{"id":"https://openalex.org/W7166315823","doi":"https://doi.org/10.1142/s0217732326501968","title":"Conformable Dirac equation for graphene: Hamiltonian formulation and exact solutions","display_name":"Conformable Dirac equation for graphene: Hamiltonian formulation and exact solutions","publication_year":2026,"publication_date":"2026-06-27","ids":{"openalex":"https://openalex.org/W7166315823","doi":"https://doi.org/10.1142/s0217732326501968"},"language":"en","primary_location":{"id":"doi:10.1142/s0217732326501968","doi":"https://doi.org/10.1142/s0217732326501968","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0217732326501968","pdf_url":null,"source":{"id":"https://openalex.org/S45468527","display_name":"Modern Physics Letters A","issn_l":"0217-7323","issn":["0217-7323","1793-6632"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-d","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Modern Physics Letters A","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5139490687","display_name":"Souheib Merad","orcid":"https://orcid.org/0009-0007-6837-1691","observed_orcids":["https://orcid.org/0009-0007-6837-1691"]},"institutions":[{"id":"https://openalex.org/I192268740","display_name":"University of Algiers Benyoucef Benkhedda","ror":"https://ror.org/011r6gp69","country_code":"DZ","type":"education","lineage":["https://openalex.org/I192268740"]}],"countries":["DZ"],"is_corresponding":false,"raw_author_name":"S. Merad","raw_affiliation_strings":["Department of Mathematics, Faculty of Sciences, University of Algiers 1, Algiers 16000, Algeria"],"raw_orcid":"https://orcid.org/0009-0007-6837-1691","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Sciences, University of Algiers 1, Algiers 16000, Algeria","institution_ids":["https://openalex.org/I192268740"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032634294","display_name":"Asma Merad","orcid":"https://orcid.org/0009-0007-1399-0743","observed_orcids":["https://orcid.org/0009-0007-1399-0743"]},"institutions":[{"id":"https://openalex.org/I4210160578","display_name":"Universit\u00e9 Djilali Bounaama Khemis Miliana","ror":"https://ror.org/05n2gzs35","country_code":"DZ","type":"education","lineage":["https://openalex.org/I4210160578"]}],"countries":["DZ"],"is_corresponding":false,"raw_author_name":"A. Merad","raw_affiliation_strings":["Department of Renewable Energies, University of Djilali Bounaama, Khemis Miliana 44225, Algeria"],"raw_orcid":"https://orcid.org/0009-0007-1399-0743","affiliations":[{"raw_affiliation_string":"Department of Renewable Energies, University of Djilali Bounaama, Khemis Miliana 44225, Algeria","institution_ids":["https://openalex.org/I4210160578"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.52163517,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"41","issue":"36","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11415","display_name":"Noncommutative and Quantum Gravity Theories","score":0.27079999446868896,"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"}},"topics":[{"id":"https://openalex.org/T11415","display_name":"Noncommutative and Quantum Gravity Theories","score":0.27079999446868896,"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"}},{"id":"https://openalex.org/T10083","display_name":"Graphene research and applications","score":0.25589999556541443,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.24719999730587006,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/conformable-fractional-derivative","display_name":"conformable fractional derivative","score":0.7799999713897705},{"id":"https://openalex.org/keywords/graphene","display_name":"graphene","score":0.7720000147819519},{"id":"https://openalex.org/keywords/dirac-equation","display_name":"Dirac equation","score":0.953000009059906},{"id":"https://openalex.org/keywords/hamiltonian-formulation","display_name":"Hamiltonian formulation","score":0.8659999966621399},{"id":"https://openalex.org/keywords/second-class-constraints","display_name":"second-class