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Physical review. B, Solid state

Dielectric approach to a dense charged boson gas at T=0

J.C. Lee

Published Oct 1, 1975• 14 citations
Publisher

Abstract

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 ${r}_{s}^{\frac{3}{4}}$. The entire treatment is fully number conserving.

DOI
10.1103/physrevb.12.2644

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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.
The Bogoliubov--de Gennes (BdG) type equations are derived for the Coulomb Bose gas describing the electrodynamics of charged bosons. The gauge invariant dielectric response function and collective excitations of a charged Bose gas at T=0 are found. We find a plasmon branch which is the same as the single-particle Bogoliubov spectrum in agreement with the textbook result. The superconducting properties of the charged Bose gas are verified. A gauge-invariant formalism for calculating the response of the supracondensate particles is proposed.
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.
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 dielectric screening function epsilon (k, omega ) of the fluid of charged bosom at zero temperature is evaluated in a range of low-to-intermediate coupling strength, in view of recent data on static screening and fluid structure from quantum Monte Carlo methods. Correlations beyond the random phase approximation are included within a class of approximations which are well known for the electron fluid, i.e. by introducing a frequency-independent local field factor to be determined through self-consistency requirements connecting various aspects of the physics contained in the dielectric function. The static dielectric function epsilon (k,0) is negative over a range of wavenumbers at all values of the density, leading to oscillatory screening of a foreign charge and to an effective long-range attraction between equi-charged impurities. Quantitative agreement with the Monte Carlo data on static screening is achieved by imposing self-consistency on the compressibility of the fluid in addition to self-consistency on the pair distribution function. Good agreement is also obtained on the pair distribution function and the correlation energy. Within the present class of approximations, the dispersion relation of longitudinal excitations takes the Feynman form, starting at the plasma frequency with a negative dispersion coefficient and going through a minimum before ending at the single-particle recoil frequency.