Influence of the contacts on the conductance of interacting quantum wires
K. Janzen, V. Meden, and K. Sch¨nhammer
o

arXiv:cond-mat/0604443v2 [cond-mat.str-el] 5 Jul 2006

Institut f¨r Theoretische Physik, Universit¨t G¨ttingen,
u
a
o
Friedrich-Hund-Platz 1, D-37077 G¨ttingen, Germany
o
We investigate how the conductance G through a clean interacting quantum wire is affected by
the presence of contacts and noninteracting leads. The contacts are defined by a vanishing twoparticle interaction to the left and a finite repulsive interaction to the right or vice versa. No
additional single-particle scattering terms (impurities) are added. We first use bosonization and the
local Luttinger liquid picture and show that within this approach G is determined by the properties
of the leads regardless of the details of the spatial variation of the Luttinger liquid parameters.
This generalizes earlier results obtained for step-like variations. In particular, no single-particle
backscattering is generated at the contacts. We then study a microscopic model applying the
functional renormalization group and show that the spatial variation of the interaction produces
single-particle backscattering, which in turn leads to a reduced conductance. We investigate how
the smoothness of the contacts affects G and show that for decreasing energy scale its deviation
from the unitary limit follows a power law with the same exponent as obtained for a system with a
weak single-particle impurity placed in the contact region of the interacting wire and the leads.
PACS numbers: 71.10.Pm, 72.10.-d, 73.21.Hb

I.

INTRODUCTION

The low-energy physics of one-dimensional (1d) metals
is not described by the Fermi liquid theory if two-particle
interactions are taken into account. Such systems fall
into the Luttinger liquid (LL) universality class1 that is
characterized by power-law scaling of a variety of correlation functions and a vanishing quasi-particle weight. For
spin-rotational invariant interactions and spinless models, on which we focus here, the exponents of the different correlation functions can be parametrized by a single
number, the interaction dependent LL parameter K < 1
(for repulsive interactions; K = 1 in the noninteracting case). As a second independent LL parameter one
can take the velocity vc of charge excitations (for details
see below). Instead of being quasi-particles the low lying excitations of LLs are collective density excitations.
This implies that impurities or more generally inhomogeneities have a dramatic effect on the physical properties
of LLs.2,3,4,5
In the presence of only a single impurity on asymptotically small energy scales observables behave as if the 1d
system was cut in two halfs at the position of the impurity, with open boundary conditions at the end points
(open chain fixed point).6,7,8 In particular, for a weak
impurity and decreasing energy scale s the deviation of
the linear conductance G from the impurity-free value
scales as (s/s0 )2(K−1) , with K being the scaling dimension of the perfect chain fixed point and s0 a characteristic energy scale (e.g. the band width). This holds
2
as long as |Vback /s0 | (s/s0 )2(K−1) ≪ 1, with Vback being a measure for the strength of the 2kF backscattering of the impurity and kF the Fermi momentum. For
smaller energy scales or larger bare impurity backscattering this behavior crosses over to another power-law scaling G(s) ∼ (s/s0 )2(1/K−1) , with the scaling dimension of
the open chain fixed point 1/K. This scenario was veri-

fied for infinite LLs6,7,8 as well as finite LLs connected to
Fermi liquid leads,9,10 a setup that is closer to systems
that can be realized in experiments. In the latter case
the scaling holds as long as the contacts are modeled to
be “perfect”, that is free of any bare and effective singleparticle backscattering, and the impurity is placed in the
bulk of the interacting quantum wire. For an impurity
placed close to perfect contacts the exponents change to
2(K − 1)/(K + 1) (close to the perfect chain fixed point)
and 1/K − 1 (close to the open chain fixed point).9,10
The role of an inhomogeneous two-particle interaction,
that is an interaction that depends not only on the relative distance of the two particles, but also the center
of mass is less well understood. In the present publication we will fill this gap. Such an inhomogeneity will
generically appear close to the interface of the interacting
quantum wire and the leads and a detailed understanding is thus essential for the interpretation of transport experiments on quasi-1d quantum wires.11 We here use two
models to study the effect of two-particle inhomogeneities
on the linear conductance. We first investigate the socalled local Luttinger liquid (LLL) model,12,13,14 that is
characterized by a spatial dependence of the LL parameters K and vc , with K = 1 and vc = vF in the leads (vF is
the noninteracting Fermi velocity). We show that regardless of the details of the spatial variation the conductance
always takes the perfect value 1/(2π) (in units such that
= 1 and the electron charge e = 1). Thus the LLL
description cannot produce any effective single-particle
backscattering from the contact region generated by an
inhomogeneous two-particle interaction. Our results generalize earlier findings obtained for step-like variations of
the LL parameters.12,13,14
We then study a microscopic lattice model with a spatially dependent nearest-neighbor interaction. Across the
contact between the left lead and the wire the interaction is turned on from zero to a bulk value U and corre-

