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\hfil Strong-coupling renormalization group\par
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\hfil in the hierarchical Kondo model\par
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\hfil Ian Jauslin
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\hfil joint with {\normalsize\bf G.~Benfatto} and {\normalsize\bf G.~Gallavotti}\par
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arXiv: \parbox[b]{1cm}{\tt\href{http://arxiv.org/abs/1506.04381}{1506.04381}\par\href{http://arxiv.org/abs/1507.05678}{1507.05678}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org/}}
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\title{Kondo model}
\begin{itemize}
\item \href{http://dx.doi.org/10.1103/PhysRev.124.41}{[P. Anderson, 1961]}, \href{http://dx.doi.org/10.1143/PTP.32.37}{[J. Kondo, 1964]}:
$$
H=H_0+V\quad\mathrm{on\ }\mathcal H=\mathcal F_L\otimes\mathbb C^2
$$
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\begin{itemize}
\item $H_0$: kinetic term of the {\it electrons}
$$
H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(\left(-\frac{\Delta}2-1\right)\,c_\alpha\right)(x)\otimes\mathds1
$$
\item $V$: interaction with the {\it impurity}
$$
V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\otimes \tau^j
$$
\end{itemize}
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\end{itemize}
\hfil\includegraphics[width=0.8\textwidth]{figs/kondo_model.pdf}\par
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\title{Kondo effect: magnetic susceptibility}
\begin{itemize}
\item Non-interacting magnetic susceptibility
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\begin{itemize}
\item Isolated impurity: $\chi^{(0)}(0,\beta)\displaystyle\mathop{\longrightarrow}_{\beta\to\infty}\infty$
\item Chain of electrons: $\displaystyle\lim_{\beta\to\infty}\lim_{L\to\infty}\frac1L\chi_e(0,\beta)<\infty.$
\end{itemize}
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\item Anti-ferromagnetic interaction: $\lambda_0<0$:
$$
\lim_{\beta\to\infty}\chi^{(\lambda_0)}(0,\beta)<\infty.
$$
\item {\it Strong-coupling} effect: the qualitative behavior changes as soon as $\lambda_0\neq0$.
\end{itemize}
\eject

\title{Previous results}
\begin{itemize}
\item \href{http://dx.doi.org/10.1143/PTP.32.37}{[J. Kondo, 1964]}: third order Born approximation.
\item \href{http://dx.doi.org/10.1088/0022-3719/3/12/008}{[P. Anderson, 1970]}, \href{http://dx.doi.org/10.1103/RevModPhys.47.773}{[K. Wilson, 1975]}: renormalization group approach
\item{\tt Remark}: \href{http://dx.doi.org/10.1103/PhysRevLett.45.379}{[N.~Andrei, 1980]}: the Kondo model (suitably linearized) is exactly solvable via Bethe Ansatz (which breaks down under any perturbation of the model).
\end{itemize}
\eject

\title{Current results}
\begin{itemize}
\item Hierarchical Kondo model: idealization of the Kondo model that has the same scaling properties.
\item It is {\it exactly solvable}: reduces the system to a 2-dimensional discrete dynamical system.
\item Kondo effect in the hierarchical model.
\end{itemize}
\hfil\includegraphics[width=150pt]{figs/sd_susc_0_28.pdf}\par
\eject

\title{Open problem}
\begin{itemize}
\item Usual approach to the Renormalization group: perturb around the uncoupled theory.
\item Cannot access strongly-coupled effects.
\item Idea: perturb around hierarchical models.
\item How? Which hierarchical models?
\end{itemize}

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