Disorder effect on the Friedel oscillations in a one-dimensional Mott
  insulator
Disorder effect on the Friedel oscillations in a one-dimensional Mott insulator
Y. Weiss, M. Goldstein and R. Berkovits
The Minerva Center, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
The Friedel oscillations resulting from coupling a quantum dot to one edge of a disordered one-
dimensional wire in the Mott insulator regime, are calculated numerically using the DMRG method.
By investigating the influence of the disorder on the Friedel oscillations decay we find that the effect
of disorder is reduced by increasing the interaction strength. This behavior is opposite to the recently
reported influence of disorder in the Anderson insulator regime, where disorder led to a stronger
decay of the Friedel oscillations.
PACS numbers: 73.21.Hb, 73.21.La, 71.45.Lr
Properties of one-dimensional systems incorporating
disorder and electron-electron interactions are the sub-
ject of many recent theoretical and experimental studies.
It is well known that when the interactions are not too
strong, the addition of disorder turns the metallic system
into an Anderson insulator (AI). However, for strong in-
teractions (i.e., when the clean system is a Mott insula-
tor) the exact effect of disorder depends on its strength,
and in general is not completely understood. While for
clean systems the Mott insulator (MI) phase is a well
studied problem1, the addition of disorder opens a few
questions, which have attracted several studies in the last
decade2,3,4,5,6.
Specifically, since most studies on disordered one-
dimensional wires concentrate on either the AI or the MI
phases, a full comparison between the two regimes is still
lacking. Nevertheless, a qualitatively different behavior
between these two regimes was demonstrated in a few
cases. For example, the effect of interactions on the per-
sistent current in one-dimensional disordered rings was
calculated in previous works7,8, and an important differ-
ence between the AI and MI phases was found. While
for strong interactions and weak disorder (MI phase) the
persistent current was reduced, for strong disorder (AI
phase) an increase of the current was found. However,
the exact diagonalization techniques which were used in
these studies, are applicable only for very small system
sizes.
The definition of a disordered MI phase should be re-
fined, since when the disorder is strong, i.e., when the
random potential felt by the electron is much larger than
any other energy scale in the problem, the MI state is
destroyed. For a weak disorder, however, it was shown
in several studies that the Mott energy gap vanishes only
when a finite disorder is introduced, so that below this
critical disorder the MI phase is stable9,10. Usually this
is not the case for a MI consisting of spinless particles,
since an Imry-Ma type of argument11 shows that the
long range order is destroyed even for an infinitesimal
disorder12. Yet, for a finite sized mesoscopic sample, the
Imry-Ma length scale might be a few orders of magnitude
larger than the sample’s size, so that the effective ground
state for a weak enough disorder remains a MI one. Such
finite one dimensional wires coupled to dots have been
recently manufactured, and signatures of a charge den-
sity wave in strong magnetic fields have been observed13.
Increasing the disorder above a critical strength changes
the MI state either to a Mott Glass or to an AI5,6.
In this paper we investigate the influence of interac-
tions on the Friedel oscillations (FO) in a disordered one-
dimensional wire, and compare this behavior between the
AI and the MI regimes. We study interacting spinless
electrons confined to a 1D wire which can be in either
its AI or MI phases. Without disorder, it is known that
in order to get a MI phase the repulsive e-e interactions
should be strong enough, while for weaker interactions
the wire is described by the Tomonaga-Luttinger liquid
(TLL) theory. The MI phase, for strong interactions,
appears for spinless 1D electrons as a 2kF charge den-
sity wave (CDW). When disorder is included, the TLL
phase switches into an AI state. However, the finite size
CDW state is expected, as noted above, to remain stable
against the application of a weak enough disorder, i.e.
to remain a MI state. For example, previous numerical
simulations have presented the long range order of such
a weakly disordered CDW2. In order to verify the exis-
tence of the CDW order in the presence of disorder for
the length scales considered, one should check the elec-
tron density of the entire system.
