X-Git-Url: http://git.treefish.org/~alex/phys/proceedings_lattice13.git/blobdiff_plain/a094d4162c1b374b851ccedfe4862bd1ff2b970b..9ecf8d2c83bb5e90aff4d6d50e3f643cdb12f1ac:/proceed.tex?ds=inline diff --git a/proceed.tex b/proceed.tex index 9ee4ed2..7083dba 100644 --- a/proceed.tex +++ b/proceed.tex @@ -1,36 +1,413 @@ \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} -\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} +\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. 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