Added results.

[phys/proceedings_lattice13.git] / proceed.tex

\documentclass{PoS}

\usepackage[intlimits]{amsmath}
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\title{Solving the sign problem of scalar, two-flavored electrodynamics 
for finite chemical potential and exploring its full phase-diagram}

\ShortTitle{Solving the sign problem of scalar electrodynamics at final chemical potential}

\author{Ydalia Delgado
\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
\\E-mail: \email{ydalia.delgado-mercado@uni-graz.at}}

\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}

\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.

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.

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}


\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).
 
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\end{thebibliography}

\end{document}