summaryrefslogtreecommitdiff
path: root/support/hypertex/tanmoy/examples/ambjorn.tex
diff options
context:
space:
mode:
Diffstat (limited to 'support/hypertex/tanmoy/examples/ambjorn.tex')
-rw-r--r--support/hypertex/tanmoy/examples/ambjorn.tex383
1 files changed, 383 insertions, 0 deletions
diff --git a/support/hypertex/tanmoy/examples/ambjorn.tex b/support/hypertex/tanmoy/examples/ambjorn.tex
new file mode 100644
index 0000000000..9ddcdea665
--- /dev/null
+++ b/support/hypertex/tanmoy/examples/ambjorn.tex
@@ -0,0 +1,383 @@
+%Paper: hep-lat/9405022
+%From: AMBJORN@nbivax.nbi.dk
+%Date: Wed, 25 May 1994 15:01:51 +0200
+\input lhyper
+\documentstyle[12pt]{article}
+\textwidth 150mm
+\textheight 235mm
+\newcommand{\rf}[1]{(\ref{#1})}
+%\newcommand{\beq}{\begin{equation}}
+%\newcommand{\eeq}{\end{equation}}
+\newcommand{\bea}{\begin{eqnarray}}
+\newcommand{\eea}{\end{eqnarray}}
+\newcommand{\g}{\gamma}
+\renewcommand{\l}{\lambda}
+\renewcommand{\b}{\beta}
+\renewcommand{\a}{\alpha}
+\newcommand{\n}{\nu}
+\newcommand{\m}{\mu}
+\newcommand{\th}{\theta}
+\newcommand{\ep}{\varepsilon}
+\newcommand{\om}{\omega}
+\newcommand{\sg}{\sigma}
+\newcommand{\k}{\kappa}
+\newcommand{\vph}{\varphi}
+\newcommand{\oh}{\frac{1}{2}}
+\newcommand{\oq}{\frac{1}{4}}
+\newcommand{\dt}{\triangle t}
+\newcommand{\fdt}{\frac{1}{\triangle t^2}}
+\newcommand{\dg}{\dagger}
+\newcommand{\non}{\nonumber \\}
+\newcommand{\tr}{{\rm Tr}\;}
+\newcommand{\ra}{\right\rangle}
+\newcommand{\la}{\left\langle}
+\newcommand{\sgn}{{\rm sgn}}
+
+\newcommand{\cD}{{\cal D}}
+\newcommand{\cS}{{\cal S}}
+\newcommand{\cM}{{\cal M}}
+\newcommand{\cK}{{\cal K}}
+\newcommand{\cT}{{\cal T}}
+\newcommand{\cN}{{\cal N}}
+\newcommand{\cL}{{\cal L}}
+\newcommand{\cO}{{\cal O}}
+\newcommand{\cR}{{\cal R}}
+%
+\newcommand{\noi}{\noindent}
+\newcommand{\no}{\nonumber}
+\newcommand{\Vqq}{V_{q\bar{q}}}
+%
+\def\void{}
+%\def\labelmark{\marginpar{\small\labelname}}
+\def\labelmark{}
+
+\newenvironment{formula}[1]{\def\labelname{#1}
+\ifx\void\labelname\def\junk{\begin{displaymath}}
+\else\def\junk{\begin{equation}\label{\labelname}}\fi\junk}%
+{\ifx\void\labelname\def\junk{\end{displaymath}}
+\else\def\junk{\end{equation}}\fi\junk\labelmark\def\labelname{}}
+
+\newenvironment{formulas}[1]{\def\labelname{#1}
+\ifx\void\labelname\def\junk{\begin{displaymath}\begin{array}{lll}}
+\else\def\junk{\begin{equation}\label{\labelname}\left.
