Full counting statistics for orbital-degenerate impurity Anderson model
with Hund’s rule exchange coupling
Rui Sakano1 , Yunori Nishikawa2 , Akira Oguri2 , Alex C. Hewson3 , and Seigo Tarucha1
1
Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo, Japan
2
Department of Physics, Osaka City University, Sumiyoshi, Osaka, Japan
3
Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom
(Dated: January 9, 2012)
We study non-equilibrium current fluctuations through a quantum dot, which includes a ferromagnetic Hund’s rule coupling, in the low-energy Fermi liquid regime using the renormalized
perturbation theory. The resulting cumulant for the current distribution in the particle-hole symmetric case, shows that spin-triplet and Kondo-spin-singlet pairs of quasiparticles are formed in the
current due to the Hund’s rule coupling and these pairs enhance the current fluctuations. In the
fully screened higher-spin Kondo limit, the Fano factor takes a value Fb = (9M + 6)/(5M + 4)
determined by the orbital degeneracy M . We also investigate crossover between the small and large
J limits in the two-orbital case M = 2, using the numerical renormalization group approach.
PACS numbers: 71.10.Ay, 71.27.+a, 72.15.Qm
The experimental study of electron transport in mesoscopic devices, subject applied bias voltages, has provided a new way of investigating inter-electron manybody effects [1–3]. For example, the theoretical prediction, that the Kondo correlation on the non-equilibrium
currents through quantum dots leads to an enhancement
of the shot noise in these systems [4–7], has stimulated
subsequent experiments [8–10].
A deeper understanding of non-equilibrium transport
in these systems can be obtained from the calculation
of the current distribution function, the higher order cumulants beyond averaged current, and the noise power.
These can be calculated from the cumulant generating function (CGF) for non-equilibrium transport. This
quantity, however, is difficult to calculate when manybody effects have to be taken into account. A recent development has been the calculation of the current probability distribution for quantum dot in the Kondo regime
[11–13] using the general formulation of the full counting statistics (FCS) [14]. It was shown that quasiparticle
singlet-pairs carry a charge 2e in the backscattering current due to the Kondo correlation and that these enhance
the non-equilibrium current fluctuations.
In this letter, we investigate the FCS for an orbitaldegenerate impurity Anderson model as a prototype
model to examine the effects of both the Hund’s rule
and Kondo correlations, and hence to deduce the timeaveraged current and shot noise. These have been experimentally investigated in vertical dots, carbon nanotubes, and double dots [9, 15, 16]. Using the renormalized perturbation theory (RPT) [17], we calculate
the zero-temperature CGF up to third order in applied
bias voltage, for arbitrary strength of the interactions at
the dot-site in the low-energy Fermi-liquid regime. We
show that the Hund’s rule correlation gives rise to spintriplet pairs and spin-singlet pairs of quasiparticles carrying charge 2e in the non-equilibrium current, which
characterize the shot noise and higher order cumulants
at low energies.
Model. Let us consider transport through a correlated
dot described by an orbital-degenerate impurity Anderson model HA = H0 + HT + HI with
∑
∑
H0 =
εkα c†kαmσ ckαmσ +
ϵd nmσ ,
(1)
mσ
kαmσ
∑ (
)
vα d†mσ ckαmσ + H.c. ,
HT =
kαmσ
HI = U
∑
ndm↑ nm↓ + W
m
+2J
∑
∑
(2)
ndmσ ndm′ σ′
m>m′ ,σσ ′
Sdm · Sdm′ .
