%%%%%%%%%% lat91.tex %%%%%%%%%%

% use this for final output
% \documentstyle{npbproc}
% and use this for draft output, with a black bar in the margin
% to warn you for lines and formulas that are too long
\documentstyle[draft]{npbproc}

% put your own definitions here

% ------------------------------------------------------------------
\begin{document}
% title of the article
\title{DYNAMIC EXPONENTS FOR POTTS MODEL CLUSTER ALGORITHMS}
% use separate \author and \address instructions
% for every author, or group of authors
\author{Paul D. Coddington}
\address{Physics Department, Syracuse University, Syracuse NY 13244, U.S.A.}
\author{Clive F. Baillie}
\address{Physics Department, University of Colorado, Boulder CO 80309, U.S.A.}

% a date field is not required
\date{}

% title and author's name as they appear in the running header
\runtitle{Dynamic Exponents For Potts Model Cluster Algorithms}
\runauthor{P.D. Coddington and C.F. Baillie} 
% the data below are for the volume and page numbers in the running header.
% Correct numbers will be pasted over the printed lines by the publisher.
\volume{XXX}  % Do not change the field.
\firstpage{1} % Do not change the field.
\lastpage{3}  % Supply the number of pages of your manuscript in the field,
              % e.g., \lastpage{10} for a 10 page manuscript.

% ------------------------------------------------------------------
\begin{abstract}
We have studied the Swendsen-Wang and Wolff cluster update algorithms
for the Ising model in 2, 3 and 4 dimensions. The data indicate simple
relations between the specific heat and the Wolff autocorrelations,
and between the magnetization and the Swendsen-Wang autocorrelations.
This implies that the dynamic critical exponents are related to the
static exponents of the Ising model. We also investigate the possibility
of similar relationships for the Q-state Potts model.
\end{abstract}

% put the front matter information on paper
\maketitle

% ------------------------------------------------------------------

The Monte Carlo cluster update algorithms of Swendsen and Wang (SW)~\cite{SW}
and Wolff~\cite{Wolff}
can dramatically reduce critical slowing down in computer simulations of spin 
models (for reviews of cluster algorithms, see
refs.~\cite{Sokalrev,reviews}).
There is little theoretical understanding of the dynamics of these 
algorithms, although a rigorous bound on the dynamic critical exponent
$z$ for the SW algorithm for Potts models has been derived~\cite{Li}.
% It is not understood why the SW and Wolff algorithms give similar values
% of $z$ for the 2-$d$ Potts model~\cite{us},
% but have very different behavior for other models, such as the Ising
% model in more than two dimensions~\cite{WIsing,Tamayo}.
%
In order to better understand these algorithms, 
we are doing numerical simulations of the Ising model in 2, 3 and 4 
dimensions using the SW and Wolff algorithms.
We measured the time correlation function $\rho(t)$ for the energy and 
extracted the integrated autocorrelation time $\tau_{int,E}$ \cite{Sokalrev}.
The dynamic critical exponent $z_{int,E}$ is given by 
${\tau_{int,E} \sim L^{z_{int,E}}}$, 
where $\tau_{int,E}$ for the different lattice sizes is measured at the 
infinite volume critical point.
A detailed account of the methods we use to do the measurements, fits and
error estimates, is given in Ref. \cite{us}.

Results for $\tau_{int,E}$ are shown in Fig. 1.
For $d=3$ we have used a log-log plot, with 
the straight lines representing $\chi^2$ fits to a power law, while
for $d=2$ and 4 we show a log-linear plot, with 
straight lines representing $\chi^2$ fits to a logarithm. 
Note that for $d=4$ we plot $\log \tau_{int,E}^{SW}$ rather than 
$\tau_{int,E}^{SW}$, since the SW autocorrelations increase as a power of $L$.
In Fig. 1 we also include the measured value of the specific heat
$C_H$, scaled by an appropriate factor, in order to show that the bound of 
Li and Sokal \cite{Li}
%
\begin{equation}
\tau_{int,E} \ge constant \times C_{H}, \qquad z_{int,E} \ge \alpha / \nu,
\label{Eqbound}
\end{equation}
%
is indeed satisfied by the SW algorithm. Here $\alpha$ is the critical
exponent for the specific heat, and
$\nu$ is the exponent for the correlation length.
No such bound has been proven for the Wolff algorithm,
although it appears from Fig. 1 that not only does the bound hold,
but that there may actually be equality in the exponents, i.e.
the Wolff algorithm for the Ising model seems to satisfy the
surprisingly simple relations
%
\begin{equation}
\tau_{int,E}^{W} = a + b \times C_{H}, \qquad z_{int,E}^{W} = \alpha / \nu,
\label{EqWtau}
\end{equation}
%
where $a$ and $b$ are constants. 

In Fig. 2 we plot the scaled difference 
%
\begin{equation}
( \tau_{int,E}^{W} - (a + b \times C_{H}) ) \quad / \quad \tau_{int,E}^{W}
\label{Eqdiff}
\end{equation}
%
with $a$ and $b$ chosen to minimize $\chi^2$ 
(smaller values of $L$ are excluded from the fit). 
In all cases values of $a$ and $b$ 
can be found such that the difference is zero within the errors, which
are generally of the order of $1\%$.
All the errors shown here are purely statistical (one standard deviation).

