Chapter 1: Introduction

The dynamics of quantum mechanical many-particle systems is described by the Schrödinger equation. This equation is easy to write down for general molecules and solids but quickly becomes intractable to solve, even for atoms and simple molecules. The reason for this, as we will see below, is the electron–electron interaction, which couples all electrons in the system. In order to describe these systems, one is therefore left with two possible approaches. The first approach is to treat the electron–electron interaction in an approximate manner (i.e., with a mean-field approximation or perturbatively) for all electrons in the system and solve the equations with the approximate interaction in the full Hilbert space. This method has the advantage of providing a coherent approximation for all electrons in the system but it fails when the interaction among the electrons is strong, such as in, e.g., transition metal compounds and lanthanides in which the localized 3d and 4f electrons are highly correlated. The other approach is to construct a low-energy model containing only a few electrons and energy eigenstates that are believed to be most important. The low-energy model can then be solved accurately and the role of the remaining degrees of freedom is to modify the effective parameters of the low energy model. It is this approach, which is called downfolding, that is the topic of this book.

The Hamiltonian of a molecule or a crystal consisting of N electrons moving in a potential arising from the nuclear charges and a possible external field is in the position representation given by

where

is the potential from the nuclear charges Z_n located at R_n and V_ext is an externally applied field. V_N is the interaction among the nuclear charges

In our notation, r is a combined variable for space and spin: r = (r, σ). Atomic units are used throughout this book, where

The atomic unit of energy is Hartree and one Hartree is equal to 2 Ry or 27.2 eV. In this book, we do not consider non-equilibrium systems for which the Hamiltonian is explicitly time dependent. In the occupation number representation, the electronic part of the Hamiltonian (without V_N) is given by

where

$ψ^(r)$ is the field operator that annihilates an electron at r and $ψ^†(r)$ is its conjugate that creates an electron at r.

One of the general goals of molecular and condensed matter physics is to develop methods that allow us to compute ground- and excited-state properties of materials from first-principles, that is to say, free from adjustable parameters. Solving the many-electron Hamiltonian in Eq. (1.1) is virtually impossible with the exception of small systems. The reason is due to the presence of the second term in Eq. (1.1), which describes the Coulomb interaction among the electrons. It is, however, precisely this term that gives rise to a wide variety of fascinating physical phenomena, such as magnetism, superconductivity, and many others. Direct attempts to solve the Schrödinger equation,

by expanding the wave function as a linear combination of Slater determinants are impractical because the number of configurations or Slater determinants increases exponentially. In some materials, the many-electron interaction can be well approximated by a mean field, in which case the many-electron problem reduces to a single-electron problem and can be considered to be solved. Such systems are referred to as weakly correlated systems, which include many materials, such as alkali metals and conventional semiconductors (silicon, gallium arsenide, germanium, etc.) whose valence electrons originate from s or p orbitals forming extended states. In this book, we do not consider such systems for which one-particle approximations work well.

One of the most fruitful methods in tackling the many-electron problems is the method of downfolding. Instead of indiscriminately including all possible configurations or degrees of freedom of the system, one selects a particular set of configurations, which is believed to be most important for the physical properties of interest. The effects of the rest of the Hilbert space are included in an approximate way. In quantum chemistry, this approach is known as the complete active space approach. The one-particle Hilbert space or the complete set of orbitals is divided into two: an active subspace in which orbitals can be partially occupied and a virtual subspace containing orbitals that are empty. In addition, we may also include an inactive subspace consisting of core orbitals that are usually completely occupied or frozen and these can be included in the virtual subspace. Electron configurations within the active subspace are enumerated and used as a basis to diagonalize the Hamiltonian. Electron correlations obtained within the active subspace are conventionally referred to as “static correlations.” These are correlations that originate from interaction among configurations with relatively large weights and with energies close to each other. Configurations involving virtual orbitals have relatively small weight and their effects can be taken into account by perturbation theory. These additional electron correlations are referred to as “dynamic correlations” since they arise from excitations to and from the virtual orbitals. The distinction between static and dynamic correlations is a matter of convention since in the full configuration interaction approach in which all configurations are considered, not just within the active subspace, the resulting correlations include both the static and dynamic ones.

