text:
\section{Introduction}

Research on heavy-electron systems has been one of central topics
in the field of strongly correlated electron physics.\cite{Kondo40,ICHE2010,SCES2013}
The origin of such heavy-electron state has been understood
on the basis of quantum critical phenomena,
\cite{review1,review2,review3,review4,review5,review6,review7}
emerging in the competing region of itinerant properties of $f$ electrons
due to the Kondo effect~\cite{Kondo1,Kondo2a,Kondo2b}
and localized nature due to the Ruderman-Kittel-Kasuya-Yosida (RKKY)
interaction.\cite{RKKY1,RKKY2,RKKY3}

The Kondo effect has been first discovered as resistance minimum
phenomena in good metals such as Cu, Ag, and Au with a small amount of
magnetic impurities such as Mn and Fe.
In the dilute system, it has been widely recognized that Kondo effect
occurs when the singlet state is formed from local magnetic moment
due to the antiferromagnetic (AF) coupling with conduction electrons.\cite{Yosida}
As for the original problem of resistance minimum phenomena
in metals with magnetic impurities,
Kondo has actually demonstrated it by quantum-mechanical calculations
for scattering amplitude of electrons due to magnetic impurities.
\cite{Kondo1,Kondo2a,Kondo2b}

On the other hand, there appears long-range interaction between localized
electrons via conduction electrons.
This is called the RKKY interaction,\cite{RKKY1,RKKY2,RKKY3}
which enhances the localized nature of $f$ electrons.
As a result of the competition between the Kondo effect and the RKKY
interactions, quantum criticality appears between AF and metallic phases,
when we control the coupling constant $\rho J_{\rm cf}$,
where $J_{\rm cf}$ denotes the AF coupling between localized and
conduction electrons and $\rho$ is the density of states at the Fermi level.
The picture has been summarized in the Doniach's phase diagram,\cite{Doniach}
which has been a guiding principle in the heavy-electron physics to
discover the unconventional superconductivity mediated by
quantum critical fluctuations.

Along this line, the heavy-electron phenomena and unconventional superconductivity
have been intensively and extensively investigated in the Ce compounds
with one $f$ electron for Ce$^{3+}$ ion,
since the pioneering discovery of
superconductivity in CeCu$_2$Si$_2$.\cite{Steglich}
Recently, the research on heavy-electron state and superconductivity in
Yb compounds has been activated on the basis
of the electron-hole conversion picture of Ce$^{3+}$.
\cite{Nakatsuji1,Nakatsuji2,Nakatsuji3}
It has been claimed that this material exists just on the
quantum critical point at ambient pressure.
For the case of two $f$ electrons, there is a long history in the research
of U compounds.
In fact, after the discovery of heavy-electron superconductivity
in CeCu$_2$Si$_2$,
superconductivity has been found in U compounds
such as UBe$_{13}$,\cite{UBe13}
UPt$_3$,\cite{UPt3} URu$_2$Si$_2$,\cite{URu2Si2}
UPd$_2$Al$_3$,\cite{UPd2Al3} and UNi$_2$Al$_3$.\cite{UNi2Al3}
In a recent decade, superconductivity and magnetism in
Pr compounds such as PrOs$_2$Sb$_{12}$,\cite{Bauer}
PrPb$_3$,\cite{Onimaru1,Onimaru2} and
PrT$_2$X$_{20}$ (T: transition metal, X=Al and Zn)
\cite{Onimaru3,Onimaru4,Sakai1,Sakai2,Matsubayashi,Tsujimoto}
have been actively investigated.
On the basis  of the electron-hole conversion picture of Pr$^{3+}$,
it is also interesting to note Tm$^{3+}$ with twelve $f$ electrons.
In Tm$_5$Rh$_6$Sn$_{18}$, superconductivity has been
found with $T_{\rm c}=2.2$K.\cite{Kase}
Peculiar reentrant properties have been considered to be
related to the coexistence of magnetism and superconductivity.

When we look back over the history of heavy-electron materials,
we immediately notice that elements around both ends of lanthanide series
have been focused thus far.
However, recently, the heavy-electron state in Eu compounds,
just at the center in lanthanide series, have attracted renewed attention.
\cite{Mitsuda1,Hiranaka,Mitsuda2,Nakamura1,Nakamura2}
In the divalent ion of europium, seven electrons are accommodated
in the $4f$ orbitals, while in the trivalent ion of europium, we find
six $4f$ electrons.
In an $LS$ coupling scheme, due to the Hund's rules,
we obtain $J=S=7/2$ and $L=0$ for Eu$^{2+}$, where
$J$, $S$, and $L$ indicate, respectively, total angular momentum,
total spin momentum, and total orbital momentum.
On the other hand, we find $J=0$ with $S=L=3$ for Eu$^{3+}$,
which is non-magnetic.
The difference in magnetic properties between Eu$^{2+}$ and Eu$^{3+}$
ions are significant, but in general, the valence fluctuations easily occur
in Eu compounds, since the energy difference between two valence states
has been known to be small.
Thus, the valence fluctuation is one of key issues in Eu compounds.
\cite{Mitsuda2}

Quite recently, in Eu$_2$Ni$_3$Ge$_5$ and EuRhSi$_3$,
heavy-electron states have been claimed to be observed in the
temperature dependence in the measurement of resistivity.\cite{Nakamura2}
At ambient pressure, those compounds are in the AF state
at low temperatures, but under pressure such as several GPa,
the AF state is suppressed and three characteristic temperatures have
been observed in the resistivity.
The highest one was assigned as $T_{\rm v}$, which is the valence transition
temperature.
Other two low temperatures were considered as Kondo temperatures.
These results were claimed to be quite similar to those of Ce compounds.

At a first glance, it seems to be difficult to accept the similarity between
Ce and Eu compounds, but it is easy to hit upon an idea on the basis
of a $j$-$j$ coupling scheme, where $j$ denotes the total angular momentum
of one $f$ electron.
Namely, in the $j$-$j$ coupling scheme, among seven $f$ electrons,
six electrons fully occupy the sextet of $j=5/2$,
whereas one electron is accommodated in the octet of $j=7/2$,
as shown in Fig.~1(a).
Since six electrons in the sextet do not contribute to electronic properties,
one $f$ electron plays a main role for the Kondo effect,
leading to the similar behavior as that of Ce compound.
This idea has been also emphasized by the present author for
the explanation of active quadrupole degrees of freedom in Gd compound
with seven $f$ electrons in the trivalent ion state of gadolinium.\cite{Hotta1}

Here we cast a naive question:
Are there any problems to use the $j$-$j$ coupling scheme
for the Eu compounds?
In fact, in a textbook of solid state physics,
it is standard to use the $LS$ coupling scheme.
Namely, first we construct the many $f$ electron state
characterized by $S$ and $L$ due to the so-called Hund's rules.
Then, we consider the effect of spin-orbit coupling
by including the term of $\Lambda {\mib L}\cdot{\mib S}$
with a spin-orbit coupling $\Lambda$,
leading to the multiplet characterized by $J$.
For seven $f$ electrons, as mentioned above, first we obtain the state
with $S=7/2$ and $L=0$ from the Hund's rules, as shown in Fig.~1(b).
Since $L$ is zero, the ground state is characterized by $J=S=7/2$.

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig1.eps}
\caption{
Electron configurations of the seven $f$-electron state
for (a)  the $j$-$j$ coupling scheme
and (b) the $LS$ coupling scheme.
Note that up and down arrows denote, respectively,
pseudo-spin up and down electrons in (a),
while in (b), they indicate real-spin up and down electrons.
The pseudo-spin state is defined through the time-reversal relation.
}
\end{figure}

For rare-earth compounds, in general, the magnitude of Hund's rule interaction
among $f$ orbitals is a few eV, while the spin-orbit coupling
takes a value of a few thousand Kelvins.
If we should take one of the two limiting situations, it is better to choose
the $LS$ coupling scheme, as readers have learned in the standard textbook.
In this sense, it seems to be difficult to understand the Kondo effect in
the Eu compound on the basis of the $j$-$j$ coupling scheme.

However, we should remark that both of the spin-orbit coupling and
the Hund's rule interaction are finite in actual materials and
the actual situation is always in the middle of the $LS$ and $j$-$j$ coupling schemes.
Namely, the wave function of the many $f$ electron state is the mixture of
those in the $LS$ and $j$-$j$ coupling schemes.
In order to discuss the Kondo effect in the Eu compound even qualitatively,
it is essential to consider both of the spin-orbit coupling and
the Hund's rule interaction.
This point has been emphasized in the research of quadrupole susceptibility
in Gd compounds.\cite{Hotta1}

In this paper, we analyze the seven-orbital Anderson model by employing
a numerical renormalization group technique.
The local term contains Coulomb interaction, spin-orbit coupling,
and crystalline electric field (CEF) potential terms.
Here we introduce the Hund's rule interaction $U$,
the spin-orbit coupling $\lambda$, and the CEF potential $W$.
As for conduction electrons, we include only one conduction band
with a$_{\rm u}$ symmetry.
Readers may consider that a trivial result of the underscreening Kondo
effect is obtained, but such a well-known result is observed only in the
$LS$ coupling limit.
When we increase the spin-orbit coupling in this situation,
we confirm the Kondo bahavior similar to that in the Ce compound,
as expected from the $j$-$j$ coupling scheme.
An important finding in this paper is that the Kondo effect
similar to the case of $n=1$ occurs for a realistic value
of spin-orbit coupling even in the case of $n=7$,
where $n$ denotes local $f$-electron number.
When we explicitly include the cubic CEF potentials,
we discuss the multipole properties and the Kondo behavior
similar to Ce compounds.

The organization of this paper is as follows.
In Sec.~2, we provide the local $f$-electron Hamiltonian
and briefly review the change of the seven $f$-electron
states between the $LS$ and $j$-$j$ coupling schemes
by evaluating the Curie constant.
We emphasize that the transition region between them
just corresponds to the situation of actual $f$-electron materials.
In Sec.~3, we introduce the impurity Anderson model
to discuss the Kondo phenomenon for the case of $n=7$.
We also briefly explain the method used in this paper and
provide the definition of multipole susceptibilities.
In Sec.~4, we exhibit our numerical results and discuss
how an entropy changes with the decrease of a temperature.
First we clearly show the underscreening Kondo effect
in the $LS$ coupling scheme for $\lambda=W=0$.
Then, we investigate the effect of the spin-orbit coupling on
the underscreening Kondo behavior for $W=0$.
For small $V$ and $\lambda/U$ in the order of $0.1$,
we confirm the Kondo behavior characterized by
the entropy release of $\ln 2$, as found in the Ce compound.
Then, we also investigate the Kondo behavior for $W \ne 0$
and discuss the multipole susceptibility to confirm
that the relevant multipole is dipole in the present Kondo effect.
Finally, in Sec.~5, we provide a few comments on future problems
and summarize this paper.
Throughout this paper, we use such units as $\hbar=k_{\rm B}=1$.

