{"id":"https://openalex.org/W7131788459","doi":"https://doi.org/10.3390/e28030259","title":"Exact Solution and Large-Scale Scaling Analysis of the Imaginary Creutz\u2013Stark Ladder","display_name":"Exact Solution and Large-Scale Scaling Analysis of the Imaginary Creutz\u2013Stark Ladder","publication_year":2026,"publication_date":"2026-02-27","ids":{"openalex":"https://openalex.org/W7131788459","doi":"https://doi.org/10.3390/e28030259","pmid":"https://pubmed.ncbi.nlm.nih.gov/41899911"},"language":"en","primary_location":{"id":"doi:10.3390/e28030259","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030259","pdf_url":null,"source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/e28030259","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102214182","display_name":"Y. Qi","orcid":"https://orcid.org/0009-0000-9539-3561"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yunyao Qi","raw_affiliation_strings":["State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"],"raw_orcid":"https://orcid.org/0009-0000-9539-3561","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5126962101","display_name":"Heng Lin","orcid":null},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Heng Lin","raw_affiliation_strings":["State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"],"raw_orcid":"https://orcid.org/0009-0002-9213-8711","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102655273","display_name":"Quanfeng Lu","orcid":null},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Quanfeng Lu","raw_affiliation_strings":["State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5126964077","display_name":"Dan Long","orcid":null},"institutions":[{"id":"https://openalex.org/I198091727","display_name":"Tiangong University","ror":"https://ror.org/00xsr9m91","country_code":"CN","type":"education","lineage":["https://openalex.org/I198091727"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Dan Long","raw_affiliation_strings":["School of Physical Science and Technology, Tiangong University, Tianjin 300387, China"],"raw_orcid":"https://orcid.org/0009-0005-8241-8623","affiliations":[{"raw_affiliation_string":"School of Physical Science and Technology, Tiangong University, Tianjin 300387, China","institution_ids":["https://openalex.org/I198091727"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076562970","display_name":"Dong Ruan","orcid":"https://orcid.org/0000-0002-8147-2151"},"institutions":[{"id":"https://openalex.org/I4210150858","display_name":"Beijing Academy of Quantum Information Sciences","ror":"https://ror.org/04nqf9k60","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210150858"]},{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Dong Ruan","raw_affiliation_strings":["Frontier Science Center for Quantum Information, Beijing 100084, China","State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Frontier Science Center for Quantum Information, Beijing 100084, China","institution_ids":["https://openalex.org/I4210150858"]},{"raw_affiliation_string":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064222852","display_name":"Gui-Lu LONG","orcid":null},"institutions":[{"id":"https://openalex.org/I4210150858","display_name":"Beijing Academy of Quantum Information Sciences","ror":"https://ror.org/04nqf9k60","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210150858"]},{"id":"https://openalex.org/I4210156423","display_name":"National Engineering Research Center for Information Technology in Agriculture","ror":"https://ror.org/04c3j3t84","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210156423"]},{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Gui-Lu Long","raw_affiliation_strings":["Beijing Academy of Quantum Information Sciences, Beijing 100193, China","Beijing National Research Center for Information Science and Technology, Beijing 100084, China","Frontier Science Center for Quantum Information, Beijing 100084, China","State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"],"raw_orcid":"https://orcid.org/0000-0002-9023-1579","affiliations":[{"raw_affiliation_string":"Beijing Academy of Quantum Information Sciences, Beijing 100193, China","institution_ids":["https://openalex.org/I4210150858"]},{"raw_affiliation_string":"Beijing National Research Center for Information Science and Technology, Beijing 100084, China","institution_ids":["https://openalex.org/I4210156423"]},{"raw_affiliation_string":"Frontier Science Center for Quantum Information, Beijing 100084, China","institution_ids":["https://openalex.org/I4210150858"]},{"raw_affiliation_string":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5064222852","https://openalex.org/A5076562970"],"corresponding_institution_ids":["https://openalex.org/I4210150858","https://openalex.org/I4210156423","https://openalex.org/I99065089"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.2503468,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"28","issue":"3","first_page":"259","last_page":"259"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.3416000008583069,"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/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.3416000008583069,"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/T11177","display_name":"Spectroscopy