Spin- and charge-density oscillations in spin chains and quantum wires
Stefan Rommer
Department of Physics and Astronomy, University of California, Irvine, California 92697

arXiv:cond-mat/0002001v2 [cond-mat.str-el] 1 Sep 2000

Sebastian Eggert
Institute of Theoretical Physics, Chalmers University of Technology and G¨teborg University, 41296 G¨teborg, Sweden
o
o
(Submitted: December 7, 1999. Last change: June 8, 2018)
We analyze the spin- and charge-density oscillations near impurities in spin chains and quantum
wires. These so-called Friedel oscillations give detailed information about the impurity and also
about the interactions in the system. The temperature dependence of these oscillations explicitly
shows the renormalization of backscattering and conductivity, which we analyze for a number of
different impurity models. We are also able to analyze screening effects in one dimension. The
relation to the Kondo effect and experimental consequences are discussed.
PACS numbers: 71.10.Pm, 75.10.Jm, 72.15.Qm, 73.23.-b

dimensional systems such as spin-chains and interacting quantum wires (Luttinger Liquids) in order to understand the detailed effects of impurity scattering and
screening as a function of temperature. In the classic
work by Kane and Fisher1,5 it was found that a generic
impurity in a spinless Luttinger Liquid results in a renormalization of the conductivity with temperature, which
leads to a perfectly reflecting barrier at T = 0 for repulsive interactions. Interestingly, this behavior can also be
explained in terms of repeated scattering off the Friedel
oscillations, which gives an explicit expression of the
transmission coefficient in the weak coupling limit.6 Independently, the analogous renormalization behavior was
also found in the spin-1/2 chain,7 where a generic perturbation in the chain effectively renormalizes to an open
boundary condition as T → 0. However, it is possible that a special symmetry in the Hamiltonian reverses
this renormalization, which leads to resonant tunneling
in quantum wires5,8 or the healing of a two-link problem
in the spin-1/2 chain.7 The renormalization behavior in
that case is analogous to the two-channel Kondo effect.7,9
The renormalization flow can easily be tested numerically by examining the scaling of the finite size energy
gaps,7,10 but we now would like to determine the reflection coefficient directly by analyzing the induced density oscillations which are also interesting in their own
right. In addition, we also consider the density oscillations from impurity models near an edge, impurities with
a net charge or magnetic moment (Kondo-type impurities), and integrable impurities. The detailed renormalization of the impurity backscattering as well as screening
can be studied in each case by analyzing the induced density oscillations as a function of temperature, which we
determined numerically with the Transfer Matrix Renormalization Group (TMRG) for impurities.9,11,12 This allows us to make predictions for conductivity measurements in quantum wires and for Knight shift measurements in spin chains, e.g. Nuclear Magnetic Resonance
(NMR) experiments. In all cases we find a typical renormalization to a fixed point of the Luttinger Liquid model,

I. INTRODUCTION

There is growing interest in impurities in lowdimensional electron and magnetic systems spurred by
high temperature superconductivity and experimental
progress in producing ever smaller electronic structures.
There appears to be two central aspects that are studied most in this context, namely the effect of impurities on the transport properties in mesoscopic systems
on the one hand,1 and impurity-impurity interactions in
antiferromagnetic systems due to impurity induced magnetic order2 on the other hand. In this paper we show
that the charge- and spin-densities near impurities give
a great deal of information about both of those aspects
and allow us to study a number of impurity models in
one dimension in detail.
Induced density fluctuations at twice the Fermi wavevector, so-called Friedel oscillations,3 are a common impurity effect in fermionic systems, which are enhanced in
lower dimensions. There are two distinct physical effects
that can give rise to Friedel oscillations. The most common source is a simple interference effect as considered
in the original work by Friedel.3 Fermions scatter off the
impurity, resulting in a superposition of incoming and
outgoing wave-functions. Summing up the squares of the
corresponding wave-functions up to the sharp cutoff at
the Fermi wave-vector kF results in a characteristic interference pattern with a 2kF x modulation, namely the
Friedel oscillations. Clearly, this pattern can give a great
deal of information about the impurity, in particular details about the scattering process. A second source for
the 2kF x oscillations are interaction effects due to the
screening of an impurity with a net charge or a magnetic
moment. A typical example of this effect is the Kondo
screening cloud,4 which we also analyze in this paper.
The 2kF x oscillations due to screening have typically a
different characteristic amplitude as a function of x than
those due to backscattering, as we will discuss in more
detail below.
We now consider the density oscillations in one1

which is described in terms of a simple (open or periodic)
boundary condition in agreement with field theory calculations.
The rest of this paper is organized as follows. In Sec. II
we present the model Hamiltonian and review the results
for Friedel oscillations due to an open end (i.e. complete
backscattering). Different impurity models of a modified
link, two modified links, an edge impurity, Kondo impurities, and an integrable impurity are then analyzed
in detail in Sec. III. Section IV contains a description
of the numerical methods used and a critical discussion
about the possible numerical errors. We conclude with a
summary and a discussion about experimental relevance
in Sec. V.

H =

v
2

g −1 (∂x φ)2 + gΠ2 ,
φ

(6)

which can be solved by a simple rescaling of the boson with the interaction parameter g. The parameter
g and the velocity v can in principle be calculated for
any interaction strength U and chemical potential µ with
Bethe ansatz techniques.13 To lowest order in U we get
g = 1 − 2U/πv and v = 4t2 − µ2 + 2U/π, so that g < 1
for repulsive interactions.
We now want to analyze the density oscillations using
this formalism. Already from the decomposition of the
fermion field in Eq. (4) it is clear that the fermion density
may contain an oscillating component with 2kF x. To see
this explicitly we can write the charge density in quantum
wires (or equivalently the spin density Sz in spin chains)
in terms of left- and right-movers

II. THE MODEL
†
†
Ψ† Ψ = ψL ψL + ψR ψR

The standard model we are considering here are spinless interacting fermions on a one-dimensional lattice, described by the Hamiltonian
H=
i

†
†
+ei2kF x ψL ψR + e−i2kF x ψR ψL .

The first two uniform terms just represent the overall
fermion density in the bulk system, while the last two
“Friedel” terms are the density oscillations nosc we are interested in. In a system with translational invariance the
†
left- and right-moving fields are uncorrelated ψL ψR = 0
and no density oscillations are present. An impurity,
however, scatters left- into right-movers and the amplitude of the oscillations gives detailed information about
the backscattering.
As the simplest example of this effect, let us consider an open boundary, i.e. an impurity with complete
backscattering at the origin. In this case the correlation
functions can be calculated directly.7,14–18 For the particular case of the left-right correlation function at equal
space and time we find

−t(Ψ† Ψi+1 + Ψ† Ψi ) + U ni ni+1 − µni ,
i
i+1
(1)

where ni = Ψ† Ψi is the fermion density. Although
i
this Hamiltonian neglects the spin degrees of freedom
of real electrons in quantum wires, it captures the essential physics in conductivity experiments. Moreover, this
model is equivalent to the spin-1/2 chain
H=
i

J + −
− +
z z
z
(S S
+ Si Si+1 ) + Jz Si Si+1 − BSi
2 i i+1
(2)

†
ψL (x)ψR (x) ∝

where the spin operators are related to the fermion field
by the Jordan-Wigner transformation
−
Si = (−1)i Ψi exp iπ

nj ,

(3)

j

nosc ∝ sin(2kF x)

with J = 2t, Jz = U and B = µ − U .
The model in Eq. (1) can be analyzed by standard
bosonization techniques in the low-temperature limit.
For low energies we only consider excitations around the
Fermi-points ±kF and introduce left- and right-moving
fermion fields with a linear dispersion relation
Ψ(x) = e−ikF x ψL (x) + eikF x ψR (x).

