Resolution of the Nematic-to-Smectic A Universality Problem: A Computational Study of the Quasiperiodic Metal–Insulator Mapping

Resolution of the Nematic-to-Smectic A Universality Problem: A Computational Study of the Quasiperiodic Metal–Insulator Mapping | 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 Resolution of the Nematic-to-Smectic A Universality Problem: A Computational Study of the Quasiperiodic Metal–Insulator Mapping Thomas A Husmann This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9141573/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 nematic-to-smectic A (N-SmA) phase transition exhibits continuously varying critical exponents depending on molecular structure, a phenomenon that has remained unexplained for over four decades despite extensive experimental and theoretical effort. Here we show computationally that the N-SmA transition maps exactly onto the Aubry–André–Harper (AAH) metal–insulator transition at the self-dual critical point V = 2J, using only the de Gennes free energy, lattice discretization, and the generic incommensurability of smectic layer spacing and molecular length. At this critical point, the energy spectrum is a Cantor set with Hausdorff dimension Ds = 1/2, yielding the correlation length exponent ν = 2/3 and continuously varying heat capacity exponent α from 0 (3D-XY) to 2/3 (fully decoupled layers). Setting the quasiperiodic frequency to 1/ϕ (golden ratio, justified by maximal incommensurability) produces a zero-free-parameter formula α(r) = (2/3)((r−rc)/(1−rc))4 with rc = 1−1/ϕ4 =0.8541. This formula fits 11 experimental compounds spanning 40 years of published calorimetry with RMS = 0.033, reduced χ2 = 0.47, and all points within 2σ. Soft Condensed-matter Physics nematic-smecticAtransition universality Aubry–André–Harper model metal–insulator transition Cantor spectrum golden ratio liquid crystals critical exponents McMillan ratio quasiperiodic systems 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-9141573","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":607160129,"identity":"adc712e8-4379-4fd7-8a06-7700908b3086","order_by":0,"name":"Thomas A Husmann","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDUlEQVRIiWNgGAWjYBAC+xvJxyQS/jDwMEjkGH/mMTic2M9+xuDgzwYGeX6JBKxa2CTS0i0+tjHIAbWYSc+peFw4syev8EFiA4PhzDkPcGjJMf85s4HBGKSF+c+Zz5UbbvB/NgBqSdhwH6cWs9u8DQyJDSAtuW2fKzff4N0mAdJicBuHw+TfmN3m+QPUIv/G+HNu2+3cbQS1gGzhYQPbYiAN0fKWsBbpP2wM9WAtvP/Op2+ekWP+A6zlBh4tcFt4244nbwCKgG3ZgFNLWhpQpURiG4RxPHnGGbAWCcOZM3BpST4mzfPHJrFfIvnwZ942YFS2nzGT/PnHBmdUQoEEAxuGyCgYBaNgFIwC8gEAdxd0VhqhM8IAAAAASUVORK5CYII=","orcid":"","institution":"Independent Researcher","correspondingAuthor":true,"prefix":"","firstName":"Thomas","middleName":"A","lastName":"Husmann","suffix":""}],"badges":[],"createdAt":"2026-03-16 20:12:53","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-9141573/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9141573/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104854001,"identity":"7b04ee9d-7183-4d58-84bb-2717fdddcbeb","added_by":"auto","created_at":"2026-03-18 03:00:10","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":199313,"visible":true,"origin":"","legend":"","description":"","filename":"NSmAHusmann.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9141573/v1_covered_d435d37c-eae2-4ca6-ac96-3880a2c0c6de.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eResolution of the Nematic-to-Smectic A Universality Problem: A Computational Study of the Quasiperiodic Metal–Insulator Mapping\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Independent Researcher","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":"nematic-smecticAtransition, universality, Aubry–André–Harper model, metal–insulator transition, Cantor spectrum, golden ratio, liquid crystals, critical exponents, McMillan ratio, quasiperiodic systems","lastPublishedDoi":"10.21203/rs.3.rs-9141573/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9141573/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe nematic-to-smectic A (N-SmA) phase transition exhibits continuously varying critical exponents depending on molecular structure, a phenomenon that has remained unexplained for over four decades despite extensive experimental and theoretical effort. Here we show computationally that the N-SmA transition maps exactly onto the Aubry–André–Harper (AAH) metal–insulator transition at the self-dual critical point V = 2J, using only the de Gennes free energy, lattice discretization, and the generic incommensurability of smectic layer spacing and molecular length. At this critical point, the energy spectrum is a Cantor set with Hausdorff dimension Ds = 1/2, yielding the correlation length exponent ν = 2/3 and continuously varying heat capacity exponent α from 0 (3D-XY) to 2/3 (fully decoupled layers). Setting the quasiperiodic frequency to 1/ϕ (golden ratio, justified by maximal incommensurability) produces a zero-free-parameter formula α(r) = (2/3)((r−rc)/(1−rc))4 with rc = 1−1/ϕ4 =0.8541. This formula fits 11 experimental compounds spanning 40 years of published calorimetry with RMS = 0.033, reduced χ2 = 0.47, and all points within 2σ.\u003c/p\u003e","manuscriptTitle":"Resolution of the Nematic-to-Smectic A Universality Problem: A Computational Study of the Quasiperiodic Metal–Insulator Mapping","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-18 03:00:04","doi":"10.21203/rs.3.rs-9141573/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":"5cad6de4-a56f-473f-afa2-7fbe20cff4ea","owner":[],"postedDate":"March 18th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":64606839,"name":"Soft Condensed-matter Physics"}],"tags":[],"updatedAt":"2026-03-18T03:00:04+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-18 03:00:04","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9141573","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9141573","identity":"rs-9141573","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}