Spontaneous Anti-Ferromagnetic Change of State on a 2D Ising Lattice Simulation via Metropolis Monte-Carlo

The Ising Model has been a staple demonstration tool of thermal properties since 1920. It proves an attractive basis for exploring statistical mechanical properties of stochastic processes, and the power of computational methods in modeling stochastic processes. Using a nearest neighbor coupling energy model on the 2-D surface with periodic boundary conditions, it is proposed to model magnetic spin evolution in varying temperature. We randomly assign either spin up or spin down to each lattice point and utilize a Monte-Carlo approach to the Metropolis-Hastings algorithm to ascertain whether nearest neighbors are in a stationary state or not. The anti-ferromagnetic state of nearest neighbors (all neighbors having opposite spin) will only remain if the lattice temperature remains below a certain point. By initializing with random spin, applying nearest-neighbor coupling energies and periodically reducing the temperature, we will see a sudden state change where the lattice goes from randomly spinned, to perfectly anti-ferromagnetic and this will occur at a consistent state change temperature