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. 2019 Nov 25;10:5331. doi: 10.1038/s41467-019-13333-9

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© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 1 — Dimensional hierarchy of the higher-order topology. a Schematics of the undeformed (left, gapless), shrunken (middle, trivial gap), and expanded (right, topological gap) lattices. The unit cell and its cross-section for each configuration are depicted in the insets. For the planar schematics in the insets, the purple regions denote the air cavities, while the blue regions denote the air channels. Calculated acoustic band structures for b, shrunken and c, expanded lattices are presented (the Brillouin zone and unit cells are shown in the insets). Acoustic wavefunctions at the high-symmetry points of the Brillouin zone and the mirror eigenvalues are also shown in the insets. d, e Wannier bands $ν_{z} (k_{x}, k_{y})$ for the trivial and topological sonic crystals (SCs), respectively. f Higher-order bulk-boundary correspondences as manifestation of the first-, second-, and third-order topology. The trivial phase with bulk states is also presented as comparison.