Nonequilibrium theory of Coulomb blockade in open quantum dots
Piet W. Brouwer
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501∗

arXiv:cond-mat/0502518v2 [cond-mat.mes-hall] 5 Oct 2005

Austen Lamacraft

Department of Physics, Princeton University, Princeton, NJ 08544, USA†

Karsten Flensberg
Nano-Science Center, Niels Bohr Institute, Universitetsparken 5, 2100 Copenhagen, Denmark
We develop a non-equilibrium theory to describe weak Coulomb blockade effects in open quantum dots. Working within the bosonized description of electrons in the point contacts, we expose
deficiencies in earlier applications of this method, and address them using a 1/N expansion in the
inverse number of channels. At leading order this yields the self-consistent potential for the charging
interaction. Coulomb blockade effects arise as quantum corrections to transport at the next order.
Our approach unifies the phase functional and bosonization approaches to the problem, as well as
providing a simple picture for the conductance corrections in terms of renormalization of the dot’s
elastic scattering matrix, which is obtained also by elementary perturbation theory. For the case of
ideal contacts, a symmetry argument immediately allows us to conclude that interactions give no
signature in the averaged conductance. Non-equilibrium applications to the pumped current in a
quantum pump are worked out in detail.
PACS numbers: 73.23.-b,73.21.La,73.23.Hk

I.

INTRODUCTION

Coulomb blockade is the phenomenon that transport
through an almost isolated system is prohibited by the
energy cost Ec to add or remove an electron. The
Coulomb blockade can be lifted by fine tuning a gate
voltage to a point of charge degeneracy, where the energy cost of adding one electron is zero, or by raising
the bias voltage or the temperature above Ec . A fundamental question is to what extent the quantization of
charge on the system, and hence the Coulomb blockade,
is affected by the inclusion of that system in an electrical
circuit. This question has been addressed both for metal
particles coupled to electrodes via tunnel barriers, as a
function of the dimensionless conductance of the tunnel
barrier, and for semiconductor quantum dots coupled to
electrodes via ballistic point contacts, as a function of the
number of propagating channels in the point contacts. In
each case, the Coulomb blockade is lifted when the conductance of the contact to the electrodes is larger than
the conductance quantum e2 /h. Such quantum dots (or
metal particles) are referred to as ‘open’, in contrast to
‘closed’ quantum dots, which are connected to the electron reservoirs through tunneling contacts with a conductance smaller than the conductance quantum. The
present paper deals with Coulomb blockade in open semiconductor quantum dots, which we shall refer to as ‘weak
Coulomb blockade’.
In the literature, two approaches have been taken to
this problem. One theory of ‘weak Coulomb blockade’
was proposed by Flensberg1 , Matveev2 and Furusaki and
Matveev,3,4 who realized that a tractable description of
the strong quantum charge fluctuations can be formulated using the one-dimensional nature of the point con-

tacts. Whereas these theories accounted for the electronelectron interactions non-perturbatively, they neglected
the effect of electrons coherently traversing the dot or
being backscattered from inside. Extensions of the theories of Refs. 1,2,3,4 to the case of open quantum dots
with coherent scattering of electrons were given by Yi
and Kane5 for quantum dots in the quantum Hall regime
and by Aleiner and Glazman6 and Brouwer and Aleiner7
for the general case (see Ref. 8 for a review).
The other theoretical approach starts from an effective
theory in which the primary dynamic variables are the
potential differences between the quantum dot and the
electrodes.9 This approach has mainly been used to study
metal grains coupled to the reservoirs via tunnel barriers with many channels,10,11,12,13,14,15,16,17,18,19,20 , but
recently it has been applied to semiconductor quantum
dots, both without21,22,23 and with24 coherent scattering
of electrons from inside the quantum dot. The general
conclusions of both approaches are the same: For the incoherent case and ideal point contacts (no backscattering
in the contact), it was found that all charge quantization
effects completely vanishes.1,2,3,4,21 In all other cases —
contacts with a small amplitude rc for backscattering in
the contact or coherent scattering from inside the dot —
, charge quantization gives a correction to the dot’s capacitance or conductance. This correction is small, but
it becomes more important as the temperature is lowered or the number of channels in the point contacts is
reduced.
The previous works on ‘weak Coulomb blockade’ all
dealt with thermodynamic properties (capacitance) or
time-independent transport. It is the goal of the present
work to develop the general non-equilibrium theory for
Coulomb blockade in open quantum dots, accounting for

2
coherent scattering of electrons inside the dot. Our main
result is an expansion in powers of 1/N for the current
through the dot, where N is the total number of channels
in the point contacts.
Two examples of non-equilibrium problems involving
quantum interference in open and interacting mesoscopic
systems have been a particular motivation for this work.
First, the possibility of applying time dependent potentials via local gating of quantum dots led to the theoretical and experimental investigation of ‘quantum pumps’,
in which a periodic perturbation of potentials inside the
quantum dot, combined with quantum interference, leads
to a dc current through the dot.25 For quantum dots
with time dependent potentials, effects of coherent scattering inside the quantum dot, charge quantization, and
electron-electron interactions were considered separately
theoretically (see, e.g., Refs. 26,27,28,29,30,31), but not
together. The second example that motivated this work
is the experiment of Ref. 32, where the nonequilibrium
electron distribution function of a current carrying diffusive wire was measured by tunneling spectroscopy. This
allowed an experimental determination of the electronelectron collision integral, and led to the discovery of a
new mechanism of energy relaxation.33,34
Following previous studies of Coulomb blockade in
open quantum dots, the only electron-electron interaction term we consider is the capacitive charging energy
of the quantum dot. This requires relatively large quantum dots: For open quantum dots Coulomb blockade effects are small in the inverse of the number of channels N
in the contacts connecting the dot to the electron reservoirs, whereas the residual parts of the Coulomb interaction are small in the inverse dimensionless conductance
of the ‘closed’ dot.6,8,35 The ratio of these two conductances equals the ratio of the dot’s dwell time and ergodic
time, which is large for large quantum dots. This parameter justifies the use of the random matrix description
of scattering by the dot, and means that the problem is
effectively ‘zero dimensional’. The dimensionality of the
system appears only in the corrections to this picture.
Unlike for the case of linear time-independent transport, where the capacitive interaction gives corrections
to transport properties beyond the Hartree level only,
interaction corrections to nonlinear or time-dependent
transport exist already when interactions are described
by a self-consistent (Hartree) potential Vd . As pointed
out by B¨ttiker and coworkers, the reason is that, in
u
non-equilibrium or time-dependent problems, a change
in bias voltage or a change in a gate voltage potentials
may change the number of electrons on the dot, which,
in turn, changes Vd ,26,27,28
Vd (t) = −

e
C

ˆ
Ndot (t) − N .

(1.1)

ˆ
Here C is the dot’s capacitance, −e Ndot is the dot
charge, and −eN the offset charge induced by the gates.
The effect of a time-dependent or bias-dependent Hartree
potential Vd on mesoscopic fluctuations of various trans-

port properties — but neglecting the ‘weak Coulomb
blockade’ — was investigated in Refs. 30,36,37,38,39.
To leading order in 1/N , our theory recovers the selfconsistent theory with the Hartree potential (1.1). In
this sense, our result can be viewed as a formal confirmation of the self-consistent theory. Further, it gives
a controlled method to find interaction corrections that
cannot be described by means of a self-consistent potential.
A second motivation of this work is to compare and
unify the results from two different approaches to these
corrections — the ‘one-dimensional’ and ‘environmental’
formalisms. Whereas both approaches have given equal
results for the case of quantum dots without coherent
scattering from inside the dot, results for the dc conductance of coherent dots, as reported by Brouwer and
Aleiner7 and Golubev and Zaikin24 , are in disagreement.
Using a formalism closely related to that of Ref. 7, we
are able to present detailed calculations in two limits:
A formal expansion in the scattering matrix of the dot
for an arbitrary number of channels N in the contacts,
and as an expansion in powers of 1/N . The former limit
reproduces the result of Ref. 7, whereas the latter limit
agrees with Ref. 24. The formal expansion in the scattering matrix S, however, is only allowed if the probability
of coherent (back)scattering from either the quantum dot
or the contact is small, and does not describe interaction
corrections to the conductance of a fully coherent quantum dot (i.e., a dot with a unitary scattering matrix).
With the same arguments, the result for the interaction
correction to the dot’s capacitance, which was calculated
in Ref. 8 using a formal expansion in S, is not applicable
to a quantum dot with a unitary scattering matrix. The
correct theory, in the limit of large N , will be given here.
We will not compare non-perturbative results in the two
formalisms, that were obtained using refermionization4
or instanton solutions.21,23
With regard to the interaction corrections to conductance, our result is particularly simple: for a fully coherent dot there is no correction to the ensemble averaged
conductance, to leading and sub-leading order in 1/N . In
particular, this applies to weak localization, the quantum
correction to the ensemble averaged conductance that is
suppressed by the application of a magnetic field. The
reason for the absence of an interaction correction to the
ensemble averaged conductance is the following. In the
Landauer description of transport, conductance is related
to a sum of squares of moduli of elements Sij (ε) of the
scattering matrix S(ε), appropriately integrated over energy if the temperature is nonzero (see, e.g., Ref. 40).
The ensemble average of the conductance depends on the
ensemble averages |S(ε)ij |2 only. According to the random matrix theory of quantum transport, such an average is determined by symmetries only (presence or absence of time-reversal and spin-rotation symmetry). At
leading and sub-leading order in 1/N , we find that the
interaction corrections can be expressed through a renormalized scattering matrix: S(ε) → S ′ (ε) ≡ S(ε) + δS(ε)

3
[see Eq. (3.6) below]. Although this interaction correction changes the scattering matrix, and hence conductance, of a specific dot, it does not change the symmetry
of the scattering matrix ensemble, leaving the ensemble
averaged conductance (including the weak-localization
correction) unchanged. We will revisit this argument in
more detail in Sec. VII.
The outline of our paper is as follows. In Sec. II
we present a detailed discussion of the relevant existing works in the literature. This discussion will serve to
introduce the necessary concepts and notations, and to
explain the disagreement between the two existing theories of ‘weak Coulomb blockade’ in coherent quantum
dots. In Sec. III we proceed with a precise definition of
the problem and an exposition of the formalism. In the
diagrammatic language that we will introduce later, the
leading interaction correction to the current arises from a
Fock-type diagram. The idea that interaction corrections
to conductance in the presence of impurities should be
thought of as due to scattering from an Hartree-Fock potential was introduced in the work of Matveev, Yue, and
Glazman.41,42 In order to make contact to that work, we
have added to Sec. III a calculation similar in spirit to
that of Refs. 41,42 that already contains the basic structure of the interaction correction to the current. The
full calculation of the current through the quantum dot,
for the general non-equilibrium and time-dependent situation is then described in Sec. IV. Sections V and VI
contain a detailed analysis of these results for nonlinear steady-state transport and adiabatic time-dependent
transport, respectively. We conclude in Sec. VII. Finally,
appendices A, B, and C contain materials on the correlation functions in the ‘one-dimensional’ formalism, the
ensemble average, and the case of a quantum dot with
partially coherent scattering, respectively. A summary
of our results, focusing on the relation between our work
and previous approaches to ‘weak Coulomb blockade’ appeared in Ref. 43.
II.

PREVIOUS WORK

In this section, we present a more detailed overview of
the results published in the literature. This will also serve
to introduce some of the needed concepts and notations.
A schematic drawing of the system under consideration
is shown in Fig. 1. It consists of a quantum dot coupled to
two electron reservoirs by means of point contacts. The
point contacts have N1 and N2 channels each (counting
spin degeneracy). The two contacts are characterized
by energy-independent reflection matrices rc1 and rc2 , of
dimension N1 and N2 , respectively. In this notation, the
conductance of each contact is
Gi =

e2
†
gi , gi = Ni − tr rci rci , i = 1, 2.
2Ï€

(2.1)

A contact with Ni = gi , i = 1, 2 is called ballistic or
‘ideal’; otherwise the contact is called ‘nonideal’. A dot

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dot
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FIG. 1: Schematic drawing of the system under consideration: a quantum dot (center) coupled to source and drain
reservoirs (left and right) via point contacts. The two point
contacts have N1 and N2 propagating channels at the Fermi
level. Direct reflection at the point contact is described by
energy- independent reflection matrices rc1 and rc2 , so that
their dimensionless conductances are G1 = (e2 /2π )(N1 −
†
†
tr rc1 rc1 ) and G2 = (e2 /h)(N2 − tr rc2 rc2 ), respectively. Interactions in the dot are described by the charging energy
Ec .

with g1 + g2 ≪ 1 is referred to as ‘closed’; an open dot
has g1 + g2 ≫ 1. For future reference, we define the total
number of channels in the point contacts N = N1 + N2 ,
the total dimensionless conductance of the point contacts
g = g1 + g2 , and the N × N combined reflection matrix
of both contacts
rc =

rc1 0
0 rc2

.

(2.2)

The discussion of this section will be limited to the
linear dc conductance G of the quantum dot. Our discussion of the literature will be limited to the cases of a
‘coherent’ quantum dot, for which transport properties
are expressed in terms of its N -dimensional unitary scattering matrix S(ε), and an ‘incoherent’ quantum dot, for
which the dot itself is treated as a reservoir in equilibrium, with a self-consistent chemical potential.1,2 In the
‘incoherent’ limit, coherent propagation across the dot,
or backscattering from within, is not taken into account.
Neglecting interactions for electrons on the dot, for the
‘incoherent’ case the conductance is nothing but the classical series conductance Gcl = (e2 / )g1 g2 /g of the two
point contacts. For the ‘coherent’ case, the conductance
is given by the Landauer formula, which we write as
G0 =

e2 N1 N2
2Ï€
N
∂f (ε)
e2
dε −
−
2Ï€
∂ε

(2.3)
tr ΛS(ε)ΛS † (ε).

Here f (ε) = 1/(1 + exp(ε/T )) is the Fermi distribution
function and the matrix Λ is defined as
Λij =

δij
×
N

N2 for j = 1, . . . , N1 ,
−N1 for j = N1 + 1, . . . , N ,

(2.4)

The introduction of the matrix Λ is a convenient way to
separate the classical conductance for ballistic contacts

4
[the first term in Eq. (2.3)] from the quantum corrections
and the effect of nonideal contacts [second term in Eq.
(2.3)].
The Landauer approach is an intrinsically singleparticle picture, and as such cannot capture any effects
of electron-electron interaction by itself. In a quantum
dot, the most important part of the interaction6,8,35 is
described by the Hamiltonian
ˆ
ˆ
Hc = Ec Ndot − N

2

,

(2.5)

where the charging energy Ec = e /2C is related to the
ˆ
geometric capacitance C of the quantum dot, Ndot is the
number of electrons on the quantum dot, and N = CVg /e
is a dimensionless gate voltage used to set the equilibrium
value of Ndot .
The charging interaction (2.5) is responsible for the
phenomenon of Coulomb blockade in closed dots at low
temperatures: suppression of the conductance, except if
the gate voltage N is tuned to a point of charge degeneracy. As a dot is opened, a situation best realized
in lateral semiconductor dots where point contacts may
be controlled electrostatically, strong quantum charge
fluctuations lead to the progressive diminishment of the
Coulomb blockade. A theory of this ‘weak Coulomb
blockade’ was developed by Flensberg,1 Matveev,2 and
Furusaki and Matveev,3,4 for the ‘incoherent’ case. For a
quantum dot with two nearly ideal single channel point
contacts (|rc1 |, |rc2 | ≪ 1) and at temperatures T ≪ Ec ,
the interactions were found to change the conductance of
the contacts according to gj → gj + δgj , j = 1, 2, where
Γ(3/4)
Γ(1/4)

eC Ec
†
tr rcj rcj , j = 1, 2, (2.6)
Ï€T

up to corrections of order |rc |4 . Here C ≈ 0.577 is the
Euler constant. As in the non-interacting case, the conductance G of the quantum dot is simply the series conductance of the two point contacts,
G =

e2
2Ï€

1−

Γ (3/4)
Γ (1/4)

eC Ec
†
tr rc rc .
Ï€T

(2.7)

The first term is the classical conductance with N1 =
N2 = 2, accounting for spin degeneracy. The second term
shows the renormalization of the backscattering at the
contacts, see Eq. (2.6). At low temperatures the system
enters into a non-perturbative regime. We refer to Refs.
1,3,4,7 for the expressions for the interaction corrections
at different numbers of channels in the point contacts.
A different approach to the same problem was taken by
Golubev and Zaikin23 , and by Levy-Yeyati et al.22 who

sj gj Ec
,
T ≫ gEc ,
(2.8)
3T
2sj gj Ec ge1+C
= −
ln
, T ≪ gEc , j = 1, 2,
g
2Ï€ 2 T

