Influence of Fock exchange in combined many-body perturbation and dynamical mean field theory

In electronic systems with long-range Coulomb interaction, the nonlocal Fock-exchange term has a band-widening effect. While this effect is included in combined many-body perturbation theory and dynamical mean field theory (DMFT) schemes, it is not taken into account in standard extended DMFT (EDMFT) calculations. Here, we include this instantaneous term in both approaches and investigate its effect on the phase diagram and dynamically screened interaction. We show that the largest deviations between previously presented EDMFT and $GW$+EDMFT results originate from the nonlocal Fock term, and that the quantitative differences are especially large in the strong-coupling limit. Furthermore, we show that the charge-ordering phase diagram obtained in $GW$+EDMFT methods for moderate interaction values is very similar to the one predicted by dual-boson methods that include the fermion-boson or four-point vertex.

Figure 1

Bare dispersion $\varepsilon \left(\mathbf{k}\right)$ (solid black lines), nonlocal Fock self-energy ${\mathrm{\Sigma }}_{\mathrm{F}}^{\mathrm{nonloc}}\left(\mathbf{k}\right)$ (solid blue line), and renormalized dispersion $\tilde{\varepsilon }\left(\mathbf{k}\right)$ (dashed green line) along a high-symmetry path in the Brillouin zone in the $GW$+EDMFT method. Inset: Color plot of ${\mathrm{\Sigma }}_{\mathrm{F}}^{\mathrm{nonloc}}\left(\mathbf{k}\right)$ in the first Brillouin zone $(U=2.0$ and $V=0.4,\beta =100$, half filling).

Figure 2

Hopping renormalization $\delta t$ according to Eq. (13) for half filling (top) and $n=0.8$ (bottom). The solid line with triangles indicates the phase transition to the charge-ordered phase and the line with stars the transition to the Mott-insulating phase.

Figure 3

Dependence of the nearest-neighbor hopping parameter renormalization $\delta t$ on the next-nearest-neighbor interaction $V^{\prime }$ for fixed $U=2.0,t^{\prime }=1/\sqrt{2}$, and $\langle n\rangle =0.8$. Inset: Color plot of ${\mathrm{\Sigma }}_{\mathrm{F}}\left(\mathbf{k}\right)$ in a calculation with $32\times 32\mathbf{k}$-points for the $V=0.4,V^{\prime }=0.2$ case.

Figure 4

$\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},i\omega )$ (left) and $\mathrm{Re}\mathrm{\Pi }(\mathbf{q},i\nu )$ (right) (black lines for EDMFT, magenta lines for $G{W}_c$+EDMFT, red lines for $GW$+EDMFT, and blue lines for $Gv$+EDMFT). Circles, squares, and triangles show results for the $(0,0),(0,\pi )$, and $(\pi ,\pi )$ points, respectively. Top, $U=2.0$ and $V=0.4$; bottom, $U=3.0$ and $V=1.0$ $\left(\beta =100,n=1\right)$.

Figure 5

$\mathrm{Im}{G}_{\mathrm{loc}}\left(i\omega \right)$ (left) and $\mathrm{Re}{W}_{\mathrm{loc}}\left(i\nu \right)$ (right) (black lines for EDMFT, magenta lines for $G{W}_c$+EDMFT, red lines for $GW$+EDMFT, and blue lines for $Gv$+EDMFT). The top panels are for $U=2.0$ and $V=0.4$, and the bottom panels for $U=3.0$ and $V=1.0$ $\left(\beta =100,n=1\right)$.

Figure 6

Local spectral functions obtained by MaxEnt [29, 30] analytical continuation (black lines for EDMFT, red lines for $G{W}_c$+EDMFT, magenta lines for $GW$+EDMFT, and blue lines for $Gv$+EDMFT). The top panels are for $U=2.0$ and $V=0.4$, and the bottom panels for $U=3.0$ and $V=1.0$ $\left(\beta =100,n=1\right)$.

Figure 7

${\chi }^{-1}(\pi \pi ,\omega =0)$ as a function of $V$ for (from top to bottom) $U=0,U=2,U=3$, and $U=4$ (black lines for EDMFT, magenta lines for $G{W}_c$+EDMFT, red lines for $GW$+EDMFT, and blue lines for $Gv$+EDMFT). The dashed lines are an estimate of the critical $V_c$ $\left(\beta =100,n=1\right)$.

Figure 8

EDMFT, $GW$+EDMFT, and $Gv$+EDMFT phase diagrams. The black lines are EDMFT results reproduced from Fig. 5 of Ref. [15]. The $G{W}_c$+EDMFT phase boundaries, magenta triangles, are very close to the EDMFT phase boundaries. Red triangles show the $GW$+EDMFT result, blue squares the $Gv$+EDMFT result. The dashed grey line corresponds to the interaction energy estimate of $V_c,V_c^{\mathrm{int}}=U/4$ (see text) $\left(\beta =100,n=1\right)$.

Figure 9

Comparison of the $U\text{−}V$ phase diagrams with results from Ref. [52] using different variants of the dual-boson scheme (orange lines). The dashed grey line is the mean field estimate $V=U/4$. Bottom panel: Zoom on the low-$U$ region, and additional comparison to the dynamical cluster approximation (DCA) results of Ref. [57] (cyan pentagons; $N_c=8$, obtained via a linear extrapolation [57]). For all results, $\beta =50,n=1$.

Figure 10

Inverse static charge susceptibility at $Q=(\pi ,\pi )$ (left column) and dependence of the critical $V$ on the decoupling (right column) for $U=1$ (top row) and $U=3$ (bottom row) within single-site TRILEX.