The exact functionals associated with the (singlet) ground state and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as is commonly known, a convex function with respect to the density, the functional associated with the doubly excited state is found to be concave. Also, because the density-potential mapping associated with the first excited state is noninvertible, its "functional" is a partial, multivalued function composed of one concave and one convex branch that correspond to two separate domains of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are "functions" whose derivatives are each other's inverse, just as in the ground state formalism.
The exact functionals associated with the (singlet) ground state and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as is commonly known, a convex function with respect to the density, the functional associated with the doubly excited state is found to be concave. Also, because the density-potential mapping associated with the first excited state is noninvertible, its "functional" is a partial, multivalued function composed of one concave and one convex branch that correspond to two separate domains of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are "functions" whose derivatives are each other's inverse, just as in the ground state formalism.
Building on our recent study [Giarrusso and Loos, J. Phys. Chem. Lett. 14, 8780 (2023)], we explore the generalization of the ground-state Kohn-Sham (KS) formalism of density-functional theory (DFT) to the (singlet) excited states of the asymmetric Hubbard dimer at half-filling. While we found that the KS-DFT framework can be straightforwardly generalized to the highest-lying doubly excited state, the treatment of the first excited state presents significant challenges. In particular, using a density-fixed adiabatic connection, we show that the density of the first excited state lacks non-interacting v-representability. However, by employing an analytic continuation of the adiabatic path, we demonstrate that the density of the first excited state can be generated by a complex-valued external potential in the non-interacting case. More practically, by performing state-specific KS calculations with exact and approximate correlation functionals-each state possessing a distinct correlation functional-we observe that spurious stationary solutions of the KS equations may arise due to the approximate nature of the functional.
The dimerized one-dimensional Hubbard model is studied 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. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength U/t and of the alternation $\ensuremath{\delta}t$ of the hopping integrals tij ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.
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. 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.
We initiate the recently proposed w w -ensemble one-particle reduced density matrix functional theory ( w w -RDMFT) by deriving the first functional approximations and illustrate how excitation energies can be calculated in practice. For this endeavour, we first study the symmetric Hubbard dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb constrained search. Second, due to the particular suitability of w w -RDMFT for describing Bose-Einstein condensates, we demonstrate three conceptually different approaches for deriving the universal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime. Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the boundary of the functional’s domain, extending the recently discovered Bose-Einstein condensation force to excited states. Our findings highlight the physical relevance of the generalized exclusion principle for fermionic and bosonic mixed states and the curse of universality in functional theories.
The thermodynamic and two-site correlation functions for the half-filled-band isolated Hubbard dimer, dimerized, and uniform Hubbard chains for different values of the Coulomb energy U are evaluated in a unified way in the pair approximation of the cluster-variation method. The energy, specific heat, and correlation functions as functions of temperature for different ratios of $\frac{{t}^{\ensuremath{'}}}{t}$ (t, ${t}^{\ensuremath{'}}$ are intradimer and interdimer transfer integrals, respectively) and tU are obtained. Our solutions for the Hubbard dimer are exact and agree with those obtained from the partition function. Solutions for the dimerized and uniform Hubbard chains correspond to the Bethe approximations in which all two-site correlations are taken into account. It is observed that the chemical potential for the half-filled dimerized Hubbard Hamiltonian is exactly 2U at all temperatures, independent of the transfer integrals t or ${t}^{\ensuremath{'}}$.
Ensemble density functional theory (eDFT) is an exact time-independent alternative to time-dependent DFT (TD-DFT) for the calculation of excitation energies. Despite its formal simplicity and advantages in contrast to TD-DFT (multiple excitations, for example, can be easily taken into account in an ensemble), eDFT is not standard, which is essentially due to the lack of reliable approximate exchange-correlation (xc) functionals for ensembles. Following Smith et al. [Phys. Rev. B 93, 245131 (2016)], we propose in this work to construct an exact eDFT for the nontrivial asymmetric Hubbard dimer, thus providing more insight into the weight dependence of the ensemble xc energy in various correlation regimes. For that purpose, an exact analytical expression for the weight-dependent ensemble exchange energy has been derived. The complementary exact ensemble correlation energy has been computed by means of Legendre-Fenchel transforms. Interesting features like discontinuities in the ensemble xc potential in the strongly correlated limit have been rationalized by means of a generalized adiabatic connection formalism. Finally, functional-driven errors induced by ground-state density-functional approximations have been studied. In the strictly symmetric case or in the weakly correlated regime, combining ensemble exact exchange with ground-state correlation functionals gives better ensemble energies than when calculated with the ground-state exchange-correlation functional. However, when approaching the asymmetric equiensemble in the strongly correlated regime, the former approximation leads to highly curved ensemble energies with negative slope which is unphysical. Using both ground-state exchange and correlation functionals gives much better results in that case. In fact, exact ensemble energies are almost recovered in some density domains. The analysis of density-driven errors is left for future work.