One-channel conductor coupled to a quantum of resistance:
exact finite-frequency conductance and noise
Redouane Zamoum and Adeline Cr´pieux
e
Centre de Physique Th´orique, Aix-Marseille University,
e
163 avenue de Luminy, FR-13288 Marseille, France

arXiv:1202.0708v2 [cond-mat.mes-hall] 17 Mar 2012

In`s Safi
e
Laboratoire de Physique des Solides, Universit´ Paris-Sud,
e
Bˆt. 510, FR-91405 0rsay Cedex, France
a
We consider a one-channel coherent conductor with a good transmission embedded into an Ohmic
environment, impedance of which is equal to the quantum of resistance Rq = h/e2 below the RC
frequency. This choice is motivated by the mapping of this problem to a Tomonaga-Luttinger
liquid with one impurity, the interaction parameter of witch corresponds to the specific value K =
1/2, allowing for a refermionization procedure. The “new” fermions have an energy-dependent
transmission amplitude which incorporates the strong correlation effects and yields the exact dc
current and zero-frequency noise through expressions similar to those of the scattering approach.
We recall and discuss these results for our present purpose. Then we compute, for the first time, the
finite-frequency differential conductance and the finite-frequency non-symmetrized noise. Contrary
to intuitive expectation, both can not be expressed within the scattering approach for the new
fermions, even though they are still determined by the transmission amplitude. Even more, the
finite-frequency conductance obeys an exact relation in terms of the dc current which is similar
to that derived perturbatively with respect to weak tunneling within the Tien-Gordon theory, and
extended recently to arbitrary strongly interacting systems coupled eventually to an environment
or/and with a fractional charge. We also show that the emission excess noise vanishes exactly above
eV , even though the underlying Tomonaga-Luttinger liquid model corresponds to a many-body
correlated system. Our results apply for all ranges of temperatures, voltages, and frequencies below
the RC frequency, and they allow us to explore fully the quantum regime.
PACS numbers:

I.

INTRODUCTION

Laws of electrical circuits are drastically modified when
mesoscopic systems are incorporated. A coherent conductor embedded into a circuit sees the imposed voltage by the generator reduced by a fluctuating voltage
associated with the impedance of the surrounding electromagnetic environment. This gives rise to a current
reduction with a pronounced non-linearity, called zerobias anomaly (ZBA): transfer of electrons becomes inelastic as they exchange photons with the electromagnetic
environment. This phenomena, called the dynamical
Coulomb blockade (DCB), has attracted a tremendous
interest both theoretically and experimentally.1 Nevertheless, most of the works have been initially developed
by focusing on the limit of a weakly transmitting conductor, i.e., the tunneling regime.2 More recently, interest in the regime of few well-transmitting channels has
emerged. On one hand, reducing the number of channels makes more apparent the effect of the circuit, as
many channels could play as well the role of an outof-equilibrium environment. On the other hand, highly
transmitting conductors raise two interesting questions.
First, whether the charge fluctuations wash out the DCB,
and second, whether the reduction in the current is related to shot noise in the absence of the environment.

Indeed, this intuitive and attractive relation had been
proposed through a perturbative computation with respect to a very weak impedance, and has been checked
experimentally.3–5 Nevertheless, its validity domain is restricted to high enough energies, as logarithmic divergences arise: one needs to go beyond the weak feedback
action. This was particularly the case for an Ohmic environment, i.e., having an impedance Z(ω) = R at frequencies ω < ωRC = 1/RC (where R is the resistance and C
the capacitance): this situation has not only relevance
to realistic experiments but also a fundamental interest,
being related to the investigation of electronic interactions. The logarithmic divergences have been resumed
using a renormalization group (RG) scheme by Kindermann and Nazarov,6 dealing again with R ≪ Rq , where
Rq = h/e2 is the quantum of resistance. It has been
nevertheless possible, in a simultaneous and independent
work, to deal for the first time with an arbitrary value
of R: one of the authors, with Saleur,7 has shown that
the problem of a short-coherent conductor in series with
a resistance R is equivalent to the impurity problem in a
Tomonaga-Luttinger liquid8,9 (TLL) with an interaction
parameter:
K = (1 + R/Rq )−1 .

(1)

When specified to the limit of small R, this mapping had
led to recover the same results as those by Kindermann

2
and Nazarov. These findings answer the two questions
addressed above. First, the DCB still persists at good
transmission, showing up below an energy scale eVB (depending in a non-universal way on R and on transmission): it corresponds to the crossover voltage between
the so-called weak backscattering (WBS) regime at high
energy and strong backscattering (SBS) regime at low
energy.10 Secondly, the reduction in the current is related
to the noise in the presence of the environment, and not
to the noise of the isolated conductor as stated before
(see Eq. (4)).
Recent pioneering experiments by Pierre’s group11 –
where the strong feedback of an arbitrary impedance on
a one-channel edge state has been investigated for the
first time – has shown satisfactory agreement with the
theoretical predictions of Refs. 6 and 7. Interestingly,
even though one has to take strictly the limit of one channel in Ref. 7, the equivalence to a TLL thus established
seems to extend to many channels when they are treated
in a mean-field framework, provided one renormalizes the
parameter K by including the resistance of the channels,
and scales the voltage appropriately, as one can infer from
a more recent study.12
By mapping a one-channel conductor in series with
an Ohmic environment to a TLL, the parameter K of
which can be controlled by tuning R (see Eq. (1)), one
gets as well a promising alternative to test the theoretical
predictions for the impurity problem in a TLL. Indeed,
satisfactory experimental evidence has been lacking, even
though partial success has been claimed to be obtained.
In particular, as noticed in Ref. 7, the case of an environmental resistance equal to the quantum of resistance,
R = Rq , corresponds to a TLL with an interaction parameter K = 1/2. In that case, the impurity problem
in a TLL can be solved exactly in a more transparent
way compared to other values of K (see Ref. 13 for instance). Such an apparent simplicity is due to the introduction of new chiral fermions which arise from a mathematical construction without any physical entity, and
incorporate non-trivial strong correlation effects. The
non-perturbative investigation has a fundamental interest: it explores crucial issues not reached through perturbative approaches. Nevertheless, the situation with
K = 1/2 has never been achieved experimentally. Indeed, the ideal candidate to investigate the TLL’s behavior is usually provided by edge states with a constriction in the fractional quantum Hall effect (FQHE). However, as K plays the role of the fractional filling factor
ν, and as the description of the edges in terms of onechannel chiral TLL model is valid only for simple values:
ν = K = 1/(2n + 1) with n an integer, it has not been
possible to achieve a chiral TLL with K = 1/2. Other
potential candidates are provided by quantum wires and
carbon nanotubes. However three main difficulties arise
in those systems: (i) achieving only one backscattering
center; (ii) tuning the interaction parameter K; and (iii)
taking into account the connection to reservoirs and finite
size effects, where only a perturbative treatment of the