2
spondingly turned off close to the right contact. We show
that this two-particle inhomogeneity generically leads to
an effective single-particle backscattering and a reduced
conductance. To compute the latter we use an approximation scheme that is based on the functional renormalization group (fRG).10 We numerically and analytically investigate the dependence of the conductance on U ,
the length of the interacting wire N , and the “smoothness” with which the interaction is turned on and off.
For weak effective inhomogeneities we analytically show
that 1/(2π) − G displays scaling with the energy scale
δN = πvF /N set by the length of the wire. The exponent we find is consistent with the one found for a weak
single-particle impurity close to a perfect contact.9,10 We
give the energy scale up to which this scaling holds. It
depends on the bulk interaction U and the 2kF Fourier
component of the function with which the interaction is
varied close to the contacts. The latter provides a quantitative measure of the “smoothness” of the contacts. This
finding suggests a similarity between the universal lowenergy properties of a LL with a single-particle inhomogeneity and a two-particle inhomogeneity. We discuss
this similarity but also point out differences.
This paper is organized as follows. In Sec. II we study
the two-particle inhomogeneity within the LLL picture.
To motivate the LLL model in Sec. II A we present a
brief introduction into bosonization and the TomonagaLuttinger model. The transport properties of the LLL
model are investigated in Sec. II B. In Sec. III we introduce our microscopic model and give the basic equations
of the fRG approach. We then discuss our numerical results in Sec. III A and the analytical findings in Sec. III B.
We close in Sec. IV with a summary.

A.

Model and generalized wave equations

In order to also elucidate the limitations of the LLL
picture we first shortly recall the basic ideas necessary to
justify the Tomonaga-Luttinger-model1,16,17 for interacting fermions in one dimension in the homogeneous case.
We consider a system of length L with periodic boundary
conditions. In second quantization the two-body interaction reads
V =

1
2
+

v(x − x′ )δρ(x)δρ(x′ )dxdx′
1
1 2
N v (0) − v(0)N ,
˜
2L
2

(1)

where δρ(x) = ψ † (x)ψ(x) − N /L is the operator of the
particle density relative to its homogeneous value, with
N the particle number operator. The Fourier transform
of the two-body interaction is denoted as v (k). The last
˜
term is usually dropped as it only modifies the chemical potential. If the range of the two-body interaction is much larger than the mean particle distance only
particle-hole pairs in the vicinity of the two Fermi points
are present in the ground state and the eigenstates with
low excitation energy.16 This allows to linearize the dispersion around the two Fermi points and to introduce two
independent types of fermions, the right- and left-movers
with particle density operators δρ± (x) = ρ± (x) − N±


ikn x
∂  −i
e
1
ρn,α 
eikn x ρn,α =
δρα (x) =
L
∂x 2π
n
n=0

n=0

∂Φα
.
≡
∂x

(2)

As shown below the field operators Φα are convenient objects for the solution of the problem.18 In the subspace of
low energy states the Fourier components of the density
ρn,α obey the commutation relations16
II.

THE LOCAL LUTTINGER LIQUID
DESCRIPTION

In this section we discuss how sufficiently smooth contacts (perfect contacts) can be properly described in
a LLL picture. This approach was successfully used
to show that for perfect contacts the properties of the
leads and not the finite size quantum wire determine
the conductance.12,13,14 Up to now the LL parameters
were always assumed to vary step-like. Here we present
a simple derivation of this result for an arbitrary variation of the LL parameters. Also the role of impurities
in the interacting wire was investigated within the LLL
picture.9,15 It was shown that the exponent of the temperature dependence of the conductance is different for
impurities placed in the bulk and close to the contacts.9
Using the fRG this was later confirmed in a microscopic
model in which the contacts were modeled to be arbitrarily smooth.10 .

[ρm,α , ρn,β ] = αm δαβ δm,−n .

(3)

After proper normalization they take the form of boson
commutation relations.
One can write the operator of the kinetic energy as a
quadratic form of the ρ± .16 The Tomonaga model then
results from replacing δρ(x) by δρ+ (x) + δρ− (x) and N
by N+ + N− in Eq. (1). With the well known “g-ology”
generalization1 the boson part of the interaction reads
1
{2g2 (x − x′ )δρ+ (x)δρ− (x′ )
(4)
2
+g4 (x − x′ ) [δρ+ (x)δρ+ (x′ ) + δρ− (x)δρ− (x′ )]} dxdx′ .

VTL,b =

This reduces to Tomonaga’s original model for g2 ≡
g4 ≡ v, while Luttinger17 later independently studied
the model with g4 ≡ 0. He also discussed the special
case g2 (x) ∼ δ(x), which corresponds to the interaction
term in the massless Thirring model.19 This simplifies

3
various aspects but brings in infinities which have to be
removed e.g. by normal ordering. In the boson part of
the kinetic energy
δρ2 (x) + δρ2 (x) dx
+
−

Tb = πvF

(5)

the integrand has to be normal ordered. This, as above,
is usually suppressed.
Inhomogeneous Luttinger liquids can be realized by
adding an external one-particle potential or by allowing the two-body interaction to depend on both spa-

HLLL,b = π
=

π
2

≡

π
2

vF +

g4,x
2π

vF +
vN (x)

tial variables, i.e. v(x − x′ ) in Eq. (1) is replaced by
v(x, x′ ). The operator of a one-particle potential can
only be expressed in terms of the δρα if it is sufficiently smooth in real space, i.e. the external potential
has a vanishing 2kF Fourier component. Similarly only
for sufficiently smooth variations—in Sec. III B we will
specify the meaning of “sufficiently smooth”—of v with
(x + x′ )/2 the analogous steps from Eq. (1) to Eq. (4)
i.e. gi (x − x′ ) → gi (x, x′ ) are allowed without changing
the low energy physics. The standard LLL model is obtained by assuming gi (x, x′ ) = gi,x δ(x − x′ )

δρ2 (x) + δρ2 (x) +
+
−
∂Φ(+) (x)
∂x

g4,x + g2,x
2π
∂Φ(+) (x)
∂x

2

+ vJ (x)

where we have defined Φ(±) (x) ≡ Φ+ (x) ± Φ− (x), as well
as the velocities vN (x) and vJ (x). As in the homogeneous model the right and left movers are coupled by the
g2 interaction and using the Φ(±) (x) as the basic fields
simplifies the solution as Φ(ν) (x) and Φ(ν) (x′ ) commute.
Apart from a term which vanishes in the thermodynamic
limit Φ(−ν) (x′ ) and ∂Φ(ν) (x)/∂x obey canonical commutation relations after proper normalization
∂Φ(ν) (x) (−ν) ′
1
i
δL (x − x′ ) −
,Φ
(x ) =
∂x
π
L

.

