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arXiv:1104.2188 · Annalen der Physik

Ab initio calculations with the dynamical vertex approximation

A. Toschi, G. Rohringer, A. A. Katanin, Karsten Held

Published Aug 26, 2011• 37 citations
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Abstract

Abstract We propose an approach for the ab initio calculation of materials with strong electronic correlations which is based on all local (fully irreducible) vertex corrections beyond the bare Coulomb interaction. It includes the so‐called GW and dynamical mean field theory and important non‐local correlations beyond, with a computational effort estimated to be still manageable.

DOI
10.1002/andp.201100036

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•Oct 8, 2020•0 citations
Abstract We propose an approach for the ab initio calculation of materials with strong electronic correlations which is based on all local (fully irreducible) vertex corrections beyond the bare Coulomb interaction. It includes the so‐called GW and dynamical mean field theory and important non‐local correlations beyond, with a computational effort estimated to be still manageable.
While key effects of the many-body problem---such as Kondo and Mott physics---can be understood in terms of on-site correlations, non-local fluctuations of charge, spin, and pairing amplitudes are at the heart of the most fascinating and unresolved phenomena in condensed matter physics. Here, we review recent progress in diagrammatic extensions to dynamical mean-field theory for ab initio materials calculations. We first recapitulate the quantum field theoretical background behind the two-particle vertex. Next we discuss latest algorithmic advances in quantum Monte Carlo simulations for calculating such two-particle quantities using worm sampling and vertex asymptotics, before giving an introduction to the ab initio dynamical vertex approximation (AbinitioDΓΓΓA). Finally, we highlight the potential of AbinitioDΓΓΓA by detailing results for the prototypical correlated metal SrVO3_33​.
Abstract The question of spatial locality of electronic correlations beyond GW approximation is one of the central issues of the famous combination of GW and dynamical mean field theory, GW + DMFT. In this work, the above question is addressed directly (for the first time) by performing calculations with and without assumption of locality of the corresponding diagrams. For this purpose we use sc(GW + G3W2) approach where the higher order part (G3W2) is evaluated with fully momentum dependent Green’s function G and screened interaction W and with ‘local’ variant, where the single site approximation is assumed for both G and W. For all three materials studied in this work (NiO, α -Ce, LiFeAs), we have found the spatial non-locality effects to be strong. For NiO and LiFeAs they, in fact, are decisive for the proper evaluation of vertex corrections. The results of this study have direct impact on our understanding of approximations made in practical implementations of GW + DMFT method, where all diagrams beyond GW (DMFT part) are assumed to be local. Taking into account the fact that the first diagrams beyond GW represent the most important contribution also in GW + DMFT calculations, we conclude that the basic assumption of GW + DMFT, namely the locality of diagrams evaluated in the DMFT part, is not as good as it is believed to be.
We develop a diagrammatic approach with local and nonlocal self-energy diagrams, constructed from the local irreducible vertex. This approach includes the local correlations of dynamical mean-field theory and long-range correlations beyond. It allows us, for example, to describe (para)magnons and weak localization effects---in strongly correlated systems. As a first application, we study the interplay between nonlocal antiferromagnetic correlations and the strong local correlations emerging in the vicinity of a Mott-Hubbard transition.
Historically, the GW approach was put forward by Hedin as the simplest approximation to the so-called Hedin equations. In Section 2, we will derive these Hedin equations from a Feynman-diagrammatical point of view. Section 3.1 shows how GW arises as an approximation to the Hedin equations. In Section 3.2, we briefly present some typical GW results for materials, including quasiparticle renormalizations, lifetimes, and band gap enhancements. In Section 4, the combination of GW and DMFT is summarized. Finally, as a prospective outlook, ab initio dynamical vertex approximation DΓΓΓA is introduced in Section 5 as a unifying scheme for all that: GW, DMFT and non-local vertex correlations beyond.
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.
We present and implement a self-consistent DΓ\GammaΓA approach for multi-orbital models and ab initio materials calculations. It is applied to the one-band Hubbard model at various interaction strengths with and without doping, to the two-band Hubbard model with two largely different bandwidths, and to SrVO3_33​. The self-energy feedback reduces critical temperatures compared to dynamical mean-field theory, even to zero temperature in two-dimensions. Compared to a one-shot, non-self-consistent calculation the non-local correlations are significantly reduced when they are strong. In case non-local correlations are weak to moderate as for SrVO3_33​, one-shot calculations are sufficient.