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.

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