√1
4π

(∂x φ ± Πφ ) ,

g

,

(8)

πT
v sinh 2πxT /v

.

(9)

g

The Friedel oscillations are exponentially damped with
temperature, because the incoming and outgoing wavefunctions that form the interference pattern lose coherence due to temperature fluctuations. In the limit T → 0
we recover the result of Ref. 19 where a power-law decay
of the Friedel oscillation nosc ∝ 1/xg was predicted.
It is now important to realize that the fermions or spins
are still pinned to a lattice, i.e. x = Integer, which gives
interesting additional effects. In particular, at half-filling
kF = π/2 the Friedel oscillations in Eq. (9) are identically zero sin(πx) = 0 for integer x, which can easily be
understood from particle-hole symmetry (or equivalently
spin-flip symmetry). Half-filling is a natural state for the
spin chains in zero magnetic field, but a small magnetic

(4)

The chiral fermion fields can then be bosonized using the
usual bosonization rules
†
ψL/R ψL/R =

πT
v sinh 2πxT /v

so that the density oscillations are given by

i−1
z
1
Si = n i − 2 ,

(7)

(5)

where Πφ is the conjugate momenta to the boson field φ.
This results in the following boson Hamiltonian density
2

field changes the Fermi vector slightly kF = π/2 + B/v.
In that case, Eq. (9) becomes
nosc ∝ (−1)x sin(2Bx/v)

πT
v sinh 2πxT /v

increases monotonically with the reflection coefficient R.
In fact we can make a firm connection between the relative amplitudes and the reflection coefficients by considering free fermions U = 0 for which we can find the
eigenfunctions exactly even in the presence of impurities.
Clearly the eigenfunctions are given by plane wave solutions |k which contain a special mix of left- and rightmoving components due to the impurity. Just like without impurities there are in fact always two such degenerate orthogonal solutions. We found the solutions for
generic impurity models and looked at the spatial structure of the square of the wave-functions, which contains
an interference pattern of incoming and outgoing waves.
In general, we always find

g

.

(10)

Now, the Friedel oscillations are simply alternating on
the lattice and for distances below the magnetic length
scale x < v/B we can use sin(2Bx/v) → 2Bx/v so that
remarkably the oscillations actually increase with x1−g .
This effect was first observed for the Heisenberg chain
(Jz = J, g = 1/2, v = Jπ/2), where the local susceptibilities χ(x) can be written as17
χ(x) = χ0 − c (−1)x χbs (x),

(11)

| x|k |2 =

with the amplitude of the alternating part given by
√
x T
bs
.
(12)
χ (x) = √
sinh 4xT

1
1+
π

R(k) cos(2kx + 2Φ) ,

(13)

where the summation over the two degenerate solutions
is implied. Here R(k) is the ordinary k-dependent reflection coefficient which has been determined independently
according to text-book methods. Therefore, the magnitude of the interference is exactly given by the square
root of the reflection coefficient, which is maybe not too
surprising but very useful in our analysis. In particular,
when we consider the fermion density at half filling we
are really directly looking at the spatial structure of the
wave-function. We can write near half-filling (i.e. for a
small field B in the spin chain model)

Here χ0 is the bulk susceptibility in the chain20 and we
measure T in units of J. The sign was chosen so that the
(constant) overall amplitude c of the alternating part is
positive. The superscript bs indicates that the alternating susceptibility is due to backscattering. As shown in
Fig. 1 from TMRG simulations there is a characteristic
maximum because the temperature damping eventually
dominates over the increasing oscillations. Clearly, the
expression in Eq. (12) reproduces the shape of this alternating part rather well, although we have neglected
possible logarithmic corrections (multiplicative and additive), which may be responsible for the apparent shift
in the characteristic maximum in Fig. 1. The numerical TMRG results of the local susceptibility near the
open end χbs (x) will be used as the reference data for
a completely backscattering impurity in our studies in
the next section. The numerical data automatically contains all corrections due to irrelevant higher order operators. The logarithmic corrections to Eq. (12) due to
the leading irrelevant operator have a special behavior
near a boundary,21 which we have not tried to predict
for the local susceptibility, but numerically we find that
a possible multiplicative logarithmic correction for χ(x)
appears to have a negative power of ln(x). The maximum in Fig. 1 occurs at x ∝ 1/T with an amplitude
√
χalt ∝ 1/ T , which results in a characteristic feature
in NMR experiments, so that it was possible to confirm
this effect experimentally as well.22 At zero temperature
the ground state has a staggered magnetization which
has a maximum in the center of a finite chain (assuming
an odd number of sites).23 The magnetization for finite
chains with impurities has also recently been analyzed,
which resulted in interesting patterns that reveal the nature of the strong correlations in the system.24
Even for a partially reflecting impurity we expect that
the same alternating contribution as in Eq. (12) due to
backscattering is present, but with an amplitude c that

π/2+B/v

n(x) − 1/2 =

B→0 B
= v

2

| x|k | dk

π/2
2

x| π
2

= B χ0 − cR (−1)x χbs ,

(14)

where we have used the fact that the spin density for
the Heisenberg chain in a small field is just given by the
susceptibility in Eq. (11), but with a coefficient cR which
now depends on the reflection coefficient R near half filling. Together with Eq. (13) we therefore arrive at the
central result that at half-filling the reflection coefficient
is proportional to to the square of the alternating density
amplitude
R=

cR
c

2

,

(15)

where c = cR=1 is the coefficient corresponding to complete backscattering in Eq. (11). We use this formula to
estimate the reflection coefficient from the density oscillations for various impurity models in the following.

3

0.8

H = −t

i=0

(Ψ† Ψi+1 + Ψ† Ψi ) − J ′ (Ψ† Ψ1 + Ψ† Ψ0 ).
0
1
i+1
i
(17)

χ(x) − χ0

0.4

0

J’
-2

−0.4

−0.8

-1

0

1

2

3

FIG. 2. One modified link.

0

20

40

60

80

The wave-functions and reflection coefficient R(k) for
this problem can be calculated exactly, with the result
that

100

x

FIG. 1. Local susceptibility close to the open end of a
spin-1/2 chain from TMRG data for T = 0.04J compared
to Eqs. (11) and (12) with c = 0.51, which was determined
by matching the characteristic maxima.

