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 \documentclass{PoS}
 
-\title{Contribution title}
+\usepackage[intlimits]{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{dsfont}
+\usepackage{subfigure}
 
-\ShortTitle{Short Title for header}
+\title{Solving the sign problem of two flavor scalar electrodynamics at finite chemical potential}
 
-\author{Ydalia Delgado Mercado, Christof Gattringer, Alexander Schmidt
-%         \thanks{Y.D.M and A.S. are members of the doctoral training program FWF DK 1203 ''{\sl Hadrons in Vacuum, Nuclei and Stars}''. Y.D.M.  is furthermore supported by the Research Executive Agency of the European Union under Grant Agreement number PITN-GA-2009-238353 (ITN STRONGnet). This work is partly supported also by DFG SFB TRR55.}\\
-         \\
-         Institut f\"ur Physik,
-        Karl-Franzens-Universit\"at, 8010 Graz, Austria \\ \\
-        \email{ydalia.delgado-mercado@uni-graz.at} \\ \email{christof.gattringer@uni-graz.at} \\ \email{alexander.schmidt@uni-graz.at} }
+\ShortTitle{Solving the sign problem of  scalar electrodynamics at finite chemical potential}
 
-\abstract{..........................\
-          ...........................}
+\author{Ydalia Delgado
+\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
+\\E-mail: \email{ydalia.delgado-mercado@uni-graz.at}}
 
-\FullConference{31st International Symposium on Lattice Field Theory - LATTICE 2013\\
-		July 29 - August 3, 2013\\
-		Mainz, Germany}
+\author{Christof Gattringer
+\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
+\\E-mail: \email{christof.gattringer@uni-graz.at}}
 
+\author{Alexander Schmidt
+\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
+\\E-mail: \email{alexander.schmidt@uni-graz.at}}
 
+
+\abstract{
+We explore two flavor  scalar electrodynamics on the lattice, which has a 
+complex phase problem at finite chemical potential. By rewriting the action 
+in terms of dual variables this complex phase problem can be solved exactly. 
+The dual variables are link- and plaquette occupation numbers, subject to local
+constraints that have to be respected by the Monte Carlo algorithm. 
+For the simulation we use a local update as well as the newly developed
+``surface worm algorithm'', which is a generalization of the Prokof'ev Svistunov 
+worm algorithm concept for simulating the dual representation of abelian 
+Gauge-Higgs models on a lattice. We assess the performance of the two algorithms, 
+present results for the phase diagram 
+and discuss condensation phenomena.}
+
+\FullConference{XXIX International Symposium on Lattice Field Theory \\
+                 July 29 $-$ August 03 2013\\
+                 Mainz, Germany}
+ 
 \begin{document}
 
-\section{dfg}
+\section{Motivation}
 
-dfgdfg
+At finite chemical potential $\mu$ the fermion determinant of QCD becomes complex
+and can not be interpreted as a probability weight in a Monte Carlo simulation.
+This so-called "complex phase problem" or "sign problem" has considerably 
+slowed down the exploration of QCD at finite density using lattice methods.  
+Although a lot of effort has been put into solving the complex phase problem of 
+QCD (see, e.g., \cite{reviews} for recent reviews), the final goal of a proper ab-initio 
+simulation of lattice QCD at finite density has not been achieved yet.
 
-\begin{thebibliography}{99}
-\bibitem{...} 
-....
+For some models, as well as for QCD in limiting cases, it is possible to deal with the complex phase 
+problem (see, e.g., \cite{solve-sign-problem}) with different techniques. Here we use a dual 
+representation, i.e., a reformulation of the system with new degrees of freedom,
+which has been shown to be a very powerful method that can solve the complex 
+phase problem of different models \cite{dual} without making any approximation of the partition sum.  
+In the following we present another example where the dual representation can be applied successfully.  
+We consider scalar QED with two flavors, i.e., a compact U(1) gauge field coupled to two complex scalar 
+fields with opposite charge and a quartic self interaction \cite{prl}. We explore the full phase diagram as a 
+function of the inverse gauge coupling and the mass parameter,  and present some results at finite $\mu$.
 
-\end{thebibliography}
+After mapping the degrees of freedom of the system to the dual variables, the weight in the 
+partition sum is positive and real and usual Monte Carlo techniques can be applied.  However, 
+the dual variables, links and plaquettes for this model, are subject to non-trivial constraints.
