Transport and scattering in inhomogeneous quantum wires
N. Sedlmayr,1, ∗ J. Ohst,1 I. Aﬄeck,2 J. Sirker,1, 3 and S. Eggert1, 3

arXiv:1204.2565v2 [cond-mat.str-el] 7 Sep 2012

2

1
Department of Physics, University of Kaiserslautern, D-67663 Kaiserslautern, Germany
Department of Physics and Astronomy, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada
3
Research Center OPTIMAS, University of Kaiserslautern, D-67663 Kaiserslautern, Germany
(Dated: July 22, 2018)

We consider scattering and transport in interacting quantum wires that are connected to leads.
Such a setup can be represented by a minimal model of interacting fermions with sudden changes in
interaction strength and/or velocity. The inhomogeneities generally cause relevant backscattering,
so it is a priori unclear if perfect ballistic transport is possible in the low temperature limit. We
demonstrate that a conducting ﬁxed point surprisingly exists even for large abrupt changes, which in
the considered model corresponds to a velocity matching condition. The general position dependent
Green’s function is calculated in the presence of a sudden change, and is conﬁrmed numerically
with high accuracy. The exact form of the interference pattern in the form of density oscillations
around inhomogeneities can be used to estimate the eﬀective strength of local backscattering sources,
oﬀering a route to design experiments where the eﬀects of the contacts are minimized.
PACS numbers: 73.63.Nm, 71.10.Pm, 73.40.-c

The description of transport through quantum wires
has become a very well studied research area, as it directly ties together conductivity experiments1–4 with central theoretical models in one dimension, such as the
Landauer formalism5 and Luttinger liquids.6 Landauer
showed that the conductance of a single spinless noninteracting quantum channel is always ﬁnite and given
by G = (1 − R2 )e2 /h, where R is the backscattering
amplitude.5 However, interaction eﬀects change this picture dramatically as R becomes temperature dependent
through renormalization. In the pioneering work of Kane
and Fisher it was found that a local perturbation in a
repulsive Luttinger liquid is relevant, which results in
a characteristic powerlaw dependence R ∝ T g−1 where
g < 1 is the interaction parameter.7 A number of impurity models in one dimension have since been analyzed in
detail8–16 and conﬁrm that generically a local perturbation cuts the transport at low temperatures, unless the
relevant operator is forbidden by symmetry.
Nearly perfect connections between leads and wires can
be achieved for diﬀerent types of quantum wires,1–4 which
show quantized conductance at moderate temperatures.
At lower temperatures, however, scattering in the connections to the leads plays an increasingly important role.
The higher dimensional contacts are eﬀectively weakly
interacting, so there has been great interest in describing
the transport through a quantum wire attached to noninteracting leads.16–28 The minimal model for this setup
is a single channel of interacting spinless fermions, where
the interaction parameter changes along the wire. Under
the assumption that it is possible to use a hydrodynamic
Luttinger liquid description for this model,17 it would be
expected that backscattering follows a non-trivial renormalization behavior which is position dependent.18 Even
if the connections are eﬀectively free from imperfections
with a homogeneous lattice structure,4 there have to be
small regions of the wire, the junctions, where the interaction changes, which does not necessarily occur adi-

abatically and will induce intrinsic backscattering. This
immediately invites the question if it is ever possible to
create a perfect connection in the low temperature limit.
Indeed it is in general unclear how large such intrinsic
backscattering is for generic junctions and if it is even
justiﬁed to use an inhomogeneous Luttinger liquid description in the ﬁrst place for this setup, since the usual
bosonization procedure assumes a translational and scale
invariant theory.
In this paper we consider the intrinsic backscattering
in an inhomogeneous wire and show that a perfectly conducting ﬁxed point can indeed be found by adjusting parameters such as the velocities on both sides. At half ﬁlling it is also possible to estimate the bare strength of the
backscattering from the change in the local velocity ﬁeld.
The full correlation function of an inhomogeneous Luttinger liquid is calculated, which agrees with numerical
simulations to high accuracy. This conﬁrms the validity
of the inhomogeneous theory and proves that a low energy conducting ﬁxed point can exist even in the presence
of large abrupt jumps.
A hydrodynamic description of interacting fermions
with a changing interaction parameter gx leads to a generalized inhomogeneous Luttinger liquid action17,18
1/T

