

We study the Kondo model a magnetic impurity coupled to a onedimensional wire
via exchange coupling by using Wilson's numerical renormalization group
technique. By applying an approach similar to which was used to compute the
twoimpurity problem we managed to improve the poor spatial resolution of the
numerical renormalization group method. In this way we have calculated the
impurityspin conductionelectronspin correlation function which is a measure of the Kondo compensation cloud whose existence has been a longstanding problem in solidstate physics. We also present results on the temperature dependence of the Kondo correlations.

Quantum mechanical phase coherence in mesoscopic structures
is destroyed by inelastic processes, where excitations such as
spin waves, electronhole excitations, phonons, etc., are
created in the environment, leading to
dephasing and loss of quantum coherence after a time
τ_{φ}.
In some weak localization measurements of the dephasing time τ_{φ}
down to very low temperatures, a surprising saturation
of τ_{φ} has been observed.
This unexpected saturation remained a puzzle for a long time until
recently, when further experiments on mesoscopic quantum wires
confirmed that the most likely candidates to produce this surprising
behavior are magnetic impurities. These magnetic
impurities seem to be present even in samples of extreme purity, and
unavoidably lead to inelastic scattering and the dephasing.
We use the numerical renormalization group method to calculate the
singleparticle matrix elements of the manybody T matrix of the
conduction electrons scattered by a magnetic impurity at zero
temperature. Since T determines both the total and the elastic,
spindiagonal scattering cross sections, we are able to compute the
full energy, spin, and magnetic field dependence of the inelastic
scattering cross section σ_{inel}(ω). We find an
almost linear frequency dependence of σ_{inel}(ω) below the Kondo temperature T_{K}, which crosses over to a ω^{2} behavior only at extremely low energies. Our method can be generalized to other quantum impurity models.

Recently the electron dephasing and energy relaxation due to different
magnetic impurities have been extensively investigated experimentally
in thin wires and in this Letter these quantities are theoretically
studied. It was shown earlier that a magnetic impurity in a metallic
host with strong spinorbit interaction experiences a surface
anisotropy of the form H=K_{d} (n S)^{2} which causes size effects for impurities with integer spin. Here we show that the dephasing and the energy relaxation are influenced by the surface anisotropy in very different ways for integer spin having a singlet ground state. That must result also in strong size effects and may resolve the puzzle between the concentrations estimated from the two kind of experiments.
The transport in mesoscopic wires with large applied bias voltage has recently attracted great interest by measuring the energy distribution of the electrons at a given point of the wire, in Saclay. In the diffusive limit with negligible energy relaxation that shows two sharp steps at the Fermi energies of the two contacts, which are broadened due to the energy relaxation. In some of the experiments the broadening is reflecting an anomalous energy relaxation rate proportional to E^{2} instead of E^{3/2} valid for Coulomb electronelectron interaction, where E is the energy transfer. Later it has been suggested that such relaxation rate can be due to electronelectron interaction mediated by Kondo impurities. In the present paper the latter is systematically studied in the logarithmic approximation valid above the Kondo temperature. In the case of large applied bias voltage Kondo resonances are formed at the steps of the distribution function and they are narrowed by increasing the bias. An additional Korringa energy broadening occurs for the spins which smears the Kondo resonances, and the renormalized coupling can be replaced by a smooth but essentially enhanced average coupling (factor of 810). Thus the experimental data can be described by formulas without logarithmic Kondo corrections, but with enhanced coupling. In certain regions of large bias, that averaged coupling depends weakly on the bias. In those cases the distribution function depends only on the ratio of the electron energy and the bias, showing scaling behavior. The impurity concentrations estimated from those experiments and other dephasing experiments can be very different, and a possible explanation considering the surface spin anisotropy due to strong spinorbit interaction is the subject of our earlier paper.

We study how the formation of the Kondo compensation cloud influences the dynamical properties of a magnetic impurity that tunnels between two positions in a metal. The Kondo effect dynamically generates a strong tunneling impurityconduction electron coupling, changes the temperature dependence of the tunneling rate, and may ultimately result in the destruction of the coherent motion of the particle at zero temperature. We find an interesting twochannel Kondo fixed point as well for a vanishing overlap between the electronic states that screen the magnetic impurity. We propose experiments where the predicted features could be observed.


