Non-Fermi Liquid Behavior of U-impurity with $f^2$-Singlet Ground State
Kazumasa {\sc Miyake}
and Hiroaki {\sc Kusunose}$^{1}$
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\def\runtitle{Non-Fermi Liquid Behavior of U-impurity with $f^2$-Singlet Ground State}
\def\runauthor{Satoshi {\sc Yotsuhashi}, Kazumasa {\sc Miyake} and Hiroaki {\sc Kusunose}}

\title{Non-Fermi Liquid Behavior of U-impurity with $f^2$-Singlet Ground State}

\author
{Satoshi {\sc Yotsuhashi}\footnote{E-mail: yotsu@eagle.mp.es.osaka-u.ac.jp}, Kazumasa {\sc Miyake}, and Hiroaki {\sc Kusunose}$^{1}$}
\inst
{Division of Materials Physics, Department of Physical Science, Graduated School of Engineering Science,\\ Osaka University, Toyonaka, Osaka 560-8531\\
$^1$Department of Physics, Tohoku University, Sendai, 980-8578}

\recdate{\today}

\abst
{It is shown by the Wilson's numerical renormalization-group approach that Non-Fermi liquid (NFL) behaviors arise even for U-impurity with an $f^2$-singlet ground state, and all the puzzling behaviors of R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Th, Y and La, $x\le 0.07$) can be explained on its basis.
There exists an unstable NFL fixed point due to the balance between Kondo singlet and Crystalline-Electric-Field singlet ground states. For wide region in parameter space the system shows the NFL behavior as a transient phenomenon in temperature range accessible by experiment.
Results for temperature dependence of the resistivity, the magnetic susceptibility and the Sommerfeld coefficient with and without magnetic field reproduce the behaviors observed in Th$_{1-x}$U$_x$Ru$_2$Si$_2$.
Less anomalous properties of R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Y and La) can be understood along the same scenario provided that those compounds have different sets of parameters.
}

\kword
{Non-Fermi liquid, numerical renormalization-group method,
 Crystalline-Electric-Field singlet,
 R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Th, Y and La)}

\begin{document}
\sloppy
\maketitle
%
%.Introduction
%
%..non-Fermi liquid behavior in dilute U compounds
Recently, issues about the Non-Fermi liquid (NFL) behaviors in dilute U arroys have attracted much attention.
They are related to a general interest of Kondo phenomena \cite{Cox} which arise in some cases due to the existence of plural number of strongly correlated 5f electrons per U ion.
Especially, the behaviors in series of compounds R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Th, Y and La, $x\le 0.07$) have been well investigated as a single-site effect, because anomalous properties of these materials are well scaled by impurity concentrations \cite{Ami94,Maru96,Ami00}.
The latter fact makes the theoretical approach simple compared with the system in which the intersite effect is considered to play some role, such as Y$_{1-x}$U$_x$Pd$_3$\cite{Sea91,And91}.
Th$_{1-x}$U$_x$Ru$_2$Si$_2$ shows $-\ln T$ divergence both of the susceptibility $\chi$ and the Sommerfeld coefficient $\gamma$ with decreasing temperature.
However, a big puzzle still remains as discussed shortly.

The logarithmic dependence of these thermodynamic quantities itself is consistent with the prediction based on the quadrupolar Kondo model \cite{Cox87}, if it is generalized appropriately \cite{Cox96,Kusu96}.
The quadrupolar Kondo effect is expected to arise for a non-Kramers doublet ground state of $f^2$-configuration and to be mapped onto the S=1/2 two-channel Kondo model \cite{Noz80}.
The exact solution of S=1/2 two-channel Kondo model predicts the asymptotic behavior of $\gamma$, $\chi$ $\propto -\ln T$ and the resistivity $\rho$ $\propto {\rm const}\pm \sqrt{T}$ in the limit $T \to 0$ with the residual entropy $S(T \to 0)=\frac{1}{2} \ln 2$ \cite{And84,Sac91,Aff93}.
Since R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Th, Y and La) has the tetragonal symmetry and the valence of the uranium ion is expected to be mainly U$^{4+}$ (5$f^2$), many theoretical studies of this problem have been done on the basis of the two-channel Kondo model assuming the magnetic doublet ground state, $|\Gamma_5^{(2)}\pm \rangle$, in (5$f^2$) configuration \cite{Cox,Cox96,Sakai96,Suzu97,Shimi98,Shimi99}.

%..susceptibility vs resistivity
However, there exist puzzling properties about Th$_{1-x}$U$_x$Ru$_2$Si$_2$ which cannot be explained by the two-channel Kondo model.
First of all, $\chi$ shows the Curie law while $\rho$ shows large value of the order of the unitarity limit for $10$K$<T<100$K.
Namely, the susceptibility indicates the existence of the localized moment while the resistivity suggests the confinement of the localized moment by the spin polarization of conduction electrons.
This apparent inconsistency was removed by the extended two-channel Anderson model introduced and discussed by Sakai and his collaborators \cite{Sakai96,Suzu97,Shimi98}, who showed that there exists wide set of parameters which gives results consistent with the experiments of Th$_{1-x}$U$_x$Ru$_2$Si$_2$.
However, in the case of R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Y and La), $\rho$ is almost constant below 10K, which cannot be solved by such an extended two-channel Anderson model.
%..specific-heat in magnetic field
Secondly, the divergence of $\gamma$ is suppressed for applied magnetic field $H$ of the order of several Tesla in Th$_{1-x}$U$_x$Ru$_2$Si$_2$ \cite{Ami00}.
In any theories based on the two-channel Kondo model, the entropy is expected to be released at finite temperature because the magnetic field lifts the degeneracy of the $|\Gamma_5^{(2)} \pm \rangle$ doublet ground state, leading enhancement of $\gamma$ \cite{Shimi99}.
Thus, the theory based on the two-channel Kondo model with magnetic doublet configuration does not seem to be a final theory for this problem.

