AC-conductance of a quantum wire with electron-electron interaction

arXiv:cond-mat/9710053v1 [cond-mat.mes-hall] 6 Oct 1997

G. Cuniberti1 , M. Sassetti2 and B. Kramer
I. Institut f¨r Theoretische Physik, Universit¨t Hamburg
u
a
Jungiusstraße 9, D-20355 Hamburg, Germany
(Received 30 July 1997, to appear in Phys. Rev. B)
The complex ac-response of a quasi-one dimensional electron system in the one-band approximation with an interaction potential of finite range is investigated. It is shown that linear response is
exact for this model. The influence of the screening of the electric field is discussed. The complex
absorptive conductance is analyzed in terms of resistive, capacitive and inductive behaviors.
PACS numbers: 72.10.Bg, 72.20.Ht, 72.30.+q

studied10,11,12,13 . It has been found that the absorption of microwaves leads to a characteristic broadening of the conductance peaks of semiconductor quantum dots in the Coulomb-blockade region14,15 . Infrared
absorption16,17,18 of quantum dots and wires provided
mainly information on the parts of the excitation spectrum that are only weakly influenced by the interaction
due to Kohn’s theorem. However, Raman scattering from
quantum wires and dots showed signatures of the dispersion of the collective excitations19,20,21 . The absorption
of microwaves in an ensemble of metallic grains has been
investigated experimentally and theoretically in many
papers22,23 .
Absorption and scattering of electromagnetic radiation are only one possibility to measure ac-transport
properties without applying voltage and current probes
that may disturb the system’s properties severely.
More recently, other highly sophisticated, non-invasive
techniques for determining ac-conductances have been
pioneered24 . Coupling a system of about 105 mesoscopic rings to a highly sensitive superconducting microresonator the perturbation of the resonance frequencies
and quality factors has been used to determine the real
and imaginary parts of an ac-conductance. Here, the fundamental question arises, what the differences between
the “conductances” as determined by different methods
are25 .
Theoretical approaches have been developed, ranging from semi-classical rate equation approximations to
fully quantum mechanical attempts. The linear theory of ac-quantum transport has been restricted to noninteracting systems26,27,28,29,30 . How to define quantum
capacitances28 and inductances27,29 was addressed. The
ac-transport through mesoscopic structures in the presence of Coulomb interaction was considered by using a
self-consistent mean field method31,32 . This approach
strongly relies on the presence of “reservoirs”, “contacts”
and “electro-chemical potentials” which are not necessary ingredients of high frequency experiments such as
the absorption of microwaves.
Photo-induced transport through a tunnel barrier33
and tunneling through semiconductor double-barrier

I. INTRODUCTION

Experimental and theoretical investigations of the actransport in nano-structures are of profound scientific
interest since they provide insight into the behavior of
(open) quantum systems in non-equilibrium that are externally controllable within wide ranges of parameters.
In addition, possible applications of nano-structures in
future electronic devices, which will have to operate at
very high frequencies, require detailed knowledge of their
frequency and time dependent transport behavior.
Electron transport in nano-structures is very strongly
influenced by charging effects. Most striking is the
Coulomb-blockade1 of the dc-current through tiny tunnel
junctions when the bias voltage and the temperature are
smaller than the “charging energy”, EC = e2 /2C (e elementary charge, C “capacitance of the tunnel junction”).
The use of a capacitance has been justified by observing
that its values determined from EC are consistent with
those obtained from the geometry of the junction2,3 ; C
was found to be of the order of 10−15 F and much smaller
for metallic junctions4 . Interactions also dominate transport through islands between two tunnel contacts in series in a semiconductor quantum wire. The linear conductance shows pronounced peaks5 if the external chemical potential coincides with the difference between the
ground state energies of N + 1 and N electrons in the
island. In the charging model, these energies are again
given in terms of a “capacitance C ”, E(N ) = N 2 e2 /2C,
C ≈ 10−15 F. One can ask, how small a capacitance can
be without being influenced by quantum effects. The limitations of the charging model become obvious in the nonlinear transport properties: fine-structure in the currentvoltage characteristic is related to the quantum properties of the interacting electrons6,7,8,9.
In recent years, frequency dependent electrical response of systems with reduced dimensionality became
the subject of activities. These techniques are of particular interest since no current and voltage probes have
to be attached to the sample. The current response
to microwave and far-infrared radiation on the transport through semiconductor microstructures has been

1

structures34 has been considered. Charging effects in
small semiconductor quantum dots in the presence of
time-varying fields were treated30,35 . The influence of
high frequency electric fields on the linear and non-linear
transport through a quantum dot with infinitely strong
Coulomb repulsion36 was studied. Photon-assisted tunneling through a double quantum dot has been investigated by using the Keldish technique37 . The photoinduced transport through a single tunnel barrier in a
1D interacting electron system has been investigated38 .
In most of the latter works, quantum effects of the interaction have been treated only approximatively.
In view of the importance of the interaction for the
ac-properties of nano-structures, Luttinger systems are
of great interest. Here, the interaction can be taken into
account exactly. The conductance of a tunnel barrier in
a Luttinger liquid with zero-range interaction has been
shown39,40 to scale with the frequency as ω 2/g−2 (g interaction parameter, see below). At ω = 0 repulsive
electron-electron interaction suppresses tunneling completely, even for a vanishingly small potential barrier.
However, it has been also shown that for ω → 0 there
is a displacement contribution to the current which can
dominate the transport for very strong repulsive interaction and very high barrier41.
The driving voltage has been assumed ad hoc in most
of these works. Since the driving electric field is determined by the interaction between the electrons42,43 , the
current can be expected to depend on how charges induced by an external electric field are rearranged by the
interaction. Even in the limit of dc-transport through
a Luttinger system this has been argued to be so44,45 .
The dependence of the ac-properties of a tunnel barrier
in Luttinger system on the shape of the electric driving field has been investigated42 . It has been found that
the current depends only on the integral over the driving
electric field – the external voltage – only in the dc-limit,
even in the non-linear regime. One of the side results
was to confirm that linear response is exact for the ideal
Luttinger system42,46,47 . The ac-properties of the Luttinger liquid with spatially varying interaction strength
were also studied46,47,48 .
In this paper, we concentrate on the ac-transport properties of electrons described by the Luttinger model with
an interaction of finite range. The model is exactly solvable. Since its current response can be determined without approximations, answers to fundamental questions
can be found, such as (i) to identify the specific signatures of the electron-electron interaction in ac-transport,
(ii) to understand the influence of the properties of the
driving electric field, how to define conductances, and (iv)
to understand ac-transport in terms of resistive, capacitive and inductive behaviors. The latter yield quantum
analogs of impedances that are common in classical electrodynamics. Such parameters are also often used for
describing the transport in nanostructures. Therefore, a
quantum approach towards their definition is highly desirable. However, in the quantum regime, they depend

