X-Git-Url: http://git.treefish.org/~alex/phys/proceedings_lattice13.git/blobdiff_plain/9ecf8d2c83bb5e90aff4d6d50e3f643cdb12f1ac..d9fffa925e5bd54a6484370c7f4365e52265fd13:/proceed.tex?ds=inline

diff --git a/proceed.tex b/proceed.tex
index 7083dba..2f3e030 100644
--- a/proceed.tex
+++ b/proceed.tex
@@ -11,7 +11,7 @@ for finite chemical potential and exploring its full phase-diagram}
 
 \ShortTitle{Solving the sign problem of scalar electrodynamics at final chemical potential}
 
-\author{\speaker{Ydalia Delgado}
+\author{Ydalia Delgado
 \\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
 \\E-mail: \email{ydalia.delgado-mercado@uni-graz.at}}
 
@@ -19,7 +19,7 @@ for finite chemical potential and exploring its full phase-diagram}
 \\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
 \\E-mail: \email{christof.gattringer@uni-graz.at}}
 
-\author{\speaker{Alexander Schmidt}
+\author{Alexander Schmidt
 \\Institut f\"ur Physik, Karl-Franzens Universit\"at, Graz, Austria
 \\E-mail: \email{alexander.schmidt@uni-graz.at}}
 
@@ -46,23 +46,23 @@ Finally, we determine the full phase diagram of the model.
 \vspace{-1mm}
 \noindent
 At finite chemical potential $\mu$ the fermion determinant becomes complex
-and cannot be interpreted as a probability weight in the Monte Carlo simulation.
+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 QCDl.  Although many efforts have been put into
+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,
-which has been shown to be a very powerful method that can be solve the solve the complex 
+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 this proceedings we present another example where the dual representation can be applied succesfully.  We consider a compact
+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.
 
-After mapping the degrees of freedom of the system to its dual variables, the terms of the 
-partition sum are positive and real and usual Monte Carlo techniques can be applied.  However, 
+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
@@ -72,7 +72,115 @@ some technical comparison of both algorithms in addition to the physics of the m
  
 \section{Two-flavored scalar electrodynamics}
 \vspace{-1mm}
-\noindent ?????????????
+\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 
+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 
+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{eqnarray}
+&& \qquad S_\phi   
+=  \! \sum_x \!\Big(  M_\phi^2 \, |\phi_x|^2  + \lambda_\phi |\phi_x|^4  -
+\label{matteraction} \\
+&& \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}
+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 $\chi$. 
+
+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
+\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 ${\cal C}_s$ (here $\delta(n)$ denotes the Kronecker delta $\delta_{n,0}$ and $\partial_\nu f_x \equiv 
+f_x - f_{x-\widehat{\nu}}$)
+\begin{equation}
+ {\cal C}_s [j] \, = \, \prod_x \delta \! \left( \sum_\nu \partial_\nu j_{x,\nu}  \right) , \;
+\label{loopconstU1}
+\end{equation}
+which enforce the conservation of $j$-flux and of $l$-flux at each site of the lattice.
+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 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 $f_x = \sum_\nu\!\big[ |j_{x,\nu}|\!+\!  |j_{x-\widehat{\nu},\nu}|  \!+\!
+2\overline{j}_{x,\nu}\! +\! 2\overline{j}_{x-\widehat{\nu},\nu} \big]$ which is an even number. The $I_p(\beta)$
+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
+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
+accessible  to Monte Carlo techniques,  using the plaquette occupation numbers and the
+flux variables as the new degrees of freedom. 
 
 
 \section{Monte Carlo simulation}
@@ -249,8 +357,66 @@ to a $16^4$ lattice.  Data taken from \cite{swa}.} \label{auto}
 
 \section{Results}
 \vspace{-1mm}
-\noindent xxxxx
+\noindent
+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.
+
+\subsection{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}
+
+\subsection{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}
+
+\subsection{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 typical for a Silverblaze type transition.
 
+\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}
@@ -408,6 +574,33 @@ under Grant Agreement number PITN-GA-2009-238353 (ITN STRONGnet).
   [arXiv:1307.6120 [hep-lat]].
   %%CITATION = ARXIV:1307.6120;%%
 
+\bibitem{LuWe}
+M.~L\"uscher, P.~Weisz,  Nucl.\ Phys.\ B {\bf 290} (1987) 25;
+Nucl.\ Phys.\ B {\bf 295} (1988) 65;
+Nucl.\ Phys.\ B {\bf 318} (1989) 705.
+
+\bibitem{DeGaSch1} 
+  Y.~D.~Mercado, C.~Gattringer, A.~Schmidt,
+  %``Surface worm algorithm for abelian Gauge-Higgs systems on the lattice,''
+  Comp.\ Phys.\ Comm.\  {\bf 184}, 1535 (2013).
+  %[arXiv:1211.3436 [hep-lat]].
+  %%CITATION = ARXIV:1211.3436;%%
+  %5 citations counted in INSPIRE as of 16 Jul 2013
+
+\bibitem{Lang}
+ K.~Jansen, J.~Jersak, C.B.~Lang, T.~Neuhaus, G.~Vones,
+  %``Phase Structure Of Scalar Compact Qed,''
+  Nucl.\ Phys.\ B {\bf 265} (1986) 129;
+  %%CITATION = NUPHA,B265,129;%%
+  % K.~Jansen, J.~Jersak, C.~B.~Lang, T.~Neuhaus and G.~Vones,
+  %``Phase Structure Of U(1) Gauge - Higgs Theory On D = 4 Lattices,''
+  Phys.\ Lett.\ B {\bf 155} (1985) 268.
+  %%CITATION = PHLTA,B155,268;%%
+  K.~Sawamura, T.~Hiramatsu, K.~Ozaki, I.~Ichinose,
+  %``Four-dimensional CP1 + U(1) lattice gauge theory for 3D antiferromagnets: Phase structure, gauge bosons and spin liquid,''
+  arXiv:0711.0818 [cond-mat.str-el].
+  %%CITATION = ARXIV:0711.0818;%% 
+
 \end{thebibliography}
 
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