Geometric symmetries and cluster simulations

Cluster Monte Carlo methods are especially useful for applications in the vicinity of phase transitions, because they suppress critical slowing down; this may reduce the required simulation times by orders of magnitude. In general, the way in which cluster methods work can be explained in terms of global symmetry properties of the simulated model. In the case of the Swendsen–Wang and related algorithms for the Ising model, this symmetry is the plus–minus spin symmetry; therefore, these methods are not directly applicable in the presence of a magnetic field. More generally, in the case of the Potts model, the Swendsen–Wang algorithm relies on the permutation symmetry of the Potts states. However, other symmetry properties can also be employed for the formulation of cluster algorithms. Besides of the spin symmetries, one can use geometric symmetries of the lattice carrying the spins. Thus, new cluster simulation methods are realized for a number of models. This geometric method enables the investigation of models that have thus far remained outside the reach of cluster algorithms. Here, we present some simulation results for lattice gases, and for an Ising model at constant magnetization. This cluster method is also applicable to the Blume–Capel model, including its tricritical point.