Kondo screening cloud and the charge staircase in one-dimensional mesoscopic devices

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by  Rodrigo G. Pereira

Kondo Screening Cloud and the Charge Staircase in One-Dimensional Mesoscopic
Devices
Rodrigo G. Pereira,1 Nicolas Laflorencie,1,2,3 Ian Affleck,1 and Bertrand I. Halperin4
1Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada, V6T 1Z1
2Insitute of Theoretical Physics, E´cole Polytechnique F´ed´erale de Lausanne, Switzerland
3Laboratoire de Physique des Solides, Universit´e Paris-Sud, UMR-8502 CNRS, 91405 Orsay, France
4Physics Department, Harvard University, Cambridge, MA 02138, USA
(Dated: February 4, 2008)
We propose that the finite size of the Kondo screening cloud, ξK, can be probed by measuring
8
the charge quantization in a one-dimensional system coupled to a small quantum dot. When the
0
chemical potential, µ in the system is varied at zero temperature, one should observe charge steps
0
whose locations are at values of µ that are controlled by the Kondo effect when the system size L
2
is comparable to ξK. We show that, if the standard Kondo model is used, the ratio between the
n widthsof theCoulomb blockadevalleyswith oddor evennumberof electrons isa universalscaling
a function of ξK/L. If we take into account electron-electron interactions in a single-channel wire,
J this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling

7 is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded
2 in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is
controlled by thesize of theKondo screening cloud.

]
l
e PACSnumbers: 72.15.Qm,73.21.La,73.23.Hk
-
r
t
s I. INTRODUCTION L
.
t
a
m TheKondoeffectcanbedescribedasthestrongrenor-
malizationof the exchange coupling between an electron
-
d gas and a localized spin at low energies.1 Below some Vdw Vg
n characteristicenergy scale,knownasthe Kondotemper-
o
c ature TK, a non-trivial many-body state arises in which FIG. 1:′ Possible experimental setup. Vdw controls the tun-
the localizedspin formsa singletwith a conductionelec- nelingt betweenthesmalldot(ontheleft)andthewireand
[
tron. In the recent realizations of the Kondo effect,2 Vg varies the chemical potential in the wire.
1 a semiconductor quantum dot in the Coulomb blockade
v
regime plays the role of a spin S = 1/2 impurity when
6
the number of electrons in the dot is odd. The Kondo Even if the effects of electron-electron repulsion are ne-
6
1 temperatureinthesesystemsisafunctionofthecoupling glectedorsubtractedoff, eachelectronaddedto the sys-
4 between the dot and the leads and can be conveniently tem costs a finite energy because the energy levels are
. controlled by gate voltages. As a clear signature of the discrete. Consequently, the number of electrons changes
1
Kondoeffect,oneobservesthattheconductancethrough by steps as one varies the chemical potential and sharp
0
8 a quantum dot with S =1/2 is a universal scaling func- conductance peaks are observed at the charge degener-
0 tion G(T/T ) and reaches the unitary limit G = 2e2/h acy points.2 In AlGaAs-GaAs heterostructures,the level
K
: when T T for symmetric coupling to the leads.3 On spacing of a 1D system with size L 1µm is of order
v ≪ K ∼
theotherhand,oneexpectsthatinafinitesystemthein- ∆ 100µeV, large enough to be resolved experimen-
i
X frared singularities that give rise to the Kondo effect are tall∼y. Clearly, the energy levels of the 1D wire should be
r cut off not by temperature, but by the level spacing ∆ if affected by the Kondo interaction with an adjacent dot.
a
∆ T. For a one-dimensional (1D) system with length Therefore, the addition spectrum should exhibit signa-
≫
L and characteristic velocity v, the relevant dimension- tures of the finite size of the Kondo cloud.
less parameter is ∆/TK ξK/L, where ξK v/TK is We consider the geometry shown in Fig. 1. One of
∼ ≡
identifiedwiththesizeofthe Kondoscreeningcloud,the the ends of a wire of length L with a large number of
mesoscopic sized wave function of the electron that sur- electrons (such that ∆ ǫ , where ǫ is the Fermi en-
F F
≪
rounds and screens the localized spin. This large length ergy) is weakly coupled to a small quantum dot. The
scale (ξK ∼0.1−1µm for typical values of v and TK) is dot-wire tunneling is controlled by a gate voltage Vdw.
comparablewiththesizeofcurrentlystudiedmesoscopic Both the dot and the wire (which can be thought of as
devices, which has motivated several proposals that the a very long and thin dot) are in the Coulomb blockade
Kondocloudshouldmanifest itselfthroughfinite sizeef- regime. The dot has a very large charging energy and
fects in such systems.4,5,6,7 a ground state with S = 1/2. We assume that the wire
One familiar property of small metallic islands in the is very weakly connected to a reservoir and that trans-
Coulomb blockade regime is the quantization of charge. fer of a single electron between wire and reservoir could
2
bemeasuredbysomeCoulombblockadetechnique. This For J =0, the system reduces to an open chain and a
coupling is assumed weak enough so as not to affect the free spin (S =1/2). In the continuum limit, we linearize
energy levels of the wire-dot system. Experiments with the dispersion about the Fermi points and introduce the
small dots and large dots have been performed, for ex- right- and left-moving components of the fermionic field
ample, in [8] and considered theoretically in [6,7]. Ψ(x) for electrons in the wire
Here we investigate the charge quantization in the
wire-dot system using the basic Kondo model, ignoring Ψ(x)=eikFxψR(x)+e−ikFxψL(x). (3)
chargefluctuationsonthedotandassumingthewirecon-
ThefreeHamiltonianfortheopenchaininthecontinuum
tains a single channel. First we deal with the simplest
limit becomes
caseandignoreelectron-electroninteractionsinthewire.
