Added results.

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 \documentclass{PoS}
 
 \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 scalar, two-flavored electrodynamics 
+for finite chemical potential and exploring its full phase-diagram}

 
 

-\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 final 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-flavored 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 links and plaquettes, subject to non-trivial 
+constraints, which have to be respected by the Monte Carlo algorithm. 
+Therefore, for the simulation we use a local update and the surface worm algorithm (SWA). 
+The SWA is a generalization of the Prokof'ev Svistunov 
+worm algorithm concept to simulate the dual representation of abelian Gauge-Higgs models on a lattice. 
+We also assess the performance of the SWA and compare it with a local update in the dual representation. 
+Finally, we determine the full phase diagram of the model.
+}
+
+\FullConference{XXIX International Symposium on Lattice Field Theory \\
+                 July 29 $-$ August 03 2013\\
+                 Mainz, Germany}
+ 

 \begin{document}
 
 \begin{document}
 

-\section{dfg}
+\section{Motivation}
+\vspace{-1mm}
+\noindent
+At finite chemical potential $\mu$ the fermion determinant becomes complex
+and can not be interpreted as a probability weight in the Monte Carlo simulation.
+This complex phase problem has slowed down considerably the exploration of QCD
+at finite density using Lattice QCD.  Although many efforts have been put into
+solving the complex phase problem of QCD (see e.g. \cite{reviews}), the final goal
+has not been achieved yet.

 
 

-dfgdfg
+For some models or QCD in limiting cases, it is possible to deal with the complex phase 
+problem (e.g. \cite{solve-sign-problem}).  Among the different techniques, we use the dual representation,
+which has been shown to be a very powerful method that can solve the complex 
+phase problem without making any approximation of the partition sum, i.e. it is an exact method \cite{dual}.  
+In the following we present another example where the dual representation can be applied succesfully.  We consider a compact
+U(1) gauge field coupled with two complex scalar fields with opposite charge. We explore the full phase diagram
+as a function of the gauge coupling, the mass parameter and the chemical potential, which has not yet been studied in detail.
+At finite density we present some preliminary results.

 
 

