We study the current produced in a Tomonaga-Luttinger liquid by an applied bias and by weak, point-like impurity potentials which are oscillating in time. We use bosonization to perturbatively calculate the current up to second order in the impurity potent

ChargetransportinaTomonaga-Luttingerliquid:e ectsofpumpingandbias

AmitAgarwalandDiptimanSen

CenterforHighEnergyPhysics,IndianInstituteofScience,Bangalore560012,India

(Dated:February6,2008)

Jul 2007

WestudythecurrentproducedinaTomonaga-Luttingerliquidbyanappliedbiasandbyweak,point-likeimpuritypotentialswhichareoscillatingintime.Weusebosonizationtoperturbativelycalculatethecurrentuptosecondorderintheimpuritypotentials.Intheregimeofsmallbiasandlowpumpingfrequency,boththeDCandACcomponentsofthecurrenthavepowerlawdependencesonthebiasandpumpingfrequencieswithanexponent2K 1forspinlesselectrons,whereKistheinteractionparameter.ForK<1/2,thecurrentgrowslargeforspecialvaluesofthebias.Fornon-interactingelectronswithK=1,ourresultsagreewiththoseobtainedusingFloquetscatteringtheoryforDiracfermions.Wealsodiscussthecasesofextendedimpuritiesandofspin-1/2electrons.

7 PACSnumbers:73.23.-b,73.63.Nm,71.10.Pm

]llahI.INTRODUCTION

-seTheconductanceofelectronsinaquantumwirehasmbeenstudiedextensivelyinrecentyearsboththeoreti-.cally[1,2]andexperimentally[3,4,5,6,7,8].Foratawireinwhichonlyonechannelisavailabletotheelec-mtronsandthetransportisballistic(i.e.,therearenoim--puritiesinsidethewire,andthereisnoscatteringfromdphononsorfromthecontactsbetweenthewireanditsnleads),theconductanceisgivenbyG=2e2/hforin-o nitesimalbias.However,ifthereisanimpurityinsidec[thewirewhichscatterstheelectrons,thentheconduc- tanceisreduced.Foraδ-functionimpuritywithstrength

2U,weobtainG=(2e2/h)(1 U2/v2

vF),tolowestorderinU,wherev9FistheFermivelocityoftheelectrons.2Inthepresenceofinteractionsbetweentheelectrons,the4impuritystrengthUe ectivelybecomesafunctionofthe2lengththrougharenormalizationgroup(RG)equa-0tion10].TheRG owhastobecut-o atthesmallest7ofthethreelengthscalesofthesystem,namely,thewire0length,thethermallengthwhichisinverselyproportional/ttothetemperature,andalengthwhichisinverselypro-amportionaltothebiasvoltageVbias.Ifthelatterlengthscaleisthesmallestofthethree,thenthe-doftheimpuritystrengthisgivenbyUVKe ective 1

value

nisaparameterrelatedtothestrengthbias,whereKoftheinterac-tionsbetweentheelectronsaswewillseelater.Hencecothecorrectiontotheconductanceduetothecombined:ve ectoftheimpurityandtheinteractionsisgiveni G～U2VXphenomenonbias2K 2

by

.

Theofchargepumpingandrecti cationrabyoscillatingpotentialsappliedtocertainpointsinasystemhasalsobeenstudiedtheoretically[11-38]andexperimentally[39,40,41,42,Forthecaseofnon-interactingelectrons,theoreticalstudieshaveusedadi-abatictheory[17,18,19],Floquetscatteringtheory[23,24],ofthenon-equilibriumGreenfunctionformalismandtheequationofmo-tionapproachThecaseofinteractingelectronshasalsobeenstudied,usingaRGmethodforweakinterac-tions[44],andthemethodofbosonizationforarbitraryinteractions[45-55].Theanalyticalmethodsusedinthe

twocasestypicallytreattheelectronsinquitedi erentways,withanon-relativisticSchr¨odingerequationoratight-bindingmodelonalatticebeingusedinthenon-interactingcase,andamasslessDiracmodelfollowedbybosonizationbeingusedintheinteractingcase.Aclearcomparisonbetweenthenon-interactingandinteractingcasesdoesnotseemtohavebeenmadebe-fore.Weplanto llthisgapinthispaperandwillstudythee ectsofelectron-electroninteractions(ofarbitraryrepulsivestrength)ontheDCandACcomponentsofthecurrentinasystemwithabiasandtime-dependentimpurities.

InSec.II,wediscussamasslessDiracmodelfornon-interactingelectronsinthepresenceofseveralpoint-likeimpurities.WeuseFloquetscatteringtheory[23,24]tostudythepumpedandbiascurrentinthismodel.InSec.III,we ingbosonizationwethencomputetheDCandACcomponentsofthecurrentuptosecondorderintheimpuritypotentials[48,49].WeshowthatthesereducetotheresultsobtainedusingFloquetscatteringtheoryforthenon-interactingcase.Thecasesofextendedimpuritiesandofspin-1/2electronsarediscussedbrie y.WesummarizeourresultsinSec.IV.Wewillconsiderin nitelylongwiresandzerotemper-aturethroughoutthepaper;hencetherelevantenergyscalesintheproblemaresetonlybythebiasandthepumpingfrequency.

II.

MASSLESSDIRACMODELWITH

IMPURITIES

InthissectionweconsideraspinlessandmasslessDiracfermionwithnointeractionsbetweenthefermions,andusetheFloquetscatteringtheorytocomputetheto-talcurrent.TheHamiltonianinthepresenceofseveral