Charge transport in a Tomonaga-Luttinger liquid effects of pumping and bias

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

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

point-likeimpuritiesisgivenbyH

=H 0+H imp,whereH 0=

dxivF( ψ ψRR),

H imp= x

dx(x)

δ(x xp)Up(t)ψ (x)ψ(x),

p

ψ(x)=ψReikFx+ψL(x)e ikFx,

(1)

whereψLandψRarethefermionic eldoperatorsoftheleftandrightmovingelectrons,vFistheFermivelocity,andkFistheFermiwavenumberwhichoriginatesfromsomeunderlyingmicroscopicmodel.Forinstance,onemayhaveasystemofnon-relativisticelectronswitha

FermienergyEF=k2

F/(2m)andvF=kF/m.(WearesettingnianH

Planck’sconstant equaltounity).TheHamilto-0isobtainedbylinearizingthedispersionthetwoFermipointsgivenbyk=±kF.H

aroundimparisesfromthetime-dependentimpuritieswhichhavestrengthsUp(t);thisHamiltoniancouplesleftandrightmoving eldssince

ψ ψ=ψ RψR+ψ LψL+ψ RψLe i2kFx+ψ

LψRei2kFx.

(2)

Wewillassumethat

Up(t)=Upcos(ωt+φp),(3)

i.e.,allimpuritiesvaryharmonicallyintimewiththe

samefrequencyω.WewillnowuseFloquetscatteringtheoryandcarryoutaperturbativeexpansioninthedi-mensionlessquantitiesUp/vF.

Theequationsofmotioninthepresenceofasingleδ-functionimpurityδ(x xp)Upcos(ωt+φp)isasfollows:

i ψR

x

=δ(x xp)Upcos(ωt+φp)(ψR+ψLe i2kFxp),i

ψL x

=δ(x xp)Upcos(ωt+φp)(ψL+ψRei2kFxp).

(4)

Ifwede nethelinearcombinationsψ+=ψReikFxp+ψLe ikFxpandψ =ψReikFxp ψLe ikFxp,we ndthati ψ x=0,

i ψ+

x

=2δ(x xp)Upcos(ωt+φp)ψ+.

(5)

Byintegratingoveralittleregionfromxpcontinuousatthepoint x =toxpxp+ ,we ndthatψ+is,whileψ hasadiscontinuitygivenby

ivF[ψ (xp+ ) ψ (xp )]=2Upcos(ωt+φp)ψ+(xp).

(6)

2

Wewouldliketonoteherethatitisthetermsψ RψR+ψ

necessarytoretain

LψLinEq.(2)inordertohavecon-tinuityofψ+.Insomepapers,thesetermsarenottakenintoconsideration.Onethenrunsintothemathemati-calpeculiaritythatψ+andψ arebothdiscontinuousatx=xp,andthediscontinuityistakentobepropor-tionaltotheirvaluesatthatpoint;butthosevaluesareactuallyill-de nedduetothediscontinuity.

WecannowsolveEqs.(4)alongwiththebound-aryconditionsinEq.(6).Forasingleδ-functionim-purityoscillatingwithfrequencyωatx=xp,letusconsiderawavecomingfromtheleft(x<xp)withenergyE0andunitamplitude.Notethatwearemea-suringenergieswithrespecttoaFermienergy,sothatE0=0correspondstoafermionattheFermienergy.Duetotheoscillatingimpuritypotential,thewavewillbere ectedbacktotheleftwithenergyEn0),ortransmittedto≡theE0+nωandamplitudeSLL(En,Eright(x>xp)withenergyEnandamplitudeSRL(En,E0),wheren=0,±1,±2,···de nestheFloquetsidebands[23].NotethatsinceweareconsideringaDiracfermion,thereisnoupperorlowerboundtotheenergyEn,andthevelocityvFisindependentoftheenergy.(Thisisunlikethecaseofanon-relativisticfermionorafermiononalatticewherethereisalowerorupperboundtotheenergy,andthevelocityisafunctionoftheenergy).Tobeexplicit,thewavefunctionisgivenbyψR=ei(k0x E0t)forx<xp,

