The Ising Model and Phase Transitions

The Ising model in two dimensions with \( {\cal B} = 0 \) undergoes a phase transition of second order. What it actually means is that below a given critical temperature \( T_C \), the Ising model exhibits a spontaneous magnetization with \( \langle {\cal M} \rangle\ne 0 \). Above \( T_C \) the average magnetization is zero. The mean magnetization approaches zero at \( T_C \) with an infinite slope. Such a behavior is an example of what are called critical phenomena. A critical phenomenon is normally marked by one or more thermodynamical variables which vanish above a critical point. In our case this is the mean magnetization \( \langle {\cal M} \rangle\ne 0 \). Such a parameter is normally called the order parameter.