As presented at Qmath13 on 2016-10-09HEADv1.0master

15 files changed, 421 insertions, 0 deletions

diff --git a/Jauslin_Qmath_2016.tex b/Jauslin_Qmath_2016.tex
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+\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}
diff --git a/README b/README
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+* 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
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--- /dev/null
+++ b/figs/basegrid.pdf
diff --git a/figs/cellgraph.pdf b/figs/cellgraph.pdf
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index 0000000..926e2c5
--- /dev/null
+++ b/figs/cellgraph.pdf
diff --git a/figs/first_nog4D.pdf b/figs/first_nog4D.pdf
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index 0000000..221d4b0
--- /dev/null
+++ b/figs/first_nog4D.pdf
diff --git a/figs/flow.pdf b/figs/flow.pdf
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index 0000000..8d95959
--- /dev/null
+++ b/figs/flow.pdf
diff --git a/figs/global_nog4D.pdf b/figs/global_nog4D.pdf
new file mode 100644
index 0000000..04b6afa
--- /dev/null
+++ b/figs/global_nog4D.pdf
diff --git a/figs/hoppings4.pdf b/figs/hoppings4.pdf
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index 0000000..7138faa
--- /dev/null
+++ b/figs/hoppings4.pdf
diff --git a/figs/monolayer.pdf b/figs/monolayer.pdf
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index 0000000..e1d7c9a
--- /dev/null
+++ b/figs/monolayer.pdf
diff --git a/figs/second_nog4D.pdf b/figs/second_nog4D.pdf
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index 0000000..8970f30
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+++ b/figs/second_nog4D.pdf
diff --git a/figs/third_nog4D.pdf b/figs/third_nog4D.pdf
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index 0000000..b8601c3
--- /dev/null
+++ b/figs/third_nog4D.pdf
diff --git a/header.sty b/header.sty
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--- /dev/null
+++ b/header.sty
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+%%
+%% 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
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--- /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
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--- /dev/null
+++ b/presentation.sty
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+%%
+%% 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
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--- /dev/null
+++ b/toolbox.sty
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+%%
+%% 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$\square$}