Nonequilibrium transport through parallel double quantum dots in the Kondo regime

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by  Y. -F. Yang

Nonequilibrium transport through parallel double quantum dots in the Kondo regime
Y.-F. Yang and K. Held
Max-Planck-Institut fu¨r Festk¨orperforschung, 70569 Stuttgart, Germany
(Dated: February 2, 2008)
We extend a perturbative, nonequilibrium renormalization group approach to multi-orbital sys-
temsandapplyitforstudyingtransportthroughtwoparallelquantumdotscoupledelectrostatically.
6 In general, the conductance shows pronounced Kondoesque peaks at three voltages. One of these
0 peaksdisappearsif,asinsomeexperiments,oneofthedotsisdecoupledfrom oneofthetwoleads.
0 For an asymmetric coupling to the leads, also negative differential conductances are possible. This
2 is a genuine nonequilibrium effect, accompanying the Kondoesque peaks. Moreover, a criterion to
distinguish spin and orbital Kondo effect in such a system is discussed.
n
a
J PACSnumbers: 73.63.Kv,71.27.+a,72.10.Fk

7
2
I. INTRODUCTION small enough δ both orbitals are screened at the same

energy,in agreementwith scaling analysesshowing a ro-
]
l bust (marginal) SU(4) fix point.15,16,17
e Kondo physics in quantum dots has been under in-
- tense investigation in recent years. As was predicted Letusalsodistinguishherebetweenorbitalconserving
r
t theoretically,1,2 the Kondo effect can lift the Coulomb andnon-conservingdot-leadcouplings. In our paper, we
s
consider the former, experimentally realized by separate
. blockade at low temperatures, leading to a conductance
at peakatzerobias.3 Insuchexperiments,oftenmorethan leads for each dot,4 see Figure 1. The non-conserving
m one orbital is involved. This is unavoidable, if the tem- case is obtained if the quantum dots couple to a single
lead like in Ref. 5. Then the hopping processes do not
- perature or Kondo temperature is larger or comparable
d conservetheorbitalquantumnumber,whichcangiverise
to the mean orbital splitting. Some nanostructures are
n toSU(2)Kondophysicsevenifmorethanonedegenerate
also intendedly constructed in such a way, for example,
o two parallel quantum dots coupled electrostatically.4,5,6 level is involved, see Ref. 18 and references therein.
c
[ The latter allows for a controlled study of the interplay So far, we only discussed equilibrium physics. A fi-
nite voltage V leads however to genuine nonequilibrium
ofspinandorbital(upper/lowerdot)degreesoffreedom.
2 effects. For example, if V ≈δ≫T (the Kondo temper-
On the theoretical side, the N-fold degenerate situa- K
v
ature) scattering between the two subsets of levels be-
0 tion where all N (spin and orbital) levels of the Ander-
comes relevantagain. A Kondoesqueresonancedevelops
2 sonmodelhavethesameone-particleenergyε,Coulomb
at V ≈δ, visible e.g. as a peak in the conductance. But,
4 interaction U, and hopping amplitude t to the leads is
2 bestunderstood. Atstrongcoupling,theSchrieffer-Wolff atthesametime,agenuineeffectofnonequilibriumisthe
0 transformation7 then leads to the Coqblin-Schrieffer presence of decoherence processes induced by the finite
5 current. This decoherence cuts off the flow to the strong
model8 in terms ofthe exchangecoupling J. This model
0 coupling fix point even for temperatures T ≪T .2,19,20
hasSU(N)symmetryfortherotationoforbitalsandspins K
/
at and, with increasing degeneracy N, leads to a strongly Then,physicalquantitiessuchasthedifferentialconduc-
tancedependstronglyontheconfigurationofthesystem
m enhanced Kondo temperature TSU(N) ≈ De−1/(Nρ0J)
K like the asymmetry of the hopping t between quantum
- (ρ0: density of states in the leads; D: bandwidth of the dot and leads.
d leads;9 we set k = ~ = e = 1 unless these constants
n B Nonequilibrium Kondo physics is not as thoroughly
appear explicitly in the equations).
o
c Of course, we cannot expect a real quantum dot
: to have SU(N) symmetry, unless special arrangements t t
v are made.11 Hence, quite an effort was devoted in the L1 R1
i
X literature12,13,14,15,16 to study the situation where ei-
r ther a magnetic field or a difference in orbital energy 1
a
δ splits N = 4 levels into two doubly-degenerate sub- t t
sets, hence breaking the SU(4) symmetry. If δ≪TSU(4) L2 R2
K
the low-energy physics still resembles that of the SU(4)-
symmetric model. In contrast for δ≫TSU(4) one of the 2
K
levels drops out of the scaling procedure; the low-energy
physics is that of the usual SU(2)-symmetric Kondo
model. In between, there is a crossoverregion.15 FIG. 1: Sketch of the parallel double quantum dot config-
ItisalsounavoidablethatthecouplingconstantsJ are uration. Each dot is coupled to its own leads; the coupling
orbital-dependent in experiment. Numerical renormal- betweenthethedotsisonlyviatheCoulombinteraction(not
ization group (NRG) calculations13,15 suggest that for indicated in theFigure).
2
investigated theoretically because standard approaches II. NONEQUILIBRIUM PERTURBATIVE RG
for the Anderson impurity model like NRG are not
applicable. Rosch and coworkers21 recently developed
Starting point for modeling a quantum dot with N-
a perturbative, nonequilibrium renormalization group
levels should be the Anderson impurity model:
(RG) method to address nonequilibrium Kondoesque
physics.22 This approachis similar to Anderson’s23 poor
man’s scaling approach but includes nonequilibrium ef- H = H + (t c+ d +H.c.)
AIM 0 λm mkλ m
fects caused by the finite current through the system. λXmk
Isnornboendeqpuroilpiberriluymb,ythaeplouwreerneenrogrympahliyzsaictisocnanonfotthebebaabre- + εmd+mdm+ Umm′nmnm′. (1)
Xm mX6=m′
couplings. It is necessary to include the frequency de-
pendence of the vertex functions.
