A Luttinger Liquid in Box

We study a Luttinger Liquid in a finite one-dimensional wire with box-like boundary conditions by considering the local distribution of the single particle spectral weight. This corresponds to the experimental probability of extracting a single electron at

ALuttingerLiquidinaBox

FabrizioAnfuso1,2andSebastianEggert1

1

InstituteofTheoreticalPhysics,ChalmersUniversityofTechnologyandG¨oteborgUniversity,S-41296G¨oteborg,Sweden

2

PhysicsDept.,UniversityofBologna,INFMandINFN,I-40127Bologna,Italy

(Dated:February2,2008)

WestudyaLuttingerLiquidina niteone-dimensionalwirewithbox-likeboundaryconditionsbyconsideringthelocaldistributionofthesingleparticlespectralweight.Thiscorrespondstotheexperimentalprobabilityofextractingasingleelectronatagivenplaceandenergy,whichcanbeinterpretedasthesquareofanelectronwave-function.Forthenon-interactingcase,thisisgivenbyastandingwaveattheFermiwave-vector.Inthepresenceofinteractions,however,thewave-functionsobtainadditionalstructurewithasharpdepletionneartheedgesandmodulationsthroughoutthewire.Inthespinfulcase,thesemodulationscorrespondtotheseparatespin-andcharge-likeexcitationsinthesystem.

PACSnumbers:71.10.Pm,73.21.Hb,73.63.-b

arXiv:cond-mat/0302625v2 [cond-mat.str-el] 1 Mar 2003

Theproblemofaparticleinaone-dimensionalboxisaclassicexampleinalmostanyquantummechanicstext-booksinceitgivesapedagogicalintroductiontotheconceptofenergyquantizationandprovidesacompletevisualizationofthecorrespondingwave-functions.Itishoweveronlyrecentlythatthisproblemhasgainedtrueexperimentalrelevanceduetotheprogressinconstructingsmallerandmorere nedstructurestocon neelectronsindotsandwires.Itisforexamplenowpossibletoresolvetheelectronwavefunctionsina nitepieceofcarbonnanotubebyScanningTunnelingMicroscopy(STM)experiments2,3].Mostexperimentalrealizationsofone-dimensionalelectronboxescontainmanyelectronsinaFermisea,butitispossibletostudyasingleparticleexcitationontopofsuchagroundstatecon gurationandclassifythepossibleenergylevelsandwave-functions.

However,electron-electroninteractionsmayproduceinterestinge ectsinsuchone-dimensionalmany-bodysystemsandsystematicallychangetheshapeoftheelectronwave-functions.InfactsingleparticleexcitationsarenolongertheeigenstatesofaninteractingHamiltonian,butofcourseitisstillinterestingtodeterminetheprobabilityofextractingorinsertingindividualelectronsatagivenpositionandenergy.Thislocalprobabilitydensitycanbeinterpretedasthesquareoftheelectronwave-function.WethereforestudythefundamentalproblemofsingleparticleexcitationsinamanybodyinteractingFermionsystemcon nedtoaone-dimensionalboxusingtheLuttingerLiquidformalism.We ndthattheclassicexampleofaboxprovidesagainagoodvisualizationofthee ectofinteractionsonthewave-functionandtheenergyquantization.Inparticular,inadditiontotheexpectedrapidFermiwave-vectoroscillationsinthewave-functionwecanrecognizelongwavelengthmodulations,whichcorrespondtotheunderlyingboson-likeexcitationsintheLuttingerLiquid.Forrepulsiveinteractionsthewave-functionsaresharplydepletedattheedgeswithacharacteristicpower-law.Analyticexpressionsforthewave-functionsofthe rstfewlevelsarepresented.TheLuttingerLiquidformalismisawell-establishedtooltodescribeinteractingelectronscon nedtoonedimension5].InalinearizedregionaroundtheFermipointstheFermion eldcanbeexpandedintermsofleft-andright-movers

Ψ(x,t)≈eikFxψR(x,t)+e ikFxψL(x,t).

(1)

TheFouriermodesoftheleft-andright-movingFermiondensityarethenrepresentedbybosoniccreationandannihilationoperators,whiche ectivelyactby“shifting”Fermionsmstepsupordownthespectrum.InthepresenceofinteractionsitisthenpossibletosolvethemodelbyaBogoliubovtransformationwhichmixestheleft-andright-movingbosons.Thistransformationcanbedescribedbyasingle“LuttingerLiquidparameter”g,whichgivesthehyperboliccosineαandsineβoftheBogoliubovtransformation

11α=+g,β= g.(2)

ggHereg<1forrepulsivebackscatteringinteractionsandg=1fornointeractions.Forwardscatteringdoesnota ect

thisparameter,butrescalesthee ectiveFermi-velocity.

LetusnowconsiderspinlessFermionsinaone-dimensionalboxoflengthLwith xedboundaryconditionsΨ(0)=Ψ(L)=0.AfterbosonizationtheFermion eldsbecomeexponentialsofthebosonoperatorsinalinearizedregion

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