(8)

This follows from Eqs. (2) and (3). Here δL denotes the
L-periodic delta function. Neglecting the correction term
yields the following Heisenberg equations of motion for
the Φ(ν) (x, t)
∂ (+)
∂Φ(−) (x, t)
Φ (x, t) = −vJ (x)
,
∂t
∂x
∂Φ(+) (x, t)
∂ (−)
Φ (x, t) = −vN (x)
.
∂t
∂x

2

+ vF +

∂Φ(−) (x)
∂x

vN

(6)
∂Φ(−) (x)
∂x

g4,x − g2,x
2π

2

dx

(7)

2

dx ,

vc,L

a

b

x

FIG. 1: (Color online) Spatial dependence of velocity vN (x)
used for the discussion of the generalized wave equation (10).
Two different realizations (full and dashed curves) of the transition region [a, b] are shown. The shaded area shows the
incoming density δρ(x, 0).

(9)

Therefore, the Φ(ν) (x, t) obey generalized wave equations, e.g. for the field related to the change of the total
density
∂
∂Φ(+) (x)
∂ 2 (+)
Φ (x, t) − vJ (x) vN (x)
= 0 . (10)
2
∂t
∂x
∂x
The spatial derivative of the first equation in Eq. (9)
yields the continuity equation for the total charge density
δρ = δρ+ + δρ−
∂
∂Φ(−) (x, t)
∂
vJ (x)
=0
δρ(x, t) +
∂t
∂x
∂x

g2,x
δρ+ (x)δρ− (x) dx
π

(11)

which implies the conservation of the total charge Q =
δρdx. As Eqs. (10) and (11) are linear the expectation
values discussed in the following obey the same equations.
In a homogeneous system the velocities vJ and vN are
constant and the corresponding “sound velocity” called
√
the charge velocity vc is given by vc = vJ vN . It constitutes the first LL parameter characterizing the system. The second LL parameter K is defined as K ≡
vJ /vN = vc /vN . The general solution of the wave
equation for constant vc and K reads f (x − vc t) + g(x +
vc t), where f and g are arbitrary functions.

4
B.

The constant value of S follows from Eq. (12)

Computing the transmitted charge

a

For general dependencies of vJ (x) and vN (x) on the
position the generalized wave equation (10) cannot be
solved analytically. In the following we discuss properties of the solution in the thermodynamic limit when the
velocities vJ (x) and vN (x) are constant for x < a and
x > b > a but have arbitrary (bounded) variation in the
interval [a, b]. Figure 1 shows two qualitatively different
types of behavior of vN (x). The dashed curve presents
a monotonous transition between the asymptotic values
while the full curve corresponds to an “interacting wire”
region in [a, b] with a higher (constant) value of vN,w
corresponding to a stronger repulsive interaction. In order to describe the charge transport through this region
we consider an incoming density with compact support
which at t = 0 is completely to the left of a and moves
towards it with constant velocity vc,L (see Fig. 1). Before
the disturbance reaches point a the time evolution is
′
Φ(+) (x, t) = fin (x − vc,L t) , δρ(x, t) = fin (x − vc,L t) .
(12)
The total incoming charge Qin is given by Qin = fin (a) −
fin (−∞). This short time solution for Φ(+) (x, t) determines Φ(−) (x, t) up to a constant. In the long time limit
there will again be no charge in the region [a, b] with the
total transmitted charge Qtrans moving to the right with
constant velocity vc,R and the total reflected charge Qref
moving to the left with velocity vc,L , where charge conservation implies Qtrans + Qref = Qin . In the following
we show that it is possible to determine Qref /Qin without
the explicit solution for δρ(x, t) in this long time limit and
without any additional assumptions on the spatial variation of vN (x) and vJ (x). The trick is to introduce the
quantity
∞

S(t) ≡

−∞
(−)

= Φ

1 ∂Φ(+) (x, t)
dx
vJ (x)
∂t

(13)

(−∞, t) − Φ(−) (∞, t)

and to show that S is time independent. This can be
seen by discussing the expression in the second line which
follows using Eq. (9) or if one wants to argue with the
density only by taking the time derivative and using the
generalized wave equation
∞

1 ∂ 2 Φ(+) (x, t)
˙
S(t) =
dx
(14)
∂t2
−∞ vJ (x)
∞
∂
∂
vN (x) Φ(+) (x) dx
=
∂x
∂x
−∞
= vN,R δρ(∞, t) − vN,L δρ(−∞, t) = 0 ,
where we have used that the assumption about the initial
state implies that ρ(±∞, t) vanishes for all finite times.

vc,L ′
f (x − vc,L t)dx
vJ,L in
−∞
Qin
vc,L
Qin = −
.
= −
vJ,L
KL

S = −

(15)

