text:
\section{Introduction}
\label{Intro}

In spite of its twenty-five years of history, the multichannel Kondo model \cite{BlandNoz-1980,Noz-1985} is still a subject of current interest in several fields of study of condensed matter physics. Indeed, this simple impurity model exhibits a non-Fermi liquid (NFL) behavior. Originally introduced to describe metallic systems containing dilute magnetic impurities, its applications have been considerably extended. For instance, it is related to two-level systems. \cite{Zawa-1980,Zawa-1982,ZarVlad-1996} It has been proposed \cite{Zarand-1996} that materials such as $\mathrm{Pb}_{1-x}\mathrm{Ge}_{x}\mathrm{Te}$ and $\mathrm{K}_{1-x}\mathrm{Li}_{x}\mathrm{Cl}$ could be modeled using a multichannel version of the Coqblin-Schrieffer model, \cite{CoqSchri} i.e., a SU($N$)$\times$SU($K$) Kondo model. Here $N$ indicates the number of spin degrees of freedom, and $K$ is the number of channels for the conduction electrons. The multichannel Kondo model is also believed to be relevant to the problem of mesoscopic quantum dots. \cite{Furu-1995,jaz} Several additional possible realizations concern the context of heavy-fermion compounds. Recent experiments in such systems have shown the existence of a quantum phase transition \cite{Sachdev-book} from a magnetically disordered to a long-range magnetic ordered phase, driven by a change in chemical composition, pressure or magnetic field. One can cite the alloy $\mathrm{CeCu}_{5.9}\mathrm{Au}_{0.1}$, \cite{Lohn-1994} the behavior of which in the disordered phase close to the quantum critical point is clearly of NFL-type: the specific heat $C$ depends on $T$ as $C/T\sim -\ln(T/T_{0})$, the magnetic susceptibility as $\chi \sim 1-\alpha \sqrt{T}$, and the $T$-dependent part of the resistivity as $\Delta \rho \sim T$ instead of $C/T \sim \mathrm{Const.}$, $\chi \sim \mathrm{Const.}$ and $\Delta \rho \sim T^{2}$ as predicted by the Fermi liquid theory. \cite{NozPin} Similar types of behavior have been observed in other systems such as $\mathrm{CePd}_{2}\mathrm{Si}_{2}$, \cite{mathur98} and $\mathrm{YbRh}_{2}\mathrm{Si}_{2}$, \cite{trovarelli00} among others (for an extensive survey of the experimental situation concerning heavy-fermion compounds, the reader is refered to the review article of Stewart \cite{Stewart-1994}). The experimentally observed breakdown of the Fermi liquid theory addresses fundamental questions about the possible formation of novel electronic states of matter with new types of elementary excitations resulting from the presence of strong correlations among electrons. \\
\indent
Since it has been proposed the multichannel Kondo model has been studied intensively by a variety of theoretical methods, such as the numerical renormalization group (RG), \cite{Noz-1980,Kusu-1996} Bethe ansatz, \cite{Andrei-1984,TsveWi-1984} conformal field theory, \cite{Frad-1990,Tsve-1990,AfLu-PRL1991,LuAf-PRL1991} and perturbative RG. \cite{Gan-PRL,Gan-thesis} Detailed discusssions concerning the various experimental applications and theoretical aspects of the multichannel Kondo model can be found in the book of Hewson, \cite{Hewson-book} and the review article of Cox and Zawadowski. \cite{CoxZaw} \\
\indent
Recent experiments performed in the heavy-fermion compound $\mathrm{CeCu}_{5.9}\mathrm{Au}_{0.1}$ by inelastic neutron scattering have shown the existence of an anomalous $\omega/T$ scaling law for the dynamical spin suceptibility at the antiferromagnetic wavevector which persists over the entire Brillouin zone. \cite{schroder98,schroder00} Such a result indicates that the spin dynamics are critical not only at large length scales, as studied in the traditional itinerant magnetism picture, \cite{hertz76,millis93,continentino93,moriya95,lp00} but also at atomic length scales. It strongly suggests the presence of locally critical modes beyond the standard spin-fluctuation theory. An approach which has been put forward to describe such a local quantum critical point is based on extended dynamical mean field theory, with encouraging results like the prediction of a scaling law for the dynamical spin suceptibility. \cite{si01} Another theory is based on the generalization of the spin operators symmetry, from SU(2) to SU($N$), to describe the Kondo impurity spin in terms of a mixed fermionic-bosonic representation. \cite{CPT-PRB} The idea behind this mixed formulation is to account for both the extended quasiparticle excitations and the locally critical modes using the fermionic and the bosonic part of the spin, respectively. \\
\indent
In order to fix ideas, let us start with the antiferromagnetic single-channel SU(2) Kondo impurity model. It is well known that within a RG analysis, \cite{Anderson-PMS,Wilson-1974,Wilson-1975,Nozieres-1976} the flow takes the Kondo coupling $J_{\mathrm{K}}$ all the way to strong coupling (i.e., to infinity). An important aspect in the discussion of the breakdown of the Fermi liquid theory is related to the question of the stability of the strong coupling (SC) fixed point. In this case the system flows to a SC fixed point which is stable and the associated behavior of the system is that of a local Fermi liquid. \cite{Noz-LocFL1974,Noz-LocFL1975,Hewson-1980} The situation is rather different when one considers several channels of conduction electrons. In the case of a spin $S$ interacting antiferromagnetically with the spin of conduction electrons belonging to $K$ different channels, Nozi\`eres and Blandin \cite{BlandNoz-1980} have shown that the multichannel Kondo model can lead to two very different situations depending on how $K$ compares to $2S$. Their calculation is based on a second order perturbation theory in the hopping amplitude of the conduction electrons, around the SC fixed point. They analyze their results by deriving an effective coupling between the spin of the composite formed by the impurity dressed by the conduction electrons in the SC limit, and the spin of the conduction electrons on the neighboring sites. On the one hand in the underscreened regime, when $K<2S$, the effective coupling is found to be ferromagnetic and the SC fixed point is stable. In this regime an effective spin, $(S-K/2)$, is formed, resulting from the screening of the impurity spin by the conduction electrons located on the same site. The system, described by the SC fixed point, behaves as a local Fermi liquid. On the other hand in the overscreened regime, when $K>2S$, the effective coupling is found to be antiferromagnetic and hence the SC fixed point is unstable. The instability of the SC fixed point in this regime constitutes a hint for the existence of an intermediate coupling (IC) fixed point, characterized by a finite critical value of the Kondo coupling and leading to a NFL excitation spectrum. \\
\indent
In a previous work \cite{JLB-Kondo03,JLB-Kondo03Short} using a second order perturbation theory similar to the analysis of Nozi\`eres and Blandin, we have considered an antiferromagnetic single-channel SU($N$) Kondo model involving a generalized impurity spin, realized by a combination of $2S$ bosonic and $q$ fermionic degrees of freedom. Preceding studies \cite{CPT-NPB} have shown that the SC fixed point associated with this model includes two regimes, one of them characterized by an instability for a certain class of mixed symmetry impurity spins. More precisely in the large-$N$ limit, \cite{ReadNew-JPC1983} the SC fixed point is stable and describes the low-energy physics of the one conduction channel model when $q < N/2$, but becomes unstable for $q > N/2$. This result suggests that the instability of the SC fixed point can arise from the symmetry properties of the impurity spin, without any introduction of additional degrees of freedom like conduction electron channels, coupling anisotropy or else. We have investigated how the system behaves, in particular when the number of conduction electrons (filling) on the neighboring site of the impurity located at the origin varies.  We have expressed our results in terms of an effective hamiltonian describing the interactions between the impurity dressed by the conduction electrons at the origin (as obtained in the SC limit), and the conduction electrons located on the neighbouring site. By determining the effective couplings, we have shown that the SC instability,  resulting from the change in sign of the effective spin interaction, is related to the change from repulsive to attractive of the effective charge interaction. \\
\indent
In the present work we consider the multichannel version of the model described above, i.e., a SU($N$)$\times$SU($K$) Kondo model, in order to achieve an even richer situa\-tion. In the overscreened regime, the magnetic degrees of freedom of the impurity are completely quenched, and the SC fixed point is unstable. \cite{BlandNoz-1980} We present a method allowing the identification and the characterization of the IC fixed point in the overscreened regime. Our study is based on a perturbative RG method pioneered by Ander\-son, the  ``poor man's scaling'' approach. \cite{Hewson-book,Anderson-PMS} Its basic principle is to reduce the conduction electron band width, $D$, by integrating out the high-energy intermediate states, and letting the low-energy physics properties unchanged. The main difficulty for the description of spin systems is that spin operators obey neither Fermi nor Bose statistics, leading to the absence of a Wick's decomposition theorem in this context. Consequently, conventional perturbative methods such as a Feynman's diagram \- approach are prohibited. The Gaudin theorem \cite{Gaudin-1960} existing for SU(2) generators does not allow the construction of simple diagrammatic techniques directly for spin operators. We tackle this problem by making use of a method initiated by Popov and Fedotov. \cite{PopovFed} In the framework of a fermionic description of the impurity spin states, this powerful approach allows a proper handling of the fermionic constraint. It introduces a set of auxiliary chemical potentials, which eliminates the unphysical Hilbert sub-space without removing the true spin states. The overscreened regime of the Kondo model has been studied intensively by several methods, for various impurity spin representations in both the SU(2) \cite{Gan-PRL,Gan-thesis,FabriZar-1996,Shaw-1998,AffLud-NPB1991} and SU($N$) \cite{jaz,AffLud-1994,Natan-1998,CoRu,pgks} spin groups. Our work considers several of these impurity spin representations as particular limits of the mixed symmetry formulation, enabling direct comparisons with the results obtained by other methods. \\
\indent
This paper is organized as follows. In Section \ref{Sec-KondoMod} the model is introduced, and we present the formalism leading to the derivation of the partition function within a path integral approach. We adopt the Abrikosov pseudofermion formulation to describe the impurity spin states. The handling of the fermionic constraint, performed by making use of the Popov-Fedotov method, is discussed. In Section \ref{Sec-SelfEn} we use a diagrammatic method to derive the conduction electron self-energy perturbatively in powers of the Kondo coupling constant, i.e., in the weak coupling limit. The imaginary part of the conduction electron self-energy is proportional to the electronic scattering rate, which is invariant under RG transformation as any physical quantity. This invariance is exploited in Section \ref{Sec-PRenGr}, to compute the $\beta$ scaling function by making use of perturbative RG arguments, including the  Callan-Symanzik equation. This leads to the identification of the IC fixed point in the overscreened regime. The temperature dependence of various physical quantities at low $T$, controlled by a unique critical exponent, is derived. The NFL nature of the physics governed by the IC fixed point is also discussed, and our results are compared with those of previous works. Section \ref{Conclu} gives a summary of the method and main results, and is devoted to conclusions. \\
\indent
The appendices contain the technical details of the calculations. In Appendix \ref{Appen-SUNgener} we give some properties followed by the SU($N$) group generators. Appendix \ref{Appen-Popov} briefly explains the principle of the method pioneered by Popov and Fedotov. This method requires a particular derivation of the conduction electron and impurity spin pseudofermion free propagators, which is performed in Appendix \ref{Appen-FreeProp}. Appendix \ref{Appen-SpinFact} contains the calculation of the first order term spin factors showing up in the conduction electron self-energy expansion. In Appendix \ref{Appen-ThirOrd} we show exhaustively how to compute the contribution to the electronic self-energy for a given example of diagram. The other diagrams can then be calculated in a similar way.