constraints","score":0.7760000228881836},{"id":"https://openalex.org/keywords/dirac-brackets","display_name":"Dirac brackets","score":0.9919999837875366},{"id":"https://openalex.org/keywords/canonical-quantization","display_name":"canonical quantization","score":0.8259999752044678},{"id":"https://openalex.org/keywords/coordinate-scaling","display_name":"coordinate scaling","score":0.7850000262260437},{"id":"https://openalex.org/keywords/nanoribbons","display_name":"nanoribbons","score":0.7369999885559082},{"id":"https://openalex.org/keywords/landau-levels","display_name":"Landau levels","score":0.9660000205039978},{"id":"https://openalex.org/keywords/massive-dirac-fermions","display_name":"massive Dirac fermions","score":0.8199999928474426}],"concepts":[{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.7139000296592712},{"id":"https://openalex.org/C86072612","wikidata":"https://www.wikidata.org/wiki/Q5160239","display_name":"Conformable matrix","level":2,"score":0.6927000284194946},{"id":"https://openalex.org/C82601208","wikidata":"https://www.wikidata.org/wiki/Q272621","display_name":"Dirac equation","level":2,"score":0.6676999926567078},{"id":"https://openalex.org/C79955541","wikidata":"https://www.wikidata.org/wiki/Q1348371","display_name":"Dirac fermion","level":3,"score":0.5856000185012817},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.5385000109672546},{"id":"https://openalex.org/C61039578","wikidata":"https://www.wikidata.org/wiki/Q604279","display_name":"Dirac (video compression format)","level":3,"score":0.5192999839782715},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.4546999931335449},{"id":"https://openalex.org/C50575695","wikidata":"https://www.wikidata.org/wiki/Q1227393","display_name":"Dirac sea","level":4,"score":0.42719998955726624},{"id":"https://openalex.org/C54937798","wikidata":"https://www.wikidata.org/wiki/Q2024141","display_name":"Massless particle","level":2,"score":0.37860000133514404},{"id":"https://openalex.org/C104367098","wikidata":"https://www.wikidata.org/wiki/Q5454146","display_name":"First class constraint","level":5,"score":0.37380000948905945},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.35850000381469727},{"id":"https://openalex.org/C149545384","wikidata":"https://www.wikidata.org/wiki/Q5280160","display_name":"Dirac algebra","level":3,"score":0.3582000136375427},{"id":"https://openalex.org/C121770821","wikidata":"https://www.wikidata.org/wiki/Q2072471","display_name":"Hamiltonian system","level":2,"score":0.3384000062942505},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.33570000529289246},{"id":"https://openalex.org/C156785651","wikidata":"https://www.wikidata.org/wiki/Q688945","display_name":"Dirac spinor","level":4,"score":0.31049999594688416},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.3061000108718872},{"id":"https://openalex.org/C2779428274","wikidata":"https://www.wikidata.org/wiki/Q662636","display_name":"Negative energy","level":2,"score":0.28450000286102295},{"id":"https://openalex.org/C2778267785","wikidata":"https://www.wikidata.org/wiki/Q5645295","display_name":"Hamiltonian constraint","level":5,"score":0.2775999903678894},{"id":"https://openalex.org/C50953366","wikidata":"https://www.wikidata.org/wiki/Q385599","display_name":"Dirac comb","level":5,"score":0.2732999920845032},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.2703999876976013},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.2667999863624573},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.262800008058548},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.26260000467300415},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.2547999918460846}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0217732326501968","doi":"https://doi.org/10.1142/s0217732326501968","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0217732326501968","pdf_url":null,"source":{"id":"https://openalex.org/S45468527","display_name":"Modern