It is commonly accepted that the GW approximation for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approximation: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the atomic limit, in which, instead, correlation is directly responsible for the observed problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be corrected by a two-point vertex contribution to the self-energy derived from time-dependent density functional theory (TDDFT). We explain why the addition of a particle, instead, requires the use of a different approximate vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the atomic limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qualitatively. We discuss the effects of the self-screening error as well as the atomic limit using GW for the exactly solvable two-site Hubbard model.
It is commonly accepted that the GW approximation for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approximation: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the atomic limit, in which, instead, correlation is directly responsible for the observed problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be corrected by a two-point vertex contribution to the self-energy derived from time-dependent density functional theory (TDDFT). We explain why the addition of a particle, instead, requires the use of a different approximate vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the atomic limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qualitatively. We discuss the effects of the self-screening error as well as the atomic limit using GW for the exactly solvable two-site Hubbard model.
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 GW 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.
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.
Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .
Accidental accuracy and vertex corrections in GW: Exact benchmarks for the extended Hubbard model
The GW 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 GW theory on the identical Hilbert space and quantify its error as a function of local (U) and non-local (V) interaction strength. We find that the required vertex correction changes character across the phase diagram: in the weak-coupling regime the effective vertex Γeff<1, reflecting the suppression of RPA charge fluctuations by exact short-range correlations, whereas in the Mott regime Γeff grows monotonically (to ∼3 at U=8t for N=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=0, static COHSEX is accidentally exact at a single crossover U∗≈3.5t; finite V, 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 GW in correlated materials.
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.
The total energy and electron addition and removal spectra can, in principle, be obtained exactly from the one-body Green's function (GF). In practice, the GF is obtained from an approximate self-energy. In the framework of many-body perturbation theory, we derive different expressions that are based on an approximate self-energy, but that yield nevertheless, in principle, the exact exchange-correlation contribution to the total energy for any interaction strength. Response functions play a crucial role, which explains why, for example, ingredients from time-dependent density functional theory can be used to build these approximate self-energies. We show that the key requirement for obtaining exact results is the consistent combination of ingredients. Also when further approximations are made, as it is necessary in practice, this consistency remains the key to obtain good results. All findings are illustrated using the exactly solvable symmetric Hubbard dimer.