Exact results for the spin-1 Ising model on pure “square” Husimi lattices: Critical temperatures and spontaneous magnetization

We investigate the second order phase transitions of the ferromagnetic spin-1 Ising model on pure Husimi lattices built up from elementary squares with arbitrary values of the coordination number. It is shown that the critical temperatures of the second order phase transitions are driven by a single equation simultaneously on all such lattices. It is also shown that for arbitrary given value of the coordination number this equation is equivalent to the corresponding polynomial equation. The explicit form of these polynomial equations is present for the lattices with the coordination numbers z=4,6, and 8. It is proven that, at least for the small values of the coordination number, the positions of the critical temperatures are uniquely determined. In addition, it is shown that the properties of all phases of the model are also driven by the corresponding single equations simultaneously on all pure Husimi lattices built up from elementary squares. The spontaneous magnetization of the model is investigated in detail.