Many-Body Approximation Scheme beyond GW

We explore the combination of the extended dynamical mean field theory (EDMFT) with the GW approximation (GWA); the former sums the local contributions to the self-energies to infinite order in closed form and the latter handles the nonlocal ones to lowest order. We investigate the different levels of self-consistency that can be implemented within this method by comparing to the exact quantum Monte Carlo solution of a finite-size model Hamiltonian. We find that using the EDMFT solution for the local self-energies as input to the GWA for the nonlocal self-energies gives the best result.

Figure 1

The potential $\Phi$ of the Baym-Kadanoff functional [9, 19] for $(\mathrm{EDMFT}+\mathrm{GW}{)}_{\mathrm{SC}}$. $(\mathrm{EDMFT}{)}_{\mathrm{SC}}$ is obtained by restricting the exchange diagram (the second on the right-hand side) to be local in space, $i=j$. The first line by itself represents the standard $(\mathrm{GW}{)}_{\mathrm{SC}}$ scheme. In this formulation, the boson Green’s function $D$ describes the screened interaction. The boson self-energy plays a similar role as the electron-hole bubble in GWA.

Figure 2

In our model system, the one-shot GWA, using the Hartree-Fock result as input, breaks down at $U\simeq 0.825$. This is an instability against the formation of a charge density wave at $(\pi ,\pi )$ [27]. The other results are plotted as references.

Figure 3

The energies, the quasiparticle residue ($Z$), and the nonlocal contribution to the self-energy ($Z_1$) as defined in the text. The same symbol scheme defined in (a) applies to all the diagrams. $Z$ is the same for $(\mathrm{EDMFT}{)}_{\mathrm{SC}}+(\mathrm{GW}{)}_{\mathrm{PSC}}$, $(\mathrm{EDMFT}{)}_{\mathrm{SC}}+\mathrm{GW}$, and $(\mathrm{EDMFT}{)}_{\mathrm{SC}}$, since the same local self-energy is used. For $Z_1$, which reflects the spatial extension of the self-energy, we are comparing numbers at least one order smaller than those for other quantities.

Figure 4

The local electron Green’s function and self-energy at $U=1.5$. The same symbol scheme defined in (a) applies to (b). The lines connecting the points are guides to the eye. The fermion Matsubara frequency $p_n=(\pi /\beta )(2n+1)$.