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arXiv:cond-mat/0608110v3 [cond-mat.mes-hall] 7 Oct 2006

Kondo Effects in Carbon Nanotubes: From SU(4) to SU(2) symmetry

Jong Soo Lim

Department of Physics, Seoul National University, Seoul 151-747, Korea and

Department of Physics, Korea University, Seoul 136-701, Korea

Mahn-Soo Choi∗

Department of Physics, Korea University, Seoul 136-701, Korea and

Department de F´isica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain

M. Y. Choi

Department of Physics and Center for Theoretical Physics,

Seoul National University, Seoul 151-747, Korea and

Korea Institute for Advanced Study, Seoul 130-722, Korea

Rosa L´ opez

Department de F´isica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain

Ram´ on Aguado

Teor´ ıa de la Materia Condensada, Instituto de Ciencia de Materiales de Madrid (CSIC) Cantoblanco,28049 Madrid, Spain

(Dated: February 6, 2008)

We study the Kondo effect in a single-electron transistor device realized in a single-wall carbon

nanotube. The K-K′double orbital degeneracy of a nanotube, which originates from the peculiar

two-dimensional band structure of graphene, plays the role of a pseudo-spin. Screening of this

pseudo-spin, together with the real spin, can result in an SU(4) Kondo effect at low temperatures.

For such an exotic Kondo effect to arise, it is crucial that this orbital quantum number is conserved

during tunneling. Experimentally, this conservation is not obvious and some mixing in the orbital

channel may occur. Here we investigate in detail the role of mixing and asymmetry in the tunneling

coupling and analyze how different Kondo effects, from the SU(4) symmetry to a two-level SU(2)

symmetry, emerge depending on the mixing and/or asymmetry. We use four different theoretical

approaches to address both the linear and non-linear conductance for different values of the external

magnetic field. Our results point out clearly the experimental conditions to observe exclusively

SU(4) Kondo physics. Although we focus on nanotube quantum dots, our results also apply to

vertical quantum dots. We also mention that a finite amount of orbital mixing corresponds, in the

pseudospin language, to having non-collinear leads with respect to the orbital ”magnetization” axis

which defines the two pseudospin orientations in the nanotube quantum dot. In this sense, some

of our results are also relevant to the problem of a Kondo quantum dot coupled to non-collinear

ferromagnetic leads.

PACS numbers: 75.20.Hr, 73.63.Fg,72.15.Qm

I.INTRODUCTION

The first observations of Kondo effect in semiconduc-

tor quantum dots (QDs)1,2,3have spurred a great deal of

experimental and theoretical activity during the last few

years. Since these experimental breakthroughs, remark-

able achievements have been reported, including the ob-

servation of the unitary limit,4the singlet-triplet Kondo

effect,5Kondo effect in molecular conductors6, and the

Kondo effect in QDs connected to ferromagnetic7and

superconducting reservoirs,8just to mention a few.

Recently, Jarillo-Herrero et al. reported perhaps the

most sophisticated example, namely the observation of

an orbital Kondo effect in a carbon nanotube (CNT)

quantum dot (QD).9In these experiments it was shown

that the delocalized electrons of the reservoirs can screen

both the orbital pseudospin degrees of freedom in the

CNT QD (the K-K′double orbital degeneracy of the two-

dimensional band structure of graphene) and the usual

spin degrees of freedom, resulting in an SU(4) Kondo ef-

fect at low temperatures. In a recent letter,10we showed

that quantum fluctuations between the orbital and spin

degrees of freedom may indeed dominate transport at low

temperatures and lead to this highly symmetric SU(4)

Kondo effect. More recently, Sakano and Kawakami11

have studied, using the Bethe ansatz method at zero

temperature and the non-crossing approximation at finite

temperatures, the more general case where the quantum

numbers of N degenerate orbital levels are conserved, and

found new interesting features of the SU(2N)-symmetric

Kondo effect.Importantly, this is true provided that

both the orbital and spin indices are conserved during

tunneling. This poses an interesting question about the

nature of the nanotube-lead contact because, in princi-

ple, there is no special reason why the orbital degrees of

freedom in the CNT should be conserved during tunnel-

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ing.

As mentioned,

from the peculiar electronic structure of the nanotube

(NT).9,12,13The electronic states of a NT form one-

dimensional electron and hole sub-bands as a result of

the quantization of the electron wavenumber k⊥perpen-

dicular to the NT axis, which arises when graphene is

wrapped into a cylinder to create a NT. By symmetry,

for a given sub-band at k⊥ = k0 there is a second de-

generate sub-band at k⊥ = −k0. Semiclassically, this

orbital degeneracy corresponds to the clockwise (?) or

counterclockwise (?) symmetry of the wrapping modes.

A plausible explanation of why this degree of freedom

is preserved during tunneling could be that the QD is

likely coupled to NT electrodes (the metal electrodes are

deposited on top of the NT so maybe the electrons tun-

neling out of the QD enter the NT section underneath

the contacts) but this issue clearly deserves a thorough

microscopic analysis about the nature of the contacts.

The conservation of the orbital quantum number seems

more likely in the vertical quantum dots (VQD),14where

the orbital quantum number is the magnetic quantum

number of the angular momentum.

this orbital pseudospin originates

Here, we take a different route and, assuming some

degree of mixing in the orbital channel, ask ourselves

about the robustness of the SU(4) Kondo effect against

asymmetry in the couplings and/or mixing.

The rest of the paper is organized as follows: In Sec-

tion II we introduce the relevant model Hamiltonian and

classify different schemes of the lead-dot coupling. These

different coupling schemes result in different symme-

tries and hence affect significantly the underlying Kondo

physics. These effects are analyzed in the subsequent sec-

tions. Section III presents the analysis with two renor-

malization group (RG) approaches. In Sec. IV two slave-

boson approaches complement the previous results. Fi-

nally, Sec. V concludes the paper.

II.MODEL

A.Nearly Degenerate Localized Orbitals

We consider a QD with two (nearly) degenerate local-

ized orbitals which is coupled to reservoirs. As we men-

tioned before, we have in mind the experimental setup

of Ref. 9 where a highly symmetric Kondo effect was

demonstrated in a CNT QD. However, our description

could well apply to vertical quantum dot (VQD),14where

the orbitals correspond to two degenerate Fock-Darwin

states with different values of the angular momentum

quantum number. Hereafter we denote this orbital quan-

tum number by m = 1,2. The dot is then described by

the Hamiltonian

HD=

?

m=1,2

?

σ=↑,↓

ǫmσd†

mσdmσ

+

?

(m,σ)?=(m′,σ′)

Umm′nmσnm′σ′ ,(1a)

where ǫmσ is the single-particle energy level of the lo-

calized state with orbital m and spin σ, d†

the fermion creation (annihilation) operator of the state,

nmσ = d†

(m = 1,2) the intra-orbital Coulomb interaction, and

U12the inter-orbital Coulomb interaction. The effect of

the external magnetic field parallel to the symmetry axis

of the system is to lift the orbital and spin degeneracy

of the single-particle energy levels. We will denote them

by ∆orband ∆Z, respectively, so that the single-particle

energy levels ǫmσhave the form

mσ(dmσ)

mσdmσ the occupation number operator, Umm

ǫmσ= ǫ0+∆orb(δm,1−δm,2)+(∆Z/2)(δσ,↑−δσ,↓). (1b)

The precise values of the Coulomb interactions Umm′ de-

pend on the details of the system, but should be of the or-

der of the charging energy EC= e2/2C with C being the

total capacitance of the dot. In this work we focus on the

regime where the system of the localized levels is occupied

by a single electron (?

much bigger than other energy scales. In this regime the

Hamiltonian in Eq. (1a) suffices to describe all relevant

physics of our concern.

mσ?nmσ? ≈ 1, quarter filling15)

and the Coulomb interaction energy (Umm′ ∼ EC) is

B.Coupling Schemes

Kondo physics arises as a result of the interplay be-

tween strong correlations in the dot and coupling of the

localized electrons with the itinerant ones in conduction

bands. Naturally, different Kondo effects are observed

depending on the way the dot is coupled to the electrodes

and whether or not the orbital quantum number m is con-

served. Nevertheless, it turns out highly non-trivial ex-

perimentally to distinguish those different Kondo effects.

