\usepackage{dsfont}
\usepackage{subfigure}
-\title{Solving the sign problem of scalar, two-flavored electrodynamics
-for finite chemical potential and exploring its full phase-diagram}
+\title{Solving the sign problem of two flavor scalar electrodynamics at finite chemical potential}
-\ShortTitle{Solving the sign problem of scalar electrodynamics at final chemical potential}
+\ShortTitle{Solving the sign problem of scalar electrodynamics at finite chemical potential}
-\author{\speaker{Ydalia Delgado}
+\author{Ydalia Delgado
\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
\\E-mail: \email{ydalia.delgado-mercado@uni-graz.at}}
\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
\\E-mail: \email{christof.gattringer@uni-graz.at}}
-\author{\speaker{Alexander Schmidt}
+\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.
-}
+We explore two flavor 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 link- and plaquette occupation numbers, subject to local
+constraints that have to be respected by the Monte Carlo algorithm.
+For the simulation we use a local update as well as the newly developed
+``surface worm algorithm'', which is a generalization of the Prokof'ev Svistunov
+worm algorithm concept for simulating the dual representation of abelian
+Gauge-Higgs models on a lattice. We assess the performance of the two algorithms,
+present results for the phase diagram
+and discuss condensation phenomena.}
\FullConference{XXIX International Symposium on Lattice Field Theory \\
July 29 $-$ August 03 2013\\
\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,
+
+At finite chemical potential $\mu$ the fermion determinant of QCD becomes complex
+and can not be interpreted as a probability weight in a Monte Carlo simulation.
+This so-called "complex phase problem" or "sign problem" has considerably
+slowed down the exploration of QCD at finite density using lattice methods.
+Although a lot of effort has been put into solving the complex phase problem of
+QCD (see, e.g., \cite{reviews} for recent reviews), the final goal of a proper ab-initio
+simulation of lattice QCD at finite density has not been achieved yet.
+
+For some models, as well as for QCD in limiting cases, it is possible to deal with the complex phase
+problem (see, e.g., \cite{solve-sign-problem}) with different techniques. Here we use a dual
+representation, i.e., a reformulation of the system with new degrees of freedom,
+which has been shown to be a very powerful method that can solve the complex
+phase problem of different models \cite{dual} without making any approximation of the partition sum.
+In the following we present another example where the dual representation can be applied successfully.
+We consider scalar QED with two flavors, i.e., a compact U(1) gauge field coupled to two complex scalar
+fields with opposite charge and a quartic self interaction \cite{prl}. We explore the full phase diagram as a
+function of the inverse gauge coupling and the mass parameter, and present some results at finite $\mu$.
+
+After mapping the degrees of freedom of the system to the 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.
+used two different Monte Carlo algorithms: A local update algorithm \cite{z3} and an extension \cite{swa} of the
+Prokof'ev Svistunov worm algorithm \cite{worm}. In addition to discussing the physics of the model, we also present
+a comparison of the performance of the two algorithms
-\section{Two-flavored scalar electrodynamics}
-\vspace{-1mm}
-\noindent ?????????????
+\section{Scalar electrodynamics}
+
+In the conventional representation two flavor scalar electrodynamics is a model of two flavors of
+oppositely charged complex fields $\phi_x, \chi_x \in \mathds{C}$ living on the
+sites $x$ of the lattice, 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.
+
+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{equation}
+S_\phi
+\; = \sum_x \!\Big( M_\phi^2 \, |\phi_x|^2 + \lambda_\phi |\phi_x|^4 -
+\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) .
+\label{matteraction}
+\end{equation}
+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). We also allow for a quartic self interaction of the scalar fields and
+the corresponding coupling is denoted as $\lambda_\phi$. Note that for $\mu_\phi \neq 0$ (\ref{matteraction})
+is complex, i.e., in the conventional form the theory has a complex action problem.
+
+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.
