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diff --git a/Jauslin_IAS_2016.tex b/Jauslin_IAS_2016.tex new file mode 100644 index 0000000..0ebea16 --- /dev/null +++ b/Jauslin_IAS_2016.tex @@ -0,0 +1,93 @@ +\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} |