Density-matrix functional study of the Hubbard model on one- and two-dimensional bipartite lattices

The inhomogeneous Hubbard model is investigated in one and two dimensions by using lattice density-functional theory (LDFT). The ground-state energy $E=T+W$ is regarded as a functional of the single-particle density matrix $\gamma$, where $T[\gamma ]$ represents the kinetic and crystal-field energy, and $W[\gamma ]$ the interaction energy. Besides the known functional $T[\gamma ]$ we propose a simple scaling approximation to the interaction energy $W[\gamma ]$, which is based on exact results for the Hubbard dimer and on a scaling hypothesis within the domain of representability of $\gamma$. As applications we consider the Hubbard model on one- and two-dimensional bipartite lattices. Several ground-state properties are determined including the kinetic and Coulomb energy, density distribution, nearest-neighbor bond order, and charge gap. Comparison with exact Lanczos diagonalizations shows that LDFT with the scaled dimer approximation to $W[\gamma ]$ yields a quite accurate description of the interplay between correlations and charge redistributions, from weak to strong coupling regimes, and for all band fillings. Goals and limitations of the present approach are discussed.

Figure 1

Comparison between the scaled dimer functional $W_{\mathrm{sc}}[\gamma ]$ as a function of the degree of electron delocalization $g_{12}=({\gamma }_{12}-{\gamma }_{12}^{\infty })/({\gamma }_{12}^0-{\gamma }_{12}^{\infty })$ [solid curves, Eq. (13)] and the exact functional $W_{\mathrm{ex}}$ derived from Lanczos diagonalizations (symbols). Results are shown for a 1D bipartite ring having $N_{\mathrm{a}}=10$ sites, different band fillings $n=N_{\mathrm{e}}/N_{\mathrm{a}}$, and representative charge transfers $\Delta n={\gamma }_{22}-{\gamma }_{11}$. The inset figures display the corresponding relative errors $\Delta W=(W_{\mathrm{sc}}-W_{\mathrm{ex}})/(W^0-W^{\infty })$.

Figure 2

Ground-state properties of bipartite Hubbard rings having $N_{\mathrm{a}}=14$ sites and half-band filling $n=1$ as a function of the Coulomb repulsion strength $U/t$. Different values of the energy-level shift $\varepsilon$ between the sublattices are considered as indicated in panel (a). Results are given for (a) ground-state energy $E_{\mathrm{gs}}$, (b) NN bond order ${\gamma }_{12}$, (c) charge transfer $\Delta n={\gamma }_{22}-{\gamma }_{11}$, and (d) average number of double occupations per site $W/{\mathit{UN}}_{\mathrm{a}}$. The solid curves refer to LDFT using the scaling approximation $W_{\mathrm{sc}}$ [see Eq. (13)] while the symbols are the results of exact Lanczos diagonalizations.

Figure 3

Band-filling dependence of the ground-state energy of 1D Hubbard rings having $N_{\mathrm{a}}=14$ sites and different bipartite potentials $\varepsilon$. The solid curves refer to LDFT with the scaled dimer functional $W_{\mathrm{sc}}$, and the symbols refer to exact numerical results. Representative values of the Coulomb repulsion strength $U/t$ are considered as indicated in panel (a).

Figure 4

Charge gap $\Delta E_{\mathrm{c}}=E(N_{\mathrm{e}}+1)+E(N_{\mathrm{e}}-1)-2E(N_{\mathrm{e}})$ as a function of band filling $n$ in 1D Hubbard rings having $N_{\mathrm{a}}=14$ sites and different bipartite potentials $\varepsilon$. The solid lines connecting discrete points refer to LDFT with the scaled dimer functional and the symbols to exact Lanczos diagonalization. Representative values of the Coulomb repulsion strength $U/t$ are considered as indicated in panel (c). Results for $n=1$ are given in Fig. 5.

Figure 5

Charge gap $\Delta E_{\mathrm{c}}=E(N_{\mathrm{e}}+1)+E(N_{\mathrm{e}}-1)-2E(N_{\mathrm{e}})$ as a function of the Coulomb repulsion strength $U/t$ in 1D Hubbard rings having a band filling $n=1$, representative values of the bipartite potential $\varepsilon$, and (a) $N_{\mathrm{a}}=12$, (b) $N_{\mathrm{a}}=14$, and (c) $N_{\mathrm{a}}=\infty$ sites. The curves refer to LDFT with the scaled dimer functional $W_{\mathrm{sc}}$ and the symbols to exact diagonalizations. In the inset figure the ring-length dependence of $(\Delta E_{\mathrm{c}}-U)/t$ is shown for $\varepsilon /t=4$ and $U/t=4$.

Figure 6

Ground-state properties of the half-filled Hubbard model on a 2D square cluster having $N_{\mathrm{a}}=4\times 4$ sites and periodic boundary conditions ($n=N_{\mathrm{e}}/N_{\mathrm{a}}=1$). Results are given for (a) ground-state energy $E_{\mathrm{gs}}$, (b) NN bond order ${\gamma }_{12}$, (c) charge transfer $\Delta n={\gamma }_{22}-{\gamma }_{11}$, and (d) interaction energy $W$. LDFT (solid curves) and exact diagonalizations (symbols) are compared as a function of the Coulomb repulsion strength $U/t$ for representative values of the energy level shift $\varepsilon$ between the sublattices, as indicated in panel (a). The discontinuities in the exact results for ${\gamma }_{12},\Delta n$, and $W$ are a consequence of the degeneracy of the single-particle spectrum and the finite cluster size.