The Hubbard dimer: a density functional case study of a many-body problem

Area of Science:

• Computational Chemistry
• Condensed Matter Physics
• Quantum Mechanics

Background:

• Density functional theory (DFT) is a powerful quantum mechanical method for electronic structure calculations.
• Strongly correlated models, like the Hubbard model, present challenges for standard DFT approximations.
• Understanding DFT's behavior in simplified, yet strongly correlated, systems is crucial for developing better functionals.

Purpose of the Study:

• To elucidate the relationship between density functional theory and strongly correlated models.
• To analyze the performance of DFT, including the Kohn-Sham (KS) method, on the two-site Hubbard model.
• To investigate the impact of asymmetry and correlation effects on DFT predictions.

Main Methods:

• Analytical solutions for the two-site Hubbard model to obtain exact ground-state energy and occupations.
• Levy-Lieb construction to parametrize the implicit density functional.
• Benchmark calculations of KS potentials, correlation kinetic energies, and the adiabatic connection.
• Testing of approximate functionals (Hartree-Fock, LDA) and analysis of the derivative discontinuity.

Main Results:

• The Kohn-Sham method fails to reproduce the fundamental gap in the two-site Hubbard model.
• Asymmetry introduces rich variations in behavior, highlighting interplay between correlation and charge-transfer.
• Approximate functionals show varying degrees of success and limitations in capturing model physics.

Conclusions:

• The two-site Hubbard model serves as an essential testbed for understanding DFT's fundamental aspects.
• The study demonstrates specific limitations of current DFT approximations for strongly correlated phenomena.
• Insights gained are valuable for the development of more accurate and robust DFT functionals.