dτ

S0 =
0

dx

ux (∂τ φ)2
+ (∂x φ)2 ,
2gx
u2
x

(1)

where φ is a canonical bosonic ﬁeld. Here the eﬀective
velocity ux generally also changes with the interaction
strength. We will derive the corresponding correlation
function for an abrupt jump below. Additionally, however, backscattering must be considered. In particular, it
is known that the Hamiltonian density contains a leading
relevant oscillating operator given by7
√
4πφ

†
e−i2kF x ψ+ ψ− ∝ e−i2kF x e−i

,

(2)

where ψ± are left and right moving fermion ﬁelds and
kF is the Fermi-wavenumber. Normally this operator

2
can be neglected under the integral, but this is no longer
the case for inhomogeneous systems. In fact, it is exactly this operator as a local perturbation which causes
the renormalization of defects in wires7 and spin chains.8
The oscillating part of the interaction results in the same
bosonic operator,8 so that even a change in interaction
alone will induce backscattering.
In principle conductance depends on the connections
of the wire to both of the leads. We will concentrate here
on the backscattering in the case where the length of the
wire is larger than the coherence length ∝ u/T so that
it suﬃces to consider each junction separately.18 A single
junction can be characterized by changes taking place in
a small region around x = 0. The ﬁeld φx is assumed to
be slowly varying on the scale of the Fermi wavelength
so that it is possible to use an expansion of φx in Eq. (2)
and a partial integration to derive an eﬀectively local
perturbation
√

H ′ ≈ λe−i

4πφx=0

+ h.c.,

Dφe−S0 −

1/T
dτ H ′
0

(4)

with the quadratic action in Eq. (1). For an abrupt
change in the interaction parameter at x = 0 from gx<0 =
gℓ to gx≥0 = gr and similarly for the velocity ux the gen˜
eral bosonic Green’s function G(x, x′ ; τ ) = φx,τ φx′ ,0 0 is
found by matching the left and right parts. In particular,
it is possible to solve
˜
G(x, x′ ; τ ) = T

˜
eiωm τ Gm (x, x′ )

(5)

m

with
2
ωm
ux ∂
∂
−
2gx ux
∂x 2gx ∂x

˜
Gm (x, x′ ) = δ(x − x′ )

g
¯
˜
G(x, x′ ; τ ) = − ln sinh πT
π

(6)

|x| |x′ |
+
− iτ
ux
ux′

sinh πT
L[x, x′ ]gx
ln
+
π
sinh πT

|x|
ux

+

|x′ |
ux′

|x−x′ |
ux

(7)
− iτ

,

− iτ

where we have deﬁned a new eﬀective interaction param1
eter g = 2( gℓ + g1 )−1 . Here L[x, x′ ] is 1 when x and x′
¯
r
are in the same region, and 0 when they are not. The
renormalization of a local perturbation in Eq. (3) can be
determined with the help of the Green’s function by integrating out the Fourier components above a cutoﬀ Λ,6,18
which gives
1 dλ
= 1 − g.
¯
λ d ln Λ

(3)

while the oscillating operator cancels everywhere in the
uniform regions. So far we have kept only the leading relevant operator, but there are higher order terms which
will be discussed later. In general λ is complex, except
for particle hole symmetric situations. Even relatively
smooth junctions become eﬀectively sharper and sharper
under renormalization, so that the relevant backscattering is non-zero unless the amplitude λ is adjusted to zero,
which requires the ﬁne tuning of two parameters. As we
will see later it is indeed possible to identify such conducting ﬁxed points in a lattice model by an appropriate
choice of parameters on both sides of the junction. The
existence of a conducting ﬁxed point is also of relevance
for the discussion about possible charge fractionalization
in Luttinger liquids.3,29,30 If the Luttinger liquid supports
only charges e(1 ± gx )/2 related to the chiral eigenstates
of the Hamiltonian on each side of the wire, then it seems
to be impossible that backscattering can be tuned to zero.
Our results thus support the analysis in Ref. [30] that at
such a junction an arbitrary charge can be injected into
a Luttinger liquid.
In order to perform a renormalization group analysis
we have to consider the full partition function
Z=

