From 9ade5e35d8792381fd7c6faa1be03ed3bcfd552c Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@roma1.infn.it>
Date: Thu, 21 Jan 2016 23:05:30 +0000
Subject: As presented in Rome on 2016-01-22

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+\documentclass{kiss}
+\usepackage{presentation}
+\usepackage{header}
+\usepackage{toolbox}
+
+\def\upitem{\itemptchange{$\scriptstyle\blacktriangleright$}}
+\let\downitem\itemptreset
+
+\begin{document}
+\pagestyle{empty}
+\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
+
+
+\setcounter{page}1
+\pagestyle{plain}
+\title{Outline}
+\vfill
+\begin{itemize}
+\item Weak coupling: {\bf Bilayer graphene}
+\vskip30pt
+\item Strong coupling: {\bf Hierarchical Kondo model}
+\end{itemize}
+\vfill
+\eject
+
+\pagestyle{empty}
+\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
+\addtocounter{page}{-1}
+\pagestyle{plain}
+
+\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
+
+\addtocounter{page}{-1}
+\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
+
+\addtocounter{page}{-1}
+\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
+
+\pagestyle{empty}
+\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
+\addtocounter{page}{-1}
+\pagestyle{plain}
+
+\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}
-- 
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