Added Ydalia's 1st version.

[phys/proceedings_lattice13.git] / proceed.tex

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@@ -1,36 +1,413 @@
 \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{\speaker{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{\speaker{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 cannot 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 QCDl.  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 be solve the solve the complex 
+phase problem without making any approximation of the partition sum, i.e. it is an exact method \cite{dual}.  
+In this proceedings 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 terms of the 
+partition sum are 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 ?????????????

 
 

-\end{thebibliography}

 
 

-\end{document}
+\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 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).
+ 
+\begin{thebibliography}{123456}
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+%
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+  %1 citations counted in INSPIRE as of 21 Oct 2013
+%
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+%
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+%
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+  %%CITATION = PRLTA,87,160601;%%
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+  %%CITATION = ARXIV:1307.6120;%%
+
+\end{thebibliography}

 
 

+\end{document}
\ No newline at end of file