\section{Local $f$ Electron State}

\subsection{Local Hamiltonian}

First, we define the local $f$-electron Hamiltonian as
\begin{equation}
\label{Hloc}
\begin{split}
  H_{\rm loc} &=\sum_{m_1 \sim m_4}\sum_{\sigma,\sigma'}
  I_{m_1m_2,m_3m_4}
  f_{m_1\sigma}^{\dag}f_{m_2\sigma'}^{\dag}
  f_{m_3\sigma'}f_{m_4\sigma}\\
 &+ \lambda \sum_{m,\sigma,m',\sigma'}
   \zeta_{m,\sigma;m',\sigma'} f_{m\sigma}^{\dag}f_{m'\sigma'}\\
 &+ \sum_{m,m',\sigma} B_{m,m'}
      f_{m \sigma}^{\dag} f_{m' \sigma},
\end{split}
\end{equation}
where $f_{m\sigma}$ denotes the annihilation operator
for local $f$ electron with the spin $\sigma$ and $z$-component $m$
of the angular momentum $\ell=3$,
$\sigma=+1$ ($-1$) for up (down) spin,
$I_{m_1m_2,m_3m_4}$ indicates the Coulomb interaction,
$\lambda$ is the spin-orbit coupling,
and $B_{m,m'}$ denotes the CEF potential.

The Coulomb interaction $I$ is known to be expressed as
\begin{equation}
   I_{m_1m_2,m_3m_4} = \sum_{k=0}^{6} F^k c_k(m_1,m_4)c_k(m_2,m_3),
\end{equation}
where $F^k$ indicates the Slater-Condon parameter and
$c_k$ is the Gaunt coefficient.\cite{Slater}
Note that the sum is limited by the Wigner-Eckart theorem to
$k=0$, $2$, $4$, and $6$.
Although the Slater-Condon parameters should be determined
for the material from the experimental results,
we assume the ratio among the Slater-Condon parameters as
\begin{equation}
  F^0=10U,~F^2=5U,~F^4=3U,~F^6=U,
\end{equation}
where $U$ is the Hund's rule interaction among $f$ orbitals.

Each element of $\zeta$ for the spin-orbit coupling is given by
\begin{equation}
\begin{split}
    \zeta_{m,\sigma;m,\sigma}&=m\sigma/2,\\
    \zeta_{m+\sigma,-\sigma;m,\sigma}&=\sqrt{\ell(\ell+1)-m(m+\sigma)}/2,
\end{split}
\end{equation}
and zero for other cases.
The CEF potentials for $f$ electrons from the ligand ions is
given in the table of Hutchings
for the angular momentum $\ell=3$.\cite{Hutchings}
For cubic structure with $O_{\rm h}$ symmetry,
$B_{m,m'}$ is expressed using a couple of CEF parameters,
$B_4^0$ and $B_6^0$, as
\begin{equation}
\begin{split}
    B_{3,3}&=B_{-3,-3}=180B_4^0+180B_6^0, \\
    B_{2,2}&=B_{-2,-2}=-420B_4^0-1080B_6^0, \\
    B_{1,1}&=B_{-1,-1}=60B_4^0+2700B_6^0, \\
    B_{0,0}&=360B_4^0-3600B_6^0, \\
    B_{3,-1}&=B_{-3,1}=60\sqrt{15}(B_4^0-21B_6^0),\\
    B_{2,-2}&=300B_4^0+7560B_6^0,
\end{split}
\end{equation}
Note the relation $B_{m,m'}=B_{m',m}$.
Following the traditional notation,\cite{LLW}
we define $B_4^0$ and $B_6^0$ as
\begin{equation}
    B_4^0=Wx/F(4),~B_6^0=W(1-|x|)/F(6),
\end{equation}
where $x$ and $y$ specify the CEF scheme for the $O_{\rm h}$ point group,
while $W$ determines the energy scale for the CEF potential.
We choose $F(4)=15$ and $F(6)=180$ for $\ell=3$.\cite{Hutchings}

Before proceeding to the results, we provide a comment on the energy scale
for $U$, $\lambda$, and $W$,
Among them, the largest one is $U$ and its magnitude is in the order of 1 eV,
since it denotes the Hund's rule interaction among $f$ orbitals.
The next one is $\lambda$ and its magnitude is in the order of 0.1 eV,
although the precise values depend on the kind of lanthanide and actinide
ions between 0.1 and 0.3 eV.
The smallest one is $W$, since the magnitude is typically in the order of
meV, although the values depend on materials.
In any case, it is reasonable to consider
$U > \lambda \gg W$ and $\lambda/U \sim 0.1$
for actual $f$-electron compounds.

Concerning the energy unit, we define it as a half of
the conduction bandwidth in this paper, as will be mentioned later.
Throughout this paper, we set it as 1 eV, but this value is
realistic as a half of the conduction bandwidth.

\subsection{$LS$ and $j$-$j$ coupling schemes}

In the standard textbook for solid state physics,
it is recommended to employ an $LS$ coupling scheme for
$f$-electron states in the limit of $U \gg \lambda$, while
we should use a $j$-$j$ coupling scheme in the limit of $U \ll \lambda$.
Since the actual situation is found for $\lambda/U \sim 0.1$,
it seems to be better to choose always the $LS$ coupling scheme.
However, as has been emphasized in our previous papers,\cite{Hotta1,Hotta3,Hotta5}
the wavefunction of the $f$-electron state for $\lambda/U \sim 0.1$ is well
approximated by that in the $j$-$j$ coupling scheme.
The $f$-electron state is continuously changed from the $LS$ to
the $j$-$j$ coupling limits, when we change the ratio of $\lambda/U$,
but the transition region is found in the range between $\lambda/U=0.1$ and $1$.

For $f^7$-electron case, we also find such a transition in
the same parameter region of $\lambda/U$.
In order to reconfirm this point for the case of $n=7$,
we evaluate the Curie constant $C$ on the basis of $H_{\rm loc}$
for $W=0$.
We obtain the magnetic susceptibility $\chi$ as
$\chi=C/T$ and the Curie constant $C$ is given by
$C=(g_J \mu_B)^2 J(J+1)/3k_{\rm B}$.
Here $\mu_{\rm B}$ indicates the Bohr's magneton,
$J$ denotes the size of total angular momentum ${\mib J}$,
which is given by ${\mib J}={\mib L}+{\mib S}$
with total angular momentum ${\mib L}$ and total spin momentum ${\mib S}$
and $g_J$ is the Land\'e's $g$-factor.
Note that this expression for $C$ is in common with the $LS$ and $j$-$j$ coupling limits,
although the magnitude of $C$ is changed.

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig2.eps}
\caption{
Curie constant $C$ for $n=7$ as a function of $\lambda/U$ for the case of $W=0$.
We find the transition from the value for the $LS$ coupling limit to that for the $j$-$j$ coupling limit
in the region pf $0.1 < \lambda/U < 1$, which includes the parameters for actual materials.
}
\end{figure}

In the $LS$ coupling limit, for the case of $n=7$,
we obtain $J=S=7/2$ and $L=0$ due to the Hund's rules.
In this case, we find $g_J=2$, which is just equal to
the electron's $g$ factor.
Thus, we obtain the magnetic moment as $M=7 \mu_{\rm B}$
and $C=21 \mu_{\rm B}^2/k_{\rm B}$.
On the other hand, in the $j$-$j$ coupling limit,
we accommodate one $f$ electron in the octet $j=7/2$,
while the sextet $j=5/2$ is fully occupied.
In this case, the total angular momentum $J$ is, of course, $7/2$,
which is originating from one $f$ electron in $j=7/2$ octet.
The Land\'e' $g$-factor of $j=7/2$ is equal to $g_J=8/7$
and thus, we obtain $M=4 \mu_{\rm B}$ and
$C=(48/7)\mu_{\rm B}^2/k_{\rm B}$.
Due to the reduction of the magnetization,
 $C$ is changed from $21$ to $48/7$.

Concerning the value of $C$ between the $LS$ and $j$-$j$ coupling limits,
it is necessary to perform the numerical evaluation of
eigenenergies and eigenstates of $H_{\rm loc}$ for
$n=7$ and $W=0$ by changing the ratio of $\lambda/U$.
The result is summarized in Fig.~2.
As mentioned above, we find $C=21$ and $48/7$ in the limit
of $\lambda/U=0$ and $\lambda/U=\infty$, respectively,
leading to the good check of the numerical calculations.
We note the continuous decrease in $C$ from $21$ to $48/7$,
which originates from the reduction of the
magnetic moment $M$ from $M=7 \mu_{\rm B}$
in the $LS$ coupling limit to $M=4 \mu_{\rm B}$
in the $j$-$j$ coupling limit.

It is emphasized here that the reduction of $C$ occurs
in the range of $0.1 < \lambda/U < 1$,
which just corresponds to the values of actual $f$-electron materials.
In this transition region, the $f$-electron wavefunction is given
by the mixture of those in the $LS$ and $j$-$j$ coupling limits.
Thus, the properties of both limits can be observed
in the range of $0.1 < \lambda/U < 1$,
leading to the expectation of a possible explanation of
the Kondo behavior in $f^7$ electron systems.

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig3.eps}
\caption{(a) Eigenenergies vs. $x$ for $\lambda=0$, $U=1$, and $W=-10^{-3}$.
(b) Eigenenergies vs. $x$ for $\lambda=0.2$, $U=1$, and $W=-10^{-3}$.
(c) Eigenenergies vs. $x$ for $\lambda=10^{4}$, $U=1$, and $W=-10^{-3}$.
Note that (a) denotes the result in the LS coupling scheme, while (c) indicates
that in the $j$-$j$ coupling limit.
}
\end{figure}

\subsection{CEF level schemes}

Another evidence for the mixture of both limits is
found in the CEF level schemes.
In Figs.~3, we summarize the variation of the CEF level schemes,
when $\lambda$ is increased for the fixed values of $U=1$
and $W=-10^{-3}$.
Note that Fig.~3(a) indicates the result in the $LS$ coupling limit,
in which we do not find any effects of CEF potentials on the $f$-electron
low-energy states for $n=7$.
Readers may be doubtful of this result, but we should note that
$J$ is purely given by spin, i.e.,
$J=S=7/2$ in the $LS$ coupling limit for $n=7$.
When we recall the fact that the CEF potential acts on charge,
not on spin, there occurs no effect of CEF potentials on the states
with $J=S=7/2$, except for the shift of the total energy.
Thus, we still obtain the octet independent of CEF potentials.

Next we consider the $j$-$j$ coupling limit.
Here we pay our attention to Fig.~3(c) by skipping Fig.~3(b).
In Fig.~3(c), we exhibit the result for $\lambda=10^{4}$.
This is, of course, unphysical value, but we use it
for the purpose to realize the $j$-$j$ coupling limit.
As mentioned above, in this limit, we accommodate six electrons
in the sextet of $j=5/2$ and one electron in the octet of $j=7/2$.
In sharp contrast to the $LS$ coupling limit with $J=S=7/2$,
one $f$ electron in the $j=7/2$ octet feels the CEF potentials,
as shown in Fig.~3(c).
Under the cubic CEF potentials, the octet is split into
two doublets ($\Gamma_6^{-}$ and $\Gamma_7^{-}$)
and one quartet ($\Gamma_8^{-}$).
It is quite natural that the present level scheme is quite similar
to that for $J=7/2$~\cite{LLW} due to the symmetry reason,
although there are small deviations due to the difference
in the definition of the CEF parameters.

Let us turn our attention back to Fig.~3(b).
This is the result for $\lambda=0.2$, which is near
the realistic values for Eu and Gd ions.
First we notice that the energy scale of Fig.~3(b)
is smaller in one order than that of Fig.~3(c).
The CEF energy splitting is in the order of meV,
which is about 10 K.
This value is considered to be small, but the CEF excitation
in this order can be detected in the experiment.
Thus, as mentioned in the previous paper,\cite{Hotta1}
it is a challenging issue to measure the CEF excitation
in Gd and Eu compounds, which have not been expected to
show CEF excitations.