˜
by allowing a discontinuity in the derivative of Gm at
′ 17
x = x . We determine the general Green’s function to
be

(8)

We therefore expect that the eﬀective backscattering
renormalizes as a power law in the temperature R ∝
¯
T g−1 .
As a concrete lattice model we can consider spinless
fermions at half-ﬁlling
H=
x

†
1
1
−tx (ψx ψx+1 + h.c.) + Ux (nx − 2 )(nx+1 − 2 ) ,

(9)
†
where nx = ψx ψx . The corresponding interaction parameter g and the renormalized velocity u in Eq. (1) are
functions of U and t, which are known from the Bethe
ansatz.6 For small jumps and interactions we have estimated the size of λ perturbatively, which turns out to be
proportional to the corresponding renormalized velocity
ﬁeld ux
λ∝

x

e−i2kF x (ux+1 − ux ),

(10)

with 2kF = π, so to lowest order it does not matter if the
velocity change occurs due to inhomogeneous interactions
or hopping amplitudes, which may even compensate each
other. In particular, Eq. (10) suggests that a conducting ﬁxed point can be achieved for a sharp discontinuity by a velocity matching uℓ = ur , but the derivation
above is valid only for small interactions. The numerical simulations show that this condition holds even in
a strongly interacting model. Equation (10) also implies
that backscattering can be made arbitrarily small by very
slow ”adiabatic” changes.
In order to calculate the backscattering amplitude R
numerically, we use quantum Monte Carlo simulations31
on systems with an abrupt junction at x = 0, i.e. Ux<0 =
Uℓ and Ux≥0 = Ur and analogously for the hopping amplitude tx . For long system sizes L 40tx /T the boundary condition at ±L/2 becomes irrelevant. The backscattering R induces a 2kF interference pattern in the density, the so-called Friedel oscillations,13,16,32 which can

3
0.2

0.12

0.45

χ

g-1

( T/TK)
left side
right side

R(T)
0.4

g-1

0.11

(T/TK)
left side
right side

R(T)
0.1
0.09

0.35

0.1

0.08
0.3

0
-40

0.07

0.06

0.25

-30

-20

-10

0

10

20

30 x 40

(a)

PSfrag replacements

Uℓ = 0 non-interacting

Ur = 1.8tr interacting

FIG. 1: (Color online) The local response in the density χ for
a jump in interaction from Uℓ = 0 to Ur = 1.8tr at T = 0.1tr .
Circles (blue): No discontinuity in hopping, tℓ = tr . Squares
(red): The hopping on the left is adjusted to tℓ ≈ 1.518tr in
order to match the velocities on both sides. Solid lines (black)
are ﬁts to the predicted behavior in Eq. (14) for even and odd
sites separately.

be calculated directly in the simulations and also give
additional information about the correlation functions.
In particular, we consider the local oscillating density in
a half-ﬁlled lattice in response to changing the chemical
potential
χx =

∂
nx
∂µ

,

(b)

0.01

(11)

µ=0

0.1

T/t

analogously to the local magnetization and susceptibility
in spin chains.8,13,32–35
The results for the Friedel oscillations χ = χ0 +
(−1)x χalt near an abrupt junction at x = 0 are shown
in Fig. 1 for a change from Uℓ = 0 to Ur = 1.8tr at ﬁnite temperatures T = 0.1tr for two cases: In the case
that the interaction strength changes but the hopping is
equal tℓ = tr , we have uℓ < ur and strong alternating
2kF = π oscillations are observed (circles). In the second case, the hopping tℓ on the left was also increased
in order to exactly match the velocities 2tℓ = uℓ = ur ≈
3.036tr (squares). Clearly, the backscattering oscillation
is strongly suppressed with a diﬀerent position dependence, but it is not zero. As the numerical data will
show it turns out that the relevant backscattering term
is exactly zero in this case. The uniform parts of χx
on the two sides are constant and approximately given
gx
by the zero temperature result χ0 = πux , which are not
equal in either case. We have also tested other cases
with more complicated changes in hopping and interactions over three sites in order to tune λ = 0 in Eq. (10),
and in all cases backscattering is strongly suppressed.
Let us ﬁrst analyze the position dependence of the
Friedel amplitude in the case of unequal velocities, which