\section{Generalized Kondo Model and formalism}
\label{Sec-KondoMod}

In this Section we present the model, and the methods used to derive the partition function which is obtained in the functional integration formalism. In the language of Young tableaus, the impurity spin is described by a L-shaped representation, as shown in Fig.\! \ref{LShape-Fig}. The bosonic degrees of freedom correspond to the horizontal line (first row) containing $2S$ boxes, while the fermionic ones are included in the $q$ boxes of the vertical line (first column).

\begin{figure}[t]
\begin{eqnarray*}
\{ \Gamma \}~=~ \overbrace{\yng(5,1,1)}^{2S}
\begin{picture}(40,40)
\put(-9.0,0.1){$\left. \begin{array}{l} \\ \\ \\ \end{array} \right\} q$ }
\end{picture}
\end{eqnarray*}
\caption {Young Tableau description of the L-shaped irreducible representation $\{ \Gamma \}$ associated with the mixed symmetry impurity spin, realized by a combination of bosonic and fermionic degrees of freedom.}
\label{LShape-Fig}
\end{figure}

\subsection{Generalized multichannel SU($N$) single-impurity Kondo model}
\label{Sub-Kondo}

 The Hamiltonian describing the Kondo model we consider is
\begin{eqnarray}
\mathcal{H}_{\mathrm{K}} = \mathcal{H}_{0} + \mathcal{V}_{\mathrm{K}}~,
\label{HK-def}
\end{eqnarray}

\noindent where $\mathcal{H}_{0}$ is the free electronic part and $\mathcal{V}_{\mathrm{K}}$ gives the interaction between the generalized impurity spin and the conduction electrons. A SU($N$) (for $N \geq 2$) spin degeneracy is required for both the impurity and the conduction electron spins, and we suppose that there exists $K$ ($\geq 1$) electronic conduction channel(s)
\begin{eqnarray}
\mathcal{H}_{0} = \sum_{\bm{k}} \sum_{\sigma=1}^{N}  \sum_{\lambda=1}^{K}
\varepsilon _{\bm{k}} c_{\bm{k},\sigma,\lambda}^{\dagger} c_{\bm{k},\sigma,\lambda}~,
\label{H0-def}
\end{eqnarray}
where $c_{\bm{k},\sigma,\lambda}^{\dagger}$ is the creation operator of a conduction electron with momentum $\bm{k}$ and energy $\varepsilon _{\bm{k}}$, $\sigma$ and  $\lambda$ are the spin and channel indices, respectively. The second term
\begin{eqnarray}
\mathcal{V}_{\mathrm{K}} = J_{\mathrm{K}} \vec{\bm{S}}_{i} \cdot  \vec{\bm{S}}_{e}(0)~,
\label{VK-def}
\end{eqnarray}
corresponds to the antiferromagnetic Kondo interaction $J_{\mathrm{K}}$ ($\geq 0$) between the impurity spin $\vec{\bm{S}}_{i}$ located at the origin (site 0) and the spin of the conduction electrons $\vec{\bm{S}}_{e}(0)$ located at the same site. \cite{Kondo-1964}

The impurity spin states corresponding to the irreducible representation $\{ \Gamma \}$ can be expressed in terms of Abrikosov pseudofermions. \cite{Abri1965} We denote the dimension of $\{ \Gamma \}$ by $d_{ \{ \Gamma \} }$. Its general expression is given in Appendix \ref{Appen-SUNgener}. Each impurity spin state is described by a fermion. Therefore we have $d_{ \{ \Gamma \} }$ species of pseudofermions located at the origin which are respectively created and destroyed by the following operators
\begin{eqnarray}
f_{\gamma}^{\dagger} \ , \ f_{\gamma} \ , \ 1 \leq \gamma \leq d_{ \{ \Gamma \} }~.
\label{Def-fermi}
\end{eqnarray}
The number of pseudofermions at the impurity site is subject to the standard constraint
\begin{eqnarray}
\hat{n}_{f} = \sum_{\gamma=1}^{d_{ \{ \Gamma \} } } f_{\gamma}^{\dagger}  f_{\gamma} = 1~.
\label{Cons-fermi}
\end{eqnarray}
\noindent According to the structure of the SU($N$) Lie group, \cite{Licht1970} the impurity spin operator owns ($N^{2}-1$) components
\begin{eqnarray}
\vec{\bm{S}}_{i} =  \Big[ \bm{S}_{i}^{A} \Big]_{1\leq A \leq N^{2}-1}~, \nonumber
\end{eqnarray}
with
\begin{eqnarray}
\bm{S}_{i}^{A} =  \sum_{\gamma,\gamma^{'}=1}^{d_{ \{ \Gamma \} } }
f_{\gamma^{'}}^{\dagger} \bm{\Gamma}_{\gamma^{'},\gamma}^{A} f_{\gamma}~,
\label{Imp-compo}
\end{eqnarray}
where $\bm{\Gamma}^{A}$ corresponds to the ($N^{2}-1$) generators of the SU($N$) spin group.
A crucial technical aspect has to be pointed out here. Due to the shape of the generalized impurity spin that we consider which is not in the fundamental SU($N$) representation (i.e., a single box representation in the language of Young tableaus), and to the Abrikosov pseudofermion formalism that we adopt, the $\big\{ \tilde{\bm{\Gamma}}^{A} \big\}_{1\leq A \leq N^{2}-1}$ SU($N$) generators are not expanded as ($N \times N$) matrices as usual but as $\big( d_{ \{ \Gamma \} } \times d_{ \{ \Gamma \} }\big)$ ones. Such a situation already occurs in the SU(2) group if instead of considering a spin 1/2 corresponding to the SU(2) fundamental representation, we work with a spin 1, 3/2, ... \cite{GreiMull} The properties of the $\big\{ \tilde{\bm{\Gamma}}^{A} \big\}$ matrices which we will need are given in Appendix \ref{Appen-SUNgener}. It is to be noted that this way of writing the impurity spin states, although equivalent, is completely different from the method that we used in our previous work. \cite{JLB-Kondo03,JLB-Kondo03Short} In the latter case, the impurity spin states were obtained as a combination of ($2S+q-1$) particles, each box of the L-shaped representation being described by either one boson or one fermion, and the SU($N$) generators we had to consider corresponded to the fundamental representation. However the approach we now consider in terms of pure pseudofermions is much more appropriate to the functional integration formalism we develop.

\begin{figure}[t]
\begin{eqnarray*}
\{ \sigma \}~=~ \yng(1)
\end{eqnarray*}
\caption{Young Tableau description of the fundamental (irreducible) representation $\{ \sigma \}$ associated with each conduction electron.}
\label{FundaRep-Fig}
\end{figure}

As shown in Fig.\! \ref{FundaRep-Fig}, each conduction electron is described by the fundamental representation of SU($N$). The spin of the conduction electrons at the origin $\vec{\bm{S}}_{e}(0)$ is given by
\begin{eqnarray}
\vec{\bm{S}}_{e}(0) = \Big[ \bm{S}_{e}^{A}(0) \Big]_{1\leq A \leq N^{2}-1}, \nonumber
\end{eqnarray}
with the $(N^{2}-1)$ spin components given by
\begin{eqnarray}
\bm{S}_{e}^{A}(0) = \sum_{\sigma,\sigma^{'}=1}^{N}  \sum_{\lambda=1}^{K}
c_{\sigma^{'},\lambda}^{\dagger}(0) \bm{\sigma}_{\sigma^{'},\sigma}^{A} c_{\sigma,\lambda}(0)~. \nonumber
\end{eqnarray}
where $\bm{\sigma}^{A}$ corresponds to the ($N^{2}-1$) generators of the SU($N$) spin group.
In the previous expression $c_{\sigma^{'},\lambda}^{\dagger}(0)$ is the creation operator of a conduction electron of spin color $\sigma^{'}$ in the channel $\lambda$ at the impurity site, which can be rewritten in the momentum space as
\begin{eqnarray}
c_{\sigma^{'},\lambda}^{\dagger}(0) = \frac{1}{\sqrt{\mathcal{N}_{s}}} \sum_{\bm{k}} c_{\bm{k},\sigma^{'},\lambda}^{\dagger}~,
\label{Def-ElOp}
\end{eqnarray}
where $\mathcal{N}_{s}$ is the number of sites of the lattice. Using Eq.\! (\ref{Def-ElOp}), the different components of the conduction electron spin can be expressed as
\begin{eqnarray}
\bm{S}_{e}^{A}(0) = \frac{1}{\mathcal{N}_{s}} \sum_{\bm{k},\bm{k}^{'}} \sum_{\sigma,\sigma^{'}=1}^{N}  \sum_{\lambda=1}^{K}
c_{\bm{k}^{'},\sigma^{'},\lambda}^{\dagger} \bm{\sigma}_{\sigma^{'},\sigma}^{A} c_{\bm{k},\sigma,\lambda}~.
\label{El-compo}
\end{eqnarray}
The SU($N$) generators for the conduction electron spin correspond to the fundamental representation, therefore the corresponding traceless hermitian matrices $\big\{ \tilde{\bm{\sigma}}^{A} \big\}_{1\leq A \leq N^{2}-1}$ have ($N \times N$) components and satisfy the following normalization equation
\begin{eqnarray}
\mathrm{Tr}\big[ \tilde{\bm{\sigma}}^{A} \tilde{\bm{\sigma}}^{B} \big] = \frac{1}{2} \delta_{AB}~. \nonumber
\end{eqnarray}
\noindent In the well known SU(2) and SU(3) groups, $\tilde{\bm{\sigma}}^{A} = \tilde{\bm{\tau}}^{A}/2$ where $\big\{ \tilde{\bm{\tau}}^{A} \big\}$ are the Pauli and Gell-Mann matrices, respectively.

By making use of Eqs.\! (\ref{H0-def}), (\ref{VK-def}), (\ref{Imp-compo}) and (\ref{El-compo}) we obtain the Hamiltonian related to the multichannel SU($N$) $\times$ SU($K$) single-impurity Kondo model for a L-shaped Young tableau representation of the impurity spin
\begin{eqnarray}
\mathcal{H}_{\mathrm{K}} = & & \sum_{\bm{k}} \sum_{\sigma=1}^{N}  \sum_{\lambda=1}^{K}
\varepsilon _{\bm{k}} c_{\bm{k},\sigma,\lambda}^{\dagger} c_{\bm{k},\sigma,\lambda} \nonumber \\
&+& \frac{J_{\mathrm{K}}}{\mathcal{N}_{s}} \sum_{A=1}^{N^{2}-1}
\Big( \sum_{\bm{k},\bm{k}^{'}} \sum_{\sigma,\sigma^{'}=1}^{N}  \sum_{\lambda=1}^{K}
c_{\bm{k}^{'},\sigma^{'},\lambda}^{\dagger} \bm{\sigma}_{\sigma^{'},\sigma}^{A} c_{\bm{k},\sigma,\lambda} \Big) \nonumber \\
& & \hspace{14mm} \times \Big( \sum_{\gamma,\gamma^{'}=1}^{d_{ \{ \Gamma \} } }
f_{\gamma^{'}}^{\dagger} \bm{\Gamma}_{\gamma^{'},\gamma}^{A} f_{\gamma} \Big)~.
\label{HK-full}
\end{eqnarray}

\subsection{Functional integration method - Handling of the fermionic constraint}
\label{Sub-FuInt}

The partition function corresponding to $\mathcal{H}_{\mathrm{K}}$ (\ref{HK-full}) is
\begin{eqnarray}
\mathcal{Z} = \mathrm{Tr}_{\mbox{\scriptsize{spin}}} \mathrm{exp} \big[- \beta \mathcal{H}_{\mathrm{K}} \big]~,
\label{Tr-PartFu}
\end{eqnarray}
with $\beta = 1/k_{\mathrm{B}}T$. The difficulty in computing $\mathcal{Z}$ comes from the constraint expressed in Eq.\! (\ref{Cons-fermi}) which requires to perform the trace $\mathrm{Tr}_{\mbox{\scriptsize{spin}}}$ over the physical states involving one pseudofermion at the impurity site only. However the Pauli principle allows the number of pseudofermions at the origin to vary between 0 and $d_{ \{ \Gamma \} }$. More explicitly, the representation of the impurity spin with pseudofermion operators enlarges the dimensionality of the Hilbert space in which these fields are acting, compared with the physical space appropriate for the true spin operators. The unphysical states displayed in the pseudofermion formulation are then eliminated with the aid of the constraint. Therefore the evaluation of the trace in Eq.\! (\ref{Tr-PartFu}) can not be simply performed using standard perturbation approaches for instance but requires the use of further techniques.

