arXiv:0804.4108v2 [cond-mat.str-el] 14 Apr 2009

Coupling-geometry-induced temperature scales in
the conductance of Luttinger liquid wires
P. W¨chter1 , V. Meden2 , K. Sch¨nhammer1
a
o
1

Institut f¨r Theoretische Physik, Universit¨t G¨ttingen, D-37077 G¨ttingen,
u
a o
o
Germany
2
Institut f¨r Theoretische Physik A, RWTH Aachen University and JARA u
Fundamentals of Future Information Technology, D-52056 Aachen, Germany
E-mail: waechter@theorie.physik.uni-goettingen.de
Abstract. We study electronic transport through a one-dimensional, finite-length
quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position
via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (twodimensional) leads, thereby bringing theory closer towards systems resembling setups
realized in experiments. In particular, we compute the temperature dependence of the
linear conductance G of a system without bulk impurities. The appearance of new
temperature scales introduced by the lengths of overhanging parts of the leads and
the wire implies a G(T ) which is much more complex than the power-law behavior
described so far for end-coupled wires. Depending on the precise setup the wide
temperature regime of power-law scaling found in the end-coupled case is broken up
in up to five fairly narrow regimes interrupted by extended crossover regions. Our
results can be used to optimize the experimental setups designed for a verification of
Luttinger liquid power-law scaling.

PACS numbers: 71.10.Pm, 73.63.Nm, 71.10.-w

Submitted to: J. Phys.: Condens. Matter

Coupling-geometry-induced temperature scales in the conductance of LL wires

2

1. Introduction
Theoretically, it is well established that the low-energy physics of a wide class of onedimensional (1d) metallic electron systems (quantum wires) with two-particle interaction
is described by the Luttinger Liquid (LL) phenomenology [1]. One of the characterizing
properties of LL physics is the power-law scaling of a variety of physical observables
as function of external parameters. For spin-rotationally invariant and spinless systems
the corresponding exponents can all be expressed in terms of a single parameter, the
LL-parameter K. On the experimental side the situation is less clear. Much effort has
been put into measurements aiming at a verification of LL behavior by observing the
predicted power-law scaling. However, in many of those experiments sources for the
observed behavior other than LL physics cannot be ruled out in a completely satisfying
way [2].
The gap between the status of theory and experiment is partly related to an
insufficient theoretical modeling of the experimentally available setups. In many
experiments on quantum wires realized e.g. by single wall carbon nanotubes or
semiconductor heterostructures, the temperature dependence of the linear conductance
G was measured [3]. A typical experimental system in which the finite-length LL wire
is coupled via tunnel barriers to two quasi two-dimensional, stripe-like (Fermi liquid)
leads is sketched in figure 1. In the preparation the precise position of the contact
regions and its width, as well as the quality of the contacts is still difficult to control.
In contrast to the experimental geometry the theoretical description is mostly done by
considering end-contacted wires, in which the lead electrons tunnel into the end of the
wire and the leads terminate at the contacts. Here we partly bridge the gap between
the simplicity of the theoretical modeling and the complexity of the experimental setup
by considering a microscopic model of a quantum wire with 1d as well as stripe-like
(two-dimensional) leads, which arbitrarily couple to the wire. Undoubtedly, in order
to achieve direct comparison to experimental data true ab-initio simulations of the
specific experimental setup would be desirable. Yet, the available ab-initio methods
do not capture LL physics and thus cannot be used in the present context. Therefore,
microscopic modeling is the method of choice to extend the LL physics originally derived
within effective field-theories. We show that the length scales set by the overhanging
parts of the leads and the wire introduce new energy scales, which strongly affect the
temperature dependence of the conductance. The G(T ) curves become much more
complex than the ones obtained from the simplified modeling.
Throughout this paper we consider the case of spinless fermions. We expect that
our results—with the appropriate changes [2] in the equations relating scaling exponents
to the LL parameter K—directly carry over to the case of spin-1/2 electrons if the
two-particle interaction is long-ranged (unscreened) in real space (that is, if the g1,⊥
term of the g-ology classification [4] can be neglected) and spin-rotationally invariant.
This is generally assumed to be the case in single wall carbon nanotubes [5]. For
semiconductor heterostructures the range of the interaction can strongly depend on the

3

...

...

Coupling-geometry-induced temperature scales in the conductance of LL wires

Figure 1. Sketch of a typical experimental setup. The quantum wire is attached to
higher dimensional leads, which further extend as indicated by the arrows; note the
overhanging parts of the leads and the wire.

screening properties of the surrounding environment (free carriers). In the presence of a
sizable backscattering component (g1,⊥ > 0) the asymptotic low-energy regime is usually
reached via a complex crossover behavior [6, 7, 8]. We expect the same to hold for the
more realistic setups studied here, which would render the behavior of G(T ) even more
complex (and less “universal”) than described in the present work.
The most elementary understanding of the temperature dependence of the linear
conductance of a LL wire can be gained using Fermi’s Golden Rule like arguments.
The local single-particle spectral function ρ of a translationally invariant, infinite LL
scales as ρ ∼ (max{ω, T })αbulk , with ω measured relative to the chemical potential and
αbulk = (K + K −1 − 2)/2 [2]. For a semi-infinite LL and ω, T ≪ vF /x, with x being the
distance from the boundary and vF denoting the Fermi velocity, the power-law scaling
of the now x-dependent spectral weight is instead given by the boundary exponent
αend = K −1 − 1 [9, 10]. Beyond the energy scale vF /x the bulk exponent is recovered.
Fermis Golden Rule then predicts, G(T ) ∼ T αbulk for tunneling into the bulk of a LL
while tunneling into the end leads to G(T ) ∼ T αend . The LL parameter 0 < K < 1 (for
repulsive interactions; K = 1 in the noninteracting case) is model dependent [1, 2]. Only
for a few cases (for an example see below) its exact dependence on the model parameters
such as the strength of the two-particle interaction and the filling of the band is known
analytically. For small interactions parameterized by the amplitude U, K generically
goes as K = 1 − U/Uc + O([U/Uc ]2 ) with the model and parameter dependent scale
Uc . From this Taylor expansion one can infer that αend is linear in U/Uc and for small
interactions dominates over αbulk which is quadratic.
In experimental setups the particles leave the LL wire at a second contact. To
obtain the conductance of this type of system one is tempted to simply take the inverse
of the sum of the two resistances resulting from the tunnel barriers. In the presence of
inelastic scattering processes this is certainly correct. In the absence of such processes,
an approximation used in the present paper, it is however less clear if this leads to
the correct result for the conductance. In [11] and [12] it was shown using scattering
theory that for low-transmittance contacts and at temperatures sufficiently larger than
the scale vF /L, with the Fermi velocity vF and the length L of the interacting wire,
adding the two resistances and taking the invers gives the correct result. We note in
passing that this can not be extended to the case of more than two impurities [12].
The Fermi Golden Rule like considerations on the conductance of a LL wire usually

