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

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.

Data availability

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

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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).

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Authors

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.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Supplementary information

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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