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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} |