%..present study
The purpose of this paper is to solve this puzzle on the basis of a novel mechanism assuming Crystalline-Electric-Field (CEF) singlet ground state, in which the competition between the conventional Kondo singlet and CEF singlet gives rise to the NFL behavior in low but intermediate temperature regime as a transient phenomenon.
A system exhibiting such competition can be mapped to the $f^2$-impurity model with anisotropic antiferromagnetic Hund's-rule coupling and different hybridizations with conduction electrons, when CEF levels are specified by pseudo spins.
It has been recognized by recent studies \cite{Jones88,Jones89,Sakai92,Aff92,Aff95,yotsu01} that such models exhibit the NFL behavior reminiscent of two-channel Kondo model for certain critical set of parameters.
On the basis of Wilson's numerical renormalization-group (NRG) method \cite{Wilson75,Krish80,Sakai89}, we show that the NFL behaviors arise, as a transient phenomenon near the unstable fixed point, rather robustly even for parameters off the criticality in rather wide temperature region accessible by experiments.
Series of present results are consistent with those observed in R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Th, Y and La).
Especially, all the anomalous properties of Th$_{1-x}$U$_x$Ru$_2$Si$_2$ can be explained semiquantitatively for a certain set of parameters.

%
%.Model
%
%..pseudo spin representation (CEF to AF Hund)
We consider two low-lying $f^1$ doublet states out of three doublet of $j=5/2$ orbitals in the tetragonal symmetry.
These four (two doublet) states can be represented by pseudo spin:
\begin{eqnarray}
&&|\Gamma_7^{(2)}, +\rangle =-\sqrt{\frac{1}{6}}\left| \frac{5}{2}\right\rangle +\sqrt{\frac{5}{6}}\left| -\frac{3}{2}\right\rangle \equiv |\uparrow , 0 \rangle , \\
&&|\Gamma_7^{(2)}, -\rangle =\sqrt{\frac{1}{6}}\left| -\frac{5}{2}\right\rangle -\sqrt{\frac{5}{6}}\left| \frac{3}{2}\right\rangle \equiv |\downarrow, 0 \rangle , \\
&&|\Gamma_6, +\rangle = \left|+\frac{1}{2} \right\rangle \equiv |0, \uparrow \rangle, \\
&&|\Gamma_6, -\rangle = \left|-\frac{1}{2} \right\rangle \equiv |0, \downarrow \rangle,
\end{eqnarray}
where $|\uparrow, 0 \rangle$ ($|0, \uparrow \rangle$) represents the state that the pseudo spin in channel 1 (2) is up and the channel 2 (1) is empty, and notations are conventional otherwise.
In terms of these four states, the relevant CEF states in $f^2$-configuration are written in the $j$-$j$ coupling scheme as follows:
\begin{eqnarray}
&&|\Gamma_4 \rangle =\frac{1}{\sqrt{2}}\left(|2\rangle - |-2\rangle\right)=\frac{1}{\sqrt{2}}\left(|\uparrow, \downarrow \rangle - |\downarrow, \uparrow \rangle \right) \\
&&|\Gamma_3 \rangle =\frac{1}{\sqrt{2}}\left(|2\rangle + |-2\rangle\right)=\frac{1}{\sqrt{2}}\left(|\uparrow, \downarrow \rangle + |\downarrow, \uparrow \rangle \right) \\
&&|\Gamma_5^{(2)}, +\rangle =\beta | 3\rangle -\alpha | -1\rangle = |\uparrow , \uparrow \rangle , \\
&&|\Gamma_5^{(2)}, -\rangle =\beta |-3\rangle -\alpha | 1\rangle = |\downarrow , \downarrow \rangle .
\end{eqnarray}
Here, the energy levels of $|\uparrow\downarrow, 0 \rangle$ and $|0, \uparrow\downarrow \rangle$ have been omitted because of large intra-orbital Coulomb repulsion.
One of fundamental assumptions is to adopt CEF level scheme shown in Fig.1.
This level scheme can be reproduced by the pseudo-spin Hamiltonian
\begin{eqnarray}
H_{\rm Hund}=\frac{J_{\bot }}{2}[S_1^+ S_2^- +S_1^- S_2^+]+ J_z S_1^z S_2^z ,
\label{HHund}
\end{eqnarray}
where $\vec{S}_m$ denotes a pseudo-spin operator of the localized electrons in the orbital $m$.
By taking such the pseudo spin representation, 
it can be understood clearly that 
the CEF singlet competes with the Kondo singlet directly and 
we can perform calculations more accurately owing to an appearance of 
new conserved quantity, pseudo spin.
The energy separation between the singlet ($\Gamma_4$) and the doublet ($\Gamma_5^{(2)} \pm$) is set to be $\Delta$ and that between the $\Gamma_4$ and the singlet ($\Gamma_3$) is set to be $K$.
The couplings $J_{\bot}$ and $J_z$ are related to $(K,\Delta)$ as $J_{\bot }$=$K$ and $J_z$=$2\Delta -K$.
It is remarked that (\ref{HHund}) represents an anisotropic antiferromagnetic Hund's-rule coupling, which cannot exist in real spin system.