R(k) =

t4

t4 − 2t2 J ′2 + J ′4
.
− 2t2 J ′2 cos 2k + J ′4

(18)

However, once the interaction U is introduced this problem becomes highly non-trivial and the reflection coefficient renormalizes with temperature T . The interacting
system has been studied in the context of both spinless
fermions1 and the spin-1/2 chain,7 where it was found
that repulsive interactions U > 0 make the perturbation
of one link relevant, so that it renormalizes to a completely reflecting barrier as T → 0. A small weakening of
a link J ′ < t produces a relevant backscattering operator
∼
in the periodic chain of scaling dimension d = g, so that
this link effectively weakens further as the temperature
is lowered. Below a cross-over temperature TK (analogous to a Kondo-temperature) the link has weakened so
much that it is more useful to consider the problem of
two open ends that are weakly coupled, which is now
described by an irrelevant operator of scaling dimension
d = 1/g. Therefore, this coupling weakens further and
ultimately the open boundary condition represents the
stable fixed point as T → 0. The same analysis is also
true for a slight strengthening of a link J ′ > t, because
∼
in this case the two ends lock into a “singlet” state as
the effective coupling grows, and the remaining ends are
weakly coupled with a virtual coupling of order t2 /J ′
which is again irrelevant.
We consider the interacting system with U = 2t, which
we can write in terms of an SU(2) invariant spin Hamiltonian via the Jordan-Wigner transformation in Eq. (3)
with a modified Heisenberg coupling between two spins

As mentioned above there may also be 2kF x density
oscillations due to screening, so that the alternating susceptibility is in general a sum of two parts
χalt (x) ≡ χ(x) − χ0 = (−1)x χscreening (x) − cR χbs (x) .
(16)
In the case of overscreening the neighboring spins (or
electrons) overcompensate the magnetic (or electric) impurity and leave an effective impurity with opposite moment which in turn gets screened by the next nearest
neighbors and so on. This finally results in a screening
cloud. Screening is purely an interaction effect where a
2kF x density oscillation is induced by an “active” impu†
rity Hamiltonian ψL ψR Himp = 0. The 2kF x oscillations due to backscattering, however, are purely an interference effect and are even present in non-interacting
fermion systems. The special shape and the increasing
nature of the alternating part in Eq. (12) for g = 1/2
makes it possible to easily identify the contribution due
to backscattering, so that we can always separate the two
possible effects near half-filling. In what follows we therefore always use the special choice of coupling U = 2t corresponding to the Heisenberg model Jz = J. This model
can be used to demonstrate the generic behavior of impurity effects in mesoscopic systems and also gives experimental consequences for spin-chain compounds. The
Luttinger Liquid parameter takes the value g = 1/2 in
this case, which is the strongest possible interaction at
half-filling before Umklapp scattering becomes relevant.

H=J
i=0

Si · Si+1 + J ′ S0 · S1 .

(19)

We now want to analyze the density oscillation near the
impurity in order to extract the reflection coefficient as
described above. In Fig. 3 we show the amplitude of the
alternating spin density for different coupling strengths
J ′ . Clearly the shape as a function of distance x remains
largely the same as in Fig. 1 for all J ′ so that the functional dependence in Eq. (12) is still adequate, but with
an overall coefficient c which is now related to the reflection coefficient R as postulated in Eq. (15).

III. IMPURITY MODELS
A. One modified link

Maybe the simplest impurity to consider is a weak link
in the chain, i.e. a modified hopping J ′ between two sites
in the chain as shown in Fig. 2
4

0.8

as T → 0. The behavior for couplings close to the periodic fixed point (J ′ > 0.4J) is consistent with Eq. (21).
∼
For smaller couplings the cross-over temperature TK is
larger, and we see an extended region where the scaling
of the stable fixed point with J ′2 and T 2/g−2 in Eq. (20)
holds (here g = 1/2). We can also compare our results
to the findings of Matveev et al in Ref. 6 where an explicit formula for the transmission coefficient was given
1−R ∝ [(D/T )2α R0 /(1−R0) + 1]−1 in terms of the noninteracting reflection coefficient R0 in Eq. (18), a cut-off
D, and a small interaction parameter α = 1/g − 1. Unfortunately, the interaction parameter is large in our case
α = 1 so that this formula does not quantitatively agree
with our findings in Fig. 4. Qualitatively, their result
looks rather similar, but we observe a sharper renormalization at low temperatures near the unstable fixed point
(J ′ > 0.4J). Indeed we find that the region where the
∼
famous scaling in Eq. (20) is valid turns out to be extremely narrow for J ′ > 0.4J.
∼
Another aspect is the high temperature behavior where
the non-interacting reflection coefficient in Eq. (18)
should be approached6 . This is indeed the case near the
unstable fixed point J ′ > 0.4J where the non-interacting
∼
value is quickly reached with high accuracy. However,
near the stable fixed point (J ′ < 0.4J) we find that the
∼
reflection coefficient can renormalize even well below the
non-interacting value, so that the interactions actually
enhance the conductivity at higher temperatures in this
case. The reason for this unexpected behavior is that
the cross-over temperature is larger than the cut-off near
the stable fixed point TK ≫ J, so that the renormalization may continue beyond the bare coupling constants at
higher temperatures.

0.4

(−1)

x+1

alt

χ (x)

0.6

0.2

0

0

20

40

60

x
FIG. 3. Envelope of the alternating susceptibility of the
one-link impurity at T = 0.04J for J ′ /J = 0.0, 0.2, 0.4, 0.6, 0.8
from above.

The reflection coefficient is directly related to the
renormalization behavior above. The basic idea behind
renormalization is to use an effective Hamiltonian with
renormalized parameters as a function of T . To estimate the reflection coefficient it is therefore possible to
make a simplified but intuitive analysis by using the free
fermion result in Eq. (18), but with a renormalized cou˜
pling strength J ′ (T ). Below the cross-over temperature
T < TK , the effective potential is small and given by the
renormalization behavior of the leading irrelevant opera˜
tor J ′ (T ) ∝ J ′ T 1/g−1 . This results in
1 − R ∝ J ′2 T 2/g−2 ,

(20)

which is the universal behavior near the stable fixed point
as first predicted in Ref. 1. Above the cross-over temperature T > TK the renormalization behavior is better
described by a relevant operator on the periodic chain
˜
giving J − J ′ (T ) ∝ (J − J ′ )T g−1 . From this result it
would even seem that we can recover the periodic chain in
the high temperature limit, but it is of course important
to realize that the renormalization is no longer possible
above a cutoff of order J. For an initial bare coupling
J ′ ∼ J very close to the unstable fixed point TK ≪ J we
therefore find that the effective coupling stops renormal˜
izing at its bare value J ′ → J ′ for large T . In summary,
the temperature dependence above TK is not as universal
as in Eq. (20), but we may still write
R ∝ (J − J ′ )2 ,

1

R 0.5

0

(21)

0

0.1

0.2

0.3

0.4

0.5

T/J

for J ′ ∼ J and T > TK .
It is now straightforward to extract the relative coefficient cR /c in Eq. (14) from the numerical data by simply
dividing the amplitude of the alternating part for each
coupling J ′ in Fig. 3 by the reference data of χbs for the
open chain. According to Eq. (15) the square of this relative coefficient then gives the reflection coefficient. Fig. 4
shows the results for the temperature dependent reflection coefficient from our TMRG data. The renormalization to a perfectly reflective barrier can clearly be seen

FIG. 4. Reflection coefficient R of one modified link for
J ′ /J = 0.1, 0.2, 0.4, 0.6, 0.8 from above. The lines are guides
for the eye.