+
+
+
+\vskip2mm
+\noindent
+{\bf Dual representation:} A detailed derivation of the dual representation for the one flavor
+model is given in \cite{swa} and the two flavor version we consider here simply uses two copies
+of the representation of the matter fields. The dual variables for the first flavor will be denoted by
+$j_{x,\nu}, \overline{j}_{x,\nu}$, while $l_{x,\nu}$ and $\overline{l}_{x,\nu}$ are used for the second flavor.
+The dual representation of the partition sum for scalar QED
+with two flavors of matter fields is given by
+\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 ${\cal C}_s$,
+\begin{equation}
+ {\cal C}_s [j] \, = \, \prod_x \delta \! \left( \sum_\nu \partial_\nu j_{x,\nu} \right)\; , \; \;
+ {\cal C}_s [l] \, = \, \prod_x \delta \! \left( \sum_\nu \partial_\nu l_{x,\nu} \right) , \;
+\label{loopconstU1}
+\end{equation}
+which enforce the conservation of $j$-flux and of $l$-flux at each site of the lattice
+(here $\delta(n)$ denotes the Kronecker delta $\delta_{n,0}$ and $\partial_\nu f_x \equiv
+f_x - f_{x-\widehat{\nu}}$).
+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
+fact that $j$- and $l$-flux enter with opposite sign is due to the opposite charge of the two
+flavors.
+
+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
+\begin{equation}
+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] \; ,
+\end{equation}
+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
+\begin{equation}
+P_\phi (2n) \; = \; \int_0^\infty dr \, r^{2n+1}
+\, e^{-M_\phi^2\, r^2 - \lambda_\phi r^4} = \sqrt{\frac{\pi}{16 \lambda}} \, \left(\frac{-\partial}{\partial M^2}\right)^{\!n} \;
+e^{\, M^4 / 4 \lambda} \left[1- erf(M^2/2\sqrt{\lambda})\right] \; .
+\end{equation}
+These integrals are related to derivatives of the error function and we evaluate them numerically and
+pre-store them for the Monte Carlo simulation. 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 and 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
+
+Because the dual variables are subject to non-trivial constraints, they cannot be modified randomly during
+the update. Here we use two strategies: A local update, referred to as LMA (local Metropolis algorithm),
+which consists of three types of steps: Steps where we change plaquettes bounded by matter flux, steps where
+we change the plaquettes on 3-cubes, and steps where we propose double lines of matter flux around the temporal
+direction. These changes are built such that the constraints remain intact for each individual step and the
+tests of the LMA are reported in \cite{prl,z3,swa}.
+
+Another possibility is to use an extension of the worm
+algorithm \cite{worm}, the so called surface worm algorithm \cite{swa}, which we refer to as SWA. Here initially
+the constraints are violated at a single link and the SWA subsequently propagates this defect on the lattice
+until the defect is healed in a final step. For both the LMA and the SWA the unconstrained $\overline{l}$ and
+$\overline{j}$ variables are updated with conventional Metropolis steps.
+Here we present results for both algorithms and
assess their performance.
-First, we start describing the LMA. It consists of the following updates:
+\subsection{Local Metropolis algorithm LMA}
+Let us begin by describing the LMA. It consists of the following update steps:
\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.
+\item A sweep for each unconstrained variable $\overline{l}$ and $\overline{j}$
+raising 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
+the link fluxes (either $j_{x,\sigma}$ or $l_{x,\sigma}$) 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 $+1$.
+by $\pm 1$ is indicated by the signs $+$ or $-$, while the flux variables $j$ ($l$) are denoted by the thin red line
+(fat blue lines for the second flavor) 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
+It consists of increasing or decreasing the occupation number of both link variables $l$ and $j$ 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
+of our 4-d 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
+parameters where the system is dominated by closed surfaces, i.e., where the link
acceptance rate is small.
\end{itemize}
\vspace*{-1mm}
\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
+$-1$ and the links $j$ (thin red links) or $l$ (fat blue links) 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}
\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}
+\caption{Cube update: Here we show the two possible changes in the plaquette occupation numbers on a 3-cube.
+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:
+\subsection{Surface worm algorithm SWA}
+
+Instead of the LMA we can use a generalization of the the worm algorithm, the SWA.
