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diff --git a/Jauslin_Qmath_2016.tex b/Jauslin_Qmath_2016.tex new file mode 100644 index 0000000..ac7649b --- /dev/null +++ b/Jauslin_Qmath_2016.tex @@ -0,0 +1,183 @@ +\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} |