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# ising - Major Project Ising Model PHZ 5156 This problem...

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Major Project: Ising Model PHZ 5156 This problem combines what we have learned about the technique of Monte-Carlo simulation with the physics of magnetic phase transitions. In a magnetic material (e.g. Ni, Fe, etc.) at high temperatures, each atom has a large local magnetic moment, but they tend to be unaligned. By contrast, at low temperatures, exchange interactions tend to align the spins and create a macroscopic magnetic moment. We will make a simple model of the interactions and then use statistical physics to describe the phase transition from the paramagnetic to ferromagnetic state. The Hamiltonian H Ω for a two-dimensional spin system is given by H Ω = - 1 2 J N i =1 N j =1 S ij ( S i +1 ,j + S i - 1 ,j + S i,j +1 + S i,j - 1 ) (1) We can also consider adding an applied external field H which adds a term to the Hamiltonian - H N i =1 N j =1 S ij . A more shorthand way to write this is H Ω = - J ij S i S j - H i S i (2) where the summation is over nearest neighbor spins and H is an externally applied field. Determine the “mean field” result for the Ising model. In other words, compare your computed result to the mean field result. Mean-field theory begins formally by

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