# Lattice Density-Functional Theory of the Single-Impurity Anderson Model

We develop a lattice density-functional theory (LDFT) of the single-impurity Anderson model. The main motivation is to gain insight into the central exchange and correlation energy functional and to apply this theory for studying more realistic and general lattice models in which electron correlationsplay a crucial role.

In this approach the single-particle density matrix *γ _{ijσ}* with respect to the lattice sites

*ij*and the spin

*σ*replaces the wave function as the basic variable of the many-body problem. As in any density functional approach it is essential to derive accurate approximations to the fundamental interaction-energy functional

*W*[

*γ*]. We have used a two-level ansatz for the purpose of computing

*W*[γ] in spin-restricted systems. This ansatz involves explicitly only the impurity orbital and a single symmetry-adapted conduction-band state. A simple analytical functional dependence of the interaction energy has been obtained on the basis of exact results by taking advantage of the invariance of

*W*[

*γ*] with respect tounitary transformations within the conduction-band states. The thus derived

*two-level approximation*(TLA) is exact in two important opposite limits: a totally degenerate conduction band and a conduction band with widely separated discrete levels.

Fig. 1 shows the interaction energy *W*^{2L} of the half-filled two-level system for singlet states as a function of the off-diagonal density matrix element *γ _{sf}*, which describes the degree of charge fluctuations between the impurity orbital and the conduction band, and for a fixed impurity occupation

*γ*. In the uncorrelated limit

_{ff}*γ*the interaction energy is given by the Hartree-Fock theory, while the strongly correlated limit

_{sf}^{0}*γ*

_{sf}^{∞}is characterized by the minimal Coulomb energy. The inset illustrates the domain of density matrices

*γ*which can be derived from the ground states of the two-level Anderson Hamiltonian.

Fig. 1: Two-level singlet interaction energy functional W^{2L} of the half-filled Anderson model as a function of the off-diagonal density matrix element γ_{sf} for γ_{ff} = 0.75. The inset illustrates the various domains of representability of the density matrix.

We consider one and two dimensional spin unpolarized systems as application of the TLA. Several ground-state properties of finite Anderson rings and periodic square clusters such as the ground-state energy, the kinetic and Coulomb contributions, the impurity valence and local moments are studied. The full range of model parameters is explored from weak to strong correlations, integer to intermediate valence, narrow to wide conduction band, and even as a function of ring length. Fig. 2 shows the ground-state interaction energy *W*_{gs} as a function of the Coulomb repulsion *U*. In the inset the degree of charge fluctuations γ_{sf} between the impurity orbital and the conduction band is illustrated as a function of the impurity level ε* _{f}*. These and other LDFT results demonstrate that the TLA is a powerful tool which accurately describes all studied properties in the regimes of strong and weak correlations, as well as the Kondo and the intermediate valence regimes.

Fig. 2: Ground-state interaction energy W_{gs} of a half-filled N = 12 orbital Anderson ring as a function of the Coulomb repulsion U. Lattice density-functional theory (LDFT) results obtained within the two-level approximation (full curve) are compared with exact Lanczos diagonalizations (crosses) and unrestricted Hartree-Fock calculations (dashed curve). The inset illustrates the degree of charge fluctuations γ_{sf} as a function of the impurity level ε_{f}.

Research on an extension of the TLA to spin polarized density matrices is currently in progress. This would allow us to investigate systems with an odd number of electrons, to quantify the effect of an external magnetic field, and to explore spin excitations.

References:

- W. Töws and G. M. Pastor,
*Lattice density-functional theory of the single-impurity Anderson model: Development and applications*, Phys. Rev. B**83**, 235101-1-16 (2011) - W. Töws and G. M. Pastor,
*Spin-polarized density-matrix functional theory of the single-impurity Anderson model*, Phys. Rev. B**86**, 245123 (2012)