add footer, work on second line of boxes
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@ -22,6 +22,7 @@
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\definecolor{fooyellow}{RGB}{234,180,0}
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\definecolor{fooyellow}{RGB}{234,180,0}
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\definecolor{lavender}{rgb}{0.71, 0.49, 0.86}
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\definecolor{lavender}{rgb}{0.71, 0.49, 0.86}
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\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36}
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\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36}
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\definecolor{footer}{cmyk}{0.739,0.288,0,0.278}
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\newcommand{\violet}[1]{\textcolor{violet}{#1}}
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\newcommand{\violet}[1]{\textcolor{violet}{#1}}
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\newcommand{\orange}[1]{\textcolor{orange}{#1}}
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\newcommand{\orange}[1]{\textcolor{orange}{#1}}
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\newcommand{\purple}[1]{\textcolor{purple}{#1}}
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\newcommand{\purple}[1]{\textcolor{purple}{#1}}
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@ -151,17 +152,17 @@
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\column{0.5}
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\column{0.5}
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\block{Similarity Renormalization Group (SRG)}{
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\block{Similarity Renormalization Group (SRG)}{
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\begin{minipage}{0.49\linewidth}
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\begin{minipage}{0.49\linewidth}
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The pillar of the SRG formalism is the flow equation
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SRG flow equation
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\begin{equation}
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\begin{equation}
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\label{eq:flowEquation}
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\label{eq:flowEquation}
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\dv{\boldsymbol{H}(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\boldsymbol{H}(s)}
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\dv{\boldsymbol{H}(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\boldsymbol{H}(s)}
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\end{equation}
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\end{equation}
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with the similarity transformed Hamiltonian
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Similarity transformed Hamiltonian
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\begin{equation}
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\begin{equation}
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\label{eq:SRG_Ham}
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\label{eq:SRG_Ham}
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\boldsymbol{H}(s) = \boldsymbol{U}(s) \, \boldsymbol{H} \, \boldsymbol{U}^\dagger(s)
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\boldsymbol{H}(s) = \boldsymbol{U}(s) \, \boldsymbol{H} \, \boldsymbol{U}^\dagger(s)
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\end{equation}
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\end{equation}
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In this work, we use Wegner's generator
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Wegner generator
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%\begin{equation}
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%\begin{equation}
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% \boldsymbol{\eta}(s) = \dv{\boldsymbol{U}(s)}{s} \boldsymbol{U}^\dagger(s) = - \boldsymbol{\eta}^\dag(s)
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% \boldsymbol{\eta}(s) = \dv{\boldsymbol{U}(s)}{s} \boldsymbol{U}^\dagger(s) = - \boldsymbol{\eta}^\dag(s)
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%\end{equation}
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%\end{equation}
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@ -172,7 +173,7 @@
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\hfill\vline\hfill
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\hfill\vline\hfill
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\begin{minipage}{0.49\linewidth}
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\begin{minipage}{0.49\linewidth}
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\begin{tikzfigure}
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\begin{tikzfigure}
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\includegraphics[width=0.9\textwidth]{SRGMatrix}
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\includegraphics[width=1.1\textwidth]{SRGMatrix}
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\end{tikzfigure}
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\end{tikzfigure}
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\end{minipage}
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\end{minipage}
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}
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}
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@ -188,18 +189,17 @@
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\column{0.65}
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\column{0.65}
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\block{SRG-$GW$}
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\block{SRG-$GW$}
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{
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{
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\begin{minipage}{0.575\linewidth}
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\begin{minipage}{0.6\linewidth}
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\begin{equation*}
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\begin{equation*}
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\begin{split}
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\begin{split}
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&\blue{\widetilde{\boldsymbol{F}}_{pq}}(s) = \delta_{pq} \blue{\epsilon^{\text{HF}}_{p}} + \sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 } \red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}} \qty[1 - e^{-((\Delta_{pr}^{\nu})^2+(\Delta_{qr}^{\nu})^2) s} ] \\
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\blue{\widetilde{\boldsymbol{F}}_{pq}}(s) &= \delta_{pq} \blue{\epsilon^{\text{HF}}_{p}} + \sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 } \red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}} \qty[1 - e^{-((\Delta_{pr}^{\nu})^2+(\Delta_{qr}^{\nu})^2) s} ] \\
