arXiv:1005.1187v1 [cond-mat.mes-hall] 7 May 2010

TOPICAL REVIEW

Charge transport through single molecules,
quantum dots, and quantum wires
S. Andergassen1 , V. Meden1 , H. Schoeller1 , J. Splettstoesser1
and M.R. Wegewijs1,2
1

Institut f¨r Theoretische Physik A, RWTH Aachen, 52056 Aachen, Germany, and
u
JARA-Fundamentals of Future Information Technology
2
Institut f¨r Festk¨rperforschung - Theorie 3, Forschungszentrum J¨ lich, 52425
u
o
u
J¨lich, Germany
u
E-mail: schoeller@physik.rwth-aachen.de
Abstract. We review recent progresses in the theoretical description of correlation
and quantum ﬂuctuation phenomena in charge transport through single molecules,
quantum dots, and quantum wires. A variety of physical phenomena is addressed,
relating to co-tunneling, pair-tunneling, adiabatic quantum pumping, charge and spin
ﬂuctuations, and inhomogeneous Luttinger liquids. We review theoretical many-body
methods to treat correlation eﬀects, quantum ﬂuctuations, nonequilibrium physics,
and the time evolution into the stationary state of complex nanoelectronic systems.

Charge transport through single molecules, quantum dots, and quantum wires

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1. Introduction
In this review we consider diﬀerent aspects of charge transport through nanoscale
devices attached to electronic reservoirs, focusing on theoretical approaches dealing
with interaction and quantum ﬂuctuation eﬀects. Transport experiments have shown
convincingly that many of these systems - from carbon nanotubes down to single
nanometer sized molecules - behave as quantum dots. A quantum dot is a conﬁned
system of electrons, which is so small that the discreteness of energy spectrum with a
level spacing ∆ǫ becomes important; it is therefore often referred to as an artiﬁcial atom.
Due to the smallness of the quantum dot Coulomb interaction between the electrons also
plays an important role: the charging energy U which can be of same order as the level
spacing. If the quantum dot is attached to reservoirs by tunneling contacts, electrons
can leave and enter the dot and the single-particle levels are therefore broadened due
to the ﬁnite lifetime of the electrons. For suﬃciently weak tunnel coupling Γ and small
temperature T , the level spacing ∆ǫ and the charging energy U can be resolved at
standard cryogenic temperatures. This opens up the possibility of performing transport
spectroscopy by measuring the diﬀerential conductance as function of gate voltage Vg
and bias voltage V . The transport characteristics allow to count the discrete charge
states of such systems, and even more complex degrees of freedom of the quantum dot
are revealed. As an example, the spin states can be identiﬁed with the help of the
Zeeman eﬀect, and also vibrational motion of the quantum dot system can be detected.
The transport behavior, however, depends strongly on the ratios of all the energy
scales, and on the way transport is induced, e.g., by static or time-dependent applied
voltages, or by measuring linear or non-linear response in the transport voltage. In
particular, for molecular scale quantum dots the excitation spectra may be quite complex
and various splittings appear. This complicates matters further but also leads to a
variety of transport eﬀects. Besides stationary properties of quantum dots, the time
evolution out of an initially prepared nonequilibrium state is another important issue
of recent interest, providing the new energy scale /t. The time evolution itself can be
used as a tool to reveal various many-body aspects. It is also of practical importance
due to the progress in the ﬁeld of solid-state quantum information processing, initiated
by the suggestion to use spin states of quantum dots as basic elements of qubits [1].
In addition to the “bare” energy scales mentioned so far, due to correlation eﬀects
new eﬀective energy scales can be generated, which describe the physics of quantum
ﬂuctuations induced by the reservoir-dot coupling. These new energy scales are complex
functions of the above “bare” energies and we denote them in the following by Tc ,
where c stands for “cutoﬀ”. Examples are the Kondo temperature describing the strong
coupling scale for spin ﬂuctuations (usually denoted by TK ) and renormalised tunneling
couplings generated by charge ﬂuctuations. These scales will dominate the measured
transport in the strong coupling regime where Tc is the largest energy scale. However,
also in the weak coupling regime quantum ﬂuctuations set on and are of particular
interest close to resonances, where they can be analysed by a perturbative treatment

Charge transport through single molecules, quantum dots, and quantum wires

3

in the renormalised couplings. To understand the new energy scales physically and
calculate the renormalised couplings, renormalisation group (RG) ideas are required.
RG transport theories reveal that interactions are of key importance for the generation
of new eﬀective energy scales by quantum ﬂuctuations. Without interactions, quantum
ﬂuctuations are well understood, they just lead to an unrenormalised broadening of the
energy conservation by the tunneling coupling. Although new eﬀects from correlations
are most pronounced in the regime of strong correlations Γ ≪ U, interesting quantum
ﬂuctuation eﬀects already start to become visible in the regime of moderate correlations
Γ ∼ U, where perturbative and mean-ﬁeld theories are often not yet applicable. Another
interesting regime is the case of a continuous level spectrum, i.e., where the quantum dot
is replaced by a one-dimensional quantum wire. In this case, quantum ﬂuctuations in
the wire itself lead to renormalisation eﬀects and Luttinger liquid physics, with typical
power-law suppression of the bulk spectral density. Similar to the case of quantum dots,
the simultaneous presence of correlations and transport through inhomogeneities leads
to very rich phenomena. A prominent example is the suppression of the spectral density
at wire boundaries, which leads to vanishing conductance at zero temperature and bias
voltage if a quantum wire is weakly coupled to two leads. This has to be contrasted to
the Kondo eﬀect in quantum dots, where conductance becomes maximal when a single
spin is placed in an “insulating” quantum dot.
In this review we will illustrate these key issues and concentrate on correlation
eﬀects in quantum dots and quantum wires. We will describe transport theories
speciﬁcally designed to treat the case of moderate to strong Coulomb interactions
and quantum ﬂuctuation eﬀects. With these methods it is possible to account for
the complex interplay of various combinations of quantum mechanical eﬀects (level
quantization, interference, spin and angular momentum), complex excitations (e.g., due
to spin-orbit interaction and/or mechanical vibrations), Luttinger liquid physics, nonequilibrium (due to static as well as time-dependent applied voltages), and the time
evolution into the stationary state. An important aspect is that these methods apply
to general models incorporating experimentally relevant microscopic details, and oﬀer
prospects for numerical implementation. Despite these common themes, the diﬀerent
sections of this review are meant to be almost self-contained and each section closes
with a short outlook. We consistently use natural units = kB = e = 1. The paper is
organized as follows.
In section 2, we will introduce a general approach to transport through quantum
dots of arbitrary complexity, aiming at the description of three terminal single-molecule
junctions. The tunneling coupling is treated perturbatively up to second order in
the coupling Γ, accounting for charge ﬂuctuation eﬀects, while not capturing spin
ﬂuctuations which start in higher order. Besides well-known level renormalisation
and cotunneling eﬀects, electron pair tunneling is shown to give rise to resonances,
even for the most basic quantum dot model. It is shown how the complexity of, e.g.,
vibrational and magnetic excitations of molecular quantum dots needs to be accounted
for in transport spectroscopy calculations.

Charge transport through single molecules, quantum dots, and quantum wires

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Section 3 uses the same perturbation theory but extends it to transport through
systems which are subject to time-dependent ﬁelds. Using the example of adiabatic
charge pumping through a single-level quantum dot, it explains why time-dependent
external ﬁelds in the adiabatic regime are of particular interest. It is shown how
time-dependent ﬁelds can be used to reveal level renormalisations induced by charge
ﬂuctuations, an eﬀect purely due to the Coulomb interaction. This oﬀers the interesting
perspective to use time-dependent ﬁelds as ﬁngerprints for the combined inﬂuence of
quantum ﬂuctuations and correlations.
The regime of strong quantum ﬂuctuations is addressed in section 4 in the linear
response regime. It is described how a recently developed functional RG (fRG) method
is a useful tool to treat generic quantum dot models. Within this method a systematic
expansion in a renormalised Coulomb interaction is performed. However, the tunneling
coupling is treated non-perturbatively, so that the strong coupling regime can be
addressed for spin as well as for charge ﬂuctuations, at least for moderate Coulomb
interactions. Two prominent examples of strong quantum ﬂuctuations are reviewed:
The Kondo eﬀect in quantum dots and the explanation of transmission phase lapses by
strong broadening and renormalisation eﬀects in multi-level quantum dots. The latter
involves a signiﬁcant progress in the insight for an important long-outstanding problem
posed by experiments.
Section 5 describes quantum ﬂuctuations in nonlinear response, i.e. at ﬁnite bias
voltage V ≫ T , together with the time evolution into the stationary state. The
basic physics of weak spin and strong charge ﬂuctuations is illustrated on recent
results for two elementary models: the anisotropic Kondo model at ﬁnite magnetic
ﬁeld and the interacting resonant level model. A real-time renormalisation group
(RTRG) method is introduced, where, complementary to the fRG scheme presented
in section 4, an expansion in a renormalised tunneling coupling is performed whereas
the Coulomb interaction is treated non-perturbatively. This allows the description of
strong correlation eﬀects Γ ≪ U for moderate tunneling.
Finally, in section 6 we describe transport properties of quantum wires. The status
of Luttinger liquid physics is reviewed, in particular in connection with the presence of
inhomogeneities, backscattering, interference eﬀects, and nonequilibrium. As in section
4, the fRG scheme is shown to be a unique method, capable of describing the whole
crossover from few- to multi-level dots up to the limit of continuous spectra in quantum
wires. Its ability to simultaneously incorporate microscopic details is of particular
importance for the description of experiments.
2. Transport through single molecule quantum dots
Molecular quantum dots present the ultimate miniaturization of quantum dot devices,
which have been realized in semi-conductor heterostructures and wires, metallic
nanoparticles, and carbon-nanotube molecular wires. Various methods have been
developed to contact nanometer size single molecules. Transport measurements in

Charge transport through single molecules, quantum dots, and quantum wires

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three terminal junctions have most clearly demonstrated that single molecules also are
quantum dots, although of a particularly complex nature, see for a review [2–4]. In view
of this a central theoretical challenge is to describe the transport through a quantum
dot of arbitrary complexity, covering the entire, broad class of systems mentioned above.
We now ﬁrst outline this general framework, which is relevant also for sections 3 and 5.
For details see [5] and the references therein.
Starting from the limit of completely opaque tunnel junction, we can specify the
exact many-body level spectrum and states |s of the quantum dot in this case,
HD =
s

Es |s s|.

(1)

Often, only a few of these states are actually accessible in an experiment. Subsequently,
we incorporate the non-zero transparency of the junction in the coupling
′

HT =
s,s′

αkσ

ss
Tαkσ |s s′ |cαkσ + h.c.

(2)

′

ss
through the amplitudes Tαkσ for an electron with spin σ to tunnel on / oﬀ the dot from
/ to one of the electronic reservoirs labeled by α = L, R. Generally, in Equation (2) all
states s and s′ are coupled which diﬀer by the addition of one electron. The reservoir α
by itself is described by

Hα =

(ǫkσ + µα )nαkσ ,

(3)

kσ

where nαkσ denotes the electron number operator and k labels the orbital states. Each
electrode is restricted to contain eﬀectively non-interacting fermion quasi-particles with
density of states ρα . We make the further statistical assumption that the electrochemical potential µα and temperature T are well deﬁned for each reservoir α separately.
The central quantity is the reduced density operator p of the dot, obtained by partial
averaging of the full density operator over the reservoirs. From p any non-equilibrium
expectation value of a local observable can be calculated, and also (see below) the
transport current. Starting from the above general model, one can derive the kinetic
equation from which the time-evolution of the dot density operator can be calculated:
d
p(t) = − iLD p(t) − i
dt

t

dt′ Σ(t, t′ ) p(t′ ).

(4)

t0

Here the Liouvillian superoperator LD acts on a density operator by commuting it with
HD , i.e., LD p(t) = [HD , p(t)]. This linear transformation generates the “free” timeevolution of the dot density matrix, i.e., in the case where it is not coupled to the
electrodes. The challenge lies in the calculation of the transport kernel Σ(t, t′ ) which
incorporates all eﬀects due to the coupling to the electrodes which is adiabatically
switched on at time t0 (t0 < t). The superoperator Σ(t, t′ ) is a non-trivial functional
of three “objects”: (i) the dot Liouvillian LD (energy diﬀerences of initial, ﬁnal and
virtual intermediate states of processes), (ii) the dot part of the tunnel operator (tunnel
vertices) and (iii) the product of the Fermi distribution functions f ((ǫkσ − µα )/T ) and
density of states ρα of all electrodes α. Importantly, compact diagrammatic rules for the

Charge transport through single molecules, quantum dots, and quantum wires

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Vg
µL

µR
TL

U

TR

Figure 1. (Color online) Sketch of a single-level quantum dot attached to two
reservoirs with µL/R = ±V /2.

calculation of this kernel in terms of these three objects are known exactly [5,6]. In this
section and section 3 we review the basic physics which emerges from the calculation of
2
the kernel by perturbation theory in the tunnel operator HT ∼ Γ to leading and next
to leading order. This is applicable to the “high-temperature” regime, where higher
order eﬀects are present but still weak. In section 4 and 5 the low-temperature physics
due to strong quantum ﬂuctuations is illustrated. In particular, in section 5 the kernel
is calculated non-perturbatively in a renormalisation-group framework. The technical
details of these two approaches can be found in [6] and [5] respectively.
We now ﬁrst review all possible basic types of transport resonances which can arise
due to tunneling processes of the ﬁrst and second order in Γ. For this purpose we
consider a single energy level with Coulomb interaction, the so-called Anderson model,
where we emphasize the need to consider a ﬁnite charging energy U [7] such that double
occupancy is possible (not only as a virtual intermediate state). This will set the stage
for the subsequent discussion of examples of transport involving more complex spectra
in section 2.2 and the time-dependent transport considered in section 3.
2.1. Transport spectroscopy: sequential, co- and pair-tunneling
Most types of transport resonances observed in quantum dot experiments can be
illustrated by the Anderson model sketched in ﬁgure 1 where a single orbital level
is coupled to electrodes α = L, R with tunnel amplitudes Tα . The strength of the
tunnel coupling is characterized by the spectral function, Γα = 2πρα |Tα |2 , whose energy
dependence is neglected. Denoting by N the electron number on the dot, the eigenstates
|s of the dot (when decoupled from the electrodes) are easily classiﬁed. For the empty
dot (N = 0) there is one state s = 0 with energy E0 = 0. For the singly occupied dot
(N = 1) there are two states s = σ =↑, ↓ with energy Eσ = εσ . The N = 1 states
are split by a magnetic ﬁeld such that s =↓ is the ground state and s =↑ the excited
state. Finally, for the doubly occupied dot (N = 2) the energy Ed = σ εσ + U of the
state s = d includes the charging energy. One can show that due to conservation of the
electron number and the spin-projection along the magnetic ﬁeld only the occupation
probabilities enter into the description of the non-equilibrium state. We collect these
into a vector p = (p0 , p↑ , p↓ , pd )T . Its time-evolution is fully described by a kinetic

Charge transport through single molecules, quantum dots, and quantum wires

7

equation with a matrix kernel Σ (t, t′ ),
d
p (t) = − i
dt

t

dt′ Σ (t, t′ ) p (t′ ) ,

(5)

−∞

where we have set t0 = −∞ (c.f. (4)). For time-independent parameters considered
here, we have Σ(t, t′ ) = Σ(t − t′ ) and the occupancies will reach a constant value in the
stationary long-time limit: p (t) → p = constant for t − t0 → ∞. With the subsidiary
condition of probability conservation s ps = 1, this state is determined uniquely by
0 = − i Σ p.

