From local to nonlocal correlations: The Dual Boson perspective

Extended dynamical mean-field theory (EDMFT) is insufficient to describe nonlocal effects in strongly correlated systems, since corrections to the mean-field solution are generally large. We present an efficient scheme for the construction of diagrammatic extensions of EDMFT that avoids the usual double-counting problem by using an exact change of variables (the Dual Boson formalism) to distinguish the correlations included in the dynamical mean-field solution and those beyond. With a computational efficiency comparable to the $\mathrm{EDMFT}+\mathrm{GW}$ approach, our scheme significantly improves on the charge order transition phase boundary in the extended Hubbard model.

Figure 1

Hedin form of the self-energy diagram in case of (a) at least one nonlocal Green's function $\tilde{G}$ and nonlocal renormalized interaction $\tilde{W}$, and (b) only local renormalized interactions $\mathcal{W}$. Straight and wave lines correspond to the Green's function and renormalized interaction.

Figure 2

Structure of the vertex corrections in theories consisted of (a) one and (b) two types of propagators. Solid straight and wave lines correspond to the Green's function and renormalized interaction of one type and the dashed lines to those of the second type, respectively.

Figure 3

Structure of the vertex corrections in different theories in case of one (top line) and two (bottom line) types of propagators. Solid straight and wave lines correspond to the Green's function and renormalized interaction of one type and the dashed lines to those of the second type, respectively.

Figure 4

Diagrammatic representation of the second- and the third-order contribution to the renormalized interaction.

Figure 5

Self-energy and polarization operator for EDMFT++ approaches. The square brackets ${[...]}_{\mathrm{nloc}}$ denote exclusion of the local part. $\mathrm{DMFT}+\mathrm{GW}$ is not listed here, it has the same diagrams as $\mathrm{EDMFT}+\mathrm{GW}$ and only differs in their choice of ${\mathcal{U}}_{\omega }$.

Figure 6

The recipe to construct an EDMFT++ theory. $\mathrm{DMFT}+\mathrm{GW}$ is obtained by taking ${\mathcal{U}}_{\omega }=U$ instead of determining it self-consistently.

Figure 7

$U-V$ phase diagram in EDMFT, DB and EDMFT++ theories at inverse temperature $\beta =50$. The dashed line shows $V=U/4$; the dot at $U=0$ shows the starting point of RPA data. CO and FL denote charge-ordered and Fermi-liquid metallic phases, respectively. The EDMFT and DB data are taken from [30]; $\mathrm{EDMFT}+\mathrm{GW}$ data practically coincides with results shown in [15, 16] papers.

Figure 8

Frequency dependence of nonlocal Re ${\overline{\mathrm{\Pi }}}_{\mathbf{q}\omega }$ for momentum $\mathbf{k}=\left(0,0\right),\mathbf{k}=\left(\pi ,\pi \right)$ for on-site interaction $U=2.3$ and the nearest-neighbor interaction $V=0.2$.

Figure 9

Local three-point vertex function ${\gamma }_{\nu \omega }$ for two bosonic frequencies ${\omega }_m=2m\pi /\beta$ with $m=0$ and $m=6$ for different values of nearest-neighbor interaction $V$ and local interaction $U=1.5$ (top line) and $U=2.3$ (bottom line).