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Physical Review A

Charged Bose Gas

Gideon Carmi, A. J. Lock

Published Mar 1, 1972• 3 citations
Publisher

Abstract

By using the Bohm-Pines and Bogoliubov formalisms as well as the random-phase approximation, the investigation of the charged Bose gas is extended into the intermediate densities for which ${r}_{s}\ensuremath{\lesssim}O(1)$. The ground state energy is essentially correlation energy, and is therefore negative as expected. The expression for the plasmon spectrum does possess an energy gap; the single-particle excitation spectrum does not. The specific-heat function is examined to first order in the temperature and is shown to approach zero from above as the temperature does the same. Finally, the expression for the pressure is shown to persist in remaining negative for all ${r}_{s}\ensuremath{\lesssim}O(1)$ at zero temperature.

DOI
10.1103/physreva.5.1447

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By using the Bohm-Pines and Bogoliubov formalisms as well as the random-phase approximation, the investigation of the charged Bose gas is extended into the intermediate densities for which rs\ensuremath≲O(1){r}_{s}\ensuremath{\lesssim}O(1)rs​\ensuremath≲O(1). The ground state energy is essentially correlation energy, and is therefore negative as expected. The expression for the plasmon spectrum does possess an energy gap; the single-particle excitation spectrum does not. The specific-heat function is examined to first order in the temperature and is shown to approach zero from above as the temperature does the same. Finally, the expression for the pressure is shown to persist in remaining negative for all rs\ensuremath≲O(1){r}_{s}\ensuremath{\lesssim}O(1)rs​\ensuremath≲O(1) at zero temperature.
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.
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 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 response of the two-dimensional charged Bose gas is investigated in the random-phase approximation. Using this temperature-dependent dielectric function we find the energy spectrum and the damping for the quasiparticles. The electrostatic potential around a test charge at T=0T=0T=0 is displayed. We also study the thermodynamic functions of the gas at very low temperatures and high densities.
By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ although with different weights. If the condensate n0=0{n}_{0}=0n0​=0, the ${\ensuremath{\omega}}_{2}$ mode only appears in the single-particle spectrum while ${\ensuremath{\omega}}_{1}$ only appears in the density-fluctuation spectrum. For finite values of n0{n}_{0}n0​, these modes are coupled and renormalized to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\ensuremath{\omega}}_{2}$) are coupled to the zero-sound density fluctuations (${\ensuremath{\omega}}_{1}$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\ensuremath{\omega}}_{1}$. The usual Bogoliubov, Hartree-Fock, and ordinary ttt-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ mode. Experimentally, it appears that the high-frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ excitation corresponds to first sound and the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.