The outline of the paper is as following: In Section Here d+m and dm (c+mkλ and cmkλ) are creation and an-
II, we extend the perturbative RG of Rosch et al.21 to nihilation operators for electrons in the dot (lead λ);
many orbitals, supplemented by the calculation of the nm =d+mdm;εm describestheone-particleenergyoflevel
decoherence rates in the Appendix. In Section III, we mwithm={σ,i}subsumingthespinandorbitalindex;
apply this method to the parallel double quantum dot Umm′ istheCoulombrepulsionwithinthedot. Thelevels
system, with each dot being coupled to separate leads. ofthe dothybridizevia tλm with non-interactingleads λ
Thereby, the first part, Section IIIA, discusses briefly described by (later, we simplify ǫλmk =ǫk)
the simplificationoftheRGEqs.forthis specialapplica-
tion. The following Sections present our results: Section H = ǫ c+ c . (2)
IIIB shows the renormalized vertex function, primarily 0 X λmk λmk λmk
λmk
of theoretical interest, Section IIIC the increase of the
decoherence rates with voltage, and Section IIID the
changeoftheoccupationinnon-equilibrium. The reader In this paper, we assume eachquantumdot levelm to
who is mainly interested in the differential conductance couple to its own lead channel which is for example the
may also directly turn to Sections IIIE and IIIF which case if the different levels correspond to different quan-
presentourresultsforthisphysicalobservableinthecase tum dots like in Ref. 4 (see also Figure 1 of the present
of symmetric and asymmetric coupling to the two leads, paperasanillustration). Butinothersituationsitispos-
respectively. Inthe caseofsymmetric couplings(Section sibletomoveanelectronphasecoherentlyfromonelevel
IIIE) we generally obtain two orbitalKondoesque peaks m via the leads to another level m′. This goes beyond
at V ≈ ±δ in the differential conductance of both dots Hamiltonian (1).
and an additional spin Kondo peak at V =0 for the dot WhiletheAndersonimpuritymodelallowsustomake
which is lower in energy. Depending on the parameters, contactwithexperiment(estimatingparameters),wead-
thepeakatV =0canbestronglysuppressedandhardly dress in the following the parameter regime ǫ ≪µ ≪
m λ
discernible. The main findings of Section IIIF for asym- Umm′+ǫm forallm,m′,λ(µλ: chemicalpotentialinlead
metric couplings are: (i) in very asymmetric situations λ)sothatchargefluctuationsaresuppressed. Then,with
the Kondoesque peaks at V ≈−δ can disappear and (ii) hn i=1electronwithinthedot,wecanmapHamil-
m m
negativedifferentialconductancesarepossibleforV &δ. tPonian(1)ontothesubspacewithoneelectroninthedot
In addition to these non-equilibrium results, the linear by a Schrieffer-Wolff transformation,7 i.e., onto the gen-
(equilibrium) conductance is analyzed in Section IIIG. eral effective Kondo Hamiltonian (neglecting a potential
A summary of the results can be found in Section IV. scattering term)
H = H + ǫ Xmm+ (Jλ1λ2 Xm1m2c+ c +J˜λ1λ2 Xm1m1c+ c ). (3)
0 X m X m1m2 λ2m2k2 λ1m1k1 m1m2 λ2m2k2 λ1m2k1
m λ1m1k1;λ2m2k2
Here, the Hubbard operators24 Xm1m2 represent the whichismuchsmallerthanU,ǫd. Note,thatform1 =m2
scattering of the local state from level m to level m . both J and J˜yield the same kind of contribution. The
2 1
When the lead’s band edge D > |ε |, the local level ε advantage of having this term twice (i.e., distributed to
m m
tohfethKeoAndnodemrsoodneli;m10pJurλi1tλy2m=odte∗lis rtenorm(alize1dto ǫ−m1in) Je˜qmλu11λam2t1ioanns.dTJhmλe11λms2a1m)eisdtishterisbiumtpiolnifiocafttihoenmof t=hemfolcloowntinrig-
m1m2 λ1m1 λ2m2 Um1m2+ǫd ǫd 1 2
andJ˜mλ11λm22 =(1−2δm1m2)t∗λ1m2tλ2m2Um1m12+ǫd whereǫd bmuotdioenl wishuiscuhailslytahlesoSeUm(Npl)oyseydmfmorettrhiecCveorqsbiolinn-oSfchHraiemffielr-
is the average level energy, neglecting the level splitting
3
(a) λ ω m k λ ω m k
1 1 2 1 2 2 2 2 (a) λ ω m k λ ω m k
1 1 2 1 2 2 2 2
λω’m k
1 λω’m k
m1 m2 m1 m2 1
m m
1 1
(b)
λ ω m k λ ω m k λ1ω1m1k1 λ2ω2m2k2 (b)
1 1 1 1 2 2 2 2 λ ω m k λ ω m k λ ω m k λ ω m k
1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
λω’mk
λω’mk λω’mk λω’mk
m m’ m m
2 1 m2 m1 m m’ m m m’ m
2 1 2 1
FIG. 2: One-loop Feynman diagrams for the renormaliza-
tion of the vertex functions (a) g˜λ1λ2 (ω) and (b) gλ1λ2 (ω). FIG. 3: Vertexcorrections and self energy diagrams beyond
The full line denotes the electronm1amn2d the dashed tmh1eml2ocal- theone-loop approximation of Figure 2.
ized state of the Kondo model. Note that the second (hole)
diagram of (b) does not contribute for (a) since it cancels
with the corresponding first (particle) diagram. For the ver-
tex renormalization, the electron line is restricted to a small
interval |dD| at theband edges, as indicated bythe dash. ~ ~ ~ ~
tonian(3) [J and J˜=J/N are then level-independent].8
Let us now generalize the perturbative RG approach
FIG. 4: Asimilar combination of vertexcorrections and self
ofRef.21tothemulti-orbitalHamiltonian(3). Thebase
energydiagramsasinFigure3emergesifwecalculategener-
quantityisthevertexfunctionwhich-atfinitevoltagesor
alized spin/orbit susceptibilities.
