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)