poster_lattice14.tex

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  77 \definecolor{WildStrawberry}{cmyk}{0,0.96,0.39,0} % important phrases
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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 %% Titel
 128 \vspace*{10mm}
 129 \begin{center}
 130 \fcolorbox{white}{white}
 131 {
 132   \begin{minipage}[b]{400mm}
 133     \begin{center}
 134       \vspace*{10mm}
 135        \Huge{\sf
 136         \textcolor{cyan}{\bf Condensation in two flavor scalar electrodynamics with non-degenerate quark masses}}\\[7mm]
 137         \Large{\bf{Alexander Schmidt} \sf{, Philippe de Forcrand, Christof Gattringer} \\ \sf{\large University of Graz}}\\\vspace{-1cm}
 138     \end{center}
 139    \vspace*{1cm}
 140   \end{minipage}
 141 }
 142 \end{center}
 143
 144
 145 \vspace{1.5cm}
 146
 147 %%%%%%%%%%%%%%%%%%%%%%% 2 columns %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 148 \begin{multicols}{2}
 149
 150
 151 %%%%%%%%%%%%%%%%%%%%%%% ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 152 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 153
 154 \large \centering{\textcolor{cyan}{\LARGE\sf The action}}
 155
 156 \vspace{1.0cm}
 157
 158 \begin{minipage}[b]{350mm}
 159
 160   In the conventional notation the lattice action is given by (the lattice constant is set to $a=1$)
 161
 162   \vspace{1cm}
 163
 164   \begin{flushleft}
 165     \small
 166     {\color{cyan}Gauge field $U_{\vec{n},\mu}$} \quad
 167     {\color{magenta}1st flavor Higgs field $\phi_{\vec{n}}^1$} \quad
 168     {\color{ForestGreen}2nd flavor Higgs field $\phi_{\vec{n}}^2$}
 169   \end{flushleft}
 170   \begin{eqnarray}
 171     S \hspace{0.1cm} & = & S_G[U] + S_H[U,\phi] \label{latac} \\ \nonumber \\
 172     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}
 173     \\
 174     S_{H} & = & \sum_{\vec{n}}\! \Bigg[ \kappa^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^2
 175     + \lambda^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^4
 176     + \kappa^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^2
 177     + \lambda^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^4 \Bigg ] \ \\
 178     &-& \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}}}
 179     + 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 \\
 180     &-& \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}}}
 181     + 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]
 182     \nonumber
 183   \end{eqnarray}
 184   \begin{flushright}
 185     \small
 186     {\color{gray}$U_{\vec{n},\mu} \in U(1)$, $\phi_{\vec{n}} \in \mathbb{C}$}
 187   \end{flushright}
 188
 189
 190   \vspace{0.2cm}
 191
 192   \vspace{0.2cm}
 193
 194   with $\beta$ the inverse gauge coupling, $\kappa^i$ the effective masses and $\lambda^i$ the Higgs coupling constants.
 195
 196   \vspace{-24pt}
 197 \end{minipage}
 198 \vspace{2.0cm}
 199
 200
 201 %%%%%%%%%%%%%%%%%%%%%%% FLUX ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 202 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 203
 204 \large \centering{\textcolor{cyan}{\LARGE\sf Flux representation of the action}}
 205
 206 \vspace{1.0cm}
 207
 208 \begin{minipage}[b]{350mm}
 209
 210   {\textcolor{cyan}{\Large\sf The basic idea}}
 211   is to expand the partition sum and perform the summation over the original degrees of freedom.
 212
 213   \vspace{0.5cm}
 214
 215   {\textcolor{cyan}{\Large\sf As an example}}
 216   we look at a single nearest neighbour term
 217   \begin{eqnarray}
 218     Z \; \propto \; e^{\phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}}}
 219     \; = \; \sum_{k_{x,\mu}}    \frac{1}{ (k_{x,\mu})!} \;
 220     \bigg[ \, \phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}} \bigg]^{\, k_{x,\mu}} \quad .
