From 2157a78b72b0de503cda8ce915df57dfc2e74b53 Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Fri, 31 Jul 2015 05:39:00 +0000 Subject: As presented at ICMP 2015 on 2015-07-30 --- Figs/beta_phase_half.pdf | Bin 0 -> 55564 bytes Figs/kondo_model.pdf | Bin 0 -> 1828 bytes Figs/susc_plot_temp.pdf | Bin 0 -> 34223 bytes Jauslin_ICMP_2015.tex | 115 +++++++++++++++++++++++++++++++++++++++++++++++ README | 32 +++++++++++++ header.sty | 17 +++++++ kiss.cls | 42 +++++++++++++++++ presentation.sty | 110 +++++++++++++++++++++++++++++++++++++++++++++ symbols.sty | 61 +++++++++++++++++++++++++ 9 files changed, 377 insertions(+) create mode 100644 Figs/beta_phase_half.pdf create mode 100644 Figs/kondo_model.pdf create mode 100644 Figs/susc_plot_temp.pdf create mode 100644 Jauslin_ICMP_2015.tex create mode 100644 README create mode 100644 header.sty create mode 100644 kiss.cls create mode 100644 presentation.sty create mode 100644 symbols.sty diff --git a/Figs/beta_phase_half.pdf b/Figs/beta_phase_half.pdf new file mode 100644 index 0000000..4ebab32 Binary files /dev/null and b/Figs/beta_phase_half.pdf differ 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/susc_plot_temp.pdf b/Figs/susc_plot_temp.pdf new file mode 100644 index 0000000..1e8f863 Binary files /dev/null and b/Figs/susc_plot_temp.pdf differ diff --git a/Jauslin_ICMP_2015.tex b/Jauslin_ICMP_2015.tex new file mode 100644 index 0000000..0e87c7d --- /dev/null +++ b/Jauslin_ICMP_2015.tex @@ -0,0 +1,115 @@ +\documentclass{kiss} +\usepackage{presentation} +\usepackage{header} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf +\large +\hfil Non-perturbative renormalization group\par +\smallskip +\hfil in a 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: {\tt1506.04381}\hfill{\tt http://ian.jauslin.org/} +\eject + +\pagestyle{plain} +\setcounter{page}{1} + +\title{Kondo model} +\begin{itemize} +\item [N.~Andrei, 1980]: +$$ +H=H_0+V +$$ +\itemptchange{$\scriptstyle\blacktriangleright$} +\begin{itemize} +\item $H_0$: kinetic term of the {\it electrons} +$$ +H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c_\alpha(x) +$$ +\item $V$: interaction with the {\it impurity} +$$ +V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\, d^\dagger_{\alpha_3}\sigma^j_{\alpha_3,\alpha_4}d_{\alpha_4} +$$ +\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 Non-perturbative} 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 beta function} \begin{itemize} \item Beta function ({\it exact}) +$$\begin{array}{r@{\ }l} +C^{[m]}=&\displaystyle1+ 3(\ell_0^{[m]})^2+9(\ell_1^{[m]})^2+9(\ell_2^{[m]})^2+324(\ell_3^{[m]})^2\\[0.3cm] +\ell_0^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\ell_0^{[m]} +18\ell_0^{[m]}\ell_3^{[m]}+3 \ell_0^{[m]}\ell_2^{[m]}+3 \ell_0^{[m]}\ell_1^{[m]} -2(\ell_0^{[m]})^2\Big)\\[0.5cm] +\ell_1^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big( \frac12\ell_1^{[m]}+9\ell_2^{[m]}\ell_3^{[m]} +\frac14(\ell_0^{[m]})^2\Big)\\[0.5cm] +\ell_2^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(2\ell_2^{[m]}+36\ell_1^{[m]}\ell_3^{[m]}+ (\ell_0^{[m]})^2\Big)\\[0.5cm] +\ell_3^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\frac12\ell_3^{[m]}+\frac14\ell_1^{[m]}\ell_2^{[m]}+\frac1{24} (\ell_0^{[m]})^2\Big). +\end{array}$$ +\end{itemize} +\eject + +\title{Flow} +\vfil +\hfil\includegraphics[width=0.8\textwidth]{Figs/beta_phase_half.pdf}\par +Fixed points: $\bm\ell^{(0)}$, $\bm\ell^{(+)}$, $\bm\ell^*$. +\eject + +\title{Kondo effect} +\begin{itemize} +\item Fix $h=0$. +\item At $\bm\ell^{(+)}$, 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/susc_plot_temp.pdf}\par + +\end{document} diff --git a/README b/README new file mode 100644 index 0000000..61a890e --- /dev/null +++ b/README @@ -0,0 +1,32 @@ +* Typeset + +In order to typeset the LaTeX presentation, run + pdflatex Jauslin_ICMP_2015.tex + pdflatex Jauslin_ICMP_2015.tex + + +* Files + + Jauslin_ICMP_2015.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/header.sty b/header.sty new file mode 100644 index 0000000..333385b --- /dev/null +++ b/header.sty @@ -0,0 +1,17 @@ +%% +%% Packages and local definitions +%% + +%% load packages +% colors +\usepackage{color} +% include figures +\usepackage{graphicx} +% extra symbols and fonts +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{dsfont} +\usepackage{bm} + +%% dark green color +\definecolor{darkgreen}{RGB}{0,128,0} diff --git a/kiss.cls b/kiss.cls new file mode 100644 index 0000000..dc1bdbc --- /dev/null +++ b/kiss.cls @@ -0,0 +1,42 @@ +%% +%% 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} + +%% default offsets: 1in, correct with \hoffset and \voffset +%\hoffset=0pt +%\voffset=0pt +%% horizontal margins +%\oddsidemargin=31pt +%\evensidemargin=31pt +%% vertical margin +%\topmargin=20pt +%% body size +%\textwidth=390pt +%\textheight=592pt +%% 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/symbols.sty b/symbols.sty new file mode 100644 index 0000000..84ae838 --- /dev/null +++ b/symbols.sty @@ -0,0 +1,61 @@ +%% +%% symbols +%% + +\def\deriv#1#2{\frac{d#1}{d#2}} +\def\pard#1#2{\frac{\partial #1}{\partial #2}} +\def\mAthop#1{\mathop{\scriptstyle#1}} + +\def\rp{\right)} +\def\lp{\left(} +\let\(\lp +\let\)\rp + +\def\ma{\\[0.2cm]} +\def\n{\\[0.5cm]} +\def\N{\\[1cm]} + +\let\mc\mathcal +\let\mbb\mathbb +\let\mf\mathfrak +\let\mds\mathds +\let\mrm\mathrm +\let\mbf\mathbf + +\def\R{\mathbb{R}} +\def\C{\mathbb{C}} +\def\tr{\mathrm{Tr}} +\let\lra\longrightarrow +\def\+{^\dagger} +\let\ge\geqslant +\let\le\leqslant + +\def\ket#1{\left|#1\right>} +\def\bra#1{\left<#1\right|} +\def\<{\left<} +\def\>{\right>} + +\def\Par{\par\penalty10000} +\def\Smallskip{\smallskip\penalty10000} +\def\Medskip{\medskip\penalty10000} +\def\Bigskip{\bigskip\penalty10000} + +\def\vvect#1{\lp\begin{array}{c}#1\end{array}\rp} +\def\hvect#1#2{\lp\begin{array}{*{#1}{c}}#2\end{array}\rp} +\def\mat#1#2{\lp\begin{array}{*{#1}{c}}#2\end{array}\rp} +\long\def\sys#1{\left\{\begin{array}l#1\end{array}\right.} + +\def\sint{\int\kern-3pt} +\def\sintb#1#2{\int_{#1}^{#2}\kern-.3cm} +\def\sIntb#1#2{\displaystyle\int_{#1}^{#2}\kern-.3cm} +\def\soint{\oint\kern-3pt} +\def\sOint{\displaystyle\oint\kern-3pt} + +\def\po{$\bullet$ } +\def\as{$\ast$ } +\def\spo{$\scriptstyle\blacktriangleright$ } + +\def\qed{\hfill$\square$} + +\def\itemsymbolt{\spo} + -- cgit v1.2.3-54-g00ecf