A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.
The inhomogeneous Hubbard model is investigated in the framework of lattice density-functional theory (LDFT) by considering the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites as the basic variable of the many-body problem. The domain of representability of ${\ensuremath{\gamma}}_{ij}$ is determined for charge-density wave states on finite bipartite lattices. Levy's constrained search of the interaction-energy functional $W[{\ensuremath{\gamma}}_{ij}]$ is numerically solved by applying the Lanczos method to an effective Hubbard-type model. The exact functional dependence of $W[{\ensuremath{\gamma}}_{ij}]$ is analyzed by varying systematically the charge transfer $\ensuremath{\Delta}n={\ensuremath{\gamma}}_{22}\ensuremath{-}{\ensuremath{\gamma}}_{11}$, the degree of electron delocalization
g12 between the sublattices, the number of sites
Na, and the band filling $n=({\ensuremath{\gamma}}_{11}+{\ensuremath{\gamma}}_{22})/2={N}_{e}/{N}_{a}$. For each $\ensuremath{\Delta}n$ the properties of
W are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, as well as in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}\ensuremath{\le}{\ensuremath{\gamma}}_{12}\ensuremath{\le}{\ensuremath{\gamma}}_{12}^{0})$. It is shown that
W follows quite closely a simple scaling behavior as a function of $\ensuremath{\Delta}n$ and ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$. The very good transferability of $W(\ensuremath{\Delta}n,{g}_{12})$ for different
Na,
n and lattice structure opens new possibilities of applying LDFT to inhomogeneous many-body models.