The model we will employ in our studies of phase transitions at finite temperature for magnetic systems is the so-called Ising model. In its simplest form the energy is expressed as $$ \begin{equation*} E=-J\sum_{< kl>}^{N}s_ks_l-{\cal B}\sum_k^Ns_k, \end{equation*} $$ with \( s_k=\pm 1 \), \( N \) is the total number of spins, \( J \) is a coupling constant expressing the strength of the interaction between neighboring spins and \( {\cal B} \) is an external magnetic field interacting with the magnetic moment set up by the spins.
The symbol \( < kl> \) indicates that we sum over nearest neighbors only. Notice that for \( J>0 \) it is energetically favorable for neighboring spins to be aligned. This feature leads to, at low enough temperatures, a cooperative phenomenon called spontaneous magnetization. That is, through interactions between nearest neighbors, a given magnetic moment can influence the alignment of spins that are separated from the given spin by a macroscopic distance. These long range correlations between spins are associated with a long-range order in which the lattice has a net magnetization in the absence of a magnetic field.