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Physical review. B, Solid state

Thermodynamic behavior of the charged Bose gas around T=0

S. R. Hore, Norman E. Frankel

Published Mar 1, 1976• 2 citations
Publisher

Abstract

We have used a low-temperature excitation spectrum to calculate the thermodynamic behavior of a charged Bose gas for $T\ensuremath{\gtrsim}0$. These results are compared with the thermodynamic behavior obtained using Foldy's $T=0$ excitation spectrum.

DOI
10.1103/physrevb.13.2242

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We have used a low-temperature excitation spectrum to calculate the thermodynamic behavior of a charged Bose gas for T\ensuremath≳0T\ensuremath{\gtrsim}0T\ensuremath≳0. These results are compared with the thermodynamic behavior obtained using Foldy's T=0T=0T=0 excitation spectrum.
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.
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.
Foldy, and later Girardeau, calculated the ground-state energy of a charged Bose gas at high density. We rederive the common first term obtained by these authors by using a nonperturbation method developed previously. Our aims are: (i) to establish the validity of this common result, which has not been proved; (ii) to establish the validity and usefulness of our nonperturbation method. We also show that our method will give the correct functional dependence of the ground-state energy on the density at low density, although the exact coefficient must await a numerical computation.
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.
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.