Implement the Metropolis algorithm for the two-dimensional Ising model using your favorite software environment. You can set the coupling constant J = 1. Play around with the number of lattice sites (use a square lattice) and the number of simulation steps needed to reach convergence. See Sethna 8.1, 8.6 and 8.7 for hints. Play with the simulation. At high temperatures, the spins should not be strongly correlated. At low temperatures the spins should align all parallel, giving a large magnetization. Roughly locate Tc , the largest temperature where distant spins remain parallel on average at T = 0. Explore the behavior by gradually lowering the temperature from just above Tc to just below Tc; does the behavior gradually change, or jump abruptly (like water freezing to ice)? Explore the behavior at T = 2 (below Tc) as you vary the external field H = ±0.1 up and down through the phase boundary at H = 0. Does the behavior vary smoothly in that case?

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Solution The Ising Model The Ising Model is a simple model of a solid that exhibits a phase transition resembling ferromagnetism In this model a spin direction is assigned to each vertex on a graph Th... View the full answer