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\hbox{}\vfil
\bf
\large
\hfil Non-perturbative 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

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\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 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.
\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 non-perturbative 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/}

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