In subsequent sections we will consider different coupling

schemes between the dot and the electrodes, show how

different physics emerges, and propose how to distinguish

them unambiguously in experiments.

The two leads α = L and R are treated as non-

interacting gases of fermions:

?

where µ denotes channels in the leads. Without loss of

generality, we assume that there are two distinguished

(groups of) channels µ = 1 and 2 in each lead. When the

leads bears the same symmetry as the dot, this channel

quantum number µ in the leads is identical to the orbital

quantum number m in the dot and will be preserved over

Hα=

k

?

µ=1,2

?

σ

ǫαkµa†

αkµσaαkµσ,(1c)

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3

FIG. 1: (color online) Schematic of a representative meso-

scopic system in question. In (a) Each of the two leads, L

and R, has two conduction bands (or “modes”), 1 and 2.

The model with two leads in (a) is equivalent in equilibrium

to the model in (b) with only one lead. The operators ckµσ

(µ = 1,2 and σ =↑,↓) are related to aLkµσ and aRkµσ by

the canonical transformation in Eq. (4). The wiggly lines in-

dicate the inter-orbital Coulomb interaction U12 whereas the

intra-orbital interaction Umm (m = 1,2) is not shown.

the tunneling of electrons from the dot to leads and vice

versa; see Fig. 1(a). Otherwise, the orbital channels be-

come mixed. The most general situation is described by

the tunneling Hamiltonian

?

HT=

?

αkµσ

Vαkµmσa†

αkµσdmσ+ h.c.

?

(1d)

and the total Hamiltonian is thus given by H = HL+

HR+ HT+ HD.

For the sake of simplicity, we assume identical elec-

trodes (ǫLkµ= ǫRkµ) together with symmetric tunneling

junctions (VLkµmσ = VRkµmσ), and ignore their k- and

σ-dependence of the tunneling amplitudes. In this way,

we consider a simplified model with Vαkµmσ= Vµ,m/√2

and define the widths

Γm≡ Γmm,Γmm′ = πρ0V∗

mVm′

(2)

with Vm≡ Vm,m, where ρ0is the density of states (DOS)

in the reservoirs. Then in equilibrium the Hamiltonian

FIG. 2: (color online) Schematics of the SU(4)-symmetric

Anderson model.

H in Eq. (1) is equivalent to H = HC+ HT+ HDwith

?

HT=Vµ,mc†

HC=

kµσ

?

εkµc†

kµσckµσ,(3a)

kµmσ

?

kµσdmσ+ h.c.

?

,(3b)

where we have performed the canonical transformation

ckµσ=aLkµσ+ aRkµσ

√2

,

bkµσ=aLkµσ− aRkµσ

√2

,

(4)

and discarded the decoupled term ǫkµb†

In the following sections we investigate the physics de-

scribed by the Hamiltonian in Eq. (3) and, in partic-

ular, clarify the role of index conservation in the sym-

metry of the underlying Kondo regime at low tempera-

tures. In order to carry out this analysis, we use four dif-

ferent approaches: the scaling theory (perturbative RG

approach), the numerical renormalization group (NRG)

method, the slave-boson mean-field theory (SBMFT),

and the non-crossing approximation (NCA).

kµσbkµσ.

III.RENORMALIZATION GROUP

APPROACHES

The renormalization group (RG) theory provides a

convenient and powerful method to study low-energy

properties of strongly correlated electron systems. Here

we take two RG approaches, the scaling theory16,17,18

and the NRG method.19,20,21,22While the scaling the-

ory is useful for qualitative understanding of the model,

a more precise quantitative analysis requires the use of

more sophisticated methods like the NRG method. This

method is known to be one of the most accurate and pow-

erful theoretical tools to study quantum impurity prob-

lems (see Appendix A).

A.SU(4) Kondo Effect

We now turn to the case where tunneling processes

conserve the orbital quantum number; see Fig. 2. In this

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4

case, the Hamiltonian reads

H =

?

+

α=L,R

?

Vm

m=1,2

?

?

kσ

εαka†

αkmσaαkmσ

?

αkmσ

a†

αkmσdmσ+ d†

mσaαkmσ

?

+ HD

(5)

or [see Eqs. (3a) and (3b)]

H =

?

kmσ

ǫkc†

kσckmσ

+

?

kmσ

Vm

?

c†

kmσdmσ+ d†

mσckmσ

?

+ HD.(6)

From the RG point of view, starting initially with

nearly degenerate levels, all the localized levels are rel-

evant for the spin and orbital fluctuations, and, as we

shall see below, contribute to the Kondo effect. To inves-

tigate the low-energy properties of the orbital and spin

fluctuations of the model, we perform the Schrieffer-Wolf

(SW) transformation and obtain an effective Kondo-type

Hamiltonian:

?

?√J1−√J2

2

+ (J1− J2)(s · S)(tz+ Tz),

where

H =

kmσ

ǫkc†

kmσckmσ+ HSU(4)

eff

− ∆ZSz− 2∆orbTz

−

?2

(1 + 4s · S)(txTx+ tyTy)

(7)

HSU(4)

eff

=J1+ J2

2

[s · S + t · T + 4(s · S)(t · T)](8)

and the exchange coupling constants Jm(m = 1,2) are

given by

?

We note that the Kondo-type effective Hamiltonian in

Eq. (7) reduces to the SU(4)-symmetric Kondo model

when V1= V2and ǫ1σ= ǫ2σ. In this case, orbitals play

exactly the same role as spins; the former are not distin-

guished from the latter.

Under the RG transformation reducing subsequently

the conduction band width D by δD, the Kondo-type

effective Hamiltonian evolves into a generic form:

Jm= V2

m

1

E+

+

1

E−

?

. (9)

Heff= Hleads− ∆ZSz− 2∆orbTz

+ 2J1(s · S)

?1

?1

2+ tz

??1

??1

2+ Tz

?

?

+ 2J2(s · S)

+1

2[J4+ 4J3(s · S)](t+T−+ t−T+)

2− tz

2− Tz

+ J5tzTz.(10)

The level splitting ∆orband ∆Z remain constant under

the RG transformation:

d∆Z

dlnD=d∆orb

dlnD= 0.(11)

The exchange coupling constants Ji (i = 1,··· ,5) are

initially given by Eq. (9) and

J3= J4=

?

J1J2,J5=1

2(J1+ J2),(12)

which, under the RG transformation, scale as

dJ1

ρ0dlnD= −2J2

dJ2

ρ0dlnD= −2J2

dJ3

ρ0dlnD= −J3(J1+ J2+ J5) −1

dJ4

ρ0dlnD= −3

dJ5

ρ0dlnD= −3J2

1− J3(J3+ J4),(13a)

2− J3(J3+ J4),(13b)

2J4(J1+ J2),(13c)

2J3(J1+ J2) − J4J5,(13d)

3− J2

4

(13e)

for D ≫ ∆orb ≥ ∆Z. For D ≪ ∆orb, it is clear from

Eq. (10) that the orbital fluctuations are frozen and only

J1is relevant, which scales as

dJ1

ρ0dlnD= −2J2

1.(14)

It implies that we recoverthe single-level Anderson model

for D ≪ ∆orb. Therefore in the remainder of this section,

we will focus on the case D ≫ ∆orb.

