Lattice density-functional theory of the attractive Hubbard model

The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy $E=T+W$ is regarded as a functional of the single-particle density matrix ${\gamma }_{ij}$ with respect to the lattice sites, where $T\left[\gamma \right]$ represents the kinetic and crystal-field energies and $W\left[\gamma \right]$ the interaction energy. Aside from the exactly known functional $T\left[\gamma \right]$, we propose a simple scaling approximation to $W\left[\gamma \right]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\gamma$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\gamma }_{ii}$, nearest-neighbor bond order ${\gamma }_{ij}$, and pairing energy $\Delta E_p$, as a function of the interaction strength $\left|U\right|/t$, onsite potential $\varepsilon /t$, and band filling $n=N_e/N_a$. Remarkable even-odd and super-even oscillations of $\Delta E_p$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.

Figure 1

Correlation between the NN bond order ${\gamma }_{12}$ and the average sublattice occupation ${\gamma }_{11}$ in the ground state of a 1D attractive Hubbard chain having alternating energy levels ${\varepsilon }_i=\pm \varepsilon /2$, $N_a=14$ sites, and half-band filling $n=1$. Different values of the local interaction $U<0$ are considered as indicated. The region enclosed by the upper bound ${\gamma }_{12}^0$ (red curve, full circles, $U=0$) and the lower bound ${\gamma }_{12}^{\infty }=0$ (open circles, $U=-\infty$) defines the domain of representability of $\gamma$.

Figure 2

(a) Interaction energy $W$ and (b) correlation energy $E_c=W-E_{\mathrm{HF}}$ of the attractive ($U<0$) and repulsive ($U>0$) Hubbard model as a function of the NN bond order ${\gamma }_{12}$, for representative charge transfers $\Delta n={\gamma }_{22}-{\gamma }_{11}$. Exact results are given for rings having $N_a=14$ sites and band filling $n=1$. Repulsive (attractive) interactions correspond to the shaded (nonshaded) area in (a) and to negative (positive) values of $E_c/U$ in (b). The red dashed curves indicate the Hartree-Fock energy $E_{\mathrm{HF}}$ corresponding to the largest representable value of ${\gamma }_{12}$ for the given $\Delta n$. The dotted curves refer to the strongly correlated repulsive limit, i.e., to the largest representable ${\gamma }_{12}$ allowing a minimal number of double occupations.

Figure 3

Scaled interaction energy $w=(W-W^{\infty })/(W^0-W^{\infty })$ of the attractive Hubbard model as a function of the degree of electron delocalization $g_{12}=({\gamma }_{12}-{\gamma }_{12}^{\infty })/({\gamma }_{12}^0-{\gamma }_{12}^{\infty })$. $W^0=E_{\mathrm{HF}}$ and ${\gamma }_{12}^0$ refer to the uncorrelated limit, while $W^{\infty }$ and ${\gamma }_{12}^{\infty }$ to the strongly correlated limit. The symbols correspond to exact numerical results $W_{\mathrm{ex}}$ for rings having an even number of sites $N_a=4$–14, half-band filling $n=1$, and different charge transfers $\Delta n={\gamma }_{11}-{\gamma }_{22}$. The curves are obtained using the dimer approximation $W_{\mathrm{sc}}$ [see Eq. (11)]. The inset figures highlight the relative errors $\Delta W=(W_{\mathrm{sc}}-W_{\mathrm{ex}})/(W^0-W^{\infty })$.

Figure 4

Charge-transfer dependence of the scaled interaction energy $w=(W-W^{\infty })/(W^0-W^{\infty })$ of the attractive Hubbard model as a function of $g_{12}=({\gamma }_{12}-{\gamma }_{12}^{\infty })/({\gamma }_{12}^0-{\gamma }_{12}^{\infty })$. Exact results are given for rings having $N_a=14$ sites, band filling $n=1$, and representative charge transfers $\Delta n={\gamma }_{11}-{\gamma }_{22}$.