δgj = −
δgj

2

δgj = −2

described the interaction Hamiltonian (2.5) through an
effective theory for the electromagnetic environment of
the junction. These authors also calculated the interaction correction to the conductance of the point contacts
for an arbitrary transparency of the contacts using 1/N
as a small parameter. They found

where s1 and s2 are the so-called Fano factors of the
contacts,
sj =

1
†
†
tr rcj rcj (1 − rcj rcj ), j = 1, 2.
gj

(2.9)

The appearance of the Fano factors is quite common to
corrections to the conductance that are perturbative in
the interaction, see Ref. 41,42. In fact, si is the only
†
quadratic function of the product rci rci that vanishes
both in the limits rci → 0 (ideal contacts) and rci → 1
(completely closed contacts). Equation (2.8) agrees with
the large-N generalization of Eq. (2.7) for nearly ideal
contacts.1,7
The results of Refs. 1,2,3,4, being non-perturbative in
Ec , are of conceptual importance, but describe the incoherent case only. They neglect entirely the possibility of
coherent backscattering from within the dot, or coherent transmission from one contact to another. Thus Eq.
(2.7) misses quantum interference effects, such as weak
localization (the small negative magnetic-field dependent
quantum interference correction to the ensemble averaged conductance) and conductance fluctuations, which
one expects to be described by the Landauer formula
(2.3) if the conductance of the contacts is large enough.
Interaction corrections to quantum interference corrections cannot be captured by a simple renormalization of
the point contact conductances; one needs to consider
the total conductance G of the quantum dot. Only at
temperatures for which relaxation processes in the dot
are dominant one expects the separate renormalizations
of the point contact conductances of Eqs. (2.6) or (2.8)
to be a sufficient description.
The first results for the ‘coherent’ case, concerning interaction corrections to transport in open dots described
by a unitary scattering matrix, were obtained by Brouwer
and Aleiner7 (see also the review Ref. 8). They found
G = G0 + GBA with

5

GBA = −

e2
2Ï€

1
Ï€
sin
Ï€
N
∞

×

dσ
t0

∞

∞

dτ2 tr S † (−τ1 )ΛS(τ2 )Λ

dτ1
0

0

(2σ + τ2 + τ1 )π 2 T 2
sinh[(σ + τ1 )πT ] sinh[(σ + τ2 )πT ]

Here t0 = π/Ec N eC is a charge relaxation time and S(τ )
is the Fourier transform of the scattering matrix S(ε),
which is defined as
S(Ï„ ) =
S † (τ ) =

dε
S(ε)e−iετ ,
2Ï€
dε †
S (ε)e−iετ .
2Ï€

(2.11)
(2.12)

For a dot with two single-channel spin-polarized point
contacts, N1 = N2 = 1, there is an additional interaction
correction to the conductance that depends explicitly on
the gate voltage N . Equation (2.10) is obtained as a
formal expansion in the scattering matrix S. Such an
expansion is controlled if S is subunitary, as is the case,
e.g., in an effective description of inelastic processes in
the dot.44 Noting that only times up to the ‘thermal time
horizon’ /T contribute to the interaction correction of
Eq. (2.10), Brouwer and Aleiner argued that their formal
expansion is also justified for unitary S if the temperature
is sufficiently high that the contribution from scattering
times
/T is small. For a chaotic dot with mean dwell
time Ï„d and ideal point contacts, this translates to the
condition T ≫ /τd .
The result of Ref. 7 captures the Landauer part of the
conductance exactly, whereas the interaction correction
(2.10) reproduces the singular behavior of Eq. (2.7) for
the scattering matrix S(τ ) = rc δ(τ ) + . . ., provided that
the scattering from inside the dot, represented by ‘. . .’,
is from far beyond the ‘thermal time horizon’. Using
known statistical properties of the scattering matrix for
chaotic quantum dots, Brouwer and Aleiner calculated
the ensemble averages of the conductance and the conductance fluctuations for T ≫ /τd , and found that both
the weak localization correction to the conductance and
the conductance fluctuations are slightly enhanced by interactions.
In this paper we question the arguments used to justify the formal expansion in the scattering matrix used
to obtain Eq. (2.10). As we will argue below, the interaction correction of Eq. (2.10) is not the correct interaction
correction to the conductance of a coherent quantum dot
at any temperature, despite the fact that it is small if
T ≫ /τd . Our main arguments will be given in the
next sections, where we present two calculations of the
interaction correction to the conductance: The first is
a first-order-in-Ec calculation of the conductance, which
differs from Eq. (2.10) if that result is expanded in Ec .

sinh[(τ1 + τ2 + σ + t0 )πT ] sinh[(σ − t0 )πT ]
sinh[(Ï„1 + t0 )Ï€T ] sinh[(Ï„2 + t0 )Ï€T ]

1/N

. (2.10)

The second is a calculation to all orders in Ec , but to
leading order in 1/N . This calculation confirms that Eq.
(2.10) is found if the amplitude for coherent scattering
from the dot is small — i.e., transport through the dot is
mainly incoherent (but not fully incoherent) —, but we
find a different interaction correction if transport is fully
coherent.
The main shortcoming of Eq. (2.10) can already be
seen noting that, although for T ≫ /τd most scattering
processes have delay time ≫ /T and thus do not contribute to the interaction correction, the remaining interaction correction is dominated by scattering processes
with delay time
/T . Hence, it is necessary that the
theory describes this time range accurately; its contribution being small is not sufficient to justify Eq. (2.10).
That Eq. (2.10) does not describe delay times
/T
accurately can be seen by considering the example of a
‘dot’ that consists of a single ballistic channel in which
there is an interaction of the form (2.5). The channel
is of length L, so the 2 × 2 scattering matrix describing
ballistic propagation from one end to the other is
S(Ï„ ) =

0 1
δ(τ − τd ),
1 0

(2.13)

where Ï„d = L/vF , vF being the Fermi velocity inside the
channel, is the time to propagate through the channel.
We assume τd ≫ t0 . Then, substituting Eq. (2.13) into
(2.10) gives a conductance that is larger than the conductance quantum e2 /2Ï€ , G = e2 /2Ï€ + GBA , with
e2 πT τd −2πT τd /
e
, T ≫ /τd ,
2Ï€
e2 π
=
,
T ≪ /τd . (2.14)
2Ï€ 8T Ï„d

GBA =
GBA

On the other hand it is well known that in the absence
of backscattering the conductance of such a system cannot deviate from the quantized value e2 /h, see, e.g., Refs.
45,46,47, so that the interaction correction to the conductance must be zero for all temperatures. Hence, although
the error made by using Eq. (2.10) is exponentially small
for τd ≫ /T , it is substantial for τd
/T .
Very recently, Golubev and Zaikin extended their environmental formalism to the case of a fully coherent dot.24
Again using 1/N as an expansion parameter, they found
G = G0 + GGZ , with

6

GGZ =

e2
Im
2Ï€ 2

dεdω −

∂f (ε)
∂ε

κ(ω)tr ΛS(ε)ΛS † (ε + ω) − ΛS(ε)S † (ε + ω)S(ε)ΛS † (ε) (1 − 2f (ε + ω)),
(2.15)

where κ(ω) is an effective interaction kernel that describes the fluctuations of the dot potential. Since these
fluctuations are small as 1/N if N is large, the correction (2.15) is a small correction to the Landauer result
G0 . Golubev and Zaikin performed the ensemble average
over S and found that the leading interaction correction
for the coherent case is the same as in the incoherent
limit (i.e., the classical combination of the corrections
Eq. (2.8)) as long as T ≫ /τd , and saturates when
T
/Ï„d . The agreement with the incoherent limit is
no surprise, since the leading-in-N contributions to the
conductance are generally insensitive to the presence of
quantum coherence.
The authors of Ref. 24 did not analyze the quantum
interference corrections to the conductance and thus attempted no comparison with the results of Refs. 7,8.
Our full calculation, which is described in the next sections and which includes quantum interference corrections, agrees with Eq. (2.15) and extends it to the case
of general time dependent transport — the focus of this
paper.
Technically, our formalism is very close to that of Refs.
7,8. Within the same formalism, we can obtain Eq. (2.10)
using a formal expansion in the scattering matrix S and
Eq. (2.15) as a formal expansion in 1/N . As we discussed above, expansion in powers of S requires a relaxation mechanism, and Eq. (2.10) can be justified only if
relaxation is strong, so that, effectively, |S(ǫ)| ≪ 1.
III.
A.

FORMALISM

Definition of the problem

The system under consideration — a quantum dot coupled to source and drain reservoirs via point contacts —
has been introduced at the beginning of Sec. II. There
are two point contacts, with N1 and N2 channels each (including spin degeneracy). Electrons on the dot interact
via the simple interaction Hamiltonian (2.5). We refer
to Refs. 6,8,35 for a microscopic justification of Eq. (2.5)
for quantum dots with a large (internal) dimensionless
conductance. In using Eq. (2.5) we ignore those parts of
the interaction describing superconducting pairing and
exchange.8
Following Flensberg1 and Matveev2 we describe the
electron dynamics in the point contacts by a onedimensional Hamiltonian. We locate the lead-dot interface at x = 0, taking dot and lead at x > 0 and x < 0
respectively, see Fig. 2. Since there is no backscattering

lead 1

quantum
dot

lead 2

x= 0
FIG. 2: Representation of the source and drain reservoirs
as one-dimensional ideal leads. The lead-dot interface is at
x = 0. The one-dimensional description (3.1) is valid from
x = −∞ up to slightly beyond x = 0.

of electrons that have left the dot through the point contacts and into the reservoirs back into the dot, we may
represent the reservoirs by extending the one-dimensional
description to all x < 0,
N

ˆ
H = ivF :
j=1

0

∂ ˆ
ˆ†
dx ψjL (x) ψjL (x)
∂x
−∞

∂ ˆ
ˆ†
ˆ
− ψjR (x) ψjR (x) : +Ec Ndot − N
∂x

(3.1)
2

,

Here the index j labels the channels in the two point contacts, including spin, N is the total number of channels
ˆ
ˆ
in the two point contacts, ψjL and ψjR are annihilation
operators for left-moving and right-moving electrons in
channel j, respectively. Because of scattering by the dot,
these are not independent fields. In the one-dimensional
Hamiltonian (3.1), the kinetic energy is linearized around
the Fermi energy, which is appropriate if the Fermi energy is not too close to a threshold at which a new channel
in the point contact is opened. The coordinate x in the
one-dimensional Hamiltonian (3.1) arises as the Fourier
transform of the one-dimensional Hamiltonian in momentum representation, see Ref. 2. Except for the immediate
vicinity of the point contact, it cannot be identified with
a distance to the lead-dot interface.
It is convenient to follow Aleiner and Glazman,6 and
write the number of electrons on the dot as
ˆ
Ndot = Nref −

0

:
j

−∞

ˆ†
ˆ
dx ψjL (x)ψjL (x)

ˆ†
ˆ
+ ψjR (x)ψjR (x) : .

(3.2)

Here Nref is the total number of electrons in the system
(reservoirs and dot), which is time independent. Now the

7
interaction does not contain the dot degrees of freedom.
Thus the relation between the operators for right moving electrons entering the quantum dot and left moving
electrons exiting the quantum dot is given by the noninteracting formula. In the general time-dependent case
one then has48

evaluated using the boundary conditions (3.3)
ˆ†
ˆ
Vij (x, y) = −2Ec ψjR (y)ψiL (x)
∞

= −2Ec
×

N

ˆ
ψjL (0, t) =
k=1

ˆ
dτ Sjk (t, t − τ )ψkR (0, t − τ ),

N

ˆ†
ψjL (0, t) =
k=1

ˆ†
dτ ψkR (0, t − τ )(S † )kj (t − τ, t),
(3.3)

where S(t, t′ ) is the scattering matrix of the dot. Notice
that the scattering matrix has two time arguments. For
time-independent transport, S(t, t′ ) is a function of the
time difference t − t′ only.
The relation (3.3), together with the one-dimensional
ˆ
Hamiltonian (3.1), Eq. (3.2) for Ndot , and the boundary conditions set by the reservoirs completely define the
problem. Notice that we never explicitly introduce the
dot degrees of freedom into the problem. If the scattering from the dot is not fully coherent, Eq. (3.3) has to
be replaced by a different boundary condition. We return to the case of incoherent scattering in section III D.
A model for partially coherent scattering is discussed in
appendix C.

B.

First order in Ec

It was shown by Matveev, Yue, and Glazman41,42 that,
at first order in the interaction, interaction corrections
to conductance in the presence of impurities may be understood in terms of the elastic scattering of electrons
from the Hartree-Fock potential created by the impurities. This gave a simple and intuitive picture of the
renormalization of backscattering in the interacting onedimensional electron gas, without recourse to sophisticated formalism. We now outline a similar calculation
for transport through a quantum dot.
For linear transport, it is sufficient to consider the
scattering off the Hartree-Fock potential in equilibrium.
With the replacement (3.2) the Hartree-Fock potential
acts in the leads, not in the dot. In equilibrium there
ˆ
is no Hartree potential as Ndot = N . For the Fock
potential we find
N

ˆ
HF =
i,j=1

0
−∞

ˆ†
ˆ
dxdy ψiL (x)Vij (x, y)ψjR (y) + h.c. ,

(3.4)
where “h.c.” denotes the hermitian conjugate, plus forward scattering terms that do not affect the linear conductance at lowest order in Ec . The Fock potentials are
written in terms of the density matrices, which may be

dτ Sij (τ )
0

iT
,(3.5)
2 sinh [πT (τ − vF [x + y] + iλ)]

where λ is a positive infinitesimal. Thus the effect of
the Coulomb interaction is to establish a non-local Fock
potential in the leads within a region of order the thermal
length vF /T from the contacts. It is now straightforward
to find the change in the scattering matrix due to this
potential
δS(ǫ) = i

dω
κ(ω) [(2f (ǫ − ω) − 1)S(ǫ − ω)
2Ï€
+ (2f (ǫ + ω) − 1)S(ǫ)S † (ǫ + ω)S(ǫ) , (3.6)

where
κ(ω) = −

Ec
,
(ω − i0+ )2

(3.7)

0+ being a positive infinitesimal. The first term in (3.6)
represents the effect of direct R → L (in to out) scattering by the Fock potential (3.4). The second corresponds
to the dot scattering R → L, followed by the Fock potential scattering L → R, and ending with the dot scattering
R → L once more. Substitution of (3.6) into the Landauer formula (2.3) immediate yields the result (2.15)
for the interaction correction to conductance, with κ(ω)
given by Eq. (3.7) above.
We may repeat the derivation for the incoherent case
that is considered in Refs. 1,2,3,4,22,23. In this case, the
inclusion of the Fock potential leads to a change of the
reflection matrix rc for direct reflection at the contacts.
Since electrons leaving the dot are incoherent with those
entering it, they do not contribute to the Fock potential
(3.5). The renormalized reflection matrix is thus
δrc (ǫ) = i

dω
κ(ω) [(2f (ǫ − ω) − 1)rc
2Ï€
†
+ (2f (ǫ + ω) − 1)rc rc rc ,

(3.8)

Taking into account the energy dependence of the renormalized reflection matrix rc , we find that the interaction
correction to the dimensionless conductance of contact j,
j = 1, 2, becomes
δgj = 2sj gj Im
× −

dε

∂f (ε)
∂ε

dω
2Ï€

(3.9)

(1 − 2f (ε + ω)) κ(ω),

where sj was defined in Eq. (2.9). Although this looks
very similar to (2.15) with the scattering matrix chosen
as S(ε) = rc , the effect of using a non-unitary scattering

8
matrix is significant. Notice that for a unitary scattering matrix, the trace in Eq. (2.15) vanishes as ω 2 , so
that the regulator in (3.7) plays no role. In (3.9) on the
other hand, the imaginary part of the integrand comes
from the region near ω = 0 and yields a finite, regulator
independent contribution to the conductance
Ec
, j = 1, 2,
δgj = −sj gj
3T

ˆ
ˆ
ˆ
H = H0 + H 1 ,
(3.10)

which coincides with the high temperature limit of Eq.
(2.8).23 Recovering this same result, together with the
low-temperature limit of (2.8), as the zero level spacing
limit of scattering from a coherent dot is subtle, see Sec.
V and Ref. 24.
With this physical picture of the interaction corrections, we now proceed with the formal development of
the theory.
C.

Effective action

We are interested in the current through the point contacts, at the dot-lead interface x = 0. The current in
channel j is
ˆ†
ˆ
ˆ†
ˆ
ˆ
Ij = −evF ψjR (0)ψjR (0) − ψjL (0)ψjL (0) .

