poster_lattice14.tex

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   2 \documentclass[portrait,a0]{a0poster}
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   4 \usepackage{etex}
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  32 \usepackage{url}
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  34 \usepackage{ctable}
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  62 % user defined commands
  63 % own commands
  64 \newcommand{\eq}[1]{(\ref{#1})}
  65 \newcommand{\fig}[1]{Fig.~{\ref{#1}}}
  66 \newcommand{\FC}{\;,}
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  73 \newcommand{\U}[1]{\mathrm{U}(#1)}
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  76
  77 \definecolor{WildStrawberry}{cmyk}{0,0.96,0.39,0} % important phrases
  78 \definecolor{CornflowerBlue}{cmyk}{0.65,0.13,0,0.4} % accentuation2
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  91 \renewcommand{\refname}{\vspace{-1.5cm}}
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  96 %\usepackage{showframe}
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 100
 101 \renewcommand{\baselinestretch}{1.1}
 102
 103
 104
 105
 106 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 107 %%%%%%%%%%%%%%%%%%%%%%% document %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 108 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 109
 110 \begin{document}
 111
 112
 113 \noindent
 114 \hspace*{-36mm}
 115 \includegraphics[scale=1.01,clip]{./hintergrund}
 116
 117
 118 \vspace*{-1140mm}
 119
 120 \vspace{-45mm}
 121 % \hfill \includegraphics[width=.1\textwidth]{../../figures/oeaw_logo}
 122 \hfill
 123 \includegraphics[width=.2\textwidth]{./fwf-logo}
 124 \hspace{-30mm}
 125 %\vspace{-40mm}
 126
 127 \vspace{1cm}
 128
 129 %% Titel
 130 \vspace*{10mm}
 131 \begin{center}
 132 \fcolorbox{white}{white}
 133 {
 134   \begin{minipage}[b]{600mm}
 135     \begin{center}
 136       \vspace*{10mm}
 137        \Huge{\sf
 138         \textcolor{cyan}{\bf Condensation in two flavor scalar electrodynamics with non-degenerate quark masses}}\\[7mm]
 139         \Large{\bf{Alexander Schmidt} \sf{, Philippe de Forcrand, Christof Gattringer} }
 140         \vspace{-1cm}
 141     \end{center}
 142    \vspace*{1cm}
 143   \end{minipage}
 144 }
 145 \end{center}
 146
 147
 148 \vspace{3cm}
 149
 150 %%%%%%%%%%%%%%%%%%%%%%% 2 columns %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 151 \begin{multicols}{2}
 152
 153
 154 %%%%%%%%%%%%%%%%%%%%%%% MOTIVATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 155 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 156
 157 \large \centering{\textcolor{cyan}{\LARGE\sf Motivation}}
 158
 159 \vspace{1.0cm}
 160
 161 \begin{minipage}[b]{350mm}
 162
 163   We study two-flavor scalar electrodynamics with two non-degenerate quark masses to find out about the characteristics of the condensation of this system induced by a finite chemical potential.
 164
 165   \vspace{-24pt}
 166 \end{minipage}
 167 \vspace{2.0cm}
 168
 169
 170 %%%%%%%%%%%%%%%%%%%%%%% ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 171 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 172
 173 \large \centering{\textcolor{cyan}{\LARGE\sf The action}}
 174
 175 \vspace{1.0cm}
 176
 177 \begin{minipage}[b]{350mm}
 178
 179   In the conventional notation the lattice action is given by (the lattice constant is set to $a=1$)
 180
 181   \vspace{1cm}
 182
 183   \begin{flushleft}
 184     \small
 185     {\color{cyan}Gauge field $U_{\vec{n},\mu}$} \quad
 186     {\color{magenta}1st flavor Higgs field $\phi_{\vec{n}}^1$} \quad
 187     {\color{ForestGreen}2nd flavor Higgs field $\phi_{\vec{n}}^2$} \quad \quad
 188     {\color{gray}$U_{\vec{n},\mu} \in U(1)$, $\phi_{\vec{n}} \in \mathbb{C}$}
 189   \end{flushleft}
 190   \begin{eqnarray}
 191     S \hspace{0.1cm} & = & S_G[U] + S_H[U,\phi] \label{latac} \\ \nonumber \\
 192     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}
 193     \\
 194     S_{H} & = & \sum_{\vec{n}}\! \Bigg[ \kappa^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^2
 195     + \lambda^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^4
 196     + \kappa^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^2
 197     + \lambda^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^4 \Bigg ] \ \\
 198     &-& \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}}}
 199     + 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 \\
 200     &-& \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}}}
 201     + 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]
 202     \nonumber
 203   \end{eqnarray}
 204
 205
 206   \vspace{0.2cm}
 207
 208   with $\beta$ the inverse gauge coupling, $\kappa^i$ the mass parameters and $\lambda^i$ the Higgs couplings.
 209
 210   \vspace{-24pt}
 211 \end{minipage}
 212 \vspace{2.0cm}
 213
 214
 215 %%%%%%%%%%%%%%%%%%%%%%% FLUX ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 216 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 217
 218 \large \centering{\textcolor{cyan}{\LARGE\sf Flux representation of the action}}
 219
 220 \vspace{1.0cm}
 221
 222 \begin{minipage}[b]{350mm}
 223
 224   {\textcolor{cyan}{\Large\sf The basic idea}}
 225   is to expand the partition sum and perform the integral over the original degrees of freedom.
