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Physical Review Letters

Quasiparticle Excitations, Bose Condensation, and the f -Sum Rule

H. R. Glyde

Published Dec 4, 1995• 8 citations
Publisher

Abstract

In a Bose fluid having a condensate, the density response has two components. One is associated with exciting single quasiparticles out of the condensate and the second is the usual two-body density response involving states above the condensate appearing in any fluid. We derive the contribution of each component to the $f$-sum rule. The single quasiparticle contribution is proportional to the condensate function ${n}_{0}(T)$ at all wave vectors Q. This simple sum rule is analogous to the Ambegaokar, Conway, and Baym sum rule giving the one-phonon contribution to the $f$-sum rule in solids. It is very useful in interpreting neutron scattering measurements of superfluid $^{4}\mathrm{He}$.

DOI
10.1103/physrevlett.75.4238

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Annalen der Physik•Feb 1, 1999•3 citations
In a Bose fluid having a condensate, the density response has two components. One is associated with exciting single quasiparticles out of the condensate and the second is the usual two-body density response involving states above the condensate appearing in any fluid. We derive the contribution of each component to the fff-sum rule. The single quasiparticle contribution is proportional to the condensate function n0(T){n}_{0}(T)n0​(T) at all wave vectors Q. This simple sum rule is analogous to the Ambegaokar, Conway, and Baym sum rule giving the one-phonon contribution to the fff-sum rule in solids. It is very useful in interpreting neutron scattering measurements of superfluid 4He^{4}\mathrm{He}4He.
Various sum rules for the density and particle operators are derived and discussed. We investigate in detail the properties of the particle state, the natural counterpart of the Feynman state describing collective density excitations. An explicit expression for the energy of the particle state is derived in terms of the interatomic potential, the two-body half-diagonal density, and the momentum distribution. Sum rules accounting for the coupling between particle and density excitations are also derived and the role of the Bose-Einstein condensation explicitly pointed out. Finally we discuss the separate contribution to the various sum rules arising from one-phonon and multiparticle excitations.
We derive and discuss sets of sum rules relating to the density fluctuation operator and to the single-particle creation and annihilation operators in a superfluid of charged bosons. The physical interpretation of the particle - particle and particle - density sum rules hinges on the single-particle excitation spectrum at long wavelengths having a gap equal to the plasma frequency in the presence of the Bose - Einstein condensate. This spectral property is shown to follow from the Hugenholtz - Pines relation for the chemical potential in terms of the half-diagonal two-body density matrix. Data on this density matrix are obtained by quantum Monte Carlo methods and are used to check the self-consistency between the Hugenholtz - Pines relation and the value of the chemical potential calculated from the ground-state energy. We also tabulate the contributions from the plasmon and from multiparticle excitations to various matrix elements and sum rules at long wavelengths.
In Chapter 3, we have shown how a Bose condensate leads to the single-particle Green's function G αβ ( Q , ω) appearing as a distinct component of the the dynamic structure factor S ( Q , ω). This feature is a direct consequence of (3.13) and is explicitly shown by the expression for S ( Q , ω) in (3.18). In the present chapter, we review the many-body diagrammatic analysis which specifies the precise relation between G αβ ( Q , ω) and S ( Q , ω) in a Bose-condensed fluid ( n 0 ≠ 0). The power of this “dielectric formalism” lies in showing that both functions mirror each other, as a result of the condensate-induced intermixing of single-particle and density excitations. At the same time, the formalism emphasizes the continuity of these excitations as the fluid passes through T λ, although these excitations have very temperature-dependent weights in G αβ ( Q , ω) and S ( Q , ω). While it is the poles of the single-particle Green's function which describe the elementary excitations, experimentally it is difficult to probe directly this field-fluctuation spectrum. In particular, it is only in a Bose-condensed fluid that one can obtain information about G αβ ( Q , ω) from inelastic neutron-scattering data on S ( Q , ω). In the absence of a condensate, there is little relation between the density-fluctuation modes and the single-particle excitations in a Bose fluid. After setting up the dielectric formalism in Section 5.1, the next two sections consider some model approximations at finite temperatures. These models are most appropriate to a dilute weakly interacting Bose gas (WIDBG), but we believe they also give crucial insight into various aspects of superfluid 4 He.
