Monte Carlo Studies of the Globally-Coupled Ising Model

Daniel Edward Shai

2007

The standard nearest neighbor Ising model is generalized to include global, long-range interactions that decay exponentially with distance. This model is used to study ferromagnetism in a variety of different lattices in 1, 2, and 3 dimensions. This computationally difficult problem is approached using Wang-Landau Monte Carlo simulation, where the density of states of a system is calculated, rather than using standard importance sampling techniques such as Metropolis Monte Carlo. Results indicate that the ferromagnetic to paramagnetic phase transition is stabilized by long-range interactions, despite the destabilizing effects of the lattice boundaries. The phase transition behavior is separated into two distinct regions. When the coupling is local, infinite lattice behavior is observed, and when the coupling is global, finite size effects are prevalent. The phase transition temperature appears to scale with the lattice packing efficiency and the total number of atoms, in the global coupling regime.