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.
The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.
An investigation is performed using normal-mode analysis on a quantum mechanical cell model whose general features are representative of 4He at low temperatures. Both random-phase and self-consistent field approximations are invoked. A Bogoliubov-type excitation energy spectrum is derived for the interacting Bose system, and the amplitudes of the system quasi-particles are obtained; these quantities are expressed in terms of zero-momentum occupation number. A relationship giving the temperature-dependent fractional occupation of the single-particle zero-momentum level is established, and special attention is devoted to the temperature extreme near absolute zero. An estimate is made of the zero-temperature sound speed corresponding to the special case of phonon excitation given by the model in low momentum. It is shown that at a critical temperature for which the zero-momentum occupation is depleted, the quasi-particles indeed become normal Bose particles. In first approximation, this critical temperature is akin in form to that obtained in treatments of ideal boson systems.
This work is concerned with a microscopic model of a nonideal boson gas. The Hamiltonian is treated in a rigorous manner free of all approximations and, in particular, the Bogoliubov approximation of replacing the zero-momentum single-particle creation and destruction operators by a
c number is entirely avoided. Utilizing a method due to Wentzel, the Hamiltonian is replaced by one involving products of two second quantized operators which, in general, can be diagonalized. This replacement is shown to be rigorously valid in the infinite volume limit for all temperatures despite the avoidance of the Bogoliubov approximation. It is found that the assumption of a positive-definite quasiparticle excitation spectrum $\ensuremath{\epsilon}(\mathrm{k})$ becomes untenable for temperatures
T less than a critical value
Tc, if the interparticle potential is described by a certain kernel which possesses positive eigenvalues only, corresponding to a sufficiently repulsive interparticle force. Assuming that $\ensuremath{\epsilon}(\mathrm{k})=0$ only for k=0, it is found that for
T<Tc the system undergoes an Einstein condensation into the k=0 single-particle state, and the integral equations characterizing the system are precisely the same as would be obtained if the Bogoliubov approximation were made, proving within the framework of this model the strict validity of this approximation procedure. The well-known criterion for an Einstein condensation
v(0)>0, where
v(k) is the Fourier transform of the interparticle potential, is shown to be a weaker statement of our eigenvalue criterion. Further, in agreement with Girardeau and Arnowitt, it is found that an energy gap separates $\ensuremath{\epsilon}(0)$ and $\ensuremath{\epsilon}(\mathrm{k})$ for
k>0. An approximation method is developed for solving the integral equations which describe those systems which undergo an Einstein condensation, and it is shown that this method can be justified if $|T\ensuremath{-}{T}_{c}|\ensuremath{\lesssim}\frac{{T}_{c}}{10}$, and in cases of short-range repulsive interparticle forces and systems for which $\ensuremath{\rho}v(0)$ possesses a certain prescribed upper bound, $\ensuremath{\rho}$ being the number of particles per unit volume. The transition temperature is found to be lower than the corresponding quantity for the ideal boson gas of the same density. A detailed discussion of the thermal properties of the system in the vicinity of
Tc is also given. In an Appendix a general type of smeared Einstein condensation is assumed, and it is found that for systems in which this property occurs $\ensuremath{\epsilon}(\mathrm{k})$ is linear for small
k. This ansatz is shown to be tenable only if
v(0)<0. Finally, by assuming an appropriate pseudopotential representation of a hard repulsive core and by including a weak, longranged attractive force between particles, $\ensuremath{\epsilon}(\mathrm{k})$ shows a nonmonotonic behavior of the same qualitative type as observed in liquid
He4.
One considers the superfluid (SF) state of a Bose liquid with a strong repulsion between bosons, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC), analogous to the Cooper condensate in a Fermi liquid with an attraction between the fermions. Such a PCC emerges in a system of bosons due to an oscillating sign-changing momentum dependence of the Fourier component of the pair interaction potential, which is characteristic of a certain family of repulsion potentials. In such cases, the Fourier component is negative in some domain of nonzero momentum transfer, which corresponds to an effective attraction. The collective effects of renormalization (``screening'') of the initial interaction lead to a suppression of the repulsion and an enhancement of the effective attraction. It is the ratio of the BEC density to the full density of the liquid
n0/n≪1 that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC,
(n−n0)/n0≪1. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid
4He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.
We give an overview of the current theory of collective modes in trapped atomic gases at finite temperatures, when the dynamics of the condensate and non-condensate must both be considered. A simple introduction is given to the quantum field formulation of the dynamics of an interacting Bose-condensed system, based on equations of motion for the condensate wavefunction and single-particle Green's functions for the non-condensate atoms. We discuss the nature of excitations in the mean-field collisionless region, including the Beliaev second-order approximation for the self-energies. We also sketch the derivation of coupled two-fluid hydrodynamic equations using a simple kinetic equation which includes collisions between condensate and non-condensate atoms.