{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T12:26:16Z","timestamp":1777983976824,"version":"3.51.4"},"reference-count":46,"publisher":"IOP Publishing","issue":"14","license":[{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/publishingsupport.iopscience.iop.org\/questions\/alternative-author-rights-policies\/"},{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/publishingsupport.iopscience.iop.org\/questions\/alternative-author-rights-policies\/"}],"funder":[{"DOI":"10.13039\/100012114","name":"Defence Research Development Canada","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100012114","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["J. Phys.: Condens. Matter"],"published-print":{"date-parts":[[2026,4,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We present a mode-resolved analysis of electron\u2013phonon interactions in lithium under extreme conditions that identifies protected regions of\n                    <jats:italic>k-<\/jats:italic>\n                    space. The Li body-centered cubic (BCC) phase exhibits topologically protected degenerate bands at the Fermi level that inhibit superconductivity by preventing gap formation via electron\u2013phonon coupling. In contrast, the 9 R and cl16 phase lacks the relevant symmetry protection. In the 9 R phase, the strong electron\u2013phonon interactions dynamically couple to degenerate valence and conduction bands opening an energy gap and lowering the ground-state energy. This stabilization energy, though small in magnitude, is comparable to the total energy difference between the BCC and 9 R phases. Through mode-resolved calculations, we identify distinctive electron\u2013phonon interaction patterns, including pole-like non-adiabatic coupling terms, localized spikes indicative of topological protection, and broad parabolic regions associated with conventional coupling. The same framework localizes the soft-mode instability in high-pressure face-centered cubic Li to a highly confined strong-coupling pocket in\n                    <jats:italic>k-<\/jats:italic>\n                    space (near\n                    <jats:italic>X<\/jats:italic>\n                    ), with rapid suppression away from that point by symmetry-driven protection. We introduce the concept of \u201ctopological superposition energy,\u201d a quantum stabilization mechanism arising from the coherent mixing of conduction and valence states mediated by lattice vibrations. Near quasi-degenerate doublets, we show that these pole-like features directly reflect large lattice nonadiabatic coupling terms, providing a quantitative link between the curvature fingerprints and interband derivative couplings. We further map the most strongly coupled doublets onto a mode-resolved spin\u2013boson Hamiltonian, using the ratio of degeneracy lifting energy and phonon energy to gauge how effectively a phonon mode can drive degeneracy lifting and gap formation across the different Li phases. This mode-resolved, non-adiabatic approach thus converts electron\u2013phonon data into a predictive\n                    <jats:italic>k-<\/jats:italic>\n                    space atlas of where symmetry blocks or enables gap formation, reconciling low-temperature and pressure-induced behavior in elemental lithium and offering a transferable recipe for other quantum materials.\n                  <\/jats:p>","DOI":"10.1088\/1361-648x\/ae585f","type":"journal-article","created":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T22:51:49Z","timestamp":1774651909000},"page":"145401","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Mode-resolved electron\u2013phonon fingerprints map topological protection in lithium beyond born-Oppenheimer"],"prefix":"10.1088","volume":"38","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-7155-390X","authenticated-orcid":true,"given":"Hang","family":"Hu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7934-4866","authenticated-orcid":false,"given":"Hsu Kiang","family":"Ooi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4742-0726","authenticated-orcid":false,"given":"Mohammad S","family":"Ghaemi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3096-931X","authenticated-orcid":true,"given":"Junan","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6652-602X","authenticated-orcid":true,"given":"Ji\u0159\u00ed","family":"Hosta\u0161","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boniface","family":"Yogendran","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8761-6478","authenticated-orcid":true,"given":"Anguang","family":"Hu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"266","published-online":{"date-parts":[[2026,4,8]]},"reference":[{"key":"cmae585fbib1","author":"Atkins","year":"2009","type":"book"},{"key":"cmae585fbib2","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1002\/andp.19273892002","type":"journal-article","article-title":"Zur Quantentheorie der Molekeln","volume":"389","author":"Born","year":"1927","journal-title":"Ann. Phys."