field-theoretical approach to the casimir-like interaction in a one-dimensional bose gas benjamin reichert资料.pdf

该【field-theoretical approach to the casimir-like interaction in a one-dimensional bose gas benjamin reichert资料 】是由【赖大文档】上传分享，文档一共【14】页，该文档可以免费在线阅读，需要了解更多关于【field-theoretical approach to the casimir-like interaction in a one-dimensional bose gas benjamin reichert资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..PHYSICALREVIEWB99,205414(2019)Field-theoreticalapproachtotheCasimir-likeinteractioninaone-dimensionalBosegasBenjaminReichert,AleksandraPetkovic,andZoranRistivojevic′LaboratoiredePhysiqueThéorique,RS,UPS,31062Toulouse,France(Received12February2019;published13May2019)Westudythe?uctuation-inducedinteractionbetweentwoimpuritiesinaweaklyinteractingone-dimensionalBosegasusingthe?eld-,,-long-rangeorderoftheBosegasand,accordingly,?,:.,whichnaturallytakesintoconsiderationthenonlin-?elddistancestheinteractiondecaysexponentiallyaspredictedtheoryisthatthevacuumshouldbeconsideredasa?uctuatinginRefs.[13,14,18],whileatlargedistancesitdecaysalge-,Casimirpredictedthattwoneutrallargecon-braicallyasfoundinRef.[17].Wecalculatedanalyticallytheductingplatesplacedinavacuumshouldattracteachotherinteractioninthecrossoverregionbetweenthesetwolimitingwithaforcedependingontheirrelativeseparation[1].?cationInthispaperwerevisittheproblemoftheinducedinter-ofthegroundstateenergyduetoconstraintsonquantum?uc-actionbetweentheimpuritiesinaone-,whichisasystematic?eld-showtheexistenceofsucheffectiveinteractionwereonlyintheoreticalapproachbasedonthepathintegralinimaginaryqualitativeagreementwiththetheoreticalpredictions[2,3].[4?7].Moregenerally,theCasimir-?-???uids,forinstance,whereWeconsideredbothcasesofzeroand?nitetemperature,incorrelationsarelongranged,theeffectiveinteractionisalsocontrasttoRef.[21].Atzerotemperaturewe?ndaresultthatexpectedtobelong-ranged[8].isidenticaltotheonefoundinRef.[21].At?nitetempera-Thankstoexperimentalprogressesmadeinthemanipula-tures,weobtaintheresultfortheinducedinteractionvalidattionofultracoldatoms,wherevarioussuper?-realized[9],theinterestintheeffectiveCasimir-likeinterac-mal?uctuationstendtodestroythequasi-long-rangeordertionbetweenimpuritiesinaquantumliquidhasbeenrenewedinaone-ordinglymodifythe[10?22].Thecaseofaone-dimensionalweaklyinteractingquantum?uctuation-inducedlongrangeinteractionturningBosegashasbeenstudiedinRefs.[13,14,17,18,21].Evenitintoanexponentialone,whichisinagreementwiththethoughthesystemexhibitsquasi-long-rangecorrelationsduegeneralpicture[8].toitssuper?uidnature,inRefs.[13,14,18]wasfoundashort-?eld-theoreticalmethodusedtodescribeRecently,inRef.