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 E0(S) of homogeneous Heisenberg models. In the one-dimensional antiferromagnetic case we propose an improved expression for E0(S), which takes into account Bethe-Ansatz data for S = 1/2. Next, we consider 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.

We calculate the ground-state energy of one and two-dimensional spatially inhomogeneous antiferromagnetic Heisenberg models for spins 1/2, 1, 3/2 and 2. Our calculations become possible as a consequence of the recent formulation of density-functional theory for Heisenberg models. The method is similar to spin-density-functional theory, but employs a local-density-type approximation designed specifically for the Heisenberg model, allowing us to explore parameter regimes that are hard to access by traditional methods, and to consider complications that are important specifically for nanomagnetic devices, such as the effects of impurities, finite-size, and boundary geometry, in chains, ladders, and higher-dimensional systems.PACS numbers: 71.15. Mb, 75.10.Jm, 75.50.Ee The study of low-dimensional spin systems is one of the central issues in the physics of correlated electrons. Model Hamiltonians of the Heisenberg type are widely used to study, e.g., antiferromagnetically coupled spin chains, ladders, and layers, 1,2 and constitute most useful models for strong correlations in undoped cuprates and manganites. Finite-size Heisenberg models, moreover, serve as paradigmatic models for the emerging field of nanomagnetism and spintronics. However, progress in the analysis of nanomagnetic devices requires capability to deal with 'real-life' complications, such as impurities of arbitrary size and location, boundaries of various shapes, and crossovers between finite and extended systems, or between one and higher dimensions. In the present paper we present a convenient and efficient numerical approach for calculating ground-state energies of antiferromagnetic Heisenberg models subjected to such complications.Traditional numerical methods are well suited for studying generic properties of homogeneous Heisenberg models, and have provided many important insights into the physics of magnetic systems. However, the very significant expenditure of computational effort required by methods such as group-theory-aided exact diagonalization, Quantum Monte Carlo (QMC), 3 or the densitymatrix renormalization group (DMRG), 4 imposes rather strict limits on the size and complexity of systems that can be studied. The mean-field approximation can, in principle, be applied to systems of almost arbitrary size and complexity, but its neglect of correlation makes it unreliable as a tool for studying many issues of current physical interest in strongly correlated systems. In ab initio calculations, density-functional theory (DFT) provides a convenient and reliable way to go beyond the mean-field (Hartree) approximation, allowing to study real systems of considerable size and complexity. 5,6,7 While DFT is in principle an exact reformulation of the many-body problem, 8 its practical implementation requires the use of approximations for the exchange-correlation energy. Among the most important such approximations is the local-density approximation (LDA), the essence of which is to use the exchange-correlation energy of the uniform electron liqui...

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Spin-density-functional theory (SDFT) is the most widely implemented and applied formulation of density-functional theory. However, it is still finding novel applications, and occasionally encounters unexpected problems. In this paper we first briefly describe a few of the latter, related to issues such as nonuniqueness, noncollinearity, and currents. In the main part we then turn to an example of the former, namely SDFT for the Heisenberg model. It is shown that time-honored concepts of Coulomb DFT, such as the local-density approximation, can be applied to this (and other) model Hamiltonians, too, once the concept of 'density' has been suitably reinterpreted. Local-density-type approximations for the inhomogeneous Heisenberg model are constructed. Numerical applications to finite-size and impurity systems demonstrate that DFT is a computationally efficient and reasonably accurate alternative to conventional methods of statistical mechanics for the Heisenberg model.

We present a mean-field calculation of the phase diagram of a simple model of localized moments, in the hexagonal uranium heavy-fermion compounds. The model considers a non-Kramers quadrupolar doublet ground state magnetically coupled with a singlet excited-state, favoring in-plane van-Vleck magnetism, as has been conjectured for UPt 3 . The Hamiltonian which defines the model is Heisenberg like in both, magnetic and quadrupolar moments. No Kondo effect physics is included in the calculations. Among our main results are: (i) for zero intersite quadrupolar coupling, the magnetic order is achieved by a first order transition above a critical intersite magnetic coupling value which becomes second order at higher coupling strengths.(ii) for finite intersite quadrupolar coupling, at temperatures below a second order quadrupolar ordering transition, the minimal magnetic coupling value is increased, but (a) the magnetic ordering temperature is enhanced above this value, and (b) the ordering of first and second order transitions in the phase diagram is reversed. By considering the general structure of the Ginsburg-Landau free energy, we argue that the Kondo effect will not modify the shape of the phase diagram, but will modify the quantitative values at which transitions occur.