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Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions

Nature Physics volume 17, pages 1239–1246 (2021)Cite this article

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Abstract

Topological phases of matter connect mathematical principles to real materials, and may shape future electronic and quantum technologies. So far, this discipline has mostly focused on single-gap topology described by topological invariants such as Chern numbers. Here, based on a tunable kagome model, we observe non-Abelian band topology and its transitions in acoustic semimetals, in which the multi-gap Hilbert space plays a key role. In non-Abelian semimetals, the topological charges of band nodes are converted through the braiding of nodes in adjacent gaps, and their behaviour cannot be captured by conventional topological band theory. Using kagome acoustic metamaterials and pump–probe measurements, we demonstrate the emergence of non-Abelian topological nodes, identify their dispersions and observe the induced multi-gap topological edge states. By controlling the geometry of the metamaterials, topological transitions are induced by the creation, annihilation, merging and splitting of band nodes. This reveals the underlying rules for the conversion and transfer of non-Abelian topological charges in multiple bandgaps. The resulting laws that govern the evolution of band nodes in non-Abelian multi-gap systems should inspire studies on multi-band topological semimetals and multi-gap topological out-of-equilibrium systems.

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Fig. 1: Conventional single-gap topology versus multi-gap non-Abelian topology.
Fig. 2: Non-Abelian topological band nodes and their evolutions.
Fig. 3: Kagome acoustic metamaterials and non-Abelian topological nodes.
Fig. 4: Dispersions of non-Abelian topological nodes.
Fig. 5: Multi-gap bulk-edge physics in kagome acoustic metamaterials.

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Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

References

  1. Kruthoff, J., de Boer, J., van Wezel, J., Kane, C. L. & Slager, R.-J. Topological classification of crystalline insulators through band structure combinatorics. Phys. Rev. X 7, 041069 (2017).

    Google Scholar 

  2. Po, H. C., Vishwanath, A. & Watanabe, H. Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).

    Article  ADS  Google Scholar 

  3. Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

    Article  ADS  Google Scholar 

  4. Wu, Q., Soluyanov, A. A. & Bzdusek, T. Non-Abelian band topology in noninteracting metals. Science 365, 1273–1277 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  5. Ahn, J., Park, S. & Yang, B.-J. Failure of Nielsen–Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle. Phys. Rev. X 9, 021013 (2019).

    Google Scholar 

  6. Bouhon, A. et al. Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe. Nat. Phys. 16, 1137–1143 (2020).

    Article  Google Scholar 

  7. Yang, B.-J. & Nagaosa, N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nat. Commun. 5, 4898 (2014).

    Article  ADS  Google Scholar 

  8. Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  9. Lenggenhager, P. M., Liu, X., Tsirkin, S. S., Neupert, T. & Bzdusek, T. From triple-point materials to multiband nodal links. Phys. Rev. B 103, 121101 (2021).

    Article  ADS  Google Scholar 

  10. Zhu, Z., Winkler, G. W., Wu, Q., Li, J. & Soluyanov, A. A. Triple point topological metals. Phys. Rev. X 6, 031003 (2016).

    Google Scholar 

  11. Kim, J., Kim, H.-S. & Vanderbilt, D. Nearly triple nodal point topological phase in half-metallic GdN. Phys. Rev. B 98, 155122 (2018).

    Article  ADS  Google Scholar 

  12. Yang, Y. et al. Topological triply degenerate point with double Fermi arcs. Nat. Phys. 15, 645–649 (2019).

    Article  Google Scholar 

  13. Johansson, N. & Sjoqvist, E. Optimal topological test for degeneracies of real Hamiltonians. Phys. Rev. Lett. 92, 060406 (2004).

    Article  ADS  Google Scholar 

  14. Bouhon, A., Bzdusek, T. & Slager, R.-J. Geometric approach to fragile topology beyond symmetry indicators. Phys. Rev. B 102, 115135 (2020).

    Article  ADS  Google Scholar 

  15. Tiwari, A. & Bzdusek, T. Non-Abelian topology of nodal-line rings in PT-symmetric systems. Phys. Rev. B 101, 195130 (2020).

    Article  ADS  Google Scholar 

  16. Unal, F. N., Bouhon, A. & Slager, R.-J. Topological Euler class as a dynamical observable in optical lattices. Phys. Rev. Lett. 125, 053601 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  17. Po, H. C., Watanabe, H. & Vishwanath, A. Fragile topology and Wannier obstructions. Phys. Rev. Lett. 121, 126402 (2018).

    Article  ADS  Google Scholar 

  18. Bouhon, A., Black-Schaffer, A. M. & Slager, R.-J. Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry. Phys. Rev. B 100, 195135 (2019).

    Article  ADS  Google Scholar 

  19. Song, Z., Elcoro, L., Regnault, N. & Bernevig, B. A. Fragile phases as affinemonoids: full classification and material examples. Phys. Rev. X 10, 031001 (2020).

