{"title":"Counting Rules of Nambu–Goldstone Modes","authors":"Haruki Watanabe","doi":"10.1146/annurev-conmatphys-031119-050644","DOIUrl":null,"url":null,"abstract":"When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu–Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be lower than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulae for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. Along the way we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb–Schultz–Mattis theorem.","PeriodicalId":7925,"journal":{"name":"Annual Review of Condensed Matter Physics","volume":" ","pages":""},"PeriodicalIF":14.3000,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1146/annurev-conmatphys-031119-050644","citationCount":"49","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annual Review of Condensed Matter Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1146/annurev-conmatphys-031119-050644","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, CONDENSED MATTER","Score":null,"Total":0}
引用次数: 49
Abstract
When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu–Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be lower than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulae for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. Along the way we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb–Schultz–Mattis theorem.
期刊介绍:
Since its inception in 2010, the Annual Review of Condensed Matter Physics has been chronicling significant advancements in the field and its related subjects. By highlighting recent developments and offering critical evaluations, the journal actively contributes to the ongoing discourse in condensed matter physics. The latest volume of the journal has transitioned from gated access to open access, facilitated by Annual Reviews' Subscribe to Open initiative. Under this program, all articles are now published under a CC BY license, ensuring broader accessibility and dissemination of knowledge.