+ 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}
+
+
+ \vspace{0.2cm}
+
+ with $\beta$ the inverse gauge coupling, $\kappa^i$ the mass parameters and $\lambda^i$ the Higgs couplings.
+
+ \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 integral 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 .
+ \end{eqnarray}
+
+ Performing the integral 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 integral}} \nonumber \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} \cite{PhysRevLett.111.141601}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+ \begin{center}
+ \includegraphics[height=22cm]{phasediagram.pdf}
+ \end{center}
+
+ \vspace{-24pt}
+\end{minipage}
+\vspace{2.0cm}
+
+%%%%%%%%%%%%%%%%%%%%%%% MASS CORRELATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf Mass correlators in the confined phase (preliminary)}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+ 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.
+
+ \vspace{-0.5cm}
+
+ \begin{center}
+ \includegraphics[height=14.5cm]{mass.pdf}
+ \end{center}
+
+ \vspace{-24pt}
+\end{minipage}
+\vspace{2.0cm}
+
+%%%%%%%%%%%%%%%%%%%%%%% CONDENSATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf Condensation (preliminary)}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+ 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.
+ \begin{center}
+ \includegraphics[height=35.8cm]{finmu_840.pdf}
+ \end{center}
+
+ \vspace{-24pt}
+\end{minipage}
+\vspace{2.0cm}
+
+%%%%%%%%%%%%%%%%%%%%%%% SUMMARY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\large \centering{\textcolor{cyan}{\LARGE\sf Summary}}
+
+\vspace{1.0cm}
+
+\begin{minipage}[b]{350mm}
+
+ 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.
+
+ \vspace{-24pt}
projects / phys / poster_lattice14.git / blobdiff