This chapter considers the interactions between the constituents of say a metal, i.e., electrons and ions. By adopting a mean field approach, the dynamics of the electrons can be obtained by perturbation theory from the properties of the non-interacting electrons, and the effective electron-electron interaction in good conductors is considered. Electron diffraction or X-ray experiments reveal the grainy character of a metal. Charge is separated spatially into two pieces: the nuclei and the tightly bound electrons, and the conduction electrons, which have their density spread throughout the solid. The mean field Hamiltonian is specified in terms of the mean field, which in turn is specified in terms of the average charge density. The diagrammatic interpretation of the degenerate free Fermi gas density response function allows us also to get a diagrammatic interpretation of the interacting Fermi gas density response function.
This chapter considers the interactions between the constituents of say a metal, i.e., electrons and ions. By adopting a mean field approach, the dynamics of the electrons can be obtained by perturbation theory from the properties of the non-interacting electrons, and the effective electron-electron interaction in good conductors is considered. Electron diffraction or X-ray experiments reveal the grainy character of a metal. Charge is separated spatially into two pieces: the nuclei and the tightly bound electrons, and the conduction electrons, which have their density spread throughout the solid. The mean field Hamiltonian is specified in terms of the mean field, which in turn is specified in terms of the average charge density. The diagrammatic interpretation of the degenerate free Fermi gas density response function allows us also to get a diagrammatic interpretation of the interacting Fermi gas density response function.
The art of being wise is the art of knowing what to overlook. William James Summary This chapter is devoted to idealized models and theoretical concepts that underlie the topics in the rest of this book. Among the most dramatic effects are the Wigner and Mott transitions, exemplified by electrons in a homogeneous background of positive charge and by the Hubbard model of a crystal. Fermi liquid theory is the paradigm for understanding quasi-particles and collective excitations in metals, building on a continuous link between a non-interacting and an interacting system. The Luttinger theorem and Friedel sum rule are conservation laws for quantities that do not vary at all with the interaction. The Heisenberg and Ising models exemplify the properties of localized electronic states that act as spins. The Anderson impurity model is the paradigm for understanding local moment behavior and is used directly in dynamical mean-field theory. The previous chapters discuss examples of experimental observations where effects of interactions can be appreciated with only basic knowledge of physics and chemistry. The purpose of this chapter is to give a concise discussion of models that illustrate major characteristics of interacting electrons. These are prototypes that bring out features that occur in real problems, such as the examples in the previous chapter. They are also pedagogical examples for the theoretical methods developed later, with references to specific sections. The Wigner transition and the homogeneous electron system The simplest model of interacting electrons in condensed matter is the homogeneous electron system, also called homogeneous electron gas (HEG), an infinite system of electrons with a uniform compensating positive charge background. It was originally introduced as a model for alkali metals. Now the HEG is a standard model system for the development of density functionals and a widely used test system for the many-body perturbation methods in Chs. 10–15. It is also an important model for quantum Monte Carlo calculations, described in Chs. 23–25. To define the model, we take the hamiltonian in Eq. (1.1) and replace the nuclei by a rigid uniform positive charge with density equal to the electron charge density n .
Abstract This chapter first discusses the quantum mechanics of non-interacting electrons in a box. The simplest model of a metal is to assume it consists of free electrons in a box, with their mutual Coulomb repulsion ‘switched off’. The atoms in these materials (Li, Na, K, Rb, and Cs) have low ionization energies, only weakly binding their single valence s electron to the nucleus. It can then be imagined that when assembled as a crystal, each atom contributes one valence electron to the formation of a kind of electron gas or quantum plasma having a number density equal to that of the atoms. On average, the nuclear charge will compensate the free or mobile electronic charge. The remainder of the chapter covers Fermi–Dirac statistics; evaluation of integrals involving the Fermi distribution function; the temperature dependence of the chemical potential; energy of an ideal Fermi gas as a function of temperature; the paramagnetic susceptibility of a Fermi gas; and a qualitative discussion of the behaviour of a Fermi gas. Sample problems are also provided at the end of the chapter.
Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.
The ways of introducing and handling renormalizations in the many-body perturbation theory are reviewed. We stress the indispensable role the technique of Green functions plays in extrapolating the weak-coupling perturbative approaches to intermediate and strong couplings. We separately discuss mass and charge renor- malizations. The former is incorporated in a self-consistent equation for the self-energy derived explicitly from generating Feynman diagrams within the Baym and Kadanoff approach. The latter amounts to self-consistent equations for two-particle irreducible vertices. We analyze the charge renormalization initiated by De Domi- nicis and Martin and demonstrate that its realization via the parquet approach may become a powerful and viable way of using the many-body diagrammatic approach reliably in non-perturbative regimes with coopera- tive phenomena induced by either strong interaction or strong static randomness. Electric and magnetic properties of solids are determined by the behavior of valence electrons weakly bound to the atoms forming the crystalline lattice. The behavior of the gas of valence electrons in solids is most profound in metals with an open Fermi surface. Conduction electrons in metals can be described in a number of situations and for various purposes quite accurately by Bloch waves, eigenstates of the Fermi-gas Hamiltonian. We know, however, that electrons even in metals are not a noninteracting gas. They are generally exposed to forces driving them out of the equilibrium state of the Fermi gas. Firstly, no metal is a pure crystal and hence the electrons do not move freely in space. They are scattered on randomly distributed impurities and lattice defects. Secondly, the electrons are neither free and their mutual repulsion cannot be neglected particularly for transition and heavy metals. It is the most important and long standing task of condensed-matter theorists to comprehend and qualitatively capture the deviations in the behavior of electrons in real metals from their idealized representation via Bloch waves. The problem with quantification of the behavior of electrons beyond the almost free-electron picture is in paucity of available tools for accomplishing this task. Even if we can reduce the description of the electron gas in metals to rather simple and generic models, we are unable to solve them exactly, except for a few limiting cases that mostly do not correspond to situations of physical interest. With the increasing power of modern computers the role of numerical solutions and numerical brute force approaches has increased. Numerical solutions are usually unbiased and can reach a rather high level of quantitative precision. In some aspects they substitute the missing exact analytic solutions. Numerical solutions are suitable and even indispensable in showing and displaying the trends and some global features of the models studied. They, however, fail in critical regions of phase transitions in the vicinity of singularities in correlation functions. In these
Abstract The study of the interacting electron gas moving in a charge-compensating background of positively charged ions is central in this book. Not only is this system a model of the solids that surround us, such as metals, semiconductors, and insulators, but historically this system played a major role as testing ground for the development of quantum field theory. In this chapter we shall study the basic properties of this system using the formalism of time-independent second quantization as developed in Chap. 1. The main emphasis will be on the non-interacting electron gas, since it will be clear that we need to develop our theoretical tools further to deal with the electron-electron interactions in full.
This chapter examines different problems that apply the field theory in condensed matter physics. It considers bosonic as well as fermionic systems and applies the techniques of the quantum field theory. The chapter provides a model describing electrons in a metal known as the electron gas model that considers a system of electrons interacting with each other via the instantaneous Coulomb force e2r. The electron gas model provides a framework from which introduces standard many-body approaches such as the Gell-Mann-Bruckner theory or random phase approximation and show how they are formulated within diagrammatic or functional integral techniques. The chapter also considers a quantum three-dimensional coulomb gas and apply the field-theoretic techniques to systems of interacting electrons. The terminology of the order parameter for the field ϕ is to indicate the treatment of the Coulomb interaction within a mean-field theory with the inclusion of Gaussian fluctuations around the mean-field value of the order parameter.