Interactions between electrons in a metal are treated in this chapter and we introduce the Cooper, Hartree and Fock terms in the perturbation expansion for the ground state energy shift. The chapter explains how to model the excitations and study them using propagators and we introduce the random phase approximation.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
The Hohenberg–Kohn theorem In Chapter 2 we explored some of the consequences of electron–electron interactions, albeit in some simple perturbative approaches and within the random phase approximation. There we found that the problem of treating these interactions is exceedingly difficult, even in the case where there is no external one–particle potential applied to the system. We have also explored some of the properties of noninteracting electrons in an external potential, in this case the periodic lattice potential. This led to the concepts of electron bands and band structure, subjects of fundamental importance in understanding the physics of metals, insulators, and semiconductors. Of course, in the real world, electrons in matter are subjected both to electron–electron interactions and to external potentials. How to include systematically and correctly the electron–electron interactions in calculations of real systems is truly a formidable problem. Why that is so is easily demonstrated. Suppose that we want to solve the problem of N electrons interacting in some external potential. The N -electron wavefunction can be expanded in Slater determinants of some suitable single-particle basis such as plane waves. We can describe the Slater determinants by occupation numbers in our second–quantized notation. Suppose furthermore that we have a basis of a total of N k plane wave states at our disposal. Here N k must be large enough that all reasonable “wiggles” of the many-body wavefunction can be included.
We especially need imagination in science. aria Mitchell Summary The many-body problem consists of two parts: the first is the non-interacting system in a materials-specific external potential; the second is the Coulomb interaction that makes the problem so hard to solve. The most straightforward idea is to use perturbation theory, with the Coulomb interaction as perturbation. This is conceptually simple, but it turns out to be difficult in practice, since the Coulomb interaction is often not small compared with typical energy differences, it is long-ranged and in the thermodynamic limit there is an infinite number of particles, contributing with an infinite number of mutual interaction processes. The present chapter outlines how one can deal with this problem. It contains an overview of facts that one can also find in many standard textbooks on the many-body problem, but that are useful to keep in mind in order to look at later chapters from a sound and well-established perspective. The many-body problem is a tough one, and it has many facets. Sorting it out is like putting together a huge puzzle. The eight introductory chapters of this book provide pieces of the puzzle, and ideas on what one might do about it. In the present chapter we choose to go in one of the possible directions, in order to arrive at something tangible. The chapter gives the general framework and the main ideas; specific approximations are the topic of Chs. 10–15. The idea is to start from an independent-particle problem and add the Coulomb interaction as a perturbation. This is not easy: first, the interaction is responsible for a rich variety of phenomena that are completely absent otherwise, such as the finite lifetime of quasiparticles, or additional structures in spectra due to the fact that a quasi-particle excitation may transfer its energy to other elementary excitations, for example plasmons. Second, because of the two-body Coulomb interaction, the problem scales badly with the number of electrons, and straightforward perturbation theory for the many-body hamiltonian with the Coulomb interaction as perturbation rapidly becomes intractable or even useless, especially in large systems. To get started, Sec. 9.1 recalls why things are not so easy. The following sections try to solve one problem after the other, starting from Sec. 9.2 where the Green's function is reformulated in a way that is appropriate for a perturbation expansion.
The ground-state wave function and energy of a finite system of interacting fermions are expanded in terms of multiple-particle excitations on an uncorrelated zero-order state. The resulting set of coupled equations constitutes a systematic variational generalization of Hartree-Fock theory. Comparison is made with many-body perturbation theory and it is shown that to any order the theory incorporates an infinite number of perturbation theory terms. Solutions of the equations for ground-state atomic systems are discussed and related to previous work using many-body perturbation theory. It is shown that the sums of perturbation terms necessary for convergence are automatically included in the equations for two-particle excitations. Application of the equations to open-shell atoms is described.
I specialize the Hartree-Fock, random-phase, and configuration-interaction approximations to a system of interacting bosons in an external potential. This approach affords a more direct connection to the methods and language of atomic-structure calculations. The derivations and equations parallel those found in atomic structure and are contrasted with the more commonly used Bogoliubov approach for bosons. I present a numerical method for solving the Hartree-Fock and random-phase equations to obtain the low-lying excitation energies for trapped alkali-metal atoms.