{"id":"https://openalex.org/W7213294844","doi":"https://doi.org/10.48550/arxiv.2609.14689","title":"BCS Gap Equation on Riemannian Manifolds: A Heat Kernel Analysis","display_name":"BCS Gap Equation on Riemannian Manifolds: A Heat Kernel Analysis","publication_year":2026,"publication_date":"2026-09-13","ids":{"openalex":"https://openalex.org/W7213294844","doi":"https://doi.org/10.48550/arxiv.2609.14689"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.14689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14689","pdf_url":"https://arxiv.org/pdf/2609.14689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.14689","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064997397","display_name":"Levent Akant","orcid":"https://orcid.org/0000-0002-1376-8129","observed_orcids":["https://orcid.org/0000-0002-1376-8129"]},"institutions":[{"id":"https://openalex.org/I4405392","display_name":"Bo\u011fazi\u00e7i University","ror":"https://ror.org/03z9tma90","country_code":"TR","type":"education","lineage":["https://openalex.org/I4405392"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Akant, Levent","raw_affiliation_strings":["Department of Physics , Bo\u011fazi\u00e7i University , Bebek , Istanbul , T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Bo\u011fazi\u00e7i University , Bebek , Istanbul , T\u00fcrkiye","institution_ids":["https://openalex.org/I4405392"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040792562","display_name":"Emine Ertu\u011frul","orcid":"https://orcid.org/0000-0001-7962-2002","observed_orcids":["https://orcid.org/0000-0001-7962-2002"]},"institutions":[{"id":"https://openalex.org/I4405392","display_name":"Bo\u011fazi\u00e7i University","ror":"https://ror.org/03z9tma90","country_code":"TR","type":"education","lineage":["https://openalex.org/I4405392"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Ertugrul, Emine","raw_affiliation_strings":["Department of Physics , Bo\u011fazi\u00e7i University , Bebek , Istanbul , T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Bo\u011fazi\u00e7i University , Bebek , Istanbul , T\u00fcrkiye","institution_ids":["https://openalex.org/I4405392"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025379450","display_name":"O. Teoman Turgut","orcid":"https://orcid.org/0000-0002-4209-1508","observed_orcids":["https://orcid.org/0000-0002-4209-1508"]},"institutions":[{"id":"https://openalex.org/I4405392","display_name":"Bo\u011fazi\u00e7i University","ror":"https://ror.org/03z9tma90","country_code":"TR","type":"education","lineage":["https://openalex.org/I4405392"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Turgut, O. Teoman","raw_affiliation_strings":["Department of Physics , Bo\u011fazi\u00e7i University , Bebek , Istanbul , T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Bo\u011fazi\u00e7i University , Bebek , Istanbul , T\u00fcrkiye","institution_ids":["https://openalex.org/I4405392"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4405392"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10037","display_name":"Physics of Superconductivity and Magnetism","score":0.6977999806404114,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10037","display_name":"Physics of Superconductivity and Magnetism","score":0.6977999806404114,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10425","display_name":"Cold Atom Physics and Bose-Einstein Condensates","score":0.05169999971985817,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.02419999986886978,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bcs-superconductivity","display_name":"BCS superconductivity","score":0.8560000061988831},{"id":"https://openalex.org/keywords/gap-equation","display_name":"gap equation","score":0.8880000114440918},{"id":"https://openalex.org/keywords/riemannian-manifolds","display_name":"Riemannian manifolds","score":0.9900000095367432},{"id":"https://openalex.org/keywords/curved-space","display_name":"curved space","score":0.6809999942779541},{"id":"https://openalex.org/keywords/heat-kernel-expansion","display_name":"heat kernel expansion","score":0.9800000190734863},{"id":"https://openalex.org/keywords/renormalization","display_name":"renormalization","score":0.9549999833106995},{"id":"https://openalex.org/keywords/curvature-effects","display_name":"curvature effects","score":0.5220000147819519},{"id":"https://openalex.org/keywords/weak-coupling-limit","display_name":"weak coupling limit","score":0.7250000238418579},{"id":"https://openalex.org/keywords/ultraviolet-divergences","display_name":"ultraviolet divergences","score":0.8510000109672546},{"id":"https://openalex.org/keywords/saddlepoint-approximation","display_name":"saddlepoint approximation","score":0.6570000052452087},{"id":"https://openalex.org/keywords/hubbard-stratonovich-transformation","display_name":"Hubbard-Stratonovich transformation","score":0.8040000200271606}],"concepts":[{"id":"https://openalex.org/C166124518","wikidata":"https://www.wikidata.org/wiki/Q1047702","display_name":"Renormalization","level":2,"score":0.6180999875068665},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5866000056266785},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.5199999809265137},{"id":"https://openalex.org/C183212220","wikidata":"https://www.wikidata.org/wiki/Q3345669","display_name":"Heat