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\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

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\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}