Observation of fractal higher-order topological states in acoustic metamaterials  

声学超材料中的分形高阶拓扑态

作　　者：Shengjie Zheng Xianfeng Man Ze-Lin Kong Zhi-Kang Lin Guiju Duan Ning Chen Dejie Yu Jian-Hua Jiang Baizhan Xia 郑圣洁;满先锋;孔泽霖;林志康;段桂菊;陈宁;于德介;蒋建华;夏百战(State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body,Hunan University,Changsha 410082,China;College of Mechanical and Electrical Engineering,Changsha University,Changsha 410022,China;Institute of Theoretical and Applied Physics,School of Physical Science and Technology&Collaborative Innovation Center of Suzhou Nano Science and Technology,Soochow University,Suzhou 215006,China)

机构地区：[1]State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body,Hunan University,Changsha 410082,China [2]College of Mechanical and Electrical Engineering,Changsha University,Changsha 410022,China [3]Institute of Theoretical and Applied Physics,School of Physical Science and Technology&Collaborative Innovation Center of Suzhou Nano Science and Technology,Soochow University,Suzhou 215006,China

出　　处：《Science Bulletin》2022年第20期2069-2075,M0004,共8页科学通报（英文版）

基　　金：supported by the National Natural Science Foundation of China(12125504,12072108,51621004,and 51905162);the Priority Academic Program Development(PAPD)of Jiangsu Higher Education Institutions;the Hunan Provincial Natural Science Foundation of China(2021JJ40626);。

摘　　要：Topological phases of matter have been extensively investigated in solid-state materials and classical wave systems with integer dimensions. However, topological states in non-integer dimensions remain almost unexplored. Fractals, being self-similar on different scales, are one of the intriguing complex geometries with non-integer dimensions. Here, we demonstrate fractal higher-order topological states with unprecedented emergent phenomena in a Sierpin? ski acoustic metamaterial. We uncover abundant topological edge and corner states in the acoustic metamaterial due to the fractal geometry. Interestingly,the numbers of the edge and corner states depend exponentially on the system size, and the leading exponent is the Hausdorff fractal dimension of the Sierpin? ski carpet. Furthermore, the results reveal the unconventional spectrum and rich wave patterns of the corner states with consistent simulations and experiments. This study thus unveils unconventional topological states in fractal geometry and may inspire future studies of topological phenomena in non-Euclidean geometries.近年来,物质的拓扑相在整数维度的固态材料和经典波系统中得到了广泛的研究.然而,非整数维度中的拓扑物态几乎仍未被探索.分形是一类具有非整数维度的有趣且复杂的几何图形,其最大的特性是在不同的尺度上具有自相似性.本文基于谢尔宾斯基地毯结构,设计出一种具有分数维度的分形声学超材料,并证明了其具有分形特征的高阶拓扑态.同时由于该声学超材料的分形几何特性,其具有丰富的拓扑边缘态和拓扑角态.有趣的是,作者发现体态、边缘态和内角态的数量与该声学系统的自相似迭代次数成指数关系,且它们的豪斯多夫维数与系统的豪斯多夫维数相同.此外,通过数值仿真和实验验证,作者揭示了该分形声学超材料的拓扑角态具有新奇的频谱特征和丰富的模态振型.本工作将有助于推动未来非欧几何中拓扑现象的研究.

关 键 词：Higher-order topological states Topological fractals Sierpinski carpet Fractal dimensions Acoustic metamaterials 

分 类 号：TB34[一般工业技术—材料科学与工程]