Ian Jauslin
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authorIan Jauslin <ian.jauslin@roma1.infn.it>2016-09-21 16:45:13 +0000
committerIan Jauslin <ian.jauslin@roma1.infn.it>2016-09-21 16:45:13 +0000
commitf16e2fa90700e12926c6b55f78b8309ac6888de3 (patch)
treead8f7a55372779c1cf1f156ae4193762a22350f2
As presented at the IAS on 2016-09-21HEADv1.0master
-rw-r--r--Jauslin_IAS_2016.tex93
-rw-r--r--README32
-rw-r--r--figs/kondo_model.pdfbin0 -> 1828 bytes
-rw-r--r--figs/sd_susc_0_28.pdfbin0 -> 27561 bytes
-rw-r--r--header.sty14
-rw-r--r--kiss.cls51
-rw-r--r--presentation.sty110
-rw-r--r--toolbox.sty32
8 files changed, 332 insertions, 0 deletions
diff --git a/Jauslin_IAS_2016.tex b/Jauslin_IAS_2016.tex
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+\documentclass{kiss}
+\usepackage{presentation}
+\usepackage{header}
+\usepackage{toolbox}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf
+\large
+\hfil Strong-coupling renormalization group\par
+\smallskip
+\hfil in the 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: \parbox[b]{1cm}{\tt\href{http://arxiv.org/abs/1506.04381}{1506.04381}\par\href{http://arxiv.org/abs/1507.05678}{1507.05678}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org/}}
+\eject
+
+\pagestyle{plain}
+\setcounter{page}{1}
+
+\title{Kondo model}
+\begin{itemize}
+\item \href{http://dx.doi.org/10.1103/PhysRev.124.41}{[P. Anderson, 1961]}, \href{http://dx.doi.org/10.1143/PTP.32.37}{[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}
+$$
+H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(\left(-\frac{\Delta}2-1\right)\,c_\alpha\right)(x)\otimes\mathds1
+$$
+\item $V$: interaction with the {\it impurity}
+$$
+V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\otimes \tau^j
+$$
+\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 \href{http://dx.doi.org/10.1143/PTP.32.37}{[J. Kondo, 1964]}: third order Born approximation.
+\item \href{http://dx.doi.org/10.1088/0022-3719/3/12/008}{[P. Anderson, 1970]}, \href{http://dx.doi.org/10.1103/RevModPhys.47.773}{[K. Wilson, 1975]}: renormalization group approach
+\item{\tt Remark}: \href{http://dx.doi.org/10.1103/PhysRevLett.45.379}{[N.~Andrei, 1980]}: the Kondo model (suitably linearized) is exactly solvable via Bethe Ansatz (which breaks down under any perturbation of the model).
+\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}: reduces the system to a 2-dimensional discrete dynamical system.
+\item Kondo effect in the hierarchical model.
+\end{itemize}
+\hfil\includegraphics[width=150pt]{figs/sd_susc_0_28.pdf}\par
+\eject
+
+\title{Open problem}
+\begin{itemize}
+\item Usual approach to the Renormalization group: perturb around the uncoupled theory.
+\item Cannot access strongly-coupled effects.
+\item Idea: perturb around hierarchical models.
+\item How? Which hierarchical models?
+\end{itemize}
+
+\end{document}
diff --git a/README b/README
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+++ b/README
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+* Typeset
+
+In order to typeset the LaTeX presentation, run
+ pdflatex Jauslin_IAS_2016.tex
+ pdflatex Jauslin_IAS_2016.tex
+
+
+* Files
+
+ Jauslin_IAS_2016.tex
+ body of the presentation.
+
+ figs :
+ figures.
+
+ header.sty :
+ packages and local definitions
+
+ presentation.sty :
+ presentation style file.
+
+ kiss.cls :
+ barebones class file
+
+
+* 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/figs/kondo_model.pdf b/figs/kondo_model.pdf
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+++ b/figs/kondo_model.pdf
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diff --git a/figs/sd_susc_0_28.pdf b/figs/sd_susc_0_28.pdf
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diff --git a/header.sty b/header.sty
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--- /dev/null
+++ b/header.sty
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+%%
+%% Load packages
+%%
+
+\usepackage{color}
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{array}
+\usepackage{etoolbox}
+\usepackage{dsfont}
+\usepackage{bm}
+
diff --git a/kiss.cls b/kiss.cls
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--- /dev/null
+++ b/kiss.cls
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+%%
+%% 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..4e07694
--- /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..b214a3e
--- /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$\square$}