impurity has been possible.14–18 Thus, a coherent onechannel conductor connected to a quantum of resistance
R = Rq , as achieved recently,11 offers an unique opportunity to explore the properties of a TLL with an impurity
at K = 1/2.
The specific value, K = 1/2, is exciting as it allows us
to obtain handy analytic expressions for dc current and
the zero-frequency noise. Beyond the stationary regime,
time-dependent transport probes even more the dynamics and tests in a precise way the underlying model.
This is the case of finite-frequency (FF) noise. In view
of the mapping in Ref. 7, one can use previous results obtained in the FQHE. On the one hand, as far as the FF
symmetrized noise is concerned, Chamon et al.19 have
computed it perturbatively with respect to the impurity strength, and non-perturbatively at the specific value
K = 1/2. On the other hand, within the framework of
the exact solution of Ref. 13, Lesage and Saleur20 have
obtained only its behavior for low frequencies or close to
the “Josephson” singularity e∗ V , where V denotes the
voltage and e∗ = Ke.21 Notice that both approaches disagree, apart from the particular value K = 1/2.
Nevertheless, it has been possible to measure experimentally the FF non-symmetrized noise22 , which is more
interesting to explore. It can be inferred from Ref. 23
dealing with edge states at simple filling factors, both in
the WBS and SBS regimes: the same results apply to a
coherent conductor with a good bare transmission τ0 connected to an arbitrary resistance at high or low enough
energies, as well as for a low transmission.24 Our aim is
to go beyond this perturbative computation, offering a
full description extending over all energy ranges below
ωc = min{ωF , ωRC }, where ωF is the frequency cutoff
associated to the fermionic degrees of freedom in the conductor. This is precisely possible at R = Rq . Thus, our
study gives the first non-perturbative results for the FF
non-symmetrized noise, without assuming neither weak
resistance, nor high or low enough energies associated
to the WBS nor SBS regimes. In addition, this offers a
benchmark for other values of the resistance.
One of the key steps within the refermionization procedure is that the chiral independent fermions have now
an energy-dependent transmission amplitude t(ω) (see
Eq. (5)) which encodes the non-trivial many-body correlations. Indeed, the associated transmission coefficient
T (ω) = |t(ω)|2 has nothing to do with the effective transmission obtained for K ≈ 1 by the RG approach,6,25 and
can not be obtained adiabatically from that in the absence of interactions, τ0 . It has been widely accepted
that transport properties of the system can be obtained
within the scattering approach using t(ω). This was
shown to be valid both for the dc current, as well as
for the full counting statistics (FCS).26 Surprisingly, our
study shows that the scattering approach fails when one
deals with time-dependent transport: the FF noise can
still be expressed in terms of t(ω), nevertheless it obeys
a different relation. The same feature occurs for another
quantity: the out-of-equilibrium FF dissipative conduc-

3
dynamical Coulomb blockade
full counting statistics
finite-frequency
fractional quantum Hall effect
renormalization group
Tomonaga Luttinger liquid
strong backscattering
weak backscattering
zero-bias anomaly
zero-frequency

TABLE I: Synthesis of the abbreviations.

tance Re[G(V, ω)] which depends on the applied dc voltage and the frequency of the superimposed modulation,
in addition to its implicit dependence on temperature.
This last quantity has started to be studied only recently in few correlated systems, where its interest has
been shown: quantum wires with an impurity27 and
Kondo problem.28 This has become possible owing to
a crucial result: Re[G(V, ω)] is given by an out-ofequilibrium Kubo type formula, i.e., by the asymmetry
between the emission and absorption FF noise.29–31 In
this present and first non-perturbative investigation, we
show that Re[G(V, ω)] does not fit with its expression
obtained within the scattering approach. Even more, it
obeys exactly a surprising relation, Eq. (14), which determines it fully from the dc current, in a way similar to that
obtained in tunnel junctions within the Tien-Gordon theory, but with a renormalized charge here.32,33 Recently,
such a relation has been generalized to full extent, including arbitrary interactions: it has been shown to be
universal to lowest order with respect to a local or spatially extended tunneling at arbitrary dimension, as well
as with respect to local or spatially extended backscattering if one-dimensional systems are considered.34,35
Notice that our results will depend on parameters such
as the frequency cutoff ωc , an effective transmission we
call τ , and the voltage scale VB , which is related in a
non-universal way to ωc and τ . Nevertheless, we will
show that the scaling laws, known to be obeyed by the
dc current, can be extended to the FF conductance and
FF noise. More precisely, all these quantities depend
only on V /VB , kB T /eVB and ω/eVB , where T denotes
the temperature.
The paper is organized as follows: In Sec. II, we review the mapping to an impurity problem in the TLL
model, which is used to take the Ohmic environment
into account, and some of its general consequences on
dc transport for arbitrary values of the environmental
resistance.7 Next, in Sec. III, we present our calculations
performed in the case R = Rq , and give the formal expressions of the current, noise and conductance in term of
transmission amplitude. In Sec. IV, we discuss in details
the dc regime (dc current, differential conductance, and

zero-frequency (ZF) noise) for which we recover known
results.36–39 In Sec. V, we explicit new results concerning the time-dependent transport: FF conductance and
FF non-symmetrized noise, obtained in all energy ranges,
and explore them in the quantum regime. We finally conclude in Sec. VI. Notice that Tab. I refers to the adopted
abbreviations and Tab. II gives a synthesis of the parameters and constants that are used.
II.

MODEL AND OUTLOOK

We consider a one-channel coherent conductor with
bare transmission τ0 , in series with a dissipative environment, the effective capacitance C of which includes
implicitly that of the conductor, thus whose impedance
reads as Z(ω) = R/(1+iωRC). However, we will restrict
to energies below ωRC = 1/RC in the following, thus one
can approximate:
Z(ω) ≈ R , for ω < ωRC .

(2)

We denote the voltage imposed by the generator by V ,
and the current through the circuit by I (see upper panel
in Fig. 1). The voltage drop across the conductor is generally different from V . In the trivial limit of τ0 = 1, it is
given simply by KV , where K is defined by Eq. (1), due
to the resistance in series of the perfect conductor and
the resistance R of the Ohmic environment. Whenever
τ0 < 1, this is no more the case, apart from the perturbative regime with respect to 1 − τ0 , when τ0 is close to
one.

Ζ(ω)

V
I

τ0
1

0.2

Ω Ωc

DCB
FCS
FF
FQHE
RG
TLL
SBS
WBS
ZBA
ZF

0.1

0

0

0.7

0.8

0.9

1

Τ
FIG. 1: Upper panel: Schematic representation of a onechannel conductor with bare transmission τ0 embedded in an
electric circuit. Lower panel: Profile of the frequency dependent transmission coefficient T of the equivalent system as a
function of frequency ω/ωc and effective transmission τ (see
Eq. (6)).

From now to the end of this section, we review the
result of Ref. 7. The one-channel conductor in series
with a resistance has been shown7 to be equivalent to an

4
R, C, Z

resistance, capacitance,
and impedance of the environment
−1
K = (1 + R/Rq )
interaction parameter of the TLL
Rq = h/e2
quantum of resistance
Gq = e2 /h
quantum of conductance
vF
Fermi velocity
a
distance cutoff of the TLL
ωF = vF /a
frequency cutoff of the TLL
ωRC = 1/RC
frequency of the RC circuit
ωc = min{ωF , ωRC } frequency cutoff of our model
τ0
bare transmission
τ
effective transmission
vB = (1 − τ )/τ
backscattering amplitude
2
eVB = 2 ωc vB
energy crossover between WBS and SBS
TABLE II: Synthesis of parameters and constants.

impurity problem in a TLL, the parameter K of which
is given by Eq. (1). Accordingly, one can use the TLL
model in the presence of backscattering with amplitude
vB , the Hamiltonian of which reads as:
H = H0 +

ωF vB iφ(t)−ieKV t/
√ e
+ h.c. ,
4π π

(3)

where H0 is the TLL Hamiltonian (for simplicity, we do
not include spin degrees of freedom). The bosonic field φ
coincides with the charge that is transferred through the
conductor: eφ(t) = Q(t). This bosonized Hamiltonian is
an effective one valid at energies below a typical energy
ωF . In combination with the condition ω < ωRC for the
mapping to hold, we denote the effective frequency cutoff
by ωc = min{ωF , ωRC }.40 In Eq. (3), the parameter K in
exp(−ieKV t/ ) results from the dc conductance of the
ballistic conductor (as can be inferred for instance from
Ref. 16). Moreover, the effective backscattering amplitude vB is not related universally to the bare transmission τ0 in the absence of the environment. Indeed one
can not, in general, express the parameters of a strongly
correlated system in terms of those without interactions,
lacking correspondence between them.41 Nevertheless, we
2
introduce an effective transmission τ = 1/(1 + vB ) keeping in mind that τ does not have to coincide with τ0 .
Of course, when τ0 = 1 one has no backscattering, and
τ = 1 too. Also, we restrict vB to be weak enough –
thus τ close to one – in order to write the backscattering Hamiltonian in its bosonized form on the r.h.s. of
Eq. (3). It is reasonable, though not well established,
that this would correspond to τ0 close to one as well. It
is however possible that the Hamiltonian in Eq. (3) could
be extended beyond that restriction.42
The important effective parameter is indeed the scaling
voltage VB ,10 which appears in the Bethe-Ansatz solution as non-universal.13 It can be roughly related to the
1/(1−K)
backscattering amplitude through eVB ≃ ωc vB
.
Indeed VB characterizes the crossover between the WBS