kernel","level":2,"score":0.5164999961853027},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5056999921798706},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.490200012922287},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.4846000075340271},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.47519999742507935},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.4018000066280365},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.39149999618530273},{"id":"https://openalex.org/C154018700","wikidata":"https://www.wikidata.org/wiki/Q898323","display_name":"Path integral formulation","level":3,"score":0.36820000410079956},{"id":"https://openalex.org/C68532491","wikidata":"https://www.wikidata.org/wiki/Q1203669","display_name":"Renormalization group","level":2,"score":0.3643999993801117},{"id":"https://openalex.org/C2778114796","wikidata":"https://www.wikidata.org/wiki/Q7575194","display_name":"Spectral gap","level":2,"score":0.36169999837875366},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.3589000105857849},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.35109999775886536},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.34630000591278076},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3215000033378601},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.30329999327659607},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.287200003862381},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.27649998664855957},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.27469998598098755},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.26570001244544983},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.2624000012874603},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.2614000141620636},{"id":"https://openalex.org/C110528137","wikidata":"https://www.wikidata.org/wiki/Q5347246","display_name":"Effective action","level":2,"score":0.25760000944137573},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.2574000060558319},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.2574000060558319}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.14689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14689","pdf_url":"https://arxiv.org/pdf/2609.14689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.14689","doi":"https://doi.org/10.48550/arxiv.2609.14689","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.14689","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.14689","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14689","pdf_url":"https://arxiv.org/pdf/2609.14689","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213294844.pdf","grobid_xml":"https://content.openalex.org/works/W7213294844.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W1570844711","https://openalex.org/W1602301868","https://openalex.org/W2102960053","https://openalex.org/W2112792121","https://openalex.org/W2168674913"],"related_works":["https://openalex.org/W7098201891","https://openalex.org/W1979141095","https://openalex.org/W3162724758","https://openalex.org/W4406219088","https://openalex.org/W3104487974","https://openalex.org/W2023656385","https://openalex.org/W1641815156","https://openalex.org/W4386601279","https://openalex.org/W2129018862","https://openalex.org/W2061340760"],"abstract_inverted_index":{"In":[0],"this":[1,146],"paper,":[2],"we":[3,57,86,126,149],"investigate":[4],"the":[5,20,24,40,43,48,54,59,71,78,83,104,115,120,133,137,141,152,156,170,176],"Bardeen-Cooper-Schrieffer":[6],"(BCS)":[7],"theory":[8],"of":[9,73,155],"superconductivity":[10],"formulated":[11],"on":[12,145],"a":[13,30,35,64,94,128,180],"three":[14],"dimensional":[15],"Riemannian":[16],"manifold.":[17],"We":[18,101],"derive":[19],"gap":[21,60,67,84,106,135,139,158],"equation":[22,61,107],"in":[23,82,119],"path":[25],"integral":[26],"picture":[27],"by":[28,34,47],"employing":[29],"Hubbard-Stratonovich":[31],"transformation":[32],"followed":[33],"saddle":[36],"point":[37],"approximation.":[38],"Under":[39],"assumption":[41],"that":[42,69],"length":[44],"scale":[45],"set":[46],"metric":[49],"is":[50],"much":[51],"larger":[52],"than":[53],"healing":[55],"length,":[56],"solve":[58],"to":[62,136],"find":[63],"slowly":[65],"varying":[66],"function":[68],"exhibits":[70],"effects":[72],"curvature.":[74],"To":[75],"rigorously":[76],"address":[77],"ultraviolet":[79],"divergences":[80],"inherent":[81],"equation,":[85],"utilize":[87],"heat":[88],"kernel":[89],"expansion":[90],"techniques,":[91],"which":[92],"provide":[93],"systematic":[95],"and":[96,111,140],"mathematically":[97],"transparent":[98],"renormalization":[99],"scheme.":[100],"explicitly":[102],"evaluate":[103],"renormalized":[105],"at":[108],"both":[109],"zero":[110],"finite":[112],"temperatures,":[113],"computing":[114],"relevant":[116],"integrals":[117],"asymptotically":[118],"weak-coupling":[121],"limit":[122],"($\u0394\\ll":[123],"\u03bc$).":[124],"Furthermore,":[125],"establish":[127],"universal,":[129],"regularization-independent":[130],"relation":[131],"connecting":[132],"finite-temperature":[134],"zero-temperature":[138],"temperature":[142,161,178],"itself.":[143],"Based":[144],"fundamental":[147],"relation,":[148],"analytically":[150],"prove":[151],"monotonic":[153],"decrease":[154],"energy":[157],"with":[159],"increasing":[160],"($\\partial":[162],"\u0394/":[163],"\\partial":[164],"T":[165],"<":[166],"0$),":[167],"successfully":[168],"recovering":[169],"standard":[171],"universal":[172],"BCS":[173],"behavior":[174],"near":[175],"critical":[177],"within":[179],"curved":[181],"space":[182],"framework.":[183]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-16T00:00:00"}