Second the symmetries of the CEF ground states are the same as those
in the $j$-$j$ coupling scheme.
It is quite natural that the ground-state multiplet is always characterized
by the total angular momentum $J=7/2$, irrespective of the values
of $U$ and $\lambda$.
Note that the magnitude of the matrix elements among the states
of $J_z$ is affected by $U$ and $\lambda$, where $J_z$ denotes the
$z$-component of  $J$.

\section{Model and Method}

\subsection{Seven-orbital Anderson model}

Now we consider the conduction electron hybridized with localized electrons.
In general, all $f$ orbitals are hybridized with conduction electrons with
the same symmetry, but it is very difficult to perform numerical calculations
by taking into account seven conduction bands.
Since the purpose here is to reveal the effect of spin-orbit coupling
on the Kondo phenomena in $f^7$ electron systems,
it is reasonable to consider the minimum model to investigate the Kondo phenomena.
In this sense, the minimum model should include one conduction band,
since we can expect the appearance of the underscreening Kondo phenomena
even for one conduction band hybridized with local $f$ orbitals.
Then, it is natural to consider $a_{\rm u}$ conduction band,
since local $a_{\rm u}$ state is non-degenerate even under the high symmetry
ligand field such as the cubic CEF potential.
Note that the local $a_{\rm u}$ state is described as
$(f^{\dag}_{m=2,\sigma}-f^{\dag}_{m=-2,\sigma})|0\rangle /\sqrt{2}$,
where $|0\rangle$ denotes the vacuum.

Then, the model is expressed as
\begin{equation}
  H=\sum_{\mib{k},\sigma} \varepsilon_{\mib{k}}
    c_{\mib{k}\sigma}^{\dag} c_{\mib{k}\sigma}
   +\sum_{\mib{k},m,\sigma} (V_m c_{\mib{k}\sigma}^{\dag}f_{m\sigma}+{\rm h.c.})
   +H_{\rm loc},
\end{equation}
where $\varepsilon_{\mib{k}}$ is the dispersion of conduction electron,
$c_{\mib{k}\sigma}$ is an annihilation operator of conduction electron
with momentum $\mib{k}$ and spin $\sigma$,
and $V_m$ is the hybridization between conduction and localized electrons.
Note that $V_{m}$ is given by $V_{m=2}=-V_{m=-2}=V$ and zero
for other components.
The energy unit is a half of the conduction bandwidth,
which is set as 1 eV throughout this paper, as mentioned above.

\subsection{Numerical renormalization group method}

For the diagonalization of the impurity Anderson model, we employ
a numerical renormalization group (NRG) method,\cite{NRG1,NRG2}
in which we  logarithmically discretize the momentum space so as
to include efficiently the conduction electrons near the Fermi energy.
The conduction electron states are characterized by ``shell'' labeled by $N$
and the shell of $N=0$ denotes an impurity site described by the local Hamiltonian.
Then, after some algebraic calculations,
the Hamiltonian is transformed into the recursion form as
\begin{eqnarray}
  H_{N+1} = \sqrt{\Lambda} H_N+\xi_N \sum_{\sigma}
  (c_{N\sigma}^{\dag}c_{N+1\sigma}+c_{N+1\sigma}^{\dag}c_{N\sigma}),
\end{eqnarray}
where $\Lambda$ is a parameter for logarithmic discretization,
$c_{N\sigma}$ denotes the annihilation operator of conduction electron
in the $N$-shell, and $\xi_N$ indicates ``hopping'' of electron between
$N$- and $(N+1)$-shells, expressed by
\begin{eqnarray}
  \xi_N=\frac{(1+\Lambda^{-1})(1-\Lambda^{-N-1})}
  {2\sqrt{(1-\Lambda^{-2N-1})(1-\Lambda^{-2N-3})}}.
\end{eqnarray}
The initial term $H_0$ is given by
\begin{equation}
  H_0=\Lambda^{-1/2} [H_{\rm loc}
  +\sum_{m\sigma}V_m(c_{0\sigma}^{\dag}f_{m\sigma}
                             +f_{m\sigma}^{\dag}c_{0\sigma})].
\end{equation}

For the calculations of thermodynamic quantities,
We evaluate the free energy $F$ for local $f$ electron in each step by
\begin{eqnarray}
   F = -T (\ln {\rm Tr} e^{-H_N/T} - \ln {\rm Tr} e^{-H_N^0/T}),
\end{eqnarray}
where a temperature $T$ is defined as $T=\Lambda^{-(N-1)/2}$
in the NRG calculation and $H_N^0$ denotes the Hamiltonian
without the hybridization term and $H_{\rm loc}$.
Then, we obtain the entropy $S_{\rm imp}$ by
$S_{\rm imp}=-\partial F/\partial T$
and the specific heat $C_{\rm imp}$ is evaluated by
$C_{\rm imp}=-T\partial^2 F/\partial T^2$.
In the NRG calculation, we keep $M$ low-energy states
for each renormalization step.
In this paper, we set $\Lambda=5$ and we keep $M=4,500$
low-energy states for each renormalization step.

\subsection{Multipole susceptibility}

For the purpose to discuss the multipole properties later,
we provide the definition of multipole operator in this subsection.
Since we discuss the effect of the spin-orbit coupling
for $\lambda/U=0$ to $\lambda/U=\infty$,
it is necessary to use the same definition both in the $LS$
and $j$-$j$ coupling schemes, irrespective of the values of $\lambda/U$.
Thus, we define the multipole as a spin-orbital density in the form of
a one-body operator from the viewpoint of
the multipole expansion of electron density in electromagnetism.
On the basis of this definition of the multipole operator,
we have developed microscopic theories
for multipole-related phenomena.
For instance, octupole ordering in NpO$_2$ has been clarified
by evaluating multipole interaction
by the standard perturbation method
in terms of electron hopping.\cite{Kubo1,Kubo2,Kubo3}
We have discussed possible multipole states of
filled skutterudites by analyzing the multipole susceptibility of
a multiorbital Anderson model based on the $j$-$j$ coupling scheme.
\cite{Hotta1,Hotta2,Hotta3,Hotta4,Hotta5,Hotta6,Hotta7,Hotta8}
We have also discussed the multipole state in actinide dioxides~\cite{Hotta9,Hotta10}
and Yb compounds.\cite{Hotta11}
Recently, a microscopic theory for multipole ordering from an itinerant picture
has been developed on the basis of a seven-orbital
Hubbard model with spin-orbit coupling.\cite{Hotta12}

The multipole operator ${\hat T}$ is expressed
in the second-quantized form as
\begin{equation}
  {\hat T}^{(k)}_{\mib{i},\gamma} = \sum_{q,m\sigma,m'\sigma'}
  G^{(k)}_{\gamma,q}
  T^{(k,q)}_{m\sigma,m'\sigma'}f^{\dag}_{\mib{i}m\sigma}f_{\mib{i}m'\sigma'},
\end{equation}
where $k$ indicates the rank of the multipole,
$q$ denotes an integer between $-k$ and $k$,
$\gamma$ is a label used to express an $O_{\rm h}$ irreducible representation,
$G^{(k)}_{\gamma,q}$ is the transformation matrix
between spherical and cubic harmonics,
and $T^{(k,q)}_{m\sigma,m\sigma'}$ can be calculated,
using the Wigner-Eckart theorem, as \cite{Inui}
\begin{equation}
 \label{Tkq}
 \begin{split}
  T^{(k,q)}_{m\sigma,m'\sigma'}
  &= \sum_{j,\mu,\mu'}
  \frac{\langle j || T^{(k)} || j \rangle}{\sqrt{2j+1}}
  \langle j \mu | j \mu' k q \rangle \\
  &\times
  \langle j \mu | \ell m s \frac{\sigma}{2} \rangle
  \langle j \mu' | \ell m' s \frac{\sigma'}{2} \rangle.
 \end{split}
\end{equation}
Here, $\ell=3$, $s=1/2$, $j=\ell \pm s$,
$\mu$ denotes the $z$-component of $j$,
$\langle j \mu | j' \mu' j'' \mu'' \rangle$ indicates
the Clebsch-Gordan coefficient,
and $\langle j || T^{(k)} || j \rangle$ is
the reduced matrix element for a spherical tensor operator
and is given by
\begin{equation}
  \label{red}
  \langle j || T^{(k)} || j \rangle=
  \frac{1}{2^k} \sqrt{\frac{(2j+k+1)!}{(2j-k)!}}.
\end{equation}
Note that $k \le 2j$ and the highest rank is $2j$.
Thus, we treat multipoles up to rank 7 for $f$ electrons
in this definition.

Here we should note that multipoles belonging to the same symmetry
are mixed in general, even if the rank is different.
Namely, the $f$-electron spin-charge density should be given by
the appropriate superposition of multipoles, expressed as
\begin{equation}
  \label{multi}
  {\hat X}=\sum_{k,\gamma} p^{(k)}_{\gamma}{\hat T}^{(k)}_{\gamma},
\end{equation}
where we redefine each multipole operator so as to satisfy
the orthonormal condition of~\cite{Kubo3}
\begin{equation}
  \label{normal}
  {\rm Tr} \Bigl\{
  {\hat T}_{\gamma}^{(k)} {\hat T}_{\gamma'}^{(k')} \Big\}
  =\delta_{kk'}\delta_{\gamma\gamma'}.
\end{equation}
In order to determine the coefficient $p^{(k)}_{\gamma}$, it is necessary
to evaluate the multipole susceptibility in the linear response theory.
Namely, $p^{(k)}_{\gamma}$ is determined by the eigenstate of the largest
eigenvalue of susceptibility matrix, given by
\begin{equation}
 \label{sus}
  \begin{split}
   \chi_{k\gamma,k'\gamma'} \!
   =\! & \frac{1}{Z}
   \sum_{i,j} \frac{e^{-E_i/T}-e^{-E_j/T}}{E_j-E_i}
   \langle i | [{\hat T}^{(k)}_{\gamma}-\rho^{(k)}_{\gamma}] | j \rangle
   \\ & \times
   \langle j | [{\hat T}^{(k')}_{\gamma'}- \rho^{(k')}_{\gamma'}]| i \rangle,
 \end{split}
\end{equation}
where $Z$ is the partition function given by $Z$=$\sum_i e^{-E_i/T}$,
$E_i$ denotes the eigenenergy of the $i$-th eigenstate
$|i\rangle$ of the Hamiltonian, $T$ is a temperature, and
$\rho^{(k)}_{\gamma}$=$\sum_i e^{-E_i/T}
\langle i |{\hat T}^{(k)}_{\gamma}| i \rangle/Z$.
The multipole susceptibility is given by
the eigenvalue of the susceptibility matrix.
By using the NRG method, we evaluate the matrix elements of
the multipole susceptibility in each NRG step.
Then, we can determined the optimized multipole state
in an unbiased manner.

Note that in this paper, we express the irreducible
representation of the CEF state by Bethe notation,
whereas for multipoles, we use short-hand notations by the
combination of the number of irreducible representation and
the parity of time reversal symmetry, ``g'' for gerade and ``u'' for
ungerade.
For O$_{\rm h}$ symmetry, we have nine independent
multipole components as 1g, 2g, 2u,
3g, 3u, 4g, 4u, 5g, and 5u.
Note that 1u does not appear within the rank 7.