Coupling-geometry-induced temperature scales in the conductance of LL wires

4

do not take the geometry of the reservoirs into account explicitely but only use their
Fermi liquid properties. For the case of end-contacted wires the above results were
confirmed within a microscopic modeling of the semi-infinite leads (1d noninteracting
tight-binding chain) and the wire (1d tight-binding chain of spinless fermions with
nearest-neighbor interaction) [11, 12]. The linear conductance shows power-law scaling
T αend for temperatures vF /N ≪ T ≪ B, with N the number of lattice sites in the
interacting wire and the B bandwidth. Other approaches to include the end-contacted
leads into the model are based on the so-called local Luttinger liquid picture [13] and
on the concept of radiative boundary conditions [14], where the wire is modeled by an
effective field-theory (bosonization) [2].
We here extend the functional renormalization group (fRG) approach used to study
the linear response transport [11] as well as the finite bias steady-state nonequilibrium
transport [15] through an end-contacted wire (described by a microscopic model) to
investigate the role of the contacts and the leads more thoroughly. The reservoirs
are modeled as two-dimensional tight-binding stripes of variable width (including the
case of 1d leads) and contacted to the 1d interacting wire at arbitrary positions. This
generically leads to overhanging parts of the wire as well as of the leads (see figure
1). We show that they set new energy scales and the temperature range of power-law
scaling of G(T ), vF /N ≪ T ≪ B, breaks up into up to five different regimes of variable
size. This leads to a temperature dependence of G(T ) which is much richer than the
simple scaling T αend , which for sufficiently long wires, holds over several decades in T
within the simplified modeling described above. Our results show that it is difficult to
find clear evidence for LL behavior of G(T ) in the generically used setups.
To fully cover all relevant temperature regimes and treat systems of experimental
length, i. e. in the micrometer range, we have developed an algorithm which allows us
to compute the conductance for systems of up to N = 105 lattice sites. Including the
stripe-like leads requires a substantial extension of the order N algorithm developed for
1d leads [16].
The rest of the paper is organized as follows. In section 2 the microscopic model
used to describe the quantum wire and the leads is introduced. In section 3 and
the appendix it is shown in detail, how to calculate the conductance through the
system using scattering theory. Section 4 is a short survey of the fRG method and
the generalizations necessary to include stripe-like leads are described. In section 5 our
results for the temperature dependence of the conductance are presented and discussed.
We conclude with a summary in section 6.
2. The Microscopic Model
We consider spinless fermions on a finite 1d lattice with nearest-neighbor hopping t and
nearest-neighbor two-body interaction U, coupled to two semi-infinite, stripe-like, and
noninteracting tight-binding leads. For simplicity all lattice constants are assumed to
be equal and chosen to be unity. The system is sketched in figure 2. The Hamiltonian

Coupling-geometry-induced temperature scales in the conductance of LL wires

5

Hwire of the N site quantum wire reads (t ≥ 0)
Hwire = Hkin + Hint
with
Hkin = − t
Hint = U

N −1

c† cj+1 + H.c.
j

j=1

N −1
j=1

c† cj −
j

1
2

c† cj+1 −
j+1

1
2

,

(1)

(†)

where cj denotes the fermionic annihilation (creation) operator in Wannier states at
site j ∈ {1, ..., N}. We here consider the case of half-filling of the leads as well as of the
wire. To assure the latter the density operator nj = c† cj in the last line of (1) is shifted
j
by −1/2. We expect similar results to hold away from half-filling.
The leads are assumed to be noninteracting
a
Hlead = −

†
a
tn,m dn,a dm,a + H.c.

,

n,m∈A

where a stands for L (left) and R (right) and the symbol A = {(nx , ny )|nx ∈ {1, ..., Na,x }
∧ny ∈ {1, ..., ∞}} denotes the set of lattice points of lead a. The extensions of the leads
in the x-direction given by Na,x are allowed to differ. The hopping matrix elements ta m
n,
are assumed to connect nearest neighbors only.
The 1d wire and the stripe-like leads are coupled via hopping terms, described by
N
a
Vj,n c† dn,a + H.c.
j

a
Hcoupl =

.

(2)

j=1 n∈A
a
The matrix elements Vj,n will be specified later.
The full Hamiltonian H of the system reads
L
R
L
R
H = Hwire + Hlead + Hlead + Hcoupl + Hcoupl

.

(3)

The model for an infinite isolated wire Hwire with interaction U between all sites can be
solved exactly using the Bethe ansatz [17]. The system is a LL for all fillings and
interactions except for half-filling at |U| > 2t, where a charge density wave forms
(U > 2t) or phase separation occurs (U < −2t). From the Bethe ansatz solution,
the LL parameter K can be computed and reads for half filling and |U| ≤ 2t [18]
K −1 =

U
2
arcos −
π
2t

.

(4)

3. Single particle scattering
3.1. General relations
The fRG procedure in the approximation scheme described in section 4 yields a frequency
independent self-energy of the interacting wire at the end of the fRG flow. This

Coupling-geometry-induced temperature scales in the conductance of LL wires

lead
tL
x

t R lead
x

...

tR
y

U

t

tL
y

6

...

...

...

...

...

...

...

...

...

...

wire

Figure 2. (Color online) Sketch of the 1d wire connected to stripe-like leads; the
crosses depict the sites of the leads, the bars those of the wire; note that the coupling
between the wire and the leads is located at arbitrary positions leading to overhanging
parts.

can be interpreted as a single-particle (scattering) potential and incorporated into a
renormalized single-particle Hamiltonian. As described in detail in [11] for end-coupled
wires with 1d leads one can compute the conductance from single-particle scattering
theory [19] using a generalized Landauer-B¨ ttiker approach [20].
u
We therefore consider the single-particle Hamiltonian
˜
h = hwire + hL + hR + hL + hR
lead
lead
coupl
coupl

.

(5)

˜
where hwire is the effective single-particle Hamiltonian for the wire at the end of the fRG
flow (which includes the generated scattering potential) and the h’s are single-particle
versions of the terms introduced in the last section. For the states in the single-particle
Hilbert space we use standard Dirac notation. In the scattering problem to be solved, the
first three terms of h are considered to be the unperturbed part h0 , while the couplings
to the leads hL + hR are considered as the perturbation h1 . We closely follow the
coupl
coupl
steps for the corresponding scattering problem with end-contacted, 1d leads [11].
The main difference of working with higher dimensional leads is the occurrence of
transverse scattering channels. The orthonormal single-particle eigenstates |k, l, a of
the isolated semi-infinite leads are standing waves with the wavenumber k ∈ [0, π] in
the y-direction and l = 1, ..., Nx,a labeling the transverse modes
2
sin(kny )
π

n, a|k, l, a =
×

2
Nx,a + 1

= ny |k nx |l

a

sin

lπ
nx
Nx,a + 1

= a n|k

a

.

The corresponding eigenenergies are
ǫa (k, l) = − 2ta cos(k) − 2ta cos
y
x

lπ
Nx,a + 1

= ǫa (k) + ǫa,l = ǫa (k) ,
with the nearest-neighbor hopping matrix elements ta and ta .
x
y

(6)

Coupling-geometry-induced temperature scales in the conductance of LL wires

7

We now consider the scattering of a particle from the left to the right lead (or vice
versa) coupled via h1 to the wire. The outgoing scattering states |k, l, a+ = |k+ a
follow from the Lippmann-Schwinger equation [21]
|k+

a

a

= |k

+ G(ǫa (k) + i0)h1 |k

a

,

(7)

where G(z) = (z − h)−1 is the resolvent of the full single-particle Hamiltonian.
If we define a as the “complement” of a this yields
¯
a
¯

m|k, +

a

a
¯

=

a
m|G(ǫa (k) + i0)|j Vj,na n|k

a

(8)

j,n

Using G = G0 + G0 h1 G with G0 (z) = (z − h0 )−1 we obtain
a
¯

m|k, +

a

a
¯

=
j,j ′
m′ ,n

a
¯ a
¯
m|G0 |m′ a Vm′ ,j ′ j ′ |G|j Vj,na n|k a .

