Cluster Expansion for the Heisenberg Ferromagnet

A cluster series is derived for the Heisenberg ferromagnet. The theory is evaluated in zero order, for two-spin clusters, and for three-spin renormalized clusters. The zero-order result is the Weiss molecular field approximation, and the two-spin cluster is identical to the constant coupling approximation. The three-spin cluster result does not coincide with any existing phenomenological cluster approximation. The origin and nature of the inconsistencies in the phenomenological cluster methods (Oguchi, Bethe-Peierls-Weiss, etc.) is discussed.