%..Hamiltonian
With this preliminary, the system in question can be described by a conventional two-channel Anderson model with {\it antiferromagnetic} Hund's-rule coupling:
\begin{eqnarray}
&&H=H_{\rm K}+H_{\rm mix}+H_{\rm f}+H_{\rm Hund}, \label{H}
\end{eqnarray}
with
\begin{eqnarray}
&&H_{\rm K}\equiv \sum_{m=1,2} \sum_{\vec{k} \sigma}\epsilon_{\vec{k}} c_{\vec{k}m\sigma}^\dagger c_{\vec{k}m\sigma}, \\
&&H_{\rm mix}\equiv \sum_{m=1,2} \sum_{\vec{k}\sigma}(V_{\vec{k},m} c_{\vec{k}m\sigma}^\dagger f_{m\sigma}+{\rm h.c}.), \\
&&H_{\rm f}\equiv \sum_{m\sigma} E_{{\rm f}m} f_{m\sigma}^\dagger f_{m\sigma}+\sum_{m \sigma} \frac{U_{m}}{2} f_{m\sigma}^\dagger f_{m\bar{\sigma}}^\dagger f_{m\bar{\sigma}} f_{m\sigma}.
\end{eqnarray}
It is noted that the hybridization $V_{{\vec{k},m}}$ depends only on its channel index $m (=1,2)$ and the inter-orbital Coulomb interaction is neglected for simplicity.
Hereafter, we take the unit of energy as $(1+\Lambda^{-1}) D / 2$ where $D$ is half the bandwidth of conduction electrons and $\Lambda$ is a discretization parameter in NRG calculation.
In the case $K$$=$$\Delta$$=$$0$, this model reduces to two independent 
conventional Anderson models.
Therefore, we can restrict ourselves to the case that each $f$-orbital is almost singly occupied by taking parameters such that  $E_{\rm f1}=E_{\rm f2}=-0.4$, $U_1=U_2=2.0$, $V_1=0.7$, and $V_2=0.34$.

%.two-impurity systems
The system described by (\ref{H}) shows the competition between the Kondo singlet and the local singlet \cite{yotsu01,Kura92} as seen in two-impurity systems \cite{Jones88,Jones89,Sakai92,Aff92,Aff95}.
Two types of ground state are possible, i.e., (i) the Kondo singlet state characterized by total phase shift being equal to $\pi$ (i.e., $\delta_1=\delta_2=\pi /2$), and (ii) the local singlet state characterized by $\delta_1=\delta_2=0$.
Then, there exists a locus of unstable fixed points characterized by $\delta_1=\delta_2=\pi/4$, and on which the NFL behavior appears.
%..phase diagram \delta. vs K_{CEF}
A critical line can be determined by the NRG calculation, across which the even-odd alternation in energy spectrum is interchanged.
For the present parameter set, the critical line is shown in Fig. 2.
When initial couplings $(K,\Delta)$ are located near the critical line, the renormalization flow passes near the NFL unstable fixed point, and then arrives at the Fermi liquid (FL) fixed point.
Namely, the NFL behavior can be seen in the observable temperature range, if $(K,\Delta)$ are close enough to the critical line.
Although we present here the results for only one parameter set, 
we have confirmed that the NFL behavior can be seen rather robustly 
at finite temperature even if the conditions, such as 
particle-hole symmetry, spin rotational symmetry, and 
identical $T_{\rm K}$'s in each orbital, are not satisfied.

%..effective S=1/2 two-channel Kondo model
Another fundamental assumption of the present theory is that Kondo temperatures of two channels are well separated as $T_{\rm K1}\gg T_{\rm K2}$ due to the difference of hybridization ($T_{\rm K1}=1.50\times 10^{-1}$ and $T_{\rm K2}=8.27\times 10^{-4}$ for $J_\perp=J_z=0$).
Thus, below $T_{\rm K1}$, $\vec{S}_1$ has been screened out by conduction electrons in channel 1, while $\vec{S}_2$ is still active as the localized moment.
As the temperature decreases further, the system sets in the critical region, where various quantities exhibit the NFL behaviors.
The characteristic energy $T_{\rm K}$ is given by that of the S=1/2 two-channel Kondo model with the exchange coupling $J\sim |V_2|^2/\max(E_{f2},U_2)$\cite{Pang91,Aff92t,yotsu01,yotsu02} because the NFL behavior arises from the balance of two couplings by which $\vec{S}_2$ interacts with the two ``conduction'' electron channels\cite{yotsu01,yotsu02}.
Below the crossover temperature $T_{\rm cr}$ at which $\chi$ saturates and $\tau^{-1}(\omega)$ ceases to change simultaneously (as shown later in Fig.4), the system flows into the local singlet ground state.
Namely, $T_{\rm cr}$ measures the ``distance'' from the critical point.
In fact, as the parameter set $(K,\Delta)$ goes away from the critical boundary, $T_{\rm cr}$ increases.
We have checked this by series of numerical calculations.
Note that the parameter set locating opposite side of the critical boundary leads to the Kondo singlet ground state.
In the present case, it is estimated that $T_{\rm K}\sim 10^{-3}$ and 
$T_{\rm cr}\sim 10^{-5}$, which is assumed to be the case of 
Th$_{1-x}$U$_x$Ru$_2$Si$_2$.
The renormalization flow in terms of the phase shift is schematically shown in Fig.3.

%
%.Results
%
%..inconsistency between $\chi$ and $\rho$
We show the relation between the magnetic susceptibility, $\chi(T)$, and the total scattering rate at $T=0$, $\tau^{-1}(\omega)$, for $K=0.14$ and $\Delta=0.08$ in Fig.4.
The temperature dependence of the resistivity may infer from the $\tau^{-1}(\omega)$, which is calculated by sum of each channel contributions $\tau_m^{-1}=|V_m|^2A_m(\omega)$, $A_m(\omega)$ being the single-particle spectral function.
The effect of magnetic field has been taken into account through the Zeeman coupling $-(g\mu_{\rm B})j_z H_z$ of each $f$-orbital with $j=5/2$ and $g=6/7$.
For $T_K<(T, \omega)<T_{K1}$, $\chi(T)$ shows the Curie-law behavior, while $\tau^{-1}(\omega)$ takes a broad maximum of the order of the unitarity limit.
Below $T_K$, $\chi(T)$ increases logarithmically and $\tau^{-1}(\omega)$ decreases.
The crossover to the local singlet ground state has not been observed in Th$_{1-x}$U$_x$Ru$_2$Si$_2$, which suggests that $T_{\rm cr}$ is well below the lowest temperature.
Behaviors shown in Fig.4 well simulates those observed in 
R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Th, Y and La) \cite{Ami00}.
It is emphasized that the adopted $K=0.14$ and $\Delta =0.08$ is not located so close to the critical line.