(9)

Only elastic scattering k, l, a → k ′ , l′ , a occurs with ǫa (k, l) = ǫa (k ′ , l′ ). This is enforced
¯
¯
a
¯
′
a
¯
by the free resolvent matrix elements m |G0 (ǫa (k, l) + i0)|m . It is easier to see this in
a mixed representation, local in the y-direction and using the transverse channel number
l′ as the additional quantum number
a
¯

my , l′ |k, +
a
¯

j,j ′
m′ ,n

a

=

a
¯ a
¯
my , l′ |G0 |m′ a Vm′ ,j ′ j ′ |G|j Vj,na n|k

= −

2π
j,j ′
m′ ,n

a
¯

¯
v a (k ′ )

a

¯ a
¯
a
k ′ |m′ a Vm′ ,j ′ j ′ |G|j Vj,n
′

×

a

n|k

a

eik my
√
.
2π

(10)

The second equality holds for my larger than the m′y involved in the coupling of the lead
and the quantum wire. To derive this relation we used the result for the resolvent of a
semi-infinite 1d chain (n = 1, 2, ..., ∞) with nearest-neighbor hopping t
−i ik(ǫ)|m−n|
1d
m|G0 (ǫ + i0)|n =
e
− eik(ǫ)(m+n)
v(ǫ)
2 sin[k(ǫ)n] ik(ǫ)m
e
,
(11)
= −
v[k(ǫ)]
where the second line holds for m ≥ n. For |ǫ| < 2t the wavenumber k(ǫ) ∈ [0, π] is
the solution of −2t cos[k(ǫ)] = ǫ and the velocity v(k) is given by 2t sin(k). In [11] only
the result for n = 1 was presented. For |ǫ| > 2t the resolvent matrix element decays
exponentially with the distance |m − n| and are thus “closed” channels in (10). This
¯
happens if |ǫa (k, l) − ǫa,l′ | < 2|ta |, and k ′ ∈ (0, π) is determined by energy conservation
¯
y
¯
¯
ǫa (k ′ ) = ǫa (k) + ǫa,l − ǫa,l′ . The velocity in (10) is given by v a (k ′ ) = 2ta sin(k ′ ). The
¯
¯
y
prefactor of the outgoing plane wave in (10) gives the transmission probability for the

Coupling-geometry-induced temperature scales in the conductance of LL wires

8

scattering k, l, a → k ′ , l′ , a
¯

|t|k a →|k′ a |2 = |t(ǫa (k), l, l′ )|2
¯
2π a ′
a
¯
¯ a
¯
k |m a Vm,j ′ j ′ |G|j Vj,na n|k
=
¯
v a (k ′ )
j,j ′

a

2

.

(12)

m,n

Taking the sum over l compatible with ǫa (k) = ǫ and over the l′ values of the
corresponding “open” channels we define [20]
Ta (ǫ) =

{l,l′ }

¯
v a (ǫ, l′ )
|t(ǫ, l, l′ )|2 .
v a (ǫ, l)

(13)

The unitarity of the S-matrix guarantees Ta (ǫ) = Ta (ǫ) = T (ǫ). This quantity
¯
determines the conductance G(T ) in the generalized Landauer-B¨ ttiker formula [20]
u
G(T ) =

e2
h

dǫ −

∂f
∂ǫ

T (ǫ) ,

(14)

where f denotes the Fermi function and e2 /h is the unit of the quantized conductance.
˜
Note that for our interacting problem the flowing self-energy and thus hwire as well as
the effective transmission probability T (ǫ) become T and N dependent (for details, see
below).
The Landauer-B¨ ttiker formula (14) holds in the abscence of inelastic processes
u
due to the two-particle interaction (that is for vanishing imaginary part of the selfenergy) [19]. Such processes do not appear in our approximate treatment of the
interaction and using (14) does thus not present an additional approximation.
For the calculation of the transmission probability one has to compute the matrix
elements j ′ |G|j of the full resolvent. This can be reduced to the inversion of a N × Nmatrix, using Feshbach projection [11]
P G(z)P =

zP − P hP − P hQ(zQ − QhQ)−1 QhP

−1

,

(15)

where P and Q = 1 − P are projection operators. Here we use as P the projection
operator Pwire = N |j j| onto the wire and Q the one on the leads.
j=1
The special geometry (mimicking a generic experimental setup) used in section
5 is shown in figure 3. Each site of the wire lying on top of a site of the lead (as
shown in the figure) is coupled to this site via a hopping term tca . The length scales
n
appearing in the system are also defined in the figure. Note that the overall length
L
R
L
R
N = Nmid + Ncon + Ncon + Nover + Nover .
The projected inverse Green function takes the form shown in figure A1 of the
appendix. We need the tridiagonal elements of this Green function to calculate the
right-hand side of the flow equations (18) and (19), see section 4. An algorithm to
compute these elements is presented in the appendix. Furthermore, to calculate the
transmission amplitude (12) we need the j, j ′ -elements of the Green function with
j ′ ∈ {N1 + 1, .., N2} and j ∈ {N3 + 1, .., N4 }, i. e. the elements connecting the left

Coupling-geometry-induced temperature scales in the conductance of LL wires
NR
con

NL
con

NR
lead

NL
lead

...

...

L

...

...

NR
over
...

...

...

...

...

...

N mid

N over
...

9

Figure 3. (Color online) Sketch of the system considered. The overlapping sites of
wire and leads are individually coupled via a hopping term. The length scales are
indicated.

and right contact region. An algorithm to efficiently compute these elements is also
briefly sketched in the appendix [22].
For simplicity, we assume the hoppings in the leads and in the wire to be equal and
define this as our unit of energy, i. e. ta = ta = t = 1.
x
y
3.2. One-dimensional leads
The appearance of new temperature scales is most easily seen in the case of oneL
R
a
dimensional leads, i.e. Ncon = Ncon = 1. For N ≫ Nlead and leads weakly coupled
to the wire, i.e. (tca )2 ≪ 1 the transmission probability T (ǫ) consists of N narrow
resonances near the eigenvalues ǫα of the wire without contacts. The integrated weight
wα of the Lorentzian resonance near ǫα is [11]
1
1
+ R
wα ≈ 4π
L
∆α ∆α

−1

.

(16)

Here the partial widths ∆a are given by
α
2
∆a = (tca )2
α
(N + 1) sin[k(ǫα )]
a
× sin2 [N a k(ǫα )] sin2 [(Nlead + 1)k(ǫα )] ,
with N a the site of the wire to which lead a couples. In the temperature regime
vF /N ≪ T ≪ B the Landauer B¨ ttiker formula simplifies to
u
e2
G(T ) ≈
h

N

α=1

wα −

∂f
∂ǫ

(17)
ǫ=ǫα

We return to this expression when discussing our numerical results.
4. The Functional Renormalization Group
In this section we present a short outline of the fRG at finite temperature T , which we
use to treat the interacting system. General aspects of the fRG can be found in [23], [24],
and [25], while its application to inhomogeneous LLs is described in [8], [11] and [16].
By comparison to results for small systems obtained by density-matrix renormalization