0.1

shows a characteristic maximum in Fig. 1 reminiscent
of the local susceptibility near ends in spin chains.32,35
However, we will show that the behavior is not exactly
of the same form as for scattering from open ends as was
conjectured before.13 In order to calculate the alternating response in the presence of the perturbation (3) with
small |λ| ∝ |ur − uℓ | we use the bosonized form of the
density (12)
1/T

√
(−1)
1
sin[ 4πφx ] (12)
nx = n0 − √ ∂x φx + const.
π
π

T/t

FIG. 2: (Color online) Eﬀective backscattering amplitude
R(T ) on a logarithmic scale extracted from the amplitude
of the density oscillations in Eq. (14) for a jump from (a)
Uℓ = 0 to Ur = 1.8t, (TK = 3.4 × 10−4 t), and (b) Uℓ = 1t to
Ur = 1.4t with t = tℓ = tr , (TK = 6 × 10−6 t). The amplitudes
are extracted from ﬁtting the local response for x < 0 (“left
side”) and x > 0 (“right side”) separately.

where the density is bosonized as
x

0.05
0.01

χalt = λ

dτ
0

√
√
∂
sin 4πφx,τ cos 4πφ0,0
∂µ

0

, (13)

where the dependence on µ is given by a shift of the
µgx
ﬁeld ∂x φ by √πux which turns the sine-dependence
into a cosine-dependence with an additional factor of
x. Using the Green’s function in Eq. (7) we arrive
1
at integrals of the form 0 dτ | sinh(X − iπτ )|−2g =
2g (sinh 2X)−g P−g (coth 2X) where Pl (z) is the Legendre
function. Therefore we ﬁnd
χalt ∝ λ

gx x
ux

√
√
dτ cos 4πφx,τ cos 4πφ0,0

1/T
0

¯
∝ λT g −1 x

ux
2πT x
sinh
T
ux

−gx

P−¯ (z)
g

0

(14)

where z = coth[2πT x/ux ]. Compared to the scattering
from an open boundary13,32 there are two notable diﬀerences: ﬁrst of all there is an additional factor in the form
of the Legendre function, which changes the shape significantly near the scatterer, but quickly approaches unity
for x
ux /πT . Second, there is an additional temper¯
ature dependent factor ∝ T g −1 which is in agreement
with the renormalization behavior predicted in Eq. (8).
As can be seen by the ﬁt in Fig. 1, the behavior (14)
describes the numerical data perfectly, where x is taken
to be the position from an eﬀective scattering center. We

4
χalt
uℓ = 0.98ur
uℓ = ur
uℓ = 1.02ur

0.01

0
PSfrag replacements

0

10

20

30

x 40

FIG. 3: (Color online) The alternating part of the density
oscillations at T = 0.1tr on the interacting side U = 1.8tr ,
where the hopping on the non-interacting side is adjusted so
that uℓ = 0.98ur , uℓ = ur , and uℓ = 1.02ur , respectively.