currents-becomesfrequencydependent. Likeinthepoor
man’sscalingapproach,weintegrateouthighenergyde-
grees of freedom in the intervals [−D,−D +|dD|] and
Another cut-off is induced by the finite voltage, driving
[D−|dD|,D], resulting in a renormalization of the ver-
the system out of equilibrium. This nonequilibrium ef-
tex. The Feynman diagramsfor this renormalizationare
fect originates from higher order diagrams such as those
shown in Figure 2 to lowest order (one-loop RG). To di-
of Figure 3. These diagrams give rise to an imaginary
rectly apply standard perturbation theory, the localized
part of the self energy and vertex corrections, even at
states of the Anderson impurity model can be expressed
low ω’s for the nonequilibrium situation. Generally, this
by pseudo Fermions, see Ref. 21. Starting point of the
will result in the following modification of the pole in
RG scheme is the unrenormalized k-independent vertex
Eq. (4):
gmλ11λm22(ω) = ρ0Jmλ11λm22, g˜mλ11λm22(ω) = ρ0J˜mλ11λm22, where we
neglecttheenergydependenceofρ0. Here,thefrequency 1 ∂ D sign(ǫ)
of the localized state is approximated by ǫ so that the dǫ
m 2∂lnD Z ǫ−∆ω+iγsign(ǫ)
vertex depends only on one ω (the average of incoming −D
and outgoing electron frequency). This is only possible ≈Θ(D− ∆ω2+γ2)≡Θγ , (5)
∆ω
in the weak coupling regime. p
Forthenonequilibriumsituation,thecalculationofthe where Θ(x)=1(0) for x≥0(<0).
diagrams of Figure 2 requires in principle the usage of For the proper calculation of the low energy cut-off γ,
Keldysh25 Green functions. Rosch et al.21 argued how- wewouldneedtocalculatediagramsasinFigure3. This
everthat, to leading orderin 1/ln(V/T ), it is sufficient goes beyond the scope of the present paper. As in Ref.
K
to keep track of the real part of gλ1λ2 (ω), as indicated 21, we follow a heuristic approach instead. To this end,
byperturbationtheory.26Thisgivems1rmis2etothesamekind first note that similar combinations of vertex corrections
of vertex renormalization per energy interval lnD as in and self energy inclusions occur when calculating gener-
the poor man’s scaling approach, i.e, at T =0: alized susceptibilities for level m1 and m2; see Figure 4.
Secondly, an incorrect prefactor of γ is not affecting the
1 ∂ D sign(ǫ) results to leading order in 1/ln(V/TK). Hence, we can
dǫ . (4)
2∂lnD Z−D ǫ−∆ω crudelyestimateγ bythedecoherenceratesγm1m2 ofthe
corresponding susceptibilities. For example, we can take
Here, ∆ω is given by the chemical potential and the the maximal γ of the levels involved (or tempera-
m1m2
change of energy of the localized state in Figure 2. A ture if higher): γ ≡ γ = max{T,γ ,γ }
m1m2m3 m1m2 m2m3
finite temperature, smears out the sharp step of the in- where m , m and m denote the incoming, intermedi-
1 2 3
tegrand in Eq. (4), cutting off the emerging logarithm. ate,andfinallocalstate. Inourcalculationfortwolevels
4
inSectionIII with SU(2)spinsymmetry,onlytwoofthe are involved. The calculation of γ via the suscepti-
m1m2
orbitals i (m = {σ,i}) can be different, and the vertices bilities is presented in the Appendix.
are symmetric w.r.t. the spin indices. We then actually
use the intra-orbital rate γ =max{T,γ ≡γ } as a cut- Having identified the cut-off γ , we can now
ii i m1m2m3
off for diagrams where only one orbital is involved and write down the RG equation for the vertex functions
theinter-orbitalγ =max{T,γ12}wheneverbothorbitals g˜mλ11λm22(ω) and gmλ11λm22(ω), corresponding to Figure 2:
∂g˜mλ11λm22(ω) = gλ1λ (ω+ ǫm1 −ǫm2)gλλ2 (ω+ ǫm1 −ǫm2)Θγm2m1m2 ,
∂lnD X m2m1 2 m1m2 2 ω+ǫm1−ǫm2−µλ
λ
∂gmλ11λm22(ω) =− gλ1λ (ω− ǫm2 −ǫm)gλλ2 (ω− ǫm1 −ǫm)Θγm2mm1
∂lnD X m1m 2 mm2 2 ω−(ǫm1+ǫm2−2ǫm)/2−µλ
mλ
ǫ −ǫ ǫ −ǫ
+ gλ1λ (ω) g˜λλ2 (ω− m1 m2)Θγm2m1m1 −g˜λλ2 (ω− m1 m2)Θγm2m2m1
Xλ m1m2 (cid:2) m1m2 2 ω−(ǫm1−ǫm2)/2−µλ m2m2 2 ω−(ǫm1−ǫm2)/2−µλ(cid:3)
ǫ −ǫ ǫ −ǫ
+ gλλ2 (ω) g˜λ1λ (ω+ m1 m2)Θγm2m2m1 −g˜λ1λ (ω+ m1 m2)Θγm2m1m1 .(6)
Xλ m1m2 (cid:2) m2m1 2 ω+(ǫm1−ǫm2)/2−µλ m1m1 2 ω+(ǫm1−ǫm2)/2−µλ(cid:3)
The equationfor g˜can be obtained from Figure 2a. The where f (ǫ)=1/(eβ(ǫ−µλ)+1) is the Fermi function for
λ
first line of the equation for g corresponds to the right lead λ. A difference to Ref. 21,30 is that we do not
diagramofFigure2b;thesecondandthirdlinestemfrom take into account a γ-broadening of the spectral func-
the particle andthe hole diagramofFigure2bwith m= tion. This is effectively described by a broadening of the
m and m = m , respectively. Note that the different Fermi functions in Ref. 21,30, resulting in corrections to
2 1
frequenciesofFigure2arerelatedbyenergyconservation subleading order in 1/ln(V/T ).
K
andthatω =(ω +ω )/227. Thek-integration(notethat
1 2 Since we consider the situation where either the
k is not conserved) over the band edge states results in
voltage-induced decoherence rate or the temperature is
a factor ρ which is already included in g. The electron
0 larger than the Kondo temperature the renormalization
statesintheleadsrangeto±Drelativethetherespective
process is cut off, instead of flowing to strong coupling.
chemical potential µ .