not only on the interaction, but also on the frequency
and the shape of the applied electric field. Together with
their microscopic definitions, it is thus important to determine the range of parameters for their validity.
The interaction potential is obtained by using a 3D
screened Coulomb potential (screening length α−1 ) and
projecting to a quasi-1D quantum wire of a finite width d.
When α−1 ≪ d we recover the Luttinger liquid with zerorange interaction. On the other hand, when α−1 ≫ d we
obtain the 1D-analog of the Coulomb interaction. The
excitation spectrum of this model shows an inflection
point at a frequency ωp that increases monotonically with
the interaction strength, and with the inverse of a characteristic length associated with the interaction49 .
By using linear response theory we obtain the complex, frequency-dependent non-local conductivity given
by the current-current correlation function. It contains
the dispersion relation of the elementary excitations, and
turns out to be independent of the temperature within
the model. There are no non-linear contributions to the
current in this model.
The conductivity describes the current response of the
system with respect to an electric field E(x′ , ω). In an
experiment, either the current as a function of an external voltage (in the dc-limit), or the absorption of electromagnetic energy at frequency ω from an external field is
determined. In both cases, not too much is known about
the internal field. Therefore, it is reasonable to search for
quantities that do not depend on the spatial form of the
field. One possibility is 50 to define the conductance Γ1
by using the absorbed power. One finds that, in the dclimit, the result is indeed independent of the shape of the
field. However, in ac-transport, the shape of the square
of the Fourier transform of the electric field appears as
a multiplicative factor. The remaining factor is the density of collective excitations of the interacting electrons.
It has a resonance at ωp . For non-interacting electrons
−1
dq/dω = vF and Γ1 (ω) reveals merely the structure of
the electric field.
That the ac-conductance depends on the shape of the
driving electric field leads to the question what the nature
of this field is in an interacting system. We show that for
the conductivity of an ideal Luttinger liquid, it is the
external electric field which has to be used since linear
response is exact. Furthermore, we will demonstrate that
this is also true for the absorptive conductance.
Having determined the absorptive part of the conductance, Γ1 (ω), the reactive part Γ2 (ω) may be obtained by
Kramers-Kronig transformation. The complex conductance Γ(ω) = Γ1 (ω) + iΓ2 (ω) relates the average current
with the voltage. The current as a function of time consists of two parts. One, ∝ Γ1 , is in phase with the driving
field. The second, ∝ Γ2 , is phase shifted by ±π/2.
When ω is small, we can expand
e2
+ γ1 ω 2 + O(ω 4 ).
(1)
h
The first term corresponds to the quantized contact
Γ1 (ω) = g

2

−1
conductance51 RK = e2 /h. It is here renormalized by
the interaction parameter39 g. The term ∝ ω 2 indicates
whether the current is capacitive (γ1 > 0) or inductive
(γ1 < 0). For Γ2 we find

Γ2 (ω) = ωγ2 + O(ω 3 ) .

the Hamiltonian for interacting Fermions with, say, periodic boundary conditions57
H = ¯ vF
h
k,s=±

(2)

Vk1 ,s1 ···k4 ,s4 c† 1 s1 c† 2 s2 ck3 s3 ck4 s4 .
k
k

+

This quantity also indicates if the system behaves capacitively and inductively, γ2 < 0 and γ2 > 0, respectively.
However, in the latter case, γ1 can still be positive indicating capacitive behavior of the real part of the conductance.
For frequencies close to the resonance, the KramersKronig transformation gives
∗
Γ2 (ω) ≈ γm (ω − ωm ),

(sk − kF )(c† cks − c† cks 0 ) +
ks
ks
(4)

k1 ,s1 ···k4 ,s4

Here, c† , cks are the creation and annihilation operators
ks
for Fermions in the states |ks of wave number k = 2πn/L
(n = 0, ±1, ±2, · · ·) in the branches s = ±, kF the Fermi
wave number, V the Fourier-transform of the electronelectron interaction, and · · · 0 denotes an average in the
ground state.
Formally, the Fermion Hamiltonian can be transformed
into a Bosonized form. For spinless particles with an interaction that depends only on the distance between the
particles, V (|x − x′ |), and taking into account only forward scattering, one obtains a bilinear form in the Boson operators which can be diagonalized by a Bogolubov
transformation56. The result is

(3)

∗
with γm > 0 when γ2 < 0 and ωm ≈ ωm (position of
maximum of Γ1 (ω)). The quantity γm indicates capacitive and/or inductive behavior close to the resonance
frequency (≈ ωp ).
The height or, equivalently, the width of the resonance
in Γ1 defines also a resistance, R. In contrast to the
contact resistance, it is truly “dissipative” and related
to the pair excitations of the Luttinger liquid. It is also
contained in γ1 , though its numerical value for small ω
is different from the one near the resonance. Generally,
we find that it is only possible to define resistances, capacitances and inductances in certain limited parameter
regions49 .
In order to observe capacitive behavior, the interaction
between the electrons should be sufficiently long range.
This is consistent with the results of a different approach
in which Coulomb blockade behavior at a tunnel barrier
between two Luttinger liquids has been discussed52 . Also
there, non-zero range of the interaction is necessary for
capacitive behavior.
In the next section, we describe briefly the model and
the dispersion relations for various interaction potentials.
We calculate the ac-conductance and study external versus internal driving fields in the section 3. Section 4 contains the identification of quantum impedances. Section
5 contains the discussion of the results.