and SBS regimes. At energies high enough compared to
eVB , one can use the perturbative RG analysis with respect to a weak impurity by Kane and Fisher.43 In the opposite limit (i.e., at energies ≪ eVB ) the wire is cut into
two pieces, with weak tunneling between them. Thus,
even though one starts from a weak impurity, lowering
the energy drives the system from the WBS behavior,
where the conductance is slightly reduced, into the SBS
regime where the conductance is suppressed. This means
that the DCB still takes place even when the effective
transmission τ is close to one,7 but below an effective
charging energy eVB . In particular, at energies much
smaller than eVB , thus in the SBS regime, it has been
shown that one recovers the P (E) theory which is rather
obtained starting from a very weak transmission.1 Even
more, it is possible to obtain non-perturbative results not
only with respect to R, but also to 1 − τ , and to describe
the whole regime of energies below ωc . Such an achievement was made possible by exploiting the Bethe-Ansatz
exact solutions of Fendley et al.13 for the impurity problem in a TLL. Within the Bethe-Ansatz solution, the FCS
can be computed exactly as well. In particular, using
these results for the current and noise, it was possible to
derive a crucial and exact relation between the derivative
of the differential conductance G(V, ω = 0) = dI(V )/dV
and the differential ZF noise S(V, ω = 0) at zero temperature:
Rq |eV | dG(V, ω = 0) = 2R dS(V, ω = 0) .

(4)

Let us remark that in the limit of small R ≪ Rq , Eq. (4)
fits perfectly with the RG equation obtained by Kindermann and Nazarov.6 Within the Bethe-Ansatz solution,
the FCS can be computed exactly as well. Using its
derivation at zero temperature,44 the mapping permits
to extend Eq. (4) to higher cumulants. It has also motivated partly the recent investigations of the FCS at finite
temperature.26,45–48
More recently, in an interesting work, Golubev et al.12
have studied a mesoscopic interacting multi-channel conductor connected to an Ohmic environment, using a
method based on Keldysh action. They have confirmed
the results presented above (of Ref. 7) in the perturbative regime they restrict to (with respect to 1 − τ ), which
corresponds to high energies in the WBS domain.49 Even
more, when treating many channels in a mean-field approach, the agreement holds too, provided the TLL’s parameter in Eq. (1) incorporates the resistance of the conductor. This shows that the mapping can be extended
to many channels, as well as its consequences discussed
above (at least at high enough energy), and motivates
further the computation done in this work.
In the following, we will consider the situation where
the resistance is equal to the quantum of resistance:
R = Rq . In that case, the TLL parameter is K = 1/2
(see Eq. (1)), and Eq. (3) is exactly solvable through a
refermionization procedure,19,36,50–53 thus one can perform a non-perturbative analysis of transport properties.
The refermionization introduces new independent chiral

5
fermions with a frequency dependent transmission amplitude:
t(ω) =

ω
,
ω + ieVB /2

voltage, I(V ) is time-independent. The details of the
calculation are presented in Appendix A. We obtain:54

(5)

I(V ) =

thus a transmission coefficient:
T (ω) = |t(ω)|2 =

4

4 2ω2
τ 2 ω2
= 2 2
,(6)
2V 2
2
+e B
τ ω + (1 − τ )2 ωc

2ω2

2
where eVB = 2 ωc vB is the energy crossover between the
WBS and SBS regimes. The profile of T (ω) is shown
in the lower panel of Fig. 1. Obviously, T (ω) is identical to 1 (perfect effective transmission) when τ = 1
whatever the frequency is. As we will show later on,
T (ω = eV / ) yields the non-linear differential conductance G(V, ω = 0) at zero temperature. In accordance
with the features concerning the crossover from WBS to
SBS, one sees that as soon as τ deviates from one, T (ω)
decreases quickly to zero at low frequency compared to
eVB / . From Eq. (6), notice that T (ω = vB ωc ) coincides
with the effective transmission τ .

III.

RESULTS

In this section, we present the formal results for the dc
current, the non-symmetrized noise, and the differential
conductance.
We first calculate the dc current, defined as the average
of the time derivative of the charge which is transferred
ˆ
˙
ˆ
through the conductor: I(V ) = I(t) = Q(t) , where I
is the current operator. Since we consider a dc applied

S(V, ω) =

e2
4π

∞
e
dωT (ω)
4π −∞
× [f ( ω − eV /2) − f ( ω + eV /2)] ,

(7)

which corresponds to the Landauer formulation of the
current, where the Fermi-Dirac distribution function is
given by f ( ω) = [1 + exp( ω/kB T )]−1 . The density of
states multiplied by the velocity is a constant, equal to
1/2π, because the energy spectrum of the new fermions
is linear too. Eq. (7) describes the behavior of the dc
current over all voltage and temperature ranges, starting from the WBS regime down to the SBS regime. We
have to recall that the model describes a strongly correlated system where interactions cannot be treated by
any mean-field approach, and that the transmission amplitude incorporates their non-trivial effects. Notice that
in Eq. (7) we have made the choice of extending the limits
of integration to plus and minus infinity. Strictly speaking, these limits should be −ωc and ωc , however we have
checked that the correction terms are negligible. In the
following, a similar choice will be made in all integrals
over frequencies.
Next, we calculate the FF non-symmetrized noise defined as the Fourier transform of the current fluctuations:
∞

S(V, ω) =

ˆ
ˆ
dteiωt δ I(0)δ I(t) ,

(8)

−∞

ˆ
ˆ
ˆ
where δ I(t) = I(t) − I . We obtain the following result
(see Appendix B for details):

∞

dω ′
±

−∞

T (ω ′ )T (ω + ω ′ ) + |t(ω ′ ) − t(ω + ω ′ )|2 /4 [1 − f ( ω ′ ± eV /2)]f ( ω ′ + ω ± eV /2)

+ T (ω ′ ) − T (ω ′ )T (ω + ω ′ ) − |t(ω ′ ) − t(ω + ω ′ )|2 /4 [1 − f ( ω ′ ± eV /2)]f ( ω ′ + ω ∓ eV /2)

A crucial and surprising observation is that this expression differs from that obtained within the scattering
approach for a one-channel conductor with an energydependent transmission55–58 (see Appendix C for more
details on that comparison), even when applied to chiral
fermions.59

(9)

form:
S(V, ω) =
+

e2
4π

±
∞

×

−∞

e
2

±

N ( ω ± eV )[I(±V ) + I(2 ω/e ± V )]

[N ( ω ± eV ) − N ( ω)]

dω ′ T (ω ′ )T (ω + ω ′ ) +

|t(ω ′ ) − t(ω + ω ′ )|2
4

×[f ( ω + ω ′ ± eV /2) − f ( ω ′ ∓ eV /2)] ,
−1

It is interesting to cast Eq. (9) under the alternative

.

