PRL 100, 036603 (2008)

PHYSICAL REVIEW LETTERS

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Effects of Fermi Liquid Interactions on the Shot Noise of an SUN Kondo Quantum Dot
P. Vitushinsky,1 A. A. Clerk,1 and K. Le Hur2
1

Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8
2
Department of Physics, Yale University, New Haven, Connecticut 06520, USA(Received 8 August 2007; published 24 January 2008)

We study shot noise in the current of quantum dots whose low-energy behavior corresponds to an
SUN Kondo model, focusing on the case N  4 relevant to carbon nanotube dots. For general N, twoparticle Fermi-liquid interactions have two distinct effects: they can enhance the noise via backscattering
processes with an N-dependent effective charge, and can also modify the coherent partition noise alreadypresent without interactions. For N  4, in contrast with the SU(2) case, interactions enhance shot noisesolely through an enhancement of partition noise. This leads to a nontrivial prediction for experiment.
DOI: 10.1103/PhysRevLett.100.036603

PACS numbers: 72.70.+m, 71.10.Ay, 72.15.Qm, 73.63.Fg

Despite being over 40 years old, the Kondo effect remains among the most studied effects in condensed matterphysics. It is a canonical example of how strong interparticle interactions can lead to nontrivial behavior; it is also
attractive as it may be controllably studied in quantum dots[1]. In spite of ultimately being the result of strong interactions, much of the low-temperature phenomenology ofthe standard SU(2) Kondo effect in quantum dots relies onan effective noninteracting model, having a resonant level(the Kondo resonance) sitting at the Fermi energy. This,
however, is not the full story: low-energy properties areproperly described by a Fermi-liquid theory having weakinterparticle interactions [2,3]. While the effects of theseinteractions in quantum dot systems can be subtle, it was
shown in Ref. [4] that they play a large role in determining
the size of current fluctuations. Fermi-liquid interactions
lead to two-particle scattering and a resulting universal
effective charge e (defined via the magnitude of thebackscattering current noise) that is larger than 1: e 
5=3e. This result is a clear sign of the interacting natureof the low-temperature Kondo state in a quantum dot.
In this Letter, we now turn to more exotic quantum
dots that realize the higher symmetry SUN Kondo model(N  2), and which are described by an SUN-symmetricFermi-liquid fixed point. Such Kondo effects involve thescreening of an effective pseudospin that incorporates both
orbital and spin degrees of freedom; various realizations of
the N  4 case have been proposed [5–7] and even measured in carbon nanotube [8] and vertical [9] dots. Here, wewill examine how Fermi-liquid interactions in these systems affect the low-voltage current noise SI . Noise incarbon nanotubes is already under experimental study[10]; also, the first observations of noise through a SU(2)Kondo quantum dot have recently been made [11]. The
picture we obtain will have relevance to the general question of how two-particle interactions affect coherent shotnoise in a mesoscopic conductor. Formulating the Fermiliquid theory in terms of scattering states and using the
Keldysh technique, we find that the situation is markedly
different from the SU(2) case. For general N, interactions
0031-9007=08=100(3)=036603(4)have two distinct effects on shot noise. The first (the only
effect for N  2) is that Fermi-liquid interactions lead to
new two-particle scattering processes; these enhance theshot noise in a Poisonnian fashion (i.e., via uncorrelated
tunneling events). Such two-particle scattering can result inthe backscattering of either one or two particles, withrespective rates 1 and 2 . We find that the effective
charges associated with these processes are functions of N:e1 =e  2T0  1  2sin2 =N  1;e2  2e1 :

(1a)
(1b)

Here, T0 is the transmission coefficient through the dot atzero temperature and voltage. Equations (1) reduce to theresults of Ref. [4] in the SU(2) case, while in the SU(4)case, both these effective charges vanish.
In addition to the scattering-induced enhancement of SI ,a second effect of Fermi-liquid interactions is to modify thepartition noise already present without Fermi-liquid corrections. Recall that in a phase-coherent conductor withtransmission coefficient T0 , partitioning of electrons intotransmitted and reflected streams results in SI / T0 1 T0 . We find that two-particle Fermi-liquid interactions
enhance this contribution, in keeping with more phenomenological studies of shot noise [12]. This effect is absent in
the ideal SU(2) case, as T0  1 [13]. In the especiallyrelevant case N  4, this partition noise modification is
the only effect of interactions; this leads to a parameter-free
prediction for the interaction enhancement of noise[cf. Eq. (17)]. Note that Fermi-liquid corrections affectonly the nonlinear in voltage parts of the current andcurrent noise; we thus will focus on these quantities.
There will also be a linear-in-voltage contribution to thenoise whose value follows immediately from the value ofthe Fermi-energy phase shift and the asymmetry of coupling to the leads.
Model.—For concreteness we consider the case of SU(4)Kondo physics in a carbon nanotube dot [6]. A small
nanotube island (i.e., the dot) is attached via tunnel junctions to two nanotube leads. In addition to spin, electrons