have conﬁrmed the position dependence for many diﬀerent temperatures and discontinuities. This unambiguously shows that the action in Eq. (1) and the Green’s
function (7) leading up to Eq. (14) are a reliable description of the problem.
An interesting detail in Fig. 1 is the maximum in the
non-interacting region where Uℓ = 0 and gℓ = 1 which is
absent for an open end but now arises due to the Legendre
function with g < 1. Physically this can be understood
¯
as a proximity eﬀect, where the behavior in a range on
the non-interacting side is inﬂuenced by the collective
excitations on the other side.
The temperature dependent amplitudes of the ﬁts to
the position-dependent part in Eq. (14) directly give the
backscattering R(T ) in Fig. 2, where we have used the
amplitude from open ends corresponding to R = 1 as
the normalization, which is known exactly.35,36 The data
conﬁrm the predicted power-law behavior at low temperatures. Note that the independent ﬁts on both sides
give roughly the same R(T ). For the larger jump in
the interaction U = 1.8tr there are deviations at higher
temperatures coming from higher order operators (left
panel). Below a characteristic temperature TK we expect the power law to break down as the stable ﬁxed
point R(0) = 1 is approached, but this energy scale could
not be reached in the simulations.
In order to ﬁnd a conducting scenario in the low temperature limit, it is interesting to analyze the case of
equal velocities uℓ = ur with strong discontinuities in
both hopping and interaction in more detail. As shown
in Fig. 1 density oscillations are still observed in this
case, but we are interested only in the contribution of
the leading relevant operator in Eq. (3), which will grow
while the temperature is lowered and must change sign
as a function of velocity at the conducting ﬁxed point.
In Fig. 3 we show the alternating part of the oscillations for a jump from Uℓ = 0 to Ur = 1.8tr , where the
hopping on the left has been adjusted to three diﬀerent

cases: uℓ = 0.98ur , uℓ = ur , and uℓ = 1.02ur . It is
quite apparent that a sign change takes place exactly at
uℓ = ur , which means that the relevant backscattering
vanishes. The remaining oscillations visible in Fig. 1 for
uℓ = ur are caused by higher order local operators in
Eq. (3).√ In particular, the next-leading terms are given
√
¯
by ∂x ei 4πφ and ei 16πφ with scaling dimension g + 1
and 4¯, respectively, which are irrelevant unless 4¯ < 1.
g
g
Away from half ﬁlling the marginal operator ∂x φx=0 will
also be present,12 but does not aﬀect the scattering to
ﬁrst order. The velocity matching rule for a conducting ﬁxed point is surprisingly simple, considering that it
is not linked to any special symmetry in this scenario.
Perfect conduction in quantum wires with impurities has
been described before in cases where a renormalization
to the periodic boundary condition ﬁxed point occurs.7
This, however, is not the case here. We ﬁnd a novel conducting ﬁxed point described by Eq. (7) which does not
correspond to a standard boundary conformally invariant
theory.
To conclude, we have studied the intrinsic backscattering present when connecting a quantum wire to a
lead. All inhomogeneities and impurities in the junction are in general relevant for repulsive Luttinger liquids, leading to a conductance which scales as G(T ) ∝
2¯−2
g
e2
], with an unusual power-law exponent
h [1 − (T /TK )
1
given by g = 2( gℓ + g1 )−1 in terms of the interaction pa¯
r
rameters on both sides of the junction gℓ , gr . We have
shown that it is possible to achieve a perfect connection,
i.e. an absence of relevant backscattering, in inhomogeneous wires by tuning other parameters such as the velocity in the lead and in the wire. The general Green’s
function was calculated in the presence of an abrupt jump
along the wire based on an inhomogeneous Luttinger liquid action, which was in excellent agreement with numerical simulations. The results suggest that the observation
of Friedel oscillations in the density along the wire, which
can be attempted by scanning probe microscopy, can be
used to analyze local scattering centers. A systematic
study of inhomogeneities and thus an experimental test of
our results seems feasible in semi-conductor heterostructures. One could start from a device where a quantum
wire adiabatically broadens into a two-dimensional electron gas.4 By applying a sharp potential barrier at the
interface a local backscattering center could then be realized allowing it to continuously tune the backscattering
amplitude λ.

Acknowledgments

We are thankful for discussions with J. Folk. This work
was supported by the DFG via the SFB/Transregio 49,
NSERC, CIfAR, and the MAINZ (MATCOR) graduate
school of excellence.

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Electronic address: sedlmayr@physik.uni-kl.de
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