(6)

As expected, here the time integral over the kernel is needed, i.e., the zero-frequency
∞
Laplace transform limz→i0 0 dteizt Σ (t, 0) = Σ. This kernel has been calculated in
closed form up to order Γ2 for an arbitrary dot model Hamiltonian in [6], allowing
arbitrarily complex transport spectra to be calculated. From a similar equation for the
measurable tunnel current from electrode α
t

Iα = −i

−∞

dt′ TrD Σα (t, t′ ) p (t′ ) → −i TrD Σα p,

(7)

the stationary current can be found. This requires an additional zero-frequency kernel
Σα which takes account of both the amount and the direction of the electrons transferred
from electrode α. Here TrD denotes the trace over the dot degrees of freedom.
In ﬁgure 2 we show exemplary results for the Anderson model of the calculations
outlined above. A complete overview of non-linear transport resonances is obtained
when plotting dI/dV (quantifying changes in the current as new transport processes
become energetically allowed) as function of the static applied bias V (which drives the
current) and gate voltage Vg (which uniformly shifts all energy levels ǫσ without changing
the quantum states) ‡. In the lower half of the panel we plot the diﬀerential conductance
dI/dV , whereas in the upper half of the panel the occupation p↑ of the spin-excited state
of the quantum dot is shown. This ﬁgure illustrates the insight provided by transport
theory, linking the measured tunnel current to what goes on in the quantum dot. To
emphasize this correspondence of the two panels, the bias axes in the lower panel is
mirrored with respect to the upper.
To the far left and right of the plot the dot is in the N = 0 and N = 2 state
respectively around zero bias. Moving towards the center from there, dI/dV shows
a peak along the ﬁrst encountered yellow skewed lines (marked “SET”) where the
condition µα = ε↓ and µα = ε↑ + U are met, respectively. Beyond this line a spin
↓ electron can now enter or leave the dot in a sequential or single-electron tunnel (SET)
process which is of ﬁrst order in Γ. In this way the N = 1 ground state s =↓ can be
reached starting from N = 0 and N = 2 respectively. Once beyond the next skew yellow
line (marked “SET*”), deﬁned by µα = ε↑ and µα = ε↓ + U, respectively, an electron
with spin ↑ can tunnel onto the dot. This ﬁrst order process makes the N = 1 excited
state s =↑ accessible, as the upper panel clearly shows.
‡ Here one assumes to a good ﬁrst approximation that the bias and gate electric ﬁelds are spatially
uniform. Corrections to this picture have been calculated in [8] for molecular junctions.

Charge transport through single molecules, quantum dots, and quantum wires

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Figure 2. (Color online) Occupation of excited spin-state, p↑ (upper panel) and
diﬀerential conductance (lower panel) for the single-level Anderson model, plotted as
a function of bias voltage V and gate voltage βVg , where β is the gate coupling factor.
The spin degeneracy is lifted by an applied magnetic ﬁeld: ε↑ − ε↓ = 50T where T
is the electron temperature. The dot is symmetrically coupled to the left and right
electrodes: ΓL = ΓR = 10−2 T = 5 × 10−5 U . The horizontal green line indicates the
inelastic cotunneling threshold, which equals the Zeeman energy. The skewed green
lines indicate where the sequential relaxation of the spin-excited state becomes possible
(COSET).

Moving further towards the center, there is a large triangular region of the dI/dV
map where only the N = 1 ground state is accessible: sequential tunneling processes out
of this state are suppressed to ﬁrst order since U ≫ T , which is referred to as Coulomb
blockade. However, a second order process, referred to as inelastic cotunneling, allows
the excited state to become occupied [9,10] when the bias exceeds the excitation energy
for a spin ﬂip, µL − µR = V > ε↑ − ε↓ . This deﬁnes a gate-voltage independent
line (horizontal), reﬂecting the fact that the charge state is left unaltered by the spinﬂuctuation process which ﬂips the spin. Below this onset voltage only elastic cotunneling
is possible, giving a smaller conductance and current. In the small triangular region
marked “ICOT” in the upper panel relaxation of the excited state is possible only by
inelastic cotunneling. Above this, in the region marked “COSET”, the spin excited state,
populated by slow inelastic excitation, can relax by a much faster sequential relaxation
process. It therefore is depleted again and the dI/dV shows a small peak as signature
of this cotunneling-assisted sequential tunneling (COSET) [11].
The above processes are well-known and have been experimentally observed in
various types of quantum dots, see, e.g., [9, 12, 13] and the reviews [2–4, 14]. However,

Charge transport through single molecules, quantum dots, and quantum wires

9

only recently, it was shown that for this elementary model yet another resonance exists
in second order in Γ [7] by taking into account all many-body states and all contributions
to the transport kernel Σ. This resonance is related to electron pair tunneling: it shows
up along a skewed line in the dI/dV map marked “PT”, the change in the contrast
indicating a step in the conductance as function of bias V . The resonance condition for
this is 2µα = σ εσ + U, indicating that the energy is paid (gained) by two electrons
coming (going) from reservoir α. This electron pair can be added to (extracted from)
the dot in one coherent process, thereby changing the charge from N = 0 to N = 2 (or
vice versa). The electro-chemical potential µα = 1 ( σ εσ + U) at which this happens
2
equals the average to the above mentioned SET resonances, i.e., due to quantum charge
1
ﬂuctuations the Coulomb energy paid (gained) per electron is reduced by 2 .
Finally, we note that the resonance positions of the sequential tunneling peaks are
slightly shifted with respect to the above mentioned values µα = ǫ↓ , ǫ↑ + U due to the
second order energy level renormalisation which is consistently accounted for. Although
this may seem unimportant here, in section 3 it will be shown that in time-dependent
transport this level renormalisation eﬀect may actually generate and dominate the full
current.
2.2. Complex excitation spectra: vibrations and spin-orbit splitting
Molecular quantum dots are clearly more complex than the above considered model,
for instance by their ability to mechanically vibrate. One important general aspect of
this complexity is that the only conserved quantity in a tunnel process is the electron
number §. In contrast, the quantum numbers of, e.g., various vibrational modes of
molecular quantum dot may change when an electron tunnels onto it. This implies
that the transport theory has to account explicitly for the density matrix oﬀ-diagonal
elements between diﬀerent excited states with the same charge number N. As is well
known, this is important already in ﬁrst order in Γ when there are degeneracies of the
energy levels Es in the same charge (N) state on the scale set by the tunneling [15–17].
In the case where there are no such degeneracies only the probabilities matter to order Γ.
However, it was shown only recently, that even in this simplest case this is no longer true
in second (or higher) order in Γ and Equation (6) can no longer be used. Instead, one
can derive from (5) an eﬀective stationary state master equation for the probabilities of
the same form as (6), but with an eﬀective kernel which contains important corrections
from the oﬀ-diagonal elements [6, 18]. An example where these corrections are found to
be very important is the Franck-Condon eﬀect. This eﬀect arises in its most elementary
form in the Anderson-Holstein model, where the dot is described by an Anderson level
(discussed in section 2.1) plus a single vibrational mode with frequency ω:
√
ω
(8)
nσ + (P 2 + Q2 ).
HD =
εσ nσ + Un↑ n↓ + λ 2Q
2
σ
σ
§ Conserved here refers to the system of reservoirs and dot, including their interaction

Charge transport through single molecules, quantum dots, and quantum wires

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Figure 3. (Color online) Diﬀerential conductance map vs. applied bias V and gate
voltage Vg (β = gate coupling factor) for a molecular level coupled to a vibrational
mode with frequency ω and strong local interaction U ≫ T, V, λ [6]. With increasing
electron vibration coupling λ, the inelastic cotunneling steps (red arrows in the λ = 2
ﬁgure indicate 1 and 2 phonon absorption) become relatively important with respect
to the suppressed SET processes. The white lines / regions correspond to negative
dI/dV , which cannot be plotted due to the logarithmic color scale. Strikingly, the
current thus displays a peak in the Coulomb blockade regime, which is very uncommon.
In all plots we have chosen ω = 40T = 104 Γ and conduction bandwidth D = 250ω.
The conductance is scaled to ΓM , the maximal sequential tunneling rate, i.e., Γ times
the maximal Franck-Condon overlap factor (which is less than 1).

√
Here Q = (b† + b)/ 2 is the vibrational coordinate normalized to the zero-point motion
√
and P = i(b† − b)/ 2 the conjugate momentum. Due to the linear coupling term ∝ λ
the vibrational mode is displaced when charging the molecule. For λ > 1 this causes
a suppression of the Franck-Condon overlap of ground-state vibrational wave functions
for two subsequent molecular charges N. As a result the tunnel rates calculated to
lowest order in Γ are suppressed and the sequential tunneling current is blocked, even
for gate voltages close to the resonance [19–21], an eﬀect also called “Franck-Condon
blockade” [22]. This implies that second order processes become of crucial importance
even at resonance [23]. The general approach developed in [6] consistently accounts
for all second order eﬀects, i.e., not just cotunneling rates as in [23]. In ﬁgure 3
we show the diﬀerential conductance calculated to second order where the vibrational
mode is shifted by several multiples of its zero-point amplitude. Whereas for λ = 1 the
standard Coulomb blockade picture is still identiﬁable, for λ = 4 it is drastically altered:
inelastic cotunneling (threshold at µL − µR = ω, 2ω, . . .) and COSET [24] resonances
proliferate in the transport spectrum. Experimentally, vibration assisted transport in
three terminal molecular devices has been reported [25–27]. Signiﬁcant Franck-Condon
blockade has been observed in suspended carbon nanotubes and compared with theory

Charge transport through single molecules, quantum dots, and quantum wires

11

in [28]. Other theoretical studies have studied the eﬀect on current noise [22, 29], the
Kondo eﬀect [30, 31], the inﬂuence of mechanical dissipation (ﬁnite Q-factor) [32, 33] or
the lack thereof [34], vibrational anharmonicity [35, 36] and interference eﬀects [35, 37].
All above works rely on the Born-Oppenheimer separation of nuclear and electronic
motion on the molecule. Transport eﬀects signaling the breakdown of this separation,
going under the generic name of (pseudo-) Jahn-Teller dynamics, have also been
calculated [38–40]. A very recent STM experiment [41] has conﬁrmed these transport
signatures of the breakdown of the Born-Oppenheimer approximation. In this work SET
resonances were measured as function of a molecular parameter, as suggested in [40], in
this case by measuring many individual molecular wires of various lengths.
Another class of examples of quantum dots with complex excitations are magnetic
molecules. Here the spin-orbit interaction results in violation of spin-selection rules for
electron tunneling and the same remarks as above apply. In particular, single molecule
magnets, reviewed in [42], oﬀer a basic example and have been of interest in view of
quantum computing [43, 44]. In such molecules the ground state has a large spin value
S > 1/2 due to strong exchange coupling of the spins of a few magnetic atoms. This
multiplet is however split by strong spin-orbit perturbations. In many cases an accurate
model for a given charge state N is
(N )

HD

2
2
2
= − DN Sz + EN Sx − Sy ,

(9)

where the uni-axial (DN ) and transverse (EN ) anisotropy parameters describe the
splitting and mixing of spin states, respectively. Recently, three-terminal measurements
on magnetic molecules have been reported, including comparison with theory [45–48],
see for a review [49]. Calculations of the sequential [50–55] and cotunneling [56–60]
regimes for models including the anisotropy have identiﬁed distinctive transport
signatures and possibly useful eﬀects. An important goal is to gain control over
the spin of a single molecule by electric means [61, 62]. Recently, the magnetic
anisotropy splittings were measured in a three terminal junction, similar to two terminal
STM measurements [63, 64]. Importantly, the magnetic ﬁeld evolution of inelastic
cotunneling [48] was measured in diﬀerent charge states. It was found that the magnetic
DN parameter depends strongly on the charge state N, which can be electrically
controlled with the gate voltage.
Beyond the issue of control over single-molecule magnetism, is the understanding
of coupled molecular magnets or magnetic centers. Apart from spin-blockade eﬀects,
expected for large spin molecular ground states [65], the eﬀect of spin-orbit interaction
on the exchange coupling between two magnetic centers inside a magnetic double
quantum dot is of interest. For instance, the spin-orbit induced Dzyaloshinskii-Moriya or
antisymmetric exchange interaction was shown to give rise to a characteristic dependence
of the transport on an externally applied magnetic ﬁeld and the polarizations of magnetic
electrodes [66].
Higher order tunneling eﬀects for magnetically anisotropic molecular quantum
dots described by (9) have attracted attention [67–69] after the prediction of a novel