(10)

where N ( ω) = [exp( ω/kB T )−1] is the Bose-Einstein
distribution function. Notice that in the trivial limit τ =
τ0 = 1 (i.e., perfect transmission), the voltage drop across

6
the mesoscopic conductor is V /2 and the FF noise reduces
to S(V, ω) = ωN ( ω)Gq whatever the temperature is,
where Gq = e2 /h is the quantum of conductance. An
interesting point is that, in the r.h.s. of Eq. (10), both
distribution functions f for fermions and N for bosons
(of the electromagnetic environment, thus electron-hole
excitations) are involved, which explains the fact that the
effective voltage is not the same in their arguments: one
has ω ± eV /2 for the function f , and ω ± eV for the
function N .
From Eq. (9), we deduce immediately the ZF noise:
∞

e2
S(V, 0) =
4π

dω
±

′

−∞

× T 2 (ω ′ )[1 − f ( ω ′ ± eV /2)]f ( ω ′ ± eV /2)
+T (ω ′ ) (1 − T (ω ′ )) [1 − f ( ω ′ ± eV /2)]f ( ω ′ ∓ eV /2) .
(11)
Contrary to what occurs for the FF noise, one recovers, in
the ZF limit, an expression in terms of the transmission
coefficient T similar to that within the scattering theory.
Let us now calculate the FF conductance which is related to the non-symmetrized noise through the exact
relation:27,29
Re[G(V, ω)] =

S(V, −ω) − S(V, ω)
.
2 ω

(12)

Reporting Eq. (9) in Eq. (12), all the terms which contain a product of transmission coefficients vanish, and we
find simply:
∞

2

Re[G(V, ω)] =

e
4hω

±

−∞

dω ′ T (ω ′ )

′

× f ( ω ± eV /2) − f ( ω + ω ′ ± eV /2) , (13)
which obeys unexpectedly the exact relation:60
Re[G(V, ω)] =

e
[I (V + 2 ω/e) − I (V − 2 ω/e)] .
4 ω
(14)

This is a central result of our paper: even though our
computation is non-perturbative, one recovers a similar
relation to that obtained within the perturbative TienGordon theory, with a renormalized charge e/2.32,33 Indeed, such a relation has been recently shown in Ref. 35
to be generally valid for all strongly correlated systems at
arbitrary dimensions, with possible coupling to an arbitrary electromagnetic environment, provided one requires
the tunneling regime. In one-dimensional systems, the
same work has shown the universality of the relation for
one or many weak impurities in the WBS regime aside
from its validity in the SBS regime or for tunneling barriers. Here, surprisingly, for a TLL with an impurity and
K = 1/2, or a conductor coupled to a quantum resistance, Eq. (14) is valid for all voltage, temperature and

frequency ranges (below ωc ) with a renormalized charge
e/2.
It is interesting as well to specify to frequencies much
larger than the applied voltage V , where the FF conductance becomes voltage-independent, thus reaches a linear
regime. In that case, Eq. (14) can be approximated simply, as in Ref. 35, by:
eI(2 ω/e)
.
ω

Re[G(V ≪ ω/e, ω)] ≈

(15)

In the two following sections, starting from the expressions derived here, we first recall some known results
of the stationary regime (Sec. IV) and next, we discuss
in details the news results of the time-dependent regime
(Sec. V).

IV.

DIFFERENTIAL CONDUCTANCE AND ZF
NOISE

In this section, we explicit in more details the dc transport in all the temperature and voltage ranges, starting
from the low temperature regime (kB T ≪ eV ) and ending with the high temperature regime (kB T ≫ eV ).
Low temperature behavior. In the limit of strictly zero
temperature, the integral over frequency in Eq. (7) can be
performed analytically and the dc current reads as:36–38
I(V ) =

Gq VB
2

V
− arctan
VB

V
VB

.

(16)

The dc current is thus given by a product of VB multiplied by a function of V /VB . This result has been used
successfully to describe recent experimental data.61 For
perfect effective transmission, τ = 1 (i.e., VB = 0), we
recover Ohm’s law: I(V ) = V /(2Rq ), whereas at τ < 1,
the dc current is reduced: DCB persists even though one
starts from a good effective transmission. Even more, one
can check, as shown in Ref. 7, that in the SBS regime, at
V ≪ VB , one recovers a power law behavior controlled by
the exponent 1 + 2R/Rq = 3 as in the P (E) theory, and
obtained rather for a weakly transmitting conductor.1
For that, one expands Eq. (16) with respect to V /VB :
I(V ≪ VB ) =

Gq V 3
2 .
6VB

(17)

In the opposite limit of the WBS regime, i.e. at V ≫ VB ,
the reduction to the perfect current, called the backscattering current:
IB (V ) =

Gq V
− I(V ) ,
2

(18)

can be expanded with respect to VB /V :
IB (V ≫ VB ) ≃

Gq VB
2

π VB
−
2
V

.

(19)

7
0.5

0.5
Τ

0.015

0.99
0.95

G V, Ω 0

G V, Ω 0

Τ

0.25
Τ
Τ

0
0

0.9

0.25

0.2

0
0

1

2

3

FIG. 2: Left panel: Differential conductance, in unit of e2 /h,
as a function of eV / ω, for different values of τ , at kB T / ωc =
0.001. Since we have eVB / ωc = 2(1−τ )/τ , the corresponding
values of VB are: eVB / ωc = 0.02 (red solid line), eVB / ωc =
0.1 (green dashed line), eVB / ωc = 0.22 (blue short dashed
line), and eVB / ωc = 0.5 (black dotted line). Right panel:
All the curves of the left graph scale to a single one when
one considers the variation with V /VB . We take kB T /eVB =
0.001.

Notice that the first term on the r.h.s. corresponds to
that obtained within the perturbative computation,38
IB (V ) ∼ V 2K−1 , which becomes voltage independent at
K = 1/2.
The differential conductance can be obtained either by
letting ω → 0 in Eq. (13), or by differentiating the dc
current of Eq. (16). At zero temperature, it reads as:
dI(V )
V2
Gq
1− 2 B 2
=
dV
2
V + VB

,

(20)

and obeys the relation:
Gq
T
2

eV
2

,

(21)

where the behavior of the transmission coefficient T is
shown on the lower panel of Fig. 1. Notice that, at τ = 1,
the differential conductance becomes constant (linear): it
is equal to Gq /2 as the ballistic conductor is in series with
a resistance R = Rq . At τ < 1, G(V, ω = 0) can formally
reach Gq /2 for V → ∞, nevertheless it stays below since
eV is limited by the cutoff ωc .
The left panel of Fig. 2 gives the differential conductance at low temperature, which shows a ZBA that is
more pronounced when τ is reduced. A surprising result
is its large sensitivity with respect to small variations of
τ in the vicinity of 1 (see the red and green curves in
Fig. 2 for example): it is due both to the rapid variations of T (ω ′ ) at frequencies low compared to ωc (when
τ is close to 1, see the lower panel of Fig. 1) which are
precisely those that give the dominant contribution to
the integral over ω ′ in Eq. (13).62 We can see that the
crossover VB varies rapidly in the vicinity of τ close to
one, thus the SBS regime is reached much more rapidly
when τ decreases. It is important to recall that all the
curves scale to the same one when one scales the voltage
by VB , as shown in the right panel of Fig. 2.
At zero-temperature, the ZF noise of Eq. (11) becomes:
S(V, ω = 0) =

Gq eVB
arctan
4

0.2 ; Ω

0

e2 Ωc

V VB

G(V, ω = 0) =

S V

0.01

eV Ωc

G(V, ω = 0) =

0.2

eΩc

0.8

0.1

I V

V
VB

−

V VB
2
V 2 + VB

.(22)

0.005

0.5

0
0.8

0.222

0.9

0

1

Τ

FIG. 3: dc current and ZF noise as a function of the effective transmission τ , for eV / ωc = 0.2, at a low temperature:
kB T / ωc = 0.001. The arrows indicate the associated values
of eVB / ωc . With a frequency cutoff of about 1 THz, the
value 0.01 eωc on the axis corresponds to a current of about
10 nA.