(3)
m>m′
Here, dmσ annihilates an electron in the dot level ϵd
with orbital m = 1, 2, · · · , M and spin σ, and ckαmσ
annihilates a conduction electron with moment k, orbital m and spin σ in lead α = L, R. Interactions U ,
W (0 ≤ W ≤ U ), and J(≤ 0) are the intra- and interorbital Coulomb repulsion, and Hund’s rule coupling, respectively. ndmσ is number of electron in the dot level
with orbital m and spin σ, and Sdm is the spin operator of electrons in orbital m of the dot. The intrinsic
level width of the dot
∑ levels owing to tunnel coupling
vα is given by Γ = α πρc vα2 with the density of state
of the conduction electrons ρc . For simplicity, the symmetric lead-dot coupling vL = vR and the particle-hole
symmetry ϵd = −U/2 − (M − 1)W are assumed. The
chemical potentials of the leads µL/R = ±V /2, satisfying
µL − µR = V (> 0), are measured relative to the Fermi
level defined at zero voltage V = 0. We take units with
h̄ = kB = e = 1 throughout this letter.
Full counting statistics. The probability distribution
P (q) of the transferred charge q across the dot during a
time interval T can provide the current correlation functions to all orders. In order to treat them systemati-
2
∑
cally, we calculate the CGF ln χ (λ) = ln q eiλq P (q)
⟨
⟩
λ
in the Keldysh formulation
[18]:
ln
χ
(λ)
=
ln
T
S
C
C
{ ∫
[
]}
λ
where SC
= TC exp −i C dt HTλ (t) + HI (t)
is the
time evolution operator for an extended Hamiltonian
λ
HA
= H0 + HTλ + HI , C is the Keldysh contour along
[t : −T /2 → +T /2 → −T /2], TC is the contour-ordering
operator, and λ is the counting field. Here, HTλ is given
by
]
∑[
HTλ =
vL eiλ(t)/2 d†mσ ckLmσ + vR d†mσ ckRmσ + H.c.(4),
kmσ
with the contour-dependent counting-field defined by
λ(t) ≡ λ∓ = ±λ for the forward and backward paths
labeled by “−” and “+”, respectively.
To calculate the CGF, we use Komnik and Gogolin’s
procedure [19] outlined below. First, a more general function
⟨ χ(λλ−⟩, λ+ ) is introduced. It is basically given by
ln TC SC
but λ∓ is formally treated as an independent
variable assigned for each contour. For the long time
limit T → ∞ where the switching effect is negligible,
the general CGF is proportional to T . Then, the derivative of the CGF with respect to λ− is given in terms of
Green’s functions as [19]
d
ln χ(λ− , λ+ )
dλ−
2 ∑∫
dω [ −iλ̄/2 λ−+
vL
0+−
= −iT
e
Gd
(ω) gkL
(ω)
2
2π
kmσ
]
0−+
− eiλ̄/2 gkL
(ω) Gλ+−
(ω) , (5)
d
0−+
with λ̄ ≡ λ− − λ+ . gkα
(ω) = i2πδ(ω − εkα )fα (ω)
0+−
and gkα (ω) = −i2πδ(ω − εkα )[1 − fα (ω)] are the lesser
and greater parts of the Green’s function for electrons in
lead α, respectively, with the fermi distribution function
fα (ω) = [e(ω−µα )/T + 1]−1 . For the long time limit T →
′
∞, the dot Green’s function is defined as Gλd νν (ω) =
∫
′
−i d(t−t′ )eiω(t−t ) ⟨TC dmσ (tν )d†mσ (t′ν ′ )⟩λ− ,λ+ . ν and ν ′
are the labels for the two Keldysh
contours.
Here, we use
⟨
⟩/
λ
a notation ⟨A(t)⟩λ ,λ ≡ TC SC
A(t) χ(λ− , λ+ ) which
−
(6)
m
with the Kronecker delta δm
Here Σrd (ω) is the self′.
