The decay of the density oscillations induced by a
defect is a long-standing problem in solid state physics.
This phenomenon, called Friedel or Ruderman-Kittel oscillations
(depending on the context),
is closely related to the singularity in the response function
for wave-vectors
close to 
.
It is expected that asymptotically, the induced density decays as
Using the DMRG, we have computed 
for a system of 200 sites
and various
interaction strengths and, as a test for the accuracy of our calculation,
for systems with M=500
and 
.
The impurity is chosen antisymmetrically for technical reasons,
.
We consider a half-filled band, i.e. 
,
where j is the distance (in units of the lattice spacing) from the defect.
A sample of our results is shown in Fig. 1,
where we plot, on a logarithmic scale, the magnitude 
versus
distance.
Clearly, it is possible to extract the exponent 
without difficulty.
We emphasize that the algebraic decay starts already at a few lattice site
sites.
Typically, we have used a basis of m=120 (200) states for the M=200 (500)
system,
and performed four sweeps through the lattice.
Figure 1: Decay of the Friedel oscillations induced by weak impurities,
located summetrically at the ends of the chain
( 
).
The increase for large j arises due to the finiteness of the chain.
The calculations are performed at half filling, N=M/2,
keeping m=120 (200) states
per block for M=200 (500).
The amplitude of the oscillation vanishes for 
.
The exponent as a function of the nearest-neighbor interaction V
is given in Fig. 2.
The exponent decreases with increasing repulsive interaction,
and increases with attractive interaction, compared to the value for
non-interacting fermions, 
(one dimension!).
Figure: The exponent vs. interaction,
for the same impurity as in Fig. 1.
The continous line is the asymptotic result [20].
Qualitatively, this trend agrees with the prediction [19]
based on the Luttinger liquid.
In a recent preprint [20],
was related to the ``dressed charge'' of the (clean) model, with the result
, where 
, related to V through
,
parameterizes the interaction.
The expression 
is also shown in Fig. 2,
and is in almost perfect agreement with our numerical data,
except for V>1 where we find that the oscillations decay
more weakly than predicted.
This seems to be related to the crossover (for a weak impurity, and V>0)
found
in [19], i.e. for the system sizes studied we may not yet be
in the asymptotic regime.
We have preliminary results showing that for a strong
impurity, tends to increase towards the asymptotic
result given in [20].