-\begin{thebibliography}{99}
-\bibitem{...} 
-....
+After mapping the degrees of freedom of the system to its 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 (LMA) \cite{z3} and an extension \cite{swa} of the
+Prokof'ev Svistunov worm algorithm \cite{worm}.   Here we present
+some technical comparison of both algorithms in addition to the physics of the model.
+ 
+ 
+\section{Two-flavored scalar electrodynamics}
+\vspace{-1mm}
+\noindent
+We here study two-flavored scalar electrodynamics, which is a model of two flavors of oppositely charged complex fields $\phi_x, \chi_x \in \mathds{C}$ living on the 
+sites $x$ and 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. Scale setting can be implemented as in any other lattice field theory 
+and issues concerning the continuum behavior are, e.g., discussed in \cite{LuWe}.
+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{eqnarray}
+&& \qquad S_\phi   
+=  \! \sum_x \!\Big(  M_\phi^2 \, |\phi_x|^2  + \lambda_\phi |\phi_x|^4  -
+\label{matteraction} \\
+&& \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) .
+\nonumber
+\end{eqnarray}
+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). The coupling for the quartic term is denoted as 
+$\lambda_\phi$. 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.
+
+Note that for $\mu_\phi \neq 0$ (\ref{matteraction}) is complex, i.e., in the
+conventional form  the theory has a complex action problem.
+
+
+\vskip2mm
+\noindent  
+{\bf Dual representation:} A detailed derivation of the dual representation for the 1-flavor
+model is given in \cite{DeGaSch1} and the generalization to two flavors is straightforward.
+The final result 
+for the dual representation of the partition sum for the gauge-Higgs model with two flavors is
+\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 ${\cal C}_s$ (here $\delta(n)$ denotes the Kronecker delta $\delta_{n,0}$ and $\partial_\nu f_x \equiv 
+f_x - f_{x-\widehat{\nu}}$)
+\begin{equation}
+ {\cal C}_s [j] \, = \, \prod_x \delta \! \left( \sum_\nu \partial_\nu j_{x,\nu}  \right) , \;
+\label{loopconstU1}
+\end{equation}
+which enforce the conservation of $j$-flux and of $l$-flux at each site of the lattice.
+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 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 $f_x = \sum_\nu\!\big[ |j_{x,\nu}|\!+\!  |j_{x-\widehat{\nu},\nu}|  \!+\!
+2\overline{j}_{x,\nu}\! +\! 2\overline{j}_{x-\widehat{\nu},\nu} \big]$ 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 $ P_\phi (2n)  =  \int_0^\infty dr \, r^{2n+1}
+\,  e^{-M_\phi^2\, r^2 - \lambda_\phi r^4} = \sqrt{\pi/16 \lambda}  \, (-\partial/\partial M^2)^n \;  
+e^{M^4 / 4 \lambda} [1- erf(M^2/2\sqrt{\lambda})]$, which we evaluate numerically and
+pre-store for the Monte Carlo. 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. 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}
+\vspace*{-1mm}
+\noindent
+Because the dual variables are subject to non-trivial constraints, they cannot be updated randomly.
+The most straight forward way to update the system is to change complete allowed objects.  In order to
+increase the acceptance rate we use the smallest possible structures.  This algorithm is called local update
+(LMA) and was used in \cite{z3,swa,prl}.  Other possibility is to use an extension of the worm
+algorithm \cite{worm}, the so called surface worm algorithm \cite{swa}.  For this model we use both algorithms and
+assess their performance.
+
+First, we start describing the LMA. It consists of the following updates:
+\begin{itemize}
+\vspace*{-1mm}
+\item A sweep for each unconstrained variable $\overline{l}$ and $\overline{k}$ 
+rising 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 $l_{x,\sigma}$ or $k_{x,\sigma}$) 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 $l$($k$) are denoted by the red(blue) lines
+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 $k$ 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 4d 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., 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 $l$ (red) or $k$ (blue) 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 changes in the plaquette occupation numbers. 
+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}
+
+\noindent
+Instead of the plaquette and cube updates we can use the worm algorithm.
+Here we will shortly describe the SWA (see \cite{swa} for a detailed description) 
+for the variable $l$ (red).
+The algorithm for the other type of link variable works in exactly the same way.
+
+The SWA is constructed by breaking up the smallest update, i.e., the plaquette update 
+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 admissible configurations are produced using 3 steps:
+\begin{enumerate}
+\item The worm starts by changing the flux by $\pm 1$ at a randomly chosen link (step 1 in Fig.~\ref{worm}).
+\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, which are chosen in such a way that the constraints are always 
+obeyed (step 2 in Fig.~\ref{worm}).
+\item 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}).
+ 
+\end{enumerate}
+A full sweep consists of $V_4$ worms using the SWA plus a sweep of the unconstraint 
+variables $\overline{l}$ and $\overline{k}$,
+and a sweep of winding loops (as explained in the LMA).
+
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=\textwidth,clip]{pics/segments}
+\end{center}
+\vspace{-4mm}
+\caption{Examples of positive (lhs.) and negative segments (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{Illustration of the worm algorithm.  See text for an explanation.} \label{worm}
+\vspace{-2mm}
+\end{figure}

 
 

-\end{thebibliography}

 
 

-\end{document}
+\section{Algorithm Assessment}
+\vspace{-1mm}
+\noindent
+For the assessment of both algorithms we used two different models, the U(1) gauge-Higgs model but couple
+only to one scalar field (see \cite{swa}) and the model presented in this proceedings.  In both cases we 
+analyzed the bulk observables (and their fluctuations): 
+$U_P$ which is the derivative wrt. $\beta$ and $|\phi|^2$ (derivative wrt. 
+$\kappa$).  First we checked the correctness of the SWA comparing the results for different 
+lattices sizes and parameters.  Examples for the one flavor model are shown in \cite{swa}.
+Fig.~\ref{obs} shows two observables for the two flavor case.  
+$\langle |\phi|^2 \rangle$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$ 
+for point ``f'' (see phase diagram) on a lattice of size $12^3 \times 60$.  
+We observe very good agreement among the different algorithms.