Charge transport through single molecules, quantum dots, and quantum wires

12

anisotropic pseudo spin Kondo eﬀect [70]. The spin-orbit induced anisotropy in (9) not
only raises a barrier which opposes spin-ﬂuctuations (due to DN ) (section 4), but it
also generates intrinsic spin-tunneling (due to EN ). This interplay leads to interesting
anisotropic eﬀective exchange interaction with electrons transported through a molecular
magnet, as will be discussed in section 5.
Besides their own interesting transport signatures, spin and vibrations can also
interact or inﬂuence each other in transport. For instance, a mechanism of vibrationinduced spin-blockade of transport was predicted for a mixed-valence dimer molecular
transistor (“double dot”) [71, 72].
Finally, we mention that level spectrum complexity may also deceive one when
analyzing experimental data. A particularly clear example is oﬀered by recent transport
data on carbon-nanotube “peapods”, i.e. fullerene molecules in a host nanotube [73].
Here weakly gate-dependent dI/dV peaks were observed which are strongly reminiscent
of inelastic cotunneling (second order in Γ, see section 2.1). However, detailed transport
calculations have conclusively shown that the complex measured spectrum can be
assigned to sequential tunneling processes only, i.e., of ﬁrst order in Γ. This deceptive
case occurs because the two coupled quantum dots, the host nanotube and the fullerene
“impurities” have a very diﬀerent gate voltage dependence. Fortunately, the signiﬁcant
hybridization of the two dots gives a characteristic interference eﬀect in the ﬁrst order
current, distinguishing it clearly form second order eﬀects. In view of applications it
is interesting that the analysis in [73] indicates that the charge state of the fullerene
impurities is electrically tunable, independently of that of the host carbon nanotube.
2.3. Outlook
In summary, we have illustrated that general methods for spectroscopy calculations
for molecular quantum dot models are essential for the experimental characterization
and ultimately the design of nanoscale quantum dot devices. The complex spectra
due to various strong many-body interactions play a crucial but complicating role,
at the same time however, enabling interesting new possibilities. An exciting future
direction important for further progress is to eﬃciently account for renormalisation
eﬀects in complex models, in particular as outlined in section 5. Also, the approach
outlined here can be extended to other transport quantities, such as spin and heat [74].
The molecule and junction models used here can be parametrized based on atomistic
electronic structure calculations, as for instance, in [17, 75, 76]. Pursuing this further is
another important step in understanding molecular quantum dots.
3. Adiabatic transport through a quantum dot due to time-dependent ﬁelds
Charge transport through quantum dots is not necessarily due to an applied static bias
voltage: in the absence of a bias, a current can be obtained by the time-dependent
modulation of ﬁelds, applied externally to a mesoscopic device. This is of interest for

Charge transport through single molecules, quantum dots, and quantum wires

13

possible future applications in nanoelectronics, where the repeated and fast operation
of a device is of interest. Indeed the mesoscopic analogue of a capacitor has been
realized [77,78] serving as a fast coherent electron source and charge pumping [79–82] is
a candidate for the realization of a quantum current standard, to name some examples.
From a fundamental point of view the nonequilibrium created by time-dependent electric
ﬁelds provides enhanced insight into the quantum properties of a nanoscale device, which
are not accessible with equilibrium methods. If the variation of the ﬁelds is slow with
respect to the internal time-scales of the system itself, the modulation is called adiabatic.
The system then remains in a quasi-equilibrium state, meaning that the system is not
brought to an excited state by the time-dependent modulation .
A prominent example for transport in the absence of a bias voltage is adiabatic
pumping: the cyclic variation of at least two of the system parameters, see ﬁgure 4,
leads to a dc charge transfer through the quantum device. In this case the pumped
charge does surprisingly not depend on the detailed time evolution of the pumping
cycle; this can be shown by formulating it in terms of geometric phases [83–85].
Theoretically, adiabatic pumping has been extensively studied in the non-interacting
regime [86–92]. In this case a scattering matrix approach is convenient [86, 93] and has
been broadly used. While indeed Coulomb interactions are of minor importance in some
setups [77, 78, 94], they play an important role in many quantum-dot systems and lead
to a variety of interesting results. Several approaches have been used to address the
problem of adiabatic pumping through interacting quantum dots in diﬀerent regimes,
including weak Coulomb interaction [95, 96], pumping in Luttinger liquids [97], and
a quite general approach using nonequilibrium Green’s functions [98–100]. The latter
works are particularly appropriate for the study of weak interaction or for the calculation
of spin and charge pumping in the Kondo regime [99] and the role of elastic and inelastic
scattering processes has been studied in detail [100]. The treatment of the tunnel
coupling is non-perturbative in these works and they are in this sense complementary
to the results presented in the following.
It turns out that the interplay of Coulomb interaction and coherent tunneling
between a quantum dot and leads, leading to an eﬀective renormalisation of the energy
level, can be at the origin of the pumping mechanism. The result is that this energy
renormalisation, which is a pure Coulomb interaction eﬀect vanishing when Coulomb
interactions are negligible, is accessible via the pumped charge.
Here we present results in the regime in which the coupling between the dot and
the leads is moderate; this means that the energy scale Γ approaches temperature T
but is still smaller than T so that charge ﬂuctuations start to become important. In
contrast, the energy scale TK of the Kondo temperature setting the scale for the onset
of spin ﬂuctuations, is taken to be much smaller than Γ and T . To approach the eﬀect
described above we use a perturbative expansion in the tunnel coupling between the
The slow variation of the externally applied ﬁelds becomes experimentally particularly important
for delicate systems as molecules, see section 2, where problems related to heating, surface excitations
(plasmons) and laser-induced chemical reactions (photo-bleaching) should be avoided.

Charge transport through single molecules, quantum dots, and quantum wires

(a)

ΓL (t)

ΓR (t)

Y(t)

14

(b)

ε (t) + U

L

ε (t)

R
X(t)

Figure 4. (Color online) (a) Sketch of a single-level quantum dot with time-dependent
parameters attached to two reservoirs. (b) Two of the system parameters are time
dependent as indicated in (a) and perform a closed cycle in parameter space.

dot and the leads up to second order in Γ, which is appropriate in this regime. The
aim is the understanding of the inﬂuence of ﬁnite and arbitrary Coulomb interaction on
the pumping characteristics and the identiﬁcation of the physical origin of the various
contributions to the pumped charge. In order to achieve this, a diagrammatic realtime technique [101–104], which has been developed to describe non-equilibrium DC
transport through an interacting quantum dot, see section (2), was extended for the
treatment of adiabatically time-dependent ﬁelds [105].
3.1. Adiabatic expansion
Even though the approach is not restricted to speciﬁc systems, here the focus is put on a
single-level quantum dot as shown in ﬁgure 4. The quantum dot has a time-dependent
level ε(t) = ε + δε(t) and is coupled to leads α = L, R with time-dependent tunnel
¯
amplitudes Tα (t). The strength of the tunnel coupling is characterized by the time∗
¯
dependent quantity, Γα (t) = 2πρα Tα (t)Tα (t) = Γα + δΓα (t), with the lead’s constant
density of states, ρα . As described in section 2, the eigenstates of the decoupled
system are given by s = {0, ↑, ↓, d}. Tracing out the lead degrees of freedom the
system is described by its reduced density matrix, where here only the occupation
probabilities p = (p0 , p↑ , p↓ , pd )T are of interest. Their time-evolution in the long-time
limit, t0 → −∞, is fully described by the generalized Master equation, Equation (5).
The goal being the description of slowly varying system parameters, it is useful to
perform an adiabatic expansion [105]: the frequency of the variation therefore has to
be much smaller than the dwell time of electrons in the system, Ω ≪ Γ. The eﬀect of
the system parameters varying slowly in time is that the density matrix of the system
at time t is not only described by the instantaneous values of the parameters, but it
lags a bit behind the parameter change. Therefore an expansion around this time t
suggests itself. To obtain such an expansion as a ﬁrst step a Taylor expansion of p(t′ )
is performed around t up to linear order in the integrand on the right hand side of
Equation (5). This is related to the fact that memory eﬀects of the kernel have to be
taken into account. Furthermore, an adiabatic expansion of the kernel Σ (t, t′ ) itself is

Charge transport through single molecules, quantum dots, and quantum wires

15

0.06

U=0.1 Γ
U=4 Γ
U=8 Γ

2

[η2/Γ ]

0.04

U=0.1 Γ
U=4 Γ
U=20 Γ
U=30Γ

Γ

L R

L

2

QΓ ,ε[η1/Γ ]

0.1

QΓ

0.05

-0.04

0
-20

-10

ε/Γ

0

10

-0.06
-40

-30

-20

ε/Γ

-10

0

10

Figure 5. (Color online) (a) Pumped charge due to a parameter variation of the dot
level and of one of the barriers as a function of the average dot level. The dominant
contribution to the pumped charge is due to ﬁrst order tunneling processes. (b)
Pumped charge due to the pure variation of the barrier strengths as a function of
the average dot level. The dominant contribution to the pumped charge is due to
second order tunneling processes only. In both plots the temperature equals T = 2Γ.
(i)

performed. The zeroth-order term, Σt (t − t′ ), is indicated with the superscript (i) for
instantaneous. The subscript t to emphasize that the system parameters X(τ ) → X(t)
are frozen at time t, i.e., the functional dependence on X(τ ) is replaced by X(τ ) → X(t).
The ﬁrst-order term is obtained by linearizing the time dependence of all parameters
d
X(τ ) with respect to the ﬁnal time t, i.e., X(τ ) → X(t) + (τ − t) dτ X(τ )|τ =t , and
retaining only linear terms in time derivatives. This linear correction to the kernel is
indicated by the superscript (a) for adiabatic. Finally, it is necessary to perform an
adiabatic expansion for the occupation probabilities in the dot,
(i)

p(t)
′

(a)

→ pt + pt

Σ(t, t ) →

(i)
Σt (t

′

−t)+
(i)

(10)
(a)
Σt (t

′

−t)

(11)

The instantaneous probabilities pt are the solution of the time-independent problem
with all parameter values ﬁxed at time t. They are obtained by solving the stationary
Master equation, Equation (6), with all parameters replaced by time-dependent
(i)
parameters evaluated at time t. Once the instantaneous probabilities pt are known,
(a)
the adiabatic corrections pt are found from the Master equation in ﬁrst order in Ω
obtained from the expansion described above. Similarly an equation for the current can
be developed, see equation (7), and adiabatically expanded, where a current kernel ΣL
takes account for the amount and the direction of electron transfer.
On top of the adiabatic expansion a perturbative expansion in the tunnel coupling
strength Γ is performed up to second order, allowing the actual evaluation of the
kernel using diagrammatic rules as elaborated in Refs. [101–105], see section 2. This
perturbative expansion is valid in the regime of moderate couplings with respect to
temperature, i.e. Γ T .

Charge transport through single molecules, quantum dots, and quantum wires

16

3.2. Pumping current
We are interested in the time-resolved pumping current as well as the charge pumped
T
through the dot per pumping cycle. The pumped charge, Q = 0 IL (t)dt, in the bilinear
response regime, i.e. for modulations with small amplitudes, is proportional to the area
spanned in parameter space, see ﬁgure 4. As a ﬁrst step the time-resolved pumping
current is calculated in lowest order in the tunnel coupling. It is straightforward to
relate the pumped current to the dynamics of the average instantaneous charge of the
(i)
dot n t and one ﬁnds
(0)

IL (t) = −

ΓL (t) d
n
Γ(t) dt

(i,0)
t

,

(12)

(i,0)

where n t denotes the instantaneous occupation in lowest order in Γ. This suggests
(i,0)
the following interpretation. As the dot occupation in lowest order, n t , is changed in
time by varying the pumping parameters the charge moves in and out of the quantum dot
generating a current from/into the leads. Note that one of the pumping parameters must
(i,0)
be the level position since n t is given by Boltzmann distributions and is therefore
(i,0)
d
independent of the tunnel-coupling strengths. This means that dt n t = 0 if the level
position is constant. The charge moving in and out of the quantum dot is divided to
the left and to the right, depending on the time-dependent relative tunnel couplings
Γα (t)/Γ(t).
The next step is to calculate the next-to-leading order correction to the lowest
order result shown in Equation (12). Also the next-to-leading order can be written in
a compact way
ΓL (t) d
d
(i,broad,L)
(i,ren)
(1)
+
nt
.
(13)
nt
IL (t) = −
dt
Γ(t) dt
The ﬁrst term of Equation (13) contains the contribution due to the correction of
the average dot occupation induced by lifetime-broadening. It contains a total time
derivative, and as parameters are periodically changing in time, it will not lead to a net
pumped charge after the full pumping cycle. The second term has the same structure
as the zeroth-order contribution, Equation (12). It can be understood as the correction
term induced by the combined inﬂuence of charge ﬂuctuations and correlation eﬀects
giving rise to a renormalisation of the level position, ε(t) → ε(t) + σ (ε(t), Γ(t), U).
The level renormalisation, σ (ε, Γ, U), is positive when the level is in the vicinity of
the Fermi energy of the leads and negative for ε + U being close to the Fermi energy.
The sign of the renormalisation thus indicates if transitions take place between empty
and singly occupied dot or between single and doubly occupied dot. The energy level
renormalisation may be time dependent via time-dependent tunnel couplings or a timedependent level. The result is that a ﬁnite charge can be pumped by means of level
renormalisation. This is a pure Coulomb interaction eﬀect, it vanishes for U → 0. We
note that correction terms from the renormalisation of the tunneling couplings do not
contribute in ﬁrst order in Γ. For the DC current, diﬀerent contributions from higher
order processes are present at the same time, which makes it challenging to identify them

