Strong enhancement of transport by interaction on contact links
Strong enhancement of transport by interaction on contact links
Dan Bohr
MIC, Department of Micro- and Nanotechnology, NanoDTU,
Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
Peter Schmitteckert
TKM, Institut für Theorie der Kondensierten Materie,
Universität Karlsruhe, D-76128 Karlsruhe, Germany.
(Dated: October 26, 2018)
Strong repulsive interactions within a one dimensional Fermi system in a two-probe configuration
normally lead to a reduced off-resonance conductance. We show that if the repulsive interaction
extends to the contact regions, a strong increase of the conductance may occur, even for systems
where one would expect to find a reduced conductance. An essential ingredient in our calculations
is a momentum-space representation of the leads, which allows a high energy resolution. Further,
we demonstrate that these results are independent of the high-energy cutoff and that the relevant
scale is set by the Fermi velocity.
PACS numbers: 73.63.Kv, 73.23.Hk, 71.10.Pm
I. INTRODUCTION
Constructing a transport theory for strongly correlated
systems is one of the major challenges of condensed mat-
ter physics. Even though many interesting ideas have
been proposed during recent years, no consensus has yet
emerged as to the general validity and applicability of the
various schemes. With this state of affairs, it is of high
importance to establish reliable benchmarks for simple
model systems, which then can be used to validate new
approaches.
Recently we presented a novel method for calculat-
ing linear response conductance1 using the density ma-
trix renormalization group (DMRG) method2. A ma-
jor challenge in this work consisted in minimizing finite
size effects, which was achieved via modified boundary
conditions. In this paper we circumvent these technical
problems by reformulating the leads in momentum space.
This approach enables us to (i) reach a much higher en-
ergy resolution (∼ 10−5) and (ii) allows for a greater
flexibility in the choice of discretization schemes.
In two recent papers Mehta and Andrei3,4 presented
nonequilibrium Bethe ansatz results for the interacting
resonant-level model (IRLM), where a single spinless
level is coupled to a left and a right lead both via a
tunneling and a density-density interaction term. How-
ever, their work currently excludes the regime of resonant
tunneling–i.e., the regime where the conductance is close
to unity.5
In this work we study the linear conductance of the
IRLM on a lattice to provide a benchmark for the uni-
versal properties of the model. In addition, we present
results for an extended model, where the central region
consists of three sites, with a similar interaction as in
the IRLM model. As we will show, this model displays
the same qualitative behavior as the IRLM. It should
be noted that despite its simplicity, the IRLM captures
much of the physics of transport through an arbitrary in-
teracting nanostructure provided that only a single level
is close to the Fermi energy of the leads, with all other lev-
els well separated and outside the voltage window within
which the transport takes place. For perfect coupling the
IRLM model corresponds to the one-dimensional model
studied by Vasseur et al.,6 and Molina et al.,7 obtained
by restricting their nanostructure to a single site. Using
the embedding method they showed that smoothing the
ramp of interaction for perfect contacts can compensate
for the decrease of transmission due to interaction on the
nanostructure. Here we go far beyond the energy resolu-
tion attained in previous work and show that interaction
on the contact links can lead to strong renormalization
effects, enhancing transport beyond the noninteracting
system.
II. METHOD AND MODELS
We use the DMRG method to evaluate the linear re-
sponse conductance of the interacting nanostructure. In
previous work1 the leads were modeled in real-space by
nearest-neighbor hopping chains. While simple to imple-
ment there are several drawbacks of this method, most
prominently the need for “damped boundary conditions”
and the resulting problem of trapping of fermions on the
Wilson chain (the damped region).1
In the present work we introduce a setup where the
leads are described in momentum space. Specifically, a
short part of the lead close to the nanostructure is rep-
resented in real-space, accounting for local (i.e., high en-
ergy) physics, while further away from the nanostruc-
ture the lead is represented in momentum space; see
Fig. 1. Since the low-energy modes of the momentum
leads are now directly coupled to the extended structure
(the nanostructure plus additional real-space sites), as il-
lustrated in Fig. 1, the trapping of fermions on the low
energy sites1 is avoided and no scaling sweeps are needed.
