Approximate Free Energies for Heisenberg Ferromagnets

A systematic discussion is given of the approximate free energies and Curie temperatures that can be obtained by using trial density matrices (which describe various possible decompositions of the ferromagnet into clusters) in a variational calculation of the free energy. Single-spin clusters lead to the molecularfield model (as is well known) and two-spin clusters yield the Oguchi pair model. The relation of the "constant-coupling" method to these approximations is clarified. A rigorous calculation using three-spin $\Lambda$ clusters is found to give results differing from those of Oguchi. Finally, the Bethe-Peierls-Weiss approximation is considered within the framework of this paper.