{"title":"Towards full instanton trans-series in Hofstadter’s butterfly","authors":"Jie Gu, Zhaojie Xu","doi":"10.1007/JHEP02(2025)099","DOIUrl":null,"url":null,"abstract":"<p>The trans-series completion of perturbative series of a wide class of quantum mechanical systems can be determined by combining the resurgence program with extra input coming from exact WKB analysis. In this paper, we reexamine the Harper-Hofstadter model and its spectrum, Hofstadter’s butterfly in light of recent developments. We demonstrate the connection between the perturbative energy series of the Harper-Hofstadter model and the vev of 1/2-BPS Wilson loop of 5d SYM and clarify the differences between their non-perturbative corrections. Taking insights from the cosine potential model, we construct the full energy trans-series for flux <i>ϕ</i> = 2<i>π</i>/<i>Q</i> and provide numerical evidence with remarkably high precision. Finally, we revisit the problem of self-similarity of the butterfly and discuss the possibility of a completed version of the Rammal-Wilkinson formula.</p>","PeriodicalId":635,"journal":{"name":"Journal of High Energy Physics","volume":"2025 2","pages":""},"PeriodicalIF":5.4000,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/JHEP02(2025)099.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of High Energy Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/JHEP02(2025)099","RegionNum":1,"RegionCategory":"物理与天体物理","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
The trans-series completion of perturbative series of a wide class of quantum mechanical systems can be determined by combining the resurgence program with extra input coming from exact WKB analysis. In this paper, we reexamine the Harper-Hofstadter model and its spectrum, Hofstadter’s butterfly in light of recent developments. We demonstrate the connection between the perturbative energy series of the Harper-Hofstadter model and the vev of 1/2-BPS Wilson loop of 5d SYM and clarify the differences between their non-perturbative corrections. Taking insights from the cosine potential model, we construct the full energy trans-series for flux ϕ = 2π/Q and provide numerical evidence with remarkably high precision. Finally, we revisit the problem of self-similarity of the butterfly and discuss the possibility of a completed version of the Rammal-Wilkinson formula.
期刊介绍：
The aim of the Journal of High Energy Physics (JHEP) is to ensure fast and efficient online publication tools to the scientific community, while keeping that community in charge of every aspect of the peer-review and publication process in order to ensure the highest quality standards in the journal.
Consequently, the Advisory and Editorial Boards, composed of distinguished, active scientists in the field, jointly establish with the Scientific Director the journal''s scientific policy and ensure the scientific quality of accepted articles.
JHEP presently encompasses the following areas of theoretical and experimental physics:
Collider Physics
Underground and Large Array Physics
Quantum Field Theory
Gauge Field Theories
Symmetries
String and Brane Theory
General Relativity and Gravitation
Supersymmetry
Mathematical Methods of Physics
Mostly Solvable Models
Astroparticles
Statistical Field Theories
Mostly Weak Interactions
Mostly Strong Interactions
Quantum Field Theory (phenomenology)
Strings and Branes
Phenomenological Aspects of Supersymmetry
Mostly Strong Interactions (phenomenology).
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