{"title":"Applications of Chebyshev polynomials and Toeplitz theory to topological metamaterials","authors":"Habib Ammari , Silvio Barandun , Ping Liu","doi":"10.1016/j.revip.2025.100103","DOIUrl":null,"url":null,"abstract":"<div><div>We survey the use of Chebyshev polynomials and Toeplitz theory for the study of topological metamaterials. We consider both Hermitian and non-Hermitian systems of subwavelength resonators and provide a mathematical framework to quantitatively explain and characterise some spectacular properties of metamaterials. Our characterisations are based on translation invariance properties of the capacitance matrices associated to the different investigated systems of resonators together with properties of Chebyshev polynomials. The three-term recurrence relation satisfied by the Chebyshev polynomials is shown to be the key to the mathematical analysis of spectra of tridiagonal (perturbed) both Toeplitz (for monomer systems) and 2-Toeplitz (for dimer systems) capacitance matrices.</div></div>","PeriodicalId":37875,"journal":{"name":"Reviews in Physics","volume":"13 ","pages":"Article 100103"},"PeriodicalIF":0.0000,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reviews in Physics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2405428325000036","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 0
Abstract
We survey the use of Chebyshev polynomials and Toeplitz theory for the study of topological metamaterials. We consider both Hermitian and non-Hermitian systems of subwavelength resonators and provide a mathematical framework to quantitatively explain and characterise some spectacular properties of metamaterials. Our characterisations are based on translation invariance properties of the capacitance matrices associated to the different investigated systems of resonators together with properties of Chebyshev polynomials. The three-term recurrence relation satisfied by the Chebyshev polynomials is shown to be the key to the mathematical analysis of spectra of tridiagonal (perturbed) both Toeplitz (for monomer systems) and 2-Toeplitz (for dimer systems) capacitance matrices.
期刊介绍:
Reviews in Physics is a gold open access Journal, publishing review papers on topics in all areas of (applied) physics. The journal provides a platform for researchers who wish to summarize a field of physics research and share this work as widely as possible. The published papers provide an overview of the main developments on a particular topic, with an emphasis on recent developments, and sketch an outlook on future developments. The journal focuses on short review papers (max 15 pages) and these are freely available after publication. All submitted manuscripts are fully peer-reviewed and after acceptance a publication fee is charged to cover all editorial, production, and archiving costs.