%..(\tau^{-1}) in typical case
The contributions of each channels $\tau_m^{-1}(\omega)$ to $\tau^{-1}(\omega)$ is shown in Fig.5.
The rather flat bump of $\tau^{-1}(\omega)$ in the region $\omega<T_{K1}$ arises from the cancellation of decreasing $\tau^{-1}_1(\omega)$ and increasing $\tau^{-1}_2(\omega)$.
The gradual decrease of $\tau_1^{-1}$ can be understood by an exchange field brought about by growing renormalized Hund's-rule coupling, which destroys the Kondo singlet in channel 1.
The further flat bottom of $\tau^{-1}(\omega)$ below $T_{\rm cr}$ 
indicates that the system flows to the local singlet ground state.

%..entropy and gamma-coefficient in typical case
The temperature dependence of the Sommerfeld coefficient $\gamma$$\equiv$$C/T$ and the entropy under the magnetic field is shown in Fig.6.
By applying the magnetic field of $10^{-3}D\sim$ several Tesla, $\gamma$ is suppressed at low temperatures $T<T_{\rm K}$ because $\gamma$ shows stronger 
divergence than $-\ln T$ in the case of $H=0$.
This divergence is caused by a transient phenomenon from 
a NFL unstable fixed point to the FL fixed point.
The change of the entropy due to the magnetic field is moderate compared with that of the two-channel Kondo model with magnetic doublet.
This is because the magnetic field has little influence on the CEF singlet ground state.
Namely, the behaviors of the present model are relatively robust against the magnetic field.

%
%.Summary
%
In summary, we have shown that the puzzling problem of Th$_{1-x}$U$_x$Ru$_2$Si$_2$ ($x\le 0.07$) can be fully understood on the basis of a mechanism characterized by Non-Fermi liquid like transient phenomenon with $f^2$-singlet ground state.
A key assumption is that the system corresponding to Th$_{1-x}$U$_x$Ru$_2$Si$_2$ is incidentally located near the critical points.
However, it is not so artificial to assume such an incident if one considers the fact that the NFL behaviors appear weakly in much more restricted temperature range for R$_{1-x}$U$_x$Ru$_2$Si$_2$ (R=Y and La) which would have different set of parameters due to the difference of conduction band structure.
Experimental results show that $T_{\rm cr}$ in the case of R=La is about $10$K while that in the case of R=Th has not been observed.
This indicates that the parameter set of La$_{1-x}$U$_x$Ru$_2$Si$_2$ would be located less close to the critical points than that of Th$_{1-x}$U$_x$Ru$_2$Si$_2$.
If $T_{\rm K}$'s are increased with unchanged CEF level structure,
the final fixed point can be changed from the CEF singlet to the Kondo singlet.
Experimentally, when high pressure is applied, these materials may show the increase of the resistivity with decreasing temperature
corresponding to the change of the fixed point.

%
%.Acknowledgement
%
We would like to thank O. Sakai and H. Amitsuka for variable comments.
One of us (S.Y.) acknowledges useful conversations with H. Maebashi.
This work is supported by a Grant-in-Aid for COE Research (No.10CE2004) of the Ministry of Education, Science, Sports, Culture and Technology.

\begin{thebibliography}{99}
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%.experiments
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\bibitem{And91} B. Andraka et al.: Phys. Rev. Lett. {\bf 67} (1991) 2886.

%.two-channel Kondo model
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\bibitem{Aff93} I. Affleck and A. W. W. Ludwig: Phys. Rev. B {\bf 48} (1993) 7297.

%.extended two-channel Anderson model
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%.channel anisotropic two-channel Kondo model
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%.two impurity systems
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\bibitem{Aff95}I. Affleck, A. W. W. Ludwig and B. A. Jones: Phys. Rev. B {\bf 52} (1995) 9528.

%.Kondo effect in f^2 singlet
\bibitem{yotsu01} S. Yotsuhashi, H. Kusunose and K. Miyake: J. Phys. Soc. Jpn. {\bf 70} (2001) 186.
\bibitem{yotsu02} S. Yotsuhashi, H. Kusunose and K. Miyake: unpublished.

%.NRG
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%.Kondo effect in f^2 singlet
\bibitem{Kura92} Y. Kuramoto: {\it Transport and Thermal Properties of f-Electron Systems}, eds. G. Oomi, H. Fujii and T. Fujita, (Plenum Press, NewYork, 1993) p. 237; Y. Shimizu, O. Sakai, Y. Kuramoto: Physica B {\bf 206\&207} (1995) 135.
\end{thebibliography}