In order to derive the effective action, a fictitious halfinfinite N -channel lead is side-coupled to the point contacts, at position x = η, where η is a positive infinitesimal. The fictitious lead couples only to electrons moving
out of the dot, i.e., to left movers. The total Hamiltonian
ˆ
H is written as

ˆ
where H0 is the Hamiltonian of the full interacting system (including the quantum dot, the reservoirs, and the
fictitious lead), with ideal coupling between the fictitious
lead and the point contact, see Fig. 3 (solid lines). Hence,
ˆ
in the system described by Hamiltonian H0 , all electrons
entering the point contact from the dot will exit through
the fictitious lead, whereas all electrons entering from the
fictitious lead will exit towards the source or drain reserˆ
voirs. The Hamiltonian H1 represents an impurity in the
contact to the fictitious lead which ensures that electrons
coming from the dot in channel j have amplitude −irj
to be transmitted coherently towards the source or drain
reservoirs (cf. dashed lines in Fig. 3) and that electrons
coming in from the fictitious lead have amplitude −irj
to be reflected into the fictitious lead,
N

ˆ†
ˆ
ˆ†
ˆ
2vF vj ψjL (0)ψjL (2η) + ψjL (2η)ψjL (0) ,

ˆ
H1 =

(3.11)

Below, the expectation value of the current is calculated using standard methods of nonequilibrium manyˆ
body theory.49 The current I(t) is related to the state of
the system at a reference time tref at which the Hamiltonian is not time-dependent. The expectation value I(t) is
then found from a thermodynamical average at the time
tref . An important issue here is that the total number of
electrons, Nref , be kept constant — especially when observables at two different values of the gate voltage N are
compared.8 In Ref. 8 this was implemented by taking this
thermodynamical average in a canonical ensemble. In our
case, however, we can take the average in a grand canonical ensemble, since, in the non-equilibrium formalism, it
is sufficient that the dynamics conserves the number of
electrons.
The one-dimensional dynamics of the Hamiltonian
(3.1) applies to the region x < 0 only; the dynamics in
the quantum dot is described by the boundary condition
(3.3). This boundary condition is difficult to implement
in the calculations, and it is advantageous to reformulate
the problem in terms of a one dimensional Hamiltonian
of the form (3.1), where the x integration extends over
the entire real axis. In this formulation of the problem,
the boundary condition (3.3), or its equivalent for a dot
in which scattering is not fully coherent, is replaced by an
effective action S that involves the fermion operators at
x = η, where η is a positive infinitesimal. Such a formulation of the problem has first been given by Aleiner and
Glazman,6 see also Ref. 8. Here we will use an equivalent
but simpler effective action formulation.

(3.12)

j=1

(3.13)
where
rj =

2vj
2 , j = 1, . . . , N.
1 + vj

(3.14)

The parameters rj (or vj ) describe how well the fictitious
lead is coupled to the point contact. In the limit rj → 1,
j = 1, . . . , N , the fictitious lead is fully decoupled. We
will take the limit rj → 1 at the end of the calculation.
In writing Eq. (3.13), the same regularization of fermion
operators has been employed as in Ref. 8.
The operator identity (3.3) applies to left-moving
fermions at the moment when they exit the dot. With
the inclusion of the fictitious lead, this means that one
should take ψL at x = 2η, not at x = 0. Hence Eq. (3.3)
ˆ
ˆ
gives a relation between ψL (2η) and ψR (0),
N

ˆ
ψjL (2η, t) =
k=1

ˆ
dτ Sjk (t, t − τ )ψkR (0, t − τ ).
(3.15)

The advantage of separating the Hamiltonian into the
ˆ
ˆ
contributions H0 and H1 is that the problem described
by the Hamiltonian H0 alone is exactly solvable, see Sec.
ˆ
III E. The effect of H1 is then treated in perturbation
theory. Using Eq. (3.15), the expectation value of the
current at time t can be expressed in terms of an effective
action S,
ˆ
Ij (t) = Tc e−iS Ij (t)

0

,

(3.16)

9

x= 0 η 2η
reservoirs

R

tref
dot

t
FIG. 4: Integration contour for Eq. (3.16).

L

to infinity and set

H0
H1

N

fictitious lead

ˆ
H0 = ivF :
j=1

reservoirs

R

dot

fictitious lead
FIG. 3: Inclusion of a fictitious lead at the point contact. Top
panel: With the Hamiltonian H0 , the fictitious lead couples
ideally to outgoing electrons (solid lines), so that all electrons
that exit the dot enter into the fictitious lead. The Hamiltonian H1 describes scattering of electrons exiting the dot
into the physical lead and backscattering of electrons coming
from the fictitious lead (dotted line). The amplitude for this
scattering process in channel j is irj , j = 1, . . . , N . Bottom
panel: If rj = 1, the fictitious lead is decoupled again and the
original problem is restored.

(3.17)

c
N

= 2vF

dt1
c

N

dτ

vj
j=1 k=1

†
× ψjL (0, t1 )Sjk (t1 , t1 − τ )ψkR (0, t1 − τ )
†
+ ψkR (0, t1 − τ )(S † )kj (t1 − τ, t1 )ψjL (0, t1 ) ,

where we dropped reference to the infinitesimal η. It
is important to note that in this effective action, the
contour-ordering occurs according to the ‘contour time’
t1 , not according to the ‘scattering delay time’ τ .
Both the current operator (3.11) and the effective action (3.17) do not contain any reference to the electrons
after they have passed through the point contact, so that
one may send the upper integration boundary in Eq. (3.1)

(3.18)
2

,

ˆ
where, as before, Ndot is expressed in terms of the fermion
fields for x < 0 only, see Eq. (3.2). In the system deˆ
scribed by the Hamiltonian H0 , the fermion fields ψR
have chemical potentials µR corresponding to the chemical potentials in the reservoirs, whereas the fermion fields
ψL have the chemical potential µL of the fictitious lead.
For the physically relevant case rj → 1, j = 1, . . . , N , the
fictitious lead is fully decoupled from the point contact.
In that limit, all physical observables become independent of the choice of µL .
The effective action (3.17) is different from the effective
action used in Refs. 6,7,8: the kernel in the latter action
is a matrix L, which is a function of the scattering matrix
S of the quantum dot.

D.

ˆ
where the current operator Ij is given by Eq. (3.11)
above, “c” denotes the Keldysh contour, see Fig. 4, Tc
denotes contour ordering along the Keldysh contour, the
time-dependence of the operators is that of the interacˆ
tion picture with respect to H0 , and the average . . . 0
denotes an average with respect to H0 at reference time
tref . The effective action S is
ˆ
dt1 H1 (t1 )

∂ ˆ
ˆ†
dx ψjL (x) ψjL (x)
∂x
−∞

∂ ˆ
ˆ†
ˆ
− ψjR (x) ψjR (x) : +Ec Ndot − N
∂x

L

S =

∞

Relaxation inside quantum dot

If the fictitious lead used for the above derivation of
the effective action is fully coupled to the point contact
(i.e.,, if rj = 0, j = 1, . . . , N ), every electron exiting the
quantum dot will escape towards the fictitious lead, not
towards the (physical) reservoirs. Setting the chemical
potential µL of the fictitious lead such that no net current
is drawn, one finds that each electron that escapes into
the fictitious lead is replaced by another one — without
phase relationship.50,51 Hence, coupling to the fictitious
lead allows for complete phase and energy relaxation of
electrons exiting the dot, just before they pass through
the point contact to the reservoirs.
The simplest model for incomplete relaxation in the
quantum dot is to use the coupling to the fictitious lead
with 0 < rj < 1. In this case, the electrons are allowed to escape into the fictitious lead and relax, but
with finite probability only. Hence imperfect coupling to
the fictitious lead describes a quantum dot with a finite
probability that the electron exits the dot without relaxation. Of course, this model is over-simplified, because
all relaxation is localized at the point contact. A more
realistic, but still phenomenological model for relaxation
in the quantum dot is discussed in appendix C.
A connection can be made between the effective action
(3.17) for general rj and the ’fully incoherent’ limit of
the problem, with nonideal contacts.1,2,3,4,22,23 Hereto,

10
the fictitious lead is interpreted as the “quantum dot’,
the scattering matrix Sij (t, t′ ) is replaced by the identity
matrix, Sij (t, t′ ) = δij δ(t − t′ ), and one identifies the
reflection matrix r with the contact’s reflection matrix
rc . In this case, energy and phase relaxation inside the
dot is still complete, but a nonzero value of the reflection
2
probabilities rj allows one to describe a nonideal coupling
between dot and reservoirs.
E.

Correlation functions

In order to formulate a perturbation theory in the effective action (3.17), we need the contour-ordered fermion
ˆ
correlation functions of the Hamiltonian H0 . The equilibrium correlators were calculated using the bosonization
method in Ref. 8. The new elements in the nonequilibrium case are few. We discuss them now in physical
terms, with the details being relegated to appendix A.
We specify the non-equilibrium conditions applied to
the system through the chemical potentials µjL (t+x/vF )
and µjR (t − x/vF ) of the left and right moving electrons.
Although the chemical potentials of the fictitious leads
should not appear in physical quantities in the rj → 1
limit, we retain them in order to illustrate how this happens.
ˆ
In the absence of scattering, i.e. setting H1 = 0, the
Hamiltonian is quadratic and a mean-field treatment is
thus exact. The average density of left moving electrons
at the contact is simply related to their chemical potential
ρjL (0, t) =
ˆ

1
µjL (t),
2Ï€vF

ρjR (t) =
ˆ

1
2Ï€vF

(3.19)

N
0

Ec −Ec N τ /π
e
θ(τ ),
imk (τ ) = δmk δ(τ − 0 ) −
Ï€

(3.20)
(3.21)

where Vd (t) was defined in (1.1). The average current at
the contact is

Ij (t) = −evF ( ρjR (0, t) − ρjL (0, t) )
ˆ
ˆ
(3.22)
e
ˆ
= −
µjR − µjL − 2Ec Ndot (t) − N .
2Ï€
ˆ
Thus with an excess charge in the dot Ndot = N , there
is a contribution from the interaction. We solve (3.20)
by writing the excess charge as an integral of the current

ρjR (t) =
ˆ

1
µjR (t)
2Ï€vF

(3.23)

N

t
Ec
dt′ ρkR (t′ ) − ρkL (t′ ) ,
ˆ
ˆ
Ï€
k=1
where we have introduced the convention that fields without a position label are taken to be evaluated at the origin. The solution of (3.23) is

−

dτ ijk (τ ) (µkR (t − τ ) − µkL (t − τ )) ,

(3.24)

in channel j is given by

where we defined the kernel

+

1
µcj (t),
2Ï€vF
µcj (t) = µjR (t) + eVd (t),

ρjR (0, t) =
ˆ

∞

µjL (t) +
k=1

For right movers, however, the presence of the charging
interaction in the lead means that the ‘electrochemical
potential’ µc is the relevant quantity

(3.25)

Ij (t) = −

e
2Ï€

N

∞

dτ ijk (τ )
k=1

0

× [µkR (t − τ ) − µkL (t − τ )] .

(3.26)

Note that in the dc bias situation Eq. (3.26) simplifies to
+

0 being a positive infinitesimal. The kernel ijk (τ ),
which appears frequently in the following development,
relates the current at the contacts to the chemical potentials in the leads, accounting properly for the effect of
the charging interaction. Note the appearance of the RC
time π/Ec N characteristic of an ideal N -channel contact.
Combining the above results, we find that the current

Ij = −

e
2Ï€

N
k=1

δjk −

1
N

(µkR − µkL ) ,

(3.27)

so that for the two-terminal set up
Ij = −

e
Λjj (∆µ1 − ∆µ2 ) ,
2Ï€

(3.28)

11
where ∆µ1,2 = µR1,2 − µL1,2 are the chemical potential
differences in the two contacts, and Λij is the matrix
defined in Eq. (2.4).
Correlators of many fermion operators will be expressed in terms of the single-fermion Green functions,
ˆ
ˆ†
− i Tc ψmL (s)ψnL (t) = GmnL (s, t)
ˆ
ˆ†
−i Tc ψmR (t)ψnR (s) = GmnR (t, s).

where λ is a positive infinitesimal and

(3.29)
(3.30)

These read
GmnL (t, s) = −
GmnR (t, s) = −

δmn T e−i(φmL (t)−φmL (s))
,
2vF sinh[πT (t − s − iλsignc (t − s))]

δmn T e−i(φmR (t)−φmR (s))
,
2vF sinh[πT (t − s − iλsignc (t − s))]
(3.31)

∞

dτ µmL (t − τ ),

φmL (t) =
0

N

∞

dτ µmL (t − τ ) +

φmR (t) =
0

(3.32)
∞

k=1

0

dτ ′ imk (τ ′ )(µkR (t − τ − τ ′ ) − µkL (t − τ − τ ′ )) ,

are the integrals of the electrochemical potentials of right and left movers. It is straightforward to see that (3.31)
is consistent with (3.19) and (3.24). The single-fermion Green function can be written as a sum of a Keldysh part,
which does not depend on the contour positions, and a delta function,
1 K
i
δmn δc (t, s),
G
(t, s) −
2 mnL
2vF
i
1
δmn δc (t, s).
GmnR (t, s) = GK (t, s) −
mnR
2
2vF
GmnL (t, s) =

(3.33)

Here we abbreviated
δc (s, t) = signc (s − t)δ(s − t).

(3.34)

Now the contour-ordered average of a product of fermion operators is calculated as
ˆ
ˆ†
ˆ
ˆ†
ˆ
ˆ†
ˆ′
ˆ†
(−i)n+m Tc ψl1 L (s1 )ψk1 L (t1 ) . . . ψln L (sn )ψkn L (tn )ψk′ R (t′ )ψl′ R (s′ ) . . . ψkm R (t′ )ψl′ R (s′ )
1
1
m
m
m
1

(−1)P +Q

=
P,Q

1

m

n

′
Gkj l′ R (t′ , s′ )
j Q(j)
Q(j)

Gli kP (i) L (si , tP (i) )
j=1

i=1

i,j

f (ti − t′ )f (si − s′ )
j
j
f (ti − s′ )f (si − t′ )
j
j

1/N

,
(3.35)

where P and Q are permutations of the numbers i = 1, . . . , n and j = 1, . . . , m, respectively, and the function f (t − t′ )
is
ln f (t − t′ ) = ln

λEC N eC
EC N
−
Ï€
Ï€

∞

dζe−EC N ζ/π ln
0

sinh[πT (t − t′ + ζ − i0+ signc (t − t′ ))]
.
sinh(πT ζ)
(3.36)

In the last equation, 0+ is a positive infinitesimal and C ≈ 0.577 is the Euler constant. We separate ln f (t − t′ ) into
a contour-independent (Keldysh) part and a contour-dependent part,
ln f (t − t′ ) = ln |f (t − t′ )| + iN κ0 (t − t′ )signc (t − t′ ),

(3.37)

12
where
κ0 (t − t′ ) =

Ï€
N

′

1 − e−Ec N (t −t)/π θ(t′ − t).

(3.38)

Here θ(x) = 1 if x > 0 and 0 otherwise.
Note that the result (3.35) is compatible with the symmetry of the Hamiltonian H0 . At first glance this is U (N ) ×
U (N ), the two factors corresponding to the right and left movers. Certainly this is the case for the non-interacting
model with Ec → 0, where (3.35) reduces to the usual Wick’s theorem result. The interaction, however, leads to
the chiral anomaly, as in the Schwinger model, that reduces the symmetry to SU (N ) × SU (N ) × U (1). Physically,
this is because the interaction can scatter right movers to left movers and vice versa. This allows a second type of
correlation function to exist, characterized by the SU (N )-invariant antisymmetric tensor εi1 i2 ...iN . We will refer to
such correlators as anomalous. They describe the scattering of right movers to left movers (and vice versa) by the
interaction.2,8 Below we list the first of these nontrivial correlators for N = 1 and for N = 2. For N = 1 one has
Ec eC f (s − t′ ) −i(2πN +φR (t′ )−φL (s))
e
,
2Ï€ 2 vF f (0)
Ec eC f (t − s′ ) −i(φL (t)−φR (s′ )−2πN )
ˆ
ˆ†
−i Tc ψL (t)ψR (s′ ) = −
e
.
2Ï€ 2 vF f (0)

ˆ
ˆ†
− i Tc ψR (t′ )ψL (s) =

(3.39)

For N = 2 the relevant anomalous correlators are
ˆ
ˆ†
ˆ
ˆ†
(−i2 ) Tc ψ1R (0, t′ )ψ1L (0, t2 )ψ2R (0, t′ )ψ2L (0, t4 )
1
3
C

=

Ec e
Ï€ 2 vF f (0)

′

′

e−i(2πN +φR1 (t1 )−φL1 (t2 )+φR2 (t3 )−φL2 (t4 )) [f (t2 , t′ )f (t4 , t′ )f (t2 , t′ )f (t4 , t′ )]
1
1
3
3

ˆ
ˆ†
ˆ
ˆ†
(−i2 ) Tc ψ1L (0, s1 )ψ1R (0, s′ )ψ2L (0, s3 )ψ2R (0, s′ )
2
4
=

Ec eC
Ï€ 2 vF f (0)

0

2
1/2

,

0

2

e

i(φL1 (s1 )−φR1 (s′ )+φL2 (s3 )−φR2 (s′ )−2πN
2
4

Notice that the ‘vacuum-angle’ describing the U (1) symmetry breaking is just 2πN .
For the perturbation theory calculation of the current,
we need contour-ordered correlators of the form

[f (s1 , s′ )f (s1 , s′ )f (s3 , s′ )f (s3 , s′ )]
2
4
2
4

1/2

.

