Johann Kroha
In diffusive Cu and Au quantum wires at finite transport voltage $U$ the non-equilibrium distribution function $f(E,U)$ exhibits scaling behavior, $f(E,U)=f(E/eU)$, indicating anomalous energy relaxation processes in these wires. We show that in nonequilibrium the Kondo effect, generated either by magnetic impurities (single-channel Kondo effect) or possibly by non-magnetic, degenerate two-level systems (two-channel Kondo effect), produces such scaling behavior as a consequence of a Korringa-like (pseudo)spin relaxation rate $\propto U$ and of damped powerlaw behavior of the impurity spectral density as a remnant of the Kondo strong coupling regime at low temperature but high bias. The theoretical, scaled distribution functions coincide quantitatively with the experimental results, the impurity concentration being the only adjustable parameter. This provides strong evidence for the presence of Kondo defects, either single- or two-channel, in the experimental systems. The relevance of these results for the problem of dephasing in mesoscopic wires is discussed briefly.
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\begin{document}


\articletitle{Kondo effect in non-equilibrium} 

\articlesubtitle{Theory of energy relaxation induced by
dynamical \\ defects in diffusive nanowires}


\author{Johann Kroha}
\affil{
Institut f\"ur Theorie der Kondensierten Materie and\\
Interfakultatives Institut f\"ur Nanotechnologie, Universit\"at 
Karlsruhe,\\ D-76128 Karlsruhe, Germany}
\email{kroha@tkm.physik.uni-karlsruhe.de}


\chaptitlerunninghead{Kondo effect in non-equilibrium }
\begin{keywords}
Single- and multi-channel Kondo effect, non-equilibrium, dephasing  
\end{keywords}


\begin{abstract}
In diffusive Cu and Au quantum wires at finite transport voltage $U$ 
the non-equilibrium distribution function $f(E,U)$ exhibits  
scaling behavior, $f(E,U)=f(E/eU)$, indicating
anomalous energy relaxation processes in these wires. 
We show that in nonequilibrium 
the Kondo effect, generated either by magnetic impurities
(single-channel Kondo effect) or possibly by non-magnetic, degenerate 
two-level systems (two-channel Kondo effect), produces such scaling behavior 
as a consequence of a Korringa-like
(pseudo)spin relaxation rate $\propto U$ and  
of damped powerlaw behavior of the impurity spectral density as a
remnant of the Kondo strong coupling regime at low
temperature but high bias. The theoretical, scaled distribution 
functions coincide quantitatively with the experimental results, 
the impurity concentration being the only adjustable parameter.
This provides strong evidence for the presence of Kondo defects,
either single- or two-channel, in the experimental systems.
The relevance of these results for the problem of 
dephasing in mesoscopic wires is discussed briefly. 
\end{abstract}

\section{Introduction}
Recently, it has become possible to determine experimentally 
\cite{pothier.97,pothier.00} the distribution function 
of quasiparticles, $f_x(E,U)$, in dependence of the particle energy $E$
in a metallic, diffusive nanowire in stationary non-equilibrium 
with a finite transport voltage $U$ applied between the ends of the wire.
The measurements were done by attaching an additional superconducting
tunneling electrode 
at a position $x$ along the wire, so that $f_x(E,U)$ could be extracted 
from the tunneling current
\begin{eqnarray}
j_t=\frac{e}{h} \gamma N_o \int dE 
\Bigl[f_x(E,U) - f^o(E + eU_t )\Bigr] N_{BCS}(E+eU_t) \ .
\label{eq:current}
\end{eqnarray}
Here $N_{BCS}$ is the peaked BCS density of states in the superconducting
electrode, $N_o$ and $\gamma$ are the approximately constant density of 
states in the
wire and the tunneling matrix element, respectively, and 
$U_t$ denotes the voltage between the wire and the tunneling electrode.
$f^o(E) = 1/({\rm e}^{E/k_BT} + 1)$ is the Fermi distribution at 
temperature $T$.  

As expected for a diffusive wire with elastic impurity scattering 
\cite{nagaev.92},  $f_x(E,U)$ has a double-step shape, corresponding to 
the two chemical potentials at $E=0$ and $E=eU$ at the ends of the wire.
In addition, the steps are rounded, indicating
inelastic scattering processes in the wire. In Cu \cite{pothier.97}
and Au \cite{pothier.00} wires, the observed rounding is such that
the quasiparticle distribution obeys a remarkable scaling property
when $U$ exceeds a certain low-energy scale, 
$eU\stackrel{>}{\sim} eU_o\approx 0.1meV$:
$f_x(E,U) = f_x(E/eU)$ \cite{pothier.97}.
One may gain phenomenological insight into the nature of the
inelastic relaxation processes by observing \cite{pothier.97} that the scaling
property of $f_x(E,U)$ implies that the equation of motion of $f_x(E,U)$,
the quantum Boltzmann equation, and consequently the inelastic 
single-particle collision rate $1/\tau$ are scale invariant as well.
Assuming a (yet to be determined) two-particle potential 
$\tilde V(\varepsilon )$
with energy transfer $\varepsilon$,
$1/\tau$ is in 2nd order perturbation theory given as 
\begin{equation}
\frac{1}{\tau (E)} \equiv \frac{1}{\tau (E/eU)} \simeq
N_o^3 \int d\varepsilon  \int d\varepsilon '
|\tilde V (\varepsilon )|^2 F \Bigl(
\frac{E}{eU},\frac{\varepsilon}{eU},
\frac{\varepsilon '}{eU} \Bigr) \ ,
\label{perturb}
\end{equation}
where $F$ is a combination of distribution functions $f_x$ ensuring
that there is only scattering from an occupied into an unoccupied state.
The experimental finding that $f_x$ and hence $1/\tau$
depend only on the dimensionless
energies, as indicated in Eq.~(\ref{perturb}), implies that in the term
on the right-hand side only dimensionless energies (normalized to $eU$)
occur, i.e.~$\tilde V(\varepsilon ) \propto 1/\varepsilon $ in the 
scaling regime $|E| \stackrel{<}{\sim}k_BU_o\stackrel{<}{\sim}eU$.
It follows that within 2nd order perturbation theory (Eq.~(\ref{perturb})) 
the energy relaxation rate has a logarithmic energy dependence,
$1/\tau (E) \propto {\rm ln}(E/eU_o)$, non-vanishing at the Fermi energy.
This anomalous behavior has raised considerable interest in the
energy relaxation measurements \cite{pothier.97}, especially
because of the possible relation of the apparently
non-vanishing energy relaxation rate to the problem of dephasing 
saturation observed at low $T$ in the magnetoresistance of
nanoscopic wires \cite{mohanty.97,gougam.00,pothier.00}. 