\begin{figure}[htbp]
\begin{center}
%\figureheight{3.5cm}
\epsfxsize=7.5cm \epsfbox{fig1.eps}
\end{center}
	\caption{CEF level scheme in $f^2$-configuration adopted in this paper.} \label{fig1}
\end{figure}
\begin{figure}[htbp]
\begin{center}
%\figureheight{5.5cm} 
\epsfxsize=7.5cm \epsfbox{fig2.eps}
\end{center}
	\caption{Phase diagram of the ground state in $K$-$\Delta$ plane. Squares represent the critical points determined by NRG calculations. The cross represents a parameter set used in this paper.} \label{fig2}
\end{figure}
\begin{figure}[htbp]
\begin{center}
%\figureheight{5.5cm}
\epsfxsize=5.5cm \epsfbox{fig3.eps}
\end{center}
	\caption{The renormalization flow below $T<T_{K1}$. The NFL behavior should be observed when the system passes near the critical point.} \label{fig3}
\end{figure}
\begin{figure}[htbp]
\begin{center}
%\figureheight{5.5cm}
\epsfxsize=7.5cm \epsfbox{fig4a.eps}
\end{center}
	\caption{Temperature dependence of the magnetic susceptibility $\chi(T)$ (triangles) and frequency dependence of the scattering rate $\tau^{-1} (\omega)$ at $T=0$ (squares).} \label{fig4}
\end{figure}
\begin{figure}[htbp]
\begin{center}
%\figureheight{5.5cm}
\epsfxsize=7.5cm \epsfbox{fig5a.eps}
\end{center}
	\caption{Contribution of each channels $\tau_1^{-1} (\omega)$ (circles) and $\tau_2^{-1} (\omega)$ (triangles) to the total scattering rate $\tau^{-1} (\omega)$ (squares).} \label{fig5}
\end{figure}
\begin{figure}[htbp]
\begin{center}
%\figureheight{5.5cm}
\epsfxsize=7.5cm \epsfbox{fig6.eps}
\end{center}
	\caption{Temperature dependence of the Sommerfeld coefficient $\gamma \equiv C/T$ under the magnetic field of $H=10^{-8}$ (squares) and $H=10^{-3}$ (circles). The inset shows temperature dependence of the entropy. The other parameter set is the same as shown in Fig.4 and Fig.5.} \label{fig6}
\end{figure}

\end{document}

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%%%                        \epsfbox{filename.ps}
%%%
%%%                        Alternatively, you can supply your own
%%%                        bounding box by
%%%
%%%                        \epsfbox[0 0 30 50]{filename.ps}
%%%
%%%                        This will not read in the file, and will
%%%                        instead use the bounding box you specify.
%%%
%%%                        The effect will be to typeset the figure as
%%%                        a TeX box, at the point of your \epsfbox
%%%                        command. By default, the graphic will have
%%%                        its `natural' width (namely the width of
%%%                        its bounding box, as described in
%%%                        filename.ps). The TeX box will have depth
%%%                        zero.
%%%
%%%                        You can enlarge or reduce the figure by
%%%                        saying
%%%
%%%                          \epsfxsize=<dimen> \epsfbox{filename.ps}
%%%                        or
%%%                          \epsfysize=<dimen> \epsfbox{filename.ps}
%%%
%%%                        instead. Then the width of the TeX box will
%%%                        be \epsfxsize and its height will be scaled
%%%                        proportionately (or the height will be
%%%                        \epsfysize and its width will be scaled
%%%                        proportionately).
%%%
%%%                        The width (and height) is restored to zero
%%%                        after each use, so \epsfxsize or \epsfysize
%%%                        must be specified before EACH use of
%%%                        \epsfbox.
%%%
%%%                        A more general facility for sizing is
%%%                        available by defining the \epsfsize macro.
%%%                        Normally you can redefine this macro to do
%%%                        almost anything.  The first parameter is
%%%                        the natural x size of the PostScript
%%%                        graphic, the second parameter is the
%%%                        natural y size of the PostScript graphic.
%%%                        It must return the xsize to use, or 0 if
%%%                        natural scaling is to be used.  Common uses
%%%                        include:
%%%
%%%                           \epsfxsize  % just leave the old value alone
%%%                           0pt         % use the natural sizes
%%%                           #1          % use the natural sizes
%%%                           \hsize      % scale to full width
%%%                           0.5#1       % scale to 50% of natural size
%%%                           \ifnum #1>\hsize\hsize\else#1\fi
%%%                                       % smaller of natural, hsize
%%%
%%%                        If you want TeX to report the size of the
%%%                        figure (as a message on your terminal when
%%%                        it processes each figure), say
%%%                        `\epsfverbosetrue'.
%%%
%%%                        If you only want to get the bounding box
%%%                        extents, without producing any output boxes
%%%                        or \special{}, then say
%%%                        \epsfgetbb{filename}.  The extents will be
%%%                        saved in the macros \epsfllx \epsflly
%%%                        \epsfurx \epsfury in PostScript units of
%%%                        big points.
%%%
%%%                        Revision history:
%%%
%%%                        ---------------------------------------------
%%%                        epsf.tex macro file:
%%%                        Originally written by Tomas Rokicki of
%%%                        Radical Eye Software, 29 Mar 1989.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Don Knuth, 3 Jan 1990.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Tomas Rokicki, 18 Jul 1990.
%%%                        Accept bounding boxes with no space after
%%%                        the colon.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 03 Dec 1991 [2.0].
%%%                        Add version number and date typeout.
%%%
%%%                        Use \immediate\write16 instead of \message
%%%                        to ensure output on new line.
%%%
%%%                        Handle nested EPS files.
%%%
%%%                        Handle %%BoundingBox: (atend) lines.
%%%
%%%                        Do not quit when blank lines are found.
%%%
%%%                        Add a few percents to remove generation of
%%%                        spurious blank space.
%%%
%%%                        Move \special output to
%%%                        \epsfspecial{filename} so that other macro
%%%                        packages can input this one, then change
%%%                        the definition of \epsfspecial to match
%%%                        another DVI driver.
%%%
%%%                        Move size computation to \epsfsetsize which
%%%                        can be called by the user; the verbose
%%%                        output of the bounding box and scaled width
%%%                        and height happens here.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 05 May 1992 [2.1].
%%%                        Wrap \leavevmode\hbox{} around \vbox{} with
%%%                        the \special so that \epsffile{} can be
%%%                        used inside \begin{center}...\end{center}
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 09 Dec 1992 [2.2].
%%%                        Introduce \epsfshow{true,false} and
%%%                        \epsfframe{true,false} macros; the latter
%%%                        suppresses the insertion of the PostScript,
%%%                        and instead just creates an empty box,
%%%                        which may be handy for rapid prototyping.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 14 Dec 1992 [2.3].
%%%                        Add \epsfshowfilename{true,false}.  When
%%%                        true, and \epsfshowfalse is specified, the
%%%                        PostScript file name will be displayed
%%%                        centered in the figure box.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 20 June 1993 [2.4].
%%%                        Remove non-zero debug setting of \epsfframemargin,
%%%                        and change margin handling to preserve EPS image
%%%                        size and aspect ratio, so that the actual
%%%                        box is \epsfxsize+\epsfframemargin wide by
%%%                        \epsfysize+\epsfframemargin high.
%%%                        Reduce output of \epsfshowfilenametrue to
%%%                        just the bare file name.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 13 July 1993 [2.5].
%%%                        Add \epsfframethickness for control of
%%%                        \epsfframe frame lines.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 02 July 1996 [2.6]
%%%                        Add missing initialization \epsfatendfalse;
%%%                        the lack of this resulted in the wrong
%%%                        BoundingBox being picked up, mea culpa, sigh...
%%%                        ---------------------------------------------
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 25 October 1996 [2.7]
%%%                        Update to match changes in from dvips 5-600
%%%                        distribution: new user-accessible macros:
%%%                        \epsfclipon, \epsfclipoff, \epsfdrafton,
%%%                        \epsfdraftoff, change \empty to \epsfempty.
%%%                        ---------------------------------------------
%%%  }
%%% ====================================================================

