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

1999/10/18 by R. López‐Sandoval, R. Lopez-Sandoval, G. M. Pastor · 46 citations
Mathematics · Physics and Astronomy · #Advanced Chemical Physics Studies #Boundary value problem #Cold Atom Physics and Bose-Einstein Condensates #Combinatorics #Density matrix #Energy (signal processing) #Lattice (music) #Mathematical physics #Mathematics #Periodic boundary conditions #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum mechanics #cond-mat.str-el

paper · pdf · doi:10.1103/physrevb.61.1764

published in Physical review. B, Condensed matter 61(3), 1764-1772 (American Physical Society) · Phys. Rev. B (1999), in press

arxiv created 1999/10/18 · openalex publication_date 2000/01/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

A density-functional theory for many-body lattice models is considered in which the single-particle density matrix \ensuremathγ_ij is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional W[\ensuremathγ_ij]. W[\ensuremathγ_ij] is expressed as the sum of Hartree-Fock energy E_HF[\ensuremathγ_ij] and the correlation energy E_C[\ensuremathγ_ij]. Exact results are obtained for E_C(\ensuremathγ₁₂) of the Hubbard model on various periodic lattices, where \ensuremathγ_ij=\ensuremathγ₁₂ for all nearest neighbors i and j. The functional dependence of E_C(\ensuremathγ₁₂) 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(\ensuremathγ₁₂) are discussed in the limits of weak (\ensuremathγ₁₂\ensuremath≃\ensuremathγ₁₂⁰) and strong (\ensuremathγ₁₂\ensuremath≃\ensuremathγ₁₂^\ensuremath∞) electronic correlations, and in the crossover region (\ensuremathγ₁₂^\ensuremath∞<~\ensuremathγ₁₂<~\ensuremathγ₁₂⁰). Using an appropriate scaling we observe that \ensuremathε_C(g₁₂)=E_C/E_HF has a pseudo-universal behavior as a function of g₁₂=(\ensuremathγ₁₂\ensuremath-\ensuremathγ₁₂^\ensuremath∞)/(\ensuremathγ₁₂⁰\ensuremath-\ensuremathγ₁₂^\ensuremath∞). The fact that \ensuremathε_C(g₁₂) 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(\ensuremathγ₁₂) for repulsive (U>0) and attractive (U<0) interactions are contrasted.

Citations

• Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v -representability problem
• Generalized Gradient Approximation Made Simple
• A local exchange-correlation potential for the spin polarized case. i
• Beyond the local-density approximation in calculations of ground-state electronic properties
• Electron correlations in narrow energy bands
• Self-Consistent Equations Including Exchange and Correlation Effects

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