{"title":"一维声子晶体中的表面阻抗和拓扑保护界面模式","authors":"A. Coutant, B. Lombard","doi":"10.1098/rspa.2023.0533","DOIUrl":null,"url":null,"abstract":"When semi-infinite phononic crystals (PCs) are in contact, localized modes may exist at their boundary. The central question is generally to predict their existence and to determine their stability. With the rapid expansion of the field of topological insulators, powerful tools have been developed to address these questions. In particular, when applied to one-dimensional systems with mirror symmetry, the bulk-boundary correspondence claims that the existence of interface modes is given by a topological invariant computed from the bulk properties of the PC, which ensures strong stability properties. This one-dimensional bulk-boundary correspondence has been proven in various works. Recent attempts have exploited the notion of surface impedance, relying on analytical calculations of the transfer matrix. In the present work, the monotonic evolution of surface impedance with frequency is proven for all one-dimensional PCs with mirror symmetry. This result allows us to establish a stronger version of the bulk-boundary correspondence that guarantees not only the existence but also the uniqueness of a topologically protected interface state. This correspondence is extended to a larger class of one-dimensional models that include imperfect interfaces, array of resonators, or dispersive media. Numerical simulations are proposed to illustrate the theoretical findings.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"48 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Surface impedance and topologically protected interface modes in one-dimensional phononic crystals\",\"authors\":\"A. Coutant, B. Lombard\",\"doi\":\"10.1098/rspa.2023.0533\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"When semi-infinite phononic crystals (PCs) are in contact, localized modes may exist at their boundary. The central question is generally to predict their existence and to determine their stability. With the rapid expansion of the field of topological insulators, powerful tools have been developed to address these questions. In particular, when applied to one-dimensional systems with mirror symmetry, the bulk-boundary correspondence claims that the existence of interface modes is given by a topological invariant computed from the bulk properties of the PC, which ensures strong stability properties. This one-dimensional bulk-boundary correspondence has been proven in various works. Recent attempts have exploited the notion of surface impedance, relying on analytical calculations of the transfer matrix. In the present work, the monotonic evolution of surface impedance with frequency is proven for all one-dimensional PCs with mirror symmetry. This result allows us to establish a stronger version of the bulk-boundary correspondence that guarantees not only the existence but also the uniqueness of a topologically protected interface state. This correspondence is extended to a larger class of one-dimensional models that include imperfect interfaces, array of resonators, or dispersive media. Numerical simulations are proposed to illustrate the theoretical findings.\",\"PeriodicalId\":20716,\"journal\":{\"name\":\"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences\",\"volume\":\"48 1\",\"pages\":\"\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2024-01-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences\",\"FirstCategoryId\":\"103\",\"ListUrlMain\":\"https://doi.org/10.1098/rspa.2023.0533\",\"RegionNum\":3,\"RegionCategory\":\"综合性期刊\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","FirstCategoryId":"103","ListUrlMain":"https://doi.org/10.1098/rspa.2023.0533","RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
引用次数: 0
摘要
当半无限声子晶体(PC)接触时,它们的边界可能存在局部模式。核心问题通常是预测它们的存在并确定其稳定性。随着拓扑绝缘体领域的迅速发展,人们开发出了强大的工具来解决这些问题。特别是,当应用于具有镜像对称性的一维系统时,体-界对应关系声称界面模式的存在是由 PC 的体特性计算出的拓扑不变量给出的,这确保了强大的稳定性。这种一维体-边界对应关系已在各种研究中得到证实。最近的尝试利用了表面阻抗的概念,依赖于传递矩阵的分析计算。在本研究中,我们证明了所有具有镜像对称性的一维 PC 的表面阻抗随频率的单调演化。这一结果使我们建立了更强版本的体界对应关系,不仅保证了拓扑保护界面态的存在性,而且保证了其唯一性。这一对应关系被扩展到包括不完美界面、谐振器阵列或色散介质在内的更大一类一维模型。我们提出了数值模拟来说明理论发现。
Surface impedance and topologically protected interface modes in one-dimensional phononic crystals
When semi-infinite phononic crystals (PCs) are in contact, localized modes may exist at their boundary. The central question is generally to predict their existence and to determine their stability. With the rapid expansion of the field of topological insulators, powerful tools have been developed to address these questions. In particular, when applied to one-dimensional systems with mirror symmetry, the bulk-boundary correspondence claims that the existence of interface modes is given by a topological invariant computed from the bulk properties of the PC, which ensures strong stability properties. This one-dimensional bulk-boundary correspondence has been proven in various works. Recent attempts have exploited the notion of surface impedance, relying on analytical calculations of the transfer matrix. In the present work, the monotonic evolution of surface impedance with frequency is proven for all one-dimensional PCs with mirror symmetry. This result allows us to establish a stronger version of the bulk-boundary correspondence that guarantees not only the existence but also the uniqueness of a topologically protected interface state. This correspondence is extended to a larger class of one-dimensional models that include imperfect interfaces, array of resonators, or dispersive media. Numerical simulations are proposed to illustrate the theoretical findings.
期刊介绍:
Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.