Charge transport through single molecules, quantum dots, and quantum wires

17

separately in an experiment. For the adiabatically pumped charge, where correction
terms associated with a renormalisation of the tunnel couplings and level-broadening
eﬀects vanish, the situation is distinctively diﬀerent. Studying adiabatic pumping is,
therefore, a convenient tool to access the energy-level renormalisation. This is most
dramatic in the case when the zeroth-order pumped current is zero, i.e., when pumping
is done by changing both tunnel couplings. In this case, the dominant contribution
to the pumped charge is due to time-dependent level renormalisation. These results
suggest that adiabatic pumping can be used to directly access the level renormalisation
in quantum dots.
The results for the pumped charge per pumping cycle are shown in ﬁgure 5, where η
is the enclosed surface in parameter space. Modulating the level position and one of the
tunnel couplings the pumped charge has a maximum contribution at the resonances,
see ﬁgure 5(a). Importantly the contribution from sequential tunneling is dominant
here. Figure 5(b) shows the pumped charge obtained by modulating the two tunnel
couplings. As described above, the pumped charge is due to level renormalisation and
therefore vanishes for vanishing Coulomb interaction. The sign change between the two
contributions at the resonances reﬂects the opposite sign of the level renormalisation for
the two resonances.
The method described here is generally formulated and it is therefore extendable
to a variety of systems in which Coulomb interaction is important and to which timedependent ﬁelds are applied. Charge and spin pumping through an interacting quantum
dot has been studied in the presence of ferromagnetic leads [106]. Pumping through
one or two metallic interacting islands with a continuous density of states has been
examined in [107]. In [108] the validity of the frequency regime has been extended to
faster modulations and in [109] the inﬂuence of quantum interference was studied on
the pumped charge through one or two quantum dots embedded in an Aharonov-Bohm
geometry. Finally this approach has been used to study the capacitive and the relaxation
properties of a driven quantum dot ﬁguring as a mesoscopic capacitor in [110].
3.3. Outlook
In this section we have presented results on adiabatic pumping, where interesting eﬀects
due to quantum charge ﬂuctuations and ﬁnite Coulomb interaction are revealed. The
underlying method shown here uses an adiabatic expansion of the Master equation based
on a real-time diagrammatic technique; this method is applicable whenever a mesoscopic
system is exposed to a certain number of adiabatically time-dependent ﬁelds. The
intriguing results found for systems with a relatively simple spectrum together with the
general formulation of the adiabatic expansion of the Master equation, motivate further
investigations. A challenging question is, e.g., to study the impact of time-dependent
ﬁelds on molecular devices with a complex energy spectrum, as introduced in section 2.
Another interesting development concerns the combination of the formalism of adiabatic
quantum pumping with renormalisation group methods, as described in sections 4 and

Charge transport through single molecules, quantum dots, and quantum wires

18

5, to describe the inﬂuence of time-dependent external ﬁelds in the regime of strong
correlations and strong quantum ﬂuctuations.
4. Quantum ﬂuctuations in linear response
In this section we will discuss the physics of strong quantum ﬂuctuations in combination
with correlation eﬀects in quantum dots. We will concentrate on the linear response and
static regime, the dependence on ﬁnite bias voltage together with the time evolution
will be discussed in section 5. One of the most prominent examples of the drastic
eﬀects of spin ﬂuctuations in quantum dots is the experimental observation of the
Kondo eﬀect [111]. In bulk solids a small amount of magnetic impurities leads to
an increased magnetic scattering of the electrons at low temperature, which results
in an increased resistance. In quantum dots, the mesoscopic realization of a single
spin coupled to two leads displays instead a zero-bias peak in the conductance [112].
The particular advantage of quantum dots is that they allow for a very ﬂexible tuning
of the parameters and can easily be extended to study the eﬀects of more complex
impurities, where orbital Kondo, quantum critical and interference eﬀects may arise.
Moreover, additional environmental degrees of freedom in presence of ferromagnetic
or superconducting reservoirs coupled to the quantum dot as well as ﬁnite-size eﬀects
aﬀect the transport properties. For spin ﬂuctuations, the characteristic energy scale Tc
is given by the Kondo temperature TK , which, for the special case of a single-level dot
√
πU
with Coulomb energy U and tunneling coupling Γ, is given by TK ∼ UΓe− 4Γ . This
scale is exponentially small for large Coulomb energy and, therefore, the Kondo eﬀect in
quantum dots is only visible for strongly coupled leads. In contrast, the importance of
charge ﬂuctuations is controlled by Γ and signatures are already signiﬁcant for Γ ∼ T .
Already in the last section, we showed how renormalised level positions can be identiﬁed
in the adiabatically pumped charge. Experimentally, eﬀects from charge ﬂuctuations
have been detected in transmission phase lapses through multi-level quantum dots in
Aharonov-Bohm geometries [113, 114]. New light was shed on this long-outstanding
puzzle by the insight that the combined inﬂuence of broadening and renormalisation
eﬀects induced by charge ﬂuctuations is responsible for this eﬀect [115, 116].
The analysis of the signatures of correlation and quantum ﬂuctuation eﬀects
requires appropriate methods for their theoretical description. The importance of the
development of analytical as well as numerical techniques for their treatment constitutes
therefore an important issue. In recent years functional renormalisation-group (fRG)
methods [117] have been established as a new computational tool in the theory of
interacting Fermi systems. These methods are particularly powerful in low dimensions.
The low-energy behavior usually described by an eﬀective ﬁeld theory can be computed
for a concrete microscopic model by solving a diﬀerential ﬂow with the energy scale
as the ﬂow parameter. Thereby also the nonuniversal behavior at intermediate scales
is obtained. For applications to quantum dots and wires the fRG scheme turned out
to be an eﬃcient approach for the description of the single-particle spectral properties

Charge transport through single molecules, quantum dots, and quantum wires

19

and the linear transport through generic models in presence of moderate correlations.
Arbitrary tunneling couplings can be considered, allowing to access the regime of strong
coupling between the reservoirs and the quantum system. Due to the ﬂexibility of
the microscopic modeling the whole parameter range can be easily explored, and more
complex geometries of multi-level quantum dots, as realized in recent experiments, can
be treated. In the following sections we will discuss applications of the fRG to the
description of strong spin and charge ﬂuctuations in quantum dots, exempliﬁed by the
discussion of the Kondo eﬀect and transmission phase lapses in multi-level quantum
dots.
4.1. The Kondo eﬀect in quantum dots
The appearance of an upturn followed by a low-temperature saturation of the resistivity
in metals containing diluted magnetic impurities was explained by Kondo in 1960 as an
enhanced scattering due to the screening of the local impurity spins by the conduction
electrons [118]. The Fermi liquid nature of the ground state [119] and the various
theoretical treatments, including Wilson’s numerical renormalisation group (NRG) [120],
make the Kondo problem one of the best understood many-body phenomenon in
condensed matter physics, as well as a paradigm for correlated electron physics [121].
In the late 90’s, the prediction by Ng and Lee [122], as well as Glazman and Raikh [112]
of the Kondo eﬀect in quantum dots led to a revival of the subject, and has been
observed in 2D GaAs/AlGaAs heterostructure quantum dots [111, 123, 124], silicon
MOSFET’s, as well as in carbon nanotubes [125] and contacted single molecules
[126, 127]. Conﬁnement and electrostatic gating enabled to investigate the Kondo
eﬀect in an artiﬁcial Anderson impurity through electrical transport measurements,
2
characterized by a unitary conductance 2 eh at zero temperature [128]. Besides providing
a paradigm for a variety of physical eﬀects involving strong electronic correlations, the
Kondo eﬀect in nanostructures allows for the realization of spintronic device constituents
as spin ﬁlters or quantum gates. The understanding and control of the transport of spinpolarized currents are of fundamental importance for the advancement of semiconductor
spintronic device technology [129].
The coupling of a quantum dot with spin-degenerate levels and local Coulomb
interaction to metallic leads gives rise to Kondo physics [121]. A small quantum dot with
large level spacing is described by the single-impurity Anderson model (SIAM) in the
regime in which only a single spin-degenerate level is relevant. At low temperatures and
for suﬃciently high barriers the local Coulomb repulsion U leads to a broad resonance
plateau in the linear conductance G of such a setup as a function of a gate voltage Vg
which linearly shifts the level positions [112, 122, 130–132]. It replaces the Lorentzian of
width Γ found for noninteracting electrons. On resonance the dot is half-ﬁlled implying
1
a local spin- 2 degree of freedom responsible for the Kondo eﬀect [121]. In the limit of
large U ≫ Γ the charge degrees of freedom can be integrated out and the spin physics is
described by the Kondo model, which will be discussed in section 5. For the SIAM the

Charge transport through single molecules, quantum dots, and quantum wires

20

zero temperature conductance is proportional to the one-particle spectral weight of the
dot at the chemical potential [133]. The appearance of the plateau of width U in the
conductance is due to the pinning of the Kondo resonance in the spectral function at the
Vg U (here Vg ≡ ǫ + U = 0 corresponds to the half-ﬁlled
chemical potential for − U
2
2
2
dot case) [121, 132]. Kondo physics in transport through quantum dots was conﬁrmed
experimentally [111, 134], and theoretically using the Bethe ansatz [130, 132] and the
NRG technique [120, 135]. However, both methods can hardly be used to study more
complex setups. In particular, the extension of the NRG to more complex geometries
beyond single-level quantum-dot systems [131, 132, 136] is restricted by the increasing
computational eﬀort with the number of interacting degrees of freedom. Alternative
methods which allow for a systematic investigation are therefore required. Here the fRG
approach is proposed to study low-temperature transport properties through mesoscopic
systems with local Coulomb correlations.
A particular challenge in the description of quantum dots is their distinct behavior
on diﬀerent energy scales, and the appearance of collective phenomena at new energy
scales not manifest in the underlying microscopic model. An example of this is the
Kondo eﬀect where the interplay of the localized electron spin on the dot and the spins
of the lead electrons leads to an exponentially small (in U/Γ) scale TK . This diversity
of scales cannot be captured by straightforward perturbation theory. One tool to cope
with such systems is the renormalisation group: by treating diﬀerent energy scales
successively, one can often ﬁnd an eﬃcient description for each one. The fermionic
fRG [137] is formulated in terms of an exact hierarchy of coupled ﬂow equations for
the vertex functions (eﬀective multi-particle interactions) as the energy scale is lowered.
The ﬂow starts directly from the microscopic model, thus including nonuniversal eﬀects
on higher energy scales from the outset, in contrast to eﬀective ﬁeld theories capturing
only the asymptotic behavior. As the cutoﬀ scale is lowered, ﬂuctuations at lower energy
scales are successively included in the determination of the eﬀective correlation functions.
This allows to control infrared singularities and competing instabilities in an unbiased
way. Truncations of the ﬂow equation hierarchy and suitable parametrizations of the
frequency and momentum dependence of the vertex functions lead to new approximation
schemes, which are devised for moderate renormalised interactions. A comparison with
exact results shows that the fRG is remarkably accurate even for sizeable interactions.
To exemplify the eﬀect of correlations, we ﬁrst focus on a quantum dot with spindegenerate levels. For simplicity only a single level with a local Coulomb repulsion U
described by the SIAM [121] is considered, see ﬁgure 1. The hybridization of the dot
2
with the leads broadens the levels on the dot by Γα = 2πTα ρα , where ρα is the local
density of states at the end of lead α = L, R assumed to be independent of frequency
here. The energy level is determined by the gate voltage Vg . Integrating out the lead
degrees of freedom, the bare Green function of the dot [121] is
1
,
(14)
G0 (iω) =
iω − Vg + i Γ sgn(ω)
2
where Γ = ΓL + ΓR . By solving the interacting many-body problem a self-energy

Charge transport through single molecules, quantum dots, and quantum wires

U/Γ=0.5
U/Γ=5
U/Γ=12.5

2

G/(e /h)

2

21

1

n

0
2
1

0
-4

-2

0
Vg/U

2

4

Figure 6. (Color online) Upper panel: conductance as a function of gate voltage for
diﬀerent values of U/Γ. Lower panel: average number of electrons on the dot.

contribution Σ(iω) is obtained dressing the bare propagator through the Dyson equation.
The fRG is used to provide a self-energy describing the physical properties of the T = 0
2
linear conductance obtained by G(Vg ) = eh πΓρ(0) [133] in terms of the dot spectral
1
function ρ(ω) = − π Im G(ω + i0+ ). In order to implement the fRG, an infrared cutoﬀ
Λ
in the bare propagator is introduced G0 (iω) = G0 (iω)Θ(|ω| − Λ). As the cutoﬀ scale Λ
is gradually lowered, more and more low-energy degrees of freedom are included, until
ﬁnally the original model is recovered for Λ → 0. Changing the cutoﬀ scale leads to an
inﬁnite hierarchy of ﬂow equations for the vertex functions. In the static approximation
the ﬂow equation for the eﬀective level position V Λ = Vg + ΣΛ reads
∂Λ V Λ =

U Λ V Λ /π
(Λ + Γ )2 + (V Λ )2
2

(15)

with the initial condition V Λ=∞ = Vg [138]. In ﬁrst approximation the two-particle
vertex U Λ ≡ U Λ=∞ = U equals the bare Coulomb interaction. At the end of the ﬂow,
2
2
the renormalised potential V = V Λ=0 determines the conductance G(Vg ) = 2 eh 4V Γ+Γ2 .
2
Results for diﬀerent values of U/Γ are shown in ﬁgure 6, together with the occupation
of the dot.
For Γ ≪ U the resonance exhibits a plateau [132]. In this region the occupation
is close to 1 while it sharply rises/drops to 2/0 to the left/right of the plateau. For
2
R
asymmetric barriers the resonance height is reduced to 2 eh (Γ4ΓL ΓR )2 [121, 132]. Focusing
L +Γ
2U

on strong couplings U ≫ Γ, the solution of the above ﬂow equation V ≃ Vg e− πΓ
describes the exponential pinning of the spectral weight at the chemical potential for
small |Vg | and the sharp crossover for a V of order U. While already the ﬁrst order
in the ﬂow-equation hierarchy captures the correct physical behavior, the inclusion of
the renormalisation of the two-particle vertex improves the quantitative accuracy of
the results. For details on the parametrization and extensions including dynamical