},{"key":"cmae585fbib3","author":"Born","year":"1998","type":"book"},{"key":"cmae585fbib4","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1038\/s41524-020-00434-z","type":"journal-article","article-title":"Predominance of non-adiabatic effects in zero-point renormalization of the electronic band gap","volume":"6","author":"Miglio","year":"2020","journal-title":"npj Comput. Mater."},{"key":"cmae585fbib5","doi-asserted-by":"publisher","first-page":"5389","DOI":"10.1073\/pnas.1701994114","type":"journal-article","article-title":"Evidence from Fermi surface analysis for the low-temperature structure of lithium","volume":"114","author":"Elatresh","year":"2017","journal-title":"Proc. Natl Acad. Sci. USA"},{"key":"cmae585fbib6","doi-asserted-by":"publisher","first-page":"1213","DOI":"10.1126\/science.1078535","type":"journal-article","article-title":"Superconductivity in dense lithium","volume":"298","author":"Struzhkin","year":"2002","journal-title":"Science"},{"key":"cmae585fbib7","doi-asserted-by":"publisher","first-page":"2135","DOI":"10.1088\/0953-8984\/9\/10\/004","type":"journal-article","article-title":"First-principles studies of the thermodynamic properties of bulk Li","volume":"9","author":"Staikov","year":"1997","journal-title":"J. Phys.: Condens. Matter"},{"key":"cmae585fbib8","doi-asserted-by":"publisher","first-page":"4028","DOI":"10.1103\/PhysRevB.59.4028","type":"journal-article","article-title":"Structural phase stability and electron-phonon coupling in lithium","volume":"59","author":"Liu","year":"1999","journal-title":"Phys. Rev. B"},{"key":"cmae585fbib9","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.111.054111","type":"journal-article","article-title":"Lithium\u2019s low-temperature phase transitions: insights into quantum lattice dynamics and superconductivity","volume":"111","author":"Racioppi","year":"2025","journal-title":"Phys. Rev. B"},{"key":"cmae585fbib10","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1038\/nature05820","type":"journal-article","article-title":"Superconductivity in lithium below 0.4 millikelvin at ambient pressure","volume":"447","author":"Tuoriniemi","year":"2007","journal-title":"Nature"},{"key":"cmae585fbib11","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1103\/PhysRevLett.53.64","type":"journal-article","article-title":"Crystal structure of lithium at 4.2 K","volume":"53","author":"Overhauser","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"cmae585fbib12","doi-asserted-by":"publisher","DOI":"10.1039\/D2CC04928G","type":"journal-article","article-title":"Solving a problem with a single parameter: a smooth bcc to fcc phase transition for metallic lithium","volume":"58","author":"Jerabek","year":"2022","journal-title":"Chem. Commun."},{"key":"cmae585fbib13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.82.184509","type":"journal-article","article-title":"Superconductivity and electron-phonon coupling in lithium at high pressures","volume":"82","author":"Bazhirov","year":"2010","journal-title":"Phys. Rev. B"},{"key":"cmae585fbib14","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.70.132504","type":"journal-article","article-title":"High-pressure band structure and superconductivity of bcc and fcc lithium","volume":"70","author":"Iyakutti","year":"2004","journal-title":"Phys. Rev. B"},{"key":"cmae585fbib15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.79.054524","type":"journal-article","article-title":"Superconductivity in lithium under high pressure investigated with density functional and Eliashberg theory","volume":"79","author":"Yao","year":"2009","journal-title":"Phys. Rev. B"},{"key":"cmae585fbib16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.96.047004","type":"journal-article","article-title":"Superconductivity and lattice instability in compressed lithium from Fermi surface hot