In order to settle this problem, we adopt a method pioneered by Popov and Fedotov.  \cite{PopovFed} As explained in Appendix \ref{Appen-Popov}, one can introduce a set of auxiliary (imaginary) chemical potentials  $\{ \mathrm{i} \mu_{n} \}_{1\leq n \leq  d_{ \{ \Gamma \} }}$ governed by a discrete distribution function $P\{ \mu \}$
\begin{eqnarray}
& &P\{ \mu \} = \frac{1}{d_{ \{ \Gamma \} }} \sum_{n=1}^{d_{ \{ \Gamma \} } }
\mathrm{exp} \left[\mathrm{i} \beta \mu_{n} \right]
\delta \big( \mu+\mathrm{i} \mu_{n} \big) \nonumber \ , \\
& &\mathrm{i} \mu_{n} = \frac{\mathrm{i} \pi}{\beta d_{ \{ \Gamma \} }} (2n-1)~.
\label{Def-Pmu}
\end{eqnarray}
A crucial property arising from Eq.\! (\ref{Def-Pmu}) is that the imaginary chemical potentials depend on temperature. The partition function can be rewritten as
\begin{eqnarray}
\mathcal{Z} = \int \mathcal{D}\mu \ P\{ \mu \} \ \mathrm{Tr}_{\mbox{\scriptsize{Fock}}} \mathrm{exp}
\Big[- \beta \big( \mathcal{H}_{\mathrm{K}}- \mu \hat{n}_{f} \big) \Big]~,
 \label{Pmu-PartFu}
\end{eqnarray}
where the trace $\mathrm{Tr}_{\mbox{\scriptsize{Fock}}}$ is performed over the entire Fock space corresponding to the fermionic impurity spin operators without any constraint, namely including any number of pseudofer\-mion(s) allowed by the Pauli principle. The contributions of the unphysical states ($n_{f} \neq 1$) to the partition function automatically cancel out one with other by making use of the imaginary chemical potentials. With Eqs.\! (\ref{Def-Pmu}) and (\ref{Pmu-PartFu}) we get
\begin{eqnarray}
& &\mathcal{Z} = \frac{1}{d_{ \{ \Gamma \} }} \sum_{n=1}^{d_{ \{ \Gamma \} } }  \mathrm{exp} \left[\mathrm{i} \beta \mu_{n} \right]  \\
& &\hspace{22mm} \times \mathrm{Tr}_{\mbox{\scriptsize{Fock}}} \mathrm{exp}
\Big[- \beta \big( \mathcal{H}_{\mathrm{K}} + \mathrm{i}\mu_{n} \hat{n}_{f} \big) \Big]~. \nonumber
\label{Sumu-PartFu}
\end{eqnarray}
In the latter equation, in contrast with Eq.\! (\ref{Tr-PartFu}) of $\mathcal{Z}$, the trace showing up in the calculation has to be performed in the grand canonical ensemble, which allows to use standard techniques, namely the functional integration, Green's function method and diagrammatic expansion. In a path integral formalism at finite temperature, \cite{NegOr,Popov-book} each Fock trace is given by
\begin{eqnarray}
& &\mathrm{Tr}_{\mbox{\scriptsize{Fock}}} \mathrm{exp}
\Big[- \beta \big( \mathcal{H}_{\mathrm{K}} + \mathrm{i}\mu_{n} \hat{n}_{f} \big) \Big] \\
&=& \int \mathcal{D}\bar{c}^{*} \mathcal{D}\bar{c} \mathcal{D}\bar{f}^{*} \mathcal{D}\bar{f}
 \mathrm{exp} \Bigg[- \int_{0}^{\beta}d\tau \big\{ \mathcal{L}_{n}(\tau) +
\mathcal{H}_{\mathrm{K}}(\tau) \big\} \Bigg]~, \nonumber
\label{Tr-Fock}
\end{eqnarray}
where $\bar{c}^{*}$, $\bar{c}$, $\bar{f}^{*}$, $\bar{f}$ are Grassmann variables associated with the fermionic operators $c^{\dagger}$, $c$, $f^{\dagger}$, $f$, respectively, and $\mathcal{L}_{n}(\tau)$ is the Lagrangian of free particles expressed as
\begin{eqnarray}
& &\mathcal{L}_{n}(\tau) = \sum_{\bm{k}} \sum_{\sigma=1}^{N}  \sum_{\lambda=1}^{K}
\bar{c}_{\bm{k},\sigma,\lambda}^{*}(\tau) \partial_{\tau} \bar{c}_{\bm{k},\sigma,\lambda}(\tau) \nonumber \\
& & \hspace{14mm} + \sum_{\gamma=1}^{d_{ \{ \Gamma \} } } \bar{f}_{\gamma}^{*}(\tau) \big( \partial_{\tau} + \mathrm{i}\mu_{n} \big) \bar{f}_{\gamma}(\tau)~.
\label{Free-Lagrang}
\end{eqnarray}
One can notice from Eq.\! (\ref{Free-Lagrang}) that a direct consequence of the introduction of the auxiliary chemical potentials is a shift of the Matsubara frequencies related to the pseudo\-fermions by a value of ($-\mathrm{i}\mu_{n}$).

Using Eqs.\! (\ref{Sumu-PartFu}) and (\ref{Tr-Fock}), we finally obtain the expression of the partition function
\begin{eqnarray}
\label{LnHK-PartFu}
\mathcal{Z} &=& \frac{1}{d_{ \{ \Gamma \} }} \sum_{n=1}^{d_{ \{ \Gamma \} } } \mathrm{exp} \left[\mathrm{i} \beta \mu_{n} \right]
\int \mathcal{D}\bar{c}^{*} \mathcal{D}\bar{c} \mathcal{D}\bar{f}^{*} \mathcal{D}\bar{f} \\
& &\hspace{15mm} \times \ \mathrm{exp} \Bigg[- \int_{0}^{\beta}d\tau
\big\{ \mathcal{L}_{n}(\tau) + \mathcal{H}_{\mathrm{K}}(\tau) \big\} \Bigg]~. \nonumber
\end{eqnarray}

\section{Calculation of the conduction electron self-energy}
\label{Sec-SelfEn}
In this Section, we expand the conduction electron self-energy perturbatively in powers of the Kondo coupling constant $J_{\mathrm{K}}$. This means that we consider the weak coupling limit. We closely follow the method developed by Gan \textit{et al.} \cite{Gan-PRL,Gan-thesis} in the special case of the SU(2) $\times$ SU($K$) Kondo model, involving an impurity of spin 1/2. We use diagrammatic techniques. \cite{Mahan,Mattuck}

\subsection{Construction of the electronic self-energy diagrams}
\label{Sub-ConstDiag}

Here we summarize the rules allowing to derive the self-energy of the conduction electron, using a diagrammatic perturbative expansion of the partition function in $J_{\mathrm{K}}$. A detailed method to build the diagrams associated with the electronic self-energy in the framework of the Kondo model is given by Silverstein and Duke, \cite{Silv-Du} following  Abrikosov. \cite{Abri1965} Although the final results will be given at $T=0$, we work in the space of (fermionic) Matsubara frequencies for technical simplicity. The rules are the following

(i) In order to get the contribution at the $m$th order in the Kondo interaction $J_{\mathrm{K}}$, i.e., $\mathcal{O} \big[(J_{\mathrm{K}})^{m} \big]$, we draw all the topologically distinct diagrams linking $m$ vertices.

(ii) Each conduction electron propagator (\ref{FreePropEl}) is represented by a solid line  \\
\begin{picture}(40,25)
\put(100.1,7.0){$\bm{k}_{\bm{1}}, \mathrm{i} \omega_{m_{1}^{e}}, \sigma_{1}$}
\put(90.1,0.2){\vector(1,0){40}}
\put(130.1,0.2){\line(1,0){35} \ .}
\end{picture}
\\

(iii) Each impurity spin pseudofermion propagator (\ref{FreePropFermi}) is drawn as a dashed line \\
\begin{picture}(40,25)
\put(110.1,10.0){$\mathrm{i} \omega_{m_{1}^{i}}, \gamma_{1}$}
\put(90.1,0.2){- - - - - - }
\put(130.1,2.7){\vector(1,0){0.5}}
\put(132.1,0.2){- - - - - \ .}
\end{picture}
\\

(iv) Each vertex \\
\begin{picture}(40,50)
\put(26.1,26.0){$\bm{k}_{\bm{1}}, \mathrm{i} \omega_{m_{1}^{e}}, \sigma_{1}$}
\put(84.1,26.0){\vector(2,-1){30}}
\put(110.1,13.0){\line(2,-1){20}}
\put(130.1,3.0){\vector(2,1){25}}
\put(150.1,13.0){\line(2,1){30}}
\put(182.1,26.0){$\bm{k}_{\bm{2}}, \mathrm{i} \omega_{m_{2}^{e}}, \sigma_{2}$}
\put(130.1,3.0){\circle*{4.9}}
\put(16.1,0.5){$\mathrm{i} \omega_{m_{1}^{i}}, \gamma_{1}$}
\put(55.1,0.2){- - - - - - }
\put(95.1,2.7){\vector(1,0){0.5}}
\put(95.1,0.2){- - - - -}
\put(130.1,0.2){- - - - - - }
\put(170.1,2.7){\vector(1,0){0.5}}
\put(170.1,0.2){- - - - -}
\put(205.1,0.5){$\mathrm{i} \omega_{m_{2}^{i}}, \gamma_{2}$ }
\end{picture}
\\ \\
contributes by a factor

\begin{eqnarray}
- \frac{\displaystyle J_{\mathrm{K}}}{\displaystyle \beta \mathcal{N}_{s}}
\delta_{m_{1}^{e}+m_{1}^{i},m_{2}^{e}+m_{2}^{i}}
\sum\limits_{A=1}^{N^{2}-1}
\bm{\sigma}_{\sigma_{2},\sigma_{1}}^{A}  \bm{\Gamma}_{\gamma_{2},\gamma_{1}}^{A}~.
\label{VertexContrib}
\end{eqnarray}

(v) The independent internal momentums, frequencies and spin indices have to be summed over.

(vi)  Each conduction electron loop gives a factor $(- K)$.

(vii) Each pseudofermion loop contributes by a factor $(-1)$.

(viii) Each diagram $\mathcal{O}\big[(J_{\mathrm{K}})^{m}\big]$ exhibits a combinatorial symmetry factor $f_{sym}$. This factor results from the non-compensation between the number of permutations of the vertex and the contribution (1/$m!$) due to Wick's theorem. \cite{ItzikZub} We show it explicitly in Figs.\! \ref{SelfEnDiags-Fig2}, \ref{SelfEnDiags-Fig3} and \ref{SelfEnDiags-Fig4}, when its value is not equal to 1.

