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\begin{document}
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\hbox{}\vfil
\bf
\large
\hfil The renormalization group\par
\smallskip
\hfil in the weak- and strong-coupling regimes
\vfil
\hfil Ian Jauslin
\rm
\normalsize

\vfil
\small
\hfil advised by {\normalsize\bf A.~Giuliani}\par
\vskip20pt
\hfil{\tt http://ian.jauslin.org/}
\eject


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\title{Outline}
\vfill
\begin{itemize}
\item Weak coupling: {\bf Bilayer graphene}
\vskip30pt
\item Strong coupling: {\bf Hierarchical Kondo model}
\end{itemize}
\vfill
\eject

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\hbox{}
\vfill
\hfil\vrule height0.5pt width0.75\textwidth\par
\vskip20pt
\hfil{\bf Bilayer graphene}\par
\vskip20pt
\hfil\vrule height0.5pt width0.75\textwidth\par
\vfill
joint with {\bf A.~Giuliani}\hfill arXiv:{\tt 1507.06024}
\eject
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\title{Bilayer graphene}
\vfill
\includegraphics[width=0.8\textwidth]{Figs/bilayer.pdf}
\vfill
\eject

\title{Bilayer graphene}
\begin{itemize}
\item Hamiltonian
$$
H=H_0+UH_I
$$
\vskip-10pt
\upitem
\begin{itemize}
\item $H_0$: Kinetic term for the electrons (hoppings between atoms).
\item $UH_I$: Short-range screened Coulomb interaction between electrons (of {\it strength} $U$):
$$
UH_I=U\sum_{x, y}v(|x-y|)\left(a_x^\dagger a_x-\frac12\right)\left(a_y^\dagger a_y-\frac12\right)
$$
\end{itemize}
\downitem
\item Non-interacting case ($U=0$): {\it integrable}.
\item Assume $|U|\ll1$: perturb.
\end{itemize}
\eject

\title{Non-interacting Hamiltonian}
\begin{itemize}
\item In Fourier space:
$$
H_0=\sum_k A_k^\dagger \hat H_0(k) A_k
$$
$A_k:=(a_{k,1}, a_{k,2}, a_{k,3}, a_{k,4})$, where 1,2,3,4 is the {\it valley index},
$$
\hat H_0(k)=
-\left(\begin{array}{*{4}{c}}
0&\gamma_1&0&\gamma_0\Omega^*(k)\\[0.2cm]
\gamma_1&0&\gamma_0\Omega(k)&0\\[0.2cm]
0&\gamma_0\Omega^*(k)&0&\gamma_3\Omega(k)e^{3ik_x}\\[0.2cm]
\gamma_0\Omega(k)&0&\gamma_3\Omega^*(k)e^{-3ik_x}&0
\end{array}\right)
$$
with $\Omega(k_1,k_2):=1+2e^{\frac32ik_1}\cos({\scriptstyle\frac{\sqrt3}2}k_2)$.
\end{itemize}
\eject

\title{Non-interacting Hamiltonian}
\begin{itemize}
\item Hopping strengths: $\gamma_0=1$, $\gamma_1=0.1$, $\gamma_3=0.034$
\end{itemize}
\hfil\includegraphics[width=\textwidth]{Figs/hoppings4.pdf}
\vfil\eject

\title{Non-interacting Hamiltonian}
\begin{itemize}
\item Eigenvalues of $\hat H_0(k)$ ({\it bands})
\end{itemize}
\hfil\includegraphics{Figs/bands_global.pdf}
\vfil\eject

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\title{Non-interacting Hamiltonian}
\begin{itemize}
\item For $|k|\gg\gamma_1$ ({\it irrelevant, superrenormalizable} regime)
\end{itemize}
\hfil\includegraphics{Figs/bands_first.pdf}
\vfil\eject

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\title{Non-interacting Hamiltonian}
\begin{itemize}
\item For $\gamma_1^2\ll|k|\ll\gamma_1$ ({\it marginal} regime)
\end{itemize}
\hfil\includegraphics{Figs/bands_second.pdf}
\vfil\eject

\addtocounter{page}{-1}
\title{Non-interacting Hamiltonian}
\begin{itemize}
\item For $|k|\ll\gamma_1^2$ ({\it irrelevant, superrenormalizable} regime)
\end{itemize}
\hfil\includegraphics{Figs/bands_third.pdf}
\vfil\eject

\title{Main result}
\vfil
\begin{framed}
If $|U|$ and $\gamma_1$ and $\gamma_3$ are small enough, then the specific ground state energy
$$
e_0:=-\lim_{\beta\to\infty}\lim_{|\Lambda|\to\infty}\frac1{\beta|\Lambda|}\log(\mathrm{Tr}(e^{-\beta H}))
$$
and the two-point correlation functions are {\it analytic} functions of $U$.
\end{framed}
\begin{itemize}
\item In other words, if $|U|$ is small enough, then the qualitative behavior of the system is similar to that at $U=0$ ({\it weak coupling}).
\end{itemize}
\eject

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\hbox{}
\vfill
\hfil\vrule height0.5pt width0.75\textwidth\par
\vskip20pt
\hfil{\bf Hierarchical Kondo model}\par
\vskip20pt
\hfil\vrule height0.5pt width0.75\textwidth\par
\vfill
\hfil joint with {\bf G.~Benfatto} and {\bf G.~Gallavotti}\par
\vskip10pt
\hfil {\scriptsize doi:{\tt 10.1007/s10955-015-1378-7} and doi:{\tt 10.1007/s10955-015-1370-2}}
\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}
$$
H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}a^\dagger_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,a_\alpha(x)
$$
\item $V$: interaction with the {\it impurity}
$$
V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2}a^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}a_{\alpha_2}(0)\otimes \tau^j
$$
\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 [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.
\end{itemize}
\eject

\title{Current results}
\begin{itemize}
\item Hierarchical Kondo model: idealization of the Kondo model that shares its 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{Hierarchical model}
\begin{itemize}
\item Imaginary time: $\psi_\alpha^\pm(t):=e^{t H_0}a_\alpha^\pm(0)e^{-t H_0}$.
\item Scale decomposition
$$
\psi_\alpha^\pm(t):=\sum_{m\leqslant0}\psi_\alpha^{[m]\pm}(t)
$$
\vfil
\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxtree.pdf}\par
\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{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}
\hfil\includegraphics[width=150pt]{Figs/sd_susc_0_28.pdf}\par
\eject

\title{Conclusion and perspectives}
\begin{itemize}
\item Two examples: {\color{blue}bilayer graphene} and the {\color{blue}hierarchical Kondo model}, which can be studied via {\it constructive}, {\it rigorous} implementations of the renormalization group technique.
\item Bilayer graphene: weak coupling.
\item Hierarchical Kondo model: strong coupling (non-trivial fixed point).
\item Extensions: full Kondo model, BCS theory, high-$T_c$ superconductivity...
\end{itemize}

\end{document}
