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arXiv:1212.3241 · Journal of Chemical Theory and Computation

Strong Correlations in Density-Functional Theory: A Model of Spin-Charge and Spin–Orbital Separations

Daniel Vieira

Published Jun 11, 2014• 7 citations
arXivHTMLPDF

Abstract

It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing one-dimensional (1D) strongly correlated systems [ Phys. Rev. B 2012 , 86 , 075132 ]. In this paper, within the density-functional theory (DFT) formalism, we extend the investigation by considering a model for the third electron fractionalization: the separation into spinons, chargons and orbitons, the last associated with the electronic orbital degree of freedom. Specifically, we deal with two exact constraints of exchange-correlation (XC) density-functionals: (i) the constancy of the highest occupied (HO) Kohn-Sham (KS) eigenvalues upon fractional electron numbers and (ii) their discontinuities at integers. By means of 1D discrete Hubbard chains and 1D H2 molecules in the continuum, we find that spin-charge separation yields almost constant HO KS eigenvalues, whereas the spin-orbital counterpart can be decisive when describing derivative discontinuities of XC potentials at strong correlations.

DOI
10.1021/ct400223d

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It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing one-dimensional (1D) strongly correlated systems [ Phys. Rev. B 2012 , 86 , 075132 ]. In this paper, within the density-functional theory (DFT) formalism, we extend the investigation by considering a model for the third electron fractionalization: the separation into spinons, chargons and orbitons, the last associated with the electronic orbital degree of freedom. Specifically, we deal with two exact constraints of exchange-correlation (XC) density-functionals: (i) the constancy of the highest occupied (HO) Kohn-Sham (KS) eigenvalues upon fractional electron numbers and (ii) their discontinuities at integers. By means of 1D discrete Hubbard chains and 1D H2 molecules in the continuum, we find that spin-charge separation yields almost constant HO KS eigenvalues, whereas the spin-orbital counterpart can be decisive when describing derivative discontinuities of XC potentials at strong correlations.
Strongly correlated electrons have been the subject of substantial theoretical interest for many years. Most work has focused on obtaining the energy in a low-cost fashion. Here, we show that even methods with good energies can yield significant "delocalization errors" that affect the orbitals and density, leading to large errors in predicting other important properties such as dipole moments. We illustrate this point by comparing existing state-of-art approaches with an accurate exchange correlation functional based on a generalised valence bond ansatz, in which orbitals and fractional occupations are treated as variational parameters via a common optimized effective potential (OEP). We show that the OEP exhibits step and peak features which, similar to the exact Kohn-Sham (KS) potential of DFT, are crucial to prevent charge delocalization. We further show that the step is missing in common approximations within reduced density matrix functional theory resulting in delocalization errors comparable to those found in DFT approximations. Finally, we explain the delocalisation error as coming from an artificial mixing of the ground state with a charge-transfer excited state which is avoided if occupation numbers exhibit discontinuities.
Electrons in one dimension display the unusual property of separating their spin and charge into two independent entities: The first, which derives from uncharged spin-1/21/21/2 electrons, can travel at different velocities when compared with the second, which is built from charged spinless electrons. Predicted theoretically in the early 1960s, spin-charge separation has attracted renewed attention since the first evidences of experimental observation, with usual mentions as a possible explanation for high-temperature superconductivity. In one-dimensional (1D) model systems, spin-charge separation causes the frequencies of Friedel oscillations to suffer a $2{k}_{F}\ensuremath{\rightarrow}4{k}_{F}$ crossover, mainly when dealing with strong correlations, where they are referred to as Wigner crystal oscillations. In nonmagnetized systems, the current density functionals that are applied to the 1D Hubbard model are not seen to reproduce this crossover, which leads to a more fundamental question: Are the Wigner crystal oscillations in 1D systems noninteracting V-representable? Or, is there a spin-independent Kohn-Sham potential that is able to yield spin-charge separation? Finding an appropriate answer to both questions is our main task here. By means of exact and density matrix renormalization group solutions, as well as an exchange-correlation potential introduced here, we show the answer to be positive. Specifically, the V-representable 4kF4{k}_{F}4kF​ oscillations emerge from attractive interactions mediated by positively charged spinless holes---the holons---as an additional contribution to the repulsive on-site Hubbard interaction.