The behavior of the FO decay length in the presence
of disorder in the AI phase, were discussed in a recent
paper14. It was shown that the effect of disorder on
the FO decay length can be described by an additional
exponential term e−x/ξ, where ξ is a characteristic de-
cay length. For a constant strength of (weak) disorder
ξ decreases as the interactions increase, i.e. the disor-
der effect is enhanced with increasing interactions. Ar-
guing that ξ is a good approximation of the mobility
localization length, it was found that it is in good accor-
dance with theoretical predictions made using the TLL
framework15,16,17, which are suitable for the weak inter-
actions regime. However, for the CDW phase umklapp
processes are important, and the above considerations
are not applicable.
In order to calculate the decay length of the weakly dis-
ordered CDW wire, we use a method similar to the one
used for the TLL regime. We couple the wire to a quan-
tum dot with a single level from one end, and study the
http://arxiv.org/abs/0704.1386v1
electrons density change in the sites nearby. The density
change, which have shown Friedel oscillations with a 2kF
wave vector and a power law decay in the metallic case
(TLL), should now present 2kF oscillations with an expo-
nential decay, since the underlying lattice state (CDW)
is an insulator. For the clean case we will show that the
exponential decay length scales as the CDW correlation
length, ζ, as predicted18. However, in the disordered case
we find an additional decay factor due to the disorder, as
in the TLL case14. By calculating this decay length we
are able to present a clear picture of the dependence of
the decay length due to disorder on interactions, in both
the AI and MI regimes. While the decay length of the
FO due to disorder is monotonically decreasing as inter-
action increases for the AI phase, for the MI phase it
is monotonically increasing. The origin of the difference
between these two regimes will be explained.
The system under investigation is the strong electron
electron interactions regime of a one-dimensional wire
with disorder. The wire is modeled by a one-dimensional
lattice of spinless fermions, moving in a random on-site
potential, and experiencing nearest neighbor repulsive in-
teractions. The Hamiltonian is
Ĥwire =
ǫj ĉ
j ĉj − t
j ĉj+1 + h.c.) (1)
j ĉj −
j+1 ĉj+1 −
where ǫj are the random on-site energies, taken from a
uniform distribution in the range [−W/2,W/2], I is the
nearest neighbor interaction strength, and t, which is the
hopping matrix element between nearest neighbors, sets
the energy unit scale. ĉ
j (ĉj) is the creation (annihila-
tion) operator of a spinless electron at site j in the wire,
and a positive background is included in the interaction
term. Such a (clean) wire undergoes a phase transition
at I = 2t between TLL and CDW. In order to study the
CDW and the weakly disordered CDW phase the inter-
action strength is taken to be strong, i.e. I > 2t.
We now introduce a quantum dot with a single orbital
at one end of the wire, by adding the following term to
the Hamiltonian:
Ĥdot = ǫ0ĉ
0ĉ0 − V (ĉ
0ĉ1 + h.c.) (2)
+I(ĉ
0ĉ0 −
1ĉ1 −
where ǫ0 describes the dot energy level. As in Ref. 14,
we take ǫ0 ≫ W and V = t.
The Hamiltonian Ĥ was diagonalized using a finite-size
DMRG method14,19, and the occupation of the lattice
sites were calculated, for different values of W and I.
The dot energy level was taken to be ǫ0 = 10t. The size
of the wire was up to L = 300 sites, which is both long
enough due to the exponential decay of the calculated
quantities, and still short enough to maintain the CDW
order for the disorder strengths taken (W/t = 0.1 and
0.2). During the renormalization process the number of
particles in the system is not fixed, so that the results
describe the experimentally realizable situation of a finite
section of a 1D wire which is coupled to a dot and to an
external electron reservoir20. Yet, the calculated density
remains close to half filling in all the calculated scenarios
(even in the presence of disorder) since the interaction
term contains a positive background, and the calculation
is done for µ = 0.
