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# Topological Wannier cycles induced by sub-unit-cell artificial gauge flux in a sonic crystal

## Abstract

Gauge fields play a major role in understanding quantum effects. For example, gauge flux insertion into single unit cells is crucial towards detecting quantum phases and controlling quantum dynamics and classical waves. However, the potential of gauge fields in topological materials studies has not been fully exploited. Here, we experimentally demonstrate artificial gauge flux insertion into a single plaquette of a sonic crystal with a gauge phase ranging from 0 to 2π. We insert the gauge flux through a three-step process of dimensional extension, engineering a screw dislocation and dimensional reduction. Additionally, the single-plaquette gauge flux leads to cyclic spectral flows across multiple bandgaps that manifest as topological boundary states on the plaquette and emerge only when the flux-carrying plaquette encloses the Wannier centres. We termed this phenomenon as the topological Wannier cycle. This work paves the way towards sub-unit-cell gauge flux, enabling future studies on synthetic gauge fields and topological materials.

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## Data availability

All relevant data are presented in detail in the manuscript and the Supplementary Information. Additional information is available from the corresponding authors through reasonable request.

## Code availability

We use the commercial software COMSOL Multiphysics to perform the acoustic wave simulations and eigenstate calculations based on finite-element methods. Reasonable requests to have the computation details can be addressed to the corresponding authors.

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## Acknowledgements

Z.-K.L., B.J., Y.L., S.-Q.W. and J.-H.J. are supported by the National Natural Science Foundation of China (grant no. 12074281), the Jiangsu Province Distinguished Professor Funding and the Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province, Soochow University. J.-H.J. thanks Z.-D. Song for many helpful discussions. He also thanks Z. H. Hang for sharing his laboratory for part of the measurements, and the Huazhong University of Science and Technology, where a large part of the manuscript was finalized, for their hospitality. Y.W. and F.L. are supported by the Guangzhou Basic and Applied Research Foundation (no. 202102020349), the Natural Science Foundation of Guangdong Province (grant no. 2020A1515010549) and China Postdoctoral Science Foundation (grant no. 2020M672615).

## Author information

Authors

### Contributions

J.-H.J. initiated the project and guided the research. J.-H.J. and Z.-K.L. established the theory. Y.W., Z.-K.L., B.J. and Y.L. performed the numerical calculations and simulations. Y.W., Z.-K.L., S.-Q.W., J.-H.J. and F.L. designed and performed the experiments. All the authors contributed to the discussions of the results and the manuscript preparation. J.-H.J., Z.-K.L. and Y.W. wrote the manuscript and the Supplementary Information.

### Corresponding authors

Correspondence to Ying Wu, Feng Li or Jian-Hua Jiang.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Materials thanks Marc Serra-Garcia and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary Information

Supplementary Figs. 1–22, Tables 1 and 2 and Discussion.

## Rights and permissions

Reprints and Permissions

Lin, ZK., Wu, Y., Jiang, B. et al. Topological Wannier cycles induced by sub-unit-cell artificial gauge flux in a sonic crystal. Nat. Mater. 21, 430–437 (2022). https://doi.org/10.1038/s41563-022-01200-w

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• DOI: https://doi.org/10.1038/s41563-022-01200-w

• ### Topological properties that can be heard

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