(3.40)

respectively. Factors Fm for the creation operators for
left-moving and right-moving fermions in channel k are
−Fj,kL and −Fj,kR , respectively.

ˆ
ˆ
ˆ
Tc Ij (t)A(t′ ) . . . A(t′ ) ,
1
n
ˆ
where the symbols A(t′ ) represent creation or annihilation operators for left or right moving fermions. There is
a simple relation between such correlators and the corresponding correlator without current operator,
ˆ
ˆ 1
ˆ n
Tc Ij (t)A1 (t′ ) . . . An (t′ )
n

ˆ 1
ˆ n
= Tc A1 (t′ ) . . . An (t′ )

Fm (t, t′ ).
m

(3.41)

IV.

CALCULATION OF THE CURRENT

m=1

Here the Fm depend on whether the corresponding operator Am is a creation or annihilation operator for left or
right moving fermions. For an annihilation operator for
left-moving and right-moving fermions in channel k, one
has
eT ∞
dτ ijk (t − t1 − τ ) coth(πT τ )
2i 0
e
(3.42)
− signc (t − t1 )ijk (t − t1 )
2

Fj,kL (t, t1 ) = −

With the help of the correlators listed in the previous
section, the perturbation expansion (3.16) can be evaluated. In this section, we first describe a full evaluation of
I(t) up to second order in the effective action (first order
for N = 1). We then reorganize the perturbation theory,
and calculate I(t) to all orders in S, but in a perturbation expansion in 1/N , which turns out to correspond to
a loop expansion.52

13
A.

To first order in the effective action one has

Expansion in effective action

To zeroth order in the effective action, one finds that
the current in channel j is given by
1
I0,j (t) = ievF GK (t, t) − GK (t, t)
jjR
jjL
2
=

e
2Ï€

ˆ
Ij,1 (t) = −i S Ij (t)

(4.1)

0

.

(4.2)

N

dt1
k=1

ijk (t − t1 ) [µkL (t1 ) − µkR (t1 )] ,

in agreement with Eq. (3.26) and with ijk (t − t1 ) defined
in Eq. (3.25) above.

I1 (t) = 4vF vIm

Substituting Eq. (3.17) for the effective action and using
the fermion correlation functions of the previous section,
one finds Ij,1 (t) = 0, except for N = 1. In that case, one
has

dτ e−2πiN S(t1 , t1 − τ )[−FL (t, t1 ) + FR (t, t1 − τ )]

dt1
c

−iEc eC f (τ )
2Ï€ 2 vF f (0)

e−i(φR (t1 −τ )−φL (t1 )) .
(4.3)

In this equation, it is important to note that all contour-dependence should depend on t1 only; the time difference τ
should not enter into the contour signs. We also note that the argument of f (Ï„ ) is always positive, so that f (Ï„ ) does
not depend on the contour position, the only the contour-dependent part of the function F that contributes to the
integration, and that the contribution from right moving fermions is zero by causality, so that
I1 (t) = −

2evEc eC
Ï€ 2 f (0)

dt1 i(t, t1 )

dτ f (τ )Re S(t1 , t1 − τ )e−i(φR (t1 −τ )−φL (t1 ))−2πiN

(4.4)

One verifies that for a time-independent situation — S(t, t − τ ) independent of t and chemical potentials µL and µR
independent of time — the current is zero.
The second order calculation proceeds similarly. We first restrict our attention to the case N > 2. Using matrix
notation, we then find
e
Ij,2 (t) = − (2vF )2
2

dt1 ds1
c

2

dτ1 dσ1

f (τ1 )f (σ1 )
f (t1 − s1 + σ1 )f (s1 − t1 + τ1 )

1/N

× tr v [ij (t, t1 )signc (t − t1 ) − ij (t, s1 )signc (t − s1 )]

× GL (s1 , t1 )S(t1 , t1 − τ1 )GR (t1 − τ1 , s1 − σ1 )S † (s1 − σ1 , s1 ).

(4.5)

Here ij is an N × N diagonal matrix, closely related to the kernel defined in Eq. (3.25),
(ij )mn (τ ) = δmn ijm (τ ).

(4.6)

At this point, we write the Green functions GR and GL as the sum of the Keldysh component and a delta function,
see Eq. (3.33). Doing this, there will be four terms: one term with delta functions for both GL and GR , two terms
with one delta function and one Keldysh Green function, and one term with two Keldysh Green functions. The first
term is easily shown to vanish. In the second and third term, the interaction function f cancels, and one finds
Ij,2,0 (t) = 2ivF e

dt1 tr v 2 ijL (t, t1 )

× GK (t1 , t1 ) −
L

dτ1 dσ1 S(t1 , t1 − τ1 )GK (t1 − τ1 , t1 − σ1 )S † (t1 − σ1 , t1 ) .
R

(4.7)

Here the last index “0” indicates that, apart from the difference between ijk (τ ) and δjk δ(τ ), Eq. (4.7) — together
with the zeroth-order result of Eq. (4.1) — reproduces precisely what one would obtain from the scattering approach
for non-interacting fermions. Finally, in the remaining term, the interaction contribution is retained. In fact, it is just

14
because of the contour-dependence of the function f that the one finds a nonzero result. We then find
2
Ij,2,ee (t) = −2eivF

t1

dt1

ds1

dτ1 dσ1

f (τ1 )f (σ1 )
f (t1 − s1 + σ1 )f (s1 − t1 + τ1 )

1/N

sin[κ0 (s1 − t1 + τ1 )]

× tr v 2 ij (t, t1 ) GK (s1 , t1 )S(t1 , t1 − τ1 )GK (t1 − τ1 , s1 − σ1 )(S † )(s1 − σ1 , s1 )
L
R
− GK (t1 , s1 )S(s1 , s1 − σ1 )GK (s1 − σ1 , t1 − τ1 )(S † )(t1 − τ1 , t1 ) .
L
R

(4.8)

For N = 2 there is an additional oscillating contribution to the current,
Ij,osc (t) = 8ev1 v2

2Ec eC
Ï€ 2 f (0)

2

Im e−2πiN

t1

dt1

dt2 dt′ dt′ sin(φ(t2 , t′ ))
1 2
1

′

′

× det[ij (t, t1 )eiφL (t1 ) S(t1 , t′ )e−iφR (t1 ) , eiφL (t2 ) S(t2 , t′ )e−iφR (t2 ) ]
1
2

× |f (t1 , t′ )f (t1 , t′ )f (t2 , t′ )f (t2 , t′ )|1/2 ,
1
2
1
2

(4.9)

where we defined
det(B(1), . . . , B(N )) =

1
N!

(−1)P B(1)k1 ,P (k1 ) . . . B(N )kN ,P (kN ) ,

(4.10)

P k1 ,...,kN

where B(j) is an N × N matrix, j = 1, . . . , N . Equations
(4.4), (4.8), and (4.9) represent the interaction corrections to the current.
This result is found to agree precisely with the linear
dc conductance (2.10) calculated in Refs. 7 and 8 in the
dc bias limit, to first order in the bias. To see this, we
set µR1 − µR2 = −eV , expand to linear order in the bias
voltage, take S(t, t − τ ) to depend on τ only, set µjL = 0
for all channels j = 1, . . . , N in the fictitious lead, use
the approximations |f (τ )| ≈ λπT /vF sinh[πT |τ + t0 |],
κ0 (τ ) = πθ(−τ − t0 )/N valid for τ ≪ t0 and τ ≫ t0 ,
and set vj = 1, so that rj = 1 (see Eq. 3.14), which corresponds to detaching the fictitious lead and thus restoring the original system. The oscillating current corrections (4.4) and (4.9) reproduce the periodic-in-N interaction corrections to the dot’s ‘electrochemical capacitance’
Cc = dQ/dV found in Ref. 6 if we set µjL = 0, choose
a time-dependent chemical potential µjR (t) = eV t for
all channels, and expand in V . [In order to recover the
non-periodic interaction correction to the capacitance reported in Ref. 8 from Eqs. (4.7) and (4.8) one has to
include time-dependencies in both µL and µR in order
to ensure that no charge escapes through the fictitious
lead.]
However, truncating at second order in the effective
action S is not a satisfactory description for a fully coherent quantum dot as there is no a priori reason why the
effective action is small in this case. This will be made
explicit in the next subsection, where we report a calculation to all orders in S (but for N large). However, that
Eqs. (4.4), (4.8), and (4.9) cannot be the correct interaction correction for a fully coherent dot becomes clear
when one realizes that the current in fact depends on the
chemical potentials µjL , j = 1, . . . , N , of the fictitious
reservoir, despite the fact that one has taken the limit

rj → 1, j = 1, . . . , N . (Note that this dependence on the
fictitious chemical potentials is not manifest in the linear response calculation of Refs. 6,7,8 because µjL = 0,
j = 1, . . . , N , in equilibrium.)
Of course, truncating the perturbation expansion at
second order in S is justified if there is a reason why
S is small. This is the case, e.g., if the rj are small,
j = 1, . . . , N . Then the fictitious lead is strongly coupled
to the point contact, and serves as a source of relaxation.
In that case the chemical potentials µjL must be chosen
such that no current flows through the fictitious lead at
any point in time. A more realistic (but still phenomenological) model of relaxation is to couple to quantum
dot itself (rather than the point contact) to a fictitious
lead with many weakly coupled channels.44,50,51,53,54,55
As shown in appendix C, this model results in an effective action S similar to (3.17), but with a sub-unitary
scattering matrix S. If relaxation is strong, S is small,
and expanding in the action is justified. For this situation, we recover the interaction corrections found in Refs.
6,7,8.

B.

Large number of channels

We have not been able to do a full calculation to all
orders in S. However, by organizing each order in the
effective action to the (formal) power of N it carries, we
have been able to calculate the current to all orders in S
while expanding in 1/N . The reason that an expansion in
1/N is possible is that the correlator (3.35) of interacting
fermions admits a systematic expansion around the noninteracting correlator, by expanding
f 1/N = 1 +

1
ln f + . . . .
N

(4.11)

15
Counting the power of 1/N is done keeping the RC time
Ï€/Ec N and the dwell time Ï„d constant, so that the only
factors 1/N arise from the expansion (4.11) and from explicit summations over the channel indices. Oscillating
contributions with an explicit dependence on the dimensionless gate voltage N , involve N th order in perturba-

1
(−i)2n S n Ij
(2n)!
1
(−2ivF )2n
=
(n!)2

Ij,n (t) =

tion theory and cannot be considered in this method.
Non-oscillating contributions to the current occur to
even order in the effective action only. The nonoscillating current contribution of order 2n in the action
reads

0

dt1 . . . dtn ds1 . . . dsn

...

′
k1 ,l1 ,k1 ,l′
1

′
kn ,ln ,kn ,l′
n

dτ1 . . . dτn dσ1 . . . dσn vk1 . . . vkN

c

′
′
× vl1 . . . vlN Sk1 ,k1 (t1 , t1 − τ1 ) . . . Skn ,kn (tn , tn − τn )(S † )l′ ,l1 (s1 − σ1 , s1 ) . . . (S † )l′ ,ln (sn − σn , sn )
n
1

ˆ†
ˆ
ˆ†
ˆ
× Tc ψk1 L (t1 )ψk′ R (t1 − τ1 )ψl′ R (s1 − σ1 )ψl1 L (s1 ) . . .
1

1

ˆ†
ˆ
ˆ†
ˆ
ˆ
× ψkn L (tn )ψk′ L (tn − τn )ψl′ R (sn − σn )ψln L (sn )Ij (t)
n

n

As before, the time differences denoted by Greek symbols do not affect the contour ordering for the fermion
operators.
The average of the product of 4n fermion operators
and one current operator in Eq. (4.12) admits a standard diagrammatic representation if the factors 1/N ln f
in the expansion (4.11) of the general average (3.35) are
considered ‘interaction lines’. Without ‘interaction lines’,
the average is simply given by Wick’s theorem: Denoting
each single-fermion Green function by a solid line, Wick’s
theorem expresses the average as a sum over partitions
in ‘bubbles’ of alternating single-fermion Green functions
for right moving and left moving fermions. One bubble
contains the current vertex i. In our formal expansion in
1/N , each bubble contributes a factor N , as it contains a
trace over the channel index. Each ‘interaction line’, on
the other hand, contributes a factor 1/N . Only diagrams
in which all ‘bubbles’ are connected by interaction lines
contribute to the current. This means that, to leading order in 1/N , the relevant diagrams have a ‘tree’ structure,
see Fig. 5: Every ‘bubble’ has precisely one connection
to the current vertex via intermediate bubbles and interaction lines. Subleading contributions to the current
contain ‘loops’.
To leading order in 1/N , we only have to examine the
‘tree’ diagrams, plus the current contribution that is zeroth order in the action, see Eq. (4.1). Combining these,
we find
e
Ij (t) =
2Ï€
+e

N

dt1
k=1

ijk (t − t1 )[µLk (t1 ) − µRk (t1 )]

dt1 tr ij (t − t1 )r2 j− (t1 ),

(4.13)

where we used Eq. (3.14) to express v in terms of the

0

.

111
000
111
000
111
000
111
000
111
000
111
000

I(t) =

(4.12)

111
000
111
000
111
000
111
000
111
000
111
000
111
000

1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000

= 1 +

111 111
000 000
111 111
000 000
111 111
000 000
111 111
000 000
111 111
000 000
111 111
000 000

+

1
2

(a)

+ ...
(b)

FIG. 5: (a) Diagrammatic representation of leading-in-1/N
contribution to the current. The current vertex i is denoted
with a filled circle, the ‘bubble’ of single-fermion Green function with a solid loop, and the tree structure with a hatched
triangle. The defining equation for the tree structure is shown
in (b), where the dotted arrow represents the interaction line.

reflection matrix r. In Eq. (4.13), we abbreviated the
contribution of the bubble by
j− (t) =

1
ivF GK (t, t) −
L
2

dτ dσS(t, t − τ )

˜
× GK (t − τ, t − σ)S † (t − σ, t)
R

(4.14)

where
˜
˜
˜
GK (t, s) = GK (t, s)e−i(φ(t)−φ(s)) .
R
R

(4.15)

˜
The phase φ(t) satisfies the self-consistent equation
˜
φ(t) = 2

dt1 κ0 (t1 − t)tr r2 j− (t1 ).

(4.16)

The quantity j− is nothing but the Landauer expression
for the difference of the current flowing into the leads
coming from the quantum dot [second term in Eq. (4.14)]
and coming from the fictitious reservoir [first term in Eq.
(4.14)].

16
the dot potential is
t N

=

+

CVd (t) =

dt′ Ij (t′ ),

(4.17)

k=j

FIG. 6: Diagrammatic representation of the effective interaction κ (thick arrow) in terms of the bare interaction line ln f
(thin arrow) and a ‘bubble’ of single-fermion Green functions
(solid loop).

