Simulation of the 2D Ising model

One of the most interesting phenomena in nature is ferromagnetism. A FM material exhibits a non-zero spontaneous magnetization in the absence of an applied magnetic field. This occurs below a well-defined critical temperature $T_c$ known as the Curie temperature. For $T > T_c$ the magnetization vanishes. Hence $T_c$ separates two phases, a disordered one for $T > T_c$, and a ferromagnetic one for $T<T_c$.

Although the Ising model is too simple, it already contains much of the physics of the FM phase transition. In order to explore the properties of this model, we need to calculate some physical quantities of interest, including the mean energy $\langle E \rangle$, the mean magnetization $\langle M \rangle$, the heat capacity $C$, and the magnetic susceptibility $\chi$.