energy of the retarded Green’s function for the dot
state Grd (ω) = [ω − ϵd + iΓ − Σrd (ω)]−1 , z = [1 −
∂Σrd (ω)/∂ω|ω=0 ]−1 is the wave function renormalization
m1 σ1 ;m2 σ2
factor, and Γm
(ω1 , ω2 , ω3 , ω4 ) is the local full
3 σ3 ;m4 σ4
four-vertex for the scattering of the electrons. Note that
these parameters are defined at equilibrium V = 0 and
λ = 0. The replacement of the bare parameters with the
renormalized ones gives the leading terms of an effective
Hamiltonian corresponding to the low-energy fixed point
of the numerical renormalization group (NRG). The perturbation theory in powers of U, W and J can be reorganized as an expansion with respect to the renormale, W
f and Je taking the free quasiparized interactions U
e −1 as the
ticle Green’s function gedr (ω) = (ω − e
ϵd + iΓ)
zero-order propagator. This procedure has enabled one
to calculate the exact form of the Green’s
function in
the absence of the counting fields Gλd (ω)λ=0 at low energies up to terms of order ω 2 , V 2 , and T 2 [3]. The
renormalized parameters can be explicitly related to the
enhancement factors of spin,[ orbital and charge
sus]
e − (M − 1)Je , z χ
ceptibilities, z χ
es = 1 + ρed (0) U
eorb =
[
]
[
]
f−U
e , zχ
e + 2(M − 1)W
f ,
1 + ρed (0) 2W
ec = 1 − ρed (0) U
(
)
e with
and the Friedel sum rule π⟨ndmσ ⟩ = cot−1 e
ϵd /Γ
e
the renormalized density of state ρed (ω) = (Γ/π)/[(ω
−
2
2
e ]. Γ
e is considered as the Kondo temperature
e
ϵd ) + Γ
e
TK = π Γ/4.
In particular, in the particle-hole symmetric case, the renormalized dot level is ϵ̃d = 0. These
relations enable one to deduce the value of renormalized
parameters in some special limits. We can also evaluate
the renormalized parameters for dots with two orbitals
(M = 2) with the NRG approach [20].
Let us now apply the RPT to the calculation of the dot
Green’s function Gλd (ω) given by the Dyson equation
[
]−1
e λ (ω) = z geλ (ω)−1 − Σ
e λ (ω)
Gλd (ω) = z G
. (7)
d
d
d
+
λ
represents an expectation for Hamiltonian HA
. Once
ln χ(λ− , λ+ ) is computed, the statistics is recovered from
ln χ(λ) = ln χ(λ, −λ).
Renormalized perturbation theory. To calculate the
low-energy Green’s function, we use the RPT outlined
below [17, 20]. The basic parameters that specify the
impurity Anderson model HA are ϵd , Γ, U , W , and J.
The low-energy properties can be characterized by the
quasiparticles with the renormalized parameters: dote = zΓ, and inlevel e
ϵd = z [ϵd + Σrd (0)], level width Γ
˜
teractions Ũ , W̃ and J defined by
1 σ1 ;m2 σ2
z 2 Γm
m3 σ3 ;m4 σ4 (0, 0, 0, 0)
=
( m1 m2 σ1 σ2
)
m1 m2 σ1 σ2
−Je δm
δ δ δ − δm
δ δ δ
3 m4 σ4 σ3
4 m3 σ 3 σ 4
[(
)
(
)(
)]
m1
m1
e + Je δm
f − J/2
e
U
+
W
1
−
δ
m2
2
( m1 m2 σ 1 σ 2
)
m1 m2 σ1 σ2
× δm4 δm3 δσ4 δσ3 − δm3 δm4 δσ3 δσ4
The zero-order part is given by [21]
]/
[
e + iΓ
e∑ f
D,
gedλ−− (ω) = ω − iΓ
l l
[
]/
e iλ̄/2 fL + iΓf
e R
gedλ−+ (ω) = iΓe
D,
[
]/
e −iλ̄/2 (1 − fL ) + iΓ
e (1 − fR ) D,
gedλ+− (ω) = − iΓe
[
]/
e + iΓ
e∑ f
gedλ++ (ω) = −ω − iΓ
D,
(8)
l l
(
)
e2 + Γ
e 2 eiλ̄/2 − 1 (1 − f ) f . The
with D = ω 2 + Γ
R
L
renormalized self-energy at T = 0 up to ω 2 , ωV and V 2
can be calculated in the second order perturbation in the
three renormalized interactions is
[
]
−i e
A(ω, V )
B(ω, V )
λ
e
Σd (ω) =
I
,
(9)
−B ∗ (−ω, −V ) A(ω, V )
e
8Γ
3
with
A(ω, V ) = [a (ω, 3V /2) + 3 a (ω, V /2)
+ 3 a (ω, −V /2) + a (ω, −3V /2)] , (10)
[
−iλ̄/2
B(ω, V ) = e
b (ω, 3V /2) + 3b (ω, V /2)
]
+3eiλ̄/2 b (ω, −V /2) + eiλ̄ b (ω, −3V /2) (11)
,
(
)
Ie = ũ2 + 2(M − 1) w̃2 + 3j̃ 2 /4
(12)
Here, a(ω, x) = − 12 (ω − x)2 sgn(−ω + x), b(ω, x) = (ω −
e /(π Γ),
e w̃ ≡ W
f /(π Γ)
e and j̃ ≡
x)2 θ(−ω + x), and ũ ≡ U
e Γ).
