The dual variables are link- and plaquette occupation numbers, subject to local
constraints that have to be respected by the Monte Carlo algorithm.
For the simulation we use a local update as well as the newly developed
-''surface worm algorithm'', which is a generalization of the Prokof'ev Svistunov
+``surface worm algorithm'', which is a generalization of the Prokof'ev Svistunov
worm algorithm concept for simulating the dual representation of abelian
Gauge-Higgs models on a lattice. We assess the performance of the two algorithms,
present results for the phase diagram
\item ``Plaquette update'':
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
+the link fluxes (either $j_{x,\sigma}$ or $l_{x,\sigma}$) 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 thin red line
+by $\pm 1$ is indicated by the signs $+$ or $-$, while the flux variables $j$ ($l$) are denoted by the thin red line
(fat blue lines for the second flavor) and we use a dashed line to indicate a decrease by $-1$ and a full line
for an increase by $+1$.
%
%
\vspace*{-1mm}
\item ``Cube update'': The plaquettes of 3-cubes
-of our 4d lattice are changed according to one of the two patterns illustrated in
+of our 4-d lattice are changed according to one of the two patterns illustrated in
Fig.~\ref{cube}.
Although the plaquette and winding loop update are enough to satisfy ergodicity,
the cube update helps for decorrelation in the region of
\end{center}
\vspace{-4mm}
\caption{Plaquette update: A plaquette occupation number is changed by $+1$ or
-$-1$ and the links $l$ (thin red links) or $j$ (fat blue links) of the plaquette are changed simultaneously. The
+$-1$ and the links $j$ (thin red links) or $l$ (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}
\begin{enumerate}
\item The worm starts by changing either the $l$ or the $j$ flux by $\pm 1$ at
-a randomly chosen link (step 1 in Fig.~\ref{worm} where a worm for $l$ fluxes starts).
+a randomly chosen link (step 1 in Fig.~\ref{worm} where a worm for $j$ fluxes starts).
\item The first link becomes the head of the worm $L_V$.
The defect at $L_V$ is then propagated through the lattice by
-attaching segments of the same kind of flux ($l$ or $j$) as the first segment,
+attaching segments of the same kind of flux ($j$ or $l$) as the first segment,
which are chosen in such a way that the constraints are always
obeyed at the link where the next segment is attached (step 2 in Fig.~\ref{worm}).
\includegraphics[width=\textwidth,clip]{pics/segments}
\end{center}
\vspace{-4mm}
-\caption{Examples of segments for the links $l$ (lhs.) and $j$ (rhs.)
+\caption{Examples of segments for the links $j$ (lhs.) and $l$ (rhs.)
in the $\nu_1$-$\nu_2$-plane ($\nu_1 < \nu_2$).
The plaquette occupation numbers are changed as indicated by the signs.
The full (dashed) links are changed by $+1$ ($-1$). The empty link shows
with respect to the chemical potential,
\begin{equation}
-n = \frac{1}{N_s^3 N_t}\frac{\partial}{\partial \mu} \ln\ Z\quad , \quad
+\langle n \rangle = \frac{1}{N_s^3 N_t}\frac{\partial}{\partial \mu} \ln\ Z\quad , \quad
\chi_{n} = \frac{1}{N_s^3 N_t}\frac{\partial^2}{\partial \mu^2} \ln\ Z\ .
\end{equation}
$\mu_\phi = \mu_\chi = \mu$ at $\beta = 0.85$ and
$M_\phi^2 = M_\chi^2 = M^2 = 5.325$ on a lattice of size $12^3 \times 60$. This point is located
in the Higgs phase and does not show any phase transition as a function of $\mu$. The bottom
-plots show the particle number $n$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$
+plots show the particle number $\langle n \rangle$ (lhs.) and its susceptibility (rhs.) as a function of $\mu$
for $\beta = 0.75$ and $M^2 = 5.73$ on a lattice of volume $12^3 \times 60$. Here we observe
a pronounced first order transition from the confining phase into the Higgs phase.
-It is obvious that in all four plots the agreement between the results from the LWA and from the
+It is obvious that in all four plots the agreement between the results from the LMA and from the
SWA is excellent.
\begin{figure}[h]
the worm algorithm has difficulties to efficiently sample the
system because it changes the link occupation number in every move, while the LMA has a sweep with only
closed surfaces. The plot on the rhs. of Fig.~\ref{auto} shows how $\overline{\tau}$ for
-$U$ is larger for the SWA than for the LMA. We remark however, that this performance issue
+$\langle U \rangle$ is larger for the SWA than for the LMA. We remark however, that this performance issue
can be overcome easily by augmenting the SWA with sweeps of cube updates as used in the LMA.
\begin{figure}[t]
%``The silver blaze property for QCD with heavy quarks from the lattice,''
Phys.\ Rev.\ Lett. 110 (2013) 122001.
%%CITATION = ARXIV:1207.3005;%%
+%
+ K.~Langfeld, B.~Lucini and A.~Rago,
+ %``The density of states in gauge theories,''
+ Phys.\ Rev.\ Lett.\ {\bf 109} (2012) 111601
+ [arXiv:1204.3243 [hep-lat]].
+ %%CITATION = ARXIV:1204.3243;%%
+ %4 citations counted in INSPIRE as of 05 Nov 2013
\bibitem{dual}
projects / phys / proceedings_lattice13.git / blobdiff