Ian Jauslin
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authorIan Jauslin <ian.jauslin@roma1.infn.it>2015-07-31 05:39:00 +0000
committerIan Jauslin <ian.jauslin@roma1.infn.it>2015-07-31 05:39:00 +0000
commit2157a78b72b0de503cda8ce915df57dfc2e74b53 (patch)
tree8f0d6f5b5ce76805e47effaa45c40f1fa0ef6102
As presented at ICMP 2015 on 2015-07-30HEADv1.0master
-rw-r--r--Figs/beta_phase_half.pdfbin0 -> 55564 bytes
-rw-r--r--Figs/kondo_model.pdfbin0 -> 1828 bytes
-rw-r--r--Figs/susc_plot_temp.pdfbin0 -> 34223 bytes
-rw-r--r--Jauslin_ICMP_2015.tex115
-rw-r--r--README32
-rw-r--r--header.sty17
-rw-r--r--kiss.cls42
-rw-r--r--presentation.sty110
-rw-r--r--symbols.sty61
9 files changed, 377 insertions, 0 deletions
diff --git a/Figs/beta_phase_half.pdf b/Figs/beta_phase_half.pdf
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diff --git a/Figs/kondo_model.pdf b/Figs/kondo_model.pdf
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diff --git a/Figs/susc_plot_temp.pdf b/Figs/susc_plot_temp.pdf
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diff --git a/Jauslin_ICMP_2015.tex b/Jauslin_ICMP_2015.tex
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+\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
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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
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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}
+