Exploiting the links between ground-state correlations and independent-fermion entropy in the Hubbard model

The ground-state properties of the half-filled Hubbard model are investigated in the framework of lattice density functional theory. The single-particle density matrix ${\gamma }_{ij\sigma }$ is regarded as the central variable of the many-body problem, where $i$ and $j$ refer to the lattice sites and $\sigma$ to the spin. The interaction-energy functional $W\left[{\gamma }_{ij\sigma }\right]$ is calculated exactly for representative finite periodic systems by performing exact Lanczos diagonalizations. The relationship between $W\left[{\gamma }_{ij\sigma }\right]$ and the entropy $S\left[{\eta }_{\mathit{k}\sigma }\right]$ of independent fermions with natural-orbital occupations ${\eta }_{\mathit{k}\sigma }$ is analyzed. A simple approximation to the interaction energy of the half-filled Hubbard model is proposed, which takes the form $W=W\left(S\left[{\eta }_{\mathit{k}\sigma }\right]\right)$. Using this functional we derive the ground-state energy, kinetic energy, average number of double occupations, charge distribution, magnetic susceptibility, and field-induced spin polarization in one-, two-, and three-dimensional periodic lattices. The limit of infinite dimensions is also explored. The accuracy of the method is assessed by comparison with available exact numerical or analytical results. Goals, limitations, and possible extensions of the domain of applicability of the functional are discussed.

Figure 1

Relation between the exact interaction energy $W\left[{\eta }_{\mathit{k}\sigma }\right]$ of the periodic half-filled Hubbard model and the independent Fermion entropy $S\left[{\eta }_{\mathit{k}\sigma }\right]$ corresponding to different natural-orbital occupation distributions ${\eta }_{\mathit{k}\sigma }$. The results are obtained from Lanczos diagonalizations on finite 1D rings having $N_a=6$ (circles), $N_a=10$ (upright triangles) and $N_a=14$ (squares), as well as for finite 2D square-lattice rectangles having $N_a=2\times 4$ (downright triangles) and $N_a=3\times 4$ (diamonds) with periodic boundary conditions. The solid symbols correspond to first NN hoppings $t_{01}$, while the open symbols also include second NN hoppings $t_{02}=t_{01}/2$.

Figure 2

Ground-state properties of the 1D Hubbard model on a ring having $N_a=14$ sites and $N_{\uparrow }=N_{\downarrow }=7$ electrons as a function of the Coulomb-repulsion strength $U/t$. The linear independent-fermion entropy (IFE) ansatz (blue full curves) is compared with exact numerical Lanczos diagonalizations (red dashed curves): (a) ground-state energy $E_0$, (b) average number of double occupations $D$ and kinetic energy $T$, (c) natural-orbital occupation numbers ${\eta }_{k\uparrow }={\eta }_{k\downarrow }$, and (d) density-matrix elements ${\gamma }_{0\delta \uparrow }={\gamma }_{0\delta \downarrow }$ between an atom $i=0$ and its $\delta \mathrm{th}$-nearest neighbor.

Figure 3

Ground-state properties of the 2D Hubbard model on a $4\times 4$ square lattice with $N_{\uparrow }=N_{\downarrow }=8$ electrons and periodic boundary conditions. Results are given for the linear independent-fermion entropy (IFE) ansatz (blue full curves) and exact Lanczos diagonalizations (crosses) as a function of the Coulomb-repulsion strength $U/t$: (a) ground-state energy $E_0$, (b) average number of double occupations $D$ and kinetic energy $T$, (c) natural-orbital occupation numbers ${\eta }_{\mathit{k}\uparrow }={\eta }_{\mathit{k}\downarrow }$, and (d) density-matrix elements ${\gamma }_{0\delta \uparrow }={\gamma }_{0\delta \downarrow }$ between an atom $i=0$ and its $\delta \mathrm{th}$-nearest neighbor, as illustrated in the inset.