spots","volume":"96","author":"Kasinathan","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"cmae585fbib17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.96.047003","type":"journal-article","article-title":"Superconductivity in lithium, potassium, and aluminum under extreme pressure: a first-principles study","volume":"96","author":"Profeta","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"cmae585fbib18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.101.220103","type":"journal-article","article-title":"Fermi surface studies of the low-temperature structure of sodium","volume":"101","author":"Elatresh","year":"2020","journal-title":"Phys. Rev. B"},{"key":"cmae585fbib19","doi-asserted-by":"publisher","first-page":"1524","DOI":"10.1038\/ncomms2524","type":"journal-article","article-title":"Topological protection of bound states against the hybridization","volume":"4","author":"Yang","year":"2013","journal-title":"Nat. Commun."},{"key":"cmae585fbib20","doi-asserted-by":"publisher","first-page":"2096","DOI":"10.1038\/s41467-019-10091-6","type":"journal-article","article-title":"Triplon band splitting and topologically protected edge states in the dimerized antiferromagnet","volume":"10","author":"Nawa","year":"2019","journal-title":"Nat. Commun."},{"key":"cmae585fbib21","doi-asserted-by":"publisher","first-page":"eaao4748","DOI":"10.1126\/sciadv.aao4748","type":"journal-article","article-title":"Observation of symmetry-protected topological band with ultracold fermions","volume":"4","author":"Song","year":"2018","journal-title":"Sci. Adv."},{"key":"cmae585fbib22","doi-asserted-by":"publisher","first-page":"7971","DOI":"10.1038\/s41467-024-52156-1","type":"journal-article","article-title":"Topological superconductivity from unconventional band degeneracy with conventional pairing","volume":"15","author":"Zhang","year":"2024","journal-title":"Nat. Commun."},{"key":"cmae585fbib23","author":"Englman","year":"1972","type":"book"},{"key":"cmae585fbib24","first-page":"84","author":"Baer","year":"2006","type":"book"},{"key":"cmae585fbib25","doi-asserted-by":"publisher","first-page":"1959","DOI":"10.1103\/PhysRev.125.1959","type":"journal-article","article-title":"Configurational instability in solids: diamond and zinc blende","volume":"125","author":"Birman","year":"1962","journal-title":"Phys. Rev."},{"key":"cmae585fbib26","first-page":"5","type":"book","article-title":"Vibronic Interactions and the Jahn-Teller Theorem","author":"Bersuker","year":"1989"},{"key":"cmae585fbib27","author":"Bersuker","year":"2006","type":"book"},{"key":"cmae585fbib28","doi-asserted-by":"publisher","first-page":"2175","DOI":"10.1016\/j.cpc.2009.06.022","type":"journal-article","article-title":"Ab initio molecular simulations with numeric atom-centered orbitals","volume":"180","author":"Blum","year":"2009","journal-title":"Comput. Phys. Commun."},{"key":"cmae585fbib29","article-title":"The Mechanical Compression Enhancement of Electronic Degeneracy in Superconducting Hydrides","author":"Anguang","year":"n.d.","type":"book"},{"key":"cmae585fbib30","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1023\/A:1008193805436","type":"journal-article","article-title":"Molden: a pre- and post-processing program for molecular and electronic structures*","volume":"14","author":"Schaftenaar","year":"2000","journal-title":"J. Comput. Aided Mol. Des."},{"key":"cmae585fbib31","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1007\/s10822-017-0042-5","type":"journal-article","article-title":"Molden 2.0: quantum chemistry meets proteins","volume":"31","author":"Schaftenaar","year":"2017","journal-title":"J. Comput. Aided Mol. Des."},{"key":"cmae585fbib32","doi-asserted-by":"publisher","DOI":"10.1088\/0953-8984\/21\/39\/395502","type":"journal-article","article-title":"QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials","volume":"21","author":"Giannozzi","year":"2009","journal-title":"J. Phys.: Condens. Matter"},{"key":"cmae585fbib33","doi-asserted-by":"publisher","DOI":"10.1063\/5.0005082","type":"journal-article","article-title":"Quantum ESPRESSO toward the exascale","volume":"152","author":"Giannozzi","year":"2020","journal-title":"J. Chem. Phys."