Extended dynamical mean-field study of the Hubbard model with long-range interactions

Using extended dynamical mean-field theory and its combination with the $GW$ approximation, we compute the phase diagrams and local spectral functions of the single-band extended Hubbard model on the square and simple cubic lattices, considering long-range interactions up to the third nearest neighbors. The longer-range interactions shift the boundaries between the metallic, charge-ordered insulating, and Mott insulating phases, and lead to characteristic changes in the screening modes and local spectral functions. Momentum-dependent self-energy contributions enhance the correlation effects and thus compete with the additional screening effect from longer-range Coulomb interactions. Our results suggest that the influence of longer-range intersite interactions is significant, and that these effects deserve attention in realistic studies of correlated materials.

Figure 1

Schematic picture of the one-band half-filled extended Hubbard model in the charge-ordered state for the square lattice (left) and simple cubic lattice (right). The full dots represent doubly occupied sites and the open dots empty sites. The red, green, and purple dots denote the NN, NNN, and 3NN sites of the black dot, respectively.

Figure 2

The paramagnetic $U\text{−}V$ phase diagrams for the single-band half-filled extended Hubbard model determined by EDMFT calculations. (a) Phase diagram for the 2D square lattice. (b) Phase diagram for the 3D simple cubic lattice. Here CO denotes a charge-ordered insulating phase, FL the metallic state, and MI the Mott insulator. The dashed lines are extrapolated FL-MI phase boundaries. The insets in (a) and (b) show the phase diagrams with axes rescaled by the bandwidth.

Figure 3

$\mathrm{Re}W(i\nu =0)$ (left $y$ axis) and $\mathrm{Re}\chi (i\nu =0)$ (right $y$ axis) as a function of $V$. $U=2.5$. (a) Results for the square lattice. (b) Results for the simple cubic lattice. The dashed lines are used to determine $V_c$ for the charge-ordering transition.

Figure 4

Quasiparticle weight $Z$ as a function of $U$. (a) Results for the square lattice, $V=0.80$. (b) Results for the simple cubic lattice, $V=0.60$. When $Z$ goes to zero, the Mott-Hubbard metal-insulator transition occurs. The corresponding $U$ is $U_c$. In (b), the dashed lines are used to guide the eyes.

Figure 5

Real part of the fully screened interactions $\mathrm{Re}W\left(i\nu \right)$ and partially screened interaction $\mathrm{Re}\mathcal{U}\left(i\nu \right)$, imaginary part of the real frequency fully screened interaction $\mathrm{Im}W\left(\nu \right)$ and partially screened interactions $\mathrm{Im}\mathcal{U}\left(\nu \right)$ for the extended Hubbard model solved by EDMFT. (a), (c), and (e) Results for the square lattice, $U=2.5$ and $V=0.8$. (b), (d), and (e) Results for the simple cubic lattice, $U=2.5$ and $V=0.6$. In this figure, SM means screening mode. In the insets of (a) and (b), the SM-resolved $\mathrm{Re}W\left(i\nu \right)$, together with the full $\mathrm{Re}W\left(i\nu \right)$ are shown for the NN case. In (c) and (d), $\mathrm{Im}W\left(\nu \right)$ for the NN case is approximated by Gaussian-type functions. The fitted results are shown in the insets. Each Gaussian peak denotes a SM. The insets in (e) and (f) show the $\mathrm{Im}\mathcal{U}\left(\nu \right)/{\nu }^2$ functions. Here $\mathrm{Im}W\left(\nu \right)$ and $\mathrm{Im}\mathcal{U}\left(\nu \right)$ are extracted using a modified maximum entropy method. See Appendix pp2 for more details.

Figure 6

Imaginary part of real frequency partially screened interactions $\mathrm{Im}\mathcal{U}\left(\nu \right)$ for the extended Hubbard model with the NN interactions solved by EDMFT. (a) Results for the square lattice. (b) Results for the simple cubic lattice. The $U$ and $V$ parameters are shown as color-filled circles in the insets. In (c) and (d), the corresponding effective screening frequencies ${\nu }_0$ are shown.

Figure 7

Comparison of the effective static interaction $\mathcal{U}\left(0\right)$ and the simple estimate $U-V$ [see Eq. (22)]. (a) Results for the 2D model with $U=2.5$. (b) Results for the 3D model with $U=2.5$.

Figure 8

Spectral functions at selected points for the single-band half-filled extended Hubbard model solved by EDMFT. (a), (c), and (e) Results for the square lattice. (b), (d), and (e) Results for the simple cubic lattice. The parameters are as follows: (a) metallic region, $U=2.5$ and $V=0.8$; (b) metallic region, $U=2.5$ and $V=0.6$; (c) Mott insulating region, $U=3.0$ and $V=1.5$; (d) Mott insulating region, $U=3.6$ and $V=1.5$; (e) “triangle” zone, $U=2.7$ and $V=1.0$; (f) “triangle” zone, $U=3.2$ and $V=0.8$. The impurity spectral functions are obtained using the analytical continuation method proposed in Ref. [31].