λ Therefore,we are at weak coupling and we can calculate
calTcuhlearteendosremlf-acloiznesdisvteernttelyxtfuongectthioenrswoifthEqγ.m(m6)′h[Eaqv.e(tAo.b7e) htλhemscaktt|eHring|λmmatrkixi,inwlhoewreestHordedre:noTˆtmλe1s1λm2t2h(ek1in,kte2r)a=c-
2 2 2 int 1 1 1 int
of the Appendix] which determine the cut-offs γ .
m1m2m3 tionbetweentheleadelectronsandthelocalstatesviaJ
and J˜, i.e., the last two terms of Eq. (3). Averaged over
the local states, this yields:
Conductance
h|Tˆλ1λ2 (k ,k )|2i=(1−δ )hn igλ1λ2 (ω)2/ρ2
m1m2 1 2 m1m2 m2 m1m2 0
To study the most important physical quantity,
+δ hn ig˜λ1λ2(ω)2/ρ2
the conductance, we employ the scattering T-matrix m1m2 X m m1m 0
Tsˆtmλa1t1λem22{(kk11m,k12})infolreatdheλ1prtoocsetsastem{okv2imng2}aninelleeacdtrλo2n. fTrohmis +δm1m2hmn6=mm1i1(gmλ11λm22(ω)+g˜mλ11λm22(ω))2/ρ20. (8)
process necessarily also involves the local state which
changesintheoppositedirection. Thechangeofthelocal
state can be expressed via spin and orbital operators in Here, we have replaced the bare coupling by the renor-
tthheenTd-menaottreixt(hseeeteh.egr.mRaelf.a1v0e)r.agBeyohf|Tˆtmhλ1e1λm2T2-(mk1a,tkr2ix)|2wi,.rw.te. hmnamliizeisdtohneea,vie.er.a,gJe o→ccugp(aωt)io/nρ0ofwliothcalωle≡vel(ǫmk.+ǫk′)/2;
thelocalstateconfigurations. Thesteadystatecondition
Ina firststep g andg˜are calculatedby Eq.(6), which
requires in terms of this averagedT-matrix:
issolvedself-consistentlytogetherwiththecut-offγ
m1m2
[Eq.(A.7)]. Withtheseg’sandg˜’s,theoccupationshn i
m
h|Tˆλ1λ2 (k ,k )|2if (ǫ )[1−f (ǫ )]= canbedeterminedviaEqs.(7)and(8)andtheconstraint
m1m2 1 2 λ1 k1 λ2 k2
X hn i=1.
λ1λ2;k1k2 m m
P
h|Tˆλ2λ1 (k ,k )|2if (ǫ )[1−f (ǫ )], (7) From the T-matrix [Eq. (8)], we can then calculate
m2m1 2 1 λ2 k2 λ1 k1
X the current from lead λ and orbital m as the difference
λ1λ2;k1k2
5
between scattering out of and into the lead: with only one electron in the dots, there is no need to
consider Hund’s coupling.
Iλm =−2π δ(ǫk−ǫk′ +ǫm′−ǫm) Using the SU(2) symmetries in both orbitals and the
λ′m′P(6=λm) abovescaling invariants,the scaling Eq.(6) become par-
kk′
fλ(ǫk)[1−fλ′(ǫk′)]h|Tˆmλλm′′(k,k′)|2i t(itchuelna,rlygi(sωim),pgl˜ei(ωfo)r, aansdymgm12e(ωtr)icdcoounpoltindgeptoentdheonleathdes
(cid:8)−fλ′(ǫk′)[1−fλ(ǫk)]h|Tˆmλ′′λm(k′,k)|2i .(9) lead indices; we can hence drop the λ, λ′ indices):
(cid:9)
∂g (ω)
∂ITλmh/e∂Vdi.fferential conductance follows as Gλm = ∂linD =−λX=±1[gi2(ω)Θγωi+λV/2+g122(ω+i2δ)Θγω1+2i(δ+λV/2)],
∂g (ω)
∂l1n2D =−g12(ω) [43gi(ω−i2δ)+21g˜i(ω−i2δ)]Θγω1+2λV−iδ,
III. TWO PARALLEL QUANTUM DOTS X 2
i;λ=±1
∂g˜(ω)
i =− g2 (ω+iδ/2)Θγ12 .
Let us now apply the perturbative RG equations to ∂lnD 12 ω+i(δ+λV/2)
X
the situation of two parallel quantum dots each coupled λ=±1
(11)
to two leads, see Figure 1 of the introduction. For the
Anderson impurity model, this dot-lead coupling is de- Eq.(11)hastobesolvedself-consistentlytogetherwith
scribed by hopping processes tλi; additionally there are thedecoherenceratesoftheAppendixwhichsimplifyfor
intra-dot and inter-dot Coulomb repulsions Uii′. We the parallel double dot system to:
study here the corresponding Kondo model which is re-
sTthriecnt,edgiλtiλo′′ tdheescsruibbsepsatcheewciothmboinneateiolenctorofntwino thhoeppdiontg. γi =πλPλ′ Zdω(cid:8)gi2(ω)fλ(ω)[1−fλ′(ω)]
processes,leavingthe number of electrons in the dotun- δ δ
changed. +g122(ω)fλ(ω+i2)[1−fλ′(ω−i2)] (12)
(cid:9)
3
A. Simplification of the RG Eq. γ12 =π2λPλ′iZdω(cid:8)[4gi2(ω)+g˜i2(ω)}fλ(ω)[1−fλ′(ω)]
δ δ
+g122(ω)fλ(ω+i2)[1−fλ′(ω−i2)] (13)
The two parallel quantum dot case corresponds to (cid:9)
quantum numbers m = {i,σ} with σ =↑ (+),↓ (−) de- These γi, or T for T >γi, cut off the Θ-functions in Eq.
notingthespinandi=1(+),2(−)thetwodots(orbitals) (11).
