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Oxford University Press eBooks

The many-body problem and the metal

Tom Lancaster, Stephen J. Blundell

Published Apr 17, 2014• 0 citations
Publisher

Abstract

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.
DOI
10.1093/acprof:oso/9780199699322.003.0044

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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.
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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.