\usepackage{dsfont}
\usepackage{subfigure}
-\title{Solving the sign problem of two flavored scalar electrodynamics at finite chemical potential}
+\title{Solving the sign problem of two flavor scalar electrodynamics at finite chemical potential}
-\ShortTitle{Solving the sign problem of two-falvored scalar electrodynamics at finite chemical potential}
+\ShortTitle{Solving the sign problem of scalar electrodynamics at finite chemical potential}
\author{Ydalia Delgado
\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
\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. For the simulation we use a local update that always obeys the constraints 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 the local update algorithm 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 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.
-
-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,
+
+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 a compact U(1) gauge field coupled with two complex scalar fields with opposite charge \cite{prl}.
-We explore the full phase diagram as a function of the inverse gauge coupling and the mass parameter,
-and present some preliminary results at finite $\mu$.
+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 its dual variables, the weight in the
+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 algorithm (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
-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
+\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. 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
+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}
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 \!
+\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) .
-\nonumber
-\end{eqnarray}
+\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). 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 $\chi$.
+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.
-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
+{\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] .
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 $\partial_\nu f_x \equiv
-f_x - f_{x-\widehat{\nu}}$)
+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 [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.
+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}
\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.
+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
\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 $I_p(\beta)$
+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 $ 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
+${\cal W}_\phi$ 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. The partition sum (\ref{Zfinal}) thus is
+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 modified randomly during the update.
-A straight forward way to sample the system is to change allowed surfaces.
-Thus we choose the smallest possible structures in order to
-increase the acceptance rate. This algorithm is called local update
-(LMA) and was used in \cite{z3,swa,prl}. Another 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.
-\subsection{Local update algorithm}
-Let us begin by 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 $\overline{j}$
-rising or lowering their occupation number by one unit.
+raising or lowering their occupation number by one unit.
%
\vspace*{-1mm}
\item ``Plaquette update'':
$p_{x,\nu\rho}$ and
the link fluxes (either $l_{x,\sigma}$ or $j_{x,\sigma}$) 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 $l$($j$) are denoted by the thin red line
-(fat 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 $l$ ($j$) 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'':
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}
\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}
-\subsection{Worm algorithm}
-\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).
-
-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 either the $l$ or $j$ flux by $\pm 1$ at
-a randomly chosen link (step 1 in Fig.~\ref{worm}, a worm for $l$ fluxes starts).
+
+\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 $l$ 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 of the same kind of flux as the first segment,
+attaching segments of the same kind of flux ($l$ or $j$) as the first segment,
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}).
+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 with the $l$ fluxes and $V_4$ worms with the $j$ fluxes,
+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).
\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}
\section{Results}
\vspace{-1mm}
\noindent
-In this section we describe the numerical analysis. We first show the assessment of both algorithms
-and then the physics of the model. In both cases we use thermodynamical observables and their fluctuations.
-We study in particular three observables: The first and second derivatives with respect to the inverse
-gauge coupling $\beta$, i.e., the plaquette expectation value and its susceptibility,
+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
\end{equation}
\noindent We also consider the particle number density $n$
-and its susceptibility which are the derivatives
+and its susceptibility which are the first and second derivatives
with respect to the chemical potential,
\begin{equation}
\chi_{|\phi|^2} = \frac{1}{N_s^3 N_t}\frac{\partial^2}{\partial (M^2)^2} \ln\ Z\ .
\end{equation}
-\subsection{Algorithm assessment}
+\subsection{Assessment of the LMA and SWA algorithms}
\noindent
-For the comparison of both algorithms we considered the U(1) gauge-Higgs model coupled
+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 are shown in \cite{swa}.
-In Fig.~\ref{obs} we can observe some examples for the two flavor case.
-The figure on the top shows
-$\langle |\phi|^2 \rangle$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$
-at $\beta = 0.85$ and $M^2 = 5.325$ on a lattice of size $12^3 \times 60$. This point is located
-in the Higgs phase and does not show any phase transition. The plot on the bottom shows
-the particle number $n$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$
-for $\beta = 0.75$ and $M^2 = 5.73$ on a lattice of volume $12^3 \times 60$. This plot shows
-the transition from the confining phase to the Higgs phase.
-We observe very good agreement between both algorithms.
+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 $n$ (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 LWA and from the
+SWA is excellent.
\begin{figure}[h]
\begin{center}
\hbox{\includegraphics[width=\textwidth,clip]{pics/aphi}}
+\vskip5mm
\hbox{\hspace{4mm}\includegraphics[width=0.97\textwidth,clip]{pics/bn}}
\end{center}
\vspace{-6mm}
\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
-$U$ is larger for the SWA than for the LMA. But this can be overcome by offering
-a sweep of cube updates.
