Blume-Capel model approximated by a sequence of generalized Husimi trees

We generalize a systematic approximation method presented by the present author earlier [Monroe, Phys. Rev. E 64, 016126 (2001)], and which was applied to Ising models with spin one-half. The generalization allows one to consider higher spin systems. In particular we consider the spin-one, Blume-Capel model on a square lattice. We obtain an approximation to the phase diagram of the system that we show is as or more accurate than any presently available. This we are able to do with a rather modest effort thereby illustrating the fact that the method gives one rather accurate results without requiring too extensive computer calculations.