\subsection{Contributions of the leading order terms to the conduction electron self-energy}
\label{Sub-CalcSelfEn}

\begin{figure}[t]
\includegraphics[width=6.0cm]{Fig3-SelfEn-diag2.eps}
\caption {Feynman diagram for the conduction electron self-energy at the second order in a perturbation theory in $J_{\mathrm{K}}$.}
\label{SelfEnDiags-Fig2}
\end{figure}

In Figs.\! \ref{SelfEnDiags-Fig2}, \ref{SelfEnDiags-Fig3}, and \ref{SelfEnDiags-Fig4} we show the diagrams which will allow us to compute the $\beta$ scaling function perturbatively. As discussed in Section \ref{Sec-PRenGr}, these digrams are sufficient to derive the critical exponent of the model. The two subscripts in each term of the self-energy expansion indicate the corresponding powers of $J_{\mathrm{K}}$ and $K$, respectively. Each diagram is factorized along two main terms. The first one is the spin factor, it results from the summation over the internal spin degrees of freedom. In Appendix \ref{Appen-SpinFact} we present the method of calculation of these spin factors for all the diagrams we consider. It requires some knowledge about the properties (detailed in Appendix \ref{Appen-SUNgener}) of the SU($N$) generators $\big\{ \tilde{\bm{\Gamma}} \big\}$  and $\big\{ \tilde{\bm{\sigma}} \big\}$ associated with the impurity and the electronic spins, respectively. The second quantity is obtained by summing over the internal momentum and frequency degrees of freedom. Due to the presence of the imaginary chemical potentials $\{ \mathrm{i} \mu_{n} \}_{1\leq n \leq  d_{ \{ \Gamma \} }}$, this part has to be computed carefully. In particular, the free propagators associated with the conduction electron and impurity spins have to be redefined, which is done in Appendix \ref{Appen-FreeProp}. In Appendix \ref{Appen-ThirOrd}, we detail the calculation of the third order diagram $\mbox{\large{$\Sigma$}}_{(3,0)\mbox{\scriptsize{a}}}^{\sigma, \sigma^{'}}$ displayed in Fig.\! \ref{SelfEnDiags-Fig3}. The other diagrams can be computed in the same way.

The single-impurity Kondo model can be used for the study of real systems of dilute (low concentration) impurities. For that purpose, we adopt the random impurity averaging, \cite{Abri1965} the principle of which is the following. A number $N_{i}$ of impurity spins are randomly distributed in the system. We only keep the linear contribution in the impurity density $n_{i} = N_{i} / \mathcal{N}_{s}$. Since the interactions between impurities are completely neglected, we recover a single-impurity Kondo model.
\begin{figure}[t]
\includegraphics[width=8.6cm]{Fig4-SelfEn-diags3.eps}
\caption {Feynman diagrams for the conduction electron self-energy at the third order in a perturbation theory in $J_{\mathrm{K}}$.}
\label{SelfEnDiags-Fig3}
\end{figure}
\indent To summarize, each of the contributions $\mbox{\large{$\Sigma$}}_{(j,l)\alpha}^{\sigma, \sigma^{'}}$ to the conduction electron self-energy corresponding to a diagram of Figs.\! \ref{SelfEnDiags-Fig2}, \ref{SelfEnDiags-Fig3}, or \ref{SelfEnDiags-Fig4} can be written as a factorized expression as follows
\begin{eqnarray}
\mbox{\large{$\Sigma$}}_{(j,l)\alpha}^{\sigma, \sigma^{'}} &=& (-1)^{b_{pf}} \cdot (- K)^{l} \cdot (N_{i}) \cdot  f_{sym} \nonumber \\
& &\times \ \mbox{Spin}\Big[ \mbox{\large{$\Sigma$}}_{(j,l)\alpha}^{\sigma, \sigma^{'}} \Big] \cdot F_{(j,l)\alpha}~,
\label{Decompo-SelfEn}
\end{eqnarray}
where $b_{pf}$ and $l$ are the numbers of pseudofermion and conduction electron loop(s), respectively, $N_{i}$ is the number of impurities in the system, $f_{sym}$ the combinatorial symmetry factor of the diagram, $\mbox{Spin}\Big[ \mbox{\large{$\Sigma$}}_{(j,l)\alpha}^{\sigma, \sigma^{'}} \Big]$ the spin term corresponding to the contribution of the spin operators which involves the summation over the internal spin degrees of freedom, and $F_{(j,l)\alpha}$ the factor resulting from the contribution of the fermionic propagators which is obtained by performing the summations over the internal momentum and Matsubara frequencies. \\
\indent We shall exclusively work at $T = 0$, allowing us to calculate the integrals in energy analytically as well as the summations over imaginary chemical potentials included in the impurity spin free propagator (\ref{FreePropFermi}). We follow the conventional cut-off scheme: a cut-off in energy is introduced for the conduction electron band, i.e., $-D \leq \varepsilon_{\bm{k}} \leq D$, with a density of states $\rho$ per spin and per channel assumed to be constant.
\vfill\eject
\begin{widetext}
At zero-temperature after analytical continuation to the real axis, we obtain the contributions of the first perturbative order terms to the imaginary part of the electronic self-energy
\begin{eqnarray}
\label{SelfEn-2Ord}
& &\mbox{Im} \Big[\mbox{\large{$\Sigma$}}_{(2,0)}^{\sigma, \sigma^{'}} \big(\omega, \omega^{'}\big) \Big]
= - \Bigg( \frac{(N+2S-1)!}{(2S)! (q-1)! (N-q)!} \cdot \frac{2S}{4 N^{2}}
\Big[    2S \big(N - 1 \big) - q \big(N + 1 \big) +  \big(N^{2}+1 \big)\Big] \Bigg)  \\
& &\hspace{35mm} \times \big( J_{\mathrm{K}} \big)^{2}\rho  \cdot n_{i}  \cdot
\left(\frac{d_{ \{ \Gamma \} }-1}{\big[d_{ \{ \Gamma \} }\big]^{2} }\right)
\cdot \mbox{Im} \Bigg\{ \int_{-D}^{+D} d \varepsilon_{1} \frac{1}{(\omega -\varepsilon_{1} + \mathrm{i} 0^{+})}  \Bigg\}
\cdot \delta_{\sigma, \sigma^{'}} \cdot \delta \big(\omega-\omega^{'}\big) \ , \nonumber \\
\label{SelfEn-3Ord}
& &\mbox{Im} \Big[\mbox{\large{$\Sigma$}}_{(3,0)\mbox{\scriptsize{a}}}^{\sigma, \sigma^{'}} \big(\omega, \omega^{'}\big) \Big] +
\mbox{Im} \Big[\mbox{\large{$\Sigma$}}_{(3,0)\mbox{\scriptsize{b}}}^{\sigma, \sigma^{'}} \big(\omega, \omega^{'}\big) \Big]  \\
&=& - \Bigg( \frac{(N+2S-1)!}{(2S)! (q-1)! (N-q)!} \cdot \frac{2S}{4 N}
\Big[    2S \big(N - 1 \big) - q \big(N + 1 \big) +  \big(N^{2}+1 \big)\Big] \Bigg) \nonumber \\
& &\hspace{4mm} \times \big( J_{\mathrm{K}} \big)^{3} \rho^{2} \cdot n_{i}  \cdot \left(\frac{d_{ \{ \Gamma \} }-1}{\big[d_{ \{ \Gamma \} }\big]^{2} }\right)
\cdot \mbox{Im} \Bigg\{ \int_{-D}^{+D} d \varepsilon_{1} d \varepsilon_{2} \
\frac{n_{\mathrm{FD}}(\varepsilon_{2})}
{(\omega -\varepsilon_{1} + \mathrm{i} 0^{+})(\varepsilon_{1} - \varepsilon_{2}) } \Bigg\}
\cdot \delta_{\sigma, \sigma^{'}} \cdot \delta \big(\omega-\omega^{'}\big)~, \nonumber \\
\label{SelfEn-4Ord}
& & \mbox{Im} \Big[ \mbox{\large{$\Sigma$}}_{(4,1)\mbox{\scriptsize{a}}}^{\sigma, \sigma^{'}}  \big(\omega, \omega^{'}\big) \Big] +
 \mbox{Im} \Big[ \mbox{\large{$\Sigma$}}_{(4,1)\mbox{\scriptsize{b}}}^{\sigma, \sigma^{'}}  \big(\omega, \omega^{'}\big) \Big] +
 \mbox{Im} \Big[ \mbox{\large{$\Sigma$}}_{(4,1)\mbox{\scriptsize{c}}}^{\sigma, \sigma^{'}}  \big(\omega, \omega^{'}\big) \Big] \\
&=& \Bigg( \frac{(N+2S-1)!}{(2S)! (q-1)! (N-q)!} \cdot \frac{2S}{8 N}
\Big[    2S \big(N - 1 \big) - q \big(N + 1 \big) +  \big(N^{2}+1 \big)\Big] \Bigg)
\cdot \big( J_{\mathrm{K}} \big)^{4}\rho^{3}  \cdot n_{i} \cdot K
\cdot \left(\frac{d_{ \{ \Gamma \} }-1}{\big[d_{ \{ \Gamma \} }\big]^{2} }\right) \nonumber \\
& &\hspace{2mm} \times \ \mbox{Im} \Bigg\{ \int_{-D}^{+D} d \varepsilon_{1} d \varepsilon_{2} d \varepsilon_{3} \
\frac{n_{\mathrm{FD}}(-\varepsilon_{1}) n_{\mathrm{FD}}(\varepsilon_{2}) n_{\mathrm{FD}}(-\varepsilon_{3})
+n_{\mathrm{FD}}(\varepsilon_{1}) n_{\mathrm{FD}}(-\varepsilon_{2}) n_{\mathrm{FD}}(\varepsilon_{3})  }
{(\omega -\varepsilon_{1} + \mathrm{i} 0^{+})
(\omega -\varepsilon_{1} +\varepsilon_{2} - \varepsilon_{3} + \mathrm{i} 0^{+})(\varepsilon_{2} - \varepsilon_{3}) } \Bigg\}
\cdot \delta_{\sigma, \sigma^{'}} \cdot \delta \big(\omega-\omega^{'}\big) ~, \nonumber
\end{eqnarray}
\end{widetext}
where $n_{\mathrm{FD}}$ is the Fermi-Dirac distribution function.

\begin{figure}[b]
\includegraphics[width=6.3cm]{Fig5-SelfEn-diags4.eps}
\caption {Feynman diagrams for the conduction electron self-energy at the fourth order in a perturbation theory in $J_{\mathrm{K}}$.}
\label{SelfEnDiags-Fig4}
\end{figure}