},{"key":"cmae585fbib34","author":"Abramowitz","year":"1972","type":"book"},{"key":"cmae585fbib35","doi-asserted-by":"publisher","first-page":"1228","DOI":"10.1103\/PhysRevLett.58.1228","type":"journal-article","article-title":"Martensitic phase transformation of single-crystal lithium from bcc to a 9R-related structure","volume":"58","author":"Smith","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"cmae585fbib36","doi-asserted-by":"publisher","DOI":"10.1063\/5.0282941","type":"journal-article","article-title":"Topologically protected quantum metallic bonding of alkali metals: from entangled electron\u2013phonon dynamics","volume":"2","author":"Hu","year":"2025","journal-title":"APL Quantum"},{"key":"cmae585fbib37","author":"Ziman","year":"1972","edition":"2nd edn","type":"book"},{"key":"cmae585fbib38","author":"Ziman","year":"2001","type":"book"},{"key":"cmae585fbib39","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1098\/rspa.1963.0202","type":"journal-article","article-title":"The specific heats below 30\u00b0K of lithium metal of various isotopic compositions and of sodium metal","volume":"276","author":"Filby","year":"1997","journal-title":"Proc. R. Soc. A"},{"key":"cmae585fbib40","doi-asserted-by":"publisher","first-page":"1189","DOI":"10.1103\/PhysRev.104.1189","type":"journal-article","article-title":"Bound electron pairs in a degenerate Fermi gas","volume":"104","author":"Cooper","year":"1956","journal-title":"Phys. Rev."},{"key":"cmae585fbib41","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1103\/PhysRev.106.162","type":"journal-article","article-title":"Microscopic theory of superconductivity","volume":"106","author":"Bardeen","year":"1957","journal-title":"Phys. Rev."},{"key":"cmae585fbib42","doi-asserted-by":"publisher","first-page":"1140","DOI":"10.1103\/PhysRev.99.1140","type":"journal-article","article-title":"Electron-phonon interaction in metals","volume":"99","author":"Bardeen","year":"1955","journal-title":"Phys. Rev."},{"key":"cmae585fbib43","doi-asserted-by":"publisher","first-page":"9197","DOI":"10.1073\/pnas.1821533116","type":"journal-article","article-title":"Emergence of topological electronic phases in elemental lithium under pressure","volume":"116","author":"Mack","year":"2019","journal-title":"Proc. Natl Acad. Sci. U.S.A."},{"key":"cmae585fbib44","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.043171","type":"journal-article","article-title":"How to pare a pair: topology control and pruning in intertwined complex networks","volume":"2","author":"Kramer","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"cmae585fbib45","doi-asserted-by":"publisher","first-page":"2544","DOI":"10.1021\/acs.nanolett.0c05117","type":"journal-article","article-title":"Intertwined topological and magnetic orders in atomically thin chern insulator MnBi2Te4","volume":"21","author":"Ovchinnikov","year":"2021","journal-title":"Nano Lett."},{"key":"cmae585fbib46","author":"Hu","year":"2025","type":"other"}],"container-title":["Journal of Physics: Condensed Matter"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:13:51Z","timestamp":1775643231000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/ae585f"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,8]]},"references-count":46,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2026,4,8]]},"published-print":{"date-parts":[[2026,4,10]]}},"URL":"https:\/\/doi.org\/10.1088\/1361-648x\/ae585f","relation":{"has-review":[{"id-type":"doi","id":"10.1088\/1361-648X\/AE585F\/v2\/review1","asserted-by":"object"},{"id-type":"doi","id":"10.1088\/1361-648X\/AE585F\/v2\/decision1","asserted-by":"object"},{"id-type":"doi","id":"10.1088\/1361-648X\/AE585F\/v1\/review2","asserted-by":"object"},{"id-type":"doi","id":"10.1088\/1361-648X\/AE585F\/v2\/review2","asserted-by":"object"},{"id-type":"doi","id":"10.1088\/1361-648X\/AE585F\/v1\/review1","asserted-by":"object"},{"id-type":"doi","id":"10.1088\/1361-648X\/AE585F\/v2\/response1","asserted-by":"object"},{"id-type":"doi","id":"10.1088\/1361-648X\/AE585F\/v1\/decision1","asserted-by":"object"}]},"ISSN":["0953-8984","1361-648X"],"issn-type":[{"value":"0953-8984","type":"print"},{"value":"1361-648X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,8]]},"assertion":[{"value":"Mode-resolved electron\u2013phonon fingerprints map topological protection in lithium beyond born-Oppenheimer","name":"article_title","label":"Article Title"},{"value":"Journal of Physics: Condensed Matter","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"\u00a9 2026 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2025-08-28","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2026-03-27","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2026-04-08","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}