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Density-Matrix functional theory of strongly-correlated lattice fermions

A density functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix gamma_{ij} as basic variable. A simple, explicit approximation to the interaction-energy functional W[gamma] of the Hubbard model is derived from exact dimer results, scaling properties of W[gamma] and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with theBethe-Ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energyand charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.


I. INTRODUCTION
First principles methods and many-body lattice models are the two main theoretical approaches to the electronic properties of matter. From the first-principles perspective, the major breakthrough in the last decades has been Hohenberg-Kohn-Sham's (HKS) density-functional theory (DFT) and the derived powerful methods of electronic-structure calculation. 1 Despite their unparalleled success in an extremely wide variety of problems, current implementations of DFT have still serious difficulties in accounting for phenomena that involve strong electron correlations as observed, for example, in heavy-fermion materials, Mott insulators or high-T c superconductors. 2 Being in principle an exact theory, the limitations of DFT have to be ascribed to the approximations used for the interaction-energy functional W [ρ( r)] and not to the underlying formalism. The development of new functionals improving the description of strong correlation effects is therefore a major current theoretical challenge.
On the other side, the physics of strongly-correlated Fermi systems is intensively studied in the framework of parametrized lattice models (e.g., Hubbard, Anderson, etc.) by using specific leading-edge many-body techniques. 2 Taking into account the universality of DFT, and its demonstrated efficiency in complex ab initio calculations, it is quite remarkable that only few investigations have been concerned so far with applying the concepts of DFT to the lattice models describing strongly correlated fermions. 3,4,5,6 In fact, already from a formal standpoint, one may expect that DFT with an appropriate Ansatz for W should be a particularly valuable many-body approach to lattice models, thus becoming a subject of theoretical interest on its own. Moreover, DFT studies on simpler universal models also provide useful new insights relevant to first principles calculations, 1 particularly since in some cases the exact solution of the many-body problem is available. 2 The purpose of this paper is to extend the scope of DFT to the description of strong electron correlations in lattice Hamiltonians and to demonstrate quantitatively for the first time the performance of lattice density-functional theory (LDFT) in one-dimensional (1D) and two-dimensional (2D) systems. Sec. II presents concisely the basic formalism of LDFT.
In this framework the ground-state properties are obtained from the solution of exact selfconsistent equations that involve derivatives of the interaction-energy functional W [γ] with respect to the single-particle density matrix γ. In Sec. III the dependence of W on the nearest-neighbor (NN) density-matrix element γ 12 is analyzed and a simple explicit approximation to W (γ 12 ) is derived for the Hubbard model. Sec. IV discusses representative applications of this Ansatz. First, the accuracy of the method is demonstrated by comparison with available exact results on the 1D Hubbard model. New results are then discussed, particularly concerning the ground-state energy and charge-excitation gap in 2D lattices.
Finally, Sec. V summarizes our conclusions.

II. LATTICE DENSITY-FUNCTIONAL THEORY
In order to be explicit we focus on the Hubbard model which is expected to capture the main physics of lattice fermions in a narrow energy band. The Hamiltonian includes nearest neighbor (NN) hoppings t ij , and an on-site interactions U (n iσ =ĉ † iσĉiσ ). The importance of electron correlations is controlled by one parameter, namely, the ratio U/t. The hopping integrals t ij are defined by the lattice structure (typically, t ij = −t < 0 for NN ij) and thus play the role given in conventional DFT to the external potential V ext ( r).
Consequently, in LDFT the single-particle density matrix γ ij replaces the density ρ( r) as basic variable, since the hopping integrals t ij are nonlocal in the sites. 4 The situation is similar to the density-matrix functional theory proposed by Gilbert for the study of nonlocal pseudopotentials. 7,8 The ground-state energy E gs and density-matrix γ gs ij are determined by minimizing the energy functional with respect to γ ij . 4 The first term is the kinetic energy associated with the electronic motion in the lattice. The second term is Levy's interaction-energy functional 9 given by where the minimization runs over all N-particle states |Ψ[γ] satisfying for all i and j. 4,5,6,7 W [γ] represents the minimum interaction energy compatible with a given γ ij . It is a universal functional in the sense that it is independent of t ij . However, note that it still depends on the type of model interaction, on the number of electrons N e or band filling n = N e /N a , and on the number of sites N a . 10 E[γ] is minimized by expressing γ ij = γ ij↑ +γ ij↓ in terms of its eigenvalues η kσ (occupation numbers) and eigenvectors u ikσ (natural orbitals) as Introducing Lagrange multipliers µ and λ kσ (ε kσ = λ kσ /η kσ ) to impose the usual constraints kσ η kσ = N e and i |u ikσ | 2 = 1, one obtains the eigenvalue equations with ε kσ < µ (ε kσ > µ) if η kσ = 1 (η kσ = 0), and In Eq. (7) self-consistency is implied by the dependence of ∂W/∂γ ijσ on η kσ and u ikσ . The present formulation is analogous to well-known results of density-matrix functional theory in the continuum. 7 However, notice the fundamental differences with respect to the KS-like approach proposed in Ref. 4 , which assumes non-interacting v-representability, and where only integer occupations are allowed. The importance of fractional orbital occupations to the description of electron correlations within density-matrix functional theory has already been stressed by Gilbert. 7 In particular for the Hubbard model, one observes that 0 < η kσ < 1 for all k, except in very special situations such as U/t = 0 or the fully-polarized ferromagnetic state. This can be understood from perturbation-theory arguments -none of the η kσ is a good quantum number for U/t = 0-and has been explicitly demonstrated in exact solutions for finite clusters or the 1D chain. 11 Therefore, the case (8) is the only relevant one in general and all ε kσ in Eq. (7) must be degenerate. Consequently, for all i and j. Note that approximations of W in terms of diagonal γ ii alone can never yield such a behavior. 8 At this point it is important to observe that the general functional, valid for all lattice structures and for all types of hybridizations, can be simplified at the expense of universality if the hopping integrals are short ranged. For example, if only NN hoppings are considered, E K is independent of γ ij for pairs of sites ij that are not NN's. In this case, the constraints (4) and (5) need to be imposed only for i = j and for NN ij. This allows to reduce drastically the number of variables and simplifies considerably the search for practical approximations to W . Moreover, in periodic lattices the groundstate γ gs ij is a translational invariant. In order to determine E gs and γ gs ij , one may then set γ ii = n = N e /N a for all sites i, and γ ij = γ 12 for all NN pairs ij. Thus, the interaction energy can be regarded as a simple function W (γ 12 ) of the density-matrix element between NN's. It should be however noted that this also implies that W loses its universal character, since the NN map and the resulting dependence of W on γ 12 are in principle different for different lattice structures. 6

