Study, using the provided program ising.f90, the ferromagnetic transition as a function of temperature for a square lattice in 2 dimensions.

Modify the program in order to calculate the unnormalized autocorrelation times for Magnetization and Energy,

$$C_{MM}(t) = < M_s M_{s+t} > - \mu_M^2, \quad C_{EE}(t) = < E_s E_{s+t} > - \mu_E^2,$$

as well as the normalized autocorrelation times

$$\rho_{MM}(t) = C_{MM}(t)/C_{MM}(0), \quad \rho_{EE}(t) = C_{EE}(t) / C_{EE}(0).$$

Choose a not too small system size which is easily treated by computer you have access to (any L larger than 10 should be fine).

Determine approximately the Curie temperature of the system looking for divergence in the magnetic susceptibility and heat capacity and compare with the Mean Field result for $T_c$ .

Verify that approaching the phase transition autocorrelation times increase and take care of this fact in choosing the lenght of the simulation in order to perform a more or less equivalent sampling at all temperatures (you will need to modify the prgram to allow different number of cycle for different temperatures).

Optional: looking to systems of different size study how the maximum values reached by the susceptibility and the specific heat increase and the estimated critical temperature moves with system size.