In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
The most of the experimental measurable quantities in many-body electron system, such as the optical absorption, luminescent spectra and exciton effect, are related to the excitation states description. The density functional theory with local density approximation (DFT-LDA) as a ground state theory, i.e. the solutions based on the Kohn-Sham equation is a powerful tool for studying the ground state properties in many-particle systems. However, the first principles computation of excited states is more complexity than ground-state calculations. A key problem is that the exchange-correlation interaction in excited states is differ from the ground-states. In recent years, nevertheless, several electronic-excitation theories have been developed. The most important theoretical and computational method include the many-body perturbation theory which is based on the quasi-particle concept and the Green function equation, the time-dependent DFT and the Bethe-Salpeter equation for describing the electron-hole interaction. Among them the central ingredient is an electron's self-energy operator Σ that describes the exchange correlation effect beyond Hartree approximation. The implementations of the above theories are unavoidable to introduce some of approximations. A good approximation for Σ is the GW approach by Hedin. It is shown from computation simulations for many real condensed matter systems that the GW approach is a successful method for electronic-excitation problems. Another effective method for excited-states is so-called screen-exchange local-density approximation (sX-LDA). It is a combination of LDA and Hartree-Fock (HF)theory, in which the intrinsic local screen-exchange interaction has been replaced by a non-local one and a generalized Kohn-Sham equation (GKS) is given. Based on a PWscf package, the sX-LDA-PWscf method has been tested and indicates that an available approach for treating band-gap problem in semiconductors. In this paper, the many-body theories for the excitation process, its development and signification are reviewed. The relation and differences consist in different theoretical methods are discussed. On the based of above discussions, the applications of many-body theory to the band-band transition (band-gap underestimation problem) and exciton effect in semiconductors are presented.
In many-body perturbation theory (MBPT) the self-energy $\ensuremath{\Sigma}=iGW\ensuremath{\Gamma}$ plays a key role since it contains all the many-body effects of the system. The exact self-energy is not known; as a first approximation one can set the vertex function $\ensuremath{\Gamma}$ to unity which leads to the GW approximation. The latter properly describes the high-density regime, where screening is important; in the low-density regime, instead, other approximations are proposed, such as the T matrix, which describes multiple scattering between two particles. Here we combine the two approaches. Starting from the fundamental equations of MBPT, we show how one can derive the T-matrix approximation to the self-energy in a common framework with GW. This allows us to elucidate several aspects of this formulation, including the origin of, and link between, the electron-hole and the particle-particle T matrix, the derivation of a screened T matrix, and the conversion of the T matrix into a vertex correction. The exactly solvable Hubbard molecule is used for illustration.
Many observables such as the density, total energy, or electric current, can be expressed explicitly in terms of the one‐body Green's function, which describes electron addition or removal to or from a system. An efficient way to determine such a Green's function is to introduce a self‐energy, which is a nonlocal and dynamic effective potential that influences the propagation of particles in an interacting system. The state‐of‐the art approximation for the self‐energy is the GW approximation, where the system to (or from) which the electron is added (or removed) is described as a polarizable, screening, medium. This is expressed by the name of the approximation: ‘GW’ stands for the one‐body Green's function G and for W , the dynamically screened Coulomb interaction. The GW approximation is very popular for the calculation of band structures in solids, and increasingly used also to describe nanostructures, clusters, and molecules. As compared to static mean‐field approximations for the effective potential, the dynamical screening of the Coulomb interaction in GW leads to a renormalization of energies, to broadening and/or to the observation of additional excitations. An analysis of the approximations that lead to the GW self‐energy, and of the underlying picture, explains the successes and the limitations of the approach. This article is categorized under: Electronic Structure Theory > Density Functional Theory Electronic Structure Theory > Ab Initio Electronic Structure Methods Theoretical and Physical Chemistry > Spectroscopy Structure and Mechanism > Computational Materials Science
Density functional theory (DFT) performs reasonably well for the determination of structural properties of many materials, but fails to predict electronic band gap values accurately. \nSuch a failure of DFT is not unexpected, since there exists no formal justification for interpreting the DFT eigenvalues as addition or removal energies of the many-body system (quasiparticle energies).\nAn alternative approach to the study of exchange-correlation effects in many-particle systems is provided by many-body perturbation theory (MBPT), which defines a rigorous approach to description of excited-state properties based on the Green's function formalism and the concept of quasiparticle electrons. Within MBPT, one can calculate the quasiparticle (QP) energies and the QP amplitudes by approximately solving, within the so-called GW approximation, the set of coupled integro-differential equations proposed by Hedin in 1965.\nThe GW method generally yields significantly better values for QP energies with respect to DFT as it accounts for the dynamic many-body effects in the electron-electron interaction going beyond the mean-field approximation of independent Kohn-Sham particles. The drawback of such an involved approach, however, is its large computational cost, which is mainly due to the evaluation of dielectric matrices, their inversion, and the solution of a non-Hermitian and nonlinear eigenvalue problem. \nThe first part of this PhD research is devoted to the implementation of an efficient and scalable coarse-grained parallel algorithms and the development of state-of-the-art methods to solve the GW equations.\nAll these techniques are then applied to the study of the quasiparticle band structures of SiO2 in the alpha quartz crystalline structure. The effects of the different approximations involved in the theory, the influence of self-consistency, and a systematic analysis of the reliability of the different plasmon-pole models are presented and discussed. \nThe second part of this PhD work is dedicated to the implementation of a methodological approach to the solution of the GW equations based on the so-called projector augmented wave method (PAW) proposed by Blochl in 1994. This new approach permits to remove many of the limitations intrinsic to the use of the pseudopotential technique widely used for ab-initio calculations, allowing one to reach GW results closer to an all-electron implementation while still maintaining computational efficiency, and flexibility. This allows us to achieve a coherent implementation of many different algorithms which enables a detailed comparison of GW calculations not yet performed until now. Results for the quasiparticle band structure and optical spectra of prototype s- and p-compounds are discussed and compared to recent studies reported in the scientific literature (when available).
Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.
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.