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PRL 100, 036603 (2008)

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PHYSICAL REVIEW LETTERS

here have an orbital degree of freedom m that can take oneof two values (m  , ); m labels two degenerate bands[6]. We take m to be conserved in lead-dot tunneling as isconsistent with experiment [8]. Because of strong on-site
Coulomb interactions, the dot can be tuned to a regime
where it is singly occupied. It then acts as an effective 4component pseudospin, characterized by both its physicalspin and its orbital index m. Further, dot-lead tunneling
becomes an effective exchangelike interaction between the
dot and conduction electron pseudospin. The resulting
model is the SU(4) Kondo model. The ground state corresponds to a singlet formed by the impurity pseudospin anda screening cloud of electrons in the leads. This model
could be easily generalized to an SUN (N > 4) Kondo
model by allowing the orbital index m to take on N=2
different values. In what follows, we will consider thegeneral SUN case, paying special attention to N  4.
Fermi-liquid theory using scattering states.—We will beinterested in the behavior of our system at energies muchsmaller than TK , the Kondo temperature. In this regime, theSUN Kondo effect is described by a local Fermi-liquid
theory [3,15], which describes both elastic scattering off
the Kondo singlet, as well as interparticle interactions
induced by virtual excitations of the singlet. Both effects
are described by the phase shift tot;  0   " fors-wave scattering of an electron with pseudospin  off the
impurity. 0 is the phase shift at the Fermi energy; its value
is determined by the Friedel sum rule to be 0  =N.
 " accounts both for the energy dependence of elastic
scattering and for inelastic processes:P"" 2  0  n0
 " 
 0

: (2)
kB TK
kB TK
kB TK
Here,  is the density of states per electron flavor and theenergy " is measured from the Fermi level. Note that we
retain the "2 dependence of the elastic phase shift; while itplays no role in the SU(2) case, it will play a critical rolefor N  4. Universality at the SUN Kondo fixed point
ensures that the ratios between , 0 , and  are fixed. The
pinning of the energy of the Kondo resonance relative to
the Fermi energy leads to the relation   N  1 [15].
For N  2, the fixed point has particle-hole symmetry,
implying 0  0. For N > 2, the Kondo resonance is above
the Fermi energy; consequently, the fixed point is notparticle-hole symmetric. For N  4, one finds 0  2 .
This follows from equating the elastic parts of the phaseshift ;tot with arctan=2"d  ", the phase shift of aresonant level (i.e., the Kondo resonance) having width and energy "d [16]. For N  4, expanding to order " yields"d  =2 and   TK =; expanding to "2 yields 2  0 .
These properties of the N  4 Kondo resonance are consistent with recent calculations [6].
Applying the Fermi-liquid picture to our quantum dot,
we may replace the dot plus two lead system by a single
one-dimensional channel described by a field operator^  x: for x < 0, this describes the left lead, while for x >0, it describes the right lead. The scattering (both elastic
and inelastic) due to the dot plus screening cloud may thenbe modeled as occurring locally at x  0. It is convenient
to work in a basis of scattering states that correspond to the
Fermi-energy phase shift 0 . This can be converted to atransmission probability T0 via T0  sin2 0 . We thus introduce scattering states L x; k and R x; k, whichcorrespond (respectively) to waves incident on the dot fromthe left or from the right lead. For simplicity, we focus on
the case where both leads are equally coupled to the dot;
deviations will be considered in [14]. Inversion symmetry
then implies that for x < 0:p
L x; k  eikF kx  1  T0 eikF kx ;(3)p
R x; k  i T0 eikF kx :(4)Letting c^ L;k , c^ R;k denote the annihilation operators forthe scattering states, the field operator may be written as^  x / Pk L x; kc^ L;k  R x; kc^ R;k . The current operator for x < 0 then takes the form:^  I^D x  I^OD x;
Ix

(5)

1  T0 )e X y0c^ R;k0  c^ R;k T0 eikk xI^D x h k;k0 ;

with (R000 c^ yL;k0  c^ L;k eikk x  R0 eikk x ;(6a)X yp ikk0 xec^ 0 c^i T0 R0 e
 H:c: :I^OD x h k;k0 ; L;k  R;k
(6b)Without Fermi-liquid corrections, the average current
would be determined by I^D , while the zero-temperature
current noise would be determined by I^OD [12].
Similar to the N  2 case [1,17], the effects of  "
may be incorporated via the Hamiltonian:XHk c^ yj;k c^ j;k  H  H0  H ;(7a)k;;jL;R