withlevelenergyǫ =−iδ/2(¯i=−iinthefollowing). The equations for the occupations [Eq. (7), taking
{iσ}
Due to the SU(2) spin symmetry in each dot, we can m1 = m2 in Eq. (8)] and the current [Eq. (9)] reduce
reducethe number ofverticeswe haveto dealwith. Fur- to
thermore, some of the vertices are connected via scaling
idnuvcatriioanntosf. vAerlttiocgees:thgeλrλt′hi(sωa)ll≡owgsλλfo′(rωt)h/e2,foglλlλo′win(ωg)r≡e- hniiλX,λ′Z dǫg122(ǫ−i2δ)fλ(ǫ)[1−fλ′(ǫ−iδ)]=(i→¯i) (14)
iσ;iσ′ i iσ;¯iσ′
gλλ′(ω)/2,g˜λλ′ (ω)≡−g˜λλ′(ω)/4,and-fromthescaling and
12 iσ,¯iσ′ ¯i
invariants-g˜λλ′ (ω)=(g˜λλ′(ω)−gλλ′(ω))/4. Themean- I = π dω [3hn ig2(ω)+g˜2(ω)][f (ω)−f (ω)] (15)
iσ;iσ′ i i iσ 8 i i i L R
ingoftheredefinedverticesbecomesclearifwewritethe R+4(cid:8)hn ig2 (ω+iδ/2)f (ω)[1−f (ω+iδ)]
HamiltonianintermsofthecouplingconstantsJ and ¯ı 12 L R
i(12) −4hn ig2 (ω−iδ/2)f (ω)[1−f (ω−iδ)]
J˜corresponding to g and g˜: i 12 R L
i(12) +4hn ig2 (ω+iδ/2)f (ω)[1−f (ω+iδ)]
¯ı 12 L L
H = (ǫ −µ )c+ c − δ(n −n ) (10) −4hniig122(ω−iδ/2)fL(ω)[1−fL(ω−iδ)] ,
X k λ λiσk λiσk 2 1 2 (cid:9)
λiσk respectively. In Eq. (15), the first line stems from the
Jλ′λ current from left channel i to right channel i. The 2nd
+ i2 Si·c+λiσkτσσ′cλ′iσ′k′ and 3rd (4th and 5th) line correspond to the current to
λλ′Xiσσ′kk′ right (left) channel¯i. The latter are balanced by similar
+ J1λ22′λ(X{1σ′}{2σ}c+λ2σkcλ′1σ′k′+H.c.) toenremosrbfriotaml ttohethreigohtthelreaids zseorot.hat the net current from
λλ′Xσσ′kk′
J˜λ′λ
+(n1−n2) 4 (c+λ1σkcλ′1σk′−c+λ2σkcλ′2σk′). B. Results: Renormalized vertex function
λλX′σkk′
Here, S denotes the spin operators in orbital i, τ the Letusnowdiscussthenumericalsolutionfortherenor-
i
vector of Pauli matrices, and n = n . Note that malized vertex functions of Eq. (6)/(11) which is pre-
i σ iσ
P
6
00..1144 is instructive to integrate Eq. (11) analytically, dividing
gg
((aa)) 11 the Eqs. by the squared coupling constant from the left
00..1122 gg22 handside (e.g.multiplying the firstline by 1/g2(ω))and
gg i
1122
00..11 ~g~g afterwards keeping the g’s on the right hand side con-
~g~g11 stant (at the initial values). This is justified as long as
00..0088 22
ωω)) the renormalization of the vertex functions stays small.
g(g( 00..0066 Butourmereaimhereistounderstandthepositionsand
widthsofthedifferentpeaks. FromEq.(11)westraight-
00..0044
forwardly get in this way:
00..0022
1
g (ω) ∼ ,(16)
00 i (ln|ω+λV/2|i +ln|ω+i(δ+λV/2)|12)
00000.....0000088888 (((((bbbbb))))) Pλ Tk0 Tk0
1
ωωωωωg()g()g()g()g() 0000000000..........00000000004646464646 g12(ω) ∼ Pi,λln|ω+(λTVk20−iδ)|12, (17)
00000.....0000022222 1
g˜(ω) ∼ , (18)
i ln|ω+i(δ+λV/2)|12
00000 (((((ccccc))))) Pλ Tk0
00000.....1111122222
00000.....11111 where |ω| ≡ ω2+max2{γ ,T}, and prefactors of or-
i i
ωωωωω))))) 00000.....0000088888 derone(like gp12/gi)beforethe logarithmictermsarene-
g(g(g(g(g( 00000.....0000066666 glected.
00000.....0000044444 By means of Eq. (16), we can now identify the origin
00000.....0000022222 of the four Kondoesque peaks in the gi’s seen in Figure
00000 5. When the frequency matches the difference to the
-----111115555500000 -----111110000000000 -----5555500000 00000 5555500000 111110000000000 111115555500000 left or right Fermi level (ω = ±V/2) a single dot (spin)
ωωωωω/////TTTTT00000 Kondoesque resonance develops; but the flow to strong
KKKKK
couplingiscutoffbythecorrespondingdecoherencerate
γ (or T). The other two peaks in g stem from the or-
FIG. 5: Renormalized vertex functions to the left lead at i i
T/T0 =0.5,V/T0 =20,δ/T0 =65 (g in the legend box bital Kondo effect. Here, the localized state changes to
K K K i(12)
stands for gLL ). Part a) is for symmetric initial couplings the other orbital and back to the original orbital. These
gλλ′ = 0.04,i(1g2λ)λ′ = 0.06 for all λ,λ′. In part b) the right processes are enhanced if the conduction electron scat-
i 12 ters to the Fermi level of the left or right lead, requiring
leadofdot2wasremoved,i.e.,parametersasina)exceptfor
ω = −iδ±V/2. Hence the resonances [and those of Eq.
gRR = gLR = 0. In part c) also the coupling of dot 1 is
2(12) 2(12) (18)] are at ω = −iδ±V/2 and the cut-off is γ . Simi-
asymmetric: gLL = 0.04,gRR = 0.06, gLL = 0.04, gRR = 0, 12
1 1 2 2 larly,the peaksforg (ω)inEq.(17)areat±V±δ (with
g1L2L=0.08. δ/2 instead of δ sinc1e2ω is the average of incom2ing and
outgoing frequency which differ by δ as g changes the
12
orbital).
sented in Figure 5. This vertex function, a frequency-
In part b) of Figure 5 the right lead of dot 2 was re-
dependent renormalization of the interactions J,J˜ be-
moved. Asscatteringtothe removedleadisnowprohib-
tween lead electrons and local states in the quantum
ited,oneoutoffourpeaksdisappearsforg andg . The
i 12
dot, is the fundamental theoreticalquantity; it is not di-
missing peak is the one at the lowest value of ω in part
rectly observable experimentally, but -together with the
a). We can hence identify the missing peak with spin
occupations- allows to calculate e.g. the current which
(g ) andorbitalscattering processes(g and g ) via the
2 1 12
will be discussed later. In the following, all energies are
rightleadofdot2. Partc)ofFigure5isforthesituation
normalized to the (equilibrium) Kondo temperature for
where also dot 1 is coupled asymmetrically. Some peaks
δ =0,i.e.,toT0 =T (δ =0).28Duetothefinitevoltage
K K are clearly suppressed.
V applied,thetwoleadsarethermalizedbytwodifferent
Fermi functions with µ = µ+V/2 and µ = µ−V/2,
L R
respectively. The quantum dot is out of equilibrium.
