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

Abstract

A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ␥ i j is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional W͓␥ i j ͔. W͓␥ i j ͔ is expressed as the sum of Hartree-Fock energy E HF ͓␥ i j ͔ and the correlation energy E C ͓␥ i j ͔. Exact results are obtained for E C (␥ 12 ) of the Hubbard model on various periodic lattices, where ␥ i j ϭ␥ 12 for all nearest neighbors i and j. The functional dependence of E C (␥ 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 (␥ 12 ) are discussed in the limits of weak (␥ 12 Ӎ␥ 12 0 ) and strong (␥ 12 Ӎ␥ 12 ϱ ) electronic correlations, and in the crossover region (␥ 12 ϱ р␥ 12 р␥ 12 0 ). Using an appropriate scaling we observe that C (g 12 ) ϭE C /E HF has a pseudo-universal behavior as a function of g 12 ϭ(␥ 12 Ϫ␥ 12 ϱ )/(␥ 12 0 Ϫ␥ 12 ϱ ). The fact that 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 (␥ 12 ) for repulsive (UϾ0) and attractive (UϽ0) interactions are contrasted.

Density-matrix functional theory of strongly correlated lattice fermions

“…4,5,6,7 W [γ] represents the minimum interaction energy compatible with a given γ ij . It is a universal functional in the sense that it is independent of t ij .…”

Reduced density-matrix functionals from many-particle theory

“…López-Sandoval and Pastor observed a pseudo-universal relation between the correlation energy and the nearest-neighbor 1-RDM elements for one-, two-and threedimensional Hubbard models [90,91]. Combined with exact results for a Hubbard dimer, they constructed an approximate RDMF, that can describe the ground-state energy of one-, two-and three-dimensional Hubbard models [92,93] at different fillings and interactions strengths as well as charge excitation gaps very well.…”

Interaction-energy functional for lattice density functional theory: Applications to one-, two-, and three-dimensional Hubbard models

“…Recent numerical studies 16 have in fact shown that W is nearly independent of system size N a , band filling n = N e /N a and lattice structure, if W is measured in units of the Hartree-Fock energy E HF = Un 2 /4 and if γ 12 is scaled within the relevant domain of representability [γ ∞ 12 , γ 0 12 ]. Here, γ 0 12 stands for the largest possible value of the NN bond order γ 12 for a given N a , n, and lattice structure.…”

“…In non-bipartite lattices away from half-band filling one may also set for simplicity a k = 0 for odd k, since the dependence on g 12 is very similar for positive and negative γ 12 , once the different domains of representability are scaled. 16,24 The uncorrelated and fully-correlated limits of W (W = E HF for g 12 = 1, and W = 0 for g 12 = 0) impose two simple conditions on the a k , namely, P n (1) = k a k = 0 and P n (0) = a 0 = 1. This defines the second-order approximation W (2) completely.…”

“…15 In previous papers we have formulated a lattice density functional theory (LDFT) of many-body models by considering the density matrix γ ij with respect to the lattice sites i and j as the fundamental variable. 16,17,18 The interaction energy W of the Hubbard model has been calculated exactly as a function of γ ij for various periodic lattices having γ ij = γ 12 for nearest neighbors (NNs) i and j. On this basis, a simple general approximation to W (γ 12 ) has been proposed which derives from exact dimer results, the scaling properties of W , and known limits.…”