We study the efficiency and theory behind various Markov chain Monte Carlo update methods (later MCMC) in the classical Heisenberg model in three dimensions. Classical Heisenberg model is a model in statistical physics that describes ferromagnetic phenomena. Classical Heisenberg model is a generalization of the Ising model where the spin is three dimensional unit vector instead of scalar -1 or 1. Both models show a second order phase transition which is the main reason we are interested in these models. The transition in our case describes the loss of magnetization of a ferromagnet as it is heated to its Curie temperature.
Monte Carlo simulating the Classical Heisenberg model uses the same MCMC update methods as the Ising model. We introduce the theoretical background of Metropolis, Overrelaxation, and Wolff single cluster updates and study the dynamic critical exponents of Metropolis and Wolff updates. Results of this study are as expected: metropolis suffers from critical slowing down near the phase transition temperature and the autocorrelation time scales to L^5.08 , where L^3 is the size of the lattice. Wolff single cluster update avoids critical slowing down and scales very well with autocorrelation time scaling to L^3.28 .
Even though Wolff update scales much better it has its downsides. Parallelization is a very important factor in modern scientific computing and Wolff update is very tricky to parallelize where as Metropolis parallelizes very well. Usually we can avoid this problem by running multiple instances of the simulation at the same time as varying simulation parameters such as the temperature. For large lattices the problems is that getting moderate results requires a lot of time.
