PHYSICAL REVIEW B 66, 153304 ~2002!

Momentum-resolved tunneling between Luttinger liquids

D. Carpentier,1 C. Peca,2 and L. Balents21Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

2Physics Department, University of California, Santa Barbara, California 93106~Received 27 November 2001; revised manuscript received 18 April 2002; published 4 October 2002!

We study tunneling between two nearby cleaved edge quantum wires in a perpendicular magnetic field. Dueto Coulomb forces between electrons, the wires form a strongly interacting pair of Luttinger liquids. Wecalculate the low-temperature differential tunneling conductance, in which singular features map out the dis-persion relations of thefractionalizedquasiparticles of the system. The velocities of several suchspin-chargeseparatedexcitations can be explicitly observed. Moreover, the proposed measurement directly demonstratesthe splintering of the tunneling electrons into a multiparticle continuum of these quasiparticles, carryingseparately charge from spin. A variety of corrections to the simple Luttinger model are also discussed. Ourresults are in agreement with recent experiments by Auslaenderet al. @Science295, 825 ~2002!#.

DOI: 10.1103/PhysRevB.66.153304 PACS number~s!: 71.10.Pm, 73.21.Hb, 73.40.Gk, 73.63.Nm

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The only universally accepted example of a non-Feliquid metallic state is the one dimensional~1D! Luttingerliquid ~LL !.1 Remarkably, the quasiparticle excitations ofLL are fractionalized, comprising a diverse set carrying sseparately from charge, and charge in fractions of the etron chargee. LL behaviorhasbeen observed experimentalin carbon nanotubes,2 through strongly energy dependentlo-cal tunneling, and more recently in GaAs quantum wirthrough power-law resonant tunneling line shapes.3 Evidenceof charge fractionalization has also been seen in shot nexperiments using fractional quantum Hall edge stat4

which are somewhat specialchiral Luttinger liquids.5 De-spite these successes, no direct experimental evidencfractionalization has ever been obtained in a nonchiralsystem. In this letter, we show that measurements ofnonlinear tunneling conductance between parallel Luttinliquids in a transverse magnetic field provide a directspec-troscopic probe of fractionalization. The results describbelow and summarized in Fig. 4 give very similar informtion to anideal photoemission experiment. Indeed, somedications of spin-charge separation were seen in photoesion spectroscopy of the quasi-1D cuprate SrCuO2.6

Tunneling spectroscopy has, however, the advantages oing possible on a single,isolated1D system and with potentially much higher resolution than photoemission.

Controlled tunneling experiments between two parawires have been recently conducted using cleaved-edge ogrowth by Auslaenderet al.7 The experimental geometry wconsider is indicated schematically in Fig. 1. The tw‘‘wires’’ are in fact confined surface states, and electriccontact is made only to the upper wire via a two-dimensioelectron gas~2DEG!. With L8@L, nearly the full electro-chemical potential drop occurs between the shorter~left! seg-ment of the upper wire and the lower wire. Because ofuniformity of the barrier, momentum along the wire is coserved during tunneling.8

We assume that the barrier is sufficiently high as to eslish a quasi-equilibrium state on either side of the barrtreating tunneling across the barrier perturbatively. A nointeracting model Hamiltonian for the system neglecting tuneling is then

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wherea51,2 labels the upper/lower wire,a5↑,↓ labels theelectron spin, andUa andma are the electrostatic and chemcal potential of theath wire, respectively. We choose bconvention to takema50 in equilibrium, so thatkFa

5A2mUa is the Fermi momentum in wirea. Neglecting theenergy dependence of the tunneling amplitudew, the Zee-man shift~see below!, and a small energy shift due to orbitamagnetic effectswithin each wire, the tunneling Hamiltoniain the presence of a magnetic fieldBz ~in the gaugeAy5Bx) is

H tun52w(a

E dx@c1a† c2aeiQx1c2a

† c1ae2 iQx#. ~2!

Here the magnetic wave vectorQ52pBd/f0 , d is thecenter-to-center distance of the wires, andf05hc/e is theflux quantum. The one-dimensional~1D! tunneling currentdensity J5 iew(c1a

† c2aeiQx2c2a† c1ae2 iQx) can be calcu-

lated directly from Fermi’s golden rule. The result, whogross features appear experimentally in Ref. 7, is shownFig. 2, takingm15V, m250. This diagram can be undestood physically by considering all processes by whichelectron can be transferred between the two wires, mov

FIG. 1. Schematic experimental geometry~from Ref. 7!.