+Therefore one has to choose a proper algorithm in order to sample the system efficiently.  In our case, we have
+used two different Monte Carlo algorithms:  A local update algorithm \cite{z3} and an extension \cite{swa} of the
+Prokof'ev Svistunov worm algorithm \cite{worm}. In addition to discussing the physics of the model, we also present
+a comparison of the performance of the two algorithms 
+ 
+ 
+\section{Scalar electrodynamics}
+ 
+In the conventional representation two flavor scalar electrodynamics is a model of two flavors of 
+oppositely charged complex fields $\phi_x, \chi_x \in \mathds{C}$ living on the 
+sites $x$ of the lattice, interacting via the gauge fields $U_{x,\sigma} \in$ U(1) sitting on the links. 
+We use 4-d euclidean lattices of size $V_4 = N_s^3 \times N_t$ with periodic 
+boundary conditions for all directions. The lattice spacing is set to 1, i.e., all dimensionful quantities 
+are in units of the lattice spacing. 
+
+We write the action as the sum, $S = S_U + S_\phi + S_\chi$, where $S_U$ is the gauge action 
+and $S_\phi$ and $S_\chi$ are the actions for the two scalars. For the gauge action we use 
+Wilson's form
+\begin{equation} 
+S_U \; = \; - \beta \, \sum_x \sum_{\sigma < \tau} \mbox{Re} \; U_{x,\sigma} U_{x+\widehat{\sigma}, \tau}
+U_{x+\widehat{\tau},\sigma}^\star U_{x,\tau}^\star \; .
+\label{gaugeaction}
+\end{equation}
+The sum runs over all plaquettes, $\widehat{\sigma}$ and $\widehat{\tau}$ denote the unit vectors in $\sigma$- and 
+$\tau$-direction and the asterisk is used for complex conjugation.  
+The action for the field $\phi$ is 
+\begin{equation}
+S_\phi   
+\; =   \sum_x \!\Big(  M_\phi^2 \, |\phi_x|^2  + \lambda_\phi |\phi_x|^4  -
+\sum_{\nu = 1}^4 \!
+\big[  e^{-\mu_\phi  \delta_{\nu, 4} } \, \phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}} 
+\, + \, 
+e^{\mu_\phi \delta_{\nu, 4}} \, \phi_x^\star \, 
+U_{x-\widehat{\nu}, \nu}^\star \, \phi_{x-\widehat{\nu}}  \big] \!  \Big) .
+\label{matteraction}  
+\end{equation}
+By $M_\phi^2$  we denote the combination $8 + m_\phi^2$, where $m_\phi$ is the bare mass
+parameter of the field $\phi$ and $\mu_\phi$ is the chemical potential, which favors forward
+hopping in time-direction (= 4-direction). We also allow for a quartic self interaction of the scalar fields and 
+the corresponding coupling is denoted as $\lambda_\phi$. Note that for $\mu_\phi \neq 0$ (\ref{matteraction}) 
+is complex, i.e., in the conventional form  the theory has a complex action problem.
+
+The action for the field $\chi$ has the same form as (\ref{matteraction}) but with complex conjugate link 
+variables $U_{x,\nu}$ such that $\chi$ has opposite charge.  $M_\chi^2$, $\mu_\chi$ and $\lambda_\chi$  
+are used for the parameters of $\chi$. 
+
+The partition sum $Z = \int D[U] D[\phi,\chi] e^{-S_U - S_\chi - S_\phi}$  is obtained by
+integrating the Boltzmann factor over all field configurations. The measures are products over
+the measures for each individual degree of freedom.
+
+
+
+\vskip2mm
+\noindent  
+{\bf Dual representation:} A detailed derivation of the dual representation for the one flavor
+model is given in \cite{swa} and the two flavor version we consider here simply uses two copies
+of the representation of the matter fields. The dual variables for the first flavor will be denoted by 
+$j_{x,\nu},  \overline{j}_{x,\nu}$, while $l_{x,\nu}$ and $\overline{l}_{x,\nu}$ are used for the second flavor.
