diff options
author | Ian Jauslin <ian.jauslin@roma1.infn.it> | 2016-10-07 22:31:25 +0000 |
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committer | Ian Jauslin <ian.jauslin@roma1.infn.it> | 2016-10-09 03:24:52 +0000 |
commit | d178209af8152764071b8743c47321447ba47c19 (patch) | |
tree | 962d95594267dfac031595e1c82c5273617f334f |
-rw-r--r-- | Jauslin_Qmath_2016.tex | 183 | ||||
-rw-r--r-- | README | 32 | ||||
-rw-r--r-- | figs/basegrid.pdf | bin | 0 -> 4326 bytes | |||
-rw-r--r-- | figs/cellgraph.pdf | bin | 0 -> 19295 bytes | |||
-rw-r--r-- | figs/first_nog4D.pdf | bin | 0 -> 106775 bytes | |||
-rw-r--r-- | figs/flow.pdf | bin | 0 -> 59736 bytes | |||
-rw-r--r-- | figs/global_nog4D.pdf | bin | 0 -> 398713 bytes | |||
-rw-r--r-- | figs/hoppings4.pdf | bin | 0 -> 146211 bytes | |||
-rw-r--r-- | figs/monolayer.pdf | bin | 0 -> 2700 bytes | |||
-rw-r--r-- | figs/second_nog4D.pdf | bin | 0 -> 107678 bytes | |||
-rw-r--r-- | figs/third_nog4D.pdf | bin | 0 -> 130317 bytes | |||
-rw-r--r-- | header.sty | 13 | ||||
-rw-r--r-- | kiss.cls | 51 | ||||
-rw-r--r-- | presentation.sty | 110 | ||||
-rw-r--r-- | toolbox.sty | 32 |
15 files changed, 421 insertions, 0 deletions
diff --git a/Jauslin_Qmath_2016.tex b/Jauslin_Qmath_2016.tex new file mode 100644 index 0000000..ac7649b --- /dev/null +++ b/Jauslin_Qmath_2016.tex @@ -0,0 +1,183 @@ +\documentclass{kiss} +\usepackage{presentation} +\usepackage{header} +\usepackage{toolbox} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil Ground state construction\par +\smallskip +\hfil of Bilayer Graphene\par +\vfil +\large +\hfil Ian Jauslin +\normalsize +\vfil +\hfil\rm joint with {\bf Alessandro Giuliani}\par +\vfil +arXiv:{\tt \href{http://arxiv.org/abs/1507.06024}{1507.06024}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}} +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\title{Monolayer graphene} +\begin{itemize} + \item 2D crystal of carbon atoms on a honeycomb lattice. +\end{itemize} +\vfill +\hfil\hskip-25pt\includegraphics[height=150pt]{figs/monolayer.pdf}\par +\eject + +\title{Bilayer graphene} +\begin{itemize} + \item 2 graphene layers in {\it AB} stacking. +\end{itemize} +\vfill +\hskip12pt\includegraphics[height=150pt]{figs/basegrid.pdf}\par +\eject + +\title{Bilayer graphene} +\begin{itemize} + \item Rhombic lattice $\Lambda\equiv\mathbb Z^2$, 4 atoms per site. +\end{itemize} +\vfill +\hskip12pt\includegraphics[height=150pt]{figs/cellgraph.pdf}\par +\eject + +\title{Hamiltonian} +\begin{itemize} + \item Hamiltonian: + $$ + \mathcal H=\mathcal H_0+UV + $$ + \item Non-interacting Hamiltonian: hoppings +\end{itemize} +\vfill +\hfil\includegraphics[width=\textwidth]{figs/hoppings4.pdf}\par +\vfill +\begin{itemize} + \item Interaction: weak, short-range (screened Coulomb). +\end{itemize} +\eject + +\title{Non-interacting Hamiltonian} +\vskip-10pt +$$ + \mathcal H_0=\sum_{k\in\hat\Lambda} + \left(\begin{array}c + \hat a_k^\dagger\\ + \hat{\tilde b}_k^\dagger\\ + \hat{\tilde a}_k^\dagger\\ + \hat b_k^\dagger + \end{array}\right)^T + \hat H_0(k) + \left(\begin{array}c + \hat a_k\\ + \hat{\tilde b}_k\\ + \hat{\tilde a}_k\\ + \hat b_k + \end{array}\right) +$$ +\vfill +$$ + \kern-10pt + \hat H_0(k):= + \left(\begin{array}{*{4}{c}} + 0&\gamma_1&0&\gamma_0\Omega^*(k)\\ + \gamma_1&0&\gamma_0\Omega(k)&0\\ + 0&\gamma_0\Omega^*(k)&0&\gamma_3\Omega(k)e^{3ik_x}\\ + \gamma_0\Omega(k)&0&\gamma_3\Omega(k)e^{-3ik_x} + \end{array}\right) +$$ +\vfill +$$ + \Omega(k):=1+2e^{-\frac32ik_x}\cos({\textstyle\frac{\sqrt3}2}k_y) +$$ +\eject + +\title{Non-interacting Hamiltonian} +\vfill +\begin{itemize} + \item Hopping strengths: + $$ + \gamma_0=1,\quad + \gamma_1=\epsilon,\quad + \gamma_3=0.33\times\epsilon + $$ + \item Experimental value $\epsilon\approx0.1$, here, $\epsilon\ll1$. +\end{itemize} +\vfill +\eject + +\title{Interaction} +$$ + V=\sum_{x,y}v(|x-y|)\left(n_x-\frac12\right)\left(n_y-\frac12\right) +$$ +\begin{itemize} + \item $\displaystyle\sum_{x,y}$: sum over pairs of atoms + \item $v(|x-y|)\leqslant e^{-c|x-y|}$, $c>0$ + \item $-\frac12$: {\it half-filling}. +\end{itemize} +\eject + +\title{Non-interacting Hamiltonian} +\begin{itemize} + \item Eigenvalues of $\hat H_0(k)$: +\end{itemize} +\vfill +\hfil\includegraphics[width=\textwidth]{figs/global_nog4D.pdf}\par +\vfill +\eject + +\title{Non-interacting Hamiltonian} +\begin{itemize} + \item $|k|\gg\epsilon$ +\end{itemize} +\vfill +\hfil\includegraphics[width=\textwidth]{figs/first_nog4D.pdf}\par +\vfill +\eject + +\title{Non-interacting Hamiltonian} +\begin{itemize} + \item $\epsilon^2\ll|k|\ll\epsilon$ +\end{itemize} +\vfill +\hfil\includegraphics[width=\textwidth]{figs/second_nog4D.pdf}\par +\vfill +\eject + +\title{Non-interacting Hamiltonian} +\begin{itemize} + \item $|k|\ll\epsilon^2$ +\end{itemize} +\vfill +\hfil\includegraphics[width=200pt]{figs/third_nog4D.pdf}\par +\vfill +\eject + +\title{Theorem} +$\exists U_0,\epsilon_0>0$, independent, such that, for $\epsilon<\epsilon_0$, $|U|<U_0$, +\begin{itemize} + \item the free energy + $$ + f:=-\frac1{|\Lambda|\beta}\log\mathrm Tr(e^{-\beta \mathcal H}) + $$ + is analytic in $U$, uniformly in $\beta$ and $|\Lambda|$, + \item the two-point Schwinger function + $$ + s_2(x-y):=\frac{\mathrm Tr(e^{-\beta \mathcal H}a_xa_y^\dagger)}{\mathrm Tr(e^{-\beta \mathcal H})} + $$ + is analytic in $U$, uniformly in $\beta$ and $|\Lambda|$. +\end{itemize} +\vfill +\eject + +\title{Renormalization group flow} +\vfill +\includegraphics[width=250pt]{figs/flow.pdf} + +\end{document} @@ -0,0 +1,32 @@ +* Typeset + +In order to typeset the LaTeX presentation, run + pdflatex Jauslin_Qmath_2016.tex + pdflatex Jauslin_Qmath_2016.tex + + +* Files + + Jauslin_Qmath_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/basegrid.pdf b/figs/basegrid.pdf Binary files differnew file mode 100644 index 0000000..8f86225 --- /dev/null +++ b/figs/basegrid.pdf diff --git a/figs/cellgraph.pdf b/figs/cellgraph.pdf Binary files differnew file mode 100644 index 0000000..926e2c5 --- /dev/null +++ b/figs/cellgraph.pdf diff --git a/figs/first_nog4D.pdf b/figs/first_nog4D.pdf Binary files differnew file mode 100644 index 0000000..221d4b0 --- /dev/null +++ b/figs/first_nog4D.pdf diff --git a/figs/flow.pdf b/figs/flow.pdf Binary files differnew file mode 100644 index 0000000..8d95959 --- /dev/null +++ b/figs/flow.pdf diff --git a/figs/global_nog4D.pdf b/figs/global_nog4D.pdf Binary files differnew file mode 100644 index 0000000..04b6afa --- /dev/null +++ b/figs/global_nog4D.pdf diff --git a/figs/hoppings4.pdf b/figs/hoppings4.pdf Binary files differnew file mode 100644 index 0000000..7138faa --- /dev/null +++ b/figs/hoppings4.pdf diff --git a/figs/monolayer.pdf b/figs/monolayer.pdf Binary files differnew file mode 100644 index 0000000..e1d7c9a --- /dev/null +++ b/figs/monolayer.pdf diff --git a/figs/second_nog4D.pdf b/figs/second_nog4D.pdf Binary files differnew file mode 100644 index 0000000..8970f30 --- /dev/null +++ b/figs/second_nog4D.pdf diff --git a/figs/third_nog4D.pdf b/figs/third_nog4D.pdf Binary files differnew file mode 100644 index 0000000..b8601c3 --- /dev/null +++ b/figs/third_nog4D.pdf diff --git a/header.sty b/header.sty new file mode 100644 index 0000000..4ebdea7 --- /dev/null +++ b/header.sty @@ -0,0 +1,13 @@ +%% +%% Load packages +%% + +\usepackage{color} +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{array} +\usepackage{etoolbox} +\usepackage{dsfont} + 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$\square$} |