H  

X  0k d^yk d^k0  ;2TK k;k0 ; k

(7b)

X
0
k  0k 2 d^yk d^k0  ;(7c)22TK  k;k0 ;
X X y

H 
(7d)
d^ d^k0  d^yq0 d^q0 0 :2
2 TK k0 ;q0 ;k;q 0 kpHere, k  @vF k and d^k  c^ L;k  c^ R;k = 2 (only thesymmetric combination enters due to inversion symmetry).
H  H0 and H represent, respectively, the elastic scattering from the impurity [to order "=TK 2 ] and the presence of two-particle scattering. This Hamiltonian
explicitly reproduces the phase shift  [Eq. (2)]; its
form follows from the usual relation between scatteringand potential matrices (see, e.g., [18]). The description

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H0  

PHYSICAL REVIEW LETTERS

PRL 100, 036603 (2008)

presented here is consistent with the results obtained inRef. [7] for N  4 using conformal field theory. As weshow, using Eqs. (7) and the current operator in Eq. (5)allows one to both recover results for the current obtained
from tot; and calculate current fluctuations.
Effect of elastic scattering.—We focus on the regime ofzero temperature, and small voltage L  R  eV
TK . As we treat the case of equal couplings to the left
and right leads, we use a symmetric voltage bias: L  R  eV=2 [i.e., the Kondo resonance position is fixed
relative to  L  R =2]. Consider first the effects ofH  H0 , which describe purely elastic scattering.
Keeping only these terms, our system is equivalent to anoninteracting system having an energy-dependent transmission coefficient T" given by



"
" 2
T"  sin2 0  
 0
:
(8)
kB TK
kB TK
We have verified through an explicit perturbative calculation in H  H0 that to order eV=TK 3 , the elastic scattering contributions to the current and noise do indeed
correspond to T". In particular, the elastic contributions
to the average current are given by [19]Ne2 Z L
hIij0d"T"
h
R



Ne2 V
2 cos20  0 sin20 eV 2
T0 

h
kB TK
12

 OeV=kB TK 4  :
(9)



Ne2 V
2 cos20  0 sin20 eV 2
T0 

h
kB TK
12

 OeV=kB TK 4  :(10)Note for N  4, there is no 2 contribution hIi.
Similarly, the elastic scattering
R contributions to the zerofrequency current noise SI 2 dthItI0i are givenby the noninteracting formula:
SI j0

2Ne3 Z L
d"T"1  T"
h
R

(11)


2Ne3 V2 cos40   0 sin40 =2
T0 1  T0  

h
12



eV 2
 OeV=kB TK 4  :
(12)
kB TK
Thus, the elastic part of  " yields V 3 terms in both hIi
and SI . In the case N  4, their ratio is simply2
1 d3 SI =dV 3 




 1:
(13)


0
2e d3 hIi=dV 3 0;V0This is just the result for a noninteracting resonant level. As
we will see, two-particle interactions modify both the sign

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and the magnitude of this ratio; this relative enhancementof noise over a noninteracting model is our key prediction
for experiment.
Effect of two-particle scattering.—We have calculated
the effect of H [Eq. (7d)] on the average current andcurrent noise perturbatively using the Keldysh approach.
Heuristically, H induces two-particle scattering between
the scattering states L , R . For L  R  eV (eV > 0)
and zero temperature, there is a range of energies " wherethe L " states are occupied, while the R " are unoccupied. As a result, three types of inelastic processes are
possible: (a) L L ! R L , (b) L R ! R R ,(c) L L ! R R . Using Fermi’s golden rule in H ,one finds that for a fixed initial and final spin configuration,
processes (a) and (b) occur at a rate 1 , while process (c)occurs at a rate 2 . These rates have no explicit-N depen1 eV eV 2
dence, and are given by 1  2 =8  2 24
h kB TK  , as inRef. [4].
The current associated with these scattering processes
does not follow immediately from the rates, as unlike theSU(2) case, T0  1 and hence the scattering states L , R
are not eigenstates of current. A long but straightforward
calculation using the Keldysh technique yields the following contribution to the current to order 2 :
hIij  Na  Nb e1 1  Nc e2 2 :(14)Here, the ‘‘effective charges’’ e1 , e2 are given in Eq. (1).
The factors Na , Nb , Nc count the multiplicity of each of thethree processes (i.e., the number of distinct initial and final
states once pseudospin is included): Na  Nb  2Nc 
NN  1. The contribution to the current given in
Eq. (14) can be combined with that in Eq. (10) to yield
the total low-voltage current to order eV=TK 3 .
There is a simple heuristic way to understand the effective charges appearing in Eq. (14): they are proportional to
the total change in the scattered-wave amplitude associatedwith a particular interaction-induced scattering process. Aninteraction-induced scattering event leads to a suddenchange in the amplitude of the scattered waves associatedwith a particular pair of scattering states. For example,
consider the 2 process, which takes a pair of L scattering
states to a pair of R scattering states (i.e., L L !
R R ); without loss of generality, consider the current in
the left lead. Initially, we have two left scattering states; theassociated scattered-wave amplitude [cf. Eq. (3)] is thus
21  T0 . In the final state, we have instead two rightscattering states; the associated scattered-wave amplitude[cf. Eq. (4)] is thus 2T0 . The 2 process thus suddenly
changes the scattered-wave amplitude by an amount22T0  1  e2 =e. A sudden change in scattered-wave
amplitude implies a delta-function blip in the current ofmagnitude e2 : It ’ e2 t. Such ‘‘blips’’ occur at a rate
2 , hence the expression in Eq. (14). A similar analysis
may be given of the 1 processes and the charge e1 . We
stress that Eq. (14) follows from a rigorous perturbative
calculation of the current.