+The dual representation of the partition sum for scalar QED 
+with two flavors of matter fields is given by
+\begin{equation}
+\hspace*{-3mm} Z = \!\!\!\!\!\! \sum_{\{p,j,\overline{j},l,\overline{l} \}} \!\!\!\!\!\!  {\cal C}_g[p,j,l]  \;  {\cal C}_s  [j] \;   {\cal C}_s  [l] \;  {\cal W}_U[p] 
+\; {\cal W}_\phi \big[j,\overline{j}\,\big] \, {\cal W}_\chi \big[l,\overline{l}\,\big]  .
+\label{Zfinal}
+\end{equation} 
+The sum runs over all configurations of the dual variables: The occupation numbers 
+$p_{x,\sigma\tau} \in \mathds{Z}$ assigned to the plaquettes of the lattice and the flux variables  $j_{x,\nu},  l_{x,\nu} \in \mathds{Z}$ and
+$\overline{j}_{x,\nu},  \overline{l}_{x,\nu} \in \mathds{N}_0$ living on the links. The flux variables $j$ and $l$ are subject
+to the constraints ${\cal C}_s$,
+\begin{equation}
+ {\cal C}_s [j] \, = \, \prod_x \delta \! \left( \sum_\nu \partial_\nu j_{x,\nu}  \right)\;  , \; \;
+ {\cal C}_s [l] \, = \, \prod_x \delta \! \left( \sum_\nu \partial_\nu l_{x,\nu}  \right) , \;
+\label{loopconstU1}
+\end{equation}
+which enforce the conservation of $j$-flux and of $l$-flux at each site of the lattice
+(here $\delta(n)$ denotes the Kronecker delta $\delta_{n,0}$ and $\partial_\nu f_x \equiv 
+f_x - f_{x-\widehat{\nu}}$).
+Another constraint,
+\begin{equation}
+ {\cal C}_g [p,j,l]  \! =\!  \prod_{x,\nu} \! \delta  \Bigg( \!\sum_{\nu < \alpha}\! \partial_\nu p_{x,\nu\alpha}  
+- \!\sum_{\alpha<\nu}\! \partial_\nu p_{x,\alpha\nu} + j_{x,\nu} - l_{x,\nu} \! \Bigg)\! ,
+\label{plaqconstU1}  
+\end{equation}
+connects the plaquette occupation numbers $p$ with the $j$- and $l$-variables. 
+At every link it enforces the combined flux of the plaquette occupation 
+numbers  plus the difference of $j$- and $l$-flux residing on that link to vanish.  The
+fact that $j$- and $l$-flux enter with opposite sign is due to the opposite charge of the two 
+flavors.
+
+The constraints (\ref{loopconstU1}) and (\ref{plaqconstU1}) restrict the admissible
+flux and plaquette occupation numbers giving rise to an interesting geometrical
+interpretation: The $j$- and $l$-fluxes form closed oriented loops made of links. The integers
+$j_{x,\nu}$ and $l_{x,\nu}$ determine how often a link is run through by loop segments, with negative
+numbers indicating net flux in the negative direction. The flux conservation 
+(\ref{loopconstU1}) ensures that only closed loops appear. Similarly, the constraint 
+(\ref{plaqconstU1}) for the plaquette occupation numbers can be seen as a continuity
+condition for surfaces made of plaquettes. The surfaces are either closed
+surfaces without boundaries or open surfaces bounded by  $j$- or $l$-flux.
+
+The configurations of plaquette occupation numbers and fluxes in (\ref{Zfinal}) come with 
+weight factors 
+\begin{eqnarray}
+{\cal W}_U[p] & = & \!\! \! \prod_{x,\sigma < \tau} \! \! \!
+ I_{p_{x,\sigma\tau}}(\beta) \, ,
+\\   
+{\cal W}_\phi \big[j,\overline{j}\big] & = & 
+\prod_{x,\nu}\! \frac{1}{(|j_{x,\nu}|\! +\! \overline{j}_{x,\nu})! \, 
+\overline{j}_{x,\nu}!} 
+\prod_x e^{-\mu j_{x,4}}  P_\phi \left( f_x \right) ,
+\nonumber
+\end{eqnarray}
+with 
+\begin{equation}
+f_x \; = \; \sum_\nu\!\big[ |j_{x,\nu}|\!+\!  |j_{x-\widehat{\nu},\nu}|  \!+\!