h
¯ ω(q)b† bq .
q

H=

(5)

q

The spectrum of the pair excitations corresponding to the
Bosonic creation and annihilation operators b† , bq is given
q
by the Fourier transform of the interaction potential58
V (q),
ω(q) = vF |q| 1 +

V (q)
.
h
¯ πvF

(6)

The particle excitations which change the total electron
number are omitted here. The number of particles is assumed to be constant, N0 = kF L/π. The dispersion relation interpolates between the limit of zero-range interaction (q → 0) where ω(k) = v | q | with the “charge sound
velocity” v ≡ vF /g, and the limit of non-interacting particles (q → ∞), ω(q) = vF |q|. The strength of the interaction is defined as
1
V (q = 0)
≡1+
.
2
g
h
¯ πvF

II. LUTTINGER LIQUID WITH LONG-RANGE
INTERACTION

(7)

Non-interacting Fermions correspond to g = 1, repulsive
interaction to g < 1.
The particle density ρ(x) can be written in terms of
the phase variable of the Luttinger model

A. Outline of the model

The Luttinger liquid is a model for the low-energy excitations of a 1D interacting electron gas53,54,55,56 . Its
excitation spectrum can be calculated analytically. Also
many of the thermodynamical and the transport properties, as the linear conductivity, can be determined, even
in the presence of perturbing potentials. The main assumption is the linearization of the free-electron dispersion relation near the Fermi level. The starting point is

ϑ(x) = i

sgn(q)
q=0

1
eϕ(q)−iqx b† + b−q
q
2L | q |

(8)

where
e2ϕ(q) =

3

vF | q |
.
ω(q)

(9)

The function E1 is the exponential integral59 .
Two limiting cases are of particular interest. When
z ≫ 1, E1 (z) ≈ exp (−z)/z. Thus, for αd ≫ 1,

With the mean particle density ρ0 = N0 /L we write
1
ρ(x) ≡ ρ0 + √ ∂x ϑ(x).
π

(10)

V (q) =

For later use in the linear response theory, we need the
coupling to the driving voltage U (x, t) and the current
density42 . The former is given by
dxρ(x)U (x, t).

(11)

−∞

(12)

V (q) ≈ −V0 ln

In order to obtain the dispersion relation explicitly, we
need a specific model for the interaction. Since we want
eventually to draw some conclusions on quantum wires
we start from a 3D screened Coulomb interaction
e−αr
,
r

(13)

(14)
C. Results for the dispersion law

2
2 − ζ2
e d ,
πd2

Results for the dispersion are shown in Fig. 1

(15)

where d represents the “diameter” of the wire.
We obtain the effective interaction potential for the
motion in the x-direction from the matrix elements of
(13) in the state (15) by performing the integrations with
respect to y and z,
2V0
V (x) = − 2
αd

∞

dζe

2
− αζ d2
2

0

d −
e
dζ

√

α2 x2 +ζ 2

ω (q) d / vF

10

g0 = 0.1

1

.

(16)

Its Fourier transform is
d2
4

2
2
V (q) = V0 e (q +α ) E1

(21)

due to the presence of the mirror charge. It is clear that
this influences the results only when α−1 ≥ D. Assume
then α = 0. The cutoff of the Coulomb tail of the potential is in this limit given by D instead of α−1 . The
results to be discussed below for αd ≪ 1 apply also for
this limit with α replaced by D−1 .

In the following, we assume for the confining wave function
ϕ(ζ) =

(20)

VD (r) = V (r) − V (|r + D|),

with V0 = e2 /4πεε0 and project on the quasi-1D states
of the quantum wire.
For the latter, we assume a parabolic confining potential in the y- and z-directions independent of x. The
corresponding states are (ζ = y 2 + z 2 ),
eikx
ψk (x, ζ) = √ ϕ(ζ) .
L

α2 + q 2 d2 .

This is the same behavior as obtained by starting from
the 1D-equivalent of the Coulomb interaction49,58,60 implying the interaction falls off as x−1 in space.
In many of the quantum wire experiments metallic
gates are present in some distance, say D, from the wire
(diameter d ≪ D). In order to discuss the changes in the
interaction induced by the presence of the gates we can
consider an infinite metal plate parallel to the wire. This
changes the interaction potential according to

B. The interaction potential

V (r) = V0

(19)

For α → ∞ but with VL ≡ 4V0 /α2 d2 = const, this is
VL δ(x), the zero-range interaction with the strength VL
of the conventional Luttinger liquid.
When αd ≪ 1 we still get the above result (18) as
long as qd ≫ 1 which implies that in x-space V (x → 0)
√
behaves as (19) but with V (x = 0) = πV0 /d. For
z → 0, E1 (z) ≈ − ln z so that for qd ≪ 1,

The electric field is E(x, t) = −∂x U (x, t). The external
∞
voltage is assumed to be given by −∞ dxE(x, t) ≡ U (t).
The current operator is defined by using the 1D continuity equation for the Heisenberg representation of the
operators,
e ˙
J(x, t) ≡ − √ ϑ(x, t) .
π

(18)

This is the Fourier transform of
1
V (x) = VL αe−α|x| .
2

∞

HU = e

4V0
1
.
2 q 2 + α2
d

d2 2
q + α2
4

αd = 0.2
αd = 0.6
αd = 1.0
αd = 2.0
αd = 3.0
g=1

0.1

.