In sections II and III, we apply field theory methods to L
calculate the charge steps for the noninteracting case in H0 =−ivF dx ψR†∂xψR+ψL†∂xψL . (4)
both limits L ≪ ξK and L ≫ ξK. In section IV, we Z0 (cid:16) (cid:17)
compute the exact charge steps by solving numerically
The open boundary conditions Ψ(0) = Ψ(L) = 0 im-
the Bethe Ansatz equations for the Kondo model. This
ply that ψ are not independent. Instead, ψ can be
L,R L
approachcomplements the field theory picture in the in-
regarded as the extension of ψ to the negative-x axis
R
termediate regime, L ξ . The collapse of the numer-
K
∼
ical results confirms that the ratio of the widths of the ψ ( x)= ψ (x). (5)
R L
charge plateaus with odd or even number of electrons is − −
a universal scaling function of ξ /L. In section V, we Wecanthenworkwithrightmoversonlyandwritedown
K
include short-range electron-electron interactions in the an effective Kondo model H =H +H in terms of ψ
0 K R
wire using the Luttinger model and discuss how the pre- only (we drop the index R hereafter)
vious results are modified. In section VI, we consider
the same problem for the geometry of a ring coupled to L ~σ
H = iv dxψ ∂ ψ+2πv λ ψ (0) ψ(0) S~, (6)
an embedded quantum dot, where the flux dependence − F † x F 0 † 2 ·
of the charge step locations can also be studied. Section Z−L
VIII presents the conclusions. where λ = 2Jsin2k /πv is the dimensionless Kondo
0 F F
coupling. For λ 1, the size of the Kondo screening
0
≪
cloud (defined in the thermodynamic limit) is exponen-
II. WEAK COUPLING tially large: ξ (v /D)e1/λ0, where D ǫ is a
K F F
∼ ≪
high-energy cutoff. The boundary condition at the weak
The tunneling between the wire and the small dot in coupling fixed point (L/ξK 0) reads
→
the setup of Fig. 1 is well described by the Anderson
model: ψ( L)=ei2δψ(L), (7)
−
L 2 where δ =k L mod π. Using the mode expansion
− F
H = −t c†jcj+1+h.c. −µc†jcj +ε0 d†σdσ i
Xj=1h (cid:16) (cid:17) i Xσ ψ(x)= eikxck, (8)
−√2L
+Un n t d c +h.c. . (1) k
d↑ d↓− ′ †σ 1σ X
Xσ (cid:16) (cid:17) we obtain the momentum eigenvalues kn = (nπ δ)/L,
Here the wire is treated as a chain with L 1 sites n Z. −
and hopping parameter t. The electron at si−te j = 1 ∈WedenotebyN(µ)thetotalnumberofelectronsinthe
is coupled to a localized state in the dot with tunneling system,includingtheoneinthequantumdot. Wearein-
amplitude t. The parameters ε and U correspond to terestedintheelementarystepsofN aroundsomeinitial
′ 0
the energy and the Coulomb repulsion for electrons in value N0 ≡N(µ∗0). At T =0, we calculate N(µ) as the
the dot, respectively. In the Coulomb blockade regime integerpartofN thatminimizesthethermodynamicpo-
t ε ε +U, the dot is singly occupied and the tential Ω(N)=E(N) µN, where E(N) is the ground
A′n≪der−son0 m∼od0el is equivalent to the Kondo model1 state energy. Defining−µℓ+21 as the value of µ where N
changes from N +ℓ to N +ℓ+1, it follows from the
0 0
H =L−2 −t c†jcj+1+h.c. −µc†jcj +Jc†1~σ2c1·S~, (2) cthhaatrgedegeneracyconditionΩ(N0+ℓ+1)=Ω(N0+ℓ)
Xj=1h (cid:16) (cid:17) i
where S~ = d (~σ/2)d is the spin operator of the electron µℓ+21 =E(N0+ℓ+1)−E(N0+ℓ). (9)
†
in the dot and J t2/ ε > 0 is the antiferromagnetic For λ =0, we set µ halfway between the highest occu-
∼ ′ | 0| 0 ∗0
Kondo coupling. In the following we assume that J is pied and the lowest unoccupied energy level of the open
independent of the chemical potential µ=eV , which in chain. In this case, N = odd: the ground state is dou-
g 0
practicemayrequirethatε andt betunedaccordingly. blydegenerate(totalspinS =1/2)andconsistsofone
0 ′ tot
3
single electron occupying the state at the Fermi surface
5
(with momentum k ). To lowest order in degenerate
N = odd F
0 perturbation theory, the ground state for λ 0 is
0
→
4
N0 δµo |GS(N =even)i=n 0c↑k†nc↓k†n|0i⊗|si, (14)
3 Y≤
-

µ) δµe where s [kF kF ]/√2is the singlet
N( 2 state b|eitw≡een| th↑ei⊗sp|i⇓nio−f|the↓dio⊗t|⇑aind the electron at
k . This state has S = 0 and S~ S~ = 3/4,
F tot el
h · i −
1 finite ξK/L welhecetrreoSn~esl.≡Sincekct†kh~σe2cKkoisndthoeetffoetcatlsopnilnyoinfvtholevceosntdhuecstpioinn
weak coupling limit P
strong coupling limit sector,the ground state energy must assume the general
0 form
0 1 2
µ µ
1/2 3/2 (µ−µ*)/∆ E(N +ℓ;λ)=E + ∆ ℓ2+f (λ ,L) +µ ℓ. (15)
0 0 0 4 s 0 ∗0
(cid:2) (cid:3)
FIG.2: (Coloronline)Chargequantizationstepsforthewire According to Eq. (12), for λ0 = 0, f0 = 0 and f1/2 = 1.
coupled to a small quantum dot. The arrows indicate the For λ0 = 0, the singlet formation lowers E(N) for N =
6
direction in which the single stepsmove as λ(L) grows. even relatively to N = odd. As a result, the Kondo
interactionsplits the double steps andgivesriseto small
plateaus with N even (Fig. 2). To O λ2 , we find
0
electron in the dot and pairs of conduction electrons in
the Fermi sea. The energy levels measured from µ∗0 are f(λ0,L) ≡ f1/2(λ0,L)−f0(λ0(cid:0),L(cid:1))
DL
ǫn =(n−1/2)∆, (10) = 1−3 λ0+λ20ln v +... , (16)
(cid:18) F (cid:19)
where∆=πv /Landnisaninteger. Theenergyofthe
F
ground state for N =N0+ℓ electrons to O(1/L) is Note the logarithmic divergence at O λ20 as L → ∞,
characteristic of the Kondo effect. We recognize the ex-
ℓ/2 pansion of the effective coupling λ(L)(cid:0) (cid:1)[ln(ξK/L)]−1
E(N0+ℓ) = E0+2 (ǫn+µ∗0), (ℓ even) in powers of the bare λ0, as expected f∼rom scaling ar-
n=1 guments. We have verified that the scaling holds up to
X
(ℓ 1)/2 O λ3 ,i.e.,thefunctionf hasnodependenceonthecut-
− 0
= E +2 (ǫ +µ ) offD buttheimplicitoneintheexpansionofλ(L). This
0 n ∗0 (cid:0) (cid:1)
n=1 isremarkablegiventhatE(N)itselfiscutoffdependent.