The infrared singular behavior of $\tilde V(\varepsilon)$, phenomenologically
deduced above, means in particular that the interaction has no 
essential momentum dependence and should be of a local origin, 
in contrast to an interaction mediated by a diffusive or dispersive 
collective mode. Therefore, it has been proposed \cite{kroha.00}
that the anomalous energy relaxation may be induced by Kondo impurities,
in particular by (nearly) degenerate atomic two-level
systems \cite{kroha.00} 
or other dynamical defects in the nanowire which can generate
a two-channel Kondo (2CK) effect \cite{nozieres.80,coxzawa.98}. 
The 2CK effect is known to have ideally a non-vanishing zeropoint
entropy $S(0) = k_B {\rm ln}\sqrt{2}$ \cite{wiegmann.83,andrei.84} 
and, hence, can cause dephasing at low energies \cite{zawa.99}. 
In fact, the electron interaction vertex mediated by a Kondo impurity
does show a $1/\varepsilon $ energy dependence. This was noted 
for the 2CK effect in the strong coupling region \cite{kroha.00}
(Fig.~\ref{fig1} a)) and independently
for the weak coupling (i.e. high energy) regime of the
magnetic, single-channel Kondo (1CK) effect \cite{glazman.00},
see also \cite{zawa.69}.
Although this analysis takes elevated single-particle energies into account,
it assumes the Kondo defect to be in thermodynamic equilibrium.  
Since, however, the effective interaction is generated dynamically
due to a Fermi edge singularity, it will be substantially modified in the
non-equilibrium situation of the energy relaxation measurements
considered here. 

Although the Kondo effect in stationary non-equilibrium
has been treated by various numerical methods before, no complete 
understanding has been reached up to now. In particular, the
scaling properties and to what extent the Kondo effect at finite 
bias is a strong coupling problem have remained unclear.
In this article we, therefore, summarize a predominantly 
analytical analysis of the 1CK and the 2CK 
non-equilibrium Kondo effect \cite{kroha.01}. It is shown that 
in nanowires with either 1CK or 2CK defects the quasiparticle distribution 
function exhibits scaling in terms of the applied bias, $eU$, when
$eU$ exceeds an intrinsic low-energy scale $eU_o$ which, up to logarithmic 
corrections, is proportional to the equilibrium Kondo temperature $T^{(0)}_K$.
The numerical solution of the quantum Boltzmann equation 
gives quantitative agreement with the experimental results discussed
above, where in the scaling regime the impurity concentration
is the only adjustable parameter. 
\begin{figure}[ht]
\centerline{
\epsfxsize=10.0cm
%\epsfysize=3.2cm
\epsfbox{NCA.eps}
}
\caption{a) Leading contributions to  the effective electron-electron
vertex induced by a Kondo impurity.  
b) Diagrammatic representation of the NCA auxiliary particle selfenergies, 
Eqs.~(\ref{NCA1})-(\ref{tmatrix}). Solid, dashed, and wiggly lines
correspond to conduction electron, pseudofermion, and slave boson
propagators, respectively (see text).}
\label{fig1}
\end{figure}
\inxx{captions,figure}


\section{A single Kondo impurity in a non-equilibrium wire}
The 1CK as well as the 2CK system can be described by a generalized
SU(N)$\times$SU(M) Anderson impurity model in the Kondo limit,  
i.e.~in terms of a quantum degree of freedom
in a low-lying level $\varepsilon_d$, coupled to the continuum of 
conduction electrons  
via a hybridization matrix element $V$,
where $N=2$ denotes the (pseudo)spin degeneracy and $M=1,2$ the number
of gegenerate, conserved conduction electron channels. 
In the auxiliary particle representation \cite{barnes.76} the model 
hamiltonian reads
\begin{eqnarray}
H=\sum _{\vec p,m,\sigma}\epsilon _p
c^{\dagger}_{pm\sigma}  c^{\phantom{\dagger}}_{pm\sigma}+
\epsilon _d \sum _{\sigma} 
f^{\dagger }_{\sigma } f^{\phantom{\dagger}}_{\sigma} 
+ V \sum _{p,m,\sigma} 
(f^{\dagger }_{\sigma} b_{\bar{m}} 
c^{\phantom{\dagger}}_{pm\sigma} + h.c.) \ ,
\label{eq:hamiltonian}
\end{eqnarray}
where $c^{\dagger}_{pm\sigma}$ is the creation operator for an electron
with momentum $\vec p$. The auxiliary fermion and boson
operators, $f^{\dagger }_{\sigma }$, 
$b^{\dagger }_{\bar m}$, create the local defect in
its quantum state $\sigma$ or in the unoccupied state, respectively.
The operator constraint 
$\hat Q = \sum _{\sigma} f^{\dagger }_{\sigma } f^{\phantom{\dagger}}_{\sigma}
        + \sum _{m}  b^{\dagger }_{\bar m} b^{\phantom{\dagger}}_{\bar m} 
\equiv 1$
ensures that the defect is in exactly one of its quantum states at any
instant of time. In the 1CK case of a magnetic Anderson 
impurity $\sigma $ denotes spin, and $m=1$ has no relevance.
For a 2CK defect, $\sigma$ is identified with 
a pseudospin, e.g. the parity of the local defect wave
function, and $m=1,2$ is the conduction electron spin
acting as the conserved channel degree of freedom. Note that in this case  
the bosonic operator $b^{\dagger }_{\bar m}$ transforms according to
the conjugate representation of SU(M) in order for the channel degree
of freedom to be conserved. The equilibrium Kondo temperature of the model 
is (to leading exponential order) $T^{(0)}_K \simeq W {\rm exp}(-1/NN_oJ)$,
with $J = V^2/|\varepsilon_d|$ the effective spin exchange coupling
and $W$ the half band width. 

The auxiliary particle representation allows for the application of 
standard Green's function methods 
and, in particular, is readily generalized \cite{hettler.94} 
to non-equilibrium by means of the Keldysh technique.
It is known \cite{cox.93} to describe the universal spectral 
properties of the 
multi-channel Kondo effect correctly both in the infrared
and in the high energy regime already within the non-crossing 
approximation (NCA), which is a selfconsistent, conserving approximation
to leading order in the effective  hybridization $\Gamma = V^2\pi N_o$. 
Thus, in the
multichannel case the NCA constitutes a correct infinite resummation 
of the logarithmic terms of perturbation theory. 
In the 1CK case {\it at finite bias} 
the logarithmic series is the same as for the 2CK effect {\it in 
equilibrium} because of the presence of two Fermi edges in the former
case \cite{kroha.01}. Thus, the NCA 
gives a correct description both of the 1CK and of the 2CK effects
in a strong non-equilibrium situation ($eU > T_K^{(0)}$) and will be used
for the explicit calculations below.    