Coupling-geometry-induced temperature scales in the conductance of LL wires

10

group and to exact results from bosonization and Bethe ansatz for the asymptotic
behavior, it was shown that the fRG in the truncation scheme used here captures the
relevant physics not only in the asymptotic low-energy regime but also on finite energy
scales [11, 16]. Towards the end of this section, we describe the extensions of the method
necessary when dealing with stripe-like leads.
Starting point of our implementation of the fRG is the introduction of an infrared
energy cutoff Λ in the Matsubara frequencies of the noninteracting propagator. This
leads to a Λ-dependence of the generating functional of the one-particle irreducible Green
functions (vertex functions). Taking the derivative of this functional with respect to Λ
leads to an exact, but infinite, hierarchy of coupled first order differential equations for
the vertex functions. We truncate this hierarchy by neglecting the m-particle vertices
with m ≥ 3. In order to reduce the numerical effort, we project the 2-particle vertex
function onto the Fermi points and parameterize it by a nearest-neighbor interaction
of amplitude U Λ [11, 16]. Finally, we neglect the feedback of the 1-particle vertex (the
self-energy) onto the flow of the 2-particle vertex function, that is the flowing effective
interaction [16]. The resulting set of equations is integrated from the initial cutoff
Λ = ∞ down to Λ = 0, at which the original, cutoff-free problem is recovered. For endcontacted, interacting wires and weak to intermediate interactions this approximate
approach was shown to provide reliable results by comparison with exact analytical and
numerical results [8, 11, 16].
Within our approximation the self-energy Σ becomes real, frequency-independent
and tridiagonal in position space. It acquires a dependence on the temperature and size
of the interacting wire and can thus be interpreted as a (T - and N-dependent) singleparticle scattering potential. This leads to an effective single-particle Hamiltonian with
renormalized onsite energies and hopping matrix elements.
As mentioned in the introduction to leading order the scaling exponent of the
tunneling conductance into an end-coupled LL is linear in U. In contrast, tunneling
into a translationally invariant (bulk) LL is characterized by an exponent which for
small U is quadratic [2]. First order scaling exponents were also found in other setups
of inhomogeneous LLs (e.g. LLs with a single impurity [9, 11, 16] and Y-junctions of
LLs; see e.g. Ref. [26] ). In all cases in which predictions for the scaling exponents of
inhomogeneous LLs from alternative approaches (e.g. effective field-theories) exist those
were reproduced to linear order using our truncated fRG method [11, 16, 26]. However,
our truncation procedure does not capture bulk LL physics, as terms of order U 2 are
only partly kept in the flow equation for the self-energy. In particular, the terms of order
U 2 ln (ω) obtained in a perturbative calculation of the self-energy of a translationally
invariant LL (evaluated at the Fermi momentum kF ), leading to the bulk LL power
laws when resummed by an RG method, are neglected in our approach. We thus expect
that, in situations which are dominated by bulk LL physics (for examples see below)
the scaling exponent vanishes within our approximation.

Coupling-geometry-induced temperature scales in the conductance of LL wires

11

The set of flow equations to be solved reads [16]
d Λ
Σ
dΛ j,j

= −

1
π

r=±1

Λ
Λ
Uj,j+r Re Gj+r,j+r (iΛ) ,

d Λ
1 Λ
Λ
Σj,j±1 = Uj,j±1 Re Gj,j±1(iΛ) ,
dΛ
π
U
UΛ
=
,
U
2+Λ2
1 + Λ − √4+Λ2 2π

(18)
(19)
(20)

where we specialized the flow equation for the effective interaction to the case of halffilling. The implementation of finite temperatures requires at each step of the flow the
replacement of the flow parameter Λ by the nearest Matsubara frequency [8].
In order to compute the right-hand side of the flow equations for ΣΛ one has to
compute the tridiagonal elements of the matrix G Λ (iΛ) inverting the given N × Nmatrix G Λ (iΛ)−1 . To cover all relevant energy scales and to treat systems of lengths in
the micrometer range we need an algorithm, which allows us to calculate these matrix
elements more efficiently than the standard order N 3 methods [27]. In the case of 1d
leads, G Λ (iΛ)−1 itself is tridiagonal and the tridiagonal elements can be computed in
order N time [16, 28]. For stripe-like leads G Λ (iΛ)−1 consists of tridiagonal blocks,
connected on the sub- and superdiagonal to full blocks, one for each contact region.
The explicit structure is depicted in figure A1 of the appendix. There we present an
algorithm, in which the problem of inverting the full N × N-matrix is split up into the
multiple inversion of each isolated block. Thus, we can readily use the order N algorithm
for the tridiagonal parts and some standard algorithm [27] for the full matrices. The
bottleneck of the method is therefore the width of the stripe-like leads, which determines
the size of the full matrices.
5. Results
5.1. One-dimensional leads arbitrarily coupled to a noninteracting wire
We first consider the conductance G(T ) through a clean quantum wire with two semiinfinite 1d leads (stripe-like lead with a width of one lattice site) coupled at arbitrary
position of the leads and the wire. This introduces four additional length scales, namely
L/R
L/R
Nlead and Nover , apart from the overall length N of the wire (see figure 3). They strongly
effect the temperature dependence of the conductance.
First we shortly comment on even-odd-effects arising from the fact that the coupling
sites in the wire as well as those in the leads can each be even or odd and so can be the
overall length of the chain. Within the notation introduced in section 3.1, the coupling
L
R
L
R
sites in the wire are Nover + 1 and Nover + 1, those in the leads Nlead + 1 and Nlead + 1.
For T → 0, the derivative of the Fermi function in (14) becomes a δ-function and the
integral is given by the effective transmission evaluated at the Fermi energy. This shows
even-odd-effects. For T → 0 the conductance reaches a nonvanishing value only if the
coupling sites are all odd. For even sites the local spectral density at the Fermi energy

Coupling-geometry-induced temperature scales in the conductance of LL wires
20

286

2

G/(e /h)

0.2

12

4

4

10 +1

10 +1

0.1

0.0 -4
10

10

-3

10

-2

T

10

-1

10

0

Figure 4. (Color online) Conductance G as function of temperature T of a
noninteracting wire of length N = 104 + 1 and coupling tcR = tcL = 0.25. Solid
L/R
line: end-contacted wire, i.e. Nover = 0 (setup shown as the right inset) obtained from
L
R
(14). Circles: Nover = 20, Nover = 286 (setup shown as the left inset) obtained from
(14). Dashed line: the same parameters as for the circles but using the approximate
analytical result (17). The vertical lines indicate the crossover scales resulting from
the overhanging parts.

vanishes, leading to G(T = 0) = 0. If, for odd coupling sites and tcL = tcR , the overall
length N is taken to be odd, G(T → 0) → e2 /h holds. For temperatures of the order of
vF /N and higher these even-odd effects vanish. We mainly consider systems with odd
N and coupling at odd sites. This implies G(T = 0) = e2 /h for tcL = tcR , which also
holds for the interacting system.
In figure 4, the conductance G(T ) of a noninteracting wire coupled at the ends is
compared to the conductance of a system coupled in the bulk. The leads are weakly
connected with tcR = tcL = 0.25 and terminate at the contacts (no overhanging parts,
L
R
L
R
Nlead = Nlead = 0). For the bulk-coupled case we chose Nover = 20, Nover = 286 (circles).
The conductance of the end-coupled wire (solid line) starts at e2 /h for T = 0, crosses
over into a plateau at a scale T ≈ vF /N (left vertical dashed line) and finally falls off
as T −1 for T ≫ B (band effect). For the bulk-coupled case the plateau region breaks
up into three clearly distinguishable regimes with the two additional crossover scales
R
L
vF /Nover and vF /Nover indicated by the other two vertical dashed lines. In the first
R
region with vF /N ≪ T ≪ vF /Nover a plateau with the same conductance as for the
end-coupled chain appears. At the two additional scales the conductance crosses over to
new plateaus with lower conductance values and finally falls off for T ≫ B. The dashed
line shows the conductance of the bulk-coupled system using the approximate analytical
expression (17) for comparison. For temperatures in the range vF /N ≪ T ≪ B we find
excellent agreement with the exact result obtained from (14).
Note that the overhanging parts of the wire and of the leads enter symmetrically in
R
L
R
L
(17). Thus, choosing Nlead = 20, Nlead = 286 and Nover = Nover = 0, gives an identical
conductance as that shown by the circles and dashed line in figure 4. Obviously this
will no longer hold in the presence of interactions in the wire.