[17]itwasshownthattheeffectiveinterac--,itwaspredictedthatitframework,-,bybetweenimpurities[17].performingaLegendretransformationoftheLandaufreeInRef.[21]-itsbehaviorwithrespecttothedistancebetweenimpurities,?-?eldGross-Pitaevskiiequa--9950/2019/99(20)/205414(14)205414-1?2019AmericanPhysicalSociety:..REICHERT,PETKOVIC,ANDRISTIVOJEVICPHYSICALREVIEWB′99,205414(2019)(dimensionless)actionishˉβWestudyone-dimensionalbosonswithcontactrepulsion1?=dτdx(ψhˉ?τψ+H?μn).(6)hˉ0theHamiltonianByμwedenotethechemicalpotentialofthesystem,Histhehˉ2gHamiltoniandensity,plex?eldH=dx(?ψ??)(?ψ?)+ψ??ψ??ψ?ψ?2m2?iθ(x,τ)ψ(x,τ)=n(x,τ)e(7)+G[n?(?/2)+n?(/2)],(1)isperiodicinimaginarytimeψ(x,0)=ψ(x,hˉβ).whereψ??andψ?arethebosonicsingle-particleoperatorsthatSinceweconsidertheweaklyinteractingBosegasandmutationrelation[ψ?(x),ψ??(x)]=weaklycoupledimpurities,wecanassumethatthedensityδ(x?x).InEq.(1),misthemassofbosonicparticles,?>0denotesthestrengthofrepulsivecontactinteractionn(x,τ)=σ+σ(x,τ),(8)0betweenthebosons,±/2locallycouplewherethe?eldσ(x,τ)accountsforthedensity?uctuationstothedensityoftheBoseliquid?n=ψ??ψ?,=0,theHamiltonian(1)correspondsworkingwithintheformalismofthegrandcanonicalensem-totheintegrableLieb-Linigermodel[23].bleandthatσ0≡σ0(μ)isafunctionofthechemicalpotential,putetheinteractionenergybetweenwhichwillbedeterminedinavariationalmanner,|σ|σ0,wecanexpandthesquarerootintoperformthiscalculationwewillevaluatetheHelmholtzEq.(7)(6):=Sc+S0+S1+S3+S4+···.(9)single-particleoperatorsas√√HereScdoesnotcontain?uctuating?elds;S0containsthe??iθ???iθ?quadratictermsinthe?uctuating?eldsσandθ,whileSψ=ne?,ψ=en?,(2)3orrections,wherethedensity?nandthephaseθ?-[?n(x),θ?(x)]=?iδ(x?x).TheHamilto-(1)cannowbeexpressedastheparametrization(8),thepartitionfunction(5)es22dependentonσ0:hˉ?2(?n?)g2H=dxn?(?θ)++dxn??S[σ0,σ(x,τ),θ(x,τ)]2m4?n2Z[σ0]=Dσ(x,τ)Dθ(x,τ)e.(10)+G[n?(/2)+n?(?/2)]+h0,(3)ouldbeconvenientlysplitasSc=whereS(0)+S(1),whereS(0)formh0=?μdxn?+L,(4a)(0)2 Sc=βLgσ02?μσ0+2βGσ0,(11)ghˉ2(1)μ=δ(x)|x→0,=?2δ(x)|x→0.(4b)Sc=βL[?μσ0+].(12)(3)ountedforthetwotermsgivenfromtheHamiltonian(3)producesthecorrectionS(1).byEq.(4a),ransformshˉσ0(?σ)2gσ20S=dτdx(?θ)2++?iσ?(1)intotheform(3).Noticethathis02m4σ22ˉhτ00formallydivergentduetothepresenceofdeltafunctions(13)andneedtoberegularized,.,bythesubstitutionδ(x)→(1/L)eikx,whereListhesizeofthesystem,whileThankstotheinvarianceunderthetranslationandtheperiod-|k|<isahigh-,itisconvenienttoworkinWenowemploythepathintegralformalism[24,25]inFourierspacewherethe?uctuating?eldsσandθaregivenordertoevaluatetheLandaufreeenergyofthesystemF=by?lnZ/=1/T,whereTisthetemperature,while(kx?ωτ)σ(x,τ)=√eσk,(14a)?S[n(x,τ),θ(x,τ)]1i(kx?