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\begin{document}
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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: {\tt1506.04381}\hfill{\tt http://ian.jauslin.org/}
\eject

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\title{Kondo model}
\begin{itemize}
\item s-d model: [P.~Anderson, 1960] [J.~Kondo, 1964]:
\itemptchange{$\scriptstyle\blacktriangleright$ }
\begin{itemize}
\item 1D chain of non-interacting spin-1/2 fermions: {\it electrons}.
\item lone spin-1/2 fermion: {\it impurity}.
\item the impurity interacts with the electron at 0.
\end{itemize}
\itemptreset
\end{itemize}
\vskip0ptplus3fil
\hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par
\vskip0ptplus3fil
\eject

\title{Kondo Hamiltonian}
\vskip5pt
\hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par
$$
H=H_0+V
$$
\begin{itemize}
\item $H_0$: kinetic term of the {\it electrons}
$$
H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c_\alpha(x)
$$
\itemptchange{$\scriptstyle\blacktriangleright$ }
\begin{itemize}
\item $c_\alpha(x)$: fermionic annihilation operator
\item $\alpha$: spin
\item $x$: site
\end{itemize}
\itemptreset
\end{itemize}
\eject


\title{Kondo Hamiltonian}
\vskip5pt
\hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par
$$
H=H_0+V
$$
\begin{itemize}
\item $V$: interaction with the {\it impurity}
$$
V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\, d^\dagger_{\alpha_3}\sigma^j_{\alpha_3,\alpha_4}d_{\alpha_4}
$$
\itemptchange{$\scriptstyle\blacktriangleright$ }
\begin{itemize}
\item $d_\alpha$: fermionic annihilation operator
\item $\sigma^j$: Pauli matrix
\item $\lambda_0>0$: {\it ferromagnetic} case
\item $\lambda_0<0$: {\it anti-ferromagnetic} case \end{itemize}
\itemptreset
\end{itemize}
\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:
$$
\chi^{(0)}(0,\beta)=\frac\beta2\mathop{\longrightarrow}_{\beta\to\infty}\infty
$$
\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}.
\item Anderson: instability of the trivial fixed point ($H_0$).
\item Wilson: numerical diagonalization at each step, and perturbative expansions around the trivial and non-trivial fixed points.
\end{itemize}
\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 With $\lambda_0<0$, the flow tends to a non-trivial fixed point, and $\chi^{(\lambda_0)}<\infty$ in the $\beta\to\infty$ limit (Kondo effect).
\item{\tt Remark}: [Andrei, 1980]: the Kondo model (suitably linearized) is exactly solvable via Bethe Ansatz.
\end{itemize}
\eject

\title{Field theory for the Kondo model}
\begin{itemize}
\item Partition function $Z:=\mathrm{Tr}(e^{-\beta H})$.
\vskip5pt
\item By introducing an extra dimension ({\it imaginary time}), $Z$ can be expressed as the {\it Gaussian} average over a {\it Grassmann} algebra:
$$
Z=\left<e^{-\int_0^\beta dt\ \mathcal V(t)}\right>
$$
where
$$
\mathcal V(t)=-\lambda_0\kern-10pt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}}\kern-10pt
\psi^+_{\alpha_1}(0,t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(0,t)\, \varphi^+_{\alpha_3}(t)\sigma^j_{\alpha_3,\alpha_4}\varphi^-_{\alpha_4}(t)
$$
with $\{\psi^\pm_\alpha(0,t),\psi^\pm_{\alpha'}(0,t')\}=0$, $\{\varphi^\pm_\alpha(t),\varphi^\pm_{\alpha'}(t')\}=0$.
\end{itemize}
\eject

\title{Hierarchical fields}
\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxes.pdf}\par
\begin{itemize}
\item For each $m<0$, we introduce fields {\it on scale $m$} indexed by an interval:
$$
\psi_\alpha^\pm(\Delta_{i,\pm}^{[m]}),\quad
\varphi_\alpha^\pm(\Delta_{i,\pm}^{[m]})
$$
where
$$\begin{array}l
\Delta_{i,-}^{[m]}:=[2^{-m}i,2^{-m}(i+\frac12))\\[0.3cm]
\Delta_{i,+}^{[m]}:=[2^{-m}(i+\frac12),2^{-m}(i+1))
\end{array}$$
\item There are 8 fields in each $\Delta_{i,\pm}^{[m]}$.
\end{itemize}
\eject

\title{Hierarchical fields}
\begin{itemize}
\item Split fields over scales:
$$
\psi_\alpha^\pm(t):=\sum_{m\leqslant0}\psi_\alpha^{[m]\pm}(\Delta^{[m]}(t)),\quad
\varphi_\alpha^\pm(t):=\sum_{m\leqslant0}\varphi_\alpha^{[m]\pm}(\Delta^{[m]}(t))
$$
\end{itemize}
\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxtree.pdf}\par
\vfil
\eject