Coupling-geometry-induced temperature scales in the conductance of LL wires
2086

0.2

86

5

6

10

G/(e /h)

13

2*10

7

2

10 +1

0.1

0.0
10

-7

10

-5

T

10

-3

10

-1

Figure 5. (Color online) Conductance G of a noninteracting wire of length N = 107 +1
L/R
L/R
and coupling tcR = tcL = 0.25. Solid line: end-coupled wire, i.e. Nlead = Nover = 0
L
5
R
obtained from the analytical result (17). Dashed line: Nover = 10 , Nover = 2 ·
R
R
106 , Nlead = 2086 and Nlead = 86 (setup shown as the inset) obtained from the
analytical result (17). The vertical lines indicate the crossover scales.
L/R

L/R

All four new temperature scales, i.e. vF /Nlead and vF /Nover can be resolved using
the analytical expression (17) for a very large system (that is for a very small lower
bound vF /N). This is shown as the dashed line in figure 5. The solid line shows the
conductance for a system coupled at the ends for comparison.
After identifying the relevance of the additional energy scales due to the overhanging
parts of the wire and the leads for noninteracting wires we turn to the interacting case.
5.2. One-dimensional leads arbitrarily coupled to an interacting wire
As described in section 4 the approximate fRG flow leads to a nontrivial static selfenergy for the interacting wire which is tridiagonal in the site indices. At half-filling
and for tunnel barriers as used here the diagonal part vanishes and the effective potential
is given by a nonlocal term, the modulated hopping Σj,j+1. If the leads are coupled to
the ends of the wire (with and without overhanging parts of the leads), the two contact
˜
barriers are the source of (nonlocal) potentials which oscillate as (−1)j (for half-filling)
and decay as 1/˜ where ˜ denotes the distance from the ends (at j = 1 and j = N,
j,
j
respectively). The prefactor of the decay is universal in the sense that it does not depend
on the strength of the tunnel barriers tcR and tcL and is the same even for tcR = tcL = 0,
that is a decoupled wire with open boundaries. The power-law decay is cut off at a scale
jT ∝ 1/T set by the temperature beyond which Σj,j+1 decays exponentially. Scattering
off this effective potential leads to the power-law scaling of the local spectral weight and
the conductance discussed in the introduction with the exponents determined by the
universal prefactor [7, 11].
A typical off-diagonal component of the self-energy Σj,j+1 of a bulk-coupled
˜
interacting wire is shown in figure 6. From the ends of the wire the (−1)j /˜ oscillations
j
with universal prefactor emerge and become exponentially damped at a distance
jT ∝ 1/T (indicated by the vertical dashed line). For clarity only one end is shown in

Coupling-geometry-induced temperature scales in the conductance of LL wires
-0.154

1086

Σj,j+1

14

286
4

10 +1

-0.158

10
0
10 0
87
12
00

j

80
0

60
0

40
0

20
0

0

-0.162

Figure 6. Off-diagonal self-energy at the end of the fRG flow for an interacting
wire with U = 0.5 and couplings tcR = tcL = 0.25 at T = 4 · 10−3 . The wire of
L
R
length N = 104 + 1 is coupled to the end of the leads (i.e. Nlead = Nlead = 0) with
L
R
Nover = 1086, Nover = 286. The part shown in the main plot is indicated by the dashed
box in the inset. Long-ranged oscillations emerge from the boundaries and from the
coupling site. The scale jT beyond which the oscillations are damped exponentially is
indicated by the vertical dashed lines.

the main plot. The coupling to the lead (at site 1087) introduces similar oscillations but
with a nonuniversal prefactor. In contrast to the situation found close to an end-contact
these oscillations do not lead to a power-law suppression of the spectral weight close to
the coupling site resulting from the local inhomogeneity. Furthermore, as discussed
in section 4 our truncated fRG procedure does not capture bulk LL power laws with
exponents of order U 2 . Thus, within our approximation the spectral weight close to the
coupling site in the bulk shows no power-law behavior at all.
Figures 7 to 9 show the conductance for several parameter sets and temperatures
larger than vF /N, i.e. temperatures in the regime in which in the noninteracting case the
plateaus emerge. The length of the wire is N = 104 +1 sites which corresponds to roughly
a micrometer taking typical lattice constants. The vertical dashed lines terminating at
the different curves indicate the new energy scales for the corresponding parameter set.
The curves for the corresponding end-coupled systems without any overhanging parts
are included in the figures for comparison (dashed lines). The effective exponent of
G(T ), computed as the logarithmic derivative of the conductance, is shown as the right
inset in each figure. The left inset shows the setups of interest.
L
R
In figure 7 we focus on wires with Nover = Nover = 0. For all three temperature
regimes,
L/R

vF /N ≪ T ≪ vF /max(Nlead ),
L/R

L/R

vF /max(Nlead ) ≪ T ≪ vF /min(Nlead ) and
L/R

vF /min(Nlead ) ≪ T ≪ B,

L
the usual power law G(T ) ∼ T αend is found. Note that for the system with Nlead = 20,
R
R
Nlead = 1086 (dotted line) the first temperature regime vF /N ≪ T ≪ vF /Nlead is too
R
small for a plateau in the effective exponent to develop. However, by reducing Nlead one
can extend this regime, such that the effective exponent tends towards αend .
Already this simple case with no overhanging parts of the wire (that is the particles

Coupling-geometry-induced temperature scales in the conductance of LL wires

15

0.2

110

exponent

0.2

0.1

0
-0.2
-0.4
-4

10

T

-2

0

10

10

2

G/(e /h)

20 1086
4

10 +1

0.03
10

-4

10

-3

10

-2

T

10

-1

10

0

Figure 7. (Color online) Main plot: Conductance G of an interacting wire with
U = 0.5 of length N = 104 + 1 as function of the temperature T . The coupling to the
L
R
leads is located at the end of the wire (Nover = Nover = 0) and has a small amplitude
L/R
L
R
cL
cR
t = t = 0.25. Dashed line: Nlead = 0. Dashed-dotted line: Nlead = 0, Nlead = 110.
L
R
Dotted line: Nlead = 20, Nlead = 1086. The vertical dashed lines terminating at
the different curves indicate the corresponding crossover scales. Left inset: Setups
studied. Right inset: Effective exponents. The solid horizontal line indicates the fRG
approximation for αend as obtained in [11].

tunnel into the end of the interacting wire) demonstrates our central result. The power
law, which for end-contacted wires of experimental length without any overhanging parts
of the leads was found to hold over several decades, is split up into significantly smaller
temperature regimes (of roughly one decade) with power-law scaling interrupted by
extended crossover regimes. Thus the three plateaus of different conductance obtained in
the noninteracting case develop into power laws with the same exponent in the presence
of two-particle interactions. For each temperature regime the appearance of the latter
can be traced back to the spatial dependence of the self-energy Σj,j+1. Over the whole
temperature range vF /N ≪ T ≪ B, when moving from the left to the right contact the
fermions have to pass the long ranged oscillations with universal amplitude originating
from the ends of the wires. For vF /N ≪ T these are well separated, such that the
simple arguments presented previously hold, implying that G(T ) ∼ T αend .
The role of the length scales resulting from overhanging parts is further exemplified
L
R
R
L
in figure 8 for systems with Nlead = Nlead = 0 and Nover = 0 but varying Nover . Beginning
L
with a purely end-coupled system (Nover = 0; dashed line) the left coupling site is moved
L
into the bulk of the wire. For temperatures T ≪ vF /Nover the particles pass both
oscillatory potentials originating from the ends of the wire and the situation of fermions
tunneling in and out of the two ends is effectively recovered. This leads to power-law
L
scaling of the conductance with exponent αend . With increasing Nover the crossover scale
L
vF /Nover decreases and the effective exponent reaches αend inside a smaller and smaller
L
L
temperature regime (see the inset). For the system with the largest Nover (Nover = 1111;
L
dashed-dotted line in figure 8), the interval [vF /N, vF /Nover ] becomes too small for the
logarithmic derivative to reach its asymptotic value (see the inset). For temperatures

Coupling-geometry-induced temperature scales in the conductance of LL wires

16

L
T
vF /Nover the oscillations originating at the left end of the wire are exponentially
damped at the position of the left contact and to the right of it. On their direct
path from the left to the right contact the fermions thus only experience the potential
originating from the right end of the wire. As discussed in the introduction one such
potential is sufficient for the spectral weight close to its origin to scale as ω αend . One
is thus tempted to conclude that G(T ) follows a power law with exponent αend also for
L
vF /Nover ≪ T ≪ B. Figure 8 shows that this is not the case. Instead, the effective
exponent seems to approach an asymptotic value which is significantly smaller than
αend (see the dashed-dotted line). The naive expectation ignores the complexity of the
scattering problem. For example, the fermions can also move to the left of the left
contact before reaching the right lead. Below we further characterize the behavior in
this regime.