In a perturbative RG analysis, we are only interested in the energy range $0 < \omega \ll D$. Consequently in the previous expressions (\ref{SelfEn-2Ord}), (\ref{SelfEn-3Ord}) and (\ref{SelfEn-4Ord}) the terms in powers of $(\omega / D)$ are neglected, and we only keep the factors generating logarithmic contributions in $\ln(\omega / D)$, which are sufficient to derive the two leading order terms of the $\beta$ scaling function as discussed in Section \ref{Obt-Beta}. Whenever necessary, we keep only the principal part of the denominator exhibiting a pole. The integrals in energy are computed in this limit following the method detailed by Gan, \cite{Gan-thesis} we get

\begin{eqnarray}
& & \mbox{Im} \Bigg\{ \int_{-D}^{+D} d \varepsilon_{1} \frac{1}{(\omega -\varepsilon_{1} + \mathrm{i} 0^{+})}  \Bigg\} = - \pi \ ,
\label{IntEn-2Ord} \\
& & \mbox{Im} \Bigg\{ \int_{-D}^{+D} d \varepsilon_{1} d \varepsilon_{2} \
\frac{n_{\mathrm{FD}}(\varepsilon_{2})}
{(\omega -\varepsilon_{1} + \mathrm{i} 0^{+})(\varepsilon_{1} - \varepsilon_{2}) } \Bigg\} \nonumber \\
&=& \pi \ln(\omega / D) \ ,
\label{IntEn-3Ord} \\
& & \mbox{Im} \Bigg\{ \int_{-D}^{+D}  \
\frac{d \varepsilon_{1} d \varepsilon_{2} d \varepsilon_{3} }
{(\omega -\varepsilon_{1} + \mathrm{i} 0^{+})
(\omega -\varepsilon_{1} +\varepsilon_{2} - \varepsilon_{3} + \mathrm{i} 0^{+}) }  \nonumber \\
& &\hspace{16mm} \times \ \frac{1}{(\varepsilon_{2} - \varepsilon_{3})} \nonumber \\
& &\hspace{16mm} \times \ \Big[ n_{\mathrm{FD}}(-\varepsilon_{1}) n_{\mathrm{FD}}(\varepsilon_{2}) n_{\mathrm{FD}}(-\varepsilon_{3}) \nonumber \\
& &\hspace{22mm} + \ n_{\mathrm{FD}}(\varepsilon_{1}) n_{\mathrm{FD}}(-\varepsilon_{2}) n_{\mathrm{FD}}(\varepsilon_{3}) \Big] \Bigg\} \nonumber \\
&=& \pi \ln(\omega / D) \ .
\label{IntEn-4Ord}
\end{eqnarray}

By incorporating the integrals (\ref{IntEn-2Ord}), (\ref{IntEn-3Ord}) and (\ref{IntEn-4Ord}) into Eqs.\! (\ref{SelfEn-2Ord}), (\ref{SelfEn-3Ord}) and (\ref{SelfEn-4Ord}) we obtain

\begin{widetext}
\begin{eqnarray}
& & \mbox{Im} \Big[ \mbox{\large{$\Sigma$}}_{ \{ \Gamma \} }^{\sigma, \sigma^{'}}  \big(\omega, \omega^{'}\big) \Big] \nonumber \\
=& & \pi \Bigg( \frac{(N+2S-1)!}{(2S)! (q-1)! (N-q)!} \Big[ 2S \big(N - 1 \big) - q \big(N + 1 \big) +  \big(N^{2}+1 \big) \Big]
\cdot \frac{2S}{4 N} \Bigg) \nonumber \\
&\times& \left(\frac{d_{ \{ \Gamma \} }-1}{\big[d_{ \{ \Gamma \} }\big]^{2} }\right)
\big( J_{\mathrm{K}} \big)^{2} \rho \cdot n_{i}  \cdot \delta_{\sigma, \sigma^{'}} \cdot \delta \big(\omega-\omega^{'}\big)  \cdot
\Bigg\{ \frac{1}{N} - \big( J_{\mathrm{K}} \rho \big) \ln(\omega / D)
+ \frac{1}{2} \big( J_{\mathrm{K}} \rho \big)^{2} K \cdot \ln(\omega / D) \Bigg\} \ .
\label{SelfEn-final}
\end{eqnarray}
\end{widetext}

\section{Perturbative Renormalization Group approach}
\label{Sec-PRenGr}

In this Section we use a perturbative RG method, the Anderson's ``poor man's scaling'' approach. \cite{Hewson-book,Anderson-PMS} The intermediate coupling (IC) fixed point associated with the model is identified, for various limits characterized by the shape of the   $\{ \Gamma \}$  representation. We start by deriving the $\beta$ scaling function, as a function of $\big(J_{\mathrm{K}}
\rho\big)$. A fixed point corresponds to a zero of the $\beta$ function. A simple analysis of the scaling function properties around a fixed point allows also to determine its stability. \cite{LeBellac} In the case of a stable IC fixed point, a rich physics characterized by a non-Fermi liquid behavior of the system (namely, anomalous temperature dependence of the specific heat, magnetic susceptibility, ...) can be expected.

\subsection{Derivation of the scaling function from the conduction electron self-energy}
\label{Obt-Beta}

In the previous Section \ref{Sub-CalcSelfEn}, we have calculated the first order terms of the conduction electron self-energy in a
perturbative approach. This yields the leading terms of the $\beta$ scaling function
\begin{equation}
\beta ( J_{\mathrm{K}} \rho ) = \frac{d (J_{\mathrm{K}} \rho )}{d (\mathrm{ln}D)}~.
\label{Beta-def}
\end{equation}
The connexion between both results from RG arguments, more precisely the Callan-Symanzik equation \cite{Callan,Symanzik} as explained below.

Without mentioning the spin dependence, we rename the imaginary part of the conduction electron self-energy $\mbox{Im} \Big[ \mbox{\large{$\Sigma$}}_{ \{ \Gamma \} }^{\sigma, \sigma^{'}} \Big]$, Eq.\! (\ref{SelfEn-final}), as  $\mbox{\large{$\Sigma$}}_{ \{ \Gamma \} }^{''}$, to have shorter notation in what follows. Following Gan \textit{et al.}, \cite{Gan-PRL,Gan-thesis} $\mbox{\large{$\Sigma$}}_{ \{ \Gamma \} }^{''}$ can be expressed in the most general way up to the fourth order in perturbation theory as
\begin{eqnarray}
\label{SelfEnRen-4ord}
& &\mbox{\large{$\Sigma$}}_{ \{ \Gamma \} }^{''} \big(\omega, D, J_{\mathrm{K}}\rho \big) \\
&=&
\frac{a}{\rho} \bigg\{ P_{0}\big( J_{\mathrm{K}} \rho \big)^{2} + P_{1}\ln(\omega / D) \big( J_{\mathrm{K}} \rho \big)^{3}
\nonumber \\
& & \hspace{5mm}
+ \ P_{2}\cdot K \cdot \ln(\omega / D) \big( J_{\mathrm{K}} \rho \big)^{4}  + P_{3}\cdot K \cdot \big( J_{\mathrm{K}} \rho \big)^{4}
\nonumber \\
& & \hspace{5mm} + \ P_{4}\ln^{2}(\omega / D) \big( J_{\mathrm{K}} \rho \big)^{4} + \mathcal{O}\Big[\big(J_{\mathrm{K}} \rho \big)^{5}\Big] \bigg\}~, \nonumber
\end{eqnarray}
where $a$, $P_{0}$, ... , $P_{4}$ are coefficients independent of $\omega$, $D$, $J_{\mathrm{K}}\rho$. $\rho \mbox{\large{$\Sigma$}}_{ \{ \Gamma \} }^{''}$ is the dimensionless scattering rate (i.e., inverse of the quasiparticle lifetime). The key point of the RG theory is that the underlying physics must be independent of the renormalization scheme, namely the cut-off $D$. Therefore the scattering rate is invariant under RG transformation. The Callan-Symanzik equation resulting from this property is then
\begin{eqnarray}
& &\Bigg[\frac{\partial}{\partial (\ln D)} + \beta\big( J_{\mathrm{K}} \rho \big) \frac{\partial}{\partial (J_{\mathrm{K}}\rho)} \Bigg]
\rho \mbox{\large{$\Sigma$}}_{ \{ \Gamma \} }^{''} \big(\omega, D, J_{\mathrm{K}}\rho \big) \nonumber \\
&=& 0~.
\label{CallSym-eq}
\end{eqnarray}
The $\beta$ scaling function is taken in its most general expansion up to the third order in powers of $J_{\mathrm{K}} \rho$
\begin{eqnarray}
\beta\big( J_{\mathrm{K}} \rho \big) &=& \kappa_{1} \big( J_{\mathrm{K}} \rho \big)^{2} + \kappa_{2} \cdot K \cdot \big( J_{\mathrm{K}} \rho \big)^{3}
\nonumber \\
& &+ \ \kappa_{3} \big( J_{\mathrm{K}} \rho \big)^{3} + \mathcal{O}\Big[\big(J_{\mathrm{K}} \rho \big)^{4}\Big]~.
\label{BetaF-def}
\end{eqnarray}
 Eqs.\! (\ref{SelfEnRen-4ord}) and (\ref{BetaF-def}) incorporated into Eq.\! (\ref{CallSym-eq}) give
\begin{eqnarray}
& &\big[2 \kappa_{1} P_{0} - P_{1} \big] \big( J_{\mathrm{K}} \rho \big)^{3} + \big[2 \kappa_{2} P_{0} - P_{2} \big] K \cdot \big( J_{\mathrm{K}} \rho \big)^{4}
\nonumber \\
& &+  \ 2 \kappa_{3} P_{0} \big( J_{\mathrm{K}} \rho \big)^{4}
+ \big[3 \kappa_{1} P_{1} - 2 P_{4} \big] \ln(\omega / D)  \cdot \big( J_{\mathrm{K}} \rho \big)^{4} \nonumber \\
& &+ \ \mathcal{O}\Big[\big(J_{\mathrm{K}} \rho \big)^{5}\Big] = 0~.
\label{CallSym-dev}
\end{eqnarray}
Eq.\! (\ref{CallSym-dev}) generates logarithmic series valid for any coupling constant $J_{\mathrm{K}}$. Therefore the coefficients in front of the terms of successive orders in $J_{\mathrm{K}}\rho$ are identically 0. The result is
\begin{eqnarray}
\label{BetaF-coeff1}
\kappa_{1} &=& \frac{P_{1}}{2 P_{0}}~, \\
\label{BetaF-coeff2}
\kappa_{2} &=& \frac{P_{2}}{2 P_{0}}~, \\
\label{BetaF-coeff3}
\kappa_{3} &=& 0~, \\
\label{BetaF-coeffP4}
 P_{4} &=& \frac{3}{2} \kappa_{1} P_{1}~.
\end{eqnarray}
The coefficients $P_{0}$, $P_{1}$ and $P_{2}$ are obtained from the imaginary part of the conduction electron self-energy, Eq.\! (\ref{SelfEn-final})
\begin{eqnarray}
\label{SelfEn-coeff0}
P_{0} &=& \frac{1}{N} ~, \\
\label{SelfEn-coeff1}
P_{1} &=& - 1 ~, \\
\label{SelfEn-coeff2}
P_{2} &=& \frac{1}{2} ~.
\end{eqnarray}
We point out here that the coefficients given by Eqs.\! (\ref{SelfEn-coeff0}), (\ref{SelfEn-coeff1}) and (\ref{SelfEn-coeff2}) are exact at this stage for any value $N$, $2S$ and $q$, without any restriction. As far as the consistency equation (\ref{BetaF-coeffP4}) is concerned, it has been checked by Gan  \cite{Gan-thesis} for an impurity spin in the fundamental representation of the SU(2) group. At each order of perturbation our study leads to the same integrals in energy displayed in Eqs.\! (\ref{IntEn-2Ord}), (\ref{IntEn-3Ord}) and (\ref{IntEn-4Ord}). The only difference between Gan's work and our generalized approach is contained in spin factors which depend on the considered group and representation describing the impurity. Since we have calculated these spin factors in an exact way, we conclude that the previous consistency equation is automatically satisfied for any group and any impurity representation we are interested in: the explicit calculation of $P_{4}$ for the generalized $\{ \Gamma \}$ representation is not required.