+\begin{array}{lll}}\fi\def\arraystretch{1.5}\junk}%
+{\ifx\void\labelname\def\junk{\end{array}\end{displaymath}}
+\else\def\junk{\end{array}\right.\end{equation}}
+\fi\junk\labelmark\def\labelname{}\def\junk{}
+\def\arraystretch{1}}
+
+\newcommand{\beq}{\begin{formula}}
+\newcommand{\eeq}{\end{formula}}
+\newcommand{\beqv}{\begin{formula}{}}
+
+\begin{document}
+\topmargin 0pt
+\oddsidemargin 5mm
+\headheight 0pt
+\headsep 0pt
+\topskip 9mm
+
+\hfill NBI-HE-94-30
+
+\hfill March 1994
+
+\begin{center}
+\vspace{24pt}
+{\large \bf Scaling with a modified Wilson action which suppresses\\
+ Z$_2$ artifacts in SU(2) lattice gauge theories}
+
+\vspace{24pt}
+
+{\sl J. Ambj\o rn } and {\sl G. Thorleifsson}
+
+\vspace{6pt}
+
+ The Niels Bohr Institute\\
+Blegdamsvej 17, DK-2100 Copenhagen \O , Denmark\\
+
+
+\end{center}
+
+\vspace{24pt}
+
+\addtolength{\baselineskip}{0.20\baselineskip}
+\vfill
+
+\begin{center}
+{\bf Abstract}
+\end{center}
+
+\vspace{12pt}
+
+\noindent
+A modified Wilson action which suppresses plaquettes which take
+negative values is used to study the scaling behavior of the
+string tension. The use of the $\b_E$ scheme gives good agreement
+with asymptotic two loop results.
+
+
+\vfill
+
+\newpage
+\tableofcontents
+\newpage
+
+
+\section{Introduction}
+The simplest and most popular choice of a gauge lattice action is
+the one proposed by Wilson, which for the group $SU(2)$ reads:
+\beq{*1}
+S_W (U_l) = \b \sum_{\Box} (1 -\oh \tr U_\Box ),
+\eeq
+where $\b = 4/g^2$, $U_l \in SU(2)$ is the link variable
+defined on the link $l \equiv (x,\m)$, while $\Box \equiv (x,\m\n)$
+refers to the location and orientation of the corresponding plaquette.
+$U_\Box$ is the standard plaquette variable:
+\beq{*2}
+U_\Box \equiv U_{x,\m\n} = U_{x,\m}U_{x+\m,\n}U^\dg_{x+\n,\m}U^\dg_{x,\n}.
+\eeq
+
+In any Monte Carlo simulation the crucial question is whether the
+field configurations generated provide an adequate representation
+of the continuum physics. The naive continuum limit can only
+be obtained for $\oh \tr U_\Box \approx 1$. Especially
+any lattice configurations which locally have some $U_\Box$'s where
+$\oh \tr U_\Box \approx -1$ are lattice artifacts which a priori
+have nothing to do with continuum configurations. A complete suppression
+of such configurations should not change any continuum physics. For
+the values of $\b$ where one performs the Monte
+Carlo simulations such local $Z_2$ fluctuations are not at all rare if
+$S_W$ is used. It is thus very important to make sure that these fluctuations
+have no impact on continuum observables. The easiest way to do so is to
+modify the lattice action in such a way that these small-scale fluctuations
+are suppressed, without influencing plaquettes where $\tr U_\Box > 0$.
+This can be done by modifying the Wilson action as follows \cite{bcm}:
+\beq{*3}
+S=S_W+S_\l,
+\eeq
+where $S_W$ is the standard Wilson action \rf{*1}, while $S_\l$ is
+\beq{*4}
+S_\l = \l \sum_{\Box} [1-\sgn( \tr U_\Box)].
+\eeq
+
+The action \rf{*3} was studied for $\l =\infty$ in \cite{mp} while
+the phase structure in the $(\b,\l)$ plane was studied in \cite{bcm}.
+For a fixed $\l$ one would expect the same continuum limit for
+$\b \to \infty$ and also the same identification $\b = 4/g^2$.
+However, as shown in \cite{bcm} there is a marked difference
+between, say, Creutz ratios, measured for $\l =0$ (the Wilson action)
+and for $\l =0.5$ in the case of $\b=2.5$.
+One would expect the difference to be even more pronounced with increasing
+$\l$. This illustrates that plaquettes with negative values may
+play an important role for the range of $\b$'s where the scaling of
+pure $SU(2)$ lattice gauge theories is usually studied, and one could
+be worried about the relation to continuum physics. The fact that one gets
+acceptable agreement with continuum scaling relations could be fortuitous
+since one would expect the action $S$ to reproduce continuum physics
+better for large $\l$. One purpose of this article is to show that
+we indeed get the correct scaling relations for large $\l$ and that there
+is no reason to expect the plaquettes with negative values which appear
+in the Wilson action to play any important role for $\b \geq 2.2$.
+Another purpose is to test whether the $\l$ modification of the action
+may improve the approach to scaling for the reasons mentioned above.
+
+\section{Numerical method}
+
+The numerical simulations were performed using a standard
+Metropolis algorithm to update the gauge fields, combined
+with an overrelaxation algorithm to
+decrease the autocorrelations. For every Metropolis sweep
+we performed 4 overrelaxation steps.
