X-Git-Url: http://git.treefish.org/~alex/phys/proceedings_lattice13.git/blobdiff_plain/a094d4162c1b374b851ccedfe4862bd1ff2b970b..d69c93e2b15e0fa4e1c63d923b0ecd0182724c8d:/proceed.tex?ds=sidebyside diff --git a/proceed.tex b/proceed.tex old mode 100644 new mode 100755 index 9ee4ed2..5556490 --- a/proceed.tex +++ b/proceed.tex @@ -1,36 +1,735 @@ \documentclass{PoS} -\title{Contribution title} +\usepackage[intlimits]{amsmath} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{dsfont} +\usepackage{subfigure} -\ShortTitle{Short Title for header} +\title{Solving the sign problem of two flavor scalar electrodynamics at finite chemical potential} -\author{Ydalia Delgado Mercado, Christof Gattringer, Alexander Schmidt -% \thanks{Y.D.M and A.S. are members of the doctoral training program FWF DK 1203 ''{\sl Hadrons in Vacuum, Nuclei and Stars}''. Y.D.M. is furthermore supported by the Research Executive Agency of the European Union under Grant Agreement number PITN-GA-2009-238353 (ITN STRONGnet). This work is partly supported also by DFG SFB TRR55.}\\ - \\ - Institut f\"ur Physik, - Karl-Franzens-Universit\"at, 8010 Graz, Austria \\ \\ - \email{ydalia.delgado-mercado@uni-graz.at} \\ \email{christof.gattringer@uni-graz.at} \\ \email{alexander.schmidt@uni-graz.at} } +\ShortTitle{Solving the sign problem of scalar electrodynamics at finite chemical potential} -\abstract{..........................\ - ...........................} +\author{Ydalia Delgado +\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria +\\E-mail: \email{ydalia.delgado-mercado@uni-graz.at}} -\FullConference{31st International Symposium on Lattice Field Theory - LATTICE 2013\\ - July 29 - August 3, 2013\\ - Mainz, Germany} +\author{Christof Gattringer +\\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria +\\E-mail: \email{christof.gattringer@uni-graz.at}} +\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 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\\ + Mainz, Germany} + \begin{document} -\section{dfg} +\section{Motivation} -dfgdfg +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. -\begin{thebibliography}{99} -\bibitem{...} -.... +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 $\mu$. -\end{thebibliography} +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 \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{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} + +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 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 $\overline{j}$ +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 $j_{x,\sigma}$ 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 $j$ ($l$) 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 $j$ 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 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., where the link +acceptance rate is small. +\end{itemize} +\vspace*{-1mm} +A full sweep consists of updating all links, plaquettes, 3-cubes and winding loops on the lattice, +offering one of the changes mentioned above and accepting them with the Metropolis +probability computed from the local weight factors. + +\begin{figure}[h] +\begin{center} +\includegraphics[width=\textwidth,clip]{pics/plaquettes} +\end{center} +\vspace{-4mm} +\caption{Plaquette update: A plaquette occupation number is changed by $+1$ or +$-1$ and the links $j$ (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} +\vspace{-2mm} +\end{figure} + +\begin{figure}[h] +\begin{center} +\includegraphics[width=0.7\textwidth,clip]{pics/cubes} +\end{center} +\vspace{-4mm} +\caption{Cube update: Here we show the 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{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 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 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 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 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 +where the segment is attached to the worm and the dotted link is the new position of the link +$L_V$ where the constraints are violated.} \label{segments} +\vspace{-2mm} +\end{figure} + +\begin{figure}[h] +\begin{center} +\includegraphics[width=\textwidth,clip]{pics/worm} +\end{center} +\vspace{-4mm} +\caption{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 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: -\end{document} +\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} +\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). 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