+2\overline{j}_{x,\nu}\! +\! 2\overline{j}_{x-\widehat{\nu},\nu} \big] \; ,
+\end{equation}
+which is an even number. The $I_p(\beta)$
+in the weights  ${\cal W}_U$ are the modified Bessel functions and the $P_\phi (2n)$ in 
+${\cal W}_\phi$  are the integrals
+\begin{equation}
+P_\phi (2n)  \; = \;  \int_0^\infty dr \, r^{2n+1}
+\,  e^{-M_\phi^2\, r^2 - \lambda_\phi r^4} = \sqrt{\frac{\pi}{16 \lambda}}  \, \left(\frac{-\partial}{\partial M^2}\right)^{\!n} \;  
+e^{\, M^4 / 4 \lambda} \left[1- erf(M^2/2\sqrt{\lambda})\right] \; .
+\end{equation}
+These integrals are related to derivatives of the error function and we evaluate them numerically and
+pre-store them for the Monte Carlo simulation. The weight factors $ {\cal
+W}_\chi$ are the same as the $ {\cal W}_\phi$, only  the parameters $M_\phi^2$,
+$\lambda_\phi$, $\mu_\phi$ are replaced by  $M_\chi^2$, $\lambda_\chi$, $\mu_\chi$. All
+weight factors are real and positive and the partition sum (\ref{Zfinal}) thus  is
+accessible  to Monte Carlo techniques,  using the plaquette occupation numbers and the
+flux variables as the new degrees of freedom. 
+
+
+\section{Monte Carlo simulation}
+ 
+Because the dual variables are subject to non-trivial constraints, they cannot be modified randomly during 
+the update. Here we use two strategies: A local update, referred to as LMA (local Metropolis algorithm),
+which consists of three types of steps: Steps where we change plaquettes bounded by matter flux,  steps where 
+we change the plaquettes on 3-cubes, and steps where we propose double lines of matter flux around the temporal 
+direction. These changes are built such that the constraints remain intact for each individual step and the 
+tests of the LMA are reported in \cite{prl,z3,swa}.  
+
+Another possibility is to use an extension of the worm
+algorithm \cite{worm}, the so called surface worm algorithm \cite{swa}, which we refer to as SWA. Here initially 
+the constraints are violated at a single link and the SWA subsequently propagates this defect on the lattice 
+until the defect is healed in a final step. For both the LMA and the SWA the unconstrained $\overline{l}$ and 
+$\overline{j}$ variables are updated with conventional Metropolis steps. 
+Here we present results for both algorithms and
+assess their performance.
+
+\subsection{Local Metropolis algorithm LMA}
+Let us begin by describing the LMA. It consists of the following update steps:
+\begin{itemize}
+\vspace*{-1mm}
+\item A sweep for each unconstrained variable $\overline{l}$ and $\overline{j}$ 
+raising or lowering their occupation number by one unit.
+%
+\vspace*{-1mm}
+\item ``Plaquette update'': 
+It consists of increasing or decreasing a plaquette occupation number
+$p_{x,\nu\rho}$ and
+the link fluxes (either $j_{x,\sigma}$ or $l_{x,\sigma}$) at the edges of $p_{x,\nu\rho}$ by $\pm 1$ as 
+illustrated in Fig.~\ref{plaquette}. The change of $p_{x, \nu \rho}$ 
+by $\pm 1$ is indicated by the signs $+$ or $-$, while the flux variables $j$ ($l$) are denoted by the thin red line
+(fat blue lines for the second flavor) and we use a dashed line to indicate a decrease by $-1$ and a full line 
+for an increase by $+1$.
+%
+\vspace*{-1mm}
+\item ``Winding loop update'': 
+It consists of increasing or decreasing the occupation number of both link variables $l$ and $j$ by 
+one unit along a winding loop in any of the 4 directions.  This update is very important because the winding loops
+in time direction are the only objects that couple to the chemical potential.
+%
+\vspace*{-1mm}
+\item ``Cube update'':  The plaquettes of 3-cubes
+of our 4-d lattice are changed according to one of the two patterns illustrated in 
+Fig.~\ref{cube}. 
+Although the plaquette and winding loop update are enough to satisfy ergodicity, 
+the cube update helps for decorrelation in the region of 
+parameters where the system is dominated by closed surfaces, i.e., where the link
+acceptance rate is small.
