Currents and correlations in Luttinger liquids and carbon nanotubes at finite temperature a

We consider problems of one dimensional interacting fermions confined to a finite size, multichannel geometry. Concentrating on Luttinger liquids and carbon nanotubes, we use nontrivial boundary conditions to represent the effect of external leads, and app

CurrentsandcorrelationsinLuttingerliquidsandcarbonnanotubesat nite

temperatureandsize:abosonizationstudy

J.-S.Caux1,2,3,A.L´opez1, andD.Suppa1

TheoreticalPhysics,UniversityofOxford,

1KebleRoad,Oxford,OX13NP,UK2

AllSoulsCollege,Oxford,OX14AL,UK

ITFA,UniversityofAmsterdam,Valckenierstraat65,

1018XEAmsterdam,TheNetherlands

(Dated:February1,2008)

1

arXiv:cond-mat/0211254v1 [cond-mat.str-el] 13 Nov 2002

3

Weconsiderproblemsofonedimensionalinteractingfermionscon nedtoa nitesize,multichan-nelgeometry.ConcentratingonLuttingerliquidsandcarbonnanotubes,weusenontrivialboundaryconditionstorepresentthee ectofexternalleads,andapplyourframeworktotransportproblemsinaJosephsonjunctionsetup.Wepresentanexactcomputationofallcorrelationfunctions,includ-ing nite-sizeandtemperaturee ects,fortwosetsofsolvableboundaryconditions.Inallcases,wecomputephysicalquantitiesliketheJosephsoncurrentandthepairingorderparameterpro le.

I.INTRODUCTION

Itisnowcommonknowledgethatone-dimensionalinteractingelectronicsystemspossessmetallicphasesthatarefundamentallydi erentfromthoseoftheirhigher-dimensionalbrethren,whichareformostpracticalpurposesextremelywelldescribedbyadaptationsandre nementsofFermiliquidtheory.TheLuttingerliquidhasbecometheparadigmfortheformersystems:thequasiparticlepoleisnomore,andinsteadone ndsonlycollectivespinandchargeexcitationslivingaroundtwoFermipoints.Themoststrikingconsequenceofinteractionsare rstthatspinandchargeexcitationsarenotboundtooneanotheranymore,andcanthustravelatdi erentvelocities.Moreover,chargefractionalizationcanoccur,whereby“fundamental”quantitiesliketheelectron’schargecanbesplitinmanypieces(forareview,seee.g.[1]).

ExperimentalrealizationsofLuttingerliquidsrangefromedgestatesinthefractionalquantumHalle ect[2](inthiscase,achiralLuttingerliquidisfound),quantumwiresinsemiconductorheterostructures[3],andperhapsmostnotablyinsingle-walledcarbonnanotubes(SWNT)[4,5,6].Thelattercanbeusedtoperformelectricaltransportexperimentsusingdi erentgeometries,forexampleinjunctionsmanipulatedbymechanicalmeans[7,8,9].“Kinks”inthenanotubesorcrossingsofnanotubescanbeproducedbyscanningtunnelingmicroscopes(STM)tips,andtheseinturnproviderealizationsofbackscatteringimpuritiesembeddedinaLuttingerliquidorofconnectionsthroughtunneljunctions.Thesejunctionscanactasrectifyingdiodesatroomtemperature,thusopeninguparoutetowardstheconstructionofnanoscaledevicesofallsorts.Anexampleofrecentlyproposedapplicationsisthatofan“entangler”,wherebytwonanotubescoupledtoasuperconductorareexpectedtoproducephysicallyseparatedentangledpairsofexcitations[10,11,12].

SingleimpuritiesembeddedinLuttingerliquidshaveattractedalotoftheoreticalinterestsincetheearlyworkofKaneandFisher[13].Inparticular,manyapproacheshavebeendevisedbasedonboundaryconformal eldtheory(CFT)[14]andboundaryintegrability[15,16].Thetrickhereisto“fold”thesystemaroundtheimpurity,therebyreplacingitwithane ectiveboundarywhosee ectsscaleeithertotheweak-orstrong-couplingregimedependingonthenatureofthebulkinteractionsinthesystem.Forexample,forabackscatteringimpurityinaLuttingerliquid,repulsiveinteractionsenhancethebackscatteringamplitude,andthereforesuppresstheconductance.

AnotherextremelyinterestingsituationtoconsideristhatofaLuttingerliquidona niteinterval.Thisclassofproblemsisaformofgeneralizationoftheprevioussingleimpurityones,inthesensethatwenowhavetwodi erentboundariestodealwith.Onethusimagineselectronsbeingtrappedbyeitheradirectphysicalcutoftheirsupport(maybeanendednanotube),orpreventedfrompropagatingfurtherbythepresenceofanexcitationgap,maybeasuperconductinggap,orvoltagegates.WeconsiderheretheproblemofaJosephsonjunctionrealizedbybridgingtwosuperconductorswitheitherquantumwiresorcarbonnanotubes.Experimentalattemptsatrealizingthisdevicecanbefoundin[17].Theoretically,thisproblemhasbeenaddressedin[18],wheretheJosephsoncurrentfor“perfect”contactswascomputed(here,byperfect,wemeanacontactsuchthatonlyAndreevre ectionoccurs;apoorcontactwouldbeonesuchthatmostofthere ectionprocessesthatoccurwouldbenormalre ectionones).Aperturbativeanalysisoftheproblemwaspresentedin[19],andane ectivemodelamenabletoboundaryCFTwasusedin[20].

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