\section{Calculation Results}

\subsection{Underscreening Kondo effect}

Let us show our numerical results.
First we consider the situation of $U \ne 0$ and $\lambda=W=0$,
which is the limit of $LS$ coupling scheme.
In this case, the ground-state multiplet is characterized by $S=7/2$
and $L=0$.
When one conduction band is hybridized with such localized state,
it has been well known that the underscreening Kondo effect occurs.
Namely, one of seven electron among the octet forms the singlet due to the hybridization
with single conduction electron band, while another six electrons are still localized.
Thus, even after the underscreening Kondo effect, the septet remains.
Here we use the Kondo temperature, even when the underscreening
Kondo effect occurs.

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig4.eps}
\caption{(Color online)
(a) Entropy $S_{\rm imp}$ vs. temperature $T$ for $U=1$ and $\lambda=W=0$.
The hybridization $V$ is changed between $0.5$ and $2.0$.
(b) Specific heat $C_{\rm imp}$ vs. temperature for the same parameters in (a).
(c) The peak temperature $T_{\rm p}$ (solid symbols) vs. $V$.
The red curve denotes $\ln T_{\rm p}=b-a/V^2$ with appropriate constants $a$ and $b$.
}
\end{figure}

In Fig.~4(a), we show the temperature dependence of entropy
for several values of $V$.
For $V$=0.5, we find the entropy $\ln 8$ in the present temperature range.
If we decrease the temperature, we expect the change of the entropy, but
it is difficult to obtain the results with a reliable precision.
Then, we increase the value of $V$ to elevate the Kondo temperature.
As expected, we find the gradual decrease in the entropy when $V$ is increased.
For $V$=2.0, the entropy immediately becomes $\ln 7$ in the present
temperature range, since the Kondo temperature is high enough.

Although it is difficult to define the Kondo temperature only from Fig.~4(a),
it is possible to define $T_{\rm K}$ as a peak temperature in the
specific heat, which is formed by the entropy change from $\ln 8$ to $\ln 7$.
The results are shown in Fig.~4(b).
Note that the absolute values of the specific heat are suppressed,
since the entropy release is relatively small in this case.
Tiny fluctuations in the specific heat are observed,
since the magnitude of the specific heat is relatively small in this case and
the effect of numerical precision appears.
For $V=0.5$, we find $C_{\rm imp}$ is zero and no peak is found in the region
of $T<10^{-3}$.
When $V$ is increased, the value of $C_{\rm imp}$ is gradually enhanced.
For $V=0.8$ and $0.9$, we find the broad peak structure in the specific heat.
These peaks are considered to originate from the entropy change
from $\ln 8$ to $\ln 7$.
For $V=1.0$, we observe a weak shoulder structure around at $T=10^{-4}$,
but the peak structure is smeared.
For $V=2.0$, $C_{\rm imp}$ becomes immediately zero and no peak structure
can be found for $T<10^{-3}$.

In oder to confirm that the entropy change and the peak formation in the specific heat
are indications of the Kondo effect, we plot the peak temperature $T_{\rm p}$ in Fig.~4(c).
We can define the peaks in the region of $0.75 < V < 1$, but we find that $T_{\rm p}$
is well fitted by $b{\rm exp}(-a/V^2)$ with appropriate constants $a$ and $b$.
This is expected from the well-known formula
for the Kondo temperature $T_{\rm K}=D{\rm exp}[-1/(\rho J_{\rm cf})]$,
where $D$ is the half of the conduction electron band.
Note that $D$ is set as unity in this paper.
In the single-band Anderson model, we can obtain $J_{\rm cf}=4V^2/I$,
where $I$ denotes the on-site Coulomb interaction.
In the present case, it is difficult to derive such a simple form of $J_{\rm cf}$,
since we consider the complicated orbital-dependent interactions through
the Slater-Condon parameters.
However, we can deduce that $J_{\rm cf}$ can be obtained, in any case,
in the second-order perturbation in terms of $V$.
We expect that $T_{\rm p}$ is in proportion to ${\rm exp}(-a/V^2)$
with an appropriate constant $a$.
We also introduce another constant $b$ to fit $T_{\rm p}$ as $b{\rm exp}(-a/V^2)$.
From the result in Fig.~4(c), we conclude that the underscreening Kondo effect
actually occurs and it is characterized by the entropy change from $\ln 8$ to $\ln 7$
due to the screening of one spin by single conduction band.

\subsection{Effect of spin-orbit coupling}

Next we include the spin-orbit coupling.
Namely, we consider the finite values of $\lambda$ for $U=1$ and $W=0$.
Depending on the values of $V$, we consider two typical situations
for large and small values of $V$ such as $V=3.0$ and $0.6$.
From Fig.~4(a), the underscreening Kondo effect occurs in the
high-temperature region as $T>0.1$ for $V>2.0$,
while the octet still remains in the temperature range for $V<0.6$.

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig5.eps}
\caption{(Color online)
(a) Entropy $S_{\rm imp}$ (solid square) and specific heat $C_{\rm imp}$
(solid triangle) for $\lambda=0.1$, $U=1$, $W=0$, and $V=3.0$.
(b) The peak temperature $T_{\rm p1}$ (solid circle) and the excitation energy
$\Delta E_2$ (solid line) of the two-site system as functions of $\lambda$.
Inset shows the schematic view for the energy levels of the two-site system,
in which the septet is split into one singlet and two triplets.
}
\end{figure}

In Fig.~5(a), we show the typical results for entropy and specific heat
for $\lambda=0.1$,  $U=1$, and $W=0$ with the large value of $V$
such as $V=3.0$.
As expected from the results in Figs.~4, we observe the plateau of $\ln 7$
for $T<0.1$.
When we further decrease the temperature, we find the release of entropy
$\ln 7$, leading to a large peak in the specific heat, as shown in Fig.~5(a).
In the present case, we find such a peak at $T \approx 10^{-3}$.
Here we define the peak temperature as $T_{\rm p1}$.
We are interested in the origin of this entropy change, but
it seems to be irrelevant to the Kondo effect.
Intuitively, the entropy change in this case is rather rapid
in comparison with the that in the Kondo effect.
A more concrete discussion can be done by the direct comparison
of the relevant energy scales.
In one word, this is the level splitting due to the hybridization.

At present we do not explicitly include the CEF potential, but
it has been well known that the level splitting occurs
due to the effect of the hybridization between localized and
conduction electrons.
In fact, as shown in Fig.~5(b), the peak temperature $T_{\rm p1}$
is well fitted by $\Delta E_2$, where $\Delta E_2$ denotes the
excitation energy between the ground state singlet and the first excited
triplet in a two-site system, as shown in the inset of Fig.~5(b).
Note that the two-site system is composed of impurity site and one conduction site,
which are connected by the hybridization $V$.
For the case of $\lambda=0$, we find that the ground state is septet.
Due to the combination of $S=7/2$ at impurity site and $S=1/2$ at the conduction site,
we obtain $S_{\rm tot}=4$ and $3$, but the state with $S_{\rm tot}=3$
becomes the ground state with septet.
Here $S_{\rm tot}$ denotes the magnitude of total spin moment.
When we include the effect of $\lambda$, the level splitting due to the hybridization
becomes significant and $\Delta E_2$ is increased with the increase of $\lambda$.
The temperature to characterize the entropy release of $\ln 7$ is deduced to
be determined by $\Delta E_2$, as shown in Fig.~5(b).

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig6.eps}
\caption{(Color online)
(a) Entropy $S_{\rm imp}$ (solid square) and specific heat $C_{\rm imp}$
for $\lambda=0.2$, $U=1$, $W=0$, and $V=0.6$.
(b) The peak temperature $T_{\rm p2}$ (solid circle), the Kondo temperature $T_{\rm K}$
(solid square with solid line), and the excitation energy $\Delta E_3$ (broken curve)
of the three-site system as functions of $\lambda$.
Inset shows the schematic view of the energy levels of the three-site system,
in which the octet is split into one quartet and two doublets.
}
\end{figure}

Next we consider the case with small $V$.
In Fig.~6(a),  we depict the entropy and specific heat
for $\lambda=0.2$, $U=1$, $W=0$, and $V=0.6$.
As expected from the discussion in Figs.~5, at high temperatures
as $10^{-3}<T<10^{-1}$, we find a plateau of $\ln 8$
originating from the local octet of $J=7/2$.
For the case of $\lambda=0$, only one electron spin is screened by
one conduction band, leading to the entropy change of $\ln 8$ to
$\ln 7$, but in the present case with $\lambda=0.2$, first we find
the entropy change from $\ln 8$ to $\ln 2$ by the level splitting
due to the hybridization with conduction electrons.
Then, the remained $\ln 2$ from the local doublet is eventually
released by the Kondo effect.

We claim that the result in Fig.~6(a) is important.
The underscreening Kondo effect in the $f^7$-electron system
has been well understood and probably, it has not been considered to
be an intriguing phenomenon.
However, under the effect of the spin-orbit coupling,
the entropy of $\ln 8$ from the local octet due to $J=S=7/2$ is
released in the two step.
Thus, in the specific heat, we observe two peak temperatures.
The higher and lower ones are defined as
$T_{\rm p2}$ and $T_{\rm K}$, respectively.

In Fig.~6(b), we plot  $T_{\rm p2}$ and $T_{\rm K}$ as functions of
$\lambda$.
Both temperatures are increased monotonically up to $\lambda=0.5$,
but for $\lambda>0.5$, those seem to be almost constant,
even if we change $\lambda$.
The higher peak temperature $T_{\rm p2}$ is found to be
well fitted by $\Delta E_3$, which is the excitation energy
in the three-site system, including one impurity and two conduction sites.
The ground state of the three-site system is found to be characterized
by $S_{\rm tot}=7/2$ for the case of $\lambda=0$.
When we include the spin-orbit coupling, the octet ground state
is split into three, two doublets and one quartet.
Among them, one doublet becomes the ground state and the quartet
is the first excited state.
The energy difference between them is here defined as $\Delta E_3$.
Again this is the level splitting due to the effect of hybridization.
Note that the $\lambda$ dependence of $\Delta E_2$ is similar to
that of $\Delta E_3$, but the magnitude is found to be different
from $T_{\rm p2}$.

As we observed in Fig.~6(b), the higher peak $T_{\rm p2}$ is
scaled by $\Delta E_3$.
The lower peak is considered to be the Kondo temperature $T_{\rm K}$.
We find that $T_{\rm K}$ is increased monotonically up to $\lambda=0.5$
and it becomes constant for $\lambda > 0.5$.
In the limit of $\lambda=\infty$, the $j$-$j$ coupling scheme becomes
exact and we know that one electron is accommodated in $j=7/2$ octet.
In such a limit, as emphasized in the section of introduction,
the hybridization of one $j=7/2$ electron with the conduction band
leads to the Kondo effect.
For enough large $\lambda$, we easily deduce that the situation is not
so changed from that in the limit of $\lambda=\infty$ and thus,
it seems natural that the present $T_{\rm K}$ is constant
in the region of large $\lambda$.

However, it is surprising that the same $T_{\rm K}$ as in the $j$-$j$ coupling
limit is obtained even for $\lambda=0.5$, which is not considered to
be large enough.
When we decrease $\lambda$, $T_{\rm K}$ is decreased rapidly,
but we still find $T_{\rm K}$ in the present temperature range
for $\lambda=0.1$.
Thus, we arrive at an important conclusion that the picture of the
Kondo effect on the basis of the $j$-$j$ coupling scheme is
applicable even for the realistic situation with the spin-orbit coupling
in the order of $\lambda/U=0.1$ for $f^7$ electron systems.
The component of the $j$-$j$ coupling scheme in the many-electron
wave function is more persevering than we have naively expected.

