X-Git-Url: http://git.treefish.org/~alex/phys/poster_lattice14.git/blobdiff_plain/9bbbb764996009dceaa53089cc486481d5f6e609..6b7830840a1224625e380d198b5b6ad1fd65702d:/poster_lattice14.tex?ds=inline

diff --git a/poster_lattice14.tex b/poster_lattice14.tex
index 0071ad8..3cb7127 100644
--- a/poster_lattice14.tex
+++ b/poster_lattice14.tex
@@ -3,6 +3,8 @@
 %
 \usepackage{etex}
 
+\usepackage[svgnames]{xcolor}
+
 \usepackage{color,colortbl,times,graphicx,multicol}
 \usepackage{psboxit,epsfig,wrapfig,boxedminipage}
 \usepackage[absolute]{textpos}
@@ -145,48 +147,152 @@
 %%%%%%%%%%%%%%%%%%%%%%% 2 columns %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{multicols}{2}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%% Chapter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% \fcolorbox{black}{kfug-yellow}
-% {
-%   \begin{minipage}[b]{350mm}
-%     \begin{center}
-%       \vspace*{7mm}
-%       \large \centering{\textcolor{black}{\LARGE\sf \bf{U(1) Lattice Gauge-Higgs Model}}}
-%     \end{center}
-%     \vspace*{1mm}
-%   \end{minipage}
-% }
 
-\large \centering{\textcolor{cyan}{\LARGE\sf Action}}
+%%%%%%%%%%%%%%%%%%%%%%% ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf The action}}
 
 \vspace{1.0cm}
 
 \begin{minipage}[b]{350mm}
 
-  The {\bf continuum action} of scalar electrodynamics is given by
-  \begin{equation}
-    S = \int{d^4x} \left(\frac{1}{4} |F_{\mu \nu}|^2 + |(\partial_\mu + ieA_\mu)\phi|^2 + m^2(\phi^* \phi) + \lambda(\phi^* \phi)^2\right) \quad , 
-  \end{equation}
-  where $e$ is the gauge coupling, $m$ the mass of the complex scalar $\phi$ and $\lambda$ the Higgs coupling constant.
+  In the conventional notation the lattice action is given by (the lattice constant is set to $a=1$)
 