e For λ = 0, it corresponds to an extension of Ref.
J/(π
3 to the orbital-degenerate Anderson model.
Results and discussion. Integrating Eq. (5) respect
with λ− , the CGF is calculated at T = 0 up to V 3 ,
as ln χ(λ) = F0 + F1 + F2 with
F0 =
2M T
2π
∫
V /2
−V /2
[
(
)]
dω ln 1 + T (ω) eiλ − 1 ,
(
)
F1 = T Pb1 e−iλ − 1 ,
(13)
(
)
F2 = T Pb2 e−i2λ − 1 .(14)
Here, F0 describes free-quasiparticle tunneling via the
renormalized level with transmission probability T (ω) =
e 2 /(ω 2 + Γ
e 2 ), which carries single charge e. F and F
Γ
1
2
represent the scattering process due to the residual intere, W
f and J.
e From the coefficient of the countactions U
3
M
ing field in F1 and F2 , it is clear that Pb1 = 12π
Ĩ V 2
e
Γ
M V3
Ĩ 2 are the probability of the singleand Pb2 = 6π
e
Γ
and paired-quasiparticle backscattering processes carrying charge e and 2e, respectively. Then, the nth-order
dn
cumulant Cn = (−i)n dλn ln χ(λ) is readily derived as
Cn = T [Iu δ1n + (−1)n (Pb0 + Pb1 + 2n Pb2 )] , (15)
where Iu = 2M V /(2π) is the linear-response current, and
M V3
Pb0 = 12π
represents the probability of the singlee
Γ2
quasiparticle backscattering processes due to the renormalized revel. Especially, the factor 2n characterizing
cumulant of interacting electrons (15) indicates existence
of spin- or orbital-singlet and spin-triplet pairs of quasiparticles in the current. The cumulant for the SU(2M )
case where U = W and J = 0 was previously obtained in
a similar form [11–13]. In the present result, the Hund’s
coupling enters through the Pb1 and Pb2 as well as the
Kondo energy scale Γ̃. It gives rise to not only the spin
triplet pairs in current but also varies the couplings of the
spin sector ũ and the orbital sector w̃ in the Kondo correlation, which causes the crossover seen in the current
fluctuation as discussed below.
In the series of cumulant Cn , the first two coefficients
C1 and C2 give time-averaged current I = C1 /T and shot
noise S = 2C2 /T , respectively. The higher order ones,
Cn for n ≥ 3, are determined by these two, and are automatically generated by Eq. (15). To investigate current
fluctuation, we consider the Fano factor of the shot noise
for the backscattering current Ib ≡ Iu − I,
(
)/(
)
Fb ≡ S/(2Ib ) = 1 + 9Ie
1 + 5Ie .
