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,