+Here we only shortly describe the SWA and refer to \cite{swa} for a detailed description.
+
+The SWA is constructed by breaking up the smallest update elements of the LMA, i.e., the plaquette updates,
+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 SWA then transports this defect on the
+lattice until it closes with a final step that heals the constraint.
+The admissible configurations are generated using 3 elements:
+
\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 worm starts by changing either the $l$ or the $j$ flux by $\pm 1$ at
+a randomly chosen link (step 1 in Fig.~\ref{worm} where a worm for $j$ fluxes starts).
+
\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}).
+attaching segments of the same kind of flux ($j$ or $l$) as the first segment,
+which are chosen in such a way that the constraints are always
+obeyed at the link where the next segment is attached (step 2 in Fig.~\ref{worm}).
+
+\item Attaching segments 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}) to heal the violated constraint.
\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).
+A full sweep consists of $V_4$ worms with $l$ fluxes and $V_4$ worms with $j$ fluxes,
+plus a sweep of the unconstrained
+variables $\overline{l}$ and $\overline{j}$,
+and a sweep of winding loops (as explained for 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.)
+\caption{Examples of segments for the links $j$ (lhs.) and $l$ (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
\includegraphics[width=\textwidth,clip]{pics/worm}
\end{center}
\vspace{-4mm}
-\caption{Illustration of the worm algorithm. See text for an explanation.} \label{worm}
+\caption{A simple example for an update with the surface worm algorithm.
+See the text for a detailed explanation of the steps involved.} \label{worm}
\vspace{-2mm}
-\end{figure}
+\end{figure}
-\section{Algorithm Assessment}
+\section{Results}
\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.
+In this section we discuss the results from the numerical analysis. We first show
+an assessment of both algorithms and compare their performance. Subsequently
+we discuss the physics of scalar QED at finite density and present the phase diagram.
+In both cases we use thermodynamical observables and their fluctuations. In particular
+we use the following observables which can be evaluated as simple derivatives of
+$\ln Z$ in both the conventional and the dual representations:
+
+\vskip5mm
+\noindent
+The first and second derivatives with respect to the inverse gauge coupling $\beta$, i.e.,
+the plaquette expectation value and its susceptibility,
+
+\begin{equation}
+\langle U \rangle = \frac{1}{6 N_s^3 N_t}\frac{\partial}{\partial \beta} \ln\ Z\quad , \quad
+\chi_{U} = \frac{1}{6 N_s^3 N_t}\frac{\partial^2}{\partial \beta^2} \ln\ Z\ .
+\end{equation}
+
+\noindent We also consider the particle number density $n$
+and its susceptibility which are the first and second derivatives
+with respect to the chemical potential,
+
+\begin{equation}
+\langle n \rangle = \frac{1}{N_s^3 N_t}\frac{\partial}{\partial \mu} \ln\ Z\quad , \quad
+\chi_{n} = \frac{1}{N_s^3 N_t}\frac{\partial^2}{\partial \mu^2} \ln\ Z\ .
+\end{equation}
+
+\noindent Finally, we analyze the derivatives with respect to $M^2$,
+
+\begin{equation}
+\langle |\phi|^2 \rangle = \frac{1}{N_s^3 N_t}\frac{\partial}{\partial M^2} \ln\ Z\quad , \quad
+\chi_{|\phi|^2} = \frac{1}{N_s^3 N_t}\frac{\partial^2}{\partial (M^2)^2} \ln\ Z\ .
+\end{equation}
+
+\subsection{Assessment of the LMA and SWA algorithms}
+\noindent
+For the comparison of our two algorithms we considered the U(1) gauge-Higgs model coupled
+with one (see \cite{swa}) and two scalar fields (as described here).
+First we checked the correctness of the SWA comparing the results for different
+lattices sizes and parameters. Examples for the one flavor model were presented
+in \cite{swa}.