X
+ǫ +µ , (ℓ odd) (11) Basedonthis, weconjecturethatf is auniversalscaling
(ℓ+1)/2 ∗0
functionofξ /L. ItfollowsfromEqs. (9), (15)and(16)
K
where E0 =E(N0). Using Eq. (10), we get that the charge steps occur at
E(N0+ℓ;λ0 =0)=E0+∆(cid:18)ℓ42 +s2(cid:19)+µ∗0ℓ, (12) µℓ+12∆−µ∗0 =(cid:22)2ℓ(cid:23)+ 21 − 34(−1)ℓλ(L), (17)
wheres=0forℓevenands=1/2forℓodd. Itiseasyto
where x is the floor function or the integer part of x,
seethat,asweraiseµ>µ ,N jumpswheneverµcrosses ⌊ ⌋
∗0 so that
anenergylevelǫ . Moreover,thereareonlydoublesteps
n
due to the spin degeneracy of the single-particle states, ℓ/2 , ℓ even
ℓ/2 = (18)
i.e. µ2m+12 = µ2m+23. As a result, N(µ)−N0 takes on ⌊ ⌋ (cid:26)(ℓ−1)/2 , ℓ odd.
only evenvalues andN is alwaysodd (see dashedline in
We define the ratio (see Fig. 2)
Fig. 2).
We obtain the charge steps in the weak coupling limit δµo
R
by calculating the correction to the ground state energy ≡ δµe
using perturbation theory in λ [5]. In the limit λ 0,
0 0 → E(N0+3) 2E(N0+2)+E(N0+1)
thegroundstateforN oddisstilldoublydegenerateand = −
E(N +2) 2E(N +1)+E(N )
we write 0 − 0 0
1+f(ξ /L)
K
= . (19)
|GS(N =odd)i= c↑k†nc↓k†n|0i⊗|γi, (13) 1 f(ξK/L)
−
n 0
Y≤
From Eq. (16), we have that in the weak coupling limit
forthevaluesofk definedbelowEq. (8). Thespinstate
n
of the dot is γ = , . For N even, there is one f(ξ /L 1) 1 3[ln(ξ /L)] 1, (20)
K K −
| i |⇑i |⇓i ≫ ≈ −
4
andthe ratiobetweenthe widthofoddandevensteps is The charge steps now appear at
R ξK 1 2ln ξK 1lnln ξK +const, µℓ+21 −µ∗0 = ℓ+1 +( 1)ℓπξK, (27)
L ≫ ≈ 3 L − 3 L ∆ 2 − 8 L
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (21) (cid:22) (cid:23)
where we included the subleading lnln term in the ex- from which we obtain
pression of the effective Kondo coupling at scale L. A
ξ πξ
similar scaling function was found in [7] for the singlet- R K 1 K. (28)
L ≪ ≈ 4 L
triplet gap (for fixed N) in the weak coupling limit. (cid:18) (cid:19)
Atthispointwewouldliketocommentabouttheeffect
of particle-hole symmetry breaking for this wire-dot ge-
III. STRONG COUPLING
ometry. FortheAndersonmodelofEq. (1),particle-hole
symmetryisabsentifthesystemisawayfromhalf-filling.
When λ , the spin ofthe dot forms a singletwith
We can account for this by adding to the Hamiltonian a
→∞
one conduction electron and decouples from the other
scattering potential term
electrons in the chain. In this limit the Kondo cloud
is small (ξK/L → 0). The strong coupling boundary Hp =2πvFVψ†(0)ψ (0), (29)
conditionsreflecttheπ/2phaseshiftfortheparticle-hole
symmetric case where V is of order the bare Kondo coupling λ0 1.1
≪
This term is strictly marginal and can be treat by first-
ψ( L)= ei2δψ(L). (22) order perturbation theory in both weak and strong cou-
− −
pling limits. Its effect is simply to shift the position of
Note the minus sign relative to Eq. (7). This implies
the charge steps by
that the k eigenvalues are shifted with respect to weak
coupling: k =(nπ+π/2 δ)/L. Atthestrongcoupling
fixed pointnwe recover the−spin degeneracy of the single- µℓ+12 →µℓ+12 +V∆. (30)
particle states. The shifted energy levels are Since the shift is independent of the parity of ℓ, the po-
tential scattering term does not lift the spin degener-
ǫ =n∆, (23)
n acy of the charge steps and has no effect on the ratio
as measured from the original µ . The ground state en- R=δµo/µe.
∗0
ergy for N =N +ℓ is
0
ℓ2 1 2 IV. BETHE ANSATZ RESULTS
E(N +ℓ)=E(N )+∆ + s +µ ℓ, (24)
0 0 " 4 (cid:18) − 2(cid:19) # ∗0
In order to calculate the scaling property of R =
where again s = 0 for ℓ even and s = 1/2 for ℓ odd. δµo/δµe for any L/ξ we use the Bethe ansatz (BA)
K
The ground state consists now of a singlet plus pairs of solution of the one-channel Kondo problem.10 We start
electrons in the wire, thus N(µ) is always even. Fig. 2 withahalf-filledbandofN 1conductionelectrons(N
0 0
−
illustrateshowthechargestaircaseevolvesmonotonically odd) coupledto a localizedimpurity spin, corresponding
from weak to strong coupling. toasystemsizeL=(N 1)/2inunits wheretheBethe
0
locWaleFeexrpmloirleiqtuhiedltihmeiotrkyF.−91T≪heξKide≪a iLs tbhyatwvoirrktiunaglotruatna- saentsaetqzucaultt-ooffopnaer(asmeeet[e1r0−D])., rSeliantceedtthoetBheAbsaonlduwtiiodnthc,ains
sitions of the singlet at x = 0 induce a local interaction be obtained for any filling factor, we can add particles
in the spin sector of the conduction electrons. The lead- one by one to the system (N =N , N +1, N +2,...)