0.1

(17)

4

1

10

qd

FIG. 1. Double logarithmic plot of the dispersion relation
ω(q) of the Luttinger model with g0 = 0.1 and different ranges
αd (α inverse potential range in position space, d diameter of
quantum wire).

FIG. 3. The resonance frequency ωp and the corresponding wave number, qp (insert top right) as a function of the
−2
strength of the interaction, (g0 − 1)−1 of the Luttinger liquid with interaction of finite range as indicated. Insert bottom
left: scaling function h(αd).

−2
for interaction parameter g0 = 0.1 (V0 ≡ ¯ πvF (g0 −
h
1)) and various αd.
There is a crossover between the interacting and the
non-interacting limits for q → 0 and q → ∞, respectively,
at the intermediate wave number qp corresponding to the
characteristic frequency ωp . It is related to the finite
range of the interaction in the wave-number space. For
zero-range interaction the dispersion becomes linear, and
ω(q) = vF q/g.

For a broad range of g0 the frequency ωp and qp decay
−1/2
−1
as g0 and g0
, respectively. The data for ωp obey the
−2
scaling law (β0 ≡ g0 − 1)
ωp (β0 ; αd) = ωp (β0 h−2 (αd); 0)h(αd) .

(22)

The scaling function h(αd) is shown in the bottom left
insert of Fig. 3 and it is proportional to the limit of ωp
for infinitesimally small interaction.

vF dq/dω

2.0
αd = 1.0

III. LINEAR RESPONSE

1.5

In this section we outline the calculation of the conductance with linear response theory.

1.0
g0 = 0.1

A. The conductivity

g0 = 0.3
g0 = 0.5
0.5

g0 = 0.7

Using the above current, (12), one obtains the complex
conductivity, σ(x − x′ ; t − t′ ) which relates the current at
a given point x at time t with the driving electric field
Eext (x′ , t′ ),

g0 = 1.0

0.0
0

5

10

15

20

25

ωd / vF 30

FIG. 2. The density of pair excitations, dq/dω for αd = 1,
different g0 .

∞

J(x, t) =

dx′

−∞

Fig. 2 shows the excitation density dq/dω for various
g0 and αd = 1.
The frequency ωp and the corresponding wave number
−2
qp as a function of (g0 − 1)−1 are shown in Fig. 3 for
various αd.

−∞

While the non-locality of the conductivity is unimportant
in the dc-limit, it is crucial for time-dependent transport.
By assuming that the electric field is concentrated only
near a given point, say x′ = x0 , we get
t

J(x, t) =

qpd/2

ωp d/ 2vF

−∞

h(αd)
3
2
1
0
0.0

1
0.0001

0.5

1.0

0.001

1.5

2.0

0.01

0.1

1

(g0-2-1)-1 10

dt′ σ(x − x0 ; t − t′ )Uext (t′ )

(24)

with the voltage drop Uext (t′ ) = dx′ Eext (x′ , t′ ) dropping only near x0 . We see that σ(x − x0 , t − t′ ) plays
then the role of a “conductance” that relates the current at some point x in the system with a voltage dropping at some other point. By using near-field microscopy,
one could possibly perform such a non-local experiment.
However, it seems to us that it is very hard to measure
the current locally in a quantum wire, especially in the
region of high frequency.
By Fourier transformation, (23) is equivalent to

αd = 0
αd = 0.2
αd = 0.6
αd = 1.0
αd = 2.0
αd = 3.0

4

dt′ σ(x − x′ ; t − t′ )Eext (x′ , t′ ) .
(23)

100

10

t

100

J(q, ω) = σ(q, ω)Eext (q, ω) .

(25)

The conductivity kernel can be expressed either by the
current-current correlation function or, by using the continuity equation, as
5

σ(q, ω) ≡ σ1 (q, ω) + iσ2 (q, ω) =

−iωe2
R(q, ω) ,
q2

one can use the external field for the calculation of the
current. The results will be used to derive absorptive and
reactive conductances.
The dielectric response function, which describes the
dynamical screening of a charge, is defined as

(26)

with the charge-charge correlation function
i
R(x, t) = − Θ(t) [ρ(x, t), ρ(0, 0)] .
h
¯

(27)
ε(q, ω) =

Here, · · · ≡ Tr (exp (−βH) · · ·) /Tr exp (−βH) is the
usual thermal average (temperature ∝ β −1 ). By using
the expression (10) we find the exact result
R(q, ω) =

q2
vF
.
2 (q) − (ω + i0+ )2
h
¯π ω

Utot (q, ω) = Uext (q, ω) + Usc (q, ω)
(28)

−∞

˙
dt′ γ(t − t′ )ϑ(x, t′ ) = L(x, t)

1
= 1 − V (q)R(q, ω) .
ε(q, ω)

∞
−∞

′

′

(35)

By using the result (28) for R(q, ω) we find the explicit
expression
ε(q, ω) =

ω 2 (q) − ω 2
,
2
ω0 (q) − ω 2

(36)

with the dispersion of the non-interacting electrons ω0 (q).
Equations (33) and (34) imply that

(29)

subject to an effective external force with the Fourier
transform
e
1
L(x, ω) = − √
π σ(x = 0, ω)

(34)

is the total potential and Uext , Usc are the external
and screening potential, respectively. Within the linear
screening model, the dielectric response function can be
written in terms of the charge-charge correlation function
as

A most remarkable feature of the above result becomes
transparent by applying the imaginary-time path integral
approach. With this, the time-dependent non-linear response to an electric field of arbitrary spatial shape of a
Luttinger system was calculated recently42 . It was found
that the average of the phase field (8) obeys the equation
of motion of a Brownian particle with mass M → 0
t

(33)

where

B. Analogy with a Brownian particle

¨
M ϑ(x, t) +

Uext (q, ω)
,
Utot (q, ω)

Etot (q, ω) = Eext (q, ω) [1 − V (q)R(q, ω)]
≡ Eext (q, ω)F (q, ω) .