In the remainder of this 
section we will consider a single Kondo defect at position
$x$ in a diffusive wire of length $L$ with a bias voltage $U$
applied between its ends.  
The generalization to the experimental situation of a dilute ensemble
of Kondo impurities will be treated in the following section.
The derivations apply both to the single and to the multi-channel case.
For purely elastic scattering in the wire 
the electronic distribution function at the position of the
impurity is 
\begin{eqnarray}
f_x(E,U)=\frac{x}{L} f^o(E+eU) +\Bigl(1- \frac{x}{L} \Bigr) f^o(E)\ .
\end{eqnarray}
Denoting the momentum integrated greater ($+$) and lesser ($-$) 
Keldysh Green's functions for the conduction electrons at position $x$ by
$G_{c,x}^{+}(E) \equiv \sum_p G_{c,x}^{+}(E,p)=2\pi i\, f_x(E,U)$ $N_o(E)$, 
\quad $G_{c,x}^{-}(E) \equiv \sum_p G_{c,x}^{-}(E,p)=$
$- 2\pi i(1-f_x(E,U))\, N_o(E)$, and similarly
the ``greater'', ``lesser'' and retarded auxiliary particle propagators by
$G_{f,b}^{+,-,r}(\omega )$, the non-equilibrium NCA equations then read 
(Fig.~\ref{fig1} b))
\begin{eqnarray}
\frac{G^{\pm}_f(\omega)}{|G^{r}_{f}(\omega)|^{2}} &=& 
   - M \Gamma  \int \frac{d\epsilon}{2\pi i}\, 
     G^{\pm}_{c,x}\, (- \epsilon ) G^{\pm}_b(\omega + \epsilon) 
\label{NCA1} \\
\frac{G^{\pm}_b(\omega)}{|G^{r}_{b}(\omega)|^{2}} &=& 
   + N \Gamma  \int \frac{d\epsilon}{2\pi i}\, 
     G^{\mp}_{c,x}\, (  \epsilon ) G^{\pm}_f(\omega + \epsilon) \ .
\label{NCA2}
\end{eqnarray}
This set of non-linear equations is closed by the Kramers-Kroenig relations, 
$G_{f,b}^{r}(\omega ) =\int d\varepsilon/(2\pi i)\, G_{f,b}^{-}(\omega )/
(\omega-\varepsilon+i0)$, which follow from analyticity and the fact
that the auxiliary particle Green's functions
have only forward in time propagating parts.
The conduction electron t-matrix due to the Kondo impurity is
\begin{eqnarray}
t_{c,x}^{\pm}(E ) =  
   - \frac{\Gamma}{\pi N_o(0)} \int \frac{d\epsilon}{2\pi i}\, 
     G^{\pm}_{f}\, (E +  \epsilon ) G^{\mp}_b(\epsilon) \ . 
\label{tmatrix}
\end{eqnarray}


It is crucial 
to determine the low-energy scales of the problem in non-equilibrium.
These may be determined from perturbation theory. 
At bias $eU$ the Kondo
temperature, definied as the breakdown scale of perturbation theory, is
suppressed, e.g. in the middle of the wire ($x=1/2$), as
\begin{eqnarray}
T_K (eU)= \sqrt{T_{K}^{(0)\ 2} + \Bigl(\frac{eU}{2k_B}\Bigr)^2}-\frac{eU}{2k_B}
\quad \stackrel{eU\gg k_BT^{(o)}_{K}} {\simeq} 
\quad \frac{k_BT_{K}^{(0)\ 2}}{eU} 
\label{eq:TK_eU}
\end{eqnarray}
In addition, a Korringa-like spin relaxation rate 
$1/\tau _s \propto eU$ appears,
because in non-equilibrium there is finite phase space available for
scattering even at $T=0$,
\begin{eqnarray}
\frac{1}{\tau_s(eU)} = 
%- 2 {\rm Im} \Sigma _f ^r(0) = 
\frac{1}{4\pi}\ NM\ (N_oJ)^2 \ eU  < eU\ .
\label{eq:tau_eU}
\end{eqnarray}
Because of this inelastic relaxation rate, the logarithmic singularities
of perturbation theory ($T=0$) are shifted by $1/\tau_s$ to the 
complex frequency plane, and the Kondo scale disappears eventually for large
bias $eU$ and is replaced by the inelastic rate
$1/\tau _s$: The low-temperature scale is $T_o(eU)={\rm max}
[T_K(eU),1/\tau _s(eU)]$.
The crossover from the Kondo to the Korringa scale occurs at a bias
\begin{eqnarray}
eU_o =   \frac {2}{N_o J}\sqrt{\frac{\pi}{NM}}\; k_BT_K^{(0)} =
         2N\sqrt{\frac{\pi}{NM}}\;
         k_BT_K^{(0)}\; {\rm ln}\Bigl( \frac{W}{T^{(0)}_K} \Bigr) \ .
\label{eq:Ucross}
\end{eqnarray}
This means that in the large bias regime, $eU>eU_{o}$, the low-energy
scale of the problem is proportional to the external bias itself.

We now turn to the scaling analysis of the non-equilibrium Kondo
equations (\ref{NCA1})--(\ref{tmatrix}).
It is well known \cite{muha.84,kroha.97}, that at $T=0$ in equilibrium
the exact auxiliary particle propagators exhibit
infrared powerlaw divergencies $G^{-}_{f,b}(\omega ) \propto  
\Theta (\omega )\ \omega ^{-\alpha_{f,b}}$, where the exponents
$\alpha _{f,b}$ are due to an orthogonality catastrophy between
the occupied and the unoccupied impurity state.
In the 2CK case the exact exponents are reproduced by NCA \cite{cox.93}
and depend in a characteristic way 
on the spin degeneracy $N$ and the channel number $M$, 
$\alpha _f = M/(N+M)$, $\alpha _b = N/(N+M)$ with $ \alpha _f+\alpha _b =1$.
In the large bias regime ($eU > eU_o$), 
the NCA equations are valid both in the 1CK and in the 2CK case, 
as mentioned above. The behavior around the Fermi edges may
be determined by defining the complex frequency variables
$z=\omega - i/2\tau_s(eU)$ etc.~and rewriting the NCA
equations for $z\to 0$ and $z\to eU$ 
as coupled differential equations in analogy 
to the equilibrium case \cite{muha.84}. We obtain damped powerlaw
behavior of the auxiliary fermion propagator for energies 
$|\omega | \stackrel{<}{\sim} 1/\tau _s$ and of the auxiliary
boson propagator for energies around the two
Fermi edges $|\omega | \stackrel{<}{\sim} 1/\tau _s$, 
$|\omega -eU| \stackrel{<}{\sim} 1/\tau _s$ where the
damping constant is $1/\tau _s$. The exponents are modified
in the large bias regime to $\alpha _f' = 2M/(N+2M)$, 
$\alpha _b' = N/(N+2M)$, where the additional factor of 2
originates from the fact that there are two separated Fermi
edges. The form of $\alpha _f'$, $\alpha _b'$ 
is reminiscent of an effective doubling of the channel number $M$
induced by the two Fermi edges. This behavior is confirmed by
the numerical evaluation of the problem and is 
consistent with a recent perturbative renormalization
group analysis \cite{coleman.00} of the problem. The fact that
powerlaw behavior (although damped) 
instead of logarithmic behavior persists
even at large bias may be seen as a remnant of the strong
coupling behavior of the Kondo problem in non-equilibrium.
It is crucial for scaling in terms of $eU$ to occur:
Putting the powerlaw behavior of the auxiliary propagators
back into Eqs.~(\ref{NCA1})--(\ref{tmatrix}), using 
$\alpha _f' +\alpha _b' =1$ and the important fact that 
the low-energy scale $T_o$ and 
the damping rate of the powerlaw behavior themselves scale
with the external bias, $T_o\ 1/\tau _s \propto eU$, it follows
immediately that these equations depend only on dimensionless
energies $\omega /eU$, $E/eU$ etc., i.e. they are scale invariant
in the regime $eU > eU_o =  2N\sqrt{\frac{\pi}{NM}}\;
k_BT_K^{(0)}\; {\rm ln}\Bigl( \frac{W}{T^{(0)}_K} \Bigr)$
(Eq.~\ref{eq:Ucross}).
It is seen that in experiments
one may extract the equilibrium Kondo temperature $T_K^{(0)}$
from the lower breakdown scale of the scaling behavior.  
   