©2002 The American Physical Society04-1

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BRIEF REPORTS PHYSICAL REVIEW B66, 153304 ~2002!

from an occupied to an unoccupied state. Geometrically,2 is obtained by drawing the locus of values ofV,Q forwhich one of the four Fermi points lies on the other parabwhen shifted vertically and horizontally byV andQ, respec-tively.

Zero biasfeatures occur when~A! Q56(kF11kF2), ~B!Q56(kF12kF2). It is in the low-energyregion of theV-Qplane near these four features thatuniversalfeatures arise inthe interacting system, to which we now turn. Focusing othe the low-bias regime, we first decompose the electfields into right and left moverscaa(x)5cRaa(x)eikFax

1cLaa(x)e2 ikFax, which are described by the LuttingeHamiltonian~taking ma50 for simplicity!

H052 i(aE dxvFa@cRaa

† ]xcRaa2cLaa† ]xcLaa#, ~3!

wherevFa5kFa /m. In general, these right- and left-movinFermions undergo a diverse set of scattering processesdiated by the Coulomb interaction~screened by the 2DEG!.A systematic study of these terms9 reveals an important simplification when, as approximately true experimentally,7 thewidth W of a typical cleaved edge quantum wire is larcompared to the Fermi wavelength (kFaW@1), though evenwhen this condition fails our theory is exact at low energie9

We also expect the 2D screening lengthls*W. These prop-erties imply a strong suppression of two-electron bascattering processes. We therefore adopt a forward scattemodel retaining only the strongest~unsuppressed! interac-tions

H int51

2 (ab

E dx na~x!Vabnb~x!, ~4!

with H5H01H tun1H int , and na5(acRaa† cRaa

1cLaa† cLaa . Equation~4! neglects momentum dependen

of the forward-scattering interactions, and is henvalid for eV&\vFa /ls . The interactionsVab can be roughlyestimated as V11.V22.(2e2/e)ln(2ls/W), V12.(2e2/e) ln(ls/d) where e is the electron charge,e thewires dielectric constant. Experimentally,d2W!W,7 so theinterwire interactionV12 is not negligible. Thus the tunnelinconductance does not in fact probe the spectral propertie

FIG. 2. Tunneling current in theV-Q plane for the noninteracting model. Regions of nonzero current are shaded, anddI/dV hasdelta-function singularities along the boundaries between shaand unshaded regions.

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two decoupled1DEG’s. Nevertheless, all properties of thinteracting HamiltonianH01H int can be calculatedexactlyby bosonization~LL theory!.

From now on we will focus on the current at zero temperature around the point A of Fig. 2. An analogous desction can be obtained around each of the low bias pointsFig. 2.9 To lowest order in perturbation inw, we can writethe current densityJ as the sumJ52euwu2(J12J2), whereJ1 ,J2 are positive functions which satisfy~time-reversalsymmetry! J1(Q,V)5J2(2Q,2V), and J1 (J2) is non-zero only for V.0 (V,0). Around point A,q5Q2(kF1

1kF2) is small, and

J15RE2`

1`

dxE0

`

dt ei [(V1 id)t1qx]CER→L~x,t→e1 i t !,

where CER→L is the Euclidean correlation functio

CER→L(x,t)5^cR1

† cL2(x,t)cL2† cR1(0,0)&. LL theory then

gives

CER→L~x,t!5

a02u112u221

~2p!2 )a51,2

~vsat2eaix !21/2

3~vcat2eaix !2h2ua~vcat1eaix !1h2ua,

~5!

whereea5(21)a11, anda0 ~of orderW) is a small distancecutoff for the LL description. Fractionalization is evident fomally in Eq. ~5! through the multiple branch points charaterized by distinct charge and spin velocitiesvc1/2 andvs1/2.For two independent wires (V1250), it is well known thath51/4 andu1 ,u2.1/4. In the present case, the strong inteactions between the wires makeh interaction dependent, anu1 ,u2 can take values smaller than 1/4. Indeed in the castwo coupled identical wires (vF15vF25vF ,V115V22), thesetake the values u151/@4A112(V111V12)/(pvF)#, u2

5A112(V112V12)/(pvF)/4, h50. General but nonillumi-nating formula are postponed to Ref. 9.

Letting x→tu in Eq. ~5! and integrating overt gives

J1~Q,V!}G~122u122u2!3ReE2`

1`

du h~u!, ~6!

where the complex functionh(u) is defined by

h~u!5@d2 i ~V1qu!#2u112u221CER→L~u,i 1e!.