\subsection{Analysis of the intermediate coupling fixed point}
\label{IC-Analys}

Following the results of previous Section \ref{Obt-Beta}, the  $\beta$ scaling function (\ref{Beta-def}) obtained in a perturbative theory calculation in $J_{\mathrm{K}}$ up to the third order is given by
\begin{eqnarray}
\beta\big( J_{\mathrm{K}} \rho \big) &=& -\frac{N}{2} \big( J_{\mathrm{K}} \rho \big)^{2}
+ \frac{NK}{4} \big( J_{\mathrm{K}} \rho \big)^{3} \nonumber \\
& &+ \ \mathcal{O}\Big[\big(J_{\mathrm{K}} \rho \big)^{4}\Big].
\label{Beta-result}
\end{eqnarray}
One of the main results of this work is that Eq.\! (\ref{Beta-result}) is valid for any value of the parameters $2S$ and $q$ characterizing the L-shaped representation of the impurity spin, provided that the perturbative expansion in $J_{\mathrm{K}}$ is valid as discussed in the next paragraph.

The critical value $J^{*}$ taken by the Kondo coupling at the IC fixed point is obtained from the zero of the $\beta$ function, i.e., $\beta\big( J^{*} \rho \big) = 0$. We get
\begin{eqnarray}
J^{*} \rho = \frac{2}{K} + \mathcal{O}\left[\frac{1}{K^{2}} \right]~.
\label{J-IntFix}
\end{eqnarray}
The latter expression for $J^{*}$ allows one to determine the domain of validity of the perturbative expansion. The RG perturbative approach we use is valid in the weak coupling regime only. The IC fixed point has to be in the vicinity of the free electron fixed point of the system. Consequently, the perturbative expansion of the $\beta$ function is trustworthy exclusively when $J^{*}$ is close to 0, which means by using Eq.\! (\ref{J-IntFix}) $K \gg 1$. Since each conduction electron loop which gives a factor of ($-K$) contains two interacting vertices at least, and each term $J^{*} \rho$ contributes by a factor of $1/K$, we conclude that the diagrams displayed in Figs.\! \ref{SelfEnDiags-Fig2}, \ref{SelfEnDiags-Fig3}, and \ref{SelfEnDiags-Fig4} include all the contributions to the electronic self-energy up to the order $\mathcal{O}\big[1/K^{3}\big]$ at the IC fixed point. This argument allows one to check the self-consistency of the calculation. Another restriction is due to the domain of validity of the perturbative expansion. As previously noticed by Abrikosov-Migdal \cite{AbriMig} and Affleck-Ludwig, \cite{AffLud-NPB1991} the fourth order term in the expansion of the $\beta$ function, $\mathcal{O}\Big[\big( J_{\mathrm{K}} \rho \big)^{4}\Big]$, contains a coefficient proportional to the eigenvalue of the quadratic Casimir operator (\ref{Casi-Gam-red}) related to the impurity spin representation. In order that the latter contribution which has not been taken into account here remains negligible  compared to the second and third order contributions, we must also suppose that $N$, $2S$ and $q$ are small compared to $K$. To summarize, the domain of validity of the scaling function perturbative expansion is given by
\begin{equation}
K \gg 1, ~N \ll K, ~2S \ll K, ~q \ll K~.
\label{PRG-limit}
\end{equation}
Consequently the IC fixed point describes an overscreened impurity, as discussed originally by Nozi\`eres and Blandin. \cite{BlandNoz-1980} The single-channel case turns out to be out of reach of the present study.

In addition to the value of $J^{*}$, another key quantity which can be drawn from the $\beta$ function is its slope $\Delta$ at the IC fixed point, namely
\begin{equation}
\Delta = \left. \frac{d \beta}{d (J_{\mathrm{K}} \rho )} \right]_{J_{\mathrm{K}} = J^{*}}
= \frac{N}{K} + \mathcal{O}\left[\frac{1}{K^{2}} \right]~.
\label{DeltaUniv}
\end{equation}
In the limit given by Eq.\! (\ref{PRG-limit}) that we consider, $\Delta$ is automatically positive therefore the IC fixed point is stable. \cite{LeBellac} The slope $\Delta$ determines the critical exponent of the model. It is a strong mark of the universality concept characteristic of the Kondo effect in the low-energy regime. \cite{Loram-1970,Andrei-RMP1983} This regime describes the system at temperatures that are low compared to the bandwidth $D$. In that case the physics of the system is governed by a stable IC fixed point, and the temperature dependence of the various physical quantities at low $T$ obeys a power law in $T / T_{\mathrm{K}}$ which is entirely controlled by $\Delta$ ($T_{\mathrm{K}}$ being the Kondo temperature).

The Kondo temperature is the only relevant energy scale in the problem, it is generated dynamically by the interactions with the impurity spin. It plays an essential role in the Kondo effect: $T_{\mathrm{K}}$  characterizes a crossover between the weak coupling regime and the low-energy regime. This temperature scale is invariant under the scaling process. One can easily derive it by inverting and integrating the $\beta$ function expression. \cite{Hewson-book} By using Eqs.\! (\ref{Beta-def}) and (\ref{Beta-result}) we get
\begin{equation}
k_{\mathrm{B}} T_{\mathrm{K}} \sim D (J_{\mathrm{K}} \rho)^{K/N} \exp{\bigg[{-\frac{2}{N(J_{\mathrm{K}} \rho)}}\bigg]}~.
\label{KondoTemp}
\end{equation}

\begin{figure}[t]
\begin{eqnarray*}
& &\yng(5,1,1)~~~~~;~~~~\yng(1)~~~~;~~~~\yng(5)~~~~;~~~~\yng(1,1,1) \\ \\
& &\hspace{7mm} \mathrm {(a)} \hspace{18mm} \mathrm {(b)} \hspace{16.5mm} \mathrm {(c)} \hspace{17mm} \mathrm {(d)}
\end{eqnarray*}
\caption {Irreducible SU($N$) representations of the impurity spin considered in the present work:
(a) L-shaped ; (b) fundamental, $2S=q=1$ ; (c) symmetric, $q=1$ ; (d) antisymmetric, $2S=1$.}
\label{ImpSpinRep-Fig}
\end{figure}
As mentioned above, the results we have obtained are valid in the overscreened regime for any value of the parameters  $N$, $2S$ and $q$, provided that Eq.\! (\ref{PRG-limit}) is satisfied. The impurity spin representations that are adressed by the present study are displayed in Fig.\! \ref{ImpSpinRep-Fig}. We conclude that {all the Kondo models defined by these different impurity spin representations belong to the same universality class in the overscreened regime, which is characterized by the following low-temperature behaviors \cite{Gan-thesis}
\begin{eqnarray}
&-&\mathrm{resistivity~due~to~Kondo~scattering:} \nonumber \\
& &~~\delta \rho_{\mathrm{e}}(T) = \rho_{\mathrm{e}}(T=0) - \alpha
\bigg( \frac{T}{T_{\mathrm{K}}} \bigg)^{\Delta} ~, \nonumber \\
&-& \mathrm{impurity~specific~heat~:}~C_{\mathrm{imp}}(T) \sim \bigg( \frac{T}{T_{\mathrm{K}}} \bigg)^{2 \Delta}
 ~, \nonumber \\
&-& \mathrm{magnetic~susceptibility~:}~\chi_{\mathrm{imp}}(T) \sim \frac{1}{T}
\bigg( \frac{T}{T_{\mathrm{K}}} \bigg)^{2 \Delta} ~. \nonumber
\end{eqnarray}
We can notice that the system described by an IC fixed point exhibits non-Fermi liquid properties. As in the original single-channel Kondo model characterized by an exactly screened spin 1/2 impurity, the resistivity increases when the temperature is lowered. Denoting by $\vec{\bm{S}}_{ef\!f}$ the effective spin of the impurity screened by the conduction electrons and $\langle ~ \rangle$ the thermal average, we have \cite{Hewson-book}
$\left\langle \big( \vec{\bm{S}}_{ef\!f} \big)^{2} \right\rangle \sim T \chi_{\mathrm{imp}}(T) \sim T^{2 \Delta} \rightarrow 0$
as $T \rightarrow 0$ which indicates that the IC fixed point is associated with a ground state corresponding to a spin singlet. In the overscreened regime the impurity spin is completely quenched by conduction electrons.

The value of the universal critical exponent $\Delta$ we have obtained in Eq.\! (\ref{DeltaUniv}) is in agreement with the result of Gan \textit{et al.} who considered a spin 1/2 in the SU(2) group. \cite{Gan-PRL,Gan-thesis} The case of larger spins ($S > 1/2$) in the SU(2) group has also been studied, \cite{FabriZar-1996,Shaw-1998} leading to similar results. Eq.\! (\ref{DeltaUniv}) allows one to recover the overscreened limit  of the exact solution of the multichannel Kondo model, obtained by using the Bethe ansatz for the symmetric impurity spin representations of SU($N$), \cite{jaz} and boundary conformal field theory in the case of an arbitrary impurity spin representation of the SU(2) \cite{AffLud-NPB1991} and  SU($N$) \cite{AffLud-1994} groups. It also corroborates the large-$N$ studies of the SU($N$)$\times$SU($K$) Kondo model involving either the fundamental \cite{CoRu} or the antisymmetric \cite{pgks} impurity spin representations, although the $K \sim N$ limit considered in these works is different from ours.

\section{Summary and conclusions}
\label{Conclu}

In this paper, we have studied a multichannel SU($N$) Kondo model, involving a mixed symmetry impurity spin representation. Namely the  generalized impurity spin is realized by a combination of $2S$ bosonic and $q$ fermionic degrees of freedom, in the presence of $K$ conduction electron channels. The study has been carried out by making use of a perturbative renormalization group technique, the Anderson ''poor man's scaling'' approach. The main difficulty for the description of spin systems is that spin operators obey neither Fermi nor Bose statistics, leading to the absence of Wick's decomposition theorem in this context. Consequently conventional perturbative methods are prohibited. We have used the Abrikosov pseudofermion formulation to describe the impurity spin states, and tackled the problem by making use of a method initiated by Popov and Fedotov. Its basic principle consists in introducing a set of auxiliary chemical potentials which eliminates the unphysical Hilbert sub-space keeping the physical spin states only, and therefore enables a proper handling of the fermionic constraint.

The partition function has been obtained within a path integral approach. The general expression of the conduction electron self-energy imaginary part has been derived perturbatively in powers of the Kondo coupling constant, i.e., by considering the weak coupling limit. At each order of a perturbative calculation the derivation involves two steps, namely the calculation of the spin factor and the computation of the fermionic propagator contribution which includes the Popov-Fedotov auxiliary chemical potentials. The imaginary part of the conduction electron self-energy is proportional to the electronic scattering rate, which is a physical quantity and therefore is invariant under renormalization group transformation. The Callan-Symanzik equation can be applied, and yields the derivation of the two leading terms of the $\beta$ scaling function, i.e., up to the order $\mathcal{O}\big[1/K^{2}\big]$. We have established that this truncated expansion of the $\beta$ function is valid in the overscreened (Nozi\`eres - Blandin) regime, for any value of  $N$, $2S$ and $q$ providing that these parameters remain small compared with $K$. The intermediate coupling (IC) fixed point describing the low-energy physics of the model has been identified in the overscreened regime. The critical exponent $\Delta$, corresponding to the slope of the $\beta$ function towards $J_{\mathrm{K}} \rho$ at the IC fixed point, is given by
\begin{equation}
\Delta = \frac{N}{K} + \mathcal{O}\left[\frac{1}{K^{2}} \right]~. \nonumber
\end{equation}
$\Delta$ governs the temperature dependence of the various physical quantities at low $T$ which are of non-Fermi liquid type, as discussed in Section \ref{Sec-PRenGr}. The IC fixed point describes the system in its ground state which is a spin singlet: in the overscreened regime the impurity spin is completely quenched by the conduction electrons located on its site.