′

dx E(x , ω)σ(x − x , ω),

(37)

Using the conductivity (26) with (28) and comparing
with (36) we see that

(30)
σ(q, ω) =

and a damping term γ(t) which is given by the non local
conductivity. The Fourier transform of γ(t) is
1
e2
.
γ(ω) =
π σ(x = 0, ω)

∞
−∞

dx′

t
−∞

(38)

Here, σ0 is the conductivity of the non-interacting electrons. This implies that

(31)

Etot (q, ω)σ0 (q, ω) = Eext (q, ω)σ(q, ω) .

¿From the solution of the equation of motion the current
is found,
J(x, t) =

σ0 (q, ω)
.
ε(q, ω)

(39)

If we calculated the conductivity from the response to the
total field, the conductivity would turn out to be that of
non-interacting electrons.
This result is true as long as one can use linear screening. It implies also that the voltage drop at frequency ω
is the same for both fields

dt′ σ(x − x′ , t − t′ )E(x′ , t′ ) . (32)

Linear response is exact for the Luttinger liquid.

∞

Utot (ω) =
C. The driving electric field

dxEtot (x, ω) = Uext (ω)

(40)

−∞

since for any finite non-zero frequency ε(q → 0, ω) = 1.
For a monochromatic external field,

In this section, we investigate the influence of screening
on the response to an external electric field. In particular,
we discuss the dc-limit and show that the two limits ω →
0 and q → 0 cannot be interchanged. It will turn out that
for the Luttinger model, where linear response is exact,

Eext (q, t) = Eext (q) cos ωt ,
one finds the result
6

(41)

Etot (q, t) = Eext (q) [ReF (q, ω) cos ωt
+ImF (q, ω) sin ωt]

in space and time. Physically, it is the absorption constant for electromagnetic radiation. Using (42) one can
show that the absorbed power for a monochromatic time
dependence is the same for both fields due to the time
average. We obtain

(42)

which means that there is a phase shift between the total
and the external field.
A final remark concerns the static limit. While we have
for any non-zero frequency ε(q → 0, ω) = 1, we find for
ω=0
lim ε(q, 0) = lim

q→0

q→0

ω 2 (q)
1
= .
2
ω0 (q)
g

Pω =

1
.
1 + V (q)/¯ vF π
h

Γ1 (ω) = vF

L(q) ≡

Γ2 (ω) =
=

(45)

−∞

We define the average
T

Using the Laplace transform

we obtain P = lims→0 sP (s). The absorptive conductance can then be defined by
P
U2

(52)

dω ′
−∞

Γ1 (ω ′ )
ω − ω′

∞

1
4πU 2

ext

dq Imσ(q, ω) | Eext (q) |2 ,

−∞

(53)

ge2
L
h

gω
vF

,

(54)

e2
L
h

ω
vF

.

(55)

This reflects that for large q the dispersion is not influenced by an interaction of a finite range.

(47)

0

Γ1 =

∞

1
P
π

Γ1 (ω) ≈

dte−st P (t)

.

−∞

(46)

0

∞

2U 2 ext

2

dxeiqx Eext (x)

the same as without interaction26 , except for the renormalization of the prefactor and the Fermi velocity with
the interaction strength g and g −1 , respectively. In the
general case of an interaction potential of finite range we
get asymptotically (ω → ∞)

∞

P (s) =

∞

1

Γ1 (ω) =

−∞

dtP (t) .

(51)

it contains information about phase shifts.
For a zero-range interaction the conductance becomes

dxJ(x, t)Etot (x, t)

1
T →∞ T

e2
dq
L(q(ω))
h
dω

By a Kramers-Kronig transformation, we can define
also a reactive conductance

∞

P = lim

(50)

with the Fourier transformed auto-correlation function of
the external electric field

Since it is very difficult to detect experimentally the
non-local conductivity some average has to be performed.
One possibility is to use the absorbed power, P (t), in order to define the conductance. This appears to be a natural choice if one wants to describe infrared or microwave
experiments,

dqJ(q, t)Etot (−q, t) .

(49)

This gives the expression
(44)

D. Absorptive and reactive conductances

1
=
2π

dq Reσ(q, ω) | Eext (q) |2 .

vF e2
[δ(ω − ω(q)) + δ(ω + ω(q))] .
2¯
h

Reσ(q, ω) =

The limits ω → 0 and q → 0 cannot be inter-changed.
The latter result has been used recently, in order to
explain that in quantum wires the dc-conductance is
not renormalized by the interaction44,45 (see also (44)).
We are discussing here frequency dependent properties.
Thus, we always obtain for small frequencies a conductance that is renormalized by the interaction since we
consider the limit of infinite system length.

P (t) =

−∞

With (28) we find for the real part of the conductivity

(43)

By inserting the dispersion relation of the Luttinger
model into (28) one obtains
Etot (q) = Eext (q)

∞

1
4π

(48)

ext

with Uext (t) = dxEext (x, t). It is independent of the
amplitude of the external field but depends on its shape
7

to simulate the frequency behavior. The circuit shown in
Fig. 5 contains the minimum set of elements that are necessary for reproducing both the low-frequency behavior
and near the resonance.

0.4

hΓ/e

2

0.3

Γ1

L

0.2

R

C

0.1

Γ2
0.0

-0.1
0

8

16

24

32

ω d / vF 40

L0

FIG. 4. Real and imaginary part, Γ1 (ω) and Γ2 (ω), of the
ac-conductance of a Luttinger wire with finite range interaction; αd = 1, g = 0.1, range of the electric field ℓ/d = 1/4.

R0

FIG. 5. Classical circuit for simulating the frequency behavior of the complex conductance.

