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 $GW$, 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) $T$-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 $T$-matrix approximation by accounting for low-order pp vertex corrections.
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 GW, 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) T-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 T-matrix approximation by accounting for low-order pp vertex corrections.
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.
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.
Starting with Hedin's equations, simple expressions for the irreducible self-energy are derived. The derivation with vertex effects included in the self-energy results in a number of terms beyond G W such as second-order screened exchange (the term that also gives rise to vertex/excitonic effects in the polarization) and an infinite series describing correlations between the added-particle (removed-hole) and the excited electrons and holes: the screened T -matrix channels. Two-body correlated propagation is considered, with the third propagating freely, however, three-body correlations are discussed and these can be added hierarchically to the method. For electron-hole propagation, the reducible polarization is calculated, which results in an expression for the self-energy that can be solved analytically without the need for widely used expensive numerical methods for frequency integration or having to adopt the plasmon-pole approximation. The method requires diagonalization of the usual particle-hole matrix—derived from the Bethe-Salpter equation—and also a particle-particle + hole -hole matrix that has a similar structure; with the second-order exchange present in all channels. The method goes beyond current widely used methods that adopt the G W approximation, whilst being in nature and requiring relatively modest extensions to existing functionality, e.g., it could be implemented in the sophisticated quasiparticle self-consistent framework. It is anticipated the method will significantly improve on the accuracy of current state-of-the-art calculations, improving our understanding of processes in simple and strongly correlated systems, with the additional contributions demanding a similar cost to methods that include vertex effects in the polarization only. Published by the American Physical Society 2024
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.
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 T 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 T-matrix approximation to the self-energy in a common framework with GW. This allows us to elucidate several aspects of this formulation, including the origin of, and link between, the electron-hole and the particle-particle T matrix, the derivation of a screened T matrix, and the conversion of the T matrix into a vertex correction. The exactly solvable Hubbard molecule is used for illustration.
The equation of motion for the Green function is combined with the Bethe-Salpeter equation for the scattering amplitude yielding a concise and formally closed system of three equations that encapsulates the essence of Green function theory. Two of the three equations formally resemble a Dyson-like relation. We prove that this formally simple set is exactly equivalent to Hedin's equations. Our derivation therefore constitutes an alternative to Hedin's derivation which is based on functional derivatives. Furthermore, we briefly discuss how approximations can be introduced as a hierarchy of approximations to the four-point Green function.