The contour of integration in Eq.~6! can then be de-formed according to Fig. 3. Ash(u) vanishes faster thanuuu22 at infinity, we are reduced to the integral ofh(u) onthe contour 3~see Fig. 3!. More explicitly in the situationwherevs1 ,vs2,vc1,vc2 , and forq.0, the remaining inte-gral can be written as the sum of real integrals

J1} (a51

3

u~V2qva!sinfaE2Min(va11 ,V/q)

2vauh~u!udu,

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BRIEF REPORTS PHYSICAL REVIEW B66, 153304 ~2002!

with v15vs2 ,v25vc1 ,v35vc2 ,v451`, and f15p/2,f25p(u12h11/2),f35p(u11u211/2). Similar expres-sions forJ1 andJ2 can be derived for any other order of thvelocities, and for the otherV.0 points of Fig. 2.9 A typicalresult is presented in Fig. 4.

The density plot of the differential conductance~perunit length! G5dJ/dV in Fig. 4 directly exhibits evidenceof electron fractionalization. First, nonanalytic features apear along rays~3 per quadrant! whose slope gives the twcharge and spin velocities. Second,dJ/dV is nonzerofor any bias above the thresholduVu.v* uqu, with v*5 Min(vs2 ,vc1 ,vc2)Q(qV) 1 Min(vs1 ,vc1 ,vc2)Q(2 qV).Both these properties can be understood from kinema

FIG. 3. Integration contour and cuts forV,q.0. The originalintegral ofh(u) defined on2`,1` ~contour 1! is modified into aintegral on contour 213.

FIG. 4. Density plot of the ideal differential conductancepoint A of Fig. 2 without disorder~top! and with disorder and weakthermal rounding~bottom!. Dashed black lines indicate the dispesion relations of the spin~outer lines! and charge~inner lines! ex-citations. Both plots are shown for values approximately approate to cleaved edge overgrowth:u150.26,u250.51,h50.25, withvelocitiesvs151,vc151.2,vs251,vc251.6 in units ofvF1.

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Each tunneling event corresponds to a transfer of charge6eand spin61/2 from one wire to the other, accompaniedthe addition of momentumq and energyeV. This creates acombination of ~six total! fractional ‘‘chargons’’ and‘‘spinons.’’ The final state with the least energy for a givemomentumq gives all the momentum to the slowest apprpriate ‘‘particle,’’ which determines the threshold. Moreovesince the final states are six-particle excitations, kinemaallows any energy abovev* uqu, explaining the nonzeroweight in dJ/dV as a multiparticle continuum. Detailed expressions for the singularities along the various rays in Figcan be extracted from Eq.~7!.9 Somewhat similar results fo1d-2d tunneling were predicted theoretically in Ref. 10.

If the Fermi velocities in the two wires are not too diffeent, both charge velocities are generally larger. Thenspin/Fermi velocities form the lower envelope for substanweight in the differential conductance plot. Most of thweight, however, lies higher, in the vicinity of the rays dfining the charge velocities. This situation appears toscribe recent experiments in Ref. 7. If, however, the tFermi velocities are very different, the slower of the twcharge velocities may lie below the larger of the two spvelocities. In these experiments the Luttinger parameteg5vs /vc in each wire is estimated to beg.0.7.

We now turn to a discussion of the numerous effectsout of the above treatment. The two most significant corrtions can be treatedexactly. First, the spectral density irounded on a scale set by temperature~the rounding can becalculated exactly to describe detailed line shapes9!. Experi-mentally, fore2/vF of O(1), LL behavior is expected to bmanifested for eV,eF;10220 mV in cleaved edgesamples. Thus for experiments withT in the K range, thethermal rounding is minimal except at very low bias. Secofor kFaW@1, the dominant impurity process is elasticfor-ward scattering. This is easily included by convolvinG(V,q) ~in q) calculated above with a Lorentzian of halwidth Dq[1/l el , and has several consequences. The sinlarities at V.0 are rounded over an energy width 1/tel;vF,i / l el . Nonvanishing weight also appears outside thenematically allowed region. At low bias and temperatuthis weight is itself singular and displays a pronounced lbias conductance dip. In particular, foreV,k BT,v* q!1/tel , G(V,q,l el);uwu2@ l el /(11q2l el

2 )#TbFb(eV/kBT),with 0,b52u112u221,1, where Fb(X) is thewell-known2 scaling function for point-contact tunnelingsatisfyingFb(0)51 andFb(X);Xb for X@1.

