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Monte Carlo Studies of the Globally-Coupled Ising Model

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Daniel Edward Shai

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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.