 
 

+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=\textwidth,clip]{pics/f}
+\includegraphics[width=\textwidth,clip]{pics/f}
+\end{center}
+\vspace{-2mm}
+\caption{Observables $\langle |\phi|^2 \rangle$ (lhs.) and $\chi_\phi$ (rhs.) 
+as a function of $\mu$ for point f on a $12^3 \times 60$ lattice size.
+We compare results from the SWA (circles) and the LMA (crosses).} \label{obs}
+\vspace*{-2mm}
+\end{figure}
+
+\noindent
+In order to obtain a measure of the computational effort, we compared the normalized 
+autocorrelation time $\overline{\tau}$ as defined in \cite{swa} of the SWA and LMA for 
+the one flavored 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 variable is not very low (eg. lhs. of Fig.~\ref{auto}).  
+On the other hand, when the constrained links have a very low acceptance rate 
+the worm algorithm has difficulties to efficiently sample the 
+system because it modifies 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
+$U_P$ is larger for the SWA than for the LMA.  But this can be overcome by offering
+a sweep of cube updates.
+
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=\textwidth,clip]{pics/u2}
+\end{center}
+\vspace{-4mm}
+\caption{Normalized autocorrelation times $\overline{\tau}$ for 2 different set
+of parameters.  Left: parameters close to a first order phase transition. 
+Right: low acceptance rate of the variable $l$.  Both simulations correspond
+to a $16^4$ lattice.  Data taken from \cite{swa}.} \label{auto}
+\vspace*{-2mm}
+\end{figure} 
+
+
+\section{Results}
+\vspace{-1mm}
+\noindent
+One of the main results of these studies so far and already published in \cite{prl} is the full phase diagram of the considered model in the $\beta$-$M^2$ plane at $\mu=0$ and some selected chemical potential driven phase transitions of the measured observables. For the sake of completeness we here again want to show the obtained phase diagram, but as a proceedings-extra also present some plots which show the shifting of the phase-boundaries at $\mu \neq 0$ and measurements of the dual occupation numbers.
+
+\subsection{Phase-diagram at $\mu=0$}
+\noindent
+We studied the different transition lines in Fig.~\ref{phasediagram} using finite size analysis of the measured observables $\langle U \rangle$ and $\langle |\phi|^2 \rangle$ and the corresponding susceptibilities, finding that the phase boundary separating  Higgs- and
+confining phase is 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 conventional results for related
+models \cite{Lang}.
+\begin{figure}[h]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=75mm,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}
+
+\subsection{Phase-boundaries at $\mu \neq 0$}
+\noindent
+In Fig.~\ref{muphases} we plot the observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$, $\langle n \rangle$ as function of $\beta$ and $M^2$ for four different values of the chemical potential $\mu=0,0.5,1,1.5$.
+
+\noindent
+The phase-transition from the confining phase to the Coulomb phase shown in Fig.~\ref{phasediagram} is characterized by $\langle U \rangle$ growing larger across the transition but no significant changes in the other observables, which is the reason why the confinement-Coulomb transition can only be seen in the $\langle U \rangle$-plots.
