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\begin{title}
On generalized cluster algorithms for frustrated spin models
\end{title}
\author{P.D. Coddington and L. Han}
\begin{instit}
Physics Department, Syracuse University, Syracuse, NY 13244, USA
\end{instit}
\receipt{1 January 1992}
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\begin{abstract}
Cluster update algorithms have proved to be very effective for many 
different spin models, however they fail for frustrated spin systems.
Recently a generalized cluster algorithm was introduced which works
extremely well for the fully frustrated Ising model on a square lattice. 
However we have applied this method to the triangular and cubic lattice
fully frustrated Ising model and found that it works poorly,
producing clusters which are too large.
\end{abstract}
\pacs{PACS numbers 05.50.+q, 75.40.Mg}

\narrowtext

\section{INTRODUCTION}
\label{sec:intro}

The cluster update algorithm of Swendsen and Wang~\cite{SW} provides
a great improvement in the efficiency of Monte Carlo simulations over the
Metropolis algorithm~\cite{metrop} and other local update algorithms
by dramatically reducing critical slowing down~\cite{sokal88,sokal90} in
certain ferromagnetic and anti-ferromagnetic spin models. However the 
Swendsen-Wang algorithm fails for frustrated spin models, which are the 
systems for which critical slowing down is the most extreme. The reason for
the failure of the algorithm is that the clusters which are created are
much too large at the critical point, with the algorithm often producing
a cluster which encompasses a large fraction (if not all) of the sites in
the lattice, so that flipping the spins in this cluster is virtually the
same as a trivial global spin flip.

Recently Kandel, Ben-Av and Domany (KBD) introduced a generalized cluster
algorithm~\cite{kandel90,kandel91} which works very well for the two 
dimensional fully frustrated Ising model~\cite{villain}. 

In section 2 we present the method we used for implementing the KBD general
cluster method for a particular spin model. We apply this to the fully 
frustrated Ising model on a triangular lattice (section 3) and a cubic 
lattice (section 4).


\section{FORMULATING A GENERAL CLUSTER ALGORITHM}
\label{sec:algorithm}


\section{THE TRIANGULAR LATTICE ISING ANTIFERROMAGNET}
\label{sec:triangle}

We first attempted to generalize the KBD cluster algorithm to a very simple
fully frustrated model, the Ising anti-ferromagnet on a triangular
lattice. This model is very similar to the fully frustrated Ising model on
a square lattice, being in the same universality class and also having a 
critical point at zero temperature~\cite{wannier,stephenson}.



\section{THE 3-D FULLY FRUSTRATED ISING MODEL}
\label{sec:FFI}

We next attempted to apply the generalized cluster Monte Carlo idea to
the three dimensional fully frustrated Ising model \cite{BMB,grest,diep}.
This is

\section{CONCLUSIONS}
\label{sec:conclusion}


\acknowledgments

We would like to thank Ofer Biham, Danny Kandel and Enzo Marinari for 
valuable discussions.
This work was sponsored in part by Department of Energy grants
DE-FG03-85ER25009 and DE-AC03-81ER40050, 
and by a grant from the IBM corporation.

\begin{references}
\bibitem{kandel90}
D. Kandel, R. Ben-Av and E. Domany, Phys. Rev. Lett. {\bf 65}, 941 (1990)
\bibitem{kandel91}
D. Kandel and E. Domany, Phys. Rev. A {\bf 43}, 8539 (1991)
\bibitem{binder}
{\it Monte Carlo Methods in Statistical Physics, Topics in Current 
Physics 7, Second Edition}, Ed. K. Binder (Springer-Verlag, Berlin, 1986);
{\it Applications of the Monte Carlo Method in Statistical Physics, Topics in 
Current Physics 36, Second Edition}, Ed. K. Binder (Springer-Verlag, Berlin,
1987).
\bibitem{metrop}
N. Metropolis {\it et al.}, J. Chem. Phys. {\bf 21}, 1087 (1953).
\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{SWrev}
J.-S.~Wang and R.~H.~Swendsen, Physica A {\bf 167}, 565 (1990).
\bibitem{villain}
J. Villain, J. Phys. C {\bf 10}, 1717 (1977).
\bibitem{wannier}
G.H. Wannier, Phys. Rev. {\bf 79}, 357 (1950).
\bibitem{stephenson}
J. Stephenson, J. Math. Phys. {\bf 11}, 420 (1970).
\bibitem{BMB}
D. Blankschtein, M. Ma and A.N. Berker, Phys. Rev. B {\bf 30}, 1362 (1984).
\bibitem{grest}
G.S. Grest, J. Phys. C {\bf 18}, 6239 (1985).
\bibitem{diep}
H.T. Diep, P.Lallemand and O. Nagai, J. Phys. C {\bf 18}, 1067 (1985).
\bibitem{sokal88}
A.~D.~Sokal, in {\em Computer Simulation Studies in Condensed Matter
Physics:  Recent Developments}, eds.\ D.~P.~Landau {\em et al.}
(Springer-Verlag, Berlin-Heidelberg, 1988).
\bibitem{sokal90}
A.~D.~Sokal, 
in Proc. of the International Conference on Lattice Field Theory,
Tallahassee, October 1990,
Nucl. Phys. B (Proc. Suppl.) {\bf 20}, 55 (1991).
\bibitem{ising}
See for example B.~M.~ McCoy and T.~T.~Wu, {\em The Two-Dimensional Ising
Model,} (Harvard University Press, Cambridge, Mass., 1973).

\end{references}

\mediumtext
\begin{table}
\setdec 0.00
\caption{The Poisson ratio defined as the ratio of lateral
contraction to longitudinal expansion for uniaxial stress.
 Experimental values are given for comparison.}
\begin{tabular}{lcclcc}
  &\multicolumn{2}{c}{$\sigma$}& &\multicolumn{2}{c}{$\sigma$}\\
 &Predicted&Observed$^{\rm a}$& &Predicted&Observed$^{\rm a}$\\
\tableline
Cu& \dec 0.48 &\dec 0.36 &Al&\dec 0.47 &\dec 0.33 \\
Ag& \dec 0.48 &\dec 0.37 &Tl&\dec 0.47 &\dec 0.35 \\
Au& \dec 0.48 &\dec 0.36 &Sn&\dec 0.46 &\dec 0.33 \\
Mg& \dec 0.47 &\dec 0.35 &Pb&\dec 0.46 &0.40--0.45 \\
Zn& \dec 0.47 &\dec 0.25 &Pb&\dec 0.49 &\dec 0.43 \\
 & & &K&\dec 0.49 &\dec 0.44 \\
\end{tabular}
\label{table6}
\tablenotes{$^{\rm a}$Here we also cite a reference}
\end{table}
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