In this lecture we have introduced the Ising model, one of the classical models often studied in various branches of physics. It consists of a lattice in d dimensions (we choose d=2 here), where the sites are occupied with single spins. Each of them has two possible orientations - along the positive and negative z-axis and they experience nearest-neighbour interactions. These interactions support the alignment of the spins at low temperatures, while at high temperatures the effects of fluctuations take over. The model is well- suited to qualitatively understand the transition between the ferromagnetic and the paramagnetic phase in, e.g., metals.
We firstly analysed the model with help of analytical methods in the framework of the mean field approximation. In particular we studied the spontaneous magnetisation, M, stemming from the spin alignment, as a function of the temperature, T. There we found a phase transition, like in the case of percolation, signalled by a steep decline of M around a critical temperature. We then implemented the Metropolis algorithm for a better quantitative picture. This lends itself to the simulation of such systems. We found, again, a phase transition, but at quite a different critical temperature, with a different critical exponent.