It is convenient to define the reduced variables ji ≡

Ji/J1(i = 2,··· ,5) and rewrite the RG equations (13)

as

dj2

dx= −j2+2j2

dj3

dx= −j3+j3(1 + j2+ j5) + (j4/2)(1 + j2)

2 + j3(j3+ j4)

dj4

dx= −j4+(3j3/2)(1 + j2) + j4j5

2 + j3(j3+ j4)

dj5

dx= −j5+

2 + j3(j3+ j4)

2+ j3(j3+ j4)

2 + j3(j3+ j4),(15a)

,(15b)

,(15c)

3j2

3+ j2

4

(15d)

with x = ln(ρ0J1), while J1obeys the scaling equation

1

(ρ0J1)2

d(ρ0J1)

dlnD

= −2 − j3(j3+ j4).(16)

The RG equations (15) have two fixed points: one de-

scribing the SU(4) Kondo physics

j2= j3= j4= j5= 1(17)

and the other describing the usual SU(2) Kondo physics

j2= j3= j4= j5= 0,(18)

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5

−5−4−3

log(ρ0J1)

−2−10

0

0.2

0.4

0.6

0.8

1

1 − J2/J1

Γ2/Γ1= 0

Γ2/Γ1= 1

0

0.2

0.4

0.6

0.8

1

0 2040

∞

log(ρ0J1)

1 − J2/J1

Γ2/Γ1= 0.5

Γ2/Γ1= 0.6

Γ2/Γ1= 0.7

Γ2/Γ1= 0

SU(2)

Γ2/Γ1= 1

SU(4)

FIG. 3: (color online) RG flows for different values of Γ2/Γ1

with Γ1 fixed.

both with J1= ∞ as indicated in Fig. 3. Linearizing the

RG equations (15) around the fixed points, one can easily

show that both the SU(2) and SU(4) Kondo fixed points

are stable (there is one marginal parameter at the SU(4)

fixed point). However, as indicated as a dashed semicircle

in Fig. 3(b), the radius of convergence is finite while the

fixed point itself is located at infinity. This implies that in

priciple, the SU(4) Kondo fixed point cannot be reached

for arbitrarily small values of 1 − Γ2/Γ1. However, as

illustrated in Fig. 3(a), in the region of physical interest

for sufficiently small values of 1 − Γ2/Γ1, the scaling be-

havior is essentially governed by the SU(4) Kondo fixed

point (see also Fig. 5). More importantly, for sufficiently

small values of 1 − Γ2/Γ1, the SU(2) fixed point governs

the physics only at extremely low energies. This suggests

that the SU(4) Kondo signature can be observed exclu-

sively at relatively higher energy scales (of the order of

the Kondo temperature), as in the experiment reported

recently.9

At B?= 0 and Γ1= Γ2≡ Γ0, the RG equations (13)

reduce to a single equation

dJ1

ρ0dlnD= −4J2

1.(19)

-6

-4-2024

ω / Γ0

0

0.2

0.4

0.6

0.8

1

πΓ0 Atot(ω)

SU(2)

SU(4)

FIG. 4: Comparison of the SU(2) and SU(4) Kondo model.

Comparing this with the corresponding one in Eq. (14)

for the usual single-level Anderson model, we note that

the Kondo temperature is enhanced exponentially:

TSU(4)

K

∼ exp(−1/4ρ0J1)(20)

with respect to the SU(2) Kondo temperature

TSU(2)

K

∼ exp(−1/2ρ0J1). (21)

The perturbative RG analysis discussed above, whose va-

lidity is guaranteed only for ρ0Ji ≪ 1, turns out to be

qualitatively correct in a wide region of the parameter

space and provides a clear interpretation of the model.

To confirm the perturbative RG analysis and make quan-

titative analysis, we use the NRG method (described in

Appendix A), the results of which are summarized in

Figs. 4–5. There the total spectral density

?

which provides direct information on the linear conduc-

tance23, is plotted.

The spectral density shows a peak near the Fermi en-

ergy, corresponding to the formation of the SU(4) Kondo

state (see Fig. 4). The peak width, which is much broader

than that for the SU(2) Kondo model (represented by the

dotted line), demonstrates the exponential enhancement

of the Kondo temperature mentioned above. Another re-

markable effect is that the SU(4) Kondo peak shifts away

from ω = EF = 0 and is pinned at ω ≈ TSU(4)

be understood from the Friedel sum rule,24which in this

case gives δ = π/4 for the scattering phase shift at EF.

Accordingly, the linear conductance at zero temperature

is given by G0= 4(e2/h)sin2δ = 2e2/h. It is interesting

to recall that the Friedel sum rule gives the same linear

conductance also for the two-level SU(2) Kondo model.

Thus, neither the enhancement of the Kondo tempera-

ture nor the linear conductance can distinguish between

the SU(4) and the two-level SU(2) Kondo effects. This

Ad(E) =

σ

?

mm′

πΓmm′Am′m;σ(E),(22)

K

. This can

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6

-0.2-0.10 0.10.2 0.3

ω / Γ0

0

1

2

3

Ad(ω)

Γ2 / Γ1 = 1.0

0.9

0.8

0.7

0.6

0.5

(a)

-0.2-0.1 0 0.10.20.3

ω / Γ1

0

1

2

3

Ad(ω)

Γ2 / Γ1 = 1.00

0.99

0.98

0.97

0.96

0.95

(b)

FIG. 5:

density Ad(E) for different values of coupling asymmetry

Γ2/Γ1. The parameter values are: ǫ0 = −0.8D, Γ1 = 0.1D,

Umm′ = 8D, and ∆orb= ∆Z = 0.

(color online) NRG results of the total spectral

FIG. 6: (color online) Model with finite mixing between or-

bital quantum numbers.

can be achieved only by studying the influence of a paral-

lel magnetic field in the nonlinear conductance, as shown

in Ref. 9

B. Effects of Mixing of Orbital Quantum Numbers

To examine the stability of the SU(4) Kondo phenom-

ena against orbital mixing, we consider the model (see

Fig. 6):

H =

?

kσ

εkc†

kσckσ+

?

k ¯ mσdmσ+ d†

kmσ

V0

?

c†

kmσdmσ+ d†

mσckmσ

?

+

?

kmσ

VX

?

c†

mσck ¯ mσ

?

+ HD,(23)

where the indices imply¯1 = 2 and¯2 = 1 and V0≡ V1,1=

V2,2and VX≡ V1,2= V2,1.

If we rewrite the Hamiltonian in the form

?

+ d†

mσ{V0ckmσ+ VXck ¯ mσ} + HD,

it now becomes clear that, in the pseudospin language,

a finite amount of orbital mixing corresponds to having

non-collinear leads with respect to the orbital “magne-

tization” axis which defines the pseudospin orientations

m = 1 and m = 2 in the dot. In other words, each

confined electron (with defined pseudospin) couples to a

linear combination of pseudospins and, as a result, be-

comes rotated in the pseudospin space by the angle de-

fined by tanφ = VX/V0.

mixing VX= V0, the tunneling electrons lose completely

information about their pseudospin orientation. In this

limit, one recovers the spin Kondo physics [with SU(2)

symmetry] of a two-level Anderson model [see the next

subsection and Eq. (36)]. For zero mixing (VX= 0), the

model reduces to the SU(4)-symmetric model of Eq. (6)

(with tunneling amplitudes which do not depend on the

orbital index).

After the RG transformation of the Anderson-type

model in Eq. (23) until the single-particle energy levels

are comparable with the conduction band width (when

the charge fluctuations are suppressed), the SW trans-

formation gives

H =

kσ

εkc†

kσckσ+

?

kmσ

{V0c†

kmσ+ VXc†

k ¯ mσ}dmσ

(24)

Note that for the maximal

Heff=

?

1 −JX

?JX

J0

?

?