= SRL(En,E0)ei(knx Ent)forx>xp,n

ψL=

SLL(En,E0)ei( knx Ent)

for

x<xp,

n=0

forx>xp,

(7)

wherekn=En/vF.Similarly,wecanconsiderawavecomingfromtherightwithenergyE0andunitampli-tude;itwillbere ectedbacktotherightwithamplitudeSRR(En,E0)ortransmittedtotheleftwithamplitudeSLR(En,E0).Letussimplifythenotationbyde ning

rL,n=SLL(En,E0),tL,n=SLR(En,E0),

tR,n=SRL(En,E0),

rR,n=SRR(En,E0).(8)

Duetounitarity,wehavetherelations

[|rL,nn

|2+|tR,n|2]=1,

[|rR,nL,nn|2+|t|2]=1.

(9)

Thedi erentFloquetscatteringamplitudesrα,nandtα,ncanbefoundbyusingtheboundaryconditionsinEq.(6).Wewillconsiderthecaseofseveralimpuri-tieslabeledbytheindexpasinEq.(1).Tosimplifyourcalculations,wewillassumethatω(xp)/vFaresmallforallpairsofimpurities xr)/vFandE0(xp xrpandr;the rstconditioncorrespondstotheadiabaticlimit,

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

whilethesecondconditionimpliesthatweareonlycon-sideringstatesclosetotheFermienergy.Keepingtermsonlyupto rstorderinUp/vF,we ndthatonlythe rstFloquetsidebandsareexcited,and

tL,1=tR,1=

i

2vF

Upeiφp.

p

rL,1=

i

p+φp)2vU.

F

pei(2kFxp

ri

R,1=

kFxp+φp)2v,

(10)

F

Upei( 2p

Wealso ndthattheunitarityrelationsinEq.(9)are

satis edup

tosecond

orderinUp/vF,andthereforetL,0andtR,0aregivenby

|tL,0|2=1 |rR,1|2 |rR, 1|2 ||ttL,1t=1 |r|2 |tL, 1|2,

|R,0|2L,1|2 |rL, 1|2 R,1|2 |tR, 1|2,

(11)tothatorderinUp/vF.NotethattheamplitudesgiveninEqs.(10-11)areallindependentofE0undertheap-proximationsthatwehavemade.

Thedcpartofthecurrentin,say,therightleadisgivenby[23]IR,dc=q

∞

dE0

∞

3

[|tR,0|2+|tR,1|2+|tR, 1|2]

+

qω

2π2π

+

qω0

2πv2Fxrp)sin(φrp),(13)

F

UpUrsin(2kp<r

wherexrp=xr xpandφrp=φr φp.Eq.(13)showsthee ectsofabias(ω0)andharmonicallyoscillatingpotentials(ω).Forthepurepumpingcasewithω0=0,Eq.(13)agreeswiththeresultspresentedinRef.[38];notethatthepumpedcurrentdependsonsin(φrp).

Itisinterestingtonotethatthe rsttermisjusttheballisticconductanceofacleanwiremultipliedbythebias,thesecondtermisacorrectiontothecleancasebe-causeofthepresenceofimpurities,andthethirdtermisthepumpedcurrent.Inthenon-interactingcase,thebiascomponentandthepumpedcomponentseparateout,butfortheinteractingcase,thecurrentinvolvespowersofω0±ω.

III.

BOSONIZATIONCALCULATIONOFBACKSCATTEREDCURRENTA.

Backscatteringcurrentoperator

Wenowcomputethecurrentinasystemofinteract-ingelectronsusingthebackscatteringcurrentoperatorintroducedinRefs.[45,46,47,48].

Letustaketheimpuritypotentialstobeabsentattimet= ∞;thentheyaregraduallyswitchedon.Attheinitialtime,H0commuteswiththenumberoperators

oftheleftmovingandright LandN

movingfermions,N

Rrespectively.Intheabsenceofanyimpuritypoten-tials,alltherightmoversoriginateintheleftreservoirwhichismaintainedatthechemicalpotentialµL,andalltheleftmoversoriginateintherightreservoirmain-tainedatthechemicalpotentialµR.Hence,thesystemisinitiallydescribedinthegrandcanonicalensemblebythechemicalpotentialsµLandµRwhicharethecientsofthenumberoperatorsN

coe -LandN Rrespectively.Wewillworkintheinteractionrepresentation,takingthechemicalpotentialstobepartoftheinteraction.Thisin-troducestimeiµdependencesintothefermionicoperatorsψLψ→RandψLeLt

andandψRψψ →ψReiµRt.Theoperators

ψLappearingininHimp(seeEqs.(1)and(2))L

thereforeRpickupfactorsofe±iω0t.

