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diff --git a/Jauslin_PhD_2016.tex b/Jauslin_PhD_2016.tex new file mode 100644 index 0000000..05c4b9e --- /dev/null +++ b/Jauslin_PhD_2016.tex @@ -0,0 +1,276 @@ +\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} |