HSU(4)

eff

?JX

+JX

?

J0HSU(2)

eff

+ J0

J0

1 −

J0

(1 + 4s · S)(tx+ Tx)

+ 2JX(txTx),(25)

where

HSU(4)

eff

= J0[s · S + t · T + 4(s · S)(t · T)]

corresponds to the SU(4) Kondo model and

(26)

HSU(2)

eff

= 2J0s·S(1+2tx)(1+2Tx)+J0(tx+Tx). (27)

the SU(2) Kondo model. The exchange coupling con-

stants J0and JX, respectively, are given by

?

J0= |V0|2

1

E+

+

1

E−

?

,JX= |VX|2

?

1

E+

+

1

E−

?

(28)

.

One can already grasp an idea about the effects of the

mixing JX(i.e., ΓX) of the orbital quantum numbers by

considering the two limiting cases, JX = 0 (no mixing)

and JX= J0(maximal mixing), of the effective Hamilto-

nian (25). In the case of no mixing (JX= 0), the effective

Hamiltonian (25) reduces to the SU(4)-symmetric Kondo

model in Eq. (26), which has been already discussed in

the previous section: The Kondo temperature is given

by TK∼ Dexp(−1/4J0). When the mixing is maximal

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FIG. 7: (color online) RG flows in case that there is a finite

amount of mixing of the orbital quantum numbers.

(JX= J0), on the other hand, the effective Hamiltonian

becomes HSU(2)

eff

in Eq. (27).

Under the RG procedure, the effective Hamiltonian

(25) transforms to the general form

Heff= J1s · S + [J3(tzTz) + 4J2(s · S)(tzTz)]

+ [J4(txTx+ tyTy) + 4J6(s · S)(txTx+ tyTy)]

+ [J5(txTx− tyTy) + 4J7(s · S)(txTx− tyTy)]

+ [J9+ 4J8(s · S)(tx+ Tx)] , (29)

where the exchange coupling constants are initially given

by

J1= J0+ JX,J2= J3= J0− Jx,

J4= J6= J0,J5= J7= JX,

J9= J8=

?

J0JX.(30)

Under the RG transformation, they scale as

dJ1

ρ0dlnD= −J2

dJ2

ρ0dlnD= −2J1J2− 2J4J6+ 2J5J7,

dJ3

ρ0dlnD= −J2

dJ4

ρ0dlnD= −J3J4− 3J2J6,

dJ5

ρ0dlnD= J3J5+ 3J2J7,

dJ6

ρ0dlnD= −J2J4− 2J1J6− J3J6− 4J2

dJ7

ρ0dlnD= J2J5− 2J1J7+ J3J7− 4J2

dJ8

ρ0dlnD= −2J1J8− 2J6J8− 2J7J8,

1− J2

2− 2J2

6− 2J2

7− 8J2

8,(31a)

(31b)

4+ J2

5− 3J2

6+ 3J2

7, (31c)

(31d)

(31e)

8,(31f)

8,(31g)

(31h)

and

dJ9

dlnD= 0.(32)

As before [see Eq. (15)], it is convenient to work with the

reduced coupling constants ji≡ Ji/J1. In terms of these

reduced constants, the RG equations read

dj2

dx= −j2+

dj3

dx= −j3+

dj4

dx= −j4+

dj5

dx= −j5−

dj6

dx= −j6+

dj7

dx= −j7−

dj8

dx= −j8+

2j2+ 2j4j6− 2j5j7

1 + j2

j2

1 + j2

j3j4+ 3j2j6

1 + j2

j3j5+ 3j2j7

1 + j2

j2j4+ 2j6+ j3j6+ 4j2

1 + j2

j2j5− 2j7+ j3j7− 4j2

1 + j2

2j8+ 2j6j8+ 2j7j8

1 + j2

2+ 2j2

4− j2

2+ 2j2

6+ 2j2

5+ 3j2

6+ 2j2

7+ 8j2

6− 3j2

7+ 8j2

8

,(33a)

7

8

,(33b)

2+ 2j2

6+ 2j2

7+ 8j2

8

,(33c)

2+ 2j2

6+ 2j2

7+ 8j2

8

,(33d)

8

2+ 2j2

6+ 2j2

7+ 8j2

8

,(33e)

8

2+ 2j2

6+ 2j2

7+ 8j2

8

,(33f)

2+ 2j2

6+ 2j2

7+ 8j2

8

(33g)

together with

1

(ρ0J1)2

d(ρ0J1)

dlnD

= −(1 + j2

2+ 2j2

6+ 2j2

7+ 8j2

8).(34)

The RG equations (33) again have two fixed points,

one associated with the SU(2) Kondo effect and the other

with the SU(4) Kondo effect; see Fig. 7. The RG flow

diagram in Fig. 7 is reminiscent of that in Fig. 3. Both

fixed points are stable. However, since the radius of con-

vergence of the SU(4) Kondo fixed point is finite, the

SU(4) Kondo fixed point cannot be reachable even for

arbitrarily small mixing VX [see Fig. 7(b)]. However, in

Page 8

8

-0.4 -0.20 0.20.4

0.6

ω / Γ0

0

0.2

0.4

0.6

0.8

1

Ad(ω)

ΓX / Γ0 = 0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

(a)

-0.4 -0.200.20.4

0.6

ω / Γ0

0

0.1

0.2

Ad(ω)

(b)

0.01

0.02

0.03

0.04

0.05

ΓX / Γ0 = 0

FIG. 8: NRG results for the effects of the finite mixing of

the orbital quantum number m. The parameter values are:

ǫ0 = −0.8D, Γ0 = 0.08D, Umm′ = 8D, ∆orb= 32TSU(4)

∆Z = 2TSU(4)

K

.

K

, and

the region of interest, the physics is essentially governed

by the SU(4) Kondo fixed point for sufficiently small VX

[see Fig. 7(a)]. Therefore, the SU(4) Kondo physics are

in principle unstable against both the orbital quantum

number anisotropy 1−Γ2/Γ1and the orbital mixing ΓX.

For sufficiently small values of those, however, the SU(4)

Kondo physics still determines the transport properties

except at extremely low energy scales. As addressed al-

ready, this suggests that to observe indications of the

SU(4) Kondo physics exclusively, one has to investigate

the properties at relatively higher energies (of the or-

der of the Kondo temperature). This is confirmed and

demonstrated in the NRG results summarized in Fig. 8.

We will also see below that there is no way to distinguish

the two-level SU(2) Kondo physics and the SU(4) Kondo

physics experimentally by means of linear conductance.

C.Two-Level SU(2) Kondo Effect

As pointed out in the previous subsection, at the max-

imum mixing (V0 = VX) the physics becomes that of

the two-level SU(2) Anderson model (see Fig. 9). In this

case, the only degree of freedom which is conserved dur-

FIG. 9:

symmetric Anderson model.

(color online) Schematic of the two-level SU(2)-

ing tunneling is the spin and the total Hamiltonian reads

H =

?

+

α=L,R

?

?

kσ

εαka†

αkσaαkσ

αkmσ

Vm

?

a†

αkσdmσ+ d†

mσaαkσ

?

+ HD

(35)

or equivalently [see Eq. (4)]

H =

?

kσ

εkc†

kσckσ

+

?

kmσ

Vm

?

c†

kσdmσ+ d†

mσckσ

?

+ HD.(36)

As the scaling theory of the Kondo-type Hamiltonian

obtained from the two-level Anderson model has been

developed in detail in Refs. 25 and 26, we here focus on

the first stage, which highlights the difference between

the two-level SU(2)-symmetric Anderson model and the

SU(4)-symmetric Anderson model. Finally, the physical

arguments based on the perturbative RG theory will be

examined quantitatively by means of the NRG method.

