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