Correlation eﬀects on resonant tunneling in one-dimensional quantum wires
V. Meden,1 T. Enss,2 S. Andergassen,2 W. Metzner,2 and K. Sch¨nhammer1
o

arXiv:cond-mat/0403655v2 [cond-mat.mes-hall] 11 Nov 2004

1

Inst. f. Theoret. Physik, Universit¨t G¨ttingen, Friedrich-Hund-Platz 1, D-37077 G¨ttingen, Germany
a
o
o
2
Max-Planck-Institut f¨r Festk¨rperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany
u
o
We study resonant tunneling in a Luttinger liquid with a double barrier enclosing a dot region.
Within a microscopic model calculation the conductance G as a function of temperature T is determined over several decades. We identify parameter regimes in which the peak value Gp (T ) shows
distinctive power-law behavior. For intermediate dot parameters Gp behaves in a non-universal way.

A decade ago resonant tunneling through a double barrier embedded in a Luttinger liquid was intensively studied theoretically.[1, 2, 3] At temperature T = 0, in the
limit of an inﬁnite system, and for symmetric barriers
the conductance G was predicted to show a series of inﬁnitely sharp resonances if the gate voltage Vg applied to
the dot region is varied.[1] For spinless fermions, on which
we focus, the peak conductance Gp at resonance voltage
Vgr is given by e2 /h if the interacting wire (denoted “the
wire” in the following) is connected by “perfect” contacts
to non-interacting semi-inﬁnite leads as will be the case
here. Increasing T resonances are still present but acquire
a non-vanishing width w(T ) ∝ T 1−K ,[1] with the interaction dependent Luttinger liquid parameter K (K = 1
for non-interacting particles). We here restrict ourselves
to the case 1/2 ≤ K ≤ 1. For ∆Vg = |Vg − Vgr | = 0, G(T )
scales to zero as T 2αB ,[1] with αB = 1/K − 1. This also
holds for an asymmetric double barrier regardless if the
conductance is evaluated at or oﬀ resonance.
Using second order perturbation theory in the barrier
height the T > 0 deviations of Gp (T ) from e2 /h were
determined to scale as T 2K .[2] At further increasing T
a regime of uncorrelated sequential tunneling (UST) was
predicted[4] based on a perturbative analysis in the inverse barrier height, where Gp (T ) ∝ T αB −1 and w(T ) ∝
T . This regime is bounded from above by the level spacing of the dot ∆D = πvF /ND , where vF denotes the
Fermi velocity, ND the number of lattice sites in the dot,
and the lattice constant was chosen to be 1. Within the
same perturbative approach and for ∆D < T < B, where
B is the bandwidth, Gp (T ) increases as T 2αB for increasing T .[2, 4] In contradiction to the temperature dependence following from UST, in transport experiments on
carbon nanotubes a suppression of Gp with decreasing
T was observed for T
∆D .[5] Using another approximation scheme “correlated sequential tunneling” (CST)
was predicted to replace UST.[6] It leads to Gp ∝ T 2αB −1
which was argued to be consistent with the experimental data. This consideration has stimulated a number of
theoretical works.[7, 8, 9, 10]
Quantum Monte Carlo (QMC) results[10] for Gp (T )
were interpreted to be consistent with CST for T ∆D .
In Refs. [8, 9] a “leading-log” type of resummation for
the eﬀective transmission at the chemical potential was
applied and led to important insights. In this approach