Figure 9

Spectral functions for the Hubbard model with NN interactions away from half-filling. (a)–(f) Results for the square lattice. (g)–(i) Results for the cubic lattice. In the left column, the impurity spectral functions $A\left(\omega \right)$ are shown. In the middle and right columns, we show the screened interaction $W\left(i\nu \right)$ and corresponding $\mathrm{Im}W\left(\nu \right)$. The parameters are as follows: (a)–(c) $U=2.4,V=0.2$, 2D lattice. (d)–(f) $U=3.6,V=1.0$, 2D lattice. (g)–(i) $U=2.5,V=0.2$, 3D lattice. (j)–(l) $U=3.6,V=1.0$, 3D lattice.

Figure 10

$V\text{−}\mu$ phase diagrams for the single-band extended Hubbard model with NN interactions, determined by EDMFT calculations. Here, $\delta \mu =\mu -U/2$. (a) and (c) show results for the 2D square lattice which at half-filling is in the FL or MI regime. (b) and (d) show similar results for the 3D simple cubic lattice. The black dashed lines in (a) and (b) show the location of $W\left(0\right)=0$, i.e., on the right side of this boundary, the static screened interaction is negative.

Figure 11

${\Sigma }_{\mathrm{nonloc}}(k,i{\omega }_0)$ for the extended Hubbard model from $GW$ + EDMFT. (a)–(f) Results for the square lattice, $U=2.5,V=0.80$. (g)–(l) Results for the simple cubic lattice, $U=2.5,V=0.60$. We only show the $k_z=0$ plane. (a)–(c) and (g)–(i) $\mathrm{Re}{\Sigma }_{\mathrm{nonloc}}(k,i{\omega }_0)$. (d)–(f) and (j)–(l) $\mathrm{Im}{\Sigma }_{\mathrm{nonloc}}(k,i{\omega }_0)$. The green curves in (a)–(c) and (g)–(i) denote the EDMFT Fermi surface.

Figure 12

Imaginary part of the local self-energy function $\mathrm{Im}\Sigma \left(i\omega \right)$ for the extended Hubbard model solved with EDMFT and $GW$ + EDMFT. (a) Results for the square lattice, $U=2.5$ and $V=0.8$. (b) Results for the simple cubic lattice, $U=2.5$ and $V=0.6$.

Figure 13

Real part of the fully screened interactions $\mathrm{Re}W\left(i\nu \right)$ and partially screened interaction $\mathrm{Re}\mathcal{U}\left(i\nu \right)$, and imaginary part of the real frequency fully screened interaction $\mathrm{Im}\mathcal{U}\left(\nu \right)$ and partially screened interactions $\mathrm{Im}\mathcal{U}\left(\nu \right)$ for the extended Hubbard model solved by $GW$ + EDMFT. (a), (c), and (e) Results for the square lattice, $U=2.5$ and $V=0.8$. (b), (d), and (e) Results for the simple cubic lattice, $U=2.5$ and $V=0.6$. In this figure, SM means screening mode. In the insets of (a) and (b), the SM-resolved $\mathrm{Re}W\left(i\nu \right)$, together with the full $\mathrm{Re}W\left(i\nu \right)$ are shown for the NN case. In (c) and (d), $\mathrm{Im}W\left(\nu \right)$ for the NN case is approximated by Gaussian-type functions. The fitted results are shown in the insets. Each Gaussian peak corresponds to a SM. The insets in (e) and (f) show the $\mathrm{Im}\mathcal{U}\left(\nu \right)/{\nu }^2$ functions. Here, $\mathrm{Im}W\left(\nu \right)$ and $\mathrm{Im}\mathcal{U}\left(\nu \right)$ are extracted using a modified maximum entropy method. See Appendix pp2 for more details.

Figure 14

Spectral functions at selected points for the single-band half-filled extended Hubbard model solved by $GW$ + EDMFT. (a), (c), and (e) Results for the square lattice. (b), (d), and (e) Results for the simple cubic lattice. The parameters are as follows: (a) metallic region, $U=2.5$ and $V=0.8$; (b) metallic region, $U=2.5$ and $V=0.6$; (c) Mott insulating region, $U=3.0$ and $V=1.5$; (d) Mott insulating region, $U=3.6$ and $V=1.5$; (e) “triangle” zone, $U=2.7$ and $V=1.0$; (f) “triangle” zone, $U=3.2$ and $V=0.8$. The impurity spectral functions are obtained using the analytical continuation method proposed in Ref. [31].

Figure 15

Benchmarks for the maximum entropy method for retarded interaction $\mathcal{U}\left(i\nu \right)$. The exact spectra for $\mathrm{Im}\mathcal{U}\left(\nu \right)/\nu$ are generated using classic Gaussian model. They are converted into $\mathcal{U}\left(\tau \right)$, and then processed by the proposed maximum entropy method. In the simulations, we assume $\beta =100$ and $U-U_{\mathrm{scr}}=2.0/\pi$.

Figure 16

Benchmarks for the maximum entropy method and Padé approximation against numerical noise. (a) Results obtained by maximum entropy method. (b) Results obtained by Padé approximation. In the calculations, we set $\beta =100$ and $\alpha =1.288$. The $\delta$ parameter is used to control the strength of data noise. Please see the text for the details.