Abstract
The common intuitive method for finding minimum energy spin states in Ising or classical vector-spin models consists of ordering spins in pairs in a manner that is consistent when one considers the spin ordering of the whole crystal. This method fails whenever there are competing interactions. It is shown how to directly extend this method so that such failure is no longer inevitable. The applicability of the extended or cluster method to various problems is investigated, and comparisons with the generalized Luttinger-Tisza methods is made. It is found that the two approaches are complementary, although there are problems in which both fail. In cases where both approaches work (which include the Ising problems considered by Luttinger), the cluster method provides the intuitively clearer derivation.
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