As a model for liquid helium II, we study the Bose–Einstein gas with a two-body interaction potential of the form ~δ(r − a), where r is the interparticle separation. Excitation spectra for various values of a are calculated using the Brueckner–Sawada approach based on the concept of the T matrix. However, unlike the work of Brueckner and Sawada and many other related works that followed, we take into account the dependence of the T matrix on the center-of-mass momentum of the interacting particles. Excitation spectra calculated with and without this dependence, using two different expressions for the two-particle propagator, indicate the validity of the Brueckner and Sawada center-of-mass approximation for physically interesting values of a. The difference between the two spectra is found to increase as a increases.
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy
E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where
n is the density and
f0 is the scattering length for the assumed two-body interaction between the bosons. The coefficients
a and
b are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient
b is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
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.
By the method of ``modes'' introduced by Sawada, the excitation spectra of a fermion system with singular interactions have been obtained in three cases; (1) with one additional particle above and and one hole below the Fermi surface, (2) with one additional particle above the Fermi surface, and (3) with two particles above the Fermi surface. The argument holds not only for the system of low density (nuclear matter with a hard-core repulsion), but also for the case of the interacting electron gas in the high-density region. The results have been compared with the Brueckner-Goldstone perturbation-expansion formulas, by using diagrams in the second- and third-order of the expansion in interaction strength. We show that in the approximation of our treatment, the occupation probability function at temperature T has the same form as the Fermi distribution function.
The Bassichis–Foldy model of a simple interacting boson is solved numerically and the results are compared with those obtained by the Bogoliubov approximation and by the Brueckner–Sawada t-matrix formalism. In the normal region, contrary to the widely held view, the Brueckner–Sawada approximation for the energy of the ground state is not reliable for strong, well-behaved, repulsive forces. The Bogoliubov approximation, on the other hand, remains valid for a wide range of values of the coupling constant. In the inverted region, the attractive force causes a population inversion in the levels of the system. For this case a modified Brueckner–Sawada approximation is developed. This method is applied to the calculation of the transition point and the energies of the ground and the first excited states of the system. Here most of the predictions of the modified Brueckner–Sawada approximation are quite accurate. By a simple change in the Bassichis–Foldy model it is shown that even, for two bosons there can be a phase transition. In this model, the derivative of the ground state energy with respect to the coupling constant is discontinuous at the transition point.
An iterative method is used in a self-consistent calculation, at T=0, of the normal Σ11 and anomalous Σ12 self-energy parts, the boson polarization operator Π on the “mass shell,” and the quasiparticle spectrum E(p) in a superfluid Bose liquid with an interaction-quenched single-particle Bose–Einstein condensate (BEC). The calculation is based on a system of “truncated” integral equations for Σ11 and Σ12 with allowance for terms of first order in the density n0/n≪1 of the BEC and with the “bare” interaction between bosons taken in the form of the repulsive pseudopotential in the “semitransparent spheres” model, for which the Fourier component of the pseudopotential is an oscillatory sign-varying function of the momentum transfer. By fitting with a single adjustable parameter—the amplitude of the initial repulsive pseudopotential—one can achieve completely satisfactory agreement of the theoretical quasiparticle spectrum E(p) with the measured spectrum of elementary excitations in superfluid helium from neutron-scattering experiments over a wide momentum range (0⩽p⩽pmax≃4 Å−1).