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

A density-functional theory for many-body lattice models is considered in w hich the single-particle density matrix gamma ij is the basic variable. Eig envalue 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[gam ma ij] Exact results are obtained for E-C(gamma(12)) of the Hubbard model o n various periodic lattices, where gamma ij = gamma(12) for all nearest nei ghbors 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. T he infinite one-dimensional chain and one-, two-, or three-dimensional fini te clusters with periodic boundary conditions are considered. The propertie s 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)(i nfinity)) electronic correlations, and in the crossover region (gamma(12)(i nfinity)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) - gamm a(12)(infinity))/(gamma(12)(0) - gamma(12)(infinity)). The fact that epsilo n(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 accurat ely from finite-cluster calculations. Finally, the behaviors of E-C(gamma(1 2)) for repulsive (U>0) and attractive (U<0) interactions are contrasted.