From d178209af8152764071b8743c47321447ba47c19 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@roma1.infn.it>
Date: Fri, 7 Oct 2016 22:31:25 +0000
Subject: As presented at Qmath13 on 2016-10-09

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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}
-- 
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