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arXiv:cond-mat/0305690 · Physical review. B, Condensed matter

Spin-distribution functionals and correlation energy of the Heisenberg model

Valter L. Líbero, Klaus Capelle

Published Jul 30, 2003• 21 citations
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Abstract

We analyze the ground-state energy and correlation energy of the Heisenberg model as a function of spin, both in the ferromagnetic and 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 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 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 us to utilize the results obtained for homogeneous systems in inhomogeneous situations too. 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.

DOI
10.1103/physrevb.68.024423

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We analyze the ground-state energy and correlation energy of the Heisenberg model as a function of spin, both in the ferromagnetic and 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){E}_{0}(S)E0​(S) of homogeneous Heisenberg models. In the one-dimensional antiferromagnetic case we propose an improved expression for E0(S),{E}_{0}(S),E0​(S), which takes into account Bethe-ansatz data for S=1/2.S=1/2.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 us to utilize the results obtained for homogeneous systems in inhomogeneous situations too. 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 describe how density-functional theory, well-known for its many uses in ab initio calculations of electronic structure, can be used to study the ground state of inhomogeneous model Hamiltonians. The basic ideas and concepts are discussed for the particular case of the Heisenberg model. As representative applications, illustrating scope and limitations of the procedure, we calculate the ground-state energy of one-, two- and three-dimensional antiferromagnetic Heisenberg models in the presence of boundaries and of impurities in the bulk and at the surfaces. Correlations are shown to lift degeneracies present in the mean-field approximation. Comparison with exact (brute force) diagonalization shows that the density-functional results are a significant improvement over the mean-field ones, at negligible extra computational cost.
An extension of density-functional theory, designed to treat spin-density waves and antiferromagnetic systems, is presented. The nonlocal nature of the antiferromagnetic correlations and possible noncollinearity in spin space are incorporated via an additional fundamental variable, the staggered density, which supplements the spin densities of conventional density-functional theory. Inclusion of this variable is justified by both physical and methodological considerations. We prove the corresponding Hohenberg-Kohn theorem, derive the pertinent Kohn-Sham equations, and present several approximate functionals depending explicitly on the staggered density. As a first test the formalism is applied to two simple model systems, a one-dimensional electron gas with a short-range interaction, and the three-dimensional electron gas with Coulomb interactions. These calculations serve to test the developed formalism, but also already allow us to draw a number of conclusions regarding the stability and nature of possible spin-density wave states in homogeneous electron systems.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.
We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.