\documentclass{ian-presentation}

\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{color}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{array}
\usepackage{dsfont}

\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\includegraphics[height=140pt]{monolayer.pdf}\par
\eject

\title{Bilayer graphene}
\begin{itemize}
  \item 2 graphene layers in {\it AB} stacking.
\end{itemize}
\vfill
\hfil\includegraphics[height=140pt]{bilayer.pdf}\par
\eject

\title{Hamiltonian}
\begin{itemize}
  \item Model for the electrons.
  \item Hamiltonian:
  $$
    \mathcal H=\mathcal H_0+U\mathcal V
  $$
  \item $\mathcal H_0$: kinetic term: hoppings (tight-binding approximation).
  \item $U\mathcal V$: interaction: weak, short-range (screened Coulomb).
\end{itemize}
\eject

\title{Lattice structure}
\begin{itemize}
  \item Rhombic lattice $\Lambda\equiv\mathbb Z^2$, 4 atoms per site.
\end{itemize}
\vfill
\hfil\includegraphics[height=140pt]{bilayer_cell.pdf}\par
\eject

\title{Hoppings}
\vfill
\hfil\includegraphics[height=140pt]{bilayer.pdf}
\vfill
\eject

\addtocounter{page}{-1}
\title{Hoppings}
\vfill
\hfil\includegraphics[height=140pt]{hoppings_0.pdf}
\vfill
\eject

\addtocounter{page}{-1}
\title{Hoppings}
\vfill
\hfil\includegraphics[height=140pt]{hoppings_1.pdf}
\vfill
\eject

\addtocounter{page}{-1}
\title{Hoppings}
\vfill
\hfil\includegraphics[height=140pt]{hoppings_3.pdf} \vfill
\eject

\addtocounter{page}{-1}
\title{Hoppings}
\vfill
\hfil\includegraphics[height=140pt]{hoppings.pdf}
\vfill
\eject

\title{Non-interacting Hamiltonian}
$$
  \begin{array}{r@{\ }>{\displaystyle}l}
    \mathcal H_0:=&
    -\gamma_0\sum_{\displaystyle\mathop{\scriptstyle x\in\Lambda}_{j=1,2,3}}\left(
      {\color{red}a_x^\dagger b_{x+\delta_j}}
      +
      {\color{red}b_{x+\delta_j}^\dagger a_x}
      +
      {\color{red}\tilde b_x^\dagger \tilde a_{x-\delta_j}}
      +
      {\color{red}\tilde a_{x-\delta_j}^\dagger\tilde b_x}
    \right)
    \\[1cm]
    &-\gamma_1\sum_{ x\in\Lambda}\left(
      {\color{green}a_x^\dagger \tilde b_x}
      +
      {\color{green}\tilde b_x^\dagger a_x}
    \right)
    \\[0.75cm]
    &-\gamma_3\sum_{\displaystyle\mathop{\scriptstyle x\in\Lambda}_{j=1,2,3}}\left(
      {\color{blue}\tilde a_{x-\delta_1}^\dagger b_{x-\delta_1-\delta_j}}
      +
      {\color{blue}b_{x-\delta_1-\delta_j}^\dagger\tilde a_{x-\delta_1}}
    \right)
  \end{array}
$$
\vfill
\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}
$$
  \mathcal V=\sum_{x,y}v(|x-y|)\left(n_x-\frac12\right)\left(n_y-\frac12\right)
$$
\begin{itemize}
  \item $\displaystyle\sum_{x,y}$: sum over pairs of atoms
  \item $n_x\equiv\alpha_x^\dagger\alpha_x$
  \item $v(|x-y|)\leqslant e^{-c|x-y|}$, $c>0$
  \item $-\frac12$: {\it half-filling}.
\end{itemize}
\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: for $\alpha,\alpha'\in\{a,b,\tilde a,\tilde b\}$,
  $$
    s(x-y):=\frac{\mathrm{Tr}(e^{-\beta \mathcal H}\alpha'_x\alpha_y^\dagger)}{\mathrm{Tr}(e^{-\beta \mathcal H})}
  $$
  is analytic in $U$, uniformly in $\beta$ and $|\Lambda|$.
\end{itemize}
\vfill
\eject

\title{Free model}
\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{Free model}
\begin{itemize}
  \item Eigenvalues of $\hat H_0(k)$:
\end{itemize}
\vfill
\hfil\includegraphics[width=\textwidth]{bands.pdf}\par
\vfill
\eject