Charge transport through single molecules, quantum dots, and quantum wires

22

properties we refer to Refs. [138–142]. In presence of a magnetic ﬁeld the Kondo
resonance splits into two peaks with a dip in G(Vg ) at Vg = 0, providing a deﬁnition of
the Kondo scale as the magnetic ﬁeld required to suppress the total conductance to one
half of the unitary limit. For the single dot at T = 0 the conductance and transmission
phase are related by a generalized Friedel sum rule to the dot occupancy [121] by
2
G = 2 eh sin2 ( π n ) and α = π n . For gate voltages within the conductance plateau
2
2
the dot ﬁlling is 1 and the phase is π . The description of transmission phases for more
2
complex setups will be discussed in the next subsection.
4.2. Mesoscopic to universal crossover of the transmission phase of multi-level
quantum dots
The following application to a multi-level quantum dot illustrates the strength of the fRG
approach, the ﬂexibility and simple implementation, in the description of the intriguing
phase-lapse behavior observed in experiments by the group of Heiblum at the Weizman
Institute [113, 114]. The transmission amplitude and phase T = |T |eiα of electrons
passing through a quantum dot embedded in an Aharonov-Bohm geometry showed
a series of peaks in |T | as a function of a plunger gate voltage Vg shifting the dot’s
single-particle energy levels. Across these Coulomb blockade peaks α(Vg ) continuously
increased by π, as expected for Breit-Wigner resonances. In the valleys the transmission
phase revealed “universal” jumps by π for large dot occupation numbers. In contrast,
for small dot ﬁllings the appearance of a phase lapse depends on the dot parameters in
the “mesoscopic” regime. From the theoretical side, the observed behavior is captured
by an fRG computation of the transport through a multi-level quantum dot with local
Coulomb interactions [115, 116]. An essential aspect in the description of the generic
gate-voltage dependence of the transmission lies in the feasibility of a systematic analysis
of the whole parameter space. The interaction is taken into account approximately, but
a comparison to numerically exact NRG data for special parameters proves the fRG
results to be reliable as long as the interaction parameter and the number of almost
degenerate levels do not become too large simultaneously. The results are shown in
ﬁgure 7, for spin-degenerate levels we refer to Ref. [116].
Assuming level spacings of the order of the level width or smaller for large dot
ﬁllings (similar to a hydrogen atom with decreasing level spacing for increasing quantum
number) and well separated levels for dots occupied by only a few electrons, the
obtained results are consistent with the experimental ﬁndings in both regimes. In
the “mesoscopic” regime the Coulomb blockade yields an increase in the separation
of the transmission peaks. The behavior in the valleys depends on the details of the
level-lead couplings, and can be continuous or discontinuous with a phase lapse of π
(see upper panels in ﬁgure 7). For several almost degenerate levels in the “universal”
regime the hybridization leads to a single broad and several narrow eﬀective levels (Dicke
eﬀect [143]). In presence of the Coulomb repulsion, the gate-voltage dependence of the
broad level is signiﬁcantly reduced, in contrast to the narrow levels crossing the broad

Charge transport through single molecules, quantum dots, and quantum wires

23

α/π

4

0
1

α/π

|T|

MESO.
δε/Γ=10

0
2

-24

0

0
1

24

|T|

UNIV.
δε/Γ=0.1

0

-6

0

6

Vg / Γ
Figure 7. (Color online) Transmission amplitude |T (Vg )| and phase α(Vg ) for U/Γ = 1
and N = 4 equidistant levels with spacing δǫ . Decreasing δǫ /Γ leads to a crossover
from mesoscopic to universal behavior.

level close to the chemical potential. The combination of these eﬀects induces well
separated transmission peaks, absent without interaction. The “universal” phase lapse
can be understood as a result of the Fano eﬀect of the eﬀective renormalised levels,
where π phase lapses appear in resonance phenomena (see lower panels in ﬁgure 7).
The mechanism leading to the phase lapses reﬂects also in the dot level occupancies,
the broad level being ﬁlled and emptied via the narrow levels [144, 145]. The detailed
understanding of the charging of a narrow level coupled to a wide one in presence of
interactions is of interest also in connection with experiments on charge sensing [146].
From an fRG analysis emerges a single-parameter scaling behavior of the characteristic
energy scale for the charging of the narrow level with an interaction-dependent scaling
function [147].
4.3. Outlook
The described fRG approach presents a reliable and promising tool for the investigation
of correlation and quantum ﬂuctuation eﬀects in quantum dots, the computational eﬀort
being comparable to a mean-ﬁeld calculation. In contrast to the latter the fRG does
not lead to unphysical artefacts. Besides the Kondo eﬀect and transmission phases in
multi-level dots, as explained above, also the competition between the Kondo eﬀect
and interference phenomena in more complex quantum-dot systems can be described
[138,148–151]. Furthermore, correlation eﬀects for a large number of single-particle levels

Charge transport through single molecules, quantum dots, and quantum wires

24

can be discussed up to the limit of one-dimensional quantum wires. These applications
involve inhomogeneous Luttinger liquids [152–155], which will be discussed in section
6, as well as the inﬂuence of Luttinger liquid leads coupled to a quantum dot [156].
Challenging extensions and theoretical developments of the fRG involve the analysis of
inelastic processes to describe dynamical properties, and the inclusion of higher-order
contributions to access the physics at large U/Γ. A fundamental question concerns the
combination with the Keldysh formalism to describe nonequilibrium problems addressed
in the next section.
5. Quantum ﬂuctuations in nonequilibrium and time evolution
In this section we discuss quantum ﬂuctuations in quantum dots or molecules in the
presence of a ﬁnite bias voltage together with the time evolution into the stationary
state. As in the previous section the quantum ﬂuctuations are induced by the coupling
to the reservoirs. If the maximum of temperature T and the distance to resonances
δ decreases, correlations eﬀects from the Coulomb interaction lead to an increased
renormalisation of the couplings and the excitation energies hi of the quantum dot
(like e.g. magnetic ﬁeld, level spacing, single-particle energy, etc.). Renormalisation
group (RG) methods, which expand systematically in the reservoir-system coupling (in
contrast to the fRG method described in the previous section 4, where an expansion in
the Coulomb interaction is performed), show that these renormalisations are typically
logarithmic or power laws. In nonequilibrium or for the time evolution, the voltage
V and the inverse time scale 1/t are two new energy scales, which can cut oﬀ the
RG ﬂow. New physical phenomena emerge, which we will illustrate in this section by
considering very basic two-level quantum dots, like two spin states (Kondo model) or
two charge states (interacting resonant level model (IRLM)). Conceptually, one has to
distinguish between the two diﬀerent regimes of strong and weak quantum ﬂuctuations.
In strong coupling, an expansion in the renormalised coupling is no longer possible,
and, except for the case of strong charge ﬂuctuations in the IRLM or for the case
of moderate Coulomb interactions (which can be treated with fRG methods), the
nonequilibrium case remains a fundamental yet unsolved problem. Especially for the
case of strong spin ﬂuctuations, represented by the nonequilibrium Kondo model or the
nonequilibrium single-impurity Anderson model, an analytical or numerical solution is
still lacking. Even very basic questions, like e.g. the splitting of the Kondo peak in
the spectral density by a ﬁnite bias voltage are not yet clariﬁed. In weak coupling,
where a controlled expansion in the renormalised coupling is possible, many results
are already known. The simplest ansatz is to take the bare perturbation theory, as
explained in section 2, use a Lorentzian broadening of the energy conservation (induced
by relaxation and decoherence rates), and replace the bare couplings and excitation
energies by the renormalised ones obtained from standard equilibrium poor man scaling
(PMS) RG equations cut oﬀ by the maximum Λc = max{T, hi , V, 1/t} of all physical
energy scales. However, it turns out that this approach is not suﬃcient. In contrast

Charge transport through single molecules, quantum dots, and quantum wires

25

to temperature, the bias voltage is not an infrared cutoﬀ since it tunes the distance to
resonances, where new physical processes, like e.g. cotunneling (see section 2), can occur.
Therefore, the correct cutoﬀ is not the voltage but the distance δi to resonances, where
RG enhanced contributions occur. Furthermore, at resonance δi = 0, these contributions
are cut oﬀ by relaxation and decoherence rates Γi , i.e. one should use the energy scale
|δi + iΓi | as cutoﬀ parameter for the couplings. An RG approach has to be developed to
reveal how the various cutoﬀ parameters inﬂuence the couplings. It turns out that the
rates Γi are transport rates, which have to be determined from a kinetic (or quantum
Boltzmann) equation, in contrast to rates describing the decay of a local wave function
into a continuum. Therefore, the RG has to be combined with kinetic equations. The
transport rates Γi depend also on voltage and are important parameters to prevent the
system from approaching the strong coupling regime for voltages larger than the strong
coupling scale Tc . Furthermore, it turns out that terms can occur in the renormalised
perturbation theory which are not present in the bare one, like e.g. the renormalisation
of the magnetic ﬁeld in linear order in the coupling for the Kondo model (similar eﬀects
can also happen in higher orders for generic models). For the time evolution, these
terms are of particular interest in the short-time limit, since in this regime they are cut
oﬀ by the inverse time scale 1/t and lead to universal time evolution. At large times,
non-Markovian parts of the dissipative kernel of the kinetic equation lead to many other
interesting eﬀects. Among them are unexpected oscillation frequencies involving the
voltage, unexpected decay rates, power-law behavior, and a diﬀerent cutoﬀ behavior at
resonances compared to stationary quantities. In particular, for metallic reservoirs with
a constant density of states, it can be shown that all local physical observables decay
exponentially accompanied possibly by power-law behavior. Since all these eﬀects occur
already for very basic two-level systems, it has to be expected in the future that many
more interesting eﬀects will be found with respect to the physics of quantum ﬂuctuations
in nonequilibrium.
From a technical point of view the description of correlated quantum dots in the
presence of quantum ﬂuctuations in nonequilibrium is a very challenging problem and a
playground for the development of new analytical and numerical methods. Concerning
numerical methods, numerical renormalisation group (NRG) with scattering waves
[157], time-dependent NRG (TD-NRG) [68, 158–160], time-dependent density matrix
renormalisation group (TD-DMRG) [161, 161–165], quantum Monte Carlo (QMC) with
complex chemical potentials [166], QMC in nonequilibrium [167–169], and iterative
path-integral approaches (ISPI) [170] have been developed. However, the eﬃciency
of these methods is often not satisfactory in the regime of either strong Coulomb
interaction, large bias voltage, or long times. Exact analytic solutions exist for some
special cases [162, 171–174] and scattering Bethe-ansatz methods have been applied
to the IRLM [175]. Other analytical methods are mainly based on RG approaches.
Besides conventional PMS [176], improved frequency-dependent RG schemes [177–179],
ﬂow equations [180], and fRG methods [141, 142, 181–184] have been used. Whereas
the latter expands systematically in the Coulomb interaction, we will introduce in this

Charge transport through single molecules, quantum dots, and quantum wires

26

section a formally exact RG method, which expands systematically in the reservoirdot coupling. It uses the perturbation theory described in section 2 and allows for a
direct calculation of the kernel of the kinetic equation (4) in Liouville space, the socalled real-time RG method (RTRG) [185–187], which recently has been technically
optimized and formulated in pure frequency space (RTRG-FS) [5]. The method has
been successfully applied to calculate analytically all static and dynamical aspects of
weak spin ﬂuctuations for the anisotropic Kondo model at ﬁnite magnetic ﬁeld [188–190]
and strong charge ﬂuctuations for the IRLM [191]. Many aspects of these models can
also be derived with other methods, which will be mentioned in the following at the
appropriate places.
5.1. Real-time RG in frequency space (RTRG-FS)
The aim of RTRG-FS is to combine the diagrammatic expansion in Liouville space,
as described in section 2, with RG to resum systematically infrared divergences. We
consider a time translational invariant system and use as an initial condition at t0 = 0
that the dot and the reservoirs are decoupled. The kinetic equation (4) can be written
in Laplace space as
i
p(t = 0) ,
(16)
p(z) =
˜
z − Leﬀ (z)
D
∞

where p(z) = 0 dt eizt p(t) is the reduced density matrix of the dot in Laplace space
˜
eﬀ
˜
˜
and LD (z) = LD + Σ(z), where Σ(z) denotes the Laplace transform of the kernel
Σ(t − t′ ) = Σ(t, t′ ). The eﬀective dot Liouville operator Leﬀ (z) contains all reservoir
D
degrees of freedom and is of dissipative nature. The qualitative dynamics can be
analyzed from the analytic structure of p(z), which is an analytic function in the upper
˜
half of the complex plane with poles and branch cuts only in the lower half. The
i
single poles located at zp = hi − iΓi correspond to exponential decay with oscillation
frequencies hi and decay rates Γi . From the pole at z = 0, the stationary state can
be obtained via the solution of Leﬀ (i0+ )p = 0. Due to non-Markovian terms arising
D
from the z-dependence of the Liouvillian, the weight of these poles is changed by Zfactors and branch cuts can occur. For a constant density of states in the leads (normal
metallic case) it can be shown generically [190, 191] that the Z-factor leads to universal
i
short-time behavior, whereas the branch cuts give rise to an exponential decay ∼ e−izb t
i
accompanied by power-law behavior. zb denotes the position of the branching point,
which is shifted from some pole position by multiples of the electrochemical potentials of
the reservoirs leading to oscillation frequencies involving the voltage. From the point of
view of error correction schemes in quantum information processing it is quite important
to understand these corrections to Markovian behavior [192–194].
It is unique to the RTRG-FS method that it provides direct access to the important
quantity Leﬀ (z). By systematically integrating out the energy scales of the reservoirs
D
step by step, a formally exact RG equation can be derived for Leﬀ (z)Λ , where all reservoir
D
energy scales beyond Λ are included. This RG equation is coupled to other RG equations

Charge transport through single molecules, quantum dots, and quantum wires

µL

Jz /

S

Jz /

µR

(a)

µL

ΓL

ε

UL

ΓR
UR

27

µR

(b)

Figure 8. (color online) Two fundamental quantum dot models. (a) is the Kondo
model, a spin- 1 coupled via exchange couplings Jz,⊥ to two reservoirs. (b) is the IRLM,
2
a spinless 1-level quantum dot coupled via tunneling rates ΓL,R and Coulomb couplings
UL,R to two reservoirs. The electrochemical potentials are given by µL/R = ±V /2

for the couplings. Similar schemes can be developed for the calculation of the transport
current [5] and correlation functions [189]. All RG equations involve resolvents similar
to the one occurring in (16) where z is replaced by Λ together with other physical energy
scales. As a consequence, it can be shown that, besides temperature, each term of the
RG equation has a speciﬁc cutoﬀ scale Λi , which is generically of the form
Λi = |E +

j

nj µαj − hi + iΓi | ≡ |δi + iΓi | .

