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.
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.
The previous chapter is devoted to the formulation of DMFT and applications that are exact in the limit of infinite dimensions; however, in finite dimensions this is only a single-site approximation. The present chapter is devoted to clusters and other methods to treat correlation between sites. The equations are derived in a unified way, applicable to a single site or a cluster. For a cluster with more than one site there are various ways to choose the boundary conditions and the embedding procedure, and special care must be taken to satisfy causality and translation invariance. A few selected applications illustrate different techniques and results for Hubbard-type models. 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.
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.
The main focus of this thesis is the detailed investigation of computational methods to tackle strongly correlated materials in which a rich variety of exotic phenomena are found. A many-body problem with sizable electronic correlations can no longer be explained by independent-particle approximations such as density functional theory (DFT) or tight-binding approaches. The influence of an electron to the others is too strong for each electron to be treated as an independent quasiparticle and consequently those standard band-structure methods fail even at a qualitative level. One of the most powerful approaches for strong correlations is the dynamical mean-field theory (DMFT), which has enlightened the understanding of the Mott transition based on the Hubbard model. For realistic applications, the dynamical mean-field theory is combined with various independent-particles approaches. The most widely used one is the DMFT combined with the DFT in the local density approximation (LDA), so-called LDA+DMFT. In this approach, the electrons in the weakly correlated orbitals are calculated by LDA while others in the strongly correlated orbitals are treated by DMFT. Recently, the method combining DMFT with Hedin's GW approximation was also developed, in which the momentum-dependent self-energy is also added. In this thesis, we discuss the application of those methodologies based on DMFT. First, we apply the dynamical mean-field theory to solve the 3-dimensional Hubbard model in Chap. 3. In this application, we model the interface between the thermodynamically coexisting metal and Mott insulator. We show how to model the required slab geometry and extract the electronic spectra. We construct an effective Landau free energy and compute the variation of its parameters across the phase diagram. Finally, using a linear mixture of the density and double-occupancy, we identify a natural Ising order parameter which unifies the treatment of the bandwidth and filling controlled Mott transitions. Secondly, we study the double-counting problem, a subtle issue that arises in LDA+DMFT. We propose a highly precise double-counting functional, in which the intersection of LDA and DMFT is calculated exactly, and implement a parameter-free version of the LDA+DMFT that is tested on one of the simplest strongly correlated systems, the H2 molecule. We show that the exact double-counting treatment along with a good DMFT projector leads to very accurate and total energy and excitation spectrum of H2 molecule. Finally, we implement various versions of GW+DMFT, in its fully self-consistent way, one shot GW approximation, and quasiparticle self-consistent scheme, and studied how well these combined methods perform on H2 molecule as compared to more established methods such as LDA+DMFT. We found that most flavors of GW+DMFT break down in strongly correlated regime due to causality violation. Among GW+DMFT methods, only the self-consistent quasiparticle GW+DMFT with static double-counting, and a new method with causal double-counting, correctly recover the atomic limit at large H-atom separation. While some flavors of GW+DMFT improve the single-electron spectra of LDA+DMFT, the total energy is best predicted by LDA+DMFT, for which the exact double-counting is known, and is static.
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.
Ab-initio calculations are dominated by density functional theory (DFT), which provides an efficient and accurate description of the electronic structure for most materials. For materials with strong correlations, however, many of the available density functionals yield poor results. Most well known is the case of transition metal oxides, for which most density functionals produce a qualitatively incorrect description. There is a quest to improve the description by borrowing from methods specifically designed for strongly correlated materials. One of the method which is widely used to study strongly correlated electron system in the thermodynamic limit is the dynamical mean-field theory (DMFT). For calculations pertaining to real materials, hybrid approaches like DFT+DMFT are commonly used. We consider reduced density-matrix functional theory (rDMFT) to be a useful framework for a rigorous formulation of such hybrid theories. The link from rDMFT to many-particle wave functions has been established by Levy's constrained-search algorithm on the one hand. The link to many-body perturbation theory and Green's function, on the other hand, has been provided via the Luttinger-Ward functional. The development in the field of rDMFT proceeded analogously to the development of DFT. In order to avoid the full complexity of an explicit many-body description, most density-matrix functionals are not extracted from the exact expressions i.e. via Levy's constrained-search or Green's function based methods . Rather, one proceeds analogously to the development of density functionals, namely by searching models for the density-matrix functional, that capture the most essential physical effects while having an algebraic dependence on the density matrix. Among such models, the most prominent ones are the Müller functional, Sharma functional and Marques-Lathiotakis functional. One of the major arguments in favor of density-matrix functionals is that one of the most simple functionals, the Müller functional, seems to provide a correct description of the bond-dissociation problem, for which common density functionals fail. Despite the successes, the common density-matrix functionals also reproduce a number of features in a qualitatively incorrect manner. Several such discrepancies of the standard state of the art density-matrix functionals are discussed in details in this thesis. The objective of this thesis is to study the performance of the class of commonly used model density-matrix functionals and to set the stage for the desire to develop the framework for an approximate scheme to evaluate new class of functionals from exact Green's function based method.
In this chapter, we present a bird’s-eye view of dynamical mean field theory (DMFT) and its extensions to highly correlated systems. We first demonstrate the elementary introduction to DMFT on a single site with detailed derivations and then a solution of the simple interacting models: the Hubbard model and the Falicov–Kimball model on infinite dimensional Bethe lattice using DMFT. Thereafter, we focus on two kinds of non-local extensions of single-site DMFT - (a) cluster dynamical mean field theory (CDMFT) on real space and (b) dynamical cluster approximation (DCA) on momentum space. Finally, we conclude the chapter with the application of DMFT combining with density functional theory (DFT) in the real-material systems (LDA+DMFT).