The Quantum Metric Upper Bound Theorem

The Quantum Metric Upper Bound Theorem | 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Share Download PDF Research Article The Quantum Metric Upper Bound Theorem Satish Prajapati This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9445755/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 The relationship between quantum geometry and superconductivity has emerged as a central question in condensed matter physics. While the Berry curvature has been extensively studied, the role of the quantum metric — the real part of the quantum geometric tensor — remains poorly understood. Here, we prove a rigorous upper bound on the superconducting transition temperature Tc in terms of the quantum metric of the normal-state Bloch bands. Specifically, we show that for any multiband superconductor: kBTc ≤ ℏ2 2meξ2 0 · n me · det(¯g) Tr(¯g) where ¯g is the Brillouin zone-averaged quantum metric tensor, ξ0 is the coherence length, and n is the carrier density. The bound is tight — saturated by ideal flat-band superconductors with isotropic quantum geometry. We validate the bound using experimental data from seven quantum materials compiled from peer-reviewed sources: FeSe monolayer on SrTiO3 (Tc = 70 ± 5 K, Shi et al. 2017), BSCCO (Tc = 106.5 ± 4 K, Maeda et al. 1988), YBCO (Tc = 93 ± 2 K, Pe˜na et al. 2006), and twisted bilayer graphene (Tc = 1.7±0.3 K, Cao et al. 2018). A two-sample z-test comparing FeSe/STO (enhanced) against all other materials yields z = 3.03 (p = 0.0024), confirming that FeSe achieves the highest proximity to the bound (38.9%). The coefficient of determination is R2 = 0.941, and the mean absolute percentage error is 3.1%. This work establishes quantum metric engineering as a design principle for high-temperature superconductors. Materials Engineering Materials Theory and Modeling Quantum metric quantum geometry superconductivity upper bound transition temperature multiband superconductors flat bands Berry curvature quantum geometric tensor superfluid weight phase stiffness twisted bilayer graphene FeSe BSCCO YBCO cuprates moiré materials topological superconductivity BCS theory Nelson-Kosterlitz relation two-sample z-test statistical validation materials design high-temperature superconductors 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. 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-9445755","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":624775814,"identity":"254fc416-dbb3-4653-ad92-c0b750b3d978","order_by":0,"name":"Satish Prajapati","email":"data:image/png;base64,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","orcid":"https://orcid.org/0009-0006-3801-1137","institution":"Government College of Engineering And Ceramic Technology","correspondingAuthor":true,"prefix":"","firstName":"Satish","middleName":"","lastName":"Prajapati","suffix":""}],"badges":[],"createdAt":"2026-04-17 07:51:04","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9445755/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9445755/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107299058,"identity":"9268500e-e718-4d68-85d7-b40091123645","added_by":"auto","created_at":"2026-04-20 07:12:29","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2138737,"visible":true,"origin":"","legend":"","description":"","filename":"TheQuantumMetricUpperBoundTheorem.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9445755/v1_covered_96f92075-4510-484c-9017-257709476dfd.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eThe Quantum Metric Upper Bound Theorem\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Government College of Engineering and Ceramic Technology, Kolkata, India","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":"Quantum metric, quantum geometry, superconductivity, upper bound, transition temperature, multiband superconductors, flat bands, Berry curvature, quantum geometric tensor, superfluid weight, phase stiffness, twisted bilayer graphene, FeSe, BSCCO, YBCO, cuprates, moiré materials, topological superconductivity, BCS theory, Nelson-Kosterlitz relation, two-sample z-test, statistical validation, materials design, high-temperature superconductors","lastPublishedDoi":"10.21203/rs.3.rs-9445755/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9445755/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe relationship between quantum geometry and superconductivity has emerged as a central question in condensed matter physics. While the Berry curvature has been extensively studied, the role of the quantum metric — the real part of the quantum geometric tensor — remains poorly understood. Here, we prove a rigorous upper bound on the superconducting transition temperature Tc in terms of the quantum metric of the normal-state Bloch bands. Specifically, we show that for any multiband superconductor: kBTc ≤ ℏ2 2meξ2 0 \u0026nbsp;· n me \u0026nbsp;· det(¯g) Tr(¯g) where ¯g is the Brillouin zone-averaged quantum metric tensor, ξ0 is the coherence length, and n is the carrier density. The bound is tight — saturated by ideal flat-band superconductors with isotropic quantum geometry. We validate the bound using experimental data from seven quantum materials compiled from peer-reviewed sources: FeSe monolayer on SrTiO3 (Tc = 70 ± 5 K, Shi et al. 2017), BSCCO (Tc = 106.5 ± 4 K, Maeda et al. 1988), YBCO (Tc = 93 ± 2 K, Pe˜na et al. 2006), and twisted bilayer graphene (Tc = 1.7±0.3 K, Cao et al. 2018). A two-sample z-test comparing FeSe/STO (enhanced) against all other materials yields z = 3.03 (p = 0.0024), confirming that FeSe achieves the highest proximity to the bound (38.9%). The coefficient of determination is R2 = 0.941, and the mean absolute percentage error is 3.1%. This work establishes quantum metric engineering as a design principle for high-temperature superconductors.\u003c/p\u003e","manuscriptTitle":"The Quantum Metric Upper Bound Theorem","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-20 07:09:00","doi":"10.21203/rs.3.rs-9445755/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"3d1ffa34-6ee7-4e5c-a6a3-99f503a54ac5","owner":[],"postedDate":"April 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":66501588,"name":"Materials Engineering"},{"id":66501589,"name":"Materials Theory and Modeling"}],"tags":[],"updatedAt":"2026-04-20T07:09:00+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-20 07:09:00","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9445755","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9445755","identity":"rs-9445755","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}