 221     \nonumber
 222   \end{eqnarray}
 223
 224   Performing the summation over $\phi^i$ our partition sum no longer depends on the fields $\phi^i$
 225   \begin{eqnarray*}
 226     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) \\
 227     &=& \sum_{\{k,l\}} \sum_{\{U\}} \; e^{-S_G(U)} \underbrace{\sum_{\{\phi\}} F(U,\phi,k,l)}_{\textnormal{perform this summation}} \quad .
 228   \end{eqnarray*}
 229
 230   {\textcolor{cyan}{\Large\sf Finally}}
 231   we end up with a real and positive partition sum plus constraints for the dual degrees of freedom
 232   \begin{eqnarray*}
 233     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 .
 234   \end{eqnarray*}
 235
 236   \vspace{0.2cm}
 237
 238   %\begin{center}
 239     %\includegraphics[height=13cm]{dofs.pdf}
 240   %\end{center}
 241
 242   \vspace{-24pt}
 243 \end{minipage}
 244 \vspace{2.0cm}
 245
 246
 247 %%%%%%%%%%%%%%%%%%%%%%% PHASE DIAGRAM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 248 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 249
 250 \large \centering{\textcolor{cyan}{\LARGE\sf Phase diagram}}
 251
 252 \vspace{1.0cm}
 253
 254 \begin{minipage}[b]{350mm}
 255
 256   \begin{center}
 257     \includegraphics[height=25cm]{phasediagram.pdf}
 258     \cite{PhysRevLett.111.141601}
 259   \end{center}
 260
 261   \vspace{-24pt}
 262 \end{minipage}
 263 \vspace{2.0cm}
 264
 265 %%%%%%%%%%%%%%%%%%%%%%% MASS CORRELATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%
 266 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 267
 268 \large \centering{\textcolor{cyan}{\LARGE\sf Mass correlators in the confined phase}}
 269
 270 \vspace{1.0cm}
 271
 272 \begin{minipage}[b]{350mm}
 273
 274   For the fundamental correlators $F_1$ and $F_2$, as expected, we see no plateaus. 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.
 275
 276   \begin{center}
 277     \includegraphics[height=28cm]{mass.pdf}
 278   \end{center}
 279
 280   \vspace{-24pt}
 281 \end{minipage}
 282 \vspace{2.0cm}
 283
 284 %%%%%%%%%%%%%%%%%%%%%%% CONDENSATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 285 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 286
 287 \large \centering{\textcolor{cyan}{\LARGE\sf Condensation}}
 288
 289 \vspace{1.0cm}
 290
 291 \begin{minipage}[b]{350mm}
 292
 293   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.
 294
 295   \begin{center}
 296     \includegraphics[height=35cm]{finmu_840.pdf}
 297   \end{center}
 298
 299   \vspace{-24pt}
 300 \end{minipage}
 301 \vspace{2.0cm}
 302
 303 %%%%%%%%%%%%%%%%%%%%%%%%%% Acknowledgments %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 304
 305 \hrule
 306 \vspace{1.0cm}
 307 \large \centering{\textcolor{cyan}{\Large\sf Acknowledgments}}
 308
 309 \vspace{1.0cm}
 310
 311 \begin{minipage}[b]{350mm}
 312 This work was supported by the Austrian Science Fund, FWF, through the Doctoral
 313 Program on {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16).
 314 \end{minipage}
 315
 316 %%%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 317
 318 %\hrule
 319 \vspace{1.7cm}
 320 \large \centering{\textcolor{cyan}{\Large\sf References}}
 321
 322 \vspace{-1.0cm}
 323
 324 \begin{minipage}[b]{350mm}
 325     %\begin{multicols}{2}
 326
 327       % \hrulefill
 328       \vspace{-8cm}
 329       \footnotesize
 330       \bibliographystyle{plain}
 331       \bibliography{bib}
 332       \vspace{-3cm}
 333
 334     %\end{multicols}\vspace{-24pt}
 335   \end{minipage}
 336
 337 \end{multicols}
 338 \end{document}