We start with the case in which no disorder is consid-
ered (W = 0), so that the ground state of the CDW is
twofold degenerate. This degeneracy is broken, however,
once a pinning impurity, denoted by ǫ
0 → 0
+, is cou-
pled to one end of the wire, and the wire shows a 2kF
modulation20. The particle density of such a state, in the
j-th site of the wire, will be denoted by N0j . When the
pinning impurity is replaced by a dot level with ǫ0 ≫ ǫ
the particle density in the wire (to be denoted as Nj)
is changed by an oscillating 2kF term. 2kF oscillations
in the density difference were also obtained in the TLL
phase (Ref. 14), where the density without the quantum
dot is flat, and the deviation of the population from this
flat density once the lead is coupled to the dot shows
Friedel oscillations. Here one should notice that the ref-
erence state (without the dot) does not have a flat parti-
cle distribution, but rather has a CDW 2kF oscillations.
Coupling the dot results in a new CDW state, which has
also 2kF oscillations, but with a different amplitude. The
difference between these two states has a 2kF oscillation,
which has an exponential decay from its value at the edge
of the wire.
5 10 15 20 25 30
-0.05
FIG. 1: Typical oscillations for a clean sample with L = 280
for a CDW with I/t = 2.5. The upper panel shows Nj (cir-
cles) and N0j (squares), and the lower panel presents their
difference ∆Nj .
In order to calculate the density difference between the
cases when the quantum dot is coupled or uncoupled to
the wire, one defines the density change in site j as
∆Nj = Nj −N
j , (3)
and studies the dependence of ∆Nj on j for different
parameters. A typical result of Nj vs. N
j , and the re-
sulting ∆Nj , showing the 2kF oscillations caused by the
dot orbital at j = 0, is presented in Fig. 1.
When W 6= 0, the CDW ground state is no longer de-
generate, and the infinitesimal pinning impurity is not
required. The disorder itself pins the CDW to different
places on the lattice, with the ability to break the long
range order of the clean CDW by localized solitons, with
a density which depends on the disorder strength2. Yet,
when a dot level with ǫ0 ≫ W is connected to one side of
the wire, the local effect in its vicinity is stronger than the
pinning caused by the disorder. This results in a change
of the particle density near the dot, and this change de-
creases with distance. It turns out that the definition of
∆Nj in Eq. (3) is suitable for the disordered case as well,
since it cancels out the disorder pinning effects which are
the same for the two cases, isolating the density fluctua-
tions created by the dot.
A typical picture of ∆Nj for a disordered CDW sample
is presented in Fig. 2. Whereas the upper panel shows the
density of the two similar systems, one which is coupled
to the quantum dot and the other is not, the lower panel
presents the difference between these two densities, and
the decay of the oscillations can be clearly seen.
5 10 15 20 25 30
FIG. 2: Typical oscillations for a single disordered sample
with L = 280, W = 0.1 and ǫ0 = 10, for a CDW with I = 3.
The upper panel shows Nj (circles) and N
j (squares), and
the lower panel presents their difference ∆Nj .
Since the CDW is an insulating phase, the decay of
the 2kF oscillations without disorder is supposed to be
exponential and the characteristic length is the CDW
correlation length2, i.e., ∝ exp(−x/ζ). In Fig. 3 such an
exponential decay of ∆Nj is shown on a semi-log scale
for various interaction strengths. An exact Bethe Ansatz
solution18 of our model gives the relation between the
correlation length and the interaction as
ζ ∼ exp(π/
I/(2t)− 1). (4)
The correlation lengths extracted from the DMRG re-
sults are presented with a fit to the exact formula in the
inset of Fig. 3. As can be seen, for I not very close to
the TLL-CDW transition point (which occurs at I = 2t),
the results fit the theory very well.
10 20 30 40j
0.0001
2 3 4 5I
I=2.5
FIG. 3: The oscillations decay in the CDW regime for various
interaction strengths and without disorder (note the semi-log
scale). As the interaction increases, the correlation length
decreases and the decay is faster. Inset: the inverse correla-
tion length of the CDW state for various interaction strengths
(symbols) fitted to the theory prediction Eq. (4).
For W 6= 0, ∆Nj is averaged over 100 realizations,
for which we expect a sampling error of the order of one
percent. Assuming that the disorder adds another expo-
nential term to the oscillations decay, which is thus pro-
portional to exp(−x/ζ − x/ξ), there are two competing
length scales - the decay length due to disorder (ξ) vs.
the correlation length (ζ). For strong interactions and
weak disorder ζ ≪ ξ so that the disorder effect is hardly
seen, but increasing the disorder or decreasing the in-
teraction strength should result in a combination of the
two exponential decays. The DMRG results, presented
in Fig. 4, show the disorder effect on the oscillations de-
cay. For I = 2.5 and I = 3 one can see faster decay
for the disordered samples with W = 0.1. For stronger
interaction larger disorder is required in order to affect
the decay.