˜
The overall phase factor that appears in GK (t, s) can
mR
be written in terms of the electrochemical potential µc
t

˜
˜
φmR (t) + φ(t) − φmR (s) + φ(s) =

dt′ µcm (t′ ).

s

Here µcm = µm + eVd [the plus sign arises because the
interaction is in the leads, not the dot, cf. Eq. (3.21)] and

κ(t, s) = κ0 (t, s) − 2vF

where C is the dot’s geometrical capacitance and we used
the specific form of κ0 (t′ − t) and ijk (τ ). Note that in
the limit rj → 1 the µmL (t) drop out of (4.13). Equation
(4.17) is just the same as (1.1), written in terms of the
currents. Hence, to leading order in 1/N , our formalism
recovers the self-consistent theory for the effect of the
Coulomb interaction on transport through quantum dots.
To subleading order in 1/N , one has to solve diagrams
with one loop. The ‘interaction lines’ in these diagrams
are replaced by an effective interaction, which is an RPAlike series, see Fig. 6. Denoting the effective interaction
by κ(t, s), one finds that κ(t, s) satisfies the self-consistent
equation

dt1 dτ1 dσ1 (κ0 (t, t1 − τ1 ) − κ0 (t, t1 − σ1 ))

˜
× tr r2 S(t1 , t1 − τ1 )GK (t1 − τ1 , t1 − σ1 )S † (t1 − σ1 , t1 )κ(t1 , s).
R

(4.18)

Note that κ(t, s) = 0 if t ≥ s and that κ(t, s) is real.
The one-loop interaction correction to the current can be represented by four diagrams, see Fig. 7. The first and
second diagram in Fig. 7 represent a correction to j− ,
j− (t) =

1
ivF GK (t, t) −
L
2
2
+ vF Im

˜
dτ dσS(t, t − τ )GK (t − τ, t − σ)S † (t − σ, t)
R

ËœR
ds1 dτ1 dσ1 κ(s1 , t − τ1 )S(t, t − τ1 )GK (t − τ1 , s1 − σ1 )S † (s1 − σ1 , s1 )

× (1 − r2 )GK (s1 , t) + r2
L

ËœR
dτ2 dσ2 S(s1 , s1 − τ2 )GK (s1 − τ2 , t − σ2 )S † (t − σ2 , t) ,

(4.19)

whereas the third and fourth diagram represent an additional renormalization of the distribution function for right
˜
moving fermions, which can be represented by a change in the relation between to GR and GR ,
˜
˜
˜
GK (t, s) = GK (t, s)e−i(φ(t)−φ(s))+α(t,s) .
R
R

(4.20)

For the calculation of α(t, s), the regularization of the fermion operators at the impurity site is important, which
makes the actual calculation rather cumbersome. The final result reads
1 2
α(t, s) = − vF
2

dt1 dt2 (κ(t1 − t) − κ(t1 − s))(κ(t2 − t) − κ(t2 − s))tr

GK (t1 , t2 )GK (t2 , t1 )
L
L

− (1 − r2 )GK (t1 , t2 ) + r2
L

˜
dτ1 dσ2 S(t1 , t1 − τ1 )GK (t1 − τ1 , t2 − σ2 )S † (t2 − σ2 , t2 )
R

× (1 − r2 )GK (t2 , t1 ) + r2
L

ËœR
dτ1 dσ2 S(t2 , t2 − τ2 )GK (t2 − τ2 , t1 − σ1 )S † (t1 − σ1 , t1 )

Equations (4.13), (4.16), (4.18), (4.19), (4.20), and (4.21)

.

(4.21)

give the solution of the problem up to sub-leading order

17
where Λ was defined in Eq. (2.4) above. We will be using
the Fourier transform of the scattering matrix and the
Green functions,
∞

dtS(t)eiεt ,

(5.2)

dtS † (t)eiεt ,

S(ε) =

(5.3)

0

S † (ε) =

0
−∞

FIG. 7: Diagrammatic representation of subleading-in-1/N
corrections to the current

in 1/N .
Note that the leading corrections to the theory are
in the form of a modified expression for the current j− ;
Equation (4.16) and, hence, the relation (4.17) between
Ij (t) and the self-consistent potential Vd remains unchanged.
The interaction correction to j− , contained in the second and third line of Eq. (4.19), consists of a term proportional to r2 and a term proportional to 1 − r2 . When
expanding in the action and truncating the expansion at
second order, as is done in Sec. IV A and in Refs. 7 and
8, one recovers the first term only, with prefactor unity
instead of 1 − r2 . However, upon setting r2 → 1, which
is the relevant limit for a coherent quantum dot with a
unitary scattering matrix, the prefactor of that first term
vanishes, and the interaction correction to j− is given by
the second term instead. It is this replacement of the
first term by the second term in the interaction correction that is the essential difference between the theories
of Refs. 7 and 8 on the one hand and that of this work
and Ref. 24 on the other hand.
V.

STEADY STATE TRANSPORT

In this section we analyze the results of the previous
section for the case of steady state transport through the
quantum dot. We assume that the chemical potentials
in the reservoirs and the gate voltages defining the shape
of the dot are time independent. This implies that the
scattering matrix S(t, t − τ ) is independent of t and the
Green functions GL (t, t−τ ) and GR (t, t−τ ) are independent of t. Current conservation then implies that there is
no net current into the dot, j Ij = 0 – see Eq. (3.27).
This allows us to focus our calculation on the weighted
difference of currents in the left and right leads,
N

Λj Ij ,

I=
j=1

(5.1)

GK
mnR,L (ε) =
=

iεt
dtGK
mnR,L (t)e

i
δmn (2fR,L (ε) − 1),
vF

(5.4)

where f is the distribution function.

A.

Fully coherent dot

We first consider the case r2 = 1, corresponding to a
fully coherent dot.
˜
˜
We first consider the function φ. By Eq. (4.16), φ
depends on tr j− only. For the time independent case,
one has

tr j− = −

˜
N ∂φ
.
2π ∂t

(5.5)

˜
Calculating the derivative ∂ φ/∂t from Eq. (4.16) one then
finds
˜
∂φ
N
= −
∂t
Ï€
˜
∂φ
=
,
∂t

dt1

˜
∂κ0 (t − t1 ) ∂ φ
∂t
∂t
(5.6)

˜
which implies that the derivative ∂ φ/∂t is left undetermined by the self-consistency equation (4.16); any time˜
independent value for ∂ φ/∂t is a solution of the selfconsistency equation. This is no problem for steady state
˜
transport; in the previous section we have seen that φ is
determined once the time-dependence of bias and gate
voltages is taken into account. In this section, we will set
˜
φ = 0 from now on.
The effective interaction function κ can be solved from
Eq. (4.18) using Fourier transform,

18

κ(ω) = κ0 (ω) κ0 (ω) −

i
Ï€

−1

dεtr S(ε − ω)(fR (ε − ω) − fR (ε))S † (ε)

.

Here κ0 (ω) is the Fourier transform of κ0 (τ ),
κ0 (ω) =

dτ κ0 (τ )eiωτ .

(5.7)

This effective interaction is the same as that obtained by Golubev and Zaikin.24 We further need to calculate the
function α(t − s), see Eq. (4.20). Fourier transforming Eq. (4.21) one has
α(τ ) = −

1
2Ï€ 2

dω(1 − cos(ωτ ))|κ(ω)|2

× ωN coth(ω/2T ) −

±

(5.8)

dεtr S(ε ± ω)fR (ε ± ω)S † (ε ± ω)S(ε)(1 − fR (ε))S † (ε) .

Finally, with these results the current through the dot is
e
e
tr ΛµR +
dεtr ΛS(ε)fR (ε)S † (ε)
2Ï€
2Ï€
e
dωdεκ(ω)tr Λ S(ε)fR (ε + ω)S † (ε + ω)S(ε)(1 − fR (ε))S † (ε)
+ 2 Im
2Ï€
+ S(ε)(1 − fR (ε + ω))S † (ε + ω)S(ε)fR (ε)S † (ε)
e
dωdε|κ(ω)|2 tr ΛS(ε)[2fR (ε) − fR (ε − ω) − fR (ε + ω)]S † (ε)
+
16Ï€ 4

I = −

× ωN coth

ω
−
2T

±

dε′ tr S(ε′ ± ω)fR (ε′ ± ω)S † (ε′ ± ω)S(ε)(1 − fR (ε′ ))S † (ε′ ) .

(5.9)

In order to find the linear conductance, we set µR = −ΛeV , µL = 0, expand in V , and find I = GV , with
G =

e2
e2
∂f (ε)
tr ΛS(ε)ΛS † (ε)
(5.10)
tr Λ2 −
dε −
2Ï€
2Ï€
∂ε
e2
∂f (ε)
+ 2 Im
κ(ω)tr ΛS(ε)ΛS † (ε + ω) − ΛS(ε)S † (ε + ω)S(ε)ΛS † (ε) (1 − 2f (ε + ω)).
dεdω −
2Ï€
∂ε

The same equation was derived previously by Golubev and Zaikin using a different method.24
Equation (5.9) gives the current for one particular realization of the scattering matrix S(ε). In order to find the
average and variance of the current or the conductance for an ensemble of quantum dots, we need to average S(ε)
over the appropriate ensemble of scattering matrices. Details on the averaging procedure can be found in Appendix
B. For large N , the average of a product of traces may be calculated as the product of the averages. This means that
the interaction kernel κ(ω) may be averaged separately, with the result
κ(ω) =

Ï€
1 + iωτd
.
giω iωτd [1 + (π/Ec gτd )(1 + iωτd )]

(5.11)

In Eq. (5.11), τd is the mean dwell time for electrons entering the dot. In terms of the dot’s mean level spacing ∆
and the total conductance g = g1 + g2 of the point contacts, one has
Ï„d =

2Ï€
.
g∆

(5.12)

(If levels are spin degenerate, ∆ is half the mean spacing between spin-degenerate levels.) Performing the ensemble
average for the current, one then finds, to leading and sub-leading order in N and in the absence of time-reversal

19
symmetry,
I

e
e
†
†
†
tr ΛµR +
dε[gtr Λrc f (ε)rc + tr Λ(1 − rc rc ) tr f (ε)(1 − rc rc )]
2Ï€
2Ï€ g
(iωτd )2
e
dεdωκ(ω)
+ 2 2 Im
2Ï€ g
(1 + iωτd )2

= −

(5.13)

†
†
†
× gtr Λrc f (ε + ω)rc (1 − rc rc ) tr (1 − f (ε))(1 − rc rc )

†
†
†
+ gtr Λrc (1 − f (ε + ω))rc (1 − rc rc ) tr f (ε)(1 − rc rc )
†
†
†
†
− tr Λ(1 − rc rc ) tr f (ε + ω)(1 − rc rc )rc rc tr (1 − f (ε))(1 − rc rc )
†
†
†
†
− tr Λ(1 − rc rc ) tr (1 − f (ε + ω))(1 − rc rc )rc rc tr f (ε)(1 − rc rc ) ,

From Eq. (5.13) we conclude that there is no interaction correction to the average current for ideal point contacts.
Setting µR = −eV Λ and expanding Eq. (5.13) to first order in the bias voltage V , we find the ensemble average of
the linear dc conductance,
G =

e2 g1 g2
2Ï€ g

1 − Im

dω

2g 2 (1

(g2 s1 + g1 s2 )τd [(ω/T ) − sinh(ω/T )]
+ iωτd )[1 + (π/Ec gτd )(1 + iωτd )] sinh2 (ω/2T )

Performing the frequency integral in the limit /τd ≪
T ≪ Ec g one finds24
2(s1 g2 + s2 g1 ) Ec ge1+C
e2 g1 g2
1−
,
ln
G =
2Ï€ g
g2
2Ï€ 2 T
(5.15)
which is the same result as for a quantum dot without
coherent scattering from inside the dot, see Eq. (2.8). In
the low temperature limit T ≪ /τd ≪ Ec g one finds a
saturation of the interaction correction,24
G =

e2 g1 g2
2(g2 s1 + g1 s2 ) Ec gτd
1−
. (5.16)
ln
2Ï€ g
g2
Ï€

For the quantum interference corrections we have performed calculations for the linear conductance with ideal
point contacts only. Weak localization is a small difference δG of the ensemble averaged conductance with and
without magnetic field. We find that the interaction correction to weak localization is zero, so that δG is given
by the non-interacting theory,56,57
δG = −

νs e 2 N 1 N 2
,
2Ï€ N 2

(5.17)

where νs denotes the spin degeneracy.
Closed form expressions for the interaction correction
to the conductance fluctuations could be obtained for
the limiting cases T ≪ /τd ≪ N Ec and /τd ≪ T ≪
N Ec , for which we find, in the absence of time-reversal
symmetry
2

var G =

νs e 2 N 1 N 2
2
2
1 − (πT τd )
2Ï€ N 2
3
Ec N Ï„d
4Ï€
.
ln
× 1+
Ec N 2 Ï„d
Ï€
(5.18)

.

(5.14)

and
var G =

νs e 2 N 1 N 2
2Ï€ N 2

2

Ï€
6T Ï„d

1−

2Ï€
5T N Ï„d

, (5.19)

respectively. For intermediate temperatures, the final result still contains two energy integrations, which are easily integrated numerically. Figure 8 shows the interaction
correction to the variance as a function of temperature
for the value π/Ec N τd = 0.1. In the presence of timereversal symmetry, the variance of the conductance a factor two higher than in the absence of time-reversal symmetry. We conclude that, at zero temperature, there is no
interaction correction to the conductance fluctuations, to
leading order in 1/N . At small but finite temperatures,
interactions lead to a small decrease of the conductance
fluctuations.
It is legitimate to ask how one can distinguish between
the interaction correction to weak localization and the
conductance fluctuations and the second order interaction corrections to the classical [i.e., of order N ] conductance or the second order quantum interference corrections in the non-interacting theory, as both are of order
1/N . With respect to the comparison to the second order interaction correction to the classical conductance, an
answer is readily given: this interaction correction does
not depend on the magnetic field and has no mesoscopic
fluctuations as a function of, e.g., Fermi energy or magnetic field. A formal method to distinguish the interaction corrections discussed above from second-order quantum interference corrections follows after introducing the
number of ‘orbital channels’ N o = N/νs . Then one finds
that the interaction correction is a 1/N correction, while
the second-order quantum interference correction is of
order N/(N o )2 . For the conductance fluctuations in the
absence of time-reversal symmetry this problem does not
arise, as there are no second order quantum interference
corrections in this case.40,56,57

20

1

theory of Refs. 1,3,4. For the linear conductance one then
finds

0.8

G =

var g

0.6

e2
2Ï€

2
N1 N2
Ec N e1+C
†
− tr Λ2 rc rc 1 +
ln
N
N
2Ï€ 2 T
(5.21)

0.4

if ln(Ec /πT ) ≪ N/2 but T ≪ Ec N and

0.2
0

xN

-0.2
0

G =

0.5

1

2πΤ/Ν∆

1.5

2

FIG. 8:
The interaction correction to the variance of
the conductance, made dimensionless by writing G =
(νs e2 /2π )(N1 N2 /N 2 )g. The solid curve shows the noninteracting contribution to the conductance fluctuations; the
dashed curve shows the interaction correction for π/Ec N τd =
0.1, multiplied by N to make it fit on the same scale as the
non-interacting contribution.

The absence of an interaction correction to weak localization, and to the conductance fluctuations at zero temperature, is perhaps one of the most striking predictions
of our theory. In Section VII, we will give an argument to
explain why such a renormalization leaves these ensemble averaged quantities unchanged, using the observation
that the form of the conductance correction derives from
the renormalized scattering matrix Eq. (3.6).

B.

Incoherent dot

The opposite limit of a quantum dot without coherent
reflection from inside the dot can be obtained from the
general formalism of Sec. IV by setting S(t, t − τ ) → δ(τ )
and interpreting r as the reflection matrix rc of the point
contacts. In this case, the distribution function of the left
moving fermions represents the distribution function of
electrons leaving the quantum dot. Self-consistent determination of the corresponding chemical potential µL is
problematic, however, because that would require knowledge of the average dwell time inside the dot, which is not
contained in this model.
For small rc , the results of Sec. IV A can be used. In
this case one finds
e
†
tr Λ(1 − rc rc )µR
(5.20)
2Ï€
1/N
f (0)2
1 2 ∞
dτ sin(κ0 (−τ ))
+ eivF
2
f (τ )f (−τ )
0

I = −

†
× tr Λrc rc (GK (τ )GK (−τ ) − GK (τ )GK (−τ )).
R
L
L
R

For N = 2 there is an additional oscillating interaction
correction to the current. Equation (5.20) is the nonequilibrium generalization of the original linear response

e2
2Ï€
×

N1 N2
1
1
− 1/2 Γ 1 −
N
N
2Ï€
Ec N eC
Ï€2 T

2/N
†
tr Λ2 rc rc cos

Γ

1
1
−
N
2

Ï€
N

(5.22)

if ln(Ec /πT ) ≫ N/2 (but temperature larger than a suitable lower limit3,4,8 ).
For large N , one recovers the same expressions as Golubev and Zaikin, see Eq. (2.8) and Ref. 23, starting from
the results of Sec. IV B.

VI. TIME-DEPENDENT TRANSPORT:
ADIABATIC APPROXIMATION AND LINEAR
RESPONSE

In this section we will consider the case of a fully coherent dot only. We will limit ourselves to a theory of
adiabatic and linear response transport — the response
to either a slowly varying internal potential or to a small
bias voltage, but not both —, and consider the average
and variance of the current only.
We will present our final results in terms of the Fourier
transform of the scattering matrix,
S(t; ε) =

dτ eiετ S(t + τ /2, t − τ /2),

S † (t; ε) =

dτ eiετ S † (t + τ /2, t − τ /2).