+For all observables it can be seen that the phase-boundaries in general become more pronounced at higher chemical potential and for the Higgs-Coulomb transition the transition type may 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. 
+\begin{figure}[h]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=130mm,clip]{pics/muphases}
+\caption{We show the observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$, $\langle n \rangle$ as function of $\beta$ and $M^2$ for different $\mu = 0,0.5,1,1.5$. It can be seen how the phase boundaries change with increasing chemical potential.}
+\label{muphases}
+\end{figure}
+
+\subsection{Dual occupation numbers}
+\noindent
+The dual reformulation of a problem 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 exciting about rewriting to dual variables, we here want to present an example.
+
+\noindent
+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 $\mu\simeq2.65$ for the larger and $\mu\simeq2.7$ for the smaller lattice, telling us that the Higgs phase is tilted towards the confining phase in $\mu$-direction. Below the critical value of the chemical potential both $\langle U \rangle$ and $\chi_U$ are independent of $\mu$, which is typical for a Silverblaze type transition.
+
+\noindent
+Then in Fig.~\ref{occutrans} we show the occupation numbers of all dual link variables $\bar{j}$, $\bar{l}$, $j$, $l$ and dual plaquette variables $p$ just below (top) and above (bottom) the critical chemical potential $\mu_c$. Here blue links/plaquettes depict positive occupation numbers, green links/plaquettes depict negative occupation numbers and links/plaquettes with $0$-occupation are spared out. It can be seen that below $\mu_c$ links and plaquettes are hardly occupied, while above $\mu_c$ they are highly occupied. In that sense the Silverblaze transition shown in Fig.~\ref{occutrans_plaq} can be understood as condensation phenomenon, which is a new perspective on the underlying physics we gained from the dual reformulation of the problem.
+
+\begin{figure}[h]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=130mm,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}[h]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=130mm,clip]{pics/occutrans}
+\caption{Dual link occupation numbers $\bar{j}$, $\bar{l}$, $j$, $l$ and dual plaquette occupation numbers $p$ just below (top) and above (bottom) the transition from the confining- to the Higgs-phase shown in the previous plot.}
+\label{occutrans}
+\end{figure}
+
+\section*{Acknowledgments} 
+\vspace{-1mm}
+\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. 
+This work was supported by the Austrian Science Fund, 
+FWF, DK {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16)
+and by the Research Executive Agency (REA) of the European Union 
+under Grant Agreement number PITN-GA-2009-238353 (ITN STRONGnet).
+ 
+\begin{thebibliography}{123456}
+\bibitem{reviews}
+  P.~Petreczky,
+  %``Review of recent highlights in lattice calculations at finite temperature and finite density,''
+  PoS ConfinementX {\bf } (2012) 028
+  [arXiv:1301.6188 [hep-lat]].
+  %%CITATION = ARXIV:1301.6188;%%