Even atT50, effectsnot included in the LL model canbroaden the spectral function. First, consider the effectsnonforward-scattering interactions between electrons atedge. In the generic situation,k F1Þk F2, and charge/spin-density-wave coupling~e.g., HCDW5V CDWcR1

† cL1cL2† cR2!

between the two wires is forbidden. The dominant residterm is then the ‘‘exchange’’ interaction~we reserve the usuaterm ‘‘back scattering’’ for impurity effects discussed below!

within a single wire, e.g., Hex, i52l ivFicRi† sW cRi

•cLi† sW cLi , where l i characterizes the dimensionless ba

scattering strength in wirei. In the experimentally relevansituation l i;(kFiW)21!1, so this is a weak interaction

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BRIEF REPORTS PHYSICAL REVIEW B66, 153304 ~2002!

Formally, such exchange interactions are known to be mginally irrelevant in the renormalization group sense, ansimple perturbative~bosonized ‘‘phonon’’ self-energy! esti-mate suggests a resulting ‘‘lifetime’’ scaling approximatelinearly with energy v, (1/tex,i)

2;l i2v2/@c1

1c2l i2 ln2(e F,i /v)#, wherec1/2 are order one constants. Th

magnitude of 1/tex,i , however, is small, due both to thsmallness ofl i and the additional logarithmic suppressionlow energies. Other ‘‘internal’’ corrections to the forwardscattering model, such as band curvature, are strongly ievant, and a similar self-energy estimate shows that tcontribute only negligible 1/tcurve;O(v2/eF) terms to thelifetime.

Electrons in the wires also interact with those in the b2DEG. Expressing the Coulomb interaction in the basiswire and bulk states, one finds several distinct scattechannels. Most interesting are charge6e decay processes, iwhich an electron/hole in one of the wires decays intoelectron/hole in the 2DEG and an electron/hole pair. Becathe 2DEG is a Fermi liquid, however, strong phase-sprestrictions reduce the rate of such decays at low energcrudely, 1/te;cev

2/eF , where theO(1) constantce de-pends upon the density of the 2DEG, etc. InterestinglykF,bulk,3kF,i , such processes are kinematically forbidd(ce50). Similar, albeit slightly less restrictive kinematconditions reduce the phase space for charge62e decay, inwhich Cooper pairs~hole pairs! are scattered between thwires and 2DEG. All such processes in which chargetransferred are further reduced by LL orthogonality catasphe effects. Finally, there are forward-scattering processewhich no charge but only energy and small momentatransferred between the wires and 2DEG. Because the Clomb potential suppresses long-wavelength charge fluctions in the 2DEG, we expect this also to be a weak effeand will explore it in greater detail in a future publication9

In addition to the elastic forward scattering discussabove, disorder also mixes surface and extended state

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lowing conduction atV5T50. A lower bound on the broadening of spectral features in the tunneling conductanceprovided by the experimental determination of the 1D-2scattering lengthl 1D-2D'6m in Ref. 11; hence 1/t1D-2D.v F,i / l 1D-2D. The latter is a lower bound to the elastic cotribution, since there is also a small-momentum elastic coponent which contributes to the broadening but notl 1D-2D ~orthe longer elastic ‘‘back scattering’’ lengthl B defined in Ref.11!.

Two additional physical effects have not been taken iaccount above. First, Zeeman coupling leadsexactly9 to twosuperimposed copies of the LL results withQ shifted left andright by DQ↑/↓

A 56(gmBB/2)(v F1211v F2

21) for point A andDQ↑/↓

B 56(gmBB/2)(v F1212v F2

21) for point B. Second, direct2DEG-wire tunneling provides a second, albeit weaker, cduction channel,10,12but can be distinguished experimentalsince it contributes at low bias for the full rangekF22kF,bulk,uQu,kF21kF,bulk instead of at the singular pointA,B.

Finally, we comment on the validity of the perturbativtreatment of tunneling. ForqÞ0, the tunneling perturbationis formally an~infinitely! irrelevant operator. Thus for sufficiently weak tunneling at nonzeroq the perturbative treat-ment is accurate. Atq50, however, tunneling isrelevant: atan energye, the effective tunneling amplitude isw(e)5w(e0 /e)3/22u12u2, where the cutoffe0;e F,i . Equatingthis to e0, we estimate the cut-off energyeV*;e0(w/e0)2/(322u122u2). For eV,v F,iq,eV* , the perturba-tive treatment breaks down. Physically, in this regimeexpect coherent motion between the two wires. Due tonecessary complication of including dissipation in the leawe therefore postpone the analysis of this limit to a futupublication.9

L.B. was supported by NSF Grant No. DMR–998525and the Sloan and Packard foundations. D.C. was suppoby the ITP through Grant No. NSF–DMR–9528578. Cwas supported by Grant No. PRAXIS/BD/18554/98.

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