+Lattice size
+$12^4$ with periodic boundary conditions was used and we
+measured
+all Wilson loops up to the size $6 \times 6$.
+In order to improve the statistics we modified the
+configurations using the Parisi trick
+\cite{ppf} before measuring. Unlike in the case of
+$\lambda = 0$ the integration involved in the Parisi trick
+had to be performed numerically and was thus costly in
+computer time. But the gain in the statistics did more than
+compensate for that.
+
+The choice of parameters was $\lambda = 10$, which was
+sufficient to suppress all negative plaquettes, and
+$\beta \in [1.0,2.5]$.
+We found a scaling window for $\beta \in
+[1.5,1.9]$ and concentrated the simulations in
+that interval. Usually 1000 sweeps where used for
+thermalization and the number of sweeps used
+for measuring, for various $\beta$,
+is shown in table 1. Measurements were performed
+every fifth sweep and the errors were estimated by
+using the jackknife method.
+
+\begin{table}
+\begin{center}
+\begin{tabular}{c|c|c}
+$\beta$ & No. of sweeps & No. of measurements \\ \hline
+1.5 & 55250 & 11050 \\
+1.6 & 65250 & 13050 \\
+1.7 & 63750 & 12750 \\
+1.8 & 88750 & 17750 \\
+1.9 & 45000 & 9000
+
+
+
+\end{tabular}
+\caption{Statistics collected at various $\beta$. In all measurements
+$\lambda = 10$}
+\end{center}
+\end{table}
+
+
+
+
+
+\section{Scaling of the string tension}
+The $\b$-values in the scaling window are even smaller
+than the ones used for the standard Wilson action. If one wants to
+test scaling by comparing with the perturbative two-loop result
+using the identification $\b = 4/g^2$
+it is doomed to fail. We clearly need a suitable effective coupling
+which we can use instead of $g^2$. Here we will use the
+so-called $\b_E$ scheme \cite{parisi,kp,smm}. This scheme is simple
+and well suited to deal with ``perturbations'' of the Wilson
+action, like the ones given by \rf{*3}-\rf{*4}. Let us define
+\beq{*5}
+\b_E = \frac{3}{4} \frac{1}{1-\la \oh \tr U_\Box \ra}.
+\eeq
+{}From weak coupling expansions ($\b \to \infty,~~\l$ fixed), it is seen that
+\beq{*6}
+\b_E \to \b~~~~~{\rm for}~~~~ \b \to \infty.
+\eeq
+It is consistent to use the $g_E=4/\b_E$ in the two loop $\b$-function
+since $g_E$ is a function of $g$ and the two first coefficients of the
+$\b$-function are invariant under a non-singular coupling constant
+redefinition. The use of $\b_E$ in the two loop $\b$-function
+is known to diminish scaling violations in a variety of situations:
+The string tension in $SU(2)$ and $SU(3)$ gauge theories,
+the mass gap in the $SU(2)$ and $SU(3)$ chiral models and the
+deconfining transition temperature $T_c$, again for $SU(2)$ and $SU(3)$
+gauge theories, just to mention some. We will now show that the method
+works well for the modified Wilson action with $\l=10$.
+
+Let us first note that for $\l =10$ the $\b$-range $1.4-1.9$ is mapped
+into the $\b_E$-range $1.837-2.104$, which should compared to the
+corresponding map for the ordinary Wilson action where the
+$\b$-range $2.2-2.5$ is mapped into the $\b_E$-range $1.741-2.154$.
+{}From the measurements of Wilson loops mentioned in the last section
+we extract the string tension using the simplest method: If $W(R,T)$
+denotes a $R \times T$ Wilson loop we first form
+\beq{*7}
+V_{eff} (R,T) = \log [W(R,T)/W(R,T-1)].
+\eeq
+For $T \to \infty$ $V_{eff} (R,T)$ should go to the ground state static
+quark potential $\Vqq(R)$. We have available $T \leq 6$ and the results
+agree within error bars for $T=5-6$. We have used these values as $\Vqq(R)$
+for $R \leq 6$. Next $\Vqq(R)$ is fitted to the form:
+\beq{*7a}
+\Vqq(R) = C - \frac{E}{R} + \sg R
+\eeq
+and the error bars for $\sg$ are mainly coming from
+systematic errors arising
+from fits with various lower cuts in $R$. It is clear that with the
+limited range of $R$ and $T$'s available to us nothing will be gained
+by applying some of the many more elaborate schemes of fitting which are
+available.