 226
 227   \vspace{0.5cm}
 228
 229   {\textcolor{cyan}{\Large\sf As an example}}
 230   we look at a single nearest neighbour term
 231   \begin{eqnarray}
 232     Z \; \propto \; e^{\phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}}}
 233     \; = \; \sum_{k_{x,\mu}}    \frac{1}{ (k_{x,\mu})!} \;
 234     \bigg[ \, \phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}} \bigg]^{\, k_{x,\mu}} \quad .
 235   \end{eqnarray}
 236
 237   Performing the integral over $\phi^i$ our partition sum no longer depends on the fields $\phi^i$
 238   \begin{eqnarray}
 239     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) \\
 240     &=& \sum_{\{k,l\}} \sum_{\{U\}} \; e^{-S_G(U)} \underbrace{\sum_{\{\phi\}} F(U,\phi,k,l)}_{\textnormal{perform this integral}} \nonumber \quad .
 241   \end{eqnarray}
 242
 243   {\textcolor{cyan}{\Large\sf Finally}}
 244   we end up with a real and positive partition sum plus constraints for the dual degrees of freedom
 245   \begin{eqnarray}
 246     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 .
 247   \end{eqnarray}
 248
 249   \vspace{0.2cm}
 250
 251   %\begin{center}
 252     %\includegraphics[height=13cm]{dofs.pdf}
 253   %\end{center}
 254
 255   \vspace{-24pt}
 256 \end{minipage}
 257 \vspace{2.0cm}
 258
 259
 260 %%%%%%%%%%%%%%%%%%%%%%% PHASE DIAGRAM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 261 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 262
 263 \large \centering{\textcolor{cyan}{\LARGE\sf Phase diagram} \cite{PhysRevLett.111.141601}}
 264
 265 \vspace{1.0cm}
 266
 267 \begin{minipage}[b]{350mm}
 268
 269   \begin{center}
 270     \includegraphics[height=22cm]{phasediagram.pdf}
 271   \end{center}
 272
 273   \vspace{-24pt}
 274 \end{minipage}
 275 \vspace{2.0cm}
 276
 277 %%%%%%%%%%%%%%%%%%%%%%% MASS CORRELATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%
 278 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 279
 280 \large \centering{\textcolor{cyan}{\LARGE\sf Mass correlators in the confined phase (preliminary)}}
 281
 282 \vspace{1.0cm}
 283
 284 \begin{minipage}[b]{350mm}
 285
 286   The masses of the bound states $U_1$ and $U_2$ are split because we set the effective masses of the two flavours to different values.
 287
 288   \vspace{-0.5cm}
 289
 290   \begin{center}
 291     \includegraphics[height=14.5cm]{mass.pdf}
 292   \end{center}
 293
 294   \vspace{-24pt}
 295 \end{minipage}
 296 \vspace{2.0cm}
 297
 298 %%%%%%%%%%%%%%%%%%%%%%% CONDENSATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 299 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 300
 301 \large \centering{\textcolor{cyan}{\LARGE\sf Condensation (preliminary)}}
 302
 303 \vspace{1.0cm}
 304
 305 \begin{minipage}[b]{350mm}
 306
 307   We here show different observables as function of $\mu=\mu_1=\mu_2$. The dotted lines show the masses $U_1$ and $U_2$ determined from the plots above. For the observables $\langle\phi^*\phi\rangle$ and $\langle n \rangle$ red symbols belong to flavor 1 and green symbols to flavor 2.
 308   \begin{center}
 309     \includegraphics[height=35cm]{finmu_840_highstat-crop.pdf}
 310   \end{center}
 311
 312   \vspace{-24pt}
 313 \end{minipage}
 314 \vspace{2.0cm}
 315
 316 %%%%%%%%%%%%%%%%%%%%%%% SUMMARY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 317 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 318
 319 \large \centering{\textcolor{cyan}{\LARGE\sf Summary}}
 320
 321 \vspace{1.0cm}
 322
 323 \begin{minipage}[b]{350mm}
 324
 325   Although we studied the condensation of the system with two non-degenerate quark masses, we do not see two seperate condensation points, as we would have expected in first place. At the moment we are doing further simulations to better understand the finite mu transition of the system and the consequences of having two different quark masses.
 326
 327   \vspace{-24pt}
 328 \end{minipage}
 329 \vspace{2.0cm}
 330
 331 %%%%%%%%%%%%%%%%%%%%%%%%%% Acknowledgments %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 332
 333 \hrule
 334 \vspace{1.0cm}
 335 \large \centering{\textcolor{cyan}{\Large\sf Acknowledgments}}
 336
 337 \vspace{1.0cm}
 338
 339 \begin{minipage}[b]{350mm}
 340 This work was supported by the Austrian Science Fund, FWF, through the Doctoral
 341 Program on {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16).
 342 \end{minipage}
 343
 344 %%%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 345
 346 %\hrule
 347 \vspace{1.7cm}
 348 \large \centering{\textcolor{cyan}{\Large\sf References}}
 349
 350 \vspace{-1.0cm}
 351
 352 \begin{minipage}[b]{350mm}
 353     %\begin{multicols}{2}
 354
 355       % \hrulefill
 356       \vspace{-8cm}
 357       \footnotesize
 358       \bibliographystyle{plain}
 359       \bibliography{bib}
 360       \vspace{-3cm}
 361
 362     %\end{multicols}\vspace{-24pt}
 363   \end{minipage}
 364
 365 \end{multicols}
 366 \end{document}