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CRC Press eBooks

Elementary Excitations

Philippe Nozières, David Pines

Published Mar 5, 2018• 5 citations
Publisher

Abstract

The density fluctuation excitation spectrum is continuous, since all particle transitions allowed by the Pauli principle will contribute. This chapter shows that the quasi-particle, as a bare particle moving together with the surrounding distortion brought about by particle interactions. The importance, or indeed the existence, of the mechanism, depends on the quasi-particle energies and the phase space available for the decay process. Higher order collision processes can also limit the lifetime of a quasi-particle. In contrast to the single quasi-particle excitations, the multi-particle excitations will be spread over a continuous range of energies, as one allows the momentum of any single quasi-particle component to vary. For an arbitrary particle interaction, there is no a priori region of momenta for which quasi-particles exist as well-defined elementary excitations of the system. In contrast to the single quasi-particle excitations, the multi-particle excitations will be spread over a continuous range of energies, as one allows the momentum of any single quasi-particle component to vary.

DOI
10.1201/9780429492679-2

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The density fluctuation excitation spectrum is continuous, since all particle transitions allowed by the Pauli principle will contribute. This chapter shows that the quasi-particle, as a bare particle moving together with the surrounding distortion brought about by particle interactions. The importance, or indeed the existence, of the mechanism, depends on the quasi-particle energies and the phase space available for the decay process. Higher order collision processes can also limit the lifetime of a quasi-particle. In contrast to the single quasi-particle excitations, the multi-particle excitations will be spread over a continuous range of energies, as one allows the momentum of any single quasi-particle component to vary. For an arbitrary particle interaction, there is no a priori region of momenta for which quasi-particles exist as well-defined elementary excitations of the system. In contrast to the single quasi-particle excitations, the multi-particle excitations will be spread over a continuous range of energies, as one allows the momentum of any single quasi-particle component to vary.
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The wave propagation might be appropriately called quasi-particle sound, since in the long wave-length limit the only density fluctuations of importance are those produced by exciting a single quasi-particle from the condensate. Quasi-particle sound has essentially the same physical origin as zero sound in a Fermi liquid. The quasi-particle sound velocity will be temperature dependent, since the energy of a given quasi-particle depends on the thermal excitation of other quasi-particles. Superfluid hydrodynamics is richer than its Fermi liquid counterpart because one can have relative motion of the thermal quasi-particles and the condensate. This chapter describes the propagation of first and second sound in the hydrodynamic limit and then go on to a consideration of quasi-particle sound in the collisionless regime. A wide variety of irreversible phenomena which have as their physical origin collisions between the thermally-excited quasi-particles.
For a system of many bosons, quasi-particles are defined as simple excitations having energies equal to those predicted by the Bogoliubov and Valatin–Butler theories. Scattering between these elementary excitations is caused by the residual interaction. This results in a change in the energies of quasi-particles, and a finite lifetime for the excitations. The force between two quasi-particles is determined in terms of the boson–boson potential, and the distribution function for different excitations. Perturbed single particle energies are also calculated within the framework of the reaction matrix.
We begin our study of the superfluid Bose liquid with a brief review of the experimental and theoretical background of the liquid 4 He, the only Bose liquid found naturally in the laboratory. We then consider, in Chapter 2 , the quasi-particle excitations and show how at T = 0 the macroscopic occupation of the zero momentum state leads, in the long wavelength limit, to an identical excitation spectrum for quasi-particles and density fluctuations. The spectrum consists in phonons with a velocity given by the macroscopic velocity of sound. We turn in Chapter 3 to a consideration of the observed elementary excitation spectrum of He II. We discuss both the experimental measurements and the theories that have thus far been proposed to explain them.
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.
In a one-dimensional weakly interacting Bose-Fermi mixture, one branch of elementary excitations is well described by the Bogoliubov spectrum. Here we use the microscopic theory to study the decay of such quasiparticle excitations. The main scattering process which leads to their decay is the backscattering of a Bogoliubov quasiparticle off the Fermi sea, where a particle-hole pair is excited. For a low-momentum quasiparticle (phonon) of momentum qqq, we find that the decay rate scales as q3{q}^{3}q3 provided qqq is smaller than the Fermi momentum kF{k}_{F}kF​, while in the opposite case the decay behaves as q2{q}^{2}q2. If the ratio of the masses of fermions and bosons is equal to the ratio of the boson-fermion and the boson-boson interaction strengths, the decay rate changes dramatically. It scales as q7{q}^{7}q7 for q<kFq<{k}_{F}q<kF​, while we find q6{q}^{6}q6 scaling at q>kFq>{k}_{F}q>kF​. For a high-momentum Bogoliubov quasiparticle, we find a constant decay rate for q<kFq<{k}_{F}q<kF​, while it scales as 1/q1/q1/q for q>kFq>{k}_{F}q>kF​. We also find an analytic expression for the decay rate in the crossover region between low and high momenta. The decay rate is a continuous, but nonanalytic function of the momentum at q=kFq={k}_{F}q=kF​. In the special case when the parameters of our system correspond to the integrable model, we observe that the decay rate vanishes.