When the incoming density hits the transition region
[a, b] it is partially reflected and transmitted, where the
detailed behavior is very different for the two realizations
of vN (x) shown in Fig. 1. For sufficiently large times the
total charge in the transition region goes to zero and
Φ(+) (x, t) takes the form

 fref (x + vc,L t) , for x < a
const. ,
for a < x < b (16)
Φ(+) (x, t) =
f
trans (x − vc,R t) , for x > b .
With the definition of S in Eq. (13) this yields
vc,L a ′
f (x + vc,L t)dx
vJ,L −∞ ref
vc,R ∞ ′
ftrans (x − vc,r t)dx
−
vJ,R b
Qref
Qtrans
=
−
.
KL
KR

S =

(17)

The comparison of Eqs. (15) and (17) as well as charge
conservation leads to the result derived earlier assuming
a stepwise variation of the LL parameters12,13,14
Qref =

2KR
KL − KR
Qin , Qtrans =
Qin . (18)
KL + KR
KL + KR

Our result shows that the properties of the transition
region play no role at all. The ratio Qtrans /Qin is positive,
while Qref /Qin has no definite sign.
From this result one can easily obtain the linear conductance G through the system. We first discuss a homogeneous system with constant LL parameters KL and
vc,L (the use of the index L becomes clear later). We
consider a current free initial state in which the density
is increased by δρ0 in the left half of the infinite system.
Then half of the additional density moves to the right
with velocity vc,L which corresponds to the current
j=

∂ρ
1
1
vc,L δρ0 = vc,L
δµ0 ≡ G δµ0
2
2
∂µ

(19)

in the central region extending linearly with time. Here µ
denotes the chemical potential. With ∂ρ/∂µ = 1/(πvN )
this yields for the homogeneous system
Ghom =

KL
.
2π

(20)

We now switch to an inhomogeneous system as in Fig. 1.
In analogy to the initial condition in the homogeneous
case we raise the density for x < a by δρ0 . The stationary
current is then obtained by multiplying the result for the

5
homogeneous case by the fraction of transmitted charge
through [a, b], computed in Eq. (18). For the conductance
this leads to
Ginhom =

2KR KL 1
,
KL + KR 2π

(21)

which again is independent of the details of the transition
region.
For an interacting quantum wire in region [a, b] attached to noninteracting leads (KL = KR = 1) the conductance is 1/(2π) independently of how quickly the LL
parameters vary spatially near the contact points a and
b. This shows that the LLL description cannot produce
one-particle backscattering from the contact region generated by an inhomogeneous two-particle interaction. As
we will discuss next using a microscopic model and the
fRG approach contacts defined by a vanishing interaction
to the left and a positive finite interaction to the right (or
vice versa) generically produce an effective single-particle
backscattering. This clearly shows the limitations of the
LLL picture. With the approximation to describe the
two-body interaction as a quadratic form in the bosefields the dependence of the conductance on the sharpness of the transition is lost. In contrast, our microscopic
model directly allows to study the transition from smooth
to abrupt contacts with the concomitant change of the
conductance.

III.

A MICROSCOPIC MODEL

As our microscopic model we consider the spinless
tight-binding Hamiltonian with nearest-neighbor hopping τ > 0 and a spatially dependent nearest-neighbor
interaction Uj,j+1 = Uj+1,j
∞

H = −τ

c† cj + c† cj+1
j+1
j
j=−∞

N −1

+
j=1

Uj,j+1 (nj − 1/2) (nj+1 − 1/2) , (22)

where we used standard second-quantized notation with
c† and cj being creation and annihilation operators on
j
site j, respectively, and nj = c† cj the local density operj
ator. The interaction acts only between the bonds of the
sites 1 to N , which define the interacting wire. We later
take
Uj,j+1 = U h(j) , j = 1, 2, . . . , N − 1 ,

(23)

with a function h that is different from 1 only in regions
close to the contacts at sites 1 and N . We here mainly
consider the half-filled band case n = 1/2. In the interacting region the fermions have higher energy compared
to the leads. To avoid that the interacting wire is depleted (implying a vanishing conductance) we added a

compensating single-particle term. It can be included as
a shift of the local density operator by 1/2. This ensures that the average filling in the leads and the wire is
n = 1/2. It is important to have a definite filling in the
interacting wire as the LL parameters depend on n. For
general fillings the required shift of the density depends
on n and the interaction.10
The model Eq. (22) with nearest neighbor interaction
U across all bonds (not only the once within 1 and N )
is a LL for all U and fillings, except at half filling for
|U | > 2τ .20 At n = 1/2 the LL parameter K is given
by20
K=

2
U
arccos −
π
2τ

−1

,

(24)

for |U | ≤ 2τ . To leading order in U/τ this gives
K =1−

U
+ O [U/τ ]2 .
πτ

(25)

At temperature T = 0 the linear conductance of the
system described by Eq. (22) can be written as21
G(N ) =

1
|t(0, N )|2
2π

(26)

with the effective transmission |t(ε, N )|2 = (4τ 2 − [ε +
µ]2 )|G1,N (ε, N )|2 . The one-particle Green function G has
to be computed in the presence of interaction and in
contrast to the noninteracting case acquires an N dependence. To compute G and thus G we use a recently
developed fRG scheme.23 The starting point is an exact
hierarchy of differential flow equations for the self-energy
ΣΛ and higher order vertex functions, where Λ ∈ (∞, 0]
denotes an infrared energy cutoff which is the flow parameter. We here introduce Λ as a cutoff for the Matsubara frequency. A detailed account of the method is
given in Ref. 10. We truncate the hierarchy by neglecting
the flow of the two-particle vertex only considering ΣΛ ,
which is then energy independent. The self-energy ΣΛ=0
at the end of the fRG flow provides an approximation for
the exact Σ. This approximation scheme and variants of
it were successfully used to study a variety of transport
problems through 1d wires of correlated electrons. In
particular, in all cases of inhomogeneous LLs studied the
exponents of power-law scaling were reproduced correctly
to leading order in U .10,24,25,26
On the present level of approximation the fRG flow
equation for the self-energy reads
∂ Λ
1
Σ ′ =−
∂Λ 1 ,1
2π