no signature of CST was found, but it was criticized for
only being valid in the limit of small interaction. That
the “leading-log” method indeed does not capture all interesting interaction eﬀects away from K → 1 can be seen
considering the single barrier case for which the conductance for diﬀerent temperatures and barrier heights can
be collapsed onto a single curve by using a one-parameter
scaling ansatz.[1] While the resulting scaling function is
known to depend on K,[1] the “leading-log” approach
leads to the non-interacting K = 1 scaling function independently of the interaction strength chosen.[11, 12]
In Ref. [7] a ND = 1 double barrier model in which
chiral fermions carry the current was treated for K =
1/2 by refermionization. After ﬁne tuning the Coulomb
coupling across the barriers Gp (T ) was predicted to show
the non-interacting behavior, i.e. e2 /h − Gp (T ) ∝ T 2 for
T → 0 and Gp (T ) ∝ T −1 in the UST regime. This
ﬁnding is surprising and not supported by any of the
results obtained for more general models using the above
mentioned methods. The physics of this particular model
thus seems to be non-generic.
We study the problem applying the fermionic functional renormalization group (fRG) method to the model
of spinless fermions on a lattice with hopping t = 1 and
nearest neighbor interaction U > 0 at half ﬁlling.[12, 13,
14] By using a method which can be applied for any barrier height and shape but is restricted to weak to intermediate interactions our study is complementary to the one
using perturbation theory in either the barrier heights
or the inverse barrier heights.[2, 4] Considering the single barrier case and comparing with exact asymptotic
and numerical results we have shown previously that our
method leads to meaningful results even for fairly large
U
2.[12, 13, 14] In Ref. [12] it was shown that for
K = 1/2 our data for G collapse onto the exactly known
scaling function of the local sine-Gordon model.[1] Without any further approximations the fRG can be applied
to the double barrier problem and we thus expect that
regimes with universal power-law scaling and the related
exponents can reliably be determined. Accurate results
can also be obtained for regions of dot parameters not investigated so far in which Gp (T ) shows non-universal behavior, i.e. a behavior which cannot be characterized as a
power-law scaling with an exponent expressible in terms
of K. Studying these turns out to be relevant for the

2
interpretation of the QMC data[10] and the experimental results.[5] In contrast to the other methods which can
only be applied in certain temperature ranges the fRG
provides reliable results for G(T ) on all energy scales. For
large barriers and large as well as small dots we identify a
variety of temperature regions in which Gp (T ) shows distinctive power-law behavior with diﬀerent exponents expressible in terms of K. For intermediate barrier height,
intermediate dot sizes, and T
∆D , Gp (T ) decreases
with decreasing T . The behavior of Gp (T ) in this temperature regime depends on ND and the height of the
barriers and is thus non-universal. Whereas our results
are consistent with the QMC data, we do not observe
signatures of CST.
The model is given by the Hamiltonian
∞

c† cj+1 + h.c. + Vl njl −1 + Vr njr +1
j

H=−
j=−∞
jr

N −1

U j nj −

nj +

+Vg
j=jl

j=1

1
2

nj+1 −

1
2

, (1)

with 1 ≪ jl < jr ≪ N , in standard second-quantized
notation. The lattice can be divided in three parts: (1)
the wire with nearest neighbor interaction Uj across the
bonds (j, j +1) with j ∈ [1, N −1]; (2) the dot (embedded
in the wire) on sites j ∈ [jl , jr ] (ND = jr − jl + 1) which
can be shifted in energy by the onsite energy Vg and is
separated by barriers of strength Vl/r from the rest of the
lattice; (3) the non-interacting leads on sites j < 1 and
j > N . Besides site impurities we also consider hopping
impurities as barriers. In this case the interaction [in the
last term of Eq. (1)] and hopping [in the ﬁrst term of Eq.
(1)] across the bonds jl −1, jl and jr , jr +1 are set to zero.
Furthermore the second and third terms in the ﬁrst line
of Eq. (1) are replaced by −tl c†l −1 cjl − tr c†r cjr +1 + h.c.
j
j
with the reduced hopping matrix elements tl/r < 1. As in
the other theoretical studies mentioned we avoid eﬀects
of the contacts. We thus model “perfect” contacts by
a smooth spatial variation of the interaction Uj close to
the sites 1 and N . Starting at site 1 in our calculations
the interaction was turned on in an arctan-shape over the
ﬁrst 100 sites and similarly turned oﬀ approaching site
N .[12] For T = 0, vanishing barriers, and up to N = 104
this gives 1 − G/(e2 /h) < 10−6 . The constant bulk value
of the interaction is denoted by U .
In linear response and the absence of vertex corrections
the conductance is given by[15]
G(T ) = −(e2 /h)

2

|t(ε, T )|2 ∂ε f (ε/T ) dε ,

(2)

−2

˜
with |t(ε, T )|2 = (4 − ε2 )|G1,N (ε, T )|2 and f being the
˜
Fermi function. The one-particle Green function Gi,j has
to be computed in the presence of interaction and leads.
To this end we extend a recently developed fRG scheme,