(17)

Here, E is the real part of the Laplace variable, nj are integer numbers, and µα denotes
the electrochemical potential of reservoir α. It shows that the cutoﬀ scale is given by the
distance δi to resonances. Furthermore, it provides the generic proof that, at resonance
δi = 0, the cutoﬀ scale is given by the corresponding rate Γi . This issue was under
debate for some time because it was speculated that electrons tunneling in and out via
the same reservoir correspond to low-energy processes, which could possibly lead to a
strong coupling ﬁxed point even in the presence of a ﬁnite bias voltage [195]. However,
it was argued that voltage-induced decay rates prevent the system from approaching
the strong coupling regime [177, 196, 197]. The microscopic inclusion of decay rates as
cutoﬀ scales into nonequilibrium RG methods was achieved within RTRG [185–187],
ﬂow equation methods [180], and RTRG-FS [5].
5.2. Applications
The two models used to illustrate the basic physics of spin and charge ﬂuctuations are
sketched in ﬁgure 8. One model is the Kondo model at ﬁnite magnetic ﬁeld h already
discussed in section 4, where a spin-1/2 couples via anisotropic exchange couplings Jz/⊥
to the spins of two reservoirs. We have assumed a symmetric coupling to the leads and
note that during the exchange it is also allowed that a particle is transferred between the
reservoirs. The model results from the Coulomb blockade regime of a quantum dot with
one level, where charge ﬂuctuations are frozen out and only the spin can ﬂuctuate. This
leads to an eﬀective band width of the reservoirs of the order of the charging energy U.
Anisotropic exchange couplings can be realized for a molecular magnet, see section 2.
The other model is the IRLM, where the quantum dot consists of a single spinless energy
level at position ǫ. The dot interacts with the reservoirs via tunneling processes, which

Charge transport through single molecules, quantum dots, and quantum wires

28

are characterized by the tunneling rates Γα = 2πρα |Tα |2 , with α = L, R. In addition,
there is a Coulomb interaction uα between the ﬁrst site of the reservoir leads and the
quantum dot, which are characterized by the dimensionless parameter Uα = ρα uα . In
the following we denote the bare parameters by a super-index (0) .
The two models have in common that due to spin/charge conservation the eﬀective
Liouvillian has the same matrix structure. There are three nonzero poles of the resolvent
1
±
(16) at zp = −iΓ1 and zp = ±h − iΓ2 . For the Kondo model, Γ1/2 corresponds to the
spin relaxation/decoherence rate and h is the renormalised magnetic ﬁeld. For the
IRLM, Γ1 is the charge relaxation rate, Γ2 describes the broadening of the local level,
and h ≡ ǫ is the renormalised level position. Denoting the two eigenstates of the dot
′
by ± ≡↑, ↓≡ 1, 0, the matrix elements Leﬀ (i0+ )ss,s′s′ = −iss′ W −s of the Liouvillian
D
involve the rates W s for the process −s → s. The stationary occupations are given
by ps = W s /W , which contain already most of the interesting nonequilibrium physics.
Similar rates can be deﬁned to calculate the current. We consider temperature T = 0
from now on to reveal the physics of quantum ﬂuctuations.
5.2.1. Kondo model. We consider ﬁrst the stationary case in the weak coupling regime
Λc = max{V, h} ≫ TK , where the Kondo temperature TK ≡ Tc is the strong coupling
scale for spin ﬂuctuations. The renormalised couplings from PMS cut oﬀ at the
scale Λc are denoted by Jz/⊥ . For the isotropic case, they are explicitly given by
√
(0)
J = 1/(2 ln(Λc /TK )), with TK ∼ D J (0) e−1/(2J ) (D ∼ U denotes the eﬀective band
width of the reservoirs).
In lowest order in Jz/⊥ , one obtains a “golden rule” like expression for the rates [188]
π 2
W s = J⊥ {4hδs− + (V − h)θΓ2 (V − h)} ,
(18)
2
where h, V > 0 and θΓ (ω) = 1/2 + (1/π) arctan(ω/Γ) is a step function broadened by
Γ. The latter corresponds to a Lorentzian broadening of the energy conservation law
by quantum ﬂuctuations. The spin relaxation/decoherence rates and the renormalised
magnetic ﬁeld are given by
π 2
Γ1 = J⊥ {4h + (V − h)θΓ12 (V − h)} ,
(19a)
2
π
1
2
(19b)
Γ2 = Γ1 + V Jz ,
2
2
where Γ12 = Γ1 − Γ2 and
(0)
h = 1 − Jz + Jz h(0)
1
Λc
Λc
2
2
+ (V − h)J⊥ ln
− hJ⊥ ln
|h + iΓ12 | 2
|V − h + iΓ12 |

,

(20)

where unimportant terms ∼ O(J 2) without a logarithm have been left out for h.
The expression for the rates W s up to O(J 2) can be interpreted as follows. At
high energies Λ > Λc , the voltage and magnetic ﬁeld are not relevant and the exchange
couplings are renormalised according to the PMS equations. This leads to an eﬀective
band width Λc for the reservoirs with renormalised exchange couplings cut oﬀ at Λc .

Charge transport through single molecules, quantum dots, and quantum wires

29

1.95

g

1.94

1.93

1.92

1.91
0

0.5

1

(0)

h

1.5

/V

Figure 9. g-factor g = 2 dh/dh(0) for the isotropic Kondo model with V = 10−4 D,
TK = 10−8 D.

The precise value of Λc is not important since the renormalisation is logarithmic. After
this step one uses lowest order perturbation theory with broadened energy conservation
leading to the result (18) for the rates. However, this interpretation does not work for
the rates in O(J 3 ) and it fails for the renormalised magnetic ﬁeld h already in O(J) and
O(J 2 ). We see that h contains a term linear in Jz , which does not occur in perturbation
theory, and was also discussed e.g. in [198]. It shows that it is generically not possible
to calculate coeﬃcients of certain orders in the renormalised couplings by comparing
with bare perturbation theory. In addition, we ﬁnd logarithmic corrections in O(J 2 ).
As was already discussed generically by Equation (17), they occur at resonances h = 0
or V = h. As was emphasized in [176], they are of O(J 2 ln J) at resonance and remain a
perturbative correction for J ≪ 1. They occur because the exchange couplings are not
completely cut oﬀ by Λc but partially by smaller cutoﬀ scales. The calculation of the
prefactor of the logarithmic terms is quite subtle and follows from the structure of the
RG equations, where each term has its own cutoﬀ scale Λi , see Equation (17). Similar
logarithmic corrections occur in O(J 3 ) for the rates and for Γ1/2 . In turns out that
the logarithmic terms for h and Γ1/2 contain an unexpected cutoﬀ scale Γ12 = Γ1 − Γ2 .
This is generic for all quantities entering the time evolution. It occurs because the pole
positions of the resolvent (16) have to be calculated self-consistently.
Logarithmic enhancements have been studied for the magnetic susceptibility and for
the conductance also in [178, 179, 199], using slave particle and ﬂow equation methods.
The logarithmic terms in the magnetic ﬁeld have ﬁrst been calculated in [188] using
RTRG-FS. They lead to a suppression of the renormalised g-factor as function of h/V
at h = 0 and h = V , i.e. a nonequilibrium induced eﬀect, see ﬁgure 9. Experimentally,
it is proposed to be measured by using a three-lead setup, where the third lead is a
weakly coupled probe lead [188]. A similar interesting nonequilibrium eﬀect occurs
for the logarithmic enhancement of the magnetic susceptibility χ(h/V ) at h = 0, as
ﬁrst proposed in [178]. All spin-spin correlation functions were calculated analytically
in [189] using RTRG-FS. Here, the frequency variable of the correlation function enters

Charge transport through single molecules, quantum dots, and quantum wires

30

Equation (17) as well. At resonances, kink structures and logarithmic terms were found
for the imaginary and real part of response functions, respectively.
In [190], the time evolution has been calculated from the inverse Laplace transform
of Equation (16). In the short-time limit t ≪ Λ−1 , the PMS equations are cut oﬀ by
c
t
the energy scale 1/t leading to time dependent exchange couplings Jz/⊥ . In O(J) only
the Z-factor from the linear z-dependence of Leﬀ (z) is important. In terms of the Pauli
D
matrices σ, the local density matrix can be written as p(t) = 1 + S (t) σ with
2
t
(0)
S (t) = Zt S (0) = 1 − 2(Jz − Jz )

S (0) .

(21)

Inserting the solution of the PMS equations, one obtains universal logarithmic (power
law) time evolution for the isotropic (anisotropic) case. A similar result has been found
for the longitudinal spin dynamics in [200, 201] for the special case of the ferromagnetic
Kondo model, which was also conﬁrmed by TD-NRG calculations [68]. In [190] it was
also shown that the short-time behavior of the conductance can be calculated from the
t
golden rule expression and replacing the exchange couplings by Jz/⊥ .
In the long-time limit t ≫ Λ−1 , the cutoﬀ scale for the PMS couplings is given by
c
Λc . In leading (Markovian) order one obtains the usual exponential behavior from the
single poles of the resolvent (16), where the longitudinal/transverse spin decays with
Γ1/2 and the transverse spin oscillates with h. Interesting non-Markovian corrections
i
i
occur in O(J 2 ), where logarithmic contributions ∼ J 2 (z − zb ) ln(Λc /(z − zb )) in Leﬀ (z)
D
lead to branch cuts. The branching point of the logarithm is generically given by
i
j
zb = zp + nV , i.e. is shifted by multiples of the voltage from some pole position. As
j
a consequence, an exponential behavior ∼ J 2 e−izp t e−inV is obtained, accompanied by
power-law behavior ∼ 1/t2 from the branch cut integral. The result explains why the
voltage occurs generically in the oscillation frequency, which is consistent with exact
solutions at special Thoulouse points of two-level systems [173]. Furthermore, it shows
that all terms decay exponentially but with unexpected decay rates and oscillation
frequencies relative to the Markovian terms. This is illustrated in ﬁgure 10 by the
time evolution of the transverse spin for the strongly anisotropic case Γ2 ≫ Γ1 , which
is typical for molecular magnets. For short times, one obtains the expected result of
oscillations with the magnetic ﬁeld and decay with the spin decoherence rate Γ2 . These
terms decay quickly for large decoherence rate. For longer times, a crossover is obtained
to an oscillation with the voltage V and a decay with the opposite spin relaxation rate Γ1 .
Finally, we mention that close to resonances, i.e. for |δ| ≪ Λc , 1/t, with |δ| = |V − h|, h,
logarithmic terms ∼ J 2 δt ln |(δ + iΓ12 )t| occur for the time evolution of the transverse
spin. Similar to the logarithmic terms in h and Γ1/2 , they are cut oﬀ by Γ12 = Γ1 − Γ2 .
5.2.2. Interacting resonant level model For the IRLM the PMS equations lead to a
(0)
renormalised tunneling rate, given by the power-law Γα = Γα (D/Λ)gα , with exponent
2
gα = 2Uα − β Uβ . Cutting oﬀ this scale at Tc = ΓL + ΓR , one deﬁnes self-consistently
the strong coupling scale Tc for the importance of charge ﬂuctuations. The level position
(0)
ǫ ≡ ǫ(0) and the Coulomb interaction Uα ≡ Uα remain unrenormalised [202]. In lowest

Charge transport through single molecules, quantum dots, and quantum wires

31

log10(|Sx(t)|)

0

c=0.2, Γ1/Γ2=0.24
c=0.25, Γ1/Γ2=0.1
c=0.3, Γ1/Γ2=0.04

-5

-10
0

1

2

TK t

3

4

5

Figure 10. (Color online) |Sx (t)| ≡ |Sx (t)| in the anisotropic Kondo model for
(0)
(0)
V = 2h(0) = 100 TK and various values of the anisotropy c2 = (Jz )2 − (J⊥ )2 , with
Sx (0) = 1/2, Sy (0) = 0. The dips have their origin in the oscillations of Sx (t).

order in Γα , one obtains a golden-rule like expression for the rates
Ws =

s
Wα

,

α

s
Wα = Γα θΓ2 (s(µα − ǫ)) ,

(22)

where µL/R = ±V /2 are the electrochemical potentials of the leads. The energy
broadening is given by Γ2 = Γ1 /2 with the charge relaxation rate Γ1 =
α Γα .
Since a single tunneling process changes the charge, there is a unique cutoﬀ parameter
Λα = |µα − ǫ + iΓ2 | for the PMS tunneling rates. This cutoﬀ scale will be used in
c
the following to deﬁne Γα . For |µα − ǫ| ∼ Tc , we are in the regime of strong charge
ﬂuctuations. We note the essential diﬀerence to the Kondo model, where several
cutoﬀ parameters are relevant. The current in lead α can be calculated from the rate
2
+
−
equation Iα = eh 2π(Wα p0 − Wα p1 ) with the occupation probabilities p1 = W + /Γ1 and
p0 = W − /Γ1 . These results are quite speciﬁc to the IRLM and are due to its elementary
form. They have been conjectured in [202] and later were conﬁrmed microscopically
by RTRG-FS [191] and fRG [184]. It is surprising that these results are consistent
with NRG [202] and TD-DMRG [162] calculations even for values of Uα ∼ O(1),
and capture all features obtained by ﬁeld theoretical [162, 174] and scattering Bethe
ansatz [175] approaches. In particular, negative diﬀerential conductance is obtained for
large voltages due to the power-law suppression of the tunneling rates.
Deﬁning Λc = max{ΛL , ΛR } and expanding in Uα , one ﬁnds logarithmic
c
c
enhancements close to the resonances Λα = 0
c
g