B. Two modified links

We now consider the impurity of two neighboring modified links in the chain as shown in Fig. 5. For the in5

backscattering in Eq. (12). Interestingly, the two contributions have opposite sign, so that the density oscillations vanish at a special distance from the impurity, but
then increase again due to the backscattering contribution. This behavior is shown in Fig. 6 together with a
fit to the two contributions in Eq. (23). The special distance at which the density oscillations vanish grows as
we approach the stable fixed point (J ′ → J or T → 0).
As already with the one-link problem, we use again the
numerical open chain data as a reference for χbs instead
of the more simplified analytical form of the backscattering contribution in Eq. (12) since this minimizes the
corrections due to irrelevant operators. However, even
the analytical form in Eq. (12) gives very good fits, so
that none of our our findings are affected by this choice.

teracting case U = 2t we can again write this model in
terms of a Heisenberg spin chain model
H=J
i=−1,0

Si · Si+1 + J ′ S0 · (S−1 + S1 ) .

(22)

This type of impurity may correspond to a charge island
that is weakly coupled to a mesoscopic wire or to doping
in a quasi-one dimensional compound where one atom in
the chain has been substituted. We have recently considered this type of impurity in the context of doping in
spin-1/2 compounds and as a simple experimental example of the two channel Kondo effect.9 In this section we
analyze the induced density oscillations in more detail,
especially in connection with the reflection coefficient.
J’
-3

J’

-2
-1
0
1
2
FIG. 5. Two modified links.

0.2

3

TMRG data
fit

0.1
alt

χ (x)

The model in Eq. (22) is equally simple as the onelink impurity, but the renormalization behavior is known
to be quite different.7 Already for the non-interacting
case at half filling the system shows a resonant behavior
with perfect transmission R = 0, so that this corresponds
to the simplest case of resonant tunneling considered by
Kane and Fisher5,8 (at half-filling the impurity potential
is automatically tuned to the resonant condition). With
interactions U = 0 the reflection coefficient is no longer
exactly zero, but shows nontheless a renormalization to
perfect transmission as T → 0 in sharp contrast to the
one-link impurity. This difference in renormalization behavior is easily explained by the different parity symmetry of the problem (namely site- instead of link-parity).
For a small perturbation from a periodic chain J ′ ∼ J
the leading operator is now irrelevant with scaling dimension of d = 1 + g, so that a perfectly transmitting chain is
the stable fixed point. For small couplings J ′ > 0 on the
∼
other hand, the leading perturbing operator is marginally
relevant, and the situation is similar to the two channel
Kondo effect where the two ends of the chain play the
role of two independent channels.7,9
Apart from the renormalization behavior there is another key difference between the one- and two-link impurities: In the two-link impurity model there is an “active”
impurity site that carries a spin or charge degree of freedom, which in turn must be screened by the surrounding
system. Therefore, the density oscillations are no longer
simply determined by the backscattering in Eq. (12), but
there is also a so-called screening cloud induced in the
system. From perturbation theory in the leading irrelevant operator the functional dependence of this screening
cloud can be calculated9 and the total alternating density
χalt is a sum of two contributions

0
−0.1
−0.2
0

20

40

60

x
FIG. 6. Alternating part of the local susceptibility for the
two-link impurity for T /J = 0.04 and J ′ /J = 0.6. Fit to
Eq. (23).

It is now straightforward to extract the reflection coefficient from the numerical data with the help of Eq. (15)
and Eq. (23) as shown in Fig. 7. Below a cross-over
temperature TK depending on J ′ the reflection coefficient clearly decreases and eventually approaches perfect
transmission as T → 0. Above TK the renormalization of
the reflection coefficient is rather weak and converges to
a finite constant (never approaching complete reflection
as the temperature increases).

χalt (x) = cI (−1)x ln[coth(xT )] − cR (−1)x χbs (x), (23)
where the first term is the induced screening cloud while
the second term is the familiar contribution due to
6

1

havior of the two channel Kondo fixed point. Note, that
although the coefficient cI is finite as T → 0, the screening cloud itself diverges logarithmically with − ln(xT ),
which is a clear indication of the famous over-screening
in the two channel Kondo effect. As we approach the
unstable fixed point the order of limits becomes crucial: For zero coupling there is no screening cloud at
all limT →0 limJ ′ →0 cI = 0, while for zero temperature
the coefficient becomes infinite limJ ′ →0 limT →0 cI = ∞.
Remarkably, exactly at zero temperature a minute perturbation therefore induces an infinite screening cloud,
although this behavior occurs in an unphysical limit.

R 0.5

0

0

0.1

0.2

0.3

0.4

0.5

T/J
C. Impurity at the edge

FIG. 7. Reflection coefficient R of the two-link impurity
for J ′ /J = 0.05, 0.1, 0.2, 0.4, 0.6, 0.8 from above. The lines
are guides for the eye.

Another category of impurities we can consider are imperfections near the end of a chain. In this case the
boundary always gives complete backscattering, but as
we will see the impurity can still give interesting effects
on the density oscillations. The simplest case to consider
is a modified link at the edge of a chain as depicted in
Fig. 9. For the interacting case U = 2t it is again useful to write the Hamiltonian in terms of the Heisenberg
spin-chain model

Equally interesting is the induced screening cloud. In
this case, the coefficient cI approaches a constant as
T < TK as it should, since this contribution was determined from perturbation theory around the stable fixed
point. Above the cross-over temperature, however, this
contribution vanishes quickly. This behavior is shown
in Fig. 8: In general the behavior of the coefficient cI
vs. J ′ is temperature dependent and cI increases as the
temperature is lowered. However, as T ≪ TK all curves
approach a limiting value, which gives a universal behavior as a function of J ′ (thick line).

H=J

i=1

0.6

Si · Si+1 + J ′ S0 · S1 .