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig7.eps}
\caption{(Color online)
Entropy $S_{\rm imp}$ (solid square) and specific heat $C_{\rm imp}$ (solid triangle)
for (a) $x=1.0$, (b) $x=0.0$, and (c) $x=-1.0$.
Other parameters are set as $\lambda=0.2$, $U=1$, $W=-10^{-4}$, and $V=0.05$.
}
\end{figure}

\subsection{Effect of CEF potential}

Thus far, we have considered the situation without the CEF potential.
As mentioned above, due to the effect of the hybridization,
the level splitting has been found to occur.
Next we include explicitly the CEF potentials.
In Figs.~7, we summarize the results of entropy and specific heat
for three CEF ground states, which are controlled by $x$.
As easily understood from Fig.~3(b),
the cases of $x=1.0$, $0.0$, and $-1.0$ correspond to the ground states of
$\Gamma_6^{-}$, $\Gamma_8^{-}$, and $\Gamma_7^{-}$, respectively.
Since $V$ is chosen to be very small here, we find the localized CEF states in
the present temperature range.
The residual entropy is $\ln 4$ for $\Gamma_8^-$ quartet,
while it is $\ln 2$ for $\Gamma_6^-$ or $\Gamma_7^-$ doublet.

The entropy release from $\ln 8$ to $\ln 4$ or $\ln 2$ occurs
and we find the peak in the specific heat at the corresponding temperature.
The height of the peak depends on the released entropy, but
the position of the peak is almost the same for three cases.
The peak position is around $10^{-4}$ and this scale is determined
by the CEF level splitting.
As understood from Fig.~3(b), for $\Lambda=0.2$, $U=1$, and $W=-0.001$,
the order of the CEF level splitting is in the order of $10^{-3}$.
In the present calculation, we set $W=-10^{-4}$, and thus,
the order of the CEF level splitting is smaller in one order.
In this sense, it is quite natural that the peak position appears
around at $T=10^{-4}$ in common among Figs.~7(a)-7(c).

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig8.eps}
\caption{(Color online)
Multipole susceptibilities for (a) $x=1.0$, (b) $x=0.0$, and (c) $x=-1.0$.
Other parameters are set as $\lambda=0.2$, $U=1$, $W=-10^{-4}$, and $V=0.05$.
}
\end{figure}

Next we consider the multipole properties.
After we faithfully diagonalize the multipole susceptibility matrix eq.~(\ref{sus}),
we obtain the eigenvalues and eigenvectors.
The multipole state with the largest eigenvalue is considered to
be realized and the corresponding eigenvectors denote the
optimized multipole state given by the mixture of multipole
components with different ranks in the same symmetry group.

First we consider the case without the CEF potential, i.e., $W=0$.
The results are not shown here, but the optimized multipole is
always dipole for $\lambda=0$, since $J$ is purely equal to
total spin moment $S=7/2$.
When we increase the value of $\lambda$, we observe the
increase of other multipole components and in the $j$-$j$ coupling limit,
one electron in the $j$=7/2 octet carries all multipoles higher than dipole.
Note that in a periodic system, the optimized multipole is affected by
the lattice structure, electron hopping amplitude, and interactions.

Next we include the CEF potential.
In Figs.~8, we show the eigenvalues of multipole susceptibility matrix
in each NRG step for the same parameters in Figs.~7.
When we see the entropy and specific heat, it is difficult to
find the difference in the cases for
$\Gamma_6^-$ and $\Gamma_7^-$ ground states,
corresponding to Figs.~7(a) and 7(c), respectively.
In the plateau of $\ln 8$, there are no significant differences
among three cases, but around at the temperature of the entropy release,
it is possible to detect the different multipole states for three cases.
For the case of $\Gamma_8^-$ ground state,
since $\Gamma_8^-$ includes orbital degrees of freedom,
it carries higher-rank multipoles.
On the other hand, $\Gamma_6^-$ and $\Gamma_7^-$ states
carry only 4u multipoles, mainly characterized by dipole.

However, for $\Gamma_6^-$ and $\Gamma_7^-$ ground states,
we can observe the difference in the multipole state with
the second largest eigenvalue.
Namely, they are 3g quadrupole and 5g quadrupole, respectively,
for Figs.~8(a) and 8(c).
The multipole with the largest eigenvalue is always 4u dipole,
but the Curie constant for the multipole susceptibility is slightly different.
Namely, in the $j$-$j$ coupling limit, we obtain the dipole moment
as $(4/3) \mu_{\rm B}$ and $(12/7) \mu_{\rm B}$ for $\Gamma_6^-$
and $\Gamma_7^-$ ground states, respectively.
This difference appears in the values of $T\chi$ at the lowest
temperatures in the present calculations.

\begin{figure}[t]
\centering
\includegraphics[width=8.0truecm]{fig9.eps}
\caption{(Color online)
(a) Entropy $S_{\rm imp}$ (solid square) and specific heat $C_{\rm imp}$ (solid triangle)
for $\lambda=0.2$, $U=1$, $W=-10^{-3}$, $x=-1.0$, and $V=0.6$.
(b) Multipole susceptibilities for the same parameters in (a).
(c) The peak temperature $T_{p2}$ (solid circle) and
the Kondo temperature $T_{\rm K}$ (solid square)
as functions of $V$.
The solid curve denotes $\Delta E_3$, whereas the horizontal broken line
indicates the local CEF excitation energy.
The dotted curve denotes the curve of $\ln T_{\rm K}=b-a/V^2$
with appropriate constants $a$ and $b$.
}
\end{figure}

Thus far, we have considered the situation with small $V$ such as $V=0.05$,
in order to focus on the localized properties of $f^7$-electron systems.
Next we consider the case with lager $V$ for the purpose to
visualize the Kondo phenomenon.
When we increase the values of $V$ for Figs.~7(a)-7(c),
we notice that the Kondo effect occurs for the case of
$\Gamma_7^-$ ground state, while for the cases of
$\Gamma_6^-$ and $\Gamma_8^-$ ground states, we cannot
observe the Kondo behavior even if we increase the magnitude of
$V$ up to $3.0$.
We remark that the $a_{\rm u}$ conduction band is hybridized only with
$\Gamma_7^-$ state,
since it includes $(f^{\dag}_{m=2,\sigma}-f^{\dag}_{m=-2,\sigma})|0\rangle /\sqrt{2}$,
while $\Gamma_6^-$ has no component of $f^{\dag}_{m=2,\sigma}|0\rangle$ and
$f^{\dag}_{m=-2,\sigma})|0\rangle$ and $\Gamma_8^-$ state contains
the component of $(f^{\dag}_{m=2,\sigma}+f^{\dag}_{m=-2,\sigma})|0\rangle /\sqrt{2}$.
\cite{Onodera}
Thus, in the following, we consider only the case of the $\Gamma_7^-$ ground state.

In Fig.~9(a), we depict the temperature dependence of entropy and specific heat
for $\lambda=0.2$, $U=1$, $W=-10^{-3}$, $x=-1.0$, and $V=0.6$.
We find the plateau of $\ln 8$ around at $T=10^{-2}$
and the entropy is changed to $\ln 2$ around at $T=10^{-3}$,
forming the peak at $T_{\rm p2}$ in the specific heat.
Then, the entropy $\ln 2$ is eventually released around
at $T=10^{-7}$, leading to the Kondo temperature $T_{\rm K}$.
The behavior of the entropy and specific heat are essentially the same as
that in Fig.~6(a) without the CEF potential.
This is nor surprising, since the level splitting due to the hybridization plays
the same role as that due to the CEF potentials.

In Fig.~9(b), we plot the multipole susceptibility
for the same parameters in Fig.~9(a).
For $T > T_{\rm K}$, we find
qualitatively the same behaviors as those in Fig.~8(c).
Around at the Kondo temperature, the Curie constant for
4u dipole gradually becomes zero, suggesting the screening
of 4u dipole moment by the conduction electrons.
Namely, the standard Kondo effect is considered to occur
in this case.

When we change the values of $V$, we plot the peak temperature
$T_{\rm p2}$ and the Kondo temperature $T_{\rm K}$ in Fig.~9(c).
Again it is found that $T_{\rm p2}$ is well scaled by $\Delta E_3$,
but we note that $\Delta E_3$ converges to the local
CEF excitation energy.
Note here that the temperature is discrete in the NRG calculation,
since it is defined as $T=\Lambda^{-(N-1)/2}$,
where $\Lambda$ is the cut-off and $N$ is the NRG step.
Thus, the peak temperature $T_{\rm p2}$ includes an error-bar
in the order of $\sqrt{\Lambda}$.
Due to the above reason, for small $V$, the solid circles seem to
scatter around the horizontal broken line.

However, for $V>1.0$, $T_{\rm p2}$ is clearly larger than the
CEF excitation energy and it seems to be scaled by $\Delta E_3$.
For small $V$, the level splitting due to the hybridization is
small and $\Delta E_3$ is almost determined by the CEF excitation
energy.
When $V$ is increased, the hybridization effect on the level splitting
overcome the CEF potentials and $\Delta E_3$ begins to deviate from
the local CEF excitation energy.
We emphasize that $T_{\rm p2}$ correctly follow the above behavior
of $\Delta E_3$.

Finally, we discuss the Kondo temperature.
As shown by the dotted curve in Fig.~9(c),
$T_{\rm K}$ is well fitted by $b{\rm exp}(-a/V^2)$ with appropriate constants
$a$ and $b$.
This fact suggests that $T_{\rm K}$ actually indicates the Kondo temperature.
Note that for $V>1.0$, $T_{\rm K}$ becomes so high that we cannot observe
the Kondo effect in the present temperature range.

\section{Discussion and Summary}

In this paper, we have discussed the Kondo effect in $f^7$-electron systems
on the basis of the seven-orbital Anderson model by using the NRG technique.
Note that we have considered the single $a_{\rm u}$ conduction band.
We have clarified that our understanding on the Kondo effect
in the Ce compound also works in the  $f^7$-electron system
with the spin-orbit coupling.
If we are simply based on the $j$-$j$ couping scheme,
the result seems to be almost obvious.
Namely, in the limit of large spin-orbit coupling, we accommodate one
electron in $j$=7/2 octet, while $j$=5/2 sextet is fully occupied.
When one electron in the octet is hybridized with conduction band,
it is possible to understand the Kondo effect in a similar way
as that for the Ce compounds.

From the qualitative viewpoint, the $j$-$j$ coupling scenario may work
for the understanding of the Kondo effect in Eu compounds.
However, it is unclear whether the scenario is really valid for actual compounds
with the finite value of the spin-orbit coupling.
In particular, in actual rare-earth materials, the effect of Coulomb
interaction is generally stronger that that of the spin-orbit coupling.
Thus, naively speaking, it is difficult to accept even qualitatively the scenario
based on the $j$-$j$ coupling scheme in the rare-earth compounds.
It has been the motivation of the present paper to clarify this point.
Our results have clearly suggested that the Ce-compound-like Kondo phenomena
should occur for $f^7$-electron systems
in the region of $\lambda/U$ in the order of $0.1$,
which is the value of actual materials.

