Front Matter

Electronic structure theory has reached a stage at which it is possible to make a theoretical prediction of a wide range of physical properties of real materials by means of theoretical analyses and computational simulations. At the heart of the electronic structure is the electron-electron interaction which determines materials properties and is responsible for the emergence of collective phenomena, such as superconductivity, Kondo effects, and many others. Materials may be classified according to the strength of the electron correlations arising from the electron-electron interaction. Alkali metals and conventional semiconductors whose valence states are dominated by the itinerant s and p electrons are generally regarded as weakly correlated. For these systems, well-established one-particle descriptions within the Kohn-Sham density function theory and many-body perturbation theory, such as the GW approximation, usually work well.

In recent decades, many new materials with fascinating properties have been discovered and synthesized. The valence states of these materials are dominated by semi-itinerant or localized 3d and 4f electrons originating from the transition metals and the lanthanides. These materials are referred to as correlated due to the strong electron-electron interaction that cannot be treated in a mean-field fashion. It has become evident that for these correlated materials, new methods are needed. Dynamical mean-field theory in combination with density functional theory and many-body perturbation theory has proven to be a method of choice for these systems. This approach necessitates the process of downfolding from the full Hilbert space to the subspace of the correlated electrons.

This book describes the fundamental concepts and theories behind the downfolding methods illustrated with examples from real materials. The aim of many-body theory is not to find exact solutions of the many-body problem, since the basic equation is already known but is not feasible to solve except for very simple systems. The aim is rather to construct approximations sufficiently accurate to yield predictive results that can be compared with experiment and used to understand the physical mechanism behind certain phenomena. The downfolding approach should be seen in this spirit.

The first few chapters describe the fundamental tools, starting with the Green function theory and the well-established density functional theory, followed by the GW approximation and the dynamical mean-field theory. The concept of downfolding is then introduced and the constrained random-phase approximation method, which is core to the downfolding methods, is described in detail. The following chapters deal with practical schemes for solving downfolded Hamiltonian in combination with first-principles methods. Special focus is put on the recently developed GW+EDMFT, a merging of the GW method and the extended dynamical mean-field theory. The book closes with an outlook of promising recent developments and future directions.

This book is aimed at graduate students and researchers in the field of electronic structure. Familiarity with theories covered in standard many-body textbooks such as those by Fetter and Walecka (2003) as well as by Negele and Orland (1998) is beneficial but not entirely mandatory. While the book is not fully self-contained, an effort is made to make it as self-contained as possible.

We would like to acknowledge our colleagues who have contributed in forming our understanding of the topics covered in the book through various discussions and collaborations with reservation that the list may be incomplete: Bernard Amadon, Vladimir Anisimov, Silke Biermann, Stefan Blügel, Lewin Boehnke, Michele Casula, Christoph Friedrich, Antoine Georges, Olle Gunnarsson, Masatoshi Imada, Krister Karlsson, Alexander Lichtenstein, Cyril Martins, Takashi Miyake, Francesco Petocchi, Lucia Reining, Rei Sakuma, Jan Tomczak, and Philipp Werner.

We would also like to acknowledge financial support from the Swedish Research Council (VR) and the Knut and Alice Wallenberg (KAW) Foundation for our research on which the topics of the book are based.

F.A. would like to thank his former supervisors and mentors who have introduced and taught him the subjects described in this book and provided him with opportunities to work on the topics, in chronological order: Malcolm Stott, Ulf von Barth, Carl-Olof Almbladh, the late Lars Hedin, Olle Gunnarsson, Ole Andersen, and Kiyoyuki Terakura.

AFM

antiferromagnetic

AIM

Anderson impurity model

AMF

around mean-field

BIS

Bremsstrahlung isochromatic spectroscopy

CCSD

coupled-cluster singles doubles

CI

configuration interaction

cLDA

constrained local density approximation

COHSEX

Coulomb hole and screened exchange

cRPA

constrained random-phase approximation

CT-QMC

continuous-time quantum Monte Carlo

DCI

dynamical configuration interaction

DFT

density functional theory

DMET

density matrix embedding theory

DMFT

dynamical mean-field theory

DOS

density of states

DΓA

dynamical vertex approximation

EDMFT

extended dynamical mean-field theory

FLAPW

full-potential linearized augmented plane-wave

FLEX

fluctuation exchange

FLL

fully localized limit

GGA

generalized gradient approximation

GWA

GW approximation

HOMO

highest occupied molecular orbital

KS

Kohn–Sham

KVC

Kondo volume collapse

LAPW

linearized augmented plane-wave

LDA

local density approximation

LMTO

linearized muffin-tin orbital

LSDA

local spin density approximation

PBE

Perdew, Burke, and Ernzerhof

PES

photoemission spectroscopy

QMC

quantum Monte Carlo

QSGW

quasiparticle self-consistent GW

QUADRILEX

quadruply irreducible local approximation

RPA

random-phase approximation

SEET

self-energy embedding theory

TRILEX

triply irreducible local expansion