The Hamiltonian of a two-dimensional Ising model is: Use periodic boundary conditions.

For this entire assignment, take h = 0.

Use or modify the program you wrote for an earlier problem to do the following simulations.

1. Simulate a 16 × 16 lattice with J = 1.0. Begin with giving each spin a random value of +1 or -1. Scan the temperature from 5.0 to 0.5 at intervals of ΔT = -0.5. Use about 100 or 1000 MC sweeps for each temperature, depending on the speed of your computer. Make a table of the results and plots of m vs. T, E vs. T, c vs. T, and χ vs. T. Does it look as if the critical temperature is at the mean field value of kBTc = 4J?

2. Repeat the sweeps, but this time choose the temperature range to concentrate on the peaks in the specific heat and the magnetic susceptibility. Estimate the location and height of the peak in specific heat and magnetic susceptibility. These will be estimates of the critical temperature.

3. And now for something completely different.

Do a simulation of a 32 × 32 lattice with negative temperature, and print out the spins at the end of the run. Begin with the spins in a random state (T = ∞), and simulate for 100 sweeps at T = -1.0. What do you see?