The concept of the active subspace is also ubiquitous in condensed matter physics. In the tight-binding method (Slater and Koster, 1954), one chooses a set of orbitals on each atom in a unit cell and these orbitals form the valence band states thought to be most important for the physics of interest. The tight-binding description is a suitable starting point for systems in which the valence states are rather localized. Systems in which the valence states are dominated by 3d transition metals and 4f lanthanides are prime examples. The valence bands are narrow since electrons are not so free to move around the crystal leading to a large onsite Coulomb interaction comparable to or larger than the kinetic energy. When the valence bands are partially filled, one expects many configurations similar in energy of equal importance so that a single Slater determinant or a single-particle description would not, in general, be sufficient. A widely used model to account for the strong onsite Coulomb interaction is the Hubbard model (Hubbard, 1963) which, in its simplest form with only one orbital per atom and one atom per unit cell, is given by

where i denotes an atomic site, ɛ_iσ is the onsite energy with spin σ, μ is the chemical potential, t_ij is the hopping parameters from site i to site j describing the kinetic energy, $n^iσ=c^iσ†c^iσ$ is the number operator counting the number of electrons of spin σ on site i, and U is the parameter describing the onsite Coulomb interaction when two electrons of opposite spin reside on the same site i. The first term is included to allow for the possibility of varying the electron number but otherwise it may be omitted. The Hubbard model may be thought of as a downfolded version of the full Hamiltonian in which the effects of the rest of the bands are lumped into the parameter U. In the language of static and dynamic correlations, one might say that static correlations are included within the active subspace defined by the Hubbard Hamiltonian, whereas those bands not included in the Hubbard model form the virtual subspace and account for the dynamic correlations that are embodied in parameter U. This parameter represents a screened Coulomb interaction where the screening arises from transitions between the active and virtual subspaces.

Unlike in quantum chemistry, where the active subspace corresponding to a molecule has a manageable size, the active subspace corresponding to the Hubbard model of a periodic lattice with partially filled bands is infinite, which renders the quantum chemical approach of diagonalizing the Hamiltonian within the active subspace inapplicable. Direct attempts to solve the Hubbard Hamiltonian run into the problem of having an unrealistically large number of electron configurations. Except for a highly specific one-dimensional case (Lieb and Wu, 1968), the solutions to the two- and three-dimensional Hubbard models are not known, even for the simplest case of one orbital per site.

Solving the Hubbard model in an approximate way necessitates further downfolding. One goes one step down and reduces the active subspace from the lattice to an atomic site so that electron configurations within a given atomic site constitute the active subspace. Diagonalizing the Hamiltonian within this active subspace becomes manageable since the number of configurations is finite and relatively small, depending on the number of electrons and orbitals on each site. From the point of view of a given site, the rest of the orbitals on other sites now constitute the virtual subspace. The strict division between active and virtual subspaces without allowing exchanges of electrons between them may work well when the coupling between them is weak, for example as in systems in which the active subspace is formed by the highly localized 4f orbitals of the lanthanides. Indeed, for these systems, the excitation spectra are atomic-like. However, a more realistic approximation should take into account the possibility of dynamic correlations due to electron density fluctuations between the active and virtual subspaces.

Downfolding to an atomic site effectively reduces the Hubbard Hamiltonian to the Anderson impurity model (AIM) (Anderson, 1961) which, for an impurity with one orbital, is given by

where $f^$ and $f^†$ are the annihilation and creation operators associated with the impurity orbital, $n^fσ=f^σ†f^σ†$⁠, $c^i$ and $c^i†$ are the annihilation and creation operators of the conduction orbitals, V_jσ is the hybridization matrix element between the impurity and the conduction orbitals, and U is the Coulomb repulsion energy when two electrons reside in the impurity orbital. Without the hybridization term (last term), the AIM reduces to an atomic problem of the impurity (first two terms) and a one-particle problem of the conduction electrons (third term). Both of these can be solved analytically for the one-orbital and one-conduction-orbital case or numerically for the multi-orbital case. The subspace of the impurity or atomic problem may be thought of as the active space, which describes the static electron correlations, whereas the conduction orbitals can be regarded as the virtual subspace which, in the case of the Hubbard model, is the subspace formed by the orbitals on the rest of the sites other than the atomic one. The coupling of the impurity orbital to the conduction orbitals describes the dynamic correlations, which are associated with the charge and spin density fluctuations in the impurity site. When the hybridization term is switched on, the problem becomes highly non-trivial but it is much more manageable compared with the original lattice problem. By integrating the degrees of freedom of the virtual subspace into an effective action, this impurity problem can be solved numerically within the Green function formalism by a number of techniques, such as the path integral quantum Monte Carlo method (Gull et al., 2011).

Although the impurity problem by itself is interesting, we wish, however, to solve the original lattice problem of the Hubbard Hamiltonian. A method that has proven successful in approximately solving the lattice Hubbard model is dynamical mean-field theory (DMFT) (Georges et al., 1996), which is a Green function approach. DMFT is based on the idea of solving the Anderson impurity model on each site augmented with a self-consistency condition on the local Green function. After solving for the Green function of the impurity, the corresponding impurity self-energy is extracted, and in order to restore the lattice periodicity, it is used as an approximate self-energy of the lattice. It is approximate because the self-energy is local and has no components between two different sites, and hence, it has no lattice momentum dependence. The Green function of the lattice is, therefore, obtained from a local self-energy, which is a fundamental assumption in DMFT. The self-consistency condition requires that the impurity Green function coincides with the local projection of the lattice Green function.

