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Journal of Physics C Solid State Physics

Dispersion and damping in normal Fermi liquids

J. S. Thakur

Published Jan 10, 1983• 0 citations
Publisher

Abstract

An expression for the response of the system, valid up to the second order in the general potential, has been derived by developing the Green function by an equation-of-motion method. In the long-wavelength limit, contact with other theories can be made in a transparent manner. The expression for the width contains contributions from direct and exchange processes, the latter contribution being exactly half of the former. The decay of zero sound into two-particle-two-hole pair excitations is found to vary as q 2 . The dispersion relation up to the second order in the potential is an upward parabola.

DOI
10.1088/0022-3719/16/1/017

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An expression for the response of the system, valid up to the second order in the general potential, has been derived by developing the Green function by an equation-of-motion method. In the long-wavelength limit, contact with other theories can be made in a transparent manner. The expression for the width contains contributions from direct and exchange processes, the latter contribution being exactly half of the former. The decay of zero sound into two-particle-two-hole pair excitations is found to vary as q 2 . The dispersion relation up to the second order in the potential is an upward parabola.
Landau showed that a nonideal Fermi gas at zero temperature can sustain collective oscillations of the acoustical type (zeroth sound). The altenuation of zeroth sound in a dilute Fermi gas with repulive intereacitions was considered. The problem was formulted interms of the Green function G which describes the paropation of density fluctuations through the system. The simplest approximation of G led to Landau's dispersion law which wae analyzed in some detail. The contribution of the phonons' zero point oscillations to the ground- state energy was estimated and found to lead to a term which has an essential singuarty at the origin of the coupling constant plane. The energy and width of the phonon are given by the poles in the spectral representation of G, and the location of these poles was determined from the Fredholm solution of an apparoximate integral equation satisfied by G. In this way it is showa that ths width, divided by the displacement of the collective state above the single- particle continunna of the free gas, vanishes linearly in the long wavelength limit. It is also shows that in the limit of extreme dilution the correct damping; can be obtained by merely taking the finite lifetimemore » of single-particles excitations into account and ignoring the dissipative effects of the noninstantaneous interactions betweein the particles in the medium. Finally, it is alao argued that the Fredholm method is the natural tool for discussing many problems in the theory of collective motion. (auth)« less
We propose a new method to calculate the particle-hole Green's function for an infinite system of fermions in the long-wavelength limit of the random-phase approximation. Contrary to the standard approach the present method does not require approximating the residual particle-hole interaction by a sum of separable terms. The method can also be applied to the Landau theory of Fermi liquids. We illustrate the method by evaluating the response function of nuclear matter in the long-wavelength limit for two different isospin channels.
We bosonize the low energy excitations of Fermi liquids in any number of dimensions in the limit of long wavelengths. The bosons are a coherent superposition of electron-hole pairs and are related with the displacements of the Fermi surface in some arbitrary direction. A coherent-state path integral for the bosonized theory is derived and it is shown to represent histories of the shape of the Fermi surface. The Landau theory of Fermi liquids can be obtained from the formalism in the absence of nesting of the Fermi surface and singular interactions. We show that the Landau equation for sound waves is exact in the semiclassical approximation for the bosons.
We derive the quantum Boltzmann equation (QBE) by using generalized Landau-interaction parameters, obtained through the nonequilibrium Green's function technique. This is a generalization of the usual QBE formalism to non-Fermi liquid (NFL) systems, which do not have well-defined quasiparticles. We apply this framework to a controlled low-energy effective field theory for the Ising-nematic quantum critical point, in order to find the collective excitations of the critical Fermi surface in the collisionless regime. We also compute the nature of the dispersion after the addition of weak Coulomb interactions. The zero angular momentum longitudinal vibrations of the Fermi surface show a linear-in-wavenumber dispersion, which corresponds to the zero sound of Landau's Fermi liquid theory. The Coulomb interaction modifies it to a plasmon mode in the long-wavelength limit, which disperses as the square-root of the wavenumber. Remarkably, our results show that the zero sound and plasmon modes exhibit the same behaviour as in a Fermi liquid, although an NFL is fundamentally different from the former.