An orbital-dependent correlation energy functional Ec to be accompanied by the exact exchange energy functional Ex is proposed for applications of density-functional theory (DFT). The present Ec comprises spin-antiparallel and spin-parallel contributions, Ecσ−σ and Ecσσ. Ecσ−σ is a modification of the spin-antiparallel component of the Hartree energy functional with a factor of gσ−σ (r, r′)−1 and Ecσσ a modification of the spin-parallel component of the same energy functional with gcσσ(r, r′) where gσ−σ(r, r′) (or gcσσ(r, r′)) is the spin-antiparallel (or the correlational part of the spin-parallel) coupling-constant-averaged pair correlation function. The present orbital-dependent gσ−σ(r, r′) and gcσσ(r, r′) fulfill the symmetric property, the Pauli principle and the sum rules. In the limit of uniform density the two correlation functions are reduced to the very accurate analogues of the electron liquid that involve long-, intermediate-, and short-range correlations as well as their exchange counterparts. It is stressed that the correlation energy functional Ec in DFT should by its very nature be defined as a functional only of occupied Kohn-Sham orbitals and occupied Kohn-Sham energies for the purpose of employing the optimized potential method (OPM) to evaluate the correlation potential υc(r). The present scheme for Ec, if applied to finite systems after making a suitable change in the treatment of long-range correlation, can give the correct asymptotic form of υc(r) of order r−4 for large r as well as the van der Waals potential.
Standard approximations for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified and extended to states with both fractional charge and fractional spin to give a much more stringent condition: the exact energy functional is a plane, linear along the fractional charge coordinate and constant along the fractional spin coordinate with a line of discontinuity at the integer. Violation of this condition underlies the failure of all known approximate functionals to describe the gaps in strongly correlated systems. It is shown that explicitly discontinuous functionals of the density or orbitals that go beyond these currently used smooth approximations is the key for the application of density functional theory to strongly correlated systems.
We generalize the exact strong-interaction limit of the exchange-correlation energy of Kohn-Sham density functional theory to open systems with fluctuating particle numbers. When used in the self-consistent Kohn-Sham procedure on strongly interacting systems, this functional yields exact features crucial for important applications such as quantum transport. In particular, the steplike structure of the highest-occupied Kohn-Sham eigenvalue is very well captured, with accurate quantitative agreement with exact many-body chemical potentials. While it can be shown that a sharp derivative discontinuity is present only in the infinitely strongly correlated limit, at finite correlation regimes we observe a slightly smoothened discontinuity, with qualitative and quantitative features that improve with increasing correlation. From the fundamental point of view, our results obtain the derivative discontinuity without making the assumptions used in its standard derivation, offering independent support for its existence.
Standard approximations for the exchange-correlation (XC) functional in Kohn-Sham density functional theory (KS-DFT) typically lead to unacceptably large errors when applied to strongly-correlated electronic systems. Partition-DFT (PDFT) is a formally exact reformulation of KS-DFT in which the ground-state density and energy of a system are obtained through self-consistent calculations on isolated fragments, with a partition energy representing the \textit{inter}-fragment interactions. Here we show how typical errors of the local density approximation (LDA) in KS-DFT can be largely suppressed through a simple approximation, the generalized overlap approximation (GOA), for the partition energy in PDFT. Our method is illustrated on simple models of one-dimensional strongly-correlated linear hydrogen chains. The GOA, when used in combination with the LDA for the fragments, improves the LDA dissociation curves of hydrogen chains and produces results that are comparable to those of spin-unrestricted LDA, but without breaking the spin symmetry. GOA also induces a correction to the LDA electron density that partially captures the correct density dimerization in strongly-correlated hydrogen chains. Moreover, with an additional correction to the partition energy, the approximation is shown to produce dissociation energies in quantitative agreement to calculations based on the Density Matrix Renormalization Group method.