As for the dc current, the ZF noise is given by a product
of VB multiplied by a function of V /VB . One can check
that the ZF noise obeys Eq. (4), with R = Rq , which
confirms that the ZBA is exactly linked to the ZF shotnoise in the presence of the electromagnetic environment.
Figure 3 shows the dc current and the ZF noise as
functions of τ , at low temperatures compared to voltage,
kB T ≪ eV : the current shows an increasing behavior,
whereas the ZF noise is non monotonous. This is due
to the fact that the current is expressed as the integral
of the transmission coefficient, see Eq. (7), and that the
integral expression of the ZF noise of Eq. (11) contains
a term proportional to T (ω ′ )(1 − T (ω ′ )). For VB > V
(i.e., τ < 0.9 using the parameters of Fig. 3), the current
and ZF noise curves converge to the same value. Thus,
the Fano factor – given by the ratio S(V, ω = 0)/|I(V )|
– tends to e, the value obtained in the Poissonian regime
with independent transfer events of charge e trough the
mesoscopic conductor. For VB ≪ V , the transfer events
are no more independent but the noise is still close to
zero because of the (1 − T (ω ′ )) factor. In addition, the
Fano factor – defined this time by the ratio S(V, ω =
0)/|IB (V )|, where IB is the backscattering current given
by Eq. (18) – is equal to e∗ = e/2. In the FQHE at
ν = K = 1/(2n + 1), the Fano factor would be given
by fractional charge e∗ = Ke. In the DCB context, this
renormalization is related to the dc conductance Gq /2
without backscattering.
Intermediate temperature behavior. The left panel of
Fig. 4 shows the differential conductance at zero voltage as a function of temperature for various values of
τ . When the temperature increases, the ZBA is suppressed, a behavior similar to what is obtained within the
P (E) theory.1,63 Again, all these curves coincide when
one considers their variation with respect to kB T /eVB
(see right panel of Fig. 4). Fig. 5 shows that the current and the ZF noise increase both monotonously with
τ , and that the noise is larger that the current due to
thermal fluctuations. In the intermediate temperature

8
0.5

G V 0, Ω 0

Τ
Τ

0.9

Τ

0
0

0.95

Τ

0.25

0.99

0.8

0.1

G V 0, Ω 0

0.5

0.25

0
0

0.2

1

2

3

kBT eVB

k B T Ωc

FIG. 4: Left panel: Differential conductance, in units of e2 /h,
as a function of temperature T , in units of ωc /kB , for different values of τ , at V = 0. Right panel: All the curves of
the left graphic scale to a single one when one considers the
variation with respect to kB T /eVB .

regime kB T ≈ eV , the ZF noise has a totally different
dependence on τ in comparison to its behavior at low
temperature due again to an increasing contribution of
the thermal noise. However, the total noise is not a direct
superposition of shot noise and thermal noise, but results
from a more complicated interplay between them.16,23,27
0.06

I V

0.2

eΩc
S V

0.04

0.2 ; Ω

Eq. (24) is similar to that of the non-linear conductance
on voltages V ≪ VB , as one can see from differentiating Eq. (17). This confirms, as generally expected, that
temperature and voltage play symmetric roles. More precisely, the differential conductance at both finite temperatures and voltages obeys a scaling law: G ∼ T α F (V /T ),
where F (x ≪ 1) ≃ xα and F (x ≫ 1) → constant value.
This leads in particular to the same power law behavior
with respect to max{kB T, eV }. While this is valid in the
SBS regime as we have just shown, with α = 2, it turns
out that this scaling behavior is violated in the opposite
limit of WBS as we discuss in detail now. More precisely,
one has to consider the backscattering conductance, defined as the differential of the backscattering current in
Eq. (18): GB = Gq /2−G. Using Eq. (23) at kB T ≫ eVB
yields:
GB =

πGq eVB
,
16kB T

in accordance with the behavior VB T 2K−2 at arbitrary
K. We see, however, that this dependence on temperature is different from that on voltage obtained by expanding Eq. (20) in the WBS regime (V ≫ VB ) (see also
Eq. (19)):

0

e2 Ωc

GB =

0.02
0.222

0.5

0
0.8

0

0.9

1

Τ

FIG. 5: dc current and ZF noise as a function of the effective
transmission τ , for eV / ωc = 0.2, and kB T / ωc = 0.2. The
arrows indicate the associated values of eVB / ωc .

High temperature behavior. In the equilibrium regime
(i.e., for kB T ≫ eV ), the dc current of Eq. (7) becomes
strictly linear in V , and the linear conductance takes the
value:36–39
G = G(V = 0, ω = 0)
Gq
eVB
=
1−
Ψ′
2
4πkB T

1
eVB
+
2 4πkB T

, (23)

where Ψ(x) = Γ′ (x)/Γ(x), and Γ is the Gamma function.
In particular, in the SBS, at kB T ≪ eVB , one recovers
the power law:
G=

(25)

2
2π 2 Gq kB T 2
.
2V 2
3e B

(24)

This result is in accordance with the generic behavior of the TLL in the SBS regime,38,43 where one has
G ∼ T 2/K−2 , as K = 1/2 here. It is also similar to that
obtained within the P (E) theory in the tunneling regime,
even though the conductor is well-transmitting here.7 Notice that this dependence on temperatures kB T ≫ eVB in

2
Gq VB
.
2
2V

(26)

As we have already commented concerning Eq. (19),
for generic values of K, the lowest order expansion of
the backscattering current is given by IB ∼ VB V 2K−1 ,
leading to a differential conductance proportional to
(2K − 1)VB V 2K−2 , which cancels at K = 1/2. This
can be seen as well by differentiating the constant term
on the r.h.s. of Eq. (19): thus one needs the next term,
yielding to the first non-zero contribution in Eq. (26).

V.

FF CONDUCTANCE AND FF NOISE

In this section, we discuss the dynamical properties
of our system by performing plot analysis of the formal
expressions of the FF conductance Re[G(V, ω)] and the
FF non-symmetrized noise S(V, ω) obtained in Sec. III.
Both quantities depend not only of the voltage V and frequency ω, but also on temperature. More precisely, they
can be cast into a scaling form, being a function of V /VB ,
ω/eVB , and kB T /eVB . All the dependencies can be obtained exactly, provided these energies are below ωc , but
we have to make restrictions on the number of curves presented. Here we will focus first on low temperatures compared to voltage and frequency, kB T ≪ { ω, eV }, thus
both out-of-equilibrium and quantum regime, allowing
for ω to be of the order or greater than eV . Then we
will consider intermediate and high temperatures compared to the voltage.

9
˜
Here Θ is the Heaviside function, V = V /VB and ω =
˜
ω/eVB . Notice that the FF noise is an even function
of V . Because of this parity, we will specify only to
eVB
2 ω ± eV
Gq
positive values of the voltage. We recall that the FF
1−
.
arctan
Re[G(V, ω)] =
2
4 ω ±
eVB
non-symmetrized noise at positive (negative) frequencies
corresponds to emission (absorption) noise.64
(27)
An important fact we observe is that the emitted noise
For τ = 1, it reduces to the perfect conductance – equal
vanishes exactly above eV . In the perturbative regime,
to Gq /2 – whereas for τ < 1, it acquires a frequency deone usually expects that the emitted noise vanishes above
pendence as shown on Fig. 6. Since Re[G(V, ω)] is an
eKV .23 Indeed, it has been shown universally for a weak
even function of ω, it is plotted at positive frequencies
tunneling junction (between strongly correlated systems
only. This result is interesting in the sense that the FF
in arbitrary dimensions and with possible coupling to an
conductance has a non monotonous behavior. It exhibits
environment) that the FF noise associated to tunneling
a minimum (see the left panel of Fig. 6) at a frequency
vanishes above qV , where q is the transferred charge.29
depending on both V and VB which is fixed by the canNevertheless, when higher order processes with respect to
cellation of the derivative ∂ω Re[G(V, ω)]. When τ detunneling are taken into account, one expects the implicreases, this non monotonous behavior disappears, and
cation of a many-body complicated process, and energy
the FF conductance increases regularly with frequency.
conservation at the level of one particle can not be used
All these curves scale to the same one when one conany more. In strongly correlated systems or in a considers their variation with respect to ω/eVB , at fixed
ductor coupled to an environment, we are not aware of
V /VB and kB T /eVB (see right panel of Fig. 6). Notice
any non-perturbative statement about the existence of a
that because of the finite value of the voltage, the FF convalue of the frequency ω over which the emitted noise
ductance acquires a non-zero value – even though small
vanishes. However, we obtain an exact result here: the
– at zero frequency. This has been checked by zooming
emitted noise vanishes strictly at frequencies ω > eV ,
around ω = 0 (not shown).
whatever the values of V and VB are. This is due to the
Bose-Einstein distribution functions of Eq. (10) which
0.5
0.5
reduce, in the zero temperature limit, to the Heaviside
Τ 0.99
functions of Eq. (29). Their presence is related to the
Τ 0.95
photon exchange between the conductor and the envi0.25
0.25
Τ 0.9
ronment corresponding to electron-hole type excitations.
Τ 0.8
One can think of two possible scenarios: (i) The new
0
0
fermions are independent. For non-interacting electrons,
0
1
2
3
0
0.1
0.2
0.3
Ω eVB
scattering theory predicts that the emitted noise vanishes
Ω Ωc
above eV . Even though the FF noise in our case does not
obey the same relation with respect to the transmission
FIG. 6: Left panel: FF conductance, in units of e2 /h, as a
function of ω/ωc , for different values of τ , at eV / ωc = 0.2
amplitude t(ω), one can imagine that a similar fact holds.
and kB T / ωc = 0.001. Right panel: All the curves of the left
(ii) The second scheme is that the system cannot emit at
graphic scale to a single one when one considers their variation
higher frequencies compared to what the generator can
with ω/eVB . We take V /VB = 0.2 and kB T /eVB = 0.001.
afford, and not compared to the voltage across the conductor V /2 if it was perfect. If this is plausible, it opens
We turn now our attention to the FF noise. At T = 0,
the question as to whether such a result is universal for
the integral in Eq. (9) can be performed analytically and
the TLL with an arbitrary value of the parameter K,
the FF non-symmetrized noise reads:
thus for other values of the resistance R. Equivalently,
for ω < −eV , we have S(V, −ω) = 0, and Eq. (12)
V
ω
S(V, ω) = Gq eVB F
,
(28)
,
reduces to:29
VB eVB
S(V, ω < −eV / ) = −2 ωRe[G(V, ω)] ,
(30)
where the dimensionless function F is given by:
Re G V 0.2, Ω