0.2
exponent

0.2

-0.4

4

0.1

10 +1

-4

-3

10

10

-2

-1

10

10

0

10

T

2

G/(e /h)

0
-0.2

0.04
10

-4

10

-3

10

-2

T

10

-1

10

0

L
R
R
Figure 8. (Color online) As in figure 7 but with Nlead = Nlead = 0, Nover = 0 and
L
L
L
for different Nover . Dashed line: Nover = 0. Dotted line: Nover = 11. Dashed-doubleL
L
dotted line: Nover = 111. Dashed-dotted line: Nover = 1111.

L
R
As a third example in figure 9 we show results obtained for Nlead = Nlead = 0
(no overhanging parts of the leads) and different positions of the coupling to the wire
L/R
L/R
Nover . In the regime vF /N ≪ T ≪ vF /max(Nover ), the conductance follows the power
law G(T ) ∼ T αend since the oscillations of Σj,j+1 from the two boundaries reach beyond
R
the positions of the contacts. For Nover = 1086 (dotted line) this regime becomes too
L/R
L/R
small for the power law to develop. For vF /max(Nover ) ≪ T ≪ vF /min(Nover ), the
oscillations from the boundary, originating at the site of the longer overhanging part,
L/R
are already cut off as the exponential damping sets in at jT ∝ 1/T , i.e. jT ≪ max(Nover ).
In this regime the curves seem to show power-law behavior with an exponent smaller
than αend similar to the behavior discussed in connection with figure 8. In the third
L/R
regime vF /min(Nover ) ≪ T ≪ B, the oscillations from the boundaries do not reach
the region between the left and right contact (due to the exponential damping) and
no power-law behavior can be found (see the conductance plateau of the dotted and
dashed-dotted line in the main part of figure 9). This is in accordance with the fact
that the oscillations of Σj,j+1 originating from the bulk contacts do not have the same

Coupling-geometry-induced temperature scales in the conductance of LL wires

17

effect on the spectral weight as the ones coming from the boundaries of the wire (no
power-law suppression). We expect the conductance in this temperature regime to show
power-law scaling with the bulk LL exponent αbulk ∼ U 2 not captured by our truncated
fRG procedure.
0.2

286
4

exponent

0.2

20

0

-0.2
10 +1
-0.4
1086 10
20

0.1

-4

-2

0

10

10

2

G/(e /h)

T

0.04
10

-4

10

-3

10

-2

T

10

-1

10

0

L
R
Figure 9. (Color online) The same as in figure 7 but with Nlead = Nlead = 0
L/R
L
and coupling into the bulk. Dashed line: Nover = 0. Dashed-dotted line: Nover =
R
L
R
20, Nover = 286 Dotted line: Nover = 20, Nover = 1086.

To obtain a better understanding of the apparent second exponent smaller than
αend found above, figure 10 shows the logarithmic derivative of G(T ) in the temperature
L
L
R
regime vF /maxNover ≪ T ≪ B for a system with Nover = 1110 and Nover = 0. Curves
for different interaction strengths U are plotted. For U ≤ 0.5 (double-dashed-dotted line
and below) one is tempted to conclude that G scales like T αend /2 . The lower horizontal
line shows αend /2 for U = 0.5 numerically calculated within the fRG-implementation
used here [11]. However, for U > 0.5 the plateau of the logarithmic derivative becomes
strongly tilted, such that no definite statement about any power law can be made
(compare the solid horizontal line showing αend /2 for U = 1 with the dashed curve).
Attempts to treat the underlying scattering problem, e.g. with phase-averaging [12]
or methods developed for Y-junctions [26] have not yet led to a consistent picture.
Furthermore, in this temperature regime G(T ) might be altered if bulk LL behavior is
properly introduced.
The conductance through “mixed” systems, that is those with overhanging parts
of the quantum wire as well as overhanging parts of the leads, shows an even richer
temperature dependence. As in the noninteracting case (see figure 5) up to five different
temperature regimes and the corresponding crossover regimes might appear in
[vF /N, B]. The behavior sufficiently away from the characteristic energy scales set by
the lengths of the overhanging parts can be understood as explained above.
These considerations show that for a generic experimental system with four
overhanging parts all having different lengths and not all of them being several orders
of magnitude smaller than the total length of the wire it will be very difficult (if not
impossible) to observe clear indications of a single power law in G(T ). We next show

Coupling-geometry-induced temperature scales in the conductance of LL wires

18

that considering stripe-like leads, as being realized in the experiments, does not improve
this situation.
5.3. Stripe-like leads arbitrarily coupled to a quantum wire
We here extend our analysis to stripe-like leads, as realized in many experiments. For
this setup the parameter space grows linearly with increasing the number of coupling
c
sites, e.g. for five sites at each contact, we can freely choose ten couplings tjL/R . To
c
c
prevent a proliferation of parameters we therefore focus on equal couplings, tjL/R = t L/R .
Figure 11 shows the conductance of a noninteracting quantum wire with Nmid =
L/R
R
L
R
L
963, Nover = Nover = 18, Nlead = Nlead = 0 and for varying width of the leads Ncon .
0.6

exponent

1110
0.4

4

10 +1

0.2

0.0
10

-4

10

-3

10

-2

T

10

-1

10

0

L
Figure 10. (Color online) System as in figure 8 with Nover = 1110. Effective exponents
L
in the regime vF /Nover ≪ T ≪ B for different interaction strengths U . From bottom to
top: U = 0.25; 0.5; 0.75; 1.0; 1.5. The horizontal lines show the fRG approximation for
αend (U = 0.5)/2 and αend (U = 1.0)/2 as obtained in [11]. The vertical line indicates
the crossover scale.

18

1

963

18

2

G/(e /h)

0.8
0.6
0.4
0.2

10

-3

10

-2

T

10

-1

Figure 11. (Color online) Conductance G as function of temperature T of a
noninteracting wire with stripe-like leads of different width. All wire-lead couplings
L
R
are set equal: tcL = tcR = 0.25. The setup is shown in the inset: Nover = Nover = 18,
L
R
Nlead = Nlead = 0 and Nmid = 963. From bottom to top, the width of the contact
L
R
region is Ncon = Ncon = 1, 3, 5, 7, 9, 11, 15, 21. The vertical lines indicate the crossover
scales.

Coupling-geometry-induced temperature scales in the conductance of LL wires
20
2

G/(e /h)

20
286
4
10 +1

286
4

0.5

10 +1
5

5

19

5

5

0.3

0.1 -4
10

10

-3

T

10

-2

Figure 12. (Color online) As in figure 11 but with length Nmid = 104 + 1 and
L
R
L
R
width Ncon = Ncon = 5 of contact region. Dashed-dotted line: Nlead = 0, Nlead = 0
L
R
L
and Nover = 20, Nover = 286 (setup shown in left inset). Dashed line: Nlead = 20,
R
L
R
Nlead = 286 and Nover = Nover = 0 (setup shown in right inset). Solid line:
L
R
L
R
L
R
Nlead = Nlead = 0 and Nover = Nover = 0. Dotted line: Ncon = Ncon = 1,
L
R
L
R
Nlead = Nlead = 0 and Nover = Nover = 0 for comparison. The vertical lines indicate
the crossover scales.
L/R

