Ian Jauslin
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authorIan Jauslin <ian.jauslin@roma1.infn.it>2016-01-21 23:05:30 +0000
committerIan Jauslin <ian.jauslin@roma1.infn.it>2016-01-21 23:05:30 +0000
commit9ade5e35d8792381fd7c6faa1be03ed3bcfd552c (patch)
treeb2c5cc1d1a3236207af2d989b55f3d1a197122fb
As presented in Rome on 2016-01-22HEADv1.0master
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-rw-r--r--Jauslin_PhD_2016.tex276
-rw-r--r--README35
-rw-r--r--header.sty15
-rw-r--r--kiss.cls51
-rw-r--r--presentation.sty110
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diff --git a/Jauslin_PhD_2016.tex b/Jauslin_PhD_2016.tex
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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}
diff --git a/README b/README
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--- /dev/null
+++ b/README
@@ -0,0 +1,35 @@
+* Typeset
+
+In order to typeset the LaTeX presentation, run
+ pdflatex Jauslin_PhD_2016.tex
+ pdflatex Jauslin_PhD_2016.tex
+
+
+* Files
+
+ Jauslin_PhD_2016.tex
+ body of the presentation.
+
+ Figs :
+ figures.
+
+ header.sty :
+ packages and local definitions
+
+ presentation.sty :
+ presentation style file.
+
+ kiss.cls :
+ barebones class file
+
+ toolbox.sty :
+ useful commands
+
+
+* Comment on the presentation style
+
+This presentation is typeset using the 'presentation.sty' style file, in which
+the margins and sizes are set to make the output look like a presentation, and
+several useful commands are defined.
+
+The LaTeX 'beamer' package is not used.
diff --git a/header.sty b/header.sty
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--- /dev/null
+++ b/header.sty
@@ -0,0 +1,15 @@
+%%
+%% Load packages
+%%
+
+\usepackage{color}
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{array}
+\usepackage{etoolbox}
+\usepackage{dsfont}
+\usepackage{framed}
+\usepackage{bm}
+
diff --git a/kiss.cls b/kiss.cls
new file mode 100644
index 0000000..7f0029f
--- /dev/null
+++ b/kiss.cls
@@ -0,0 +1,51 @@
+%%
+%% Barebones class declaration
+%%
+
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{kiss}
+
+\setlength\paperheight {297mm}
+\setlength\paperwidth {210mm}
+
+%% fonts
+\input{size11.clo}
+\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
+\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
+\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
+\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
+\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
+\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
+\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
+
+%% something is wrong with \thepage, redefine it
+\gdef\thepage{\the\c@page}
+
+%% array lines
+\setlength\arraycolsep{5\p@}
+\setlength\arrayrulewidth{.4\p@}
+
+
+%% default offsets: 1in, correct with \hoffset and \voffset
+%\hoffset=0pt
+\hoffset=-50pt
+%\voffset=0pt
+\voffset=-72pt
+%% horizontal margins
+%\oddsidemargin=31pt
+%\evensidemargin=31pt
+%% vertical margin
+%\topmargin=20pt
+%% body size
+%\textwidth=390pt
+\textwidth=460pt
+%\textheight=592pt
+\textheight=704pt
+%% header size and margin
+%\headheight=12pt
+%\headsep=25pt
+%% footer size
+%\footskip=30pt