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PHYSICAL REVIEW LETTERS

The utility of the interpretation of Eq. (14) in terms ofeffective charges becomes apparent when we now turn tothe interaction contribution to SI . Based on this interpretation, we would expect a Poissonian contribution to SI :
SI j;D  2Na  Nb e1 2 1  2Nc e2 2 2 :(15)

This is precisely what we find in our full calculation when
we sum all contributions independent of I^OD [cf. Eq. (6b)];in the case where T0 ! 1 or T0 ! 0, these are the only
contributions, as I^OD  0. Thus, interaction-induced scattering events enhance the shot noise in a simple way,
namely, via an uncorrelated Poisson process. For N  2,
these contributions reduce to what was found in Ref. [4].
For N  4, e1 and e2 are zero, and these scattering processes make no contribution.
We now turn to the second interaction-induced contribution to the noise, namely, the contribution from diagrams
involving I^OD [cf. Eq. (6b)]. Recall that I^OD is responsible
for the partition shot noise already present without Fermiliquid corrections [i.e., first term in Eq. (12)]. We find that
two-particle interactions enhance this noise; summing all
diagrams involving I^OD to order 2 :8SI j;OD  2N  1  NN  1
3


3
eV
eV 2
T0 R0 2
:
(16)
h
kB TK
As will be discussed in Ref. [14], the dominant effect here
is an enhancement of noise due to an interaction-induced
enhancement of the particle-hole density of states. Notethat in the N  2 case this contribution vanishes, as there isno partition noise in this case. Also note that it is impossible to express this contribution purely in terms of thescattering rates 1 and 2 introduced above.
Focusing now on the experimentally relevant case N 4, we may again compare the nonlinear-in-V dependenciesof the current noise and current. Combining all contributions [Eqs. (10), (12), and (14)–(16)], we find2  152 2
1 d3 SI =dV 3 


(17)

 :


3
0
2e d3 hIi=dV 3 V0This ratio is modified by a factor of 2=3 compared to thenoninteracting expectation [cf. Eq. (13)]. This universal
interaction-induced noise enhancement is a central resultof this Letter; it is a stringent test of the theory involving
three Fermi-liquid parameters. We derived Eq. (17) for asymmetric dot-lead coupling; for a small coupling asymmetry x and small residual potential scattering (with phaseshift r1), the ratio in Eq. (17) becomes (to leading
order) 2=3  c1 r  c2 x2 , where c1 , c2 are constants oforder unity [14]. We stress again that in the SU(4) case,Fermi-liquid interactions enhance shot noise through a
modification of the coherent partition noise. This is in

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contrast to the SU(2) case, where the modification is due
to interaction-induced scattering events.
In conclusion, we have shown how Fermi-liquid interactions affect current noise through a SUN Kondo quantum dot, identifying two distinct physical mechanisms. In
the experimentally relevant N  4 case, we show that twoparticle interactions modify the shot noise in a nontrivial
and universal manner; this serves as a robust prediction forexperiment. Our approach in this Letter is also well suited
to study the effect of interactions on current noise in avariety of mesoscopic conductors [20].
We thank T. Kontos for useful discussions and C. Mora
for a very constructive conversation. A. C. and K. L. H.
acknowledge support from NSERC and CIFAR.

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