The vertical lines indicate the temperature scales vF /Nover and vF /N, which turned out
to be relevant for 1d leads. They seem to be of equal importance for stripe-like leads
(see below). The overall shape of the curves is unaltered by increasing the width of the
leads. For large numbers of lead channels G approaches the unitary conductance e2 /h
of a single channel wire. This trend can be counteracted by reducing the strength of
the lead-wire coupling.
In the case of 1d leads we identified two pronounced even-odd-effects. The
dependence on the total length of the wire is also found for stripe-like leads. The
vanishing local spectral density at even sites of the wire gives rise to the second evenodd-effect. For stripe-like leads one coupling site of the wire has to be odd and this
effect vanishes as the leads become broader.
Next we further investigate the relevance of the energy scales identified for 1d leads
starting with the noninteracting case. Figure 12 shows the conductance for similar
L/R
configurations as in figure 4, but with Ncon = 5. For comparison, the results for the
L
R
L
R
L
R
end-coupled system with Nlead = Nlead = 0, Nover = Nover = 0, and Ncon = Ncon = 5 are
included. Overall, the conductance is considerably higher than in figure 4 as expected
L
from the results shown in figure 11. At the temperature scales vF /N, vF /Nover/lead , and
R
vF /Nover/lead , the conductance subsequently crosses over to lower plateaus. However,
L
R
the complete equivalence of Nover/lead and Nover/lead is lost even for the non-interacting
case as the geometric equivalence is no longer present.
Figure 13 shows a typical off-diagonal component of the self-energy for the case
of stripe-like leads coupled to the bulk of an interacting wire. For comparison to the
case of 1d leads the parameters are chosen the same as in figure 6. The oscillations
emerging from the boundaries are expectedly unaffected by the width of the leads. The

-0.154

12
00

20

286

10
8
11 7
07

1086

Σj,j+1

10
00

j

80
0

60
0

40
0

20
0

0

Coupling-geometry-induced temperature scales in the conductance of LL wires

4

10 +1

-0.158

-0.156

11

87
10

07

-0.162

Σj,j+1

11
20

j

11
10

10
90

10
80

10
70

-0.16

11
00

-0.158

Figure 13. (Color online) Off-diagonal self-energy at the end of the fRG flow for an
interacting wire with U = 0.5 at T = 4 · 10−3 . All wire-lead couplings are set equal:
L
R
tcR = tcL = 0.25. The setup is shown in the inset: Ncon = 5, Ncon = 1, Nmid = 104 + 1,
L
R
L
R
Nlead = Nlead = 0 and Nover = 1086, Nover = 286. The part shown in the upper plot
is indicated by the dashed box in the inset. Long-ranged oscillations emerge from the
boundaries and from all coupling sites. The lower plot is a magnification of the left
contact region.

L
contact region itself (of size Ncon ; lower panel in the figure) contains a superposition of
oscillations emerging from all coupling sites. The symmetric decay of these oscillations
results from the fact that we chose all couplings to be equal. The oscillations originating
from the left contact to the left and the right contact to the right however still fall off
as the inverse distance up to a scale jT ∝ 1/T beyond which they decay exponentially.
As for 1d leads these oscillations do not induce a power-law suppression of the spectral
density within our fRG approximation. Because of these similarities the temperature
dependence of G for an interacting wire with stripe-like leads can be traced back to the
case of 1d leads.
In figure 14 we focus on end-coupled systems (no overhanging parts) with tcL =
cR
t = 0.1 (weak coupling). The only energy scales emerging are vF /N and B. For
vF /N ≪ T ≪ B, the conductance scales as ∼ T αend . The closeness of the results
L/R
for Ncon = 5 and 6 shows the minor importance of the even-odd effect. Figure 15
shows systems with tunneling into the end of the wire but with overhanging leads. For
comparison the end-coupled case without overhanging leads is shown as the dashed line.
As in the case of 1d leads the universally decaying oscillations from the ends of the
system induce power laws G(T ) ∼ T αend in the three temperature regimes introduced
by the overhanging parts. For a wire of roughly a micrometer in length it is difficult
to clearly resolve these power laws due to the extended crossover regions and the upper
(bandwidth B) and lower (cutoff scale vF /N) bounds of any power-law scaling. We
L
R
thus choose Nlead = Nlead in order to reduce the number of regimes to two. Even in
this case the two power laws cannot be resolved for a single parameter set (compare the
dashed-dotted and dotted lines).

Coupling-geometry-induced temperature scales in the conductance of LL wires
0.6

21

exponent

0.2

4

10 +1

0.15

0.1
-4
10

0.1

-3

-2

10

10

2

G/(e /h)

T

5

11
6

0.01
10

1
-4

10

-3

10

T

-2

Figure 14. (Color online) Main plot: Conductance G of an interacting wire with
interaction strength U = 0.5 as function of the temperature T coupled to stripe-like
leads of different width. The couplings to the leads are set equal: tcR = tcL = 0.1. The
L
R
L
R
setup is shown in the left inset: Nmid = 104 + 1, Nlead = Nlead = 0, Nover = Nover = 0
L
R
and Ncon = Ncon = 1; 6; 5; 11 (from bottom to top as indicated in the plot). The
vertical dashed lines indicate the crossover scales. Right inset: Effective exponents.
The solid horizontal line indicates the fRG approximation for αend as obtained in [11].
Line styles as in main part.
332

332

0.2

4

10 +1

2

G/(e /h)

0.06

5
100

0.04

5
100

exponent

0.1

0
-0.2
-0.4
-4

-3

10

-2

10

10

-1

10

T

4

10 +1
0.02

10

-4

10

-3

T

10

-2

L
R
Figure 15. (Color online) As in figure 14 but with different Nlead = Nlead . Dashed
L
R
L
R
line: Nlead = Nlead = 0. Dotted line: Nlead = Nlead = 100. Dashed-dotted line:
L
R
Nlead = Nlead = 332.

The case of leads coupled to the bulk of the interacting wire and terminating at
the contacts is shown in figure 16. As anticipated from the noninteracting case (figure
R
L
12), the temperature scales introduced by the additional length scales Nover and Nover
are still present. In all three temperature regimes
L/R
vF /N ≪ T ≪ vF /max(Nover ),

L/R
L/R
vF /max(Nover ) ≪ T ≪ vF /min(Nover ) and

L/R
vF /min(Nover ) ≪ T ≪ B,

the conductance G(T ) is governed by the same behavior as in the case of 1d leads. In the
first regime the oscillations from the boundaries strongly superpose the coupling sites,

Coupling-geometry-induced temperature scales in the conductance of LL wires
4

10 +1

0

4

-0.2
-4
10

10 +1

-3

-2

10

10

-1

10

T

2

G/(e /h)

0.1

1086 -0.1

1086
0.06

0.2

exponent

0.1 1110

22

0.04

0.02
10

-4

10

-3

T

10

-2

L
R
Figure 16. (Color online) As in figure 14 but with different Nover = Nover . Dashed
L
R
L
R
line: Nover = Nover = 0. Dashed-dotted line: Nover = Nover = 1086. Dash-dot-dotted
L
R
line: Nover = 1110, Nover = 0.

thus we find G(T ) ∼ T αend . Note that to better resolve the second and third regime, this
first regime is chosen such small in the curves shown that the asymptotic value of the
exponent is not reached. The second regime is exemplified by the dashed-double-dotted
L/R
curve. After crossing over at scale vF /Nover , the conductance seems to follow a power
law. However, as explained in detail in the section on 1d leads, this can not definitely
be confirmed and the behavior might be altered in the presence of the bulk LL power
law. In the third regime, the oscillations in the self-energy from the boundaries are cut
off by the exponential fall-off at scale ∝ 1/T before reaching the contact region, thus, no
power-law behavior can be found within the truncated fRG (figure 16, dashed-dotted
line). We expect the conductance in this regime to show scaling with the bulk LL
exponent if computed with an improved approximation which properly describes also
bulk LL behavior.
The conductance of “mixed systems” with overhanging parts of the leads and the
wire again shows an even richer temperature dependence which can be understood by
combining the cases studied above.
6. Summary
In the present study we were aiming at a more realistic modeling of transport through
interacting 1d quantum wires showing LL physics by considering stripe-like leads of
variable width coupled to the wire at arbitrary positions via tunnel barriers. We showed
that the overhanging parts of the leads and the wire, which generically appear in
experimental setups designed for a verification of LL power-law scaling, strongly modify
the temperature dependence of the linear conductance. For end-contacted wires of
experimental length with leads terminating at the contacts the range of possible powerlaw scaling (with exponent αend ) bound by vF /N and B extends over roughly four
decades. In the presence of up to four additional energy scales induced by the lengths
of the four overhanging parts and the corresponding extended crossover regimes the

