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\hfil Non-perturbative renormalization group\par
\smallskip
\hfil in a hierarchical Kondo model\par
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\hfil Ian Jauslin
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\vfil
\small
\hfil joint with {\normalsize\bf G.~Benfatto} and {\normalsize\bf G.~Gallavotti}\par
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arXiv: {\tt1506.04381}\hfill{\tt http://ian.jauslin.org/}
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\title{Kondo model}
\begin{itemize}
\item [N.~Andrei, 1980]:
$$
H=H_0+V
$$
\itemptchange{$\scriptstyle\blacktriangleright$}
\begin{itemize}
\item $H_0$: kinetic term of the {\it electrons}
$$
H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c_\alpha(x)
$$
\item $V$: interaction with the {\it impurity}
$$
V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\, d^\dagger_{\alpha_3}\sigma^j_{\alpha_3,\alpha_4}d_{\alpha_4}
$$
\end{itemize}
\itemptreset
\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
\itemptchange{$\scriptstyle\blacktriangleright$}
\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 Non-perturbative} effect: the qualitative behavior changes as soon as $\lambda_0\neq0$.
\end{itemize}
\eject

\title{Previous results}
\begin{itemize}
\item [J.~Kondo, 1964]: third order Born approximation.
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\item [P.~Anderson, 1970], [K.~Wilson, 1975]: renormalization group approach
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\begin{itemize}
\item Sequence of effective Hamiltonians at varying energy scales.
\item For anti-ferromagnetic interactions, the effective Hamiltonians go to a {\it non-trivial fixed point}.
\end{itemize}
\itemptreset
\item{\tt Remark}: [N.~Andrei, 1980]: the Kondo model (suitably linearized) is exactly solvable via Bethe Ansatz.
\end{itemize}
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\title{Current results}
\begin{itemize}
\item Hierarchical Kondo model: idealization of the Kondo model that shares its scaling properties.
\item It is {\it exactly solvable}: the map relating the effective theories at different scales is {\it explicit} (no perturbative expansions).
\item For $\lambda_0<0$, the flow tends to a non-trivial fixed point, and $\chi^{(\lambda_0)}<\infty$ in the $\beta\to\infty$ limit (Kondo effect).
\end{itemize}
\eject

\title{Hierarchical beta function} \begin{itemize} \item Beta function ({\it exact})
$$\begin{array}{r@{\ }l}
C^{[m]}=&\displaystyle1+ 3(\ell_0^{[m]})^2+9(\ell_1^{[m]})^2+9(\ell_2^{[m]})^2+324(\ell_3^{[m]})^2\\[0.3cm]
\ell_0^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\ell_0^{[m]} +18\ell_0^{[m]}\ell_3^{[m]}+3 \ell_0^{[m]}\ell_2^{[m]}+3 \ell_0^{[m]}\ell_1^{[m]} -2(\ell_0^{[m]})^2\Big)\\[0.5cm]
\ell_1^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big( \frac12\ell_1^{[m]}+9\ell_2^{[m]}\ell_3^{[m]} +\frac14(\ell_0^{[m]})^2\Big)\\[0.5cm]
\ell_2^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(2\ell_2^{[m]}+36\ell_1^{[m]}\ell_3^{[m]}+ (\ell_0^{[m]})^2\Big)\\[0.5cm]
\ell_3^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\frac12\ell_3^{[m]}+\frac14\ell_1^{[m]}\ell_2^{[m]}+\frac1{24} (\ell_0^{[m]})^2\Big).
\end{array}$$
\end{itemize}
\eject

\title{Flow}
\vfil
\hfil\includegraphics[width=0.8\textwidth]{Figs/beta_phase_half.pdf}\par
Fixed points: $\bm\ell^{(0)}$, $\bm\ell^{(+)}$, $\bm\ell^*$.
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\title{Kondo effect}
\begin{itemize}
\item Fix $h=0$.
\item At $\bm\ell^{(+)}$, the susceptibility diverges as $\beta$.
\item At $\bm\ell^*$, the susceptibility remains finite in the $\beta\to\infty$ limit.
\end{itemize}
\hfil\includegraphics[width=150pt]{Figs/susc_plot_temp.pdf}\par

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