\title{Hierarchical propagators}
\begin{itemize}
\item Moments:
$$\begin{array}{r@{\ }l}
\left<\psi_\alpha^{[m]-}(\Delta_{i,-\eta}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\eta}^{[m]})\right>=&\eta 2^m\\[0.3cm]
\left<\varphi_\alpha^{[m]-}(\Delta_{i,-\eta}^{[m]})\varphi_\alpha^{[m]+}(\Delta_{i,\eta}^{[m]})\right>=&\eta
\end{array}$$
\item Full propagator:
$$
\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>=\mathrm{sign}(t-t') 2^{m_{t,t'}},\quad
\left<\varphi_\alpha^{-}(t)\varphi_\alpha^{+}(t')\right>=\mathrm{sign}(t-t')
$$
\item For the (non-hierarchical) Kondo model:
$$\begin{array}{r@{\ }l}
\left<\psi_\alpha^{-}(0,t)\psi_\alpha^{+}(0,t')\right>\approx&\sum_m 2^{m}g_\psi^{[0]}(2^m(t-t')),\\[0.3cm]
\left<\varphi_\alpha^{-}(t)\varphi_\alpha^{+}(t')\right>\approx&\sum_m g^{[0]}_\varphi(2^m(t-t'))
\end{array}$$
\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, $e^{-\mathcal V_{i,\pm}^{[m]}}$ is a polynomial in the fields of order $\leqslant 8$.
\end{itemize}
\eject

\title{Hierarchical beta function}
\begin{itemize}
\item The computation of the beta function reduces to computing the average of a degree-$16$ polynomial.
\item 4 running coupling constants $\ell_0,\cdots,\ell_3$:
$$
e^{-\int dt\ \mathcal V^{[m]}(t)}
=1+\sum_{i,\eta}\sum_{n=0}^3\ell_n^{[m]}O_{n}^{[\leqslant m]}(\Delta_{i,\eta})
$$
\end{itemize}
\eject

\title{Hierarchical beta function}
\begin{itemize}
\item Beta function ({\it exact})
$$\begin{array}{r@{\ }l}
C^{[m]}=&\displaystyle1+ 3\ell_0^2+9\ell_1^2+9\ell_2^2+324\ell_3^2\\[0.3cm]
\ell_0^{[m-1]}=&\displaystyle\frac1C\Big(\ell_0 +18\ell_0\ell_3+3 \ell_0\ell_2+3 \ell_0\ell_1 -2\ell_0^2\Big)\\[0.5cm]
\ell_1^{[m-1]}=&\displaystyle\frac1C\Big( \frac12\ell_1+9\ell_2\ell_3 +\frac14\ell_0^2\Big)\\[0.5cm]
\ell_2^{[m-1]}=&\displaystyle\frac1C\Big(2\ell_2+36\ell_1\ell_3+ \ell_0^2\Big)\\[0.5cm]
\ell_3^{[m-1]}=&\displaystyle\frac1C\Big(\frac12\ell_3+\frac14\ell_1\ell_2+\frac1{24} \ell_0^2\Big).
\end{array}$$
\end{itemize}
\eject

\title{Hierarchical beta function}
\begin{itemize}
\item Beta function ({\it exact})
$$\begin{array}{r@{\ }l}
C^{[m]}=&\displaystyle1+ 3\ell_0^2+9\ell_1^2+9\ell_2^2+324\ell_3^2\\[0.3cm]
\color{blue}\ell_0^{[m-1]}=&\color{blue}\displaystyle\frac1C\Big(\ell_0 +18\ell_0\ell_3+3 \ell_0\ell_2+3 \ell_0\ell_1 -2\ell_0^2\Big)\\[0.5cm]
\color{darkgreen}\ell_1^{[m-1]}=&\color{darkgreen}\displaystyle\frac1C\Big( \frac12\ell_1+9\ell_2\ell_3 +\frac14\ell_0^2\Big)\\[0.5cm]
\color{red}\ell_2^{[m-1]}=&\color{red}\displaystyle\frac1C\Big(2\ell_2+36\ell_1\ell_3+ \ell_0^2\Big)\\[0.5cm]
\color{darkgreen}\ell_3^{[m-1]}=&\color{darkgreen}\displaystyle\frac1C\Big(\frac12\ell_3+\frac14\ell_1\ell_2+\frac1{24} \ell_0^2\Big).
\end{array}$$
{\color{red}relevant}, {\color{blue}marginal}, {\color{darkgreen}irrelevant}
\end{itemize}
\eject

\title{Flow}
\vfil
\hfil\includegraphics[width=0.8\textwidth]{Figs/beta_phase.pdf}\par
Fixed points: $\bm\ell^{(0)}$, $\bm\ell^{(+)}$, $\bm\ell^*$, $\bm\ell^{(-)}$.
\eject

\title{Fixed points}
\begin{itemize}
\item $\bm\ell^{(0)}$: unstable.
\item $\bm\ell^{(+)}$: ferromagnetic ($\lambda_0>0$).
\item $\bm\ell^*$: anti-ferromagnetic ($\lambda_0<0$).
\end{itemize}
\eject

\title{Susceptibility}
\begin{itemize}
\item Add magnetic field $h$ on the impurity.
\item New term in the potential:
$$
-h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{\alpha,\alpha'}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha'} \varphi^-_{\alpha'}(t).
$$
\item 9 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 $\bm\ell^{(+)}$, 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
\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 100 Feynman diagrams.
\item By adding the field on the entire chain (open problem), this number increases to 1089.
\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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