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