+

ω=±Λ 2,2′

Λ
eiω0 G2,2′ (iω) Γ1′ ,2′ ;1,2 ,

(27)
where the lower indices 1, 2, etc. label single-particle
states, Γ1′ ,2′ ;1,2 is the bare antisymmetrized two-particle
interaction and
−1
G Λ (iω) = G0 (iω) − ΣΛ

−1

,

(28)

6
-2

10

-3

1-2πG

with the noninteracting propagator G0 . For the initial condition of the self-energy flow see below. As our
single-particle basis we later use Wannier states [as in the
Hamiltonian (22)] as well as momentum states.

10

-4

10

-5

10

-6

Numerical results

In the real space Wannier basis the self-energy matrix
is tridiagonal and the set of coupled differential equations
(27) reads (with j ∈ [1, N ])
d Λ
1
Σ =−
dΛ j,j
2π

+

l=±1 ω=±Λ

Λ
Uj,j+l Gj+l,j+l (iω) eiω0 , (29)

d Λ
Uj,j±1
Σ
=
dΛ j,j±1
2π

exponent

A.

10
-0.06
h1(j)
h2(j)
h3(j)

-0.07
-0.08
-4

10

-3

10

-2

10
δN/τ

-1

10

+

ω=±Λ

Λ
Gj,j±1 (iω) eiω0 .(30)

To derive these equations we have used that
¯
′ ′
′
′
′
′
Γj1 ,j2 ;j1 ,j2 = Uj1 ,j2 (δj1 ,j1 δj2 ,j2 − δj1 ,j2 δj2 ,j1 )

(31)

¯
with Uj1 ,j2 = Uj1 ,j1 +1 (δj1 ,j2 −1 + δj1 ,j2 +1 ). The initial
condition of ΣΛ at Λ = ∞ is10,23
Σ∞ = − (Uj−1,j + Uj,j+1 ) /2 ,
j,j
Σ∞
j,j±1 = 0 .

(32)

Even for very large N (earlier results were obtained for
up to N = 107 )10,24 the system Eqs. (29) and (30) can
easily be integrated numerically starting at a large but
finite Λ0 . Due to the slow decay of the right-hand-side
(rhs) of the flow equation the integration from ∞ to Λ0
gives a nonvanishing contribution on the diagonal of the
self-energy matrix that does not vanish even for Λ0 → ∞
and that exactly cancels the term in Eq. (32) such that

(33)

For half filling the Hamiltonian (22) is particle-hole
symmetric. For N odd and symmetric h(j), that is
h(j) = h(N − 1 − j), on which we focus here, it is furthermore invariant under inversion at site j = (N + 1)/2. Together these symmetries lead to a vanishing conductance
as was discussed earlier.27,28 We thus only consider even
N . The two extreme cases of (a) an abrupt turning on
and off of the interaction h(j) = 1 for j = 1, 2, . . . , N − 1,
and zero otherwise, and (b) a very smooth variation of
h(j) were considered earlier. In case (a) 1/(2π) − G increases for fixed N and increasing24,28 U as well as for
fixed U and increasing N .24 Using our fRG scheme it was
shown that for asymptotically large N , G vanishes as
G ∼ (δN /τ )2(1/K−1) ,

reached for fairly large U , e.g. U = 1.5τ . As discussed
in the introduction scaling of G with the same exponent
as in Eq. (34) is found for a single impurity in an otherwise perfect chain. This suggests that the low-energy
physics of a correlated quantum wire with two contact inhomogeneities due to the two-particle interaction is similar to that of a wire with a single impurity (single-particle
term). Here we will further investigate this relation.
Already at this stage it is important to note that the
similarity is not complete. As shown elsewhere29 G(T )
for two contacts, fixed U , fixed large N , and decreasing
temperature T vanishes as (for T ≫ δN )
G ∼ (T /τ )1/K−1 ,

ΣΛ0 = 0 ,
j,j
ΣΛ0
j,j±1 = 0 .

FIG. 2: (Color online) Upper panel: The conductance 1 −
2πG as a function of δN /τ for U/τ = 0.2, contacts of m =
7 lattice sites, and three different contact functions hi (j) of
increasing smoothness. Lower panel: The effective exponent
(logarithmic derivative) of the data. The thin solid line shows
the exponent 2(K − 1)/(K + 1) with K(U/τ = 0.2) = 0.9401.