\section{Solving the kinetic equation}


\begin{figure}[ht]
\centerline{
{\epsfxsize=\linewidth
%\epsfysize=3.2cm
\epsfbox{f_CuAu.eps}}
}
\caption{Scaled distribution functions $f_x(E,U)$ in nanowires. ---
Left panel: Two Cu nanowires
($L=1.5\mu m$, $D=65cm^2/s$, $x/L=0.5$ and 
$L=5.0\mu m$, $D=45cm^2/s$, $x/L=0.5$).
Black curves: Experimental data for 0.05mV$\leq U\leq$0.3mV 
in steps of 0.05mV \cite{pothier.97}. 
The two curves showing deviations from
scaling are at small voltages.
Light curves: Theory for 2CK impurities in the scaling regime, $eU > eU_o$.
Fitted 2CK impurity concentration: $c_{imp} \approx 
8\cdot 10 ^{-6}$/(lattice unit cell). --- 
Right panel: Two Au nanowires
(wire 1 and wire 2 of Ref.~\cite{pothier.00}),
measured at different positions $x/L$. Open circles: Experimental data;
Solid lines: Theory for magnetic impurities in the scaling regime.
Fitted magnetic impurity concentration: $c_{imp} \approx
2\cdot 10 ^{-4}$/(lattice unit cell).
} 
\label{fig2}
\end{figure}
\inxx{captions,figure}


The remaining task is to calculate the electronic
non-equilibrium distribution function $f_x(E,U)$ explicitly
at an arbitrary position $x$ along the wire 
in the presence of a dilute but finite density of 1CK or 2CK defects
$c_{imp}$.

The Boltzmann quantum kinetic equation for the momentum $\vec p$
dependent ``greater'' conduction electron Green's function 
$G_{c, (x,t)}^{+} (E,\vec p)$, 
which contains information about the distribution function, reads
\cite{landau.97}
\begin{eqnarray}
\Bigl[ \frac{\partial }{\partial t} +
\nabla _x \cdot \frac{\vec p}{m}
+ e\vec E (\vec x) \cdot \nabla _p\Bigr]
G_{c, (x,t)}^{+} (E,\vec p) =  {\cal C}
\bigl( \{ G_{c,(x,t)}^{+}(E,\vec p) \} , c_{imp} \bigr) , \nonumber\\
\label{boltz} 
\end{eqnarray}
where ${\cal C}$ is the collision integral induced by inelastic 
scattering off Kondo defects and
$\vec E(\vec x)$ are the external electric fields, including
those generated by static, random impurities in the wire.
Exploiting the fact that the current in the disordered system is diffusive,
\begin{equation}
\vec j _x(E) =\sum _p \frac{\vec p}{m}  
G_{c, x}^{+} (E,\vec p)  = - D \nabla _x 
\rho _x(E) = - D \nabla _x  \sum _p 
G_{c, x}^{+} (E,\vec p) \ ,
\end{equation}
with $D$ the diffusion coefficient,
and summing in Eq.~(\ref{boltz}) over quasiparticle momenta $\vec p$,
the electric field term drops out as a surface term, and we obtain
in the stationary case a diffusive kinetic equation \cite{nagaev.92}
for the impurity averaged distribution function 
$\overline{f}_x(E,U)= \sum _p G_{c, x}^{+} (E,\vec p)/$ $2\pi i N_o$
as function of the quasiparticle energy $E$ and position $x$, 
\begin{equation}
-D \nabla _x^2 \overline{f}_x(E,U) =  {\cal C}
\bigl( \{ \overline{f}_x(E,U) \} , c_{imp} \bigr) \ .
\label{boltzdiff}
\end{equation}
After momentum integration, the collision integral is given in terms of the 
impurity t-matrix (Eq.~\ref{tmatrix}) to arbitrary order in the
electron-impurity coupling $\Gamma$ by
\begin{equation}
{\cal C}(\{ \bar f _x(E,U) \}, c_{imp} ) = \frac{c_{imp}}{2\pi N_o} 
[ t_{c,x}^-(E) G_{c,x}^+(E) - t_{c,x}^+(E) G_{c,x}^-(E) ] \ .
\label{eq:collint} 
\end{equation}
We have solved the set of non-linear integro-differential
equations (\ref{NCA1})--(\ref{tmatrix}), (\ref{boltzdiff}),
(\ref{eq:collint}) numerically with the boundary conditions 
$\overline{f}_{x=0}(E,U)$ $ = f^o(E)$,
$\overline{f}_{x=L}(E,U) = f^o(E+eU)$ and that the electron system should be
in local equilibrium with the dynamical impurity.
The results are shown in Fig.~\ref{fig2}.
The theoretical $\overline{f}_{x}(E/eU)$
curves show scaling behavior in the regime $eU>eU_o$ (Eq.~(\ref{eq:Ucross}))
for bias voltages $U$ varying within a factor of 4 to 9, 
depending on the high energy parameters of the model Eq.~(\ref{eq:hamiltonian}).
There is excellent
quantitative agreement between theory and experiment for all samples.
The experimental data for Cu and Au wires
can be fitted equally well by the 1CK or by the 2CK theory, with 
very similar defect concentrations in the two cases. As an example,
Fig.~\ref{fig2} shows the fits of the Cu data to the 2CK theory and of the
Au data to the 1CK theory only. 
The agreement is consistent with the conjecture that the 
anomalous energy relaxation in those wires is mediated either 
by 1CK or by 2CK defects . 



