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CRC Press eBooks

Neutral Fermi Liquids

David Pines, Philippe Nozières

Published Mar 9, 2018• 0 citations
Publisher

Abstract

For real fermion systems, the particle interaction and the exclusion principle act simultaneously; we are thus led to study degenerate Fermi liquids, in which both effects are important. This chapter explores Landau's semiphenomenological point of view and the applications of the theory to neutral systems such as degenerate liquid. It focuses on the Landau theory of Fermi liquids, which shows clearly all the important features brought about by the interaction. Landau's theory corresponds to an expansion in powers of the fraction of excited particles. Within the two restrictions, the theory is rigorous. It lends itself to a large number of problems: equilibrium properties, transport properties, response to long wavelength external perturbations. Landau's theory is not a "universal panacea": its well-defined limitations should always be kept in mind. Moreover, it uses (and provides) far more information than any experiment will ever sample. The chapter concludes with a brief summary of the range of validity and main achievements of Landau's theory.

DOI
10.4324/9780429492662-2

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Cambridge University Press eBooks•May 16, 2013•0 citations
For real fermion systems, the particle interaction and the exclusion principle act simultaneously; we are thus led to study degenerate Fermi liquids, in which both effects are important. This chapter explores Landau's semiphenomenological point of view and the applications of the theory to neutral systems such as degenerate liquid. It focuses on the Landau theory of Fermi liquids, which shows clearly all the important features brought about by the interaction. Landau's theory corresponds to an expansion in powers of the fraction of excited particles. Within the two restrictions, the theory is rigorous. It lends itself to a large number of problems: equilibrium properties, transport properties, response to long wavelength external perturbations. Landau's theory is not a "universal panacea": its well-defined limitations should always be kept in mind. Moreover, it uses (and provides) far more information than any experiment will ever sample. The chapter concludes with a brief summary of the range of validity and main achievements of Landau's theory.
Landau (1958) gave a rigorous microscopic derivation of his phenomenological Fermi -liquid theory starting from the first principles Hamiltonian of strongly interacting Fermions. The derivation is based on an asymptotic expansion in small parameters such as q/kF, bc.>/EF, kT/EF, where q and c.> represent the wave numbers and frequencies of real excitations of the system. Landau showed that the dynamics of these low-lying excitations (quasiparticles and collective modes) is governed by classical transport equations with all quantum many-body effects absorbed into effective interaction parameters. Landau's approach has been generalized to superfluid Fermi liquids and superconductors by Leggett (1965) and, going beyond linear response, by Eliashberg (1971). Their ideas merged into a recent review on the quasiclassical approach to superfluid 3 He by Serene and Rainer (1983). I will use the notation of this review throughout this paper. Landau's transport theory together with its offsprings will be called 'quasiclassical theory in the following. In this context it is important to mention that Eilenberger (1968) and Larkin and Ovchinnikov (1968) have shown that the complete theory of superconductors can be formulated as a quasiclassical transport theory. Surprisingly, this theory permits calculating microscopic properties, e.g. the quasiparticle excitation spectrum, which are predominantly governed by quantum effects.
The macroscopic theory of Fermi liquids we have developed in the preceding chapters possesses the advantage of being exact for any normal liquid, irrespective of its density or law of interaction, so long as the temperature is sufficiently low and the phenomena under discussion involve both long wavelengths and low frequencies. It is, of course, incomplete, since it can provide no account of microscopic phenomena involving short wavelengths and large excitation energies, or of equilibrium properties such as the ground state energy and pair distribution function. In contrast, microscopic theories are able to describe both macroscopic and microscopic phenomena. They are, however, subject to one of the following limitations: They are exact for only a limited range of particle density or interaction, one which rarely corresponds to physical reality. The extent to which they provide an approximate account of a real physical system can rarely be estimated with any precision.
We will consider the normal state of interacting Fermi particles without any longrange order. The normal state at low temperatures is called the Fermi liquid and is considered to be the system of free quasi-particles that is continuously connected with free Fermi gas. The concept of the Fermi liquid was introduced and developed by L. D. Landau. Landau's Fermi liquid theory, which concentrates rich contents into a simple theory, is a good example to solve many-body problems. In this chapter, we introduce the basis and main contents of the Fermi liquid. Principle of continuity The basis of the Fermi liquid theory is the principle of adiabatic continuity, which connects free Fermi gas with the Fermi liquid by introducing gradually an interaction among particles. There is a one-to-one correspondence between two states before and after the introduction of the interaction. States belonging to the same symmetry do not cross each other, and new states with the interaction can be represented by the old quantum number. Since the distribution of free Fermi gas is given by the Fermi distribution function n( k ) , that of the Fermi liquid is also written n ( k ). The state denoted by k , σ in the Fermi liquid is called the quasi-particle. For the system to be described by quasi-particles, the following condition is necessary. Let us consider the basis of the Fermi liquid theory following the explanation of the Landau theory given by Nozières and Anderson.
Abstract This chapter analyses Landau’s Fermi liquid theory. A fundamental postulate of Landau’s Fermi liquid theory is that quasiparticle energies are a function of the momentum, p, and that the momentum, pF, corresponding to the Fermi energy continues to be given by a free particle expression. Thus, the quasiparticle states are in one-to-one correspondence with those of an ideal Fermi gas and each quasiparticle has a spin quantum number, σ ± 1/2. The remainder of the chapter deals with the quasiparticle interaction function; sum rules; and collisionless collective modes, which are defined as a collective mode as a time- (or frequency) and space- (or wave vector) dependent oscillation of the fluid involving a coordinated motion of its constituent particles. Sample problems are also provided at the end of the chapter.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.