(34)

with the energy scale δN = πvF /N . The scale on which
the asymptotic low-energy scaling sets in strongly depends on U and even for up to N = 106 it was only

(35)

that is with an exponent that is only one-half of the one
found in the δN scaling Eq. (34). This is in clear contrast
to the temperature dependence of G for a single impurity
in an otherwise perfect chain. In this case the exponent
2(1/K−1) is found in the scaling with any infrared energy
scale, e.g. T , δN , and Λ.
For a given N and U it is always possible to find a
sufficiently smooth function h(j) such that 1/(2π) − G
is smaller then an arbitrarily small upper bound [case
(b)].10,24 In that case the contacts are perfect and the
effect of impurities in the interacting wire connected to
Fermi liquid leads can be studied. Further down we will
give a quantitative definition of the meaning of “sufficiently smooth”.
Here we study the dependence of the conductance on
N , U , and the function h(j) by numerically solving the
flow equations (29) and (30) with the initial condition
Eq. (33). In Fig. 2 we show 1 − 2πG as a function
of δN /τ ∼ 1/N on a log-log scale (upper panel) for
U/τ = 0.2 and three different h(j) of increasing smoothness. In our numerical calculations for simplicity we focus

on contact functions h(j) that are symmetric around the
middle of the interacting wire (equal contacts) and therefore only give their definition for the first half of the wire
(with j ∈ [1, N − 1])

1-2πG

7

h1 (j) = 1 , j = 1, . . . , N/2 ,

and

j = 1, . . . , (m − 1)/2
 2(j/m)2 ,
h3 (j) = 1 − 2(j/m − 1)2 , j = (m + 1)/2, . . . , m − 1 ,
 1,
j = m, . . . , N/2

where m is odd and measures the length of the contact
region. In Fig. 2 we chose m = 7. The lower panel of
Fig. 2 shows the logarithmic derivative of the data. A
plateau in this figure corresponds to power-law scaling of
the original data with an exponent given by the plateau
value. For fixed δN , 1 − 2πG decreases quickly with increasing smoothness of h(j). For sufficiently small δN /τ
all curves follow power-law scaling. The smoother the
contact the smaller δN /τ has to be before the scaling sets
in. The numerical exponent is close to 2(K − 1)/(K + 1)
(shown as the thin solid line), with K from Eq. (24).
The latter is the exponent found for a system with perfect contacts and a weak single-particle impurity placed
in the contact region. We also studied other values for
the length of contact m and found similar results. The
deviation 1 − 2πG from the unitary limit decreases if m
is increased while all other parameters are fixed.
Within our approximation scheme we map the manybody problem on an effective single-particle problem
with the energy independent ΣΛ=0 as an impurity
potential.10,23 During the fRG flow the interplay of the
two-particle inhomogeneity and the bulk interaction generates an oscillating self-energy with an amplitude that
decays slowly away from the two contacts. This is in close
analogy to the case of a single impurity in an otherwise
perfect chain.10,23,30 Scattering off this effective potential
leads to the reduced conductance.
The U dependence of 1 − 2πG and the effective exponent is shown in Fig. 3 for h2 (j). At fixed δN the deviation of the conductance from the unitary limit increases
with U . For all U/τ shown we find power-law behavior with exponents that are close to 2(K − 1)/(K + 1)
(again shown as the thin solid lines). The larger U/τ
the larger is the deviation of the numerical exponent and
2(K − 1)/(K + 1). This is not surprising as our approximate scheme can only be expected to capture exponents
to leading order in U/τ .10
For later reference we note that power-law behavior is
not only found in the δN (that is 1/N ) dependence of
1 − 2πG but also in the dependence on the fRG flow parameter Λ at a large fixed N . In this case G is computed
using Eq. (26) with the Green function at scale Λ. This
is shown in Fig. 4 where we compare the dependence on
δN /τ and Λ/τ for U = 0.2 and h2 (j). Scaling with Λ/τ

exponent

j = 1, . . . , m − 1
,
j = m, . . . , N/2

-0.1
U/τ=0.6
U/τ=0.4
U/τ=0.2

-0.2
-4

-3

10

-2

10

10
δN/τ

-1

10

FIG. 3: (Color online) Upper panel: The conductance 1−2πG
as a function of δN /τ for the contact function h2 (j), contacts
of m = 7 lattice sites, and U/τ = 0.2, 0.4, 0.6. Lower panel:
The effective exponent (logarithmic derivative) of the data.
The thin solid lines show the exponents 2(K − 1)/(K + 1)
with K(U/τ = 0.2) = 0.9401, K(U/τ = 0.4) = 0.8864, and
K(U/τ = 0.6) = 0.8375.

-4

1-2πG

j/m ,
1,

-4

10

10

-5

10 0

exponent

h2 (j) =

-3

10

-0.05
-0.1
-0.15 -5
10

δN/τ
Λ/τ
-4

10

-3

-2

10
10
δN/τ, Λ/τ

-1

10

0

10

FIG. 4: (Color online) Upper panel: The conductance 1−2πG
as a function of δN /τ and Λ/τ (with N = 216 sites) for the
contact function h2 (j), contacts of m = 7 lattice sites, and
U/τ = 0.2. Lower panel: The effective exponent (logarithmic derivative) of the data. The thin solid line shows the
exponents 2(K − 1)/(K + 1) with K(U/τ = 0.2) = 0.9401.

holds roughly down to the scale δN /τ = πvF /(τ N ) set
by the length of the interacting wire. Beyond this scale
G saturates.
We now analytically confirm the power-law scaling of
1/(2π) − G for weak effective inhomogeneities and compute the exponent to leading order in the bulk interaction.

B.

Analytical results

Numerically we found that 1/(2π) − G shows powerlaw scaling in both effective low-energy cutoffs δN and Λ,

8
as long as the scaling variable considered is sufficiently
larger than the other fixed energy scale. We note that the
same holds for the temperature T as the variable.31 In all
cases the same exponent is found. Analytically the easiest access to scaling is by considering N → ∞, T = 0, and
taking Λ as the variable. We will here follow this route.
For the case of a single impurity with bare backscattering
amplitude Vback in an infinite LL (no Fermi liquid leads)
a similar calculation was performed within the fRG23 to
show that for small Λ
−U/(πτ )

ΣkF ,−kF /τ ∼ (Vback /τ ) (Λ/τ )

.

