{"title":"Topological Quantum Numbers in Quasicrystals","authors":"Johannes Kellendonk","doi":"10.1002/ijch.202400027","DOIUrl":null,"url":null,"abstract":"<p>We provide an overview on the theory of topological quantum numbers from the point of view of non-commutative topology. Topological phases are described by <i>K</i>-groups of <i>C</i>*-algebras. The algebras are constructed from the set of positions of the nuclei of the materials we want to study. Topological quantum numbers are Chern numbers of <i>K</i>-group elements. Maps between <i>K</i>-groups which are of algebraic topological origin provide the means to obtain relations between different topological quantum numbers as, for instance, in the bulk edge correspondence. We present simple aperiodic examples related to quasicrystals to illustrate the theory.</p>","PeriodicalId":14686,"journal":{"name":"Israel Journal of Chemistry","volume":"64 10-11","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Chemistry","FirstCategoryId":"92","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/ijch.202400027","RegionNum":4,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We provide an overview on the theory of topological quantum numbers from the point of view of non-commutative topology. Topological phases are described by K-groups of C*-algebras. The algebras are constructed from the set of positions of the nuclei of the materials we want to study. Topological quantum numbers are Chern numbers of K-group elements. Maps between K-groups which are of algebraic topological origin provide the means to obtain relations between different topological quantum numbers as, for instance, in the bulk edge correspondence. We present simple aperiodic examples related to quasicrystals to illustrate the theory.
我们从非交换拓扑学的角度概述了拓扑量子数理论。拓扑相由 C* 矩阵的 K 群描述。这些代数是由我们要研究的材料的原子核位置集合构建的。拓扑量子数是 K 群元素的切尔数。拓扑代数的 K 群之间的映射为获得不同拓扑量子数之间的关系提供了方法，例如在体边对应关系中。我们列举了与类晶体有关的简单非周期性例子来说明这一理论。
期刊介绍：
The fledgling State of Israel began to publish its scientific activity in 1951 under the general heading of Bulletin of the Research Council of Israel, which quickly split into sections to accommodate various fields in the growing academic community. In 1963, the Bulletin ceased publication and independent journals were born, with Section A becoming the new Israel Journal of Chemistry.
The Israel Journal of Chemistry is the official journal of the Israel Chemical Society. Effective from Volume 50 (2010) it is published by Wiley-VCH.
The Israel Journal of Chemistry is an international and peer-reviewed publication forum for Special Issues on timely research topics in all fields of chemistry: from biochemistry through organic and inorganic chemistry to polymer, physical and theoretical chemistry, including all interdisciplinary topics. Each topical issue is edited by one or several Guest Editors and primarily contains invited Review articles. Communications and Full Papers may be published occasionally, if they fit with the quality standards of the journal. The publication language is English and the journal is published twelve times a year.
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