(16)
We first discuss the weak coupling limit u, w, |j| ≪
1, where the renormalized parameters can be replaced
as
u, w,
e e
j) → ( (u, w, j) and
) [ (e
)] z → 1 −
(
3 − 41 π 2 u2 + 2(M − 1) w2 + 3j 2 /4 with scaled interactions u ≡ U/(πΓ), w ≡ W/(πΓ), and j ≡ J/(πΓ).
Substituting these into Eq. (16), the result for the second
order perturbation in the bare interactions is produced.
In particular, the Fano factor for non-interacting limit
u = w = j = 0 naturally becomes unity which represents
single charge transport given by Eq. (13).
In the large Hund’s coupling limit |j| ≫ 1, the dot
state is restricted to the highest spin state, for which we
obtain the Kondo coupling HK from the Anderson model
HA in the form
∑ †
HK = JK
ckmσ σσσ′ ck′ mσ′ · Sd
(17)
kk′ σσ ′ m
−1
2
2
with JK = 4(vL
+∑
vR
) [U − (M
√ − 1)J] , the Pauli ma′
trix σσσ , ckmσ = α ckαmσ / 2, and the S = M/2 spin
operator for the dot-site Sd . Namely, the fully screened
Kondo effect for higher spin S = M/2 (≥ 1) occures.
The Wilson ratio R ≡ z χ
es for this Kondo model can be
deduced exactly in the form R = 2(M + 2)/3, following
Yoshimori [20, 22]. In addition, the charge and orbital
fluctuations are suppressed χ
ec → 0 and χ
eorb → 0 in the
limit j → −∞. Therefore, the renormalized parameters
take the value (e
u, w,
e e
j) → (1, 0, −2/3). Then, through
Eq. (16), we find the Fano factor in this limit,
Fb →
9M + 6
9S + 3
=
.
5M + 4
5S + 2
(18)
m1
Furthermore, the δm
-dependence seen in Eq. (6) van2
ishes owing to the restored rotational-symmetry in the
orbital sector, and we find the vertex function of the
Kondo model (17),
m1 σ1 ;m2 σ2
z 2 Γm
(0, 0, 0, 0)
3 σ3 ;m4 σ4
)
4TK [( m1 m2 σ1 σ2
m1 m2 σ1 σ2
→
δm4 δm3 δσ4 δσ3 − δm
δ δ δ
3 m4 σ 3 σ 4
3
( m1 m2 σ 1 σ 2
)]
m1 m2 σ1 σ2
+2 δm
δ δ δ − δm
δ δ δ
. (19)
3 m4 σ4 σ3
4 m3 σ3 σ4
The explicit values of the Fano factor in this limit for several M are shown in the TABLE I. For comparison, the
Fano factor in the SU(2M ) Kondo limit u = w → ∞, j =
0, Fb → (M +4)/(M +2) [6, 13], are also shown in the TABLE I. Increasing orbital degeneracy M in the SU(2M )
Kondo limit, the system approaches the mean field limit
Fb → 1 [13]. In contrast, for large |j|, the renormalized interactions ũ, w̃ and j̃ converge to the values independent of orbital degeneracy M . As M increases, the
number of coupled orbitals with the Hund’s coupling J
4
M
(a) S = M/2 Kondo
(b) SU(2M ) Kondo
(a)
1
5/3
2
12/7
3/2
4
7/4
4/3
→∞
→ 9/5
→1
(b) u=w=3
0.5
0
u=w=3.0
u=w=2.0
u=w=1.0
u=w=0.25
u=w=0
1.4
1.2
1
0
-0.5
-1
-1.5
-2
0.5
0
~
(3/2) j
~
Γ/ Γ
(c) u=w=0
-0.5
-1
0
-0.5
-1
-1.5
-2
j
FIG. 1. (Color online) (a) The Fano factor for several choices
of Coulomb repulsion u = w as a function of scaled Hund’s
coupling j. (b) The renormalized parameters for u = w = 3
and (c) u = w = 0, as a function of j.
increases. It results an enhancement of the spin fluctuation, as seen in the expression of z χ
es or the interaction
factor of the self-energy given in Eq. (12). Therefore, we
conclude that the Fano factor increases with M , through
the enhancement of the quasiparticle-pair scatterings due
to the residual Hund’s coupling j̃.
