poster_lattice14.tex

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 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 summation 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     \nonumber
 236   \end{eqnarray}
 237
 238   Performing the summation over $\phi^i$ our partition sum no longer depends on the fields $\phi^i$
 239   \begin{eqnarray*}
 240     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) \\
 241     &=& \sum_{\{k,l\}} \sum_{\{U\}} \; e^{-S_G(U)} \underbrace{\sum_{\{\phi\}} F(U,\phi,k,l)}_{\textnormal{perform this summation}} \quad .
 242   \end{eqnarray*}
 243
 244   {\textcolor{cyan}{\Large\sf Finally}}
 245   we end up with a real and positive partition sum plus constraints for the dual degrees of freedom
 246   \begin{eqnarray*}
 247     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 .
 248   \end{eqnarray*}
 249
 250   \vspace{0.2cm}
 251
 252   %\begin{center}
 253     %\includegraphics[height=13cm]{dofs.pdf}
 254   %\end{center}
 255
 256   \vspace{-24pt}
 257 \end{minipage}
 258 \vspace{2.0cm}
 259
 260
 261 %%%%%%%%%%%%%%%%%%%%%%% PHASE DIAGRAM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 262 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 263
 264 \large \centering{\textcolor{cyan}{\LARGE\sf Phase diagram} \cite{PhysRevLett.111.141601}}
 265
 266 \vspace{1.0cm}
 267
 268 \begin{minipage}[b]{350mm}
 269
 270   \begin{center}
 271     \includegraphics[height=22cm]{phasediagram.pdf}
 272   \end{center}
 273
 274   \vspace{-24pt}
 275 \end{minipage}
 276 \vspace{2.0cm}
 277
 278 %%%%%%%%%%%%%%%%%%%%%%% MASS CORRELATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%
 279 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 280
 281 \large \centering{\textcolor{cyan}{\LARGE\sf Mass correlators in the confined phase (preliminary)}}
 282
 283 \vspace{1.0cm}
 284
 285 \begin{minipage}[b]{350mm}
 286
 287   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.
 288
 289   \vspace{-0.5cm}
 290
 291   \begin{center}
 292     \includegraphics[height=14.5cm]{mass.pdf}
 293   \end{center}
 294
 295   \vspace{-24pt}
 296 \end{minipage}
 297 \vspace{2.0cm}
 298
 299 %%%%%%%%%%%%%%%%%%%%%%% CONDENSATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 300 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 301
 302 \large \centering{\textcolor{cyan}{\LARGE\sf Condensation (preliminary)}}
 303
 304 \vspace{1.0cm}
 305
 306 \begin{minipage}[b]{350mm}
 307
 308   We here show different observables as function of $\mu$. The dotted lines show the masses $U_1$ and $U_1$ determined from the plots above.
 309
 310   \begin{center}
 311     \includegraphics[height=35.8cm]{finmu_840.pdf}
 312   \end{center}
 313
 314   \vspace{-24pt}
 315 \end{minipage}
 316 \vspace{2.0cm}
 317
 318 %%%%%%%%%%%%%%%%%%%%%%% SUMMARY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 319 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 320
 321 \large \centering{\textcolor{cyan}{\LARGE\sf Summary}}
 322
 323 \vspace{1.0cm}
 324
 325 \begin{minipage}[b]{350mm}
 326
 327   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.
 328
 329   \vspace{-24pt}
 330 \end{minipage}
 331 \vspace{2.0cm}
 332
 333 %%%%%%%%%%%%%%%%%%%%%%%%%% Acknowledgments %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 334
 335 \hrule
 336 \vspace{1.0cm}
 337 \large \centering{\textcolor{cyan}{\Large\sf Acknowledgments}}
 338
 339 \vspace{1.0cm}
 340
 341 \begin{minipage}[b]{350mm}
 342 This work was supported by the Austrian Science Fund, FWF, through the Doctoral
 343 Program on {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16).
 344 \end{minipage}
 345
 346 %%%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 347
 348 %\hrule
 349 \vspace{1.7cm}
 350 \large \centering{\textcolor{cyan}{\Large\sf References}}
 351
 352 \vspace{-1.0cm}
 353
 354 \begin{minipage}[b]{350mm}
 355     %\begin{multicols}{2}
 356
 357       % \hrulefill
 358       \vspace{-8cm}
 359       \footnotesize
 360       \bibliographystyle{plain}
 361       \bibliography{bib}
 362       \vspace{-3cm}
 363
 364     %\end{multicols}\vspace{-24pt}
 365   \end{minipage}
 366
 367 \end{multicols}
 368 \end{document}