Have no fear of perfection; you'll never reach it. Marie Curie Summary This chapter is devoted to the combination of many-body perturbation and dynamical mean-field approaches to build more powerful first-principles methods. The GWA and corresponding approximations to the Bethe–Salpeter equation are very successful for one-particle and optical spectra of materials, especially for delocalized states. Dynamical mean-field methods are a set of techniques using an auxiliary embedded system to treat interacting electrons in localized atomic-like states that lead to strong correlation and large temperature dependence due to low-energy excitations. The combination holds promise for feasible approaches to calculate spectra, correlation functions, thermodynamic properties, and many other properties for a wide range of materials. 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.
Have no fear of perfection; you'll never reach it. Marie Curie Summary This chapter is devoted to the combination of many-body perturbation and dynamical mean-field approaches to build more powerful first-principles methods. The GWA and corresponding approximations to the Bethe–Salpeter equation are very successful for one-particle and optical spectra of materials, especially for delocalized states. Dynamical mean-field methods are a set of techniques using an auxiliary embedded system to treat interacting electrons in localized atomic-like states that lead to strong correlation and large temperature dependence due to low-energy excitations. The combination holds promise for feasible approaches to calculate spectra, correlation functions, thermodynamic properties, and many other properties for a wide range of materials. 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.
Think globally, act locally. Summary Dynamical mean-field theory is designed to treat systems with local effective interactions that are strong compared with the independent-particle terms that lead to delocalized band-like states. Interactions are taken into account by a many-body calculation for an auxiliary system, a site embedded in a dynamical mean field, that is chosen to best represent the coupling to the rest of the crystal. The methods are constructed to be exact in three limits: interacting electrons on isolated sites, a lattice with no interactions, and the limit of infinite dimensions d → ∞ where mean-field theory is exact. This chapter is devoted to the general formulation, the single-site approximation where the calculation of the self-energy is mapped onto a self-consistent quantum impurity problem, and instructive examples for the Hubbard model on a d → ∞ Bethe lattice. Further developments and applications are the topics of Chs. 17–21. One of the most rewarding features of condensed matter theory is the ability to address difficult problems from different points of view. The preceding Chs. 9–15 present an approach based on perturbation expansions in the Coulomb interaction. In particular, the GW approximation for the self-energy has proven to be extremely successful in describing electronic spectra of many materials, as described in Ch. 13. The methods can be applied to the ordered states of materials with d and f states, for example, ferromagnetic Ni and anti-ferromagnetic NiO, as described in Secs. 13.4 and 20.7. However, the GW and related approximations have difficulties in treating cases with degenerate or nearly degenerate states and low-energy excitations. Present-day methods do not describe phenomena like the fluctuations of local moments in a ferromagnetic material above the Curie temperature or the insulating character of NiO in the paramagnetic phase; more generally, they have difficulty describing strong correlation. The topic of this chapter and Chs. 17–21 is dynamical mean-field theory, which is also a Green's function method in which the key quantity is the self-energy. However, it is designed to treat strong interactions for electrons in localized atomic-like states, such as the d and f states in transition metals, lanthanide and actinide elements and compounds.
In the previous chapters DMFT has been developed as a methodology for calculation of dynamical and thermodynamic properties of many-body systems, with applications to model systems. In order for DMFT to be a general method for quantitative calculations, there must be systematic procedures to derive all the ingredients from first principles. In this chapter we consider the most prominent examples, materials with partially filled 3 d , 4 f , or 5 f localized, atomic-like states, and set up the characteristic form of the hamiltonian. This is the basis for DFT+DMFT methods discussed here and used in the examples in Ch. 20, and for the combination of DMFT with many-body perturbation theory in Ch. 21. Simpler techniques, including self-interaction correction, DFT+U, and DFT+Gutzwiller, are also useful and are summarized in the last sections of this chapter. 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.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. 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 ( r 1 , r 2 , …): 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 Q 0 , 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 F m [ Q ] that can then be evaluated at Q = Q 0 . 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 n 0 ( r ) and total energy E 0 .