0 0 0
ing irrelevantoperatorthat perturbs the strongcoupling and compute the corresponding energies. This has been
fixed point and respects SU(2) symmetry is donebysolvingnumericallythecoupledBAequations,10
using a standard Newton-Rapson method. In Fig. 3 we
H = 2π2 vF2 ψ (0)~σψ(0) 2, (25) present results obtained for the ratio R = δµo/δµe with
FL †
− 3 T 2 N =51, 101, 201, 501, 1001, 2001and13differentval-
K (cid:20) (cid:21) 0
ues of the Kondo coupling in the range 0.06 J 0.8.
where the prefactor is fixed such that the impurity sus- ≤ ≤
The universalscaling curve has been obtained by rescal-
ceptibility is χ = 1/(4T ). This interaction lowers
imp K ing the x-axis L L/ξ (J) in order to get the best
the groundstate energy when there is an odd number of → K
collapse of the data. The entire crossover curve is ob-
remaining conduction electrons (S = 1/2, or s = 0),
el tained, ranging from the strong coupling L ξ to the
thus splitting the charge steps in the strong coupling ≫ K
weak coupling regime L ξ . Both strong coupling
limit. If we introduce the function f as in (15) and (16), ≪ K
[Eq. 28)] and weak coupling [Eq. (21)] results are per-
we find to lowest order in 1/T
K fectlyreproduced. Asdisplayedintheinset(a)ofFig.3,
πξ BA results agreewith Eq.(21) using only one fitting pa-
f(ξK/L)= 1+ K +O (ξK/L)2 . (26) rameter: const 0.33. TheKondolengthscale,shownin
− 2L ≃
(cid:0) (cid:1)
5
weak coupling J=0.06 where V(x) is the screened Coulomb potential and
J=0.07
10 JJ==00..018 ρσ(x)≡Ψ†σ(x)Ψσ(x)isthedensityofelectronswithspin
15 J=0.125 σ. In the low-energy limit, we expand Ψ(x) in terms of
(a) J=0.15
10 J=0.2 right and left movers as in Eq. (3) and write
1 J=0.3
J=0.4
5 J=0.5 ρ (x) = ρ (x)+ρ (x)
) J=0.6 σ R,σ L,σ
K 0 J=0.7
ξ/ 0.1 10-9 10-6 10-3 100 J=0.8 +[ei2kFxψL†σ(x)ψRσ(x)+h.c.], (34)
L
(
R 0.01 1012 F0.r3o1m*e cxopl(lπa/pcs)e where ψR/Lσ(x) ≡ ψR†/Lσ(x)ψR/Lσ(x). Separating the
ξ 108 processes that involve small momentum transfer (q 0)
K ≈
0.001 110040 (b) strong c fcroonmvetnhteiobnaaclkgs-coaltotgeyrinngotoantieosn(q ≈2kF),weobtaininthe
0 5 10 15 20 o
0.0001 10-9 10-6 10L-3/ξ1/J 100 103 upling HI = Xσ Z0Ldxng24k (cid:2)gρ2R,σ+ρ2L,σ(cid:3)
K +g2kρR,σρL,σ+ 42⊥ [ρR,σρR,−σ+ρL,σρL,−σ]
FIG. 3: (Color online) Universal ratio R = δµo/δµe as a +g2 ρR,σρL, σ +Hex, (35)
scaling function of ξK/L. Bethe ansatz results obtained for ⊥ − }
various systems sizes (N0 = 51, 101, 201, 501, 1001, 2001) where g = g = g = g = V˜(0) for a spin-
2 2 4 4
and13differentvaluesoftheKondoexchangeJ,indicatedby independeknt inter⊥actionpkotential⊥, with V˜(0)the Fourier
different symbols. For each value of J, the system lengths L
transform of V(x) at momentum q = 0. The term H
have been rescaled L→L/ξK(J) in order to obtain the best ex
contains the backscattering interaction processes. As we
collapse of thedata using the strong coupling curve Eq. (28)
will discuss below, this interaction is marginally irrele-
(dashed red line) as a support for the rest of the collapse.
The weak coupling regime for L ≪ ξK, enlarged in the in- vant in the sense of the renormalization group. If we
set (a),isdescribed bytheweak couplingexpansionEq.(21) neglect Hex for a moment, the resulting Hamiltonian is
with constant≃0.33 (continuousbluecurve). Inset(b): The exactly solvable by bosonization.12 For open boundary
Kondo length scale, extracted from the universal data col- conditions,wecanworkwithrightmoversonaringwith
lapse of the main panel (black squares), is described by the size 2L and the boundary conditions of Eq. (5). After
exponential fit Eq. (31) with ξ0≃0.31 (dashed green line). introducing charge and spin densities
ρ ρ
R, R,
the inset (b) of Fig. 3, displaysthe expected exponential ρR,c/s = ↑√±2 ↓, (36)
behavior10
one finds that the charge and spin degrees of freedom
ξ (J)=ξ eπ/c, (31) decouple. The effective model for electrons in the wire
K 0
with zero Kondo coupling is the Luttinger model with
with c = 2J/(1 3J2/4). Fitting ξ to this expression open boundary conditions11
K
−
gives ξ 0.31 (see inset (b) of Fig. 3) which is in very
0
good agr≃eement with the expected value of ξ0 = 1/π, H0+HI =HLL+Hex, (37)
resulting1 fromthe impuritysusceptibilitynormalizedto
where (setting µ =0)
1/(4TK). ∗0
πv πv
H = c Nˆ2+ s(Sˆz)2+ v q b b , (38)
LL 4K L L ν| | †qν qν
c
V. LUTTINGER LIQUID EFFECTS qX>0
ν=c,s
Letusnowincludetheeffectofelectron-electroninter- where Nˆ and Sˆz are the zero modes for charge and spin
actions in the wire. We consider the Hamiltonian11
excitations, v and v are the velocities of the collective
c s
charge and spin modes created by the bosonic operators
H =H +H +H . (32)
0 I K b†qc/s, and Kc is the Luttinger parameter for the charge
sector. We assume spin SU(2) symmetry and set the
The kinetic energy part H is given, in the low energy
0
Luttinger parameter for the spin sector to be K = 1.
