6 \times 6$ (a) topological and (b) nontopological systems. The edge band is present in the bulk band gap in (a), but absent in (b).
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Figure 4
(a)–(d) Patterns of modes. Colors represent the oscillation amplitudes of the spinners, estimated from slow motion movies by eyes. Stars mark actuated spinners. In (a),[(b)], an edge [bulk] mode is revealed at 23.6 Hz [19.5 Hz] in the edge [bulk] band for the $4 \times 4$ topological system. In (c),[(d)], an edge mode is revealed at 23.7 Hz [24.5 Hz] for the $6 \times 4$ [ $8 \times 4$ ] topological system. See Supplemental Material for movies [28]. (e) Normalized oscillation amplitude of the $(n_{1}, n_{2})$ spinner versus its distance from the right edge, $12 - n_{1}$ , when a single $(12, n_{2})$ spinner $(n_{2} = 1, \dots, 6)$ at the right edge is actuated for the $12 \times 6$ topological system. (f) Symbols: semilogarithmic plot of (e) for even $12 - n_{1}$ . Lines: linear fittings, leading to an average localization length $ξ = 1.5 \pm 0.3$ , consistent with the theoretical range of $ξ (k)$ , $0.9 \sim 3.1$ .
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Figure 5
(a) Illustration of the SSH ring with fifteen spinners and a seam experimentally studied. See the caption for Fig. 1. The purple line represents the extra coupling at a magnet distance of 6.5 mm for the chiral symmetry. (b) Red (blue) line: Edge (bulk) mode spectrum experimentally obtained by actuating and measuring at the spinner No. 1 (No. 2) for the SSH ring shown in (a). See Supplemental Material for the movie of a seam mode [28]. (c) Tubular model system with thirteen spinners in the azimuthal direction and an antiphase boundary seam, obtained by joining the $(1, n_{2})$ and $(13, n_{2} + 1)$ spinners of a planar system like Fig. 1 with $N_{1} = 13$ . The gray balls represent fixed spinners interacting with the spinners at the seam. The magnification shows the antiphase boundary seam with the weak azimuthal couplings on both sides (the blue lines), and the couplings with the fixed spinners (the purple lines). (d) Band structure for the tubular system in (c). The blue lines represent the bulk bands and the red line the flat antiphase boundary seam band.
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