There are various ways that functionals can be employed to generate auxiliary systems and useful approximations by limiting the range of the functions input to the functional. Each of the functionals of the density, Green's function, or self-energy are defined over a specified domain D, e.g., ФG] and F[∑] are defined for all functions that have the required analytic properties for a Green's function or self-energy (see Sec. H.1). Here we consider limiting the domain to a subset of D denoted by D′. The first examples below derive expressions that provide insights concerning the link of non-locality and frequency dependence, and that lead to useful approximations in the spirit of optimized effective potentials. This is followed by an approach to find an interacting auxiliary system where the full many-body problem can be solved with no approximations, thus providing a self-energy that is exact within the restricted domain of the auxiliary system.

Auxiliary system to reproduce selected quantities

Let us first consider an auxiliary system that reproduces exactly some quantity of interest. This is similar to the Kohn–Sham approach, but it is more general and it leads to explicit expressions for the functionals in terms of Green's functions [1163].

Suppose we are interested in a quantity P that is part of the information carried by the Green's function G, symbolically expressed as P = p{G}. An example is the density where the “part” to be taken is the diagonal of the one-particle G: p{G} = n(r) = −iG(r, r, t′−t = 0+) or −G(r, r, τ = 0−). Consider an auxiliary system that has the same bare Green's function G0 as the original one, but a different effective potential, or a self-energy ∑P that leads to an auxiliary Green's function GP.

This chapter is dedicated to the calculation of the two-particle correlation function L, which is one of the central quantities in this book. It contains a wealth of experimentally accessible information: optical spectra, electron energy loss spectra, and the dynamic structure factor that is measured for example by inelastic X-ray scattering, the energy of doubly charged defects, and much more. Moreover, many-body perturbation theory can be formulated in terms of the dynamically screened Coulomb interaction W, which is derived directly from the electron–hole correlation function. The GW approximation is the most prominent example of an approximation based on W. Finally, the total energy can be expressed exactly in terms of L, because the Coulomb interaction vc is a two-body interaction.

The two-particle correlation function is formally given by the Dyson-like Bethe– Salpeter equation derived in Sec. 10.3. The present chapter starts in Sec. 14.1 with a brief reminder of the links between L and measurable quantities, since in this chapter we are interested in the spectra that are derived from L. Some important formal relations are recalled in Sec. 14.2. The simplest approximation for L is the random phase approximation introduced in Sec. 11.2. Section 14.3 is dedicated to the study of spectra calculated in the RPA: which physics is contained in this approximation? What are its strengths and limitations? We will see that optical spectra, for example, often require a description beyond the RPA.

The shortcomings of the RPA motivate the effort to go beyond. The two-particle correlation function contains information about the propagation of two particles, which can be electron–hole pairs, or two electrons or two holes, as explained in Sec. 5.8.

This chapter is devoted to representative examples that bring out the range of phenomena observed in materials with strong local atomic-like interactions, and the types of properties that can be calculated with various methods. The foremost examples are elements and compounds of the series of transition elements with d and f states that are localized as depicted in Fig. 19.1. As emphasized in the previous chapter, the theory must deal with the complexities of many competing interactions: direct Coulomb, exchange, spin–orbit, crystal field splitting, hybridization with other orbitals, and other effects, all of which may be essential for understanding any particular material.

Many-body perturbation theory, especially the GW approximation and beyond, is a firstprinciples method that can be applied directly to ordered states at low temperature. Several examples of applications to transition metal systems are given in Ch. 13 and referred to here to provide a unified picture. Density functional theory and approximate static mean-field approximations also provide useful results and insights. However, many of the most striking phenomena can be understood only by taking into account strong correlation, including large renormalization of electronic states, satellites in excitation spectra, magnetic phase transitions, metal–insulator transitions, and other phenomena. Many phenomena can be understood only if temperature is taken into account. Dynamical mean-field theory brings this capability, but at a price because it is feasible to treat explicitly only a small subset of the degrees of freedom of the electrons. All the results presented here are done in the single-site approximation (single-site DMFA), except the two-site cluster for VO2 in Sec. 20.6.

In this chapter we return to the goal of robust first-principles methods that can treat materials and phenomena, such as the examples of Ch. 2: from covalent semiconductors like silicon to lanthanides with partially filled f shells like cerium; from optical properties in ionic materials like LiF to metal–insulator transitions in transition metal compounds, and much more. A natural approach is Green's function methods; however, it is very difficult to have one method that treats all cases. In fact, this is not the goal. We seek understanding as well as results. We seek combinations of methods that are each firmly rooted in fundamental theory, with the flexibility to take advantage of different capabilities and provide robust, illuminating predictions for the properties of materials.