\section{Relation to dephasing}
For an ideal 2CK system in equilibrium, the non-vanishing quasiparticle 
scattering rate $1/\tau (E)$ crosses over to a pure dephasing 
rate $1/\tau _{\varphi}$,
as the quasiparticle energy approaches the Fermi energy, $E,T\to 0$
\cite{zawa.99}. However, because of the finite level splitting $\Delta$
in real systems, one expects an upturn of $\tau _{\varphi}$
at the lowest $T\stackrel {<}{\sim} \Delta /k_B$, with an intermediate
plateau for $\Delta /k_B \stackrel {<}{\sim} T \stackrel {<}{\sim} T_K^{(0)}$
\cite{zawa.99}. 
Such behavior can also be expected from magnetic (1CK) Kondo impurities 
in combination with a phonon contribution, because the spin
flip rate reaches a maximum at $T\simeq T_K^{(0)}$ \cite{muha.71}. 
This has been observed in the Kondo system AuFe \cite{bergmann.87}  
and most recently in clean Au wires presumably containing a small 
concentration of Fe impurities \cite{pothier.00}.
One might, therefore, conjecture that 1CK or 2CK defects could be the
origin of the dephasing time saturation observed in magnetotransport
measurements of weak localization \cite{mohanty.97,gougam.00,pothier.00}. 
This assumption is indeed supported by
several coincidences between the dephasing time measurements and the  
results on the non-equilibrium distribution function: (1) The
dephasing time $\tau_{\varphi}$ extracted from magnetotransport
experiments \cite{gougam.00,mohanty.97} 
is strongly material, sample, and preparation dependent.
This suggests a non-universal dephasing mechanism, like dynamical
defects, which is not inherent to the electron gas.
(2) The dephasing time in Au wires is generically shorter 
than in Cu wires \cite{gougam.00}.
This is consistent with the fact that the estimates
for the dynamical impurity concentration $c_{imp}$, obtained from the
fit of the present theory to the experimental distribution functions,
is substantially higher in Au than in Cu wires (Fig.~\ref{fig2}).
(3) In Ag wires one observes neither dephasing saturation 
nor $E/eU$ scaling of the distribution function \cite{pierre.00}. 
This is consistent with the assumption that dephasing saturation and
anomalous energy relaxation in the nanoscopic wires have the same
origin and that there are no Kondo defects present in the Ag samples.


\section{Concluding remarks}

We have analysed the single- as well as the two-channel Kondo effect 
in a stationary non-equilibrium situation.
It was found that a Korringa-like, inelastic spin relaxation rate
appears which at large bias sets the low-energy scale of the problem. 
Nevertheless, a remnant of the strong coupling Kondo fixed point
persist even at large bias, which manifests itself by damped
powerlaw behavior of the local spectral density. The latter leads
to scaling of the non-equilibrium distribution function
$f_x(E,U)=f_x(E/eU)$ as an experimentally observable signature.
The present theory yields quantitative agreement with the
experimental results for all samples measured, the density of 
Kondo defects in the wire being the only 
adjustable parameter in the scaling regime.
This strongly suggests that the anomalous energy relaxation
might be caused by either 1CK or 2CK defects. However, it was shown
that the scaling property of the distribution function does not
distinguish between 1CK or 2CK impurities. Further experimental tests,
like application of a magnetic field, will be required for that 
purpose.

\section*{Acknowledgements}
It is a pleasure to thank A. Zawadowski, H. Pothier, B. L. Al'tshuler,
D. Esteve, J. v. Delft, and P. W\"olfle for stimulating and fruitful discussions.
The members of the 
Saclay group have provided experimental data prior to publication
which is gratefully acknowledged. 
This work is supported by DFG through SFB 195.



\begin{chapthebibliography}{1}

\bibitem{pothier.97} H. Pothier, S. Gu\'eron, Norman. O. Birge,
D. Esteve and M. H. Devoret, Phys. Rev. Lett. {\bf 79}, 3490 (1997);
Z. Phys. B {\bf 104}, 178 (1997). 

\bibitem{pothier.00} F.~Pierre, H.~Pothier, D.~Esteve, M.~H.~Devoret,
A.~B.~Gougam, N.~O.~Birge, this volume, cond-mat/0012038.

\bibitem{nagaev.92} K. E. Nagaev, Phys. Lett. A {\bf 169} 103 (1992);
Phys. Rev. B {\bf 52}, 4740 (1995).   

\bibitem{mohanty.97} P. Mohanty, E.M.Q. Jariwala and R. A. Webb,
Phys. Rev. Lett. {\bf 78}, 3366 (1997).

\bibitem{gougam.00}
A. B. Gougam, F. Pierre, H. Pothier, D. Esteve and Norman O. Birge,
J. Low Temp. Phys. {\bf 118}, 447 (2000).

\bibitem{kroha.00} 
J. Kroha, Adv. Solid. State. Phys. {\bf 40}, 267 (2000).

\bibitem{nozieres.80} P. Nozi\`eres and A. Blandin, 
                      J. Phys. (Paris) {\bf 41}, 193 (1980).

\bibitem{coxzawa.98}  For a comprehensive overview and references
see D. L. Cox and A. 
Zawadowski, Adv. Phys. {\bf 47} (5), 599--942 (1998). 

\bibitem{wiegmann.83} P. B. Wiegmann and A. M. Tsvelik,
Pis'ma Zh. eksp. teor. Fiz. {\bf 38}, 489 (1983) 
[JETP Lett. {\bf 38}, 591 (1983);
Adv. Phys. {\bf 32}, 453 (1983).  

\bibitem{andrei.84} N. Andrei and C. Destri, Phys. Rev. Lett. {\bf 52},
364 (1984).

\bibitem{zawa.99} A. Zawadowski, J. v. Delft and D. Ralph, 
Phys. Rev. Lett. {\bf 83}, 2632 (1999).

\bibitem{glazman.00}
A.~Kaminski and L.~I.~Glazman, cond-mat/0010379.

\bibitem{zawa.69}
J. S\'olyom and A. Zawadowski, Z. Phys. {\bf 226}, 116 (1969).

\bibitem{kroha.01} J. Kroha and A. Zawadowski, in preparation.

\bibitem{barnes.76}
S. E. Barnes, J. Phys. {\bf F 6}, 1375 (1976); {\bf F 7}, 2637 (1977).


\bibitem{hettler.94} M. H. Hettler, J. Kroha and S. Hershfield,
Phys. Rev. Lett. {\bf 73}, 1967 (1994); Phys. Rev. B {\bf 58}, 5649 (1998).

\bibitem{cox.93} D. L. Cox and A. E. Ruckenstein, Phys. Rev. Lett. {\bf 71},
1613 (1993).


\bibitem{muha.84} E. M\"uller-Hartmann, Z. Phys. B {\bf57}, 281 (1984).

\bibitem{kroha.97} 
J. Kroha and P. W\"olfle, Phys. Rev. Lett., {\bf 79}, 261 (1997).


\bibitem{coleman.00}  
P.~Coleman, C. Hooley and O. Parcollet, cond-mat/0012005.

\bibitem{landau.97}
E. M Lifshitz and L. P Pitaevskii, in {\it Landau and
Lifshitz Course of Theoretical Physics}, Vol. 10: {\it Physical Kinetics},
Chapt. X (Butterworth-Heinemann, Oxford, 1997). 

\bibitem{muha.71}
E. M\"uller-Hartmann and G. T. Zittartz, Phys. Rev. Lett. {\bf 26}, 428 (1971).

\bibitem{bergmann.87}
R. P. Peters, G. Bergmann and R. M. M\"uller,
Phys. Rev. Lett. {\bf 58}, 1964 (1987).

\bibitem{pierre.00} 
F. Pierre, H. Pothier, D. Esteve and M. H. Devoret,
J. Low Temp. Phys. {\bf 118}, 437 (2000).