limit, by Eq. (4). The short-range interactions are de- s
From Eq. (38), we have that the energy of the ground
scribed by
state for ℓ extra electrons (neglecting H ) is
ex
1
HI = 2 dx dx′ρσ(x)V(x−x′)ρσ′(x′), (33) E(N0+ℓ,λ0 =0)=E0+ πvc ℓ2+ πvs s2, (39)
Xσσ′ Z Z 4KcL L
6
The noninteracting case in Eq. (12) corresponds to the 2k terms become products of spin and charge op-
F
v = v = v and K = 1. For repulsive interactions, erators. However, for open boundary conditions we can
c s F c
K <1 and v >v . For weak interactions, we have the useEq. (5)andthe entire spindensityatx=0 becomes
c c s
perturbative results from bosonization proportional to J~ (0). For this reason, spin-charge sep-
R
aration is preserved (up to irrelevant operators) for the
1/2
v 2V˜(0) − geometry of Fig. 1 with a finite Kondo interaction. The
F
Kc = 1+ (40) bulkKondointerationsonlyimportanteffectsonthespin
vc ≈" πvF #
sector, in the low energy limit, are to modify somewhat
the spin velocity, v and to introduce the marginally ir-
and v =v . s
s F
relevant bulk interaction of Eq. (41). The finite size
The term H on the rhs of Eq. (37) is sometimes
ex
energies still break up into a sum of spin and charge
called the marginally irrelevant bulk interaction. This
parts. The charge part is exactly as in Eq. (39). In the
is entirely in the spin sector and is analogous to the ex-
spinpart,thedimensionlessKondocouplingisdefinedas
change interaction in the constant interaction model.2
λ = 2Jsin2(k ε)/πv , with v replacing v in contrast
The operator can be written as 0 F s s F
with Eq. (6). The Kondo coupling is still marginally
relevant for K =1. We can then write
H = 2πg v dxJ~ (x) J~ (x), (41) s
ex 0 s L R
− · πv πv
Z E(N +ℓ,λ )= c ℓ2+ sf (λ,g,L). (47)
0 0 s
4K L 4L
where J~L/R ≡ ψL†/R(~σ/2)ψL/R. The bare coupling con- c
stant g is proportional to V˜(2k ) (backward scatter- Thescalingfunctionsfs(x),s=0,1/2,arethesameones
0 F
ing process).13 It is typically very small if the screening thatwecalculatedignoringLuttingerliquidinteractions,
length is much larger than kF−1. The renormalized cou- avapnatrtbfurlokminttheeraccotniotrni.buTtioonlofwroesmt othrdeemr ainrgλinaalnlydigrr,ewlee-
pling g(L) obeys the renormalization group (RG) equa-
have
tion
dg g3 f(λ,g,L) = f1/2(λ,g,L) f0(λ,g,L)
= g2 +..., (42) −
dl − − 2 3
= 1 g(L) 3λ(L). (48)
− 2 −
where l = ln(DL/v ). Precisely the same interaction
s
appears in a spin chain (see, for example, [14]). The Note that both the bulk marginal operator and the
solution to O(g2) is Kondo interaction reduce the energy when N is even.
TheRGequationfortheKondocouplingλismodified
g(l) g0 . (43) by the marginal bulk operator16
≈ 1+g l
0
dλ
=λ2+gλ+.... (49)
Therefore, g(L) 1/ln(L) vanishes in the limit L . dl
∼ →∞
The correction to the energy of the lowest energy state
due to H is14 Solvingthis equationinthe presence ofthe secondterm,
ex
one finds
3πv
δE(N +ℓ,λ =0)= sg(L)s2+O(g2). (44) 2ln(L/L )
0 0 1
− 2L λ(L)= , (50)
ln2(ξ /L ) ln2(L/L )+2ln(ξ /L )
K 1 1 K 1
−
The Kondo effect has also been studied in Luttinger
liquids.15 Onecrucialpointisthattheimpurityspincou- where ξK is defined such that λ(L = ξK) = 1, and L1
ples only to the spin degrees of freedom of the Luttinger is a characteristic length scale for the bulk marginal in-
liquid when the impurity spin is at one end. This is not teraction defined by g(L) 1/ln(L/L1). If the bare g0
true if the spin couples far from the end. The Kondo is small, then L1 (vs/D≈)e−1/g0 vs/D. In the limit
∼ ≪
interaction is as before L1 0 we recover the scaling function λ = λ(ξK/L)
→ ∼
[ln(ξ /L)] 1. In general, we expect λ = λ(ξ /L,g(L)).
K − K
~σ
HK =JΨ†(ε)2Ψ(ε)·S~, (45) Ethqe.b(a4r9e)Kaolsnodoimcpoulipeslinagn16u,1n7usual dependence of ξK on
gwehneerrealε, t∼heksF−p1in≪denLsit(yε o=pe1raitnorthise lattice model). In ξ L exp c+ 2 ln vs +ln2 vs +c2 ,
K 1
∼ "− sλ (cid:18)DL1(cid:19) (cid:18)DL1(cid:19) #
~σ ~σ (51)
Ψ†(x)2Ψ(x)∼J~L+J~R+ ei2kFxψL† 2ψR+h.c. . (46) wherec is a positiveconstantofO(1). Inthe limitL1
(cid:18) (cid:19) ≪
v /D we recover the usual result ξ (v /D)e 1/λ0. In
s K s −
∼
Whenwe bosonize,the left andrightspindensities J~L/R the limit λ → 0 with g held fixed this becomes ξK ∝
canbe expressedentirely interms ofthe spin boson,but econst/√λ0.
7
Inthe chargestaircase(see againFig. 2), the widthof 1 J=0.8
the plateaus is normalizedby the chargeadditionenergy 1-3/2*g(1000) J=0.7
J=0.6
∆ = πv /K L. The charge steps for ξ L are given J=0.5
c c c K J=0.4
by ≫ J=0.3
0.5 J=0.2
J=0.15
Spin chain Kondo model J=0.125
µℓ+12 −µ∗0 = ℓ + 1 (−1)ℓ 1 uf ξK,g(L) , JJ==00..108
∆c (cid:22)2(cid:23) 2 − 4 (cid:20) − (cid:18) L (cid:19)(cid:21) ξ)K JJ==00..0076
(52) / 0
L Free electrons
where (
f
u v K /v <1 (53)
s c c
≡ -0.5
and f is given by Eq. (48). As the charge and spin
addition energies are different for u = 1,18 there are no
-1+3/2*g(1000)
6
double steps in the interacting case even for λ 0. The
→ -1
ratiobetweenoddandevenstepsincludinginteractionis 10-9 10-6 10-3 100 103
L/ξ
1+uf(ξ /L,g(L)) K
R˜ = K . (54)
1 uf(ξ /L,g(L))
− K FIG. 4: (Color online) Effect of the marginally irrelevant
Thus we see that the main effect of the screened bulk bulk operator on the scaling function f(ξK/L,g(L)) calcu-
Coulomb interactions is to suppress R˜ somewhat due to lated from the Bethe ansatz. The symbols correspond to the
noninteracting case, g(L)=0, as in Fig. 3. The black circles
the reduction of the parameter u from 1. As u 0 (for
→ areobtainedwiththeBetheansatzsolutionofthespinchain
very strong Coulomb interactions) the even-odd effect Kondomodel (60) for L=1000 (g(1000)≃0.115).
disappears entirely and we see simple Coulomb block-
ade type steps. For cleaved edge overgrowth quantum
wires, u 0.5.19,20,21 For carbon nanotubes, one finds, R andR˜ (1+u)/(1 u). Inthe strongcoupling
typically,≈u 0.1.22 (For further discussionof the exper- lim→it,∞R 0 an→d R˜ (1 u−)/(1+u).