This is an active area with on-going developments; the topics of this chapter are not meant to be exhaustive, but rather to formulate basic principles and indicators of possible paths for future work. The approach developed here can be used with any method that partitions a system into parts, treats the parts using techniques that may be different for the different parts, and then constructs the solution for the entire system. The formulation in terms of Green's functions, self-energies, and screened interactions is already given in Ch. 7; here we put this together with the practical methods developed in Chs. 9–20. If the procedures are judiciously chosen, the results have the potential to be more efficient, more accurate, and more instructive than a comparable calculation with a single method.

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 essential features of DMFT are brought out in the previous chapter in the simplest form: the single-site approximation in which correlation between sites can be neglected. In this case the self-energy due to interactions can be calculated using an auxiliary system of a single site embedded in an effective medium that represents the surrounding crystal. The resulting many-body problem is equivalent to an Anderson impurity model that must be solved self-consistently. The results are exact for infinite dimensions d → ∞; however, in any finite dimension, i.e., real systems, this is just a mean-field approximation analogous to the Weiss mean field in magnetic systems or the CPA for disordered alloys.

There are, however, many important properties and phenomena that require us to go beyond mean-field approximations. For example, basic questions concerning the nature of a metal–insulator transition can be established only by quantitative assessment of the correlations between different sites. Is a single site sufficient (the central idea in the arguments by Mott discussed in Sec. 3.2) or is correlation between sites (for example, an ordered antiferromagnetic state) essential for interactions to lead to an insulating state? A complete theory must establish the range of correlation needed to understand the mechanisms and to make quantitative calculations. A striking example that requires a k-dependent self-energy is the opening of a “pseudogap” in only part of the Brillouin zone in a high-temperature superconductor, as illustrated in Fig. 2.16.

This chapter is devoted to methods that go beyond the single-site approximation to take into account interactions and correlations between electrons on sites.

Many physical quantities can be expressed as functionals, i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction (r1, r2, …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions.

In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E[Q] of this quantity. The variational character of E allows one to derive equations that determine Q0, the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals Fm[Q] that can then be evaluated at Q = Q0. This strategy is outlined in Fig. 8.1.

To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n(r) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E[n] leads to equations to determine the actual ground-state density n0(r) and total energy E0.

Up to this point, dynamical mean-field theory has been developed for model systems with Hubbard-type hamiltonians, where the hopping matrix elements and interactions are considered as parameters. For problems with strong interactions, DMFT leads to renormalized bands, satellites in the one-particle spectra, magnetic phase transitions and local moments, metal–insulator transitions, and many other phenomena observed in real materials, as brought out in Ch. 2. The purpose of this chapter is to provide the framework for a quantitative theory of these materials.

These are difficult problems, experimentally, theoretically, and computationally, and the first step is to identify the aspects of the materials that must be treated explicitly as the local strongly interacting system, and the “rest” that are treated by a different method. This division of the problem is useful because a real material involves many degrees of freedom, i.e., the electronic states must be described in a basis sufficiently large to describe all the relevant states. However, the part that can be treated by non-perturbative solvers (Ch. 18) is limited by the computational cost that scales rapidly with the number of states included in the embedded site or cluster. Thus one must choose a small number of localized orbitals that inevitably are non-unique. This may seem like a step backward for quantitative methods; however, it opens a window for methods to treat these interesting problems and it presents a challenge to overcome the limitations.

The GW approximation to the self-energy is a surprisingly simple formula, with a clear physical content as discussed in Ch. 11, and the potential for efficient calculations of electronic properties beyond independent-particle methods as described in Ch. 12. It is now time to look at results for realistic systems and ask: which properties can be described successfully by the GWA, and for which materials? And what can we learn from the results?

The selection presented in this chapter covers only a fraction of what could be cited. The GWA has become a very popular method – for valuable reasons, as the chapter shows: having a good approximation to the self-energy allows one to calculate accurate band structures, electron addition and removal spectra, densities, density matrices, total energies, and many related properties. GWA calculations lead to qualitative and often quantitative agreement with experimental findings. There exist many more applications, but here we have selected examples meant to illustrate content, strong and weak points of the GWA, in its various flavors such as perturbative or non-perturbative calculations. The examples are chosen to stress certain aspects, and they are not necessarily the first or the most recent calculations. A time span of several decades is covered. As a consequence, the computational scheme that has been used, for example the level of self-consistency from G0W0 to fully self-consistent GW, is not always the same. However, the quality of the results is always sufficient to support the message that the example is supposed to illustrate.

One can find, for instance, a huge number of GW bandgap calculations in the literature: for our purpose, it is sufficient to give just a few representative examples. Far fewer results exist for spectral functions and satellites, although the dynamically screened interaction W contains electron–hole and plasmon excitations.

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, vc]-functional of the bare vc. 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.