Iftherewerenoimpurities,therewouldbeacur-rent owingtotheleftgivenbyI0=q2Vbias/(2π)=qω0/(2π).Inthepresenceofimpurities,someofthis

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

currentisbackscatteredtotheright.Thetotalcurrent owingtotherightisgivenbyI= I0+Ibs,whereIbsisthecorrectiontothecurrentduetobackscatteringbytheimpurities.Thebackscatteredcurrentisde nedasI dN bs(t)=q

R2

dxdyρ(x)V(x y)ρ(y),(19)

whereV(x)isarealand

evenfunction

of

x

,

andtheden-sityρ=ψ ψisgiveninEq.(2).WecanwriteEq.(19)inasimplewayifV(x)issoshortrangedthattheargu-mentsxandyofthetwodensity eldscanbesetequal

4

ingtheanticommu-tationrelationsbetweenthefermion elds,weobtain

Hint=g2

dxψ RψRψ

LψL,(20)whereg2isrelatedtotheFourierasg2=V

transformofV(x)(0) V (2kF).De ningaparameterγ=g2/(2πvF),wehavetherelations

K=

1 γ

2v

φ

2

φ

2παη Rη Lei2√

2πα

η Lη R

e

i2

√(2π)2[(xp xr)2 (v(t′ t) iα)2]K

(24)

forallvaluesofK.

Forthenon-interactingcasewithK=1,wecaneval-uatetheabovegroundstateexpectationvaluedirectlywithoutusingbosonization.Weusethesecondquan-tizedexpressionsforthefermion elds,

ψ ∞

dk

R=

∞

ik( x vFt)2π

aLke,

(25)

wherethecreationandannihilationoperatorssatisfythe

anticommutationrelations{aRk,a

k k′).ThegroundstateRk′|0 }is=annihilated{aLk,aLk′2πδ(}by

=aRk,aLkfork>0andbya Rk,a

Lkfork<0.Wethen ndthatthegroundstateexpectationvalueagreeswiththeresultgiveninEq.(24)forK=1andv=vF.

Ingeneral,thebackscatteredcurrenthastwoparts:oneindependentoftimewhichwecallIdc,andtheothervaryingwithtime,withfrequency2ωtosecondorderinUp,whichwecallIac.Iacdoesnotcontributetoany

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

chargetransferasitsaverageoveracycleiszero.Inthenextfewsubsections,wecalculatetheexpectationvalueofthebackscatteredcurrentforvariouscasesandstudythemindi erentlimits.Tosimplifyourcalculations,weagainassumethatωxrp/vandω0xrp/varesmallandthatω≥0.Itwillbeconvenienttode nethecombinations

ω+=ω0+ω,

andω =ω0 ω.

(26)

C.

Singleimpurity

ThiscasehasbeendiscussedinRef.[48];werepro-ducetheresultshereforthesakeofcompleteness.SomedetailsofthecalculationsareprovidedintheAppendix.

IppqUp

2bs,dc

=

v

×[sgn(ω+)|ω+|2K 2K 2

1+sgn(ω )|ω |2K 1],

(27)

IppqUp

2bs,ac=

v

×[sgn(ω+)|ω+ 2K 2

ωt+|2K 1

×cos(22φp+sgn(ω+)πK)

+sgn(ω )|ω+2 φ|2K 1

×cos(2ωtp sgn(ω )πK)],

(28)

wheresgn( )≡1if >0,0if =0and 1if <0.InEqs.(27-28),wenotethatthecurrentsbecomelargeinthelimitω0tiveexpansionin→powers±ωifKof<Up1/breaks2.Hencedownthewhenperturba-ω0iscloseto±ω[48].Theregionofvalidityoftheperturba-tiveexpansioncanbeestimatedusingaRGanalysisasdiscussedbelow.

