The Hubbard dimer: a density functional case study of a many-body problem
Abstract
This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of density functional theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy–Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the 'failure' of the Kohn–Sham (KS) method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree–Fock and Bethe ansatz local density approximation) to show their successes and limitations. Here, we also discuss and illustrate the concept of themore »
- Authors:
-
- Univ. de Oviedo (Spain); Nanomaterials and Nanotechnology Research Center, Oviedo (Spain)
- Univ. of California, Irvine, CA (United States)
- Publication Date:
- Research Org.:
- Univ. of California, Irvine, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES)
- OSTI Identifier:
- 1595136
- Grant/Contract Number:
- FG02-08ER46496; FIS2012-34858; DGE-1321846
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Physics. Condensed Matter
- Additional Journal Information:
- Journal Volume: 27; Journal Issue: 39; Journal ID: ISSN 0953-8984
- Publisher:
- IOP Publishing
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; density functional theory; Hubbard model; strongly correlated electron systems
Citation Formats
Carrascal, D. J., Ferrer, J., Smith, J. C., and Burke, K. The Hubbard dimer: a density functional case study of a many-body problem. United States: N. p., 2015.
Web. doi:10.1088/0953-8984/27/39/393001.
Carrascal, D. J., Ferrer, J., Smith, J. C., & Burke, K. The Hubbard dimer: a density functional case study of a many-body problem. United States. doi:https://doi.org/10.1088/0953-8984/27/39/393001
Carrascal, D. J., Ferrer, J., Smith, J. C., and Burke, K. Thu .
"The Hubbard dimer: a density functional case study of a many-body problem". United States. doi:https://doi.org/10.1088/0953-8984/27/39/393001. https://www.osti.gov/servlets/purl/1595136.
@article{osti_1595136,
title = {The Hubbard dimer: a density functional case study of a many-body problem},
author = {Carrascal, D. J. and Ferrer, J. and Smith, J. C. and Burke, K.},
abstractNote = {This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of density functional theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy–Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the 'failure' of the Kohn–Sham (KS) method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree–Fock and Bethe ansatz local density approximation) to show their successes and limitations. Here, we also discuss and illustrate the concept of the derivative discontinuity. Useful appendices include analytic expressions for density functional energy components, several limits of the exact functional (weak- and strong-coupling, symmetric and asymmetric), various adiabatic connection results, proofs of exact conditions for this model, and the origin of the Hubbard model from a minimal basis model for stretched H2.},
doi = {10.1088/0953-8984/27/39/393001},
journal = {Journal of Physics. Condensed Matter},
number = 39,
volume = 27,
place = {United States},
year = {2015},
month = {9}
}
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