+\end{itemize}
+\vspace*{-1mm}
+A full sweep consists of updating all links, plaquettes, 3-cubes and winding loops on the lattice,
+offering one of the changes mentioned above and accepting them with the Metropolis 
+probability computed from the local weight factors.
+
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=\textwidth,clip]{pics/plaquettes}
+\end{center}
+\vspace{-4mm}
+\caption{Plaquette update: A plaquette occupation number is changed by $+1$ or
+$-1$ and the links $j$ (thin red links) or $l$ (fat blue links) of the plaquette are changed simultaneously. The
+full line indicates an increase by +1 and a dashed line a decrease by $-1$. 
+The directions $1 \le \nu_1 < \nu_2 \le 4$
+indicate the plane of the plaquette.} \label{plaquette}
+\vspace{-2mm}
+\end{figure}
+
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=0.7\textwidth,clip]{pics/cubes}
+\end{center}
+\vspace{-4mm}
+\caption{Cube update: Here we show the two possible changes in the plaquette occupation numbers on a 3-cube. 
+The edges of the 3-cube are parallel to the directions $1 \leq \nu_1 < \nu_2 < \nu_3 \leq 4$.} \label{cube}
+\vspace*{-2mm}
+\end{figure}
+
+\subsection{Surface worm algorithm SWA}
+ 
+Instead of the LMA we can use a generalization of the the worm algorithm, the SWA.
+Here we only shortly describe the SWA and refer to  \cite{swa} for a detailed description.
+
+The SWA is constructed by breaking up the smallest update elements of the LMA, i.e., the plaquette updates, 
+into smaller building blocks called ``segments'' (examples are shown in Fig.~\ref{segments}), used to build 
+larger surfaces on which the flux and plaquette variables are changed. In the SWA the constraints are temporarily 
+violated at a link $L_V$, the head of the worm, and the two sites at its endpoints. The SWA then transports this defect on the
+lattice until it closes with a final step that heals the constraint.
+The admissible configurations are generated using 3 elements:
+
+\begin{enumerate}
+
+\item The worm starts by changing either the $l$ or the $j$  flux by $\pm 1$ at 
+a randomly chosen link (step 1 in Fig.~\ref{worm} where a worm for $j$ fluxes starts).
+
+\item The first link becomes the head of the worm $L_V$.
+The defect at $L_V$ is then propagated through the lattice by 
+attaching segments of the same kind of flux ($j$ or $l$) as the first segment, 
+which are chosen in such a way that the constraints are always 
+obeyed at the link where the next segment is attached (step 2 in Fig.~\ref{worm}).
+
+\item Attaching segments the defect is propagated through the lattice until the worm decides to
+end with the insertion of another unit of link flux at $L_V$ (step 3 in Fig.~\ref{worm}) to heal the violated constraint.
+ 
+\end{enumerate}
+A full sweep consists of $V_4$ worms with $l$ fluxes and $V_4$ worms with $j$ fluxes, 
+plus a sweep of the unconstrained 
+variables $\overline{l}$ and $\overline{j}$,
+and a sweep of winding loops (as explained for the LMA).
+
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=\textwidth,clip]{pics/segments}
+\end{center}
+\vspace{-4mm}
+\caption{Examples of segments for the links $j$ (lhs.) and $l$ (rhs.) 
+in the $\nu_1$-$\nu_2$-plane ($\nu_1 < \nu_2$).
+The plaquette occupation numbers are changed as indicated by the signs. 
+The full (dashed) links are changed by $+1$ ($-1$). The empty link shows
+where the segment is attached to the worm and the dotted link is the new position of the link
+$L_V$ where the constraints are violated.} \label{segments}
+\vspace{-2mm}
+\end{figure}
+
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=\textwidth,clip]{pics/worm}
+\end{center}
+\vspace{-4mm}
+\caption{A simple example for an update with the surface worm algorithm.  
+See the text for a detailed explanation of the steps involved.} \label{worm}
+\vspace{-2mm}
+\end{figure} 
+
+
+\section{Results}
+\vspace{-1mm}
+\noindent
+In this section we discuss the results from the numerical analysis.  We first show 
+an assessment of both algorithms and compare their performance. Subsequently 
+we discuss the physics of scalar QED at finite density and present the phase diagram.  