Physics Letters A","issn_l":"0217-7323","issn":["0217-7323","1793-6632"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","dongbi-d","jufo-1","norway-1","poland-70","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Modern Physics Letters A","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"sustainable_development_goals_aurora":[{"score":0.7432273626327515,"id":"https://metadata.un.org/sdg/16","display_name":"Peace, justice, and strong institutions"}],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1491016379","https://openalex.org/W2014935324","https://openalex.org/W2021978747","https://openalex.org/W2028260084","https://openalex.org/W2035533039","https://openalex.org/W2058122340","https://openalex.org/W2063067239","https://openalex.org/W2079148328","https://openalex.org/W2080931732","https://openalex.org/W2089523877","https://openalex.org/W2099631417","https://openalex.org/W2101112255","https://openalex.org/W2105685140","https://openalex.org/W2126421651","https://openalex.org/W2136216301","https://openalex.org/W2155422709","https://openalex.org/W2308534901","https://openalex.org/W2472805732","https://openalex.org/W2604601957","https://openalex.org/W2618049591","https://openalex.org/W2947200988","https://openalex.org/W3121378234","https://openalex.org/W3184136447","https://openalex.org/W3206290823","https://openalex.org/W3208084143","https://openalex.org/W3217221055","https://openalex.org/W4298222025","https://openalex.org/W4416935056"],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"formulate":[4],"the":[5,38,44,61,68,80,88,101,107,118,122,154,161,169,179,184,190,213],"low-energy":[6],"Dirac":[7,40,46,83,126,138],"equation":[8,41,47],"for":[9,103,133,149,212],"graphene":[10],"using":[11],"conformable":[12,25,39,123,191,214],"derivatives,":[13],"introducing":[14],"reference":[15],"length":[16],"and":[17,115,124,146,198],"time":[18],"scales":[19,163],"to":[20,29,32,85,201],"ensure":[21],"dimensional":[22],"consistency.":[23],"The":[24,64,157,187],"derivative":[26],"is":[27,54,95,181],"shown":[28],"be":[30],"equivalent":[31],"a":[33,74,140,150,194,202,209],"coordinate":[34,196],"scaling,":[35],"which":[36],"maps":[37],"exactly":[42],"onto":[43],"ordinary":[45,125],"in":[48,60,153,160,183,206],"scaled":[49,62,155,185],"coordinates.":[50,63,156,186],"A":[51],"Hamiltonian":[52,93],"formulation":[53],"developed":[55],"via":[56],"Dirac\u2019s":[57],"constraint":[58,76],"analysis":[59],"consistency":[65],"conditions":[66],"of":[67,82,110,113,174],"constrained":[69],"system":[70],"are":[71,131,172],"examined,":[72],"revealing":[73],"second-class":[75],"structure":[77],"that":[78],"requires":[79],"introduction":[81],"brackets":[84],"properly":[86],"define":[87],"reduced":[89],"phase":[90],"space.":[91],"This":[92],"derivation":[94],"physically":[96],"meaningful":[97],"because":[98],"it":[99,200],"provides":[100],"framework":[102],"canonical":[104],"quantization,":[105],"ensures":[106],"correct":[108],"counting":[109],"physical":[111,210],"degrees":[112],"freedom,":[114],"rigorously":[116],"establishes":[117],"unitary":[119],"equivalence":[120],"between":[121],"pictures.":[127],"Exact":[128],"analytical":[129],"solutions":[130],"obtained":[132],"free":[134],"massless":[135],"fermions,":[136,139],"massive":[137],"nanoribbon":[141,162],"with":[142],"infinite-mass":[143],"boundary":[144],"conditions,":[145],"Landau":[147,170],"levels":[148,171],"field":[151,180,205],"uniform":[152,182],"energy":[158],"spectrum":[159],"as":[164,193],"[Formula:":[165,175,216],"see":[166,176,217],"text],":[167],"while":[168],"independent":[173],"text]":[177],"when":[178],"work":[188],"clarifies":[189],"deformation":[192],"local":[195],"scaling":[197],"links":[199],"power-law":[203],"strain":[204],"graphene,":[207],"providing":[208],"interpretation":[211],"parameter":[215],"text].":[218]},"counts_by_year":[],"updated_date":"2026-10-14T16:39:01.820549","created_date":"2026-06-28T00:00:00"}