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diff --git a/Jauslin_Princeton_2017.tex b/Jauslin_Princeton_2017.tex new file mode 100644 index 0000000..2218b6d --- /dev/null +++ b/Jauslin_Princeton_2017.tex @@ -0,0 +1,327 @@ +\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} |