From: Alex Schmidt Date: Tue, 22 Oct 2013 16:34:03 +0000 (+0200) Subject: Added conventional and dual representation of model. X-Git-Url: http://git.treefish.org/~alex/phys/proceedings_lattice13.git/commitdiff_plain/d04fee83c59ce66d5b8e732b3facb4ad8450748b?ds=sidebyside Added conventional and dual representation of model. --- 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} \ No newline at end of file