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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.

References

  1. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  2. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  3. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  4. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  ADS  Google Scholar 

  5. Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    Article  ADS  Google Scholar 

  6. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscattering immune topological electromagnetic states. Nature 461, 772–775 (2009).

    Article  ADS  Google Scholar 

  7. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).

    Article  Google Scholar 

  8. Poo, Y., Wu, R., Lin, Z., Yang, Y. & Chan, C. T. Experimental realization of self-guiding unidirectional electromagnetic edge states. Phys. Rev. Lett. 106, 093903 (2011).

    Article  ADS  Google Scholar 

  9. Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).

    Article  ADS  Google Scholar 

  10. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    Article  ADS  Google Scholar 

  11. Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).

    Article  ADS  Google Scholar 

  12. Chen, W.-J. et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nat. Commun. 5, 6782 (2014).

    Google Scholar 

  13. Wu, L.-H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).

    Article  ADS  Google Scholar 

  14. Xu, L., Wang, H.-X., Xu, Y.-D., Chen, H.-Y. & Jiang, J.-H. Accidental degeneracy and topological phase transitions in two-dimensional core-shell dielectric photonic crystals. Opt. Express 24, 18059–18071 (2016).

    Article  ADS  Google Scholar 

  15. Lu, L., Joannopoulos, J. D. & Soljacic, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    Article  ADS  Google Scholar 

  16. Chen, Z. G. et al. Accidental degeneracy of double Dirac cones in a phononic crystal. Sci. Rep. 4, 4613 (2014).

    Article  Google Scholar 

  17. Yang, Z. J. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).

    Article  ADS  Google Scholar 

  18. Ni, X. et al. Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow. New J. Phys. 17, 053016 (2015).

    Article  ADS  Google Scholar 

  19. Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015).

    Article  Google Scholar 

  20. Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alu, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).

    Article  ADS  Google Scholar 

  21. He, C. et al. Acoustic topological insulator and robust one-way sound transport. Nat. Phys. 12, 1124–1129 (2016).

    Article  Google Scholar 

  22. Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369 (2017).

    Article  Google Scholar 

  23. Li, F. et al. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30–34 (2017).

    Article  Google Scholar 

  24. He, H. et al. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. Nature 560, 61–64 (2018).

    Article  ADS  Google Scholar 

  25. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  26. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).

    Article  ADS  Google Scholar 

  27. Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).

    Article  ADS  Google Scholar 

  28. Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F. & Brouwer, P. W. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett. 119, 246401 (2017).

    Article  ADS  Google Scholar 

  29. Song, Z. D., Fang, Z. & Fang, C. (d–2)-Dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017).

    Article  ADS  Google Scholar 

  30. Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).

    Article  ADS  Google Scholar 

  31. Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized mircowave quadrupole insulator with topological protected corner states. Nature 555, 346–350 (2018).

    Article  ADS  Google Scholar 

  32. Imhof, S. et al. Topolectrical circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).

    Article  Google Scholar 

  33. Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).

    Article  ADS  Google Scholar 

  34. Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018).

    Article  Google Scholar 

  35. Ma, G. & Sheng, P. Acoustic metamaterials: from local resonances to broad horizons. Sci. Adv. 2, e1501595 (2016).

    Article  ADS  Google Scholar 

  36. Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).

    Article  ADS  Google Scholar 

  37. Jackiw, R. & Rebbi, C. Solitons with fermion number 1/2. Phys. Rev. D 13, 3398–3409 (1976).

    Article  ADS  MathSciNet  Google Scholar 

  38. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).

    Article  ADS  Google Scholar 

  39. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  ADS  Google Scholar 

  40. Jiang, J.-H. & Wu, S. Spin susceptibility and helical magnetic order at the edges/surfaces of topological insulators due to Fermi surface nesting. Phys. Rev. B 83, 205124 (2011).

    Article  ADS  Google Scholar 

  41. Kharitonov, M. Interaction-enhanced magnetically ordered insulating state at the edge of a two-dimensional topological insulator. Phys. Rev. B 86, 165121 (2012).

    Article  ADS  Google Scholar 

Download references

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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