Several types of impurity spin representations have been considered in previous works and can be viewed as special limits of the mixed symmetry formulation we have used. Hence direct comparisons with our study have been made possible. Our results are in agreement with those obtained by other methods, like Bethe ansatz and boundary conformal field theory. We have established that the Kondo models defined successively by fundamental, symmetric, antisymmetric and mixed symmetry impurity spin representations yield exactly the same low-energy physics in the overscreened regime. To summarize the present work gives a unified view of the overscreened Kondo regime, for a large set of representations of the impurity spin.
It is also possible to consider arbitrary impurity spin representations, beyond the cases mentioned here. For example one can think about representations of the SU($N$) group with rectangular-shaped Young tableaux, as discussed in Ref.\! \onlinecite{Natan-1998}. The method we present can be generalized to study the case of an arbitrary SU($N$) representation. No additional conceptual difficulty has to be expected. The spin factors associated with such a representation, which are present in the conduction electron self-energy, have to be computed following the derivation displayed in Appendix \ref{Appen-SpinFact}. The traces over $\tilde{\bm{\Gamma}}$ matrices we have to consider involve products of two or three generators in the impurity spin representation, and therefore the spin factors can be derived exactly for any irreducible representation of SU($N$). \cite{Okubo-1982,Schell-1999,Okubo-1977-2} However the computation of these spin factors as a function of the Young tableau parameters describing a completely arbitrary impurity spin representation certainly requires extremely lengthy algebra, as the complexity of Eqs.\!  (\ref{TrGam3-fin}), (\ref{Contrib-Spin3}) and (\ref{Contrib-Spin3b}), which are restricted to the L-shaped case we consider, suggests it easily. \\
\indent Finally we mention the case of the single-channel SU($N$) Kondo model involving a mixed symmetry impurity spin, which we considered in a previous work. \cite{JLB-Kondo03,JLB-Kondo03Short} In such a case the L-shaped impurity exhibits a rather surprising duality: it is exactly screened if one considers the fermionic component, but underscreened towards the bosonic one. An interesting problem remains. Namely, the understanding of the low-energy physics in the regime characterized by an instability of the strong coupling fixed point turns out to be out of reach of the present work. This issue deserves further investigations. It might have important future applications for the Kondo lattice problem, \cite{Doniach-1977} with potential implications for the description of heavy-fermion compounds. In order to gain some insight it would be interesting to carry out a non-perturbative study. Unfortunately it is quite a difficult task. Indeed well-known methods usually efficient to study the Kondo problem, as numerical renormalization group or Bethe ansatz techniques, have severe limitations in the context of the mixed symmetry representation. For example, the simplest impurity that displays an IC fixed point in the single-channel case implies a large Hilbert space which restricts the efficiency of a numerical renormalization group study. The difficulties encountered by the Bethe ansatz are of different nature: due to the mixed symmetry properties of the impurity spin representation, the model with Kondo coupling only is not integrable. \cite{Natan-PCom}

\begin{acknowledgements}

We would like to thank Natan Andrei for highly valuable comments and discussions at several steps of this work. We are grateful to Marijana Kir\'{c}an for a careful reading of the manuscript. D.B. would like to acknowledge extremely fruitful discussions with Alex Hewson, Naoto Nagaosa, Philippe Nozi\`eres and Roland Zeyher.

\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


text:
Q: How to show column name on admin site in django? Admin site user table   I'm extending my admin table on django default admin table. After adding a column, when I go to the admin site, I can only see the username column name on the user section. But when click it, it shows the column name and ready to put some fields. All I want is to show all the column name when visiting to the user table.
Im just a newbie in Django Framework, I read some documents and ways to extend my admin table. All I want is to add column name on existing admin table.
#models.py
from django.db import models
from django.contrib.auth.models import AbstractUser

class User(AbstractUser):
    contact_number = models.CharField(max_length=250, blank=True)

#views.py

from django.shortcuts import render
from .models import User
from django.urls import reverse_lazy
from django.views import generic
from .forms import NewUserCreationForm
from django.http import HttpResponseRedirect

class SignUp(generic.CreateView):
    form_class = NewUserCreationForm
    success_url = reverse_lazy('login')
    template_name = 'signup.html'

def userView(request):
    users = User.objects.all()
    return render(request,'registration/view_users.html',
        {"all_users":users})

#admin.py
from django.contrib import admin
from django.contrib.auth.admin import UserAdmin
from .models import User
from .forms import NewUserCreationForm, NewUserChangeForm

class NewUserAdmin(UserAdmin):
    add_form = NewUserCreationForm
    form = NewUserChangeForm
    model = User
    list_display =
    ['username']

admin.site.register(User,NewUserAdmin)

I want to show all the column on the admin site.

A: list_display = ['username'] will only display username field, so you need add contact_number field also, like this:
list_display = ['username', 'contact_number']
Add all the columns to list_display field which you want to display.
You could also try this for all fields:
list_display = [field.name for field in User._meta.get_fields()]

This will display list in sorted order.

redpajama_set_name: RedPajamaStackExchange


hellllooooo and good eve to you all. I'm just home after a really lovely workshop at Pink Tulip. Our project to day was a 12 x 12 shadow box and our task was to take the signature sheet from the 'Twas the Night Before Chistmas' and transform it into a three dimensional scene, just like the one on the Long Winters Nap paper. And we did it, with the help of a little chipboard and some snappy pairs of scissors. I think that's one of my favourite features of Graphic 45 papers, you can bring them to life and lift them off the page to make gorgeous home decor projects.
Wonderful project - the ladies look as if they had a super time.
wow fab and gorgeous Clare.
Afternoon Clare, Woweeeee !! these projects look stunning, and the girlies all look so proud of themselves, well done All !!
This is stunning , wish I had been there!


text:
Human behaviour remains key risk to protecting data
S&P improves its assessment of outlook for Guernsey
Industry welcomes new construction contract
Two senior promotions at Grant Thornton
Advice for businesses on secondary pensions
Thirty two personal breaches were reported to The Office of the Data Protection Authority (ODPA) in the two months up to 26 August 2019.
Eighteen of the breaches were due to personal data being sent, via email or post, to the wrong person. The remaining fourteen were through criminal activity, hacking, personal data being accessed inappropriately, the disclosure of personal data when not authorised to do so, or personal data being lost.
Emma Martins, Data Protection Commissioner, said there was a human aspect to personal data breaches.
"What is striking from this period's statistics is that all the breaches reported to us were due to human action, whether deliberate or accidental," she said.
"There was not a single incidence of system error. We must all recognise that it is people's awareness, attitudes, behaviour and choices that often pose the biggest risk to the protection of personal data, rather than our IT systems. Because of this, my office is laser-focused on raising everyone's appreciation and awareness of data protection, in the hope that we can create positive cultural change around how people think, and feel, about taking care of personal data."
Part of this awareness-raising is the ODPA's decision to take part in this year's Global Privacy Enforcement Network (GPEN) 'Privacy Sweep' for the first time. This international intelligence-gathering exercise examines a different theme each year and in 2019, the focus is on how data breach notifications are handled.
Mrs Martins added: "We will be contacting a sample of local organisations directly, asking them to respond to a short survey from GPEN later this month. Honest responses to the survey are encouraged, as it is only through honesty that an accurate snapshot of the challenges organisations face can be taken, from which we can all learn lessons. Building a culture of honest and constructive learning can help us all to work towards higher standards of compliance."
Pictured top: Emma Martins.
« Return to Business
Inflation up 0.1% on 2018