≈
imental possibilities see Sec. VII.) For ξ L, → → −
K A nonzero g leads to a weak violation of scaling be-
≫ 0
cause f is not only a function of ξ /L but also of g(L).
1+u 3uλ(L) 3ug(L) K
R˜ − − 2 . (55) The effective g(L) is typically small in quantum wires,
≈ 1 u+3uλ(L)+ 3ug(L) except at very low electron densities, and could in prac-
− 2
ticebetreatedasafittingparameter. Wecanwritedown
Now consider the strong coupling limit, ξK L. The a general expansion for f in powers of g:
≪
Fermi liquid interaction is the same as in Eq. (25), but
with ξK given by Eq. (51). In contrast with Eq. (48), f(ξK/L,g)=f0(ξK/L)+gf1(ξK/L)+O(g2). (59)
themarginaloperatornowlowersthegroundstateenergy
when N is odd, because in the strong coupling limit the It follows from Eqs. (48) and (56) that f (ξ /L 1)=
1 K
≫
number of free electrons in the wire in N 2. Thus we 3/2+O(λ(L)) and f (ξ /L 1)=3/2+O(ξ /L).
1 K K
− − ≪
have in this case
3 πξ
K
f(ξ /L,g(L))= 1+ g(L)+ . (56)
K
− 2 2L
The charge steps for ξ L are given by Eq. (52) with FIG.5: Schematicpicturefor an open Heisenbergchain cou-
K
f taken from Eq. (56). T≪he ratio becomes pled to a spin impurity (arrow) at the left boundary via a
weak antiferromagnetic exchangeJ′ (dashed bond).
K
1 u+ 3ug(L)+uπξK
R˜ = − 2 2L . (57) In fact, the function f(ξ /L,g) can be calculated in
1+u 3ug(L) uπξK K
− 2 − 2L the framework of a Heisenberg spin chain model with a
weak coupling J at the end of the chain16,17
If we set g =0, we can express the ratio R˜ in terms of K′
the ratio calculated for the noninteracting case
L 1
=J − S~ S~ + , =J S~ S~ . (60)
1 u+(1+u)R H 1 i· i+1 Himp Himp K′ imp· 1
R˜ = − , (58) i=1
1+u+(1 u)R X
−
This model is depicted in Fig. 5. Its low energy limit is
whereR=R(ξK/L)isthescalingfunctionshowninFig. thesameasthespinsectoroftheLuttingerliquid.16,17,23
3. For u = 1, the function R˜ is finite at both weak and In this case the bulk marginalcoupling has a bare value,
strong cou6 pling fixed points. In the weak coupling limit, g ,ofO(1). TheKondocouplingisproportionaltoJ .16
0 K′
8
Since the charge sector separates from the spin sector where ε ∼ kF−1 ≪ L. In the symmetric case it is conve-
anyway, in the Luttinger liquid Kondo model, it follows nient to label the eigenstate of the open chain by their
that the needed function f(ξ /L,g(L)) can be deter- symmetry under the parity transformation x L x.
K
→ −
mined from the spin chain model. Expanding Ψ(x) as in Eq. (3), we have
The function f(ξ /L,g(L)) can be evaluated via the
K Ψ(ε)+Ψ(L ε)=eikFε[ψ (ε)+e ikFLψ (L ε)]
energydifference betweenthe groundstatesforevenand − R − L −
odd chain length, of total spin 0 and 1/2: +eikF(L−ε)[ψR(L ε)+e−ikFLψL(ε)].(65)
−
πv This motivates defining the even and odd fields
s
E E f. (61)
0− 1/2 → 4L ψ (x) e ikFLψ (L x)
R − L
ψ (x)= ± − . (66)
e,o
One can extract this quantity exactly using the √2
BA solution17 of the model (60) by computing
Thisway,forzerofluxH involvesonlytheevenchannel
1[E(L+2)+E(L) 2E(L+1)]withLevenforvarious K
2 − since
KondocouplingstrengthsJ . Thishasbeenachievedfor
K′ Ψ(ε)+Ψ(L ε)
L = 1000 as shown in Fig. 4, where one can see the full − =eikFεψe(ǫ)+eikF(L−ε)ψe(L ǫ).
scaling function with the two limiting cases √2 −
(67)
1 3g if L/ξ 0 For J = 0, the system is equivalent to an open chain
f →(cid:26)−1−+ 322g if L/ξKK →→∞. (62) aHnadmailtforeneiasnpionf.EInq.te(4rm) issosfimevpelny5and odd fields, the free
Inordertoobtainthescalingfunctionf ofthespinchain
L
uwKniotunhsduLoalm=eoxd1pe0ol0ni0ne,nttwhiaeelehsnaqtduiraetroreecrgooimnotveeboretfhLJa/Kv′ξiKo→rsfhoξorKwtnhuesiniKnFgoingtdh.4oe TheopenHb0o=un−daivrFycZo0nddixtio(cid:0)nψse†i∂nxtψhee+weψao†k∂cxoψuop(cid:1)l.inglim(6i8t),
length scale (51). In order to make things quantitative, Ψ(0)=Ψ(L)=0, imply
we used the following conversion17 for ξ (J )
K K′ ψ (L)= e ikFLψ (0). (69)
e/o − e/o
∓
ξ =ξ exp π 1/J 1 . (63) Because of the relative minus sign in Eq. (69), the even
K 0 K′ − and odd channels are nondegenerate. The energy levels
(cid:18) q (cid:19)
alternate between the even and channels. We assume
with16,17 ξ = √e/π. It is reassuring to note that, even that the lowest energy single-particle state in the band
0 2
for this large value of bare coupling g0, the effects of the is even under parity. In this case, for N0 = 4p+1, p
bulk marginal operator on the cross-over function f are integer, the lowest unoccupied state belongs to the even
fairly small for L=1000. We might expect these effects channel. The energy levels relative to µ∗0 for J = 0 are
to be even smaller for the Luttinger model with realistic ǫen =(2n+1/2)∆andǫon =(2n−1/2)∆,with∆=πvF/L
parameters chosen to describe the systems discussed in and n integer.