\end{chapthebibliography}



\end{document}







it follows by rescaling 
the integration variables $\varepsilon$, $\varepsilon '$ in terms of 
$eU$ that the effective electron-electron interaction has an energy 
dependence 
 
and demanding scale invariance

Therefore, the experimental results about the scaling property of $f$
imply that $\tilde F$ depends only on the
dimensionless energies as displayed in Eq. (\ref{perturb}). 
Demanding scale invariance
for $1/\tau$ with respect to $eU$, i.e. making the frequency integrals
dimensionless, implies a characteristic energy dependence of the 
interaction and the single particle scattering rate,
$\tilde V(\varepsilon ) \propto 1/\varepsilon $ \cite{pothier.97}
and $1/\tau (E) \propto -{\rm ln} (E/E_o)$, for energies less than $E_o$. 






\section{Introduction}
Two-level systems (TLS) have been known to exist in disordered
solids for a long time. Their signatures have been observed in
the anomalous thermodynamic properties of glasses, which may be
explained by a flat distribution of level splittings. In metals,
slow two-level fluctuators are evidenced by telegraph noise 
in the conductance \cite{telegraph}. A new physical phenomenon, the two-channel
Kondo (2CK) effect \cite{nozieres.80,coxzawa.98}, arises when a fast, 
energetically (nearly)
degenerate TLS is embedded in a metal, where the local impurity degree
of freedom couples to the continuum of conduction electrons via
an exchange interaction. The channel degree of freedom, conserved
by this interaction, is the magnetic conduction electron spin,
which is always degenerate in the presence of time reversal symmetry.

The 2CK effect exhibits striking non-Fermi liquid behavior 
\cite{andrei.84,wiegmann.83,schlottmann.93} with 
a non-vanish\-ing single-particle scattering rate at the Fermi energy
$\varepsilon_F$ for temperature $T=0$, logarithmic behavior of the 
linear specific heat coefficient and the susceptibility, 
and a non-analytic, universal correction to the electronic density
of states, $\Delta N(E)/(B\sqrt{T}) = h(x)$, where $B$ is a non-universal 
constant, $x:=|E-\varepsilon_F|/T$ and $h(x)$ is a universal scaling 
function with $h(x)=\sqrt{x}$ for $x\gg 1$ \cite{affleck.93,hettler.94}. 
Signatures of the latter have been observed
as zero bias conductance anomalies (ZBA) of nano point contacts
\cite{ralph.92,ralph.94,upad.97}. 
As suggested in Ref. \cite{zawa.99}, the finite 
single-particle scattering rate at the Fermi energy would provide a natural
explanation for the saturation of the dephasing time $\tau _\varphi$
at low temperatures,
which has recently been observed \cite{mohanty.97} in magnetotransport 
measurements of weak localization in disordered wires.


However, the actual existence of the 2CK effect in nature has remained
a controversial issue, partly because the ZBA could also be explained
qualitatively \cite{wingreen.95}, although not quantitatively
\cite{hettler.94,upad.97,ralph.95,altshuler.98}, by the 
Al'tshuler-Aronov diffusion-enhanced Coulomb interaction \cite{altaronov.79}, 
partly because the physical realization of degenerate TLS is poorly
understood. Therefore, it is essential to develop unambiguous 
methods for detecting 2CK physics in mesoscopic systems. 


Recently, it has been demonstrated in a landmark experiment performed
by the Saclay ``Quantronics'' group \cite{pothier.97}
that unique information about the electronic interactions
in a metal may be extracted from the distribution function
$f(E,U)$ of quasiparticles with energy $E$ when the system is driven
far away from equilibrium by a transport voltage $U$: The shape of
$f(E,U)$ is determined by the energy dependence of relaxation processes
which tend to equilibrate the system. It has been 
found that in diffusive Cu and Au 
nanowires the distribution function obeys a scaling form,
$f(E,U)=f(E/eU)$, implying a non-Fermi liquid interaction
\cite{pothier.97,pothier.99}. 

In this article we discuss the close relation between this 
peculiar scaling property, which we will call
``$E/eU$ scaling'' for brevity, 
and non-Fermi liquid behavior. It is then shown that 
the 2CK effect obeys $E/eU$ scaling and reproduces quantitatively the
measured distribution functions, while any other type of interaction
is ruled out. This provides the strongest case to date for the 
physical realization of the 2CK effect in Cu and Au nanowires
induced by TLS. We briefly discuss its relation \cite{gougam.00}
to the problem of the dephasing time saturation at low temperatures
\cite{mohanty.97}.


\section{Experiment and non-Fermi liquid signature}

The distribution function $f(E,U)$ was measured \cite{pothier.97}
in an experimental 
setup where a non-equilibrium current was driven through a Cu nanowire
contacted by two reservoirs at chemical potentials $0$ and $eU$,
respectively (Fig.~\ref{expsetup}). In addition, a superconducting Al
tunneling junction was attached at a position $x$ along the wire,
the Al slab being in equilibrium with itself. For a voltage
$V$ across the junction the tunneling current is given by 
\begin{equation}
j_{tunnel} = \frac{e}{\hbar} |t|^2 \sum _{\sigma}
\int dE \bigl[ f(E) - f^o(E+eV) \bigr]
N_{Cu,\sigma}(E)\; N_{sc}(E+eV)\ ,
\label{jtunnel}
\end{equation}
where $t$ is the (energy independent) tunneling matrix element,
$f^o(E)$ is the Fermi distribution function in the superconductor,
and $N_{Cu}$, $N_{sc}$ denote the density of states in the wire and in 
the superconductor, respectively. Since for the voltages used in the
experiment $N_{Cu}$ is flat and the BCS density of states $N_{sc}$
is measured independently, the non-equilibrium distribution $f(E)$ in the Cu 
wire can be extracted from this expression. The electronic transport in the
wire is diffusive with diffusion coefficient $D$. Length $L$ and thickness $d$
of the wire are such that the Fermi surface is three dimensional
and the Coulomb and phonon scattering times are large compared to the
electronic diffusion time $\tau _D = L^2/D$ through the wire,
so that equilibration due to these processes can be neglected
\cite{pothier.97}.
In this situation, assuming purely elastic scattering, one expects
the distribution function at a given position $x$ along the wire 
to be a linear superposition of the Fermi functions in the reservoirs,
\begin{equation}
f^{elast}_x(E,U) = \bigl(1-\frac{x}{L}\bigr) f^o(E+eU) + \frac{x}{L} f^o(E)\ ,
\label{felast}
\end{equation}
since there is no energy exchange within the electron system or between
the electron system and the lattice. Eq.~(\ref{felast}) is a solution of
the diffusive Boltzmann equation (\ref{boltz}) \cite{nagaev.92}, 
when the collision 
integral vanishes (see section 3).
This situation is to be distinguished from the
hot electron regime, where local equilibration occurs due to inelastic
processes \cite{kozub.95}. 