+
+$\log \sg(\b_E)$ is finally plotted in fig. 1 against $\b_E$. The curve
+shown is the one obtained from the two-loop $\b$-function, according to
+which the scaling of the lattice string tension
+$\sg(\b_E) = \sg_{cont} a^2(\b_E)$ is governed by the two-loop scaling
+of the lattice spacing $a(\b_E)^2$:
+\beq{*8}
+ a^2(\b_E) = \Lambda^{-2}
+\left( \frac{6\pi^2}{11} \b_E\right)^{102/121}
+\exp\left[-\frac{6\pi^2}{11} \b_E\right].
+\eeq
+It is clear that scaling is reasonable well satisfied in the given
+range of $\b_E$.
+
+It is interesting to compare these results with the corresponding
+ones obtained by using the ordinary Wilson action. The range of $\b$
+which gives the same range of $\b_E$ as were used for the
+modified action will be $2.3 \le \b \le 2.5$. In this range it is
+well known that naive application of scaling does not work particularly
+well, and to get a fair comparison with the results for the modified
+action we use again the $\b_E$ coupling constant transformation.
+We now follow the same method as in the case of the modified action
+and extract the string tension\footnote{The data used
+for $\b=2.4$ and 2.5 are taken from \cite{gutbrod}.}. The result is
+shown as a function of $\b_E$ in fig. 1. The use of $\b_E$ has
+improved the scaling considerable compared to
+the use of $\b$ as a coupling constant, as already mentioned, and it
+is apparent from fig.1 that agreement with the two-loop scaling
+is as good for the Wilson action as for the modified action.
+
+
+\section{Conclusions}
+The suppression of negative valued plaquettes should not change the
+continuum limit of pure $SU(2)$ lattice gauge theories since these
+configurations are pure lattice artifacts. Naively one might even
+expect a smoother approach to the continuum limit if such a suppression
+is implemented. The modified Wilson action \rf{*3} indeed suppresses
+negative valued plaquettes for large values of $\l$ and it is possible
+to approach continuum physics for much smaller values of $\b$. However,
+for these small values of $\b$ the relation between the continuum
+coupling constant $g^2$ and $\b$ is not a simple one and in order to
+extract the scaling behavior one has to use a modified coupling constant
+closer reflecting the physics of the system. The $\b_E$-scheme is such
+a prescription and using it we have found good agreement with continuum
+physics. From these results it seems clear that
+the theory with modified Wilson action
+belongs to the same universality class as the theory defined by the
+ordinary Wilson action when $\b$ is sufficiently large. The worry
+mentioned in \cite{bcm} is thus ruled out. In addition it seems
+that the approach to scaling is not dramatically improved
+compared to the situation for ordinary Wilson action. Indirectly
+this indicates that negative plaquette excitations play no important
+role in the scaling region for the ordinary Wilson action, at least
+when we discuss observables like the string tension. However,
+one would expect that the modified Wilson action is
+considerable better when it comes to the measurements of
+topological objects like instantons.
+
+\vspace{24pt}
+
+\noindent
+{\bf Acknowledgement} We thank V. Mitrjushkin for discussions and
+many useful suggestions. Part of the computations were performed
+at UNI-C and were made possible by a grant from the
+Danish Ministry of Research and Technology.
+
+
+
+\vspace{24pt}
+
+\addtolength{\baselineskip}{-0.20\baselineskip}
+
+
+\begin{thebibliography}{99}
+\bibitem{bcm}V. G. Bornyakov, M. Creutz and V.K. Mitrjushkin, Phys.Rev. D44
+(1991) 3918.
+\bibitem{mp}G. Mack and E. Pietarinen, Nucl.Phys. B205, (1982) 141.
+\bibitem{parisi}G. Parisi, Proceedings of the XX$^th$ conference
+on high energy physics, Madison 1980.
+\bibitem{kp}F. Karsch and R. Petronzio, Phys.Lett. 139B (1984) 403.
+\bibitem{smm}S. Samuel, O. Martin and K. Moriarty, Phys.Lett. 153B
+(1985) 87.
+\bibitem{ppf}G. Parisi, R. Petronzio and F. Rapuano, Phys.Lett. 126B
+(1983) 418.
+\bibitem{gutbrod}F. Gutbrod, Z.Phys. C37 (1987) 143.
+\end{thebibliography}
+
+
+\vspace{36pt}
+
+\noindent
+{\bf Figure 1:} The measured string tension using the modified
+action (upper curve) and the ordinary Wilson action (lower curve)
+as functions of $\beta_E$.
+
+\end{document}