and Quantum Chemical Studies","score":0.11100000143051147,"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.10589999705553055,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear 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/scaling","display_name":"Scaling","score":0.7354000210762024},{"id":"https://openalex.org/keywords/imaginary-time","display_name":"Imaginary time","score":0.5827999711036682},{"id":"https://openalex.org/keywords/gravitational-singularity","display_name":"Gravitational singularity","score":0.5536999702453613},{"id":"https://openalex.org/keywords/wave-function","display_name":"Wave function","score":0.5286999940872192},{"id":"https://openalex.org/keywords/exact-solutions-in-general-relativity","display_name":"Exact solutions in general relativity","score":0.47290000319480896},{"id":"https://openalex.org/keywords/thermodynamic-limit","display_name":"Thermodynamic limit","score":0.44589999318122864},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4410000145435333},{"id":"https://openalex.org/keywords/spectrum","display_name":"Spectrum (functional analysis)","score":0.4361000061035156},{"id":"https://openalex.org/keywords/spectral-gap","display_name":"Spectral gap","score":0.4056999981403351}],"concepts":[{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.7354000210762024},{"id":"https://openalex.org/C88850056","wikidata":"https://www.wikidata.org/wiki/Q3274262","display_name":"Imaginary time","level":5,"score":0.5827999711036682},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.5536999702453613},{"id":"https://openalex.org/C113603373","wikidata":"https://www.wikidata.org/wiki/Q2362761","display_name":"Wave function","level":2,"score":0.5286999940872192},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.47290000319480896},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4697999954223633},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.44769999384880066},{"id":"https://openalex.org/C54021167","wikidata":"https://www.wikidata.org/wiki/Q1103484","display_name":"Thermodynamic limit","level":2,"score":0.44589999318122864},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4410000145435333},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.4361000061035156},{"id":"https://openalex.org/C2778114796","wikidata":"https://www.wikidata.org/wiki/Q7575194","display_name":"Spectral gap","level":2,"score":0.4056999981403351},{"id":"https://openalex.org/C112675119","wikidata":"https://www.wikidata.org/wiki/Q2992284","display_name":"Periodic boundary conditions","level":3,"score":0.39399999380111694},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.37439998984336853},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.36550000309944153},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.35989999771118164},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3596999943256378},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.3537999987602234},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.34689998626708984},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.3458000123500824},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.3449999988079071},{"id":"https://openalex.org/C121130766","wikidata":"https://www.wikidata.org/wiki/Q192510","display_name":"Wavenumber","level":2,"score":0.3386000096797943},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.32120001316070557},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.3140000104904175},{"id":"https://openalex.org/C80845027","wikidata":"https://www.wikidata.org/wiki/Q15144062","display_name":"Continuous spectrum","level":2,"score":0.29670000076293945},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.2962999939918518},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.2879999876022339},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.2777000069618225},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.27230000495910645},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.250900000333786}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e28030259","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030259","pdf_url":null,"source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:41899911","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41899911","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:85cbed76f2ac4e50bd5ec85fdcb12aa1","is_oa":true,"landing_page_url":"https://doaj.org/article/85cbed76f2ac4e50bd5ec85fdcb12aa1","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 28, Iss 3, p 259 (2026)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:13025370","is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC13025370/","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e28030259","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28030259","pdf_url":null,"source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1581485471","display_name":null,"funder_award_id":"62131002","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":80,"referenced_works":["https://openalex.org/W1978986114","https://openalex.org/W1988591348","https://openalex.org/W1991050432","https://openalex.org/W1995654939","https://openalex.org/W1997323544","https://openalex.org/W2007583263","https://openalex.org/W2011301426","https://openalex.org/W2031656208","https://openalex.org/W2047540262","https://openalex.org/W2054294534","https://openalex.org/W2067123857","https://openalex.org/W2072703702","https://openalex.org/W2088245140","https://openalex.org/W2131762505","https://openalex.org/W2273068609","https://openalex.org/W2297830556","https://openalex.org/W2480854438","https://openalex.org/W2538420542","https://openalex.org/W2739884744","https://openalex.org/W2744045251","https://openalex.org/W2744151324","https://openalex.org/W2769934158","https://openalex.org/W2781491230","https://openalex.org/W2790056669","https://openalex.org/W2807779697","https://openalex.org/W2908266731","https://openalex.org/W2913834835","https://openalex.org/W2916939830","https://openalex.org/W2930665417","https://openalex.org/W2935949992","https://openalex.org/W2945565493","https://openalex.org/W2947450731","https://openalex.org/W2962942350","https://openalex.org/W2967690853","https://openalex.org/W2968367765","https://openalex.org/W2976994215","https://openalex.org/W2979035204","https://openalex.org/W2994887904","https://openalex.org/W2998801211","https://openalex.org/W3003257820","https://openalex.org/W3003545277","https://openalex.org/W3008895341","https://openalex.org/W3010555848","https://openalex.org/W3020326296","https://openalex.org/W3034071862","https://openalex.org/W3035965352","https://openalex.org/W3037850448","https://openalex.org/W3041604294","https://openalex.org/W3048931798","https://openalex.org/W3083758530","https://openalex.org/W3171178784","https://openalex.org/W4221080129","https://openalex.org/W4286964939","https://openalex.org/W4307887695","https://openalex.org/W4310986895","https://openalex.org/W4323671569","https://openalex.org/W4365450877","https://openalex.org/W4367312220","https://openalex.org/W4380481328","https://openalex.org/W4382940253","https://openalex.org/W4386032182","https://openalex.org/W4386067772","https://openalex.org/W4386162888","https://openalex.org/W4386828607","https://openalex.org/W4387342339","https://openalex.org/W4387502972","https://openalex.org/W4387743473","https://openalex.org/W4390662045","https://openalex.org/W4391679227","https://openalex.org/W4391734191","https://openalex.org/W4392737321","https://openalex.org/W4401332073","https://openalex.org/W4401713839","https://openalex.org/W4406799491","https://openalex.org/W4407712061","https://openalex.org/W4412403160","https://openalex.org/W4413113141","https://openalex.org/W4414383205","https://openalex.org/W4414977787","https://openalex.org/W4416389191"],"related_works":[],"abstract_inverted_index":{"We":[0,37,96],"present":[1],"an":[2,15,186],"analytical":[3],"solution":[4,32],"for":[5,33,199],"the":[6,21,30,34,60,71,77,83,91,105,119,124,130,148,152,157,166,190],"complex":[7],"spectrum":[8,42],"of":[9,44,52,126,151,160,165],"a":[10,24,39,49,65,87,112,196],"Creutz":[11],"ladder":[12],"subject":[13],"to":[14,23,140],"imaginary":[16,114,154,206],"Stark":[17,207],"potential.":[18],"By":[19],"mapping":[20],"system":[22,141],"momentum-space":[25,35,92],"differential":[26],"equation,":[27],"we":[28,133],"derive":[29],"closed-form":[31],"wavefunctions.":[36],"identify":[38],"distinct":[40],"cross-shaped":[41],"consisting":[43],"discrete":[45,61],"localized":[46],"sectors":[47,62],"and":[48,170],"continuous":[50],"branch":[51,74,128],"asymptotically":[53,72],"real":[54,73],"states.":[55,162],"Our":[56,193],"derivation":[57],"reveals":[58],"that":[59,98,117,174],"arise":[63],"from":[64],"global":[66],"phase":[67,188],"winding":[68],"condition,":[69],"whereas":[70],"emerges":[75],"when":[76],"energy":[78,115],"magnitude":[79],"is":[80],"smaller":[81],"than":[82],"inter-cell":[84],"hopping":[85],"strength,":[86],"regime":[88],"in":[89,129,181,189,203,212],"which":[90],"wavefunction":[93,120],"develops":[94],"singularities.":[95],"demonstrate":[97],"these":[99,161,175],"singularities":[100],"prevent":[101],"standard":[102],"quantization;":[103],"instead,":[104],"open":[106],"boundary":[107],"conditions":[108],"are":[109],"satisfied":[110],"via":[111],"size-dependent":[113,179],"component":[116],"regulates":[118],"decay.":[121],"To":[122],"investigate":[123],"properties":[125],"this":[127],"thermodynamic":[131,191],"limit,":[132],"perform":[134],"large-scale":[135],"finite-size":[136],"scaling":[137,164],"analysis":[138],"up":[139],"sizes":[142],"L\u223c109.":[143],"The":[144],"numerical":[145],"results":[146,194],"confirm":[147],"power-law":[149],"decay":[150],"residual":[153],"energy,":[155],"supporting":[156],"asymptotic":[158],"reality":[159],"Furthermore,":[163],"inverse":[167],"participation":[168],"ratio":[169],"fractal":[171],"dimension":[172],"indicates":[173],"states,":[176],"while":[177],"exhibiting":[178],"localization":[180],"finite":[182],"systems,":[183],"evolve":[184],"into":[185],"extended":[187],"limit.":[192],"establish":[195],"theoretical":[197],"framework":[198],"understanding":[200],"spectral":[201],"transitions":[202],"systems":[204],"with":[205,209],"potentials,":[208],"potential":[210],"realizations":[211],"photonic":[213],"frequency":[214],"synthetic":[215],"dimensions.":[216]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2026-02-27T00:00:00"}