Ydalia:

[phys/proceedings_lattice13.git] / proceed.tex

diff --git a/proceed.tex b/proceed.tex
index 12e77681f304f33968b88b05394b66881a4a7fb8..bef9b30ff3bbacb636beb8de9ffb20f4d55a6b8f 100644 (file)
--- a/proceed.tex
+++ b/proceed.tex
@@ -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 $p_{x, \nu \rho}$ 
 $p_{x,\nu\rho}$ 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}
 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
 \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}
 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}


@@ -337,10 +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
 \subsection{Algorithm assessment}
 \noindent
 For the comparison of both algorithms we considered the U(1) gauge-Higgs model coupled

-with two (as described above) 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}.
 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
 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


@@ -356,7 +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}
 \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}
 We compare results from the SWA (circles) and the LMA (triangles).} \label{obs}
 \vspace*{-2mm}
 \end{figure}


@@ -380,7 +382,7 @@ a sweep of cube updates.
 \end{center}
 \vspace{-4mm}
 \caption{Normalized autocorrelation times $\overline{\tau}$ for 2 different set
 \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}
 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}


@@ -400,7 +402,7 @@ models \cite{Lang}.
 \begin{figure}[h]
 \centering
 \hspace*{-3mm}
 \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 $\chi_n$.}
 \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 $\chi_n$.}


@@ -417,7 +419,7 @@ For all observables it can be seen that the phase-boundaries in general become m
 \begin{figure}[h]
 \centering
 \hspace*{-3mm}
 \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}
 \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}


@@ -431,19 +433,20 @@ In Fig.~\ref{occutrans_plaq} we plot the plaquette expectation value $\langle U
 $\langle U \rangle$ and $\chi_U$ are independent of $\mu$, which is characteristic for a Silverblaze type transition \cite{cohen}.
 
 \noindent
 $\langle U \rangle$ 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}
 
 \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}
 \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}
 \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}