 \immediate\write16{This is `epsf.tex' v2.7 <25 October 1996>}%
%
\newread\epsffilein    % file to \read
\newif\ifepsfatend     % need to scan to LAST %%BoundingBox comment?
\newif\ifepsfbbfound   % success?
\newif\ifepsfdraft     % use draft mode?
\newif\ifepsffileok    % continue looking for the bounding box?
\newif\ifepsfframe     % frame the bounding box?
\newif\ifepsfshow      % show PostScript file, or just bounding box?
\epsfshowtrue          % default is to display PostScript file
\newif\ifepsfshowfilename % show the file name if \epsfshowfalse specified?
\newif\ifepsfverbose   % report what you're making?
\newdimen\epsfframemargin % margin between box and frame
\newdimen\epsfframethickness % thickness of frame rules
\newdimen\epsfrsize    % vertical size before scaling
\newdimen\epsftmp      % register for arithmetic manipulation
\newdimen\epsftsize    % horizontal size before scaling
\newdimen\epsfxsize    % horizontal size after scaling
\newdimen\epsfysize    % vertical size after scaling
\newdimen\pspoints     % conversion factor
%
\pspoints = 1bp        % Adobe points are `big'
\epsfxsize = 0pt       % default value, means `use natural size'
\epsfysize = 0pt       % ditto
\epsfframemargin = 0pt % default value: frame box flush around picture
\epsfframethickness = 0.4pt % TeX's default rule thickness
%
\def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}%
   \global\def\epsfurx{540}\global\def\epsfury{720}%
   \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket
   \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}%
%
% We use \epsfgetlitbb if the user specified an explicit bounding box,
% and \epsfnormal otherwise.  Because \epsfgetbb can be called
% separately to retrieve the bounding box, we move the verbose
% printing the bounding box extents and size on the terminal to
% \epsfstatus.  Therefore, when the user provided the bounding box,
% \epsfgetbb will not be called, so we must call \epsfsetsize and
% \epsfstatus ourselves.
%
\def\epsfgetlitbb#1#2 #3 #4 #5]#6{%
   \epsfgrab #2 #3 #4 #5 .\\%
   \epsfsetsize
   \epsfstatus{#6}%
   \epsfsetgraph{#6}%
}%
%
\def\epsfnormal#1{%
    \epsfgetbb{#1}%
    \epsfsetgraph{#1}%
}%
%
\def\epsfgetbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
    \openin\epsffilein=#1
    \ifeof\epsffilein
        \errmessage{Could not open file #1, ignoring it}%
    \else                       %process the file
        {%                      %start a group to contain catcode changes
            % Make all special characters, except space, to be of type
            % `other' so we process the file in almost verbatim mode
            % (TeXbook, p. 344).
            \chardef\other=12
            \def\do##1{\catcode`##1=\other}%
            \dospecials
            \catcode`\ =10
            \epsffileoktrue         %true while we are looping
            \epsfatendfalse     %[02-Jul-1996]: add forgotten initialization
            \loop               %reading lines from the EPS file
                \read\epsffilein to \epsffileline
                \ifeof\epsffilein %then no more input
                \epsffileokfalse %so set completion flag
            \else                %otherwise process one line
                \expandafter\epsfaux\epsffileline:. \\%
            \fi
            \ifepsffileok
            \repeat
            \ifepsfbbfound
            \else
                \ifepsfverbose
                    \immediate\write16{No BoundingBox comment found in %
                                    file #1; using defaults}%
                \fi
            \fi
        }%                      %end catcode changes
        \closein\epsffilein
    \fi                         %end of file processing
    \epsfsetsize                %compute size parameters
    \epsfstatus{#1}%
}%
%
% Clipping control:
\def\epsfclipon{\def\epsfclipstring{ clip}}%
\def\epsfclipoff{\def\epsfclipstring{\ifepsfdraft\space clip\fi}}%
\epsfclipoff % default for dvips is OFF
%
% The special that is emitted by \epsfsetgraph comes from this macro.
% It is defined separately to allow easy customization by other
% packages that first \input epsf.tex, then redefine \epsfspecial.
% This macro is invoked in the lower-left corner of a box of the
% width and height determined from the arguments to \epsffile, or
% from the %%BoundingBox in the EPS file itself.
%
% This version is for dvips:
\def\epsfspecial#1{%
     \epsftmp=10\epsfxsize
     \divide\epsftmp\pspoints
     \ifnum\epsfrsize=0\relax
       \special{PSfile=\ifepsfdraft psdraft.ps\else#1\fi\space
                llx=\epsfllx\space
                lly=\epsflly\space
                urx=\epsfurx\space
                ury=\epsfury\space
                rwi=\number\epsftmp
                \epsfclipstring
               }%
     \else
       \epsfrsize=10\epsfysize
       \divide\epsfrsize\pspoints
       \special{PSfile=\ifepsfdraft psdraft.ps\else#1\fi\space
                llx=\epsfllx\space
                lly=\epsflly\space
                urx=\epsfurx\space
                ury=\epsfury\space
                rwi=\number\epsftmp
                rhi=\number\epsfrsize
                \epsfclipstring
               }%
     \fi
}%
%
% \epsfframe macro adapted from the TeXbook, exercise 21.3, p. 223, 331.
% but modified to set the box width to the natural width, rather
% than the line width, and to include space for margins and rules
\def\epsfframe#1%
{%
  \leavevmode                   % so we can put this inside
                                % a centered environment
  \setbox0 = \hbox{#1}%
  \dimen0 = \wd0                                % natural width of argument
  \advance \dimen0 by 2\epsfframemargin         % plus width of 2 margins
  \advance \dimen0 by 2\epsfframethickness      % plus width of 2 rule lines
  \vbox
  {%