+%% margin size and margin
+%\marginparwidth=35pt
+%\marginparsep=10pt
diff --git a/presentation.sty b/presentation.sty
new file mode 100644
index 0000000..e55928f
--- /dev/null
+++ b/presentation.sty
@@ -0,0 +1,110 @@
+%%
+%% Presentation style
+%%
+
+%% can call commands even when they are not defined
+\def\safe#1{%
+\ifdefined#1%
+#1%
+\else%
+{\color{red}\bf?}%
+\fi%
+}
+
+
+%% paper size
+\setlength\paperheight{240pt}
+\setlength\paperwidth{320pt}
+
+%% body size
+% height=paperheight-2xtopmargin-footskip
+\textheight=208pt
+% width=paperwidth-2xoddsidemargin
+\textwidth=272pt
+
+%% margins
+\voffset=-1in
+\hoffset=-1in
+\oddsidemargin=24pt
+\evensidemargin=24pt
+\topmargin=8pt
+\headheight=0pt
+\headsep=0pt
+\marginparsep=0pt
+\marginparwidth=0pt
+\footskip=16pt
+\skip\footins=0pt
+
+%% reset skips
+\parskip=0pt
+\parindent=0pt
+\baselineskip=0pt
+\AtBeginDocument{
+}
+
+%% footer
+\def\ps@plain{
+ \def\@oddhead{}
+ \def\@evenhead{\@oddhead}
+ \def\@oddfoot{\tiny\hfill\thepage/\safe\slidecount\hfill}
+ \def\@evenfoot{\@oddfoot}
+}
+\def\ps@empty{
+ \def\@oddhead{}
+ \def\@evenhead{\@oddhead}
+ \def\@oddfoot{}
+ \def\@evenfoot{\@oddfoot}
+}
+
+%% save total slide count
+\AtEndDocument{
+ \immediate\write\@auxout{\noexpand\gdef\noexpand\slidecount{\thepage}}
+}
+
+\pagestyle{plain}
+
+%% title of slide
+\def\title#1{
+ \hfil{\bf\large #1}\par
+ \hfil\vrule width0.75\textwidth height0.3pt\par
+ \vskip5pt
+}
+
+%% itemize
+\newlength\itemizeskip
+% left margin for items
+\setlength\itemizeskip{20pt}
+% item symbol
+\def\itemizept{\textbullet}
+\newlength\itemizeseparator
+% space between the item symbol and the text
+\setlength\itemizeseparator{5pt}
+
+\newlength\current@itemizeskip
+\setlength\current@itemizeskip{0pt}
+\def\itemize{
+ \vfil
+ \addtolength\current@itemizeskip{\itemizeskip}
+ \leftskip\current@itemizeskip
+}
+\def\enditemize{
+ \addtolength\current@itemizeskip{-\itemizeskip}
+ \par\leftskip\current@itemizeskip
+ \vfil
+}
+\newlength\itempt@total
+\def\item{
+ \settowidth\itempt@total{\itemizept}
+ \addtolength\itempt@total{\itemizeseparator}
+ \par
+ \vfil
+ \hskip-\itempt@total\itemizept\hskip\itemizeseparator
+}
+
+\def\itemptchange#1{
+ \let\itempt@prev\itemizept
+ \def\itemizept{#1}
+}
+\def\itemptreset{
+ \def\itemizept{\itempt@prev}
+}
diff --git a/toolbox.sty b/toolbox.sty
new file mode 100644
index 0000000..04c8acd
--- /dev/null
+++ b/toolbox.sty
@@ -0,0 +1,32 @@
+%%
+%% A collection of useful commands
+%%
+
+%% can call commands even when they are not defined
+\def\safe#1{%
+\ifdefined#1%
+#1%
+\else%
+{\color{red}\bf?}%
+\fi%
+}
+
+
+%% larger skip
+\newskip\hugeskipamount
+ \hugeskipamount=24pt plus8pt minus8pt
+\def\hugeskip{\vskip\hugeskipamount}
+
+
+%% stack relations in subscript or superscript
+\def\mAthop#1{\displaystyle\mathop{\scriptstyle #1}}
+
+%% array spanning the entire line
+\newlength\largearray@width
+\setlength\largearray@width\textwidth
+\addtolength\largearray@width{-10pt}
+\def\largearray{\begin{array}{@{}>{\displaystyle}l@{}}\hphantom{\hspace{\largearray@width}}\\[-.5cm]}
+\def\endlargearray{\end{array}}
+
+%% qedsquare
+\def\qed{\penalty10000\hfill\penalty10000$\square$}