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arXiv:2203.05029 · Physical review. B./Physical review. B

Self-consistency in GWΓ formalism leading to quasiparticle-quasiparticle couplings

Carlos Mejuto-Zaera, Vojtěch Vlček

Published Oct 26, 2022• 40 citations
arXivHTMLPDF

Abstract

Within many-body perturbation theory, Hedin's formalism offers a systematic way to iteratively compute the self-energy $\mathrm{\ensuremath{\Sigma}}$ of any dynamically correlated interacting system, provided one can evaluate the interaction vertex $\mathrm{\ensuremath{\Gamma}}$ exactly. This is, however, impossible, in general, for it involves the functional derivative of $\mathrm{\ensuremath{\Sigma}}$ with respect to the Green's function. Here, we analyze the structure of this derivative, splitting it into four contributions and outlining the type of quasiparticle interactions that each of them generate. Moreover, we show how, in the implementation of self-consistency, the action of these contributions can be classified into two: A quantitative renormalization of previously included interaction terms and the inclusion of qualitatively distinct interaction terms through successive functional derivatives of $\mathrm{\ensuremath{\Gamma}}$ itself. Implementing this latter type of self-consistency can extend the validity of perturbative approximations based on Hedin's equations toward the high interaction limit, as we show in the example of the Hubbard dimer. Our analysis also provides a unifying perspective on the perturbation theory landscape, showing how the T-matrix approach is completely contained in Hedin's formalism.