The surprising simplicity of this result led us to look for a 
similar relation for the SW algorithm. 
The relative average size of the largest SW cluster, 
${m = <|c_{SW}^{max}|> / L^d}$, is an estimator of the magnetization
\cite{Binder},
and the exponent $\beta / \nu$ for the divergence of the
magnetization has values which are similar
to our values for $z_{int,E}^{SW}$.
%
We thus scaled the SW autocorrelations by multiplying them by $m$.
This is also shown in Fig. 1 (note that the results are also scaled 
by an additional arbitrary constant, so that these points are not entangled
with others in the plots).
For ${d = 4}$ the results are very close to a constant, while for ${d = 3}$ 
they seem to approach a constant as $L$ increases. 
In two dimensions the scaled autocorrelations are not constant, but they
fit very well (much better than the unscaled data) to a logarithm, 
as can be seen in Fig. 1(a).
The results therefore support the assertion that
%
\begin{equation}
m \hskip 0.5em\relax \tau_{int,E}^{SW} = a + b \times \log L, \qquad z_{int,E}^{SW} = \beta / \nu.
\label{EqSWtau}
\end{equation}
%

Better data is needed to test these conjectures, since it is possible
that they are merely good approximations. Clearly a theoretical 
understanding of these results is also very desirable.
The relations \ref{EqWtau} and \ref{EqSWtau} are certainly not general 
results, since for the 2-$d$ 3-state Potts model we find that
${z^{W} > \alpha / \nu}$ and ${z^{SW} > \beta / \nu}$ \cite{Li,us}.
Also, it is quite surprising that these empirical relations imply that 
$z^{SW}$ is not equal to $z^{W}$ for the 2-$d$ Ising model, whereas the
two appear to be equal for the 2-$d$ 3-state Potts model. We have therefore
attempted to find similar simple relations for the 3-state Potts model.
The measured value of $z$ for this model is approximately 0.55(1) \cite{us},
which is very close to the value of $(\alpha + \beta) / \nu = 8/15 
\approx 0.533$. It is possible that this result also holds for the 4-state
model, although there the situation is less clear, since there are large
logarithmic corrections to scaling~\cite{Li}. 

In order to test this conjecture, we have scaled the autocorrelations (for
both the Wolff and SW algorithms) by the magnetization $m$, and then 
attempted to find a linear relation between these scaled autocorrelations 
and the specific heat. In both cases we find that the data is consistent
with such a relation for lattices of size $16^2$ to $256^2$ within the 
errors (which are of the order of $1\%$). Again, better data is required
to properly check this conjecture, and a similar analysis of data for the
4-state model would be very useful. 


% ------------------------------------------------------------------
% the bibliography comes at the end
\begin{thebibliography}{9}
\bibitem{SW}
R.H. Swendsen and J.-S. Wang, Phys. Rev. Lett. {\bf 58}, 86 (1987)
\bibitem{Wolff}
U. Wolff, Phys. Rev. Lett. {\bf 62}, 361 (1989)
\bibitem{Sokalrev}
A.D. Sokal, in: {\it Computer Simulation Studies in Condensed 
Matter Physics: Recent Developments}, eds. D.P. Landau, K.K. Mon
and H.-B. Sch\"uttler (Springer-Verlag, Berlin-Heidelberg, 1988);
A.D. Sokal, Proc. of the International Conference on Lattice Field Theory,
Tallahassee, (Oct. 1990), Nucl. Phys. B (Proc. Suppl.) {\bf 20}, 55 (1991)
\bibitem{reviews}
U. Wolff, Proc. of the Symposium on Lattice Field
Theory, Capri, (Sept. 1989), Nucl. Phys. B (Proc. Suppl.) {\bf 17}, 93 (1990);
J.-S. Wang and R.H. Swendsen, Physica A {\bf 167}, 565 (1990);
C.F. Baillie, Int. J. of Mod. Phys. C {\bf 1}, 91 (1990)
\bibitem{Li}
X.-J. Li and A.D. Sokal, Phys. Rev. Lett. {\bf 63}, 827 (1989)
\bibitem{us}
C.F. Baillie and P.D. Coddington, Phys. Rev. B {\bf 43}, 10617 (1991)
% \bibitem{WIsing}
% U. Wolff, Phys. Lett. {\bf B228}, 379 (1989)
% \bibitem{Tamayo}
% P. Tamayo, R.C. Brower and W. Klein, J. Stat. Phys. {\bf 58}, 1083 (1990)
% \bibitem{Wu}
% F.Y. Wu, Rev. Mod. Phys. {\bf 54}, 235 (1982)
% \bibitem{MCRGthree}
% G.S. Pawley {\it et al.}, Phys. Rev. B {\bf 29}, 4030 (1984)
% \bibitem{MCRGfour}
% H.W.J. Bl\"ote and R.H. Swendsen, Phys. Rev. B {\bf 22}, 4481 (1980)
\bibitem{Binder}
M. D'Onorio De Meo, D.W. Heermann and K. Binder, J. Stat. Phys. {\bf 60}, 585 (1990)

\end{thebibliography}

\begin{figure}
\vspace{18cm}    %  for example
    \caption[fig1]{$\tau_{int,E}$ for the Wolff and SW algorithms for the
    Ising model in (a) 2-$d$, (b) 3-$d$ and (c) 4-$d$. Also shown is the
    specific heat $ C_H$  and $\tau^{SW}_{int,E}$ scaled by the average 
    maximum cluster size $m$.}
    \label{figure1}
\end{figure}

\begin{figure}
\vspace{18cm}    %  for example
    \caption[fig2]{The scaled difference between the Wolff autocorrelations
    and a linear function of the specific heat for the Ising model 
    in (a) 2-$d$, (b) 3-$d$ and (c) 4-$d$.
    The values of $a$ and $b$ are chosen so as to minimize $\chi^2$.}
    \label{figure2}
\end{figure}

\end{document}