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&\qq{with} \Delta_{pr}^{\nu} = \teal{\epsilon^{GW}_{p}} - \teal{\epsilon^{GW}_{r}} \pm \orange{\Omega_{\nu}} \\
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\Delta_{pr}^{\nu} &= \teal{\epsilon^{GW}_{p}} - \teal{\epsilon^{GW}_{r}} \pm \orange{\Omega_{\nu}} \\
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\\
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\\
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&\violet{\widetilde{\Sigma}_{pq}^{\SRGGW}}(\omega;s) = \\
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\violet{\widetilde{\Sigma}_{pq}^{\SRGGW}} &= \sum_{i\nu} \frac{e^{-(\Delta_{pi}^{\nu})^2 s}\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}e^{-(\Delta_{qi}^{\nu})^2 s}}{\omega - \teal{\epsilon^{GW}_{i}} + \orange{\Omega_{\nu}}} + \sum_{a\nu} \frac{e^{-(\Delta_{pa}^{\nu})^2 s}\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}e^{-(\Delta_{qa}^{\nu})^2 s}}{\omega - \teal{\epsilon^{GW}_{a}} - \orange{\Omega_{\nu}}}
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&\sum_{i\nu} \frac{\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}e^{-((\Delta_{pi}^{\nu})^2+(\Delta_{qi}^{\nu})^2) s}}{\omega - \teal{\epsilon^{GW}_{i}} + \orange{\Omega_{\nu}}} + \sum_{a\nu} \frac{\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}e^{-((\Delta_{pa}^{\nu})^2+(\Delta_{qa}^{\nu})^2) s}}{\omega - \teal{\epsilon^{GW}_{a}} - \orange{\Omega_{\nu}}}
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\end{split}
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\end{split}
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\end{equation*}
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\end{equation*}
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\end{minipage}
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\end{minipage}
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\begin{minipage}{0.425\linewidth}
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\begin{minipage}{0.4\linewidth}
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\begin{tikzfigure}
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\begin{tikzfigure}
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\includegraphics[width=\textwidth]{fig1.pdf}
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\includegraphics[width=\textwidth]{fig1.pdf}
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\end{tikzfigure}
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\end{tikzfigure}
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@ -210,8 +210,8 @@
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\block{Functional form of the qs$GW$ and SRG-qs$GW$}
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\block{Functional form of the qs$GW$ and SRG-qs$GW$}
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{
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{
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\begin{minipage}[t]{0.275\linewidth}
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\begin{minipage}[t]{0.24\linewidth}
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\vspace{2.5cm}
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\vspace{3cm}
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\begin{equation*}
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\begin{equation*}
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\begin{split}
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\begin{split}
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&\boldsymbol{\Sigma}^{\text{qs}GW}_{pq}(\eta) = \\
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&\boldsymbol{\Sigma}^{\text{qs}GW}_{pq}(\eta) = \\
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@ -221,14 +221,14 @@
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\end{equation*}
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\end{equation*}
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\end{minipage}
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\end{minipage}
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\begin{adjustbox}{valign=t}
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\begin{adjustbox}{valign=t}
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\begin{minipage}[t]{0.45\linewidth}
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\begin{minipage}[t]{0.5\linewidth}
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\begin{tikzfigure}
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\begin{tikzfigure}
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\includegraphics[width=0.8\textwidth]{fig2.pdf}
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\includegraphics[width=0.82\textwidth]{fig2.pdf}
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\end{tikzfigure}
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\end{tikzfigure}
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\end{minipage}
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\end{minipage}
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\end{adjustbox}
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\end{adjustbox}
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\begin{minipage}[t]{0.275\linewidth}
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\begin{minipage}[t]{0.26\linewidth}
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\vspace{2.5cm}
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\vspace{3cm}
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\begin{equation*}
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\begin{equation*}
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\begin{split}
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\begin{split}
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&\boldsymbol{\Sigma}^{\text{SRG-qs}GW}_{pq}(s) = \\
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&\boldsymbol{\Sigma}^{\text{SRG-qs}GW}_{pq}(s) = \\
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@ -270,7 +270,6 @@
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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 863481).}
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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 863481).}
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\end{columns}
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\end{columns}
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\node [above right,outer sep=0pt,minimum width=\paperwidth,align=center,draw,fill=blue!30] at (bottomleft) {Blabla};
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\node [above right,outer sep=0pt,minimum width=\paperwidth,align=left,draw,fill=footer] at (bottomleft) {\Large \textcolor{white}{\textbf{amarie@irsamc.ups-tlse.fr \hspace{50cm} arXiv:2303.05984}} };
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\end{document}
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\end{document}
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