C. Results: Decoherence rates
Here and in the following figures, the initial coupling
constantswhicharenotexplicitly mentionedfollowfrom
Figure6showsthe threedecoherenceratesγ ,γ ,and
1 2
the ones of the Figure caption since they stem from the
γ whichneedtobecalculatedself-consistentlytogether
same t’s of the Anderson impurity. For example, gLR = 12
12 with the RG Eqs. As in Eqs. (16)-(18) it is instruc-
g1L2L g1RR/g1LL. We always set g˜iλλ′ =0 initially. tive to calculate γ from the unrenormalizedg’s (notself-
Topunderstand the four peak structure of Figure 5, it consistently/inlowestorderapproximation). Atlargeor-
7
bitalsplitting|δ|≫max{T,|V|},thisyieldsforsymmet- In between the two extremes |V| ≪ |δ| and |V| ≫
4
ric coupling to left/right lead (Figure 6 around zero): |δ|, the transitionfrom a strong orbital decoherence rate
of dot 2 due to the high level energy to strong current
|V| induced decoherences in both dots is clearly visible at
γ ≈4πg2max{T, },
1 1 4 V ≈±δ in Figure 6.
γ ≈4πg2 |δ|, (19) Neglecting the finite orbital splitting, we can also get
2 12
thehigherorderasymptoticbehaviorforthedecoherence
γ ≈2πg2 |δ|.
12 12 rates analogous to Ref. 19:
The decoherence rate for dot 2, i.e., γ2, and the intra- π V 2
γ ∼ [1+ +...]. (21)
orbital rate γ12 are both proportional to |δ| as there are i 4(ln V )2 ln V
many decoherence processes, reflecting the instability of TK0 TK0
the highenergylevelofdot2. Decoherenceprocessesfor
dot 1 on the other hand become only available at finite
D. Results: Non-equilibrium occupation
biasvoltages. Forverylarge|δ|≫T0,the systemshows
K
the usual single-orbital (spin) Kondo effect. Note that,
for small V, we have γ <T0. One might envisage that Thenumericalresultsforthedifferentelectronoccupa-
1 K
thissignalsaflowtostrongcouplingand,hence,abreak- tion of the two orbitals is shown in Figure 7 for different
down of the RG Eqs. However, for the large splitting δ orbital splittings δ. To ensure that the RG approach is
the real Kondo temperature is very much reduced. It valid, i.e., that we stay in the weak coupling regime, we
actually is only ≈1/100T0 here,muchsmaller thanthe use here and in the following a rather high temperature
cut-off. K T/TK0 = 2. Such temperatures are actually relevant in
manyexperimentswhereoftenthelowestaccessibletem-
Let us now discuss the large voltage range of Figure
peraturesareofthe orderofthe Kondotemperature. At
6. For |V| ≫ |δ|, all curves show a similar asymptotic
lowbiasvoltageV <δ,onlythelowerorbitalisoccupied
behavior:
for sufficient large δ, as is to be expected for δ ≫T. At
|V| high bias voltage V > δ, the electrons tend to stay in
γ ≈4π[(g )2+(g )2]max{T, },
i i 12 4 both dots with the same probability;hn1−n2i decreases
3 |V| (20) with increasing V.
γ ≈2π [ (g )2+(g˜)2+(g )2]max{T, }.
12 i i 12
4 4
X
i
E. Results: Conductance for symmetric coupling
While allγ ’s arenow proportionalto |V|, we see in Fig-
i
ure 6 that nonetheless γ ≪ |V|. This is due to the
i Let us now turn to the physical observable of inter-
small prefactors ∼ g2 in Eq. (20). The current which
est, the differential conductance through the two dots
has similar prefactors causes the decoherence processes.
G = ∂I (V)/∂V (i = 1,2), presented in Figure 8. The
For sufficiently large V, all γ ’s become larger than tem- i i
i conductanceconsists ofthe intra-dotspinscatteringand
perature; the current induced decoherence exceeds the
the inter-dot orbitalscattering. For δ =0, a pronounced
temperature effect.
peak is found at V = 0, characteristic of the orbitally
enhanced Kondo effect. The flow to strong coupling is,
however, cut off by temperature. If the orbital degener-
888
γ acy is lifted, the peak at V = 0 is suppressed. For dot
777 γγ21 2, it finally disappears at large δ when this dot is de-
666 12 populated. For dot 1, we still have a spin Kondo effect
at large δ, but with a very much reduced Kondo tem-
555
000KKK perature TSU(2) ≪ T0. Because also TSU(2) ≪ T, the
TTT 444 K K K
γγγ/// peak at V = 0 is very much smeared out. At finite δ,
333
two new peaks occur at V = ±δ, corresponding to or-
222 bital scattering. For T/T0 = 2, we study an interplay
K
111 between temperature-induced decoherence and current-
induceddecoherence. Forsmallδ andV thedecoherence
000
---111000000 ---555000 000 555000 111000000 is solely due to temperature, whereas for largerthe volt-
VVV///TTT000 agesitismainlycurrentnoise. Fortheδ =20andδ =40
KKK
curves in Figure 8, the γ decoherence rate is at least at
1
FIG. 6: Scattering rates as a function of bias voltage V at the smaller voltages temperature-induced. In contrast,
T/T0 = 0.2 and δ/T0 = 40. The initial couplings are sym- theimportantinter-orbitaldecoherencerateγ andalso
K K 12
′ ′ ′
metric: g1λλ =0.04, g2λλ =0.01, g1λ2λ =0.06. For large bias γ2 are not even for smaller voltages.
voltage, the scattering rates are larger than T (dashed line) If we decrease temperature, the main driving source
but much smaller than V. fordecoherenceisthevoltage-inducedcurrentnoise. The
8
in energy. The emerging single-dot (spin) Kondo effect
11
at V ≈ 0 is somewhat suppressed, see Figure 9 c). In
contrast in part a) and b), it is the weaker coupled dot
00..88
which is lower in energy (dot 2). Then the spin Kondo
peak for dot 2 is hardly discernible, see Figure 9 d).
>> 00..66 IntheexperimentsofRef.4,theinter-dotCoulombin-
22
nn teractionU ismuchsmallerthantheintra-dotCoulomb
--11 12
<n<n 00..44 interaction Uii. Hence, the initial inter-dot coupling is
stronger g (∼ 1 − 1) > g (∼ 1 − 1), as in
12 U12+ǫd ǫd i Uii+ǫd ǫd
Figures 8 and 9.
