Lattice density functional theory of the single-impurity Anderson model: Development and applications

A lattice density functional theory (LDFT) of the single-impurity Anderson model is presented. In this approach the basic variable is the single-particle density matrix ${\gamma }_{\mathit{ij}}$ with respect to the lattice sites and the fundamental unknown functional is the Coulomb interaction energy $W[\gamma ]$. Using general symmetry properties, a two-level ansatz for $W[\gamma ]$ in spin-restricted systems is proposed which involves explicitly only the impurity orbital and a single symmetry-adapted conduction-band state. A simple analytical functional dependence of $W[\gamma ]$ is derived on the basis of exact results of this two-level problem. The resulting approximation is shown to be exact in two important opposite limits: a totally degenerate conduction band and a conduction band with widely separated discrete levels. Applications to finite rings having $N⩽100$ atoms yield very accurate results for ground-state properties such as the kinetic, interaction, and total energy, as well as the occupation and magnetic moment of the local impurity orbital. This holds for all considered interaction strengths, from weak to strong correlations, as well as in the Kondo and intermediate valence regimes. One concludes that the present two-level approximation provides an appropriate framework for investigating subtle electron-correlation effects of the Anderson model within LDFT.

Figure 1

Two-level singlet interaction-energy functional $W^{2\mathrm{L}}$ of the half-filled Anderson model as a function of the off-diagonal density matrix element ${\gamma }_{\mathit{sf}}$. Results are given for (a) ${\gamma }_{\mathit{ff}}=0.75$, (b) ${\gamma }_{\mathit{ff}}=1.0$, and (c) ${\gamma }_{\mathit{ff}}=1.25$. ${\gamma }_{\mathit{sf}}^{\infty }$ refers to the limit of strong correlations and ${\gamma }_{\mathit{sf}}^0$ to the uncorrelated limit. Notice that $W^{2\mathrm{L}}$ is also defined for ${\gamma }_{\mathit{sf}}<0$ and that it depends only on $\left|{\gamma }_{\mathit{sf}}\right|$.

Figure 2

Domain of representability of the single-particle density matrix $\gamma$ for the half-filled two-level problem (singlet subspace) in terms of the occupation of the localized impurity orbital ${\gamma }_{\mathit{ff}}$ and the degree of charge fluctuations ${\gamma }_{\mathit{sf}}$. Only the density matrices $\gamma$ satisfying ${\gamma }_{\mathit{sf}}^{\infty }⩽\left|{\gamma }_{\mathit{sf}}\right|⩽{\gamma }_{\mathit{sf}}^0$ are ground-state representable (pure-state $v$ representable). The domains of pure-state and ensemble $N$ representability coincide and are simply given by $\left|{\gamma }_{\mathit{sf}}\right|⩽{\gamma }_{\mathit{sf}}^0$.

Figure 3

Ground-state properties of a $N_o=12$ orbital Anderson ring as a function of the Coulomb repulsion $U$: (a) kinetic energy $T_{\mathrm{gs}}$, (b) interaction energy $W_{\mathrm{gs}}$, (c) ground-state energy $E_{\mathrm{gs}}$, (d) occupation of the impurity orbital ${\gamma }_{\mathit{ff}}$, (e) degree of charge fluctuations ${\Gamma }_{\mathit{sf}}=(\sum _{\mathbf{k}}|{\gamma }_{\mathbf{k}f}|{}^2){}^{1/2}$, and (f) impurity magnetic moment ${\mu }_f$. Lattice density functional theory (LDFT) results obtained within the two-level approximation (full curves) are compared with exact Lanczos diagonalizations (crosses) and unrestricted Hartree-Fock calculations (dashed curves). The inset in (b) illustrates the corresponding single-particle spectrum, where the twofold degeneracy of most conduction-band orbitals is indicated by double lines. The inset of (d) shows $\langle {\mathrm{n̂}}_{f\uparrow }{\mathrm{n̂}}_{f\downarrow }\rangle$ for small $U$ and different $V_{\mathit{sf}}$, while the inset of (f) shows the broken-symmetry spin-polarization ${\gamma }_{\mathit{ff}\uparrow }-{\gamma }_{\mathit{ff}\downarrow }$ of the $f$ orbital in the unrestricted Hartree-Fock approximation.

Figure 4

Ground-state properties of a $N_o=12$ orbital Anderson ring as a function of the impurity energy ${\varepsilon }_f$: (a) interaction energy $W_{\mathrm{gs}}$, (b) ground-state energy $E_{\mathrm{gs}}$, (c) occupation of the impurity orbital ${\gamma }_{\mathit{ff}}$, (d) degree of charge fluctuations ${\Gamma }_{\mathit{sf}}=(\sum _{\mathbf{k}}|{\gamma }_{\mathbf{k}f}|{}^2){}^{1/2}$, and (e) local impurity moment ${\mu }_f$. LDFT (full curves) and exact Lanczos diagonalizations (crosses) are compared for $V_{\mathit{sf}}/t=0.4$ and $U/t=8$.

Figure 5

Ground-state properties of a $N_o=12$ orbital Anderson ring as a function of the conduction-band hopping integral $t$: (a) interaction energy $W_{\mathrm{gs}}$, (b) degree of charge fluctuations ${\Gamma }_{\mathit{sf}}=(\sum _{\mathbf{k}}|{\gamma }_{\mathbf{k}f}|{}^2){}^{1/2}$, (c) impurity-orbital occupation ${\gamma }_{\mathit{ff}}$, and (d) local impurity moment ${\mu }_f$. LDFT (full curves) and exact Lanczos diagonalizations (crosses) are compared for ${\varepsilon }_f/V_{\mathit{sf}}=-1.0$ ($V_{\mathit{sf}}>0$) and two different Coulomb repulsion parameters, $U/V_{\mathit{sf}}=20$ and $U/V_{\mathit{sf}}=5$.

Figure 6

Ground-state properties of Anderson rings as a function of the number of sites $N_a$ (even $N_o=N_a+1$ orbitals). The model parameters are $t>0$, ${\varepsilon }_f/t=-0.4$, $V_{\mathit{sf}}/t=0.4$, $U/t=2$, and $N_e=N_o=N_a+1$. Results are given for (a) the interaction energy $W_{\mathrm{gs}}$, (b) the degree of charge fluctuations ${\Gamma }_{\mathit{sf}}=(\sum _{\mathbf{k}}|{\gamma }_{\mathbf{k}f}|{}^2){}^{1/2}$, and (c) the occupation of the impurity orbital ${\gamma }_{\mathit{ff}}$. The inset in (a) shows the ground-state energy $E_{\mathrm{gs}}$, while the inset in (b) shows the size dependence of the conduction-band Fermi level ${\varepsilon }_F$. For $N_a⩽13$ the LDFT results (dots) are compared with exact Lanczos diagonalizations (crosses). The lines between the dots are a guide to the eye. The dashed vertical lines highlight the results for $N_a=3$ and $N_a=11$.

Figure 7

(a) Ground-state interaction energy $W_{\mathrm{gs}}$ and (b) local moment ${\mu }_f$ of the $N_a=3\times 3$ atoms Anderson square lattice as a function of the Coulomb repulsion $U$. In the lattice the additional impurity orbital is located at the center while the system parameters are $t>0$, ${\varepsilon }_f/t=-0.4$, $V_{\mathit{sf}}/t=0.4$, and $N_e=N_a+1=10$. The LDFT results (dots) are compared with exact Lanczos diagonalizations (crosses). In the insets the corresponding electronic properties for the $N_a=7\times 7$ atom lattice having an additional impurity orbital at the center and $N_e=50$ electrons are presented.