http://arxiv.org/abs/0704.1917v3
2PSfrag replacements
Nanostructure, MS sites
Optional real space sites, MAdd sites
Mandatory real space sites, 2 sites
Left Lead Right LeadME = MS + MAdd
tDottDot tDot
VVV γVγV
tktk t tt
FIG. 1: (Color online) Schematics of the nanostructure ex-
tended by real space sites and attached to momentum-space
leads.
This enables much higher energy resolution, and in the
current work we resolve resonances of widths O(10−5).
By virtue of the momentum representation of the leads
the discretization scheme can be chosen arbitrarily to
suit the problem at hand. In the present work we use a
logarithmic discretization to cover a large energy range,
while switching to a linear discretization for the lowest-
energy states in order to describe Fermi-surface physics
accurately. The linear discretization on the low-energy
scale allows for a better representation of the low-energy
physics relevant for transport properties–i.e., excitations
created by η.
The models considered in this work are the IRLM and
the natural extension of this model to resonant linear
chains, defined by the Hamiltonians
HRS =
µg ĉ
j ĉj −
j,j−1∈SE
tj ĉ
j ĉj−1 + h.c.
j,j−1∈SE
n̂j −
n̂j−1 −
, (1)
HMS =
i∈L,R
i ĉi, (2)
HT = −
tk ĉ
ĉ1 +
tk ĉ
+ h.c., (3)
where ĉ
and ĉℓ are the (spinless) fermionic creation and
annihilation operators at site ℓ, n̂ℓ = ĉ
. HRS , HMS ,
and HT denote real-space, momentumspace, and tunnel-
ing between real- and momentum-space Hamiltonians,
respectively. The symbols S and SE denote the nanos-
tructure and the extended nanostructure (the full real-
space chain), respectively. The indices 1 and ME denote
the first and last site in SE . The general setup and the
specific values of the hopping matrix elements tj and the
interactions Vj are indicated in Fig. 1, and note specif-
ically the interactions on the contact links, γV . The
coupling tk of the extended real-space structure to the
momentum leads is chosen in such a way that in the case
of a cosine band it corresponds to a nearest-neighbor hop-
ping chain in real-space with a hopping parameter of t.
In the following we measure all energies in units of t = 1.
0.000 0.001 0.002 0.003
IRLM, t’=0.01, γ=1.0
V=0.0 
w=2.0 10
V=0.01
w=2.1 10
V=0.03
w=3.2 10
V=1.0
w=2.1 10
V=5.0
w=8.9 10
V=25.0
w=1.0 10
0.000 0.010 0.020 0.030
IRLM, t’=0.03, γ=1.0
V=0.0 
w=1.8 10
V=0.03
w=2.1 10
V=0.1
w=2.5 10
V=1.0
w=1.0 10
V=5.0
w=4.5 10
V=25.0
w=9.0 10
FIG. 2: (Color online) Conductance versus gate potential for
the interacting resonant level model for a contact hopping of
(a) t′ = 0.01 and (b) t′ = 0.03 and contact interaction ranging
from zero to 25. To each set of DMRG data a Lorentzian of
half width 2w has been added as a guide to the eye. The leads
are described with a cosine band between ±2 such that the
Fermi velocity is vF = 2. In contrast to intradot interaction
the contact interaction enhances the conductance and shows
a nonmonotonic behavior versus contact interaction.
For a single-site nanostructure and γ = 1 this model
reduces to the IRLM. The properties of the leads are
defined by the band structure ǫk, which can take any
form. In this work we use either the cosine band, ǫk =
−2 cos(k), or the linear band, ǫk = 2k. D is a cutoff
parameter such that the Fermi velocity vF = 2 is kept
constant in all work presented here, and the band ranges
between energies −D and D. Throughout this work we
use the notion of “contact interaction” for interaction on
the link between the nanostructure and the leads.
