Cusp singularity in mean field Ising model

An entropy of the Ising model in the mean field approximation is derived by the Hamilton-Jacobi formalism. We consider a grand canonical ensemble with respect to the temperature and the external magnetic field. A cusp arises at the critical point, which shows a simple and new geometrical aspect of this model. In educational sense, this curve with a cusp helps students acquire a more intuitive view on statistical phase transitions.