Similarly to the AI phase, the extra decay length can
be extracted by fitting, for each value of I, the W 6= 0
curve multiplied by ex/ξ to the W = 0 one. Such a
rescaling is presented in the inset of Fig. 4.
As can be seen in Fig. 5, the decay length extracted
for the disordered MI regime increases as a function of
the interaction strength (for 2t < I <∼ 3.5t), an opposite
behavior to the AI case (I < 2t). Results for stronger
values of I are not shown, since for too strong interactions
10 20 30 40j
0.001
10 20 30 40j
0.001
FIG. 4: The decay of the oscillations of a disordered CDW
with I = 2.5, 3 and 3.5 (top to bottom, note the semi-log
scale). The lines correspond to the clean sample result, and
the symbols to the averaged disordered data. For W = 0.1
(circles) the disorder effect is clearly seen for I = 2.5 and
I = 3 but not for I = 3.5 in which ξ is much larger than the
correlation length ζ. For W = 0.2 (squares) ξ is small enough
to affect the decay even for I = 3.5. Inset: multiplying ∆Nj
by ex/ξ collapses the disordered data on the clean curves.
0 1 2 3 4
300 300
600 600
900 900
1200 1200
1500 1500
2 2.5 3 3.5 4
FIG. 5: The decay length due to disorder (ξ) in the TLL
(I < 2t) and in the CDW (I > 2t) phases as a function of the
interaction strength. The data for the TLL phase was taken
from Ref. 14. Inset: zoom into the CDW regime.
the correlation length is very small, and thus the estimate
of ξ is less accurate.
These results point out that as the interaction strength
increases in the MI phase, the disorder effect decreases.
In the AI phase, on the other hand, the disorder effect
is enhanced with increasing interactions. The difference
between these two behaviors results from the difference
in the ground states of the two phases in the clean case.
In our model there is a competition between the kinetic
energy (the hopping term) and the potential (the inter-
action). The hopping term prefers the existence of a flat
particle distribution whereas the interaction term prefers
a CDW-like form. For different values of I the results
of that competition are different: for I < 2t (the TLL
phase) the hopping term wins, and the distribution is
flat, while for I > 2t (the CDW phase) a CDW starts to
form.
Inside the clean TLL phase, as I increases, the CDW
fluctuations are stronger. Yet, the average density profile
in the ground state remains flat because of the hopping
term. But when disorder is introduced, the flat density
state becomes less favorable than a state with a fluctu-
ating density, the latter being preferred by both the dis-
order and the interactions. For a constant disorder, as
the interactions become stronger, these fluctuations are
enhanced, so the disorder effect increases.
In the CDW phase, on the other hand, without dis-
order, the interaction wins over the hopping, and the
ground state has a CDW form. Turning on the disorder
might change the particle distribution, e.g. by allowing
an electron to move into a site with lower on-site energy,
but this results in raising the interaction energy. As the
interaction strength gets stronger, the probability of such
a process decreases, so that the actual effect of the dis-
order is getting weaker.
In conclusion, while the decay length of the 2kF os-
cillations envelope due to disorder is monotonically de-
creasing in the AI phase, we have shown that it is mono-
tonically increasing in the disordered MI phase. The dif-
ference between these two regimes is explained by the
difference between the ground states of the clean sam-
ples in each case. In the AI phase the pure ground state
is flat, and both the disorder and the interactions try
to introduce fluctuations in it. In the MI phase, on the
other hand, the pure ground state oscillates with a 2kF
wave vector, and these oscillations are enhanced by the
interactions and reduced by the disorder. As a result,
the disorder effect (for a constant disorder strength) is
getting weaker as the interactions are enhanced.
Acknowledgments
Support from the Israel Academy of Science (Grant
877/04) is gratefully acknowledged.
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