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Physical review. B./Physical review. B

Vertex function compliant with the Ward identity for quasiparticle self-consistent calculations beyond GW

Alexey A. Tal, Wei Chen, Alfredo Pasquarello

Published Apr 8, 2021• 33 citations
Publisher

Abstract

We extend the quasiparticle self-consistent approach beyond the $GW$ approximation by using a range-separated vertex function. The developed approach yields band gaps, dielectric constants, and band positions with an accuracy similar to highest-level electronic-structure calculations without exceeding the cost of regular quasiparticle self-consistent $GW$. We introduce an exchange-correlation kernel that accounts for the vertex over the full spatial range. In the long range it complies with the Ward identity, while it is approximated through the adiabatic local density functional in the short range. In this approach, the renormalization factor is balanced and the higher-order diagrams are effectively taken into account.

DOI
10.1103/physrevb.103.l161104

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We extend the quasiparticle self-consistent approach beyond the GWGWGW approximation by using a range-separated vertex function. The developed approach yields band gaps, dielectric constants, and band positions with an accuracy similar to highest-level electronic-structure calculations without exceeding the cost of regular quasiparticle self-consistent GWGWGW. We introduce an exchange-correlation kernel that accounts for the vertex over the full spatial range. In the long range it complies with the Ward identity, while it is approximated through the adiabatic local density functional in the short range. In this approach, the renormalization factor is balanced and the higher-order diagrams are effectively taken into account.
We present self-consistent GW calculations for a quasi-one-dimensional model semiconductor, using multipole representations for W, the screened interaction, and G, the electron Green function. In the case of G, we distinguish between the quasiparticle contribution and the so-called incoherent contribution. We consider various strengths of the external potential. For two different starting points of the self-consistency cycle, local-density approximation and Hartree-Fock, the band gaps converge to the same values. The self-consistent GW band gaps differ very little from the Hartree-Fock band gaps. There is considerable disagreement with quantum Monte Carlo calculations for the same model, indicating the importance of vertex corrections.
On the basis of the self-consistent calculation scheme for the electron self-energy Σ with the use of the three-point vertex function Γ always satisfying the Ward identity, we find that the obtained quasi-particle dispersion in the normal state in gapped systems, such as semiconductors, insulators, and molecules, is well reproduced by that in the one-shot GW (or G0W0) approximation. In calculating the superconducting transition temperature Tc, we also find a similar situation; the result for Tc in the gauge-invariant self-consistent framework including the effect of Γ satisfying the Ward identity is different from that in the conventional Eliashberg theory (which amounts to the GW approximation for superconductivity) but is close to that in the G0W0 approximation. Those facts indicate that the G0W0 approximation actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner and thus we can understand that the G0W0 approximation is better than the fully self-consistent GW one in obtaining some of physical quantities.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
We study the effects of local vertex corrections to the self energy of the electron gas. We find that a vertex derived from time-dependent density-functional theory can give accurate self energies without including the explicit time dependence of the exchange-correlation potential provided, however, that a proper decay at large momentum transfer (large q) is built into the vertex function. (The local-density approximation for the vertex fails badly.) Total energies are calculated from the Galitskii-Migdal formula and it is shown that a proper large-q behavior, results in a close consistency between the chemical potentials derived from these energies and those obtained directly from the self energy. We show that this internal consistency depends critically on including the same vertex correction in both the self-energy and the screening function. In addition the total energies become almost as accurate as those from elaborate quantum Monte-Carlo (QMC) calculations. We also study the accuracy and utility of the functional for the total energy proposed by Luttinger and Ward and a generalization by Almbladh, von Barth, and van Leeuwen. For the electron gas, even the simplest and readily evaluated approximations to these functionals yield total energies of similar quality as those of QMC calculations. The functionals depend on the one-electron Green's function and the screened Coulomb interaction and already rather crude approximations to these quantities produce accurate energies thus demonstrating the insensitivity of the functionals to their arguments. Different ways of incorporating vertex corrections beyond the GWGWGW level are studied in simple, exactly soluble polaron-like models. We study models of a structureless core electron coupled to valence electrons and a local polaron model by Cini, Hewson and Newns. Our model results indicate that the first vertex correction alone will in general not suffice to improve the spectrum away from the quasi-particle peak. By including a subsequence of Mahan's fractal vertex series, however, we obtain results with correct physical properties which agree better with exact model results.
Accidental accuracy and vertex corrections in GWGWGW: Exact benchmarks for the extended Hubbard model
The GWGWGW approximation is the standard tool for quasiparticle predictions in materials, yet its regime of validity in correlated systems remains poorly quantified, because \textit{ab initio} vertex corrections are computationally prohibitive. Using exact diagonalization of the half-filled extended Hubbard model on finite rings as a numerically exact reference, we construct the corresponding model-space GWGWGW theory on the identical Hilbert space and quantify its error as a function of local (UUU) and non-local (VVV) interaction strength. We find that the required vertex correction changes character across the phase diagram: in the weak-coupling regime the effective vertex Γeff<1Γ_{\rm eff} < 1Γeff​<1, reflecting the suppression of RPA charge fluctuations by exact short-range correlations, whereas in the Mott regime ΓeffΓ_{\rm eff}Γeff​ grows monotonically (to ∼3\sim 3∼3 at U=8tU=8tU=8t for N=6N=6N=6), reflecting the local dynamical self-energy structure required to open the Hubbard gap. Vertex corrections in the electron-hole (polarizability) channel are shown to \emph{worsen} the gap error, indicating that reproducing the Mott gap requires dynamical self-energy structure rather than improved screening. For V=0V=0V=0, static COHSEX is accidentally exact at a single crossover U∗≈3.5 tU^* \approx 3.5\,tU∗≈3.5t; finite VVV, through non-local Fock exchange, splits this point into a double-crossover window that collapses toward weak coupling. Even at the crossover, however, the exact spectral function retains Hubbard-band structure that no static functional reproduces, so gap agreement does not imply functional accuracy. These results yield quantitative diagnostics for the reliability of GWGWGW in correlated materials.
Theoretical studies of semiconductors and band insulators are usually based on variants of the GWGWGW method without full self-consistency, like single-shot G0W0G^0W^0G0W0 or quasiparticle self-consistent GWGWGW. Fully self-consistent GWGWGW provides a poor description of the gap size and electronic structure due to the lack of vertex corrections. While it is hard to predict at which order corrections can be neglected, local vertex corrections to all orders can be consistently included by combining GWGWGW with extended dynamical mean field theory (EDMFT). Here, we show that \textit{ab initio} multitier GWGWGW+EDMFT calculations, which achieve full self-consistency in a suitably defined low-energy space, provide an accurate description of semiconductors and band insulators, comparable to the established methods which are typically used for this class of materials. Our results demonstrate that the range of applicability of GWGWGW+EDMFT extends to weakly correlated systems, and they imply that despite the weak correlations, local vertex corrections are an important ingredient in the diagrammatic treatment of semiconductors and band insulators.