+In both cases we use thermodynamical observables and their fluctuations. In particular 
+we use the following observables which can be evaluated as simple derivatives of 
+$\ln Z$ in both the conventional and the dual representations:  
 
-\end{document}
+\vskip5mm
+\noindent
+The first and second derivatives with respect to the inverse gauge coupling $\beta$, i.e., 
+the plaquette expectation value and its susceptibility,
 
+\begin{equation}
+\langle U \rangle = \frac{1}{6 N_s^3 N_t}\frac{\partial}{\partial \beta} \ln\ Z\quad , \quad
+\chi_{U} = \frac{1}{6 N_s^3 N_t}\frac{\partial^2}{\partial \beta^2} \ln\ Z\ .
+\end{equation}
+
+\noindent We also consider the particle number density $n$ 
+and its susceptibility which are the first and second derivatives 
+with respect to the chemical potential,
+
+\begin{equation}
+\langle n \rangle  = \frac{1}{N_s^3 N_t}\frac{\partial}{\partial \mu} \ln\ Z\quad , \quad
+\chi_{n} = \frac{1}{N_s^3 N_t}\frac{\partial^2}{\partial \mu^2} \ln\ Z\ .
+\end{equation}
+
+\noindent Finally, we analyze the derivatives with respect to $M^2$,
+
+\begin{equation}
+\langle |\phi|^2 \rangle = \frac{1}{N_s^3 N_t}\frac{\partial}{\partial M^2} \ln\ Z\quad , \quad
+\chi_{|\phi|^2} = \frac{1}{N_s^3 N_t}\frac{\partial^2}{\partial (M^2)^2} \ln\ Z\ .
+\end{equation}
+
+\subsection{Assessment of the LMA and SWA algorithms}
+\noindent
+For the comparison of our two algorithms we considered the U(1) gauge-Higgs model coupled
+with one (see \cite{swa}) and two scalar fields (as described here).  
+First we checked the correctness of the SWA comparing the results for different 
+lattices sizes and parameters.  Examples for the one flavor model were presented  
+in \cite{swa}. 
+
+In Fig.~\ref{obs} we now show some examples for the two flavor case. The top figures 
+of Fig.~\ref{obs} show 
+$\langle |\phi|^2 \rangle$ (lhs.) and the corresponding susceptibility (rhs.) as a function of 
+$\mu_\phi = \mu_\chi = \mu$ at $\beta = 0.85$ and 
+$M_\phi^2 = M_\chi^2 = M^2  = 5.325$ on a lattice of size $12^3 \times 60$.  This point is located
+in the Higgs phase and does not show any phase transition as a function of $\mu$.  The bottom
+plots show the particle number $\langle n \rangle$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$
+for $\beta = 0.75$ and $M^2 = 5.73$ on a lattice of volume $12^3 \times 60$.  Here we observe
+a  pronounced first order transition from the confining phase into the Higgs phase. 
+It is obvious that in all four plots the agreement between the results from the LMA and from the 
+SWA is excellent.
+
+\begin{figure}[h]
+\begin{center}
+\hbox{\includegraphics[width=\textwidth,clip]{pics/aphi}}
+\vskip5mm
+\hbox{\hspace{4mm}\includegraphics[width=0.97\textwidth,clip]{pics/bn}}
+\end{center}
+\vspace{-6mm}
+\caption{Observables for the two flavor model as a function of $\mu$ for different 
+parameters on a $12^3 \times 60$ lattice.
+We compare results from the SWA (circles) and the LMA (triangles).} \label{obs}
+\vspace*{-2mm}
+\end{figure}
+
+\noindent
+In order to obtain a measure of the computational effort, in \cite{swa} we compared the normalized 
+autocorrelation time $\overline{\tau}$ of the SWA and LMA for 
+the one flavor model for different volumes and parameters.  We concluded that,
+the SWA outperforms the local update near a phase transition and if
+the acceptance rate of the constrained link variables is not very low (e.g., lhs.\ of Fig.~\ref{auto}).  
+On the other hand, for parameter values where the constrained links have a very low acceptance rate 
+the worm algorithm has difficulties to efficiently sample the 
+system because it changes the link occupation number in every move, while the LMA has a sweep with only
+closed surfaces. The plot on the rhs. of Fig.~\ref{auto} shows how $\overline{\tau}$ for
+$\langle U \rangle$ is larger for the SWA than for the LMA.  We remark however, that this performance issue
+can be overcome easily by augmenting the SWA with sweeps of cube updates as used in the LMA.