   \vspace{1cm}
 
-  In the conventional notation the {\bf lattice action} is given by (the lattice constant is set to $a=1$)
+  \begin{flushleft}
+    \small
+    {\color{cyan}Gauge field $U_{\vec{n},\mu}$} \quad 
+    {\color{magenta}1st flavor Higgs field $\phi_{\vec{n}}^1$} \quad
+    {\color{ForestGreen}2nd flavor Higgs field $\phi_{\vec{n}}^2$}
+  \end{flushleft}
   \begin{eqnarray}
-    S &=& S_G[U] + S_H[U,\phi] \label{latac} \\ \nonumber \\
-    S_G &=& -\beta \sum_{x,\nu < \rho} \Re{\left(U_{\nu\rho}(x)\right)}, \quad \beta=\frac{1}{2e^2} \nonumber \\ \nonumber \\
-    S_H &=& \sum_x \left[- \frac{1}{2} \sum_{\mu=1}^4 \left( \phi(x)^* U_\mu(x) \phi(x+\hat{\mu}) + \phi(x)^* U_\mu(x-\hat{\mu})^*\phi(x-\hat{\mu})\right) \right . \nonumber \\ 
-    && \quad\quad\;\, + \left . \kappa \phi(x)^*\phi(x) + \lambda\left(\phi(x)^*\phi(x)\right)^2 \right], \quad \kappa = \frac{m^2+8}{2} \nonumber \quad . \nonumber
-  \end{eqnarray} 
-
-% \begin{wrapfigure}{r}{0.5\textwidth}
-%   \begin{center}
-% 		\includegraphics[width=0.49\columnwidth]{sine}
-%   \end{center}
-%   \caption{This is the sine function.}\label{fig1}
-% \end{wrapfigure}
-
-\vspace{-24pt}
+    S \hspace{0.1cm} & = & S_G[U] + S_H[U,\phi] \label{latac} \\ \nonumber \\
+    S_G & = & -  \beta \sum_{\vec{n}} \sum_{\mu < \nu}  Re \; {\color{cyan}U_{\vec{n},\mu} \, U_{\vec{n} + \hat{\mu}, \nu} \, U_{\vec{n} + \hat{\nu}, \mu}^\star \, U_{\vec{n},\nu}^\star} 
+    \\
+    S_{H} & = & \sum_{\vec{n}}\! \Bigg[ \kappa^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^2
+    + \lambda^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^4  
+    + \kappa^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^2
+    + \lambda^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^4 \Bigg ] \ \\
+    &-& \sum_{\vec{n}}\! \Bigg[ \sum_{\mu}\! \Bigg( e^{\delta_{\mu 4} \mu^1}{\color{magenta}{\phi^1_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n},\mu}} \,  {\color{magenta}\phi^1_{\vec{n}+\widehat{\mu}}}   
+    + e^{-\delta_{\mu 4} \mu^1} {\color{magenta}{\phi^1_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n} - \widehat{\mu},\mu}^\star} \,  {\color{magenta}\phi^1_{\vec{n}-\widehat{\mu}}} \Bigg) \Bigg] \nonumber \\
+    &-& \sum_{\vec{n}}\! \Bigg[ \sum_{\mu}\! \Bigg( e^{\delta_{\mu 4} \mu^2}{\color{ForestGreen}{\phi^2_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n},\mu}^\star} \,  {\color{ForestGreen}\phi^2_{\vec{n}+\widehat{\mu}}}   
+    + e^{-\delta_{\mu 4} \mu^2} {\color{ForestGreen}{\phi^2_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n} - \widehat{\mu},\mu}} \,  {\color{ForestGreen}\phi^2_{\vec{n}-\widehat{\mu}}} \Bigg) \Bigg]
+    \nonumber
+  \end{eqnarray}
+  \begin{flushright}
+    \small
+    {\color{gray}$U_{\vec{n},\mu} \in U(1)$, $\phi_{\vec{n}} \in \mathbb{C}$}
+  \end{flushright}
+
+
+  \vspace{0.2cm}
+
+  \vspace{0.2cm}
+
+  with $\beta$ the inverse gauge coupling, $\kappa^i$ the effective masses and $\lambda^i$ the Higgs coupling constants.
+
+  \vspace{-24pt}
+\end{minipage}
+\vspace{2.0cm}
+
+
+%%%%%%%%%%%%%%%%%%%%%%% FLUX ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf Flux representation of the action}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+  {\textcolor{cyan}{\Large\sf The basic idea}} 
+  is to expand the partition sum and perform the summation over the original degrees of freedom.
+
+  \vspace{0.5cm}
+
+  {\textcolor{cyan}{\Large\sf As an example}} 
+  we look at a single nearest neighbour term
+  \begin{eqnarray}
+    Z \; \propto \; e^{\phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}}}
+    \; = \; \sum_{k_{x,\mu}}    \frac{1}{ (k_{x,\mu})!} \; 
+    \bigg[ \, \phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}} \bigg]^{\, k_{x,\mu}} \quad .
+    \nonumber
+  \end{eqnarray}
+
+  Performing the summation over $\phi^i$ our partition sum no longer depends on the fields $\phi^i$
+  \begin{eqnarray*}
+    Z \; = \; \sum_{\{\phi\}} \sum_{\{U\}} \; e^{-S_G(U)-S_H(U,\phi)} &=& \sum_{\{\phi\}} \sum_{\{U\}} \; e^{-S_G(U)} \sum_{\{k,l\}} F(U,\phi,k,l) \\
+    &=& \sum_{\{k,l\}} \sum_{\{U\}} \; e^{-S_G(U)} \underbrace{\sum_{\{\phi\}} F(U,\phi,k,l)}_{\textnormal{perform this summation}} \quad .
+  \end{eqnarray*}
+
+  {\textcolor{cyan}{\Large\sf Finally}}
+  we end up with a real and positive partition sum plus constraints for the dual degrees of freedom
+  \begin{eqnarray*}
+    Z \; = \; \sum_{\{k,l\}} \sum_{\{p\}} FB(k,l,p) = \hspace{-0.5cm} \sum_{\{p, k^1, l^1, k^2, l^2\}} \hspace{-0.5cm} {\cal W}(p,k,l) \, {\cal C}_B(p,k^1,k^2) \, {\cal C}_F(k^i) \quad .
+  \end{eqnarray*}
+
+  \vspace{0.2cm}
+
+  \begin{center}
+    \includegraphics[height=13cm]{dofs.pdf}
+  \end{center}
+
+  \vspace{-24pt}
+\end{minipage}
+\vspace{2.0cm}
+
+
+%%%%%%%%%%%%%%%%%%%%%%% PHASE DIAGRAM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf Phase diagram}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+  \begin{center}
+    \includegraphics[height=20cm]{phasediagram.pdf}
+    \cite{PhysRevLett.111.141601}
+  \end{center}
+
+  \vspace{-24pt}
+\end{minipage}
+\vspace{2.0cm}
+
+%%%%%%%%%%%%%%%%%%%%%%% MASS CORRELATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf Mass correlators}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+  \begin{center}
+    \includegraphics[height=28cm]{mass.pdf}
+  \end{center}
+
+  \vspace{-24pt}
+\end{minipage}
+\vspace{2.0cm}
+
+%%%%%%%%%%%%%%%%%%%%%%% CONDENSATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf Condensation}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+  \begin{center}
+    %\includegraphics[height=28cm]{mass.pdf}
+  \end{center}
+
+  \vspace{-24pt}
 \end{minipage}
 \vspace{2.0cm}
 
@@ -209,20 +315,20 @@ Program on {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16).
 \vspace{1.7cm}
 \large \centering{\textcolor{cyan}{\Large\sf References}}
 
-\vspace{1.0cm}
+\vspace{-1.0cm}
 
 \begin{minipage}[b]{350mm}
-    \begin{multicols}{2}
+    %\begin{multicols}{2}
 
-%\hrulefill
-\vspace{-8cm}
-\footnotesize
-% \bibliographystyle{h-physrev}
-% \bibliography{lgt.bib}
-\vspace{-3cm}
+      % \hrulefill
+      \vspace{-8cm}
+      \footnotesize
+      \bibliographystyle{plain}
+      \bibliography{bib}
+      \vspace{-3cm}
 
-	\end{multicols}\vspace{-24pt}
-\end{minipage}
+    %\end{multicols}\vspace{-24pt}
+  \end{minipage}
 
 \end{multicols}
 \end{document}
