Screening and nonlocal correlations in the extended Hubbard model from self-consistent combined $GW$ and dynamical mean field theory

We describe a recent implementation of the combined $GW$ and dynamical mean field method ($GW$+DMFT) for the two-dimensional Hubbard model with onsite and nearest-neighbor repulsion. We clarify the relation of the $GW$+DMFT scheme to alternative approaches in the literature, and discuss the corresponding approximations to the free-energy functional of the model. Furthermore, we describe a numerically exact technique for the solution of the $GW$+DMFT equations, namely, the hybridization expansion continuous-time algorithm for impurity models with retarded interactions. We compute the low-temperature phase diagram of the half-filled extended Hubbard model, addressing the metal-insulator transition at small intersite interactions and the transition to a charge-ordered state for stronger intersite repulsions. $GW$+DMFT introduces a nontrivial momentum dependence into the many-body self-energy and polarization. We find that the charge fluctuations included in the present approach have a larger impact on the latter than on the former. Finally, within the $GW$+DMFT framework, as in extended DMFT, the intersite repulsion translates into a frequency dependence of the local effective interaction. We analyze this dependence and show how it affects the local spectral function.

Figure 1

Diagrammatic expansion of the electron-electron vertex. From left to right, top to bottom: Bare electron-electron interaction vertex ${\Gamma }_{\mathrm{ee}}^{\left(0\right)}$. Fully boldified ${\Sigma }_{\mathrm{ee}}$. Expansion of the full electron-electron vertex ${\Gamma }_{\mathrm{ee}}$.

Figure 2

Diagrammatic expansion of the electron-boson vertex. From left to right, top to bottom: Bare electron-boson interaction vertex ${\Gamma }_{\mathrm{eb}}^{\left(0\right)}$. Fully boldified ${\Sigma }_{\mathrm{eb}}$. Fully boldified $\Pi$. Expansion of the full electron-boson vertex ${\Gamma }_{\mathrm{eb}}$.

Figure 3

Illustration of a Monte Carlo configuration in the segment representation. The top figure corresponds to a configuration with one spin-up and one spin-down segment, each representing a time interval marked by a creation operator (empty circle) and an annihilation operator (full circle). The overlap of the segments, corresponding to the width of the hashed region, yields the weight due to the instantaneous interaction $U$. The retarded interaction $\mathcal{D}(\tau -{\tau }^{\prime })$ is represented by the blue curved lines. The bottom panel represents the weight of this configuration after integration of the retarded interaction over the segments. The dashed blue lines correspond to the interaction $K({\tilde{\tau }}_i-{\tilde{\tau }}_j)$ between creation and annihilation operators.

Figure 4

Computational scheme (HS-$UV$ decoupling).

Figure 5

Phase diagram of the $U$-$V$ Hubbard model for HS-$UV$ (red), and HS-$V$ (green) decoupling at $\beta =100$. The top panel shows the EDMFT result, and the bottom panel compares the $GW$+EDMFT result for the HS-$V$ scheme to EDMFT.

Figure 6

Imaginary-frequency data for the EDMFT calculations at $U=2.2$ and indicated values of $V$ (HS-$UV$ scheme). (a) $\mathrm{Im}{G}_{\text{loc}}\left(i{\omega }_n\right)$. (b) $\mathrm{Im}{\Sigma }_{\text{loc}}\left(i{\omega }_n\right)$. (c) $\mathrm{Re}{W}_{\text{loc}}\left(i{\nu }_n\right)$. (d) $\mathrm{Re}{\Pi }_{\text{loc}}\left(i{\nu }_n\right)$.

Figure 7

Partially screened $\mathcal{U}(\omega =0)$ as a function of $U$ (HS-$UV$ scheme).

Figure 8

Influence of $V$ on the effective local interaction for $U=2.2$ (HS-$UV$ scheme). Upper panel: $\mathrm{Re}{W}_{\mathrm{loc}}\left(\omega \right)$. Lower panel: $\mathrm{Im}{W}_{\mathrm{loc}}\left(\omega \right)$.

Figure 9

Static screened interactions in the HS-$UV$ scheme. (a) Fully screened $W_{\text{loc}}(\omega =0)$ as a function of $U$. (b) $W_{\text{loc}}(\omega =0)$ as a function of $V$.

Figure 10

Screening parameters. Upper panel: $\langle \omega \rangle$ as a function of $V$. Lower panel: $\lambda$ as a function of $V$ (HS-$UV$ scheme). Inset: ${\left(\frac{d\lambda }{dV}\right)}^2$ as a function of $U$.

Figure 11

Local spectral function $A_{\text{loc}}\left(\omega \right)$ (HS-$UV$ scheme).

Figure 12

$GW$ nonlocal contribution in the HS-$UV$ scheme. Top: $\mathrm{Im}{\Sigma }_{\text{nonlo}c}^{GW}(k,i{\omega }_{n=0})$. Middle: $\mathrm{Re}{\Sigma }_{\text{nonloc}}^{GW}(k,i{\omega }_{n=0})$. Bottom: $\mathrm{Re}{\Pi }_{\text{nonloc}}^{GW}(k,i{\nu }_{n=0})$.

Figure 13

Comparison of various polarizations for $U=1.5$, $V=0.4$ (HS-$UV$ scheme).

Figure 14

Convergence properties for $U=2$, $V=0.4$ (HS-$UV$ scheme). EDMFT corresponds to the converged EDMFT result, EDMFT+$GW$${}_i$ denotes the observables corresponding to the $i$th iteration with nonlocal self-energies, with EDMFT as starting input. Upper panel: $\mathrm{Im}{G}_{\text{loc}}\left(i{\omega }_n\right)$. Lower panel: $\mathrm{Re}{W}_{\text{loc}}\left(i{\nu }_n\right)$.

Figure 15

Influence of the self-consistency for $U=2$, $V=0.4$ (HS-$UV$ scheme). (a) $\mathrm{Im}{G}_{\text{loc}}\left(i{\omega }_n\right)$. (b) $\mathrm{Re}{W}_{\text{loc}}\left(i{\nu }_n\right)$. (c) $A_{\text{loc}}\left(\omega \right)$ (obtained from MaxEnt continuation).

Figure 16

$\text{Im}{\chi }_{\text{loc}}^{l\text{-DMFT}}$ in the ($\omega$,$U$) plane for $V=0$. The poles have been artificially broadened by an imaginary factor $\eta =0.01$. Black line: EDMFT result for $\langle \omega \rangle$ as a function of $U$ (for $V=0$).

Figure 17

Sketch of transitions in generic spectra for the Hubbard model at various interaction strengths.