(6.1)

In the adiabatic approximation, the Fourier transform
S(t; ε) satisfies the unitarity condition
(1 − (i/2)Dt,ε )S(t; ε)S † (t; ε) = 1,

(6.2)

where Dt,ε AB = (∂A/∂t)(∂B/∂ε) − (∂A/∂ε)(∂B/∂t).
To leading order in the adiabatic approximation, the
Fourier transform S(t; ε) is equal to the ‘frozen’ scattering matrix, taken by fixing the internal potentials at the
value they have at time t. However, the ‘frozen’ scattering matrix is unitary, whereas S(t; ε) is not. Since our
final expression for the current will be manifestly of first
order in combined linear response and adiabatic approximation, we can neglect the difference between S(t; ε) and
the ‘frozen’ scattering matrix in our final expressions.
We will set the chemical potential µL of the electrons
in the fictitious reservoir equal to zero for all times. In
the adiabatic approximation and for linear transport, the

21
Keldysh Green function GK (t, s) depends on the time
mR
difference t − s only, m = 1, . . . , N , and reads

the equation for κ(t, s) becomes

T
δmn
(6.3)
vF
e−i(µmR (t)+eVd (t))τ
×P
,
sinh(Ï€T Ï„ )

GK (t + τ /2, t − τ /2) = −
mnR

κ(t, s) = κ0 (t − s) +
′

Ec
∂Vd
= −
∂t
Ï€
+

t

t

dt′ κ(t2 , s)
2

′

∂f (ε − µc )
∂ε

tr

(6.4)

This equation has no reference to the distribution functions in the lead or the time dependence of the scattering
matrix. Hence we find that the solution is of the equilibrium form κ(t − s), the Fourier transform of which
is given in Eq. (5.11). For the interaction correction
α(t + τ /2, t − τ /2), we note that it is an even function of
Ï„ . This implies its ensemble average must vanish if we
consider the adiabatic approximation and linear response
only.
We are now in a position to calculate the self-consistent
potential Vd and the current Ij . The self-consistent equation for the potential Vd reads

∂µc
∂S † (t; ε)
∂S † (t; ε)
S(t; ε)
+ iS(t; ε)
∂t
∂ε
∂t
∂
∂µc ∂
[S(t; ε)S † (ε + ω)] ,
dωκ(ω)(2f (ε + ω − µc ) − 1)
+
∂t ∂ε ∂t

dε −
1
Re
Ï€

t2

dt2

× e−EC N (t2 −t)/π e−(t2 −t2 )/τd .

where Vd is the dot potential, see Eq. (4.17).
Calculating the interaction function κ and the interaction correction α in the adiabatic limit is problematic. The reason is that both quantities involve long
time scales during which one cannot assume that the
derivative of the scattering matrix is constant (as one
does in the adiabatic approximation). However, since κ
and the interaction correction α contain traces, they are
self-averaging, and we can replace them by their ensemble averages. Taking the ensemble average in the defining equation (4.18) for κ(t, s), using the known probability distributions of time-dependent scattering matrices,48

e

EC N
Ï€

i

(6.5)

where the distribution function f is evaluated with respect to the electrochemical potential
µc = eVd +

1
N

µkR .

(6.6)

k

The first terms give the mean-field contribution to Vd . The last term gives the interaction correction. Upon taking
the ensemble average, which is appropriate if Vd enters as a variable in the expression for the current, the last term
vanishes.
For the current we find
Ij (t) =

C ∂Vd
e
+
N ∂t
2Ï€
× j(t; ε) +

dε −
1
Im
Ï€

∂f (ε − µc )
∂ε

tr

δjk −

1
N

dωκ(ω) (2f (ε + ω − µc ) − 1) S(t; ε)S † (t; ε + ω)(j(t; ε + ω) − j(t; ε)) ,

(6.7)

where we abbreviated
j(t; ε) = µR − S(t; ε) µR − i

∂
∂µc ∂
−i
∂t ∂ε
∂t

S † (t; ε).

(6.8)

In the expressions for Ij (t) and ∂Vd (t)/∂t one may replace the Fourier transformed scattering matrix S(t; ε) by
the ‘frozen’ scattering matrix. Then a time derivative ∂/∂t is replaced by (∂X/∂t)(∂/∂X), where X is a (set of)
parameter(s) that characterizes the potentials in the dot.
In Eq. (6.7) the terms proportional to µR give the dc conductance of the dot, cf. Eq. (5.10). The first line of Eq.
(6.7) is nothing but the Landauer formula with a self-consistent potential in the dot; the last lines give the correction
from interactions beyond the mean-field level. The self-consistent potential Vd was omitted from Eq. (5.10) because
it cannot be determined from the steady state formulation of the problem only. The terms proportional to the time
derivative give the emissivity −dQ/dX of the quantum dot. Again, the first two lines of Eq. (6.7) correspond to the
emissivity according to the self-consistent (Hartree) theory of B¨ttiker and coworkers28 whereas the remaining lines of
u
that equation give the correction from interactions beyond the mean-field level. The governing equation (6.5) for the

22
self-consistent potential Vd (t) is the same as that of the Hartree theory of Ref. 28. It depends both on (changes in)
the dot potential and the chemical potential in the lead. After taking the ensemble average there are no interaction
corrections to the Hartree theory for Vd .
Although the above expression for the current was derived in linear response and in the adiabatic approximation,
it can be used to calculate the electrochemical capacitance Cc of the quantum dot, the derivative of the charge on the
dot to a uniform shift if chemical potential µR . This is possible because, for the Hamiltonian H0 , slowly increasing the
chemical potential of all channels simultaneously corresponds to a time-independent small shift of the electrochemical
potential, see Eq. (3.32). Hence, the time derivatives of the phases carried by the bare Green functions GR and GL
differ only by a small amount, and linear response is meaningful. Considering the rate of change of the charge on the
dot for the case in which there is no explicit time dependence of the dot scattering matrix, one finds from Eq. (6.7)
dQ
∂Vd
= −C
,
dt
∂t

(6.9)

where ∂Vd /∂t can be calculated from Eq. (6.5). Upon changing all chemical potentials for incoming electrons simultaneously by an amount −eV (t), we find26
dQ
dV
= C −1 + (e2 dn/dε)−1

Cc =

−1

,

(6.10)

where dn/dε is the density of states of the dot,
dn
1
=
dε
2Ï€

dε −

∂f (ε − µc )
∂ε

tr iS(ε)

∂S † (ε)
1
+ Re
∂ε
Ï€

dωκ(ω)(2f (ε + ω − µc ) − 1)

∂
S(ε)S † (ε + ω) .
∂ε
(6.11)

2
ν 2 τd
var (dn/dε) = s 2
2Ï€

22
1+
3N

1
0.8
0.6

var c

Here, the first term is the Hartree expression for the density of states of an open quantum dot;26,27 the second
term is the interaction correction.
The ensemble average of the interaction correction to
the density of states, and hence of the interaction correction to the electrochemical capacitance, is zero. It has,
however, mesoscopic fluctuations. Calculating the interaction correction to the variance of the density of states,
we find, for zero temperature,

0.4

xN

0.2
,

(6.12)

0
0

0.5

from which one concludes, for /τd ≪ Ec N ,
var Cc =

2
π 2 e 4 νs
4N 4 Ï„ 2
2Ec
d

1+

22
3N

.

(6.13)

Equations (6.12) and (6.13) are valid in the absence
of time-reversal symmetry. With time-reversal symmetry, the variance is a factor two larger. The first term
is the variance of the mean-field contribution to the
capacitance,36,58 the second term is the interaction correction. Figure 9 shows the temperature dependence of
the variance of the electrochemical capacitance.
If the quantum dot is operated as a quantum pump,
two shape-defining parameters X1 and X2 are varied periodically in time, whereas the bias voltages are kept zero.
We consider a harmonic time dependence for the param-

1

2πΤ/Ν∆

1.5

2

FIG. 9:
The variance of the mean-field contribution
to the capacitance, made dimensionless by writing Cc =
2
2Ï€e2 c/(Ec N 2 Ï„d ) (solid curve) and the interaction correction
to the variance of the electrochemical capacitance (dashed),
as a function of temperature. The interaction correction to
the variance has been multiplied by N in order to fit on the
same vertical scale.

eters X1 and X2 ,
X1 (t) = δX1 sin(ωt),
X2 (t) = δX2 sin(ωt + φ).

(6.14)

The charge pumped in one cycle Qpump is defined as the
integral of the current in one of the point contacts. Inte-

23
grating over one cycle, one then finds, for small pumping
amplitudes30
Qpump = (πδX1 δX2 sin φ)2
∂ dQpump
∂ dQpump
−
×
∂X1 dX2
∂X2 dX1

. (6.15)

[For large amplitudes δX1 and δX2 one has to average
Eq. (6.15) over the area enclosed in the two-dimensional
space spanned by the parameters X1 and X2 .]
Since the total current is conserved, we may calculate Qpump as the integral of I(t) =
j Ij (t)Λjj over

e
dQpump
= −
dX
2Ï€
+

dε −

1
Re
Ï€

∂f (ε)
∂ε

tr

iΛS(ε)

4
4N1 N2 νs
N4

1+

2
N

(δX1 δX2 sin φ)2 ,(6.17)

irrespective of the presence or absence of a time-reversal
symmetry breaking magnetic field. Here we have used
the standard convention (within random matrix theory)
to relate the effect of the shape-defining parameters X1
and X2 to a parametric motion of the scattering matrix.
The first term in Eq. (6.17) is the leading non-interacting
contribution to the variance;30,59 the second term is the
interaction correction. For temperatures T ≫ /τd we
find
var Qpump = (πδX1 δX2 sin φ)2
4
N 1 N 2 νs
Ï€
×
1+
4Ï„
12T N d
T N Ï„d

, (6.18)

where, again, the first term is the leading non-interacting
contribution to the variance of the pumped current59,60
and the second term is the interaction correction.
VII.

∂S † (ε)
∂X

dωκ(ω)(2f (ε + ω) − 1)tr ΛS(ε)

Using the results of appendix B to perform the ensemble
average, we find that, at zero temperature and for small
pumping amplitudes, the variance of the pumped charge
is given by
var Qpump =

one period, i.e., (N2 /N ) times the integral over the current through the first point contact minus (N1 /N ) times
the current in the second point contact. The ensemble
average Qpump is zero for symmetry reasons. When
calculating the variance of the pumped charge, we note
that, up to corrections of order 1/N 2 , one may replace
the average of a product of traces by the product of the
averages, with the exception of the two traces that contain the matrix Λ. Their average is zero, and one needs
to consider the average of the product. This means that
all terms proportional to ∂Vd /∂t can be dropped, and
one may set

CONCLUSION

In this paper we have presented a systematic theory
to evaluate the effects of electron-electron interactions
on quantum transport through open quantum dots, accommodating arbitrary bias and time dependence of the
dot’s scattering matrix S(t, t′ ). Our result takes the form
of an expression for the sample-specific current through

∂S † (ε)
∂S † (ε + ω)
− S † (ε + ω)S(ε)
∂X
∂X

.

(6.16)

a quantum dot in terms of S(t, t′ ). This expression
has been obtained using a systematic expansion in 1/N ,
where N is the total number of channels in the point
contacts connecting the dot to the electron reservoirs.
To leading order in 1/N , our theory reproduces what
one finds if the interactions were treated in the Hartree
approximation. The subleading-in-1/N corrections represent contributions to the current through the dot that
cannot be described by means of a self-consistent potential.
Upon taking the average over an ensemble of chaotic
quantum dots, we have been able to calculate the interaction corrections to weak localization, universal conductance fluctuations, capacitance fluctuations, and to the
pumped current in case the quantum dot is operated as
a quantum pump. For a quantum dot with ideal point
contacts, we found that the interaction correction to the
weak localization correction to the conductance vanishes,
as well as the interaction correction to the conductance
fluctuations at zero temperature. There is a negative interaction correction to the conductance fluctuations at
finite temperature. There are positive interaction corrections to the capacitance fluctuations and to the fluctuations of the pumped current, but no corrections to their
average. All interaction corrections are small as 1/N in
comparison to the non-interacting (i.e., Hartree) contributions.
The problem of ‘weak Coulomb blockade’ in open
quantum dots has been addressed previously in the
literature.7,8,24 Our findings for the quantum interference
corrections, as well as our expression for the samplespecific dc conductance and capacitance, differ from
those of Refs. 7,8. To explain the difference, we have
shown that the formal expansion in the scattering matrix, used in Refs. 7,8, cannot be used to describe a co-

24
herent quantum dot. We believe that the expressions
presented here, which were obtained using a systematic
expansion in 1/N , are correct. Our sample-specific expression agrees with that obtained in Ref. 24, although
Ref. 24 does not consider the quantum corrections to the
conductance.
The absence of an interaction correction to weak localization and to the conductance fluctuations at zero
temperature can be understood using a simple argument:
Since the charging interaction cannot lead to dephasing7
— it corresponds to a time-dependent uniform shift of
the dot’s potential —, its only effect is a renormalization
S(ε) → S ′ (ε) of the scattering matrix S(ε) of the dot.
To leading order in perturbation theory, such a renormalization was calculated explicitly in Sec. III B and in Ref.
24. Like the probability distribution functional P [S(ε)] of
the ‘bare’ energy-dependent scattering matrix S(ε), the
probability distribution functional P [S ′ (ε)] of the renormalized scattering matrix S ′ (ε) is invariant under left
and right multiplication of S ′ (ε) with energy-independent
unitary matrices U and V ,61,62,63,64
P [S ′ (ε)] = P [U S ′ (ε)V ].

(7.1)

In the presence of time-reversal symmetry, U = V T . The
statistical invariance (7.1), together with the same invariance for P [S(ε)], implies that the distributions of both
S(ε) and S ′ (ε) for a fixed energy ε are those of the circular ensemble from random matrix theory.40,65 Since the
weak localization correction to the conductance and the
conductance fluctuations at zero temperature do not depend on statistical correlations between scattering matrices S(ε) at different energies, we conclude that these are
given by the circular ensemble averages,56,57 and, hence,
that there is no interaction correction. There is an interaction correction, however, to the conductance fluctuations at a finite temperature, which do involve such
correlations. The above argument does not apply to a
quantum dot with nonideal leads, for which the invariance property (7.1) does not hold, or to a quantum dot
with relaxation, for which the scattering matrix is subunitary and the invariance property (7.1) is not sufficient
to fix the distribution.
How reliable is the expansion in 1/N and how significant are the interaction corrections, as they are a factor 1/N smaller than the Hartree contributions? Recalling that N is the total number of channels in the
point contacts connecting the dot and the electron reservoirs, not counting spin degeneracy, one has N = 4 for
a dot with two spin-degenerate single-mode point contacts. Although N = 4 cannot be classified as ‘large’,
we believe that for N = 4 the 1/N expansion should
give a reliable estimate of the interaction correction,
whereas, on the other hand, the interaction corrections
should still be significant (if they are nonzero). Measurements of the conductance distribution in single-mode
quantum dots have shown excellent agreement with the
(non-interacting) random matrix theory down to the lowest temperatures.66 This experimental observation is in

agreement with our theory since, to leading order in 1/N
and at zero temperature, we find no interaction correction to the average and variance of the conductance. The
argument following Eq. (7.1) is extremely suggestive and
may hint that the agreement with random matrix theory is not restricted to the approximations of this paper,
though we caution that higher orders in the interaction
are presumably not simply described by an effective S ′ (ε)
due to inelastic processes. As long as the Fermi Liquid
picture holds, however, one may expect this description
to be restored at zero temperature.
The interaction correction we calculated here is the result of the capacitive interaction only. For large N , it is
a factor ∼ 1/N smaller than the leading non-interacting
(Hartree) contribution to the transport properties and
their quantum interference corrections. We have not considered other contributions to the electron-electron interaction. These were omitted from the Hamiltonian of
the quantum dot because they are much smaller than
the capacitive interaction if the dimensionless conductance of the dot is large.6,8,35 The interactions we omitted will give additional corrections to the dot’s transport
properties, e.g., because they lead to dephasing. To the
best of our knowledge, a microscopic theory of the effect
these residual interactions on the transport properties of
a chaotic quantum dot still has to be developed.
In addition to the interaction corrections studied here,
which are not periodic in the dimensionless gate voltage N , there exist periodic-in-N interaction corrections
to the dot’s capacitance and conductance.6,7,8 These occur to N th order in perturbation theory, which rules out
a large-N approach. Periodic-in-gate-voltage interaction
corrections have been studied using an expansion in the
scattering matrix.6,8 Although we have shown that such
an expansion is not justified for aperiodic interaction corrections, we must leave the question of its validity for the
periodic-in-N corrections unanswered.
Before closing, we would like to make two remarks on
the relation of our work to that of Golubev and Zaikin.24
First, Golubev and Zaikin consider both the case of a
‘voltage biased’ and a ‘current biased’ quantum dot. The
latter case corresponds to a quantum dot in series with
a large ohmic resistor. In Ref. 24 it is shown that the
interaction corrections in the two cases are qualitatively
different. Our calculation, in which the chemical potentials of the electrons coming in through the point contacts
are kept fixed, corresponds to the ‘voltage biased’ case.
Second, Golubev and Zaikin argue that the sole effect of
interactions as a renormalization of the reflection matrix
of the point contacts rc only. While this is true for an
incoherent dot,41,42,67,68 our calculation shows that this
is not correct for a coherent dot: When quantum interference corrections are taken into account, the renormalization of the scattering matrix is more complicated, and
involves more than one energy. This is manifested, e.g.,
in the existence of an interaction correction to the conductance fluctuations at finite temperature.

