The neutral system consisting of charged-bosons and -fermions has been investigated in terms of density parameter expansion at the high-density limit. The ground state energy of this system is calculated and accords with the result obtained in our previous work. The elementary excitations in the system determined from the poles of the boson Green's function are the two collective excitations in the low momentum region : acoustic- and optical-phonon like. The life-time of the acoustic phonon approaches infinite as s/v_f →0 where s and v_f arc the sound velocity and Fermi velocity, respectively, while that of the optical pbonon is infinite.
The neutral system consisting of charged-bosons and -fermions has been investigated in terms of density parameter expansion at the high-density limit. The ground state energy of this system is calculated and accords with the result obtained in our previous work. The elementary excitations in the system determined from the poles of the boson Green's function are the two collective excitations in the low momentum region : acoustic- and optical-phonon like. The life-time of the acoustic phonon approaches infinite as s/v_f →0 where s and v_f arc the sound velocity and Fermi velocity, respectively, while that of the optical pbonon is infinite.
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 ground state energy and elementary excitations of a charged gas of bosons at high densities are examined by use of the method developed by Bogoliubov for boson gases. It is conjectured, but not herein established, that this method yields exact results in the high-density limit analogous to those obtained by Gell-Mann and Brueckner, and by Sawada, in the corresponding case of a charged fermion gas. The ground state energy is essentially correlation energy, and is therefore negative, and its magnitude varies as the one-fourth power of the density at high densities. The elementary excitations have for low momenta the energy appropriate to plasma waves, and for high momenta the energy appropriate to single-particle excitation. There is therefore an energy gap, suggesting that the gas is both a superfluid and a superconductor at low temperatures. At low densities the behavior of a charged gas is independent of statistics; hence, such a gap must disappear as the system is expanded.
The ground state of a charged Bose gas is studied throughout a large range of densities in a self-consistent-field approximation formalism which includes the short-range correlations between bosons through a local-field correction. Numerical self-consistent calculations have been carried out in order to determine the static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. Hence, from $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ we have obtained the pair-correlation functions, the ground-state energy which essentially is the correlation energy, the pressure of the gas, and, from Feynman's relation, the energy spectrum of elementary excitations. The response of the system to a static impurity charge has also been investigated. In the high-density limit our calculations show the same results as those given by Bogoliubov's perturbation theory. In the intermediate-density region, corresponding to the strongly coupled system, our results are in very good agreement with recent calculations based on the hypernetted-chain integral equation as well as Monte Carlo computations.
We investigate the properties of a dilute boson-fermion mixture at zero temperature. The ground-state energy of the system is calculated to second order in the Bose-Fermi coupling constant. The Green's function formalism is applied to obtain results for the effect of Bose-Fermi interactions on the spectrum of phonon excitations, the momentum distribution, the condensate fraction, and the superfluid density of the bosonic component. A quantitative discussion of these effects is presented in the case of mixtures with attractive interspecies interaction.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of N particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of N particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.
The maximum deviation of the radial distribution function g(r) from its asymptotic value, $g(\ensuremath{\infty})=1$, is used as an expansion parameter in calculating the properties of a degenerate boson system. The uniform limit, defined by the condition $1\ensuremath{-}g(0)\ensuremath{\ll}1$, holds at low densities under appropriate constraints on the Fourier transforms of the interaction potential and also at high densities for the charged-boson system. In the uniform limit a procedure based on (1) a Jastrow-type trial function, (2) the Wu-Feenberg functional for the kinetic energy, (3) the Kirkwood superposition approximation, or the more accurate Abe form for the three-particle distribution function, yields the Bogoliubov formulas for the ground-state energy and the excitation energies of a degenerate or nearly degenerate boson system. Results for the ground state of the high-density charged-boson system include $\frac{〈\mathrm{KE}〉}{N}=\ensuremath{-}\frac{1}{5}\frac{〈\mathrm{PE}〉}{N}+O({{r}_{s}}^{0}),$ $\frac{E}{N}\ensuremath{\leqq}\ensuremath{-}\frac{0.8031}{{{r}_{s}}^{\frac{3}{4}}}+0.028.$ Numerical results are also computed at intermediate and moderately low densities ($0.01\ensuremath{\leqq}{r}_{s}\ensuremath{\leqq}100$).