As we integrate out the electronic states in the ranges

[−D,−(D−δD)] and [D−δD,D] of the conduction band,

the dot Hamiltonian (1a) evolves as

HD=

?

mσ

ǫmσd†

mσdmσ− t

?

σ

?

d†

1σd2σ+ d†

2σd1σ

?

+

?

(m,σ)?=(m′,σ′)

Umm′nmσnm′σ′

(37)

with other terms in the total Hamiltonian (36) kept un-

changed. Notice here the appearance of a new term in

t, i.e., a direct transition between the two orbitals m = 1

and 2. Scaling of the parameters ǫmσand t are governed

by the RG equations

dǫmσ

dlnD= −2

πΓm

(38)

and

dt

dlnD= −2

π

?

Γ1Γ2,(39)

respectively.

Page 9

9

012345

−4

−2

0

2

4

D/Γ

ǫd/Γ

ǫ∗

d≪ −Γ

|ǫ∗

d| ≪ Γ

ǫ∗

d≫ +Γ

FIG. 10: (color online) Scaling of the single-particle energy

level ǫd, to be compared with ǫmσ in Eq. (38), of the single-

level Anderson model. ǫ∗

quantity.

d= ǫd(D = Γ) is a scale-invariant

The RG equation (38) for the single-particle energy

levels ǫmσis the same as that in the usual single-level An-

derson model17,18(the corresponding RG flow diagram is

shown in Fig. 10). However, due to the direct transition

t emerging from the RG equation (39), ǫmσare not rele-

vant to the Kondo effect [they are not the eigenvalues of

HDin Eq. (37)]. To find the relevant energy level(s) di-

rectly involved in the Kondo effect, one may diagonalize

HDin Eq. (37) by means of the canonical transformation

?d+,σ

d−,σ

?

=

?

cos(θ/2) sin(θ/2)

−sin(θ/2) cos(θ/2)

??d1σ

d2σ

?

,(40)

where the angle φ is defined by the relation

tanθ ≡

t

ǫ2− ǫ1.(41)

The dot Hamiltonian in Eq. (37) then takes the form

HD=

?

µ=±

?

σ

ǫµd†

µσdµσ+

?

(m,σ)?=(m′,σ′)

Umm′nmσnm′σ′

(42)

with

ǫµ=±=1

2(ǫ1+ ǫ2) ∓1

2

?

(ǫ1− ǫ2)2+ t2.(43)

At the same time, the canonical transformation in

Eq. (40) also changes the coupling term in the total

Hamiltonian in Eq. (36) to

HT =

?

µ=±

?

kσ

Vµ

?

c†

kσdµσ+ d†

µσckσ

?

(44)

02468 10

−10

−8

−6

−4

−2

0

D/Γ0

ǫd/Γ0

ǫ+

ǫ−

Γ+≈ 2Γ0

0246810

−10

−8

−6

−4

−2

0

D/Γ0

ǫd/Γ0

ǫ+

ǫ−

Γ+≈ Γ0

FIG. 11:

symmetric Anderson model. The arrowed lines indicate RG

flows of the effective single-particle energy levels ǫ± [see

Eq. (43)] and the widths of the shadowed regions around ǫ±

the RG flow of Γ± [see Eq. (46)]. The mean of Γ1 and Γ2 is

denoted by Γ0.

(color online) Scaling of the two-level SU(2)-

with V±defined by

?V+

Accordingly, the tunneling rates Γ± ≡ πρ0|V±|2of the

effective orbital levels ǫ±,σare given by

V−

?

=

?

cos(θ/2) sin(θ/2)

−sin(θ/2) cos(θ/2)

??V1

V2

?

.(45)

Γ±=1

2(Γ1+Γ2)±

?

Γ1Γ2sinθ+1

2(Γ1−Γ2)cosθ. (46)

Figure 11 shows the scaling of ǫ±(arrowed lines) and

Γ±(widths of the shadowed regions around ǫ±), governed

by Eqs. (38), (39), (43), and (46). Note that the effective

single-particle energy levels ǫ±always repel each other,27

and the hybridization Γ+(Γ−) of the lower (upper) level

ǫ+ (ǫ−) always increases (decreases). Essential in this

scaling property of the two-level Anderson model is the

direct transition t between the orbitals m = 1 and 2,

mediated by the conduction band.

The scaling of ǫ± and Γ± stops when the lower level

ǫ+ becomes comparable with D (ǫ+≃ D); see Fig. 11.

Then the charge fluctuations are highly suppressed and

the occupation of the lower level becomes close to unity

(?n+? ≈ 1). Therefore, only the lower level ǫ+ gets in-

volved in the Kondo physics, and hence the resulting

Page 10

10

Kondo effect is identical to the usual SU(2) Kondo ef-

fect. To be more specific, let us consider the two limiting

cases, |ǫ1− ǫ2| ≫ Γ0and |ǫ1− ǫ2| ≪ Γ0, assuming

|Γ1− Γ2| ≪ Γ0≡ (Γ1+ Γ2)/2. (47)

Since t ∼ Γ0, one has

θ ≈

?

π/2,

0,

|ǫ1− ǫ2| ≪ Γ0;

|ǫ1− ǫ2| ≫ Γ0,

(48)

or equivalently,

Γ+≈

?

2Γ0,

Γ0,

|ǫ1− ǫ2| ≪ Γ0;

|ǫ1− ǫ2| ≫ Γ0.

(49)

This implies that when the two orbital levels are nearly

degenerate (|ǫ1−ǫ2| ≪ Γ0), the Kondo temperature17,18

is enhanced exponentially:

TK≃1

2

?

2Γ0Dexp

?

+πǫ0

2Γ0

?

, (50)

[with ǫ0 ≡ (ǫ1+ ǫ2)/2] compared with the single-level

case (i.e., |ǫ1− ǫ2| ≫ Γ0)

?

In the limit of nearly degenerate levels (|ǫ1−ǫ2| ≪ Γ0),

the upper level ǫ−is located at distance smaller than Γ+

from the lower level ǫ+ [(ǫ−− ǫ+) ? Γ+; see the up-

per panel in Fig. 11] and the transition from ǫ+ to ǫ−

is allowed in general. Indeed, this effect can be taken

into account by a proper SW transformation includ-

ing both levels and scaling of the resulting Kondo-like

Hamiltonian,25,26and gives rise to a bump structure at

ω = ∆eff above the Fermi energy EF of the leads, with

∆eff given by27(with ǫ1σ= ǫ2σinitially)

T0

K≃1

2

Γ0Dexp

?

+πǫ0

Γ0

?

. (51)

∆eff∼2Γ0

π

lnD

Γ0

(52)

in the single-particle excitation spectrum Ad(ω) in

Fig. 12; see below.

Again, all the interpretations made above on the ba-

sis of the perturbative RG are confirmed with the NRG

method.Figure 12 shows the total spectral density

Ad(E). One can see that as ∆orbincreases with ∆Z= 0,

the Kondo peak gets sharper, i.e., the enhancement of

the Kondo temperature TK in Eq. (50) diminishes for

∆orb≥ Γ0; see Fig. 12(a). Notice that the bump above

the Fermi energy originates from the excitation via the

transition from the lower level ǫ+to the higher one ǫ−,

and is thus located at E = ∆eff [see Eq. (52]. When we

allow ∆Z finite as well, the Kondo peak then splits into

two because of the Zeeman splitting.28,29

-0.4-0.20 0.20.4

0.6

0.81

E/Γ0

0

1

2

Ad(E)

∆orb = 0.5Γ0

∆orb = Γ0

∆orb = 2Γ0

∆orb = 0

(a)

∆Z = 0

-0.4 -0.20 0.20.4

0.6

0.81

E / Γ0

0

1

2

Ad(E)

∆orb = Γ0 / 2

∆Z = 0

∆Z = TK

∆Z = 2TK

∆Z = 10TK

(b)

FIG. 12: (color online) Total single-particle excitation spec-

trum Ad(ω) with (a) only the orbital degeneracy lifted

(∆orb ?= 0, ∆Z = 0) and (b) both the orbital and spin de-

generacies lifted (∆orb,∆Z ?= 0). The short vertical arrows

indicate the transition from ǫ+ to ǫ−, whose excitation en-

ergy is given by ∆eff [see Eq. (52)]. The parameter values

are: ǫ0 = −0.8D, Γ0 = 0.1D, and Umm′ = 8D.

