From f16e2fa90700e12926c6b55f78b8309ac6888de3 Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Wed, 21 Sep 2016 16:45:13 +0000 Subject: As presented at the IAS on 2016-09-21 --- Jauslin_IAS_2016.tex | 93 ++++++++++++++++++++++++++++++++++++++++++ README | 32 +++++++++++++++ figs/kondo_model.pdf | Bin 0 -> 1828 bytes figs/sd_susc_0_28.pdf | Bin 0 -> 27561 bytes header.sty | 14 +++++++ kiss.cls | 51 +++++++++++++++++++++++ presentation.sty | 110 ++++++++++++++++++++++++++++++++++++++++++++++++++ toolbox.sty | 32 +++++++++++++++ 8 files changed, 332 insertions(+) create mode 100644 Jauslin_IAS_2016.tex create mode 100644 README create mode 100644 figs/kondo_model.pdf create mode 100644 figs/sd_susc_0_28.pdf create mode 100644 header.sty create mode 100644 kiss.cls create mode 100644 presentation.sty create mode 100644 toolbox.sty diff --git a/Jauslin_IAS_2016.tex b/Jauslin_IAS_2016.tex new file mode 100644 index 0000000..0ebea16 --- /dev/null +++ b/Jauslin_IAS_2016.tex @@ -0,0 +1,93 @@ +\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 new file mode 100644 index 0000000..8de3846 --- /dev/null +++ b/README @@ -0,0 +1,32 @@ +* 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 new file mode 100644 index 0000000..589cf74 Binary files /dev/null and b/figs/kondo_model.pdf differ diff --git a/figs/sd_susc_0_28.pdf b/figs/sd_susc_0_28.pdf new file mode 100644 index 0000000..e4e23a3 Binary files /dev/null and b/figs/sd_susc_0_28.pdf differ diff --git a/header.sty b/header.sty new file mode 100644 index 0000000..cf4d854 --- /dev/null +++ b/header.sty @@ -0,0 +1,14 @@ +%% +%% 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 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..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$\square$} -- cgit v1.2.3-54-g00ecf