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\centerline{\bf Computer Simulation of Spin Models}

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\centerline{Paul Coddington}
\centerline{Physics Department}
\centerline{Syracuse University}
% \centerline{Paul Coddington, Physics Department, Syracuse University}

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The study of spin models of magnetic materials is important not only for
understanding the properties of these materials, but also for the insights
it provides to the general theory of phase transitions and critical
phenomena. Much of our understanding of spin models comes from computer
simulations using Monte Carlo methods. Recently new algorithms have been
introduced by Swendsen and Wang, and by Wolff, which greatly increase the 
computational efficiency of these simulations by dramatically reducing 
critical slowing down, which is the tendency for a Monte Carlo algorithm
to take many iterations to produce a statistically independent value near 
the critical point. These new ``cluster'' algorithms allow much larger spin 
systems to be studied. The dynamics of the algorithms themselves are also 
of interest.

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There are currently three areas of study which are being pursued:

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1) Parallel cluster algorithms.

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   Cluster algorithms are non-local, updating clusters of spins at a time,
rather than single spins. The non-locality and irregularity of the clusters
means that unlike the standard local algorithms, the cluster algorithms are
very difficult to implement efficiently on large parallel computers. Some
algorithms have been developed for MIMD machines such as the NCUBE hypercube,
as well as SIMD machines such as the Connection Machine, but much more work
needs to be done to improve the rather poor efficiency of these algorithms,
especially in the case of SIMD machines.

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2) Study of cluster algorithms.

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   There is very little theoretical knowledge about how cluster algorithms 
work so well, and computer simulations allow us to obtain accurate numerical
data to compare different algorithms, in particular their autocorrelation
lengths and dynamical critical exponents. This data will also help in the 
construction and testing of theories of how these algorithms work. New 
variations of the cluster algorithms are also being investigated, especially 
those which might be more efficiently implemented on parallel computers.

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3) Physics of spin models.

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   The cluster algorithms allow us to simulate much larger systems with much
better statistics, thus enabling us to study their properties in more detail.
In particular, the use of the parallel algorithms mentioned above to implement
cluster algorithms on large parallel machines will allow us to do much better
than standard vector machines such as CRAYs, since the algorithm does not
vectorize at all.

   Here we are interested in studying phase transitions in the Q-state Potts
model, in particular the transition from second order to weakly first-order 
to strongly first order as Q is increased, and methods of unambiguously
determining the order of a weakly first-order transition. Other problems 
include the study of the XY and O(3) models at large lattice sizes. For the
XY model, recent computations have shown an apparent inconsistency in the 
measured
and theoretical critical exponents which needs to be resolved, while for the 
O(3) model the expected scaling behavior has not yet been seen for the lattice 
sizes currently able to be simulated.

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