25
Acknowledgments

We would like to thank I. Aleiner, K. Matveev, and
L. Glazman for discussions. This work was supported
by the NSF under grant no. DMR-0334499 and by the
Packard Foundation.

vF
ˆ
H0 =
4Ï€

APPENDIX A: BOSONIZATION

In this section we calculate the contour-ordered correlation functions of the fermion fields in the Hamiltonian
ˆ
H0 of Eq. (3.18). The bosonization method allows for an
exact calculation of the fermion correlation functions for
ˆ
ˆ
the Hamiltonian H0 , despite the fact that H0 is quadratic
ˆ
in the fermion operators:69 H0 is rewritten in terms boson fields ϕjL (x) and ϕjR (x). In the bosonized language
ˆ
ˆ
ˆ
H0 is diagonalized easily, and the fermion correlators can
be found from the boson correlation functions.
The boson fields ϕjL (x) and ϕjR (x) are related to the
ˆ
ˆ
ˆ
ˆ
fermion fields ψjL (x) and ψjR (x) as
ηj
ˆ
ˆ
ˆ
ψjL = √
e−iϕjL , j = 1, . . . , N,
2πλvF
ηj
ˆ
ˆ
ˆ
eiϕjR , j = 1, . . . , N.
ψjR = √
2πλvF

(A1)
(A2)

They obey the commutation rules
[ϕjL (x), ϕLi (y)] = −iπδij sign(x − y),
ˆ
ˆ
[ϕjR (x), ϕRi (y)] = iπδij sign(x − y),
ˆ
ˆ
[ϕjR (x), ϕLi (y)] = −iπδij .
ˆ
ˆ

There are no boundary terms at x = −∞ because the
interaction extends over the entire lead. The reference
electron number Nref has been absorbed in the definition
of normal-ordering of the fermion operators. Then, once
expressed in terms of the boson fields, the Hamiltonian
ˆ
H0 becomes quadratic,
N

∞
−∞

j=1

1 ∂
Ï•jL (x),
ˆ
2π ∂x
1 ∂
ˆ†
ˆ
: ψjR (x)ψjR (x) : =
Ï•jR (x).
ˆ
2π ∂x
ˆ†
ˆ
: ψjL (x)ψjL (x) : =

The Hamiltonian is supplemented with the condition
that the chemical potential of left-moving fermions moving towards the dot-lead interface at x = 0 is equal
to µjL (t)/2πvF and that the density of right-moving
fermions is equal to µjR (t)/2πvF . We generate the electron densities corresponding to these chemical potentials
by the inclusion of a time-dependent forward-scattering
potential in the Hamiltonian. This potential is located at
a distance a from the dot-lead interface at x = 0. (The
precise value of a is not important for the subsequent
considerations, as long as a > 0.) In boson language it
has the form

−

N

mjL (t)
j=1

vF
2Ï€

(A6)

Hence, the number of electrons on the quantum dot is
simply expressed as the sum of left-moving and rightmoving boson fields at x = 0,
1
ˆ
Ndot = −
2Ï€

∂ ϕjL (x)
ˆ
∂x

N

mjR (t)
j=1

x→a

∂ ϕjR (x)
ˆ
∂x

, (A9)
x→−a

where mjL and mjR are the integrals of the chemical
potentials µjL and µjR , respectively,
t+a/vF
−∞
t+a/vF

(A4)

(A5)

2

(A8)

(A3)

where the sign signc is defined with respect to the ordering along the integration contour c.
The (normal-ordered) densities of left and right moving
fermions are proportional to the derivatives of the boson
fields,

∂ ϕjR
ˆ
∂x

+

2
N
Ec 
+ 2
(ϕjL (0) + ϕjR (0) + 2πN ) .
ˆ
ˆ
4Ï€ j=1

mjL (t) =
Tc ηj (t), ηi (s) = δij signc (t − s) ,
ˆ
ˆ

2



vF
ˆ
H ′ (t) =
2Ï€

In Eq. (A2) ηj = ηj is a Majorana fermion, {ˆj , ηi } =
ˆ
ˆ†
η ˆ
2δij , and λ is a cut-off time that is taken to zero at the
end of the calculation. Since the Majorana fermions do
not enter into the Hamiltonian, their average is given by

∂ ϕjL
ˆ
∂x

dx

mjR (t) =

dt′ µjL (t′ ), j = 1, . . . , N,
dt′ µjR (t′ ), j = 1, . . . , N. (A10)

−∞

We assume that the system is in equilibrium, µjL (t) =
µjR (t) = 0, j = 1, . . . , N for times t smaller than a reference time tref .
In order to find correlation functions of fermion creation and annihilation operators, expressions of the form
Tc ea need to be averaged over the boson fields, where
a is an arbitrary linear combination of the boson fields
ϕjL and ϕjR , j = 1, . . . , N , at different times. Since
the Hamiltonian is quadratic in the boson fields, such an
average can be calculated as
exp(ˆ) = exp
a

a +
ˆ

1
2

a2 − a
ˆ
ˆ

2

,

(A11)

N

(Ï•jL (0) + Ï•jR (0)).
ˆ
ˆ
j=1

(A7)

which means that only contour-ordered correlators of up
to two boson fields at different times are needed. Hence,

26
it remains to calculate the correlation functions of the
boson fields.
For this calculation, it is necessary that we transform
the boson fields such that the charge mode is separated
from modes that do not charge the quantum dot. The
corresponding basis change reads
oµj ϕjR ,

oµj ϕjL , ϕµR =

ϕµL =

(A12)

where

oµj mjR (t).

oµj mjL (t), mµR (t) =

mµL (t) =
j

j

j

j

(A17)

where the transformed fields carry a Greek index and the
original fields carry a roman index. Here the oµj are real
numbers satisfying
oµj oνj = δµν ,

oµj oµi = δij .

In the transformed basis there is no charging interaction for the modes µ = 2, . . . , N . Solving the Heisenberg
evolution equation for the fields ϕµL and ϕµR ,
ˆ
ˆ

µ

j

and
o1j = N −1/2 , j = 1, . . . , N.

(A13)

With this choice, the transformed fields ϕ1L and ϕ1R
ˆ
ˆ
describe a mode that charges the quantum dot, whereas
all other fields ϕµL and ϕµR , µ = 2, . . . , N , do not involve
ˆ
ˆ
charging of the dot. The transformed boson fields ϕµL
ˆ
and ϕµR satisfy the same boson commutation relations
ˆ
as the original fields ϕjL and ϕjR ,
ˆ
ˆ
[ϕµL (x), ϕνL (y)] = −iπδµν sign(x − y),
ˆ
ˆ
[ϕµR (x), ϕνR (y)] = iπδµν sign(x − y),
ˆ
ˆ
[ϕµR (x), ϕνL (y)] = −iπδµν .
ˆ
ˆ

(A14)

In terms of the transformed fields, the boson Hamiltonian
ˆ
ˆ
H0 and the time-dependent perturbation H ′ read
vF
ˆ
H0 =
4Ï€
+
vF
ˆ
H′ =
2Ï€
−

N

∞

∂ ϕµL
ˆ
∂x

dx
−∞

µ=1

Ec N
4Ï€ 2

mµL (t)
vF
2Ï€

∂ ϕµL (x)
ˆ
∂x

N

mµR (t)
µ=1

+

2

∂ ϕµR
ˆ
∂x

2Ï€N
ϕL,1 (0) + ϕR,1 (0) + √
ˆ
ˆ
N

N
µ=1

2

,

ϕµL (x, t) = ϕµL (x + vF (t − tref ), tref )
ˆ
ˆ
+ mµL (t + (x − a)/vF )θ(a − x),
ϕµR (x, t) = ϕµR (x − vF (t − tref ), tref )
ˆ
ˆ
− mµR (t − (x + a)/vF )θ(a + x), (A18)

2

, (A15)

x→a

∂ ϕµR (x)
ˆ
∂x

∂ ϕµL (x, t)
ˆ
∂ ϕµL (x, t)
ˆ
= vF mµL (t)δ(x − a) + vF
,
∂t
∂x
∂ ϕµR (x, t)
ˆ
∂ ϕµL (x, t)
ˆ
= vF mµR (t)δ(x + a) − vF
,
∂t
∂x
we obtain an expression for ϕ(x, t) in terms of boson fields
ˆ
at the reference time tref ,

(A16)

(Recall that mµL (t) = mµR (t) = 0 for t < tref .) For
the transformed mode µ = 1, the charging interaction
is important. In this case, the Heisenberg equation of
motion reads

x→−a

∂ ϕ1L (x, t)
ˆ
∂ ϕ1L (x, t) Ec N
ˆ
= vF m1L (t)δ(x − a) + vF
+
∂t
∂x
Ï€
∂ ϕ1R (x, t)
ˆ
∂ ϕ1L (x, t) Ec N
ˆ
= vF m1R (t)δ(x + a) − vF
−
∂t
∂x
Ï€

2Ï€N
ϕ1L (0) + ϕ1R (0) + √
ˆ
ˆ
N
2Ï€N
ϕ1L (0) + ϕ1R (0) + √
ˆ
ˆ
N

θ(−x),
θ(−x).

27
and has the solution
ϕ1L (x, t) = ϕ1L (x + vF (t − tref ), tref ) + m1L (t + (x − a)/vF )θ(a − x)
ˆ
ˆ
−

Ec N
Ï€

t

tref

′

dt′ θ(−x)θ(t′ − t − x/vF )e−Ec N (t−t )/π

2Ï€N
,
× ϕ1R (−vF (t′ − tref ), tref ) − m1R (t′ − a/vF ) + ϕ1L (vF (t − tref ), tref ) + m1L (t′ − a/vF ) + √
ˆ
ˆ
N
ϕ1R (x, t) = ϕ1R (x − vF (t − tref ), tref ) − m1R (t + (x + a)/vF )θ(a + x)
ˆ
ˆ
−

Ec N
Ï€

t

tref

′

′

dt′ θ(x)θ(t − x/vF − t′ )e−Ec N (t−x/vF −t )/π + θ(−x)e−Ec N (t−t )/π

2Ï€N
.
× ϕ1R (−vF (t′ − tref ), tref ) − m1R (t′ − a/vF ) + ϕ1L (vF (t − tref ), tref ) + m1L (t′ − a/vF ) + √
ˆ
ˆ
N
(A19)
According to Eqs. (A18) and (A19), the effect of the time-dependent potentials is to shift the boson fields by a real
number. Such a shift affects the average of the boson fields,
ϕµL (0, t) = mµL (t − a/vF ),
ˆ

(A20)

2Ï€N
Ec N
ϕµR (0, t) = −mµR (t − a/vF ) − √ δµ,1 +
ˆ
δµ,1
Ï€
N

t

′

tref

dt′ e−Ec N (t−t )/π [m1R (t′ − a/vF ) − m1L (t′ − a/vF )] ,

but not the connected two-point correlators. Hence, the connected two-boson correlation functions are the same as
in equilibrium. They can be obtained by analytical continuation from the imaginary-time correlators obtained in
Refs. 6,8 or they can be calculated directly from the solution of the equation of motion. In the latter method one
first calculates the spectral density A(t, t′ ). Since Eqs. (A18) and (A19) express all boson operators in terms of the
operators at the same reference time tref , the spectral density can be found from the equal-time commutation relations
(A14). We only need the spectral density at x = 0,
AµL,νL (t, t′ ) = [ϕµL (0, t), ϕνL (0, t′ )]−
ˆ
ˆ
= −iπδµν sign(t − t′ ),
′
AµL,νR (t, t ) = [ϕµL (0, t)ϕνR (0, t′ )]−
ˆ
ˆ
∞
Ec N
δµ,1
dζe−Ec N ζ/π sign(t′ − t − ζ) ,
= iπδµν 1 −
Ï€
0
AµR,νL (t, t′ ) = [ϕµR (0, t)ϕνL (0, t′ )]−
ˆ
ˆ
∞
Ec N
dζe−Ec N ζ/π sign(t − ζ − t′ ) ,
δµ,1
= −iπδµν 1 −
Ï€
0
AµR,νR (t, t′ ) = [ϕµR (0, t), ϕνR (0, t′ )]−
ˆ
ˆ
= −iπδµν sign(t − t′ ).

(A21)

The two-boson correlation functions are then found using the fluctuation-dissipation theorem, see, e.g., Ref. 70. The
resulting expressions for the averages and the connected correlation functions of the original boson fields ϕLj and
ˆ
Ï•Rj , j = 1, . . . , N are
ˆ
t

Ï•jL (t) =
ˆ

dt′ µjL (t′ )
t

ϕjR (t) = −
ˆ

dt′ µjL (t′ ) −

(A22)
2Ï€N
−
N

t

N

dt′
k=1

ijk (t − t′ )[µkR (t′′ ) − µkL (t′′ )].

(A23)

where iik was defined in Eq. (3.25), and
λπT /i
sinh [Ï€T
− t′′ − iλsignc (t′ − t′′ ))]
′
′′
= Tc Ï•iR (t )Ï•jR (t ) ,
ˆ
ˆ
1
iπ
ln f (t′′ , t′ ) − A
Tc ϕiL (t′ )ϕjR (t′′ ) = δij signc (t′ − t′′ ) −
ˆ
ˆ
2
N
= Tc ϕiR (t′′ )ϕjL (t′ ) ,
ˆ
ˆ
Tc ϕiL (t′ )ϕjL (t′′ ) = δij ln
ˆ
ˆ

(t′

+ A + δij −

1
N

B

(A24)

28
where A and B are positive constants that are taken to
infinity at the end of the calculation and f is given in
Eq. (3.36). Calculation of the fermion correlators of Sec.
III E using the rule (A11) is straightforward now.

involves scattering from inside the dot that has fluctuations. In order to manifestly separate the two contributions, S is parameterized as72
†
†
†
S = rc + (1 − rc rc )1/2 S0 (1 + rc S0 )−1 (1 − rc rc )1/2 , (B2)

APPENDIX B: ENSEMBLE AVERAGE

In this section we present some material relevant for
performing the averages over an ensemble of microscopically different but macroscopically equivalent quantum
dots.
We consider the statistical distribution of the scattering matrix for an ensemble of chaotic quantum dots. The
dots are placed in a magnetic field. The magnetic field
strength is described by the dimensionless parameter α,
which is proportional to the magnetic flux Φ through the
dot,
α=

eΦ κ vF l
hc L2 ∆

1/2

,

(B1)

where ∆ is the mean spacing between spin-degenerate
levels, L its linear size, l the elastic mean free path inside the dot, and κ a numerical constant of order unity.
One has κ = 4π/15 for a diffusive sphere of radius L and
κ = π/2 for a diffusive disk of the same radius71 . If the
electron motion inside the dot is ballistic, but with diffusive boundary scattering, l is replaced by 5L/8 and πL/4
for the cases of the sphere and the disk, respectively.
The shape of the dot is controlled by two gate voltages,
which are described by the dimensionless parameters X1
and X2 . The normalization of the parameters X1 and
X2 is the same as in Ref. 48.
Two types of scattering processes contribute to the
scattering matrix: direct reflection at the contacts, which
is described by the energy-independent reflection matrix
rc , and scattering from inside the quantum dot. The matrix rc is a property of the contacts and is not subject to
mesoscopic fluctuations; it is only the contribution that

Sij (t, t − τ )Sij (s, s − σ)∗

=

1
δ(τ − σ)θ(τ )
N Ï„d


Ï„

× exp −

0

dτ ′ 

where S0 has the same statistics as the scattering matrix
of a chaotic quantum dot ideal contacts.
For electrons with spin (and without spin-orbit scattering), the scattering matrix takes the form
S = So ⊗ 1 2,
1

(B3)

where 1 2 is the 2 × 2 unit matrix (acting on the spin
1
degrees of freedom) and S o is a unitary matrix of size
N o = N/2 representing scattering of the orbital degrees
of freedom. A similar decomposition applies to the matrices rc and S0 . The results below are for the distribution
o
o
P (S0 ) of the orbital scattering matrix S0 , in the limit
of large N . For simplicity of notation we will drop the
subscript “0” and the superscript “o”.
For large N , only moments of the distribution that
contain as many factors of S as of its complex conjugate S ∗ and in which the indices of factors S and S ∗
are pairwise equal are nonzero.73 This implies that all
odd moments of P (S) are zero. Since the probability
of S is close to Gaussian for large N , with small nonGaussian corrections, P (S) is well characterized by its
moments. The second moment characterizes the Gaussian part of the distribution; Higher moments represent
the non-Gaussian corrections to P (S).
In order to specify P (S), it is sufficient to consider
generic moments in which all (pairs of) indices are different. Moments in which two or more pairs of indices
coincide are fully determined by the invariance of the ensemble under basis changes in the leads.73 For the general time-dependent case, we need the second moment
Sij (t, t − τ )Sij (s, s − σ)∗ for i = j only, which was calculated in Ref. 48,

1
+ (α(t − ξ) − α(s − ξ))2 + 2
Ï„d

j

(B4)


(Xj (t − ξ) − Xj (s − ξ))2  .

