Observation of ultrametricity in photonic spin glasses

Observation of ultrametricity in photonic spin glasses | 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 Physical Sciences - Article Observation of ultrametricity in photonic spin glasses Neda Ghofraniha, Josivanir Gomes Camara, Stefano Ferretti, Silvia Gentilini, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5433512/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract Hierarchy is the architecture of complex systems [1]. While Euclidean metric describes homogeneous and static systems, ultrametric geometry is for heterogeneous ensembles, hierarchic relations and dynamic contexts. Applications spread across all fields: communication technologies, information retrieval, clusterwise regression, analysis and synthesis of narratives and emotions [2]. The paradigmatic model describing complex systems is the spin glass theory, according to which the topology of the many metastable energy states is intrinsically ultrametric, as a consequence of the breaking of their symmetry [3]. Despite the importance, a real experimental observation of the universal behaviour has not yet been reported, due to the difficulty of finding a physical system that displays all the characteristics of spin glasses. Random lasers have recently been proposed as excellent photonic counterparts of spin glasses with robust replica symmetry breaking features [4–6]. Here, we report the ultrametric structure of the replica space with clustered domains in random lasers, as predicted in the Parisi Ansatz [7]. We show that the number of the states forming isosceles triangles in the metric space increases through the laser threshold giving rise to the construction of a genealogical tree. Moreover, from the hierarchical topology of the states we obtain a direct observation of the complex energy landscape with evident breaking of ergodicity in the glassy regime. Our results, as well as being a milestone in complexity science, fuel the lively research into physical systems which in recent years has been leading to the creation of optical computers and quantum annealers [8–12]. Physical sciences/Physics Physical sciences/Physics/Optical physics Spin Glass Ultrametricity Replica Symmetry Breaking Random Laser Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Under Review 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-5433512","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Physical Sciences - Article","associatedPublications":[],"authors":[{"id":381501480,"identity":"c38365cf-4f57-4a83-b991-4602fdbfbb2f","order_by":0,"name":"Neda Ghofraniha","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0002-4049-8187","institution":"Institute for complex systems","correspondingAuthor":true,"prefix":"","firstName":"Neda","middleName":"","lastName":"Ghofraniha","suffix":""},{"id":381501481,"identity":"540555c0-fe27-44e8-8e50-f06e43f363cc","order_by":1,"name":"Josivanir Gomes Camara","email":"","orcid":"","institution":"Polytechnic School, University of Sao Paulo","correspondingAuthor":false,"prefix":"","firstName":"Josivanir","middleName":"Gomes","lastName":"Camara","suffix":""},{"id":381501482,"identity":"800f8a3c-18f3-46c3-bc93-b552d880bb42","order_by":2,"name":"Stefano Ferretti","email":"","orcid":"","institution":"CNR, Istituto dei Sistemi Complessi","correspondingAuthor":false,"prefix":"","firstName":"Stefano","middleName":"","lastName":"Ferretti","suffix":""},{"id":381501483,"identity":"a51d0caa-23bc-4b68-920c-6beaec03123c","order_by":3,"name":"Silvia Gentilini","email":"","orcid":"","institution":"CNR, Istituto dei Sistemi Complessi","correspondingAuthor":false,"prefix":"","firstName":"Silvia","middleName":"","lastName":"Gentilini","suffix":""},{"id":381501484,"identity":"0c1a9001-4267-40ca-8d62-3488ae147ad0","order_by":4,"name":"Davinson Da Silva","email":"","orcid":"","institution":"Faculdade de Tecnologia de Sao Paulo","correspondingAuthor":false,"prefix":"","firstName":"Davinson","middleName":"Da","lastName":"Silva","suffix":""},{"id":381501485,"identity":"c7258b04-9f22-4983-bf47-f617ffea6cda","order_by":5,"name":"Claudio Conti","email":"","orcid":"https://orcid.org/0000-0003-2583-3415","institution":"Sapienza University of Rome","correspondingAuthor":false,"prefix":"","firstName":"Claudio","middleName":"","lastName":"Conti","suffix":""}],"badges":[],"createdAt":"2024-11-11 16:12:43","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5433512/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5433512/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":73235101,"identity":"6ebee52f-cd5d-4e02-b7fa-39797786fe87","added_by":"auto","created_at":"2025-01-08 04:44:33","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2410992,"visible":true,"origin":"","legend":"Article File","description":"","filename":"RLUltrametricity.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5433512/v1_covered_56aa105c-1668-43fa-af89-0366b799dcd8.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Observation of ultrametricity in photonic spin glasses","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"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":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Spin Glass, Ultrametricity, Replica Symmetry Breaking, Random Laser","lastPublishedDoi":"10.21203/rs.3.rs-5433512/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5433512/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHierarchy is the architecture of complex systems [1]. While Euclidean metric describes homogeneous and static systems, ultrametric geometry is for heterogeneous ensembles, hierarchic relations and dynamic contexts. Applications spread across all fields: communication technologies, information retrieval, clusterwise regression, analysis and synthesis of narratives and emotions [2].\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe paradigmatic model describing complex systems is the spin glass theory, according to which the topology of the many metastable energy states is intrinsically ultrametric, as a consequence of the breaking of their symmetry [3]. Despite the importance, a real experimental observation of the universal behaviour has not yet been reported, due to the difficulty of finding a physical system that displays all the characteristics of spin glasses.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eRandom lasers have recently been proposed as excellent photonic counterparts of spin glasses with robust replica symmetry breaking features [4–6].\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eHere, we report the ultrametric structure of the replica space with clustered domains in random lasers, as predicted in the Parisi Ansatz [7]. We show that the number of the states forming isosceles triangles in the metric space increases through the laser threshold giving rise to the construction of a genealogical tree. Moreover, from the hierarchical topology of the states we obtain a direct observation of the complex energy landscape with evident breaking of ergodicity in the glassy regime. 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