IV.SLAVE-BOSON TREATMENT

In order to confirm our previous results and obtain

analytical expressions for intermediate mixing, we also

use slave boson techniques.

approach, which provides a good approximation in the

strong coupling limit T ≪ TK, allows us to obtain an-

alytical expressions for the Kondo temperature and the

Kondo peak position for arbitrary mixing. Our SBMFT

results are complemented with the NCA, which takes into

account both thermal and charge fluctuations in a self-

consistent manner.

In particular, the SBMF

At equilibrium it is convenient to change into a rep-

resentation in terms of the symmetric (even) and anti-

symmetric (odd) combinations of the localized and de-

localized orbital channels.30Thus the even-odd trans-

formation consists of ak,1(2),σ = (ckeσ± ickoσ)√2 and

d1(2)σ= (deσ± idoσ)/√2. In this basis the Hamiltonian

Page 11

11

in Eq. (3) reads

H =

?

Unν↓nν↑+ Uneno+ Ve

σ,ν=e,o

?

ǫkνc†

kν,σckν,σ+

?

σ,ν=e,o

ενσd†

ν,σdν,σ

+

ν=e,o

?

c†

ko,σdoσ+ h.c.

ke,σ

?

c†

ke,σdeσ+ h.c

?

(53)+ Vo

?

ko,σ

??

,

where, again we have taken V0 = V1,1 = V2,2, VX =

V1,2 = V2,1, Um,m′ = U, and εν,σ = ε0,σ. The occu-

pation per channel and spin is given by nνσ = d†

and the total occupation per channel is nν =?

Ve ≡ V0+ VX and Vo ≡ V0− VX.

malize the total tunneling rate, we take, for the diago-

nal and off-diagonal tunneling amplitudes, V0= V cosφ

and VX = V sinφ, namely Ve = V (cosφ + sinφ) and

νσdνσ

σnνσ.

The tunneling amplitudes for each channel are given by

In order to nor-

Vo= V (cosφ−sinφ). Notice that one needs φ ∈ [0,π/4]

in order to have always Vopositively defined. There ex-

ist two very different situations, namely (i) φ = 0, where

there are only tunneling processes that conserve the or-

bital index, and (ii) φ = π/4, where the mixing and no

mixing tunneling amplitudes are the same.

Now we write the physical fermionic operator as a

combination of a pseudofermion operator and a boson

operator as follows: dνσ = b†fνσ, where the pseud-

ofermion operator fν,σ annihilates one “occupied state”

in the νth localized orbital and the boson operator b†cre-

ates an “empty state”. Quite generally, the intra-/inter-

Coulomb interaction is very large and we can safely take

the limit U → ∞.

?

bital or different orbitals. This constraint is treated with

a Lagrange multiplier.

This fact enforces the constraint

νσfνσ+ b†b = 1 that prevents the accommodation

of two electrons at the same time in either the same or-

νσf†

HSB=

?

σ,ν=e,o

ǫkνc†

kν,σckν,σ+

?

σ,ν=e,o

ε0,σf†

ν,σfν,σ+Vν

√N

?

k,σ,ν=e,o

?

c†

kν,σb†fν,σ+ h.c.

?

??

+ λ

ν,σ

f†

ν,σfν,σ+ b†b − 1

?

. (54)

Notice that we have rescaled the tunneling amplitudes

Ve(o) →

expansion (where N is the total degeneracy of the lo-

calized orbital).

Our next task is to solve this Hamiltonian, which is

rather complicated due to the presence of the three op-

erators in the tunneling part and the constrain. In order

to do this we employ two approaches that describe two

different physical regimes. The first one is the SBMFT

approach which describes properly the low-temperature

strong coupling regime. This SBMFT provides a good

approximation in the deep Kondo limit, namely in case

that only spin fluctuations are taken into account. The

NCA, on the other hand, takes into account both ther-

mal and charge fluctuations in a self-consistent manner.

It is well known that the NCA fails in describing the low-

energy strong coupling regime. Nevertheless, the NCA

has proven to give reliable results at temperatures down

to a fraction of TK.

√NVe(o) according to the spirit of a 1/N-

A. Slave-boson Mean-Field Theory

We begin with the discussion of the mean-field approx-

imation of Eq. (54). The merit of this approach is its

simplicity while capturing the main physics in the pure

Kondo regime. It has been successfully applied to inves-

tigate the out-of-equilibrium Kondo effect31,32,33,34and

double quantum dots,35,36,37,38,39just to mention a few.

This approach corresponds to taking the lowest order

O(1) in the 1/N expansion, where the boson operator

b(t) is replaced by its classical (nonfluctuating) average,

i.e., b(t)/√N → ?b?/√N ≡˜b, thereby neglecting charge

fluctuations. In the limit N → ∞, this approximation

becomes exact. The corresponding mean-field Hamilto-

nian is given by

?

+

ν,k,σfν,σ+ h.c.

HMF=

σ,ν=e,o

ǫkνc†

kν,σckν,σ+

?

ν,σ

˜ ε0,σf†

ν,σfν,σ

?

ν,k,σ

?˜Vνc†

??

?

+ λ

ν,σ

f†

ν,σfν,σ+ N|˜b|2− 1

?

,(55)

where˜Vν = Vν˜b = Vν?b? and ˜ ε0,σ = ε0,σ+ λ are the

renormalized tunneling amplitude and the renormalized

orbital level, respectively. The two mean-field parameters

˜b and λ are to be determined through the mean-field

equations, which are the constraint

?

ν,σ

?f†

ν,σ(t)fν,σ(t)? + N|˜b|2= 1(56)

Page 12

12

and the equation of motion for the boson field

?

ν,k,σ

˜Vν

?

c†

kν,σ(t)fν,σ(t)

?

+ λN|˜b|2= 0. (57)

The Green function for the ν (= e,o) localized orbital

and the corresponding lesser lead-orbital Green func-

tion are given by G<

and G<

Expressing the mean-field equations in terms of these

nonequilibrium Green functions, we obtain Eqs. (56) and

(57) in the form

ν,σ(t − t′) = −i?f†

kν,σ(t′)fν,σ(t)?, respectively.

ν,σ(t′)fν,σ(t)?

ν,σ,kν,σ(t − t′) = −i?c†

?

?

σ

G<

ν,σ(t,t) + N|˜b|2= 1, (58a)

ν,k,σ

˜VνG<

ν,σ;kν,σ(t,t) + λN|˜b|2= 0. (58b)

In order to solve the set of mean-field equations, we

proceed as follows: First, we employ analytic continua-

tion rules to the equation of motion for the time-ordered

Green functions Gt

Gt

notes the time-ordering operator along a complex time

contour.40Second, we use the equation of motion tech-

nique to relate the lead-orbital Green function with the

orbital Green function. Finally, we rewrite the mean-field

equations in the frequency domain (taking ε0,σ= ε0):

ν,σ(t−t′) = −i?TC{f†

kν,σ(t′)fν,σ(t)}?, where TCde-

νσ(t′)fνσ(t)}? and

ν,σ;kν,σ(t−t′) = −i?TC{c†

|˜b2| −1

N

?

?