+
+In Fig.~\ref{obs} we now show some examples for the two flavor case. The top figures
+of Fig.~\ref{obs} show
+$\langle |\phi|^2 \rangle$ (lhs.) and the corresponding susceptibility (rhs.) as a function of
+$\mu_\phi = \mu_\chi = \mu$ at $\beta = 0.85$ and
+$M_\phi^2 = M_\chi^2 = M^2 = 5.325$ on a lattice of size $12^3 \times 60$. This point is located
+in the Higgs phase and does not show any phase transition as a function of $\mu$. The bottom
+plots show the particle number $\langle n \rangle$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$
+for $\beta = 0.75$ and $M^2 = 5.73$ on a lattice of volume $12^3 \times 60$. Here we observe
+a pronounced first order transition from the confining phase into the Higgs phase.
+It is obvious that in all four plots the agreement between the results from the LMA and from the
+SWA is excellent.
\begin{figure}[h]
\begin{center}
-\includegraphics[width=\textwidth,clip]{pics/f}
-\includegraphics[width=\textwidth,clip]{pics/f}
+\hbox{\includegraphics[width=\textwidth,clip]{pics/aphi}}
+\vskip5mm
+\hbox{\hspace{4mm}\includegraphics[width=0.97\textwidth,clip]{pics/bn}}
\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{-6mm}
+\caption{Observables for the two flavor model as a function of $\mu$ for different
+parameters on a $12^3 \times 60$ lattice.
+We compare results from the SWA (circles) and the LMA (triangles).} \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,
+In order to obtain a measure of the computational effort, in \cite{swa} we compared the normalized
+autocorrelation time $\overline{\tau}$ of the SWA and LMA for
+the one flavor 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 acceptance rate of the constrained link variables is not very low (e.g., lhs.\ of Fig.~\ref{auto}).
+On the other hand, for parameter values where 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
+system because it changes 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.
+$\langle U \rangle$ is larger for the SWA than for the LMA. We remark however, that this performance issue
+can be overcome easily by augmenting the SWA with sweeps of cube updates as used in the LMA.
\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}
+\caption{Normalized autocorrelation times $\overline{\tau}$ for the observables $\langle U \rangle$ and
+$\langle |\phi|^2 | \rangle$ for two different sets
+of parameters for the one flavor model. Left: Parameter values close to a first order phase transition.
+Right: A parameter set characterized by a low acceptance for matter flux. Both simulations
+were done on $16^4$ lattices, with data taken from \cite{swa}.} \label{auto}
\vspace*{-2mm}
-\end{figure}
+\end{figure}
+\subsection{Physics results}
+So far one of the main physics results of our studies of 2-flavor scalar QED
+(already published in \cite{prl}) is the full phase diagram of the considered
+model in the $\beta$-$M^2$ plane (using $M_\phi^2 = M_\chi^2 = M^2$)
+at $\mu=0$ and the analysis of phase
+transitions driven by the chemical potential $\mu_\phi = \mu_\chi = \mu$
+when starting from the different
+phases of the model. For the sake of completeness we here again show the
+$\mu = 0$ phase diagram, and then present new results for the observables
+in the $\beta$-$M^2$ plane at several values of $\mu > 0$, which illustrate the
+shift of the phase-boundaries at $\mu > 0$, i.e., the positions of the critical surfaces.
+In addition we show that some of the transitions at finite $\mu$ can be seen as
+condensation phenomena of the dual occupation numbers.
+
+\subsubsection*{Phase diagram at $\mu=0$}
+
+The results for the phase diagram at $\mu = 0$ are summarized in Fig.~\ref{phasediagram}. The various phase
+boundaries were determined from the observables $\langle U \rangle$ and $\langle |\phi|^2 \rangle$ and the
+corresponding susceptibilities. We found that the phase boundary separating Higgs- and
+confining phase is of 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 results for related
+models \cite{Lang} studied in the conventional formulation.
