Density-matrix functional theory of the Hubbard model:: An exact numerical study

A density-functional theory for many-body lattice models is considered in which the single-particle density matrix gamma ij is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional W[gamma ij]. W[gamma ij] is expressed as the sum of Hartree-Fock energy E-HF[gamma ij] and the correlation energy Ec[gamma ij] Exact results are obtained for E-C(gamma(12)) of the Hubbard model on various periodic lattices, where gamma ij = gamma(12) for all nearest neighbors i and j. The functional dependence of E-C(gamma(12)) is analyzed by varying the number of sites N-a, band filling N-e, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of E-C(gamma(12)) are discussed in the limits of weak (gamma(12) similar or equal to gamma(12)(0)) and strong (gamma(12) similar or equal to y(12)(infinity)) electronic correlations, and in the crossover region (gamma(12)(infinity)less than or equal to gamma(12)less than or equal to gamma(12)(0)). Using an appropriate scaling we observe that epsilon(C)(g(12)) =E-C/E (HF) has a pseudo-universal behavior as a function of g(12) = (gamma(12) - gamma(12)(infinity))/(gamma(12)(0) - gamma(12)(infinity)). The fact that epsilon(C)(g(12)) depends weakly on N-a, N-e, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of E-C(gamma(12)) for repulsive (U>0) and attractive (U<0) interactions are contrasted.