(36)

−π

+

Λ
dq dq ′ eiω0 Gq,q′ (iω) Γk′ ,q′ ;k,q ,

(38)
with
Γk′ ,q′ ;k,q =

′
′
′
′
1
ei(q−q ) − ei(q−k ) + ei(k−k ) − ei(k−q )
2π
˜
×U (k + q − k ′ − q ′ )
(39)

˜
U (k)

(40)

∂ Λ
(1)
(2)
Σ ′ = Fk′ ,k (Λ) + Fk′ ,k Λ, ΣΛ ,
∂Λ k ,k

(41)

with

′
U ˜
h(k − k ′ ) eik + e−ik
2πτ

ω=±Λ

q,q′

.

(44)

=

˜
U h(k) =

U
2π

N −1

eijk
j=1

iN k

U e −e
2π 1 − eik
1
N →∞
−→ U δ2π (k) .
2

j=1

˜
Here h(k) denotes the Fourier transform of the function
h(j) that contains the shape of the turning on and off
of the interaction. In the present section we do not assume special symmetry properties of h(j). Thus the two
contacts might be different, as it is generically the case
in experiments. At Λ = Λ0 all matrix elements of ΣΛ0
are zero. We now expand the rhs of Eq. (38) to first
order in ΣΛ . This expansion is justified as long as ΣΛ
remains small (it is certainly small at the beginning of
the fRG flow). The flow equation then becomes an inhomogeneous, linear differential equation

=−

dq dq ′ Γk′ ,q′ ;k,q
−π

=

Uj+1,j eijk

1
2π

π

ik

˜
= U h(k) .

(1)

(43)
.

Before further evaluating this term we take N → ∞ and
compute Γk′ ,q;k,q in this limit. As we will see in this case
the information about the smoothness of h(j) is lost on
the rhs of Eq. (44). It nevertheless enters the solution of
(1)
the differential equation via the inhomogeneity Fk′ ,k in
Eq. (41). As we are only interested in the leading divergent behavior of the linear part on the rhs of Eq. (41)
this procedure is justified. To be specific we first assume
that the interaction is turned on and off abruptly, that is
h(j) = 1 for j = 1, 2, . . . , N − 1 and zero otherwise. To
obtain the Fourier transform of the two-particle interaction we have to perform

N −1

Fk′ ,k (Λ) = −

1
2π

× G0 ΣΛ G0

and the Fourier transform of the interaction
1
˜
U (k) =
2π

Λ
1− √
2 + 4τ 2
Λ

The term on the rhs of the linearized flow equation that
contains ΣΛ is

(37)

where −2U/(πτ ) is the leading order approximation of
the weak impurity exponent 2(K − 1) [see Eq. (25)]. This
scaling holds as long as the rhs stays small. For analytical
calculations it is advantageous to switch from the real
space basis to the momentum states.
In the momentum state basis the flow equation (27) is
given by

ω=±Λ

U ˜
h(−2kF )e−ikF
πτ

(2)

K
− G ∼ |Vback /τ |2 (Λ/τ )−2U/(πτ ) ,
2π

π

(1)

FkF ,−kF (Λ) = −

Fk′ ,k (Λ, ΣΛ ) = −

Within the Born approximation this leads to

∂ Λ
1
Σk′ ,k = −
∂Λ
2π

where we used ξk = −2τ cos (k) − µ and Eq. (39). The
+
factor eiω0 was dropped as the integration starts at
Λ0 < ∞. Later we will primarily be interested in the
kF , −kF matrix element (backscattering) of Σ. For these
momenta the rhs simplifies to

π

dq
ω=±Λ

−π

Γk′ ,q;k,q
iω − ξq

Λ
1− √
2 + 4τ 2
Λ

,(42)

(45)

With this result the two-particle vertex simplifies to
Γk′ ,q;k,q

N →∞

−→

U
[cos(k − k ′ ) − cos(q − k ′ )]
π
×δ2π (k + q − k ′ − q ′ ) .
(46)

Up to a factor 1/2 the vertex is then equivalent to the one
obtained for a homogeneous nearest neighbor interaction.
(2)
With this vertex Fk′ ,k (Λ, ΣΛ ) Eq. (44) reads
(2)

Fk′ ,k (Λ, ΣΛ ) = −

U
2π 2

− cos(q − k ′ )]

π
ω=±Λ

−π

dq[cos(k − k ′ )

Σq+k−k′ ,q
.
(iω − ξq+k−k′ )(iω − ξq )

(47)

The differential equation (41) can be solved by the variation of constant method. To apply this we first determine the solution of the homogeneous equation. The
scaling can be extracted if only the leading singular con(2)
tribution (for Λ → 0) of Fk′ ,k (Λ, ΣΛ ) Eq. (47) at k ′ = kF

9
and k = −kF is kept. Following the same steps as in
Ref. 23 we find
∂
ΣΛ
∂Λ kF ,−kF

U 1
ΣΛ
2πτ Λ kF ,−kF

hom

≈−

hom

/τ ≈ c0 (Λ/τ )−U/(2πτ ) ,

hom

,

(48)

with the solution
ΣΛF ,−kF
k

(49)

where c0 is a dimensionless constant. The singular part
of the solution of the inhomogeneous linear differential
equation is then given by
ΣΛF ,−kF /τ ≈ c(Λ) (Λ/τ )−U/(2πτ ) ,
k

(50)

with
Λ

dΛ′

c(Λ) =
Λ0

1
′
ΣΛF ,−kF
k

(1)

FkF ,−kF (Λ′ ) .