The measured distribution functions showed rounding of the Fermi steps
as compared to Eq.~(\ref{felast}) and obeyed scale invariance with
respect to the transport voltage $U$,  
$f(E,U)=f(E/eU)$, when $U$ exceeded a certain 
low energy scale, $eU \stackrel{>}{\sim} 0.1meV$ \cite{pothier.97}.
Deviations from scaling were observed again for voltages larger than
a high energy scale $E_o \simeq 0.5meV$. 
The latter may be explained by reservoir 
heating effects or by the electrons coupling to additional degrees of freedom 
at high energies. We now deduce the non-Fermi liquid
signature from the scaling property. The latter implies that the equation of
motion for $f(E)$, the Boltzmann equation and, as a consequence, the inelastic
single-particle collision rate $1/\tau (E)$ are scale invariant.
Assuming a (yet to be determined) two-particle potential $\tilde V(\varepsilon )$
with energy transfer $\varepsilon$,
$1/\tau$ is given in 2nd order perturbation theory as \cite{remark}
\begin{equation}
\frac{1}{\tau (E)} \equiv \frac{1}{\tau (E/eU)} \simeq
N_{Cu}(0)^3 \int d\varepsilon  \int d\varepsilon '
|\tilde V (\varepsilon )|^2 \tilde F \Bigl(
\frac{E}{eU},\frac{\varepsilon}{eU},
\frac{\varepsilon '}{eU} \Bigr) \ . 
\label{perturb}
\end{equation}
Here $\tilde F$ is a combination of distribution functions $f$ guaranteeing
that there is only scattering from an occupied into an unoccupied state.
Therefore, the experimental results about the scaling property of $f$
imply that $\tilde F$ depends only on the
dimensionless energies as displayed in Eq. (\ref{perturb}). 
Demanding scale invariance
for $1/\tau$ with respect to $eU$, i.e. making the frequency integrals
dimensionless, implies a characteristic energy dependence of the 
interaction and the single particle scattering rate,
$\tilde V(\varepsilon ) \propto 1/\varepsilon $ \cite{pothier.97}
and $1/\tau (E) \propto -{\rm ln} (E/E_o)$, for energies less than $E_o$. 
These infrared singularities indicate a breakdown of Fermi liquid theory
within the 2nd order perturbation theory argument applied here.

\section{\hspace*{-0.1cm}Two-channel Kondo effect and scaling in non-equilibrium}
The strong infrared divergence of the two-particle 
potential, $\tilde V (\varepsilon )
\propto 1/\varepsilon $, deduced above, is not explained by any conventional 
interaction, including the Al'tshuler-Aronov interaction
\cite{altaronov.79}. It must, therefore,
be generated by an infinite resummation of logarithmic terms
obtained in perturbation theory due to the presence of a Fermi edge. 
In this section the 2CK effect is briefly reviewed. We then show that the
effective electron-electron vertex, mediated by a 2CK impurity, has a
$1/\varepsilon $ divergence and calculate the resulting 
distribution functions away from equilibrium. 

The 2CK effect arises whenever a local, energetically degenerate
two-level degree of freedom (pseudospin $\tau=\pm 1/2$)
is coupled to a system of two identical
conduction electron bands or channels via a pseudospin exchange interaction,
which, however, conserves the channel degree of freedom. 
As for the physical realization of the pseudospin, it has been 
suggested \cite{coxzawa.98} that it might arise from the
degenerate positions of an atom in a symmetrical double well potential,
from a rotational degree of freedom of a lattice defect or group of atoms,
or from sliding kinks on screw dislocations in a lattice 
\cite{sethna.00}. The channel degree of freedom is then the magnetic
electron spin $\sigma =\pm 1/2$, which is necessarily degenerate because of
time reversal symmetry (Kramers degeneracy).

In order to describe non-equilibrium situations,
it is useful to represent this system in terms of the low-energy
physics of a SU(2)$_{\rm pseudospin}$$\times$SU(2)$_{\rm channel}$
Anderson impurity model in the Kondo limit,
i.e. in terms of a doubly degenerate local level $E_d<0$,
coupled via a hybridization $v$ to two identical conduction channels, 
with conduction electron
operators $c^{\phantom{\dag}}_{k\tau\sigma }$, $c^{\dag}_{k\tau\sigma }$. 
The dynamics of the two-level degree of freedom (pseudospin) 
is described by a fermionic field operator $d^{\dag}_{\tau\sigma}$, with
the restriction 
that the local level must not be doubly occupied at any time.
The latter is implemented exactly by decomposing the local
operator into auxiliary fermion $f^{\dag}_{\tau }$
and boson $b^{\phantom{\dag}}_{\bar\sigma}$ operators,  
$d^{\dag}_{\tau\sigma} = f^{\dag}_{\tau } b^{\phantom{\dag}}_{\bar\sigma}$,
supplemented by the operator constraint
$\hat Q = \sum _{\tau} f^{\dag}_{\tau } f^{\phantom{\dag}}_{\tau }
+\sum _{\sigma} b^{\dag}_{\bar\sigma} b^{\phantom{\dag}}_{\bar\sigma} \equiv 1$
 \cite{barnes.76}. Thus, we describe the 2CK system by the Anderson hamiltonian 
in auxiliary particle representation,
\begin{equation}
H = \sum _{k\sigma\tau} 
\varepsilon _k c^{\dag}_{k\tau\sigma } c^{\phantom{\dag}}_{k\tau\sigma }
+E_d \sum _{\tau} f^{\dag}_{\tau } f^{\phantom{\dag}}_{\tau }
+v \sum _{k\sigma\tau}( f^{\dag}_{\tau } b^{\phantom{\dag}}_{\bar\sigma}
                       c^{\phantom{\dag}}_{k\tau\sigma } +h.c.) \ .
\label{hamiltonian}
\end{equation}
Note that the boson field $b_{\bar\sigma}$ transforms with respect to the
adjoint representation of SU(2)$_{\rm channel}$ (denoted by the index
$\bar\sigma$).
% in order for $H$ to preserve the channel index.
By integrating out $ b^{\phantom{\dag}}_{\bar\sigma}$
in the Kondo regime ($\langle\sum _{\tau}
 f^{\dag}_{\tau } f^{\phantom{\dag}}_{\tau } \rangle \simeq 1$, $|E|\ll |E_d|$),
$H$ is reduced to a 2CK model where the exchange coupling $J=|v|^2/|E_d|$
is mediated by the boson propagator and, as required, conserves the
channel degree of freedom $\sigma$. 
The above auxiliary particle 
representation is known to describe the infrared behavior of the 2CK
effect correctly \cite{cox.93}
already on the level of a leading-order selfconsistent
perturbation theory in the hybridization $v$, the non-crossing
approximation (NCA) \cite{bickers.87}.  
It also allows for the application of standard Green's function methods 
and, in particular, is readily generalized \cite{hettler.94} 
to non-equilibrium by means of the Keldysh technique.