+
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=\textwidth,clip]{pics/u2}
+\end{center}
+\vspace{-4mm}
+\caption{Normalized autocorrelation times $\overline{\tau}$ for the observables $\langle U \rangle$ and 
+$\langle |\phi|^2 | \rangle$ for two different sets
+of parameters for the one flavor model.  Left: Parameter values close to a first order phase transition. 
+Right: A parameter set characterized by a low acceptance for matter flux.  Both simulations  
+were done on $16^4$ lattices, with data taken from \cite{swa}.} \label{auto}
+\vspace*{-2mm}
+\end{figure}
+
+\subsection{Physics results}
+So far one of the main physics results of our studies of 2-flavor scalar QED  
+(already published in \cite{prl}) is the full phase diagram of the considered 
+model in the $\beta$-$M^2$ plane (using $M_\phi^2 = M_\chi^2 = M^2$)
+at $\mu=0$ and the analysis of phase 
+transitions driven by the chemical potential $\mu_\phi = \mu_\chi = \mu$ 
+when starting from the different 
+phases of the model. For the sake of completeness we here again show the 
+$\mu = 0$ phase diagram, and then present new results for the observables 
+in the $\beta$-$M^2$ plane at several values of $\mu > 0$, which illustrate the 
+shift of the phase-boundaries at $\mu >  0$, i.e., the positions of the critical surfaces.
+In addition we show that some of the transitions at finite $\mu$ can be seen as
+condensation phenomena of the dual occupation numbers.
+
+\subsubsection*{Phase diagram at $\mu=0$}
+
+The results for the phase diagram at $\mu = 0$ are summarized in Fig.~\ref{phasediagram}. The various phase 
+boundaries were determined from the observables $\langle U \rangle$ and $\langle |\phi|^2 \rangle$ and the 
+corresponding susceptibilities. We found that the phase boundary separating  Higgs- and
+confining phase is of strong first order, the line separating confining- and Coulomb phase is  of weak
+first order, and the boundary between Coulomb- and Higgs phase is a continuous transition. 
+Our results for the $\mu = 0$ phase diagram are in qualitative
+agreement with the results for related
+models \cite{Lang} studied in the conventional formulation.
+
+\begin{figure}[h]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=85mm,clip]{pics/phasediagram}
+\caption{Phase diagram in the $\beta$-$M^2$ plane at $\mu = 0$. We show
+the phase boundaries determined from the maxima of the susceptibilities $\chi_U$ and $\chi_{\phi}$ and the
+inflection points of $\chi_n$.}
+\label{phasediagram}
+\end{figure}
+
+\begin{figure}[p]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=\linewidth,clip]{pics/muphases}
+\caption{The observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$, and 
+$\langle n \rangle$ as a function of $\beta$ and $M^2$ for different chemical 
+potentials $\mu = 0.0,\,0.5,\,1.0$ and $1.5$. It can be seen how the phase 
+boundaries shift with increasing chemical potential.}
+\label{muphases}
+\end{figure}
+
+\begin{figure}[t]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=\linewidth,clip]{pics/occutrans_plaq}
+\caption{We here show the plaquette expectation value $\langle U \rangle$ and the corresponding suscpetibility $\chi_U$ as function of the chemical potential, for two different volumes $12^3\times60$ and $16^3\times60$.}
+\label{occutrans_plaq}
+\end{figure}
+\begin{figure}[b]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=\linewidth,clip]{pics/occutrans}
+\caption{Link occupation numbers $\bar{j}$, $\bar{l}$, $j$, $l$ and plaquette occupation numbers $p$ for values of $\mu$
+just below (top) and above (bottom) the critical value $\mu_c$ for the transition from the confining- to the Higgs-phase.}
+\label{occutrans}
+\end{figure}
+
+
+\subsubsection*{Phase boundaries at $\mu > 0$}
+
+As a first step in the determination of the phase boundaries as functions of all three parameters $\beta, \, M^2$ and $\mu$, 
+in Fig.~\ref{muphases} we plot the observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$ and $\langle n \rangle$ as functions 
+of $\beta$ and $M^2$ for four different values of the chemical potential $\mu=0.0,\, 0.5,\, 1.0$ and $1.5$.