(51)

hom

For small Λ the function c(Λ) is non-singular and given
by
lim c(Λ) = −

Λ→0

U˜
¯
h(−2kF )e−ikF c
τ

(52)

inhomogeneity this is replaced by the square of the bulk
interaction and the square of the 2kF Fourier component
(backscattering) of the function h(j) with which the interaction is turned on and off. The smaller this component the smaller is the perturbation due to the two
contacts. Therefore the smoothness of the turning on
and off is directly measured by ˜
h(−2kF ). The presence
of the factor |U/τ |2 explains why in the numerical study
the weak inhomogeneity exponent for larger U/τ is only
observable for fairly smooth contacts, that is h(j)’s with
small 2kF component. For larger U/τ and fairly abrupt
contacts the rhs of Eq. (54) becomes too large already
on intermediate energy scales and no energy window for
scaling is left. For Λ ≪ Λc defined in Eq. (55) the inhomogeneity is effectively large and Eq. (34) holds.
We here considered the half-filled band case, but also
other fillings can be studied following the same steps.23
The analysis Eqs. (38) to (54) also holds if in addition weak single-particle impurities are placed close to
0
the contacts, with the only difference that ΣΛF ,−kF now
k
has a non-vanishing initial condition set by the backscattering of the bare impurity. The prefactor in Eq. (54)
is determined by either the square of the single impurity
2
2
backscattering amplitude or |U/τ | |h(−2kF )| depending on the relative size.

with c being a constant of order 1. This gives
¯
ΣΛF ,−kF /τ ≈ −
k

U˜
¯
h(−2kF )e−ikF c (Λ/τ )−U/(2πτ ) (53)
τ

and with the Born approximation the final result
1
˜
− G ∼ |U/τ |2 h(−2kF )
2π

2

(Λ/τ )−U/(πτ ) .

(54)

To obtain the singular part of the solution of the homowe assumed
geneous differential equation ΣΛF ,−kF
k
hom
that the interaction is turned on and off abruptly. One
can show that the parts of h(j) with a smooth variation
of finite length do not contribute to the singular part of
Eq. (44). Thus the same singular part is found independently of how the interaction is varied and Eq. (54) is
valid for general h(j).
The power-law scaling Eq. (54) holds for Λc ≪ Λ ≪ τ
with a scale Λc set by
|U/τ |

2

˜
h(−2kF )

2

(Λc /τ )−U/(πτ ) ∼ 1/(4π) .

(55)

To leading order in U/τ the exponent U/(πτ ) agrees with
the exponent 2(K − 1)/(K + 1) [see Eq. (25)] found for a
single weak impurity placed close to a perfect (that is arbitrarily smooth) contact.9,10 We thus have analytically
shown, that with respect to the scaling exponent weak
single-particle and weak two-particle inhomogeneities are
indeed equivalent. The leading order exponent is furthermore consistent with the numerical results of the last
section.
For the single impurity case the prefactor of the power
2
law is given by |Vback /τ | . In case of the two-particle

IV.

SUMMARY

In the present paper we have investigated the role of
contacts, defined by an inhomogeneous two-particle interaction, on the linear conductance through an interacting 1d quantum wire. The wire and contacts were
assumed to be free of any bare single-particle impurities.
We first showed that within the LLL picture the contacts
are always perfect, that is the conductance is 1/(2π) independent of the strength of the interaction, the length of
the wire, and the spatial variation of the LL parameters.
Earlier only step-wise changes of the LL parameters were
considered. We then studied the problem within a microscopic lattice model. Similar to the case of a single impurity the interplay of the two-particle inhomogeneity and
the bulk interaction generates an oscillating self-energy
with an amplitude that decays slowly away from the contacts. Scattering off this effective potential leads to the
discussed effects. We showed that within the microscopic
modeling 1/(2π) − G increases with increasing bulk interaction U , increasing wire length N , and decreasing
smoothness. The measure for smoothness is given by
˜
h(−2kF ), that is the backscattering Fourier component
of the spatial variation h(j) of the two-particle interaction. As long as the inhomogeneity stays effectively small
1/(2π) − G shows power-law scaling with an exponent
that is consistent with 2(K − 1)/(K + 1), the scaling exponent known for the case of a wire with a single impurity
placed close to one of the two smooth contacts.
In experiments on quantum wires the leads are electronically two- or three-dimensional. Depending on the

10
systems studied the contacts are either regions in which
the system gradually crosses over from higher dimensions
to quasi 1d or the contact regions extend over a finite part
of the wire (see the experiments on carbon nanotubes).11
This shows that our simplified description—1d leads, spatially dependent interactions close to end contacts, no explicit single-particle scattering at the contacts—provides
only an additional step towards a detailed understanding
of the role of contacts in transport through interacting
quantum wires, such as carbon nanotubes and cleaved

1

2
3
4
5

6

7
8

9

10

11

12
13
14
15
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edge overgrowth samples.11

Acknowledgments

We thank S. Jakobs and H. Schoeller for very fruitful
discussions. The authors are grateful to the Deutsche
Forschungsgemeinschaft (SFB 602) for support.

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