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# Observation of higher-order topological acoustic states protected by generalized chiral symmetry

## Abstract

Topological systems are inherently robust to disorder and continuous perturbations, resulting in dissipation-free edge transport of electrons in quantum solids, or reflectionless guiding of photons and phonons in classical wave systems characterized by topological invariants. Recently, a new class of topological materials characterized by bulk polarization has been introduced, and was shown to host higher-order topological corner states. Here, we demonstrate theoretically and experimentally that 3D-printed two-dimensional acoustic meta-structures can possess nontrivial bulk topological polarization and host one-dimensional edge and Wannier-type second-order zero-dimensional corner states with unique acoustic properties. We observe second-order topological states protected by a generalized chiral symmetry of the meta-structure, which are localized at the corners and are pinned to ‘zero energy’. Interestingly, unlike the ‘zero energy’ states protected by conventional chiral symmetry, the generalized chiral symmetry of our three-atom sublattice enables their spectral overlap with the continuum of bulk states without leakage. Our findings offer possibilities for advanced control of the propagation and manipulation of sound, including within the radiative continuum.

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## Data availability

Data that are not already included in the paper and/or in the Supplementary Information are available on request from the authors.

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## Acknowledgements

The work was supported by the Defense Advanced Research Projects Agency under the Nascent programme with grant number HR00111820040, and by the National Science Foundation with grant numbers CMMI-1537294, EFRI-1641069 and DMR-1809915. Research was carried out in part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the US Department of Energy, Office of Basic Energy Sciences, under contract number DE-SC0012704.

## Author information

Authors

### Contributions

All authors contributed extensively to the work presented in this paper.

### Corresponding authors

Correspondence to Andrea Alù or Alexander B. Khanikaev.

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### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary Sections 1–11, Supplementary Figures 1–9, Supplementary References 1–8

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Ni, X., Weiner, M., Alù, A. et al. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nature Mater 18, 113–120 (2019). https://doi.org/10.1038/s41563-018-0252-9

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