Evaluating the renormalized parameters with the NRG
approach, we investigate J-dependent crossover of the
Fano factor for two orbital case (M = 2). The Fano
factor for u = w and u = 3 > w as a function of
Hund’s coupling j are shown in Fig. 1(a) and Fig. 2(a),
respectively. The crossover is observed in the Fano factor
around the coupling corresponding to the Kondo tempere
ature |j| ∼ tK ≡ Γ/(4Γ).
At large |j|, the Fano factor
for any u ≥ w converges to a value 12/7 given by Eq.
(18), as discussed above. However, at small |j|, the values of the Fano factor depends on Coulomb repulsions
Fb
(a)
1
1.7
(b) u=3,w=0
0.5
1.6
u=3.0,w=0
u=3.0,w=2.0
u=3.0,w=2.8
u=3.0,w=2.9
u=w=3.0
1.5
0
-0.1 -0.2 -0.3 -0.4 -0.5
j
(b)
1
0.8
1.6
~
Γ/Γ
u=3.0
1.5 u=2.0
u=1.0
1.4
0.6
u=0.5
0.4 u=1.0
u=2.0
0.2 u=3.0
u=0.5
0
1.3
0
0.2
0.4
0.6
w/u
0.8
1
0
0.2
0.4
0.6
0.8
1
w/u
-0.5
j
~
u
~
w
1.7
FIG. 3. (Color online) (a)The Fano factor and (b) the renormalized level width for no Hund’s coupling j = 0 and several intra-orbital Coulomb replusion u, as a function of interorbital Coulomb replusion w(≤ u).
1
1.8
1.6
Fb
3
33/19
7/5
(a)
Fb
TABLE I. The Fano factor Fb = C2 /Ib in (a) the large Hund’s
coupling j → −∞ or the S = M/2 Kondo limit, and (b) the
SU(2M ) Kondo limit u = w → ∞, j = 0.
0
u~
~
w
~
(3/2)
~ j
Γ /Γ
-0.5
-1
0
-0.05
-0.1
-0.15
-0.2
j
FIG. 2. (Color online) (a) The Fano factor for several
Coulomb interaction w(≤ u = 3) as a function of scaled
Hund’s coupling j. (b) The renormalized parameters for
(u, w) = (3, 0) as a function of j.
u and w. The Fano factor and the renormalized levelwidth (or the Kondo temperature) at j = 0 for several
intra-orbital Coulomb interaction u as a function of interorbital Coulomb interaction w are shown in Fig. 3 (a) and
(b), respectively. With increase of inter-orbital Coulomb
repulsion w, the Fano factor varies from the SU(2) value
to the SU(4) value around (u − w) ∼ tK .
The crossover seen in the Fano factor is a consequence
of the behavior of the renormalized parameters, which
enter through Eq. (16). Thus, the feature discussed above
is also seen in the renormalized parameters, which for
(u, w) = (3, 3), (0, 0) and (3, 0) are shown in Figs. 1 (b)
and (c), and Fig. 2 (b), respectively, as a function of j.
Particularly, in Fig. 1 (b), we see a general trend that
the spin coupling of the Kondo correlation u
e increases,
whereas the orbital coupling w
e decreases, with increase
of the ferromagnetic coupling −j. These renormalized
parameters converge to the value (ũ, w̃, j̃) = (1, 0, −2/3)
in the large Hund’s coupling limit. Even for (u, w) =
(0, 0), the Hund’s coupling enhances the spin coupling ũ
as shown in Fig. 1(c).