Abstract In this chapter we introduce some basic concepts of superfluid 4He that we will draw on in Chapters 8-12. Liquid 4 He has a long and rich history and there are many concepts that are implicit in the field today. We begin with a sketch of the unique mass and heat flow character of superfluid 4He. The assumptions of the two-fluid model used successfully to describe these properties are noted in §7.1. We introduce Bose-Einstein condensation in a Bose gas and the extension of this to interacting superfluid 4 He in §7.2. This leads naturally to the one-body density matrix which is the intermediate scattering function in the impulse approximation,We introduce a number of basic properties such as the static structure factor, the pair correlation function, and the momentum distribution which can be calculated from first principles using Monte Carlo and path integral Monte Carlo (PIMC) methods. We conclude by noting the two current descriptions of the characteristic excitation in superfluid 4He. The first is a density excitation, based on the Landau and Feynman view, with suitable modifications at higher Q to take account of correlations among atoms. In the second, the density excitation is separated into a part due to the excitation of single particles out of the condensate to states above the condensate and a part due to regular density excitations involving only states above the condensate. In the latter picture the condensate appears explicitly in S(Q, ω).
Recently there has been renewed interest in second sound in superfluid Bose and Fermi gases. By using two-fluid hydrodynamic theory, we review the density response $\chi_{nn}(\bq,\omega)$ of these systems as a tool to identify second sound in experiments based on density probes. Our work generalizes the well-known studies of the dynamic structure factor $S(\bq,\omega)$ in superfluid 4^44He in the critical region. We show that, in the unitary limit of uniform superfluid Fermi gases, the relative weight of second vs. first sound in the compressibility sum rule is given by the Landau--Placzek ratio $\lp\equiv (\bar{c}_p-\bar{c}_v)/\bar{c}_v$ for all temperatures below TcT_cTc​. In contrast to superfluid 4^44He, $\lp$ is much larger in strongly interacting Fermi gases, being already of order unity for T∼0.8TcT \sim 0.8 T_cT∼0.8Tc​, thereby providing promising opportunities to excite second sound with density probes. The relative weights of first and second sound are quite different in $S(\bq,\omega)$ (measured in pulse propagation studies) as compared to $\mathrm{Im}\chi_{nn}(\bq,\omega)$ (measured in two-photon Bragg scattering). We show that first and second sound in $S(\bq,\omega)$ in a strongly interacting Bose-condensed gas are similar to those in a Fermi gas at unitarity. However, in a weakly interacting Bose gas, first and second sound are mainly uncoupled oscillations of the thermal cloud and condensate, respectively, and second sound has most of the spectral weight in $S(\bq,\omega)$. We also discuss the behaviour of the superfluid and normal fluid velocity fields involved in first and second sound.
The spectrum of noncondensate excitations in neutron scattering on bosons is obtained in the framework of the Bogoliubov models both for liquid 4He and a dilute gas. The problem is solved using a path-integral representation of the partition function of the system. We describe the influence of scattering of neutrons on a Bose condensate in a stationary (time-independent) picture in the Gibbs equilibrium ensemble. This influence is a stationary boson response, and it depends on the initial neutron momentum k, transfer momentum p, and the neutron-boson interaction λ, which is related to the scattering length. The contribution of the neutrons to the initial Bogoliubov spectrum is found to be important for “quasi-elastic” scattering on the noncondensate, while the contribution of deep inelastic scattering is small; no contribution from elastic scattering on the Bose condensate is found. In the case of liquid Helium, the response is unlikely to be observable for all values p. On the other hand, for a gas one may expect a visible effect, in particular for a small momentum transfer p and a small density of the Bose condensate ϱ.