Eqs.(27-28)implythatforthepurewithω0=0,Ipppp

pumpingcase

bs,dc=Ibs,ac=0.Forasingleimpurity,therefore,chargepumpingdoesnotoccur,whetherornotthereareinteractionsbetweentheelectrons.Forthe

purebiascasewithω=0andφp=0,wehaveIpp

Ippbs,ac～Upω20K 1bs,dc+2.Thusthebackscatteringcorrectiontotheconductancegivenby Ibs,dc/Vbias= qIbs,dc/ω0is

proportionaltoU22K 2

InthepresencepVofbias.

bothbiasandpumping,thecorrec-tiontothedi erentialconductance G= q Ibs,dc/ ωgrowslargeasUwithresultsp2

|ω±|2K 2forω+orωconsistentbasedonRGcalculations →0.This[9,10].isNamely,thepresenceofinteractionsbetweentheelec-tronse ectivelymakestheimpuritystrengthUpafunc-tionofthelengthscale;thisisdescribedbytheRGequa-tiondUp/dlnL=(1 K)Up,to rstorderinUp(L).HencethevalueofUp(L)atalengthscaleLisrelatedtoitsvalueUpde nedatamicroscopiclengthscale(say,

5

α)asUp(L)=(L/α)1 KUp.Inourcase,thelengthscaleLissetbyv/|ω+ritystrengthUp(L)therefore|orv/increases|ω |.Theas(e ectivev/|ω±|)1impu- KUpforω+orω[Up(L)]2～U p2

|ω→±|20,K and2.Thisthedivergencecorrectionmust Gbegrowscuto as

when Gbecomesoforder1,inunitsofq2/(2π).Restor-ingtheappropriatedimensionfulquantities,weseethattheaboveRGanalysisandperturbativeexpansionarevalidaslongasUp/v<<(α|ω±|/v)1 K.

D.

Severalimpurities

Wenowconsiderthecaseofseveralimpuritieslocatedatxpwiththephasesoftheoscillatingpotentialsbeingφp.Weagainde nexrpandφrpasinEq.(13).Thebackscattered currentcanbewrittenasIbs=ofINext,we ndthat

bsaregiven pppIbs+IprThedcandacpartspp

p<rbs.intheprevioussubsection.IprUr

bs,dc

=

qUpv

×[sgn(ω+)|ω+|2K 2K 2

1cos(2kFxrp+φrp)

+sgn(ω )|ω |2K 1cos(2kFxrp φrp)],

(29)

Ipr=qUpUr

2K 2

bs,ac

v cos(2kFxrp)

×[sgn(ω+)|ω+ωt+|2K 1

×cos(2φp+φr+sgn(ω+)πK)

+sgn(ω )|ω |2K 1

×cos(2ωt+φp+φr sgn(ω )πK)].

(30)

Forthepurepumpingcasewithω0=0,weseethatIprω2K 1sin(2kFxrp)sin(φrp),whileIprbs,dc～[49]bs,ac=0.Eq.(29)di ersfromtheresultsgiveninRef.duetothetermsinvolving2kFxrp.

WenotethatthecurrentsgiveninEqs.(27-28)and(29-30)allreversesignifwechangeω0→ ω0andxp xpforallp.Thisisanaturalconsequenceofparity→reversal,i.e.,interchangeofleftandright.

ThedcpartsgiveninEqs.(27)and(29)becombined togiveatotalcurrentIbs,dc= canpppIIprbs,dc+p<rbs,dc,Ibs,dc=

q

2K 2

v

×[sgn(ω+)|ω+|

2K 1

|

Upei(2kFxp+φp)2

p

|+sgn(ω )|ω |2K 1|

Upei(2kFxp φp)p

|2].

(31)

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

Theaboveexpressionsuggeststhatthecurrentwillbemaximizedifeither2kFxp+φpor2kFxpallp.Thismeansthatthe φphasthesamevalueforpotentialsinEq.(3)shouldbeoftheformU

pcos(ωt 2kFxp)orUpcos(ωt+2kFxp);thisdescribesapotentialwavetravelingtotherightortotheleft.Suchawavehasbeenstudiedextensivelyforthecaseofnon-interactingelectrons;seeRefs.[20,21,22,31,32,33,34,38]and[40,41,42,43].

