{"id":"https://openalex.org/W2339839892","doi":"https://doi.org/10.1103/physrevb.93.205112","title":"Robust topological degeneracy of classical theories","display_name":"Robust topological degeneracy of classical theories","publication_year":2016,"publication_date":"2016-05-09","ids":{"openalex":"https://openalex.org/W2339839892","doi":"https://doi.org/10.1103/physrevb.93.205112","mag":"2339839892"},"language":"en","primary_location":{"id":"doi:10.1103/physrevb.93.205112","doi":"https://doi.org/10.1103/physrevb.93.205112","is_oa":false,"landing_page_url":"https://doi.org/10.1103/physrevb.93.205112","pdf_url":null,"source":{"id":"https://openalex.org/S4210237713","display_name":"Physical review. B./Physical review. B","issn_l":"2469-9950","issn":["2469-9950","2469-9969","2469-9977"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-a","jufo-2","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320261","host_organization_name":"American Physical Society","host_organization_lineage":["https://openalex.org/P4310320261"],"host_organization_lineage_names":["American Physical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physical Review B","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1511.07867","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020356891","display_name":"Mohammad-Sadegh Vaezi","orcid":"https://orcid.org/0000-0003-3436-4825","observed_orcids":["https://orcid.org/0000-0003-3436-4825"]},"institutions":[{"id":"https://openalex.org/I204465549","display_name":"Washington University in St. Louis","ror":"https://ror.org/01yc7t268","country_code":"US","type":"education","lineage":["https://openalex.org/I204465549"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mohammad-Sadegh Vaezi","raw_affiliation_strings":["Department of Physics, Washington University, St. Louis, Missouri 63160, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Washington University, St. Louis, Missouri 63160, USA","institution_ids":["https://openalex.org/I204465549"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5077522382","display_name":"Gerardo Ort\u00edz","orcid":"https://orcid.org/0000-0003-3254-4494","observed_orcids":["https://orcid.org/0000-0003-3254-4494","https://orcid.org/0009-0000-3516-7730"]},"institutions":[{"id":"https://openalex.org/I4210119109","display_name":"Indiana University Bloomington","ror":"https://ror.org/02k40bc56","country_code":"US","type":"education","lineage":["https://openalex.org/I4210119109","https://openalex.org/I592451"]},{"id":"https://openalex.org/I592451","display_name":"Indiana University","ror":"https://ror.org/01kg8sb98","country_code":"US","type":"education","lineage":["https://openalex.org/I592451"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gerardo Ortiz","raw_affiliation_strings":["Department of Physics, Indiana University, Bloomington, Indiana 47405, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Indiana University, Bloomington, Indiana 47405, USA","institution_ids":["https://openalex.org/I4210119109","https://openalex.org/I592451"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083751736","display_name":"Zohar Nussinov","orcid":"https://orcid.org/0000-0001-7947-7829","observed_orcids":["https://orcid.org/0000-0001-7947-7829"]},"institutions":[{"id":"https://openalex.org/I204465549","display_name":"Washington University in St. Louis","ror":"https://ror.org/01yc7t268","country_code":"US","type":"education","lineage":["https://openalex.org/I204465549"]},{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL","US"],"is_corresponding":true,"raw_author_name":"Zohar Nussinov","raw_affiliation_strings":["Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel","Department of Physics, Washington University, St. Louis, Missouri 63160, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel","institution_ids":["https://openalex.org/I53964585"]},{"raw_affiliation_string":"Department of Physics, Washington University, St. Louis, Missouri 63160, USA","institution_ids":["https://openalex.org/I204465549"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5083751736"],"corresponding_institution_ids":["https://openalex.org/I204465549","https://openalex.org/I53964585"],"apc_list":null,"apc_paid":null,"fwci":0.4136,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.53656496,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":87,"max":93},"biblio":{"volume":"93","issue":"20","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11804","display_name":"Quantum many-body systems","score":0.9175000190734863,"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"}},"topics":[{"id":"https://openalex.org/T11804","display_name":"Quantum many-body systems","score":0.9175000190734863,"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/T11682","display_name":"Advanced Condensed Matter Physics","score":0.04430000111460686,"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/T10657","display_name":"Topological Materials and Phenomena","score":0.008799999952316284,"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/topological-degeneracy","display_name":"topological