Strong correlations in density-functional theory: A model of spin-charge and spin-orbital separations
It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing strongly correlated density distributions in one dimension. In this manuscript, 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 one-dimensional (1D) discrete Hubbard chains and 1D Hydrogen 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.
💡 Research Summary
The paper addresses two exact constraints that any exchange‑correlation (XC) functional in density‑functional theory (DFT) must satisfy: (i) the highest‑occupied Kohn‑Sham (KS) eigenvalue ε_H remains constant when the electron number is fractional, and (ii) ε_H exhibits a discontinuous jump at integer electron numbers, reflecting the derivative discontinuity of the exact XC potential. In one‑dimensional (1D) systems, strong electron correlations often lead to the phenomenon of spin‑charge separation, where an electron fractionalizes into a spin‑carrying spinon and a charge‑carrying chargon. The authors extend this concept by introducing a third fractionalization channel—spin‑orbital separation—where the orbital degree of freedom is carried by an “orbiton.”
To test the impact of these fractionalizations on the two constraints, the authors study two model systems: (a) discrete 1D Hubbard chains with varying on‑site interaction U/t, and (b) continuous 1D hydrogen molecules (H₂) where the inter‑nuclear distance controls the correlation strength. Exact reference data are obtained using density‑matrix renormalization group (DMRG) for the Hubbard chains and high‑level configuration‑interaction calculations for the H₂ molecule. The KS potentials are then reverse‑engineered from the exact densities, allowing a direct comparison of ε_H behavior under different functional approximations.
When only spin‑charge separation is incorporated into the XC functional, the calculated ε_H stays nearly flat across fractional electron numbers, thereby satisfying constraint (i). This result confirms that the spinon‑chargon picture correctly captures the linear interpolation of total energy with respect to particle number, a hallmark of the exact functional. However, the same approximation fails to produce the required jump in ε_H at integer electron numbers; the derivative discontinuity is severely underestimated, especially in the strong‑correlation regime (large U/t or stretched H₂).
Introducing the spin‑orbital (orbiton) degree of freedom remedies this deficiency. By adding an auxiliary potential that mimics the effect of orbiton excitations, the authors obtain ε_H curves that are flat in the fractional regime yet display a pronounced step at the integer electron number. The magnitude of the step matches the exact derivative discontinuity extracted from the reference calculations. This demonstrates that the orbital component of electron fractionalization is essential for reproducing the correct XC potential behavior when electron correlations are strong.
The analysis yields several key insights. First, spin‑charge separation alone is sufficient to enforce the constancy of the HO KS eigenvalue for fractional occupations, confirming its relevance for the “piecewise linearity” condition. Second, the derivative discontinuity—a subtle many‑body effect tied to the gap between the ionization potential and electron affinity—requires the additional orbiton channel. Third, the combined spinon‑chargon‑orbiton framework provides a unified picture that simultaneously satisfies both exact constraints, something that conventional local‑density or generalized‑gradient approximations cannot achieve.
Beyond the specific 1D models, the authors argue that similar fractionalization mechanisms may be operative in higher‑dimensional strongly correlated materials such as Mott insulators, cuprate superconductors, and low‑dimensional organic conductors. Consequently, future XC functional development should consider incorporating explicit spin‑charge‑orbital separation terms, possibly through non‑local or orbital‑dependent kernels, to capture both piecewise linearity and derivative discontinuities.
In summary, the paper demonstrates that a model incorporating spin‑charge and spin‑orbital separations can accurately reproduce the two exact XC constraints in strongly correlated 1D systems. The spin‑charge part ensures a constant highest‑occupied KS eigenvalue for fractional electron numbers, while the spin‑orbital part is decisive for generating the correct derivative discontinuity at integer occupations. This work provides a compelling blueprint for designing next‑generation XC functionals capable of handling strong correlation effects that are beyond the reach of traditional semilocal approximations.