+\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}
+\hbox{\includegraphics[width=\textwidth,clip]{pics/aphi}}
+\vskip5mm
+\hbox{\hspace{4mm}\includegraphics[width=0.97\textwidth,clip]{pics/bn}}
+\end{center}
+\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, 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 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 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
+$\langle U \rangle$ 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 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 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.
+
+\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}
+\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) 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 / blobdiff