00..22
And, as in Figures 8 and 9 b), the single-dot (spin)
Kondo peak at V = 0 is much weaker (if at all dis-
00
cernible)thanthe pronouncedorbitalKondoesquepeaks
--110000 --5500 00 5500 110000
at V ≈±δ.
00
VV//TT
KK
FIG.7: Differenceoftheoccupationofthetwoorbitals,hn −
1
n i, vs. bias voltage V at different orbital splitting δ/T0 =
2 K F. Results: Conductance for asymmetric coupling
60,40,20,10,4,1(fromtoptobottom),temperatureT/T0 =
K
′ ′
2, and symmetric barecouplings gλλ =0.04, gλλ =0.06.
i 12
Let us now discuss the situation of strongly differ-
ent(initial/unrenormalized)couplingsgλ1λ2 totheright
m1m2
00..22 and the left lead: gRR 6= gLL . This is typically the
0.4 (a) m1m2 m1m2
00..1155 case in experiments,4 and can in extreme cases give rise
0.3 δ/T0 = 0 00..11 to averydifferentphysicalbehaviorcomparedto thatof
G0 K 10 00..0055 the symmetric coupling cases.
δ)/ 20 00 In Ref. 4, this asymmetry is even so strong that one
G(1 0.2 40 --6600 --3300 00 3300 6600 of the dots is effectively decoupled from the right lead
(gRR = gLR = 0), albeit still coupled to the left lead
0.1 2(12) 2(12)
via gLL. Such a situation is shown in Figure 10. Since
2
dot 2 is decoupled from the left lead, the orbital Kondo
0
effect is now only possible at V =+δ, not any longer at
0.15
0000....4444 ((((bbbb))))
V = −δ. This can be understand by means of Figure
0.1 11. At V =+δ, we canmovean electronfromthe Fermi
0000 0000....3333
GGGG energy of the left lead into dot 2 and, conservingenergy,
δδδδ()/()/()/()/2222 0000....2222 0.05 the electron from dot 1 to the Fermi energy of the right
GGGG 0 lead, and vice versa. In the RG equations we hence get
0000....1111 -60 -30 0 30 60 largecouplings,resultinginanorbitalKondoesquepeak.
Ifthecouplingtotheleadswassymmetricwecouldhave
0000
----88880000 ----66660000 ----44440000 ----22220000 0000 22220000 44440000 66660000 88880000 similarscatteringprocessesmovingtheelectronfromdot
VVVV////TTTT0000 2 to the right lead’s Fermi energy and an electron from
KKKK
the left lead’s Fermi energy to dot 1 at V = −δ. How-
FIG. 8: Differential conductance G vs. bias voltage V for
′ i ′ ever,sincegRR =0thissecondscatteringprocessisnot
symmetric initial couplings gλλ = 0.04, gλλ = 0.06, tem- 2(12)
perature T/T0 = 2, and diffeirent orbital s1p2littings δ. Here possible, as was already argued in Ref. 4. Therefore, in-
K
and in the following figures, G = 2e2 denotes the conduc- steadof two conductance peaks as in Figures 8 and9 we
0 h see only one at V = +δ in Figure 10, which presents re-
tancequantum. Inset: temperaturedependenceofthediffer-
ential conductance for δ/T0 = 20 (T/T0 = 2: dashed line; sultsforsuchstronglyasymmetriccouplings. Thenearly
K K
T/T0 =1: solid line ; T/T0 =0.1: dotted line). indiscernible spin Kondo peak and the missing peak at
K K
V ≈ −δ qualitatively agree with experiments.4 As for
symmetriccoupling,theheightoftheorbitalKondopeak
at V =±δ decreases with increasing δ.
changewithtemperatureisstudiedintheinsetofFigure
An interesting phenomena in the case of asymmetric
8, showing the sharpening of the Kondoesque peaks. At couplingsoccursifgRR =0asbeforebutgRR ≫gLL: As
thelowesttemperatureT/T0 =0.1,wereachedthelimit 2 1 1
K shown in Figure 12, we find a negative differential con-
wherethevoltage-induceddecoherenceisdominatingev-
ductancewhichisalwayspresentaftertheorbitalKondo
erywhere, except for the central peak in Figure 8a).
peak, i.e., for V > δ. Generally such negative conduc-
In Figure 9, we show how the conductances change if tancesoccurwhenthehoppingparametersoftheAnder-
thecouplingsofthetwodotstotheleadsaredifferent. In sonmodelfulfill |t |≫|t |, |t |≪|t |. Howcanwe
R1 L1 R2 L2
part c) and d), the stronger coupled dot (dot 1) is lower understand this unexpected negative conductance?
9
0.4
(a) δ/T0 = - 6 ((cc)) δδ//TT00 == 66
K KK
-10 1100
0.3 -20 2200
-40 4400
0
G
δ)/ 0.2
(1
G
FIG. 9: Differential conduc-
0.1
tance G vs. bias voltage V
i
forsymmetricinitialcouplings
0 gλλ′ = 0.04, gλλ′ = 0.01,
1 2
′
((((bbbb)))) ((((dddd)))) gλλ = 0.06, temperature
12
T/T0 = 2, and different or-
K
0000
GGGG bital splittings δ.
δδδδ)/)/)/)/ 0000....1111
((((2222
GGGG
0000
----88880000 ----66660000 ----44440000 ----22220000 0000 22220000 44440000 66660000 88880000 ----66660000 ----44440000 ----22220000 0000 22220000 44440000 66660000 88880000
VVVV////TTTT0000 VVVV////TTTT0000
KKKK KKKK
00..33
((aa)) δ/T0 = - 6 (((ccc))) δ/T0 = 6
K K
-10 10
-20 20
00..22
-40 40
00
GG
V)/V)/
G(G(11 00..11 FIG. 10: Part (a) and (c)
are the same as in Figure 9
butnowforasymmetricinitial
00 couplings gLL = 0.04, gRR =
1 1
11111 0.01, g2LL = 0.01, g2RR = 0,
(((((bbbbb))))) (((((ddddd))))) and g1L2L = 0.12. The Kon-
doesque peak at V = −δ is
00000.....55555 now missing. Instead of the
>>>>> conductance G2 which is zero,
22222
nnnnn part (b) and (d) show the dif-
----- 00000
nnnnn11111 ference in the occupation of
<<<<<
the two dots.