+  %3 citations counted in INSPIRE as of 21 Oct 2013
+%  
+  G.~Aarts,
+  %``Complex Langevin dynamics and other approaches at finite chemical potential,''
+  PoS LATTICE {\bf 2012} (2012) 017
+  [arXiv:1302.3028 [hep-lat]].
+  %%CITATION = ARXIV:1302.3028;%%
+  %3 citations counted in INSPIRE as of 08 Apr 2013
+  
+\bibitem{solve-sign-problem}
+  D.~Sexty,
+  %``Simulating full QCD at nonzero density using the complex Langevin equation,''
+  arXiv:1307.7748 [hep-lat].
+  %%CITATION = ARXIV:1307.7748;%%
+  %4 citations counted in INSPIRE as of 21 Oct 2013
+%
+  S.~Chandrasekharan,
+  %``Fermion Bag Approach to Fermion Sign Problems,''
+  Eur.\ Phys.\ J.\ A {\bf 49} (2013) 90
+  [arXiv:1304.4900 [hep-lat]].
+  %%CITATION = ARXIV:1304.4900;%%
+  %1 citations counted in INSPIRE as of 21 Oct 2013
+%
+  G.~Aarts, P.~Giudice, E.~Seiler and E.~Seiler,
+  %``Localised distributions and criteria for correctness in complex Langevin dynamics,''
+  Annals Phys.\  {\bf 337} (2013) 238
+  [arXiv:1306.3075 [hep-lat]].
+  %%CITATION = ARXIV:1306.3075;%%
+  %4 citations counted in INSPIRE as of 21 Oct 2013
+%
+  G.~Aarts, L.~Bongiovanni, E.~Seiler, D.~Sexty and I.~-O.~Stamatescu,
+  %``Controlling complex Langevin dynamics at finite density,''
+  Eur.\ Phys.\ J.\ A {\bf 49} (2013) 89
+  [arXiv:1303.6425 [hep-lat]].
+  %%CITATION = ARXIV:1303.6425;%%
+  %6 citations counted in INSPIRE as of 21 Oct 2013
+%
+  M.~Cristoforetti, F.~Di Renzo, A.~Mukherjee and L.~Scorzato,
+  %``Monte Carlo simulations on the Lefschetz thimble: taming the sign problem,''
+  Phys.\ Rev.\ D {\bf 88} (2013) 051501
+  [arXiv:1303.7204 [hep-lat]].
+  %%CITATION = ARXIV:1303.7204;%%
+  %4 citations counted in INSPIRE as of 21 Oct 2013
+%
+  J.~Bloch,
+  %``A subset solution to the sign problem in simulations at non-zero chemical potential,''
+  J.\ Phys.\ Conf.\ Ser.\  {\bf 432} (2013) 012023.
+  %%CITATION = 00462,432,012023;%%
+%
+  M.~Fromm, J.~Langelage, S.~Lottini, O.~Philipsen,
+  %``The QCD deconfinement transition for heavy quarks and all baryon chemical potentials,''
+  JHEP {\bf 1201} (2012) 042.
+  %  [arXiv:1111.4953 [hep-lat]].
+  %%CITATION = ARXIV:1111.4953;%%
+%  
+  M.~Fromm, J.~Langelage, S.~Lottini, M.~Neuman, O.~Philipsen,
+  %``The silver blaze property for QCD with heavy quarks from the lattice,''
+  Phys.\ Rev.\ Lett. 110 (2013) 122001.
+  %%CITATION = ARXIV:1207.3005;%%
+  
+  
+\bibitem{dual}
+  A.~Patel, Nucl.~Phys. B {\bf 243} (1984) 411;
+  Phys.\ Lett.\  B {\bf 139} (1984) 394.
+  %
+  T.~DeGrand and C.~DeTar, 
+  Nucl.\ Phys.\  B {\bf 225} (1983) 590. 
+  %   
+  J.~Condella and C.~DeTar,
+  %``Potts flux tube model at nonzero chemical potential,''
+  Phys.\ Rev.\  D {\bf 61} (2000) 074023,
+  [arXiv:hep-lat/9910028].
+  %%CITATION = PHRVA,D61,074023;%%
+%
+  C.~Gattringer and T.~Kloiber,
+  %``Spectroscopy in finite density lattice field theory: An exploratory study in the relativistic Bose gas,''
+  Phys.\ Lett.\ B {\bf 720} (2013) 210
+  [arXiv:1212.3770 [hep-lat]].
+  %%CITATION = ARXIV:1212.3770;%%
+  %2 citations counted in INSPIRE as of 21 Oct 2013
+%
+  T.~Sterling, J.~Greensite,
+  %``Portraits Of The Flux Tube In Qed In Three-dimensions: A Monte Carlo Simulation With External Sources,''