redpajama_set_name: RedPajamaCommonCrawl


text:
British Governors-competoid.com
EVENTS DURING GOVERNORS-GENERAL OF BRITISH INDIA
Competoid.com
Warren Hastings (1772-85)
Introduced Quinquennial settlement of land revenue in 1772 farming out lands to the highest bidder.
Later on he reversed annual to settlement (1777) on the basis of open auction to the highest bidder.
Made appointments of Collectors and other Revenue officials. Codified Hindu and Muslim laws.
Trial of Nand Kumar and his judicial murder in 1775.
Foundation of Asiatic Society of Bengal with the help of William Jones in 1784.
Dual Government was abolished
Associated with the Chait Singh (Banaras Raj) Affair 1778.
Alter his return to England in 1785, impeachment proceedings were initiated against him in the house of lords.
But after a long trial of 7 years he was eventually acquitted.
Dual Government was abolished by him in 1772.
Code of Gentoo Laws
In 1776, Manu's law was translated into English as Code of Gentoo Laws.
Creation of laws
In 1791, William Jones and Colebrook prepared Digest of Hindu Laws.
Fatwa-i-Alamgiri was also translated in 1772, created a Diwani and Faujdari Adalat.
The Regulating Act 1773, appointed him the first Governor General along with four counciIors-Clavering. Francis, Monson and Barwell.
First Anglo Maratha War took place.
Wilkins translated Gila and Hitopadesa in English. In 1780, James August Hickey started a weekly paper called Bengal Gazette or Calcutta General Advertiser.
Syed Ghulam Hussain wrote Siyar-uI-Mutqherin.
Lord Cornwallis (1786-93)
In 1791.Sanskrit College in Benaras, founded by Jonathan Duncan.
The Permanent Settlement introduced in 1793. The police System was Introduced.
The district Faujdari Adalats presided over abolished and in their place four Circuit Courts were established presided by European covenanted servants.
The Sadr Nizamat Adalat also replaced. Cornwallis code was introduced.
Lord Wellesley (1798-1805)
Described himself a Bengal tiger.
Created the Madras Presidency after the annexation of the Kingdoms of Tanjore and Carrratie. Introduced the System of Subsidiary Alliance.
The States that signed the alliance were first Hyderabad in 1798 and then Mysore, Tanjore, Awadh, Peshwa, Bhonsle, Srndhia, Jodhpur, Jaipur. Mceheri. Bundi, Bharatpur, Berar.
Signed treaty of Bassein with the Peshwa and fought Second Angl-Maratha War.
Lord Minto (1807-13)
Signed treaty of Amritsar in 1809 between Ranjit Singh and the English.
Sent in 1808. Mission Malcom to Persia and that of Eliphinston to Kabul.
Lord Hastings (1813-24)
He was made Marquess of Hastings due to his success in the Gorkha war of the Anglo Nepalese war.
Signed treaty of Sagauli in 1816 annexed his territories in the Bombay Presidency after the third Anglo-Maratha War (1818).
Introduction of Ryotwari Settlement in Madras Presidency by governor, Thomas Munro (1820). Mahalwari (Village Community) System of land revenue was made in North-West province by James Thomson.
Lord Amherst (1824-28)
Received by the Mughal emperor (Akbar II) on terms of equality in 1827.
Signed treaty of Yandaboo in 1826 with lower Burma or Pegu by which British merchants were allowed to settle in southern coast of Burma and Rangoon.
Lord Willian Bentinck (1828-35)
First Governor General of India by Government of India Act 1833. Known as benevolent Governor General.
Banned practice of sati in 1829,suppressed Thugs in 1830 (military operation led by William Sleeman).
Banned female infanticide. Created the Province of Agra in 1834.
Land Revenue Settlement in North
English was to be Court language In Higher Court but Persian continued in Lower Courts.
The regulation of 1833 on land revenue settlement by Mertins Bird (called father of Land Revenue Settlement in North).
Use of fields maps and filed registers were prescribed for the first time.
Inquiry into titles of Inam lands 1830 annexation of Cachar took place
Sir Charles Metcalfe (1835-36)
Abolished restriction on press.
He was called 'Liberator of Press.
Lord Auckland (1836-42)
Tripartite Treaty was signed between the company, Ranjit Singh and Shah Shuja.
First Afghan War started Mandavi State was annexed.
Lord Ellenborough (1842-44)
Annexation of Sind to British Empire (1843).
Charles Napier was replaced by Major Outram as the President in Sind.
Lord Dalhousie (1848-56)
Introduced the policy of ' Doctrine of Lapse' or 'Law of Escheat' which postulated that Indian states having no natural heir would be annexed to the British Empire.
The Indian states thus annexed were Satara (1848) Jaitpur and Sambhalpur (1849), Baghat(1850), Udaipur(1852), Jhansi (1853)and Nagpur(1854).
Introduced Wood's dispatch known as the Magna Carta of English education in India prepared by Charles Wood the President of the Board of Control in 1854.
Development of railways-laid
Boosted up the development of railways-laid the first railway line in 1853 from Bombay to Thane and second from Calcutta to Raniganj.
Gave a great impetus to Post and Telegraph.
Telegraph lines were laid (first line from Calcutta to Agra).
Organised a separate Public Works Department by divesting the military board of this power.
Lord Canning (1856-62)
Revolt of 1857,Queen Victoria's Proclamation and passing of the Indian Act of 1858.
Doctrine of Lapse started by Lord Dalhousie was withdrawn in 1859.
Foundation of the Universities of Calcutta Bombay and Madras in 1857.
Indigo revolt in Bengal in 1859-60.
Lord Elgin (1862-63)
Supressed the Wahabi Movement Famine Commission was constituted.
Followed a policy of rigid non interference in Afghanistan called ' Palicy of Masterly inactivity".
Set up High Courts in Calcutta Bombay and Madras (1865).
Lord Mayo (1864-72)
Wahabi and Kuka Movement was active.
Mayo was murdered.
Lord Lytton (1876-80)
Passing of the Royal Title Act of 1876 and the assumption of the title empress on India ( Qaiser-i-hind) by Queen Victoria.
Holding of Delhi Durbar in 1877.
Passing of the Arms Act of 1878.
Policy of Masterly inactivity towards Afghanistan Was replaced by forward policy.
Lord Ripon (1880-84)
Passing of the First Factory Act in 1881 for the welfare of child labour.
Repeal of Vernacular Press Act in 1882.
Foundation of Local self government ( 1882), passing of Local self government acts in various Provinces during the period 1883-84.
Holding of first decennial and regular census in 1881, which put the total population at 254 million.
Lord Dufferin (1884-88)
Foundation of Indian National Congress.
Third Anglo-Burmese war. In 1886, upper Burma was annexed. Aitchison Committee.
Commented on the Congress being a microscopic minority.
Lord Lansdowne (1888-94)
Passed the Age of Consent Act in 1891 which forbade marriage of gut below 12.
In 1891, Second Factory Act was passed.
In 1892, Indian Council Act passed.
In 1893, Dufand Mission was sent to Kabul.
Lord Elgin II (1894-99)
Famine in part of Rajasthan-Lyall Commission appointed to look into the cause of famines.
Lord Curzon (1899-1905)
Creation of a new Province called the North-West Frontier Province.
Appointment of Police Commission under Sir Andrew Frazer to inquire into the Police Administration in 1902-03.
It recommended enhancement in salaries and creation of a Department of Criminal Intelligence.
Appointment of Universities Commission in 1902 under Sir Thomas Releigh and passing of Indian Universities Act 1904.
Lord Minto ll (1905-10)
S.P. Sinha was appointed a member of Governor General's Council Act of 1909.
In 1906, Arundale Committee on Political reforms submitted its report.
Lord Hardinge (1910-16)
Coronation durbar in 1911 at Delhi in honor of George V. A separate state of Bihar and Orissa was created in 1911.
A bomb was thrown as he was entering Delhi at Chandini Chowk.
Capital Shifted to Delhi in 1911, In 1912 Islington Committee on Civil Services.
Lord Reading (1921-26)
Foundation of Rashtriya Swayam Sevak Sangh by K B. Hedge var at Nagpur in 1925.
Beginning of Indianisation of the officers cadre of the Indian Army.
Railway Budget was separated from General Budget in 1921.
Lord Irwin (1926-31)
Popularly known as Christian Viceroy.
Appointment of the Indian States Commission under Harcourt Butler ( 1927) to recommend measures for the establishment of better relations between the Indian States and the Central government.
Royal Commission on Agriculture constituted in 1927.
Deepavali declaration (1929) that India would be granted dominion status in due Course.
Gandhi leaves for London to participate in Second Round Table Conference.
Lord Willingdon (1931-36)
Communal award in 1932.
Third Round Table Conference in 1932. White Paper on political reforms in India was published in 1933.
Burma and Aden were separated from the British Empire in 1935. Orissa, Bihar and Sind were made new States.
Lord Linlithgow (1936-43)
August offer by the Viceroy in which he declared the dominion status as the ultimate goal of British policy in India.
At its Haripura Session (1939), Congress declared Poorna Swaraj, ideal to cover native states and British India.
In 1940, individual Civil Disobedience movement.
In 1942, Cripps Mission came. C. Raj gopalchan formula proposed.
Lord Wavell (1943-47)
Wavell Plan, Shimla Conference.
Congress represented by Maulana Azad. RIN mutiny (1946).
Interim Government has formed (September 2, 1946).
Prime Minister of Britian Clement Atlee announced independence of India by June 1948 (February 20,1947) Cabinet mission (1946). 16th August, 1946, Muslim League begins 'Direct Action Day'.
Lord Mountbatten (1947-48)
Earlier proposed plan Balkhan i.e., total dismemberment of the Indian territory, but soon left the idea.
Made it clear to the princely States that they shall not be granted separate independence and they will have to join either India or Pakistan.
His plan to make India free on August 15, 1947 is also known as June 3rd plan.
C. Rajagopalchari (1948-50)
Fitst Indian Governor General.
British Governors
This quiz has been prepared with questions related to British Governors. These question were asked in various competitive exams across India. You can practice these questions to gain more knowledge.
Start Quiz Next Slide


Oppo A71 (2018) smartphone was launched in January 2018. This cellphone comes with a 5.20-inch touchscreen display with a resolution of 720 pixels via 1280 pixels.
The Oppo A71 (2018) is powered by means of 1.8GHz octa-core Qualcomm Snapdragon 450 processor and it comes with 2GB of RAM. The phone packs 16GB of inner storage that can be accelerated up to 256GB via a microSD card. As a long way because the cameras are involved, the Oppo A71 (2018) comes with a thirteen-megapixel number one digital camera on the rear and a five-megapixel front shooter for selfies.
The Oppo A71 (2018) runs Android 7.1 and is powered by using a 3000mAh. It measures 148.10 x 73.80 x 7.60 (top x width x thickness) and weigh 137.00 grams.
The Oppo A71 (2018) is a twin SIM (GSM and GSM) phone that accepts Nano-SIM and Nano-SIM. Connectivity alternatives include wi-fi, Bluetooth, USB OTG, FM, 3G and 4G. Sensors at the smartphone encompass Compass Magnetometer, Proximity sensor, Accelerometer and Ambient mild sensor.


Looking for Nevada Commercial Real Estate? Search these Nevada Commercial Property listing services and resources to find exactly what you're looking for!
Summerlin Las Vegas real estate, homes, commercial, high rise condo,second homes, investment properties.
AllPages.com Yellow Pages provides business listings (name, address, phone, fax) for all categories covered under Commercial & Industrial, Nevada (NV).
Design & Consulting Services, Inc.
The blog covers a wide variety of topics related to real estate. It provides tips for buying and selling residential and commercial properties including homes, offices, industrial property, condominium, hotels, malls, retail stores, farm land, warehouses and more. To know more, visit http://realestatemonitor.net/ and http://realestatemonitor.net/category/buy-and-sell/.
Online Las Vegas Real Estate resource for buyers, sellers, and investors. Including home search of Las Vegas, North Las Vegas, Henderson, Summerlin, Green Valley, and Northwest Las Vegas.


Cookouts, Picnics, Parties and more, summer is prime time for fun get-togethers with friends and family. Mojitos, Margaritas, Spiked Lemonades and Teas: if you are looking for a tasty, delicious, summertime cocktail recipe, try on of the 25 outstanding summer libations listed here!
Summertime is a great time to enjoy a tall, cool, adult libation! Whether sitting on the deck, lounging in the pool, partying with friends, or simply enjoying a fun, refreshing cocktail after a hard day at work, summertime cocktails can really hit the spot. Unique in that they are generally cooler, generally fruitier, a summer cocktail sure can hit the spot on a hot day.
Have I got a lot of cocktail recipes for you! If what is listed is not "quite" what you are seeking, take a look at my Cocktails, Wine and Liquor page for even more delightful adult beverages to enjoy!


text:
Butterfield may refer to:

 Butterfield (surname)
 Butterfield Market

Places

 Butterfield, Dublin, a suburb and townland of Dublin, Ireland
 Butterfield Green, Luton, England

United States
 Butterfield, Minnesota
 Butterfield, Missouri
 Butterfield, Texas
 Butterfield Township, Michigan
 Butterfield Township, Minnesota
 Butterfield, Illinois

Fiction
 Butterfield, a fictional town in Kansas in L. Frank Baum's 1909 novel The Road to Oz
 Butterfield, a fictional butler of Watkyn Bassett in Wodehouse's Jeeves stories
 Brian Butterfield, overweight salesman character from The Peter Serafinowicz Show

Other uses
 Butterfield Bank, an international bank based in Bermuda
 Butterfield & Butterfield, a large auction house based in San Francisco
 BUtterfield 8, a 1960 film with Elizabeth Taylor and Laurence Harvey based on a novel of the same name by John O'Hara
 Butterfield Overland Mail, a stagecoach service in the United States operating from 1858 to 1861
 Butterfield Overland Mail, a.k.a. Butterfield Stage, Stagecoach service in the US (1858–1861)
Butterfield station, a light rail station in Sacramento, California, US
 Butterfield and Swire, a British trading company from the 1860s to 1960s

redpajama_set_name: RedPajamaWikipedia


1. Trying to write today's post I find I can't not mention Mark E. Smith, leader singer / head honcho of The Fall. As you may have read, he died yesterday at the age of 60. Others have described him and his influence far better than I ever could. I'm glad I got to see The Fall play at Bodega back in 2015. They are kind of special, by which I mean they're quite something.
2. My friend Nik wrote about my work Slipped Monochrome #2. As he said to me, it took his writing to an unexpected place. Find out where over here.
5. These paintings by Ryan Crotty look rather nice. Whoever took the photos has done a great job. (And for you sports fans, no, not that Ryan Crotty).


I've been using Matomo on basic web projects, but I'm now on a project where I would like to monitor data transmitted by IoT devices, i.e : the number of people in a room or the number of people who used a door.
I'm not sure how to proceed with the HTTP API (all my data is already redirected to a backend).
Any idea on how to realize something like this ?


Resurrection - Sin:ergy Vs PureFilth!
I write this with a heavy heart. This will be the last ever gig I play. You all know my ears are really damaged from work and music and the time has come to stop.
I want to say a big thank you to everyone who has supported me over the last 12 years. I hope you can join me for one last time.
Expect the biggest line up I've put on since I began promoting in 2008.
CELL 200 which is part of the EGG. Has a funktion one sound system and that underground feel us hard housers love.