Since the purpose of this paper has been to clarify the appearance of
the Kondo phenomena in $f^7$-electron systems from the conceptual
viewpoint, we have not discussed the quantitative explanation of
actual Eu compounds.
In order to promote further the study of the Kondo effect in Eu compounds,
it is inevitable to include the effect of valence fluctuations,
which has been ignored in this paper.
It has been pointed out that the valence fluctuations are involved
in the heavy-electron formation in Eu compounds.\cite{Mitsuda2}
Interesting properties induced by critical valence fluctuations
have been actively investigated by Watanabe and Miyake.\cite{Miyake}

A way to investigate the effect of valence fluctuations on the Kondo phenomena
in $f^7$-electron systems is to include further the inter-site Coulomb
interaction between impurity and conduction sites.
Although for the standard impurity Anderson model without orbital degeneracy,
it is easy to include such an inter-site repulsion,
in the present seven-orbital Anderson model,
there are several possibilities to consider such orbital-dependent
inter-site repulsion.
Thus, the first step to promote the research of the effect of valence fluctuations
on the Kondo phenomena is to construct the valid model including
inter-site repulsions.
It is one of future problems.

We have considered one $a_{\rm u}$ conduction band,
but it can be validated, for instance, in cage-structure compounds
such as filled skutterudites,
since the band-structure calculations have revealed that
the main conduction band composed of pnictogen $p$ electrons
possess $a_{\rm u}$ symmetry.\cite{Harima}
If Eu ion becomes divalent in the cage of filled skutterudites,
it is interesting to investigate Eu-based filled slitterudites.
Another cage-structure can be also the candidate of the research
of the Kondo effect in $f^7$-electron systems.
If it is possible to synthesize Eu-based 1-2-20 compounds,
it may be interesting.
As for the theoretical task, it is necessary to increase the
number of conduction bands.
For instance, we consider $e_{\rm u}$
conduction bands with double degeneracy.
Since they are hybridized with $\Gamma_8^-$ states,
it may be possible to obtain the two-channel Kondo
effect in the Eu compounds.
It is another interesting future problem.

In summary,
we have analyzed the seven-orbital Anderson model by employing
the NRG technique.
When we have included the spin-orbit coupling in the situation,
we confirm the Kondo behavior similar to that in the Ce compound
with increasing the magnitude of the spin-orbit coupling,
as expected from the $j$-$j$ coupling scheme.
An important point is that the Kondo effect
similar to the case of $n=1$ occurs for realistic values
of the spin-orbit coupling even in the case of $n=7$.
Even when we include the cubic CEF potentials,
we have also found the Kondo behavior similar to Ce compounds
for some CEF parameter regions.

\section*{Acknowledgement}

The author thanks K. Hattori, K. Kubo, Y. \=Onuki, and K. Ueda
for discussions on heavy-electron systems.
The computation in this work has been partly done using the facilities of the
Supercomputer Center of Institute for Solid State Physics, University of Tokyo.

redpajama_set_name: RedPajamaArXiv


Children were walking on air today after Turner Prize-winning artist Jeremy Deller unveiled his life-size bouncy castle… of Stonehenge.
The 20ft-high inflatable, called Sacrilege, is modelled on the prehistoric monoliths and was opened to the public on Glasgow Green as part of the 18-day Glasgow International Festival of Visual Arts Festival.
It was designed using detailed plans of the Salisbury monument and took two months to make thanks to the efforts of workers at Inflatable World Leisure, who Mr Deller said built the first ever bouncy castles in the UK.
His giant inflatable is one of the highlights of a festival programme featuring more than 130 artists at almost 50 venues across the city.
Mr Deller, who won the Turner Prize in 2004, said: 'It has taken two months to put together so it is wonderful to finally see it up and being used by the public.
'Stonehenge is a part of our history and it is such an iconic structure that I wanted to recreate it as accurately as I could.
After appearing in Glasgow, the castle – the artist's first major work in Scotland – will be taken on a tour of the UK.
Other highlights at the Glasgow festival include solo shows by Glasgow-based 2009 Turner Prize winner Richard Wright, Adrian Wiszniewski and Karla Black.
The festival also includes the first UK show called Triumph, an installation of more than 2,500 discarded sporting trophies collected by Polish-born Aleksandra Mir, and an exhibition focused on Glasgow's Socialist Sunday School movement that flourished in the early 20th century.
More than 90 per cent of the work on show during the 18-day festival is either new or previously unseen in the UK.
There will also be a range of newly commissioned works drawing on other artistic disciplines such as dance, film and music.
Teacher Lynda Darrock, 31, visited the bouncy castle with children from Annette Street Primary School in Govan, Glasgow.
She said: 'The children thought it was absolutely amazing. They were talking about it all day, I even had a go myself.
'They keep asking if we are going back.
'Jeremy spent lots of time talking to the children beforehand and afterwards asking if they enjoyed it. He was brilliant.
The Wiltshire monument was completed around 4,500 years ago and is believed to have taken around 35 years to complete.
The largest of the gigantic upright stones weighs about 40 tons – the equivalent of an articulated lorry.
A Time Team dig (for the Channel 4 show) in 2009 established that Stonehenge was built around the same time as Durrington Walls, another henge, or circular earthwork, two miles away.
The two adjacent henges were part of the same complex, with Durrington Walls the location for a massive Neolithic village that housed the workers who built Stonehenge.
The Time Team suggested that this site housed up to 4,000 people, which would have made it the largest Neolithic settlement in north-west Europe.
While the circle at Durrington Walls represented life and the land of the living, Stonehenge, encircled by burial mounds, represented the land of the dead, the team claimed.
The two were connected by the River Avon and the procession route from one to the other represented the transition from life to death.
It is thought that the stones used at Stonehenge were moved from Marlborough Downs, about 20 miles to the north.
Digs suggest that the area around the stone circle was used to bury the cremated remains of hundreds of people.
Other experts believe that it was a place for healing.
Meanwhile, a study earlier this week suggested Stonehenge could have been designed with acoustics in mind like a Greek or Roman theatre.
A team of researchers from the University of Salford spent four years studying the historic site's acoustic properties in a bid to crack the mystery of why it was built.
While they could not confirm the exact purpose of the stones, the researchers did find the space reacted to acoustic activity in a way that would have been noticeable to the Neolithic man.
'Stonehenge is very well known, but people are still trying to find out what it was built for and we thought that doing this research would bring an element of archaeology that so far hasn't been looked at,' lead researcher, Bruno Fazenda said.
He added the new area of acoustic science, named archaeoacoustics, could be helpful in the archaeological interpretation of important buildings and heritage sites, some of which may not exist in their original form, such as in the case of Stonehenge.
Because the site in Wiltshire is in a derelict state, researchers travelled to Maryhill in the U.S. where a full-sized concrete reconstruction of Stonehenge was built in 1929 as a memorial to the soldiers of WWI.
They were able to make proper acoustic measurements that allowed an investigation into striking acoustic effects such as echoes, resonances and whispering gallery effects.
The second phase consisted in the creation of a full 3D audio-rendition of the space using a system comprised of 64 audio channels and loudspeakers especially developed at the University of Salford based on Wave Field Synthesis.
This system enables an accurate and immersive recreation of what Stonehenge would have sounded like.
Salford's clever academics, who once took me shopping in a virtual supermarket – you sat in an armchair wearing a helmet and a glove – have now recreated the sound of Stonehenge.
We are nowhere nearer cracking the mystery of the monument as a result; but who would want to be? Apart from all the mountains of remaindered books of theories, a puzzle solved is never as gripping as a conundrum still under way.
But the four-year project by Dr Bruno Fazenda and colleagues atHuddersfield and Bristol universities, has established how the shouts, speeches, songs or sacrificial screams would have sounded, whatever material they may have contained. The method has been a painstaking piece of 'archaeoacoustics', a relatively new discipline which reveals the sound quality of buildings from the past.
Stonehenge is very well known, but people are still trying to find out what it was built for and we thought that doing this research would add an element of archaeology that so far hasn't been looked at. It's a new area of acoustic science and it could be very helpful in the archaeological interpretation of important buildings and heritage sites, some of which may not exist in their original form, such as in the case of Stonehenge.
The number of missing bits at the famous stone circle by the A303 in Wiltshire was an obvious problem, and tests there by the team produced only a limited number of weak echoes and no noticeable reverberation. Ancient Britons would not have been terribly overawed by this, if the monument was built to impress. But luckily there is a full-sized replica in the United States.
It was possible to make proper acoustic measurements that allow an investigation into striking effects such as echoes, resonances and whispering gallery effects. The data gathered does not unequivocally reveal whether the site was designed with acoustics in mind, like Greek or Roman theatres. It nevertheless shows that the space reacted to acoustic activity in a way that would have been noticeable to the Neolithic man.
This system give us an accurate and immersive recreation of what Stonehenge would have sounded like. We can not only see ourselves surrounded by the stones using virtual reality, but we can also listen how the stone structure would have enveloped people in a sonic experience. It is as if we can travel back in time and experience the space in a more holistic way.
There's more on the subject on a separate and excellent website called Sonic Wonders which has all manner of acoustic revelations. Here's a clapping experiment on YouTube courtesy of them – background details are here.
Einstein once declared that 'the most beautiful thing we can experience is the mysterious', and this statement is particularly fitting as regards Stonehenge. It was declared a World Heritage Site in 1986 and retains its intensely spiritual and otherworldly ambience, despite the number of tourists it attracts. If you choose to visit this extraordinary site, ensure you also spend some time delving into the captivating areas that surround Stonehenge. Explore picturesque Windsor and beguiling Bath.
Modern archaeological techniques and a number of recent digs have helped to form new theories about the stones that constitute Stonehenge. However, their definitive use stays as enigmatic as ever; it is suffused with a continuing sense of mystery. Stonehenge stands as an authoritative witness to the once extraordinary civilisations of the Stone and Bronze Ages of around 3,700 BC to 1,600 BC. Similarly, its surrounding areas stand as testament to these mighty cultures.
There is no place on earth quite like Stonehenge. Its brooding, pervasive mysticism, its intense spirituality and its endless aura of mystery all combine to make it an inimitable, idiosyncratic landmark. Take a Stonehenge tour and discover just why it continues to fascinate and perplex people from all over the world.
You may not know that Stonehenge is not the only ancient site in this region. Yet, in fact, a mere 25 miles north of Stonehenge is the impressive Avebury complex, which is a robust contender to be the most grandiose of all the residual prehistoric earthworks in Europe. The Avebury stone circle is actually much greater than that of Stonehenge – but the stones are smaller. A strong benefit that comes of visiting Avebury is that you can in fact touch its stones. Woodhenge, which consists of a circle of wooden posts, is an even more obscure and oft overlooked landmark.
Theories as to the use of Stonehenge range from the eminently believable to the absurd. In the twentieth century, both Fred Hoyle and Gerald Hawkins fascinatingly argued that not only was Stonehenge used as an observatory, but also to calculate future astronomical events such as eclipses. Yet it is probably von Däniken's theory about Stonehenge that has led to the most furore, conflict and general disputation. Däniken claims that either extraterrestrials, or humans who were aided by extraterrestrials, built Stonehenge. He then goes on to argue that Stonehenge is an exact replica of our solar system, including the asteroid belt.
Why not merge a visit to Stonehenge with visits to its nearby districts, which hold their own captivating and idiosyncratic attractions? A combination Bath and Stonehenge tour from London is one of the most extraordinary tours you can do. Bath is, without a doubt, one of the most absorbing destinations to explore in the UK. An especially great number of visitors assemble to view the extraordinary sunrise at the summer solstice.


