X-Git-Url: http://git.treefish.org/~alex/phys/proceedings_lattice13.git/blobdiff_plain/9ecf8d2c83bb5e90aff4d6d50e3f643cdb12f1ac..d04fee83c59ce66d5b8e732b3facb4ad8450748b:/proceed.tex

diff --git a/proceed.tex b/proceed.tex
index 7083dba..7e18791 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}
@@ -408,6 +516,19 @@ 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
+
 \end{thebibliography}
 
 \end{document}
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