     Next: 12.6.2 Cluster Algorithms Up: 12.6 Cluster Algorithms for Previous: 12.6 Cluster Algorithms for

## 12.6.1 Monte Carlo Calculations of Spin Models

The goal of computer simulations of spin models  is to generate configurations of spins typical of statistical equilibrium and measure physical observables on this ensemble of configurations. The generation of configurations is traditionally performed by Monte Carlo  methods such as the Metropolis  algorithm [Metropolis:53a], which produce configurations with a probability given by the Boltzmann distribution , where is the action, or energy, of the system in configuration , and is the inverse temperature. One of the main problems with these methods in practice is that successive configurations are not statistically independent, but rather are correlated with some autocorrelation time, , between effectively independent configurations.

A key feature of traditional (Metropolislike) Monte Carlo algorithms is that the updates are local (i.e., one spin at a time is updated), and its new value depends only on the values of spins which affect its contribution to the action, that is, only on local (usually nearest neighbor) spins. Thus, in a single step of the algorithm, information about the state of a spin is transmitted only to its nearest neighbors. In order for the system to reach a new effectively independent configuration, this information must travel a distance of order the (static or spatial) correlation length . As the information executes a random walk around the lattice, one would suppose that the autocorrelation time . However, in general, , where z is called the dynamical critical exponent. Almost all numerical simulations of spin models have measured for local update algorithms. (See also Sections 4.3, 4.4, 7.3, 12.6, and 14.2).

For a spin model with a phase transition,  as the inverse temperature approaches the critical value, diverges to infinity so that the computational efficiency rapidly goes to zero! This behavior is called critical slowing down (CSD), and until very recently it has plagued Monte Carlo simulations of statistical mechanical systems, in particular spin models, at or near their phase transitions. Recently, however, several new ``cluster algorithms'' have been introduced which decrease z dramatically by performing nonlocal spin updates, thus reducing (or even eliminating) CSD and facilitating much more efficient computer simulations.     Next: 12.6.2 Cluster Algorithms Up: 12.6 Cluster Algorithms for Previous: 12.6 Cluster Algorithms for

Guy Robinson
Wed Mar 1 10:19:35 EST 1995