 0  5  10  15  20  25
Local density, t’=0.01
nd, µg=0.0005
nc, µg=0.0005
nd, µg=0.0002
nc, µg=0.0002
FIG. 3: (Color online) Site occupation nd of the resonant level
and nc of the real-space sites attached to the level vs the link
interaction in the IRLM for t’=0.01 and two different gate
voltages.
III. RESULTS
The aim of this work is to study the effect of contact
interaction. It is known from previous work1 that strong
repulsive interactions within the nanostructure lead to
suppression of the transport off resonance due to the for-
mation of a density-wave-like state on the dot.
In Fig. 2 we show results for the conductance versus
gate potential for different couplings to the leads and
different contact interactions for the IRLM (γ = 1). The
calculations have been performed with typically 130 sites
in total, ME = 10 real-space sites and 120 momentum-
space sites. Due to the symmetry of the band, we use
a discretization that is symmetric around ǫF = 0, and
further use identical discretization of the two leads. To
represent the “large” energy span in the band we use 20
logarithmically scaled sites, and thereafter use 10 linearly
spaced sites to represent the low-energy scale correctly.
In the DMRG calculations presented we used at least
1300 states per block and 10 finite lattice sweeps. To
each set of DMRG results in Fig. 2 is added a Lorentzian
of half width 2w as a guide to the eye.
As the interaction is turned up the width of the reso-
nance is increased far beyond the noninteracting result,
up to an order of magnitude larger; e.g., for t′ = 0.01 and
V = 1 the resonance width is increased by a factor of 10.
However, for a larger interaction V > vF = 2, transport
is suppressed, and for very large interactions the width
even becomes smaller than the noninteracting resonance.
A similar nonmonotonic behavior is observed by Borda
et al.8 using a perturbative calculation and is opposite to
the one originally reported by Mehta and Andrei3 which,
however, has been corrected in an erratum.5 Where pre-
ceding work3,4,5,8 failed to reach the unitary limit, we
demonstrate that indeed the resonant value remains uni-
tary.
Further by changing the bandwidth D for linear bands
we have verified that the relevant energy scale is the
Fermi velocity vF of the leads, while the bandwidth D
0.00 0.01 0.02 0.03 0.04 0.05
ERLM, MS=3, tDot=0.5, t’=0.05 
V=0.0, γ=0.0
w=2.5 10
V=2.0, γ=0.0
w=1.7 10
V=2.0, γ=0.5
w=8.7 10
V=2.0, γ=1.0
w=1.3 10
FIG. 4: (Color online) Conductance versus gate potential for
a resonant three site chain. To each set of DMRG results a
Lorentzian of half width 2w has been added as a guide to the
eye. The leads are described by a cosine band between ±2
such that vF = 2. The interdot interaction suppresses the
transport while the contact interaction is seen to enhance the
transport.
does not influence the conductance, as long as D ≫ V ;
compare Fig. 5.
Borda et al.8 conclude in their work that “in the case
of repulsive interaction the site next to the occupied d
level is empty and thus that electron can easily jump
to the conduction band”, while for attractive interaction
fermions accumulate close to the impurity. From that
reasoning we would expect an asymmetric conductance
curve depending on whether the impurity is filled or de-
pleted. However, this would violate particle-hole symme-
try of the model. In Fig. 3 we plot the site occupation
nd of the resonant level and the averaged site occupa-
tion nc of the left and right real-space sites attached to
the level. The occupations are plotted versus the contact
link interaction for the interacting resonant-level model,
and for two different gate voltages. The site occupation
of the resonant level and the neighboring sites are both
enhanced by the repulsive interaction as long as interac-
tion is in the range that enhances the conductance. For
stronger interaction the site occupancy of the resonant
level is indeed reduced; however, this is the regime where
the conductance is reduced. We would like to remark
that in the noninteracting case and for a weak contact,
t′ ≪ 1, the site occupations of the real-space sites in the
leads changes only slightly with gate voltage and are all
very close to half filling. Thus it seems that the densi-
ties of the hybridizing lead levels are not the determining
quantity for the interaction-induced changes of transport
properties.
