Ground State of Interacting Spins

The question is posed of how the ground state of the Heisenberg Hamiltonian $H=-\Sigma F_{\mathrm{ij}}{\mathbf{S}}_i·{\mathbf{S}}_j$ depends on the magnitude $s$ of $N$ interacting spins, particularly in the case of long-ranged oscillatory interactions $F_{\mathrm{ij}}$. It is discussed whether fixing the geometry and the bond strengths $F_{\mathrm{ij}}$ suffices to determine the nature of the spin correlations in the ground state, and a review is given of known instances when this is the case; those are special situations in ferromagnetism and antiferromagnetism when qualitative ground-state properties such as "lack of nodes" can be proved to be independent of $s$. These are valuable examples for the application of semiclassical methods, which are strictly valid only for $s\to \infty$ and depend on the convergence of a series in powers of $s^{-1\mathrm{}}$. But these examples are, after all, only special cases, and it is argued that, in general, the nature of the ground state can depend sensitively on $s$. The following situation is considered in some detail: An oscillatory interaction which leads to a ferromagnetic ground state in the correspondence limit $s\gg 1$, but for which the ferromagnetic state of saturation magnetization may be unstable for small quantum mechanical spins, e.g., $s=\frac{1}{2} or 1$. Two distinct types of interaction are considered which lead to this result, and it is seen that the ferromagnetic instability is a consequence, not so much of the long range of the interaction as of the presence of some relatively strong antiferromagnetic (negative) bonds. However, the variational approach which is used casts no light on the nature of the true ground state or of the thermal properties, problems which are increasingly interesting in these instances when semiclassical procedures are seen to fail.