    \hrule height \epsfframethickness depth 0pt
    \hbox to \dimen0
    {%
      \hss
      \vrule width \epsfframethickness
      \kern \epsfframemargin
      \vbox {\kern \epsfframemargin \box0 \kern \epsfframemargin }%
      \kern \epsfframemargin
      \vrule width \epsfframethickness
      \hss
    }% end hbox
    \hrule height 0pt depth \epsfframethickness
  }% end vbox
}%
%
\def\epsfsetgraph#1%
{%
   %
   % Make the vbox and stick in a \special that the DVI driver can
   % parse.  \vfil and \hfil are used to place the \special origin at
   % the lower-left corner of the vbox.  \epsfspecial can be redefined
   % to produce alternate \special syntaxes.
   %
   \leavevmode
   \hbox{% so we can put this in \begin{center}...\end{center}
     \ifepsfframe\expandafter\epsfframe\fi
     {\vbox to\epsfysize
     {%
        \ifepsfshow
            % output \special{} at lower-left corner of figure box
            \vfil
            \hbox to \epsfxsize{\epsfspecial{#1}\hfil}%
        \else
            \vfil
            \hbox to\epsfxsize{%
               \hss
               \ifepsfshowfilename
               {%
                  \epsfframemargin=3pt % local change of margin
                  \epsfframe{{\tt #1}}%
               }%
               \fi
               \hss
            }%
            \vfil
        \fi
     }%
   }}%
   %
   % Reset \epsfxsize and \epsfysize, as documented above.
   %
   \global\epsfxsize=0pt
   \global\epsfysize=0pt
}%
%
%   Now we have to calculate the scale and offset values to use.
%   First we compute the natural sizes.
%
\def\epsfsetsize
{%
   \epsfrsize=\epsfury\pspoints
   \advance\epsfrsize by-\epsflly\pspoints
   \epsftsize=\epsfurx\pspoints
   \advance\epsftsize by-\epsfllx\pspoints
%
%   If `epsfxsize' is 0, we default to the natural size of the picture.
%   Otherwise we scale the graph to be \epsfxsize wide.
%
   \epsfxsize=\epsfsize{\epsftsize}{\epsfrsize}%
   \ifnum \epsfxsize=0
      \ifnum \epsfysize=0
        \epsfxsize=\epsftsize
        \epsfysize=\epsfrsize
        \epsfrsize=0pt
%
%   We have a sticky problem here:  TeX doesn't do floating point arithmetic!
%   Our goal is to compute y = rx/t. The following loop does this reasonably
%   fast, with an error of at most about 16 sp (about 1/4000 pt).
%
      \else
        \epsftmp=\epsftsize \divide\epsftmp\epsfrsize
        \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp
        \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp
        \epsftmp=\epsfysize
        \loop \advance\epsftsize\epsftsize \divide\epsftmp 2
        \ifnum \epsftmp>0
           \ifnum \epsftsize<\epsfrsize
           \else
              \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp
           \fi
        \repeat
        \epsfrsize=0pt
      \fi
   \else
     \ifnum \epsfysize=0
       \epsftmp=\epsfrsize \divide\epsftmp\epsftsize
       \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp
       \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp
       \epsftmp=\epsfxsize
       \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2
       \ifnum \epsftmp>0
          \ifnum \epsfrsize<\epsftsize
          \else
             \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp
          \fi
       \repeat
       \epsfrsize=0pt
     \else
       \epsfrsize=\epsfysize
     \fi
   \fi
}%
%
% Issue some status messages if the user requested them
%
\def\epsfstatus#1{% arg = filename
   \ifepsfverbose
     \immediate\write16{#1: BoundingBox:
                  llx = \epsfllx\space lly = \epsflly\space
                  urx = \epsfurx\space ury = \epsfury\space}%
     \immediate\write16{#1: scaled width = \the\epsfxsize\space
                  scaled height = \the\epsfysize}%
   \fi
}%
%
%   We still need to define the tricky \epsfaux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}%
\global\def\epsfatend{(atend)}%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.
%
%   If we find a line
%
%   %%BoundingBox: (atend)
%
%   then we ignore it, but set a flag to force parsing all of the
%   file, so the last %%BoundingBox parsed will be the one used.  This
%   is necessary, because EPS files can themselves contain other EPS
%   files with their own %%BoundingBox comments.
%
%   If we find a line
%
%   %%BoundingBox: llx lly urx ury
%
%   then we save the 4 values in \epsfllx, \epsflly, \epsfurx, \epsfury.
%   Then, if we have not previously parsed an (atend), we flag completion
%   and can stop reading the file.  Otherwise, we must keep on reading
%   to end of file so that we find the values on the LAST %%BoundingBox.
\long\def\epsfaux#1#2:#3\\%
{%
   \def\testit{#2}%             % save second character up to just before colon
   \ifx#1\epsfpercent           % then first char is percent (quick test)
       \ifx\testit\epsfbblit    % then (slow test) we have %%BoundingBox
            \epsfgrab #3 . . . \\%
            \ifx\epsfllx\epsfatend % then ignore %%BoundingBox: (atend)
                \global\epsfatendtrue
            \else               % else found %%BoundingBox: llx lly urx ury
                \ifepsfatend    % then keep parsing ALL %%BoundingBox lines
                \else           % else stop after first one parsed
                    \epsffileokfalse
                \fi
                \global\epsfbbfoundtrue
            \fi
       \fi
   \fi
}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsfempty{}%
\def\epsfgrab #1 #2 #3 #4 #5\\{%
   \global\def\epsfllx{#1}\ifx\epsfllx\epsfempty
      \epsfgrab #2 #3 #4 #5 .\\\else
   \global\def\epsflly{#2}%
   \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi
}%
%
%   We default the epsfsize macro.
%
\def\epsfsize#1#2{\epsfxsize}%
%
%   Finally, another definition for compatibility with older macros.
%
\let\epsffile=\epsfbox
\endinput

