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+\documentclass{kiss}
+\usepackage{presentation}
+\usepackage{header}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf
+\large
+\hfil Strong-coupling renormalization group\par
+\smallskip
+\hfil in a 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}{\tt1506.04381\par1507.05678}\hfill{\tt http://ian.jauslin.org/}
+\eject
+
+\pagestyle{plain}
+\setcounter{page}{1}
+
+
+\title{Kondo model}
+\begin{itemize}
+\item [P.~Anderson, 1960], [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 Magnetic susceptibility: response to a magnetic field $h$:
+$$
+\chi(h,\beta):=\partial_hm(h,\beta).
+$$
+($m(h,\beta)$: magnetization).
+\item Isolated impurity:
+$$
+\chi^{(0)}(0,\beta)=\beta\mathop{\longrightarrow}_{\beta\to\infty}\infty
+$$
+\item Chain of electrons: Pauli paramagnetism:
+$$
+\lim_{\beta\to\infty}\lim_{L\to\infty}\frac1L\chi_e(0,\beta)<\infty.
+$$
+\end{itemize}
+\eject
+
+\title{Kondo effect: magnetic susceptibility}
+\begin{itemize}
+\item Turn on the interaction: $\lambda_0\neq0$. Impurity susceptibility $\chi^{(\lambda_0)}(h,\beta)$.
+\item Ferromagnetic interaction ($\lambda_0>0$):
+$$
+\lim_{\beta\to\infty}\chi^{(\lambda_0)}(0,\beta)=\infty.
+$$
+\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 [J.~Kondo, 1964]: third order Born approximation.
+\vskip0pt plus3fil
+\item [P.~Anderson, 1970], [K.~Wilson, 1975]: renormalization group approach
+\itemptchange{$\scriptstyle\blacktriangleright$ }
+\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 (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}: 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{Field theory for the Kondo model}
+\begin{itemize}
+\item By introducing an extra dimension ({\it imaginary time}), the partition function $Z:=\mathrm{Tr}(e^{-\beta H})$ can be expressed as the {\it Gaussian} average over a {\it Grassmann} algebra:
+$$
+Z=\mathrm{Tr}\left<e^{-\int_0^\beta dt\ \mathcal V(t)}\right>
+$$
+\vskip-10pt
+\item Potential:
+$$
+\mathcal V(t)=-\lambda_0\kern-10pt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}}\kern-10pt
+\psi^+_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(t)\tau^j
+$$
+with $\{\psi^\pm_\alpha(t),\psi^\pm_{\alpha'}(t')\}=0$.
+\item $\left<\cdot\right>$ is defined by its second moment $\left<\psi^-_{\alpha_1}(t_1)\psi^+_{\alpha_2}(t_2)\right>$.
+\end{itemize}
+\eject
+
+\title{Hierarchical model}
+\begin{itemize}
+\item Replace $\psi_\alpha^\pm(t)$ in $\mathcal V(t)$ by
+$$
+\psi_\alpha^\pm(t):=\sum_{m\leqslant0}\psi_\alpha^{[m]\pm}(t)
+$$
+where $\psi_\alpha^{[m]\pm}(t)$ is {\it constant} over the ``time'' intervals $\Delta_{i,\pm}^{[m]}$:\par
+\vfil
+\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxes.pdf}\par
+\item There are 4 fields in each $\Delta_{i,\pm}^{[m]}$.
+\item Moments:
+$$
+\left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m
+$$
+\end{itemize}
+\eject
+
+\title{Full propagator}
+\begin{itemize}
+\item Moments:
+$$
+\left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m
+$$
+\item Full propagator:
+$$
+\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t')
+$$
+\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxtree.pdf}\par
+\vfil
+\eject
+
+\title{Comparison with the Kondo model}
+\item Hierarchical model:
+$$
+\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t')
+$$
+\item For the (non-hierarchical) Kondo model:
+$$
+\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>\approx\sum_m 2^{m}g_\psi^{[0]}(2^m(t-t'))
+$$
+where $g^{[0]}_\psi$ is odd and decays faster than any power.
