From f16e2fa90700e12926c6b55f78b8309ac6888de3 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@roma1.infn.it>
Date: Wed, 21 Sep 2016 16:45:13 +0000
Subject: As presented at the IAS on 2016-09-21

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+\documentclass{kiss}
+\usepackage{presentation}
+\usepackage{header}
+\usepackage{toolbox}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf
+\large
+\hfil Strong-coupling renormalization group\par
+\smallskip
+\hfil in the hierarchical Kondo model\par
+\vfil
+\hfil Ian Jauslin
+\rm
+\normalsize
+
+\vfil
+\small
+\hfil joint with {\normalsize\bf G.~Benfatto} and {\normalsize\bf G.~Gallavotti}\par
+\vskip10pt
+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/}}
+\eject
+
+\pagestyle{plain}
+\setcounter{page}{1}
+
+\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]}:
+$$
+H=H_0+V\quad\mathrm{on\ }\mathcal H=\mathcal F_L\otimes\mathbb C^2
+$$
+\itemptchange{$\scriptstyle\blacktriangleright$}
+\begin{itemize}
+\item $H_0$: kinetic term of the {\it electrons}
+$$
+H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(\left(-\frac{\Delta}2-1\right)\,c_\alpha\right)(x)\otimes\mathds1
+$$
+\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}
+\itemptreset
+\end{itemize}
+\hfil\includegraphics[width=0.8\textwidth]{figs/kondo_model.pdf}\par
+\eject
+
+\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}
+\itemptreset
+\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}
+
+\end{document}
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