X-Git-Url: http://git.treefish.org/~alex/phys/proceedings_lattice13.git/blobdiff_plain/fcf0e96d40c7d11e72e34263dc6b8ebf10257a8b..8e422e13d7ddcd89704668a0ccf688f2dfeea749:/proceed.tex?ds=inline

diff --git a/proceed.tex b/proceed.tex
index 5d1181d..bef9b30 100644
--- a/proceed.tex
+++ b/proceed.tex
@@ -6,7 +6,7 @@
 \usepackage{dsfont}
 \usepackage{subfigure}
 
-\title{Solving the sign problem of two-flavored scalar electrodynamics at finite chemical potential}
+\title{Solving the sign problem of two flavored scalar electrodynamics at finite chemical potential}
 
 \ShortTitle{Solving the sign problem of two-falvored scalar electrodynamics at finite chemical potential}
 
@@ -27,11 +27,10 @@
 We explore two-flavored 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 links and plaquettes, subject to non-trivial 
-constraints, which have to be respected by the Monte Carlo algorithm. 
-bvFor the simulation we use a local update that always obeys the constraints and the surface worm algorithm (SWA). 
+constraints, which have to be respected by the Monte Carlo algorithm. For the simulation we use a local update that always obeys the constraints and the surface worm algorithm (SWA). 
 The SWA is a generalization of the Prokof'ev Svistunov 
 worm algorithm concept to simulate the dual representation of abelian Gauge-Higgs models on a lattice. 
-We also assess the performance of the SWA and compare it with a local update in the dual representation. 
+We also assess the performance of the SWA and compare it with the local update algorithm in the dual representation. 
 Finally, we determine the full phase diagram of the model.
 }
 
@@ -64,7 +63,7 @@ After mapping the degrees of freedom of the system to its dual variables, the we
 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
+used two different Monte Carlo algorithms:  A local update algorithm (LMA) \cite{z3} and an extension \cite{swa} of the
 Prokof'ev Svistunov worm algorithm \cite{worm}.   Here we present
 some technical comparison of both algorithms in addition to the physics of the model.
  
@@ -186,13 +185,14 @@ flux variables as the new degrees of freedom.
 \vspace*{-1mm}
 \noindent
 Because the dual variables are subject to non-trivial constraints, they cannot be modified randomly during the update.
-An straight forward way to sample the system is to change allowed surfaces.  
+A straight forward way to sample the system is to change allowed surfaces.  
 Thus we choose the smallest possible structures in order to
 increase the acceptance rate.  This algorithm is called local update
-(LMA) and was used in \cite{z3,swa,prl}.  Other possibility is to use an extension of the worm
+(LMA) and was used in \cite{z3,swa,prl}.  Another possibility is to use an extension of the worm
 algorithm \cite{worm}, the so called surface worm algorithm \cite{swa}.  For this model we use both algorithms and
-asses their performance.
+assess their performance.
 
+\subsection{Local update algorithm}
 Let us begin by describing the LMA. It consists of the following updates:
 \begin{itemize}
 \vspace*{-1mm}
@@ -205,7 +205,8 @@ It consists of increasing or decreasing a plaquette occupation number
 $p_{x,\nu\rho}$ and
 the link fluxes (either $l_{x,\sigma}$ or $j_{x,\sigma}$) 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 $l$($j$) are denoted by the red(blue) lines
+by $\pm 1$ is indicated by the signs $+$ or $-$, while the flux variables $l$($j$) are denoted by the thin red line
+(fat blue lines)
 and we use a dashed line to indicate a decrease by $-1$ and a full line for an increase by $+1$.
 %
 \vspace*{-1mm}
@@ -234,7 +235,7 @@ probability computed from the local weight factors.
 \end{center}
 \vspace{-4mm}
 \caption{Plaquette update: A plaquette occupation number is changed by $+1$ or
-$-1$ and the links $l$ (red) or $j$ (blue) of the plaquette are changed simultaneously. The
+$-1$ and the links $l$ (thin red links) or $j$ (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}
@@ -252,6 +253,7 @@ the directions $1 \leq \nu_1 < \nu_2 < \nu_3 \leq 4$.} \label{cube}
 \vspace*{-2mm}
 \end{figure}
 
+\subsection{Worm algorithm}
 \noindent
 Instead of the plaquette and cube updates we can use the worm algorithm.
 Here we will shortly describe the SWA (see \cite{swa} for a detailed description).
@@ -309,7 +311,7 @@ $L_V$ where the constraints are violated.} \label{segments}
 \noindent
 In this section we describe the numerical analysis.  We first show the assessment of both algorithms
 and then the physics of the model.  In both cases we use thermodynamical observables and their fluctuations.
-We study in particular three observables:  the first and second derivatives with respect to the inverse
+We study in particular three observables:  The first and second derivatives with respect to the inverse
 gauge coupling $\beta$, i.e., the plaquette expectation value and its susceptibility,
 
 \begin{equation}
@@ -336,10 +338,10 @@ n  = \frac{1}{N_s^3 N_t}\frac{\partial}{\partial \mu} \ln\ Z\quad , \quad
 \subsection{Algorithm assessment}
 \noindent
 For the comparison of both algorithms we considered the U(1) gauge-Higgs model coupled
-with two (as described previously) and with only one scalar field \cite{swa}.  
+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 are shown in \cite{swa}.
-Fig.~\ref{obs} shows two observables for the two flavor case.
+In Fig.~\ref{obs} we can observe some examples for the two flavor case.
 The figure on the top shows 
 $\langle |\phi|^2 \rangle$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$ 
 at $\beta = 0.85$ and $M^2 = 5.325$ on a lattice of size $12^3 \times 60$.  This point is located
@@ -355,7 +357,8 @@ We observe very good agreement between both algorithms.
 \hbox{\hspace{4mm}\includegraphics[width=0.97\textwidth,clip]{pics/bn}}
 \end{center}
 \vspace{-6mm}
-\caption{Observables as a function of $\mu$ for different parameters on a $12^3 \times 60$ lattice.
+\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}
@@ -379,7 +382,7 @@ a sweep of cube updates.
 \end{center}
 \vspace{-4mm}
 \caption{Normalized autocorrelation times $\overline{\tau}$ for 2 different set
-of parameters.  Left: parameters close to a first order phase transition. 
+of parameters for the one flavor model.  Left: parameters close to a first order phase transition. 
 Right: low acceptance rate of the variable $l$.  Both simulations correspond
 to a $16^4$ lattice.  Data taken from \cite{swa}.} \label{auto}
 \vspace*{-2mm}
@@ -399,7 +402,7 @@ models \cite{Lang}.
 \begin{figure}[h]
 \centering
 \hspace*{-3mm}
-\includegraphics[width=75mm,clip]{pics/phasediagram}
+\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 $\chi_n$.}
@@ -416,7 +419,7 @@ For all observables it can be seen that the phase-boundaries in general become m
 \begin{figure}[h]
 \centering
 \hspace*{-3mm}
-\includegraphics[width=130mm,clip]{pics/muphases}
+\includegraphics[width=\linewidth,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}
@@ -427,22 +430,23 @@ The dual reformulation of a problem makes it possible to look at the same physic
 
 \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 \cite{cohen}.
+$\langle U \rangle$ and $\chi_U$ are independent of $\mu$, which is characteristic for a Silverblaze type transition \cite{cohen}.
 
 \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.
+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}
+\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}[h]
 \centering
 \hspace*{-3mm}
-\includegraphics[width=130mm,clip]{pics/occutrans}
+\includegraphics[width=\linewidth,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}