Figure 4

Ground-state properties of the half-filled 2D Hubbard model on a $4\times 4$ triangular lattice with periodic boundary conditions as functions of the Coulomb-repulsion strength $U/t$. Results are given for the linear independent-fermion entropy (IFE) ansatz (blue full curves) and exact Lanczos diagonalization (crosses): (a) ground-state energy $E_0$, (b) average number of double occupations $D$ and kinetic energy $T$, (c) natural-orbital occupation numbers ${\eta }_{\mathit{k}\uparrow }={\eta }_{\mathit{k}\downarrow }$, and (d) density-matrix elements ${\gamma }_{0\delta \uparrow }={\gamma }_{0\delta \downarrow }$ between an atom 0 and its $\delta \mathrm{th}$-nearest neighbor (see inset). The open circles in (c) and (d) show the average of the exact results over the $\mathit{k}$ vectors having the same single-particle energy ${\varepsilon }_{\mathit{k}}$.

Figure 5

Ground-state properties of the half-filled Hubbard model on hypercubic lattices having $d=1$–3 dimensions and $d\to \infty$ as functions of the Coulomb-repulsion strength $U/t$. Results are given for the (a) ground-state energy $E_0$, (b) average number of double occupations $D$, and (c) kinetic energy $T$. The curves were obtained in the linear independent-fermion entropy (IFE) approximation to LDFT. The symbols correspond either to the exact Bethe-ansatz solution [43] (crosses, 1D) or to numerical quantum Monte Carlo simulations [76, 77, 78] (open circles and triangles, 2D). For each dimension $d$ the NN hopping integral $t_d$ is scaled as $t_d=t/\sqrt{d}$ in order that the second moment of the local density of states $w_2=2d{t}_d^2$ is independent of $d$. The inset in (c) shows the relative difference $\mathrm{\Delta }X=|X^{\mathrm{ex}}-X^{\mathrm{IFE}}|/|X^{\mathrm{ex}}|$ where $X$ stands for the total energy $E_0$, the kinetic energy $T$, and the average double occupations $D$ in the 1D case.

Figure 6

Single-particle density-matrix elements ${\gamma }_{0\delta \uparrow }={\gamma }_{0\delta \downarrow }$ between an atom $i=0$ and its $\delta \mathrm{th}$-nearest neighbor in the ground state of the half-filled 1D Hubbard model, as obtained by using the linear IFE approximation. In (a), ${\gamma }_{0\delta \sigma }$ is given for odd $\delta$ as a function of the Coulomb-repulsion strength $U/t$. Electron-hole symmetry implies ${\gamma }_{0\delta \sigma }=0$ for even $\delta$. In (b), ${\gamma }_{0\delta \sigma }$ is shown as a function of $\delta$ for representative values of $U/t$.

Figure 7

Ground-state properties of the half-filled Hubbard model on the infinite 2D triangular lattice: (a) total energy $E_0$ and (b) average number of double occupations $D$ and kinetic energy $T$, as functions of the Coulomb-repulsion strength $U/t$. The full curves are obtained by using the linear independent-fermion entropy (IFE) approximation to LDFT, while the crosses correspond to the exact diagonalization of a finite $4\times 4$ plaquette with periodic boundary conditions.

Figure 8

Ground-state properties of the one-dimensional half-filled Hubbard model as functions of the spin polarization per atom $n_{\uparrow }-n_{\downarrow }=2S_z/N_a$ for representative Coulomb-repulsion strengths $U/t$: (a) total energy $E$, (b) average number of double occupations $D$, and (c) kinetic energy $T$. The curves were obtained using the IFE approximation to LDFT, while the crosses in (a) for $U/t=3$ and 10 are results taken from Ref. [83].

Figure 9

Ground-state magnetization $M=2S_z$ of the one-dimensional half-filled Hubbard model as a function of the applied magnetic field $B$, for different values of the Coulomb repulsion $U$, as obtained by using the IFE approximation. In the inset, the zero-field susceptibility is shown as a function of $U/t$, where ${\chi }_P$ stands for the Pauli susceptibility. The solid curve shows the present IFE approximation, while the dashed curve shows exact results from Ref. [83].