Lattice density-functional theory of the attractive Hubbard model
Résumé
The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT).
The ground-state energy E = T + W is regarded as a functional of the single-particle density matrix γij with
respect to the lattice sites, where T [γ ] represents the kinetic and crystal-field energies and W[γ ] the interaction
energy. Aside from the exactly known functional T [γ ], we propose a simple scaling approximation to W[γ ],
which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within
the domain of representability of γ . As applications, we consider one-, two-, and three-dimensional finite and
extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is
investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies,
as well as for the density distribution γii , nearest-neighbor bond order γij , and pairing energy �Ep, as a function
of the interaction strength |U|/t, onsite potential ε/t , and band filling n = Ne/Na . Remarkable even-odd and
super-even oscillations of �Ep are observed in finite rings as a function of band filling. Comparison with exact
Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very
good quantitative description of the properties of the model in the complete parameter range, thus providing a
significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.