-----00000.....55555
-----11111
-----8888800000 -----6666600000 -----4444400000 -----2222200000 00000 2222200000 4444400000 6666600000 8888800000 -----6666600000 -----4444400000 -----2222200000 00000 2222200000 4444400000 6666600000 8888800000
VVVVV/////TTTTT00000 VVVVV/////TTTTT00000
KKKKK KKKKK
The basic picture is that –although higher in energy– the process involving the transfer of the dot 2 electron
dot 2 is increasingly occupied for V > δ. Counterintu- to the right lead and simultaneously that of an electron
itively, it becomes even more occupied than dot 1, see fromtheleftleadtodot1. Theamplitudeforthisprocess
thelowerpartofFigure12. Thereasonforthisinversion is by a factor of |gLR/gLR| (≈ |t /t ||t /t | ≪ 1)
12 21 L1 R1 R2 L2
of the occupation (in comparison to the equilibrium oc- smallerthanthatofFigure10. Hence,thelocalizedstate
cupation) can be understood from the sketch in Figure is nearly trapped in dot 2, hn −n i < 0, for large bias
1 2
11. The scattering process shown in the Figure is ener- voltage.
getically possible: Because V −δ > 0, we can transfer Altogether the behavior of the conductance in Figure
an electron from the Fermi level of the left lead to the 12 can be explained as following: At V ≈δ a new, inter-
empty states above the Fermi level of the right lead and orbital channel opens for conductance to the left lead of
stillpaytheenergyδ tochangethelocalstatefromdot1 dot 1, the one displayed in Figure 11. This explains the
to dot 2. But the reverse process, starting with an elec- strong enhancement of G for V ≈ δ; the orbital Kondo
1
tronfromtherightlead’sFermilevel,isnotpossiblesince peakinFigure12. WithafurtherincreaseofV,however,
δ−V <0. All the other processesfor changing the local thereisadramaticdecreaseofhn −n iandhenceofthe
1 2
state back from dot 2 to dot 1 are much smaller because number of electrons in dot 1 (hn i). Because of this, the
1
of |tL1|/|tR1|≪1 and |tR2|/|tL2|≪1. Take for example contribution to the current involving solely dot 1, i.e.,
10
000...444
(((aaa)))
000...333
δ/T0 = 6, T/T0 = 1
K K
y 000...222 6, 3
g GGG000 20, 1
Ener (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) V G(V)/G(V)/G(V)/111 000...111 20, 3
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)δ 21(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ---00000000........1111100011155555 (((((bbbbb)))))
1 2 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) >>>>>
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 22222
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) nnnnn
1 2 -----11111 00000
nnnnn
<<<<<
-----00000.....55555
FIG.11: ScatteringprocesswhichconnectstheFermienergy
of the left lead (at +V/2) and dot 2 (upper level) and that
-----11111
of the right lead (at −V/2) with dot 1 (lower level); there is -----3333300000 -----2222200000 -----1111100000 00000 1111100000 2222200000 3333300000
a separate Fermi levelfor each dot as indicated. Theprocess VVVVV/////TTTTT00000
KKKKK
displayedresults,togetherwiththereversescatteringprocess,
inaKondoesqueresonanceatV ≈δ. ForV >δthescattering FIG. 12: Same as on the right hand side of Figure 10 but
process shown is still possible but not thereverse one. now for gLL = 0.01 < gRR = 0.04, and gLL = 0.06. (gLL =
1 1 12 2
0.01, gRR = 0 as before); shown are two different values of
1
δ and temperature. For these parameters, the differential
the first line in Eq. (9), rapidly decreases. This decrease conductanceis negative after the Kondoesquepeak at V ≈δ
which can be explained by the inversion of the occupation
dominates for V > δ, resulting in a negative differential
numbers,shown in part (b).
conductance.
Negative conductances for quantum dots with many
levelshavebeenobservedexperimentally31anddescribed
Thisdifferentbehaviorallowsustoidentifythechange
theoretically,32 albeitin the sequentialtunneling regime.
oftheconductancewithδ,i.e.,∂G(δ)/∂δ,asagoodcan-
Our results show that similar effects are also possible in
didate to determine whether we have a spin or orbital
the Kondo regime.
Kondo effect. As Figure 13 b) shows, this quantity is
symmetric for the spin Kondo effect but asymmetric for
the orbital Kondo effect. Note, that ∂G(δ)/∂δ is often
G. Results: Linear conductance accessibleexperimentally;thetwodotsneedonlybecon-
trollable by different gate voltages for changing δ.
Substantial simplifications arise in the linear response
regime for small voltages since there are no nonequilib-
rium effects any more like the current induced decoher- IV. CONCLUSION
ence rate γ and the frequency dependence of the ver-
i
tex. This linear conductance can be used as an indica- In conclusion, we reported on the extension of the
tor whether we have a spin or an orbital Kondo effect, nonequilibrium perturbative RG scheme to multi-orbital
corresponding to the limits g ≫ g and g ≫ g , re- applications. The method keeps the simplicity of the
i 12 12 i
spectively. Inbothcases,thelinearconductancefordot1 poor man’s scaling approach, but can be used for
increasesatδ ≈0,sincethendot1becomesoccupiedand nonequilibrium transport by including the frequency de-
conducting. But while at larger values of δ (δ/T0 ≫ 1) pendence of the vertex. However, it is restricted to the
K
thespinKondoeffectstillworks,theorbitalKondoeffect weakcouplingregime. Eithercurrentortemperaturehas
breaks down in the linear conductance regime. In Fig- to cut off the flow to the strong coupling fix point.
ure 13 a), the conductance has hence a plateau form in Specifically, we calculated conductances for the case
the former case, while it has a peak form for the orbital of two quantum dots coupled only electrostatically. For
Kondo effect. The reason for this dramatic decrease of SU(2)spinsymmetryonespin(atV ≈0)andtwoorbital
the conductance is simply that the RG flow of the inter- Kondoesque peaks (at V ≈±δ) are found, in agreement
orbital coupling (and that for the high energy dot) is with the expectations. For the typical experimental sit-
cut-offatanenergyscale∼δ,preventingastrongrenor- uation that the intra-dot Coulomb interaction is much
malization and, hence, suppressing the conductance at stronger than the inter-dot Coulomb interaction (trans-
larger values of δ. latingtoinitialcouplingsg ≫g ,g )however,the spin
12 1 2