Re G V 0.2, Ω

Low temperature behavior. At T = 0, using Eqs. (14)
and (16), the FF conductance reads as:

1
˜ ˜
F (V , ω ) = −˜ Θ(−˜ ) +
ω
ω
8

±

˜
± Θ(−˜ ± V )arctan(V )
ω ˜

+ 3Θ(−˜ ) − Θ(−˜ ∓ V ) arctan 2˜ ± V
ω
ω ˜
ω ˜
+

1
8˜
ω

±

− Θ(−˜ ) + Θ(−˜ ± V )
ω
ω ˜

˜
ω ˜
× ln 1 + V 2 − ln 1 + (2˜ ∓ V )2

. (29)

which means that at negative frequencies below −eV /
and zero temperature, the absorption FF noise is governed by the FF conductance.
It is also interesting to express the FF nonsymmetrized excess noise which is often measured experimentally, both because zero-point fluctuations are
not easy to measure (apart from a pioneering work in
Ref. 65), and in order to subtract undesirable sources of
noise. It is given by:
∆S(V, ω) = S(V, ω) − S(V = 0, ω) .

(31)

˜ ˜
∆S(V, ω) = Gq eVB [F (V , ω) − F (0, ω)] ,
˜

(32)

where the function F is given by Eq. (29).
0.0006
Τ

0.9

Τ

0
0.3

0.95

Τ

0.004

0.8

0.1

0

0.0002

3

2

1

0

Ω eVB

FIG. 7: Left panel: Non-symmetrized excess noise at negative
frequency, in units of e2 ωc , as a function of ω/ωc , for different
values of τ , at eV / ωc = 0.2 and kB T / ωc = 0.001. Right
panel: Non-symmetrized excess noise at negative frequency,
in units of e2 VB / , as a function of ω/eVB , at V /VB = 0.2
and kB T /eVB = 0.001.

S V 0.2, Ω

0.8

Τ

0.9

Τ

0.95
0.99

0.001

0.9

Τ

0.8

0.001

S V 0.2, Ω

S V 0.2, Ω

0.95

Τ

0.1

0.2

Ω Ωc

0.3

2

3

kBT eVB

0.1 0.2 0.3

0.005

0
0.3 0.2 0.1

0

0.1 0.2 0.3

Ω eVB

0.00004
V

V

2 VB

0.00002

VB

0.1

0.2

0

0
0

0

Ω Ωc

0.99

1

0.01
Τ

Τ

Τ

0
0

FIG. 9: Left panel: FF conductance, in units of e2 /h, as a
function of kB T / ωc , for different values of τ , at V = 0 and
ω/ωc = 0.1. Right panel: All the curves of the left graphic
scale to a single one when one considers their variations with
respect to kB T /eVB . We take V = 0 and ω/eVB = 0.1.

0
0.3 0.2 0.1
Τ

0.2

0.25

k B T Ωc

Ω Ωc

0.002

0.8

0.5

0.0004

0
0.2

0.9

Τ

0.1

0.95

Τ

0
0

0.99

Τ

0.25

0.99

S V 0.2,Ω

S V 0.2, Ω

Τ

Τ

S V 0.2, Ω

0.008

0.5

Re G V 0, Ω 0.1

At zero temperature, it reads:

Re G V 0, Ω 0.1

10

0

0.3

Ω eVB

FIG. 8: Left panel: Non-symmetrized excess noise at positive
frequency, in units of e2 ωc , as a function of ω/ωc , for different
values of τ , at eV / ωc = 0.2 and kB T / ωc = 0.001. Right
panel: Non-symmetrized excess noise at positive frequency,
in units of e2 VB / , as a function of ω/eVB , at V /VB = 0.2
and kB T /eVB = 0.001.

In the left panels of Figs. 7 and 8 is plotted the excess
noise at a very low temperature66 at positive frequency
and negative frequency respectively. The red curve corresponds to a value of VB ≪ V , i.e. to the WBS regime.
In that case one sees, according to Eq. (29), that the dependence close to ω = ±e∗ V = ±eV /2 is due mainly to
the arctan function: this explains the step whose width
is controlled by VB . The step is smoothed out at higher
VB (i.e., lower τ ) in the green and blue dashed curves of
Fig. 7. However, the asymmetry between absorbed and
emitted excess noise is always present whatever the value
of the effective transmission is. This asymmetry has been
shown to be related, in a universal way, to non-linearity,
using Eq. (12).29 Here, the non-linearity is induced by the
electromagnetic environment. All these curves at different τ scale to a unique one as it is shown in the right
panels of Figs. 7 and 8 where ∆S is plotted as a function
of ω/eVB when the ratios V /VB and kB T /eVB are fixed.
We have to draw attention to the fact that this requires
us to change simultaneously all energy scales while VB
changes.

FIG. 10: Left panel: Non-symmetrized excess noise, in units
of e2 ωc , as a function of ω/ωc , for different values of τ . We
use the values eV / ωc = 0.2, and kB T / ωc = 0.2. Right
panel: Non-symmetrized excess noise, in units of e2 VB / , as
a function of ω/eVB , at V /VB = 0.2 and kB T /eVB = 0.2.

Intermediate temperature behavior. Next, we look at
the effect of the temperature on the FF conductance and
the FF noise. The left panel of Fig. 9 shows that the
FF conductance increases monotonously with temperature. Again, all the curves scale to a single one when one
considers the variation with respect to kB T /eVB at fixed
ω/eVB (see the right panel of Fig. 9). The FF excess
noise at intermediate temperatures is plotted on the left
panel of Fig. 10. However, the asymmetry of the FF excess noise is still visible in that regime, and we observe its
enhancement when the effective transmission decreases.
Again, all the curves scale to a single one when one considers the variation with ω/eVB at fixed values of V /VB
and kB T /eVB (see the right panel of Fig. 10).
High temperature behavior. At high temperatures compared to voltage, kB T ≫ eV , we have checked, using
Eqs. (10) and (14), that the FF non-symmetrized noise
is related to the FF conductance and becomes voltage
independent in that limit according to the fluctuationdissipation theorem:

S(V ≪ kB T /e, ω) = 2 ωN ( ω)Re[G(V ≪ kB T /e, ω)] .
(33)