A most important feature of the result (51) is the factorization into a part that depends only on the internal
properties of the interacting electron system, dq/dω, and
a part that contains the information about the shape of
the electric field. Only in the limit of vanishing frequency,
the shape of the latter is unimportant26,46 . In general,
the ac-response depends on the spatial properties of the
electric field42 which is certainly determined by the interactions.
Most remarkably, the temperature does not enter the
result, although T = 0 was assumed in the derivation.
This is due to the linearization of the spectrum. As long
as this assumption is justified, the response of the Luttinger liquid is independent of temperature.
A typical result for Γ(ω) is shown in Fig. 4. The Fourier
transform of the electric field has been assumed to be
a Gaussian E(x) = E0 exp (−2x2 /ℓ2 ). If the range of
∗
the electric field is zero, the zero of Γ2 (ω), ωm , and the
position of the maximum of Γ1 (ω), ωm , do not agree.
∗
However, as soon as ℓ is finite, ωm ≈ ωm for a wide
region of parameters.
In the Coulomb case, α = 0, Γ1 ∝ (ln ω)−1/2 for ω → 0,
due to the logarithmic singularity of the dispersion for
q → 0.

Its complex impedance Z(ω) is given by
Z −1 (ω) =

iωC
1
.
+
1 + iωRC − ω 2 LC
R0 + iωL0

(56)

The resistance R0 is fixed to be the resistance at zero
frequency, h/ge2 . The circuit shows a resonance near
ω0 = (LC)−1/2 with a width depending on R.
At low frequency, the real and imaginary parts of
Z −1 (−ω), Γ1 (ω) and Γ2 (ω), respectively, behave as
−1
Γ1 (ω) = R0 + γ1 ω 2 ,

Γ2 (ω) = γ2 ω

(57)

L0
2 .
R0

(58)

with
γ1 = RC 2 −

L2
0
3,
R0

γ2 = −C +

The circuit is defined to behave “capacitively” if simultaneously γ1 > 0 and γ2 < 0. If simultaneously γ1 < 0 and
γ2 > 0 the circuit is clearly “inductive”. If γ2 = 0, i.e.
there is no phase shift between current and voltage, γ1
indicates capacitive or inductive behavior depending on
whether R/R0 is larger or smaller than 1, respectively.
Note that until O(ω 2 ) the inductance L does not play any
role. Whether the circuit behaves capacitively or inductively near the resonance is therefore to a certain extent
independent of its behavior at small frequency.

IV. “QUANTUM IMPEDANCES”

In this section, we analyze the results for the complex
conductance presented above with respect to “resistive”,
“capacitive” and “inductive” behavior. We compare the
Luttinger system with an equivalent classical circuit.

B. Low frequency

We have extracted the parameters γ1 and γ2 (cf. (1)
and (2)) which characterize the low-frequency behavior
of the conductance from the ac-conductance of the Luttinger liquid.

A. Impedance network

Our system of interacting electrons shows a resonance
in the ac-conductance. It can be useful to consider a circuit of capacitances, inductances and resistances, in order
8

By decreasing αd, i.e. increasing the range of the interaction, the end points of the two trajectories at g = 0 are
shifted to higher values of ℓ/d. The region of capacitivelike behavior increases at the expense of the inductive
region for increasing interaction strength.
¿From the behaviors deep in the capacitive and inductive regions, one can deduct formulas for equivalent
inductances and capacitances. However, these are not
always unique, although their scaling properties with the
parameters of the system are. For instance, in the region denoted by “L” in Fig. 7 γ2 is given by the second
term in (59) only. √
This leads, by comparing with (58),
to define L0 = hℓ/ πe2 vF , independently of the interaction parameter. Qualitatively the same result is obtained
when using the expression for γ1 . Apart from a different
numerical prefactor, the scaling of the inductance with ℓ
is the same. This is related to the fact that the behavior
of Γ2 at low frequency is determined via the KramersKronig transformation to the behavior of Γ1 also at high
frequency. The latter depends strongly on the shape of
the field. Only if this is assumed in order to reproduce
the ω −2 -behavior of the classical circuit, one can expect
γ1 and γ2 to yield the same L0 .
In the capacitive region, where g ≪ 1, the second terms
on the right hand sides of (59) can be neglected. The first
terms can be used to define an equivalent capacitance C
and a dissipative resistance R by comparing with (58).
For αd ≪ 1, in the limit of Coulomb interaction, we
obtain from γ2

0.10

γ1
0.05

γ
0.00

-0.05

d/O= 4
d/O= 6
d/O= 8
d/O=10
d/O= 12

-0.10

γ2

-0.15
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

g 1.0

2
FIG. 6. The parameters γ1 and γ2 (in units of e2 d2 /hvF and
e d/hvF , respectively) which characterize the low frequency
behavior of the ac-conductance of the Luttinger liquid as a
function of the interaction parameter g for different ranges of
the electric field, ℓ.
2

By assuming, as above, the electric field to be the
Gaussian distributed, we obtain the explicit results (in
units of e2 /h with V (0) = V (q = 0))
γ1 =

g 3 d2 3(1 − g 2 )
2
4vF
2

γ2 = −

2vF
π

∞

dqe−q

4V0
ℓ2
−1 − 2
α2 d2 V (0)
d
2 2

ℓ /4

0

(59)
g ℓ
1
1
+ √
− 2
ω 2 (q) ωg (q)
vF π
2

e2 4β0
1
,
h αvF (1 − 2β0 ln αd)2

(60)

h 3 1
3/2
(1 − 2β0 ln αd)
.
e2 32 β0

(61)

C≈

where ωg (q) = vF |q|/g. Figure 6 show γ1 and γ2 as a
function of g. Depending on the range of the field, ℓ, the
behavior changes from capacitive (small ℓ, γ1 > 0, γ2 <
0) to inductive (large ℓ, γ1 < 0, γ2 > 0). As functions
of g, γ1 and γ2 change also signs. Always, this change of
sign happens at a smaller value of g for γ2 . The “phase
separation” lines defined by γ1 (ℓ, g1 ) = γ2 (ℓ, g2 ) = 0 are
shown in Fig. 7 for αd = 1.