+
+The phase-transition from the confining phase to the Coulomb phase shown in Fig.~\ref{phasediagram} 
+is characterized by a rapid increase of  $\langle U \rangle$ across the transition but does not give rise to 
+significant changes in the other observables (compare the top row of plots in Fig.~\ref{muphases}). 
+This behavior persists also at finite $\mu$ and  the 
+confinement-Coulomb transition can only be seen in the $\langle U \rangle$-plots.
+
+The transition between the Higgs- and the confinig phase is characterized by a strong first order discontinuity in all observables 
+(except for $\langle n \rangle = 0$ at $\mu = 0$), a feature that persists for all our values of $\mu$. Also the transition between the Higgs- and the 
+Coulomb phase is seen in all observables. It is obvious from the plots, that with increasing $\mu$ all three transitions become more pronounced in 
+all variables they are seen in, and the Higgs-Coulomb transition might even change from crossover to first order. Still, the shown results 
+have to be considered preliminary and more detailed studies will be necessary to draw final conclusions. 
+
+\subsubsection*{Dual occupation numbers}
+ 
+The dual reformulation of lattice field theories makes it possible to look at the same physics from a different perspective
+by studying the dynamics of the dual degrees of freedom instead of the conventional ones. 
+This being a feature we find especially interesting about the dual formulation, we here present an example where a transition 
+manifests itself as the condensation of dual variables.
+ 
+Let us first look at the transition using the standard observables. In Fig.~\ref{occutrans_plaq} we 
+plot the plaquette expectation value $\langle U \rangle$ and the corresponding susceptibility $\chi_U$ 
+as function of the chemical potential, for two different volumes $12^3\times60$ and $16^3\times60$. 
+We see that for the larger volume the transition is shifted slightly towards lower chemical potential, 
+but the volume dependence seems to be reasonably small. The parameters $\beta$ and $M^2$ are 
+fixed to $\beta=0.75$ and $M^2=5.73$. Increasing the chemical potential takes us from the confining- 
+to the Higgs phase where we cross the phase boundary 
+at some critical value of $\mu$, which is $\mu_c\simeq2.65$ 
+for the larger of the two lattices. Below the critical value of the chemical potential both 
+$\langle U \rangle$ and $\chi_U$ are independent of $\mu$, which is characteristic for a Silver Blaze type of transition \cite{cohen}.
+At $\mu_c$ a strong first order transition signals the entry into the Higgs phase.
+ 
+In Fig.~\ref{occutrans} we have a look at the same transition, by now showing typical configurations of the dual variables 
+just below (top) and above (bottom) the critical chemical potential $\mu_c$.
+In particular we show snapshots of the occupation numbers of all dual link variables $\bar{j}$, $\bar{l}$, $j$, 
+$l$ and dual plaquette variables $p$.  Here blue links/plaquettes depict positive occupation numbers, 
+green links/plaquettes depict negative occupation numbers and links/plaquettes with $0$-occupation 
+are not shown. It can be seen that below $\mu_c$ links and plaquettes are hardly occupied, 
+while above $\mu_c$ their occupation is abundant. In that sense the Silver Blaze transition of Fig.~\ref{occutrans_plaq} 
+can be understood as a condensation phenomenon of the dual variables, which is a new perspective on the underlying 
+physics we gained from the dual reformulation of the problem.
+
+\section*{Acknowledgments} 
+\noindent
+We thank Hans Gerd Evertz 
+for numerous discussions that helped to shape this project and for 
+providing us with the software to compute the autocorrelation times. 
+We also acknowledge interesting discussions with Thomas Kloiber 
+on aspects of the dual formulation for charged scalar fields. 
+This work was supported by the Austrian Science Fund, 
+FWF, DK {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16). Y.~Delgado is supported by
+the Research Executive Agency (REA) of the European Union 
+under Grant Agreement number PITN-GA-2009-238353 (ITN STRONGnet) and by {\it Hadron Physics 2}. 
+Furthermore this work is partly supported by DFG TR55, ``{\sl Hadron Properties from Lattice QCD}'' 
+and by the Austrian Science Fund FWF Grant.\ Nr.\ I 1452-N27.   
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+\end{document}
\ No newline at end of file
