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

TL;DR

This paper develops a density-matrix functional theory for the Hubbard model, deriving exact correlation energies for various lattice structures and analyzing their behavior across different correlation regimes.

Contribution

It introduces an exact numerical approach to compute correlation energies in lattice models using density-matrix functional theory, including analysis of finite clusters and scaling properties.

Findings

Exact correlation energies for Hubbard model on various lattices

Pseudo-universal scaling behavior of correlation energy

Contrast of correlation energies for repulsive and attractive interactions

Abstract

A density functional theory for many-body lattice models is considered in which the single-particle density matrix is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional W, which is expressed as the sum of Hartree-Fock energy and the correlation energy E_C. Exact results are obtained for E_C of the Hubbard model on various periodic lattices. The functional dependence of E_C is analyzed by varying the number of sites, band filling 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 are discussed in the limits of weak and strong electronic correlations, as well as in the crossover region. Using an appropriate scaling we observe a pseudo-universal behavior which suggests that the correlation energy of extended systems could be obtained quite accurately from finite cluster calculations. Finally, the behavior of E_C for repulsive (U>0) and attractive (U<0) interactions are contrasted.