The strong renormalization of the resonance width and
the non-monotonic behavior is, however, not specific to
the IRLM. In Fig. 4 we show results for the center peak
of a three-site nanostructure. Without a contact interac-
-1.0 -0.8 -0.6 -0.4 -0.2
Non particle-hole symmetric IRLM, t’=0.1, V=1.0, gV=1.0
D=1.0
D=2.0
D=4.0
D=10.0
w=4.7 10
FIG. 5: (Color online) Conductance versus gate potential
for a single site nanostructure without particle-hole symme-
try with a contact interaction of V = 1.0 and a contact
hopping of t′ = 0.1 for a linear band with cutoff parame-
ter D = 1.0, 2.0, 4.0, and 10.0 and constant Fermi velocity,
vF = 2. The conductance is independent of the cutoff. The
solid line is a fit with a Lorentzian of half width w= 4.7 10−2.
tion we find that the intradot interaction V = 2.0 = 4tDot
leads to a suppression of the transport in agreement with
previous results.1 As in the single-level case already a
small contact interaction increases again the width of
the resonance at zero gate potential. The enhancement
of the conductance by a contact interaction is stronger
than the corresponding suppression by the intradot inter-
action. Therefore we conjecture that the enhancement of
conductance due to the contact interaction is a universal
feature, which should also be present in other systems.
These findings may also be relevant for disordered struc-
tures, where repulsive interaction was found to enhance
transport in the case of strong disorder.9
Finally we have considered a non-particle-hole-
symmetric IRLM to address the question of parame-
ter renormalization versus bandwidth cutoff. The non-
particle-hole-symmetric model is defined by replacing the
(n̂j −
) terms in HRS by n̂j . The results are shown in
Fig. 5. It is clearly seen from the calculation that vary-
ing the cutoff over an order of magnitude does not change
the resonance, providing the interaction is not cut off by
the band. Neither the position nor the width of the res-
onance peak is influenced by the change of the cutoff
D, which is in contrast to the the renormalization group
flow that follows from the nonequilibrium Bethe ansatz.3
There, all transport quantities depend on the cutoff D
and the conductance changes with the cutoff. While it is
often difficult to compare a field theoretical model, like
the IRLM of Mehta and Andrei, with a lattice model,
we can at least conclude that the RG flow found in their
work is absent in our model with regularized (tight bind-
ing) leads and that the relevant energy scale is the Fermi
velocity.
IV. SUMMARY
A normal paradigm in transport calculations is to make
a principal division between transport region, the nanos-
tructure or “molecule”, and leads, where all correlation
effects are excluded from the leads.
In this work we have investigated the influence of an
interaction on the contact between a nanostructure and
the leads in a simple tight binding model. Using the
nonperturbative DMRG method to evaluate the linear
conductance we have demonstrated that a contact inter-
action significantly influences the transport properties. A
repulsive interaction smaller or comparable to the Fermi
velocity in the leads enhances the conductance, while a
large interaction leads to a suppression of the conduc-
tance. Our work shows that even a slight spread of
the interaction on the contacts influences the transport
strongly. This demonstrates that particular care should
be taken in treating the contacts correctly, especially re-
garding the interaction.
Acknowledgments
D.B. acknowledges support from the HPC-EUROPA
under Project No. RII3-CT-2003-506079, supported by
the European Commission. This work also profited from
Project 710 of the Landesstiftung Baden-Württemberg
and partial support through project B2.10 of the DFG
Center for Functional Nanostructures. Parts of the com-
putations were performed on the XC1 and XC2 at the
SSC Karlsruhe. The discussion of the the site occupa-
tion is attributed to an anonymous referee.
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