\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 flavored scalar electrodynamics at finite chemical potential}
-\ShortTitle{Solving the sign problem of scalar electrodynamics at final chemical potential}
+\ShortTitle{Solving the sign problem of two-falvored scalar electrodynamics at finite chemical potential}
\author{Ydalia Delgado
\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
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).
+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 a local update in the dual representation.
+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.
}
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 solve the complex
-phase problem without making any approximation of the partition sum, i.e. it is an exact method \cite{dual}.
-In the following 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.
+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 $\mu$.
After mapping the degrees of freedom of the system to its 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
+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.
\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
+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
assess their performance.
-First, we start describing the LMA. It consists of the following updates:
+\subsection{Local update algorithm}
+Let us begin by describing the LMA. It consists of the following updates:
\begin{itemize}
\vspace*{-1mm}
-\item A sweep for each unconstrained variable $\overline{l}$ and $\overline{k}$
+\item A sweep for each unconstrained variable $\overline{l}$ and $\overline{j}$
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
+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 $l$($k$) are denoted by the red(blue) lines
+by $\pm 1$ is indicated by the signs $+$ or $-$, while the flux variables $l$($j$) 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
+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.
%
\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 $l$ (red) or $j$ (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}
+\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)
-for the variable $l$ (red).
-The algorithm for the other type of link variable works in exactly the same way.
+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''
$L_V$, 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 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 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
+attaching segments of the same kind of flux 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}).
\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 the $l$ fluxes and $V_4$ worms with the $j$ 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 $l$ (lhs.) and $j$ (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
\vspace{-4mm}
\caption{Illustration of the worm algorithm. See text for an explanation.} \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
+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,
+
+\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 derivatives
+with respect to the chemical potential,
+
+\begin{equation}
+n = \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{Algorithm assessment}
+\noindent
+For the comparison of both algorithms we considered the U(1) gauge-Higgs model coupled
+with two (as described above) and with only one scalar field \cite{swa}.
+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.
+Fig.~\ref{obs} shows two observables for the two flavor case.
+The figure on the top shows
$\langle |\phi|^2 \rangle$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$
-for point ``f'' (see phase diagram) on a lattice of size $12^3 \times 60$.
-We observe very good agreement among the different algorithms.
+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.
\begin{figure}[h]
\begin{center}
-\includegraphics[width=\textwidth,clip]{pics/f}
-\includegraphics[width=\textwidth,clip]{pics/f}
+\hbox{\includegraphics[width=\textwidth,clip]{pics/aphi}}
+\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 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}
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
+$U$ is larger for the SWA than for the LMA. But this can be overcome by offering
a sweep of cube updates.
\begin{figure}[t]
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}
+\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.
-\section{Results}
-\vspace{-1mm}
-\noindent xxxxx
+\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
+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}.
+\begin{figure}[h]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=75mm,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}
+
+\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]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=130mm,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.}
+\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]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=130mm,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]
+\centering
+\hspace*{-3mm}
+\includegraphics[width=130mm,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.}
+\label{occutrans}
+\end{figure}
\section*{Acknowledgments}
\vspace{-1mm}
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).
-
+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.
+
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\end{thebibliography}
\end{document}
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projects / phys / proceedings_lattice13.git / blobdiff