Method for Determining Ground-State Spin Configurations

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Abstract
The method of Luttinger and Tisza for finding the rigorous minimum of a quadratic form subject to certain strong constraints is generalized. In the extended method, one still minimizes the quadratic form with respect to a single weak constraint, which however now contains adjustable parameters. In determining the ground state for the classical Heisenberg exchange energy, some cases involving crystallographically nonequivalent spins can now be handled. The following applications are made. The ground state for a linear chain with two different types of spins is obtained. We then prove that in the cubic spinel the Néel configuration is the ground state if it is locally stable—that is, it is never metastable. This result was assumed in a recent perturbation theory of spin configurations. Finally, a similar result concerning the Yafet-Kittel triangular configurations in noncubic spinels is discussed. In the course of the analysis it is shown that the ground state is always a spiral for any lattice in which the spins are equivalent.

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