Electronic structure of materials from first principles is essential for one-to-one comparisons between theories and experiments. However, the many-body effects resulting from electron correlation and their interplay with other degrees of freedom pose a great challenge to design an ab initio approximate method with good accuracy and efficiency at the same time. In this thesis, we present our recent developments of ab initio many-body theory for solids based on the finite-temperature Green’s function formalism. We begin by reviewing the basic concepts and definitions of realistic electronic Hamiltonians (Ch. 2) as well as Green’s functions at finite temperature (Ch. 3). Overviews of self-consistent many-body perturbation theories and quantum embedding theories are then presented in Chs. 4 and 5 as complementary materials for the following chapters. In Ch. 6, we present algorithms and implementation details for the finite-temperature self-consistent GW (scGW) method in Gaussian-type orbitals for solids. Our implementation is based on the finite-temperature Green’s function formalism in which all equations are solved on the imaginary axis, without resorting to analytical continuation during the self-consistency. No quasiparticle approximation is employed and all matrix elements of self-energies are explicitly evaluated. We show agreement with other, differently formulated finite-temperature scGW implementations. By migrating computationally intensive calculations to GPUs, we obtain scalable results on large supercomputers with nearly optimal performance. Inclusion of relativistic effects in scGW is discussed in Ch. 7. Our formulation of relativistic scGW is based on the exact two-component formalism with one-electron approximation (X2C1e) and non-relativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and their interplay with electron correlation without adjustable parameters. We examine the effect of the X2C1e by comparison to the established four-component formalism and reach excellent agreement. The simplicity of X2C1e enables construction of higher order theories, such as embedding theories, on top of perturbative calculations. In Ch. 8, we use self-energy embedding theory (SEET) to study the spectra of the prototypical undistorted cubic perovskites SrVO3 and SrMnO3. In the strongly correlated metallic SrVO3, we find that the usual attribution of the satellite peaks at -1.8 eV to Hund or Hubbard physics in the t2g orbitals is inconsistent with our calculations. In the strongly correlated insulator SrMnO3 we recover insulating behavior due to a feedback effect between the strongly correlated orbitals and the weakly correlated environment. Our calculation shows a systematic convergence of spectral features as the space of strongly correlated orbitals is enlarged, paving the way to a systematic parameter-free study of correlated perovskites. Lastly, in Ch. 9, we apply the Green’s function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems in strongly correlated solids within the framework of SEET. We describe the implementation details of our Green’s function coupled cluster (GFCC) solver, investigate its performance, and highlight potential advantages and problems on examples of impurities created for antiferromagnetic MnO and paramagnetic SrMnO3. GFCCSD provides satisfactory descriptions for weakly and moderately correlated impurities with sizes that are intractable by existing accurate impurity solvers such as exact diagonalization. However, when correlations become strong, the singles and doubles approximation used in GFCC could lead to instabilities in searching for the particle number present in impurity problems.
Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.
It is a characteristic of wisdom not to do desperate things. Henry David Thoreau Summary The essence of a mean-field method is to replace an interacting many-body problem with a set of independent-particle problems having an effective potential. It can be chosen either as an approximation for effects of interactions in an average sense or as an auxiliary system that can reproduce selected properties of an interacting system. The effective potential can have an explicit dependence on an order parameter for a system with a broken symmetry such as a ferro- or antiferromagnet. Mean-field techniques are vital parts of many-body theory: the starting points for practical many-body calculations and often the basis for interpreting the results. This chapter provides a summary of the Hartree–Fock approximation, the Weiss mean field, and density functional theory that have significant roles in the methods described in this book. Mean-field methods denote approaches in which the interacting many-body system is treated as a set of non-interacting particles in a self-consistent field that takes into account some effects of interactions in some way. In the literature such methods are often called “one-electron”; however, in this book we use “non-interacting” or “independent-particle” to refer to mean-field concepts and approaches. We reserve the terms “one-electron” or “one-body” to denote quantities that involve quantum mechanical operators acting independently on each body in a many-body system. Mean-field approaches are relevant for the study of interacting, correlated electrons because they lead to approximate formulations that can be solved more easily than more sophisticated approaches; when judiciously chosen, mean-field solutions can yield useful, physically meaningful results, and they can provide the basis and conceptual structure for investigating the effects of correlation. The particles that are the “bodies” in a many-body theory can be the original particles with their bare masses and interactions, or, most often, they may be the solutions of a set of mean-field equations chosen to facilitate the solution of the many-body problem. A large part of many-body theory in condensed matter involves the choice of the most appropriate independent particles. Hence, it is essential to define clearly the particles that are created and annihilated by the operators and in which the many-body theory is formulated.