(24)

J’
J’/J=0.1
J’/J=0.2
J’/J=0.4
J’/J=0.6
J’/J=0.8

0.2

cI

0.4

0.1

cI

0

0

0.1

0.2

0.3

0

0.2

0.4

0.6

0.8

2

3

4

5

Just like the two-link impurity was related to the twochannel Kondo problem, we can identify the field theory
description of the edge impurity model with the regular one-channel Kondo problem. There are two possible
fixed points: The case J ′ = 0 corresponds to the unstable fixed point of a decoupled spin at the end of a chain
with a marginally relevant perturbation for J ′ > 0. The
∼
case J ′ = J corresponds to the completely screened spin,
which is a stable fixed point with a leading irrelevant
operator of scaling dimension d = 2. Just like in the ordinary Kondo effect both fixed points are represented by
the same boundary condition and differ only by a simple
π/2 phase shift on the fermions. (The infinite coupling
fixed point J ′ → ∞ is also stable, but is actually absolutely equivalent to the J ′ = J fixed point since both
cases represent a π/2 phase shift on the fermions by removing or adding a site, respectively). For intermediate
couplings the phase shift Φ takes on values between 0 and
π/2 which will be reflected in the backscattering contribution of the density oscillations as we will see below.
A screening cloud for the impurity spin at the end
should also be present in this model, but with a different

0.4

T/J

0

1

FIG. 9. Edge impurity.

0.2

0

∞

1

J’/J
FIG. 8. Coefficient cI vs J ′ of the two link impurity for
different temperatures T /J = 0.2, 0.1, 0.04, 0.025, 0.0167, 0.01
from below. For J ′ ≈ J and/or low temperatures cI approaches a universal T -independent curve (thick line). Inset:
cI vs T . The lines are guides for the eye.

The competing contributions in Eq. (23) have the opposite renormalization behavior: Above TK backscattering is constant, while the screening cloud is reduced
which is the open chain behavior. Below TK on the other
hand backscattering is reduced, while the coefficient for
the induced screening cloud is constant, which is the be-

7

π/2

behavior than for the overscreened case in Eq. (23). Instead we find that the leading operator that causes the
screening cloud is the same as that for an edge magnetic
field in the xxz-chain which has been analyzed in Ref. 25,
so we can use the corresponding result for the shape of
the induced screening cloud. Taking into account finite
temperatures and the phase shift on the fermions we can
write for the density oscillations
√
(−1)x T
alt
− cos(πx + 2Φ) c χbs (x), (25)
χ (x) = cI
sinh(4xT )

T=0.01J

Φ

π/4
T=0.25J

0

where the first term is the induced screening cloud, while
the second term is the backscattering contribution in
Eq. (12) but with a phase shift Φ. However, the coefficient c always takes the value corresponding to complete
backscattering in Eq. (11). There is also an implied shift
of 2Φ/π in the argument of χbs , which we used for a selfconsistent fitting. The effective boundary condition in
the continuum limit is therefore technically between two
lattice sites (although it is not really that meaningful to
define locations on the scale of less than a lattice spacing
in the continuum limit theory anyway).
Figure 10 shows the envelope of the alternating part of
the susceptibility for temperature T = 0.04J and different couplings J ′ , which always fits well to the superposition in Eq. (25). At the fixed points J ′ = 0 and J ′ = J
there is no screening, but the backscattering contribution
has opposite signs due to the π/2 phase shift.

0

0.2

0.4

0.8

The screening cloud coefficient cI again approaches a
constant as we lower the temperature below TK as shown
in Fig. 12. Although formally the behavior looks similar
to the over-screened case of the two link problem in Fig. 8
it is important to realize that now the screening cloud in
Eq. (25) is finite as T → 0 and drops off with 1/x (while
in the two link case the screening cloud was divergent
with ln xT ).
J’/J=0.1
J’/J=0.2
J’/J=0.4
J’/J=0.6
J’/J=0.8

4

10
cI

3
2

TMRG data
fit

0.4

1

FIG. 11. Phase shift of alternating part of the edge impurity for T /J = 0.25, 0.1, 0.04, 0.02, 0.01 from below. The lines
are guides for the eye.

0.8

1

cI

0

5

0

0.1

0.2

T/J

(−1)

(x+1)

alt

χ (x)

0.6

J’/J

0

−0.4

0

0

−0.8

0.2

0.4

0.6

0.8

1

J’/J
0

20

40

60

FIG. 12. Coefficient cI vs. J ′ of the edge impurity for
T /J = 0.2, 0.1, 0.04, 0.02, 0.133, 0.01 from below. The lines
are a guide for the eye.

x

FIG. 10. Alternating susceptibility for the edge impurity
at T /J = 0.04 for J ′ /J = 0, 0.1, 0.2, 0.4, 0.6, 1.0 from above.
Fits to Eq. (25).

D. Generalized two link impurity

It is now straightforward to extract the screening cloud
amplitude cI and the phase shift Φ from our numerical
data for all temperatures and couplings J ′ . As expected
we find that the phase shift increases with J ′ and renormalizes to larger values as the temperature is lowered as
shown in Fig. 11. In the limit of very low temperatures
the jump to the stable fixed point value Φ = π/2 becomes
more abrupt as a function of J ′ .

It is now instructive to summarize the findings of the
three impurity models in the previous subsections by considering one generalized two link impurity model that is
not symmetric as shown in Fig. 13
H=J
i=−1,0

Si · Si+1 + J1 S−1 · S0 + J2 S0 · S1 .

(26)

The three impurity cases above can be identified easily:
8

• J2 = J1 = J one modified link in Eq. (19)

J2

• J1 = J2 = J two modified links in Eq. (22)

1

• J1 = 0, J2 = J edge impurity in Eq. (24)
The density oscillations for the more general model in
Eq. (26) are much more complex than in the special cases,
so that a detailed analysis of this effect is not always useful. The renormalization behavior on the other hand is
straightforward and can be read off from what we already
know about the special cases.

J1
-3

0

J2

0

-2
-1
0
1
2
3
FIG. 13. Generalized two-link impurity.

A weak coupling J1 > 0 and J2 > 0 to an additional
∼
∼
site is always marginally relevant, so that the open chain
with a decoupled impurity site is unstable for any antiferromagnetic coupling (i.e. negative hopping probability).
The periodic chain on the other hand is only stable for
the special site-parity symmetric case J1 = J2 , where the
renormalization behavior is analogous to the two channel
Kondo effect. In general, however, one of the two couplings is larger and renormalizes to unity, absorbing the
spin. The smaller coupling is then irrelevant as in the
one-weak problem, so that the stable fixed point is an
open chain with an absorbed impurity site J1 = J, J2 = 0
(or J2 = J, J1 = 0) in most cases, except for a site-parity
symmetric impurity or two ferromagnetic coupling constants. The complete renormalization flow is summarized
in Fig. 14 where the possible fixed points are indicated by
the black dots. In cases where the coupling diverges to infinity a singlet forms, and we can therefore again describe
the system by one of the four finite fixed points in the
figure. Interestingly, the more stable fixed points always
have a lower ground state degeneracy, in accordance with
the g-theorem.26 The phase diagram in Fig. 14 is valid
for all interaction strengths 0 < U ≤ 2t as long as the
system is half-filled.

1

J1

FIG. 14. Renormalization flow diagram.

E. Spin-1 impurity

We now turn to a magnetic impurity in the chain with
spin Simp = 1 given by the Heisenberg Hamiltonian
H =J
i=0

Si · Si+1 + J ′ Simp · (S0 + S1 ) .

