The dynamic structure function $\mathcal{S}(k, \ensuremath{\omega})$, as a function of wave vector
k and frequency $\ensuremath{\omega}$, is calculated rigorously for a simple Bose gas at
T=0, The dielectric formulation is used to satisfy general symmetry requirements for a Bose system, and the calculation is carried out in the first approximation beyond Bogoliubov taking into account the three-phonon process. An analytic solution in terms of elliptic integrals is obtained for the width of $\mathcal{S}(k, \ensuremath{\omega})$ valid for arbitrary
k and $\ensuremath{\omega}$. Qualitatively, $\mathcal{S}(k, \ensuremath{\omega})$ at finite but small
k is found to have a square-root behavior near the threshold frequency for two-phonon production, a peak at the elementary excitation frequency, and a long power-law tail at high frequencies. Illustrative numerical results are presented for the width of $\mathcal{S}(k, \ensuremath{\omega})$, the imaginary part of the phonon spectrum, and $\mathcal{S}(k, \ensuremath{\omega})$ itself. Finally, the impulse approximation and the extraction of the condensate density
n0 from $\mathcal{S}(k, \ensuremath{\omega})$ in the large-
k limit are discussed.
This paper is concerned with the application of sum rules satisfied by the density correlation function to the calculation of properties of the interacting Bose system, particularly liquid
He4 and the hard-core gas. In addition to the two well-known sum rules which have been extensively discussed in the literature, we derive a third useful relation which directly involves the two-body interaction. The assumption that a single excitation exhausts all the sum rules (which is known to lead to the Feynman theory when only the first two sum rules are used) now provides, with the help of the additional sum rule, an explicit integral equation for the liquid structure factor
S(k). This equation can be solved quite easily for the hard-core gas, and we find the low-lying excitation spectrum to have the form ${{\ensuremath{\omega}}_{\mathrm{hc}}}^{2}(k)={k}^{2}{{v}_{T}}^{2}+{\left(\frac{{k}^{2}}{2m}\right)}^{2}+4\left(\frac{{k}^{2}}{2m}\right)\left[\left(\frac{P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{1}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{1}(0))+\left(m{{v}_{T}}^{2}\ensuremath{-}\frac{2P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{2}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{2}(0))\right],$ where
vT,
P,
〈N〉〈KE〉 and
a are, respectively, the sound velocity, pressure, kinetic energy per particle, and core diameter.
P1 and
P2 are sinusoidal functions given in the text. Although the sum rules themselves are exact, there are several difficulties associated with the use, even in the low-
k limit, of the sharp resonance assumption in all such integral relations. In spite of these difficulties, which are discussed briefly, we have pursued the consequences of the sharp-resonance approximation to obtain an integral equation for the liquid structure factor
S(k) together with the above closed-form solution for the hard-core gas excitation spectrum. The speculation that the approximation becomes essentially exact in the low-
k limit provides, through the third sum rule, an expression for the sound velocity in terms of the two-body potential
v(r) and the radial distribution function
S(r). We apply this relation to the case of liquid
He4 and discuss extensions of the single-resonance approximation which can be made with the help of the third sum rule.
An analytic expression is presented for the elementary excitation spectrum of the Bose-Einstein condensate of a trapped boson system in the weakly interacting, low-density limit. Explicit analytic formulas for the elementary excitation spectrum are obtained for harmonic-oscillator traps. These formulas provide information about the behavior of the elementary excitation levels as a function of the number of atoms and their interaction strength for a given trap geometry. They also provide a low-density benchmark for results of fully numerical calculations.
In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where
lnf(r) has an infinite range. The other factor is assumed to have this same form but with
lnf(r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors
k, the structure factor
S(k) is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature
T,
S(k) tends to the constant value
mc2kBT, where
m is the mass of the particles and
c the velocity of propagation of the phonons. The momentum distribution
nk has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-
k behavior of both
S(k) and
nk is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.
The paradoxes and disparities in the contemporary microscopic theory of superfluid helium (He–II) are discussed along with possible ways of resolving them by taking pair correlations of He4 atoms into consideration. It is shown that most paradoxes are associated with the commonly accepted initial assumption concerning the dominating role of single-particle Bose condensate (SPBC) in the quantum microstructure of the superfluid component ρs. The existence of intensive SPBC leads to a strong hybridization of the elementary excitation branches and to a common dispersion law for all boson branches, which is identified with the quasiparticle spectrum E(p) observed experimentally from slow neutron scattering in liquid helium. However, the stability of this spectrum during a transition through the λ-point and the large value of the gap in the vicinity of the “rotonic” minimum contradict both the Landau theoretical criterion of superfluidity and the small value of experimentally measured critical velocity. At the same time, a strong interaction between particles in the Bose liquid He4 strongly suppresses the SPBC which amounts to less than 1% of all He4 atoms and hence cannot be the main constituent of the superfluid component, unlike the case of a weakly nonideal Bose gas. Moreover, for a quite strong attraction between particles in a certain region of the momentum space, bound pairs of bosons can be formed in the superfluid Bose liquid, and a coherent pair condensate (CPC) analogous to the Cooper pair condensate in superconductors may appear. Such a strong CPC may completely suppress the weak SPBC. In this case, the one-particle spectrum ε(p) of elementary excitations does not hybridize with the collective (two-particle) spectrum and does not appear in the structure of the dynamic form factor S(p,ε), i.e., does not coincide with the spectrum measured from neutron scattering. The dispersion of one-particle spectrum is defined by the momentum dependence of the pair order parameter Ψ̃(p) and may have a minimum or a point of inflection at p≠0. This peculiarity in the one-particle spectrum of a Bose liquid with CPC but without SPBC vanishes together with Ψ̃(p) at the temperature Tc=Tλ of the phase transition from the superfluid to the normal state (unlike the rotonic minimum in the collective spectrum), while the corresponding critical velocity vc=min[ε(p)/p] vanishes at the λ-point in accordance with the Landau criterion and the experimental data. The assumption that the strong “Cooper-like” CPC is responsible for the quantum structure of the superfluid component ρs is confirmed indirectly by the successful application of the Justrow approximation (based on strong pair correlations) for describing the properties of liquid He4 and quantum liquid mixtures He3–4He on one hand, and by an anomalously large effective mass of He3 impurity atoms in He4, which is approximately equal to total mass of He3 and He4 atoms, thus pointing to the existence of helium atoms in superfluid liquid He–II. The value of the superfluid velocity circulation quantum in the Onsager–Feynman vortices in a Bose liquid with CPC but without SPBC is discussed as well as the critical velocities of superfluid He4 in ultrathin films and channels in which the creation and motion of quantum vortices are ruled out, and the quasiparticle spectrum undergoes dimensional quantization.