From f500b7cb0daef7b3a491d8f0b8720b6951f27a3e Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Thu, 24 Mar 2016 16:23:49 +0000 Subject: At presented at the University of Zurich on 2016-03-24 --- Figs/beta_phase.pdf | Bin 0 -> 54214 bytes Figs/beta_phase_half.pdf | Bin 0 -> 55564 bytes Figs/hierarchical_boxes.pdf | Bin 0 -> 31871 bytes Figs/hierarchical_boxtree.pdf | Bin 0 -> 31763 bytes Figs/kondo_model.pdf | Bin 0 -> 1828 bytes Figs/sd_phase.pdf | Bin 0 -> 47801 bytes Figs/sd_susc_0_005.pdf | Bin 0 -> 51078 bytes Figs/sd_susc_0_02.pdf | Bin 0 -> 49924 bytes Figs/sd_susc_0_28.pdf | Bin 0 -> 27561 bytes Figs/susc_plot_temp.pdf | Bin 0 -> 34223 bytes Jauslin_Zurich_2016.tex | 280 ++++++++++++++++++++++++++++++++++++++++++ README | 32 +++++ header.sty | 19 +++ kiss.cls | 42 +++++++ presentation.sty | 110 +++++++++++++++++ symbols.sty | 61 +++++++++ 16 files changed, 544 insertions(+) create mode 100644 Figs/beta_phase.pdf create mode 100644 Figs/beta_phase_half.pdf create mode 100644 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b/Figs/sd_susc_0_28.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_Zurich_2016.tex b/Jauslin_Zurich_2016.tex new file mode 100644 index 0000000..1868fa7 --- /dev/null +++ b/Jauslin_Zurich_2016.tex @@ -0,0 +1,280 @@ +\documentclass{kiss} +\usepackage{presentation} +\usepackage{header} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf +\large +\hfil Strong-coupling 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: \parbox[b]{1cm}{\tt1506.04381\par1507.05678}\hfill{\tt http://ian.jauslin.org/} +\eject + +\pagestyle{plain} +\setcounter{page}{1} + + +\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} +$$ +H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(\left(-\frac{\Delta}2-1\right)\,c_\alpha\right)(x)\otimes\mathds1 +$$ +\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 Magnetic susceptibility: response to a magnetic field $h$: +$$ +\chi(h,\beta):=\partial_hm(h,\beta). +$$ +($m(h,\beta)$: magnetization). +\item Isolated impurity: +$$ +\chi^{(0)}(0,\beta)=\beta\mathop{\longrightarrow}_{\beta\to\infty}\infty +$$ +\item Chain of electrons: Pauli paramagnetism: +$$ +\lim_{\beta\to\infty}\lim_{L\to\infty}\frac1L\chi_e(0,\beta)<\infty. +$$ +\end{itemize} +\eject + +\title{Kondo effect: magnetic susceptibility} +\begin{itemize} +\item Turn on the interaction: $\lambda_0\neq0$. Impurity susceptibility $\chi^{(\lambda_0)}(h,\beta)$. +\item Ferromagnetic interaction ($\lambda_0>0$): +$$ +\lim_{\beta\to\infty}\chi^{(\lambda_0)}(0,\beta)=\infty. +$$ +\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 (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}: 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{Field theory for the Kondo model} +\begin{itemize} +\item By introducing an extra dimension ({\it imaginary time}), the partition function $Z:=\mathrm{Tr}(e^{-\beta H})$ can be expressed as the {\it Gaussian} average over a {\it Grassmann} algebra: +$$ +Z=\mathrm{Tr}\left +$$ +\vskip-10pt +\item Potential: +$$ +\mathcal V(t)=-\lambda_0\kern-10pt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}}\kern-10pt +\psi^+_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(t)\tau^j +$$ +with $\{\psi^\pm_\alpha(t),\psi^\pm_{\alpha'}(t')\}=0$. +\item $\left<\cdot\right>$ is defined by its second moment $\left<\psi^-_{\alpha_1}(t_1)\psi^+_{\alpha_2}(t_2)\right>$. +\end{itemize} +\eject + +\title{Hierarchical model} +\begin{itemize} +\item Replace $\psi_\alpha^\pm(t)$ in $\mathcal V(t)$ by +$$ +\psi_\alpha^\pm(t):=\sum_{m\leqslant0}\psi_\alpha^{[m]\pm}(t) +$$ +where $\psi_\alpha^{[m]\pm}(t)$ is {\it constant} over the ``time'' intervals $\Delta_{i,\pm}^{[m]}$:\par +\vfil +\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxes.pdf}\par +\item There are 4 fields in each $\Delta_{i,\pm}^{[m]}$. +\item Moments: +$$ +\left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m +$$ +\end{itemize} +\eject + +\title{Full propagator} +\begin{itemize} +\item Moments: +$$ +\left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m +$$ +\item Full propagator: +$$ +\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t') +$$ +\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxtree.pdf}\par +\vfil +\eject + +\title{Comparison with the Kondo model} +\item Hierarchical model: +$$ +\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t') +$$ +\item For the (non-hierarchical) Kondo model: +$$ +\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>\approx\sum_m 2^{m}g_\psi^{[0]}(2^m(t-t')) +$$ +where $g^{[0]}_\psi$ is odd and decays faster than any power. +\end{itemize} +\eject + +\title{Hierarchical beta function} +\begin{itemize} +\item Compute $Z$ by $\mathcal V^{[0]}(t):=\mathcal V(t)$ +$$ +e^{-\int dt\ \mathcal V^{[m-1]}(t)}:=\left_m +$$ + +\item Effective potential: +$$ +\int dt\ \mathcal V^{[m]}(t)=\sum_{i=1}^{2^{-m}}\mathcal V^{[m]}_{i,-}+\mathcal V^{[m]}_{i,+} +$$ +\item Iteration +$$ +\left_m=\prod_{i=1}^{2^{-m}}\left_m +$$ +\item By anti-commutation of the fields, $e^{-(\mathcal V_{i,-}^{[m]}+\mathcal V_{i,+}^{[m]})}$ is a polynomial in the fields of order $\leqslant 8$. +\end{itemize} +\eject + +\title{Hierarchical beta function} +\begin{itemize} +\item $\mathcal V^{[m]}$ is parametrized by 2 real numbers ({\it running coupling constants}) $\ell_0^{[m]},\ell_1^{[m]}$: +$$\begin{array}{r@{\ }>{\displaystyle}l} +\frac{e^{-\scriptstyle\int dt\ \mathcal V^{[m]}(t)}}{C^{[m]}} +=&1+\frac{\ell_0^{[m]}}2\int dt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}} +\psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\tau^j\\[0.5cm] +&+\frac{\ell_1^{[m]}}2\int dt\left(\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}} +\psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\right)^2 +\end{array}$$ +\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{Susceptibility} +\begin{itemize} +\item Add magnetic field $h$ on the impurity. +\item New term in the potential: +$$ +-h \sum_{j\in\{1,2,3\}}\bm\omega_j \tau^j +$$ +\item 6 running coupling constants. +\item The susceptibility can be computed by deriving $C^{[m]}$ with respect to $h$. +\end{itemize} +\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} +\eject + +\title{Susceptibility} +\begin{itemize} +\item $\lambda_0=-0.28$ +\end{itemize} +\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_28.pdf}\par +\eject + +\addtocounter{page}{-1} +\title{Susceptibility} +\begin{itemize} +\item $\lambda_0=-0.02$ +\end{itemize} +\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_02.pdf}\par +\eject + +\addtocounter{page}{-1} +\title{Susceptibility} +\begin{itemize} +\item $\lambda_0=-0.005$ +\end{itemize} +\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_005.pdf}\par +\eject + +\title{Open questions} +\begin{itemize} +\item Magnetic field on the chain as well. This requires defining the hierarchical model to reflect the $x$-dependence of $\psi(x,t)$. +\item Rigorous renormalization group analysis for the Kondo model (non-hierarchical). +\item The exact solvability of the hierarchical Kondo model is merely a consequence of the fermionic nature of the system. Other fermionic hierarchical models can be studied to investigate other strong-coupling phenomena, e.g. high-$T_c$ superconductivity. +\end{itemize} +\eject + +\title{Epilogue: {\tt meankondo}} +\begin{itemize} +\item The computation in the $h$-dependent case requires computing many Feynman diagrams ($\approx100$). +\item Software to perform the computation: {\tt meankondo}. +\item {\tt meankondo} can be configured to study any fermionic hierarchical model. +\end{itemize} +\hfil{\tt http://ian.jauslin.org/software/meankondo/} + +\end{document} + + + + + diff --git a/README b/README new file mode 100644 index 0000000..e725790 --- /dev/null +++ b/README @@ -0,0 +1,32 @@ +* Typeset + +In order to typeset the LaTeX presentation, run + pdflatex Jauslin_Zurich_2015.tex + pdflatex Jauslin_Zurich_2015.tex + + +* Files + + Jauslin_Zurich_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..23529cc --- /dev/null +++ b/header.sty @@ -0,0 +1,19 @@ +%% +%% 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} +% extra array related commands +\usepackage{array} + +%% 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