which has been shown to provide excellent results at
T = 0,[12, 13, 14] to ﬁnite temperatures. The starting
point of this approach is an exact hierarchy of diﬀerential ﬂow equations for the self-energy ΣΛ and higher order vertex functions, with an energy cutoﬀ Λ as the ﬂow
parameter. The hierarchy is truncated by neglecting nparticle vertices with n > 2, and the 2-particle vertex
is parametrized approximately by a renormalized nearest neighbor interaction U Λ . The spatial dependence of
the renormalized impurity potential, described by ΣΛ , is
however fully kept. This procedure is perturbative in the
2-particle interaction, but non-perturbative in the impurity strength. Within our approximate scheme ΣΛ has no
imaginary part, which implies the vanishing of the vertex corrections to the conductance formula (2).[15] The
˜
Green function G1,N entering Eq. (2) is obtained from
the Dyson equation with ΣΛ=0 .
As a test of our method at ﬁnite T we ﬁrst discuss results for the single barrier problem. We ﬁnd the expected
scaling behavior of the conductance (for an example see
the solid line in the upper panel of Fig. 2).[1, 12] From
the suppression of G(T ) ∝ T 2αB , the exponent αB can
be read oﬀ. Within our truncation scheme we obtain
an approximation αRG (U ) for αB (U ). Consistent with
B
the exponents of the 1/N -scaling determined in Ref. [12]
at T = 0 we ﬁnd αRG (0.5) = 0.165, αRG (1) = 0.35,
B
B
and αRG (1.5) = 0.57 in good agreement with the exB
act exponents αB (0.5) = 0.1608, αB (1) = 1/3, and
αB (1.5) = 0.5398 known from the Bethe ansatz solution
of the bulk model.[16]
For the double barrier problem we start the discussion
with the case of strong, symmetric barriers. In Fig. 1
G(Vg ) is shown for ND = 6, U = 0.5, Vl/r = 10 and
diﬀerent T . In all ﬁgures the wire length is chosen to
be N = 104 sites in rough agreement with the length of
carbon nanotubes accessible to transport experiments.[5]
For suﬃciently small T resonance peaks are clearly developed. For T → 0, Gp (Vgr ) tends towards e2 /h. Because
of N < ∞ even at T = 0 the peaks have ﬁnite width. For
RG
all peaks the T = 0 width scales to zero as N K −1 ,[1]
with the fRG approximation K RG for K. The diﬀerent
height and width of the peaks at ﬁxed T is a ﬁnite band
eﬀect. In the following we always study the behavior of
the resonance closest to Vg = 0. We have veriﬁed that
Gp (T ) for the other peaks shows a similar T -dependence.
In the upper panel of Fig. 2 Gp (T ) is shown for
U = 0.5, Vl/r = 10, and ND = 2, 6, 100. The lower panel
contains the logarithmic derivative of Gp (T ) from which
the exponents of possible power-law behavior can be read
oﬀ directly. Several temperature regimes can be distinguished. For T larger than the bandwidth, Gp (T ) ∝ T −1 .
For T < ∆D , marked by arrows in Fig. 2, down to a lower
RG
bound T ∗ discussed below we ﬁnd Gp (T ) ∝ T αB −1 as
indicated in the lower panel of Fig. 2 by the dashed line.
This is the regime of UST,[4] in which the integral Eq.
(2) leading to Gp (T ) is dominated by a single peak of

3

0.02

T=0.3
T=0.1
T=0.03
T=0

2

G(Vg)/(e /h)

0.03

0.01

0.00
-3

-2

-1

0
Vg

1

2

3

FIG. 1: The conductance as a function of gate voltage for
ND = 6, U = 0.5, Vl/r = 10, N = 104 , and diﬀerent T .
ND=100

10
10

-2

-3

ND=50
10

0
-0.5
-1 -4
10

ND=30

ND=10

0

2

exponent

10

-1

Gp(T)/(e /h)

2

Gp(T)/(e /h)

10

ND=2

ND=6

0

T → 0, Gp (T )/(e2 /h) saturates at 1. Due the ﬁniteness
of the wire for T ≪ ∆W , with ∆W = πvF /N , we ﬁnd
1 − Gp (T )/(e2 /h) ∝ T 2 . For small ND and intermediate
to small barriers at increasing T this behavior is followed
RG
by a regime with 1 − Gp (T )/(e2 /h) ∝ T 2K .[2] All the
above scaling regimes we also ﬁnd for U = 1 and 1.5.
Temperature regimes with power-law behavior and the
same scaling exponents were obtained using a ﬁeld theoretical model and perturbation theory in Vl/r as well as
1/Vl/r .[2, 4] The agreement of the results obtained from
two complementary approximate methods and for diﬀerent models constitutes a strong argument for the correctness of the physics obtained. These considerations put
very tight limits on the parameters for which a scaling
regime with exponent 2αB − 1 could occur.[6] As an advantage of our method results for Gp on all energy scales
down to T = 0 can be obtained within one approximation scheme and for more realistic lattice models. Furthermore the fRG allows for extensions to non-perfect
contacts which have to be considered when comparing
with experiments.[5]