Λc
D α
1 + 2Uα ln
+ O(U 2 )
.
(23)
Γα =
Λc
|µα − ǫ + iΓ2 |
It is important to notice that if the level is in resonance with one of the reservoirs
it is not with the other, i.e. exactly at resonance the cutoﬀ scale is Γ2 for one rate
and the voltage V for the other rate. This has to be contrasted to speculations that
√
both rates are cut oﬀ by the geometric average Γ2 V [203], which leads to incorrect
power-law exponent for the on-resonance current as function of the voltage. Therefore, a
Γ(0)
α

Charge transport through single molecules, quantum dots, and quantum wires

32

microscopic determination of the precise cutoﬀ scales from nonequilibrium RG methods
is very essential.
The time evolution into the stationary state has been calculated for the IRLM
in [191]. The basic features already found for the Kondo model were recovered, showing
that these eﬀects even hold for models with charge ﬂuctuations. It turns out that the
oscillation frequencies are generically given by hi + j nj µαj , with integer numbers nj ,
i.e. the excitation energies hi of the dot are shifted by arbitrary multiples of the chemical
potentials of the reservoirs. For the IRLM it turns out that the algebraic part of the
time-evolution is ∼ (1/t)1−gα , i.e. the exponent depends on the Coulomb interaction
strength.
5.3. Outlook
The status of the ﬁeld of quantum ﬂuctuations in nonequilibrium for strongly correlated
quantum dots is that powerful techniques have been developed to study the weakcoupling limit in a controlled way. Elementary models of spin and charge ﬂuctuations
are well understood in this limit and the methods can now be applied to more complex
quantum dot models, such as discussed in section 2.2. However, two fundamental
issues are still open. First, the case of strong spin ﬂuctuations, represented in its most
elementary form by the isotropic Kondo model in the limit max{T, h, V } ∼ TK , is one
of the most fundamental unsolved problems. The case of strong charge ﬂuctuations
at resonances has so far only been understood for the IRLM, where it seems that
the broadening of the level together with a PMS renormalisation of the tunneling
couplings captures the essential physics. Whether this holds also for more complicated
models has to be tested in the future. Secondly, most of the methods used to
describe nonequilibrium properties of quantum dots are parametrized by the many-body
eigenstates of the isolated quantum dot. Therefore, they can not be extended easily to
larger systems like multi-level quantum dots or quantum wires. Two exceptions are
the ﬂow-equation method [180] and nonequilibrium fRG methods. [141, 142, 181–184].
In fRG, as already described in section 4, a perturbative expansion in the Coulomb
interaction is used and the vertices are parametrized in terms of single-particle levels.
Therefore, although fRG is restricted to the regime of moderate Coulomb interactions,
the potential of the method lies particularly in the possibility to treat multi-level
quantum dots and quantum wires. Preliminary studies have tested the fRG for the IRLM
and the single-impurity Anderson model (SIAM). For the IRLM, a static version was
used, where the frequency dependence of the Coulomb vertex was neglected [184, 191].
An agreement was found with RTRG-FS and TD-DMRG [162]. Recently, a dynamic
scheme with frequency-dependent 2-particle vertices has been developed and was used
to analyze the nonequilibrium SIAM in the regime of strong spin ﬂuctuations [141,142].
A good agreement with TD-DMRG, ISPI and QMC results was obtained for moderate
Coulomb interactions [204]. As will be described in the next section, the static version
of nonequilibrium fRG has also been applied to quantum wires and it is a challenge for

Charge transport through single molecules, quantum dots, and quantum wires

33

future research to generalize the dynamic fRG scheme to quantum wires as well.
6. Correlation eﬀects in quantum wires
6.1. An introduction to Luttinger liquid physics
A diﬀerent type of correlation physics than discussed so far is found in (quasi) onedimensional (1d) quantum wires at low temperatures. By this we mean electron systems
conﬁned in two spatial directions such that only the lowest 1d subband is occupied.
Compared to the quantum dots considered above they contain many correlated degrees
of freedom and the single-particle level spacing of the wire becomes the smallest energy
scale (“almost” continuous spectrum; see below). The dominant quantum ﬂuctuations
are now driven by the interaction U itself. In fact, below we will ﬁrst discuss isolated,
translationally invariant wires. Still, as discussed in the main part of this section,
the physics becomes even more interesting if the coupling to leads (or the coupling
between two wires) is included. Quantum wires are realized in single-walled carbon
nanotubes [205, 206], speciﬁcally designed semiconductor heterostructures [207, 208],
and in atom chains which form on certain surfaces [209]. A diﬀerent class of quasi
1d systems are highly anisotropic (chain-like) bulk materials (for a recent review see
e.g. [210]). Compared to the ﬁrst type of systems it is less clear how a single wire of
this class can be incorporated as an element in an electronic transport nanodevice and
they will thus play a minor role in the present discussion.
Although being far from experimentally realizable, translationally invariant models
of 1d electrons with sizeable two-particle interaction (Coulomb repulsion) were already
studied in the ﬁfties and sixties. Exactly solvable models were constructed leading
to a rather quick (compared to higher-dimensional correlated systems) gain in the
understanding of correlation eﬀects. Tomonaga showed that all excitations of a 1d
correlated electron system at (asymptotically) low energies are collective in nature
(“plasmons”) rather than single-particle-like as in the three dimensional counterpart
(Landau quasi-particles) [211]. In particular, collective spin and charge density
excitations travel with diﬀerent speed. Quite often this diﬀerence in velocities of
collective excitations is mistaken as being characteristic for 1d systems although it might
also occur in higher dimensional Fermi liquids. Only when adding, that no additional
quasi-particle excitations are possible spin-charge separation becomes a unique feature
of 1d correlated electrons. As a second (related) interaction eﬀect certain correlation
functions display power-law behavior as a function of energy ω and momentum k as ﬁrst
shown for the single-particle momentum distribution function n(k) by Luttinger [212]
and Mattis and Lieb [213]. Using the concepts of statistical physics one can view 1d
chains of electrons as being (quantum) critical, although the exponents depend on the
details of the underlying single-particle model (band structure, band ﬁlling) as well as on
the interaction strength. S´lyom [214] and Haldane [215] showed using renormalisation
o
group (RG) arguments that the physics of the two exactly solvable models—being

Charge transport through single molecules, quantum dots, and quantum wires

34

quite similar they are nowadays considered as a single model, the Tomonaga-Luttinger
model—is generic for 1d interacting electrons as long as correlations do not drive the
system out of the metallic phase (e.g. into a Mott insulator phase). This led to the term
Luttinger liquid (LL) physics for the above correlation eﬀects. In particular, Haldane
argued that all the exponents appearing in correlation functions of a spin-rotational
invariant system with repulsive two-particle interaction can be expressed in terms of
a single model parameter dependent number K < 1, with K = 1 for noninteracting
electrons. Together with the velocities of the charge- and spin-density excitations vc and
vs , K completely determines the low-energy physics. For a given microscopic model the
strategy to obtain the low-energy physics is thus the following: one has to determine
the three Luttinger liquid parameters as functions of the model parameters (ways to
achieve this are e.g. discussed in [216]) and plug them into the analytic expressions for
correlation functions computed within the Tomonaga-Luttinger model.
Despite the intense eﬀort (see e.g. [205–210]) a commonly accepted experiment
which shows LL behavior beyond any doubts is still pending. This has to be contrasted
to the situation in quantum dots discussed earlier where the appearance of Kondo physics
as a consequence of local correlations has convincingly been shown (see above). The
lack of clear cut experiments is partly related to the status of the theory. LL theory
only makes predictions for the asymptotic low-energy behavior but does not provide
the scales on which this sets in. In the construction of the Tomonaga-Luttinger model
all microscopic scales as e.g. given by the band curvature [217] and the shape of the
two-particle interaction [218] are disregarded. One can imagine that the upper energy
scale Tc beyond which LL physics can be found becomes so small that power laws
are masked by other scales such as the single-particle level spacing set by the length
of the wire. A profound comparison to experiments requires a knowledge of Tc and
possible lower bounds for LL physics while methods which allow to extract these scale
for microscopic models are rare. It is thus mandatory to develop new methods which
capture LL physics and can directly be applied to microscopic models. Further down,
we will return to this issue. We note in passing that the standard ab initio method—
density functional theory—cannot be applied as the existing approximation schemes fail
when it comes to LL physics.
When using a quantum wire as part of a low-temperature nanodevice one has to
be alert that correlation eﬀects might alter the electronic properties. To understand
this let us ﬁrst imagine an electron tunneling from the tip of a scanning tunneling
microscope into the bulk of a 1d quantum wire. Using Golden Rule-like arguments it
becomes clear that the diﬀerential conductance dI/dV computed from the tunneling
current I is determined by the product (more precisely the convolution) of the singleparticle spectral function ρ(ω) (the “local density of states”) of the wire and the tip.
As the latter is constant at small energies, dI/dV displays the LL power law of the
wire spectral function ρ(ω) ∼ |ω − µ|αbulk , with the wires chemical potential µ and
αbulk = (K + K −1 − 2)/4 [216]. The exponent changes to αend = (K −1 − 1)/2 if the
electron tunnels into the end of a LL as it is e.g. the case in end-contacted wires [216].

Charge transport through single molecules, quantum dots, and quantum wires

35

Evidently, tunneling in and out of either the bulk or the end of a wire plays a crucial
role when using it in a nanodevice.
Secondly, we imagine a local inhomogeneity in the quantum wire. It might either
appear (in a diﬃcult to control way) in the process of producing the wire or might
intentionally (in a controlled way) be introduced to design a functional device. As
examples for the latter one can think of double barriers [219] deﬁning a dot region with
energy levels that can be tuned via a backgate (see the discussion on transport through
quantum dots) or junctions of several quantum wires [220, 221]. One can easily imagine
that because of the collective nature of the excitations in a LL any inhomogeneity with
a (single-particle) 2kF -backscattering component, with kF being the Fermi momentum,
strongly aﬀects the low-energy physics. A ﬁrst step to substantiate this expectation is to
compute the 2kF-component of the static density-density-response function χ(q) of the
Tomonaga-Luttinger model [222]. For a noninteracting 1d electron system it diverges
logarithmically when q approaches 2kF (Lindhard function). The divergence is enhanced
and becomes power-law like χ(q) ∼ |q − 2kF |K−1 if the interaction is turned on. This
shows that the inhomogeneity strongly couples to the system and linear response theory
breaks down. This physics was further investigated applying diﬀerent RG methods to
diﬀerent models [223–227]. These studies show that even a single impurity at low
energy scales eﬀectively acts as if the chain is cut at the position of the impurity
with open boundary conditions at the two end points. E.g. for temperatures T → 0
the linear conductance G(T ) across the impurity is suppressed in a power-law fashion
G(T ) ∼ T 2αend as can again be understood using Fermis Golden Rule and the powerlaw scaling of the local spectral function at the end of a quantum wire (tunneling
from end to end of two LLs). Remarkably, the exponent is independent of the bare
scattering potential. Also a Breit-Wigner resonance of G as a function of the level
position (modiﬁed by) Vg showing up in the double-barrier geometry is strongly altered
by the interaction. In case of a “perfect” resonance with peak conductance e2 /h (per
spin), that is for equal left and right barriers (which is experimentally diﬃcult to realize),
the resonance width becomes zero, while the resonance completely disappears, that is
G → 0 for all Vg , in all other cases.
The discussed physics of inhomogeneous LLs can be understood in terms of an
eﬀective single particle problem. Using RG techniques one can show that the interplay
of a local inhomogeneity and two-particle correlations leads to an eﬀective oscillatory
and slowly decaying (∼ 1/x, with the distance x from the inhomogeneity) single-particle
scattering potential of range 1/δ, where δ is the largest of the relevant energy scales
(e.g. temperature). The wave-length of the potential is set by the (common) chemical
potential µ of the leads. Scattering oﬀ this (so-called Wigner-von Neumann potential)
leads to the discussed power laws in transport with the exponent given by the amplitude
of the potential. [224–227]
In the remaining part of this section on coherent transport through quantum wires
we describe four examples of recent attempts (by three of the present authors together
with varying colleagues) to gain a detailed understanding of the interplay of correlations

G/(2e2/h)

Charge transport through single molecules, quantum dots, and quantum wires

10−2

10−4

U′
U′
U′
U′
U′
10−3

10−2
T

36

=0
= 0.25
= 0.5
= 0.75
=1

10−1

100

Figure 11. (Color online) Temperature dependence of the linear conductance for the
extended Hubbard model with 104 sites and a single site impurity (“δ-impurity”) of
strength V = 10, for a Hubbard interaction U = 1 and various choices of U ′ ; the
density is n = 1/2, except for the lowest curve, which has been obtained for n = 3/4
and U ′ = 0.65 (leading to a very small backscattering interaction); the dashed line is
a power law ﬁt for the latter parameter set.

and local inhomogeneities in systems with diﬀerent geometries. We believe that our
insights should be kept in mind when designing transport setups to search for LL physics.
They might also be of relevance when quantum wires (in the above deﬁned sense) are
used as elements in future nanoelectronic devices. In this case correlation eﬀects might
either be used to enhance the functionality or must be tuned away if they corrupt the
latter.
In all the examples we use the functional renormalisation group (fRG) method [137]
(for a brief review see also [228]) to (approximately) treat the Coulomb interaction. It
was already introduced in section 4. Compared to other methods it has the advantages
that it (i) can directly be applied to microscopic models (continuum or lattice; ﬂexible
modeling), (ii) captures all energy scales (not restricted to the asymptotic low-energy
regime), and (iii) can be applied to systems with many correlated degrees of freedom.
In the present context (iii) means chains of realistic length (in the micrometer range)
coupled to eﬀectively noninteracting (Fermi liquid) leads. The aspects (i) and (ii)
together assure that Tc and additional energy scales aﬀecting LL physics can be
investigated. As mentioned in section 4, the approximations inherent to the fRG
approach are justiﬁed up to moderate renormalised two-particle interactions. In the
implementation (static self-energy) used for quantum wires (i) the LL exponents come
out correctly (only) to leading order in the two-particle interaction and (ii) inelastic
processes generated by the two-particle interaction are neglected. We will return to the
latter in the outlook.