+\end{itemize}
+\eject
+
+\title{Hierarchical beta function}
+\begin{itemize}
+\item Compute $Z$ by $\mathcal V^{[0]}(t):=\mathcal V(t)$
+$$
+e^{-\int dt\ \mathcal V^{[m-1]}(t)}:=\left<e^{-\int dt\ \mathcal V^{[m]}(t)}\right>_m
+$$
+
+\item Effective potential:
+$$
+\int dt\ \mathcal V^{[m]}(t)=\sum_{i=1}^{2^{-m}}\mathcal V^{[m]}_{i,-}+\mathcal V^{[m]}_{i,+}
+$$
+\item Iteration
+$$
+\left<e^{-\int dt\ \mathcal V^{[m]}(t)}\right>_m=\prod_{i=1}^{2^{-m}}\left<e^{-(\mathcal V^{[m]}_{i,-}+\mathcal V_{i,+}^{[m]})}\right>_m
+$$
+\item By anti-commutation of the fields, $e^{-(\mathcal V_{i,-}^{[m]}+\mathcal V_{i,+}^{[m]})}$ is a polynomial in the fields of order $\leqslant 8$.
+\end{itemize}
+\eject
+
+\title{Hierarchical beta function}
+\begin{itemize}
+\item $\mathcal V^{[m]}$ is parametrized by 2 real numbers ({\it running coupling constants}) $\ell_0^{[m]},\ell_1^{[m]}$:
+$$\begin{array}{r@{\ }>{\displaystyle}l}
+\frac{e^{-\scriptstyle\int dt\ \mathcal V^{[m]}(t)}}{C^{[m]}}
+=&1+\frac{\ell_0^{[m]}}2\int dt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}}
+\psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\tau^j\\[0.5cm]
+&+\frac{\ell_1^{[m]}}2\int dt\left(\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}}
+\psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\right)^2
+\end{array}$$
+\end{itemize}
+\eject
+
+\title{Hierarchical beta function}
+\begin{itemize}
+\item Beta function ({\it exact})
+$$\begin{array}{r@{\ }l}
+C^{[m]}=&\displaystyle1+ \frac32(\ell_0^{[m]})^2+9(\ell_1^{[m]})^2\\[0.3cm] \ell_0^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\ell_0^{[m]} +3 \ell_0^{[m]}\ell_1^{[m]} -(\ell_0^{[m]})^2\Big)\\[0.5cm]
+\ell_1^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\frac12\ell_1^{[m]}+\frac18(\ell_0^{[m]})^2\Big)
+\end{array}$$
+\end{itemize}
+\eject
+
+\title{Flow}
+\vfil
+\hfil\includegraphics[width=0.8\textwidth]{Figs/sd_phase.pdf}\par
+Fixed points: 0 (stable), $\bm\ell^*$ (marginal in $\ell_0$ and stable in $\ell_1$)
+\eject
+
+\title{Susceptibility}
+\begin{itemize}
+\item Add magnetic field $h$ on the impurity.
+\item New term in the potential:
+$$
+-h \sum_{j\in\{1,2,3\}}\bm\omega_j \tau^j
+$$
+\item 6 running coupling constants.
+\item The susceptibility can be computed by deriving $C^{[m]}$ with respect to $h$.
+\end{itemize}
+\eject
+
+\title{Kondo effect}
+\begin{itemize}
+\item Fix $h=0$.
+\item At $0$, the susceptibility diverges as $\beta$.
+\item At $\bm\ell^*$, the susceptibility remains finite in the $\beta\to\infty$ limit.
+\end{itemize}
+\eject
+
+\title{Susceptibility}
+\begin{itemize}
+\item $\lambda_0=-0.28$
+\end{itemize}
+\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_28.pdf}\par
+\eject
+
+\addtocounter{page}{-1}
+\title{Susceptibility}
+\begin{itemize}
+\item $\lambda_0=-0.02$
+\end{itemize}
+\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_02.pdf}\par
+\eject
+
+\addtocounter{page}{-1}
+\title{Susceptibility}
+\begin{itemize}
+\item $\lambda_0=-0.005$
+\end{itemize}
+\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_005.pdf}\par
+\eject
+
+\title{Open questions}
+\begin{itemize}
+\item Magnetic field on the chain as well. This requires defining the hierarchical model to reflect the $x$-dependence of $\psi(x,t)$.
+\item Rigorous renormalization group analysis for the Kondo model (non-hierarchical).
+\item The exact solvability of the hierarchical Kondo model is merely a consequence of the fermionic nature of the system. Other fermionic hierarchical models can be studied to investigate other strong-coupling phenomena, e.g. high-$T_c$ superconductivity.
+\end{itemize}
+\eject
+
+\title{Epilogue: {\tt meankondo}}
+\begin{itemize}
+\item The computation in the $h$-dependent case requires computing many Feynman diagrams ($\approx100$).
+\item Software to perform the computation: {\tt meankondo}.
+\item {\tt meankondo} can be configured to study any fermionic hierarchical model.
+\end{itemize}
+\hfil{\tt http://ian.jauslin.org/software/meankondo/}
+
+\end{document}
+
+
+
+
+