DOI
10.1103/physrevb.106.165129

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Within many-body perturbation theory, Hedin's formalism offers a systematic way to iteratively compute the self-energy \ensuremathΣ\mathrm{\ensuremath{\Sigma}}\ensuremathΣ of any dynamically correlated interacting system, provided one can evaluate the interaction vertex \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ exactly. This is, however, impossible, in general, for it involves the functional derivative of \ensuremathΣ\mathrm{\ensuremath{\Sigma}}\ensuremathΣ with respect to the Green's function. Here, we analyze the structure of this derivative, splitting it into four contributions and outlining the type of quasiparticle interactions that each of them generate. Moreover, we show how, in the implementation of self-consistency, the action of these contributions can be classified into two: A quantitative renormalization of previously included interaction terms and the inclusion of qualitatively distinct interaction terms through successive functional derivatives of \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ itself. Implementing this latter type of self-consistency can extend the validity of perturbative approximations based on Hedin's equations toward the high interaction limit, as we show in the example of the Hubbard dimer. Our analysis also provides a unifying perspective on the perturbation theory landscape, showing how the T-matrix approach is completely contained in Hedin's formalism.
Self-consistent solutions of Hedin's equations (HE) for the two-site Hubbard model (HM) have been studied. They have been found for three-point vertices of increasing complexity (Γ = 1 (GW approximation), Γ1 from the first-order perturbation theory, and the exact vertex Γ(E)). Comparison is made between the cases when an additional quasiparticle (QP) approximation for Green's functions is applied during the self-consistent iterative solving of HE and when QP approximation is not applied. The results obtained with the exact vertex are directly related to the present open question-which approximation is more advantageous for future implementations, GW + DMFT or QPGW + DMFT. It is shown that in a regime of strong correlations only the originally proposed GW + DMFT scheme is able to provide reliable results. Vertex corrections based on perturbation theory (PT) systematically improve the GW results when full self-consistency is applied. The application of QP self-consistency combined with PT vertex corrections shows similar problems to the case when the exact vertex is applied combined with QP sc. An analysis of Ward Identity violation is performed for all studied in this work's approximations and its relation to the general accuracy of the schemes used is provided.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
In many-body perturbation theory (MBPT) the self-energy $\ensuremath{\Sigma}=iGW\ensuremath{\Gamma}$ plays a key role since it contains all the many-body effects of the system. The exact self-energy is not known; as a first approximation one can set the vertex function $\ensuremath{\Gamma}$ to unity which leads to the GW approximation. The latter properly describes the high-density regime, where screening is important; in the low-density regime, instead, other approximations are proposed, such as the TTT matrix, which describes multiple scattering between two particles. Here we combine the two approaches. Starting from the fundamental equations of MBPT, we show how one can derive the TTT-matrix approximation to the self-energy in a common framework with GWGWGW. This allows us to elucidate several aspects of this formulation, including the origin of, and link between, the electron-hole and the particle-particle TTT matrix, the derivation of a screened TTT matrix, and the conversion of the TTT matrix into a vertex correction. The exactly solvable Hubbard molecule is used for illustration.
Hedin's equations provide an elegant route to compute the exact one-body Green's function (or propagator) via the self-consistent iteration of a set of nonlinear equations. Its first-order approximation, known as GWGWGW, corresponds to a resummation of ring diagrams and has shown to be extremely successful in physics and chemistry. Systematic improvement is possible, although challenging, via the introduction of vertex corrections. Considering anomalous propagators and an external pairing potential, we derive a self-consistent set of closed equations equivalent to the famous Hedin equations but having as a first-order approximation, the particle-particle (pp) TTT-matrix approximation, where one performs a resummation of the ladder diagrams. This pp version of Hedin's equations offers a way to go systematically beyond the TTT-matrix approximation by accounting for low-order pp vertex corrections.
In this chapter, shortcomings of the GW approximation are analyzed, and different ways are proposed to go beyond. The approaches fall into three broad classes: first, one can iterate Hedin's equations that link the self-energy, the polarizability, the screened Coulomb interaction, and the vertex function with the Dyson equation for the one-body Green's function. Second, one can use other Dyson equations like in the T-matrix approach, and third, one can make an ansatz for the one-body Green's function inspired by physical considerations or insight from models, in particular the idea of electron–boson coupling that leads to the cumulant expansion. Applications are given for illustration. Some more ideas and links to subsequent chapters provide an outlook. The GW approximation is a widely used approach in computational electronic structure. However, the previous chapters show some severe limitations. One example is systems with low-energy excitations, as illustrated by the Hubbard dimer in the dissociation limit, where bonding and antibonding orbitals become degenerate. This can be seen in Sec. 11.7. Another example is satellite series, for example in the photoemission spectrum of silicon in Sec. 13.1. Moreover, even in a situation where the approximation is well justified, it induces some error, though maybe small, and one eventually wants to do better. The full self-energy functional is in principle known: it is the Luttinger–Ward sum of skeleton diagrams derived from Eq. (11.35). However, this is an infinite sum, and no closed form of the result is known. Hence, the task is not to invent new diagrams: it is to make a choice of diagrams or combinations of diagrams, which are evaluated fully or in an approximate way. This is the framework of the present chapter. In some cases we will add diagrams to the GWA ones, which gives rise to vertex corrections ; in other cases some of the GWA diagrams are neglected while different ones are taken into account. An example is the T-matrix approximation, which sums ladder diagrams instead of bubbles. In general one tries to combine this with the GWA, especially in extended systems where screening is important. Therefore the GWA plays a prominent role, and the whole chapter is called beyond GW . The chapter contains advanced material, that is in part exploratory and refers to current research. It is intended to make links to communities outside the GW world, and to other chapters.
In many ways the solution to the Hedin equations represents an exact solution to the many body problem. However, for most systems of practical interest, the solution to the Hedin equations is rendered nearly numerically intractable because the Hedin equations are of functional derivative form. Integral equations, on the other hand, are much more numerically tractable, than functional derivative equations, as they can often be solved iteratively. In this work we present a systematic set of integral equations (with no functional derivatives) - Hedin approximations I, II, III, IV etc. - whose solutions converge to the solutions of the exact Hedin equations. The Hedin approximations are well suited to iterative numerical solutions (which we also describe). Furthermore Hedin approximation I is just the GW approximation (as such this work may be viewed as a systematic improvement of the GW approximation). We present a systematic study of the Hedin equations for zero dimensional field theory (which, in particular, is a method to enumerate Feynman diagrams for field theories in arbitrary dimensions) and show better and better convergence to the solutions of the Hedin equations for higher and higher Hedin approximations, with Hedin approximations I, II and III being explicitly studied. We, in particular, show that the higher Hedin approximations capture more and and more Feynman diagrams for the self energy. We also show that already Hedin approximation II captures more diagrams than the state of the art diagrammatic vertex corrections approach. Furthermore Hedin approximation III is a near perfect match to the exact solutions of the Hedin equations, at least in the zero dimensional case, and enumerates a large number of Feynman diagrams.