+  Nucl.\ Phys.\ B {\bf 220} (1983) 327.
+  %%CITATION = NUPHA,B220,327;%%
+%
+  M.~Panero,
+  %``A Numerical study of confinement in compact QED,''
+  JHEP {\bf 0505} (2005) 066.
+  %[hep-lat/0503024].
+  %%CITATION = HEP-LAT/0503024;%%
+%
+  V.~Azcoiti, E.~Follana, A.~Vaquero, G.~Di Carlo,
+  %``Geometric Algorithm for Abelian-Gauge Models,''
+  JHEP {\bf 0908} (2009) 008.
+%  [arXiv:0905.0639 [hep-lat]].
+  %%CITATION = ARXIV:0905.0639;%%
+%
+  T.~Korzec, U.~Wolff,
+  %``A worm-inspired algorithm for the simulation of Abelian gauge theories,''
+  PoS LATTICE {\bf 2010} (2010) 029.
+  %[arXiv:1011.1359 [hep-lat]].
+  %%CITATION = ARXIV:1011.1359;%%
+%
+  P.N.~Meisinger, M.C.~Ogilvie,
+  %``The Sign Problem, PT Symmetry and Abelian Lattice Duality,''
+  arXiv:1306.1495 [hep-lat].
+  %%CITATION = ARXIV:1306.1495;%%
+  
+\bibitem{z3}
+  C.~Gattringer and A.~Schmidt,
+  %``Gauge and matter fields as surfaces and loops - an exploratory lattice study of the Z(3) Gauge-Higgs model,''
+  Phys.\ Rev.\ D {\bf 86} (2012) 094506
+  [arXiv:1208.6472 [hep-lat]].
+  %%CITATION = ARXIV:1208.6472;%%
+  %8 citations counted in INSPIRE as of 21 Oct 2013
+  
+\bibitem{swa}
+  Y.~D.~Mercado, C.~Gattringer and A.~Schmidt,
+  %``Surface worm algorithm for abelian Gauge-Higgs systems on the lattice,''
+  Comput.\ Phys.\ Commun.\  {\bf 184} (2013) 1535
+  [arXiv:1211.3436 [hep-lat]].
+  %%CITATION = ARXIV:1211.3436;%%
+  %6 citations counted in INSPIRE as of 21 Oct 2013
+
+\bibitem{worm}
+  N.~Prokof'ev and B.~Svistunov,
+  %``Worm Algorithms for Classical Statistical Models,''
+  Phys.\ Rev.\ Lett.\  {\bf 87} (2001) 160601.
+  %%CITATION = PRLTA,87,160601;%%
+  
+\bibitem{prl}
+  Y.~D.~Mercado, C.~Gattringer and A.~Schmidt,
+  %``Dual lattice simulation of the U(1) gauge-Higgs model at finite density - an exploratory proof-of-concept study,''
+  Phys.\ Rev.\ Lett.\  {\bf 111} (2013) 141601
+  [arXiv:1307.6120 [hep-lat]].
+  %%CITATION = ARXIV:1307.6120;%%
+
+\bibitem{LuWe}
+M.~L\"uscher, P.~Weisz,  Nucl.\ Phys.\ B {\bf 290} (1987) 25;
+Nucl.\ Phys.\ B {\bf 295} (1988) 65;
+Nucl.\ Phys.\ B {\bf 318} (1989) 705.
+
+\bibitem{DeGaSch1} 
+  Y.~D.~Mercado, C.~Gattringer, A.~Schmidt,
+  %``Surface worm algorithm for abelian Gauge-Higgs systems on the lattice,''
+  Comp.\ Phys.\ Comm.\  {\bf 184}, 1535 (2013).
+  %[arXiv:1211.3436 [hep-lat]].
+  %%CITATION = ARXIV:1211.3436;%%
+  %5 citations counted in INSPIRE as of 16 Jul 2013
+
+\bibitem{Lang}
+ K.~Jansen, J.~Jersak, C.B.~Lang, T.~Neuhaus, G.~Vones,
+  %``Phase Structure Of Scalar Compact Qed,''
+  Nucl.\ Phys.\ B {\bf 265} (1986) 129;
+  %%CITATION = NUPHA,B265,129;%%
+  % K.~Jansen, J.~Jersak, C.~B.~Lang, T.~Neuhaus and G.~Vones,
+  %``Phase Structure Of U(1) Gauge - Higgs Theory On D = 4 Lattices,''
+  Phys.\ Lett.\ B {\bf 155} (1985) 268.
+  %%CITATION = PHLTA,B155,268;%%
+  K.~Sawamura, T.~Hiramatsu, K.~Ozaki, I.~Ichinose,
+  %``Four-dimensional CP1 + U(1) lattice gauge theory for 3D antiferromagnets: Phase structure, gauge bosons and spin liquid,''
+  arXiv:0711.0818 [cond-mat.str-el].
+  %%CITATION = ARXIV:0711.0818;%% 
+
+\end{thebibliography}

 
 

+\end{document}
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