Sec. VII. Wecanaccountforthemagneticfluxthroughthe ring
byintroducingaphaseinthehoppingbetweenthequan-
tum dot and the two leads.5 This does not affect the
VI. FLUX DEPENDENCE IN QUANTUM Hamiltonian of the open chain in (68), but the Kondo
RINGS interaction is modified to
~σ
The effect of the Kondo screening cloud on the charge HK =J[Ψ†(ε)+eiαΨ†(L−ε)]2[Ψ(ε)+e−iαΨ(L−ε)]·S~,
steps of a 1D system is not unique to the wire geome- (70)
try. One interesting alternative is to suppose that the where α 2πΦ/Φ0 and Φ0 = hc/e is the flux quantum.
≡
quantum dot is embedded in a ring with circumference The Kondo interaction in terms of even and odd fields
L (inset of Fig. 6). This geometry offers the possibility reads
of looking at the dependence of the charge steps on the α α
H = 2πv λ cos ψ (0) isin ψ (0)
magnetic flux Φ threading the ring, which is related to K F 0 2 e† − 2 o†
thepersistentcurrentfortheembeddedquantumdot.5,24 ~σ hα α i
cos ψ (0)+isin ψ (0) S~, (71)
Inthissectionwefocusagainonthenoninteractingcase. ×2 2 e 2 o ·
For simplicity, we assume that the coupling between the h i
whereλ =4Jsin2(k ε)/πv . Ifparticle-holesymmetry
dot and the leads is parity symmetric. We can write the 0 F F
is broken we have to add the potential scattering term
Hamiltonian (for zero flux) as H =H +H , where H
0 K 0
is the same as in Eq. (4) and the Kondo interaction is α α
Hp = 2πvFV cos 2ψe†(0)−isin 2ψo†(0)
~σ αh α i
H =J[Ψ (ε)+Ψ (L ε)] [Ψ(ε)+Ψ(L ε)] S~, (64) cos ψ (0)+isin ψ (0) , (72)
K † † − 2 − · × 2 e 2 o
h i
9
2.5 (see dashed lines in Fig. 6)
∆
µ)/*0 Φ Vg µℓ+21 −µ∗0 = ℓ + 1 + 1+cos α+π ℓ
- µ7/2 ∆ (cid:22)2(cid:23) 2 (cid:20) (cid:18) (cid:22)2(cid:23)(cid:19)(cid:21)×
µ l+1/2 1.5 µ5/2 ×(cid:20)V − 34(−1)ℓλ(L)+O λ2 (cid:21). (74)
( (cid:0) (cid:1)
s µ
p Notethat,althoughpotentialscatteringgivessomeweak
3/2
e
st 0.5 fluxdependencetotheenergylevels,itdoesnotlifttheir
e µ spin degeneracy. Only the term due to the Kondo inter-
g 1/2
r actionalternatesbetweenℓevenandℓoddandtherefore
a
h lifts the spin degeneracy. Hence the main effect of the
c
Kondo interaction in the weak coupling limit is to split
-0.5
0 0.5 1 1.5 2 the charge steps that are degenerate for λ = 0, namely
α/2π = Φ/Φ µ and µ with ℓ even. We define
0 ℓ+1/2 ℓ+3/2
FIG. 6: (Color online) Flux dependence of the charge steps δµn ≡µ2n+32 −µ2n+21, (75)
intheringwithanembeddedquantumdot(inset),assuming
nopotentialscattering(V =0). Dashedlines: weakcoupling where n is an integer. From Eq. (74) we obtain
regime (ξK ≫ L); solid lines: strong coupling regime (ξK ≪
L). δµ (α) 3
n = [1+( 1)ncosα]λ(L)+O λ2 . (76)
2∆ 4 −
(cid:0) (cid:1)
For symmetric coupling to the leads only the even chan-
whereV O(λ0)isthedimensionlesscouplingconstant. nel couples to the impurity when α = 2mπ, m integer.
∼
This operator is strictly marginal in the noninteracting As a result, the steps in the odd channel (corresponding
case. We will assume V 1 and treat Hp using first tooddvaluesofninEq. (76))donotsplit. Theopposite
≪
order perturbation theory. happens when α=(2m+1)π (see Fig. 6).
For λ=0, the energy levels are flux-independent, due Inthe strongcouplinglimit,theπ/2phaseshiftmodi-
to the open boundary conditions. In the weak coupling fiestheboundaryconditionsinsuchawaythatthetrans-
limit λ(L) 1, perturbation theory in the Kondo inter- mission across the dot becomes perfect.3,25,26 For zero
actionofEq≪. (71)forgeneralαyieldsadifferentsplitting flux, the phase shift in the even channel implies
for even and odd channels. As a result, the spin part of
the ground state energy depends on ℓ mod 4. To O(λ2), ψe/o(L)=e−ikFLψe/o(0). (77)
0
we find
Withthesenew,periodicboundaryconditions,theright-
and left-moving components are decoupled
ℓ2
E(N +ℓ) = E +µ ℓ+∆ +s2
0 0 ∗0 (cid:18)4 (cid:19) ψR/L(L)=e−ikFLψR/L(0). (78)
3s2∆(1 cosα˜ )λ(L)
ℓ
− − Thesystemisthenequivalenttoanideal1Dring,whose
3s2(ln2)∆cosα˜ (1 cosα˜ )λ2(L)
− ℓ − ℓ eigenstates can be labeled as right or left movers. The
+3∆ 1+cos2α λ2(L)+O λ3 , (73) degeneracy of right and left-moving energy levels at flux
16 mπ implies charge steps of magnitude 4 in the strong
(cid:0) (cid:1) (cid:0) (cid:1) couplinglimit, ascanbeseeninFig. 6. Aswedidinthe
weakcouplinglimit,wecanintroducethefluxinthehop-
where α˜ α+(ℓ+1)π/2; s = 0 for ℓ even and s =
ℓ ≡ ping from x = L to x = 0. Since the ring is now closed,
1/2 for ℓ odd. In Eq. (73) it is implicit that a λ2lnL
0 the unperturbed energy levels are flux dependent,27
termisgeneratedbyperturbationtheorywiththecorrect
coefficient to recover the expansion of λ(L).
