Self-dual Vortices in the Generalized Abelian Higg

arXiv:hep-th/9209110v228Sep1992SNUTP92–79hepth@xxx/9209110Self-dualVorticesintheGeneralizedAbelianHiggsModelwithIndependentChern-SimonsInteractionChanjuKimDepartmentofPhysics,SeoulNationalUniversitySeoul151-742,KoreaAbstractSelf-dualvortexsolutionsarestudiedindetailinthegeneralizedabelianHiggsmodelwithindependentChern-Simonsinteraction.Forspecialchoicesofcouplings,itreducestoaMaxwell-Higgsmodelwithtwoscalarfields,aChern-Simons-Higgsmodelwithtwoscalarfields,orothernewmodels.Weinvestigatethepropertiesofthestaticsolutionsandperformdetailednumericalanalyses.FortheChern-Simons-Higgsmodelwithtwoscalarfieldsinanasymmetricphase,weprovetheexistenceofmultisolitonsolutionswhichcanbeviewedashybridsofChern-SimonsvorticesandCP1lumps.Wealsodiscusssolutionsinasymmetricphasewiththehelpofthecorrespondingexactsolutionsinitsnonrelativisticlimit.Themodelinterpolatingallthreemodels—Maxwell-Higgs,Chern-Simons-Higgs,andCP1models—isdiscussedbriefly.Finallywestudythepossibilityofvortexsolutionswithhalf-integervorticityinthespecialcaseofthemodel.Numericalresultsarenegative.I.INTRODUCTIONInthelastfewyearstherehasbeenmuchinterestinthe(2+1)-dimensionalabelianHiggsmodelsandtheCP1modelwithaChern-Simonsterminthegaugefieldaction,partlybecauseoftheirpossiblerelevanceincondensedmatterphysics[1].Thesemodelsallowclassicalvortexlikesolutionsandtheanalysisofthemhasbecomequiteactiverecently[2,3].Hereparticularlyinterestingaretheso-calledself-dualsystems.Sincethediscoveryoftheself-dualchargedvortexsolutionsinrel-ativisticpureChern-Simons-Higgssystem[4],severalotherself-dualsystemsinclud-ingtheChern-Simonstermarenowknown[5,6,7].Ofsomeinterestamongtheseisthegeneralizedself-dualabelianHiggsmodelwithindependentChern-Simonsin-teraction,discussedinRefs.[6,7].Ithastwoabeliangaugefields,oneofwhichhasonlytheMaxwelltermasitskinetictermandtheotheronlytheChern-Simonsterm.ItmaybeconsideredasarealisticmodelifoneidentifiestheMaxwellfieldastheelectromagneticfieldandtheChern-Simonsfieldasaneffectivefieldarisinginacondensedmattersystem.Besides,aswewillseebelow,itcontainsalinearizedversionoftheCP1modelwithorwithoutaChern-SimonstermasspecialcasesandthereforeinterpolatesabelianHiggssystemsandtheCP1model.Thepresentpaperisdevotedtothedetailedstudyofstaticsolutionsinthismodel,whichwasnotgiveninRef.[6].Thismodelconsistsoftwocomplexscalarfields,aneutralscalarfield,andtwogaugefields.Eachcomplexscalarfieldcouplestobothgaugefieldsandsotherearefourgaugecouplingsingeneral.But,withsomespecialchoicesmadeforthesecouplings,themodelreducestosimplersystems.Forexample,forsomeparticularchoices[6]itbecomesadecoupledsumofself-dualMaxwellHiggsandself-dual1Chern-SimonsHiggssystems.Moreinterestingly,itispossibletoobtainasysteminwhichtwoscalarfieldsinteractwithaMaxwellorChern-Simonsgaugefield(butnotwithboth),whilepossessinganadditionalglobalSU(2)symmetry.ThiskindofmodelwithaMaxwellgaugefieldhasattractedmuchattentionrecently[8]becauseitadmitstopologicalvortexsolutionsinspiteofhavingasimplyconnectedvacuummanifold,andaclassofsolutionsobtainedhereinterpolatesordinaryGinzburg-LandauvorticesandCP1lumps[8].WeconsiderthecorrespondingmodelwithaChern-Simonsgaugefield,andfindthatithasnontopologicalvortexsolutionsinasymmetricphaseaswellastopologicalvorticesinanasymmetricphase(asintheMaxwellcase).AgainaclassofsolutionsobtainedinthiscaseapproximateCP1lumps,andtheexistenceofmultivortexsolutionscanbeshownbyadaptingthemethodofWang[9].Therearealsomoregeneralself-dualsystemsbasedonthesamesetoffieldsbutwithnoadditionalglobalsymmetry.Ifbothgaugefieldscoupletoscalarfieldsnontrivially,anewkindofmodelsareobtained[6].Ifoneofthetwocomplexscalarfieldsdecouplesandtheothercouplestobothgaugefields,onlyabrokenphaseispossibleandtopologicalvorticesexist.WefindthatthenatureofthevortexsolutionsisverysimilartothoseoftheGinzburg-Landausystem,theChern-Simonsfieldplayingaminorroleonly.Wewillbrieflydiscussitsextensiontothetwocomplex-scalar-fieldcaseinsuchawaythatthereisanadditionalglobalSU(2)symmetry.Complicatedbutinterestingisthecase(f)ofRef.[6],whereallfieldscoupletooneanothernontrivially.Inref[6],itwasnotedthattheremightbesolutionswithahalf-integervorticity(asregardseachofthetwogaugefields).Wewillreporttheresultofournumericalwork,whichisnegative.Thispaperisorganizedasfollows.InSec.II,webrieflyreviewthemodeltobe2consideredbyus,followingRef.[6].SectionIIIisthemajorpartofthepaper.Westudystaticsolutionsforvariousinterestingcasesofthismodel,basedonanalyticalandnumericalmeans.WesummarizeourfindingsinSec.IV.Intheappendix,aU(1)self-dualMaxwell-Chern-SimonsHiggssystemwithadditionalglobalSU(2)symmetryisgiven.Thismodelmightbeinterestingbecausethesolutionsinterpolateallthreecases—Ginzburg-Landau,Chern-Simons,andCP1solitons.II.THEMODELWeconsiderthe(2+1)-dimensionalsystemdescribedbythefollowingLagrangian[6]L=−14FμνFμν+12κεμνρaμ∂νaρ+|Dμχ|2+|Dμψ|2+12(∂μN)2−U(χ,ψ,N),(1)whereDμχ=[∂μ−i(e1aμ+e2Aμ)]χ,Dμψ=[∂μ−i(e3aμ+e4Aμ)]ψ,andU(χ,ψ,N)=12(e2|χ|2+σ1e4|ψ|2−u2)2+e21κ2(e1|χ|2+σ1e3|ψ|2−σ2κe2e1N−v2)2+e23κ2|ψ|2(e1|χ|2+σ1e3|χ|2−σ2κe4e3N−v2)2.(2)Here,σ1andσ2mayassumethevalues±1independently,χa