III. INTERACTION-ENERGY FUNCTIONAL FOR THE HUBBARD MODEL
Given a self-consistent scheme that implements the variational principle, the challenge is to find a good, explicit approximations to the interaction-energy functional. W [γ] may be determined exactly for small clusters by using numerical methods that perform the constrained minimization explicitly. 6 For a Hubbard dimer with N e = N a = 2 a straightforward analytical calculations yields which represents the minimum average number of double occupations for a given degree of electron delocalization, i.e., for a given γ 12 (U > 0). Despite its simplicity, Eq. (ii) For γ 12 = γ 0 12 , the underlying electronic state Ψ[γ 0 12 ] is a single Slater determinant and therefore W (γ 0 12 ) = E HF = n 2 U/4, where E HF is the Hartree-Fock energy. Moreover, ∂W/∂γ 12 = ∞ for γ 12 = γ 0 12 , since γ gs 12 < γ 0 12 already for arbitrary small U/t, as expected from perturbation theory.
(iv) γ ∞ 12 represents the largest NN bond order that can be obtained under the constraint of vanishing Coulomb energy. A lower bound for γ ∞ 12 is given by the bond order γ FM 12 in the fully-polarized ferromagnetic state which is formed by occupying the N e lowest singleparticle states of the same spin (n ≤ 1). Note that the ground-state γ gs 12 always satisfies γ ∞ 12 ≤ γ gs 12 ≤ γ 0 12 even though, for n = 1, it is possible to construct N e -electron states having |γ 12 | < |γ ∞ 12 |. In order to derive a simple approximation to W (γ 12 ) that preserves the previous general properties we take advantage of its scaling properties. Exact numerical studies 6 have shown that W (γ 12 ) depends weakly on N e , N a and lattice structure if it is measured in units of E HF and if γ 12 is scaled within the relevant domain of representability [γ ∞ 12 , γ 0 12 ]. Physically, this means that the relative change in W associated to a given change in the degree of electron localization g 12 = (γ 12 −γ ∞ 12 )/(γ 0 12 −γ ∞ 12 ) can be regarded as nearly independent of the system under study. A good general approximation to W (γ 12 ) can then be obtained by applying such a scaling to the functional dependence extracted from a simple reference system which already contains the fundamental relationship between localization and correlation. We therefore derive an approximate W (γ 12 ) taking its functional dependence from the exact result for the Hubbard dimer given by Eq. (10). In this way one obtains where E HF , γ 0 12 and γ ∞ 12 are system specific [see (i)-(iv) above]. In practice, γ ∞ 12 may be approximated by the ferromagnetic fully-polarized γ FM 12 which is calculated, as γ 0 12 , by integration of the single-particle spectrum.   11 One observes that the proposed approximation follows W ex (γ 12 ) quite closely all along the crossover from weak correlations (large W/U and γ 12 ) to strong correlations (small W/U and γ 12 ). This is remarkable, taking into account the simplicity of Eq. (11) and the strong band-filling dependence of E HF , γ 0 12 , and γ ∞ 12 . The quantitative discrepancies between Eq. (11) and W ex (γ 12 ) remain small in the complete domain of representability of γ and for all band fillings: |W − W ex |/E HF ≤ 0.063 for all γ 12 and n. Consequently, a good general performance of the method can be expected already at this stage. In the following section several applications of LDFT are discussed by using Eq. (11) as approximation to the interaction-energy functional.