ǫR/L(α)=(2n 1/2 α/π)∆, (79)
n − ±
The charge step locations can be calculated using Eq.
(9). The last term in Eq. (73) is independent of ℓ and and exhibit zigzag lines with level crossings at α = mπ,
drops out when taking the difference E(N +ℓ+ 1) m integer. The energy levels in Eq. (79) are measured
0
−
E(N +ℓ). Tofirstorder,thepotentialscatteringtermin from µ defined in the weak coupling limit. The shift of
0 ∗0
Eq. (72)simplyincreasestheenergybyV∆(1+cosα)for 1/2 is due to the phase shift of π/2 for the states in
−
each electron added in the even channel and by V∆(1 the even channel at α=0 and can be understood in the
−
cosα) for each electron added in the odd channel. The followingway. ConsiderthelatticemodelwithN 1sites
−
charge steps in the weak coupling limit are located at and a symmetric Kondo coupling between the electrons
10
at the ends of the chain, at j =1 and j =N 1, and to Fig. 6).
−
the impurity spin at j =0,
µ4n+1±21 −µ∗0 = 2n+ 1 1 πξK
∆ 2 ± ∓ 4L
L 2
−
H = −t c†jcj+1+h.c. α−mπ 2+ πξK 2, (84)
Xj=1(cid:16) (cid:17) ∓s π 4L
(cid:18) (cid:19) (cid:18) (cid:19)
~σ
+J c†1+c†N−1 2 c1+cN−1 ·S~. (80) µ4n+3±21 −µ∗0 = 2n+ 3 α−mπ 2+ πξK 2
(cid:16) (cid:17) (cid:16) (cid:17) ∆ 2 ∓s π 4L
(cid:18) (cid:19) (cid:18) (cid:19)
ForJ =0andathalf-filling,thereareN 1freeconduc- 2 2
tion electrons in the chain. The hopping−Hamiltonian is 2 α−mπ + πξK . (85)
invariant under the particle-hole transformation ± s(cid:18) π (cid:19) (cid:18) 8L (cid:19)
For α mπ, m odd, the charge steps are located at
≈
cj →(−1)jc†j. (81)
µ4n+1±12 −µ∗0 = 2n+ 1 α−mπ 2+ πξK 2
For J = 0, particle-hole symmetry is broken by the ∆ 2 ∓s π 4L
(cid:18) (cid:19) (cid:18) (cid:19)
6
KondointeractionifN isoddandpreservedifN iseven,
2 2
α mπ πξ
since under particle-hole transformation 2 − + K , (86)
± s π 8L
(cid:18) (cid:19) (cid:18) (cid:19)
c +c c +( 1)Nc . (82) µ4n+3±12 −µ∗0 = 2n+ 3 1 πξK
1 N 1 →− 1 − N 1 ∆ 2 ± ∓ 4L
− −
h i
2 2
α mπ πξ
K
− + . (87)
As we chose to fix µ∗0 so that N0 = 4p+1 is odd, the ∓s π 4L
chargestepsinthestrongcouplinglimitarenotsymmet- (cid:18) (cid:19) (cid:18) (cid:19)
ricaboutµ∗0. However,thechargestepsmustbesymmet- Note that two charge steps remain degenerate at each
ric aboutµ∗0±∆/2,which correspondto values ofchem- point of level crossing α = mπ. This is because for α =
ical potential with an even number of electrons in the mπ either the single-particle states in the odd channel
weakcouplinglimit. Noticethat,ifwestartfromN0 odd (for m even) or the states in the even channel (for m
intheweakcouplinglimit,thereisavalueofλ(L) O(1) odd) decouple from the spin of the dot. As a result, the
∼
(for L ∼ξK) at which the charge step µ21 crosses µ∗0 for Fermi liquid interaction reduces to
α = 0. This means that, for fixed µ = µ and α = 0,
∗0
the ground state at the strong coupling fixed point has 2π2 v2 ~σ 2
an even number of electrons (N = N0+1 = 4p+2, i.e. HFL =− 3 TF ψe†,o(0)2ψe,o(0) , (88)
N 1 = 4p free electrons in the ring plus two in the K (cid:26) (cid:27)
0
−
singlet). where the index e applies to m even and the index o
Usingthestrongcouplingboundaryconditions,thelo- tomodd. Thetwodegeneratechargestepcorrespondto
calFermiliquidinteractionanalogoustoEq. (25)canbe addingelectronstothestatewhoseenergyisnotmodified
by the Fermi liquid interaction.
written
If the potential scattering is nonzero, the degener-
acy between the even and odd channel at α = mπ is
π2 v2
HFL = − 6 TKF ψR(0)+e−iαψL(0) † ltiefrtaedct.ioFnoarnξdKt/hLe p∼otVent≪ial1sc,abttoetrhintghecaFnebrmeitrleiqauteidd uins--
~σn(cid:2) (cid:3)2 ing degenerate perturbation theory near α mπ. If
×2 ψR(0)+e−iαψL(0) . (83) ξK/L ≪ V ∼ O(1), the Fermi liquid interact≈ion can be
(cid:27) addedontopoftheunperturbedenergylevelsofanideal
(cid:2) (cid:3)
ringwithadeltafunctionpotentialatx=0. Inanycase,
The main effect of this interaction is to lift the degen- theresultisthat,duetoavoidedlevelcrossings,potential
eracy between right and left movers at α = mπ and to scattering suppresses the flux dependence of the charge
split two out of the four values of µℓ+12. We calculate steps in the limit ξK/L ≪ 1. However, a feature that
the correction to the ground state energy using degen- survives when V = 0 is that the splitting of the double
6
erate perturbation theory to O(1/T ) for V = 0. The steps in the even(odd) channelfor α mπ with m even
K
≈
calculation is similar to the weak coupling limit of the (m odd) is proportional to ξ /L.
K
side-coupled quantum dot5 and is presented in detail in Thissignificantdifferenceinthefluxdependenceofthe
Appendix A. The charge steps (as labeled in the weak charge steps, between the weak and strong Kondo cou-
coupling limit) for α mπ, m even, are given by (see pling limits, is consistent with previous results for the
≈