ν,σ

?

dǫ

2πiG<

νσ(ǫ) =

1

N,

(59a)

λ|˜b|2+1

N

ν,σ

?

dǫ

2πiG<

νσ(ǫ)(ε0− ˜ ε0) = 0. (59b)

The integrals in Eq. (59) can be carried out analytically

by introducing a Lorentzian cutoff ρ(ǫ) = D(ǫ2+ D2)−1

for the DOS in the leads and the lesser orbital Green

function G<

Γν|˜b|2and the Fermi distribution function f(ǫ):

?

D

1

N− |˜b|2,

2˜Γe

πNRe

D

ν(ǫ) = 2i˜Γνf(ǫ)/[(ǫ − ˜ ε0)2+˜Γ2

ν] with˜Γν =

2

πNIm ln

?

˜ ε0+ i˜Γe

??

+

2

πNIm

?

ln

?

˜ ε0+ i˜Γo

D

??

=

(60a)

??

?

ln

?

˜ ε0+ i˜Γe

??

+2˜Γo

πNRe

?

ln

?

˜ ε0+ i˜Γo

D

= −λ|˜b|2. (60b)

In the deep Kondo limit, where N−1− |˜b|2≈ N−1and

−λ ≈ ε0, these equations obtain the forms:

?

D

=π

2,

?

D

=πε0

2

Im ln

?

˜ ε0+ i˜Γe

??

+ Im

?

ln

?

˜ ε0+ i˜Γo

D

??

(61a)

??

ΓeRe ln

?

˜ ε0+ i˜Γe

??

+ ΓoRe

?

ln

?

˜ ε0+ i˜Γo

D

,(61b)

where Γν = Γν/N is the original rate for the ν (= e,o)

channel. Using the parametrization Ve= V (cosφ+sinφ)

and Vo= V (cosφ−sinφ), the tunneling rates read Γe=

πV2ρ(1+sin2φ) = Γ(1+sin2φ) and Γo= ρ = πV2ρ(1−

sin2φ) = Γ(1 − sin2φ). Taking sin2φ = β with β ∈

[0,1] (notice that 0 ≤ sin2φ ≤ 1 for φ ∈ [0,π/4]), we

parametrize the even and odd rates as Γe= (1+β)Γ and

Γo= (1 − β)Γ, respectively. Accordingly, the case β ?= 0

accounts for the process where even and odd channels are

not coupled equally to the lead electrons or equivalently,

the process where the orbital index is not conserved. In

terms of the new notation, the mean-field equations can

be written in a compact way:

ln

?

˜ ε0+ i˜Γe

D

?

+ ln

?

˜ ε0+ i˜Γo

D

?

+ ln

?

˜ ε2

˜ ε2

0+˜Γ2

0+˜Γ2

πε0

Γe+ Γo,

e

o

?β/2

= iπ

2+

(62)

or equivalently,

[˜ ε0+i˜Γe][˜ ε0+i˜Γo]

?

˜ ε2

˜ ε2

0+˜Γ2

0+˜Γ2

e

o

?β/2

= iD2eπε0(Γe+Γo)−1,

(63)

the real and imaginary parts of which read

[˜ ε2

0−˜Γe˜Γo]

?

?

˜ ε2

˜ ε2

0+˜Γ2

0+˜Γ2

e

o

?β/2

?β/2

= 0, (64a)

˜ ε0(˜Γe+˜Γo)

˜ ε2

˜ ε2

0+˜Γ2

0+˜Γ2

e

o

= D2eπε0(Γe+Γo)−1.(64b)

It is obvious that Eq. (64a) has the solution ˜ ε0 =

±

Eq. (64b). Substituting this result in Eq. (64b), we ar-

rive after some algebra at

?˜Γe˜Γo, among which only the positive root satisfies

|˜b|2=

D

√2

1

NΓ

(1 − β)

(1 + β)

β−1

4

β+1

4

e(πε0/2)(Γe+Γo)−1.(65)

Using the previous result, we may define the Kondo tem-

Page 13

13

perature for each channel as:41

Te

K≡

?

?

˜ ε2

0+˜Γ2

e=(1 − β)

(1 + β)

β−1

4

β−1

4

De(πε0/2)(Γe+Γo)−1,

To

K≡˜ ε2

0+˜Γ2

o=(1 − β)

(1 + β)

β+1

4

β+1

4

De(πε0/2)(Γe+Γo)−1, (66)

and obtain the renormalized level position:

˜ ε0=

D

√2e(πε0/2)(Γe+Γo)−1 (1 − β)

β+1

4

(1 + β)

β−1

4

.(67)

Equations (66) and (67), which are the main results of

this section, give the Kondo temperatures and level posi-

tion for arbitrary mixing β. Note that Γe+Γo= 2Γ does

not depend on β and therefore the Kondo temperature

depends on the orbital mixing only through the prefac-

tor. While Te

down to zero as β → 1 (maximum mixing). Similarly, ˜ ε0

goes from TK ≡ NΓ|˜b|2(β = 0) = (D/√2)exp(πε0/4Γ)

to zero, in agreement with the Friedel sum rule. From

the above results, we conclude that the odd orbital be-

comes decoupled at maximum mixing, where we are left

with SU(2) Kondo physics arising from spin fluctuations

in the even orbital channel. This SU(4)-to-SU(2) tran-

sition as mixing increases is illustrated in Fig. 2, where

the SBMFT parameters are plotted versus β.

Now we are in position to calculate transport prop-

erties. For this purpose, it is more convenient to write

SBMFT equations in the matrix form:

?

λ|˜b|2+1

N

Kchanges very little with β, To

Kreduces

|˜b2| −1

N

dǫ

2πiTrˆG<(ǫ) =

dǫ

2πiTr{ˆΣrˆG<(ǫ) +ˆΣ<ˆGa(ǫ)} = 0, (68b)

1

N,

(68a)

?

where we are back to the original basis and the trace

includes also the sum over spin indices. Here,ˆG<is the

lesser matrix orbital Green function, which is related to

the advancedˆGaand retardedˆGrmatrix Green functions

through the expression

ˆG<=ˆGaˆΣ<ˆGr

(69)

withˆΣ<being the lesser matrix self-energy. The explicit

expressions for these matrices are

ˆGa(ǫ) =

1

(ǫ − ˜ ε0− iTK)2+ β2T2

?ǫ − ˜ ε0− iTK

K

(70)

×

iβTK

iβTK

ǫ − ˜ ε0− iTK

?

andˆGrgiven by direct complex conjugation ofˆGa). The

lesser matrix self-energy reads

ˆΣ<= −2i[fL(ǫ) + fR(ǫ)] ×

whereas in the same way the retarded matrix self-energy

is

?

TK

βTK

βTK

TK

?

(71)

ˆΣr= −iˆΓ = −i

?

TK

βTK

βTK

TK

?

. (72)

Inserting Eqs. (70) and (71) in Eq. (69), we obtain the

lesser orbital Green function:

ˆG<=

−iTK

K(ǫ − ˜ ε0)2+ (β2− 1)2T4

(ǫ − ˜ ε0)4+ 2(1 + β2)T2

K

?

(ǫ − ˜ ε0)2+ TK(1 − β2)

β?(ǫ − ˜ ε0)2− TK(1 − β2)?

β?(ǫ − ˜ ε0)2− TK(1 − β2)?

(ǫ − ˜ ε0)2+ TK(1 − β2)

?

. (73)

Using the explicit expressions of the self-energies and the

nonequilibrium Green functions, we write Eq. (68b) in

the simple form:

λ|˜b|2+1

N

?

dǫ

2πiTrˆG<(ǫ)(ǫ − ˜ ε0) = 0.(74)

Solving in a self-consistently way Eqs. (68a) and (74)

for each dc bias Vdc, one gets the behavior of the two

renormalized parameters in nonequilibrium conditions.36

The electrical current has in appearance the same form

as the conventional Landauer-B¨ uttikercurrent expression

for noninteracting electrons:

I =2e

?