Anunusualphenomenonoccursiftheinteractionsaresu cientlystrong,i.e.,ifK<1/2.Ifthereisnobias,theDCpartofthecurrentgenerallygoesasω2K 1whichincreasesasωdecreases.However,itisclearthatifωwasexactlyzero(time-independentimpurities),thenthecurrentwouldalsobezero.Thesetwostatementsimplythatthecurrentmustbeanon-monotonicfunctionofω,andmusthaveatleastonemaximumatsomevalueofω.FindingthelocationofthemaximumrequiresustogobeyondthelowestorderperturbativeresultsofthisFIG.1:(Coloronline)DCpartofthebackscatteredcur-rentasafunctionofthebiasω0forseveralimpurities,whenω±aresmall.Thered(dot),magenta(dash),blue(dash-dot)andblack(solid)linesshowtheresultsforK=1/4,1/2,3/4and1respectively.Wehave|Ptaken+φpUpei(2kFxpp)|2/|P

i(2kpUpeFxp φp)|2=2:1.

Figure1showsthedcpartofthebackscatteredcur-rentasafunctionoftheappliedbias,fora xednon-zerovalueofthepumpingfrequencyω,assumingthatωandω0aresmall.WehaveusedtheexpressioninEq.(31)toplotthevalueofIbs,dc(ω0)/Ibs,dc(ω0=0)asafunctionofω0/ω,forfourdi erentvaluesoftheparameter| pUpe

i(2kK=1/4,1/2,3/4Fxp+φp)2

example.ForK=1|/|/4, and1,takingtheratiothepUpei(2kFxp φp)currentdiverges|2=2:1asanatω0=±ωasmentionedabove.Wealsonotethelinearandpiece-wiseconstantdependencesofthecurrentonω0forK=1

6

and1/2respectively;thisisdiscussedinSubsec.III.Ebelow.

Ifwerelaxtheassumptionsthatωxrp/vandω0xrp/varesmall,thentheexactexpressions(uptosecondorderintheimpuritypotentials)fortheacanddccomponentofthebackscatteredcurrentaregivenby

Ipr=

qUpUr

√bs,dc

4πv2Γ(K)

(ω

αv

×[sgn 1/2 K

+)|ω+|K 1/2JK 1/2(|ω+xrp×+sgncos(2(ωk|/v)Fxrp+φrp)

k )|ω |K 1/2JK 1/2(|ω×cos(2 xrp|/v)Fxrp φrp)],

(32)

Iprbs,ac

=

qUpUr

√4πv2Γ(K)cos( αv

×[sgn(ω πK)

1/2 K

cos(2kFxrp)

+)|ω+|K 1/2JK 1/2(|ω+xrp×+sgncos(2(ωωt+φ|/v)p+φr+sgn(ω+)πK)

)|ω |K 1/2JK 1/2(|ω+φ xrpr+×{|cos(2ωωt+φ sgn(ω|/v)p )πK)

+|K 1/2J1/2 K(|ω+xrp|/v)

sin(2 |ωtω +|K 1/2J1/2 K(|ω×φ xrp|/v)}p+φr)].(33)

BesselfunctionJisdiscussedintheAppendix;us-apowerseriesexpansiongiventhere,wecanshowEqs.(32-33)reducetoEqs.(29-30)inthelimitrp/v→0.WenotethattheexpressionsinEqs.(27-donotchangeifwerelaxtheassumptionsthatωand0aresmall.

Figure2showsthedcpartofthebackscatteredcurrentasafunctionoftheappliedbiasforthecaseoftwoim-purities,labeled1and2,takingU1=U2,2kFx12=π/2,φ12= π/4,andωx12/v=1;thusωandω0arenotsmall,incontrasttothecaseshowninFig.1.WehaveusedtheexpressionsinEqs.(27)and(32)toplotthevalueofIbs,dc(ω0)/Ibs,dc(ω0=0)asafunc-tionofω0/ω,forfourdi erentvaluesoftheparameterK=1/4,1/2,3/4and1.WeseesomeoscillationsinFig.2duetotheappearanceoftheBesselfunctionsinEq.(32).ForK=1/4,weagainseedivergencesatω0=±ω.

