Self-Consistent Correlation Method (S.C.C.M.) Applied to Ideal Ising and Heisenberg Magnetic Systems

Abstract

The theory to be described here may be classified as a cluster theory of ferromagnetism. It is presently restricted to lattices with nearest-neighbor interactions, but is unrestricted in spin magnitude. Its point of departure from previous cluster theories is that it represents the effect of the other spins of the lattice on the cluster as a temperature-dependent exchange coupling, rather than as a molecular field. A second significant departure is to impose a consistency condition on the spin-spin correlation functions of the cluster rather than on the spin averages as in previous theories. This new cluster theory can be motivated from quite fundamental considerations, and achieves a significant improvement over previous cluster theories, especially in the region just above the Curie temperature. Given a particular lattice, the cluster chosen for application of the theory consists of a central spin pair plus the additional spins and bonds necessary to form all the next most direct linked paths between the central spin pair. In the cluster Hamiltonian the central coupling has the constant value J while all the other couplings are taken to have a temperature-dependent value J′ (T) which will be evaluated self-consistently. J′ (T) can be understood physically as the direct coupling J plus an additional coupling caused by the correlative effect of spins outside the cluster acting on the cluster spins. The S.C.C.M. Hamiltonian can be derived more rigourously by writing the density matrix for the entire spin system and formally summing over all spin variables except those of the cluster. From this reduced density matrix it is possible to infer the form of the corresponding Hamiltonian. Apart from small terms which vanish at high temperatures, this is the S.C.C.M. Hamiltonian. The method of determining J (T) for any particular case brings out the second feature of the theory. By demanding that the correlation between the central pair be equal to that of any other nearest neighbor pair, J (T) can be uniquely determined for any temperature. Since the average energy, the free energy, and the specific heat of the lattice are related to the nearest-neighbor correlation function, these properties are immediately obtainable from the cluster calculation. If the correlation function is calculated with an external magnetic field added, then the dc magnetic susceptibility and average magnetization can also be determined. A complete presentation of the theory and results for the Heisenberg and Ising square and simple cubic lattices of spin one-half are contained in a paper by this author to be published elsewhere.