Demonstration of Acoustic Higher-Order Topological Stiefel-Whitney Semimetal

Xiao Xiang, Yu-Gui Peng, Feng Gao, Xiaoxiao Wu, Peng Wu, Zhaoxian Chen, Xiang Ni, and Xue-Feng Zhu

Phys. Rev. Lett. 132, 197202 – Published 10 May 2024

Abstract

The higher-order topological phases have attracted intense attention in the past years, which reveals various intriguing topological properties. Meanwhile, the enrichment of group symmetries with projective symmetry algebras redefines the fundamentals of topological matter and makes Stiefel-Whitney (SW) classes in classical wave systems possible. Here, we report the experimental realization of higher-order topological nodal loop semimetal in an acoustic system and obtain the inherent SW topological invariants. In stark contrast to higher-order topological semimetals relating to complex vector bundles, the hinge and surface states in the SW topological phase are protected by two distinctive SW topological charges relevant to real vector bundles. Our findings push forward the studies of SW class topology in classical wave systems, which also show possibilities in robust high- $Q$ -resonance-based sensing and energy harvesting.

Received 20 May 2023
Accepted 12 April 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.197202

Physics Subject Headings (PhySH)

Acoustic metamaterials Phononic crystals

Semimetals

Topology

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiao Xiang¹, Yu-Gui Peng^1,*, Feng Gao¹, Xiaoxiao Wu ^2,3, Peng Wu¹, Zhaoxian Chen⁴, Xiang Ni^5,†, and Xue-Feng Zhu ^1,‡

¹School of Physics and Innovation Institute, Huazhong University of Science and Technology, Wuhan 430074, China
²Quantum Science and Technology Center and Advanced Materials Thrust, The Hong Kong University of Science and Technology (Guangzhou), Nansha, Guangzhou 511400, Guangdong, China
³Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
⁴College of Engineering and Applied Sciences, Collaborative Innovation Center of Advanced Microstructures, and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210023, China
⁵School of Physics, Central South University, Changsha 410083, China

^*Corresponding author: ygpeng@hust.edu.cn
^†Corresponding author: nixiang@csu.edu.cn
^‡Corresponding author: xfzhu@hust.edu.cn

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Issue

Vol. 132, Iss. 19 — 10 May 2024

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Images

Figure 1
Tight-binding model of 3D HOTSWS. (a) The schematic of the HOTSWS model, where each unit cell has four sites with different hoppings. The blue and red sticks indicate the positive and negative hoppings, respectively. For the unit cell in the bottom without dimerization, we set $t = 2.1$ , $J_{1} = 2.1$ , $J_{2} = 2.1$ . After breaking $L_{z}$ , we obtain the unit cell of HOTSWS, where $t = 2.1$ , $J_{1} = 1$ , $J_{2} = 2.1$ . (b) Evolution process from DPs to nodal loops on the zero-energy surface in $k$ space. (c) The band structure of HOTSWS along the high-symmetry line of the first BZ. The right panel shows the first BZ where the nodal loops are depicted by cyan rings, and the two Stiefel-Whitney charges of $w_{1}$ and $w_{2}$ by the perpendicular red and yellow rings.
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Figure 2
Acoustic realization of the HOTSWS. (a) A sonic crystal comprising hexagonal acoustic resonators and coupling tubes, in which the blue and red coupling tubes denote the positive and negative couplings, respectively. Up panel inset: the primitive cell of HOTSWS in a toy model. Bottom panel inset: the configuration of one primitive cell in an acoustic system. (b) A sphere centered at $K$ point in the first BZ, which wraps a nodal loop. The blue arrows (N wave vectors) mark the circle at a fixed azimuthal angle in $k$ space. Right panel: Wilson loop spectra. (c) Distribution of eigenstates of HOTSWS. (d) Intensity fields in HOTSWS at 5515 Hz and 5506 Hz, showing the presence of hinge and surface states. (e) The band structure of HOTSWS depicted in white color is calculated from the TBM and the hinge states are marked by the red lines. The thermal diagram shows the dispersion calculated by Fourier transformation from the simulated pressure field.
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Figure 3
Surface states in HOTSWS. (a) A schematic of acoustic HOTSWS. The top layer, bottom layer, and vertical layer are highlighted. In experiments, the sources are placed at Source 1 and Source 2 on the top and bottom layer, marked by the blue and red stars, aiming at exciting the surface states. (b) The distribution of surface states on the top layer, where acoustic intensity fields are majorly localized at the sites of Type-A atoms. The blue box covers a unit cell of HOTSWS, in which the Type-A and Type-B atoms are denoted by brown and green dots. The right panel shows the acoustic intensity in one unit cell, where acoustic fields are localized in the cavities connected with positive interlayer couplings. (c) The measured intensity field distributions on the top and bottom layers. (d) The transmission spectra measured at Ports $P_{1}$ and $P_{2}$ , marked by the blue and gray circles in (c).
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Figure 4
Hinge states in HOTSWS. (a) The section in the $x - z$ plane of the sample. Locations of the sound sources (Source 3 and Source 4) are marked by the purple and yellow stars, respectively. (b),(c) Measured acoustic intensity field distributions for the hinge states when the sources are placed on the (b) top layer (Source 3) and (c) bottom layer (Source 4). (d) The transmission spectra measured at the Ports $P_{1}$ , $P_{2}$ , $P_{3}$ (Type-A atom), and $P_{4}$ (Type-B atom) marked by the circles in (b), reveal the properties of hinge states, surface states, and bulk states in the frequency domain.
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