X-Git-Url: http://git.treefish.org/~alex/phys/proceedings_lattice13.git/blobdiff_plain/d04fee83c59ce66d5b8e732b3facb4ad8450748b..e25a6f9c13b726931d60f5958d3c96be8427acb8:/proceed.tex?ds=sidebyside

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@@ -357,8 +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}
@@ -529,6 +587,20 @@ Nucl.\ Phys.\ B {\bf 318} (1989) 705.
   %%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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