(27)

as shown in Fig. 15. In the previous impurity models
in Sections III A-III D it was always possible to interpret the Heisenberg Hamiltonians equally well in terms
of mesoscopic systems and electrons hopping on the lattice by identifying the spin-1/2 impurity in terms of an
extra site or charge island. However, for the spin-1 impurity in Eq. (27) no meaningful interpretation in terms
of spinless fermions is possible. On the other hand this
impurity model has important implications for doping in
quasi one-dimensional spin-1/2 compounds, so that we
find it useful to discuss it here.

S imp=1
J’
-2

-1

0

J’
1

2

3

FIG. 15. The spin-1 impurity.

Similar to the impurity models in Sections III B and
III C we find again that the field theory language is analogous to a Kondo impurity model. The two ends of the
spin-chain play the role of the two channels coupled to
a spin-1 impurity. A small antiferromagnetic coupling
is therefore marginally relevant and the renormalization
flow goes to the strong coupling limit. The stable fixed
point is given by an open spin chain with two sites re-

9

20

moved and a decoupled singlet containing the spin-1 and
the two end spins (J ′ → ∞).
Just like the edge impurity in Sec. III C this Kondotype model is an exactly screened impurity. The shape
of the screening cloud is again given by that of an edge
magnetic field25 just like in Eq. (25)
√
(−1)x T
alt
χ (x) = cI
− cR (−1)x χbs (x),
(28)
sinh(4xT )

cR

−1

0

alt

χ (x)
(x+1)

(−1)

2

0

1

2

FIG. 17. Coefficient cI of the spin-1 impurity for T /J =
0.2, 0.1, 0.04, 0.025, 0.0167, 0.0133, 0.01 from below. Inset:
Backscattering coefficient cR . The dashed lines are a guide
for the eye.

More interesting are the experimental consequences for
Knight shift experiments in doped spin-1/2 chain compounds (as for example Ni doping in CuO chains). For
that case we can predict an interesting NMR spectrum
with a characteristic feature (sharp edge) corresponding
to the maximum in the alternating susceptibility. Such
a sharp edge has been observed before in NMR experiments on spin-1/2 chain compounds with non-magnetic
defects.22 In that case the sharp edge broadens with a
√
1/ T behavior as discussed in Sec. II. For the magnetic spin-1 impurities a sharp edge from the maximum
in the backscattering part may also be present, but it
depends on if the temperature is above or below TK how
this feature changes. Above TK the backscattering part
becomes weaker as the temperature is lowered, but the
induced screening cloud increases, so that the sharp kink
may vanish in a quickly broadening line-shape from the
screening cloud as shown in the left part of Fig. 18. Below TK on the other hand, the screening has saturated
and the backscattering contribution dominates again (albeit with a phase shift). Therefore, the kink feature in
the NMR spectrum will sharpen further as the tempera√
ture is lowered and widen with the usual 1/ T behavior
as shown in the right part of Fig. 18. The detailed Tdependence can be predicted for any particular value of
J ′ of an actual experimental compound.

TMRG data
fit

−1

40

1

J’/J

0

20

0

J’/J

0.5

0

0
−0.5

cI 10

1

−1.5

T/J=0.2
T/J=0.1
T/J=0.04
T/J=0.02
T/J=0.01

0.5

where the first term is again the induced screening cloud,
while the second term is the backscattering contribution
in Eq. (12). As shown in Fig. 16 the fits to this expression are excellent (again using the open chain data as
a reference for χbs ). The coefficient cI for the induced
screening cloud again approaches a constant for temperatures below TK which results in a universal curve as
T → 0 as shown in Fig. 17. The backscattering coefficient is an indication of the effective phase shift and
changes sign depending on the temperature and coupling
strength. From Fig. 16 it is clear that the backscattering coefficient cR is positive for small coupling strengths
J ′ (or equivalently high temperatures) and negative for
larger coupling strengths J ′ (or equivalently lower temperatures). The renormalization of cR is explicitly shown
in the inset of Fig. 17. As T → 0 the jump of cR to
negative values happens at smaller J ′ and becomes very
sharp.

−0.5

1

60

x

FIG. 16. Envelope of alternating part at T /J = 0.04
for the spin-1 impurity.
From above at x > 20:
∼
J ′ /J = 0, 0.05, 0.1, 0.2, 0.4, 0.8, 4.0. Fits to Eq. (28).

10

Eq. (29) was of course constructed in a way to avoid all
backscattering, but it is remarkable that even the induced
alternating part from the magnetic impurity vanishes exactly, i.e. no conventional screening takes place.
Nonetheless, the impurity spin is somehow reduced
from a spin-1 to an effective spin-1/2 as the temperature
is lowered. This can be explicitly seen from the impurity
susceptibility in small magnetic fields

NMR signal

T/J=0.2
T/J=0.067
T/J=0.04
T/J=0.02

CCurie
(30)
T
where we have assumed some type of Curie-law. At high
temperatures the impurity susceptibility must follow the
Curie-law for a spin-1 CCurie = 2/3, while at low temperatures a Curie-law for a spin-1/2 CCurie = 1/4 has been
predicted up to logarithmic corrections.28 In Fig. 20 we
plot the temperature dependent Curie constant (i.e. the
impurity susceptibility times temperature). It appears
that the asymptotic value CCurie = 1/4 is indeed approached with logarithmic corrections as T → 0. The fit
in the figure is
z
Simp = B

B

B

FIG. 18. NMR signal of the spin-1 impurity for J ′ /J = 0.1
(left) and J ′ /J = 1.4 (right).

F. Integrable impurity model

Finally, we would like to consider a more exotic impurity model which has been especially constructed to
preserve the integrability of the entire system.27 We consider here the simplest non-trivial example of such an
impurity model which corresponds to an impurity spin
with Simp = 1 that is coupled in a special way to two
sites in the chain. The corresponding Hamiltonian has
been set up in Ref. 27
H=J
i=0

Si · Si+1 −

7J
9 S0

· S1

CCurie =

1
1
ln (ln(2π/T )/b)
+
+a
4 8 ln(2π/T )
ln(2π/T )2

(31)

with a = 1.62 and b = 1.32.

0.45

(29)
CCurie

+ 4J [(S0 + S1 ) · Simp + {S0 · Simp , S1 · Simp }] ,
9
where Simp is the external spin-1 impurity and {, } denotes the anticommutator.

TMRG data
fit

0.35

S imp = 1
0.25

-2

-1

0

1

2

3

0

0.1

0.2
T

0.3

0.4

FIG. 20. Susceptibility of the spin-1 in the integrable model
multiplied by temperature. Fit to Eq. (31).

FIG. 19. The integrable impurity.

