Added conventional and dual representation of model.

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@@ -1,36 +1,534 @@
 \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 $\chi$. 

 
 

-\end{thebibliography}
+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).

 
 

-\end{document}
+\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 xxxxx
+
+
+\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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+%
+  T.~Sterling, J.~Greensite,
+  %``Portraits Of The Flux Tube In Qed In Three-dimensions: A Monte Carlo Simulation With External Sources,''
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+  %%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
+
+\end{thebibliography}

 
 

+\end{document}
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