[phys/proceedings_lattice13.git] / proceed.tex

\documentclass{PoS}

\usepackage[intlimits]{amsmath}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{dsfont}
\usepackage{subfigure}

\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{\speaker{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{\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{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.

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.

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 ?????????????


\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. 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;%%

\end{thebibliography}

\end{document}