Dual-Space Invariance as a Universal Criterion for Multifractal Critical States

Dual-Space Invariance as a Universal Criterion for Multifractal Critical States | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Dual-Space Invariance as a Universal Criterion for Multifractal Critical States Tong Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9018016/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In Anderson localization, eigenstates of disordered quantum systems are broadly classified as extended, localized, or critical. Although critical states exhibit multifractal character, a precise and operational criterion for their identification remains an open challenge, as Lyapunov exponents in real space cannot uniquely distinguish them from extended states. Here we address this challenge by asserting that critical states are uniquely characterized by an emergent dual-space invariance between position and momentum space. Building on the Liu--Xia criterion of the simultaneous vanishing of Lyapunov exponents ($\gamma=\gamma_m=0$), we show that this dual-space invariance extends beyond Lyapunov exponents and governs wavefunction scaling, revealing a fundamental property inaccessible from either space alone. Through numerical simulations, we demonstrate that the inverse participation ratio exhibits matching scaling behavior in position and momentum space for critical states, in sharp contrast to extended and localized states, which display a pronounced asymmetry between the two spaces. This dual-space invariance provides a direct, robust, and universal criterion for identifying multifractal critical states. Our results establish a fundamental principle of Anderson criticality and open new avenues for its detection in modern quantum simulation platforms. Hard Condensed-matter Physics Theoretical Physics Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9018016","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":599835992,"identity":"56febfed-063c-406e-b8ee-79e25440792f","order_by":0,"name":"Tong Liu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVRIiWNgGAWjYDAC9gYQacPACKYZmAnr4OE5DKLSQFpAuojRIuEMosD6iNRiL8F/8HHBr/N5zDNyjz9gqLBObGA/ewC/LdLNzMYz+24XM87IS2xgOJOe2MCTl4Bfi8xhNmnentuJjTNyDBsY2w4nNkjwGBDwSzJIyzmoln/EauH5cQCqpYEYLWcOGxvzNiQnNva8S5yRcCzduI0nB78W9vbGh495/tglbmzPPfDhQ421bD/7GfxawICxjYHBsIGHgSEByGEjrB4E/jAwyDPwEKd2FIyCUTAKRh4AAJs3RcgHkuYwAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Tong","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2026-03-03 08:39:57","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9018016/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9018016/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104401473,"identity":"ff33931d-316e-4d2b-9ea2-78c2af49c720","added_by":"auto","created_at":"2026-03-11 12:12:46","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":431883,"visible":true,"origin":"","legend":"","description":"","filename":"2411.09067v6.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9018016/v1_covered_3f549c46-dff0-4abf-94bc-dad80073b82c.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eDual-Space Invariance as a Universal Criterion for Multifractal Critical States\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9018016/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9018016/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn Anderson localization, eigenstates of disordered quantum systems are broadly classified as extended, localized, or critical. Although critical states exhibit multifractal character, a precise and operational criterion for their identification remains an open challenge, as Lyapunov exponents in real space cannot uniquely distinguish them from extended states.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eHere we address this challenge by asserting that critical states are uniquely characterized by an emergent dual-space invariance between position and momentum space. Building on the Liu--Xia criterion of the simultaneous vanishing of Lyapunov exponents ($\\gamma=\\gamma_m=0$), we show that this dual-space invariance extends beyond Lyapunov exponents and governs wavefunction scaling, revealing a fundamental property inaccessible from either space alone.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThrough numerical simulations, we demonstrate that the inverse participation ratio exhibits matching scaling behavior in position and momentum space for critical states, in sharp contrast to extended and localized states, which display a pronounced asymmetry between the two spaces.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThis dual-space invariance provides a direct, robust, and universal criterion for identifying multifractal critical states. 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