A closer analysis of this model28 showed that the thermodynamics at low temperatures were in fact described
by a periodic spin chain with one additional site and an
asymptotically free impurity spin with S = 1/2, so that
it appears that the original spin-1 has somehow been partially absorbed by the chain. From a field theory point
of view it was later shown that this type of impurity corresponds in fact to an unstable fixed point which can
only be reached by an artificial tuning of the coupling
parameters.29
We are now interested in what kind of density oscillations might be observable from such an impurity. Interestingly, we found that the density oscillations were
identically zero at all temperatures as if the system was
translationally invariant. The impurity Hamiltonian in

IV. NUMERICAL METHOD

The numerical method we have used here is based on
the Density Matrix Renormalization Group (DMRG)30
applied to transfer matrices. While the ordinary DMRG
considers the properties of individual eigenstates in a
finite system, we are interested in the thermodynamic
limit, namely properties of an infinite system at finite
temperatures. This can be achieved by the Transfer
Matrix Renormalization Group (TMRG),12 which we
adapted especially for impurities9,11 as we will review
briefly. We consider the partition function Z of the models in Eqs. (1) and (2). After the standard Trotter decomposition, we obtain for an infinite system (L → ∞)
11

L/2

Z = lim trTM
M→∞

L/2

→ lim λM ,

z
Sj

(32)

M→∞

sz
ψM |TM (j)|ψM
1
z
,
tr Sj e−βH →
Z
λM

ǫ = 0.06 exp(−58T ),

sz
ψM |TM (TM )j/2 Timp |ψM
j/2+1

λM

ψM |Timp |ψM

.

(35)

(36)

where we have kept 64 states in the TMRG simulations. For interacting fermions the suppression also has
the exponential dependence in Eq. (35), but the energy scale ǫ is in general dependent on the interaction
U . For the Heisenberg model an independent analysis
of the free energy hinted at a value of approximately
ǫ = 0.02 exp(−34T ), but the value in Eq. (36) is more
reliable and gives a relatively good estimate of the error
for all interaction strengths. We chose to correct our data
for the alternating fermion densities by dividing out the
factor in Eq. (35) together with the estimate in Eq. (36)
in all cases presented above. However, the use of this
correction or the particular choice of the error ǫ makes
no qualitative difference in any of our findings, since the
temperature suppression always dominates (i.e. the energy scale in Eq. (36) is always smaller than the temperature). Another important energy scale is the finite
magnetic field B that is used in the simulations (i.e. how
close the system is to half filling). We typically used a
value of B = 0.003 which makes the magnetic length
scale in Eq. (10) always negligible compared to the finite
temperature correlation length.

(33)

where ψM | and |ψM are the left and right target states
for the eigenvalue λM . So far we have considered a translational invariant system.
We now introduce a generic impurity which modifies
one of the transfer matrices TM → Timp . Even in the
presence of impurities the thermodynamics of the system is entirely determined by the highest eigenvalue λM
and corresponding eigenstate of the pure transfer matrix TM which always appears with an infinite power in
the partition function in Eq. (32). The measurement of
the spin (or charge) density near the impurity is again
straightforward. For the spin density at a distance of j
sites from the impurity we write
z
Sj =

∝ (1 − ǫ)j = exp(−jǫ)

where ǫ depends only on temperature. This exponential
suppression with the distance from the boundary is again
a consequence of the fact that the incoming and outgoing waves lose coherence but this time due to error fluctuations. However, the corresponding energy scale from
the truncation error is always smaller than the temperature in our case. We observe that the suppression error
in Eq. (35) is actually very systematic, so that we can
even correct our data very well using Eq. (35). For free
fermions we find to high accuracy the following dependence of the error on temperature

where TM is the transfer matrix with M time-slices. In
the limit of infinite system size only the largest eigenvalue
λM determines the thermodynamics of the system, which
we find numerically. We start with small time-steps so
that the Trotter-error is negligible, and successively increase the number of time-slices M to reach lower temperatures. At each step the dimension of TM increases
so we keep only the most important states to describe
the state with the highest eigenvalue λM by using the
DMRG algorithm with some modifications for asymmetric matrices.11 A measurement of the local spin-density
at site j for example is straightforward, since we can just
z
absorb the measuring operator Sj into one of the transfer
sz
matrices TM → TM
z
Sj =

osc

(34)

Since we step-wise approximate the transfer matrix,
it is important to make a careful error-analysis. The
error due to the Trotter approximation is the simplest
to estimate since it is just proportional to the square
of the time-step τ = 1/T M . We found that a value of
τ = 0.05/J makes this error negligible compared to the
DMRG truncation errors. To estimate the truncation
errors we can compare our results to the exact solution
of the free fermion Hamiltonian in Eq. (1) with U = 0.
The structure of the transfer matrix is not fundamentally
changed by taking U = 0 so that the truncation error will
be of the same order as for U = 0. Keeping 64 states we
find for the local response of the spins closest to typical
impurities a relative error of less than 10−4 for T > 0.04,
less than 10−3 for 0.02 < T < 0.04 and a relative error of
less than 10−2 for temperatures 0.01 < T < 0.02. However, already from Eq. (34) it is clear that the spin and
charge densities far away from the impurity will contain a
larger error. Each transfer matrix contains a small error
ǫ which then gets exponentiated in Eq. (34) and hence
z
the oscillating part of the density Sj is suppressed exponentially with distance j

V. CONCLUSION

We have considered a number of impurity models
and were able to extract detailed information about
the backscattering amplitude, the backscattering phaseshift, and the impurity screening effects by examining
the Friedel oscillations. The results for the various impurities have direct and indirect implications for a large
number of theoretical models and experimental systems
as we will summarize below.
A. Kondo-type impurities

Kondo impurity problems are maybe the most famous
examples of impurity renormalization effects ever since
the classic work by Wilson.31 Many of the impurity models we have considered here are analogous to Kondo impurity problems in terms of the field theory language.
12

0.4

In particular, the field theory description of a Heisenberg
chain is the same as that of the spin-channel for a spin-full
electron field (while the charge excitations are neglected).
Moreover, it is known that coupling the open end of a
Heisenberg chain to an impurity spin produces the same
impurity operators as in the real Kondo problem.7,32 The
number of channels in the equivalent Kondo problems is
given by the open ends that the impurity spin is connected to (e.g. the two link impurity in Sec. III B is analogous to the two channel S=1/2 Kondo problem). It
is important to realize that the Heisenberg spins in the
chains that we consider here have different expressions
in terms of the boson fields than the real electron spins
in the full three dimensional Kondo problems. Nonetheless, we can still use our models to gain some insight into
the central aspects of renormalization, scaling, cross-over
temperature, and screening clouds.
We have shown that the Kondo-type impurities indeed
show the expected renormalization to a screened impurity spin. In particular, we have found a diverging screening cloud (and vanishing backscattering) for the overscreened case in Sec. III B, while the exactly screened
cases in Sec. III C and III E are characterized by a finite
screening cloud and a phase shift in the backscattering
as T → 0.
To analyze the renormalization process more quantitatively it is important to introduce the concept of scaling.
It can be expected that the impurity introduces a new
energy scale that depends on the initial bare coupling
constants. Commonly this energy scale is referred to as
the cross-over temperature TK . By making use of scale
invariance it is then possible to describe the renormalization process universally in terms of the single parameter T /TK . In particular, impurity properties like the
impurity susceptibility are described by a universal scaling function χimp = f (T /TK )/T , which is valid for all
T and TK below the cut-off. This behavior was demonstrated explicitly before for the two weak link problem9
and works for all Kondo-type impurities in this paper
(not shown). In fact it is possible to extract the crossover temperature TK up to an arbitrary overall scale explicitly by collapsing the data according to the scaling
analysis.9,11 We have determined TK this way as a function of coupling J ′ in each case as shown in Fig. 21 (up
to an arbitrary overall scale). The Kondo temperature
shows the same exponential dependence for small J ′
TK ∝ exp(−0.85J/J ′ )

two link impurity
spin−1 impurity
edge impurity
spin−1 coupled to one end
exp(−0.85J/J’)

0.3

TK/J

0.2

0.1

0

0

0.1

0.2

0.3 0.4
J’/J

0.5

0.6

FIG. 21. Crossover temperature TK of four different
Kondo-type impurities. TK has been multiplied by arbitrary
constants in order to compare the four cases.

