%
\usepackage{etex}
+\usepackage[svgnames]{xcolor}
+
\usepackage{color,colortbl,times,graphicx,multicol}
\usepackage{psboxit,epsfig,wrapfig,boxedminipage}
\usepackage[absolute]{textpos}
%%%%%%%%%%%%%%%%%%%%%%% 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}
\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}
projects / phys / poster_lattice14.git / commitdiff