DMFT has been combined with density functional theory (DFT) within the local density approximation (LDA) or its improvements, leading to the LDA+DMFT method (for a review, see, e.g., Kotliar et al., 2006), which has been successfully applied to calculate the electronic structure of strongly correlated materials. The description of this class of materials, as the name indicates, requires theories beyond the mean-field level due to strong electron–electron interaction, which cannot be approximated by a simple one-particle form as in Eq. (1.2). One famous example is the high-temperature superconductors, which have eluded intense research in explaining the mechanism behind the superconductivity since their discovery in the latter half of the 1980s (Bednorz and Mueller, 1986). Interest in these materials was recently rekindled with the discovery of another class of superconducting materials based on iron compounds (Kamihara et al., 2008) and the nicklates (Li et al., 2019, 2020; and Zeng et al., 2020) with a similar structure as the original high-temperature superconductors discovered in 1986. To explain the mechanism behind superconductivity, good knowledge of the Fermi surface and the electronic states at low energy around the Fermi level is crucial. Among other intriguing phenomena in strongly correlated materials are the metal-to-insulator transition, colossal magneto-resistance, and heavy fermion behavior. A common denominator permeating these intriguing phenomena is the sensitivity of strongly correlated materials upon small changes in parameters, such as temperature and pressure. Thus, for example, a small change in pressure can turn an insulator into a metal (Imada et al., 1998).

One of the main achievements of DMFT is to provide a coherent description of the metal-to-insulator transition in which both phases are treated on equal footing within the same framework. There are, however, a number of limitations of DMFT. One of these is the assumption of local self-energy. If we were to uncover the mechanism of, for example, superconductivity in the high-temperature superconductors with the help of DMFT, we would be faced with the fundamental problem of describing the long-range correlations of the electrons forming the Cooper pairs. These long-range correlations are outside the domain of DMFT. The lack of long-range self-energy has also prevented the applications of DMFT to systems with extended states. The second fundamental problem of DMFT when used in combination with the LDA is the problem of double counting. Part of the impurity self-energy calculated within DMFT is already included in the LDA exchange-correlation potential. There is, however, no well-defined way of removing this double counting. The third fundamental problem is the use of the adjustable parameter U in the Hubbard model and in the corresponding impurity problem. Determination of the Hubbard U from first-principles is very important because it provides a link between the Hubbard model and the real material that the model is supposed to represent. An unambiguous determination of U allows for calculations of material-specific properties that can be compared directly with experiments. Indeed, one of the aims of materials science is to develop a practical scheme which allows us to compute the electronic structure of materials with high precision entirely from first principles without using adjustable parameters. The absence of adjustable parameters is crucial because it permits us to make quantitative predictions of materials properties without any prior knowledge other than the atom types and positions. It is a dream in materials science to able to design materials with desirable properties by means of computer simulations. Computer simulations can also help experimentalists by providing clues as to which materials are likely to possess the desired properties. With ever increasing computer power, computer simulations play an increasingly important role. A recent development in atomistic spin dynamics illustrates the power of computer simulations (Eriksson et al., 2017; Eriksson, 2018; and Hellsvik et al., 2019). However, one should not forget the crucial role that model investigations play in understanding fundamental phenomena in condensed matter physics. Model studies often form a foundation and a starting point for first-principles calculations. Combining model and first-principles approaches is, therefore, highly desirable.

This book describes a method for going beyond DMFT to take into account dynamic long-range correlations by means of many-body perturbation theory. A highly successful many-body perturbation theory is the GW approximation (GWA) (Hedin, 1965; Hedin and Lundqvist, 1969; Aryasetiawan and Gunnarsson, 1998; and Onida et al., 2002), widely used for calculating the electronic structure of weakly to moderately correlated materials. A combination of DMFT and GWA leads to the GW +DMFT method (Biermann et al., 2003; Sun and Kotliar, 2002). The book starts with the basic concept of DFT in Chap. 2, placing emphasis on its deficiencies when used as a practical scheme for treating strongly correlated materials and its important role as a starting point in many-electron Green function theory. Chapter 3 deals with fundamental concepts of Green function theory and Chaps. 4 and 5 elaborate upon the well-established GWA and DMFT. The Löwdin downfolding idea is described in Chap. 6, both from the Hamiltonian and Green function points of view. Chapter 7 explains in detail the constrained random-phase approximation (cRPA) method (Aryasetiawan et al., 2004) for downfolding a full Hilbert space to a low-energy subspace and examples are provided to illustrate the method. Chapter 8 describes several methods for solving the impurity problem. This is followed in Chap. 9 by a detailed description of the multitier GW+DMFT scheme (Boehnke et al., 2016; and Nilsson et al., 2017) and the last chapter summarizes the book and presents an outlook for future development. The emphasis of the book as a whole is on concepts and theories, but applications to some prototypical materials will be presented as illustrations.