2-D Ising Model Consider the 2-D Ising Model on a square lattice (with 4 nearest neighbors) and interaction strength J=2 at zero external magnetic field (except for part (d)). Use temperature units in which the Boltzmann constant KB=1. Python (a) Write a program to numerically study the 2-D Ising Model with periodic boundary conditions (N=n2: total number of spins). Calculate and plot the magnetization per spin, (M) /N = (s), as a function of temperature for T ~ 0 - 2T and n=20 (allowing for enough Monte Carlo sweeps to reach equilibrium and then average over time), and determine the critical tempera- ture as well as the critical exponent B in (M) x (To - T) . (Hint: To extract S and To, concentrate on the temperature region T ~ (0.9 -1)Tc. For several trial values of To, plot log(M) vs. log(T. - T), take as the best estimate the case where a straight-line fit appears most appropriate, and read off B; perform this procedure for two lattice sizes, n=20 and 40, to check for finite-size effects). (b) Compute and plot the thermodynamic energy per spin, (E) /N, for T = 0-2T. and n=20. Give a physical interpretation of what you find above To. (c) Calculate (and plot) the specific heat per spin, C(T) /N, for 3 different lat- tice sizes (say, n=10,30,100), using the fluctuation-dissipation theorem, C = (AE)?/(KBT2). Make sure to use sufficient temperature resolution to accu- rately determine the maximum in C(T) /N for each n. Can you verify the expected n dependence of Cmar/ N o log(n)? (d) Introduce an external magnetic field, H, into your code (use units in which the magnetic moment (=1) and calculate (and plot) the dependence of the magnetization on H for two temperatures, T = 1.5T. and T = Tc/3. Use fine-enough steps in H near 0 to convince yourself that the transition in M( H) is smooth and of first order, respectively. Do you see a hysteresis effect in the latter case