ωτ)Z=Dn(x,τ)Dθ(x,τ)e,(5)θ(x,τ)=√eθk,(14b)Lhˉβk205414-2:..FIELD-THEORETICALAPPROACHTOTHE?PHYSICALREVIEWB99,205414(2019)withk=(k,ω)andwhereω≡ωj=2πj/hˉβ(j∈Z)istheInFourierspace,?eldsarerealinrealspace,onehasσ=σ?andθ=θ?S=θ(q,p)σθθ?kk?,inthesesummations,k,q,pthek=0modeisremovedsuchthatallthex-independentσpartiscontainedinσ[.(8)].Inprincipleoneshould+3(k,q,p)σkσqσpδk+q+p,0,0√imposeacutoffonthemomentumsummationinorderθhˉqptopreventlarge?,however,explicitly3(q,p)=?√,βL2mdisplayedinthesumsonlywhenneededtoregularizethe√,thedivergenthˉk2+q2+p2σ(k,q,p)=?√,(22)termsareabsorbedintherenormalizationprocedureandthe3βL48mσ2limit→∞(14),thequadraticactionwhileS4takestheform(13)esS4=4(k,q,p,r)σkσqσpσrδk+q+p+r,0,k,q,p,r1?1σqS0=(σk,θk)Gk,q.(15)1k2+q2+p2+r22k,qθq4(k,q,p,r)=.(23)βL96mσ30encodesthecorrelationfunctionsThepartoftheactionthatdescribestheinteractionwiththeimpuritieshastheformσkσqσkθqGk,q=GθkσqθkθqS1=dτ[σ(/2,τ)+σ(?/2,τ)].(24)??hˉhˉ2k2σ0hˉm?hˉωSinceσk=0atk=0,theterminvolvingdxσ(x,τ)=0in=?mε2?δk+q,0.(16)hˉ2ω2+√2βHeretheaverage...iswithrespecttothequadraticactionS1=1(k)σkδω,0,1(k)=√Gcos(k/2).(25)khLˉ1?S0[σ0,σ,θ]Atweakinteractiononecanusethescalinganalysis[27]A=DθDσAe,(17)Z0toshowthatS0,S3,andS4correspondtothe?rstthreetermsoftheexpansionoftheactionwithrespecttothesmallwhere1/401/4√parameterγ,whereS0∝γ,S3∝γ,andS4∝γ.Hereγ=mg/hˉ2nˉ=DθDσe?S0[σ0,σ,θ].(18)√0smallboson-impuritycoupling,Gg/γ,onecanshowthatthedepletionofthebosondensityduetoimpuritiesisThespectrumofelementaryexcitationsinEq.(16),weperformtwo√√expansions,oneinγandanotherinGγ/gasweexplain2222,theresulthˉkhˉkεk=gσ0+.(19)fortheinducedinteractionwillcoveralldistancessincewem2maredealingwithasymptoticallyexactBogoliubovdispersionNoticethatthenonlinearityappearingintheBogoliubovspec-(19).trumisduetothequantumpressureterm∝(?σ)2inEq.(13).Thistermisusuallydiscardedinalow-,.,,?rst?ndingtheLandaufreeenergyattheountfortheanharmonicmean-ountingforitsleadingquantumtermsthatdescribetheinteractionbetweenBogoliubovquasi-?nallyperformtheLegendretransformtoob-particles[26,27].Thecubicandquarticanharmonictermsintainthe(Helmholtz),respectively,Thepartitionfunctionofasystemallowsustoobtainall,theLandaufreehˉ(?σ)2energyF(μ)isgivenbyS3=dτdxσ(?θ)2?σ,(20)2m4σ20βF(μ)=?lnZ,(26)hˉ22S4=3dτdxσ(?σ).(21)wherethepartitionfunctionZisgivenbyEq.(5).After8mσ0introducingtheparametrization(8),thepartitionfunction205414-3:..REICHERT,PETKOVIC,ANDRISTIVOJEVICPHYSICALREVIEWB′99,205414(2019)esafunctionalofσ0[seeEq.(10)].Thegrandpotentialarbitraryfunctionofkandq,wouldbeconsideredasaloop(μ,σ0)isde?nedbytherelationintegralwhiletheintegralover√,itisconvenientβ(μ,σ0)=?lnZ[σ0],(27)toexpressthegrandpotentialastheloopexpansionwherethepartitionfunctionZ[σ0]isgivenbyEq.(10).UnliketheLandaufreeenergythatdep