    Google Scholar 

  20. Song, Z. D., Elcoro, L. & Bernevig, B. A. Twisted bulk-boundary correspondence of fragile topology. Science 367, 794–797 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  21. Peri, V. et al. Experimental characterization of fragile topology in an acoustic metamaterial. Science 367, 797–800 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  23. Slager, R.-J., Mesaros, A., Juričić, V. & Zaanen, J. The space group classification of topological band-insulators. Nat. Phys. 9, 98–102 (2013).

    Article  MATH  Google Scholar 

  24. Bouhon, A. & Black-Schaffer, A. M. Global band topology of simple and double Dirac-point semimetals. Phys. Rev. B 95, 241101 (2017).

    Article  ADS  Google Scholar 

  25. Slager, R.-J. The translational side of topological band insulators. J. Phys. Chem. Solids 128, 24 (2019).

    Article  ADS  Google Scholar 

  26. Fang, C., Gilbert, M. J. & Bernevig, B. A. Bulk topological invariants in noninteracting point group symmetric insulators. Phys. Rev. B 86, 115112 (2012).

    Article  ADS  Google Scholar 

  27. Cornfeld, E. & Carmeli, S. Tenfold topology of crystals: unified classification of crystalline topological insulators and superconductors. Phys. Rev. Res. 3, 013052 (2021).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  29. Xiao, M., Chen, W.-J., He, W.-Y. & Chan, C. T. Synthetic gauge flux and Weyl points in acoustic systems. Nat. Phys. 11, 920–924 (2015).

    Article  Google Scholar 

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

    Article  Google Scholar 

  31. Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369–374 (2017).

    Article  Google Scholar 

  32. Li, F., Huang, X., Lu, J., Ma, J. & Liu, Z. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30–34 (2017).

    Article  Google Scholar 

  33. 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 

  34. Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. Nat. Mater. 18, 108–112 (2019).

    Article  Google Scholar 

  35. Ni, X., Weiner, M., Alu, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).

    Article  Google Scholar 

  36. Zhang, X. et al. Second-order topology and multidimensional topological transitions in sonic crystals. Nat. Phys. 15, 582–588 (2019).

    Article  Google Scholar 

  37. Ma, G., Xiao, M. & Chan, C. T. Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281–294 (2019).

    Article  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. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).

    Article  ADS  Google Scholar 

  40. Guo, Q. et al. Experimental observation of non-Abelian topological charges and edge states. Nature 594, 195–200 (2021).

    Article  ADS  Google Scholar 

  41. Alexander, G. P., Chen, B.-G., Matsumoto, E. A. & Kamien, R. D. Colloquium: disclination loops, point defects, and all that in nematic liquid crystals. Rev. Mod. Phys. 84, 497–514 (2012).

    Article  ADS  Google Scholar 

  42. Liu, K., Nissinen, J., Slager, R.-J., Wu, K. & Zaanen, J. Generalized liquid crystals: giant fluctuations and the vestigial chiral order of I, O and T matter. Phys. Rev. X 6, 041025 (2016).

    Google Scholar 

  43. Volovik, G. E. & Mineev, V. P. in Basic Notions of Condensed Matter Physics 392–401 (CRC Press, 2018).

  44. Beekman, A. J. et al. Dual gauge field theory of quantum liquid crystals in two dimensions. Phys. Rep. 683, 1–110 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Bzdusek, T. Euler class of a Pair of Energy Bands on a Manifold with a Boundary Publicly available mathematica code (ResearchGate, 2019).

  46. Bouhon, A., Lange, G. F. & Slager, R. J. Topological correspondence between magnetic space group representations and subdimensions. Phys. Rev. B 103, 245127 (2021).

    Article  ADS  Google Scholar 

  47. Ahn, J., Kim, D., Kim, Y. & Yang, B.-J. Band topology and linking structure of nodal line semimetals with Z2 monopole charges. Phys. Rev. Lett. 121, 106403 (2018).

    Article  ADS  Google Scholar 

  48. Lange, G. F., Bouhon, A. & Slager, R. J. Subdimensional topologies, indicators and higher order boundary effects. Phys. Rev. B 103, 195145 (2021).

    Article  ADS  Google Scholar 

  49. Hahn, Th. (ed.) International Tables for Crystallography. Volume A: Space-Group Symmetry Online edition (IUCr, 2006); https://it.iucr.org/Ab/

  50. Hahn, Th. (ed.) International Tables for Crystallography. Volume E: Subperiodic Groups Online Edition (IUCr, 2010); https://it.iucr.org/E/

Download references

Acknowledgements

B.J., Z.-K.L. and J.-H.J. are supported by the National Natural Science Foundation of China (grant no. 12074281) and Jiangsu Distinguished Professor Funding. X.Z. and B.H. are supported by the National Natural Science Foundation of China (grant no. 12074279), the Major Program of Natural Science Research of Jiangsu Higher Education Institutions (grant no. 18KJA140003). The work at Soochow University is also supported by Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. R.-J.S. acknowledges funding from the Marie-Skłodowska-Curie programme under EC grant no. 842901, the Winton programme as well as Trinity College at the University of Cambridge. F.L. is supported by the Natural Science Foundation of Guangdong Province (no. 2020A1515010549).