Physically, the presence of two identical
conduction electron channels leads to an overscreening of the impurity 
pseudospin below the Kondo temperature
$T_K = \bigl( 2 \Gamma/\pi \bigr) {\rm exp}
\bigl(- \pi |E_d|/2\Gamma\bigr)$, $\Gamma = \pi |v|^2 N_{Cu}(0)$,
as elucidated in Fig.~\ref{2CKcartoon}. This results in a non-vanishing
zero-point entropy $S(T=0)=k_B{\rm ln}\sqrt{2}$
\cite{andrei.84} and is closely
related to the finite single-particle decay rate $1/\tau (E=0,T=0) >0$
predicted by the 2CK effect.

Below the Kondo scale $T_K$, 
the auxiliary particle propagators $G_f$, $G_b$ are known to 
exhibit infrared power-law behavior in equilibrium,  
$G_f(\omega ) \propto \omega ^{-\alpha _f}$,
$G_b(\omega ) \propto \omega ^{-\alpha _b}$ for $\omega 
{\stackrel {<}{\sim}}T_K$ \cite{muha.84}, where the exact 
\cite{cox.93,affleck.93} relation 
$\alpha _f + \alpha _b = 1$ is characteristic for the multichannel
Kondo effect ($\alpha _f = M/(M+N)$, $\alpha _b = N/(M+N)$ for pseudospin
degeneracy $N$ and channel number $M$). Using power counting arguments,
it follows that each fermion (boson) propagator contributes a power
$-\alpha _f$ ($-\alpha _b$) and each frequency integral contributes
a power 1 to the frequency dependence of 
the effective electron-electron vertex $\gamma (\omega )$,
shown in Fig.~\ref{eff_int}. Thus, the 2CK electron-electron 
\begin{figure}[ht]
\centerline{
\epsfxsize=10.0cm
%\epsfysize=3.2cm
\epsfbox{eff_int.eps}
}
\caption{Leading contributions to  the effective electron-electron
vertex induced by a 2CK impurity. Solid, dashed, and wiggly lines
correspond to conduction electron, pseudofermion, and slave boson
propagators, respectively.}
\label{eff_int}
\end{figure}
\inxx{captions,figure}
vertex depends on the frequency transfer $\omega $ as
$\gamma (\omega) \propto \omega ^{1-2(\alpha _f + \alpha _b)} = \omega ^{-1}$
\cite{remark2}. On the perturbative level, this implies scale invariance 
of the inelastic scattering rate $1/\tau (E)$ with respect to
the transport voltage $U$, as discussed in section 2. A self-consistent
resummation of these terms in non-equilibrium shows that the scale 
invariance persists up to single-particle energies $E$ of the order of 
$eU$ \cite{kroha.00}. 
It may be shown that the range of $eU$ values for which scaling
behavior is obeyed is bounded from below by the Kondo temperature $T_K$.   
For large bias voltage, deviations from scaling occur when $eU$ becomes
comparable to the band cutoff or to the local level $|E_d|$.














































cond-mat0102185.extracted_bib
\bibitem{pothier.97} H. Pothier, S. Gu\'eron, Norman. O. Birge,
D. Esteve and M. H. Devoret, Phys. Rev. Lett. {\bf 79}, 3490 (1997);
Z. Phys. B {\bf 104}, 178 (1997). 


\bibitem{pothier.00} F.~Pierre, H.~Pothier, D.~Esteve, M.~H.~Devoret,
A.~B.~Gougam, N.~O.~Birge, this volume, cond-mat/0012038.


\bibitem{nagaev.92} K. E. Nagaev, Phys. Lett. A {\bf 169} 103 (1992);
Phys. Rev. B {\bf 52}, 4740 (1995).   


\bibitem{mohanty.97} P. Mohanty, E.M.Q. Jariwala and R. A. Webb,
Phys. Rev. Lett. {\bf 78}, 3366 (1997).


\bibitem{gougam.00}
A. B. Gougam, F. Pierre, H. Pothier, D. Esteve and Norman O. Birge,
J. Low Temp. Phys. {\bf 118}, 447 (2000).


\bibitem{kroha.00} 
J. Kroha, Adv. Solid. State. Phys. {\bf 40}, 267 (2000).


\bibitem{nozieres.80} P. Nozi\`eres and A. Blandin, 
                      J. Phys. (Paris) {\bf 41}, 193 (1980).


\bibitem{coxzawa.98}  For a comprehensive overview and references
see D. L. Cox and A. 
Zawadowski, Adv. Phys. {\bf 47} (5), 599--942 (1998). 


\bibitem{wiegmann.83} P. B. Wiegmann and A. M. Tsvelik,
Pis'ma Zh. eksp. teor. Fiz. {\bf 38}, 489 (1983) 
[JETP Lett. {\bf 38}, 591 (1983);
Adv. Phys. {\bf 32}, 453 (1983).  


\bibitem{andrei.84} N. Andrei and C. Destri, Phys. Rev. Lett. {\bf 52},
364 (1984).


\bibitem{zawa.99} A. Zawadowski, J. v. Delft and D. Ralph, 
Phys. Rev. Lett. {\bf 83}, 2632 (1999).


\bibitem{glazman.00}
A.~Kaminski and L.~I.~Glazman, cond-mat/0010379.


\bibitem{zawa.69}
J. S\'olyom and A. Zawadowski, Z. Phys. {\bf 226}, 116 (1969).


\bibitem{kroha.01} J. Kroha and A. Zawadowski, in preparation.


\bibitem{barnes.76}
S. E. Barnes, J. Phys. {\bf F 6}, 1375 (1976); {\bf F 7}, 2637 (1977).



\bibitem{hettler.94} M. H. Hettler, J. Kroha and S. Hershfield,
Phys. Rev. Lett. {\bf 73}, 1967 (1994); Phys. Rev. B {\bf 58}, 5649 (1998).


\bibitem{cox.93} D. L. Cox and A. E. Ruckenstein, Phys. Rev. Lett. {\bf 71},
1613 (1993).



\bibitem{muha.84} E. M\"uller-Hartmann, Z. Phys. B {\bf57}, 281 (1984).


\bibitem{kroha.97} 
J. Kroha and P. W\"olfle, Phys. Rev. Lett., {\bf 79}, 261 (1997).



\bibitem{coleman.00}  
P.~Coleman, C. Hooley and O. Parcollet, cond-mat/0012005.


\bibitem{landau.97}
E. M Lifshitz and L. P Pitaevskii, in {\it Landau and
Lifshitz Course of Theoretical Physics}, Vol. 10: {\it Physical Kinetics},
Chapt. X (Butterworth-Heinemann, Oxford, 1997). 


\bibitem{muha.71}
E. M\"uller-Hartmann and G. T. Zittartz, Phys. Rev. Lett. {\bf 26}, 428 (1971).


\bibitem{bergmann.87}
R. P. Peters, G. Bergmann and R. M. M\"uller,
Phys. Rev. Lett. {\bf 58}, 1964 (1987).


\bibitem{pierre.00} 
F. Pierre, H. Pothier, D. Esteve and M. H. Devoret,
J. Low Temp. Phys. {\bf 118}, 437 (2000).