Coupling-geometry-induced temperature scales in the conductance of LL wires

23

regions of possible scaling become so small that the power laws are not visible. This
shows the difficulties to clearly confirm power-law behavior in transport experiments
with LL wires.
One might speculate that the situation becomes even worse if inelastic two-particle
scattering processes neglected in our approximate treatment of the interaction (and
also in bosonization) are taken into account. They set a new upper energy scale B ∗
for the simple scaling behavior considered here which is presumably smaller than the
band width B. Power laws might further be obscured by the presence of relaxation and
dissipation in the leads, not included in our calculations.
The optimal setup to observe the boundary exponent αend is an end-contacted,
long wire with no overhanging parts of wire or leads for which one can expect scaling
to hold for vF /N ≪ T ≪ B ∗ . Although our approximation does not include the bulk
LL exponent the results indicate that the best setup to observe αbulk is a long wire with
leads which both couple to the bulk of the wire and which terminate at the contacts.
Our approach allows for more general systems with e.g. leads of asymmetric widths,
random wire-lead couplings or screened interactions in the contact regions.
Acknowledgments
The authors are grateful to the Deutsche Forschungsgemeinschaft (SFB 602) for support.
Appendix
Calculation of resolvent matrix elements
In order to compute the right-hand side of the flow equations (18) and (19) for the
self-energy, one needs the tridiagonal elements of the Green function G. In the case
of 1d leads G −1 itself is tridiagonal and these elements can be computed in order N
time [16, 28] with N being the length of the wire. In the geometry with stripe-like
leads, introduced in section 3.1, the inverse Green function is of the structure shown in
figure A1. Large tridiagonal blocks (the middle and overhanging parts of the wire) are
connected on the sub- and superdiagonal to (small) full blocks (the contact regions).
We can exploit the tridiagonal structure of large parts of this matrix enabling us to
treat lattice systems with up to 105 sites‡. To achieve this, we make multiple use of the
projection formalism introduced in section 3.1 to split the problem such that we only
have to invert matrices of size and structure of the isolated blocks 1 to 5 (figure A1).
This allows us to use the effective order N time algorithm for the tridiagonal parts and
some standard algorithm [27] for the (small) full matrices. Thus, the bottleneck of this
algorithm is the size of the full matrices.
‡ Our calculations were performed on a standard desktop PC. Thus the maximum system size reachable
can be easily extended by using more elaborate hardware.

Coupling-geometry-induced temperature scales in the conductance of LL wires
N1

N2

N3

24

N4 N5= N

1

..

1111
00002
1111
0000
1111
0000
1111
0000
1111
0000

..

3

1111
0000
..
1111
0000
1111
0000
1111
0000
4

1111
0000
1111
0000
1111
0000
1111
0000 5

..

Figure A1. Structure of the inverse Green function for stripe-like leads: block 1,3 and
5 are tridiagonal (left, right overhang and middle part of wire), block 2 and 4 are full
matrices (left and right contact region), the dots represent the connecting elements;
(Nj , Nj ) is the lower right element of the j-th block

In the first step of the algorithm the full Hamiltonian h appearing in the Green
function G(z) = (z − h)−1 is split into five smaller parts hj , referring to block j ∈
{1, .., 5} respectively, and elements hj,j+1 = βj,j+1|Nj Nj + 1| + H.c. with j ∈ {1, .., 4}
con
connecting block j and j + 1 (with βj,j+1 ∈ R), i.e.
4

5

hj,j+1
con

hj +

h=
j=1

(A.1)

j=1

We next define the projection operators Pj on each block. Then we project the full
Green function G onto the block, whose elements we want to compute, thereby reducing
the effort of inverting the full matrix to the inversion of an effective matrix of size and
structure of block j of the inverse Green function. We will exemplify this by calculating
the elements of block 1 as well as those elements on the sub- and superdiagonal
connecting block 1 and 2. Making subsequent use of the projection formula (15), we
obtain
P1 GP1 =

2
˜
z − h1 − β1,2 |N1 N1 + 1|G (2) |N1 + 1 N1 |

˜
P2 G (2) P2 =

2
˜
z − h2 − β2,3 |N2 N2 + 1|G (3) |N2 + 1 N2 |

˜
P3 G (3) P3 =

2
˜
z − h3 − β3,4 |N3 N3 + 1|G (4) |N3 + 1 N3 |

˜
P4 G (4) P4 =

2
˜
z − h4 − β4,5 |N4 N4 + 1|G (5) |N4 + 1 N4 |

˜
P5 G (5) P5 = (z − h5 )−1

−1

−1

−1

−1

,

,

,

,

Coupling-geometry-induced temperature scales in the conductance of LL wires

25

−1

˜
with G (i) = z − 5 hj + 4 hj,j+1
. We calculate the elements of block 1 of the
j=i
j=i con
Green function by going through this hierarchy from bottom to top, having only to invert
matrices of size and structure of the isolated blocks at each step. The computations for
the other blocks of G can be performed similarly.
˜
The element N1 + 1|G|N1 can be calculated by splitting h into h = 5 hj +
j=1
4
hj,j+1 and h1,2 to write
con
j=2 con
˜ ˜
˜ ˜
˜
G = G + Gh1,2 G + Gh1,2 Gh1,2 G
con
con
con
˜
˜
with G = z − h
with

−1

. After projecting G onto the relevant block P1 GP2 , one ends up

N1 + 1|G|N1 =

˜
˜
β1,2 N1 |G|N1 N1 + 1|G|N1 + 1
4
˜
˜
1 − β1,2 N1 |G|N1 2 N1 + 1|G|N1 + 1

2
˜
˜
× 1 + β1,2 N1 |G|N1 N1 + 1|G|N1 + 1

2

.

˜
The relevant elements of G can be efficiently calculated using the algorithm introduced
above for the blocks. Again the calculation of the other elements Nj + 1|G|Nj can be
performed analogously.
For the calculation of the transmission probability (12) we need all matrix elements
′
j |G|j with j ′ ∈ {N1 + 1, .., N2} and j ∈ {N3 + 1, .., N4 } of the Green function G −1 . We
use an algorithm which only requires the inversion of matrices of size and structure of
˜
the isolated blocks 1 to 5. We split the full Hamiltonian (A.1) into blocks h = 5 hj
j=1
and connecting elements hcon = 4 hj,j+1 and decompose
j=1 con
˜ ˜
˜ ˜
˜
G = G + Ghcon G + Ghcon Ghcon G
˜
˜
with G = z − h

−1

. Multiplication with projection operators P2 and P4 yields

˜
P2 GP4 = P2 G (β2,3 |N2 N2 + 1| + β1,2 |N1 + 1 N1 |)
˜
G (β3,4 |N3 N3 + 1| + β4,5 |N4 + 1 N4 |) GP2 .

˜
˜
The elements of G can be easily computed as the disconnected blocks of G −1 can
be inverted individually. The elements N2 + 1|G|N3 , N1 |G|N3 , N3 + 1|G|N4 ,
N1 |G|N4 +1 can be calculated by splitting h into blocks and bonds and projecting onto
the blocks needed. Details of the rather lengthy procedure can be found elsewhere [22].
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