Observation of Topological $p$-Orbital Disclination States in Non-Euclidean Acoustic Metamaterials

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Observation of Topological $p$ -Orbital Disclination States in Non-Euclidean Acoustic Metamaterials

Ying Chen, Yuhang Yin, Zhi-Kang Lin, Ze-Huan Zheng, Yang Liu, Jing Li, Jian-Hua Jiang, and Huanyang Chen

Phys. Rev. Lett. 129, 154301 – Published 3 October 2022

Abstract

Disclinations—topological defects ubiquitously existing in various materials—can reveal the intrinsic band topology of the hosting material through the bulk-disclination correspondence. In low-dimensional materials and nanostructure such as graphene and fullerenes, disclinations yield curved surfaces and emergent non-Euclidean geometries that are crucial in understanding the properties of these materials. However, the bulk-disclination correspondence has never been studied in non-Euclidean geometry, nor in systems with $p$ -orbital physics. Here, by creating $p$ -orbital topological acoustic metamaterials with disclination-induced conic and hyperbolic surfaces, we demonstrate the rich emergent bound states arising from the interplay among the real-space geometry, the bulk band topology, and the $p$ -orbital physics. This phenomenon is confirmed by clear experimental evidence that is consistent with theory and simulations. Our experiment paves the way toward topological phenomena in non-Euclidean geometries and will stimulate interesting research on, e.g., topological phenomena for electrons in nanomaterials with curved surfaces.

Received 13 February 2022
Accepted 6 September 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.154301

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic wave phenomena Topological defects

Topological materials

Acoustic techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ying Chen^1,*,†, Yuhang Yin^2,*, Zhi-Kang Lin^3,*, Ze-Huan Zheng², Yang Liu³, Jing Li², Jian-Hua Jiang ^3,‡, and Huanyang Chen ^2,§

¹Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
²Department of Physics and Pen-Tung Sah Institute of Micro-Nano Science, Xiamen University, Xiamen 361005, China
³Institute of Theoretical and Applied Physics, School of Physical Science and Technology, and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, 1 Shizi Street, Suzhou 215006, China

^*These authors contributed equally to this work.
^†Corresponding author. yingchen@hqu.edu.cn
^‡Corresponding author. jianhuajiang@suda.edu.cn
^§Corresponding author. kenyon@xmu.edu.cn

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Issue

Vol. 129, Iss. 15 — 7 October 2022

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Images

Figure 1
(a) A 2D honeycomb lattice consists of the $A$ -type (yellow) and $B$ -type (blue) sublattice sites. The shaded region denotes a $2 π / 3$ sector (enclosed by the magenta dashed lines). (b) and (c) Schematics of the conic (hyperbolic) lattice constructed by removing (adding) a $2 π / 3$ sector from (into) the honeycomb lattice, leading to a disclination with the Frank angle $Ω = - (2 π / 3)$ ( $2 π / 3$ ). $G$ represents the Gaussian curvature of the structure.
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Figure 2
(a) Honeycomb lattice with acoustic $p$ orbitals (top panel; also depicted in the main panel). ${\vec{e}}_{1}$ , ${\vec{e}}_{2}$ , and ${\vec{e}}_{3}$ denote three hopping vectors. Upper right: the Brillouin zone. Lower right: a unit cell of the acoustic metamaterial (gray denotes air regions) with geometric details. (b) and (c) Simulated band structures of acoustic metamaterials with different parameters: (b) $r_{A} = r_{B} = 12 mm$ , (c) $r_{A} = 13.5 mm$ and $r_{B} = 10.5 mm$ . Cyan curves represent the $p_{x}$ and $p_{y}$ bands, while red curves denote the $p_{z}$ bands. Insets: band structures from the tight-binding model: (b) $ϵ_{A x, y} = ϵ_{B x, y} = 0$ , $ϵ_{A z} = ϵ_{B z} = 0.3$ , (c) $ϵ_{A x, y} = - 1.8$ , $ϵ_{A z} = - 2.1$ and $ϵ_{B x, y} = 2$ , $ϵ_{B z} = 2.8$ . In panel (c), acoustic phase profile (arrows indicate the phase winding) at the $K$ point (black dot) is also depicted. (d) Dispersions for a supercell with zigzag edge boundaries between two acoustic metamaterials I and II. The radii are $r_{A} = 12.3 mm$ and $r_{B} = 11.7 mm$ for I ( $r_{A}$ and $r_{B}$ are interchanged for II), respectively. Right: acoustic pressure profile for the edge state at the red triangle in the brown curve. Green curve is for the edge states at the opposite edge. $l = 3 1.0 mm$ and $d = 5.6 mm$ .
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Figure 3
(a) Simulated acoustic eigenfrequencies around the topological band gap in the conic lattice with $r_{A} = 13.5 mm$ , $r_{B} = 10.5 mm$ , and $d = 5.6 mm$ . Topological disclination states in groups I and II are labeled by the purple dots. Inset shows the acoustic pressure profile of a disclination state. (b)–(e) Top views of the simulated acoustic pressure profiles for the disclination states. (f) Photograph of the fabricated acoustic structure for the conic lattice and the experimental setup. In the lower panels, the pump-probe configurations I and II are shown where the stars (dots) denote the positions of the acoustic source (detector). Purple dashed lines indicate the mirror symmetry lines. (g) Measured acoustic pressure $p$ for the pump-probe configurations I and II at different frequencies. (h) Measured acoustic pressure profiles $| p |$ (absolute value) around the disclination core (cavities are depicted as circles) for two disclination states, $Ψ_{I, O}$ and $Ψ_{II, E}$ , under resonant excitation conditions.
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Figure 4
(a) Simulated acoustic eigenfrequencies around the topological band gap in the hyperbolic lattice with $r_{A} = 13.5 mm$ , $r_{B} = 10.5 mm$ , and $d = 5.6 mm$ . Topological disclination states are labeled by the golden dots. Inset shows the acoustic pressure profile of a disclination state. (b)–(e) Top views of the acoustic pressure profiles of the disclination states. (f) Photograph of the fabricated acoustic structure for the hyperbolic lattice. Lower panels show the disclination and bulk pump-probe setups (labeled as DS and BS, respectively) where the stars (dots) denote the positions of the acoustic source (detector). (g) Measured acoustic pressure $| p |$ (absolute value) for the disclination and bulk pump-probe setups versus the excitation frequency. Inset gives the measured acoustic pressure profile for the cavities (depicted as circles) around the disclination core at the excitation frequency of 9.20 kHz. (h) Measured responses for the $D_{1} S$ and $D_{2} S$ pump-probe setups.
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