which is independent of the electric field and diverges for
infinite interaction range, α → 0. By comparing with the
first term of γ1 we obtain the dissipative resistance
R≈

For αd > 1, we find
C≈

1.0

β0 + z 2 ℓ 2
e2 d
β0
exp
×
2
h vF (z 2 + β0 )3/2
8
d2

L

O/d

× 1−Φ

0.8

ℓ
2d

β0 + z 2
2

(62)

with z = αd/2 and Φ the error function. Here, the capacitance depends on the electric field, for instance,

0.6

0.4

C=

C
0.2

e2 g(1 − g 2 )
C1
h
vF α

(63)

with C1 = 2 for g/ℓα ≫ 1 and C1 = (8 2/π)(g/αℓ) for
g/ℓα ≪ 1. The corresponding dissipative resistances are

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

g 1.0

R=

FIG. 7. “Phase trajectories” in the (ℓ/d)-g plane separating capacitive from inductive behaviors of the Luttinger liquid
for αd = 1.

h
g
R1
e2 1 − g 2

(64)

with R1 = 3/8 and R1 = (3π/256)(αℓ/g)2 for g/ℓα ≫ 1
and g/ℓα ≪ 1, respectively.
9

C. Near resonance

V. DISCUSSION

As seen in fig. 4 the absorptive conductance shows a
resonance near the frequency ωp . For this, the interaction has to be very strong, i.e. g ≪ 1. Furthermore, the
range of the electric field should not be too large (see
below). Due to the scaling (cf. (22)) the results in the
Coulomb and Luttinger limits are closely related. Indeed, in contrast to the limit ω → 0, in the Coulomb
limit αd ≪ 1 the conductance does not show any singular behavior near the resonance frequency ωm . Since near
the resonance the zero-frequency resistance does not play
any role, R0 and L0 are neglected when fitting C, L and
R to the ac-conductance. The parameters of the circuit
can be obtained by fitting to the resonance frequency,
and the width and the height of the resonance.
−1
For small range of the electric field, ℓ ≪ qp , we find

We considered the ac-transport properties of quantum
wire with finite range interaction. The linear response
theory was found to be exact, consistent with earlier
work, as a result of the linearization of the dispersion
relation. The dependence of the current on the electric
field is given by the microscopic non-local conductivity.
However, the latter is not very useful when aiming at a
description of experiments. What is needed is a description in terms of externally accessible macroscopic quantities, as for instance given by Ohm’s law. Such a relation
can also be found in the present, non-local quantum region. By assuming the electric field to be non-zero near a
given point xj and the current to be detected by a probe
at a point xi one finds Γ(ω) ≡ Γij (ω) = σ(xi − xj , ω).
By generalizing to several probe positions x1 . . . xp one
obtains

e2 3/2
β λ,
hvF 1
h 1/2
L∝
β λ,
vF e2 1
h 1/2
R ∝ 2 β1
e

C∝

(65)

e2 λ2 2
β
hvF ℓ 1
h
L∝
ℓ
vF e2
h 1 ℓ3
R∝ 2
.
e β1 λ3

(71)

j

(66)

Such an approach has been used recently32 in order to
generalize the Landauer dc-approach to finite frequency.
In the present work, the non-local conductances Γij (ω)
are natural results of the response theory when suitable
assumptions are made for the shape of the electric field.
However, it seems to us that in the ac-regime this approach is not necessarily appropriate since it may be very
difficult to apply experimentally ac-fields locally.
We find it more suitable to define a global average conductance via the time average of the absorbed power. If
we consider a monochromatic electric driving field, this
absorptive ac-conductance can be considered as the real
part of a complex conductance. It provides information
about the magnitude of the current in phase with the
electric field. The imaginary part of the ac-conductance
provides information about the phase shift between current and voltage. It was obtained by a Kramers-Kronig
transformation from the real part.
We found that the absorptive ac-conductance factorizes into a product of the density of excitations and the
Fourier-transformed auto-correlation function of the external electric field. A question related to this is whether
the electric field to be used is the external one or whether
one has to use the internal electric field that contains
screening contributions. We found that one can use
the external electric field in order to obtain the acconductance of the interacting system. If one uses the
total field, the same result is obtained for the conductance. In any case, the result for ω = 0 depends on the
spatial shape of the field.
Only in the dc-limit, the conductance becomes independent of the shape of the applied electric field and depends only on the applied external voltage. It is renormalized by the interaction parameter, Γ1 = ge2 /h. This
does not contradict other recent results which indicate
that for a Luttinger system of a finite length connected

(67)

with β1 = g, λ = α−1 and β1 = g0 , λ = d for αd ≫ 1
and αd ≪ 1, respectively.
If, on the other hand, the range of the electric field is
−1
large, ℓ > qp , the resonance becomes smaller and very
broad. The parameters of the circuit depend here again
on the field range. We find
C∝

Γij (ω)Uj (ω) .

