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arXiv:1510.03425 · Physical review. B./Physical review. B

Anomalous scaling and breakdown of conventional density functional theory methods for the description of Mott phenomena and stretched bonds

Zu‐Jian Ying, Valentina Brosco, Giorgia Maria Lopez, Daniele Varsano, Paola Gori‐Giorgi, J. Lorenzana

Published Aug 29, 2016• 25 citations
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Abstract

Density functional theory provides the most widespread framework for the realistic description of the electronic structure of solids, but the description of strongly correlated systems has remained so far elusive. We consider a particular limit of electrons and ions in which a one-band description becomes exact all the way from the weakly correlated metallic regime to the strongly correlated Mott-Hubbard regime. We provide a necessary condition a density functional should fulfill to describe Mott-Hubbard behavior in this one-band limit and show that it is not satisfied by standard and widely used local, semilocal, and hybrid functionals. We illustrate the condition in the case of few-atom systems and provide an analytic approximation to the exact exchange-correlation potential based on a variational wave function which shows explicitly the correct behavior, combining in a neat way lattice and continuum methods.

DOI
10.1103/physrevb.94.075154

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Density functional theory provides the most widespread framework for the realistic description of the electronic structure of solids, but the description of strongly correlated systems has remained so far elusive. We consider a particular limit of electrons and ions in which a one-band description becomes exact all the way from the weakly correlated metallic regime to the strongly correlated Mott-Hubbard regime. We provide a necessary condition a density functional should fulfill to describe Mott-Hubbard behavior in this one-band limit and show that it is not satisfied by standard and widely used local, semilocal, and hybrid functionals. We illustrate the condition in the case of few-atom systems and provide an analytic approximation to the exact exchange-correlation potential based on a variational wave function which shows explicitly the correct behavior, combining in a neat way lattice and continuum methods.
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We study the Mott insulating phase of the one-dimensional Hubbard model using a local-density approximation (LDA) that is based on the Bethe Ansatz (BA).Unlike conventional functionals, the BA-LDA has an explicit derivative discontinuity.We demonstrate that, as a consequence of this discontinuity, the BA-LDA yields the correct Mott gap, independently of the strength of the correlations.A convenient analytical formula for the Mott gap in the thermodynamic limit is also derived.We find that in one-dimensional quantum systems the contribution of the discontinuity to the full gap is more important than that of the band-structure gap, and discuss some consequences this finding has for electronic-structure calculations.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.
Reduced density matrix functional theory for the case of solids is presented and an exchange-correlation functional based on a fractional power of the density matrix is introduced. We show that compared to other functionals, this produces more accurate behavior for total energies as a function of particle number for finite systems. Moreover, it captures the correct band-gap behavior for conventional semiconductors, as well as strongly correlated Mott insulators, where a gap is obtained in the absence of any magnetic ordering.