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Second-order topology and multidimensional topological transitions in sonic crystals

Nature Physics volume 15, pages 582–588 (2019)Cite this article

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Abstract

Topological insulators with unique edge states have revolutionized the understanding of solid-state materials. Recently, higher-order topological insulators (HOTIs), which host both gapped edge states and in-gap corner/hinge states, protected concurrently by band topology, were predicted and observed in experiments, unveiling a new horizon beyond the conventional bulk-edge correspondence. However, the control and manifestation of band topology in a hierarchy of dimensions, which is at the heart of HOTIs, have not yet been witnessed. Here, we propose theoretically and observe experimentally that tunable two-dimensional sonic crystals can be versatile systems to visualize and harness higher-order topology. In our systems, the two-dimensional acoustic bands mimic the quantum spin Hall effect, while the resultant one-dimensional helical edge states are gapped due to broken space-symmetry and carry quantized Zak phases, which then lead to zero-dimensional topological corner states. We demonstrate that topological transitions in the bulk and edges can be triggered independently by tuning the geometry of the sonic crystals. With complementary experiments and theories, our study reveals rich physics in HOTIs, opening a new route towards tunable topological metamaterials where novel applications, such as the topological transfer of acoustic energy among two-, one- and zero-dimensional modes, can be achieved.

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Fig. 1: Bulk, edge and corner states in SOTIs.
Fig. 2: Gapped helical edge states and edge pseudo-spins.
Fig. 3: Measurements of topological corner states.
Fig. 4: Topological transitions in a hierarchy of dimensions.

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Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

X.J.Z., Y.T., B.X., M.-H. L. and Y.-F.C. are supported by the National Key R&D Program of China (2017YFA0303702 and 2018YFA0306200) and the National Natural Science Foundation of China (grants nos. 11625418, 11890700 and 51732006). H.-X.W., Z.-K.L. and J.-H.J. are supported by the National Natural Science Foundation of China (grant no. 11675116), the Jiangsu distinguished professor funding and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). X.J.Z. thanks H. Ge and S. Yu for their support and assistance with experimental measurements, and Z. Chen for his help with simulations. J.-H.J. thanks H.-Y. Kee and J. Sipe for helpful discussions.

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Author notes

  1. These authors contributed equally: Xiujuan Zhang, Hai-Xiao Wang.

Authors and Affiliations

  1. National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing, China

    Xiujuan Zhang, Yuan Tian, Biye Xie, Ming-Hui Lu & Yan-Feng Chen

  2. School of Physical Science and Technology, and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou, China

    Hai-Xiao Wang, Zhi-Kang Lin & Jian-Hua Jiang

  3. Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, China

    Ming-Hui Lu & Yan-Feng Chen

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Contributions

J.-H.J., M.-H.L. and Y.-F.C. conceived the idea and guided the research. X.J.Z., H.-X.W. and Z.-K.L. performed the numerical simulations. J.-H.J., H.-X.W. and Z.-K.L. performed theoretical analyses. X.J.Z. and Y.T. performed experimental measurements. All authors contributed to discussions of the results and manuscript preparation. J.-H.J., X.J.Z. and M.-H.L. wrote the manuscript.

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Correspondence to Ming-Hui Lu or Jian-Hua Jiang.

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Journal peer review information: Nature Physics thanks Romain Fleury and other anonymous reviewer(s) for their contribution to the peer review of this work.

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Zhang, X., Wang, HX., Lin, ZK. et al. Second-order topology and multidimensional topological transitions in sonic crystals. Nat. Phys. 15, 582–588 (2019). https://doi.org/10.1038/s41567-019-0472-1

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