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Physical Review A

Theory of a Charged Boson Gas Using the Random-Phase Approximation

J.C. Lee

Published Apr 1, 1972• 2 citations
Publisher

Abstract

The Brandow boson many-body theory is used to calculate the generalized dielectric constant within the random-phase approximation. The generalized dielectric constant obtained this way agrees with the Bogoliubov theory in the high-density limit and thus yields all of the well-known Bogoliubov results. The momentum distribution function for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\ne}0$ is also calculated. The treatment is fully number conserving.

DOI
10.1103/physreva.5.1925

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Progress of Theoretical Physics•Dec 1, 1974•1 citation
The Brandow boson many-body theory is used to calculate the generalized dielectric constant within the random-phase approximation. The generalized dielectric constant obtained this way agrees with the Bogoliubov theory in the high-density limit and thus yields all of the well-known Bogoliubov results. The momentum distribution function for k\ensuremath→\ensuremath≠0\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\ne}0k\ensuremath→​\ensuremath=0 is also calculated. The treatment is fully number conserving.
The Brandow boson linked-cluster expansion has been used to redo the Foldy calculation of the Bogoliubov theory for a charged boson gas. The particle-number conservation is treated exactly. The ring diagrams have been summed using the same techniques as used by Gell-Mann and Brueckner for the electron gas. The next-most-divergent diagrams (twisted rings) have also been summed. These diagrams contain a renormalized hole line and thus show explicitly the dynamics of the condensate which is treated only in an average way in the Bogoliubov theory. The results for the ground-state energy and the condensation fraction agree exactly with Foldy in the high-density limit.
The generalized dielectric constant is calculated to the order next to the Bogoliubov approximation. This is done by using the analogy between the condensed boson system and a fictitious fermion system with spin degeneracy equal to the total number of particles (instead of two). From the zero of the dielectric constant, we have calculated the first-order correction to the Bogoliubov plasmon energy and the half linewidth of the plasmon states. The real part of the solution is examined from a graphical view point to show that up to the specified order of approximation there exists only one mode of elementary excitation. The screening of the system to a static impurity charge is shown to be exponential at a long distance. The response of the system to a static transverse vector potential shows a perfect Meissner effect at long- and short-wavelength limits. We have also examined the diagrammatic structure for the number of particles in condensate. We show that the series contains only terms of integer powers of rs34{r}_{s}^{\frac{3}{4}}rs43​​. The entire treatment is fully number conserving.
This is the first of two papers in which the high-density charged Bose gas at zero temperature is treated by two independent and self-contained methods. In this paper, we are concerned with the excitation spectrum of the system. An analysis of the formal and physical aspects of the theory is followed by a simple numerical calculation. We apply the usual field-theoretic formulation for a Bose system. The Green's functions and other response functions, or propagators, are analyzed, emphasizing the role of the dielectric constant. The general features of the excitation spectrum are described. To apply and to further illustrate the formalism, the low-lying levels are numerically determined to the order of rs34{{r}_{s}}^{\frac{3}{4}}rs​43​ in units of ${\ensuremath{\omega}}_{\mathrm{pl}}$ (the next order beyond the Bogoliubov approximation). rs{r}_{s}rs​ is the ratio of the average interparticle distance to the Bohr radius and ${\ensuremath{\omega}}_{\mathrm{pl}}$ is the plasma frequency.
A closed form for the dielectric function of a charged Bose gas is found in the random-phase approximation. This dielectric constant is used to explore the quasiparticle energy spectrum (i.e., the allowed modes of oscillation of the gas), the damping of these quasiparticles, and the long-ranged form of the electrostatic potential around a test charge in the gas. We do this around the three temperature regions of interest; namely, T=0T=0T=0, $T=\ensuremath{\infty}$, and T=TcT={T}_{c}T=Tc​, where Tc{T}_{c}Tc​ is the transition temperature of the gas.
A multi-component, many-particle system with Coulomb interaction is described in terms of a generalized dielectric constant ε(q, ω). The ε(q, ω) is calculated by the use of the equation-of-motion of a correlation function obtained within the random phase approximation (R.P.A.). For the Fermi, Bose, Bose and Bose, and Bose and Fermi systems, the ε(q, ω) obatined above leads directly to the result of Gell-Mann and Brueckner, Foldy, Bassichis, and Ginoza and Kanazawa, respectively. As for the Fermi and Fermi system, it has two collective excitations: plasma wave and acoustic phonon wave. The latter is well-defined only for such a special choice of parameters specifying the system as vF1 ≫s > vF2 and further, for the choice of parameters as vF1 ≫s≫vF2, agrees with the acoustic phonon wave of an electronion system treated by the Thomas-Fermi approximation. For the parameters chosen otherwise, such a wave is unstable or does not exist. In the expressions of the conditions above, s is the velocity of acoustic phonon wave and vF1 and vF2 are Fremi velocities in one component and in the other, respectively. The above results show that in the weak coupling limit, the results obtained by treating Bose components consistently by the Bogoliubov method are equivalent to those obtained with the R.P.A.