degeneracy","score":0.9819999933242798},{"id":"https://openalex.org/keywords/topological-quantum-order","display_name":"topological quantum order","score":0.8159999847412109},{"id":"https://openalex.org/keywords/riemann-surfaces","display_name":"Riemann surfaces","score":0.925000011920929},{"id":"https://openalex.org/keywords/genus","display_name":"genus","score":0.6759999990463257},{"id":"https://openalex.org/keywords/ground-state-degeneracy","display_name":"ground state degeneracy","score":0.9490000009536743},{"id":"https://openalex.org/keywords/u-1-gauge-theory","display_name":"U(1) gauge theory","score":0.9929999709129333},{"id":"https://openalex.org/keywords/nonlocal-entanglement","display_name":"nonlocal entanglement","score":0.7870000004768372}],"concepts":[{"id":"https://openalex.org/C199793620","wikidata":"https://www.wikidata.org/wiki/Q7825030","display_name":"Topological degeneracy","level":5,"score":0.7647547721862793},{"id":"https://openalex.org/C84883863","wikidata":"https://www.wikidata.org/wiki/Q7825777","display_name":"Toric code","level":4,"score":0.6930580139160156},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.659476637840271},{"id":"https://openalex.org/C148908050","wikidata":"https://www.wikidata.org/wiki/Q11681505","display_name":"Topological entropy in physics","level":3,"score":0.6279616951942444},{"id":"https://openalex.org/C45646460","wikidata":"https://www.wikidata.org/wiki/Q17165159","display_name":"Topological order","level":3,"score":0.5839572548866272},{"id":"https://openalex.org/C2777727622","wikidata":"https://www.wikidata.org/wiki/Q5251772","display_name":"Degeneracy (biology)","level":2,"score":0.5649952292442322},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.5624130964279175},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5061420798301697},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.47624921798706055},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.47404003143310547},{"id":"https://openalex.org/C92542899","wikidata":"https://www.wikidata.org/wiki/Q7661340","display_name":"Symmetry protected topological order","level":4,"score":0.47355690598487854},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.43553197383880615},{"id":"https://openalex.org/C69523127","wikidata":"https://www.wikidata.org/wiki/Q4480008","display_name":"Ground state","level":2,"score":0.433530330657959},{"id":"https://openalex.org/C77223115","wikidata":"https://www.wikidata.org/wiki/Q634781","display_name":"Topological quantum number","level":2,"score":0.3971741795539856},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.3835468292236328},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2984178960323334},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.24240192770957947},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11237096786499023},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C60644358","wikidata":"https://www.wikidata.org/wiki/Q128570","display_name":"Bioinformatics","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1103/physrevb.93.205112","doi":"https://doi.org/10.1103/physrevb.93.205112","is_oa":false,"landing_page_url":"https://doi.org/10.1103/physrevb.93.205112","pdf_url":null,"source":{"id":"https://openalex.org/S4210237713","display_name":"Physical review. B./Physical review. B","issn_l":"2469-9950","issn":["2469-9950","2469-9969","2469-9977"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-a","jufo-2","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320261","host_organization_name":"American Physical Society","host_organization_lineage":["https://openalex.org/P4310320261"],"host_organization_lineage_names":["American Physical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physical Review B","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1511.07867","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1511.07867","pdf_url":"https://arxiv.org/pdf/1511.07867","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":"mag:2174251276","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1511.07867","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1511.07867","doi":"https://doi.org/10.48550/arxiv.1511.07867","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1511.07867","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":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1511.07867","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1511.07867","pdf_url":"https://arxiv.org/pdf/1511.07867","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":[{"id":"https://openalex.org/G8522391738","display_name":"Theoretical Approaches to Multi-Scale Complex