The Best Picture category is always the headliner of all the categories and in a surprise, only eight films were nominated for Outstanding Best Picture. On June 24, 2009, the Academy announced that they would expand the category from five films to ten. Since that time, only twice has there been a year where all the slots available were filled (2009 and 2010). The biggest snubs for the category this year are the absences of If Beale Street Could Talk and First Man from the category. Beale Street is Academy Award winner Barry Jenkins' (2016's Moonlight) adaptation of the novel by James Baldwin. The movie has been showered with praise and recognition by critics worldwide. Actress Regina King who is nominated by the Academy for Best Supporting Actress has won the award for her work in this movie at the Critics Choice Awards and the Golden Globes and is considered by many to be the lock for the Oscar. Beale Street was even nominated for Best Picture with both the Globes and Critics Choice yet failed to receive recognition in any of the ten spots allowed by the Academy.
First Man is a giant snub on a level all of its own. The film failed to garner any recognition in any of the major categories including Claire Foy for Best Supporting Actress and was relegated to four nominations in Sound Editing, Sound Mixing, Visual Effects and Production Design. While it is an honor to be nominated and despite its debatable absence from the Best Picture category, the biggest snub for movie appears to be its absence from the Best Original Score category where it won the award at the Critics Choice and Golden Globes among other places. For First Man to not even be nominated in the category seems to be an extreme oversight.
There are two other snubs that deem recognition. Even though he is not slated to be the winner, Michael B. Jordan's performance as the Erik Killmonger in Black Panther was not even recognized in the category. His spot in the awards appears to be taken by Sam Rockwell's performance as George W. Bush in Vice. Despite the large disparity in screen time between the two characters, it appears that Disney's push for recognition for their domestic record-setting movie could only garner enough support for Black Panther to become the first superhero film to receive a Best Picture nomination, not for any of its actors or actresses. Jordan appears to be racking up near misses in nominations. He was previously overlooked for his work in 2013's Fruitvale Station and last year's Creed. Hopefully, Ruth Carter or Kendrick Lamar can pull off at least one win in the Costume Design or and Best Original Song categories to secure a win for Black Panther.
Finally, Peter Farrelly separated from his brother to be the director of the movie Green Book. The Farrelly Brothers were known for inane comic gold as Dumb and Dumber, Kingpin and There's Something About Mary. Not the highest in cinema but very entertaining fare. In directing Green Book, Peter Farrelly has achieved a new cinematic height. Green Book was complicated and smart, tragic and hilarious. It touched the serious topics of racism and homosexuality (two topics valued by the Academy) and brought them to the fore of conversations in a way that few other movies have. Many believe that Farrelly would never win the award but his turn of nature and style and the vision he presented in his work is akin to the change that Jordan Peele exhibited his breakout hit Get Out and deserved recognition. In this case, the recognition is an honor in and of itself.
That being said, many people are deserving of a nomination for their work and it is a huge delight to welcome and celebrate Spike Lee's first Academy Award nomination for directing BlackkKlansman. In his 40 year career, Spike Lee has only been nominated twice before this year, once for Best Screenplay (Do The Right Thing) and once for Best Documentary (Four Little Girls). With his three nominations this year (Directing, Best Picture, and Best Screenplay), Mr. Lee is finally receiving the recognition that he has been so deserving and many thought he would never receive from the Academy, short of the honorary award he received in 2016. Take note Michael B., your time will come.


The South Pacific Gyre is the world's largest expanse of oligotrophic ocean and supports communities of endemic gadfly petrels Pterodroma spp., yet little is known about their foraging ecology in this nutrient-poor environment. We tracked Murphy's petrels Pterodroma ultima with geolocators from Henderson Island, Pitcairn Islands, for 2 consecutive years (2011 to 2013). During pre-laying exodus, petrels travelled south and southwest of the colony, with males travelling further than females to more productive waters. During incubation, birds foraged at the southern and eastern edges of the Gyre, with some travelling over 4800 km from the colony, the greatest recorded foraging range of any breeding seabird. During non-breeding, the petrels migrated to the Subarctic Gyre in the North Pacific to forage in cool, mesotrophic waters. Habitat models revealed that these birds do not have clear preferences for oceanographic (such as fronts or eddies) or topographic (seamounts) features, generally favouring deep and unproductive waters. Analyses of activity patterns indicated Murphy's petrels are amongst the most active of all seabirds, particularly during incubation when they spent ca. 95% of their time at sea in flight. The birds did not appear to forage during darkness, but flight activity peaked at dawn, particularly during non-breeding, suggesting they feed on mesopelagic prey that are diel vertical migrants. At-sea protection for such a wide-ranging species would require management at huge spatial scales, and hence in the short term, the principal focus for conservation should be on eliminating the immediate threat from invasive mammals at breeding sites.


I take my Halloween costumes very seriously. It takes a lot of prep-time to make sure that everything is perfect. It's important that you never break character once in costume. If you do, the illusion is broken and Halloween loses all of its magic.
For instance, this year I'm going as a private eye. So, for the last two months, I've been reading detective novels in order to learn the lingo. Because of the intensive nature of this training, the speech patterns have become writ into my brain and I'll be able to make wry, cynical comments laced with slang like, "frail" and "moll" and "dame" at the drop of a hat. Consider it a kind of voluntary brainwashing!
I'm also going to need the right clothes. So I've spent my time in the Goodwill and thriftshops. Not only are the clothes cheaper there, but they're more likely to be authentic than something that you'd find in a normal retail setting. If you can't find a pair of suspenders that would be found on a real 1940's private detective, then you'll need to start from scratch.
I'm also shaving off the goatee, plucking my eyebrows, cutting my hair and engaging in some light cosmetic surgery to better fit the model of a guy in the 1940's.
Once I get my firearms license and buy a Luger from the rather shady, elderly German gentleman that lives over in JP, my costume will be all set.
Just remember: It's no fun if you don't take your Halloween costume deadly serious. Spend that extra time to research the fabrics that would be used in your mad scientist costume. A lab coat made of 100% polyester wouldn't be safe around an errant Bunsen burner! Be sure to read up on original Vlad the Impaler so that you know whether or not your accent truly depicts how Count Dracula would speak, were he alive today.
Because when it's time to be scary, authenticity is what truly counts.


Looking for a restaurant in Oklahoma City which offers Breakfast and Lunch? Aurora Breakfast, Bar & Backyard is the answer. Situated in the historic Plaza district, the restaurant follows the scratch-kitchen idea by offering food that is made absolutely from scratch. The ingredients are fresh and sourced locally.
Breakfast is considered to be the most important meal of the day. This is the reason; Chef Henry Boudreaux offers a menu which is high on energy and packed with nutritional value. The items on the menu combine nutrition with taste so that you get an incentive to eat healthy food.
The breakfast and lunch menu contains Housemade Scones, Yoghurt Bowl, Banana Pancakes, Tiramisu Pancakes, Grapefruit Brulee, Overnight Oats, Pecan Waffles, French Toast, Crab and Asparagus Omelette, Savory Lentil Hummus Bowl, Pulled Pork Hash, Aurora Burger, Open-Faced BLT and a lot more. There are meals available too which contain a mix of bacon, eggs, sausages, and bread.
There is a special menu for kids which consists of Pancakes with choco chips, whipped cream, sprinkles, maple syrup, waffles, PB&J and Yoghurt Bowl. There is a Grab N' Go menu as well which offers wraps, pastries, coffee, biscuits, yogurt bowl and overnight oats.
If you qualify as a senior, you can get a 10% senior discount on all purchases from Aurora Breakfast, Bar & Backyard. By activating the senior discount, you can get a hearty meal at an attractive rate. Make your breakfast special by ordering anything from the amazing menu of this restaurant. Breakfast does not need to be boring any longer!
Visit Aurora Breakfast, Bar & Backyard to enjoy breakfast, lunch, and brunch in this charming restaurant and receive the senior discount today!