E.

K=1and1/2

WenowdiscussthespecialcasesK=1and1/2wheretheexpressionsforsomepartsofthecurrentssimplifyconsiderably.

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

FIG.2:(Coloronline)DCpartofthebackscatteredcurrentasafunctionofthebiasω0fortwoimpurities,whenω±arenotsmall.Thered(dot),magenta(dash),blue(dash-dot)andblack(solid)linesshowtheresultsforK=1/4,1/2,3/4and1respectively.WehavetakenU1=U2,2kFx12=π/2,φ12= π/4,andωx12/v=1.

Fornon-interactingfermionswithK=1,we ndfromEqs.(27-28)thatinthesingleimpuritycase,

Ippp

bs,dc=

qU2

4πv2ω0cos(2ωt+2φp).(34)

F

ThetotalcurrentisgivenbyI= I0+Ipp+Ipp

bs,dcbs,ac,

2I=

qω0

vF

.(35)

Thisisconsistentwiththefactthatthetransmission

probabilityacrossastaticpoint-likebarrierofheightUis1 (U/vF)2uptoorderU2.Forthecaseofseveralimpurities,we ndfromEqs.(29-30)that

IprUr

bs,dc=

qUp2πv2ω0cos(2kFxrp)cos(2ωt+φp+φr).

F

(37)

Notethatthedcpartofthecurrentisgivenbyalinearcombinationofthepurebiaspartandthepurepumpingpart,anditagreeswiththeexpressiongiveninEq.(13).ForK=1/2,wecanobtainthedi erentpartsofthecurrentsbytakingthelimitK→1/2inEqs.(27-28)

7

(29-30).We ndthat

ppqUp

2=

4παv

[sgn(ω+)cos(2kFxrp+φrp)+sgn(ω )cos(2kFxrp φrp)],

ppqUp

2=

ω+

π

ln|

4παv

[(sgn(ω+)+sgn(ω ))cos(2ωt+φp+φr)+

2

ω

|sin(2ωt+φp+φr)].

(38)

ThustheDCpartsofthecurrentsdonotdependontheprecisevaluesofωandω0iftheyareunequal,andtheyhavea nitediscontinuitywhenωcrosses±ω0.

Toconclude,weseethatthedcpartsofthecurrentsarelinearfunctionsofω0,ωforK=1,andarepiecewiseconstantfunctionsofω0,ωforK=1/2.

F.

Extendedimpurities

TheanalysisinSubsec.III.Dcanbereadilygener-alizedtothecasewherethereisanextendedregionof

oscillatingpotentials[50].LetusreplacethediscretesetofpotentialsgiveninEq.(3)byanoscillatingpotentialofthefollowingform

U(t)=

dxU(x)cos[ωt+φ(x)].(39)WethenseefromEq.(31)thatthedcpartofthe

backscatteredcurrentisgivenby

Iq

bs,dc=

v

×[sgn(ω+)|ω+|2K 2K 2

1

| dxU(x)ei[2kFx+φ(x)]|2

+sgn(ω )|ω |2K 1

| dxU(x)ei[2kFx φ(x)]|2].

(40)

tosecondorderinU(x).Forthepurepumpingcasewithω0=0,we ndthat

I

q

bs,dc=× v

dxdx′U(x)U 2K 2

ω2K 1(x′)sin[2kF(x x′)]

×sin[φ(x) φ(x′)].

(41)

Eq.(41)impliesthatthechargepumpedpercycle,

Q=(2π/ω)Ibs,dc,scalesasω2K 2;forK<1,this

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

growslargeintheadiabaticlimitω→0.Inthislimit,wesawearlierthatthee ectivelength-dependentimpuritystrengthdivergesatsmallenergyscales,whichimpliesthattheimpuritypresentsaverylargebarriertotheelectronsandthetransmissioncoe cientisverysmall.Inthislimit,ithasbeenarguedinRefs.[44,47]thatthepumpedcharge Qisquantizedtobeanintegermultipleofq.