Summary. We have studied the role of the ferromagnetic Hund’s rule coupling on the FCS for the orbitaldegenerate Anderson impurity in the particle-hole symmetric case. Using the RPT, we have derived the CGF
for non-equilibrium-current distribution. The result is
asymptotically exact at low energies, and is described by
the quasiparticles of the local Fermi liquid. Specifically,
the explicit expression of the Fano factor for the shot
noise is obtained in the fully screened higher-spin Kondo
limit, which depends only on the orbital degeneracy M .
We have also investigated the crossover between the large
and small Hund’s coupling limits in the two-orbital case
(M = 2), using the NRG approach. Furthermore, the
CGF indicates that the Hund’s coupling gives rise to singlet and triplet pairs of quasiparticles carrying charge 2e
in the backscattering current and these correlated charges
characterize current fluctuation through the dot.
RS acknowledges Takeo Kato, Yasuhiro Utsumi, Kensuke Kobayashi, Yuma Okazaki, and Satoshi Sasaki for
fruitful discussion. This work was supported by the
JSPS through its FIRST program and the JSPS Grantin-Aid for Scientific Research C (No. 23540375) and S
5
(No. 19104007).
[1] M. Grobis, I. G. Rau, R. M. Potok, H. Shtrikman, and
D. Goldhaber-Gordon, Phys. Rev. Lett. 100, 246601
(2008).
[2] S. Hershfield, J. H. Davies, and J. W. Wilkins, Phys.
Rev. Lett. 67, 3720 (1991).
[3] A. Oguri, Phys. Rev. B 64, 153305 (2001).
[4] E. Sela, Y. Oreg, F. von Oppen, and J. Koch, Phys. Rev.
Lett. 97, 086601 (2006).
[5] A. Golub, Phys. Rev. B 73, 233310 (2006).
[6] C. Mora, P. Vitushinsky, X. Leyronas, A. A. Clerk, and
K. Le Hur, Phys. Rev. B 80, 155322 (2009).
[7] T. Fujii, J. Phys. Soc. Jpn. 79, 044714 (2010).
[8] O. Zarchin, M. Zaffalon, M. Heiblum, D. Mahalu, and
V. Umansky, Phys. Rev. B 77, 241303 (2008).
[9] T. Delattre, C. Feuillet-Palma, L. G. Herrmann,
P. Morfin, J.-M. Berroir, G. Fève, B. Plaçais, D. C. Glattli, M.-S. Choi, C. Mora, and T. Kontos, Nat. Phys. 5,
208 (2009).
[10] Y. Yamauchi, K. Sekiguchi, K. Chida, T. Arakawa,
S. Nakamura, K. Kobayashi, T. Ono, T. Fujii, and
R. Sakano, Phys. Rev. Lett. 106, 176601 (2011).
[11] A. O. Gogolin and A. Komnik, Phys. Rev. Lett. 97,
016602 (2006).
[12] T. L. Schmidt, A. Komnik, and A. O. Gogolin, Phys.
Rev. B 76, 241307 (2007).
[13] R. Sakano, A. Oguri, T. Kato, and S. Tarucha, Phys.
Rev. B 83, 241301 (2011).
[14] D. Bagrets, Y. Utsumi, D. Golubev, and G. Schön,
Fortschr. Phys. 54, 917 (2006).
[15] S. Sasaki, S. Amaha, N. Asakawa, M. Eto,
and
S. Tarucha, Phys. Rev. Lett. 93, 017205 (2004).
[16] U. Wilhelm and J. Weis, Physica E 6, 668 (2000).
[17] A. C. Hewson, Phys. Rev. Lett. 70, 4007 (1993).
[18] L. S. Levitov and M. Reznikov, Phys. Rev. B 70, 115305
(2004).
[19] A. Komnik and A. O. Gogolin, Phys. Rev. Lett. 94,
216601 (2005).
[20] Y. Nishikawa, D. J. G. Crow, and A. C. Hewson, Phys.
Rev. B 82, 115123 (2010).
[21] A. O. Gogolin and A. Komnik, Phys. Rev. B 73, 195301
(2006).
[22] A. Yoshimori, Prog. Theor. Phys. 55, 67 (1976).