Ab initio dynamical vertex approximation

Diagrammatic extensions of dynamical mean-field theory (DMFT) such as the dynamical vertex approximation $\left(\text{D}\mathrm{\Gamma }\text{A}\right)$ allow us to include nonlocal correlations beyond DMFT on all length scales and proved their worth for model calculations. Here, we develop and implement an Ab initio $\text{D}\mathrm{\Gamma }\text{A}$ approach (Abinitio$\text{D}\mathrm{\Gamma }\text{A}$) for electronic structure calculations of materials. The starting point is the two-particle irreducible vertex in the two particle-hole channels which is approximated by the bare nonlocal Coulomb interaction and all local vertex corrections. From this, we calculate the full nonlocal vertex and the nonlocal self-energy through the Bethe-Salpeter equation. The Abinitio$\text{D}\mathrm{\Gamma }\text{A}$ approach naturally generates all local DMFT correlations and all nonlocal $GW$ contributions, but also further nonlocal correlations beyond: mixed terms of the former two and nonlocal spin fluctuations. We apply this new methodology to the prototypical correlated metal ${\text{SrVO}}_3$.

Figure 1

(Top) In addition to the Hartree term, $GW$ takes into account the (screened) exchange Feynman diagram (wiggled line: Coulomb interaction $V$; double wiggled line: screened interaction $W$; line: interacting Green's function $G$; ${\mathbf{R}}_i, {\mathbf{R}}_j$ indicate the lattice sites). (Middle) Calculation of $W$ as a screened Coulomb interaction within the RPA. (Bottom) Feynman diagrams for the DMFT self-energy $\mathrm{\Sigma }$.

Figure 2

Outline of the main equations of the Abinitio$\text{D}\mathrm{\Gamma }\text{A}$ approach. (a) The local Bethe-Salpeter equation allows extracting the local irreducible vertex ${\mathrm{\Gamma }}_{\mathrm{loc}}$ from the local generalized susceptibility ${\chi }_{\mathrm{loc}}$. (b) The local irreducible vertex (which contains the local interaction $U$) is supplemented by the nonlocal interaction $V$ resulting in the momentum-dependent irreducible vertex $\mathrm{\Gamma }$. (c) From this $\mathrm{\Gamma }$ the full vertex $F$ is obtained using the Bethe-Salpeter equation (shown is only the particle-hole channel, but the related contribution from the transversal particle-hole channel is also included). (d) Finally, the self-energy is constructed via the Schwinger-Dyson equation consisting of the vertex part (top), and the Hartree-Fock contribution (bottom). From this self-energy one can, in principle, determine an updated local vertex, closing the loop. For details and the index convention, see Secs. 2c and 2d.

Figure 3

Diagrammatic representation of (a) the crossing symmetry in Eq. (22) and (b) the swapping symmetry in Eq. (24).

Figure 4

Band structure and Fermi surface of ${\text{SrVO}}_3$ within GGA. Shown are the dispersion of the vanadium $t_{2g}$ states (left) and the Fermi surface in the $(k_x,k_y)$ plane for $k_z=0$ (right).

Figure 5

Real (left) and imaginary (right) part of the generalized susceptibility ${\chi }_{m,1111}^{\omega \nu {\nu }^{\prime }}$ in the magnetic $\left(m\right)$ channel for the 1111 orbital component at $\omega =0$. $\chi$ is related to the irreducible local vertex via Eq. (30). By summing ${\chi }_m^{\omega \nu {\nu }^{\prime }}$ over its two fermionic frequencies $\nu$ and ${\nu }^{\prime }$ one can obtain the physical local magnetic susceptibility ${\chi }_m^{\omega }$, as, e.g., in Ref. [90].

Figure 6

Abinitio$\text{D}\mathrm{\Gamma }\text{A} \mathbf{k}\text{-dependent}$ self-energies and spectral functions for ${\text{SrVO}}_3$. Shown are the imaginary (top) and real (middle) part of the self-energy and the corresponding spectral function (bottom) for the $k$ points $\mathrm{\Gamma }=(0,0,0)$ (first column), $X=(0,\pi ,0)$ (second column), and $M=(\pi ,\pi ,0)$ (third column).

Figure 7

(a) Real and (b) imaginary parts of the Abinitio$\text{D}\mathrm{\Gamma }\text{A}$ self-energy $\mathrm{\Sigma }(\mathbf{k},i{\nu }_0)$ at the first Matsubara frequency ${\nu }_0$ (c) scattering rate ${\gamma }_{\mathbf{k}}$ and (d) quasiparticle weight $Z_{\mathbf{k}}$ in the $k_z=0$ plane for the $d_{xy}$ orbital [99].