cond-mat0109072.extracted_bib
\begin{thebibliography}{99}
\bibitem{Cox} D. L. Cox, A. Zawadowski: {\it Exotic Kondo Effects in Metals: Magnetic ions in a crystalline electric field and tunnelling centres} (Taylor \& Francis, London, 1999).
%.experiments
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\bibitem{Maru96} K. Marumoto et al.: Phys. Rev. B {\bf 54} (1996) 12194.
\bibitem{Ami00} H. Amitsuka et al.: Physica B {\bf 281\&282} (2000) 326.
\bibitem{Sea91} C. L. Seaman et al.: Phys. Rev. Lett. {\bf 67} (1991) 2882.
\bibitem{And91} B. Andraka et al.: Phys. Rev. Lett. {\bf 67} (1991) 2886.

%.two-channel Kondo model
\bibitem{Cox87} D. L. Cox: Phys. Rev. Lett. {\bf 59} (1987) 1240.
\bibitem{Cox96} D. L. Cox and M. Jarrell: J. Phys.: Condens. Matter {\bf 8} (1996) 9825.
\bibitem{Kusu96} H. Kusunose et al.: Phys. Rev. Lett. {\bf 76} (1996) 271.
\bibitem{Noz80} P. Nozi\`eres and A. Blandin: J. Phys. {\bf 41} (1980) 193.
\bibitem{And84} N. Andrei and C. Destri: Phys. Rev. Lett. {\bf 52} (1984) 364.
\bibitem{Sac91} P. D. Sacramento and P. B. Schlottmann: Phys. Rev. B {\bf 43} (1991) 13294.
\bibitem{Aff93} I. Affleck and A. W. W. Ludwig: Phys. Rev. B {\bf 48} (1993) 7297.

%.extended two-channel Anderson model
\bibitem{Sakai96} O. Sakai et al.: Solid State Commun. {\bf 99} (1996) 461.
\bibitem{Suzu97} S. Suzuki, O. Sakai and Y. Shimizu: Solid State Commun. {\bf 104} (1997) 429.
\bibitem{Shimi98} Y. Shimizu, O. Sakai and S. Suzuki: J. Phys. Soc. Jpn. {\bf 67} (1998) 2395.
\bibitem{Shimi99} Y. Shimizu, O. Sakai and S. Suzuki: Physica B {\bf 259-261} (1999) 366.

%.channel anisotropic two-channel Kondo model
\bibitem{Pang91} H. -B. Pang and D. L. Cox: Phys. Rev. B {\bf 44} (1991) 9454.
\bibitem{Aff92t} I. Affleck et al.: Phys. Rev. B {\bf 45} (1992) 7918.

%.two impurity systems
\bibitem{Jones88}B. A. Jones, C. M. Varma and J. W. Wilkins: Phys. Rev. Lett. {\bf 61} (1988) 125.
\bibitem{Jones89}B. A. Jones and C. M. Varma: Phys. Rev. B {\bf 40} (1989) 324.
\bibitem{Sakai92}O. Sakai and Y. Shimizu: J. Phys. Soc. Jpn. {\bf 61} (1992) 2333.
\bibitem{Aff92}I. Affleck and A. W. W. Ludwig: Phys. Rev. Lett. {\bf 68} (1992) 1046.
\bibitem{Aff95}I. Affleck, A. W. W. Ludwig and B. A. Jones: Phys. Rev. B {\bf 52} (1995) 9528.

%.Kondo effect in f^2 singlet
\bibitem{yotsu01} S. Yotsuhashi, H. Kusunose and K. Miyake: J. Phys. Soc. Jpn. {\bf 70} (2001) 186.
\bibitem{yotsu02} S. Yotsuhashi, H. Kusunose and K. Miyake: unpublished.

%.NRG
\bibitem{Wilson75}K. G. Wilson: Rev. Mod. Phys. {\bf 47} (1975) 773.
\bibitem{Krish80} H. R. Krishna-Murthy, J. W. Wilkins and K. G. Wilson: Phys. Rev. B {\bf 21} (1980) 1003; {\bf 21} (1980) 1044.
\bibitem{Sakai89}O. Sakai, Y. Shimizu and T. Kasuya: J. Phys. Soc. Jpn. {\bf 58} (1989) 3666.

%.Kondo effect in f^2 singlet
\bibitem{Kura92} Y. Kuramoto: {\it Transport and Thermal Properties of f-Electron Systems}, eds. G. Oomi, H. Fujii and T. Fujita, (Plenum Press, NewYork, 1993) p. 237; Y. Shimizu, O. Sakai, Y. Kuramoto: Physica B {\bf 206\&207} (1995) 135.
\end{thebibliography}