More interesting in the context of the density oscillations is maybe the scaling of the screening cloud. As the
screening cloud we define that part of the alternating density that is induced by the magnetic impurity, labeled by
cI in Eqs. (23), (25), and (28). In Ref. 4 it was postulated
that the screening cloud in the real Kondo effect should
be a function of the scaling variables xT and T /TK . In
our cases we can make a similar argument except that we
need to include an overall factor T g−1 to account for the
dimensionality of the correlation functions. We therefore
obtain the following scaling law
χscreening = T g−1 f (xT, T /TK ).

(38)

Indeed we find that the shape of the screening cloud is
not affected by TK and can always be expressed as a
function of the scaling variable xT . The coefficient cI
must therefore be a function of T /TK multiplied by appropriate powers of T . As an example we can take the
two link problem at g = 1/2 with the screening cloud
given in Eq. (23), √
where the coefficient can be written
as cI = f (TK /T )/ T with some function f . In Fig. 22
we replot the coefficient cI analogous to Fig. 8 but with
the argument replaced by TK /T instead of J ′ . The inset shows that the data indeed collapses if multiplied by
√
T as implied by Eq. (38). The solid line in Fig. 8 there√
fore is proportional to 1/ TK and diverges exponentially
with J ′ according to Eq. (37). Similar arguments can be
made for the coefficients cI in the screening clouds of the
exactly screened cases in Eqs. (25) and (28), except that
cI = f (TK /T )/T and the solid line is proportional to
1/TK in that case.

(37)

as shown in Fig. 21 (coming from the same marginally
relevant operator at the unstable fixed point in all cases).
The underscreened case of a spin-1 coupled to the end of
one chain has also been included in Fig. 21 for completeness.

13

0.6

C. Impurities in Mesoscopic systems
0.05

Finally, our analysis also allows us to draw important conclusions for transport measurements in one dimensional mesoscopic structures. This is probably the
first time that the conductivity could be explicitly extracted from numerical data for Luttinger Liquid type
models. Not surprisingly, we found that a generic impurity indeed renormalizes to complete backscattering as
the temperature is lowered, and we also could explicitly
observe the “healing effect” in the symmetric resonant
tunneling case as predicted by Kane and Fisher.1,5,8 Our
numerical results not only confirm the asymptotic powerlaws, but also give a quantitative estimate of the conductivity for all temperatures and impurity strengths. For
a generic impurity with little or intermediate backscattering we find that the asymptotic scaling region turns
out to be extremely narrow. For impurities with strong
backscattering we find that the conductivity is enhanced
by interactions at higher temperatures.
One obvious question is how those results can be generalized to spinful electron systems and carbon nanotubes.
A number of works have addressed the question of impurities in spinful wires5,6,33,34 and found a richer structure since renormalization takes place in both the spin
and the charge channels. However, if realistic SU(2) invariant interactions are assumed the generic behavior is
very similar to the spinless case, so that we expect that
our results for the reflection coefficient carry over in a
straight forward fashion. The shape and amplitude of
the density oscillations, however, will in general be very
different for spinful electron systems. For carbon nanotubes it has been shown that the Friedel oscillations
impose a characteristic pattern that can be observed with
scanning tunneling microscopy.35 For spinful wires it is
expected that the Friedel oscillations from an open end
can reveal the nature of the spin-charge separation in
real space.36 Although our results do not allow for quantitative predictions of the density oscillations in spinful
systems, we generally expect that strong, long-range density oscillations should be present from backscattering in
one-dimension. One experimental consequence of those
oscillations is that the measurement through a lead close
to an impurity is very sensitive to the exact location.
Previous studies have shown that even the distance between two leads can play a crucial role.37 The current
may be strongly enhanced or depleted, depending on if
the distance to the impurity is a multiple of 2kF x or not.
Especially interesting are therefore experiments with an
adjustable lead such as a tunneling tip. The direct observation of those oscillations could give detailed information about both the nature of the impurity and also
about the interactions in the system.

1/2

T cI

0.4

cI

0

0

0.2

0

10

20

TK/T

0

10
TK/T

20

FIG. 22. The coefficient cI for the two link impurity in
Eq. (23) as a function of TK /T for different temperatures
T /J = 0.2, 0.1, 0.04, 0.025, 0.0167, 0.01 from below.

B. Doping in spin chains

Our results also have immediate experimental consequences for impurities in spin-chain compounds such as
KCuF3 or Sr2 CuO3 . The spin density oscillations are directly linked to the local Knight shifts (susceptibilities)
close to the corresponding impurities, which can be measured by standard NMR techniques or muon spin resonance. NMR experiments have already successfully detected the sharp feature corresponding to the maximum
in Fig. 1 from open boundaries due to non-magnetic defects that were naturally present in the crystal.22 We now
propose to use intentional doping with magnetic or nonmagnetic impurities to see the predicted renormalization
effects. Impurities of one or two modified links in the
chain can possibly be created by doping the surrounding
non-magnetic atoms in the crystal at link or site parity
symmetric locations. The spin-1 impurities in Sec. III E
could be produced in a more straightforward way by substituting Cu ions by Ni ions in the corresponding compounds. In Sec. III E we discussed explicitly how the
renormalization effects for spin-1 impurities would show
up in an actual experiment. Similar arguments can also
be made for the two link9 or one link impurities by simply
using the analytic form of the corresponding alternating
spin densities with the coefficients cR and cI that we have
calculated.
In general we find a strong enhancement of the antiferromagnetic order near impurities. This enhancement can
also be observed in higher dimensions2 and may have important consequences for impurity-impurity interactions.
In one dimension this effect is strongest, but the complex
functional dependence we found here is often beyond the
intuitive explanation in terms of valence bond states.2

14

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ACKNOWLEDGMENTS

S.R. acknowledges support from the Swedish Foundation for International Cooperation in Research and
Higher Education (STINT). S.E. is thankful for the support from the Swedish Natural Science Research Council
through the research grants F-AA/FU 12288-301 and SAA/FO 12288-302.

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15