IV. RESULTS AND DISCUSSION
In Fig. 2  observe that the accuracy of the calculated E gs is not the result of a strong compensation of errors since a similar accuracy is achieved for the kinetic and Coulomb energies separately.
Indeed, as shown in Fig. 3, both local moments S 2 i = 3 (n i↑ −n i↓ ) 2 and kinetic-energy renormalizations are also very well reproduced as a function of U/t. Moreover, notice that no artificial symmetry breaking is required in order to describe correctly the correlation-induced localization, as it is often the case in other approaches (e.g., antiferromagnetic spin-density wave for n = 1). For n ≤ 0.8, the LDFT results are almost indistinguishable from the exact ones. Even the largest quantitative discrepancies, found for n = 1 and intermediate U/t, are acceptably small (e.g., |E gs − E ex gs |/t = 0.044 for U/t = 4). For U ≫ t and n = 1 we obtain E gs ≃ −αt 2 /U with α  For instance, for U/t = 1 one obtains |E gs − E ex gs |/|E ex gs | = 4.4 × 10 −3 , and for U/t = 4 |E gs − E ex gs |/|E ex gs | = 9.8 × 10 −2 . Results with similar precision are found for a the triangular 2D lattice. In this case, using also a N a = N e = 3 × 4 cluster with periodic boundary conditions, we find |E gs − E ex gs |/|E ex gs | = 1.7 × 10 −4 for U/t = 1, and |E gs − E ex gs |/|E ex gs | = 6.6 × 10 −2 for U/t = 4. For both lattice structures |E gs − E ex gs | decreases quite rapidly away from half-band filling as in the 1D chain (see Fig. 2). LDFT, combined with Eq. (11) for W (γ 12 ), provides a correct description of electron correlations in different dimensions and lattice structures.
The charge-excitation or band gap is a property of considerable interest in strongly correlated systems which measures the insulating or metallic character of the electronic spectrum as a function of U/t and n. It can be directly related to the discontinuity in the derivative of the kinetic and correlation energies per site with respect to electron density n. 14 Therefore, the determination of ∆E c constitutes a much more serious challenge than the calculation of E gs , particularly in the framework of a density-functional formalism. At half-band filling, ∆E c = 0 in the uncorrelated limit (U/t = 0) and it increases with increasing U/t.
the energy of the bottom of the single-particle band (E b = −4t for a 1D chain and E b = −8t for the 2D square lattice). Fig. 6 presents LDFT results for ∆E c in 1D and 2D Hubbard models (n = 1). Comparison with the Bethe-Ansatz results 11 and with available QMC calculations 13 shows a good overall agreement. However, a more detailed analysis reveals that in the 1D case the gap is significantly overestimated for U/t ≪ 1. Here we obtain ∆E c ∝ (U/t) 2 , while the exact solution shows that for the infinite chain ∆E c increases much more slowly, namely, exponentially in −t/U. This discrepancy reflects the difficulty to describe long range-effects using an interaction-energy which functional dependence is derived from the dimer. Thus, it is possible that a similar overestimation of the gap at small U/t may also affect our results on 2D lattices. For larger U/t the accuracy improves rapidly as electron localization starts to set in, and the relative error in ∆E c vanishes. Therefore, the development of a Mott insulator with increasing U/t is described correctly.
Finally, we would like to comment briefly on a few other applications: (i) Dimerized chains with hoppings t ± δt have been investigated by allowing for alternations of γ 12 in Eq. (11).
One observes that the precision of the results improves systematically with increasing dimerization. For example, for N a = N e = 12 and U/t = 4, we find |E gs − E ex gs |/|E ex gs | = 3.3 × 10 −2 , 1.4 × 10 −2 , and 2.6 × 10 −3 for δt/t = 0, 1/2, and 3/4, respectively. The non-dimerized case, shown in detail in Fig. 2 An accurate approximation to W (γ 12 ) has been also derived for the attractive (negative U) Hubbard model in an analogous way as for U > 0. For a 1D ring with N a = N e = 12 we find |E gs − E ex gs |/|E ex gs | = 1.2 × 10 −3 , 7.5 × 10 −3 , and 1.5 × 10 −4 for |U|/t = 1, 4, and 64, respectively. These results show that LDFT describes electronic correlations correctly also when intra-atomic pairing is favored. Systematic investigations along these lines are currently in progress and will be published elsewhere. 12

V. CONCLUSION
A new density-functional approach to lattice-fermion models has been developed that is by all means independent of the homogeneous electron gas. A simple approximation to the interaction-energy functional is derived for the Hubbard model, which provides with a unified description of correlations in all interaction regimes from weak to strong coupling. Results for the ground-state energy and charge-excitation gap of 1D and 2D systems demonstrate the ability of lattice density functional theory to describe quantitatively the subtle competition between kinetic charge fluctuations and correlation-induced localization. The scope of DFT is thereby extended to the limit of strong electron correlations.
Several interesting directions open up with potential implications in various related areas.
For example, one may explore more general approximations to W [γ] and one may apply the present approach to richer physical situations such as low-symmetry systems, disorder, magnetic impurities, or multiband Hamiltonians. These developments should be relevant to the study of lattice fermion models and also in view of a DFT description of strong correlations from first-principles. K