Ising model

is a model in statistical mechanics to describe a field of two-state variables (spins) with local (nearest neighbor) interaction. The engery of the system is described by the Hamiltonian: $$E\left(\right\{s_i\left\}\right)=-J\sum _{⟨i,j⟩}{s}_i{s}_j-h\sum _i{s}_i$$ where $s_i=\pm 1$ is the spin at site $i$, $J$ is the coupling constant between neighboring spin pairs $⟨i,j⟩$, and $h$ is the external field.

In the following simulation of a 2D Ising model, we assume the coupling constant $J=1$ without losing generality. The temperature of the system is give by $T={\left(k\beta \right)}^{-1}$ where $k$ is the Boltzmann constant.