Charge transport in a Tomonaga-Luttinger liquid effects of p(3)

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