Figure 5

Ground-state properties of bipartite Hubbard rings having $U<0$, $N_{\mathrm{a}}=14$, sites and half-band filling $n=1$, as a function of the attractive interaction strength $\left|U\right|/t$. Different values of the energy-level shift $\varepsilon$ between the sublattices are considered as indicated in (c). Results are given for (a) ground-state energy $E_{\mathrm{gs}}$, (b) charge transfer $\Delta n={\gamma }_{22}-{\gamma }_{11}$, (c) NN bond order ${\gamma }_{12}$, and (d) average number of double occupations per site $W/U{N}_a$. The symbols correspond to exact Lanczos diagonalizations and the solid curves to LDFT using the dimer approximation $W_{\mathrm{sc}}$ [see Eq. (11)].

Figure 6

Ground-state properties of the one-dimensional attractive Hubbard model at half-band filling as a function of the interaction strength $\left|U\right|/t$. Different values of the alternating energy-level shift $\varepsilon$ between the sublattices are considered as indicated in (c). Results are given for (a) ground-state energy $E_{\mathrm{gs}}$, (b) charge transfer $\Delta n={\gamma }_{22}-{\gamma }_{11}$, (c) NN bond order ${\gamma }_{12}$, and (d) average number of double occupations per site $W/U{N}_a$. The solid curves correspond to LDFT using the scaling approximation $W_{\mathrm{sc}}$ [see Eq. (11)], the symbols to accurate numerical results obtained using the density-matrix renormalization-group method, and the red dashed curves with asterisks to the BCS approximation for $\varepsilon /t=0$.

Figure 7

Ground-state properties of the two-dimensional attractive Hubbard model at half-band filling (square lattice) as a function of the interaction strength $\left|U\right|/t$. Results are given for (a) ground-state energy $E_{\mathrm{gs}}$, (b) charge transfer $\Delta n={\gamma }_{22}-{\gamma }_{11}$, (c) NN bond order ${\gamma }_{12}$, and (d) average number of double occupations per site $W/U{N}_a$, as obtained by using LDFT and the scaling approximation $W_{\mathrm{sc}}$ [see Eq. (11)]. Different values of the energy-level shift $\varepsilon$ between the sublattices are considered as indicated in (c).

Figure 8

Ground-state energy of attractive Hubbard rings having $N_a=14$ sites as a function of the number of electrons $N_e$. Different bipartite potentials $\varepsilon$ and interaction strengths $\left|U\right|$ are considered. The symbols correspond to exact diagonalizations, while the solid lines connecting discrete points refer to LDFT with the scaled dimer functional $W_{\mathrm{sc}}$.

Figure 9

Binding energy per atom $E_B/N_a$ of the (a) one-dimensional and (b) two-dimensional attractive Hubbard model as a function of band filling $n$. The curves are obtained by using LDFT with the scaled dimer functional $W_{\mathrm{sc}}$ for representative values of the interaction strength $\left|U\right|/t$. The symbols in (a) are the corresponding exact Bethe-ansatz results.

Figure 10

Pairing energy $\Delta E_p$ as a function of the number of electrons $N_e$ in 1D attractive Hubbard rings having $N_a=14$ sites. The symbols refer to exact Lanczos diagonalizations and the solid lines connecting small dots to LDFT with the scaled dimer functional. Representative values of the interaction strength $\left|U\right|$ and bipartite potential $\varepsilon$ are considered.

Figure 11

Pairing energy $\Delta E_p$ as a function of the interaction strength $\left|U\right|/t$ for the attractive Hubbard model on (a) rings having $N_a=14$ sites and (b) the 1D chain, 2D square lattice, and 3D simple-cubic lattice. The solid curves correspond to LDFT with the scaled dimer functional. In (a) the symbols are exact numerical results. In (b) the crosses refer to the exact Bethe-ansatz solution and the open circles to the BCS approximation (1D). In the insets, small values of $\left|U\right|/t$ are highlighted.

Figure 12

Pairing energy $\Delta E_p$ of the half-filled attractive Hubbard model on a Bethe lattice as a function of $1/(z-1)$, where $z$ is the local coordination number (see inset).