Charge transport through single molecules, quantum dots, and quantum wires

37

6.2. The role of two-particle backscattering
In 1d systems at low energies and incommensurate ﬁllings only forward (momentum
transfer q ≈ 0) and backward (momentum transfer q ≈ 2kF ) two-particle scattering is
active. All other processes are suppressed due to momentum conservation and phase
space restrictions (Fermi surface consists of only two points). The relative importance of
the two processes is determined by the (real-space) range of the two-particle interaction,
which in turn is set by the screening properties of the “environment”, e.g. the substrate
on which the carbon nanotube is placed. If screening is strong, the two-particle
interaction becomes short ranged in real space and backscattering is sizable and vice
versa. Backscattering processes involving a spin up and a spin down electron are not
part of the Tomonaga-Luttinger model. For a translationally invariant LL it was shown
using an RG analysis [214] that these processes do not modify the asymptotic lowenergy physics (they are “RG irrelevant”), but that they become irrelevant only on a
scale which is exponentially small in the bare backscattering strength (they ﬂow to zero
only logarithmically). From this one can expect that the LL power laws only occur on
exponentially small scales Tc also for inhomogeneous wires. For the Hubbard model with
open boundaries it was shown that Tc ∼ exp (−πvF /U), where U is the local Coulomb
repulsion and vF the Fermi velocity [229]. In ﬁgure 11 this behavior is exempliﬁed for
the T -dependence of the linear conductance G of a wire with a single impurity described
by the extended Hubbard model [155]. This lattice model consists of a standard tightbinding chain with nearest-neighbor hopping t, a local two-particle interaction U as
well as a nearest-neighbor one U ′ . Throughout the rest of this section we use t as the
unit of energy (that is we set t = 1). For a ﬁxed band ﬁlling the relative strength
of the forward and backward scattering can be modiﬁed by varying the ratio U ′ /U.
The considered size of N = 104 lattice sites corresponds to wires in the micrometer
range, which is the typical size of quantum wires available for transport experiments.
For U ′ = 0 (strong backscattering) due to logarithmic corrections the conductance
increases as a function of decreasing T down to the lowest temperatures in the plot,
Tc being smaller than the latter. For increasing nearest-neighbor interactions U ′ the
(relative) importance of backscattering decreases and a suppression of G(T ) at low T
becomes visible, but in all the data obtained at quarter-ﬁlling n = 1/2 the suppression
is much less pronounced than what one expects from the asymptotic power law with
exponent 2αend . By contrast, the suppression is much stronger and follows the expected
power law more closely if parameters are chosen such that two-particle backscattering
becomes negligible at low T , as can be seen from the conductance curve for n = 3/4 and
U ′ = 0.65 in ﬁgure 11. The value of K for these parameters almost coincides with the
one for another parameter set in the plot, n = 1/2 and U ′ = 0.75, but the behavior of
G(T ) is completely diﬀerent. To avoid interference eﬀects discussed next, in the present
setup the two leads are coupled “adiabatically” to the wire, such that in the absence
of the impurity the conductance would be 2e2 /h for all temperatures smaller than the
band width [227].

Charge transport through single molecules, quantum dots, and quantum wires

38

One can conclude that for wires in the micrometer range clear power laws can
only be observed if screening is weak, leading to small two-particle backscattering
[155, 224–226]. Note that at T ∼ πvF /N ﬁnite size eﬀects set in (the level structure
becomes apparent), as can be seen at the low T end of some of the curves in the ﬁgure.
6.3. Overhanging parts
In most transport experiments the wires are not end-contacted and the leads do not
terminate at the contacts. This has to be contrasted to the modeling in which almost
exclusively end-to-end contacted wires are considered. It is thus crucial to understand
how overhanging parts of the wire and the leads alter the transport characteristics.
Before tackling this problem we have to discuss transport through an interacting wire
with two contacts—above we only mentioned the single contact case with powerlaw scaling G(T ) ∼ T αend . One might be tempted to argue, that two contacts can
be understood as two resistors in series, which then have to be added leading to
G(T ) ∼ T αend also for a (clean) wire with two contacts to semi-inﬁnite leads. Although
the result is correct, as can be shown using (coherent) scattering theory [227, 230], this
simple argument ignores, that adding of the resistance (of resistors in series) only holds
if the transport is incoherent, which is not the case at low temperatures as considered
here.
We here exemplify the role of overhanging parts by considering overhanging leads.
In ﬁgure 12 G(T ) is shown for a spinless tight-binding wire with nearest-neighbor
interaction U ′ coupled to two semi-inﬁnite leads via tunneling hoppings TL/R . For
spinless models the low-energy LL physics can be described by K and the charge
velocity vc . Several correlation functions are again characterized by power-law scaling.
Compared to models with spin the analytic dependence of exponents on K is modiﬁed
(see e.g. [216]). Three setups with diﬀerent overhanging parts are considered (see the left
L/R
inset). As becomes clear each non-zero number of overhanging lattice sites Nlead infers a
L/R
new energy scale πvF /Nlead and the simple power law (with exponent αend ; dashed line)
is divided in subsections separated by extensive crossover regimes. If clearly developed
the power laws in the subsections all have exponent αend (see the inset of ﬁgure 12).
Overhanging parts of the wire lead to the same eﬀect [231]. There is no reason to believe
that adding the spin degree of freedom will change this. The same holds for the two
eﬀects discussed next and we thus stick to the spinless lattice model for the rest of this
section.
One can conclude that in a generic experimental setup with overhanging parts the
power law is much more diﬃcult to observe than suggested by most theoretical studies
considering end-to-end coupling.
6.4. A wire with two contacts and a single impurity
We now exemplify the very rich interference eﬀects which occur when considering
coherent linear transport through interacting wires with several inhomogeneities. A

Charge transport through single molecules, quantum dots, and quantum wires

39

0.2

110

exponent

0.2

0.1

0
-0.2
-0.4
-4

10

T

-2

0

10

10

2

G/(e /h)

20 1086
4

10 +1

0.03
10

-4

10

-3

10

-2

T

10

-1

10

0

Figure 12. (Color online) Main plot: Linear conductance G of an interacting wire
with U ′ = 0.5 of length N = 104 + 1 as function of the temperature T . The coupling to
the leads is located at the end of the wire and has a small amplitude TR = TL = 0.25.
L/R
Dashed line: Overhanging parts Nlead = 0, that is end-to-end contacted wire. DashedL
R
L
R
dotted line: Nlead = 0, Nlead = 110. Dotted line: Nlead = 20, Nlead = 1086. The
vertical dashed lines terminating at the diﬀerent curves indicate the corresponding
crossover scales. At the lowest scale T ∼ 10−4 the power law is cut oﬀ by the level
spacing ∼ 1/N of the disconnected wire. Left inset: Setups studied (excluding the
end-to-end contacted one). Right inset: Eﬀective exponents obtained from taking the
logarithmic derivative of G(T ). The solid horizontal line indicates αend .

simple setup of this class is a wire with two end-to-end tunneling contacts to semiinﬁnite leads and an additional impurity in its bulk. As mentioned in the last subsection
the concept of “adding resistances” cannot be applied in the present context. Using
averaging over the (two) scattering phases one can show that for suﬃciently large T (of
√
the order 10−2 in ﬁgure 13) the total linear conductance is given by G ∼ G1 G2 G3
with the conductances Gi of the individual barriers. For the present setup we have
G1/3 (T ) ∼ T αend and G2 (T ) ∼ T 2αend , leading to G(T ) ∼ T 2αend ¶. As is shown in
ﬁgure 13 the dashed-dotted curve obtained by phase averaging nicely follows the bare
data (solid and dashed line) down to Tp ≈ O(10−2 ) and a power law with the expected
exponent develops for Tp
T ≪ D. For T < Tp details of the relative energy-level
structure of the two “dots” deﬁned by the three barriers matter and might even lead to
resonances (see the solid line). The “dots” energy-level structure is set by the position
of the bulk impurity and the solid and dashed line in ﬁgure 13 display data obtained
for diﬀerent positions. The example clearly shows that interference due to multiple
inhomogeneities can set a lower bound to power-law scaling of the conductance.
6.5. Transport at ﬁnite bias voltages
As our ﬁnal example we study transport at ﬁnite bias voltages. In particular we
consider the steady state current through a clean wire end-to-end coupled to two
¶ Accidentally the exponent 2αend would also follow from the unjustiﬁed adding of resistances.

Charge transport through single molecules, quantum dots, and quantum wires

40

-3

2

G/(e /h)

10

10

-4

-5

10 -3
10

-2

-1

10

10

0

10

T
Figure 13. (Color online) Linear conductance G of an interacting tight-binding wire
with tunnel couplings TL = TR = 0.1, an additional hopping impurity (single bond
with reduced hopping) of strength t′ = 0.1, and nearest-neighbor interaction U ′ = 1 of
length N = 104 as function of the temperature T . The solid and dashed curve show
data obtained for diﬀerent positions of the bulk impurity. Dashed-dotted line: result
obtained by phase averaging.

leads (reservoirs) which are hold on diﬀerent electrochemical potentials µL/R = ±V /2.
Applying a generalization of the fRG to non-equilibrium transport [183] one can
show that in the limit of high tunnel barriers the eﬀective single-particle potential
generated by the interplay of the contacts and the interaction is a superposition
of two decaying oscillations with wave-lengths set by µL and µR . The respective
2
2
2
2
amplitudes are proportional to TL/R /(TL + TR ) = ΓL/R /(ΓL + ΓR ), with Γα ∼ Tα
deﬁned as in the preceding sections. Accordingly, the non-equilibrium local spectral
function close to either of the two contacts displays power-law suppressions at µL and
µR , ρ(ω) ∼ |ω − µL/R |αL/R , with exponents proportional to the respective couplings
αL/R = ΓL/R αend /(ΓL + ΓR ). In contrast to the linear response regime the exponents
now dependent on the strength of the inhomogeneities. This behavior is shown in
ﬁgure 14. The spectral weight remains ﬁnite even at µL/R as the power-laws are cut
oﬀ by the single-particle level spacing ∼ 1/N of the disconnected wire. A transport
geometry in which the two power laws show up was proposed in [183].
6.6. Outlook
This short review on correlation eﬀects in 1d quantum wires gives a ﬂavor of the rich
and partly surprising physics resulting from the interplay of the two-particle interaction
and local (single-particle) inhomogeneities. An important ingredient missing in the
current description using the (approximate) fRG for microscopic models are inelastic
scattering processes resulting from the (Coulomb) interaction and leading to decoherence
and dephasing at intermediate to large energies. We note that they are also only partly
kept in the standard modeling which is based on the Tomonaga-Luttinger model. One
can expect that inelastic processes set an upper energy scale Tc for power-law scaling
smaller than the maximal scale given by the band width. They will also play a prominent

Charge transport through single molecules, quantum dots, and quantum wires

41

~ρ(ω)

2

1

αL=αend/5
αR=4αend/5

0
-0.1

µR

0
ω

µL

0.1

Figure 14. (Color online) Solid line: Local spectral function (close to the left contact)
in the steady-state of an interacting tight-binding wire of length N = 2·104 with tunnel
couplings TL = 0.075, TR = 0.15, and nearest-neighbor interaction U ′ = 0.5 as function
of ω. The voltage is V = 0.1

.
role when further considering non-equilibrium transport. First attempts to include
such inelastic two-particle scattering for quantum dots in and out of equilibrium were
mentioned in section 4 and are discussed in [139–141].
List of symbols
In this section we provide a list of the most relevant symbols used:
∆ǫ
Γ
Vg
V
dI/dV
t
TK
U
= kB = e = 1
T
|s
HD
HT
ss′
Tαkσ
σ
α = L, R
Hα
nαkσ
k

level spacing
tunnel coupling
gate voltage
bias voltage
diﬀerential conductance
time
Kondo temperature
charging energy
units
temperature
exact many-body level spectrum and states
dot Hamiltonian
tunneling Hamiltonian
tunnel amplitudes
spin
reservoirs
reservoir Hamiltonian
electron number
label of the orbital state

Charge transport through single molecules, quantum dots, and quantum wires
ρα
µα
p(t)
LD
Σ(t, t′ )
Iα
ǫσ
N
ω
DN
λ
Ω
Q
G(0)
Σ(iω)
G
δi
zi
h
Jz/⊥
Ws
χ
Λ
Λc
K
vc/s
kF , vF
U′
αbulk
αend

42

density of states
electro-chemical potential
dot density operator
dot Liouville operator
transport kernel
tunnel current
level energy
charge number
frequency
magnetic parameter
electron vibration coupling
dwell time of electrons in the system
pumped charge
(bare) Green function of the dot
self-energy
conductance
distance to resonances
poles of the reduced dot density matrix p in Laplace space
˜
magnetic ﬁeld
exchange couplings
rates for the process −s → s
susceptibility
ﬂow parameter
cutoﬀ
Luttinger liquid parameter
velocities of the charge/spin-density excitations
Fermi momentum, Fermi velocity
nearest-neighbor interaction
power-law exponent of the wire spectral function
power-law exponent of an electron tunneling into the end of a LL

Acknowledgments
We acknowledge all our collaborators in the various works presented in this review.
This work is supported by the DFG-FG 723, FG 912, SPP-1243, the NanoSci-ERA, the
Ministry of Innovation NRW, the Helmholtz Foundation and the FZ-J¨ lich (IFMIT).
u
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