Spin-distribution functionals and correlation energy of the Heisenberg model

We analyse the ground-state energy and correlation energy of the Heisenberg model as a function of spin, both in the ferromagnetic and in the antiferromagnetic case, and in one, two and three dimensions. First, we present a comparative analysis of known expressions for the ground-state energy $E_0(S)$ of {\it homogeneous} Heisenberg models. In the one-dimensional antiferromagnetic case we propose an improved expression for $E_0(S)$, which takes into account Bethe-Ansatz data for $S=1/2$. Next, we consider {\it inhomogeneous} Heisenberg models (e.g., exposed to spatially varying external fields). We prove a Hohenberg-Kohn-like theorem stating that in this case the ground-state energy is a functional of the spin distribution, and that this distribution encapsulates the entire physics of the system, regardless of the external fields. Building on this theorem, we then propose a local-density-type approximation that allows to utilize the results obtained for homogeneous systems also in inhomogeneous situations. We conjecture a scaling law for the dependence of the correlation functional on dimensionality, which is well satisfied by existing numerical data. Finally, we investigate the importance of the spin-correlation energy by comparing results obtained with the proposed correlation functional to ones from an uncorrelated mean-field calculation, taking as our example a linear spin-density wave state.