-\section{Results}
-\vspace{-1mm}
-\noindent xxxxx
+\begin{figure}[h]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=85mm,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}
+\begin{figure}[p]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=\linewidth,clip]{pics/muphases}
+\caption{The observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$, and
+$\langle n \rangle$ as a function of $\beta$ and $M^2$ for different chemical
+potentials $\mu = 0.0,\,0.5,\,1.0$ and $1.5$. It can be seen how the phase
+boundaries shift with increasing chemical potential.}
+\label{muphases}
+\end{figure}
+
+\begin{figure}[t]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=\linewidth,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}[b]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=\linewidth,clip]{pics/occutrans}
+\caption{Link occupation numbers $\bar{j}$, $\bar{l}$, $j$, $l$ and plaquette occupation numbers $p$ for values of $\mu$
+just below (top) and above (bottom) the critical value $\mu_c$ for the transition from the confining- to the Higgs-phase.}
+\label{occutrans}
+\end{figure}
+
+
+\subsubsection*{Phase boundaries at $\mu > 0$}
+
+As a first step in the determination of the phase boundaries as functions of all three parameters $\beta, \, M^2$ and $\mu$,
+in Fig.~\ref{muphases} we plot the observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$ and $\langle n \rangle$ as functions
+of $\beta$ and $M^2$ for four different values of the chemical potential $\mu=0.0,\, 0.5,\, 1.0$ and $1.5$.
+
+The phase-transition from the confining phase to the Coulomb phase shown in Fig.~\ref{phasediagram}
+is characterized by a rapid increase of $\langle U \rangle$ across the transition but does not give rise to
+significant changes in the other observables (compare the top row of plots in Fig.~\ref{muphases}).
+This behavior persists also at finite $\mu$ and the
+confinement-Coulomb transition can only be seen in the $\langle U \rangle$-plots.
+
+The transition between the Higgs- and the confinig phase is characterized by a strong first order discontinuity in all observables
+(except for $\langle n \rangle = 0$ at $\mu = 0$), a feature that persists for all our values of $\mu$. Also the transition between the Higgs- and the
+Coulomb phase is seen in all observables. It is obvious from the plots, that with increasing $\mu$ all three transitions become more pronounced in
+all variables they are seen in, and the Higgs-Coulomb transition might 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.
+
+\subsubsection*{Dual occupation numbers}
+
+The dual reformulation of lattice field theories 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 interesting about the dual formulation, we here present an example where a transition
+manifests itself as the condensation of dual variables.
+
+Let us first look at the transition using the standard observables. 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 $\mu_c\simeq2.65$
+for the larger of the two lattices. Below the critical value of the chemical potential both
+$\langle U \rangle$ and $\chi_U$ are independent of $\mu$, which is characteristic for a Silver Blaze type of transition \cite{cohen}.
+At $\mu_c$ a strong first order transition signals the entry into the Higgs phase.
+
+In Fig.~\ref{occutrans} we have a look at the same transition, by now showing typical configurations of the dual variables
+just below (top) and above (bottom) the critical chemical potential $\mu_c$.
+In particular we show snapshots of the occupation numbers of all dual link variables $\bar{j}$, $\bar{l}$, $j$,
+$l$ and dual plaquette variables $p$. Here blue links/plaquettes depict positive occupation numbers,
+green links/plaquettes depict negative occupation numbers and links/plaquettes with $0$-occupation
+are not shown. It can be seen that below $\mu_c$ links and plaquettes are hardly occupied,
+while above $\mu_c$ their occupation is abundant. In that sense the Silver Blaze transition of Fig.~\ref{occutrans_plaq}
+can be understood as a condensation phenomenon of the dual variables, which is a new perspective on the underlying
+physics we gained from the dual reformulation of the problem.
\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.
+We also acknowledge interesting discussions with Thomas Kloiber
+on aspects of the dual formulation for charged scalar fields.
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).
-
+FWF, DK {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16). Y.~Delgado is supported by
+the Research Executive Agency (REA) of the European Union
+under Grant Agreement number PITN-GA-2009-238353 (ITN STRONGnet) and by {\it Hadron Physics 2}.
+Furthermore this work is partly supported by DFG TR55, ``{\sl Hadron Properties from Lattice QCD}''
+and by the Austrian Science Fund FWF Grant.\ Nr.\ I 1452-N27.
+
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%
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projects / phys / proceedings_lattice13.git / blobdiff