?

dǫ

2πT (ǫ,Vdc)[fL(ǫ) − fR(ǫ)] .(75)

Here caution is needed for a correct interpretation of

Eq. (75), since it contains “many-body” effects via the

renormalized parameters that have to be determined for

each Vdcin a self-consistent way. As a result, the trans-

mission T (ǫ,Vdc) possesses, in contrast with the noninter-

acting case, nontrivial behavior with voltage. The non-

linear conductance is calculated by direct differentiation

of Eq. (75) with respect to the bias voltage: G ≡ dI/dVdc.

Page 14

14

00.2 0.40.60.81

β

0

0.2

0.4

0.6

0.8

1

ε0/TK

~

SU(4)

00.2 0.40.60.81

β

0

0.5

1

1.5

2

TK

TK

e/TK

o/TK

SU(4)

SU(4)

(a)

(b)

FIG. 13:

Kondo physics as obtained from SBMFT: As the orbital mix-

ing is increased, the SU(4) Kondo effect reduces to the SU(2)

spin Kondo effect. This is reflected by (a) the position of the

Kondo resonance as well as by (b) the reduction of the odd

Kondo temperature down to zero. See the main text.

(Color online) Transition from SU(4) to SU(2)

In the limit Vdc→ 0 (at equilibrium), the linear conduc-

tance G0is given by the well-known expression:

G0=2e2

hT (0),(76)

where the transmission is

T (ǫ) = Tr{ˆGaˆΓˆGrˆΓ}.(77)

Finally, inserting Eqs. (70) and (72) in Eq. (77), one ar-

rives at the explicit formula for the transmission

T (ǫ) =

2T2

K

?(1 + β2)(ǫ − ˜ ε0)2+ T2

K(β2− 1)2?

(ǫ − ˜ ε0)4+ 2(1 + β2)T2

K(ǫ − ˜ ε0)2+ (β2− 1)2T4

K

, (78)

which is the main result of this part. It is remarkable

that the linear conductance G0 does not depend on β.

In particular, for the SU(4) Kondo model (β = 0), the

transmission takes the simple form:

T (ǫ) =

2T2

K

(ǫ − ˜ ε0)2+ T2

K

. (79)

In this case the resonance is pinned at ǫ = ˜ ε0= TKwith

the width given by TK; this leads to T (0) = 1 and in

consequence G0= 2e2/h. For β = 1 corresponding to the

two-level SU(2) Kondo model, Eq. (78) reduces to

T (ǫ) =

4T2

K

(ǫ − ˜ ε0)2+ 4T2

K

,(80)

which leads to the resonance at ǫ = ˜ ε0 = 0 and again

G0 = 2e2/h from T (0) = 1. As pointed out, this fact

makes both Kondo effects indistinguishable through the

linear conductance measurement.

All these features are clearly observed in Fig. 14, where

the transmission for different amounts of mixing, i.e.,

different values of β is plotted. For β = 0 the trans-

mission peak is located at TK as expected, whereas for

β = 1 this moves towards ǫ = 0. During the transition

from the SU(4) to the two-level SU(2) Kondo model, the

transmission gets narrower and develops a “cusp”, signal-

ing competition between even and odd channels. This is

manifested by the following from of the transmission:

T (ǫ) =

(1 + β)2T2

K

(ǫ − ˜ ε0)2+ T2

K(1 + β)2+

(1 − β)2T2

K

(ǫ − ˜ ε0)2+ T2

K(1 − β)2. (81)

Note that both channels are resonant at the same energy

˜ ε0but have different widths (˜Γeand˜Γo), which explains

the “cusp” behavior. Here we speculate that finite split-

ting δε originating from charge fluctuations42(not in-

cluded at the SBMFT level) would give rise to two split

resonances for β ?= 0, namely, ˜ ε0→ ˜ ε±

This is confirmed in the next section, where we present re-

sults obtained from full NCA calculations including fluc-

tuations. Eventually, for β = 1 the competition does not

exist and the transmission does not display the cusp.

0= δε ±

?˜Γe˜Γo.

Page 15

15

-10

-5

0

5

10

ε/ΤΚ

SU(4)

0

0,5

1

1,5

2

T( ε)

β=0

β=0.5

β=0.7

β=0.8

β=0.9

β=1

0

0,5

β

1

0,75

0.5

TK/TK

SU(4)

FIG. 14: (Color online) Equilibrium SBMFT result: Trans-

mission T (ǫ) as a function of the frequency for several values

of β. The left inset displays the Kondo temperature versus β.

B.Non-crossing approximation Method

The SBMFT suffers from two drawbacks: 1) It leads

always to local Fermi liquid behavior and 2) there arises

a phase transition (originating from breakdown of the lo-

cal gauge symmetry of the problem) that separates the

low-temperature state from the high-temperature local

moment regime. The latter problem may be corrected

by including 1/N corrections around the mean-field so-

lution. The non-crossing approximation (NCA)43,44,45is

the lowest-order self-consistent, fully conserving, and Φ

derivable theory in the Baym sense46which includes such

corrections. Without entering into details of the theory,

we just mention that the boson fields in Eq. (54), treated

as averages in the previous subsection (b(t)/√N →

?b?/√N ≡˜b), are now treated as fluctuating quantum

objects. In particular, one has to derive self-consistent

equations of motion for the time-ordered double-time

Green function (with subindexes omitted):

iG(t,t′) ≡ ?Tcf(t)f†(t′)?,

iB(t,t′) ≡ ?Tcb(t)b†(t′)?,(82)

or in terms of their analytic pieces,

iG(t,t′) = G>(t,t′)θ(t − t′) − G<(t,t′)θ(t′− t),

iB(t,t′) = B>(t,t′)θ(t − t′) + B<(t,t′)θ(t′− t).(83)

A rigorous and well-established way to derive these

equations of motion was introduced,47and related to

other non-equilibrium methods such as the Keldysh

method.40Here, we just present numerical results of the

NCA equations for our problem and refer interested read-

ers to Refs. 38,43,44,45 for details.

In particular, the DOS is given by

?

ρ(ε) = −1

π

ν=e,o,σ

Im[Ar

νσ(ε)], (84)

FIG. 15: (Color online) NCA results: Density of states around

ε = 0 at T = 0.25TK (left) and T = TK (right) for several

values of β. The inset shows a close-up of the β = 0.5 curve

(thick dashed), together with the individual even and odd

channel contributions (thin dashed).

where Ar

Greens function:

νσ(ε) is the Fourier transform of the retarded

Ar

νσ(t) = Gr

ν,σ(t)B<(−t) − G<

νσ(t)Ba(−t).(85)

The DOS for several values of β at two different temper-

atures is plotted in Fig. 15. Interestingly, the cusp be-

havior of Fig. 14 in the previous subsection becomes split

for the even and odd channels. This is illustrated in the

inset, where the curve corresponding to β = 0.5 is plot-

ted together with the individual even and odd channel

contributions. As we anticipate, the presence of charge

fluctuations induces splitting of ε0 due to the different

renormalization arising from different couplings for the

even and odd channels Γe/o[see Eqs. (38) and (39)].

V. CONCLUSION

We have considered the single-electron transistor

(SET) device with the CNT QD or VQD as the cen-

tral island in the Kondo regime. In particular we have

examined the case where the CNT QD or VQD has a

high symmetry so that the orbital quantum numbers are

conserved through the system. Emphasis has been paid

on how different Kondo physics, the SU(4) Kondo effect

or the two-level SU(2) Kondo effect, emerges depending

on the extent to which the symmetry is broken in real-

istic situations. Employed are four different theoretical

approaches: the scaling theory, the NRG method, the

SBMFT, and the NCA method to address both the lin-