10

-3

10

-2

10

-1

10

0

10

1

10

-1

T

1
exponent

FIG. 2: Upper panel: Gp (T ) for U = 0.5, N = 104 , Vl/r = 10,
and ND = 2 (circles), 6 (squares), and 100 (diamonds). The
arrows indicate ∆D . The solid curve shows G(T )/2 for a
single barrier with V = 10 and U = 0.5, N = 104 . Lower
panel: Logarithmic derivative of Gp (T ). Solid line: 2αRG ;
B
dashed line: αRG − 1; dashed-dotted line: 2αRG − 1.
B
B

0.5
0
-0.5
-1 -4
10

10

-3

10

-2

10

-1

10

0

10

1

T

|t(ε, T )|2 , but T is still much larger than the width of
this peak. Peaks in Gp (Vg ) are clearly separated and
the width w(T ) ∝ T can be read oﬀ. The size of the
UST regime grows with increasing ND and shifts towards
smaller T . For ND ≥ 4 we ﬁnd non-monotonic behavior of Gp (T ). For ND
30 the region of decreasing
conductance (with decreasing T ) can be described by a
power-law with exponent 2αRG (solid line in the lower
B
panel of Fig. 2).[2] For these T many peaks of |t(ε, T )|2
contribute to the integral. For Vl/r ≫ 1 and ND ≫ 1,
i.e. in the parameter regime with distinctive power-law
scaling, the non-monotonic behavior can be understood
using Kirchhoﬀ’s law. The solid line in the upper panel
of Fig. 2 displays G(T )/2 obtained for a single barrier
with V = 10 and U = 0.5. The comparison shows that
the conductance of the double barrier problem for T up
to the local minimum can be determined by adding up resistances of the related single barrier case. We do not observe indications of CST with exponent 2αRG − 1, shown
B
as the dashed-dotted line in the lower panel of Fig. 2. For

FIG. 3: Upper panel: Gp (T ) for U = 1.5, N = 104 , ND = 10
with Vl/r = 0.8 (circles), ND = 30 with Vl/r = 1.5 (squares),
and ND = 50 with Vl/r = 0.8 (diamonds). The arrows indicate ∆D . Lower panel: Logarithmic derivative of Gp (T ).
Solid line: 2αRG ; dashed line: αRG − 1.
B
B

We next discuss the case of weak to intermediate barriers, dots of a few ten lattice sites, and larger U . In this
part of the parameter space we ﬁnd non-universal behavior of Gp (T ) which so far has not been studied theoretically and provides an alternative interpretation of the
QMC data.[10] It might also be relevant in connection
with experiments[5] since it is not clear that the experimental dot parameters fall in the regime of universal scaling. In Fig. 3 Gp (T ) is shown for U = 1.5, ND = 10 with
Vl/r = 0.8 (circles), ND = 30 with Vl/r = 1.5 (squares),
and ND = 50 with Vl/r = 0.8 (diamonds). These parameter sets are chosen to be close to the set used in
the QMC calculations.[10] In the limit of weak barriers
and small dots (circles) neither the power-law with the
exponent 2αB nor the one with αB − 1 are clearly de-

4
veloped. For increasing ND or increasing Vl/r the UST
regime gets more pronounced. For weak to intermediate barriers no plateau with exponent 2αB is established
(see the lower panel). The behavior in the regime of decreasing Gp (T ) (for decreasing T ) depends on Vl/r and
ND and is thus non-universal. It might be identiﬁed incorrectly as a power-law with an exponent signiﬁcantly
smaller than 2αB , i.e. in the vicinity of 2αB − 1.
Our fRG results for Gp (T ) in the case of asymmetric
barriers and the oﬀ resonance scaling of G(T ) for both
symmetric and asymmetric barriers will be presented in
Ref. [17]. For later reference we mention that in all these
RG
cases we obtain T 2αB for T → 0 as expected.[1]
Using the projection operator onto the subspace of the
one-particle states of the dot and the orthogonal complement to the left and right of it the result |t(ε, T )|2 =
˜
(4 − ε2 )|G1,N (ε, T )|2 can be written in a form which is
useful for obtaining a deeper understanding of the different scaling regimes discussed above and to work out
the diﬀerences of our approach to the “leading-log” approximation. As an example we take the case of hopping
barriers and ND = 1. Then one obtains
|t(ε, T )|2 =