G.

Spin-1/2electrons

Forspin-1/2electronsinonedimension,thephe-nomenonofspin-chargeseparationoccursifthereareinteractionsbetweentheelectrons.Thespinandchargedegreesoffreedomcanbeseparatelybosonized[56,58].Thetwobosonictheoriesarecharacterizedbythepa-rameters(Ks,vs)and(Kc,vc)respectively.ForasystemwithSU(2)rotationalinvariance,Ks=1.ThegroundstateexpectationvalueinEq.(24)thentakestheform

0|ψ σR(xp,t′)ψσL(xp,t′)ψ σL(xr,t)ψσR(xr,t)

|0

～

1

[(xp xr)2 (vc(t′ t) iα)2]Kc/2

,(42)

whereσ=↑,↓isthespinlabel.Theappearanceoftwodi erentvelocities,vsandvc,andtwodi erentexpo-nents,1/2andKc/2,inEq.(42)makestheexpressionsforthebackscatteredcurrentrathercomplicated.How-ever,wecan ndthepowerlawofthedependenceofthecurrentsonthefrequenciesbyasimplescalingargu-ment.Withtheapproximationsmadeearlier,ωxrp/vs,candω0xrp/vs,cchangedfrom1→/(t0,′ wet)2KseeinthatEq.the(24)timeto1dependence/(t′ t)Kc+1hasinEq.(42).Thedependencesofthebackscatteredcurrentsonthefrequenciesthereforechangefrom|ω0ω|Kcinthespin-1/2±case.ω|2K 1inthespinlesscaseto|ω0±SinceKcispositiveingeneral,thecurrentnolongerdivergesasω0→±ω.

IV.

DISCUSSION

Wehaveconsideredthee ectsofabiasandanumberofweakandharmonicallyoscillatingpotentialsonchargetransportinaTomonaga-Luttingerliquid.Wehavecom-putedthebackscatteredcurrenttosecondorderintheamplitudesofthepotentials.Formostofourresults,wehaveassumedtheoscillationfrequencyandthebiastobesmall,butwehaverelaxedthatassumptioninEqs.(32-33).ForourassumptionofaDiracfermionwithalineardispersiontobevalidforanexperimentallyreal-izablesystem,wemustofcourseassumethatωandω0aresmallcomparedtothebandwidthoftheelectrons.We ndthatthebackscatteredcurrentismaximizedforatravelingpotentialwaveinwhichthepositionsand

8

phasesoftheoscillatingpotentialsarerelatedinalinear

way.Forspinlesselectrons,iftheinteractionsaresu -cientlyrepulsivewithK<1/2,thebackscatteredcur-rentdivergesforspecialvaluesofthebias,namely,forω0the→correction±ω.Fortoanytherepulsivedi erentialinteraction,conductancewithdivergesK<1,forω0liaritywhich→±ωarises.Finally,whenweseveralhaveimpuritiespointedoutareapresentpecu-andK<1/2;namely,thecurrentmustingeneralbeanon-monotonicfunctionofthepumpingfrequencywhenthereisnobias.

Itwouldbeusefultogeneralizeourresultstothecaseofoneormorestrongimpuritypotentials,orweaktunnelingsbetweentwoTomonaga-Luttingerliquids;thetechniqueofbosonizationcanbeusedinsuchsituationsalso.

Acknowledgments

A.A.thanksCSIR,IndiaforaJuniorResearchFel-lowship.D.S.thanksSourinDasandSumathiRaoforstimulatingdiscussions.WethankDST,Indiafor nan-cialsupportundertheprojectsSR/FST/PSI-022/2000andSP/S2/M-11/2000.

APPENDIXA:SOMEMATHEMATICAL

FORMULAE

Weneedtoevaluateintegralsoftheform

∞

dτ

exp(±i τ)

x

(τ2 x2)K

=

√2

2√

2x

(τ2 x2)K

=

2

ν

∞n

(z/2)2n

2

( 1)

n=0

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

9

andtherelation

Yν(z)=

1

,

sin(πz)√

Γ(z)Γ(z+1/2)=2z 1Γ(2z).

2

(A5)

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