Ji (ω) =

(68)
(69)
(70)

Remarkably, the capacitance obtained here scales in the
same way with the system parameters as the one obtained in the limit of low frequency, cf. (63). Also the
inductance scales as for ω → 0, although L = L0 . The
dissipative resistance, however, scales differently and depends much stronger on ℓ than L and C. This reflects,
that the dissipative resistance is much more sensitive to
the width and the height of the resonance than L and
C. The product of the latter is fixed by the resonance
frequency which is quite insensitive to the range of the
electric field in a broad region of field ranges. If we assume L = L0 the scaling behavior of C near the resonance follows directly from ωm = (LC)−1/2 . Thus, the
somewhat astonishing result of this is that one can identify a region of parameters for the 1D electron system, in
which the scaling properties of L and C are independent
of the frequency, though the numerical prefactors can be
different.
10

to Fermi liquid leads the conductance is not renormalized by the interaction. We can argue that the interaction influences transport for system lengths L above
q −1 (ω), the wave length of the charge density wave.
When L < q −1 (ω) the effect of the interaction can be
neglected61,62 . Since we are considering the thermodynamic limit, this region is outside the range of the validity
of our model.
There is a resonance in the absorptive conductance at
a frequency which corresponds to the inverse of a characteristic length scale of the interaction. In the limit of a
1D Coulomb interaction the latter is given by the cutoff
length d. For the Luttinger limit, it is the interaction
range α−1 . Its height and width is influenced by the
electric field, but its position is not. However, when having in mind a “realistic” situation, a word of caution is
here in order. If one wants to use the Luttinger model as
a model for a quantum wire, the parameters should be
such that (i) linearization of the free-electron dispersion
is a good approximation, i.e. q < kF , and (ii) interband
transitions are not important for the absorptive conductance, m∗ ω < hd−2 (m∗ effective mass). We have found
above that qp ∝ d−1 in the Coulomb limit. The Fermi
energy should be smaller than the interband energy distance. Therefore, the Fermi wave number is restricted to
kF < d−1 . Then, qp ≈ kF , and corrections to linearization should be taken into account. Since furthermore
ωm ∝ vF /d ∝ h/m∗ d2 also interband transitions could
become important near the resonance. For the Luttinger
limit of the interaction the situation does not improve.
One needs also to take into account interband mixing induced by the interaction for a proper description of the
frequency region near the resonance.
However, at low frequency, and if the Fermi level
is well below the onset of the second lowest subband,
interaction-induced mixing of the bands can be neglected,
and interband transitions are unimportant. Here, the
real and the imaginary part of the conductance depend
quadratically and linearly on ω, respectively. The signs
of the prefactors of these terms indicate capacitive- or
inductive-like behavior of the system. By comparing the
frequency behavior of the microscopic model with that of
a “minimal” classical circuit of capacitances, inductances
and resistances we find microscopic expressions for these
quantities. They are, however, not unique. Their validity is restricted to certain parameter regions. Only the
scaling properties are the same. Astonishingly, one can
identify a parameter region, in which the scaling properties with the parameters of the system are the same for
low frequency and near the resonance. This fact gives us
some confidence that although the frequency region near
the resonance is somewhat out of the range of validity of
the model certain general features of the results remain
valid.
In particular, the impedances depend in general
strongly on the shape of the applied electric field. There
is a competition between the range of the field and the
range of the interaction, which determines whether the

system behaves as a capacitor (ℓα ≪ 1) or as an inductor (ℓα ≫ 1). In the former case, depending on the ratio
between the interaction parameter g and ℓα, the capacitance and the dissipative resistance may (g/ℓα ≪ 1) or
may not (g/ℓα ≫ 1) depend on the field range. The
infinite range of the interaction removes the dependence
on the range of the field, such that one can interpret the
impedances as genuine properties of the system that are
only determined by the parameters of the microscopic
model.
It is particularly interesting to note that the model allows to identify two conceptually different “resistances”.
In the dc-limit, we have the equivalent of the “contact
resistance”, h/ge2 , which is however renormalized by the
interaction parameter. Near the resonance frequency, one
can define a “dissipative resistance”, approximately given
by the inverse of the height of the resonance. The former depends on if and how the system is connected with
the “outside world” via contacts. It changes to h/e2 if
the system length is smaller than the wave length of the
charge density wave. In contrast, we expect that the lat−1
ter is not changed as long as L > qp ∝ d. By comparing
with the classical circuit one also notes that the dissipative resistance is also present at low frequency, though it
scales differently with the system parameters.

VI. CONCLUSION

How to calculate the frequency dependent lowtemperature electronic quantum transport properties of
nanostructures when interaction effects are important is
presently a subject of strong debate. Since experiment
can access this regime by using modern fabrication technology, developing theoretical concepts for the treatment
of transport processes taking into account interactions
is important. In addition, future information technology will require high-frequency signal transportation in
nano-scale systems. Proper theoretical understanding of
the underlying microscopic processes will be essential for
developing this technology in the far future.
In order to obtain insight into some of the peculiar
microscopic features of this transport region, we have investigated the ac-response of a simple 1D model of an
interacting electron system. We found that in spite of
the simplicity of the model the definition of transport
parameters provides already non-trivial questions to be
solved. The transport parameters depend on the purpose for which they are supposed to be used. For instance, when asking for the current in some probe as a
response to an electric field applied to another probe it
is the non-local conductivity, which has to be used as
the “conductance”. On the other hand, when being interested in the absorption of electromagnetic radiation,
an average of the conductivity provides an “absorptive
conductance”.
By starting from the “absorptive conductance”, we

11

14

have made an attempt to define for our system the quantum counterparts of the impedance parameters used in
the transport theory of classical circuits. We found that
they cannot be defined uniquely in terms of the parameters of the microscopic model, in accordance with the
above general statement. However, certain scaling properties remain valid in astonishingly large regions of parameters. They even resemble the scaling properties of
the corresponding classical definitions. This encourages
to find similar quantities for more complicated systems as
tunnel barriers and multiple tunnel barriers in the presence of interactions.
Useful discussions with Andrea Fechner are gratefully
acknowledged. This work has been supported by Istituto
Nazionale di Fisica della Materia, by the Deutsche Forschungsgemeinschaft via the Graduiertenkolleg “Nanostrukturierte Festk¨rper” and the SFB 508 of the Univero
sit¨t Hamburg, by the European Union within the HCM
a
and TMR-programmes, and by NATO.
1
on leave of absence from Dipartimento di Fisica, INFM,
Universit` di Genova, Via Dodecaneso 33, I–16146, Gena
ova.
2
on leave of absence from Istituto di Fisica di Ingegneria, INFM, Universit` di Genova, Via Dodecaneso 33,
a
I–16146, Genova.

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