11
VI.

CONCLUSION

In this paper, we have studied both stationary
and time-dependent transport properties of a welltransmitting one-channel conductor embedded in an
Ohmic environment with a quantum of resistance Rq =
h/e2 . We have taken advantage of the mapping of this
problem to a TLL with a particular value of the interaction parameter K = 1/2. This has allowed us to obtain results which are non-perturbative with respect to
the resistance of the environment R (here being equal
to Rq , see Eq. (1)), and which describes all regimes of
voltages, frequencies and temperatures below the cutoff
ωc . The results are controlled by a unique scaling and
non-universal parameter VB which depends on both the
properties of the conductor and the environment. While
the dc properties can as well be derived exactly for other
values of R (analytically at zero temperature, otherwise
one would need numerical methods), using the BetheAnsatz solution for the impurity problem in the TLL,
the particular choice of R = Rq has allowed us to obtain
the first non-perturbative results for both the FF noise
and the FF conductance. For the latter, we have used its
universal relation to the asymmetry of the emission and
absorption noise, via an out-of-equilibrium fluctuationdissipation theorem type formula (see Eq. (12)) derived
in Ref. 31.
We have first recalled the results for the dc transport
obtained in a TLL with parameter K = 1/2 in order
to apply them to our present problem and to discuss
them in more details. The key procedure is based on
refermionization, i.e., casting the non-trivial strong correlation effects into new chiral fermions with an energydependent transmission coefficient T (ω). This cannot be
obtained by any “adiabatic” evolution from the effective
transmission τ (which explains our choice for a different
notation). The dc current through the mesoscopic conductor takes a form similar to that within the scattering
theory, applied to the new fermions: it is expressed as the
integral over energy of T (ω) times the difference of the
Fermi-Dirac distribution functions of the left and right
reservoirs. Even for a good effective transmission τ , one
recovers the DCB at energies below the voltage VB . The
reduction to the dc current has a power law with different exponents when voltages are high or low compared
to VB , corresponding to the WBS and SBS regimes. In
particular, in the latter limit, one recovers the behavior predicted within the P (E) theory for V ≪ VB for a
weakly transmitting conductor, even though we consider
a well transmitting one here. In these two opposite limits, the ZF noise becomes Poissonian, with a Fano factor
respectively given by e/2 or e. The former renormalization is related to the conductance of the conductor in
the perfect limit, being in series with the environmental
quantum resistance.
The ZF noise obeys as well the same expression in
terms of T (ω) as within the scattering approach. Nevertheless, this does not hold for the FF noise even though

it can be expressed in terms of the transmission amplitude t(ω) : this is a surprising and crucial result of our
paper. Another interesting fact is that the emitted FF
noise vanishes strictly above eV at zero temperature, a
fact common to non-interacting electrons, which is not
obvious to expect within the underlying strongly correlated system. We have shown that the FF conductance
does not obey the scattering approach formulation for the
new fermions. Rather, it obeys a simple relation that involves dc currents. This kind of relation was previously
shown for tunneling barriers within the Tien-Gordon theory, or in the FQHE at simple fillings.23 Recently it has
been extended to arbitrary filling factors, and even more
has been shown to be universal either for tunneling barriers in arbitrary dimensions, as well as weak barriers in
one dimension.35 It is quite remarkable that such a relation extends to arbitrary regimes within the TLL model
at K = 1/2. It turns out that the FF conductance has
a non-monotonous behavior with respect to frequency,
having a minimum at a frequency that depends both on
the applied voltage and the scaling voltage VB , and the
value of which is reduced when the effective transmission
τ decreases.
We have shown that both the FF conductance and the
FF noise have a universal behavior at different τ when
voltages, temperatures and frequencies are all divided by
the same scaling voltage VB . This extends the result obtained for the differential conductance, valid for arbitrary
values of the parameter K – thus of the resistance R in
our problem – where we expect to get the same scaling
behavior for time-dependent transport as well.
The coherent conductor connected to an Ohmic environment offers a unique framework to realize a TLL
with a tunable parameter K, and a unique possibility
to realize K = 1/2. The present experiments on that direction are very promising.11 Beyond this issue, our work
provides benchmark results for time-dependent transport
in various fundamental problems: the DCB phenomena,
other strongly correlated systems where the special value
K = 1/2 has been studied fully,48 and more generally
those where one solves the interacting problem in terms
of new independent particles, such as within the BetheAnsatz methods for the impurity problem in TLL.13 We
have highlighted novel counter-intuitive facts, in particular, we have shown that one can not systematically apply
the scattering approach to those new independent particles.

Acknowledgments

I.S. is grateful to A. Anthore, S. J´zouin, F.D. Parmene
tier, F. Pierre for their insightful remarks and to S. Carr,
B. Dou¸ot, D. Est`ve, C. Glattli, P. Joyez, H. Saleur,
c
e
P. Simon, J.R. Souquet, E. Sukhorukov and A. Zaikin for
fruitful discussions; she acknowledges financial support of
ANR under contract ANR-09-BLAN-0199-01. A.C. and
R.Z. thank T. Martin for preliminary discussions about

12
˜
˜
that ψ † (x, t)ψ(x, t) = 0. The Klein factors η and η al˜
low us to have the proper commutation relation for the
new fermionic operators.

the refermionization procedure.
Appendix A: Calculation of the current

R

The mapping of Ref. 7 applies to the effective action
once degrees of freedom apart from those at x = 0 are
integrated out. It is more convenient, for the present
computation, to consider the extended TLL Hamiltonian
over space coordinate x, with an impurity located at x =
0 (see Fig. 11):
H =

vF
4π
+

˜
dx[(∂x φ(x, t))2 + (∂x φ(x, t))2 ]
ωF eVB iφ(0,t)−ieV t/2
e
+ h.c. ,
32π 3

(A1)

˜
where the bosonic fields φ and φ are related to the initial
bosonic fields describing right (R) and left (L) movers
through:
1
φ(x, t) = √ (φR (x, t) + φL (x, t)) ,
(A2)
2
1
˜
(A3)
φ(x, t) = √ (φR (x, t) − φL (x, t)) ,
2
√
where ψL,R = ηL,R eiφL,R (x,t) / 2πa are the initial
fermionic operators. a is the distance cutoff of the TLL
theory. The Klein factors ηL,R allow to have the proper
commutation relation for the fermionic operators. Notice
that in Eq. (3), the bosonic field refers to φ(t) ≡ φ(0, t).
The average current through the conductor is given
by:42
I = evF ρR − ρL ,
ˆ
ˆ

(A4)

where vF is the Fermi velocity. The operators ρR and ρL
ˆ
ˆ
refer to the densities of right and left moving electrons in
TLL reservoirs (see Fig. 11). They are related to the new
fermionic operators introduced in the refermionization
procedure through the relations:
˜
˜
ψ † (x, t)ψ(x, t) + ψ † (x, t)ψ(x, t)
ρR (x, t) =
ˆ
,
(A5)
2
˜
˜
ψ † (−x, t)ψ(−x, t) − ψ † (−x, t)ψ(−x, t)
,
ρL (x, t) =
ˆ
2
(A6)
√
˜
where ψ(x, t) = ηeiφ(x,t) / 2πa and ψ(x, t) =
√
˜
iφ(x,t)
ηe
˜
/ 2πa are the new fermionic fields associated to
˜
the bosonic fields φ and φ: ψ(x, t) is affected by backscattering and by the applied voltage (see Eq. (A1)), whereas
˜
ψ(x, t) is neither affected by backscattering, nor by the
applied voltage. In Eq. (A4), the average value is defined
as the average over the scattering state19 minus the average at equilibrium (i.e., V = 0), since we have to take the
normal order52 in the products of operators that appear
in Eqs. (A5) and (A6). Thus, we immediately conclude

R

VB

L

L

x
0

FIG. 11: Schematic representation of the left and right prop2
agating channels in the reservoirs. eVB = 2 ωF vB is the
energy scale that characterizes backscattering of electrons at
the conductor position x = 0.

