add footer, work on second line of boxes

This commit is contained in:
Antoine Marie 2023-03-27 22:32:26 +02:00
parent 08538b12eb
commit b92deb622e
2 changed files with 17 additions and 18 deletions

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

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