(3)
4Γl (ε, T )Γr (ε, T )
2

2

[ε − Vg − Ωl (ε, T ) − Ωr (ε, T )] + [Γl (ε, T ) + Γr (ε, T )]

˜
˜
with Γl = t2 Im G0l −1,jl −1 , Ωl = t2 Re G0l −1,jl −1 and simj
j
l
l
˜
ilar expressions for l → r. The Green function G0 (ε, T )
i,j
Λ=0
is obtained by considering Σ
as an eﬀective potential
outside the dot and setting tl/r = 0.
For Vg = 0, ε − Vg − Ωl (ε, T ) − Ωr (ε, T ) vanishes at
ε = 0 and we ﬁnd a resonance. Due to the generalized
Breit-Wigner form of Eq. (3) for tl = tr the transmission
at resonance is 1 independent of the dependence of Γl/r
on its arguments and the parameters. It is important to
note that in this case and for T → 0, (i) Γl/r (ε = 0, T )
does not follow the single barrier scaling T αB but instead
saturates (even for N → ∞). For asymmetric barriers
RG
with tr < tl < 1 and at resonance Γr (ε, T ) ∝ T αB
RG
while (ii) Γl (ε, T ) increases as T −αB . Similar scaling
is found for the ε and 1/N dependence. These powerlaws are cut-oﬀ by the largest of the scales T , ε, and
∆W . Inserting the expressions for Γl/r into Eq. (3) leads
to a power-law suppression of |t(ε, T )|2 with exponent
RG
2αRG which directly gives Gp (T ) ∝ T 2αB mentioned
B
above. We next study the oﬀ resonance case with Vg = 0
and tl = tr ≤ 1 including the single barrier situation
(tl/r = 1). Because of the symmetry in the following we
suppress the indices l/r. We focus on T = 0 and consider
∆W ∝ 1/N as the relevant energy scale. Then the integral in Eq. (2) can be performed and |t(0, 0)|2 is directly
related to G. For small Vg we (iii) ﬁnd Γ(0, 0) ∝ const.
RG
and Vg + 2Ω(0, 0) ∝ Vg N 1−K , which combines to the
RG
expected 1 − G/(e2 /h) ∝ N 2(1−K ) . For large Vg , (iv)

RG

Γ(0, 0) ∝ t2 N −αB and Vg + 2Ω(0, 0) ∝ Vg which leads
RG
to G ∝ N −2αB . The observations (i)-(iv) are not captured by the parameterization of the eﬀective transmission |t(ε, T )|2 used in the “leading-log” method.[8, 9, 11]
In this approach the important energy dependence of the
real part Ω [see (iii) and (iv)] does not appear.
Using Eq. (3) and its generalization for ND > 1[17] the
UST regime can be understood quite simply.[18] We here
focus on symmetric barriers. Approaching from large T
UST sets in at a scale at which only one peak of |t(ε, T )|2
contributes to the integral in Eq. (2) (respectively for T
smaller than the bandwidth at ND = 1). As can be seen
in Fig. 2 this scale is signiﬁcantly smaller than ∆D . For
RG
these T ≫ |ε| we ﬁnd Γ(ε, T ) ∝ t2 T αB /ND . PerformRG
ing the integral in Eq. (2) leads to Gp (T ) ∝ T αB −1 and
w(T ) ∝ T . This scaling continues down to the temperRG
ature T ∗ ∝ (t2 /ND )1/(1−αB ) at which the width of the
2
peak in |t(ε, T )| is equal to T . This deﬁnes the lower
bound of the UST regime.
In conclusion, applying the fRG we obtained a comprehensive picture of resonant tunneling in a microscopic
model for a Luttinger liquid with 1/2 ≤ K ≤ 1. Depending on the dot parameters we found distinctive power-law
scaling of Gp (T ), but also regimes with non-universal behavior. No signatures of CST were found.
We thank X. Barnab´-Th´riault, R. Egger, M. Grifoni,
e
e
A. Komnik, D. Polyakov, M. Thorwart, and especially
H. Schoeller for discussions.

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u
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[17] T. Enss et al., in preparation.
[18] In contrast to what is found in the model of Ref. [7] for
not too small barriers we observe indications of UST with
exponent αB −1 also for ND = 1. At K = 1/2, i.e. αB = 1
this leads to a plateau in Gp (T ).