From Eqs. (A5) and (A6), we understand that ρR and
ˆ
ρL depend on the position x, however, the average of the
ˆ
difference ρR − ρL , which appears in the current, does
ˆ
ˆ
not depend on the position, since the average of the total
current is a conserved quantity (it is the reason why we
do not keep the x dependency in the current).
Using this Hamiltonian, one can write the equations
of motion for the fields. The solution for the fermionic
˜
field ψ, associated with the free bosonic field, is that of
free propagating electrons, whereas the solution for ψ is
given by19 (from now, we take x > 0):
1
ψ(−x, t) = √
2πaωF

∞

aω e

−i(ω+ eV ) vx −iωt
2
F

dω ,

−∞

(A7)
1
ψ(x, t) = √
2πaωF

∞

bω e

i(ω+ eV
2

) vx
F

−iωt

dω ,

−∞

(A8)
where the operator bω = t(ω)aω + r(ω)a† is a combina−ω
tion of the annihilation and creation operators, a† and
ω
aω , which obey the commutation relation {aω , a† ′ } =
ω
δω,ω′ . The transmission and reflection amplitudes read
as:
2 ω
,
2 ω + ieVB
ieVB
r(ω) =
.
2 ω + ieVB
t(ω) =

(A9)
(A10)

With the help of the solutions given by Eqs. (A7) and
(A8), one can calculate the average over the product of
the two fermionic fields:
1
2πvF
× [f ( ω − eV /2) − f ( ω)] ,

∞

ψ † (−x, t)ψ(−x, t) =

dω
−∞

(A11)

and,
∞
1
dω
2πvF −∞
× (2T (ω) − 1) [f ( ω − eV /2) − f ( ω)] , (A12)

ψ † (x, t)ψ(x, t) =

13
where T = t∗ t. We have used the following average values: aω aω′ = 0, and a† aω′ = f (ω − eV /2)δω,ω′ , with
ω
f is the Fermi-Dirac distribution function. Reporting
Eqs. (A11) and (A12) into Eqs. (A4) and (A5), we finally obtain Eq. (7).

We obtain S2 = S3 with:
1
2
2πvF

S2 =

∞

−r(ω ′ )r∗ (ω ′ )r(ω + ω ′ )r∗ (ω + ω ′ ) .

dteiωt ψ † (−x, 0)ψ(−x, 0)ψ † (−x, t)ψ(−x, t) ,

ψ † (x, 0)ψ(x, 0)ψ † (x, t)ψ(x, t) =

dteiωt ψ † (x, 0)ψ(x, 0)ψ † (−x, t)ψ(−x, t) ,

×

−∞
∞

dteiωt ψ † (−x, 0)ψ(−x, 0)ψ † (x, t)ψ(x, t) ,
−∞
∞

×e

dω1

(B1)
The determination of S1 is rather simple since it
involves contributions which are not affected by the
backscattering:

×

dω1

×e

dω2

dω3
F

1
4
4π 2 a2 ωF

dω4 a† 1 aω2 a† 3 aω4
ω
ω

i(−ω1 +ω2 −ω3 +ω4 ) vx +i(ω3 −ω4 )t

.

dω3

dω4 b† 1 bω2 b† 3 bω4
ω
ω

i(−ω1 +ω2 −ω3 +ω4 ) vx +i(ω3 −ω4 )t
F

−∞

ψ † (−x, 0)ψ(−x, 0)ψ † (−x, t)ψ(−x, t) =

dω2

1
4
4π 2 a2 ωF

.

(B7)

Applying Wick’s theorem, we obtain:

dteiωt ψ † (x, 0)ψ(x, 0)ψ † (x, t)ψ(x, t) .

S4 =

(B6)

Next, we calculate S4 which involves bω and b† operaω
tors since:

−∞
∞

S3 =

dω ′ f ( ω + ω ′ − eV /2)

× t(ω ′ )t∗ (ω ′ )t(ω + ω ′ )t∗ (ω + ω ′ )

To calculate the FF non-symmetrized noise, we need
to evaluate the following correlators (x > 0):

S2 =

−∞

× [1 − f ( ω ′ − eV /2)]

Appendix B: Noise calculation

S1 =

∞

S4 =

1
2
2πvF

We use Wick’s theorem to calculate the correlator
. After successive integrations over frequencies and times, we obtain :

−∞

dω ′ f ( ω + ω ′ − eV /2)

× [1 − f ( ω ′ − eV /2)]

× |t(ω ′ )|2 |t(ω + ω ′ )|2 + |r(ω ′ )|2 |r(ω + ω ′ )|2

−2t(ω ′ )t∗ (ω + ω ′ )r∗ (ω ′ )r(ω + ω ′ )
+
±

(B2)

∞

f ( ω + ω ′ ± eV /2) (1 − f ( ω ′ ∓ eV /2))

× |t(ω ′ )|2 |r(ω + ω ′ )|2
+t(ω ′ )t∗ (ω + ω ′ )r∗ (ω ′ )r(ω + ω ′ )

.

(B8)

a† 1 aω2 a† 3 aω4
ω
ω

S1 =

1
2
2πvF

∞
−∞

dω ′ f ( ω + ω ′ − eV /2)

× [1 − f ( ω ′ − eV /2)] .

(B3)

The calculation of S2 and S3 mix aω and bω operators
since:
ψ † (x, 0)ψ(x, 0)ψ † (−x, t)ψ(−x, t) =
×
×e

dω1

dω2

dω3

1
4
4π 2 a2 ωF

dω4 b† 1 bω2 a† 3 aω4
ω
ω

i(−ω1 +ω2 −ω3 +ω4 ) vx +i(ω3 −ω4 )t
F

,

(B4)

and
ψ † (−x, 0)ψ(−x, 0)ψ † (x, t)ψ(x, t) =
×
×e

dω1

dω2

dω3
F

1
4
4π 2 a2 ωF

dω4 a† 1 aω2 b† 3 bω4
ω
ω

i(−ω1 +ω2 −ω3 +ω4 ) vx +i(ω3 −ω4 )t

Finally, the FF non-symmetrized noise of Eq. (9) is
2
obtained from S(V, ω) = e2 vF [S1 + S2 + S3 + S4 ]/4,
where we replace, everywhere it appears, r(ω) by 1−t(ω).

.

(B5)

Appendix C: Comparison to the results obtained
within the scattering theory

In this Appendix, we compare the expressions of current, FF conductance and noise that we have established
to those obtained in the framework of the scattering theory.
It has been already noted that the expressions of the
dc current given by Eq. (7) and the ZF noise are formally identical to those derived from the scattering approach (i.e., they are expressed as integrals of the transmission coefficient times Fermi Dirac distribution functions), even though the non-trivial many-body effects are
encoded into this transmission. We show here that the
FF conductance and the FF noise, even though expressed
in terms of the transmission and reflection amplitudes,

14
do not obey the relation derived within the scattering
approach.
To make a comparison with the FF noise given in the

S(V, ω) + S(V, −ω) =

e2
4π

literature,55–58 we need to symmetrize the noise given by
Eq. (9):

∞

dω ′
−∞

T (ω ′ )T (ω + ω ′ ) + |t(ω ′ ) − t(ω + ω ′ )|2 /4 F++ (ω, ω ′ ) + F−− (ω, ω ′ )

+ T (ω ′ ) − T (ω ′ )T (ω + ω ′ ) − |t(ω ′ ) − t(ω + ω ′ )|2 /4 F+− (ω, ω ′ ) + F−+ (ω, ω ′ )

,

(C1)

where Fss′ (ω, ω ′ ) = ± f ( ω ′ ± ω ± seV /2)[1 − f ( ω ′ ± s′ eV /2)]. Using our notations, the symmetrized noise of
a coherent one-channel coherent conductor with an energy-dependent transmission T can be obtained within the
scattering approach:57
Sscattering (V, ω) + Sscattering (V, −ω) =

e2
4π

∞

dω ′
−∞

T (ω ′ )T (ω + ω ′ ) + |t(ω ′ ) − t(ω + ω ′ )|2 F++ (ω, ω ′ )

+T (ω ′ )T (ω + ω ′ )F−− (ω, ω ′ ) + T (ω ′ )(1 − T (ω + ω ′ ))F+− (ω, ω ′ ) + T (ω + ω ′ )(1 − T (ω ′ ))F−+ (ω, ω ′ )

The main difference between Eqs. (C1) and (C2) resides into the factors in front of F++ and F−− which are
identical in Eq. (C1) but not in Eq. (C2). The same

1

2

3

4

5

6

7
8
9
10

11

12

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44

45

46

47
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55
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63

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be performed analytically and the derivative of the differential conductance according to τ can be obtained
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2
2
τ )e2 V 2 2 ωc /(τ 2 e2 V 2 + 4(1 − τ )2 2 ωc )2 . Since eV ≪ ωc ,
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