\title{Perturbation theory}
\begin{itemize}
  \item Trotter formula:
  $$
    \kern-10pt
    \frac{\mathrm{Tr}(e^{-\beta(\mathcal H_0+U\mathcal V)})}{\mathrm{Tr}(e^{-\beta\mathcal H_0})}=
    \left<\mathbb T \exp\left(-U\int_0^\beta\kern-5pt dt\ \mathcal V(t)\right)\right>_0
  $$
  \item ``Imaginary time'':
  $$
    \mathcal V(t):=e^{t\mathcal H_0}\mathcal Ve^{-t\mathcal H_0}
  $$
  \vskip-2pt
  \item ``Non-interacting Gibbs measure'':
  $$
    \left<A\right>_0:=\frac{\mathrm{Tr}(e^{-\beta\mathcal H_0}A)}{\mathrm{Tr}(e^{-\beta\mathcal H_0})}
  $$
\end{itemize}
\vfill
\eject

\title{Perturbation theory}
\begin{itemize}
  \item Matsubara frequency: for $\alpha\in\{a,b,\tilde a,\tilde b\}$,
  $$
    \hat\alpha_{k_0,k}:=\int dt\ e^{ik_0t}e^{t\mathcal H_0}\hat\alpha_ke^{-t\mathcal H_0}.
  $$

  \item Non-interacting Gibbs measure: ``Gaussian'', and singular: for $\alpha,\alpha'\in\{a,b,\tilde a,\tilde b\}$,
  $$
    \hat s^{(0)}_{\alpha',\alpha}(k_0,k):=\left<\hat\alpha'_{k_0,k}\hat\alpha_{k_0,k}^\dagger\right>_0=(-ik_0\mathds 1+\hat H_0(k))^{-1}_{\alpha',\alpha}
  $$
\end{itemize}
\vfill
\eject

\title{Scaling}
\begin{itemize}
  \item Eigenvalues of $\hat H_0(k)$:
\end{itemize}
\vfill
\hfil\includegraphics[width=\textwidth]{bands.pdf}\par
\vfill
\eject

\addtocounter{page}{-1}
\title{Scaling}
\begin{itemize}
  \item First regime: $|k|\gg\epsilon$:
\end{itemize}
\vfill
\hfil\includegraphics[width=240pt]{bands_first.pdf}\par
$$
  \hat s^{(0)}(k_0,k)\sim(|k_0|+|k|)^{-1}
$$
\vfill
\eject

\addtocounter{page}{-1}
\title{Scaling}
\begin{itemize}
  \item Second regime: $\epsilon^2\ll|k|\ll\epsilon$:
\end{itemize}
\vfill
\hfil\includegraphics[width=240pt]{bands_second.pdf}\par
$$
  \hat s^{(0)}(k_0,k)\sim\left(|k_0|+\frac{|k|^2}\epsilon\right)^{-1}
$$
\vfill
\eject

\addtocounter{page}{-1}
\title{Scaling}
\begin{itemize}
  \item Third regime: $|k|\ll\epsilon^2$:
\end{itemize}
\vfill
\hfil\includegraphics[height=100pt]{bands_third.pdf}\par
$$
  \hat s^{(0)}(k_0,k)\sim(|k_0|+\epsilon|k|)^{-1}
$$
\vfill
\eject

\title{Renormalization group}
\begin{itemize}
  \item Scale decomposition: scale $h\leqslant 0$:
  $$
    \hat s^{(0)}(k_0,k)\sim 2^{-h}
  $$

  \item Scale by scale integration:
  $$
    \left<A\right>_0=\left<\cdots\left<\left<A\right>_{0,0}\right>_{0,-1}\cdots\right>_{0,h}\cdots
  $$

  \item Effective potential: $\mathcal V_h(t)$:
  $$
    \kern-10pt
    \left<\mathbb T\exp\left(-U\int dt\ \mathcal V_h(t)\right)\right>_{0,h}
    ``="\ \mathbb T\exp\left(-U\int dt\ \mathcal V_{h-1}(t)\right)
  $$
\end{itemize}
\vfill
\eject


\title{Renormalization group flow}
\vfill
\includegraphics[width=250pt]{flow.pdf}

\end{document}