For the time-independent case, we need to go calculate the corrections to the Gaussian distribution. Using the
short-hand notation S(1) = S[ε(1), α(1), X1 (1), X2 (1)], the relevant moments are
Sij (1)Sij (2)∗
Sij (1)Skj (2)∗ Skl (3)Sil (4)∗
Sij (1)Sij (2)∗ Skl (3)Skl (4)∗
Sij (1)Skj (2)∗ Skl (3)Sml (4)∗ Smn (5)Sin (6)∗

=
=
=
=

W1 (1, 2),
W2 (1, 2, 3, 4),
W1 (1, 2)W1 (3, 4) + W1,1 (1, 2, 3, 4),
W3 (1, 2, 3, 4, 5, 6),

(B5)
(B6)
(B7)
(B8)

29
where all indices i, j, k, l, m, and n are different. In addition to the moments listed above, we need moments in
∗
which the indices of a factor Sij or Sij are interchanged. These follow from the above moments using the identity
Sij (ε, X1 , X2 , α) = Sji (ε, X1 , X2 , −α). The function W1 was calculated in Refs. 74,75; the function W2 was calculated
in Ref. 48. Repeating the calculations of Ref. 48 for W1,1 and W3 , one finds, for large N ,
W1 (1, 2) =

1
F1 (1, 2)

(B9)

F2 (1, 2, 3, 4)
,
F1 (1, 2)F1 (3, 2)F1 (3, 4)F1 (1, 4)
1
F2 (1, 2, 3, 4)F2 (1, 4, 3, 2) F2 (1, 2, 1, 4)F2(1, 4, 3, 4)
W1,1 (1, 2, 3, 4) =
+
F1 (1, 2)2 F1 (3, 4)2
F1 (1, 4)F1 (3, 2)
F1 (1, 4)2
F2 (1, 2, 3, 2)F2 (3, 2, 3, 4) F3 (1, 2, 1, 4, 3, 4) F3 (1, 2, 3, 4, 3, 2)
,
−
−
+
F1 (3, 2)2
F1 (1, 4)
F1 (3, 2)
1
F2 (1, 2, 3, 4)F2 (1, 4, 5, 6)
W3 (1, 2, 3, 4, 5, 6) =
F1 (1, 2)F1 (3, 2)F1 (3, 4)F1 (5, 4)F1 (5, 6)F1 (1, 6)
F1 (1, 4)
F2 (1, 2, 5, 6)F2 (3, 4, 5, 2) F2 (3, 4, 5, 6)F2 (1, 2, 3, 6)
+
+
− F3 (1, 2, 3, 4, 5, 6) ,
F1 (5, 2)
F1 (3, 6)
W2 (1, 2, 3, 4) = −

(B10)

(B11)

(B12)

where
n

Fn (1, . . . , n) = N 1 − i

2

n
m

(−1) ε(m)τd +
m=1

m

(−1) α(m)
m=1

2

2

n
m

(−1) Xj (m)

+2
j=1

.

(B13)

m=1

With the help of these moments, the ensemble averages of Secs. V and VI can be done using the diagrammatic
technique of Ref. 73.

APPENDIX C: QUANTUM DOT WITH
RELAXATION

A highly simplified model for relaxation is to consider
the model of Sec. III C for r < 1. In this appendix,
we describe a more realistic (but still phenomenological)
model for relaxation in the quantum dot.
In this model, the quantum dot itself (rather than the
point contact) is coupled to a fictitious lead, see Fig.
10. This fictitious lead is connected to a reservoir, the
chemical potential of which is chosen such that no net
current flows through this lead. For that reason, the fictitious lead is referred to as ‘voltage probe’. This model

was originally proposed by B¨ ttiker,50,51 and has been
u
applied by various authors to describe relaxation effects
in chaotic quantum dots.44,53,54,55,66 In order to model
spatially distributed relaxation, the voltage probe has
M ≫ 1 channels, each coupled to the quantum dot via a
tunnel barrier with transmission probability ΓV ≪ 1.44
The relaxation rate is set by the product M ΓV .
Electrons in the physical point contact are labeled “R”
and “L” as before. We label electrons entering the dot
from the voltage probe by “V”, and electrons exiting the
dot towards the voltage probe by “W”, see Fig. 11. The
scattering matrix of the dot now has dimension (N + M ),
and relates the fields ψR and ψV to the fields ψE and ψW ,

N

ˆ
ψjL (0, t) =
k=1

M

ˆ
dτ Sjk (t, t − τ )ψkR (0, t − τ ) +

M

ˆ
ψjW (t) =
k=1

k=1

ˆ
dτ sjk (t, t − τ )ψkV (t − τ ),

N

ˆ
˜
dτ Sjk (t, t − τ )ψkV (t − τ ) +

k=1

ˆ
dτ sjk (t, t − τ )ψkR (0, t − τ ).
˜
(C1)

The fields ψV and ψW are evaluated at the interface between dot and the voltage probe reservoir.
In order to derive an effective action for the quantum dot with voltage probe attached to it, we again attach a
fictitious lead at the location of the point contact, as described in Sec. III C. The Hamiltonian is written as the sum
ˆ
ˆ
ˆ
ˆ
ˆ
H = H0 + H1 , where H0 and H1 represent ideal coupling of the fictitious lead and an impurity at the contact to the

30
voltage
probe

x= 0

quantum
dot

voltage probe

x= 0

V
reservoirs R

W
dot

L

FIG. 10: Top panel: schematic drawing of a quantum dot
with a voltage probe. Bottom panel: layout of labeling of
fermion fields “R”, “L”, “V”, and “W”.

ˆ
fictitious lead, respectively. For H1 we take Eq. (3.13) with rj = 1, j = 1, . . . , N . We then use Eq. (C1) to eliminate
ˆ
the fields ψL from the expression for H1 and arrive at the effective action
S = 2vF

dt1
c

dτ
mn

+ 2vF

dt1

ˆ†
ˆ
ψmL (0, t1 )Smn (t1 , t1 − τ )ψnR (0, t1 − τ ) + h.c.

dτ

c

mp

ˆ†
ˆ
ψmL (0, t1 )smp (t1 , t1 − τ )ψV p (t1 − τ ) + h.c. .

(C2)

The following formal manipulations closely follow those of Ref. 6. Since the current Ij , j = 1, . . . , N , does not
ˆ
depend on the fields ψV k , k = 1, . . . , M , we can integrate these out. This results in a new effective action
S ′ = 2vF

dt1

dτ1

c

2
+ 4vF

mn

dt1 dt2
c

ˆ†
ˆ
ψmL (t1 )Smn (t1 , t1 − τ1 )ψnR (t1 − τ1 ) + h.c.

dτ1 dτ2
mnp

(C3)

ˆ†
ˆ
ψmL (t1 )smp (t1 , t1 − τ1 )GV (t1 − τ1 , t2 − τ2 )(s† )pn (t′ , t2 )ψnL (t2 ),
2

where
ˆ
ˆ†
GV (t1 − τ1 , t2 − τ2 ) = −i Tc ψV p (t1 − τ1 )ψV p (t2 − τ2 )

(C4)

is the contour-ordered Green functions for fermions coming in from the voltage probe. (Note that the contour ordering
is with respect to the contour times t1 and t2 .) We assume that the change of potentials (either external or internal)
is slow in comparison to the minimum of the relaxation time and the escape rate into the physical leads. Then, using
the gradient expansion we find
′
dτ1 dτ2 s(t1 , t1 − τ1 )GV (t1 − τ1 , t2 − τ2 )s† (t2 − τ2 , t2 ) = GV (t1 , t2 ) + δGV (t1 , t2 ),

(C5)

where δGV (t1 , t2 ) is defined as
δGV (t1 , t2 ) = −

1
∂fV (ε)
dεe−iε(t1 −t2 )
2Ï€vF
∂ε
∂s(t; ε) ∂µV ∂s(t; ε) †
s (t; ε) .
(C6)
+
∂t
∂t
∂ε

dτ1 dτ2 S(t1 , t1 − τ1 )GV (t1 − τ1 , t2 − τ1 )S † (t2 − τ2 , t2 ) +
× tr

∂S(t; ε) ∂µV ∂S(t; ε)
+
∂t
∂t
∂ε

S † (t; ε) +

31
Here fV (ε) is the distribution function in the voltage probe, and S(t; ε) and s(t; ε) are the Fourier transforms of the
scattering matrices S(t, t − τ ) and s(t, t − τ ), respectively, see Eq. (6.1). The second term in Eq. (C6) does not depend
on the contour positions of t1 and t2 . Upon substitution of Eq. (C5) into the effective action (C3), one finds
S ′ = 2vF
+

dt1

2
4vF

ˆ†
ˆ
ψmL (t1 )Smn (t1 , t1 − τ1 )ψnR (t1 − τ1 ) + h.c.

dτ1

c

mn

2
ˆ†
ˆ
ψmL (t1 )GV (t1 , t2 )ψnL (t2 ) + 4vF

dt1 dt2
c

mn

ˆ†
ˆ
ψmL (t1 )δGV (t1 , t2 )ψnL (t2 ).

t1 dt2
c

(C7)

mn

second term in Eq. (C7) is that of the Hamiltonian
E

E

L

L

ˆ†
ˆ
vm ψnL (0)ψmE (0)

ˆ
H2 = 2vF
m

ˆ†
ˆ
+ ψmE (0)ψmL (0) .
R

(C8)

R

FIG. 11: Left diagram: Schematic drawing of the fictitious
lead (E) coupling of which to the left-moving fermions through
the Hamiltonian (C8) gives the second term of the effective
action (C7). Right diagram: Definition of the fermion fields
R, E, and L after the change of variables of Eq. (C9).

At this point, we recognize that the second term in Eq.
(C7) is the same one as the effective action that one would
have obtained by coupling the left-moving fermions to a
fictitious lead with N channels, with chemical potential
µV . This situation is shown schematically in Fig. 11,
where the fermions in that fictitious lead are labeled with
the subscript “E”. The coupling that corresponds to the

S = vF

dt1

dτ1

c

2
+ vF

mn

old
new
ˆmL (x) → ψmL (x)θ(−x) + ψmE (−x)θ(x),
ˆ
ˆ
ψ
ˆ
ˆ
ˆ
ψEm (x) → i[ψmE (−x)θ(−x) − ψmL (x)θ(x), (C9)
where the Heaviside function θ(x) is defined such that
θ(0) = 1/2. For the new fermion fields, the “L” fields
have the chemical potential µV of the voltage probe,
whereas the “E” fields have the chemical potential µL
of the fictitious lead. Substituting Eq. (C9) into the remaining terms of Eq. (C7), one finds the effective action

ˆ†
ˆ†
ˆ
(ψmL (t1 ) + ψmF (t1 ))Smn (t1 , t1 − τ1 )ψnR (t1 − τ1 ) + h.c.

ˆ†
ˆ†
ˆ
ˆ
(ψmL (t1 ) + ψmF (t1 ))δGV mn (t1 , t2 )(ψnL (t2 ) + ψnE (t2 )),

dt1 dt2
c

In order to eliminate the Hamiltonian (C8) we introduce
new fermion fields according to8 (see Fig. 11)

(C10)

mn

where, as before, all fermion fields are evaluated at the position x = 0+ . Although the field ψEj , j = 1, . . . , N could,
in principle, be integrated out of the effective action, it is more convenient to keep them explicit and deal with them
at the level of the perturbation theory in S. No physical observables depend on the distribution function fL of the
field ψE .
The chemical potential µV of the voltage probe reservoir must be chosen such as to have no net current flowing
into that reservoir at any time. For the current IV in the voltage probe we find, in the adiabatic approximation,
IV (t) =

eivF
4Ï€

i
dεtr GK (t; ε) − 1 − Dt,ε
V
2

˜
˜
S(t; ε)GK (t; ε)S † (t; ε)
V

˜
˜
− s(t; ε)GK (t; ε)˜† (t; ε) − s(t; ε)GK (t; ε)S † (t; ε) − S(t; ε)GK R (t; ε)˜† (t; ε)
˜
s
˜
s
RR
RV
V

.
(C11)

32
where Dt,ε AB = (∂A/∂t)(∂B/∂ε) − (∂A/∂ε)(∂B/∂t), the superscript “K” refers to the Keldysh component, and
ˆ
ˆ†
GRR (t′ , s′ ) = −i Tc ψR (t′ )ψR (s′ ) ,
ˆ
ˆ†
GRV (t′ , s′ ) = −i Tc ψR (t′ )ψV (s′ ) ,
ˆ
ˆ
GV R (t′ , s′ ) = −i Tc ψ (t′ )ψ † (s′ ) .
V

R

(C12)

Calculation of the Green function GRR (t′ , s′ ) can be done with help of the effective action (C10) derived above. For
the Green functions GRV (t′ , s′ ) and GV R (t′ , s′ ), which keep explicit reference to field ψV , a slightly more complicated
calculation is necessary. After some algebra, we then find
˜
(GRV )ij (t′ , s′ ) = −vF i

c

˜
(GV R )ij (t, s) = −vF i

c

dt1

dτ1
m

dt1

dτ1
m

ˆ
ˆ†
ˆ†
ψRi (t)(ψmL (t1 ) + ψmE (t1 ))smj (t1 , t1 − τ1 )GV (t1 − τ1 , s) ,
ˆ
ˆ
ˆ†
GV (t, t1 − τ1 )s† (t1 − τ1 , t1 )(ψmL (t1 ) + ψmE (t1 ))ψjR (s) . (C13)
im

where the average is taken with respect to the effective
action (C10).
This concludes the presentation of the formal framework of the current calculation in the presence of relaxation. The remainder of the calculation proceeds along
the same lines as in Sec. IV. The backscattering term in
the effective action (C10) is equal to the original effective
action (3.17), but with a sub-unitary scattering matrix
S instead of a unitary scattering matrix S; the remaining terms in the effective action do not contain the fields
ψR and hence do not contribute to the interaction correction to the current. As a consequence, the expression
for the interaction correction to the current one obtains
in an expansion in the action (C10) is formally equal to
the expressions obtained in an expansion in the original
action (3.17), but with a sub-unitary scattering matrix
S(t, t − τ ) instead of a unitary scattering matrix.
There is a crucial difference with the calculation of Sec.
IV, however: The fact that the action (C10) features a
sub-unitary scattering matrix means that, for strong relaxation, the effective action is small, and an expansion
in the action is justified. This is in contrast to the case
of a fully coherent quantum dot, where we argued that
there was no small parameter that justified an expansion
in the action. We may thus conclude that the interaction
correction to the dc conductance and the capacitance calculated in Refs. 7 and 8 describe the case of a quantum
dot with a relaxation time much smaller than the mean
dwell time Ï„d . In this sense, although they do not apply
to the case of a fully coherent quantum dot, Refs. 7 and
8 do capture the first effect of coherent scattering inside
the dot at temperatures where scattering inside the dot
is still predominantly incoherent.
In light of this, the general trend of the results obtained
in Refs. 7 and 8 is that the charging interaction reduces
the effective dephasing rate: the weight of scattering processes with a short delay time is increased,8 leading to a
smaller probability to dephase.76 (Since the average delay time must remain constant, interactions must also

increase the weight of scattering processes with a long
delay time. If the dephasing time is shorter than the
average delay time, the net effect of the interactions is
to suppress dephasing.) A smaller dephasing rate corresponds to a larger weak localization correction and larger
conductance fluctuations, as was found in Refs. 7,8. A
detailed analysis of the expression for the current in the
voltage probe model for the various limiting cases, as we
did for the fully coherent quantum dot, is of little practical use, since the non-interacting (Hartree) contribution
to the current contains the dephasing rate, which is an
unknown parameter this model.

33

∗
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14

15

16

17
18

19

20

21
22

23

24

25

26

27
28

29

30
31

32

33

34

brouwer@ccmr.cornell.edu
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