Systems","funder_award_id":"1411229","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2339839892.pdf","grobid_xml":"https://content.openalex.org/works/W2339839892.grobid-xml"},"referenced_works_count":88,"referenced_works":["https://openalex.org/W253797028","https://openalex.org/W1538499532","https://openalex.org/W1551530254","https://openalex.org/W1631417646","https://openalex.org/W1649303816","https://openalex.org/W1927618417","https://openalex.org/W1931176270","https://openalex.org/W1963826472","https://openalex.org/W1975474316","https://openalex.org/W1975635086","https://openalex.org/W1975910891","https://openalex.org/W1981965174","https://openalex.org/W1984021697","https://openalex.org/W1989901548","https://openalex.org/W1990577812","https://openalex.org/W1993289500","https://openalex.org/W1995809559","https://openalex.org/W1996276022","https://openalex.org/W1997012992","https://openalex.org/W2002248243","https://openalex.org/W2006294391","https://openalex.org/W2007443476","https://openalex.org/W2008317504","https://openalex.org/W2010617683","https://openalex.org/W2014686026","https://openalex.org/W2017010391","https://openalex.org/W2018006971","https://openalex.org/W2019024022","https://openalex.org/W2022394733","https://openalex.org/W2023883372","https://openalex.org/W2025586267","https://openalex.org/W2026461529","https://openalex.org/W2026964817","https://openalex.org/W2027509539","https://openalex.org/W2028948400","https://openalex.org/W2032527773","https://openalex.org/W2032706954","https://openalex.org/W2035286449","https://openalex.org/W2037242760","https://openalex.org/W2040048470","https://openalex.org/W2041820600","https://openalex.org/W2042072697","https://openalex.org/W2042239800","https://openalex.org/W2043647075","https://openalex.org/W2046439097","https://openalex.org/W2047157549","https://openalex.org/W2047485468","https://openalex.org/W2050698597","https://openalex.org/W2051564392","https://openalex.org/W2052247596","https://openalex.org/W2054678250","https://openalex.org/W2057002866","https://openalex.org/W2057042903","https://openalex.org/W2059044209","https://openalex.org/W2059102470","https://openalex.org/W2060078445","https://openalex.org/W2063693460","https://openalex.org/W2065380813","https://openalex.org/W2066512281","https://openalex.org/W2067187311","https://openalex.org/W2077485894","https://openalex.org/W2081484203","https://openalex.org/W2086600886","https://openalex.org/W2086860422","https://openalex.org/W2091164553","https://openalex.org/W2091268741","https://openalex.org/W2094760062","https://openalex.org/W2097361916","https://openalex.org/W2104549471","https://openalex.org/W2110283120","https://openalex.org/W2113487168","https://openalex.org/W2117367729","https://openalex.org/W2121123255","https://openalex.org/W2135273380","https://openalex.org/W2141708961","https://openalex.org/W2147255455","https://openalex.org/W2150074608","https://openalex.org/W2152240519","https://openalex.org/W2157276712","https://openalex.org/W2169919279","https://openalex.org/W2172199359","https://openalex.org/W2230110036","https://openalex.org/W2329053006","https://openalex.org/W2565273911","https://openalex.org/W2949089294","https://openalex.org/W3100008668","https://openalex.org/W3141849522","https://openalex.org/W4301787874"],"related_works":["https://openalex.org/W2174251276","https://openalex.org/W2894689036","https://openalex.org/W2067748918","https://openalex.org/W4376879948","https://openalex.org/W2950353155","https://openalex.org/W7221590836","https://openalex.org/W2240767150","https://openalex.org/W2755585466","https://openalex.org/W2037242760","https://openalex.org/W2801937251"],"abstract_inverted_index":{"We":[0,50,119,151],"challenge":[1],"the":[2,5,38,53,65,68,73,148],"hypothesis":[3],"that":[4,52,94,122,142,153],"ground":[6,55,74,134],"states":[7,75],"of":[8,37,46,60,67,76,88,126,139],"a":[9,34,86,99,133],"physical":[10],"system":[11,79,149],"whose":[12],"degeneracy":[13,57,96],"depends":[14,63],"on":[15,43,64],"topology":[16,66],"must":[17],"necessarily":[18],"realize":[19],"topological":[20,95,103,160],"quantum":[21,104],"order":[22],"and":[23,32,129,155],"display":[24,132,159],"nonlocal":[25],"entanglement.":[26],"To":[27],"this":[28,77,91],"end,":[29],"we":[30],"introduce":[31],"study":[33],"classical":[35,78],"rendition":[36],"Toric":[39],"Code":[40],"model":[41],"embedded":[42],"Riemann":[44],"surfaces":[45],"different":[47],"genus":[48],"numbers.":[49],"find":[51,152],"minimal":[54],"state":[56,135],"(and":[58,137],"those":[59,138],"all":[61,140],"levels)":[62,141],"embedding":[69],"surface":[70],"alone.":[71],"As":[72],"may":[80],"be":[81],"distinguished":[82],"by":[83],"local":[84],"measurements,":[85],"characteristic":[87],"Landau":[89],"orders,":[90],"example":[92],"illustrates":[93],"is":[97,108,143],"not":[98],"sufficient":[100],"condition":[101],"for":[102],"order.":[105],"This":[106],"conclusion":[107],"generic":[109],"and,":[110],"as":[111],"shown,":[112],"it":[113],"applies":[114],"to":[115,163],"many":[116],"other":[117,130],"models.":[118],"also":[120],"demonstrate":[121],"certain":[123],"lattice":[124],"realizations":[125],"these":[127],"models,":[128],"theories,":[131],"entropy":[136],"``holographic'',":[144],"i.e.,":[145],"extensive":[146],"in":[147],"boundary.":[150],"clock":[154],"$U(1)$":[156],"gauge":[157],"theories":[158],"(in":[161],"addition":[162],"gauge)":[164],"degeneracies.":[165]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2016-06-24T00:00:00"}