+$U$ 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 for the one flavor model. 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}
-\subsection{Physics}
-One of the main results of these studies so far and already published in \cite{prl} is the full phase diagram of the considered model in the $\beta$-$M^2$ plane at $\mu=0$ and some selected chemical potential driven phase transitions of the measured observables. For the sake of completeness we here again want to show the obtained phase diagram, but as a proceedings-extra also present some plots which show the shifting of the phase-boundaries at $\mu \neq 0$ and measurements of the dual occupation numbers.
-
-\subsubsection*{Phase-diagram at $\mu=0$}
-\noindent
-We studied the different transition lines in Fig.~\ref{phasediagram} using finite size analysis of the measured observables $\langle U \rangle$ and $\langle |\phi|^2 \rangle$ and the corresponding susceptibilities, finding that the phase boundary separating Higgs- and
-confining phase is strong first order, the line separating confining- and Coulomb phase is of weak
+\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 conventional results for related
-models \cite{Lang}.
+agreement with the results for related
+models \cite{Lang} studied in the conventional formulation.
+
\begin{figure}[h]
\centering
\hspace*{-3mm}
\label{phasediagram}
\end{figure}
-\subsubsection*{Phase-boundaries at $\mu \neq 0$}
-\noindent
-In Fig.~\ref{muphases} we plot the observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$, $\langle n \rangle$ as function of $\beta$ and $M^2$ for four different values of the chemical potential $\mu=0,0.5,1,1.5$.
-
-\noindent
-The phase-transition from the confining phase to the Coulomb phase shown in Fig.~\ref{phasediagram} is characterized by $\langle U \rangle$ growing larger across the transition but no significant changes in the other observables, which is the reason why the confinement-Coulomb transition can only be seen in the $\langle U \rangle$-plots.
-For all observables it can be seen that the phase-boundaries in general become more pronounced at higher chemical potential and for the Higgs-Coulomb transition the transition type may 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.
-\begin{figure}[h]
+\begin{figure}[p]
\centering
\hspace*{-3mm}
\includegraphics[width=\linewidth,clip]{pics/muphases}
-\caption{We show the observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$, $\langle n \rangle$ as function of $\beta$ and $M^2$ for different $\mu = 0,0.5,1,1.5$. It can be seen how the phase boundaries change with increasing chemical potential.}
+\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}
-\subsubsection*{Dual occupation numbers}
-\noindent
-The dual reformulation of a problem 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 exciting about rewriting to dual variables, we here want to present an example.
-
-\noindent
-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 $\mu\simeq2.65$ for the larger and $\mu\simeq2.7$ for the smaller lattice, telling us that the Higgs phase is tilted towards the confining phase in $\mu$-direction. Below the critical value of the chemical potential both
-$\langle U \rangle$ and $\chi_U$ are independent of $\mu$, which is characteristic for a Silverblaze type transition \cite{cohen}.
-
-\noindent
-Then in Fig.~\ref{occutrans} we show the occupation numbers of all dual link variables $\bar{j}$, $\bar{l}$, $j$,
-$l$ and dual plaquette variables $p$ just below (top) and above (bottom) the critical chemical potential $\mu_c$. Here blue links/plaquettes depict positive occupation numbers, green links/plaquettes depict negative occupation numbers and links/plaquettes with $0$-occupation are spared out. It can be seen that below $\mu_c$ links and plaquettes are hardly occupied, while above $\mu_c$ they are highly occupied. In that sense the Silverblaze transition shown in Fig.~\ref{occutrans_plaq} can be understood as condensation phenomenon, which is a new perspective on the underlying physics we gained from the dual reformulation of the problem.
-
-\begin{figure}[h]
+\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}[h]
+\begin{figure}[b]
\centering
\hspace*{-3mm}
\includegraphics[width=\linewidth,clip]{pics/occutrans}
-\caption{Dual link occupation numbers $\bar{j}$, $\bar{l}$, $j$, $l$ and dual plaquette occupation numbers $p$ just below (top) and above (bottom) the transition from the confining- to the Higgs-phase shown in the previous plot.}
+\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). Y.~Delgado is supported by
the Research Executive Agency (REA) of the European Union
-under Grant Agreement number PITN-GA-2009-238353 (ITN STRONGnet), HP2 and TRR 55.
+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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projects / phys / proceedings_lattice13.git / commitdiff