Author information

Author notes

  1. These authors contributed equally: Bin Jiang, Adrien Bouhon, Zhi-Kang Lin, Robert-Jan Slager, Jian-Hua Jiang.

Authors and Affiliations

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

    Bin Jiang, Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou & Jian-Hua Jiang

  2. Nordic Institute for Theoretical Physics (NORDITA), Stockholm, Sweden

    Adrien Bouhon

  3. School of Physics, Beijing Institute of Technology, Beijing, China

    Feng Li

  4. TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge, UK

    Robert-Jan Slager

  5. Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province & Key Lab of Modern Optical Technologies of Ministry of Education, Soochow University, Suzhou, China

    Jian-Hua Jiang

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Contributions

A.B. and R.-J.S. performed the theory analysis underpinning the project. B.J. and J.-H.J. designed the metamaterials. B.J., Z.-K.L., X.Z., B.H., F.L. and J.-H.J. performed the experiments. A.B., R.-J.S. and J.-H.J. wrote the manuscript and the Supplementary Information, with input from all authors.

Corresponding authors

Correspondence to Adrien Bouhon, Robert-Jan Slager or Jian-Hua Jiang.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Schematic illustration of the rules for the momentum-space braiding of non-Abelian topological nodes and their charge conversions with the Dirac strings (DSs).

The left column shown the effect of a DS residing in the second gap on nodes in the first gap. The charge inversion processes of pulling a first-gap node through this DS or subsequently retracting the string over the other DS that connects the pair in the first gap is illustrated in the bottom panels, respectively. The right column depicts the recombination rules of Dirac strings as outlined in the main text. Every panel is merely a different representation of the same physics.

Extended Data Fig. 2 Schematic illustration of the rules for the momentum-space braiding of non-Abelian topological nodes and their charge conversions with the Dirac strings (DSs).

The left column shown the effect of a DS residing in the second gap on nodes in the first gap. The charge inversion processes of pulling a first-gap node through this DS or subsequently retracting the string over the other DS that connects the pair in the first gap is illustrated in the bottom panels, respectively. The right column depicts the recombination rules of Dirac strings as outlined in the main text. Every panel is merely a different representation of the same physics.

Extended Data Fig. 3 Obstructions to nodes annihilation through the braiding of nodes on the (Brillouin zone) torus.

In a the moving node does not cross the whole Brillouin zone, in b (AB) and b (CD) the moving node crosses the Brillouin zone along one of the cyclic direction of the torus leaving a non-contractible DS behind. The patch Euler class (\(\xi ({{{\mathcal{D}}}})\)) and the topological configuration after braiding are fully determined by the nodes charges and the Dirac strings of the initial configuration.

Extended Data Fig. 4 Multi-gap topology in kagome models.

a, Taking \((\epsilon _{{{\mathrm{A}}}},\epsilon _{{{\mathrm{B}}}},\epsilon _{{{\mathrm{C}}}})\) = (1, 0, −1) and \((t,t\prime ) = (0,0)\) gives crossing Dirac strings (DS) in both gaps (blue for first gap, red for second gap). b and c, As the next step, turning off the onsite potentials and switching on the hopping terms induces band nodes. The nodes in the second gap (filled/empty red circles indicating ± topological charges) cross the DS in the first gap, forming a stable pair as the double node at Γ (brown circle) in (c) which has finite patch Euler class ξ = 1. Meanwhile, the first gap features nodes at K points (triangles). c-i, Braiding process and transfer of band nodes from one gap to another through triple points. Band nodes and DS strings evolve such that the degeneracy at K in the first gap (blue triangles) is tuned into a double node configuration that has finite patch Euler class in the second gap (brown circles).

Extended Data Fig. 5 Triply-degenerate points at K and M, and their frame charges.

Band structure in the vicinity of the triply-degenerate points K (a) and M (b), with the encircling base loops lK and lM. c The accumulated geometric frame angle computed over the base loops lK (full line) and lM (dashed line). The triply-degenerate point at K exhibits a total frame angle of π indicating the non-Abelian frame charge k, while the triply-degenerate point at M exhibits a total frame angle of 2π indicating the non-Abelian frame charge of −1.

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Supplementary Information

Supplementary Figs. 1–10, Discussion and Table 1.

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Jiang, B., Bouhon, A., Lin, ZK. et al. Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions. Nat. Phys. 17, 1239–1246 (2021). https://doi.org/10.1038/s41567-021-01340-x

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