text:
SOURCE TANTRA BEGINNERS WEEKEND SEMINAR
Feb 10-13 • Malibu Private Estate
All Skill Levels | For Singles & Couples
Learn 22 Tantra Techniques that will transform your sexual love life
into one of ecstatic pleasure, harmony, and deep emotional connection with
Tantra Masters Charles Muir & Christy Muir, and the Source School of Tantra Team
In Ancient India, Tantric sex was taught as an art form that every conscious person must master.
Now you can learn these ancient and proven skills translated as a 21st-century art form and experience a harmonious, joyful union.
Over the weekend course, you will learn how to be a master lover with:
Breathing in lovemaking ~ key to ecstasy
Touching ~ 6 kinds of healing and erotic touch
Tired of dating? Discover the skills to attract the Beloved
Learn how your eyes, words, emotions, and vulnerability can take you to intimacy bliss!
Freeing the female orgasm ~ it's easy!* 1,001 methods of exquisite pelvic movement
Sexual awakening for men, the Secret that can Quadruple the length and power of your orgasm
Advanced lovemaking techniques you will use forever
Female sacred spot massage ritual ~ step-by-step instructions to heal, awaken, and release unlimited energy
Internal muscular exercises for increased energy & pleasure.
Tantra: The Art of Conscious Loving' seminars provide the education we never had and always needed in order to have wonderful, real and lasting love, and to rediscover the full potential of our sexual energy. Tantra: The Art of Conscious Loving is a spiritual yet practical guide to being present and intimate with the most important person in your life. Sure there was a lot of discussion of sex, but only in the context of celebrating yourself, your partner, your love, and of life itself…this should be required learning for all people."
ABOUT YOUR TEACHERS:
Tantra Master Charles Muir, Founder and Director of Source School of Tantra Yoga, has been teaching Yoga for over forty-five years and is considered the originator and pioneer of the Modern Tantra Movement in the United States. His original sacred spot massage work, Tantric insights, and experiential exercises have become the world's cornerstones for many Tantra teachers.
He has authored several Tantra books, DVDs, White Tantra and Red Tantra CDs, and is currently working on several new books in his Tantra: The Art of Conscious Loving® series. His revolutionary work has been featured in over fifty media articles, numerous television and radio appearances, and two Hollywood movies. Charles is quoted in over a hundred books on Tantra, sexuality, Yoga, and relationship. He is the originator of the Tantra: The Art of Conscious Loving® format and Source Tantra Sacred Spot Massage Techniques. To learn more about Charles Muir's work, visit http://sourcetantra.com/
Christy Rose Muir, Advanced Certified Tantra Educator® began her quest for knowledge when she began practicing Shivananda Yoga, and meditation. In the years that followed, she earned certifications in Massage Therapy, Reiki, Holistic Psychology, and Life Coaching. Christy has traveled to India on a spiritual pilgrimage, visiting 20 cities and various temples and holy sites. In 2013, she began studying Tantra Yoga with the Source School of Tantra Yoga and achieved its highest certification level. Christy is also a Certified Yoga Teacher, a Blue Belt Nia Instructor, and has trained in Russian Kettlebells and Functional Movement. And a soulful singer!
Website: https://sourcetantra.com/christy-rose-muir
Info & Sign-up: www.SOURCETANTRA.COM/BEG
No explicit sexual activity or nudity takes place in class; optional homeplay is assigned
For registration:
Contact Orion at 831-201-8466 or orion@sourcetantra.com
FRIDAY: Please arrive at the seminar by 6:45 pm on Friday night. Class will end around 10:30-11:00 PM. We will keep the singles after class for a half hour for orientation after dismissing the couples.
SATURDAY: Class will start promptly at 10 am. There will be a two hour lunch break roughly between 1:30 – 3:30 pm. Class will end at 7:30 pm so you have time for home-play. Choosing a partner for home-play is optional for singles. If you have children at home we recommend organizing child care for the evening so you can have this special time for yourself.
SUNDAY: Class starts promptly at 10 am. There's a two-hour lunch break between 1:30 – 3:30 p. We will close the class with a special ritual by 9 pm.
Tell them I sent you! Love, xoxo Mare!
Got Q's? Ask Mare #323-903-7713 | TantraMare@gmail.com
Sexual Healing : Why we Need it
I was oblivious to the orgasmic potential in my body. Unaware of any pleasure past my clitoris and even that sometimes took more effort than it was worth. For the fleeting few seconds of release. Is that all there is?
Relying on vibrators was addicting and the more I used them the more numb I became. I just didn't know what I didn't know, which is a silent epidemic among most women worldwide.
Historically, women have been taught that their bodies are owned by their partner in marriage, to give them satisfaction. It was just a chore. So to keep the marriage happy, women like my mom learned to just endure sex. She never really knew its power except as a way to manipulate men to get what she wanted and negotiate she did. Though eventually she turned bitter and stopped playing the game.
The hunger that doesn't get met, as a result, leaves us starving for the nourishment that somewhere deep inside our body needs to be fed. As a Tantric love coach, I've worked with numerous women who cannot orgasm, or so they say. Especially with penetration sex with a partner if he is on his own rhythm woman gets left in the dust as he gallops along to the finishing line.
So why are we not feeling met and what's missing I've asked myself this question so many ways this timely answer comes to me in the awakening of my own great sea of possibilities. I think I speak for many women when I say that intercourse is not so exciting, kind of numb, or even painful when I allow myself to feel without a few drinks to numb out once more… I hear that story time after time and then I realize why women "fake it". This short-circuits the experience and keeps it at a very low vibrational survival level which causes such a deep void between men and women.
If men only knew where women could take to they would bow on their knees and serve her as the Creatress of life. The womb of a woman is the creative power source that can activate into whirlwinds of swirls bursting through the body, in bubbles of light. Exhilarating absolutely! Uplifting and inspiring giving energy and receiving blessings as the gift of the exchange. Tantric orgasm is like tapping into the source of life itself.
We can meet in the middle. In a woman's flow, slower is often more valuable and rewarding like the rich maturation of fine wine.
That's where tantra comes in blending the masculine and feminine energies so we need to heal and understand each other inside and out. Your inner standing connects us to the center where we meet physically, consciously and emotionally as well as physically in the world.
Because when we meet in the middle we're both nourished in a way that leaves us better than started like that quality of love is an elixir from the heavens.
So as we think so we experience. On some level I think all women at some point in their life feel these deep yearning desires that don't get met.
Fleeting is the pleasure that stimulation of the clitoris can bring. As wonderful as it is for those few seconds, then there's a quick decline in arousal overshadowed by hypersensitivity, where it's too sensitive to touch for a while. This is much like a man feels after he's climaxed before he zones out to sleep.
But for women it's just the beginning of her sexual pleasure potential, even if she doesn't know anything beyond her clitoris, her love tunnel will nudge her every now and again to remind her that there's more pleasure inside even if it's hard to find. That was my experience until I was shown how to discover my Om zone portal of pleasure as a cosmic, multidimensional, richly delicious portal where time stands still in the quivering pulses of life.
Since then I've found that a clitoral orgasm is like the doorbell that if you bring it right, it opens the sacred gates to her temple of love inside. To get a woman to fully expand and receive the fullest effects of your love, you have got to take your time and activate all her centers and erogenous zones. From the outside in, to all the internal sweet spots too. G-Spot and OMG-Spot [aka cervix] in particular.
Tantra says that women can be be multi-orgasmic! That brings up the question why do so many women feel numb, frustrated, and disinterested in sex?
Statistics show us that nearly 70% of women have difficulties achieving a deeply satisfying orgasm with penetration! Many women are so attached to their clitoral vibrator stimulators that they don't know what they're missing on the inside.
Problem with the vibrator is that it creates a short-circuit to pleasure after it reaches a peak and eventually causes more numbness and hyper-sensitivity; unless it's used very lightly like a hot spice. Many women that I work with tell me that orgasm during intercourse is hit and miss, at best. Often it's either numb or in some degree of pain.
This can be devastating when we told it's supposed to be a place of amazing pleasure. Why is it numb?? What's wrong with this picture! What can we do about it?
It naturally takes women longer to reach their highest pleasure peaks and some may worry that their man won't last long enough for her to be satisfied and reach her highest heaven inside.
Her concern fuels the problem and makes it all the more challenging, because stress, anxiety, and worry are orgasms enemies. They activate the fight or flight response which you don't want in the bedroom.
Men are hard-wired to naturally come quick to insure that they sow their seeds and procreate the species. So when testosterone is high the urge to release is very difficult to control.
But Tantra has a solution for that to train a man to expand and become more multi-orgasmic too!
But to reach the most sublime states of intimacy, men need to learn how to at least slow down and enjoy the journey more, so his Queen can relax, open fully and receive more deeply.
I can identify with this as I was one of many women who couldn't reliably climax with a partner. I felt broken with frustration, pressure and anxiety which exasperated the situation.
It burdens the relationship when women put this responsibility entirely their partner to be their healer and awakener. Yet it's been a part of our cultural conditioning that men take the initiative in sex. In this case learning to read her body language is essential as the body expressions and positions can say much more than words. Breath and sounds of her body are good cues to follow. When you breath in sync, you start to feel each other more.
Then enter the realm of sounds and see what unfolds…
Unless your partner is a trained sexual healer, it's important for women know her own body enough that she can tap into her O-Zone within her body. Share it. Invest in it so it may grow.
Then she can educate her partner, if he's willing to follow her lead. I can assure you it's worth the journey even if it feels vulnerable and scary. That's when the deepest transformations can take place.
Here's a massage tool that can help open the gates of Heaven by releasing and dearmoring numbness and pain, with conscious love.
Cervix wand- Click on the image to see more….
Click on this link to see this tool and how to get it for yourself. Or for a friend…
Classes in Tantra Touch for Healing and Empowerment coming this Fall, 2022

redpajama_set_name: RedPajamaCommonCrawl


If i'm being totally honest, date nights are super slim around our house. It's not that we prefer them that way (who does?!), but scheduling, finding babysitters, and everything else that comes with it can be daunting at times. But with baby #3 on the way, Pat and I decided we REALLY need to make more of an effort because it's going to be even harder when he arrives! I wore this adorable dress on a recent date night and today I'm sharing the perfect Maternity Black Dresses with you!
I love little black dresses because they're versatile enough for date nights, cocktail parties, or even weddings. Many of you have asked for some great maternity dress options to wear to events like this, and I feel like black is ALWAYS the way to go. You can switch out this look with dressier heels and statement earrings for wedding attire, or even dress it down with some flats if your feet are killing you 😉 I love fancier necklines like this off shoulder style, too!
This particular black dress is from Matron Saint, and actually comes in other colors if you want to mix it up! I absolutely love the fit, and it would work after pregnancy as well (bonus points!).
I'm linking up some of my FAVORITE maternity black dresses with you below!
This helped me so much!! None of my cute clothes for me anymore and it's kind of depressing:/ now I can go out buy something I feel good in and fits!! Thanks for sharing this valid information.
Agreed with your blog. Pregnant women always feel uncomfortable with normal maternity clothes. They need something to make them look quite stylish yet comfortable. Thay's why many clothing companies are making the stylish maternity clothes in a vast amount of variety like maternity tops, bottoms, dresses, jumpsuits etc.
Shop Designer Gowns, Black And Gold Jewelry, Gold Flat Shoes, Red Bikinis Tops and more. Get this widget.


Green Party Leader Elizabeth May is among the opposition MPs who are demanding the government remove environment-related policy changes from the budget implementation bill.
Beth Wallace, who works with the National Wildlife Federation and also grew up in Battle Creek, Michigan, describes the devastation of the Enbridge oil spill into the Kalamazoo River in July 2010. Enbridge wants to build the Northern Gateway pipeline to transport Alberta tar sands oil through the Rocky Mountains to Kitimat, British Columbia on the Pacific Coast.
This presentation was part of the Think Pipeline Speaking Tour in late 2010 which visited 8 communities in B.C. including Prince George, Vanderhoof, Houston, Burns Lake, Smithers, Terrace, Prince Rupert, and Kitimat. The sponsoring organizations included Friends of the Wild Salmon, Sea to Sands Conservation Alliance, Friends of Morice Bulkley, SkeenaWild, Northwest Watch, Douglas Channel Watch, and Prince Rupert Environmental Society.


text:
Refugee family loses Australian liberty
Australia's top court has backed the conservative government's tough stand on illegal immigrants when it overruled a lower court judgment that released five children from an immigration detention centre.
29 Apr 2004 08:41 GMT
Immigrants can be held for years in the Woomera centre
The High Court on Thursday found the Family Court did not have the jurisdiction to release three sisters, aged five, nine and 11, and two brothers, aged 13 and 15, or make any other order concerning the welfare of children held in detention centres.
"Under the constitution, the Family Court, as a federal court, only has jurisdiction that parliament confers," the court said in a statement.
A spokesman for acting Immigration Minister Gary Hardgrave said the government was now considering whether the children - who have been in the care of Catholic welfare agency Centacare since their release in August - should be returned to detention.
"The implications of this complex judgment will need to be assessed before making further comment. The current arrangements for the children will remain in place for the time being," said Hardgrave's spokesman.
The five children at the centre of this legal battle, who cannot be identified, had been in detention since arriving illegally in Australia with their mother in January 2001,
claiming to be refugees from Afghanistan's Taliban government.
"The High Court's decision is unfortunate"
Greg Barns,
spokesman, A Just Australia
Although their father, who arrived in Australia in October 1999, was granted a temporary protection visa, the mother and children were refused visas on the basis they were from Pakistan and not Afghanistan and were awaiting deportation.
The father was later taken into detention after a review of his case.
But the family's case hit the spotlight in June 2002 when two of the boys escaped from a remote detention centre at Woomera in the Australia's harsh outback and spent three weeks on the run.
They were released in August the following year by the Family Court decision amid strong lobbying by pro-refugee groups.
"The High Court's decision is unfortunate," said Greg Barns, spokesman for refugee group A Just Australia. "The children in question have suffered enough as a result of
government policy. It would be grossly inhumane to return them to detention."
Australia has one of the world's toughest immigration regimes, detaining boat people, illegal arrivals or anyone overstaying their visa in secure centres while their cases are handled - which can take years if appeals are involved.
International human rights groups, the United Nations and Australia's opposition Labour party have condemned the detention of asylum seeking children in these fenced camps, which have been plagued by riots, hunger strikes and suicide bids.


Welcome to my Web page! Thanks for the opportunity to let you know a bit about me.
I'm originally from the East Coast, but came to California to attend medical school at Loma Linda University. I married a local, and was fortunate to find a job in this lovely town, so it appears I'm here to stay. We enjoy living here in a house surrounded by lots of trees, proximity to family, a good church, and nearby church school for our children. I come from a close-knit family of scientists. Two of my three brothers have Ph.D.s in biology; my husband also has a Ph.D in biology. Our children don't seem interested in careers in science, but they are such cute redheads, we let it slide.
I'm board certified in obstetrics and gynecology, and have been in practice over twenty-five years. Ask what I like most about my practice, and I'll tell you its spending time talking, teaching, and learning from the women who are my patients. I would be delighted to be your physician.
I thrive on my family and friends, and am practicing part-time so I can be more available for them. My faith is important to me, and keeps me grounded. I've been a vegetarian since childhood, and eat healthfully. To stay fit, I enjoy walking. I always keep crossword puzzles, Sudoku, or Kakuro puzzles in my lab coat pocket to make sure my mind is getting exercise. I enjoy writing and reading non-fiction, travel, and music of all kinds.


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